` DEGLI STUDI DI FIRENZE UNIVERSITA Dipar Dipartimento timento di Sistemi e Informat Informatica ica
Dottorato di Ricerca in Ingegneria Informatica e dell’Automazione ING-INF/04 XVIII Ciclo
Spacecr Sp acecraft aft Attitude Dynamics and Contro Control l
Fabio Bacconi
Ph.D. Coordinator
Advisors
Prof. Edoardo Mosca
Prof. Edoardo Mosca
Prof. Alessandro Casavola
Anno Accade Accademico mico 2005–2006 2005–2006
Contents Introduction
1
Prio Priorr Lite Litera ratur turee on on Spa Space cecr craf aftt Att Attitu itude de Contr Control ol . . . . . . . . . . . . . .
2
Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
I
Ge Geo-Stationary Spacecraft Attitude Control
7
1
Dynamics of the 3D Pendulum
9
1.1 Rigi Rigid d Bodi Bodies es Mathe themati maticcal Mode odels . . . . . . . . . . . . . . . . . . 1.2
The 3D Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3
Equili ilibria of the 3D Pendulum . . . . . . . . . . . . . . . . . . . .
17
1.4 Triax iaxial ial Att Attit itud udee Con Contr trol ol Testb estbeed . . . . . . . . . . . . . . . . . . .
19
1.5 2
10
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Special Cases of the 3D Rigid Pendulum
22
2.1
Axisy isymmetric 3D Rigid Pendulum . . . . . . . . . . . . . . . . . .
23
2.2 2.2
Spin Spinni ning ng Axis Axisym ymme metr tric ic 3D Rigi Rigid d Pendu endulu lum m . . . . . . . . . . . . . 26
2.3 2.3
Hang Hangin ing g Equ Equil ilib ibri rium um of the the Sym Symme metri tricc Spi Spinn nnin ing g Top Top . . . . . . . . 28
2.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 St Stabilization of an Asymmetric 3D Rigid Pendulum
34
3.1
3D Rigid Bod Bodyy Control Torques . . . . . . . . . . . . . . . . . . . .
35
3.2 3.2
Asym Asympto ptotic tic Stab Stabil iliz izat atio ion n of of the the Hang Hangin ing g Equ Equil ilib ibri rium um . . . . . . . . 38
3.3 Experim Experiment entss on on Stabi Stabiliz lizati ation on of the Hangin Hanging g Equi Equilib libriu rium m . . . . . . 43 CONTENTS
i
3.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Attitude Control of an Asymmetric 3D Rigid Pendulum
47 48
4.1 Local Asymptotic Stabilization of an Arbitrary Equilibrium . . . . .
49
4.2
52
Attitude Control of a Geo-Stationary Spacecraft . . . . . . . . . . .
4.3 Simulation on Stabilization of an Arbitrary Equilibrium . . . . . . . 56 4.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Attitude Control of a 3D Axially Symmetric Rigid Pendulum
5.1 Global Coordinates Model of the 3D Axially Symmetric Pendulum .
59 62
64
5.2 Stabilization of the Hanging Equilibrium of the Symmetric Spinning Top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
5.3 Stabilization of the Inverted Equilibrium of the 3D Axially Symmetric
II
Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
5.4 Attitude Control of an Underactuated Geo-Stationary Spacecraft . .
75
5.5
Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MEO and LEO Spacecraft Attitude Control
6 Medium and Low Earth Orbit Satellites
82
87 89
6.1
LEO Spacecraft Attitude Model . . . . . . . . . . . . . . . . . . . .
90
6.2
LEO Spacecraft Position Model . . . . . . . . . . . . . . . . . . . .
94
6.3
LEO Spacecraft Control Problem Formulation . . . . . . . . . . . .
95
6.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
7 Attitude Control of a Low Earth Orbit Satellite
100
7.1 Command Governor Approach for LEO Spacecraft Control . . . . . 101 7.2 Simulation of a LEO Spacecraft Reconfiguration Maneuver . . . . . 106 7.3
Simulation of a LEO Spacecraft Tracking Maneuver . . . . . . . . . 108
7.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8 Attitude Control of a LEO Satellite for Large Angle Maneuvers
116
8.1
LEO Spacecraft Nonlinear Attitude Model . . . . . . . . . . . . . . 117
8.2
Hybrid Command Governor for LEO spacecraft . . . . . . . . . . . 120
8.3 Simulations of a LEO spacecraft performing Large Angle Maneuvers 123 8.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Conclusion
128
Geo-stationary Spacecraft and the 3D Pendulum . . . . . . . . . . . . . 129 Constrained MEO and LEO Spacecraft . . . . . . . . . . . . . . . . . . . 131 Research Contribution and Future Developments . . . . . . . . . . . . . 132 Bibliography
134
CONTENTS
iii
List of Figures {}
{ }
1.1 Inertial references frame I and body reference frame B
. . . .
10
{}
1.2 Sequence of Euler’s angles to be applied to frame I for obtaining
{ }
frame B o r i e n t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.3
A schematic of a 3D Rigid Pendulum . . . . . . . . . . . . . . . . .
14
1.4
Triaxial Air Bearing Testbed. . . . . . . . . . . . . . . . . . . . . .
20
2.1 A schematic of a 3D Rigid Pendulum with one axis of symmetry . .
23
2.2 Classical representation of the 2D spherical pendulum . . . . . . . .
25
2.3 A schematic of a heavy spinning top not constrained by the table .
27
3.1 Fan thrusters scheme of the Triaxial Attitude Control Testbed. . . .
37
3.2 Experimental results for the evolution of the angular velocity of the 3D pendulum in the body frame. . . . . . . . . . . . . . . . . . . .
45
3.3 Experimental results for the evolution of the components of the direction of gravity Γ in the body frame. . . . . . . . . . . . . . . . .
45
3.4 Experimental result for the evolution of the angle between the reduced attitude vector Γ(t) and the desired reduced attitude vector Γ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.1 Evolution of the closed loop system (3.2), (4.1) and (4.2) towards the unstable equilibrium. . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Evolution of the angular velocity of the 3D pendulum in the body frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
4.3 Evolution of the components of the direction of gravity Γ in the body frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
4.4 Evolution of the angle between the reduced attitude vector Γ(t) and the desired reduced attitude vector Γ0 . . . . . . . . . . . . . . . . .
58
4.5 Vertical angular momentum (solid line) and total energy (dash line) of the 3D pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.6 Motion of the vector between the pivot and the center of mass of the 3D pendulum in the inertial frame. . . . . . . . . . . . . . . . .
60
4.7 Evolution of the angle between the reduced attitude vector Γ(t) and the desired reduced attitude vector Γ0 for ω(0) = 0. . . . . . . . . .
60
5.1 Evolution of the angular velocity of the symmetric heavy top in the body frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.2 Evolution of the components of the direction of gravity Γ in the body frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.3 Vertical angular momentum (solid line) and total energy (dash line) of the symmetric top. . . . . . . . . . . . . . . . . . . . . . . . . .
81
5.4 Closed-loop trajectory of the symmetric top in the inertial frame. . .
82
5.5 Evolution of the angular velocity of the 2D pendulum in the body frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
5.6 Evolution of the components of the direction of gravity Γ in the body frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
5.7 Evolution of the angle between the reduced attitude vector Γ(t) and the desired reduced attitude vector Γ0 for ω(0) = 0. . . . . . . . . .
84
5.8 Closed-loop trajectory of the 2D pendulum in the inertial frame. . .
84
{ } erence frame {EC I } . . . . . . . . . . . . . . . . . . . . . . . . . Reference frame {B } with respect to an orbiting reference frame {O}, both moving in {EC I } . . . . . . . . . . . . . . . . . . . . .
6.1 Reference frame B motion with respect to an Earth centered ref-
6.2
7.1
90
92
Command Governor scheme. . . . . . . . . . . . . . . . . . . . . . 101
7.2 Relative position components x(k), y(k), z(k) of the LEO satellite, under the proposed LQ+CG control law (7.14). . . . . . . . . . . . 108 LIST OF FIGURES
v
7.3 Euler’s angles ψ(k), θ(k), φ(k) of the LEO satellite, under the proposed LQ+CG control law (7.14). . . . . . . . . . . . . . . . . . . 108 7.4
Input forces f i (t) generated by the proposed LQ+CG control law (7.14). The dash lines represent the constraint boundaries. . . . . . 109
7.5
Input torques τ i (t) generated by the proposed LQ+CG control law (7.14). The dash lines represent the constraint boundaries. . . . . . 109
7.6 Relative positions x(k), y(k), z(k) of the LEO satellite under LQ c o n t r o l l a w o n l y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.7 Euler’s angles ψ(k), θ(k), φ(k) of the LEO satellite under LQ control law only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.8
Input forces f i (t) required by the LQ control law. The dash lines represent the constraint boundaries. . . . . . . . . . . . . . . . . . 111
7.9
Input torques τ i (t) required by the LQ control law. The dash lines represent the constraint boundaries. . . . . . . . . . . . . . . . . . 111
7.10 Evolution of the relative position variables x, y and z of the LEO satellite with the application of LQ+CG (tracking problem) for a period of revolution. . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.11 Errors ei (t), i=1,2,3 in coordination accuracy for the position components x(k), y(k) and z(k) related to the circumference tracking maneuver under (7.14). The dash lines represent the constraint
| |
boundaries ( ei
max
= 0.1). . . . . . . . . . . . . . . . . . . . . . . 113
7.12 Values of ∆(t) and α(t) for the CG selection logic (7.14) corresponding to the maneuver. . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.13 Trajectory in the (x, y) plane under (7.14) (left) in comparison with reference signal (right). . . . . . . . . . . . . . . . . . . . . . . . . 114 7.14 Errors ei (t) in coordination accuracy for the position components x, y and z related to the circumference tracking maneuver under LQ
| |
only. The dash lines represent the constraint boundaries ( ei
max
=
0.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.1
{ } and leader orbiting reference frame {C }. At any time instant, there exists at least one {O } satisfying the small angular displacement assumption with respect to {B }. . . Set of reference frames Oi
i
8.2
118
Switching logic for attitude reconfiguration. . . . . . . . . . . . . . 123
8.3
Reconfiguration for a LEO spacecraft, requiring a large angle maneuver.124
8.4 Relative position y(t) (desired reference in dash lines), control force f 2 (t) and error e2 (t) in coordination accuracy for y(t) (constraint boundaries in dash lines) under HCG. The vertical dash lines indicate instants of switching. . . . . . . . . . . . . . . . . . . . . . . . . . 124 8.5
Euler’s angle φ(t) (desired reference in dash lines), control torque τ 3 (t) and error e3 (t) in coordination accuracy for φ(t) (constraint boundaries in dash lines) under HCG. The vertical dash lines indicate instants of switching. . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.6
Values of ∆(t) and α(t) for the HCG selection logic (7.14), (8.7). . 125
8.7 Relative position y(t), control force f 2 (t) and error e2 (t) in coordination accuracy for y(t) under LQ (constraint boundaries in dash lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
LIST OF FIGURES
vii
Introduction Attitude control has been one of the most interesting and challenging problems in aerospace systems, from the first half of the twentieth century, to present time. A modern spacecraft is a multibody system, composed of rigid bodies connected together by flexible rods. From a mechanical point of view and with the aim of controlling its orientation, however, it can be modeled as a single rigid body. As a consequence, a spacecraft can be thought as a mechanical system with three translational and three rotational degrees of freedom. Moreover, if the satellite orbits at an altitude of approximately 35.786 km above Earth’s surface, it appears stationary. Thus, only rotational motion is observed from Earth. Behavior of space vehicles on Medium (MEO) and Low Earth orbits (LEO) is quite different from the one at GEO altitudes. On a GEO orbit, dynamics is described by attitude models. Moreover, gravity does not affect rotational motion. At lower altitudes, on the contrary, also translational motion, and its effects on rotational motion, have to be taken into account. Indeed, different positions may correspond to different angular velocities, producing changes on attitude. The main goal of this dissertation is modeling and controlling the attitude of spacecraft, orbiting around Earth at GEO, MEO and LEO altitudes. Although on GEO orbits, gravity does not affect motion, we generalize the former case, introducing a new mechanical system with three rotational degrees of freedom in uniform gravity. Such a system, consists of a rigid body constrained to INTRODUCTION
1
rotate about a pivot. We refer to it as 3D Pendulum. Furthermore, we study attitude dynamics and control of rigid bodies traveling on MEO and LEO orbits. To do that, we introduce a mathematical model describing both rotations and translations.
Prior Literature on Spacecraft Attitude Control The idea of using satellites for communication purposes, goes back to 1945, when the science fiction author Arthur C. Clarke, proposed a geo-stationary orbit where placing a mechanical system, suitably oriented with respect to Earth. From that time, a huge literature has been produced on the several aspect related to the motion of space vehicles, as orbital motion and maneuvers, perturbations, and attitude acquisition. Related theory has been summarized in numerous text books on spacecraft motion and control. Two classical texts are mainly referred in this dissertation. The first one, has been published by Kaplan [Kap76] in the seventies. It analyzes spacecraft dynamics. The second one, written by Huges [Hug86] during the subsequent decade, is more focused on attitude. From a more technological perspective, especially related to space vehicles moving on MEO and LEO orbits and to their instrumentations, prominent is the text recently published by Sidi [Sid00]. Astrodynamics, on the other hand, are extensively described in [BMW71]. In particular, it illustrates in detail the classical two body and three body problems. Dynamics on MEO and LEO orbits are strictly connected to this theory. In this thesis, we study attitude control problems. Beyond the peculiar properties of space systems, listed above, we focus our attention on the mechanical properties of rigid bodies. From this point of view, related theory is especially connected to classical mechanics and geometry. Therefore, attitude models described by Goldstein et al. in [GPS80] and geometric properties presented by Marseden et al. in [MR99] and [MRS00] are fundamental. The literature on rotating mechanical systems is wide. One and two degrees of freedom rotating rigid bodies have been analyzed in several works, giving rise to pendulum models. They constitute fundamental background for studying more sophisticated structures, as three degrees of freedom rotating rigid bodies. A comprehensive
survey on pendulum literature has been presented by Furuta in [Fur03]. Related references cover the evolution of corresponding theory. Prominent is the work made by Shirieav et al. in the nineties. [SEL99], [SPLE00] and [SEL04] propose control techniques for 2D pendulum, using conserved quantities of the motion and passivity-based results. We use a similar approach in this dissertation. In particular, we refer to a 3D rotating rigid body and we use reduction theory for modeling the system. Then, we use input-output passivity, for controlling attitude. Reduction theory is described in [MRS00]. Its main aspects have been developed by Bloch et al., for controlling mechanical systems [BLM00], [BLM01]. Connections to the 2D pendulum and to the spinning heavy top, have been illustrated by Simo et al. in [SLRM92], by Wan et al., [WCB95], [WCB95], and by related references. Passivity theory has been extensively described by Ortega et al. in [OSR98]. Passivity control, lies in the more general context of energy-based control. In this thesis we make extensive use of energy-based control and Lyapunov functions. They are particularly suitable in the presence of mechanical systems. Ideas on suitable Lyapunov functions for rotating 3D rigid bodies have been provided by Burkov. In [Bur95], the author proposes a Lyapunov function for controlling the attitude of a spacecraft, with three control torques. On the other hand, control techniques based on the use of only two input torques, have been summarized by Doumtchenko and Tsiotras in [DT00]. Tsiotras et al. have provided prominent work on modeling and controlling underactuated spacecraft. See [CLT00], [WTCB94] and related references. Recent theory in spacecraft attitude dynamics and control, has been focusing its attention on small satellites placed on MEO and LEO orbits. For Earth observing missions and interferometric systems, these structures are more suitable than GEO spacecraft. If large instrumentation is required, the use of distributed systems, shared between vehicles flying in formation, has been suggested. A survey on this new theory has been illustrated by Esper et al. in [ENS+ 03]. Suitable models and control techniques have been analyzed for such systems. Due to their small dimension, constraints play a key role on their dynamics. On this subject, Veres et al. present a Model Based Predictive Control (MPC) technique for a fleet of two LEO nano-satellites, in [VGRM01]. In this disser-
INTRODUCTION
3
tation we propose an attitude control technique also based on MPC. It consists on a suitably modified version of the Command Governor (CG) approach, developed by Bemporad et al. [BCM97] and subsequently adopted in [CMA00] and [ABC+ 03].
Dissertation Outline We treat attitude models and control for rigid spacecraft. Since, dynamics of rigid bodies on GEO orbits are different from dynamics on LEO orbits, this dissertation can be broadly divided into two parts. In the first part, we face the problem of modeling and controlling attitude of geo-stationary satellites. Apart of the presence of gravity, we notice immediately that their motion can be reproduced on Earth, pivoting a rigid body on a fixed point. Thus, in Chapter 1 we introduce the new concept of 3D Pendulum. It consists exactly of a three rotational degrees of freedom rigid body, whose translations are obstructed by the pivot. Its equilibria and their stability properties are described as well. Actually, the 3D pendulum model generalizes GEO spacecraft attitude models. Moreover, it generalizes 2D pendulum, 1D pendulum and spinning top models. These aspects are developed in Chapter 2. Assumptions to be placed on the 3D pendulum model, for obtaining the mathematical description of these classical systems, are analyzed. Furthermore, advantages on using the 3D pendulum model, for controlling the equilibria of 2D pendulum and spinning top, in comparison to prior techniques, are motivated. A new mechanical system, following directly from the 3D pendulum model, is illustrated in Section 2.3. It represents a heavy top, spinning below its pivot. This system has not been studied in previous literature. Subsequently, we face the problem of stabilizing the natural equilibria of a 3D pendulum. They consist of the two configurations where direction of gravity is vertical, with respect to the rigid body. In Chapter 3, we propose passivitybased control laws for stabilizing these configurations in the general case, i.e. when the 3D pendulum is asymmetric. Techniques for stabilizing arbitrary orientations of the 3D pendulum are illustrated in Chapter 4. They represent attitude controllers that can be directly used to stabilize any attitude of geo-
stationary spacecraft. The case when the 3D pendulum is axi-symmetric, is illustrated in Chapter 5. In this case, the control laws we propose, actually stabilize the 2D pendulum and the spinning top. In particular, in Section 5.3 we show that our approach guarantees the use of a continuous state-feedback for swinging up the spherical pendulum. It is a prominent result, if compared to previous literature, where this problem is solved by a swing-up controller, singular at the desired equilibrium, a local stabilizing controller, and a technique for switching between the two. Attitude control of MEO and LEO spacecraft is faced in the second part of this dissertation. In Chapter 6 we describe the main properties of space vehicles orbiting around Earth at constant velocity, under a leader-following approach. The subsequent model, describing both rotational and translational motion, is presented. Model Based Predictive Control techniques, under the assumption of small angular displacements between the satellite and a suitably defined orbital leader reference frame, are illustrated in Chapter 7. They adopt a Command Governor approach, for taking the presence of saturation constraints into account. Indeed, satellites moving on low orbits are usually small. Thus, their thrusters cannot provide large torques. Moreover, we introduce suitable stateaccuracy constraints, usually present in real missions. The importance of considering constraints is the main difference between space systems on high orbits, as geo-stationary spacecraft, and space systems on low orbits. In this dissertation this difference is the watershed between the first and the second part. Finally, in Chapter 8, we remove the small angular displacements assumption. In order to restore results illustrated in the previous chapter, we address our control technique in a hybrid framework, proposing a Hybrid Command Governor controller. It consists of a bank of controller as those presented in Chapter 7, each one designed with respect to a reference attitude, and of a technique for switching between them.
INTRODUCTION
5
Part I
Geo-Stationary Spacecraft Attitude Control
7
Chapter
1
Dynamics of the 3D Pendulum
Three degrees of freedom (3D) rigid bodies have been extensively studied in mechanics and their dynamics equations are well known in literature. They provide the fundamental background for modeling and controlling modern space and underwater vehicles. Simply, a rigid body is a system of particles whose relative distances are fixed with time. In this chapter we introduce a particular kind of 3D rigid body: the 3D pendulum . Its dynamics is strictly related to the one of spacecraft in geo-stationary (GEO) orbits. Pendulum models have provided a rich source of examples in nonlinear dynamics and, in recent years, in nonlinear control. Several different models have been developed to analyze the properties of the uncontrolled planar 1D pendulum and spherical 2D pendulum. However, their extension to a three degrees of freedom p endulum, supported by a fixed frictionless pivot, has never surprisingly been performed. Indeed, very few publications view the pendulum as a rigid body. We first remind the main concepts describing 3D rigid body attitude models. This provides appropriate background for the subsequent introduction of the 3D pendulum. Further, we present both the complete mathematical model of this system and a simplified version of it, particularly suitable for control pur9
poses. We analyze the stability properties of the 3D pendulum, as well. Finally, we introduce a laboratory 3D pendulum developed at the Attitude Dynamics and Control Laboratory, Department of Aerospace Engineering, University of Michigan: the Triaxial Attitude Control Testbed (TACT). It will be useful in the following chapter to verify the models and their stability properties and to assess the performance of control laws proposed in the first part of this dissertation.
1.1 Rigid Bodies Mathematical Models Equations of motion of a 3D rigid body can be determined using standard arguments [GPS80], [KK73]. They are composed by the dynamics and kinematics equations describing the rate of change of the orientation of an inertial frame
{I } = {X , Y , Z} with respect to a body-fixed three versors right-oriented set {B} = {X , Y , Z }.
Figure 1.1. Inertial references frame {I } and body reference frame {B }
The dynamics equations concern the effects of forces on the motion of the body. They can be simply obtained representing the time evolution of the momentum P of the body-fixed reference frame in terms of its angular velocity. Briefly, the
{ }
time derivative of the vector P in B can be written as dP dt
dP = dt I
B
+ω
× P
{ } {}
where ω is the angular velocity of B in I . Now, dP dt
B
= J
dω = J ˙ω dt
(1.1)
with J denoting the constant inertia matrix in the body-fixed reference frame, and dP dt
I
=
d (r dt
dv d(mv) v) + m(r × ) = r × = (r × F ) = τ × mv) = m( dr × dt dt dt
with m representing the total mass of the rigid body, r the position of its center
{}
of mass in I and τ a possibly applied torque. Thus,
J ˙ω = J ω
× ω + τ
(1.2)
Equation (1.2) is referred to as the Euler’s equation and represents the dynamics of the rigid body in the general case. It is usually a simplifying assumption to
{ }
select B aligned with the principals axes of the body, or equivalently, the momentum P parallel to the angular velocity ω. In this case, all the extra-diagonal terms of J are null and the system reduces to three scalar equations. Details can be found in [GPS80], [Kap76] and [Sid00]. Kinematics equations complete the mathematical model of the 3D rigid body. They describe the geometry of the motion apart from its causes. Many different kinematics models have been proposed in mechanics literature and aerospace control. An extensive overview of these models with focus on their main differences can be found in [Hug86]. This text has also strongly inspired the connections between space vehicles control and the 3D pendulum control described in the following chapters. In this dissertation we make use of two of these models. The first one, namely the Poisson’s model , aims at describing the orientation of
{} { }
{ }
the inertial frame I in B , simply expressing its versors in terms of the B
coordinates X , Y and Z . It is useful to control the attitude of satellites in GEO orbits. For this reason, it will b e used hereafter to describe the 3D pendulum. The second model uses the Euler’s angles equations and it is suitable to control the attitude of space vehicles in medium (MEO) and low (LEO) orbits. It will be widely described in the second part of this dissertation.
{} { }
The kinematics model arises from the representation of I in B by a suitable rotation matrix R
{I } = R{B}
(1.3) 11
It follows from the fact that the configuration manifold of a rotating body lies in1 SO(3)
× R3, in the general case [MR99], being SO(3) the group of the
orthogonal matrices with determinant equal to 1:
{ ∈ Λ(3) : RR
SO(3) = R
T
= I, det(R) = 1
}
As described in detail in [Hug86] and in [Hal02], equation (1.3) is not a proper matrix product, but a notation commonly referred to as the vectrix representation . The elements of R represent the cosines directors. Indeed, the ori-
{}
entation of any axes of I can be expressed as a linear combination of the
{B} versors:
X = R11 X + R12 Y + R13 Z , Y = R21 X + R22 Y + R23 Z
and Z = R31 X + R32 Y + R33Z where X = [1, 0, 0]T , Y = [0, 1, 0]T and Z = [0, 0, 1]T in B . Now, applying equation (1.1) to the first axis of I ,
{ }
{}
gives ˙ = X X
× ω = −ωˆX
(1.4)
and analogous for Y and Z . Here ω ˆ is the skew-symmetric matric denoting the cross products between two vectors, such that a
a ˆ=
−
× b = aˆb. It corresponds to
0
−a3
a2
a3
0
−a1
a1
0
a2
(1.5)
Hence, rewriting (1.4) in terms of (1.3), we have directly R˙ = Rˆ ω
(1.6)
that is the kinematics equation of the rigid body. The Poisson’s representation is originated by writing the above expression for any component. However, this representation is redundant because of the six orthogonality constraints. In fact SO(3) is a sub-variety of dimension 3 of R9 [MR99]. The, three variables, corresponding to the three degrees of freedom of the rigid body are enough for describing its orientation in space. Several mathematical models have been developed in order to characterize R using a lower number of variables. The most common is the Euler’s angles kinematic model. It describes R in terms of the angles corresponding to three subsequent 1
The properties of SO(3) affecting control actions on rigid bodies will be discussed in the
following chapters. More details on the topology of SO(3) can be found in [MR99] and [Hat01].
rotations to apply to the inertial frame, to become coincident with the body frame. In this work we consider three successive elementary rotations defined by the yaw angle ψ, the pitch angle θ and the roll angle φ. First, impressing o
Figure 1.2. Sequence of Euler’s angles to be applied to frame {I } for obtaining frame {B } orientation
rotation of angle ψ to the Z axis of I , we have a new reference frame I :
{}
−
{ }
cos(ψ)
{I } = R {I } = ψ
sin(ψ)
0
sin(ψ) cos(ψ) 0 0
0
1
{ } I
Subsequently, we can impress to this new reference frame a second rotation of pitch angle θ giving another frame I and then a third rotation of roll angle
{ } φ to {I }, obtaining {B } = R R R {I }. Thus, R in (1.3) corresponds to cψcθ sψcφ + cψsθsφ sψsφ − cψsθcφ R = R R R = −sψcθ cψcφ − sψsθsφ cψsφ + sψsθcφ (1.7) −sθ cθsφ cθcφ
φ
T ψ
T θ
T φ
θ
ψ
where cx stands for cos(x) and sx stands for sin(x). Equation (1.6) can then be written in terms of Euler’s angles taking the first derivative of (1.7) and ˙ θ˙ and ψ. ˙ It produces three first order differential equations extracting φ, ψ˙
= ωy sin φ sec θ + ωz cos φ sec θ
θ˙
= ωy cos φ
−ω
z
sin φ
(1.8) (1.9)
φ˙ = ωx + ωy sin φ tan θ + ωz cos φ tan θ
(1.10)
known as the Euler’s angles representation . Note that these equations are valid as long as the pitch angle avoid the singularities at
±π/2.
Model (1.2), (1.6) and some variations will be used in the first part of this thesis in order to illustrate the properties of the 3D pendulum and its connections to 13
previous previous literature. literature. It is particular particularly ly suitable for controlling controlling the attitude attitude of GEO spacecraft. On the contrary, contrary, model (1.2), (1.8)-(1.10) is more convenient for attitude attitude control control of MEO and LEO satellites. satellites. It will be mostly mostly used in the second part of this thesis.
1.2 1.2
The The 3D Pen endu dulu lum m A pendulum is a rigid body supported by a fixes, frictionless pivot, acted on by gravitational forces and disturbance and control forces or moments. In the most general general case, we consider consider a pendulum pendulum without without axes of symmetry. symmetry. A schemati schematicc of a rigid pendulum pendulum is shown shown in Figure Figure 1.3. The supporting supporting pivot allows allows three
Figure 1.3. A schematic of a 3D Rigid Pendulum
degrees degrees of rotational rotational freedom freedom to the pendulum. Uniform Uniform constant constant gravity gravity is assumed. It represents the element τ in equation (1.2). Thus, the pendulum is described by at least three spacial coordinates and three rotational coordinates, suggesting the terminology 3D pendulum .
{}
Two coordinate coordinate frames, frames, are of interest interest.. The inertial inertial reference reference frame I has its origin at the pivot, the first two coordinate axes lie in the horizontal plane and the third third coordinate coordinate axis is vertical vertical in the direction direction of gravit gravity y. The origin
{ }
of the body frame B is also located located at the pivot. pivot. In this frame the inertia inertia matrix J is constant and can be simply computed from the moment of inertia of a translated coordinate frame whose origin is located at the center of mass of the pendulum, using the parallel axis theorem [HRW04]. The body-fixed vector from the pivot to the center of mass of the pendulum is denoted by ρ. The symbol g denotes the constant acceleration due to gravity. This is the data on
which the equations of motion are based. The dynamics are given by the Euler’s equation (1.2) that includes the moment due to gravity: J ˙ J ω˙ = J ω
× ω + mgρ × R
T
(1.11)
e3
where m is the total mass of the rigid body and e3 is the third unit vector in the inertial coordinate frame, namely e3 = (0, (0, 0, 1)T . Equation Equation (1.11) (1.11) and the rotational kinematics equation (1.6) define the full dynamics of a rigid pendulum in SO(3) SO (3)
× R3.
