Translational Motion Analysis of a Hexapod Walking Robot - English

PON29_IES2013IEEE INTERNATIONAL ENGINEERING SUMMIT29-31 DE OCTUBRECOATZACOALCOS, VERACRUZ, MEXICO

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Translational Motion Analysis of a Hexapod Walking Robot G. Evangelista, D. Lázaro Abstract This paper focuses on translational motion analysis of a hexapod walking robot, for this, use is made of conventions published in 2010 by “Xilun Ding, Zhiying Wang, Alberto Rovetta and J.M. Zhu”, which are support for the parameters of this study. The analysis is divided in three parts: step phases, locomotion of movement and walking gaits; in the first part, was analyzed the movement phases: Transfer and support, and how each movement is formed. The second part defines features such as: periods, duty factor, lengths, offsets, support polygon, work area, gaits and crab angle; in the walking gaits, are studied four types suggested: Tripod, quadruped, 4+2 quadruped and Pentapod. Finally, we propose an analysis formulation and an application model, concluding that it can be applied to any type of hexapod robot and can be the starting point for future research in the field of path planning for walking robots. Keywords Locomotion analysis, hexapod robot, step trajectory, tripod gait, quadruped gait, quadruped 4+2 gait, pentapod gait.

I.

the locomotion of the robot can be classified into dynamic, such as running or jumping, and static, such as walking [1]. Is because that this article focuses on the analysis of statically stable translational motion of a hexapod robot in order to define and parameterize four different gaits: Tripod, quadruped, 4+2 quadruped and pentapod.

II. STEP PHASES The step is the basic unit of displacement which compose a different types of walking, the step is itself composed of two phases of motion:  

INTRODUCTION

he term “locomotion” comes from the physical phenomenon known as motion (change of position or orientation in space). The locomotion therefore refers to a movement realized by a body for moving from one place to another or rotation thereof about a point in a plane; a body can be a person, animal or machine, and it means that locomotion varies according to the shape, structure, propulsion system or other factor that depends on the reference subject. The locomotion of the living creatures is current topic of interest, since it is intended based on related physical of their motion, understand and improve the mechanisms of propulsion and / or to establish parameters to determine what would be the appropriate positions to perform some type of movement. From the branches associated to this subject, the robotics according to architecture classifies a bio-inspired robots at: articulated, mobiles, androids, zoomorphic and hybrid. In the classification of the articulated, the hexapods (mechanical vehicles that walk on six legs), have brought considerable attention in recent decades, this is because they have significant advantages over wheeled robots to walk on uneven ground (with no continuous contact with the surface). There are several advantages of a hexapod model: high degree of flexibility in the manner in which it can move, robustness in case of failure of the extremities and the possibility of using one, two or three legs to work as hands and perform complex operations with them. The most studied problem for multi-legged robots, is how to determine the best sequence for the elevation and location of the legs in the walking cycle. From the point of view of stability,

T

G. Evangelista Adrianzén, Universidad Privada Antenor Orrego (UPAO), Trujillo, La Libertad, Perú, [email protected] D. Lázaro Cerna, Universidad Privada Antenor Orrego, Trujillo, La Libertad, Perú, [email protected]

Support Phase: The legs performs a supporting / stance cycle in contact with the displacement surface, generally formulated by a linear or trapezoidal equation [2]. Transfer Phase: The legs performs a transferring / swing cycle over the surface of displacement, generally formulated by a parabolic equation or 𝑛-degree polynomial.

III.

MOVEMENT LOCOMOTION

In order to understand the way in which the robot moves, it requires a synthesis of characteristics about step and that can produce movement:    

   

 

Support / Stance Period: Period in which it the performs the support stance, denoted as: 𝑇𝑠𝑖 Transfer/ swing period: Period in which it performs the transfer stance, denoted as: 𝑇𝑡𝑖 Step Period: Period in which it conducts a full step, denoted as 𝑇𝑖 , where: 𝑇𝑖 = 𝑇𝑠𝑖 + 𝑇𝑡𝑖 (1) Duty Factor (𝛽): Is the ratio between the period of support and step period, where: 𝑇 𝛽 = 𝑠𝑖 (2) 𝑇𝑖

Stroke Length: Is the distance traveled in the support phase. Stride Length: Is the distance that moves the center of gravity (COG) during a complete cycle of locomotion. Origin Offset: Displacement of final effector respect its origin. Support Polygon: Is formed during the support phase, this polygon is constructed from the union of points of the end effectors of legs in contact with the surface, this characteristic varies respect at gait and offset of origin. Work Area: Space formed by all the coordinates in which can be locate the end-effector. Periodic Gait: Is periodic if similar states of the same leg during successive strokes occur at the same interval for all legs, that interval being the cycle time.


  

Symmetric Gait: Is symmetric if the motion of legs of any right-left pair is exactly half a cycle out of phase. Regular Gait: Is said to be regular if all the legs have the same duty factor. Crab angle (α): It is defined as the angle from the longitudinal axis to the direction motion, which has the positive measure in the anti-clockwise.

IV.

Features: - 𝛽 = 2⁄3 in each of the legs - The stroke length is ½ of the stride length - Present a quadrilateral such a support polygon - The gait is periodic, non-symmetric and regular. C. 4+2 Quadruped gait: It is characterized by have 4 or 5 legs en every support stage [6]:

WALKING GAITS

A walking is a sequence of actions that have a moving unit to a “step”, there are many types of gaits which can adopt a hexapod robot, but these, differ in the type of approach; when is adopted from a living creature, is called biological approach, which is based on the Central Pattern Generator (CPG) [3] and when is artificially created (Physical and Kinematic Equations) is called the classical approach. The description of gaits will be from the classical approach and for this is required the numbering the legs relative to the body and displacement plane, as shown in figure 1:

168

TABLE III 4+2 QUADRUPED GAIT PHASES STAGE

SUPPORT

TRANSFER

1 2 3 4

𝑙2 -𝑙3 -𝑙4 -𝑙5 𝑙1 -𝑙2 -𝑙3 -𝑙5 -𝑙6 𝑙1 -𝑙3 -𝑙4 -𝑙6 𝑙1 -𝑙2 -𝑙4 -𝑙5 -𝑙6

𝑙1 -𝑙6 𝑙4 𝑙2 -𝑙5 𝑙3

Features: - 𝛽 = 3⁄4 in each of the legs - The stroke length is ½ of the stride length - Present a quadrilateral such a support polygon in the 1 and 3 stage - Present a pentagon such a support polygon in the 2 and 4 stage - The gait is periodic, non-symmetric and regular. D. Pentapod Gait: It is characterized by have 5 legs en every support stage [