It is important to point out that in GEO aerospace applications the dynamics are not affe affecte cted d by gravit gravity y terms terms [Kap76 [Kap76]. ]. Hence, Hence, these these system systemss are special special cases of the above model. In particular, this is the case when the center of mass of the rigid body is located at the pivot, that is ρ = 0. In the contex contextt of the rigid pendulum, we refer to this situation as the balanced case. case. It allo allows ws at viewing the geo-stationary spacecraft attitude control problem as a simplified version version of the 3D pendul p endulum um attitude control control problem. problem. Therefore, Therefore, the focus on
what follows, will be on the more general unbalanced case, viz. ρ = 0. There There are two conserv conserved ed quant quantiti ities es for the rigid rigid pendulu pendulum. m. The first one is the total energy energy, which which is the sum of the rotation rotational al kineti kineticc energy energy and the gravitational potential energy. In addition, there is a rotational symmetry of the equations of motion corresponding to the group of rotations about the vertical line through the pivot. These two results are summarized as follows. Proposition 1.1 The total energy
E = 0.5ω T J ω
− mgρ
T
RT e3
and the compone omponent nt of the angula angularr moment momentum um vecto vectorr about about the vertic vertical al axis axis through the pivot h = ωT J RT e3 are constant along motions of the rigid pendulum. The proof follows by showing that the time derivative of the total energy and the time derivative of the angular momentum component are identically zero. This can be shown shown using the dynamics equations equations (1.11) and the kinematics kinematics equation (1.6). 15
Thanks to the presence of a conserved quantity along the motion, it is possible to obtain obtain a lowe lowerr dimens dimension ional al reduce reduced d model model for the pendulu pendulum. m. In practi practice, ce, the function h can be seen as a constraint imposing the system to lie in a lower dimensional manifold. It allows at expressing one of the variables of the motion with respect to the others. A comprehensive explanation of the constants of the motion for a dynamic system and their effects on the configuration manifold can be found in [MR99]. [MR99]. Here, Here, we introduce introduce a model reduction reduction by exploiting exploiting the fact that the dynamics and kinematics can be written in terms of the reduced attitude vector Γ = RT e3 , which is the unit vector that expresses the gravity
{ }
direction in the body-fixed frame B . Proposition 1.2 The dynamics of the 3D pendulum given by (1.11) and (1.6)
induce a flow on S 2
× R3 given by the reduced dynamics J ˙ J ω˙ = J ω
× ω + mgρ × Γ
(1.12)
and the reduced kinematics Γ˙ = Γ
×ω
(1.13)
The proof follo follows ws from from the definitio definition n of the reduced reduced attitu attitude de vect vector or Γ and
demonstration that Γ = 1. Equations (1.12) and (1.13) are in non-canonical form, but they are very useful for studying the reduced dynamics of the rigid pendulum. Thus, they are used in this dissertation dissertation,, in place of the full dynamical dynamical model. It is worth to point out that the reduced model describes the orientation of the 3D rigid pendulum modulo a rotation along its third axis Z . Howeve However, r, this is not a limitation limitation in spacecraft control problems, where usually, the task is of controlling the direction of an axis, e.g. represented by an antenna, instead of the complete attitude. Finally, equations (1.12) and (1.13) can also be expressed in terms of the Euler’s angles noticing that Γ(θ, Γ(θ, φ) =
− sin θ sin φ cos θ cos φ cos θ
(1.14)
In fact, equations (1.9)-(1.10) and (1.13) using (1.14) are exactly an expression of the reduced dynamics of motion of the rigid pendulum on S 2
× R3 expressed
in terms of the Euler’s angles. Further, the kinematics can b e shown to be independent from the yaw angle ψ [MR99].
1.3
Equilibria of the 3D Pendulum To further understand the dynamics of the 3D pendulum, we study its equilibria. To simplify the analysis it is convenient to choose the body-fixed frame so that its third axis Z is aligned with the vector from the pivot to the center of mass. Thus, in B , ρ = (0, 0, l)T , where l > 0. Consequently, the gravity terms that
{ }
appear in equation (1.11) can be considerably simplified. The equilibrium solutions for the 3D pendulum satisfy ω = 0, so that the attitude of the pendulum is constant. Substituting ω = 0 in equation (1.11), we obtain ρ
×R
T
e3 = 0
The above equation implies that the 3D pendulum is in equilibrium only if the center of mass vector ρ is collinear with the gravity vector RT e3 in the body-fixed coordinate frame. This implies that ReT e3 = λe3 where λ =
±1.
For λ = 1, the center of mass vector ρ points in the direction
of the gravity vector. Similarly, for λ =
−1, the center of mass vector ρ points
in the direction opposite to the gravity vector. Now, let Re denote an attitude rotation that solves equation (1.11) and define Γ e = ReT e3 . Then, any attitude in the configuration manifold given by
{R ∈ SO(3)
¯ ψ¯ arbitrary : R = Re Q(ψ),
}
defines an equilibrium attitude corresponding to ω = 0 where
−
cos ψ¯
¯ = Q(ψ)
λ sin ψ¯ 0
λ sin ψ¯ 0 cos ψ¯
0
0
1
Hence, if Re defines an equilibrium attitude for the 3D pendulum, then a rotation of the pendulum about the gravity vector by an arbitrary angle is also an 17
equilibrium. Consequently, in SO(3) there are two disjoint equilibrium manifolds of the 3D pendulum. We define the manifold corresponding to λ = 1 in the above description as the hanging equilibrium manifold , since the center of mass is below the pivot for each attitude in the manifold. The manifold corresponding to λ =
−1 in the above description is defined as the inverted equilibrium
manifold , since the center of mass is above the pivot for each attitude in the manifold. Next, we study the equilibria of the reduced equations (1.12) and (1.13). As before, the equilibria of the system satisfies ω = 0. Substituting in (1.12), we get ρ
× Γ = 0.
Thus, we obtain two isolated equilibrium solutions of the reduced equations in S 2
× R3 given by Γ = e3, ω = 0, and by Γ = −e3, ω = 0.
Alternatively, the
two equilibrium solutions can be expressed using Euler’s angles as θ = 0, φ = 0, ω = 0 and θ = 0, φ = π, ω = 0. For each of these representations of the equilibrium solutions, the first corresponds to the hanging equilibrium where the center of mass is below the pivot and the second corresponds to the inverted equilibrium where the center of mass is above the pivot. The stability of the two isolated equilibrium solutions of the reduced equations of motion is readily assessed. The stability of the two families of equilibrium manifolds of the full equations of motions follows. The stability analysis can then be based on the reduced model. Proposition 1.3 The hanging equilibrium of the reduced dynamics of the 3D
pendulum described by equations (1.12) and (1.13) is locally stable in the sense of Lyapunov. Proof. The hanging equilibrium corresponds to the configuration ω0 = 0 and Γ0 = (0, 0, 1)T . Consider the Lyapunov function V (ω, Γ) =
1 T ω Jω + mg(l 2
T
−ρ
Γ)
V (ω0 , Γ0 ) = 0 and V (ω, Γ) > 0 elsewhere. The derivative of V along the trajectories is ˙ = ωT J ˙ω V = ωT
− mgρ Γ˙ = ω (Jω × ω + mgρ × Γ) − mgρ (Γ × ω) mgρ × Γ − mgρ Γ × ω = mg(ω ρ × Γ − ρ Γ × ω) ≡ 0 T
T
T
T
T
T
where the last equality follows from the mixed vector product property [HRW04]. Hence, V is a continuously differentiable function, positive definite and such ˙ that V (ω0 , Γ0 ) = 0 and2 V (ω, Γ)
≤ 0 ∀ (ω, Γ).
Thus, the hanging equilibrium
is locally Lyapunov stable [Kha01].
Proposition 1.4 The inverted equilibrium of the reduced dynamics of the 3D
pendulum described by equations (1.12) and (1.13) is locally unstable. Proof. The hanging equilibrium corresponds to the configuration ω0 = 0 and Γ0 = (0, 0, 1)T . Consider the Lyapunov function
−
1 V (ω, Γ) = ω T J ω + mg(l + ρT Γ) 2 V (ω0 , Γ0 ) = 0 and V (ω, Γ) > 0 elsewhere. The derivative of V along the trajectories is ˙ = ωT J ˙ω + mgρT Γ˙ = ω T (J ω V = ωT mgρ
× Γ + mgρ
T
× ω + mgρ × Γ) + mgρ Γ × ω = 2mgω (ρ × Γ)
T
(Γ
× ω)
T
where the last equality follows from the mixed vector product property [HRW04]. ˙ is not definite. Hence, V is a continuously differentiable function such that V ˙ V (ω0 , Γ0 ) = 0 and V (ω, Γ) > 0, V (ω, Γ) > 0 for some points (ω, Γ) arbitrarily close to (ω0 , Γ0 ). Thus, by Chetaev’s theorem [Kha01], the inverted equilibrium is unstable.
As a consequence of the two propositions above, one can conclude that the hanging equilibrium manifold of the full dynamics of the 3D pendulum is stable in the sense of Lyapunov and that the inverted equilibrium manifold of the full dynamics of the 3D pendulum is unstable in the sense of Lyapunov.
1.4 Triaxial Attitude Control Testbed The test-bed used for experiments in the first part of this dissertation is based on the spherical air bearing manufactured by Space Electronics, Inc., Berlin, 2
˙ ≡ 0 could be expected, since V defines the total energy of the Notice that the result V
system, that is a constant of the motion, as described in section 1.2.
19
CT, depicted in Figure 1.4. It is available at the Attitude Dynamics and Control Laboratory, Department of Aerospace Engineering, University of Michigan. We refer to this systems as the Triaxial Attitude Control Testbed (TACT).
Figure 1.4. Triaxial Air Bearing Testbed.
An aluminum sphere of diameter 11 inch floats on a thin film of air that exits holes located in the surface of the cup. Air at high constant pressure is supplied to the cup by means of a hose that passes through the center of the vertical support. Once the main components are mounted, additional masses can be added to modify the final mass distribution. Hence, the center of mass can be balanced at the rotational center (no effects from the gravity), in order to simulate a geo-stationary spacecraft, or in any other position, representing a 3D unbalanced rigid pendulum. The spherical air bearing guarantees low friction and three-dimensional motion with unrestricted roll and yaw and a maximum
±45 degrees pitch angle. A comprehensive description of the hardware related to the TACT has been provided by [BMB01]. Mathematical models for the TACT have been introduced in [CSMB01] and [CSM03]. There, a description equivalent to (1.11) and (1.6), is obtained by means of the Lagrangian equations. This geometrical approach to mechanical systems, well known in the literature, is widely described in [MR99] and [MRS00]. As illustrated in Figure 1.4, a one-piece shaft passes through the center of the sphere and extends between a pair of circular mounting plates. Extension shafts connect the circular mounting plates to larger square mounting plates. Circular
plates are used for supporting hardware instrumentation, while square plates are used for supporting fan thrusters, batteries and additional weights. Indeed, weights can be attached to the TACT, thanks to access holes cut into the plates. This allows at modifying the inertia matrix J of the TACT, before performing experiments. Once the entire system is set up, the procedure described in [CBA+ 06] can be adopted, to identify the inertia matrix of the system.
1.5
Conclusions In this chapter we have introduced the new concept of 3D rigid pendulum. It represents a three rotational degrees of freedom rigid body, supported by a frictionless pivot. We have presented the 3D pendulum model in the general unbalanced case. The special balanced case, corresponding the absence of gravity terms in the system, encompasses the geo-stationary spacecraft dynamics model. Therefore, we have addressed the problem of controlling the attitude of GEO satellites, as a simplified version of the 3D pendulum attitude control problem.
21
Chapter
2
Special Cases of the 3D Rigid Pendulum The published literature on pendulum models is very large, since they are useful for both pedagogical and research reasons. Indeed, they represent physical mechanisms that can be viewed as simplified versions of mechanical systems that arise in robotics and spacecraft. Standard pendulum models are defined by a single rotational degree of freedom, referred to as a planar rigid pendulum, or two rotational degrees of freedom, referred to as a spherical rigid pendulum. Control problems for planar and spherical pendulum models have been studied in many references from the seventies [MNF76], to the last decade. [BS96], [GD99], [AF00] and [Ang01] are only a few examples of the huge literature on this subject. A comprehensive survey on this literature is presented in [Fur03]. Prominent is the work made by Shirieav et al. in the last decade: [SEL99], [SPLE00] and [SEL04]. In these references, some geometrical concepts are applied to a spherical pendulum, suggesting passivity-based control laws [OSR98]. In this chapter we show that the classical mathematical models of the 1D pendulum, the 2D pendulum and the heavy spinning top, can be simply viewed as special cases of the 3D pendulum model described in Chapter 1. We first intro-
duce assumptions to be placed on model (1.12), (1.13) for obtaining a spherical pendulum model and then a planar pendulum model. Further, we show that, with simple assumptions, the 3D pendulum model can describe the heavy top dynamics. The corresponding dynamics seem not to have been previously studied. Finally, we study the stability properties of this new spinning top model, illustrating connections with the classical heavy top problem, widely discussed in [GPS80].
2.1 Axisymmetric 3D Rigid Pendulum We select the body-fixed reference frame B such that its versors X , Y and
{ }
Z are aligned with the principal axes of the 3D rigid body. This renders the extra-diagonal terms of the inertia matrix J null. In this case, the diagonal terms of J are referred to as the principal moments of inertia . Hence, model (1.12) using (1.14), reduces to three scalar equations, as described in Chapter 1. Moreover, we consider the axisymmetric case, where two of the principal moments of inertia of the pendulum are identical and the pivot is located on the axis of symmetry of the pendulum. This is an usual assumption while facing
Figure 2.1. A schematic of a 3D Rigid Pendulum with one axis of symmetry
attitude control problems for underactuated geo-stationary spacecraft [DT00]. In particular, we assume that the body-fixed axes are selected so that J = diag(J t , J t , J a ) and ρ = le3 where l is a positive scalar constant. Consequently, 23
equation (1.12) using (1.14), can be written in scalar form, as
− J )ω ω − mgl cos θ sin φ = (J − J )ω ω − mgl sin θ
J t ω˙ x = (J t
a
y
z
J t ω˙ y
t
z
x
a
(2.1)
J a ω˙ z = 0 From the last equation in (2.1) we see that the component of the angular velocity vector about its axis of symmetry is constant. This means that a constant ωz defines an invariant manifold for the pendulum dynamics. The resulting reduced equations of motion, for constant ωz , can be represented by equations (2.1) and (1.9)-(1.10)
− J )ω ω − mgl cos θ sin φ J ω˙ = (J − J )ω ω − mgl sin θ θ˙ = ω cos φ − ω sin φ
J t ω˙ x = (J t t
y
a
y
z
t
z
x
a
y
(2.2)
z
φ˙ = ωx + ωy sin φ tan θ + ωz cos φ tan θ Remember that the yaw motion of the pendulum is described by the kinematic equation (1.8). Hence, the evolution of ψ can be deduced from the evolution of the other variables of motion. Now, the special case ωz
≡ 0 leads to an axisymmetric rigid body whom atti-
tude dynamics are described by two angles and two angular velocities, namely, a 2D spherical pendulum .
Proposition 2.1 Assume the rigid pendulum has a single axis of symmetry and
the pivot is located on the axis of symmetry of the pendulum. The equations of motion of the rigid pendulum define an induced flow on the manifold S 2
× R2
corresponding to ωz = 0, given by the equations
−mgl cos θ sin φ = −mgl sin θ
J t ω˙ x = J t ω˙ y
θ˙ = ωy cos φ
(2.3)
φ˙ = ωx + ωy sin φ tan θ These equations are said to represent a 2D spherical pendulum. They exhibit singularities for θ =
±nπ/2.
It is worth to point out that the above representation is not the one commonly
used for describing the dynamics of the spherical pendulum. Indeed, in literature the 2D p endulum is not introduced as a rotating rigid body. Usually, the mathematical description of the system follows from the definition of the two
{}
angles q 1 and q 2 in I depicted in Figure 2.2 [SEL04]. There, q 1 is the angle
Figure 2.2. Classical representation of the 2D spherical pendulum
between the vertical and the axis of symmetry of the pendulum and q 2 is the angle representing the displacement of the center of mass of the pendulum with
{} sition of the center of mass in {I } coordinates and taking the second derivative respect to axis X of frame I . Then, the model is obtained by writing the po-
or, more simply, by writing the Lagrangian equations of the system d ∂ L ( ) dt ∂ ˙q i Here, the Lagrangian1
− ∂ ∂q L = 0
(2.4)
i
L is
L =
T
− V = 12 ml2[q ˙12 + (sin2 q 1)q ˙22]
where T and V are, respectively, the kinetic and the potential energy. Thus, the model of the 2D spherical pendulum is q¨ 1 = sin q 1 (cos q 1 )q ˙22 q¨ 2 =
q 1 q ˙1 q ˙2 −2 cos sin q 1
− gl sin q 1
(2.5)
This is the description commonly used for representing the dynamics of the 2D spherical pendulum. The fundamental difference with respect to model (2.3) 1
Lagrangian approach is widely used in mechanical systems modeling. An extensive de-
scription of this technique can be found in [GPS80], [HRW04] and many other references discussing rational mechanics.
25
is that now the singularities are in q 1 =
±nπ, i.e.
in the upward position.
This leads to several disadvantages while designing control laws for swinging up the spherical pendulum, that is a basic problem in the literature related to attitude control. For instance, in [SEL99] and [SEL04] the authors use model (2.5) and they switch to a different model in the close proximity of the upward position. It produces a discontinuous representation of the system with many related difficulties. As it will be seen in the following chapters, it could simply be avoided, by selecting the more suitable description of the mechanical system (2.3), arising from the interpretation of the 2D spherical pendulum as a rotating rigid body. Now, beyond the assumption above of ωz = 0, consider the case where ωx = 0 and φ = 0. It is straightforward to show that this leads to an invariant dynamics of the axisymmetric rigid pendulum, described as follows: Proposition 2.2 Assume the rigid pendulum has a single axis of symmetry and
the pivot is located on the axis of symmetry of the pendulum. The equations of motion of the rigid pendulum define an induced flow on the manifold S 1
× R1
corresponding to ωx = 0, ωz = 0, and φ = 0, given by the equations J t ω˙ y =
−mgl sin θ
(2.6)
θ˙ = ωy These equations are said to represent a 1D planar pendulum .
Thus, for an axially symmetric pendulum with the pivot located on the axis of symmetry, the well known 2D spherical pendulum and 1D planar pendulum, can be viewed as special cases of the 3D rotating rigid pendulum. On the other hand, if the pendulum is asymmetric, then its dynamics are general, in the sense that neither the spherical pendulum dynamics, nor the planar pendulum dynamics, arise as special cases.
2.2 Spinning Axisymmetric 3D Rigid Pendulum The 2D spherical pendulum has been shown to originate from the 3D axisymmetric rigid pendulum, under the assumption of ωz
≡ 0. The induced dynamics
corresponding to a nonzero value of ωz is fundamentally different from this case.
These dynamics seem not to have been previously studied. Thus, consider again an axisymmetric pendulum with the pivot located on its axis of symmetry. Its dynamics are described by model (2.2). Assume
ωz = k = 0. Since the third equation in (2.1) holds true, we have the following result: Proposition 2.3 Assume the rigid pendulum has a single axis of symmetry and
the pivot is located on the axis of symmetry of the pendulum. For any constant
k = 0, the equations of motion of the rigid pendulum define an induced flow on the manifold S 2
× R2 corresponding to ω = k, given by the equations J ω˙ = (J − J )k ω − mgl cos θ sin φ J ω˙ = (J − J )k ω − mgl sin θ θ˙ = ω cos φ − k sin φ z
t
x
t
a
y
t
y
a
t
x
(2.7)
y
φ˙ = ωx + ωy sin φ tan θ + k cos φ tan θ These equations represent the dynamics of a symmetric rigid body spinning steadily around its axis of symmetry under the effects of gravity, namely, an heavy spinning top [GPS80]. In particular, differently from previous literature, they extend the concept of heavy spinning top to the space “under the table”. It is depicted in Figure 2.3.
Figure 2.3. A schematic of a heavy spinning top not constrained by the table
Thus, the motion of the symmetric top can be seen as a special case of the 3D rigid pendulum dynamics. Problems related to the symmetric spinning top have been extensively studied in 27
mechanics. A wide discussion on the properties of the motion, on the conditions for precession and on the stability of the upward configuration, can be found in [GPS80]. Several papers have analyzed the properties of its dynamics as well, with particular focus on stabilizing the sleeping motion. On this subject, see [WTCB94], [WCB95] and related references. The sleeping motion consists of the situation where the top stands in the vertical configuration, steady rotating. Furthermore, the dynamics of the heavy top represent basic background in the literature of astrodynamics, for instance, to explain the Earth wobble and the precession of equinox [BMW71]. Surprisingly, none of the published works have extended the motion of the heavy top to the case where there is no table, i.e. when the top is also allowed to move under the horizontal plane. From this point of view, current theory can be considered incomplete. Further, proposed control laws for the vertical upward equilibrium, defined global as in [WCB95], are actually not global. This is due to the presence of another equilibrium for the system, represented by the downward vertical configuration. In the language introduced for the 3D rigid pendulum, this configuration consists of the hanging equilibrium of the reduced dynamics (1.12) and (1.13). On the other hand, the upward equilibrium configuration, consists of the inverted equilibrium. In order to complete the description of the dynamics of the spinning top in the new context of the 3D rigid pendulum, we study the stability of its hanging equilibrium. An almost global attitude control law for the system, will be proposed in the following chapters.
2.3
Hanging Equilibrium of the Symmetric Spinning Top For any constant velocity ωz = k, system (2.7) has two equilibria on S 2
× R2.
They are given by the two states where ωx = ωy = 0 and angles θ and φ corresponding to the upward and the downward configuration. The first one is described by θ = 0, φ = π or θ = π, φ = 0 and it is referred to as the inverted equilibrium. The second one is described by θ = 0, φ = 0 or θ = π, φ = π
and it is referred to as the hanging equilibrium. As introduced above, only the inverted equilibrium has been studied in classical mechanics, restricting the
∈
analysis to the space above the horizontal plane: θ
(π/2, 3π/2). However,
model (2.7) describes the motion on the entire space S 2
× R2, provided that
angle θ does not cross the horizontal: θ = π/2 and θ = 3π/2. Thus, we study the properties of the second equilibrium, located below the horizontal plane. It aims at completing theory related to the heavy symmetric top. In order to analyze the stability of the hanging equilibrium, we make use of a different model with respect to (2.7), representing a 3-2-1 formulation of the Euler’s angles. The reason of this selections is motivated by the fact that in model (1.8)-(1.10), the sleeping motion does not correspond to an isolated equilibrium. Indeed, when the first rotation is around axis Z , the second and the third Euler’s angles are not distinguishable. It could be expected noticing that each equilibrium is described by two different configurations. In this context, the problem can be overcame by using a 2-1-3 Euler’s angles formulation, as the one previously presented in [WCB95]. There, the inertial
{}
frame I attitude is described by three angles corresponding to three subsequent rotations, where the first one is around Y , the second one is around the first axis of the new frame and the last one is around the third axis of the obtained frame. In [WCB95] an angle ψ is introduced, corresponding to the first rotation about Y . Then, the new frame is rotated of angle θ about the first axis. Resulting reference frame is referred to as of the
F -frame
F .
Finally, a positive rotation
of angle θ about its third axis, results in the body-fixed frame.
Thus, F precesses, but does not spin with the top. The angular velocity of the 3D rigid body can then be expressed in
F ,
˙ ψ˙ cos θ, φ˙ ω = [θ,
giving
− ψ˙ sin θ]
T
It allows at writing the kinetic and potential energy in
(2.8) F
coordinates and then
at following the Lagrangian approach (2.4). The resulting model is
J t θ¨ + J t ψ˙ 2 cos θ sin θ + pφ ψ˙ cos θ
− mgl sin θ cos ψ = 0 J ψ¨ cos2 θ − 2J ψ˙ θ˙ cos θ sin θ − p θ˙ cos θ − mgl cos θ sin ψ = 0 t
t
(2.9)
φ
29
where pφ is the constant generalized momentum corresponding to φ: pφ =
∂ L = J t (φ˙ ∂ φ˙
− ψ˙ sin θ)
The rational under description (2.9), depending only on two of the three Euler’s angles, is that φ does not appear in the Lagrangian [WCB95], so it is an ignorable variable. As a consequence, the Lagrangian can be replaced by the Routhian R , ˙ Reduction theory, useful in the presence of symmetries defined by R = L pφ φ.
−
in mechanical systems, is extensively described in [MRS00]. Now, letting x1 = θ
x2 = θ˙
b=
pφ J t
x3 = ψ c=
x4 = ψ˙
2mgl J t
(2.10)
we obtain a state space representation for model (2.9): x˙ 1 = x2 2
x˙ 2 =
− x24 sin(2x1) − bx4 cos x1 + 2c sin x1 cos x3
(2.11)
x˙ 3 = x4 x˙ 4 = (2x2 x4 sin x1 + bx2 +
c 1 sin x3 ) 2 cos x1
Notice again the presence of singularities, here for θ = nπ/2, n
∈ Z.
As ex-
pected, the equilibria of the system correspond now to isolated points represented by x = [iπ, 0,jπ, 0]T . This results in the inverted equilibrium, i.e. the
−
upward vertical sleeping motion, when i j is even. Otherwise, it corresponds to the hanging equilibrium. In [WCB95] a linearized analysis around the inverted equilibrium xi = [0, 0, 0, 0]T is performed, to show that it is unstable for b2 < c. It means that the rotation of the top needs to be fast enough, to guarantee a vertical configuration. An energy based equivalent proof is provided in [GPS80]. Here, we use a similar approach, defining a suitable Lyapunov function. Indeed, it is easy to show that the linearized model around xh does not provide information on its stability, being the eigenvalues complex conjugates. Thus, we introduce a Lyapunov function based on the conserved quantities of the motion embedded in (2.11): total energy and vertical momentum. This way to proceed, previously suggested by [SLRM92], is referred to as the Energy-Casimir
method. It consists of selecting a Lyapunov function composed by Casimir functionals, i.e. differences between conserved quantities and their minimum values. Now, the total energy of (2.9) is given by E =
1 1 J t (θ˙2 + ψ˙2 cos2 θ) + J a (φ˙ 2 2
− ψ˙ sin θ)2 + mgl cos θ cos ψ
Disregarding the kinetic energy related to the rotation around the spinning axis and dividing by J t we obtain a constant of the motion: hc1 (x) =
1 2 c (x2 + x24 cos2 x1 ) + cos x1 cos x3 2 2
(2.12)
expressed in the state space variables defined above. Further, the vertical momentum is given by h = ω T J Γ with ω defined by (2.8) and Γ as introduced in Chapter 1, but expressed in
Γ=
1
0
0
0
cos θ
sin θ
0
− sin θ
cos θ
Hence,
F
coordinates: cos ψ
0
− sin ψ
0
1
0
sin ψ
0
cos ψ
0 0 1
=
−
sin ψ
sin θ cos ψ cos θ cos ψ
−J θ˙ sin ψ + J ψ˙ cos θ cos ψ sin θ + J 3(θ˙ − ψ˙ sin θ)cos θ cos ψ = = −J θ˙ sin ψ + J ψ˙ cos θ cos ψ sin θ + J b cos θ cos ψ
h=
t
t
t
t
t
Dividing by J t and using the state space representation, we have a second constant of the motion: hc2 (x) =
−x2 sin x3 + x4 sin x1 cos x1 cos x3 + b cos x1 cos x3
(2.13)
Therefore, a suitable Energy-Casimir based Lyapunov function for the system is V (x) = [hc1 (x)
− h 1(x )] + 2b [h 2(x) − h 2(x )] c
h
c
c
h
Consequently, stability of the hanging equilibrium can be studied.
Proposition 2.4 For any constant spinning velocity ωz = k = 0, the hanging
equilibrium of the axisymmetric rigid pendulum (2.7), is locally stable in the sense of Lyapunov. 31
Proof. Consider description (2.11) of the system and the following EnergyCasimir based Lyapunov function V (x) = [hc1 (x)
− h 1(x )] + 2b [h 2(x) − h 2(x )] c
h
c
c
h
1 c c b = (x22 + x24 cos2 x1 ) + cos x1 cos x3 + + ( x2 sin x3 2 2 2 2
−
(2.14)
+ x4 sin x1 cos x1 cos x3 + b cos x1 cos x3 + b) where b and c are defined by (2.10). ˙ V (xh ) = 0 and V (x)
≡ 0 ∀ x.
Indeed, (2.14) is composed by two constants of
the motion. Moreover, consider the Hessian matrix of V (x) evaluated in xh :
¨ h) = V (x
Its eigenvalues are λ1,2 =
−
λ3,4 = +
c+b2
2
0
0
1
0
b
b
−2
b
2
2 c+b2 2
0
0
√ b4 + 2b2c + c2 − 4c + 4 √ b4 + 2b2c +4 c2 − 4
b
−2
0
0 0 1
(2.15)
b2 + c + 2 4 2 4c + 4 b + c + 2 + 4 +
Simple manipulations show that λ1,2 can be negative only if b2 + 2c < 0, but this is clearly impossible, since c = 2mgl/J t > 0. Moreover, λ3,4 are always positive. Thus, all the eigenvalues of the Hessian matrix evaluated in xh are ¨ h ) is positive definite. Therefore, the hanging equilibrium is positive, i.e. V (x
Lyapunov stable [Kha01].
As a consequence of the above proposition, the inverted equilibrium cannot be globally stabilized in S 2
× R2.
Finally, in [GPS80] it is shown that the heavy top is in steady precession when p2φ
≥ 4J mgl cos θ cos φ t
Thus, since any attitude of the top below the horizontal plane is such that cos θ cos φ < 0, there the precession is always steady, no matter is the angular velocity. These dynamics are fundamentally different from those of the spherical pendulum. They open new interesting theoretical problems that seem not to be previously studied, as the research of possibly existing periodic trajectories.
2.4
Conclusions In Chapter 1 we have introduced a 3D rigid pendulum mathematical model encompassing geo-stationary spacecraft descriptions. In this chapter we have shown that this model actually embodies many other classical problems. Indeed, the 2D spherical pendulum, the 1D planar pendulum and the symmetric heavy top have been shown to represent special cases of model (1.12)-(1.13). Thus, the study of the 3D pendulum includes the study of these classical problems. Moreover, we have extended the classical description of the heavy top to the space below the horizontal plane, giving rise to novel dynamics. Finally, we have analyzed the equilibria of this new inverted top illustrating connections with current literature.
33
Chapter
3
Stabilization of an Asymmetric 3D Rigid Pendulum
In the previous chapters we have introduced several different classifications of an uncontrolled 3D rigid pendulum supported by a frictionless pivot, both in the asymmetric and axisymmetric case. Related theory is basic in spacecraft dynamics and control problems, since an asymmetric balanced 3D rigid pendulum, actually represents a generic geo-stationary satellite [Hug86], [Kap76]. Moreover, assumptions on the symmetry of the rigid body, have been taken in the literature, while facing attitude control problems for underactuated spacecraft. Indeed, a rigid body with only two control inputs satisfies the fundamental Brokett’s condition [BMS83] for continuous feedback stabilization, only if it is symmetric. Consequent model coincide with the axisymmetric balanced 3D rigid pendulum developed in Chapter 2. In this chapter we focus the attention on an asymmetric 3D rigid pendulum and we study the stabilization of its reduced model. For stabilizing the reduced model of the 3D pendulum we mean stabilizing its hanging equilibrium, i.e. pointing the Γ vector downwards. First, we make control actuation assumptions for the system, reminding briefly
some alternatives to our selection. Then, we present a simple controller consisting of angular velocity feedback. The related control law is based on damping injection and it is shown to have a dense domain of attraction. Damping in jection, as well as passivity based control [OSR98], is particularly suitable for mechanical systems, as the TACT presented in Chapter 1. Afterwards, we show experimental results on the TACT, confirming the effectiveness of the proposed controller.
3.1 3D Rigid Body Control Torques So far, we have considered the mathematical model of an uncontrolled 3D rigid body supported by a frictionless pivot, making assumptions on its geometrical structure and analyzing its dynamics. In order to consider control inputs to the system, the dynamics equation have to be modified for including moments. In particular, (1.11) can be modified as J ˙ω = Jω
× ω + mgρ × R
T
e3 + M
(3.1)
where M represents the vector control moment on the 3D rigid pendulum, expressed in the body-fixed coordinate frame. The specific form of the control moment depends on the actuation assumptions, i.e. on the hardware selected for providing torques to the system. Several different actuation structures have been studied in the literature. One important example of control actuation assumption, generalizing the development in [AF00], is the pivot acceleration in the control input. In this case the control moment is M = mρ
×R
T
u
where u denotes the pivot acceleration, expressed in the inertial coordinate frame. Such a control actuation assumption typically breaks the symmetry of the uncontrolled rigid pendulum dynamics. Consequently, the angular momentum component about the vertical axis through the pivot is not conserved and it is not possible to express the controlled dynamics in terms of the reduced attitude Γ. Another control actuation assumption, generalizing the development in [CSMB01] 35
and [CM02], is that proof mass actuators are rigidly mounted on the 3D pendulum. Although the model presented in these references include actuation dynamics, it is possible to give a simple expression for the control moments as M = mgu
×R
T
e3
ignoring the dynamics of the proof actuators. Here, the control input u denotes the incremental displacement vector due to the proof mass motion, so that the position vector of the center of mass of the total body is ρ + u, expressed in the body-fixed coordinate frame. In this case, the mass m denotes the total mass of the 3D pendulum and the proof mass actuators. It is easy to show that this type of control actuation does not invalidate conservation of the angular momentum component about the vertical axis through the pivot. Hence, the resulting controlled dynamics can still be expressed in terms of the reduced attitude vector Γ. However, such a kind of control actuation is useful only with unbalanced systems, since it makes use of the gravity vector. Thus, it is not suitable for geo-stationary spacecraft control problems. Other actuation schemes, based on reaction wheels, magnetic torques, precession and spin stabilization have been proposed. This argument is a constant subject of study. In this dissertation, besides these several actuation systems, we consider the case where M is directly provided by thrusters, as the gas jet actuators usually mounted on geo-stationary satellites. This way to proceed is inspired by the control test-bed presented in Chapter 1. There, four fan thrusters are mounted on one of the two external plates, as illustrated in Figure 1.4. Combining their forces, these thrusters are able to provide roll, pitch and you torques. Defining as F i , where i = 1,
··· , 4, the forces provided by each fan, τ
roll , τ pitch
and τ yaw
can be calculated. Their structure is depicted in Figure 3.1, where θ is the angle between the fans position and the center of mass of the rigid body and r is the distance between the center of mass of the TACT and each fan thruster. With simple calculations it is possible to show that the torques applied to the system by the fans are τ roll = (rF 1
−(rF 3 + rF 4)cos θ = −(rF 1 + rF 2 )cos θ
τ pitch = τ yaw
− rF 2 + rF 3 − rF 4)sin θ
F2 Y
r F3
X
q
CM
q Z
F4 F1
Figure 3.1. Fan thrusters scheme of the Triaxial Attitude Control Testbed.
In the case of the TACT, forces F i are generated applying suitable voltages to the fan thrusters. The relations between torques and voltages are described in [CBA+ 06]. They allows at expressing control inputs to the TACT in terms of voltages to be applied to the fans. Results obtained in this reference are promising for continuing fruitful research on the control test-bed. Beyond the relations between F i and the torques in the roll, pitch and yaw directions, we assume here that inputs to the 3D rigid body are directly the angular momentum corresponding to τ roll , τ pitch and τ yaw . Hence, the dynamics model extending (1.11) is simply J ˙ω = Jω
T
× ω + mgρ × R
e3 + u
Moreover, RT e3 can be replaced by Γ, since the angular momentum component about the vertical axis through the pivot is conserved along the trajectories of rigid body. Therefore, the control model for the fully actuated 3D rigid pendulum on which we focus our attention is
× ω + mgρ × Γ + u Γ˙ = Γ × ω
J ˙ω = Jω
(3.2)
∈ R3 is the angular velocity of the body, Γ ∈ S 2 is the reduced attitude vector representing the direction of gravity in {B } and u ∈ R3 is the where ω
angular momentum input vector, and where all the variables are expressed in
{B} coordinates. 37
3.2 Asymptotic Stabilization of the Hanging Equilibrium In this section, we develop a simple controller that makes the hanging equilibrium of the asymmetric 3D pendulum asymptotically stable. The controller is based on the observation that the control model given by equation (3.2) is input-output passive if the angular velocity is taken as the output. Passivity based control is a natural way to proceed in the presence of mechanical systems, as well as in the presence of electrical systems [OSR98]. Briefly, a system with state x, input u and output y is dissipative with respect to a supply function s(u(t), y(t)) if there exists a function H such that T
H (x(T ))
≤ H (x(0)) +
s(u(t), y(t))dt
(3.3)
0
Such a function is referred to as the storage function . When the supply rate is s(u, y) = uT y, the system is said input-output passive . In this case, it is immediate to notice that dH dt Thus, selecting u =
T
≤u
y
−y, gives a closed-loop system where dH/dt ≤ 0.
There-
fore, if H is the total energy, u drives the system to the minimum of its energy, corresponding to a stable equilibrium. In the case of the 3D rigid pendulum, the total energy 12 ωT Jω
− mgρ
T
Γ can be
selected as a storage function. Since it has a minimum at the hanging equilibrium (0, Γh ), a control law based on angular velocity feedback is suggested. Let Ψ : R3
→ R3 be a smooth function such that 2 ≤ x
1 x where 2
≥ 1
T
Ψ(x)
≤ 2x2, ∀x ∈ R3
(3.4)
> 0. Thus, we propose a class of controllers, referred to as
damping injection controllers, given by u=
−Ψ(ω)
(3.5)
where Ψ satisfies (3.4). We next show that the above family of controllers, which requires only angular velocity feedback, renders the hanging equilibrium
of a 3D asymmetric pendulum asymptotically stable, with a guaranteed domain of attraction. Lemma 3.1 Consider the fully actuated 3D asymmetric pendulum given by
(3.2). Let Ψ :
R3
→ R3 be a smooth function satisfying (3.4) and choose a
controller in the form of (3.5). Then, the hanging equilibrium (0, Γh ) is asymptotically stable.
∈ (0, 2mgl), all solutions of the closed-loop system given by (3.2) and (3.5), such that (ω(0), Γ(0)) ∈ H, where Furthermore, for every
H=
(ω, Γ)
3 R
∈ ×
1 1 S : ω T Jω + mgl Γ 2 2
ρ l
T
ρ l
− − ≤
2
Γ
2mgl
−
satisfy lim ω(t) = 0 and lim Γ(t) = Γh . t→∞
t→∞
Proof. Consider the closed loop system given by (3.2) and (3.5). We propose the following candidate Lyapunov function
− −
1 T V (ω, Γ) = ω Jω + mgl Γ 2
ρ l
T
Γ
ρ l
Note that the above Lyapunov function is positive definite on
(3.6) R3
× S 2 and
V (0, Γh ) = 0. Furthermore, the derivative along a solution of the closed-loop system given by (3.2) and (3.5) is ˙ V (ω, Γ) = ωT (J ˙ω) + mg(Γl
− ρ) Γ˙ = ω (Jω × ω + mgρ × Γ − Ψ(ω)) + mg(Γl − ρ) = −ω Ψ(ω) + mgω (ρ × Γ) − mgρ (Γ × ω) = −ω Ψ(ω) ≤ −1 ω 2 , T
T
T
T
T
(Γ
× ω)
T
T
where the last inequality follows from (3.4). Thus, V (ω, Γ) is positive definite ˙ and V (ω, Γ) is negative semidefinite on
R3
× S 2. Next, consider the sub-level set given by H = {(ω, Γ) ∈ R3 × S 2 : V (ω, Γ) ≤ 2mgl − }. Note that the compact set H contains the hanging equilibrium ˙ (0, Γ ). Since, V (ω, Γ) ≤ 0, all solutions such that (ω(0), Γ(0)) ∈ H satisfy (ω(t), Γ(t)) ∈ H for all t ≥ 0. Thus, H is an invariant set for solutions of (3.2) h
and (3.5). Furthermore, from LaSalle’s invariant set theorem, we obtain that solutions satisfying (ω(0), Γ(0))
∈ H converge to the largest invariant set in 39
{(ω, Γ) ∈ H : ω = 0}, that is ρ × Γ = 0. Therefore, either Γ(t) → ρ/l = Γ or Γ(t) → −ρ/l = Γ as t → ∞. Since (0, Γ ) ∈ H, it follows that Γ(t) → Γ as t → ∞. Thus, (0, Γ ) is an asymptotih
i
i
h
h
cally stable equilibrium of the closed-loop system given by (3.2) and (3.5), with a domain of attraction
H.
This result could be expected noticing that 2mgl is the total energy of the pendulum at the inverted equilibrium. Thus, when (ω(0), Γ(0)) is in
H, injection
of dumping can only decelerate the rigid body, towards the other equilibrium, represented by the hanging equilibrium. Different is the case when (ω(0), Γ(0)) is such that the total energy is exactly 2mgl. In this case Γ(t) can be shown to approach Γh as t
→ ∞, only if the initial kinetic energy is not zero.
The
nontrivial proof of this result is provided by the following Lemma.
Lemma 3.2 Consider the fully actuated 3D asymmetric pendulum given by
(3.2). Let Ψ :
R3
→ R3 be a smooth function satisfying (3.4) and choose a
controller as given by (3.5). Then, all solutions of the closed-loop system given
by (3.2) and (3.5), such that ω(0) = 0 and (ω(0), Γ(0))
A=
(ω, Γ)
3 R
∈ ×
1 1 S : ω T Jω + mgl Γ 2 2
ρ l
∈ A, where
T
ρ = 2mgl l
−−
2
Γ
satisfy lim ω(t) = 0 and lim Γ(t) = Γ h . t→∞
t→∞
Proof. Consider the closed loop system given by (3.2) and (3.5), and the Lya˙ punov function given in (3.6). As already shown, V (ω, Γ) = ¨ V (ω, Γ) =
−ω˙
T
Ψ(ω)
−
˙ ω Ψ(ω) = T
−ω˙
since Ψ is smooth. Furthermore, for all (ω, Γ) ω˙ = J −1 (Jω
T
−ω
T
∂ Ψ Ψ(ω) + ∂ω
is also bounded. Define N =
sup (ω, Γ)∈A
ω˙
T
T
ω
∈ A, ω is bounded and
× ω + mgρ × Γ − Ψ(ω))
Ψ(ω). Thus,
∂ Ψ Ψ(ω) + ∂ω
T
ω
<
∞.
Next, since (ω(0), Γ(0))
¨ Γ(0)) ≤ N . ∈ A, V (ω(0), Γ(0)) = 2mgl and V (ω(0),
Expanding V (ω(t), Γ(t)) in a Taylor series, gives V (ω(t), Γ(t)) = 2mgl
− ω(0)
T
Ψ(ω(0))t + R(t)
since the remainder necessarily satisfies R(t)
≤ 2mgl − 1ω(0)2 t + N t 2,
≤ N t2.
Define ¯ = min
ω(0)4
2mgl, 321
16N
.
∈ [T 1, T 2], ω(0)2 t + ¯ ≤ 0 t2 − 1 N N
It can be easily shown that for all t
where T 1 = 1
ω(0)
2
1
2N
−
1
N − 24¯ω(0) 4 1
>0
2
ω
− T 1 ≥ 1 2(0) . Choose an ∈ (0, ¯). Then, for all t ∈ [T 1, T 2], V (ω(t), Γ(t)) ≤ 2mgl − ¯ < 2mgl − and hence, (ω(t), Γ(t)) ∈ H, where H is and T 2
N
the invariant set given in Lemma 1. Thus, from Lemma 1, we obtain the result that ω(t)
→ 0 and Γ(t) → Γ , as t → ∞. h
By the two Lemmas above, the hanging equilibrium is stable, with a guaranteed domain of attraction. Indeed, when the initial conditions (ω(0), Γ(0))
∈ H, the
trajectories converge to the hanging equilibrium. Furthermore, when the initial total energy is 2mgl and the initial kinetic energy is not zero, the trajectories enter
H in finite time.
This result is obtained applying the simple damping
injection based control law (3.5). This result is summarized by next theorem.
Theorem 3.1 Consider the fully actuated 3D pendulum given by (3.2). Let
Ψ:
R3
→ R3 be a smooth function satisfying (3.4) and choose controller as in
(3.5). Then, all solutions of the closed-loop system given by (3.2) and (3.5), such that (ω(0), Γ(0))
N =
(ω, Γ)
3 R
∈N\{(0, Γ )}, where
∈ ×
i
− − ≤
1 1 S : ω T Jω + mgl Γ 2 2 2
ρ l
T
Γ
ρ l
2mgl
(3.7)
satisfy lim ω(t) = 0 and lim Γ(t) = Γh . t→∞
t→∞
41
Proof.
From Lemmas 3.1 and 3.2, we obtain the result that for every
∈ H A
∈
H and A are as defined in Lemmas 3.1 and 3.2, ω(t) → 0 and Γ(t) → Γ as t → ∞. Since, N can be written as (0, 2mgl) and (ω(0), Γ(0))
, where
N =
h
HA (
)
∈(0,2mgl )
the result follows.
It is important to point out that
N\{(0, Γ )} is an invariant set, but it is not i
the maximal domain of attraction of the hanging equilibrium. Any solution of
N\{(0, Γ )}, that does not pass through (0, Γ ), must eventually enter N\{(0, Γ )}. Hence, the maxi(3.2) and (3.5), starting from an initial condition not in i
i
i
mal domain of attraction of the hanging equilibrium is a proper superset of the invariant set
N\{(0, Γ )}. i
In the subsequent theorem, we claim that for the control given in (3.5), the equilibrium (0, Γh ) of (3.2) and (3.5) is almost globally asymptotically stable, and the domain of attraction of the hanging equilibrium is dense.
Theorem 3.2 Consider the fully actuated 3D asymmetric pendulum described
→ R3 be a smooth function satisfying (3.4) and Ψ (0) be positive definite and symmetric. Choose controller as given in (3.5). Let M denote the stable manifold of the equilibrium (0, Γ ). Then (R3 × S 2 )\M is dense in (R3 × S 2 ), and all solutions of the closed-loop system given by (3.2) and (3.5), such that (ω(0), Γ(0)) ∈ (R3 × S 2 )\M, satisfy lim ω(t) = 0 and lim Γ(t) = Γ . by (3.2). Let Ψ :
3 R
i
t→∞
Proof.
t→∞
h
Consider the closed loop system given by (3.2) and (3.5), and the
Lyapunov function given in (3.6). Since S 2 is a compact set and the Lyapunov function V (ω, Γ) is quadratic in ω, each sublevel set of V (ω, Γ) is a compact ˙ set. Furthermore, since V (ω, Γ) =
−ωTΨ(ω) ≤ 0, by LaSalle’s invariant set theorem, all solutions converge to the largest invariant set in {(ω, Γ) ∈ R3 × S 2 : ˙ V (ω, Γ) = 0}. The largest such invariant set is given by {(0, Γ )} {(0, Γ )}. Hence, it is sufh
i
ficient to show that the stable manifold of the inverted equilibrium (0 , Γi ), has dimension less than the dimension of (R3 S 2 ), i.e. five, since all other solutions
×
converge to the hanging equilibrium (0, Γh ) by invariant set theorem [Kha01]. Using linearization, it can be shown that the equilibrium (0, Γi ) of the closed loop given by (3.2) and (3.5) is unstable and hyperbolic with nontrivial stable and unstable manifolds. Hence, from theorem 3.2.1 in [GH83], it follows that the dimension of the stable manifold of (0, Γi ) is less than five, so that the domain of attraction in theorem 3.1 is dense in ( R3
× S 2).
Comparing the conclusions in the two previous theorems we notice that, theorem 3.1 provides an explicit description of a domain of attraction of the hanging equilibrium; however that domain of attraction is not maximal. In contrast, theorem 3.2 shows that the maximal domain of attraction consists of all points in (R3
× S 2) that are not in the stable manifold M of the inverted equilibrium.
However, the stable manifold of the inverted equilibrium is not explicitly described. We note that Lebesgue measure of
M is zero.
3.3 Experiments on Stabilization of the Hanging Equilibrium In this section we present experimental results on the stabilization of the hanging equilibrium of the asymmetric 3D rigid pendulum. They validate the controller proposed in the previous section, showing that (3.5) asymptotically stabilizes the hanging equilibrium with angular velocity feedback. The test-bed on which the experiments are performed, is the Triaxial Attitude Control Testbed (TACT) described above. We are able to measure the angular velocity ω of the TACT thanks to three orthogonal gyroscopes mounted on one of the two external plates, suitably calibrated. First, bias is eliminated keeping the TACT in a resting position. Then, a rotation about each axis is performed, assuring output only from the corresponding gyro, and no data from the others. In order to measure the attitude of the TACT, a triaxial accelerometer is mounted on one of the two external plates, aligned with the body-fixed axes. Before each test, the accelerometer triad is calibrated to read g on each indi43
vidual axis when aligned with the downward vertical. To do this, the TACT is rotated about each axis independently, while taking continuous reading from the accelerometer. The accelerometer axis is defined to be aligned with the downward vertical when a maximum occurs in the data. Bias and scaling factors are then adjusted such that each accelerometer outputs g at the maximum point. Actually, the reduced attitude cannot be measured directly. Indeed, accelerometer are not mounted at the center of gravity of the system. Thus, Γ must be estimated. We are able to estimate the reduced attitude using accelerometers and gyroscopes together. First we read the accelerometer measure. It gives not only the acceleration due to gravity, but also angular and centripetal acceleration terms. Centripetal acceleration can be determined indirectly by using the knowledge of the angular velocity. It correspond to ω
× (ω × r).
The angular
acceleration term is ignored in the computation of the reduced attitude, but preliminary calculations suggest that its contribution is quite small. Hence, direction of gravity is calculated as Gravity = Measure
− ω × (ω × r)
representing the Γ vector, since already expressed in body-fixed coordinates. Finally, inertia matrix J is identified as described in [CBA + 06]. Results of the application of (3.5) to stabilize the hanging equilibrium, are depicted in Figures 3.2 to 3.4. The controller used for this purpose is chosen such that Ψ(ω) = where P = 10I 3 .
−P ω
This choice of equal gains in the three feedback chan-
nels does not represent a particularly effective choice in terms of closed-loop performance.
However, our objective in the experiment is to demonstrate
closed-loop stabilization; we do not seek to tune the gains to achieve closedloop p erformance. The desired reduced attitude vector is the hanging equilibrium Γ0 = (0, 0, 1)T . The pendulum is released from the reduced attitude Γ(0) = ( 0.2936, 0.8579, 0.4217)T , with ω(0) = 0. These initial conditions are
−
−
chosen to excite all degrees of freedom of the system. As seen from experimental closed loop responses shown in Figures 3.2 and 3.3,
Angular Velocities 5 ) c e s / g e D (
0
x
ω
−5 0
5
10
15
20
25
30
35
40
5
10
15
20
25
30
35
40
5
10
15
20 Time [sec]
25
30
35
40
5 ) c e s / g e D (
0
y
ω
−5 0 100 ) c e s / g e D (
50 0
z
ω
−50 0
Figure 3.2. Experimental results for the evolution of the angular velocity of the 3D pendulum in the body frame.
Reduced Attitude Vector in Body Frame 0.5 x
0
Γ
−0.5 0
5
10
15
20
25
30
35
40
5
10
15
20
25
30
35
40
5
10
15
20 Time [sec]
25
30
35
40
1 0.5 y
0
Γ
−0.5 −1
0
1 z
Γ
0.5
0 0
Figure 3.3. Experimental results for the evolution of the components of the direction of gravity Γ in the body frame.
the angular velocity converges to zero and Γ(t) converges to Γ0 = (0, 0, 1)T as t
→ ∞.
This is clear from Figure 3.4 which shows that the response of the 45
Angle between Γ (t) and Γ in Body Frame 0
80
70
60
50 ] g e 40 D [ θ
30
20
10
0 0
5
10
15
20 Time [sec]
25
30
35
40
Figure 3.4. Experimental result for the evolution of the angle between the reduced attitude vector Γ(t) and the desired reduced attitude vector Γ0 .
angle between Γ(t) and Γ0 converges to zero as t
→ ∞. Notice that a residual
damped oscillation remains in the pitch axis angular velocity. It can be reduced adjusting the feedback control gains. Notice also that measurements during the initial transient period are especially noisy. This can be attributed to both gyros noise and an attitude estimation algorithm that ignores angular acceleration, which can add to noisy measurements during the initial transient period. The fan thrusters mounted on the TACT provide approximately 5 N m of control authority in each axis. Thrusters saturation can be an issue in these control experiments and the feedback gains need to be tuned to minimize any nonlinearities arising from saturation. This is not critical in stabilizing the hanging equilibrium of the 3D pendulum. Indeed, saturation might only produce a slower convergence. On the other hand, it has to be taken into account while stabilizing a generic attitude. Techniques to be used in the presence of saturation constraints, for stabilizing spacecraft, will be presented in the second part of this dissertation. Finally, notice that in this experiment, initial conditions are inside the guaranteed domain of attraction
N . Several experiments have been performed on the
TACT showing that under (3.5), dynamics converge to the hanging equilibrium also starting outside
3.4
N . They are presented in [SCB+05].
Conclusions In this chapter we have introduced a passivity based control law for stabilizing the hanging equilibrium of an asymmetric 3D rigid pendulum. The described control technique is based on the observation that angular velocity feedback renders the closed-loop input-output passive, with the total energy as storage function. We have shown that this control law cannot be global, due to the presence of the inverted equilibrium, but it is near-global. Finally, we have presented results of the application of the technique to the TACT. Even if the task of stabilizing the hanging equilibrium of the 3D rigid pendulum can appear quite simple, since Γ h is a minimum for the potential energy, this is fundamental for introducing the control of a generic attitude described in the following chapter.
47
Chapter
4
Attitude Control of an Asymmetric 3D Rigid Pendulum In the previous chapter we have presented a technique for stabilizing the hanging equilibrium of an asymmetric 3D pendulum, based on damping injection. The proposed control law has been inspired by the fact that the hanging equilibrium is a minimum of the potential energy of the system. In this chapter we generalize this concept, proposing a control law for an arbitrary reduced attitude vector. For controlling the reduced attitude of a 3D pendulum we mean pointing its Γ vector towards a pre-established direction. This is an usual task in satellite systems, corresponding to stabilize the orientation of spacecraft instrumentations or antennas. Thus, this problem is strictly related to geo-stationary satellites attitude control. Several local control laws for geo-stationary spacecraft, have been proposed in the literature, based on Euler’s angle error, quaternion error, Euler axis of rotation and many other techniques 1 . In particular, here we focus 1
See [Kap76] and [Sid00] for a comprehensive overview.
our attention on the approach adopted in [Bur95], for a model described by (3.2). We first show that this control law does not guarantee convergence to an arbitrary attitude. Therefore, we propose a class of controllers that suitably modify the potential energy of the rigid body, giving a closed-loop system whose potential has a minimum exactly at the desired attitude. This technique is referred to as potential shaping [OSR98]. We show that it produces a nearly global stabilization of any arbitrary attitude. Finally, we apply the proposed technique to an asymmetric 3D rigid pendulum and we depict simulation results, showing its effectiveness.
4.1 Local Asymptotic Stabilization of an Arbitrary Equilibrium As stated in the previous chapters, the mathematical model of a balanced 3D rigid pendulum actually describes the motion of a geo-stationary spacecraft. Indeed, the most common description of spacecraft evolution in composed by the Poisson’s kinematics equation (1.6) and the dynamics equation (3.1), without the gravity term mgρ RT e3 . Moreover, the reduced kinematics equation (1.13)
×
is used as a simplified version of (1.6), since it describes the evolution of RT e3 , i.e the third row of the rotation matrix R. Thus, controllers proposed in the literature of geo-stationary spacecraft, appear suitable also for stabilizing the orientation of a generic unbalanced asymmetric 3D rigid pendulum. It can be made selecting a control input of the form u=u ¯
− mgρ × R
T
e3
(4.1)
viz. canceling the effects of gravity. Besides the several control strategies described in [Sid00], here we consider a simplified version of the technique presented in [Bur95]. In this reference, a passivity based control input for complete attitude is designed, assuming no availability of angular velocity measurements. Here, we remove this assumption and we consider the reduced attitude kinematics model (1.13). It produces the 49
following control law for stabilizing an arbitrary orientation Γ 0 ¯= u
−P ω + k(Γ0 × Γ)
(4.2)
with P a positive definite matrix and k a real positive constant.
¯ as given in (4.1), Theorem 4.1 The closed-loop system (3.2) and (4.2) with u possesses an asymptotically stable equilibrium in (ω, Γ) = (0, Γ0 ) and an unstable
−
equilibrium in (ω, Γ) = (0, Γ0 ). Proof. Let Γ0 be an arbitrary reduced attitude. Note that the closed loop system given by (3.2), (4.1) and (4.2) has two equilibria for ( ω, Γ) = (0, Γ0 ) and
−
(ω, Γ) = (0, Γ0 ). Consider the following candidate Lyapunov function V (ω, Γ) =
1 T ω J ω + k(Γ 2
− Γ0)
T
(Γ
The above Lyapunov function is positive definite on
− Γ0)
R3
(4.3)
× S 2 and V (0, Γ0) = 0.
Furthermore, the derivative along a solution of the closed-loop system given by (3.2), (4.1) and (4.2) is ˙ V (ω, Γ) = ω T J ˙ω + k(Γ
− Γ0) Γ˙ = ω (−P ω + k(Γ × Γ0 )) + k(−Γ0 ) (Γ × ω) = −ω P ω + kω (Γ × Γ0 ) − kΓ0 (Γ × ω) = −ω P ω ≤ 0 T
T
T
T
T
T
T
˙ Thus, V (ω, Γ) is positive definite and V (ω, Γ) is negative semidefinite on R3 S 2 .
×
Hence, from LaSalle’s invariant set theorem, we obtain that (ω, Γ) converges to the largest invariant set in R3
× S 2 such that ω = 0, i.e. (Γ × Γ0) = 0. Moreover,
if (ω(0), Γ(0)) is such that V (ω(0), Γ(0)) < 2k, V (ω(0), Γ(0)) is an invariant set for the trajectories of the closed loop system, starting from ( ω(0), Γ(0)). Since
−
V (0, Γ0 ) = 2k, it follows that Γ(t)
→ Γ0 as t → ∞.
Thus, (0, Γ0 ) is an
asymptotically stable equilibrium of the closed-loop system given by (3.2), (4.1) and (4.2). On the other hand, consider the following Lyapunov function V (ω, Γ) =
1 T ω J ω + k(Γ + Γ0 )T (Γ + Γ0 ) 2
(4.4)
The above function is positive definite on
R3
× S 2 and V (0, −Γ0) = 0.
Fur-
˙ thermore, it is straightforward to show that V (ω, Γ) is not defined along the trajectories of the closed-loop system given by (3.2), (4.1) and (4.2). Thus,
−
(0, Γ0 ) is an unstable equilibrium of the closed-loop system given by (3.2),
(4.1) and (4.2).
Notice that the term
−P ω in (4.2) can be replaced by any function Ψ(ω) as
defined by (3.5). Indeed, it plays the role of injecting damping into the system. Hence, the above controller is strictly connected to the one presented in chapter 3, for stabilizing the hanging equilibrium Γh = ρ/l. This can also be deduced comparing equation (3.6) and equation (4.3). The fundamental difference is that an arbitrary Γ0 is not a minimum of the potential energy, unless the potential in suitably modified. In some way, this is what equations (4.1) and (4.2) do. They modify the system, for having a closed-loop whose potential energy has a minimum at Γ = Γ0 . This way to proceed, is referred to as potential shaping , in the sense that the control law actually re-shape the potential energy of the whole system. A comprehensive description of this technique, that is particularly suitable in the presence of mechanical systems, can be found in [BLM00], [BLM01] and related references. theorem 4.1 states the capability of control law (4.1) and (4.2) of stabilizing a desired attitude Γ0 , when initial condition satisfies V (ω(0), Γ(0)) < 2k, with V (ω, Γ) defined by (4.3). Unfortunately, it does not guarantee that in general, the evolution of the closed loop system converge to the desired attitude. Indeed, initial condition cannot be assured to belong to the invariant set V (ω(0), Γ(0)) < 2k. Selecting a large k for enlarging the guaranteed domain of attraction of (0, Γ0 ) is not in accordance with constraints usually present in practical problem. In fact, in real applications, thrusters cannot supply arbitrary large torques. This is especially true in aerospace applications, where minimizing consumption is also an issue. When (ω(0), Γ(0)) is not in V (ω(0), Γ(0)) < 2k, an initial condition belonging
−
to the stable manifold of (0, Γ0 ) cannot be avoided a priori. This situation is depicted in Figure 4.1 where lines represent function V (ω, Γ) level sets. Therefore, the technique proposed in [Bur95] and its simplified version described 51
G G0
V(w,G) = 2k (w(0),G(0))
0
w -G0
Figure 4.1. Evolution of the closed loop system (3.2), (4.1) and (4.2) towards the unstable equilibrium.
above, represent a local attitude control law. In order to arbitrarily enlarge the domain of attraction of the desired final attitude, one should design a controller
−
that enforces the closed-loop trajectories to move away from ( ω, Γ) = (0, Γ0 ) as soon as the closed-loop evolution approaches the unstable equilibrium. This argument leads to the nearly global control law presented in the following section.
4.2 Attitude Control of a Geo-Stationary Spacecraft In the last section, we have presented a controller that asymptotically stabilizes an arbitrary configuration of the asymmetric 3D rigid pendulum, with a guaranteed domain of attraction. Since the desired orientation of the Γ vector is arbitrary, the controller requires the knowledge of Γ in addition to ω, as it could be expected. Hence, it extends (3.5), providing an angular velocity and reduced attitude vector feedback based control. Unfortunately, the proposed controller is local. This motivates us to study the problem of designing an alternative, nearly globally stabilizing control law. It has to be noted that we aim at obtaining a nearly global controller. It means that we rule out the possibility of globally stabilizing an arbitrary attitude of the 3D asymmetric rigid body. The reason is that there exists a topological obstruction in designing a continuous time-invariant controller for global stabilization of an arbitrary reduced attitude, as widely motivated in [BB00]. This reference
represents a basic result in the theory of control for mechanical systems with rotational degrees of freedom. Indeed, in [BB00] it is shown that systems having a compact configuration manifold cannot be globally asymptotically stabilized to a rest configuration, using continuous state feedback. Since the two rotational degrees of freedom manifold S 2 is compact, this result applies for the problem at hand. Thus, we propose a controller that almost globally asymptotically stabilizes the reduced attitude vector Γ to a desired value Γ 0 . By almost global, we mean that the domain of attraction of the equilibrium is dense in
R3
× S 2. In particular, we
propose a control law that stabilizes the desired attitude Γ 0 from every initial condition in
R3
× S 2, except one point.
→ R be a C 1 monotonically increasing function such that Φ(0) = 0 and Φ(x) → ∞ as x → 1. Let Ψ : R3 → R3 be a smooth function satisfying Let Φ : [0, 1)
(3.4). We propose a class of controllers given by u=
−
Φ
1 T (Γ Γ 4 0
Theorem 4.2 Let Γ0
2
− 1)
T
(Γ0 Γ
− 1)Γ0 + mgρ
×
Γ
− Ψ(ω)
(4.5)
∈ S 2. Consider the fully actuated system given by (3.2).
Choose a controller as given in (4.5). Then (0, Γ0 ) is an equilibrium of the closed loop system (3.2) and (4.5) that is globally asymptotically stable with domain of attraction R3
× (S 2\{−Γ0}).
Proof. Consider the closed-loop system given by (3.2) and (4.5) and the candidate Lyapunov function 1 V (ω, Γ) = ω T Jω + 2Φ 2
1 T (Γ Γ 4 0
− 1)2
(4.6)
×S 2 and V (0, Γ0) = 0. Furthermore, every sub-level set of the Lyapunov function in R3 × S 2 is compact, Note that the Lyapunov function is positive definite on
R3
and the closed-loop vector field given by (3.2) and (4.5) has only one equilibrium in each sub-level set, namely (0, Γ0 ). Computing the derivative of the Lyapunov function along the solution of (3.2) 53
and (4.5) yields ˙ V (ω, Γ) = ω J ˙ω + Φ
1 T (Γ Γ 4 0
− 1) (Γ0 Γ − 1)Γ0 Γ˙ = ω (Jω × ω + mgρ × Γ − Ψ(ω) − mgρ × Γ) − ω Φ 14 (Γ0 Γ − 1)2 (Γ0 Γ − 1)(Γ0 × Γ) T
2
T
T
T
T
+Φ
= Φ
T
1 T (Γ Γ 4 0 1 T (Γ Γ 4 0
2
− 1)
−
(ΓT 0Γ
T
T
− 1)Γ0 (Γ × ω)
− 1)2 ( ω (Γ0 × Γ) + Γ0 (Γ × ω))(Γ0 Γ − 1) − ω Ψ(ω) = −ω Ψ(ω) ≤ −1 ω2 ˙ Thus, V (ω, Γ) is negative semidefinite on R3 × S 2 and hence, all solutions remain T
T
T
T
T
in the compact sub-level set given by
N = {(ω, Γ) ∈ R3 × S 2 : V (ω, Γ) ≤ V (ω(0), Γ(0))} Note that
N ⊂ R3 × (S 2\{−Γ0}). Next, from LaSalle’s invariant set theorem,
for any arbitrary initial condition, the solutions converge to the largest invariant
{
set in (ω, Γ)
∈ N : ω = 0}. Thus, from the second equation in (3.2), we obtain
that Γ is constant and from the first equation in (3.2), we obtain that either ΓT 0 Γ = 1 or Γ 0
× Γ = 0, or both. Therefore, either Γ(t) → Γ0 or Γ(t) → −Γ0 as t → ∞. However, (0, −Γ0 ) is not contained in N , hence, Γ(t) → Γ0 as t → ∞. It should be noted that the controller (4.5) is not defined if Γ = all other initial conditions in
R3
−Γ0 ∈ S 2. For
× (S 2\{−Γ0}), the controller guarantees that
the solution of the closed-loop, given by (3.2) and (4.5), asymptotically con-
·
verges to (0, Γ0 ). Indeed, thanks to its form, Φ( ) pushes the pendulum away
−Γ0 as soon as evolution approaches this orientation. It guarantees that closed-loop solution do not pass through the stable manifold of −Γ0 . from
It is insightful to view the class of controllers in (4.5) as providing a combination
·
of potential shaping, represented by the function Φ( ) and the term containing
· it may be noted that the argument of the potential function Φ(·) is proportional gravity, and damping injection, represented by the function Ψ( ). Furthermore,
to the cosine of the angle between Γ and Γ0 . The closed loop has the property
that if ω(0) = 0, then for all t
≥ 0, the angle between Γ(t) and Γ0 is bounded
above by the angle between Γ(0) and Γ 0 .
Corollary 4.1 Consider the fully actuated system given by (3.2) with controller
−Γ0. Then, for all t ≥ 0,
as in (4.5). Furthermore, let ω(0) = 0 and Γ(0) = ∠(Γ0 , Γ(t))
≤ ∠(Γ0, Γ(0))
Proof. Consider the candidate Lyapunov function (4.6), for the closed-loop ˙ system. As already shown, V (ω, Γ) =
˙ −ωTΨ(ω). Thus, since V (ω, Γ) is negative
≤ V (ω(0), Γ(0)). Thus substituting ω(0) = 0 in (4.6), we obtain the result that for all t ≥ 0, semidefinite, V (ω(t), Γ(t))
1 ω(t)T Jω(t) + 2Φ 2
1 T (Γ Γ(t) 4 0
2
− 1)
≤ 2Φ
1 T (Γ Γ(0) 4 0
2
− 1)
·
Since, the kinetic energy term is strictly non-negative and Φ( ) is a monotonic function, we obtain
|ΓT0Γ(t) − 1| ≤ |ΓT0Γ(0) − 1|,
t
≥0
Thus, ΓT 0 Γ(t)
≥ ΓT0Γ(0),
t
≥ 0,
which implies that cos(∠(Γ0 , Γ(t))) Since ∠(Γ0 , Γ(t))
≥ cos(∠(Γ0, Γ(0))) ,
t
≥0
∈ [0, π) and cos(·) is non-increasing in [0, π), the result follows.
The result stated by the previous corollary is prominent in aerospace control theory. Indeed, it assures that controller (4.5) avoid the so called unwinding phenomenon . This phenomenon, firstly exposed in [BB00], is produced by many attitude control laws proposed in the literature. It consists of the case when a control law produces an attitude response that starts close to the desired orientation Γ0 , diverges from Γ0 and then converges once again to Γ 0 . Thus, in some sense, controller (4.5) is able to impress to the asymmetric 3D pendulum, the correct direction of angular motion, that is the one reducing the 55
angle between Γ(0) and Γ 0 . Hence, controller (4.5) uses a suitable metric on S 2 , to define the distance between initial attitude and final desired attitude. Such a metric is (ΓT 0 Γ(0)
− 1). On the contrary, corollary 4.1 does not hold true
for technique (4.1) and (4.2) illustrated in previous section. There, the term describing distance between initial condition and final desired configuration is represented by (Γ 0
× Γ(0)). It is worth pointing out that the problem of defining
a correct metric on SO(3), initially faced in [WD91] and [FF94], is currently a subject of study. Thus, the proposed control law (4.5) produces a closed-loop system converging to any arbitrary desired equilibrium Γ 0 , from every initial condition on R3
× (S 2\{−Γ0}), i.e.
a nearly global control law. Moreover, (4.5) provides a
control input that avoids unwinding. Hence, in some way, (4.5) minimizes the path between (ω(0), Γ(0)) and (0, Γ0 ). This is one of the most important problems in spacecraft practical applications, where minimizing consumption while controlling attitude is a fundamental task.
4.3 Simulation on Stabilization of an Arbitrary Equilibrium In the previous section, we introduced a class of controllers that guarantee near global stabilization of the reduced system (3.2). In this section, we present simulation results for a specific controller selected from the family of controllers given by (4.5). We choose Φ(x) =
−k ln(1 − x)
where k > 0, and Ψ(x) = P x where P is a positive definite matrix. The resulting control law (4.5) is given by
T
u=
0 Γ − 1) −P ω − mgρ × Γ + k 1 −(Γ1 (Γ (Γ × Γ0 ) Γ − 1)2 T
4
(4.7)
0
Consider model (3.2) where m = 140 kg, ρ = (0, 0, 0.5)T m and J = diag(3, 40, 50) kg m2 . These parameters correspond to the ones of the TACT described above.
−
Let u be the controller (4.7) where P = 2.5I 3 and k = 5. The following figures describe the evolution of the closed loop system (3.2) and (4.7) corresponding to the inverted equilibrium Γ 0 = (0, 0, 1)T . The initial conditions are
−
ω(0) = (1, 3, 1)T rad/s and Γ(0) = (0.5, 0.7071, 0.5)T . Hence, control law (4.7) is activated while the system is moving. Simulation results in Figures 4.2 and 4.3 show that ω(t) as t
→ ∞.
→ 0, and Γ(t) → Γ0
Indeed, expressing control input u as u = u1 + u2 , the control Angular Velocity 5
] s / d a r [
0
1
ω
−5 0
50
100
150
200
250
300
50
100
150
200
250
300
50
100
150 time [s]
200
250
300
4 2
] s / d a r [
0
2
ω
−2 −4 0 4
] s / d a r [
2
3
0
ω
−2 0
Figure 4.2. Evolution of the angular velocity of the 3D pendulum in the body frame.
component u1 =
− Φ
1 T (Γ Γ 4 0
− 1)2
(ΓT 0Γ
− 1)Γ0 + mgρ
×
Γ
shapes the 3D pendulum energy for generating a closed-loop system whose potential energy has a minimum at the inverted equilibrium. In the meanwhile, the control component u2 =
−Ψ(ω)
injects damping into the systems to reduce angular velocity. This is also clearly seen from the plot of the angle between Γ( t) and Γ0 in Figure 4.4. Moreover, the vertical angular momentum approaches 0 and the total energy E
→
686.3 N
− 57
Components of
Γ in
Body Frame
1 0.5 1
0
Γ
−0.5 −1
0
50
100
150
200
250
300
50
100
150
200
250
300
50
100
150 time [s]
200
250
300
1 0.5 2
0
Γ
−0.5 −1
0
1 0.5 3
0
Γ
−0.5 −1
0
Figure 4.3. Evolution of the components of the direction of gravity Γ in the body frame.
Angle between Γ (t) and Γ
0
180
160
140
120
] g e d [ θ
100
80
60
40
20
0
0
50
100
150 time [s]
200
250
300
Figure 4.4. Evolution of the angle between the reduced attitude vector Γ(t) and the desired reduced attitude vector Γ0 .
m, i.e. the value corresponding to the inverted equilibrium, as shown in Figure 4.5.
In order to illustrate the result described in Corollary 4.1, we repeat the simulation, activating (4.7) while the system is in a rest position, i.e. ω(0) = 0. We select gains P = 2I 3 and k = 10. In this case evolution is similar to the one depicted above. Thus, we only show the plot of the angle between Γ(t) and Γ0 in Figure 4.7. As expected from Corollary 4.1, the angle between Γ(t) and Angular Momentum about Vertical Axis − Total Energy 800 Momentum Total Energy 600
400
200 E − h
0
−200
−400
−600
0
50
100
150 time [s]
200
250
300
Figure 4.5. Vertical angular momentum (solid line) and total energy (dash line) of the 3D pendulum.
Γ0 remains bounded above by the angle between Γ(0) and Γ 0 which is 120 deg. This is in contrast to Figure 4.4, where the excursion in angle exceeds 120 deg.
4.4
Conclusions In this chapter we have introduced a control law for stabilizing an arbitrary attitude Γ0 of an asymmetric 3D rigid pendulum. It is made by designing a closed-loop system whose potential energy has a minimum exactly at the desired attitude, i.e. shaping the potential. Meanwhile, damping is injected into the system for dissipating kinetic energy. The obtained control law distinguishes with respect to previous literature because it provides nearly global asymptotic stability. Indeed, evolution con59
3D Pendulum Motion: ρ /|ρ| vector in Inertial Frame
−1
−0.5
z
0
0.5
1 1 0.5
1 0.5
0 0 −0.5
−0.5 −1
x
−1
y
Figure 4.6. Motion of the vector between the pivot and the center of mass of the 3D pendulum in the inertial frame.
Angle between Γ (t) and Γ
0
140
120
100
] g e D [
80
θ
60
40
20
0
0
50
100
150 time [s]
200
250
300
Figure 4.7. Evolution of the angle between the reduced attitude vector Γ(t) and the desired reduced attitude vector Γ0 for ω (0) = 0.
verges to the desired state (ω, Γ) = (0, Γ0 ), from any initial condition except of
−
(ω(0), Γ(0)) = (0, Γ0 ). On the contrary, many controllers proposed previously
are shown to be only local. Besides the ability of nearly globally stabilize the desired attitude, the proposed scheme, has been shown to provide input torques in the direction that reduces the angle between Γ(0) and Γ 0 . Thus, it uses a suitable metric on S 2 . This is prominent in GEO-stationary spacecraft attitude control problems, since it avoids the unwinding phenomenon and then it reduces fuel consumption. Finally, we have presented simulation results, that confirm the theory. This suggests the application of a potential shaping and damping injection technique also to the special cases of the 3D rigid pendulum, illustrated in Chapter 2. It will be made in the following chapter.
61
Chapter
5
Attitude Control of a 3D Axially Symmetric Rigid Pendulum
In Chapter 2 we have shown how, with simple assumptions, the 2D pendulum, the 1D pendulum and the symmetric heavy top, can be seen as special cases of the 3D rigid pendulum. In particular we have analyzed the axisymmetric case, i.e. the case where two of the principal moments of inertia of the pendulum are identical and the pivot is located on the axis of symmetry of the pendulum model. This is an assumption usually adopted while facing attitude control problems for underactuated geo-stationary spacecraft [DT00]. Indeed, when an underactuated satellite is not symmetric with respect to the unactuated axis, the whole system does not satisfy Brokett’s condition [BMS83] for continuous feedback stabilization. In this chapter we develop controllers for a 3D pendulum, under the above assumption. Moreover, only two independent control moments are assumed to act on the pendulum. In particular, we assume that there is no control moment about the axis of symmetry of the pendulum. We first present a reduced model that forms the basis for controller design and closed-loop analysis. Such a model follows directly from equation (1.12) and
equation (1.13) when inertia matrix has the form J = diag(J t , J t , J a ). It is based on a global representation of the reduced attitude. This is an important feature of the development. Indeed, it avoids the use of Euler’s angles and other non-global attitude representations, leading to singularities, as seen in Chapter 2. Then, we treat asymptotic stabilization of an equilibrium of the reduced equations of motion of the 3D pendulum. This corresponds to stabilization of a relative equilibrium of the 3D pendulum. The relative equilibrium consists of a specified constant value of the reduced attitude and an angular velocity that is a pure spin about the pendulum’s axis of symmetry. It leads to the development of controllers that asymptotically stabilize the hanging relative equilibrium, the development of controllers that asymptotically stabilize the inverted relative equilibrium, and for the special case that there is zero angular velocity about the axis of symmetry of the pendulum, the development of controllers that asymptotically stabilize an arbitrary reduced equilibrium. We suggest a control technique based on potential shaping and damping injection as presented in Chapter 4. It is effective both for the case when the angular velocity component about the axis of symmetry is nonzero and for the case when it is zero. In the first case, our control results can be compared with other results in the literature on stabilization of heavy spinning top [SLRM92], [WTCB94], [WCB95]. In the second case, our control results can be compared with other results in the literature on stabilization of spherical pendulum [SEL99], [SEL04]. In all of these cases, our proposed stabilization results are new. Finally, we depict simulations for all the illustrated cases. Simulation results confirm that the proposed control technique provides near-global asymptotic stabilization in a direct way using relatively simple nonlinear controllers. This approach avoids the artificial need to develop a swing-up controller, a locally asymptotically stabilizing controller, and a strategy for switching between the two [AF00], [SEL04].
63
5.1 Global Coordinates Model of the 3D Axially Symmetric Pendulum In this section we introduce a reduced model for the controlled 3D axially symmetric pendulum. We also recall some important stability properties of the uncontrolled 3D axially symmetric pendulum, introduced in Chapter 2. Since the pendulum is assumed to be axially symmetric, there is no loss of generality in assuming that the inertia matrix is J = diag(J t , J t , J a ). As described in Chapter 1, let ρ = (0, 0, l)T denote again the vector from the pivot to the center of mass of the pendulum, where l is a nonzero scalar. Moreover, let ω = (ωx , ωy , ωz )T denote the angular velocity vector of the pendulum, and Γ = (Γx , Γy , Γz )T the reduced attitude vector of the pendulum, both expressed in body-fixed coordinates. Taking into account the moment due to gravity and the control moments, Euler’s equations in scalar form for the rotational dynamics of the 3D axially symmetric pendulum, are
− J )ω ω − mglΓ = (J − J )ω ω + mglΓ
J t ω˙ x = (J t
a
z
y
y
+ ux
J t ω˙ y
t
z
x
x
+ uy
a
(5.1)
J a ω˙ z = 0 Here, ux and uy denote the control moments. Besides kinematics models illustrated in Chapter 2 in terms of Euler’s angles, here we describe the kinematics of the symmetric pendulum in terms of reduced attitude vector components. Hence, we expand equation (1.13), obtaining the following three scalar differential equations Γ˙ x = Γ y ωz Γ˙ y = Γ˙ z
−Γ ω z
y
−Γ ω + Γ ω =Γ ω −Γ ω x
x
y
z
z
y
x
(5.2)
x
Since Γ = 1, the motion of the 3D axially symmetric pendulum evolves on R3
× S 2.
It is clear that the above equations cannot be asymptotically stabilized in any meaningful sense. Clearly, the third equation in (5.1) implies that the angular
velocity component about the axis of symmetry of the pendulum is ωz = c
(5.3)
where c is a constant. Reduction1 in this case is easily achieved by ignoring the third equation in (5.1) and substituting equation (5.3) into (5.2). This leads to the reduced dynamics equations
− J )ω − mglΓ = c(J − J )ω + mglΓ
J t ω˙ x = c(J t J t ω˙ y
a
a
t
+ ux
y
y
x
x + uy
(5.4)
and the reduced kinematics equations Γ˙ x = cΓy Γ˙ y =
−Γ ω z
−cΓ
x
Γ˙ z = Γ x ωy
y
+ Γz ωx
(5.5)
−Γ ω y
x
Equations (5.4)–(5.5) form a global-coordinates mathematical description of an axisymmetric 3D pendulum, with two control torques. It can be shown that this representation is equivalent to model (2.2). As a consequence, for any constant c, the motion of the axially symmetric 3D pendulum can be viewed as evolving on
R2
× S 2.
The uncontrolled equations (5.4)–(5.5) have two different equilibrium solutions, namely ωx = ωy = 0 , Γ = Γh = (0, 0, 1)T , and ωx = ωy = 0 , Γ = Γi = (0, 0, 1)T . Note that an equilibrium of (5.4)–(5.5) corresponds to a relative
−
equilibrium of equations (1.12) and (1.13). It consists of a pure spin of the 3D pendulum about its axis of symmetry. Only if the constant c = 0, a relative equilibrium solution of (5.4)–(5.5) is an ordinary equilibrium solution of the 3D pendulum.
Thus, in the general case of c = 0, an axially symmetric 3D rigid pendulum, without any control torque about its axis of symmetry, spins steadily. Consequently, only stabilization of its sleeping motions, i.e. inverted and hanging configuration, makes sense. Therefore, it is important to remind the result below, following from Proposition 2.4 and from the analysis in [GPS80]. Proposition 5.1 Consider the 3D axially symmetric pendulum given by equa-
tions (5.4)–(5.5). Then the hanging equilibrium is stable in the sense of Lya1
Consider again the reduction theory described in [MR99] and in [MRS00]
65
∈ R and the inverted equilibrium is stable in the sense of Lyapunov if and only if J 2 c2 ≥ 4mglJ . punov for all c
t
a
This background provides motivation for study of controllers that asymptotically stabilize one of the natural equilibrium solutions, either the hanging equilibrium or the inverted equilibrium, or asymptotically stabilize a new equilibrium induced by the control action.
5.2
Stabilization of the Hanging Equilibrium of the Symmetric Spinning Top
In this section we assume that the constant angular velocity is c = 0. In this case, the 3D axially symmetric pendulum described by equations (5.4)–(5.5) is effectively a heavy top, as described in Chapter 2. Hence, that terminology is used in this section. We propose two classes of feedback controllers that asymptotically stabilize the hanging equilibrium of the reduced model described by equations (5.4)–(5.5). The first result is based on feedback of the angular velocity of the top. The second result is based on feedback of both angular velocity and the reduced attitude of the top. In each case, the domain of attraction of the hanging equilibrium is shown to be nearly-global. First, we consider controllers based on feedback of the angular velocity, of the form
where ψx : R
→ R and ψ
y
sector inequalities with 2 1 x 2
:R
−ψ (ω ) = −ψ (ω )
ux =
x
x
uy
y
x
(5.6)
→ R are smooth functions satisfying the following
≥ 1 > 0
| | ≤ {xψ (x), xψ (x)} ≤ 2|x|2, x
y
∀x ∈ R
(5.7)
and ψx (0), ψy (0) > 0. Theorem 5.1 Consider the 3D axially symmetric pendulum given by the equa-
tions (5.4)–(5.5). Let (ψx , ψy ) be smooth functions satisfying (5.7) and choose ux and uy as in (5.6). Then the hanging equilibrium of (5.4)–(5.5) is asymptotically stable. Furthermore, for every
∈ (0, 2mgl), all solutions of the closed-loop
system given by (5.4)–(5.5) and (5.6), such that (ωx (0), ωy (0), Γ(0))
H=
(ωx , ωy , Γ)
∈ (R2 × S 2) : 12
J t (ωx2 + ωy2 ) + mgl Γ
− Γ 2 ≤ 2mgl − h
∈ H, t ≥ 0, and lim ω (t) = 0,
satisfy (ωx (t), ωy (t), Γ(t))
∈ H, where (5.8)
lim ωy (t) = 0 and
x
t→∞
t→∞
lim Γ(t) = Γh .
t→∞
Consider the closed-loop system given by (5.4)–(5.5) and (5.6). We
Proof.
propose the following candidate Lyapunov function 1 V (ωx , ωy , Γ) = J t (ωx2 + ωy2 ) + mgl Γ 2
−Γ
h
2
(5.9)
Note that the Lyapunov function is positive definite in R2 S 2 and V (0, 0, Γh ) =
×
˙ along a solution of the closed-loop system is 0. Furthermore, the derivative V ˙ x , ωy , Γ) = J t (ωx ω˙ x + ωy ω˙ y ) + mgl(Γ V (ω
T
−Γ ) h
Γ˙
− J )ω ω − mglΓ ω + c(J − J )ω ω + mglΓ ω + u ω + u ω + mgl[cΓ Γ − Γ Γ ω − cΓ Γ + Γ Γ ω + (Γ − 1)(Γ ω − Γ ω )] = −mglΓ ω + mglΓ ω + u ω + u ω + mgl[−Γ Γ ω + Γ Γ ω +Γ Γ ω −Γ Γ ω −Γ ω +Γ ω ] = u ω + u ω = −ω ψ (ω ) − ω ψ (ω ) ≤ −1 (ω 2 + ω2 ) ≤ 0 = c(J t
a
x
z
x
x
x
x
y
y
x
y
x
y
z
z
y
x
y
x
x
y
y
y
y
x
x
x
x
a
y
x
y
x
y
t
y
y
x
z
x
x
y
z
y
x
y
x
z
x
y
x
x
y
y
x
z
x
y
y
x
y
y
x
y
y
y
˙ where the last inequality follows from (5.7). Thus, V is positive definite and V is negative semidefinite on
R2
× S 2.
Next, consider the sub-level set given by
H=
(ωx , ωy , Γ)
∈
(R2
× S 2) : V (ω , ω , Γ) ≤ 2mgl − x
y
H contains the hanging equilibrium (0, 0, Γ ). Since ˙ V (ω , ω , Γ) ≤ 0 on H, all solutions such that (ω (0), ω (0), Γ(0)) ∈ H satisfy (ω (t), ω (t), Γ(t)) ∈ H for all t ≥ 0. Thus, H is an invariant set for the soluNote that the compact set x
h
y
x
x
y
y
tions of the closed-loop system. Furthermore, from LaSalle’s invariant set theorem, we obtain that the solu-
∈ H converge to the largest invariant set in {(ω , ω , Γ) ∈ H : (ω , ω ) = ( 0, 0)}. Next, ω ≡ ω ≡ 0 implies that Γ = Γ = 0 and Γ˙ = 0, that is Γ = ±1. Thus, as t → ∞, either Γ = Γ or ∈ H it follows that Γ = Γ as t → ∞. Γ = Γ . However, since (0, 0, Γ ) tions satisfying (ωx (0), ωy (0), Γ(0)) x
x
y
i
y
x
y
x
z
y
z
i
h
h
67
Therefore, Γh is an asymptotically stable equilibrium of the closed-loop system given by (5.4)–(5.5) and (5.6), with a guaranteed domain of attraction
H.
It is straightforward to notice that the result stated above holds true also for the complete model given by (1.12)–(1.13). Indeed, the candidate Lyapunov function 1 V 1 (ω, Γ) = [ω T Jω + mgl(Γ 2
T
−Γ ) h
(Γ
− Γ )] − J ω2 h
a
z
similar to (3.6), introduced in Chapter 3, is such that V 1 (ω, Γ) = V (ωx , ωy , Γ)
∀(ω, Γ), where V (ω , ω , Γ) is given by (5.9). x
y
Hence, V 1 (ω, Γ) has a minimum
in the hanging equilibrium (ω, Γ) = ([0, 0, c]T , Γh ) that is asymptotically stable with guaranteed domain of attraction
H.
We next strengthen the conclusion of above theorem, showing that the domain of attraction of the hanging equilibrium is almost global. Theorem 5.2 Consider the 3D axially symmetric pendulum given by the equa-
tions (5.4)–(5.5). Let (ψx , ψy ) be smooth functions satisfying (5.7) and choose ux and uy as in (5.6). Then, all solutions of the closed-loop system given by
∈ R3 × (S 2\{M}) satisfy Here M, a set of Lebesgue
(5.4)–(5.5) and (5.6), such that (ωx (0), ωy (0), Γ(0)) lim ωx (t) = 0, lim ωy (t) = 0 and lim Γ(t) = Γh .
t→∞
t→∞
t→∞
measure zero, is the stable manifold of the closed loop inverted equilibrium. Proof.
Since, the proof is very similar to process leading to theorem 3.1, here
we present only an outline of the proof. Denote
N =
(ωx , ωy , Γ)
∈ (R2 × S 2) : 12
J t (ωx2 + ωy2 ) + mgl Γ
− Γ 2 ≤ 2mgl h
Then, it can be shown that all solutions of the closed-loop system (5.4)–(5.5) and (5.6), satisfying (ωx (0), ωy (0), Γ(0))
∈ ∂ N\{(0, 0, Γ )}, enter the set H defined i
in theorem 5.1 in finite time, for some > 0. Then, from theorem 5.1 and the
N , we note that for every ∈ (0, 2mgl) and (ω (0), ω (0), Γ(0)) ∈ N\{(0, 0, Γ )}, ω(t) → 0 and Γ(t) → Γ , as t → ∞. Next, since definition of
x
i
y
h
N =
H N ∂
∈(0,2mgl )
it follows that all solutions satisfying (ωx (0), ωy (0), Γ(0)) verge to the hanging equilibrium.
∈N\{(0, 0, Γ )} coni
Furthermore, it can be shown that all solutions of the closed-loop (5.4)–(5.5) under (5.6), enter the set
N in finite time. Hence, all solutions either converge
to the inverted equilibrium, or the hanging equilibrium. Thus, it is sufficient to show that the stable manifold of the inverted equilibrium (0, 0, Γi ), has dimension less than the dimension of (R2
× S 2), i.e. four, since all other solutions
converge to the hanging equilibrium. Next, the equilibrium (0, 0, Γi ) of the closed-loop system is unstable as shown in Chapter 1. Using linearization it can be shown that its instability is hyperbolic with nontrivial stable and unstable manifolds. Thus, from theorem 3.2.1 in [GH83], it follows that the dimension of the stable manifold of (0, Γi ) is less than four, so that the domain of attraction in theorem 5.2 is ( R2 being
× S 2)\M,
M the stable manifold. Hence, the result follows.
Theorem 5.2 provides conditions under which the heavy spinning top is made asymptotically stable by feedback of angular velocity. Any controller of the form (5.6) can be viewed as providing damping injection. In theorem 5.1, the hanging equilibrium has a domain of attraction that is easily computed. In theorem 5.2, the domain of attraction is almost global. We are not aware of any prior literature on stabilizing the hanging equilibrium of an heavy top. Next, we consider controllers based on feedback of the angular velocity and the reduced attitude. These controllers provide more design flexibility than the controllers that depend on angular velocity only; hence they can provide improved closed loop performance.
Theorem 5.3 Consider the 3D axially symmetric pendulum given by equations
R be a C 1 monotonically increasing → function such that Φ(0) = 0 and Φ(x) → ∞ as x → 1. Furthermore, let (ψ , ψ )
(5.4)–(5.5) with c = 0. Let Φ : [0, 1)
x
y
be smooth functions satisfying inequality given by (5.7). Choose
− c(J − J )ω 1 + αΦ ( (Γ − 1)2 ) + mglΓ 4 = − ω + ψ (β ) − c(J − J )ω 1 + β Φ ( (Γ − 1)2 ) − mglΓ 4
ux =
−ω
x
+ ψx (α)
uy
y
a
z
y
t
+ J t (Γz
− 1)(−cΓ
x
+ Γz ωx )ψx (α)
y
a
z
y
t
x
(5.10) + J t (1
− Γ )(−cΓ z
y
+ Γz ωy )ψy (β )
x
69
where α = (Γz
− 1)Γ
y
and β = (1
− Γ )Γ . Then, (ω , ω , Γ) = (0, 0, Γ ) is an z
x
x
y
h
equilibrium of the closed-loop given by (5.4)–(5.5) and (5.10) that is asymptotically stable with domain of attraction R2
× (S 2\{Γ }). i
Proof. Consider the system represented by (5.4)–(5.5) and (5.10). We propose the following candidate Lyapunov function along a solution of the closed-loop J t V (ωx , ωy , Γ) = [ωx 2
−
J t ψx (α)] + [ωy 2 2
−
1 ψy (β )] + 2Φ (Γz 4
2
Note that the above Lyapunov function is positive definite on
2
− 1)
R2
× S 2 with
V (0, 0, Γh ) = 0.
Suppose that (ωx (0), ωy (0), Γ(0)) = (0, 0, Γi ). Computing the derivative of the Lyapunov function along a solution of the closed-loop, we obtain ˙ V
≤ −[ω − ψ (α)]2 − [ω − ψ (β )]2 − 1Φ x
x
y
y
1 (Γz 4
− 1)2
(Γz
− 1)2(Γ2 + Γ2 ) ≤ 0 x
y
˙ is negative semidefinite and hence, each solution remains in the comThus, V
K = {(ω , ω , Γ) ∈ R2 × S 2 : V (ω , ω , Γ) ≤ V (ω (0), ω (0), Γ(0))}. ˙ is negative semidefinite and Φ is monotonic with Φ (x) = 0 if x = 0, we Since V obtain that α → 0, β → 0, ω → ψ (0) = 0 and ω → ψ (0) = 0 as t → ∞. The pact set
x
y
x
y
x
y
x
x
y
y
last two properties follow from (5.7) and the Sandwiching theorem for the limit of a function. Furthermore, by LaSalle’s invariant set theorem, each solution converges to the largest invariant set
M ⊆ {(ω , ω , Γ) ∈ K : ω x
y
} ≡ ω ≡ 0, we obtain that the solution also satisfies Γ
x
= ωy = 0, α =
0, β = 0 . Since any closed-loop solution of (5.4)–(5.5) and (5.10) in ωx
y
z
M satisfies
= constant.
Next, α = β = 0 yields either Γ z = 1, in which case Γ = Γh , or it yields Γx = 0
and Γy = 0 and hence, Γ = Γh or Γ = Γi . However, since V (ωx (t), ωy (t), Γ(t)) =
V (ωx (0), ωy (0), Γ(0)), therefore Γ(t) = Γi for all t > 0. Thus Γi
∈ M.
Hence,
the only solution of the closed-loop system contained in the invariant set ωx = ωy = 0 and Γ = Γ h .
M is
Result in theorem 5.3 follows from the observation that control model (5.4)– (5.5) is input-output passive if the couple (α, β ) is taken as the output. It provides conditions under which the hanging equilibrium of the heavy top can be made almost-globally stable by feedback of the angular velocity and feedback of the reduced attitude. Any controller of the form (5.10) requires knowledge of
the axial and the transverse principal moments of inertia, mass, location of the center of mass and spin rate of the rigid body. Notice that, in equation (5.10), Φ plays the role described in theorem 4.2, i.e. it shapes the potential energy of the closed-loop system. Namely, it pushes the reduced attitude vector away from the unstable equilibrium. In the case of the hanging equilibrium stabilization described above, it is not necessary. Indeed, the same role could also be played by gravity terms, as in theorem 5.2. Here, gravity is canceled by the control law and the choice of ux and uy as in equation (5.10) is made so as to provide a desired rate of convergence to the hanging equilibrium. Moreover, it is useful for introducing the subsequent stabilization of the inverted equilibrium of the heavy top, where canceling the gravity and shaping the potential is necessary.
5.3 Stabilization of the Inverted Equilibrium of the 3D Axially Symmetric Pendulum We consider the stabilization of the inverted equilibrium of the 3D axially symmetric rigid pendulum. First, as in previous section, we assume that the con-
stant c = 0, so that the 3D axially symmetric pendulum described by equations (5.4)–(5.5) is an heavy top. Then, we assume that c = 0. In this case, equations (5.4)–(5.5) describe effectively a spherical pendulum as illustrated in Chapter 2. We propose feedback controllers that almost-globally asymptotically stabilize the inverted equilibrium of the reduced model. The result is based on feedback of both angular velocity and the reduced attitude of the top.
Theorem 5.4 Consider the 3D axially symmetric pendulum given by equations
R be a C 1 monotonically increasing → function such that Φ(0) = 0 and Φ(x) → ∞ as x → 1. Furthermore, let (ψ , ψ )
(5.4)–(5.5) with c = 0. Let Φ : [0, 1)
x
y
71
be smooth functions satisfying inequality given by (5.7). Choose ux =
−ω
x
+ ψx (α)
− c(J − J )ω t
a
y
+ J t (Γz + 1)( cΓx + Γz ωx )ψx (α)
−
1 + αΦ ( (Γz + 1)2 ) + mglΓy 4
− c(J − J )ω − J (1 + Γ )(−cΓ + Γ ω )ψ (β ) 1 + β Φ ( (Γ + 1)2 ) − mglΓ 4 where α = (1+ Γ )Γ and β = −( 1 + Γ )Γ . Then, (ω , ω , Γ) = (0, 0, Γ ) is an uy =
−ω
(5.11)
y
+ ψy (β )
a
t
z
x
t
z
y
z
y
y
x
z
y
z
x
x
y
i
equilibrium of the closed-loop given by (5.4)–(5.5) and (5.11) that is asymptotically stable with domain of attraction R2
× (S 2\{Γ }). h
Proof. Consider the system represented by (5.4)–(5.5) and (5.11) and consider the following candidate Lyapunov function J t [ωx 2
V (ωx , ωy , Γ) =
− ψ (α)]2 + J 2 [ω − ψ (β )]2 + 2Φ t
x
y
y
1 (Γz + 1)2 4
Note that the above Lyapunov function is positive definite on
2 R
× S 2 with
V (0, 0, Γi ) = 0. Suppose that (ωx (0), ωy (0), Γ(0))
∈ R2 × (S 2\{Γ }). Computing the derivative h
of the Lyapunov function along a solution of the closed-loop, we obtain ˙ V
2
2
≤ −[ω − ψ (α)] − [ω − ψ (β )] − x
x
y
y
1 (Γz + 1)2 (Γz + 1)2 (Γ2x + Γ2y ) 1 Φ 4
≤0
˙ is negative semidefinite and hence, each solution remains in the comThus, V pact set
K = {(ω , ω , Γ) ∈ R2 × S 2 : V (ω , ω , Γ) ≤ V (ω (0), ω (0), Γ(0))}. x
y
x
y
x
y
The remainder of the proof follows exactly the arguments used in theorem 5.3. The only solution of the closed-loop system (5.4)–(5.5) and (5.11) such that (Γz + 1)Γy
→ 0, (Γ
z
+ 1)Γx
→ 0, ω → ψ (0) = 0, ω → ψ (0) = 0 as t → ∞, x
x
is the inverted equilibrium (ωx , ωy , Γ) = (0, 0, Γi ).
y
y
Theorem 5.4 provides conditions under which the inverted equilibrium of the heavy spinning top can be made asymptotically stable by feedback of the angular velocity and feedback of the reduced attitude. The inverted equilibrium of the top is guaranteed to have an almost-global domain of attraction . These results can be compared with the extensive literature on stabilization of the upward sleeping motion of the top. See, e.g. [SLRM92], [WTCB94], [WCB95] and related references. Here, the main difference with respect to previous works
is that the spinning top is retained in the more general framework of the 3D pendulum. Hence, its motion can be described by global coordinates in the form of (5.5). Consequently, the results in Theorem 5.4 are substantially different from any of these cited results. We consider now the case where the axially symmetric 3D pendulum does not spin around the unactuated axis, i.e. ωz = c = 0. As reminded above, in this case the 3D pendulum is effectively a spherical pendulum and model (5.4)–(5.5) reduces easily to the dynamics equation J t ω˙ x =
−mglΓ
y
+ ux
(5.12)
J t ω˙ y = mglΓx + uy and to the kinematics equation Γ˙ x =
−Γ ω z
y
Γ˙ y = Γz ωx Γ˙ z = Γx ωy
(5.13)
−Γ ω y
x
It is easy to show that model (5.12)–(5.13) is equivalent to representation (2.3) introduced in Chapter 2. Indeed, it is sufficient to express Γ in terms of Euler’s angles. Moreover, (5.12)–(5.13) represents a description of the 2D pendulum motion, alternative to (2.5), that is the one commonly used in literature while facing 2D pendulum control problems. It can be shown that the relation between the above model and (2.5) is nonlinear and non smooth. This result could be expected, noticing that (2.3) and (2.5) have singularities in different configurations. In particular, (2.5) is not defined in the inverted equilibrium, as described in detail in Chapter 2. It leads to several control problems that here are simply avoided using model (5.12)–(5.13).
Theorem 5.5 Consider the 3D axially symmetric pendulum given by equations
R be a C 1 monotonically increasing → function such that Φ(0) = 0 and Φ(x) → ∞ as x → 1. Furthermore, let (ψ , ψ ) (5.4)–(5.5) with c = 0. Let Φ : [0, 1)
x
y
be smooth functions satisfying inequality given by (5.7). Assume ω(0) = c = 0, and let α (1 + Γz )Γx
(5.14)
β (1 + Γz )Γy 73
− (ω − ˙ − (ω − + J ψ (β )β
ux = mglΓy + J t ψx (α)α˙ uy = mglΓx
t
x
y
y
1 (Γz + 1)2 ψx (α)) + αΦ 4 1 ψy (β )) + β Φ (Γz + 1)2 4
(5.15)
˙ are obtained by differentiating (5.14) and substituting from (5.5). where α˙ and β Then, (0, 0, Γi ) is an equilibrium of the closed-loop given by (5.4)–(5.5) and (5.15) that is asymptotically stable with domain of attraction R2
× (S 2\{Γ }). Consider the system represented by (5.4)–(5.5) and (5.15). Since ω ≡
Proof.
h
c = 0, (5.4)–(5.5) reduces to (5.12)–(5.13). We propose the following candidate Lyapunov function V (ωx , ωy , Γ) =
J t [ωx 2
− ψ (α)]2 + J 2 [ω − ψ (β )]2 + 2Φ t
x
y
y
1 (Γz + 1)2 4
Note that the above Lyapunov function is positive definite on
R2
× S 2 with
V (0, 0, Γi ) = 0. The derivative of the Lyapunov function along a solution of the closed-loop, is ˙ V
≤− − [ωx
ψx (α)]2
−
[ωy
−
ψy (β )]2
−
1 Φ
1 (Γz + 1)2 (α2 + β 2 ) 4
≤0
˙ is negative semidefinite and hence, each solution remains in the comThus, V
K = {(ω , ω , Γ) ∈ R2 × S 2 : V (ω , ω , Γ) ≤ V (ω (0), ω (0), Γ(0))}. ˙ is semidefinite and from properties of Φ( ·), we obtain that α → 0, Next, since V β → 0, ω → ψ (0) = 0 and ω → ψ (0) = 0 as t → ∞. The last two equalities pact set
x
x
x
y
x
y
y
x
y
y
follow from the Sandwiching theorem for the limit of a function. Furthermore, by LaSalle’s invariant set theorem, the solution converges to the largest invariant set
M ⊆ {(ω , ω , Γ) ∈ K : ω x
y
x
} ≡ ω ≡ 0,
= ωy = 0, α = 0, β = 0 . Since
any closed-loop solution of (5.12)–(5.13) and (5.15) in
M satisfies ω
x
y
we obtain that the solution also satisfies Γ = constant. Next, α = β = 0 yields either Γ z =
−1, in which case Γ = Γ , or it yields Γ i
x
=0
≤ ∈ M. Thus (0, 0, Γ )
and Γy = 0 and hence, Γ = Γi or Γ = Γh . However, since V (ωx (t), ωy (t), Γ(t))
V (ωx (0), ωy (0), Γ(0)), therefore Γ(t) = Γh for all t
≥ 0.
h
Hence, the only solution of the closed-loop system contained in the invariant set
M is ω
x
= ωy = 0 and Γ = Γi .
Previous theorem provides conditions under which the inverted equilibrium of the spherical pendulum can be made asymptotically stable by feedback of the
angular velocity and feedback of the reduced attitude. It follows from the observation that (5.12)–(5.13) is input-output passive if α and β are taken as outputs. The proposed control law (5.15), injects damping and shapes the potential so as to generate a closed-loop system whose potential energy has a minimum in the inverted equilibrium. Notice that (5.15) is similar to (5.11) used in theorem 5.4 for stabilizing the upward sleeping motion of the heavy top. Indeed, both problems arise from the more general context of the 3D pendulum. On the other hand, they are substantially different from controllers for spherical pendulums that have appeared in prior literature [SEL99], [SPLE00], [SEL04]. Indeed, our results provide an almost-globally stabilizing controller that avoids the need to construct a swing-up controller, a locally stabilizing controller, and a switching strategy between the two. In this comparative sense, our results are direct and simple.
5.4 Attitude Control of an Underactuated Geo-Stationary Spacecraft Treatment in the last section can be easily extended to the case where the desired final attitude of the 3D axisymmetric pendulum is not represented
−
just by the inverted equilibrium Γi = (0, 0, 1), but by any possible attitude Γ0 = (Γx0 , Γy0 , Γz0 ). Apart of the difference between whole attitude control and reduced attitude control, widely described in previous chapters, this problem consists effectively of the attitude control problem for underactuated geostationary satellites [DT00]. Indeed, removing gravity terms from equations (5.4)–(5.5) gives the rotational model of a symmetric GEO spacecraft, without actuation around its axis of symmetry. Thus, we face the problem of controlling any arbitrary reduced attitude of an axially symmetric 3D pendulum, with ωz = c = 0, viz. no spinning about the axis of symmetry, and uz = 0 viz. no control action about the axis of symme-
try. Notice that, for our purposes, we assume c = 0. If ωz = c = 0, one should first apply to the system a detumbling control law . It consists of stabilizing the angular velocity to zero before controlling the attitude. Detumbling controllers 75
for geo-stationary spacecraft have been proposed in prior literature. For the problem at hand, one possibility is of breaking the symmetry of the system using proof mass as described in [CM02], then applying the control law described in [Son00] and finally restoring the symmetry, again by proof mass movement. We propose a class of controllers which almost globally asymptotically stabilize any arbitrary orientation of the spherical pendulum on
·
·
R2
× S 2 using the func-
·
tions Φ( ), ψx ( ) and ψy ( ) introduced above.
Theorem 5.6 Consider the 3D axially symmetric pendulum given by equa-
tions (5.12)–(5.13). Let Γ0 = (Γx0 , Γy 0 , Γz0 )T be a desired orientation and
R be a C 1 monotonically increasing function such that Φ(0) = 0 → and Φ(x) → ∞ as x → 1. Furthermore, let (ψ , ψ ) be smooth functions satisΦ : [0, 1)
x
y
fying inequality given by (5.7). Assume ω(0) = c = 0, and let α (ΓT 0Γ
− 1)(Γ 0Γ − Γ 0Γ ) β (Γ0 Γ − 1)(Γ 0 Γ − Γ 0 Γ ) 1 = mglΓ + J ψ (α)α˙ − (ω − ψ (α)) + αΦ (Γ0 Γ − 1)2 4 ˙ − (ω − ψ (β )) + β Φ 1 (Γ Γ − 1)2 = −mglΓ + J ψ (β )β 4 0 T
ux uy
y
t
x
x
x
t
y
y
z
x
z
z
x
x
y
y
z
y
T
T
(5.16)
(5.17)
˙ are obtained by differentiating (5.16) and substituting from where α˙ and β (5.13). Then, (0, 0, Γ0 ) is an equilibrium of the closed-loop given by (5.12)– (5.13) and (5.17) that is asymptotically stable with domain of attraction R2 (S 2
×
\{−Γ0}).
Proof. Consider the system represented by (5.12)–(5.13) and the following candidate Lyapunov function J t V (ωx , ωy , Γ) = [ωx 2
−
J t ψx (α)] + [ωy 2 2
−
1 ψy (β )] + 2Φ (ΓT Γ 4 0
2
− 1)
2
Note that the above Lyapunov function is positive definite and V (0, 0, Γ0 ) = 0. Furthermore, V (ωx , ωy , Γ) is radially unbounded on
R2
× S 2.
Calculating the
derivative of the Lyapunov function along a solution of the closed-loop (5.12)– (5.13) and (5.17), we obtain
−[ω − ψ (α)]2 − [ω − ψ (β )]2 − Φ 14 (Γ0 Γ − 1)2 (αψ (α) + βψ (β )) ≤ −[ω − ψ (α)]2 − [ω − ψ (β )]2 − 1Φ 14 (Γ0 Γ − 1)2 (α2 + β 2) ≤ 0
˙ = V
x
x
x
x
y
y
y
y
T
T
x
y
˙ is negative semidefinite and hence, each solution remains in the compact Thus, V
K = {(ω , ω , Γ) ∈ R2 × S 2 : V (ω , ω , Γ) ≤ V (ω (0), ω (0), Γ(0))}. Next, ˙ is semidefinite and from properties of Φ( ·), we obtain that α → 0, β → 0, since V ω → ψ (0) = 0 and ω → ψ (0) = 0 as t → ∞. The last two equalities follow set
x
x
y
x
x
y
y
x
y
y
from (5.7) and from the Sandwiching theorem for the limit of a function. Furthermore, by LaSalle’s invariant set theorem, the solution converges to the largest invariant set
M ⊆ {(ω , ω , Γ) ∈ K : ω x
y
x
} ≡ ω ≡ 0,
= ωy = 0, α = 0, β = 0 . Since
any closed-loop solution of (5.12)–(5.13) and (5.17) in
M satisfies ω
x
y
we obtain that the solution also satisfies Γ = constant. Next, α
T
≡ β ≡ 0 yields either Γ0 Γ = 1, in which case Γ = Γ0, or it yields Γz0 Γy = Γy0 Γz
(5.18)
Γz0 Γx = Γx0 Γz
Since Γ = Γ0 = 1, taking the square of left hand side and right hand side of (5.18), gives Γ2z0 (1
− Γ2) = Γ 2(1 − Γ20) that means Γ2 = Γ20 and hence, Γ = ±Γ 0 . Substituting Γ = ±Γ 0 in equation (5.18), we obtain that Γ = ±Γ0 . However, since V (ω (t), ω (t), Γ(t)) ≤ −Γ0 for all t ≥ 0. Thus (0, 0, −Γ0) ∈ M. V (ω (0), ω (0), Γ(0)), therefore Γ(t) = z
z
z
z
z
z
z
z
z
x
x
y
y
Hence, the only solution of the closed-loop system contained in the invariant set
M is ω
x
= ωy = 0 and Γ = Γ0 .
Theorem 5.6 shows that control law (5.17) stabilize an arbitrary reduced attitude Γ0 of an underactuated axisymmetric spacecraft, whose attitude kinematics is described by (5.13). Indeed, function Φ in (5.17) shapes the closed-loop potential energy so as to guarantee that the attitude cannot approach unstable equilibrium
−Γ0.
On the other hand, ψx and ψy provide suitable damping in-
jection. As a consequence, lim ωx (t) = 0, lim ωy (t) = 0 and lim Γ(t) = Γo t→∞
t→∞
t→∞
−
for any initial condition except of (ωx (0), ωy (0), Γ(0)) = (0, 0, Γ0 ). This result follows in a direct and easy way, retaining the problem in the framework of controlling the axisymmetric 3D rigid pendulum. The above result can be compared with other techniques presented in literature for controlling the attitude of underactuated geo-stationary satellites. See, e.g. [GE95], [CLT00] and 77
[DT00]. It is worth pointing out that (5.17), which depends on two control inputs, can be easily extended to the case where the 3D pendulum is fully actuated and asymmetric asymmetric,, as discussed discussed in Chapter Chapter 4. In this case J = (J 1 , J 2 , J 3 )T and the system is described by model (3.2). Thus, defining γ (ΓT 0Γ alon along g with (5.1 (5.16) 6),, i.e. i.e. (α,β,γ )T smooth function ψz : R ux = J 1 ψx (α)α˙ uy uz
− 1)(Γ 0Γ − Γ 0Γ ) = (Γ0 Γ − 1)(Γ × Γ0 ), introducin introducingg another another y
x
z
y
T
→ R, satisfying (5.7), and taking x
x
y
y
z
z
1 T (Γ0 Γ 1)2 + mglΓ mgl Γy 4 1 T (Γ Γ 1)2 mglΓ mgl Γx 4 0 1 T (Γ0 Γ 1)2 4
− (ω − ψ (α)) + αΦ ˙ − (ω − ψ (β )) = J 2 ψ (β )β )) + β Φ = J 3 ψ (γ )γ ˙ − (ω − ψ (γ )) )) + γ Φ y
x
− − −
−
(5.19)
we obtain that (ω, (ω, Γ) = (0, (0, Γ0 ) is an almost-global equilibrium point of the closed-loop system defined by (3.2) and (5.19), with domain of attraction (S 2
R3
×
\{−Γ0}). The proof of this result is easily obtained proceeding as in theorem
5.6, with the following candidate Lyapunov function J 1 J 2 J 3 1 2 V ( V (ω, Γ) = [ωx ψx (α)]2 + [ωy ψy (β )] )]2 + [ωz ψz (γ )] )]2 +2Φ (ΓT 0 Γ 1) 2 2 2 4
−
−
−
−
The difference between control (5.19) and control (4.5) lies in the first two terms
of (5.19), that only mean a different damping injection with respect to (4.5). Hence, properties of (4.5) are maintained. In particular, (5.19) utilizes the same metric on S 2 for representing the distance between Γ and Γ 0 . Thus, unwinding phenomenon is still avoided. Therefore, the attitude of a geo-stationary spacecraft can be controlled using (5.19) (5.19) both in the fully and in the underactu underactuated ated case. In the first situation the satellite can be asymmetric with general inertia matrix J . In presence of failures, the satellite must be made symmetric with respect to the unactuated axis, e.g. using proof mass movements movements.. We are not aware aware of other control control technique techniquess proposed in prior literature, able of controlling the attitude of a spacecraft both in the absence and in the presence of actuator failures. Simulation results on the application of the control techniques described above, are presented in the following section.
5.5 5.5
Simu Si mula lati tion on resu result ltss In this section we present simulation results on the application of the control technique techniquess described described in the previous sections. sections. First, First, we stabilize stabilize the inverte inverted d equilibrium of a heavy spinning top, using (5.11). Then, we control an arbitrary attitude of a 2D pendulum, assuming ωz = 0 and applying control law (5.17). We choose again Φ(x Φ(x) =
−k ln(1 − x)
where k > 0, and ψi (x) = pi x where pi are positive constants. Consider model (5.4)–(5.5) where m = 140 kg, kg , ρ = (0, (0, 0, 0.5)T m and J = diag(40 diag(40,, 40 40,, 50) kg m2 . These parameters correspond to the ones of the TACT,
−
once it is made symmetric with respect to the vertical. It can be done applying suitable weights to the external plates. Let ux and uy be given by controller (5.11) where ψx (α) = 3α 3 α and ψy (β ) = 3β . Moreover Moreover,, we choose choose k = 5 and and we assu assume me ωz = c = 1. Henc Hence, e, (5. (5.4) 4)–– (5.5) represents effectively a heavy top, that spins steadily at velocity ωz = 1 around around its axis axis of symmet symmetry ry.. The followin followingg figures figures show that in closed closed-loo -loop p system (5.4)–(5.5) and (5.11) the inverted equilibrium Γ i = (0, (0, 0, 1)T is as-
−
ymptotical ymptotically ly stable. Initial Initial conditions conditions are ω(0) = (1, (1, 3, 1)T rad/s and Γ(0) = (0. (0.5627 5627,, 0.8501 8501,, 0.0033)T . Notice that control law (5.11) is activated while the
−
system is moving. Simula Simulatio tion n result resultss in Figure Figuress 5.1 and 5.2 show show that that ω (t) Γ(t Γ(t)
→Γ
i
as t
→ ∞.
→
(0, (0, 0, 1)T , and
Inde Indeed ed,, the two cont contro roll compo compone nen nts u1 and u2 in
(5.11), shape the energy of the symmetric top, so as to generate a closed-loop system system whose whose potent potential ial has a minim minimum um in the invert inverted ed equili equilibri brium. um. On the other hand, the control law does not act on the third component of the angular velocity velocity. This is evident evident from Figure 5.3, where angular momentum momentum with respect to the vertical converges to ω T J Γ J Γi =
−50. The whole trajectory of the
closed-loop system, under controller (5.11), is depicted in Figure 5.4. There, the motion is drowned from the inertial frame point of view. Remember that body and inertial versors are related by B (t) = R(t)T I (t) and Γ(t Γ(t) = R(t)T e3 .
{
}
{ }
79
Angular Velocity 2 ] s / d a r [
0
1
ω
−2 −4 0
20
40
60
80
100
120
140
160
180
200
20
40
60
80
100
120
140
160
180
200
20
40
60
80
100 time [s]
120
140
160
180
200
4 ] s / d a r [
2
2
0
ω
−2 0 2 ] s / d a r [
1.5 1
3
ω
0.5 0 0
Figure 5.1. Evo Figure Evolut lution ion of the angular angular velocit velocityy of the symmetr symmetric ic heavy top in the body frame.
Components of
Γ in
{B}
1
1
Γ
0.5
0 0
20
40
60
80
100
120
140
160
180
200
20
40
60
80
100
120
140
160
180
200
20
40
60
80
100 time [s]
120
140
160
180
200
1 0.5 2
Γ
0 −0.5 0 0.5 0 3
Γ
−0.5 −1 −1.5 0
Figure 5.2. Evolut Figure Evolution ion of the compone components nts of the direction direction of gravi gravity ty Γ in the body frame.
{}
Hence, in order to depict the motion in I , it is necessary to define an initial
{}
{ }
condition also for the rotation matrix R between I and B . In the presen presentt
Vertical Momentum − Total Energy 800
700 Momentum Total Energy 600
500
400 E − h
300
200
100
0
−100
−200
0
20
40
60
80
100 time [s]
120
140
160
180
200
Figure 5.3. Vertical angular momentum (solid line) and total energy (dash line) of the symmetric top.
{}
example we have selected the following initial rotation matrix between I and
{B } R(0) =
0.8389 0.1373 0.5267
−0.5191 0.1635 −0.0889 −0.9865 0.8501 −0.0033
For the reason explained above, even if we control the third row of R, i.e. the
{ }
direction of gravity in B , Figure 5.4 illustrates its third column evolution, i.e. evolution of the vector from the pivot to the center of mass of the 3D symmetric pendulum. Notice that it converges to the upward vertical position, as expected.
Consider now the case where the axisymmetric 3D rigid pendulum does not spin about its axis of symmetry. In this case, model (5.12)–(5.13) describes effectively a spherical pendulum and, removing gravity terms, an underactuated geo-stationary spacecraft. Let ux and uy be given by controller (5.17) where, ψx (α) = 3α and ψy (β ) = 3β .
The following figures show the re-
sult of the simulation when initial condition is ω(0) = (1, 3, 0)T rad/s and Γ(0) = (0.5627, 0.8501, 0.0033)T and where parameters are the same of previ-
−
81
ous simulation:m = 140 kg, ρ = (0, 0, 0.5)T m and J = diag(40, 40, 50) kg
− √
√
− m2 .
The final desired attitude is Γ 0 = ( 1/ 5, 0, 2/ 5). As expected, angular ve3D Pendulum Motion: ρ /|ρ| vector in {I}
−1
−0.5
z
0
0.5
1 1 0.5
1 0.5
0 0 −0.5
−0.5 −1
x
−1
y
Figure 5.4. Closed-loop trajectory of the symmetric top in the inertial frame.
locity ω
→ 0 and reduced attitude Γ → Γ0 as t → ∞.
This is illustrated in
Figure 5.5 and Figure 5.6 Moreover, the unwinding phenomenon is avoided thanks to the term Φ
1 (ΓT 0Γ 4
− 1)2
in (5.17). It guarantees that the controller
push the pendulum in the correct angular direction, in such a way that ∠(Γ0 , Γ(t))
≤ ∠(Γ0, Γ(0))
It is clear observing Figure 5.7. The whole trajectory is depicted in Figure 5.8. Again, in order to represent the vector between the pivot and the center of mass of the pendulum, it is necessary to define an initial rotation matrix R(0). We select the same matrix of previous example.
5.6
Conclusions In this chapter we have introduced control laws for an axisymmetric 3D rigid pendulum, assuming the presence of only two control torques. This problem is extremely important in aerospace literature, since the related model describes
Angular Velocity 1.5 1
] s / d a r [
0.5
1
ω
0 −0.5 0
50
100
150
200
250
300
50
100
150
200
250
300
50
100
150 time [s]
200
250
300
3 ] s / d a r [ 2
ω
2 1 0
0
1 0.5
] s / d a r [
0
3
ω
−0.5 −1 0
Figure 5.5. Evolution of the angular velocity of the 2D pendulum in the body frame.
Components of
Γ in
{B}
1 0.5 1
Γ
0 −0.5
0
50
100
150
200
250
300
50
100
150
200
250
300
50
100
150 time [s]
200
250
300
1
2
Γ
0.5
0
0
1 0.5 3
Γ
0 −0.5
0
Figure 5.6. Evolution of the components of the direction of gravity Γ in the body frame.
an underatuated geo-stationary spacecraft, by simply removing gravity terms. First, we have proposed a control law based on angular velocity feedback and we 83
Angle between Gravity and Desired Attitude 120
100
80
] g e d [
60
θ
40
20
0
0
50
100
150 time [s]
200
250
300
Figure 5.7. Evolution of the angle between the reduced attitude vector Γ(t) and the desired reduced attitude vector Γ0 for ω (0) = 0.
3D Pendulum Motion: ρ /|ρ| vector in {I}
−1
−0.5
z
0
0.5
1 1 0.5
1 0.5
0 0 −0.5 x
−0.5 −1
−1
y
Figure 5.8. Closed-loop trajectory of the 2D pendulum in the inertial frame.
have shown that it can easily stabilize the hanging equilibrium of the system, both in the case where the pendulum does or does not spin steadily. In the
first case the controller stabilizes the vertical configuration of a top “under the table”, i.e. its downward configuration. Surprisingly, this problem has not been studied in previous literature. Then, we have presented a controller able to stabilize the sleeping motion of a spinning top. The proposed control technique appears simple and direct, from the 3D pendulum theory presented above. Finally, we have proposed a control law that stabilizes an arbitrary attitude of the axisymmetric pendulum, when ωz = 0. It is based on damping injection and potential shaping, thanks to the selection of suitable outputs that render closed-loop system, input-output passive. This controller has been generalized to the asymmetric case, producing a control law useful both in the fully and in the underactuated case. The possibility of designing simple continuous state-feedback controllers for stabilizing any arbitrary attitude of 2D and 3D rigid bodies, appears a prominent result in mechanics and aerospace literature. This is possible thanks to the idea of viewing all the related models as special cases of the 3D p endulum. Its global attitude representation, avoids the presence of singularities and the consequent artificial need to develop swing-up controllers, local controllers, and strategies for switching between them.
85
Part II
MEO and LEO Spacecraft Attitude Control
87
Chapter
6
Medium and Low Earth Orbit Satellites The 3D pendulum model presented in the first part of this dissertation describes the evolution of geo-stationary satellites. Since, geo-stationary spacecraft travel on orbits, whose period can be made exactly equal to the time Earth takes for rotating once about its axis, for an observer on Earth only attitude motion makes sense. Thus, apart of gravity terms, controlling the attitude of a 3D pendulum is equivalent to control the attitude of a geosynchronous satellite. Beyond the huge literature on geo-stationary spacecraft [Hug86], [Kap76], [Sid00], developed from the fifties to present time, in the last few decades large efforts have been made for studying the motion of satellites traveling on orbits at lower distance from Earth. Indeed, progresses in telecommunications have suggested the use of small space vehicles on Medium Earth orbits 1 (MEO) and Low Earth orbits2 (LEO). MEO and LEO satellites have produced a significant reduction of the costs related to launches, maneuvers and maintenance. In this chapter we present a mathematical model for MEO and LEO spacecraft dynamics. We first propose a novel attitude model for satellites orbiting at ve1 2
Orbits between 3.000 and 30.000 km above the Earth’s surface Orbits between 200 and 2.000 km above the Earth’s surface
89
locity ω0 relative to Earth. It is designed with respect to an orbital reference frame that spins at velocity ω0 about its third axis. Small angle maneuvers are taken into account. Then, we complete the mathematical model by presenting the Hill’s model . Hill’s model describes the position of a satellite with respect to a reference point on its orbit. It originates from the two body problem , well known in literature [Sid00], [BMW71], [TIH02]. The need of a position model, along with an attitude model, is due to the fact that in MEO and LEO orbits the motion is not geosynchronous. Thus, describing the position is necessary for modeling the whole system state. Finally, we introduce the problem of controlling the attitude and the position of a LEO spacecraft. Since there is no mathematical difference between MEO and LEO systems, apart of vehicles’ masses and orbiting velocities, we focus our attention on LEO satellites. For these systems, involving small spacecraft, the presence of limited thrusts cannot be left out of consideration.
6.1 LEO Spacecraft Attitude Model Consider a satellite in Low Earth Orbit, traveling around Earth at a velocity of ω0 rad/s. It consists of a rigid body, with three rotational and three translational degrees of freedom, with respect to an Earth Centered Inertial (ECI)
{ }
reference frame. Following the same arguments of Chapter 1, let B be an
R (t) 0 1 = R (t)
R (t) 0
{ECI} Figure 6.1. Reference frame {B } motion with respect to an Earth centered reference frame {EC I }
orthonormal reference frame aligned with the principal axes of the rigid body. Hence, attitude can be represented describing the evolution of the rotation ma-
{ }
{
}
trix R(t) between B and EC I . However, it is immediate to notice that this way to proceed, that in Chapter 1 leads to equations (1.2) and (1.6), is not
{ }
suitable in the present case. Indeed, as depicted in Figure 6.1, B rotates with
{
}
respect to EC I , even in the absence of external torques. On the other hand, when τ
≡ 0, R(t) can be calculated at each time instant, given its initial value
R(t0 ). In fact, in this case R(t) depends only on ω0 .
{ }
Thus, it is easier to analyze the rotations of B with respect to another ref-
{ }
{
}
erence frame O , that is not inertial and that orbits at velocity ω0 in EC I . This way to proceed is referred to as the leader following approach [Sid00], [TIH02]. It means that the rotational and translational motion of the 3D rigid body can be modeled with respect to a reference frame whose origin is fixed at the center of mass of a virtual satellite . The virtual satellite can be a moving point on the 3D rigid body’s orbit, or possibly, another spacecraft. In the latter case, the two satellites consist of a formation flying system with one leader and one follower. Fleets of satellites flying in LEO orbits, have been collecting great interest in the last few years [ENS + 03]. It has been emphasized that they can substitute large monolithic satellites, drastically reducing costs. Therefore, it is expected that a great deal of future space missions will be based on formations of satellites.
{ } {
}
For our purposes, an orbiting reference frame O = X , Y , Z is defined. It is selected with the first axis in the opposite direction of Earth center. The second axis is assumed aligned with the orbit and the third axis completes an orthonormal right oriented versors set. Consequently, its center is assumed
{ } respect to the third axis of {EC I }.
moving in a circular orbit, i.e. O exhibits a constant rotational motion with We define the corresponding angular ve-
locity as ω o = (0, 0, ω0 )T . Moreover, with the same notations of Chapter 1,
{B} = {X , Y , Z } is introduced, as depicted in Figure 6.2. The dynamics equations, concerning effects of forces on the motion of the satellite, are again represented by equation (1.2). Thus,
J ˙ω = J ω
× ω + τ
(6.1) 91
Z w0
Y
X
’
w0
Y Z
X
’
’
{B} {ECI}
{O}
Figure 6.2. Reference frame {B } with respect to an orbiting reference frame {O}, both moving in {EC I }
where J = diag(J x , J y , J z ). We describe the kinematics of the rigid body using Euler’s angles ψ, θ and φ, as introduced in Chapter 1. In particular here, they
{ } { } for this reason, relations between {O} and {B } in present chapter are identical to relations between frames {I } and {B } defined in Chapter 1. Indeed, {B } does not rotate in {O} unless external torques are applied. However, we do not are used to represent the orientation of O , with respect to B . Notice that,
use directly representation (1.8)–(1.10), since we also have to consider relations
{ }
{
}
between O and EC I . Thus, we rewrite the absolute angular velocity of
{B} as the sum of the velocity of {B} with respect to {O} plus the velocity of {O} with respect to {EC I }, all the variables expressed in {B} coordinates: ωb = ω bo + ωo
(6.2)
Further, ωbo depends on the sequence of rotations that the orbit frame has to perform in order to reach the body frame:
ωbo = ω bo + ω o
o
+ ωo o
These rotations are related to the Euler’s angles. In particular
(6.3)
{O } is the
{ }
reference frame obtained from O after a first rotation of angle ψ along the first axis and O is the one obtained from O after a second rotation of
{ }
{ }
angle θ. Hence, ω bo =
sφθ˙ + cθcφψ˙ cφθ˙
− cθsφψ˙
φ˙ + sθψ˙
(6.4)
where ψ˙ = ω o o , θ˙ = ω o
o
and φ˙ = ωbo . For brevity we have replaced cx =
cos(x) and sx = sin(x). Furthermore, also ω o needs to be expressed in body coordinates:
ωo = R
0 0 ωo
=
(sψsθ
− cψsθcφ)ω
o
(sψcφ + cψsθsφ)ωo cψcθωo
where
−
cθcφ
R=
cθsφ
sψsθcφ + cψsφ sψsφ
− sψsθsφ −sψcθ
cψcφ
sθ
− cψsθcφ
sψcφ + cψsθsφ cψcθ
(6.5)
is the rotation matrix that leads O to coincide with B and ω o is the angular
{ } { } velocity of {O} with respect to {EC I }, as defined above. Notice that (6.5) differs
from (1.7), since here we have selected a 1-2-3 sequence of rotations, while in Chapter 1 the sequence is 3-2-1. Thus,
ωb =
sφθ˙ + cθcφψ˙ + (sψsθ cφθ˙
− cψsθcφ)ω
o
− cθsφψ˙ + (sψcφ + cψsθsφ)ω
o
φ˙ + sθψ˙ + cψcθωo
(6.6)
As it will be seen in the following chapter, for the problem at hand, it is convenient the use of controllers designed for linear systems. Hence, for the time
{ }
{ } the previous expression can be linearized, considering cos(ψ) 1, sin(ψ) ψ, being, we assume small angular displacements between B and O . Therefore,
and analogous for θ and φ. It results
ωb
ψ˙
−ω θ o
θ˙ + ωo ψ φ˙ + ωo
(6.7)
93
Consequently, using (6.7) and its first derivative into (6.1) and disregarding nonlinear terms, the Euler’s equations become J x ψ¨ = (J y
− J )ω2ψ + (J + J − J )ω θ˙ + τ J θ¨ = (J − J − J )ω ψ˙ − (J − J )ω 2 θ + τ y
z
z
x
o
y
x
o
y
z
o
x
z
x
o
y
(6.8)
J z φ¨ = τ z Model (6.8) represents the linear attitude model for a spacecraft orbiting around Earth at velocity ω0 . It is equivalent at using equation (6.1) together with the first derivative of (6.7). This representation can be used for describing the rotational motion of the orbiting rigid body as long as it spans small angles. In
{ }
this case any sequence in the rotations of angles ψ, θ and φ describing O in
{B} gives the same result [GPS80]. 6.2 LEO Spacecraft Position Model Model (6.8) describes the rotational motion of a 3D rigid body traveling on a Low Earth Orbit at velocity ω0 . In order to complete the mathematical description of this kind of systems, it is necessary to introduce a position model. Indeed, in practical applications, a LEO satellite is allowed to move with respect to its orbit by means of a suitable combination of control inputs. Therefore,
{ }
the coordinates of B origin, have to be described in the moving reference
{ }
frame O . The importance of a position model, besides attitude model (6.8), is simply understood, thinking to the influence of small perturbations on the spacecraft, such as drug forces. They require some control action for moving the rigid body back to its orbit. Furthermore, a position model has to be used while facing practical problems involving LEO satellites, as the fundamental rendezvous problem between two spacecraft. In practice, all the real missions in LEO and MEO orbits, require the design of coordinated attitude and position maneuvers. Earth observing systems and space interferometers [ENS+ 03] are two examples. Since the satellite orbits in close proximity, i.e. the distance between the LEO
{
}
orbit and the EC I reference frame is much greater than the one between the satellite and the point selected as leader, a standard approach to model its
position is that of linearizing the dynamics of the spacecraft around the orbit. In the case that the reference orbit is approximately circular, the related linearized equations of motion, commonly referred as the Hill’s equations , are as follows: m¨ x = 3ω02 mx + 2ω0 my˙ + f x (6.9)
−2ω0mx˙ + f m¨ z = −ω02 mz + f
m¨ y=
y
z
where m is the mass of the satellite. In (6.9) we have defined the three Cartesian coordinates as x, y, z and the forces that act on the spacecraft as f x , f y , f z . They can be obtained starting from the two body problem [BMW71]. Briefly, once represented the position of two close objects in neighboring near-circular orbits, (6.9) is derived, linearizing the equations of motion under gravitational attraction, around one of the two orbits, selected as leader. Indeed, the distance between the two orbits is negligible with respect to the distance between the leader and Earth center. A comprehensive analytical description of this approach can be found in [Sid00] and [TIH02]. Note that in the present description, model (6.8) and model (6.9) are decoupled.
However, components x = 0 would correspond to different angular velocities of the satellite and the leader point, i.e. ω0 would not be constant. Therefore, the attitude of the vehicle would change with x. Anyway, in LEO systems, the effects of relative position on the attitude are much lower than the ones due to disturbances and they can be neglected. Furthermore, note that usually, inputs are given in the body reference frame
{B }.
For a correct use of (6.9), here they should be expressed in the orbit
{ }
frame O by means of an appropriate transformation. However, since we as-
{ }
{ } there is no difference in
sume small angle deviations between O and B
representing these variables either in body coordinates or in orbit coordinates.
6.3 LEO Spacecraft Control Problem Formulation In the following, we introduce the problem of controlling the attitude and the position of a LEO spacecraft, both defined with respect to an orbiting point, or 95
to another satellite in a leader following framework. Since equations (6.8) and (6.9) are linear, any control law designed for continuous linear systems can be applied for stabilizing a desired state or pre-designed path [Kha01]. However, in real missions, the simple stabilization problem is not important, since it does not fit with practical applications. Indeed, classical attitude and position controllers may fail because of the presence of stringent saturation constraints, always present with small space vehicles, such those involved in the present kind of systems. Thus, we focus our attention on the control problem in the presence of constraints. The same approach is adopted in the most part of the literature on LEO and MEO space systems. See, for instance, [VGRM01], [TIH02] and related references. Consider again equations (6.8) and (6.9). For our purposes, they can be summarized in the state-space time-continuous representation s(t) ˙ = Ac s(t) + Bc (u(t) + ξ (t))
(6.10)
where s is the whole state of the system, selected as ˙ ˙ φ, ˙ φ]T s = [x,x, ˙ y,y, ˙ z,z, ˙ ψ,ψ, θ,θ,
∈
12 R
As a consequence, u is composed by the six input components τ and f defined in (6.8) and (6.9). Thus, u
∈ R6. Moreover, we have added input disturbances
forces and torques, always to be considered in practical LEO space systems. Thus ξ
∈ R6.
Definition of matrices Ac and Bc in equation (6.10) is straight-
forward from (6.8) and (6.9). Now, the general state-space model given in equation (6.10) can be discretized using the well-known technique of approximately integrating the dynamics in equations (6.8) and (6.9) over one sample period, T [FPW98]. Here, Ak , Bk correspond to Ak = A(kT ), Bk = B(kT ). In the case of system (6.10), Ak = A which is a constant and Bk = B which is a constant. Therefore, the ZOH sampled dynamical model of the satellite, takes the form s(k + 1) = As(k) + B(u(k) + d(k))
(6.11)
where d(k) represents relative disturbance forces and torques accumulated dur-
∈ R6 satisfying d(k) ∈ D , ∀k ∈ Z+, with D a specified convex and compact set such that 0 6 ∈ D . ing a sampling period. We assume d(k)
We now formulate the problem of controlling a spacecraft in Low Earth Orbit. Controlling a LEO spacecraft means driving the orbiting rigid body to a desired state, i.e. solving a reconfiguration problem. Desired state is represented by a reference position relative to the leader orbit and by a reference attitude relative
{ }
to the orbiting frame O . Moreover, the satellite can be required to cover a pre-specified path, i.e. to solve a tracking problem. Finally, a LEO spacecraft can be simply required to keep its initial state, in spite of disturbances acting on it: keeping problem. These problems are especially important when the satellite is a member of a formation flying system. In this case its maneuvers have to be performed in coordination with all the other vehicles of the fleet [ENS + 03]. While facing a reconfiguration, a tracking or a keeping problem for a LEO satellite, as defined above, it has to be taken into account that maneuvers are made thanks to a combination of small jet actuators that are not able of supplying arbitrary large forces and torques. Thus, input saturation constraints of the form
|u (k)| ≤ u
imax ,
i
i = 1,
··· , 6
(6.12)
are present. Furthermore, we want to handle state accuracy constraints, i.e. state-related constraints
|y (k) − r (k)| < ε, i
i
i = 1,
··· , 6
(6.13)
∀ k ∈ Z+, with r(k) a reference signal and y(k) = Cs(k) some suitably selected output. Notice that the time-step T , is typically chosen to be small (i.e. on the order of 4
− 5 sec so that there are about 1.000 points per typical LEO orbit) to obtain
an accurate discretization. However, this results in a large number of inputs and constraints for reconfiguration or keeping problems, which can take several minutes to be solved. One method to reduce these computational times for a real-time implementation is to only allow inputs every n-(integer) time-steps and/or only enforcing the constraints every n-(integer) time-steps. For example, making it every 10 steps, greatly reduces the size of the control problem, resulting in solution times on the order of seconds. This decreased solution time is obtained at the expense of system performance because the control inputs are allowed less often (reduces efficiency) and the constraints are only tested 97
at certain time-steps (violations are possible at intermediate time-steps). This approach has been followed in [TIH02]. On the contrary, here we propose a control technique able to guarantee low numerical burdens. Model (6.11) under constraints (6.12) and (6.13) suggests the use of a model based predictive control strategy [Mos95]. In particular, in the following chapter, we propose a control strategy based on the Command Governor (CG) approach [BCM97], [CMA00]. It consists in designing a primal control law that does not take the constraints into account, and an external unit capable of taking care of constraint fulfillment by modifying, whenever necessary, the reference. Since the primal controller is designed for the linear model (6.11), any simple control strategy can be selected. In this dissertation we solve the unconstrained control problem using a simple Linear Quadratic (LQ) Regulator [Mos95]. The LQ controller is used to stabilize the system in linear regimes, viz. when the constraints are not active. To this end, by considering the following quadratic cost function +∞
J (u, s) =
(sT ( j)Qs( j) + uT ( j)Ru( j))
(6.14)
j =0
with the weights Q and R appropriately selected, e.g. for achieving, approximately, minimum energy control, the control law which minimizes (6.14) is given by u(k) =
−F s(k) + Hr(k)
(6.15)
where F = (R + B T P ∞ B)−1 BP ∞ A. Here r(k) is the reference, P ∞ is the solution of the Riccati Difference Equation (RDE) and H is such that C (I (A
−1
− BF ))
−
BH = I .
Hence, the evolution of the closed-loop system can be described by s(k + 1) = Φs(k) + Gr(k) + Gd d(k)
(6.16)
y(k) = H y s(k) Clearly, the closed-loop system evolutions under LQ state feedback (6.16) not necessarily satisfy the constraints (6.12) and (6.13) introduced above. Thus, we need to equip the overall control strategy with a unit capable of taking care of constraint fulfillment.
6.4
Conclusions In this chapter we have introduced the problem of controlling the attitude of MEO and LEO spacecraft. We have defined a LEO satellite as a nongeosynchronous 3D rigid body orbiting around Earth at a constant velocity, depending on its orbit’s altitude. First, we have illustrated the mathematical model of the system, focusing our attention on the main differences with respect to the geo-stationary case. In particular we have adopted the leader-following approach, leading to a novel linear attitude model, besides the well known Hill’s relative position model. Then, we have introduced input saturation and state accuracy constraints, that have to be taken into account while facing MEO and LEO spacecraft control problems. Finally, we have proposed the use of an MPC control technique. It will be illustrated in more detail in the following chapter.
99
Chapter
7 Attitude Control of a Low Earth Orbit Satellite
In Chapter 6 we have introduced the problem of reconfiguring and keeping the state of a LEO spacecraft, as well as the problem of making it to track a prespecified trajectory. The mathematical model of LEO satellites is relatively simple and it can be linearized, under common assumptions. Hence, it appears that any control law designed for continuous linear systems, is suitable for this problem. However, as explained in detail in Chapter 6, LEO satellites are typically small and their thrusters are not able to provide large control efforts [VGRM01], [TIH02]. Thus, in order to face practical applications, it is necessary to design control strategies that take constraints into account. Therefore, for the sake of simplicity and low numerical burden, we have proposed the use of an MPC based technique: the Command Governor approach. The main properties of such a control technique are illustrated in this chapter. Then, a CG scheme is applied to a satellite orbiting at 600 m above Earth’s surface and simulation results are depicted for illustrating the effectiveness of the technique.
7.1 Command Governor Approach for LEO Spacecraft Control In Figure 7.1 a CG control scheme with plant, primal controller and Command Governor unit [BCM97] is depicted. Equation (7.1) is a closed-loop state-space
d(k) 2 D r(k)
g(k)
CG
c(k)
y(k) ~ r(k)
u(k)
Primal Controller
Plant
c(k) 2 C
c
c
(k)
p
[] s(k)=
[ ] (k)
c
p
c
(k)
c
Figure 7.1. Command Governor scheme.
description of the plant regulated by the primal controller. s(k + 1) = Φs(k) + Gg(k) + Gd d(k) (7.1)
y(k) = H y s(k) c(k) = H c s(k) + Lg(k) + Ld d(k) In particular, s(k) (if any), g(k)
m
∈R
n
∈R
is the state which includes plant and compensator states
, which would be typically g(k) = r(k) if no constraints were
present (no CG present), is the CG output, viz. a suitably modified version of the reference signal r(k)
m
∈R
. Note that r(k) can be any desired function,
even a constant, to be tracked by system output. Moreover, d(k)
nd
∈R
is an
∈ D, ∀ k ∈ Z+, with D a specified convex and compact set such that 0 ∈ D ; y(k) ∈ R is the output, viz. a performance related signal which is required to track r(k) and c(k) ∈ R is the vector to be exogenous disturbance satisfying d(k) nd
m
nc
101
constrained. It is assumed that 1. (7.1) is asymptotically stable; 2. (7.1) is offset free, viz. H y (I
−1
− Φ)
(7.2) G = I m
The CG design problem consists of finding, at each time instant k, a command g(k) as a function of the current state s(k) and reference r(k) g(k) := g(s(k), r(k))
(7.3)
in such a way that g(k) is the best approximation of r(k) at time k, under the constraint c(k)
∈ C, ∀ k, and all possible disturbance sequences d(k) ∈ D.
Moreover, it is required that: 1. g(k)
→ rˆ whenever r(k) → r, with rˆ the best feasible approximation of r;
2. the CG have a finite settling time, viz. g(k) = rˆ for a possibly large but finite k whenever the reference stays constant after a finite time. By linearity, one is allowed to separate the effects of initial conditions and inputs from those of disturbances, i.e. s(k) = s¯(k) + s˜(k), where ¯s is the disturbancefree component (depending only on initial state and input) and ˜s depending only on disturbances. Then, denote the disturbance-free steady-state solutions of (7.1), for a constant command g(k)
≡ w, as follows
− Φ) 1Gw := H (I − Φ) 1 Gw := H (I − Φ) 1 Gw + Lw −
s¯w := (I n y¯w c¯w
y
n
c
n
−
(7.4)
−
Consider next the following set recursion
C0 C C j
∞
:= :=
C ∼L D C 1 ∼ H Φ C d
j−
c
∞
:=
j −1
Gd
D
(7.5)
j
j =0
A ∼ E is defined as {a ∈ A : a + e ∈ A, ∀ e ∈ E}. It can be shown that the sets C are non-conservative restrictions of C such that c¯( j) ∈ C , ∀ j ∈ Z+ , implies c( j) ∈ C , ∀ j ∈ Z+ . Thus, one can consider only disturbance-free where
j
∞
evolutions of the system and adopt a worst case approach . Next consider, for a small enough δ > 0, the sets: δ
C := C ∼ B W := {w ∈ R ∞
δ
where
B
δ
δ
m
: c¯w
(7.6)
δ
∈C }
is the ball of radius δ centered at the origin. In particular,
δ
W , which
we assume non-empty, is the set of all commands whose corresponding steadystate solution satisfies the constraints with margin δ . The main idea is to choose at each time step a constant virtual command δ
· ≡ w, with w ∈ W , such that the corresponding virtual evolution fulfills
v( )
the constraints over a semi-infinite horizon and its distance from the constant reference of value r(k) is minimal. Such a command is applied, a new state is measured and the procedure is repeated. In this respect we define the set
V (s)
as δ
V (s) = {w ∈ W : where
c¯( j, s, w)
∈ C , ∀ j ∈ Z+} j
j −1
c¯( j, s, w) := H c Φj s( j) +
Φj −i−1 Gw
i=0
+ Lw
(7.7)
(7.8)
has to be understood as the disturbance-free virtual evolution at time j of c
·≡
from the initial condition s at time j = 0 under the constant command v( ) w. As a consequence,
V (s) ⊂ W . δ
Moreover, if non-empty, it represents the
set of all constant virtual sequences in
W
δ
whose evolutions starting from s
satisfies the constraints also during transients. Thus, taking as a selection index a quadratic cost, the CG output is chosen according to the solution of the following constrained optimization problem g(k) = arg where Ψ = Ψ > 0 p and w
min
w ∈V (s(k ))
2Ψ := s Ψs.
w − r(k)2Ψ
(7.9)
It has been shown in [BCM97] and
[CMA00] that the CG technique described above, has the following properties.
Theorem 7.1 Let assumptions (7.2) be fulfilled. Consider system (7.1) along
V
with the CG selection rule (7.9), and let (s(0)) be non-empty. Then:
∈ Z+ and it can be obtained by solving a convex constrained optimization problem, viz. V (s(0))
1. the minimizer in (7.9) uniquely exists at each k
103
V
non-empty implies (s(k)) non-empty along the trajectories generated by the CG command (7.9); n
V ∀ s ∈ R
, is finitely determined, viz. there exists an integer j0
such that if c¯( j, s, w)
∈ C , j ∈ {0, 1, . . . j0}, then c¯( j, s, w) ⊂ C ∀ j ∈ Z+.
2. the set (s),
j
j
Such a constraint horizon j0 can be determined off-line; 3. the constraints are fulfilled
∀ k ∈ Z+;
4. the overall system is asymptotically stable; in particular, whenever r(k)
≡
r, g(k) converges in finite time either to r or to its best steady-state admissible approximation, g(k)
→ rˆ := arg min w − r2Ψ
(7.10)
w∈W δ
Consequently, by the offset free condition (7.2.2),
lim y¯(k) = rˆ, where
k →+∞
y¯ is the disturbance-free component of y.
[BCM97] also provides a comprehensive description on the algorithm for calculating the constraint horizon j0 defined above. Other details on solvability and computability of CG schemes, can be found in [GK92], [GKT95] and [GK99]. It is worth pointing out that, in order to accommodate coordination constraints in the form of (6.13) one has to generalize the definition of c(k) in (7.1) as follows c(k) = H c s(k) + Lg g(k) + Ld d(k) + Lr r(k)
(7.11)
Unfortunately, (7.11) differs from the third equation in (7.1) because of the presence of Lr r(k). Apart from cases where r(k) is constant, the condition
V (s(0)) non-empty, does not guarantee any longer that a solution exists and is admissible at each future time instant as indicated in Theorem 7.1. However, all the theorem’s stated properties can be restored acting as described below. Consider a parametric description r(α(k)) of the reference trajectory, with the introduction of a scalar parameter α(k), where α(k) = α(k
− 1) + ∆(k)
r(α(k)) can be thought as any point
m
∈R
∆(k)
∈ [0, 1]
between r(k
(7.12)
− 1) and r(k) along
the reference trajectory. In particular, one would be allowed to choose g(k) =
r(α(k)) corresponding to the nominal point r(k) in the reference trajectory, taking ∆(k) = 1, if no constraints were present. Therefore, we can substitute (7.12) into both g(k) and r(k) in (7.11) obtaining c(k) = H c s(k) + Lg r(α(k)) + Ld d(k) + Lr r(α(k)) = H c s(k) + Lr(α(k)) + Ld d(k) that produces again an equation in the form of (7.1), with r(α(k)) in place of g(k). It allows to successfully deal with the feasibility problems arising from the presence of state accuracy constraints (6.13). Consequently, we can replace (7.1) with s(k + 1) = Φs(k) + Gr(α(k)) + Gd d(k) y(k) = H y s(k)
(7.13)
c(k) = H c s(k) + Lr(α(k)) + Ld d(k) α(k) = α(k
− 1) + ∆(k)
where the parameter α(k) represents the novelty of our approach with respect to the classical CG strategy proposed in [BCM97]. The new CG problem is then ∆(k) := arg
max
r(α(k−1)+∆)∈V (s(k))
w(k) := r(α(k
∆ (7.14)
− 1) + ∆(k))
which can simply be solved via bisection. Thus, all properties pertaining to the CG approach described above are restored and the state can be driven to a desired final value, along an a priori established trajectory, without constraints violation. The same conclusion cannot be stated for other techniques proposed in literature as in [TIH02] and [VGRM01]. The modified-CG scheme illustrated above, appears particularly suitable for the problem at hand. Indeed, LEO spacecraft are often required to change their orientation and position, with respect to their nominal orbit or with respect to other satellites flying in formation. If they perform maneuvers moving along pre-designed trajectories, constraints in the form of (6.13), actually encompass many other constraints. In particular they include collision-avoidance constraints, that represent one of the main issues in formation flying control [ENS+ 03], [SBW06]. 105
Finally, notice that models (6.8) and (6.9) appear independent each other. Therefore, it seems that input commands τ and f for attitude and position can be provided separately. However, in LEO spacecraft, both translational and rotational motion are obtained by suitable combinations of the same thrusts. Moreover, attitude maneuvers and paths are usually defined in combination with position maneuvers and paths. Thus, representing (6.8) and (6.9) as a single model, as made above by (6.11), is necessary for the problem at hand. Consequently, at any time instant k, the CG control law needs to select the most restrictive ∆(k) between the one resulting from the attitude maneuver and the one resulting from the position maneuver.
7.2 Simulation of a LEO Spacecraft Reconfiguration Maneuver In the following, we present simulation results on the application of the modifiedCG technique described in previous section, to a LEO spacecraft. As illustrated
{ }
{ }
above, we assume small angle displacements between B and O , justifying linear attitude description (6.8). Moreover, we assume small distances between the orbiting 3D rigid body and its reference orbit. Hence, Hill’s model (6.9) holds true. From the arguments described in Chapter 6 and in previous section, we rewrite (6.8)–(6.9) in the more compact form (6.11) and we apply to it an LQ control law, that does not take constraints into account. Hence, system object of study becomes (6.16). Since we also want to handle saturation constraints in the form of (6.12) and state accuracy constraints in the form of (6.13), the whole system is finally described by (7.13). Thus, the control problem is effectively solved by technique (7.14). We refer to a LEO satellite that orbits around Earth at an altitude of about 600 Km, i.e. at a velocity ω0 = 0.0011rad/sec. We assume, for the rigid body a mass of 150 Kg and a diagonal inertia matrix J = diag(35, 16, 25) Kg m2 . These, are the typical dimensions of small satellites [ENS + 03]. Further we suppose the following saturation constraints:
|f |
max
= 5 10−2 N and τ max =
·
||
10−3 N m. The maximum amplitudes of disturbance forces are assumed to be
|d |
i max
= 10−3 N , i = 1, 2, 3. Moreover, the maximum amplitudes of distur-
| |
bance torques are assumed to be di
max
= 10−4 N m i = 4, 5, 6. A value of
δ = 10−6 in equation (7.6) is selected.
We first apply the technique described above to perform a reconfiguration maneuver , i.e. to change the position and the attitude of the satellite. Notice that this task can be interpreted as a tracking problem, where the reference signal to be tracked is defined only by a final state r and where constraints in the form of (6.13) are absent. A LEO spacecraft, leaving from position (x,y,z)T = (60, 5, 5)T m with re-
−
{ }
spect to the origin of O , selected as the leader orbiting point, has to be driven to the final desired position (x,y,z)T = (70, 0, 0)T m. In the meanwhile it is asked to rotate, from its initial attitude corresponding to Euler’s angles (ψ,θ,φ)T = (0.1, 0.1, 0.1)T rad in O , to a final desired attitude correspond-
−
ing to Euler’s angles (ψ , θ , φ)T
{ } = (−0.1, 0.1, 0)
T
rad. Position and attitude
evolutions under controller (7.14), are depicted in Figures 7.2 and 7.3. A constraint horizon of jo = 23 steps is determined by means of the appropriate algorithm presented in [BCM97]. Moreover, Q and R in (6.14) are selected as Q = I 12 and R = 50I 6 .
Figures 7.4 and 7.5 show input forces and torques
related to the desired maneuver. We observe that there are instants in which saturation constraints become active. Thanks to the action of the CG unit, such constraints are never violated. If the CG unit is not used, the LQ control law (6.14)-(6.15) is not able of driving the spacecraft towards the desired position due to input saturation. It results from Figure 7.6, where components x and y do not reach the desired values. The rationale is that input forces generated by the LQ controller are no longer admissible. It can be seen from Figures 7.8 and 7.9 that show how forces and thrusts actually applied, are a saturated version of those computed by the LQ control law. In this simulation, a different behavior results for angles ψ, θ and φ which reach the desired attitude in spite of constraints violations during the transient. It is depicted in Figure 7.7.
107
72
5
1
4
70
0
3 68
−1
] m [
] m [
x
y
66
2
] m [
−2
z
1
64
−3 0
62
60
−4
−1
0
1000 Time [s]
−2
2000
0
1000 Time [s]
−5
2000
0
1000 Time [s]
2000
Figure 7.2. Relative position components x(k ), y (k), z (k) of the LEO satellite, under the proposed LQ+CG control law (7.14).
0.15
0.15
0.1
0.1
0.08 0.1
0.05 0.06 0.05 ] d a r [
0
ψ
] d a r [
] d a r [
θ
φ
0.04
−0.05 0 0.02 −0.1 −0.05 0
−0.15
−0.2
0
1000 Time [s]
2000
−0.1 0
1000 Time [s]
2000
−0.02 0
1000 Time [s]
2000
Figure 7.3. Euler’s angles ψ (k), θ (k), φ(k) of the LEO satellite, under the proposed LQ+CG control law (7.14).
7.3 Simulation of a LEO Spacecraft Tracking Maneuver Next, we present simulation results of the proposed control technique, when a LEO satellite is required to reach a desired position and attitude, moving along
] N [
1 f
0.06
0.06
0.06
0.04
0.04
0.04
0.02
0.02
0.02
] N [
0
2 f
] N [
0
3 f
0
−0.02
−0.02
−0.02
−0.04
−0.04
−0.04
−0.06 0
1000 Time [s]
2000
−0.06 0
1000 Time [s]
2000
−0.06 0
1000 Time [s]
2000
Figure 7.4. Input forces f i (t) generated by the proposed LQ+CG control law (7.14). The dash lines represent the constraint boundaries.
−3
−3
x 10
] N [
1
−3
x 10
x 10
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
] N [
0
2
τ
] N [
0
3
τ
0
τ
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1
−1
−1
0
1000 Time [s]
2000
0
1000 Time [s]
2000
0
1000 Time [s]
2000
Figure 7.5. Input torques τ i (t) generated by the proposed LQ+CG control law (7.14). The dash lines represent the constraint boundaries.
a pre-designed trajectory. In this case, constraints in the form of (6.13) are present. Then, whole system is described by (7.13) and control action (7.14) 109
300
5
2
1
0 250
0 −5 200 −1
] m [
] m [
] m [
x
y
z
−10
−2 150 −15 −3 100 −20
50 0
1000 Time [s]
−4
−25 0
2000
1000 Time [s]
−5 0
2000
1000 Time [s]
2000
Figure 7.6. Rel Figure Relati ative ve positions positions x(k ), y (k ), z (k ) of the LEO satellite under LQ control law only.
] d a r [
0.15
0.25
0.1
0.1
0.2
0.08
0.05
0.15
0.06
0
] d a r [
ψ
0.1
] d a r [
θ
0.04
φ
−0.05
0.05
0.02
−0.1
0
0
−0.15
−0.05
−0.02
−0.2
0
1000 Time [s]
2000
−0.1 0
1000 Time [s]
2000
−0.04 0
1000 Time [s]
2000
Figure 7.7. Euler’ Figure Euler’ss angles ψ (k), θ (k), φ(k) of the LEO satellite under LQ control law only.
has to be applied in place of (7.9). We suppose that the spacecraft, leaving from the state reached after previous maneuver, is required to track a pre-specified circumference of radius 20 m in
5
2
0.6
0.5
0
1.5 0.4
−5 1 0.3 −10 ] N [
] N [
f
f
1
2
] N [
0.5
3 f
0.2
−15 0.1 0 −20 0 −0.5
−25
−0.1
−30 0
1000 Time [s]
−1 0
2000
1000 Time [s]
−0.2 0
2000
1000 Time [s]
2000
Figure 7.8. Figure 7.8. Inp Input ut forces forces f i (t) req requir uired ed by the LQ control control law. law. The dash lines lines represent the constraint boundaries.
−3
0.01
0.015
4
x 10
2
0.005 0.01
0 0 0.005 −2 ] N [
] N [
] N [
τ
τ
τ
−0.005
1
2
3
−4 0 −0.01 −6 −0.005 −0.015
−0.02
−8
0
1000 Time [s]
2000
−0.01
0
1000 Time [s]
2000
−10 0
1000 Time [s]
2000
Figure 7.9. Inp Figure Input ut torques torques τ i (t) req requir uired ed by the LQ con contro troll law. The dash lines lines represent the constraint boundaries.
the (x, (x, y ) orbit plan and centered in (x, ( x, y )T = (50, (50, 0)T m, i.e. to track a variable reference reference signal. signal. Further, urther, we require require z , ψ , θ and φ to maintain their initial 111
values, i.e. z = 0 m, and (ψ,θ,φ (ψ,θ,φ))T = ( 0.1, 0.1, 0)T rad (keeping).
−
Here, besides the same saturation constraints of the first example, we want to
|
− r (α(k))| < 0.1 m for each of the for i = 1, 2, 3, and |s (k) − r (α(k ))| < 0.01 rad for
consider also accuracy constraints si (k ) position position components, components, i.e.
i
i
i
each each of the attitude attitude components, components, i.e. for i = 4, 5, 6. This This kind of accurac accuracy y is required, as an example, in order to maintain the pointing of a geographic site,
{ }
despite of O motion around ECI. Results of the application of technique (7.13)-(7.14) are illustrated below. Figure 7.10 shows evolution of position variables x, y and z from initial to final state. Note that the reference signal, depicted in dash lines, is tracked by x and y, 75
25
5
70
20
4
65
15
3
60
10
2
55
5
] m [
] m [
x
y
50
1 ] m [
0
0
z
45
−5
−1
40
−10
−2
35
−15
−3
30
−20
−4
25 0
2 4 Time [s]
6 4
x 10
−25 0
2 4 Time [s]
6 4
x 10
−5
0
2 4 Time [s]
6 4
x 10
Figure 7.10. Evolu Figure Evolution tion of the relati relative ve position variables variables x, y and z of the LEO satellite with the application of LQ+CG (tracking problem) for a period of revolution.
while z (k) = z (0) is kept at any time instant k . Initial values of Euler’s angles, here here not depicte depicted, d, are kept kept as well. well. In the case case of present present simula simulatio tion, n, the active constraints are the coordination ones related to the position in the x and y directions directions.. This is evident from Figure 7.11, 7.11, where e1 and e2 approach constrain constraints ts boundar b oundaries. ies. Input variables variables,, on the contrar contrary y, are well inside their respect respectiv ivee satura saturatio tion n boundar boundaries ies.. Notice Notice that, because because of the use of the CG selection logic (7.14), the reference trajectory is tracked with variable speed,
] m [
1
e
0.15
0.15
0.15
0.1
0.1
0.1
0.05
0.05
0.05
] m [
0
2
e
] m [
0
3
e
0
−0.05
−0.05
−0.05
−0.1
−0.1
−0.1
0
2 4 Time [s]
6
0
4
x 10
2 4 Time [s]
6 4
x 10
0
2 4 Time [s]
6 4
x 10
Figure 7.11. Errors ei (t), i=1,2,3 in coordination accuracy for the position components x(k ), y (k) and z (k) related to the circumference tracking maneuver under (7.14). The dash lines represent the constraint boundaries ( |ei |max = 0.1).
lower than the planned velocity. This can be deduced from Figure 7.12 where ∆(k) is not constant and α(k) is not a line. Thus, evolution in the (x, y) plane is the one depicted on the left side of Figure 7.13, while reference evolution would be the one represented on the right side, in dash line. It is evident that the nominal trajectory r(k) is tracked slowly, with respect to planned velocity, but desired path is covered, without accuracy constraints violation. As is previous section, also in this case, if the CG unit is not used, constraints are not fulfilled. In particular here, closed-loop system (6.16), without (7.14), does not satisfy formation accuracy constraints related to the position components. On the other hand, saturation constraints are always satisfied under this maneuver. Thus, the trouble in this case is that coordination accuracy deteriorates up to a level given in Figure 7.14. Euler’s angles ψ, θ and φ, input torques ui (i=4,5,6) and attitude accuracy related to the present tracking maneuver, here not depicted, stay inside their respective bounds. 113
1
2500
0.9
0.8
2000
0.7
0.6 ) t (
1500 ) t (
0.5
∆
α
0.4
1000
0.3
0.2
500
0.1
0
0
2
4 Time [s]
0
6
0
2
4 Time [s]
4
x 10
6 4
x 10
Figure 7.12. Values of ∆(t) and α(t) for the CG selection logic (7.14) corresponding to the maneuver. 25
25
20
20
15
15
] m [
] m [ 2 r
y
10
10
5
5
0 20
40
60 x [m]
80
0 20
40
60
80
r [m] 1
Figure 7.13. Trajectory in the (x, y ) plane under (7.14) (left) in comparison with reference signal (right).
7.4
Conclusions In this chapter we have proposed a technique for controlling the attitude of LEO satellites. The proposed control scheme is based on the MPC technique,
0.25
0.25
0.25
0.2
0.2
0.2
0.15
0.15
0.15
0.1
0.1
0.1
0.05 ] m [
1
e
0.05 ] m [
0
2
e
0.05 ] m [
0
3
e
0
−0.05
−0.05
−0.05
−0.1
−0.1
−0.1
−0.15
−0.15
−0.15
−0.2
−0.2
−0.2
−0.25
−0.25
−0.25
0
1 Time [s]
2 4
x 10
0
1 Time [s]
2 4
x 10
0
1 Time [s]
2 4
x 10
Figure 7.14. Errors ei (t) in coordination accuracy for the position components x, y and z related to the circumference tracking maneuver under LQ only. The dash lines represent the constraint boundaries ( |ei |max = 0.1).
referred to as the Command Governor. It consists in designing a primal control law and an external unit capable of taking care of constraint fulfillment by modifying, whenever necessary, the reference signal. Besides classical CG scheme, well known in literature and widely discussed in [BCM97], we have proposed a parameterization of reference signal, in order to handle state accuracy constraints. Since in LEO orbits, controlling attitude usually requires simultaneously controlling relative position, we have applied the control scheme to a mathematical description of the system, combining both position and attitude variables. Then, we have shown simulation results of the application of the proposed technique, both for reconfiguring the state of a LEO satellite and for making it to cover a pre-designed trajectory. The proposed LQ+CG algorithm needs the system to be linear. Hence, we have assumed small angular displacements between initial body frame orientation (ψ(0), θ(0), φ(0)) and final desired orientation, leading to model (6.8). This assumption will be removed in next chapter.
115
Chapter
8
Attitude Control of a LEO Satellite for Large Angle Maneuvers In previous chapter, we have proposed a model predictive control technique, namely a modified version of the Command Governor approach [BCM97], for controlling the attitude of a LEO spacecraft. The presented scheme, has been shown to be effective under the assumptions of small distances and small angu-
{ }
lar displacements between the body frame B and a leader orbiting reference
{ }
frame O . This allows at describing relative position by linear model (6.9) and attitude by linear model (6.8). However, the second assumption, requiring small angular displacements between a LEO satellite body frame
{B} and its leader reference frame {O}, might
not match with some demanding applications. Experimental results show that model (6.8) has significant discrepancy with respect to the real model for Euler’s angles deviations larger than 15 deg. Thus, in this chapter we remove the small angles assumption , taking into account maneuvers requiring the space vehicle to span large angles. For instance, this is an usual request, when a LEO satellite
has to observe a large area on the Earth surface, during an Observation Mission [ENS+ 03]. On the other hand, the sake of simplicity and the need of low numerical burdens still suggest the use of the CG techniques for linear systems, adopted in Chapter 7. Thus, we propose a bank of controllers, each one designed with respect to a pre-established orbital reference frame. A hybrid control scheme is then applied. The bank of controllers is orchestrated by a supervisory logic which at each time instant selects the most appropriate CG unit, on the basis of the spacecraft state and the prescribed tracking paths.
8.1 LEO Spacecraft Nonlinear Attitude Model Consider model (6.8)-(6.9) illustrated in Chapter 6, describing attitude and position variables dynamics of a LEO spacecraft, with respect to a leader reference
{ }
frame O . As described above, the assumption of close proximity between the LEO satellite and its nominal orbit, under which model (6.9) can be used, is effective for all the real missions, even when the space vehicle is a component of a formation flying system. On the contrary, the hypothesis of small angles, that leads to system (6.8), does not hold true in some practical applications. On the other hand, the use of above model is convenient, since it guarantees the possibility of using the CG-based control law (7.14), which provides simplicity and law numerical burdens. In order to maintain representation (6.8)-(6.9) in the general case, one possibility is that of parting any possible large angle attitude maneuver, in a sequence of small angle maneuvers, defined with respect to a suitable set of reference frames
{O }. i
{ } { }
In this way, any sub-maneuver describing B in Oi , can satisfy the
small angular displacement assumption. Therefore, equation (6.8) can be used, as long as Euler’s angles ψ, θ and φ, between Oi and B , are small1 . Then,
{ }
{ }
{ }
model (6.8) can be re-written with respect to another reference frame Oj satisfying the small angular displacement assumption. Finally, at any time instant
{ }
k, B attitude can be easily described in the leader frame, in the following
{ }
{ }
{
}
{
}
denoted as C , combining B orientation in Oi (k) and Oi (k) orientation 1
Less than 15 deg, from experimental results.
117
{ }
{ }
in C . Indeed, the constant attitude of any Oi frame, can be expressed with
{ }
respect to C , by a constant rotation matrix. Clearly, it is necessary that at any time instant, there exist at least one frame
{O }, with small angular displacement with respect to {B}. An example of the i
described approach is depicted in Figure 8.1.
{O5} {O4} w0
Y
’
{O6}
{C} Z
’
...
X
Z
{B}
{O3}
’
X Y
{O1}
{O2}
Figure 8.1. Set of reference frames {Oi } and leader orbiting reference frame {C }. At any time instant, there exists at least one {Oi } satisfying the small angular displacement assumption with respect to {B }.
Thus, we consider again equation (6.4), expressing ωbo in body coordinates. As
{ } set of reference frames {O }, (i = 1, ··· , ¯i), orbiting in {EC I } along with {C }, introduced above, we denote the leader reference frame as C and we define a i
and covering the space SO(3). It means that we take a number ¯i of reference
{ }
frames Oi , in such a way that ¯i
SO(3)
⊂
Roi c ∆Roi c Oi
{
i=1
}
where Roi c is the constant rotation matrix that changes
(8.1)
{C }
in
{O } i
and
∆Roi c = ∆Roi c (ψ , θ , φ) spans angles up to 15 deg in ψ, θ and φ directions.
{ }
{ }
Moreover, any Oi is motionless in C . Hence, ω oi = ω c
∀ i = 1, ··· , ¯i
Next, considering the rotation matrix Rbc between the body frame B and the
{ }
{ }
leader frame C , we have
ω oi = Rbc
0 0 ω0
= RRoi c
{ }
0 0 ω0
{ } R is the constant rotation matrix that changes {C } in {O }. Now, since both the velocity ω0 of {C } in {EC I } and the attitude of any {O } in {C } are constant and known, we can define the angular velocity of {C } in {O } coordinates as where R is the rotation matrix between Oi and B , defined by (6.5), and oi c
i
i
i
ωc = Roi c
It follows that ωoi corresponds to
ω oi =
−
0 0
ω0
=
ωc1 ωc2 ωc3
cθcφωc1 + (cψsφ + sψsθcφ)ωc2 + (sψsφ
(8.2)
− cψsθcφ)ω 3 c
− sψsθsφ)ω 1 + (sψcφ + cψsθsφ)ω 3 sθω 1 − sψcθω 2 + cψcθω 3
cθsφωc1 + (cψcφ
c
c
c
c
c
(8.3)
Finally, adding (6.4) and (8.3), taking the first derivative of ω b in (6.2), and substituting into (6.1), yields to a nonlinear attitude model. However, we as-
{ }
{
}
sume small angular deviations between B and the selected Oi (k) , around
{ }
which B orientation is expressed at time k. Thus, cos(ψ)
1, sin(ψ) 0 and
analogous for θ and φ. Therefore, Euler’s equations (6.1) become ψ¨ = (ωc23
− ω22)J ¯1ψ + ω 3(1 + J ¯1)θ˙ + ω 1ω 2J ¯1θ ¯1 − 1)φ˙ − ω 1 ω 3 J ¯1 φ + τ 1 + ω 2 (J ¯2 − 1)ψ˙ − ω 1 ω 2 J ¯2 ψ + (ω2 − ω 2 )J ¯ θ¨ = ω 3 (J 1 3 2θ c
c
c
c
c
c
c
c
c
c
c
c
¯2 )φ˙ + ωc2 ωc3 J ¯2 φ + τ 2 + ωc1 (1 + J ¯3 )ψ˙ + ωc1 ωc3 J ¯3 ψ + ωc1 (J ¯3 φ¨ = ωc2 (1 + J
J j −J k J i
− 1)θ˙
− ω 2ω 3J ¯3θ + (ω22 − ω21)J ¯3φ + τ 3 j = k). (i,j,k = 1, 2, 3, i = c
¯i = where J
(8.4)
c
c
c
System (8.4) can be placed in (6.10) in place of (6.8). Then, all the properties 119
{ }
{ }
of the scheme presented in Chapter 7 for controlling B orientation in O ,
{ }
are restored. Clearly, at time k they hold true in a suitable Oi , satisfying the small angular displacement assumption. As soon as ψ, θ or φ become large, a technique for switching to another CG controller, has to be adopted. This
{ } ∈ (1, ··· , ¯i), by the
new controller needs to be defined in a suitable Oj , j corresponding (8.4)-(6.9) model.
{ }
A method for switching between the CG controller designed in Oi , to the one
{ }
designed in Oj is proposed in the next section. It addresses present control problem in a hybrid framework.
8.2 Hybrid Command Governor for LEO spacecraft In this section we propose a control scheme still based on the CG technique (7.14) for linear systems, even when the LEO satellite has to perform large angle maneuvers. As described above, this can be made introducing a bank of linearized models in the form of (8.4), each one representing the attitude relative
{ }
to a specific reference frame Oi . The position components are not affected by this problem and just one single model (6.9) can be used along with each member of the given set of attitude models. Consequently, referring to the orbiting rigid body, a single CG unit can be designed for each linearized model and a suitably designed supervisory unit can take care of orchestrating the switching among the CG candidates during the on-line operations. We refer to the overall technique as hybrid CG control scheme (HCG). A similar approach has been previously used in [ABC + 03]. Consider the following set of reference set-points r which are desired to be tracked without offset r
m
∈Ξ⊂R
Assume, for the problem at hand, that Ξ
δ
δ
⊂ W , with W defined in (7.6). Thus,
the requirement that all the set-points in Ξ will be tracked without error cannot be satisfied. A way to overcome this limitation is that of covering the set Ξ with a collection of
δ i
W
(i = 1,
··· , ¯i), with overlapping interior corresponding to ¯i
different CGs such that
¯i
⊂ W
δ i
Ξ
(8.5)
i=1
{W ∩ W } = 0, for at least a pair (i, j) ∈ {1, ··· , ¯i}. Clearly, CG operates properly when initial and final set-points belong to W . If the final set-point belongs to a different set W , a procedure for switching between CG δ i
and Interior
δ j
i
δ i
δ j
i
and CGj has to be defined. To this end, let us consider the output admissible set
δ i
m
n
Z ⊂ R × R
for CGi . It consists of the set of all pairs [r, s]T whose
evolutions satisfy the constraints for all t
∈ Z+.
Hence, we can define the set of all states which can be steered to feasible equilibrium points without constraints violation δ j
n
X := {x ∈ R
: [w, s]T
δ i
m
∈ Z for at least one w ∈ R } δ i
δ j
{W ∩ W } =
δ i
{X ∩ X } = 0. Thus, one can a-priori define a convenient transition reference r ∈ Interior{W ∩ W } such that Now, if (i, j) is such that Interior δ j
0 then also Interior
ij
δ i
δ j
sij
δ i
δ j
∈ Interior{X ∩ X }
where sij is the equilibrium disturbance-free steady-state corresponding to rij (using the worst case approach stated in Chapter 7). Finally, [rij , sij ]T
δ i
δ j
∈ {Z ∩ Z } and the transfer strategy is simply defined. Assume to be at instant k, be using CG and let r(k) ∈ W , r(k + 1) ∈ W with {W ∩ W } = 0. Hence, a possible switching logic is as follows: δ i
i
δ i
δ j
δ j
1. Solve and apply g(k + h) = arg
min w∈V i (s(k+h))
w − r(k)2Ψ, h = 1, . . . , h
2. At k = k + h, as soon as s(k)
δ i
δ j
∈ Interior{X ∩ X }
(8.6)
switch to CGj and solve g(k) = arg
min
w∈V j (s(k ))
w − r(k + 1)2Ψ, k ≥ k + h + 1 121
The illustrated scheme is motivated by the fact that for any s evolution will enter in Interior
δ i
δ j
n
∈R
the state
{X ∩ X } within a finite number of time in-
stants. An upper bound to this integer can be computed off-line with respect to all s
n
∈R
in a way similar to that used to determine the constraint horizon
j0 defined in Chapter 7. Then, instead of checking the set-membership condition in step 1), one could determine such an upper bound h off-line and exploit it during the on-line operations by waiting for exactly h steps before switching to step 2) above. Of course, other possibilities for the switching logic exist, which could be more effective for some applications. In all cases, the above guidelines, inspired by [GK99], allows one to retain in a hybrid general context the same stability and feasibility properties pertaining to the basic CG approach (7.14). In conclusion, a hybrid CG scheme can be adopted defining a correct decomposition of r into a well suited sequence of transition reference signals rij and a criterion for switching between the corresponding controllers. For planned missions, the transition reference sequence can be defined a priori. Further, a graph of connections and a look-up table can replace the switching criterion. In the general case, however, a switching logic has to be employed. We propose a criterion to characterize Interior
δ i
δ j
{X ∩ X } in (8.6), based on the Euclidean
norm between the state and the linearization point. The supervisor can main-
{ }
tain CGi as long as the distance between Oi and present attitude, assumed to be available, is minimal. On the contrary it can switch to the j
− th model,
{ }
centered in Oj where CGj corresponds to
j = arg
min
j ∈(1,··· ,¯ i)
(ψj , θj , φj )T
(8.7)
The criterion is depicted is Figure 8.2. Simulations results show that this switching criterion is effective in performing large angle maneuvers. Finally notice that the above control scheme easily encompasses the control technique described in Chapter 7. Indeed, when Ξ
δ
⊂ W , a single model linearized
around (ψ , θ , φ)T = (0, 0, 0)T rad, covers the entire attitude reconfiguration maneuver and no switching is required.
Figure 8.2. Switching logic for attitude reconfiguration.
8.3 Simulations of a LEO spacecraft performing Large Angle Maneuvers We refer again to a LEO satellite orbiting around Earth at a velocity ω0 = 0.0011 rad/s. It could represent, for instance, a slave satellite in an Earth Observing Formation Flying System. For the space vehicle, we assume the same mass and inertia matrix of previous examples, and the following saturation constraints: f max = 5 10−2 N , τ max = 2 10−3 N m. The maximum ampli-
||
·
||
·
| |
tudes of disturbance forces are assumed to be di
max
= 6 10−3 N , i = 1, 2, 3.
·
Moreover, the maximum amplitudes of disturbance torques are assumed to be
|d |
i max
= 2 10−4 N m, i = 4, 5, 6. A value of δ = 10−4 in equation (7.6) is
·
selected. The LEO spacecraft, leaving from positions (x,y,z)T = (0, 10, 0)T m in C , is
{ }
required to move away from the formation center, along the second axis direction, in order to reach positions (x,y,z)T = (0, 200, 0)T m. Further, it has to maintain its pointing towards position (x,y,z)T = ( 100, 0, 0)T m. This could
−
be the position of another satellite, with which the spacecraft has to exchange information. It means that the satellite is required to follow a prescribed attitude trajectory in accordance with the maneuver, consisting in a rotation around its own third axis. Thus, only the angle φ has to be modified, whereas angles ψ and θ have to be maintained constantly at zero. Besides saturation constraints 123
Z w0
w0
Y
X
{0,10,0}
{0,200,0}
{-100,0,0}
Figure 8.3. Reconfiguration for a LEO spacecraft, requiring a large angle maneuver.
|
− r (α(k))| < 0.5 m for each of for i = 1, 2, 3, and |s (k) − r (α(k))| < 0.1 rad
we want to consider accuracy constraints si (k) the position components, i.e.
i
i
i
0.0017 rad) for each of the attitude components, i.e. for i = 4, 5, 6.
(
Results of the application of the hybrid CG controller (7.14) and (8.7) to the satellite are illustrated in Figures 8.4 and 8.5.
Here, the constraint horizon
150 ] m [ y
100 50 0
0.5
1
1.5
2
2.5 4
x 10 0.05 ] N [
0
2 f
−0.05 0
0.5
1
1.5
2
2.5 4
x 10 0.5 ] m [
2
e
0
−0.5 0
0.5
1
1.5 Time [s]
2
2.5 4
x 10
Figure 8.4. Relative position y(t) (desired reference in dash lines), control force f 2 (t) and error e2 (t) in coordination accuracy for y (t) (constraint boundaries in dash lines) under HCG. The vertical dash lines indicate instants of switching.
introduced in theorem 7.1, results j0 = 51. Values of ∆(t) and α(t) relative to equation (7.12), corresponding to the maneu-
0 ] g e d [ φ
−20 −40 −60 0
0.5
1
1.5
2
2.5
−3
4
x 10
x 10
1 ] m N [
0.5 0
3
τ
−0.5 −1 0
0.5
1
1.5
2
2.5 4
x 10 0.1 ] g e d [
0.05 0
3
e
−0.05 −0.1 0
0.5
1
1.5
2
2.5
Time [s]
4
x 10
Figure 8.5. Euler’s angle φ(t) (desired reference in dash lines), control torque τ 3 (t) and error e3 (t) in coordination accuracy for φ(t) (constraint boundaries in dash lines) under HCG. The vertical dash lines indicate instants of switching.
ver, are depicted in Figure 8.6. The active constraints are the coordination ones 1 1600 0.9 1400
0.8
) t (
∆
0.7
1200
0.6
1000 ) t (
0.5
α
800
0.4 600 0.3 400 0.2 200
0.1
0 0
0.5
1 1.5 Time [s]
2
2.5 4
x 10
0 0
0.5
1 1.5 Time [s]
2
2.5 4
x 10
Figure 8.6. Values of ∆(t) and α(t) for the HCG selection logic (7.14), (8.7).
related to the position y(t). This is evident in Figure 8.4. Thus, in the present 125
example, the constraints that influence the values of ∆(t) in (7.14) are the ones related to the position accuracy. Input forces and torques, on the contrary, are well inside their respective saturation boundaries. Moreover, since the position constrains are very stringent, input forces and torques take values close to zero during the entire reconfiguration. The dash vertical lines depicted in Figures 8.4 and 8.5, indicate the time instants when the supervisor switches from the acting CG to another one, selected according to rule (8.7). In the present simulation, a different linear model has been associated to the system, for Euler’s angles of (0, 0, 0)T rad, (0, 0, 20)T rad, (0, 0, 40)T rad and so on. Again, notice that, because of the use of the HCG, the reference trajectory is tracked with a speed typically lower than the planned velocity. This is evident in Figure 8.6 where ∆(t) is less than one (∆(t) = 1 would correspond to nominal tracking) and in Figure 8.4. If the hybrid CG scheme is not used, the formation accuracy constraints related to the position are no longer fulfilled as it results from Figure 8.7.
150 ] m [ y
100 50 0
5000
10000
15000
5000
10000
15000
10000
15000
0.05 ] N [
0
2 f
−0.05 0 1 ] m [
2
e
0.5 0 −0.5 −1 0
5000 Time [s]
Figure 8.7. Relative position y(t), control force f 2 (t) and error e2 (t) in coordination accuracy for y (t) under LQ (constraint boundaries in dash lines).
8.4
Conclusions In this chapter we have proposed a technique for controlling the attitude of LEO satellites, when they are required to span large angles. The presented scheme is a natural extension of the technique illustrated in previous chapter. It consists of a bank of Command Governors controllers, each one designed with respect to a suitable attitude model. At any time instant, a supervisor selects the most appropriate CG controller from the bank, on the basis of current state and prescribed constraints. The proposed switching logic acts in such a way that the
− th CG, designed around the reference frame {O }, that is the closest to {B } in the bank. supervisor always selects the i i
The whole technique, addressing the attitude control problem for LEO spacecraft in a hybrid framework, allows at restoring the properties of the classical CG control scheme, stated in theorem 7.1. Thus, simplicity and reduced numerical burdens, proper of CG, are restored, with the only additional cost of solving (8.7) at any time instant. Effectiveness of the technique has been illustrated by simulation results.
127
Conclusion Spacecraft motion has been subject of study for many years. Prior to the development of space vehicles, scientist analyzed motion of heavy orbiting bodies, giving rise to the discipline called celestial mechanics . Nowadays, two general areas of study on the subject of spacecraft motion can be identified. The first one, namely astrodynamics , considers spacecraft motion in gravity fields. The second one is attitude control . It is concerned with motion about the vehicle’s center of mass. In this dissertation we have presented theory lying in this last field of research. In particular, we have proposed attitude mathematical models and control techniques for satellites, traveling around Earth on geo-synchronous and non geo-synchronous orbits. Behavior of satellites on geo-stationary orbits is quite different with respect the one pertaining to satellites moving on Medium and Low Earth orbits. Thus, this dissertation has been broadly divided into two parts. The first part has dealt with the problem of modeling and controlling the attitude of spacecraft on GEO orbits. Since in this case the satellite period can be made exactly equal to the time it takes the Earth for rotating once about its axis, no translational motion has been taken into account. As a consequence, we have addressed the GEO spacecraft control problem in the more general framework of controlling a rigid body, constrained to rotate about a pivot. Therefore, we have introduced the new concept of 3D pendulum . The second part of this dissertation, has dealt with the problem of modeling and controlling the attitude of spacecraft on MEO and LEO orbits. In this case,
beyond rotational motion, a rigid body translational motion is observed from Earth. Thus, we have introduced an orbital model, describing satellite rotations and translations, with respect to a non-inertial reference frame.
Geo-stationary Spacecraft and the 3D Pendulum A geo-stationary spacecraft is an orbiting rigid body, whose center of mass is motionless with respect to Earth surface. Thus, it possesses only three rotational degrees of freedom. They correspond to rotations generated by suitable combinations of thrust actions. As a consequence, its dynamics can be reproduced on Earth, pivoting a 3D rigid body on its center of mass. This argument has suggested the introduction of a new system, that we have defined in Chapter 1 as the 3D pendulum . Introduction of this new concept, has also been suggested by a laboratory process, available at the Attitude Dynamics and Control Laboratory, Department of Aerospace Engineering, University of Michigan. In this dissertation we have referred to this system as Triaxial Attitude Control Testbed . It actually consists of a 3D pendulum whose pivot can be located on the center of mass or on another arbitrary point. This second case, i.e. the unbalanced one, generalizes the geo-stationary spacecraft model, since it also considers the presence of gravity. In the first part of this thesis we have analyzed in detail, mathematical descriptions and control techniques for unbalanced 3D rigid pendulum. First of all, in Chapter 1 we have analyzed several possible attitude models based both on local and global coordinates. We have used Euler’s angles, for showing that the 3D pendulum model embodies many other classical mechanical systems. Indeed, in Chapter 2, we have proven that the spherical 2D pendulum model, the planar 1D pendulum model and the heavy top model, are special cases of the 3D pendulum model. Moreover, we have shown that a symmetric 3D pendulum can easily describe the motion of a spinning top under the table . This system has a great theoretical value. Surprisingly, it has not been studied in previous literature. On the other hand, we have shown that global coordinates are more suitable for describing the 3D pendulum motion and for relating it to attitude models for CONCLUSION
129
geo-stationary satellites, proposed in prior literature. Indeed, they avoid singularities, that are one of the main issues in controlling mechanical systems. Thus, we have provided an attitude description based on Poisson’s variables. Moreover, noticing that the momentum about the vertical is a prime integral of the system, we have performed a Routh reduction. We have obtained an attitude model lying on S 2
× R3.
It describes the evolution of the gravity direction in
body-fixed coordinates. Expressing the involved quantities in body-coordinates is particularly convenient, because in space systems, torques are provided by actuators mounted on the body. Further, in Chapter 3, we have proposed passivity based control techniques for stabilizing the natural equilibria of the system, i.e. the hanging equilibrium and the inverted equilibrium . We have faced stabilization of arbitrary attitudes, in Chapter 4. Selecting passivity-based control has been natural. Indeed, angular velocity feedback renders the closed-loop, input-output passive, with the total energy as storage function. Thus, suitably shaping the potential energy and injecting damping, stabilizes any arbitrary orientation. Contrary to previous literature, the control law that we have proposed in Chapter 4, provides nearly global asymptotic stability. Moreover, the control scheme, has been shown to generate input torques in the direction that avoids the unwinding phenomenon . Thus, it is particularly suitable in spacecraft control, since it reduces fuel consumption. Finally, in Chapter 5, we have proven that a similar continuous state feedback control law, again based on potential shaping and damping injection, can be used for re-orienting a symmetric rigid body with only two control torques. This is the classical problem of controlling underactuated GEO satellites. The possibility of designing simple continuous state-feedback controllers for stabilizing any arbitrary attitude of 2D and 3D rigid bodies, appears a prominent result in mechanics and aerospace literature. In this dissertation it has been possible thanks to the idea of viewing all the related models as special cases of the 3D pendulum. Its global attitude representation, has avoided the presence of singularities and the consequent artificial need to develop swing-up controllers, local controllers, and strategies for switching between them.
Constrained MEO and LEO Spacecraft We have faced the problem of controlling the attitude of MEO and LEO spacecraft, in the second part of this dissertation. Since they represent non-geo-synchronous 3D rigid bodies, we have proposed a mathematical model describing both attitude and position, with respect to an orbital reference frame. This way to proceed is known as the leader following approach. MEO and LEO satellites are usually small, if compared to GEO satellites. Indeed, they have been designed for performing tasks, previously demanded to large space vehicles, reducing costs for launch and maneuvers and increasing reliability. As a consequence, they cannot be equipped with powerful thrusters. Thus, in Chapter 6, we have introduced input saturation constraints. Moreover, we have described the meaning and the role of state accuracy constraints, usually to be included in keeping , reconfiguration and tracking maneuvers. Assuming small angles, the orbital model describing LEO attitude dynamics, is linear. Therefore, any control law designed for linear systems would be appropriate for stabilizing any desired attitude in the absence of constraints. Hence, the presence of constraints is actually the main issue in LEO and MEO space systems. As a consequence, in Chapter 7 we have proposed the use of an MPC technique, referred to as command governor . It consists in designing a primal control law and an external unit capable of taking care of constraint fulfillment by modifying, whenever necessary, the reference signal. It guarantees low numerical burdens, that are fundamental in real time missions. Besides classical CG scheme, we have proposed a parameterization of reference signal, in order to handle state accuracy constraints. Finally, we have extended the promising obtained results, to the case when MEO and LEO satellites are required to span large angles. In this case, their attitude orbital model is not linear. In order to restore results of Chapter 7, in Chapter 8 we have proposed a hybrid control technique. It consists of a bank of CG controllers, each one designed with respect to a suitable attitude configuration. At any time instant, a supervisor selects the most appropriate CG controller from the bank, on the basis of current state and prescribed constraints. We have shown that this control law allows at retaining all the results obtained in the presence of small angles, with the only additional price of evaluating the CONCLUSION
131
minimum of a quadratic function, at any time instant.
Research Contribution and Future Developments This dissertation has presented several results on controlling space vehicles. Both geosynchronous and non-geosynchronous satellites have been taken into account, dividing the thesis into two parts. The main contribution of the first part, is the idea of describing GEO satellites as a special case of a 3D pendulum. Although many works in literature use an attitude model similar to the one proposed in Chapter 1, the idea is new. As a consequence, it produces new mechanical systems, as the heavy top rotating below the pivot. Our control technique for attitude stabilization, based on the idea of pushing the rigid body away from unstable equilibria, by shaping the p otential, is also new. It guarantees the presence of a continuous state-feedback, almost globally stabilizing any desired attitude. In the special case of the 2D pendulum, it solves the swingup problem, using one single control law. On the contrary, previous literature proposes controllers that drive the pendulum in proximity of the upward configuration, local controllers around the upward configuration, and techniques for switching between them. Furthermore, the proposed control technique, avoids the unwinding phenomenon. Thus, it represents an optimal strategy, in the sense that it always pushes the rigid body in the right direction. The TACT, available at the Attitude Dynamics and Control Laboratory, Department of Aerospace Engineering, University of Michigan, is a natural laboratory process for testing all the control laws proposed in the first part of this dissertation. Many experiments have already been performed in the unbalanced case, confirming theoretical results. On the other hand, work still has to be done, for making experiments in the balanced case. The TACT has to be set-up for making gravity ineffective. This represents a crucial step in the present field of research, before applying the proposed controllers to real space systems. Studying strategies for selecting optimal Ψ and Φ functions, making controllers proposed in Chapter 4 and in Chapter 5 minimizing some suitable cost function, is also an interesting subject of future research. Moreover, efforts can be directed to the modification of present results to different actuation tech-
nologies, as rotation wheels or proof masses. As widely discussed above, the second part of this dissertation treats aspects of spacecraft motion, more technological than the ones analyzed in the first part. Beyond the attitude orbital model presented in Chapter 6, that appears new, the attention has been focused on constraints, that are naturally present in all the real missions. The main contribution of part II, lies on the hybrid command governor (HCG) approach illustrated in Chapter 8. Although it is based on an MPC strategy, as other control techniques proposed in previous literature, it represents a novelty in non-geostationary space systems. Indeed, it allows at facing large angle maneuvers using a combination of simple linear attitude models. Moreover, it can be directly extended to formation flying systems, just enclosing attitude and position of all the members of the formation, in the system state s(k). On the other hand, some aspects need future development. For instance, effectiveness of the switching technique, proposed in Chapter 8, has been proven only by simulation results. Proof of the properties of such a switching logic represents a first step of future research. Moreover, the study of different switching criterions, possibly associated with common Lyapunov functions, appears an interesting direction of development. The research on attitude spacecraft control is a fervent field of research. Many aspects, both theoretical and technological are under development. It is testified by the several recent works referred by this dissertation. In this thesis we have faced some of these aspects. From a theoretical point of view, we have proposed a new concept related to mechanical systems, defined 3D pendulum. From a technological point of view, we have proposed a hybrid control strategy for low altitude spacecraft. Along with the development of nano and pico-satellite, it represents a starting point for designing future formation flying systems, both orbiting around Earth and traveling in deep space.
CONCLUSION
133
Bibliography [ABC+ 03] A. Albertoni, A. Ballucchi, A. Casavola, C. Gambelli, E. Mosca, and A.L. Sangiovanni Vincentelli. Hybrid command governor approach for idle speed control in gasoline direct injection engines. In Proc. of American Control Conference , pages 773–778, 2003. [AF00]
K.J. Astrom and K. Furuta. Swinging up a pendulum by energy control. Automatica , 36(2):287–295, 2000.
[Ang01]
D. Angeli. Almost global stabilization of the inverted pendulum via continuous state feedback. Automatica , 37(7):1103–1108, 2001.
[BB00]
S.P. Bhat and D.S. Bernstein. A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon. Systems & Control Letters , 39:63–70, 2000.
[BCM97]
A. Bemporad, A. Casavola, and E. Mosca. Nonlinear control of constrained linear systems via predictive reference menagement. IEEE Transactions on Automatic Control , 42:340–349, 1997.
[BLM00]
A.M. Bloch, N.E. Leonard, and J.E. Marsden.
Controlled la-
grangians and the stabilization of mechanical systems i: The first matching theorem.
IEEE Transactions on Automatic Control ,
45(12):1–48, 2000. [BLM01]
A.M. Bloch, N.E. Leonard, and J.E. Marsden.
Controlled la-
grangians and the stabilization of mechanical systems ii: Potential
shaping. IEEE Transactions on Automatic Control , 46(10):1556– 1571, 2001. [BMB01]
D.S. Bernstein, N.H. McClamroch, and A. Bloch. Development of air spindle and triaxial air bearing testbeds for spacecraft dynamics and control experiments. In Proc. of American Control Conference , pages 3967–3972, 2001.
[BMS83]
R.W. Brockett, R.S. Millman, and H.J. Sussmann. Asymptotic stability and feedback stabilization. In Differential Geometric Control Theory , pages 181–191, 1983.
[BMW71] R.R. Bate, D.D. Mueller, and J.E. White. Fundamentals of Astrodynamics . Dover Publications, New York, USA, 1971. [BS96]
A. Bradshaw and J. Shao. Swing-up control of inverted pendulum systems. Robotica , 14:397–405, 1996.
[Bur95]
I.V. Burkov. Asymptotic stabilization of the position of a rigid body with fixed point without velocity measurements. System and Control Letters , 25:205–209, 1995.
[CBA+ 06] N.A. Chaturvedi, D.S. Bernstein, J. Ahmed, F. Bacconi, and N.H. McClamroch. Globally convergent adaptive tracking of angular velocity and inertia identification for a 3-dof rigid body. IEEE Transactions on Control Systems Technology , 14(5):841–853, 2006. [CLT00]
M. Corless, J.M. Longuski, and P. Tsiotras. A novel approach to the attitude control of axi-symmetric spacecraft. Automatica , 31(8):1099–1112, 2000.
[CM02]
S. Cho and N.H. McClamroch. Feedback control of triaxial attitude control testbed actuated by two proof mass devices. In Proceedings of 41st IEEE Conference on Decision and Control , pages 498–503, 2002.
[CMA00]
A. Casavola, E. Mosca, and D. Angeli. Robust command governors for constrained linear systems. IEEE Transactions on Automatic Control , 45:2071–2077, 2000. BIBLIOGRAPHY
135
[CSM03]
S. Cho, J. Shen, and N.H. McClamroch. Mathematical models for the triaxial attitude control testbed. Mathematical and Computer Modeling of Dynamical Systems , 9(2):165–192, 2003.
[CSMB01] S. Cho, J. Shen, N.H. McClamroch, and D.S. Bernstein. Equations of motion of the triaxial attitude control testbed. In Proc. of of 40th IEEE Conference on Decision and Control , pages 3429–3434, 2001. [DT00]
V. Doumtchenko and P. Tsiotras. Control of spacecraft subject to actuator failures: State-of-the-art and open problems. Journal of the Astronautical Sciences , 48(2,3):337–358, 2000.
[ENS+ 03]
J. Esper, S. Neeck, J.A. Slavin, J. Leitner, W. Wiskombe, and F.H. Bauer. Nano/micro satellite constellations for earth and space science. Acta Astronautica , 52:785–791, 2003.
[FF94]
O.E. Fiellstad and T.I. Fossen. Comments on “the attitude control problem”. IEEE Transactions on Automatic Control , 39(3):699– 700, 1994.
[FPW98]
G. Franklin, G. Powell, and M. Workman. Digital Control of Dynamic Systems (3rd edn). Addison-Wesley, Reading, MA, 1998.
[Fur03]
K. Furuta. Control of pendulum: From super mechano-system to human adaptive mechatronics. In Proc. of 42nd IEEE Conference on Decision and Control , pages 1498–1507, 2003.
[GD99]
F. Ghorbel and J.B. Dabney. Spherical pendulum system to teach key concepts in kinematics, dynamics, control, and simulation. IEEE Transactions on Education , 42(4):359, 1999.
[GE95]
J.M Godhavn and O. Egeland. Attitude control of underactuated satellites. In Proc. of 34nd IEEE Conference on Decision and Control , 1995.
[GH83]
J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields . Springer-Verlag, Berlin, 1983.
[GK92]
T.J. Graettinger and B.H. Krogh. On the computation of reference signal constraints for guaranteed tracking performance. Automatica , 28:1125–1141, 1992.
[GK99]
E.G. Gilbert and I.V. Kolmanovsky. Set-point control of nonlinear systems with state and control constraints: A lyapunov-function, reference governor approach. In Proceedings of the 37th IEEE Con ference on Decision and Control , pages 2507–2512, 1999.
[GKT95]
E.G. Gilbert, I. Kolmanovsky, and K. Tin Tan. Discrete-time reference governors and the nonlinear control of systems with state and control constraints. International Journal of Robust and Nonlinear Control , 1995.
[GPS80]
H. Goldstein, C. Poole, and J. Safko. Classical Mechanics . Addison Wesley, 1980.
[Hal02]
C. Hall. Spacecraft Attitude Dynamics and Control . Lecture Notes, Chapter 3, 2002.
[Hat01]
A. Hatcher. Algebraic Topology . Cambridge University Press, Cambridge, UK, 2001.
[HRW04]
D. Halliday, R. Resnick, and J. Walker. Foundametals of Physics . John Wiley & Sons, New York, USA, 2004.
[Hug86]
P.C. Huges. Spacecraft Attitude Dynamics . John Wiley & Sons, New York, USA, 1986.
[Kap76]
M.H. Kaplan. Modern Spacecraft Dynamics and Control . John Wiley & Sons, New York, USA, 1976.
[Kha01]
H.K. Khalil. Nonlinear Systems . Prantice Hall, Third Edition, New Jersey, USA, 2001.
[KK73]
D. Kleppner and R. Kolenkov.
An Introduction to Mechanics .
McGraw-Hill Science, 1973. [MNF76]
S. Mori, H. Nisihara, and K. Furuta. Control of unstable mechanical systems: Control of pendulum. International Journal of Control , 23:673–692, 1976. BIBLIOGRAPHY
137
[Mos95]
E. Mosca. Optimal, Predictive, and Adaptive Control . Prentice Hall, Englewood Cliffs, NJ, 1995.
[MR99]
J.E. Marsden and T.S. Ratiu.
Introduction to Mechanics and
Symmetry: A Basic Exposition of Classical Mechanical Systems . Springer Verlag, California, USA, 1999. [MRS00]
J.E. Marsden, T.S. Ratiu, and J. Scheurle. Reduction theory and the lagrange-routh equations. Journal of Mathematical Physics , 41(6):3379–3429, 2000.
[OSR98]
R. Ortega and H. Sira-Ramirez. Passivity-Based Control of EulerLagrange Systems: Mechanical, Electrical and Electromechanical Applications . Springer Verlag, California, USA, 1998.
[SBW06]
G.L. Slater, S.M. Byram, and T.W. Williams. Collision avoidance for satellites in formation flight. Journal of Guidance, Control, and Dynamics , 29(5):1140–1146, 2006.
[SCB+ 05]
M.A. Santillo, N.A. Chaturvedi, F. Bacconi, N.H. McClamroch, and D.S. Bernstein. Experiments on stabilization of the hanging equilibrium of a 3d asymmetric rigid pendulum. In Proceedings of IEEE Conference on Control Applications , 2005.
[SEL99]
A.S. Shiriaev, O. Egeland, and H. Ludvigsen. Global stabilization of unstable equilibrium point of pendulum. In Proc. of American Control Conference , pages 4034–4038, 1999.
[SEL04]
A.S. Shiriaev, O. Egeland, and H. Ludvigsen. Swinging up the spherical pendulum via stabilization of its first integrals. Automatica , 40(1):73–85, 2004.
[Sid00]
M.J. Sidi. Spacecraft Dynamics and Control: a Practical Engineering Approach . Cambridge University Press, Cambridge, UK, 2000.
[SLRM92] J.C. Simo, D. Lewis, T. Ratiu, and J.E. Marsden. The heavy top: a geometric treatment. Nonlinearity , 5:2253–2270, 1992.
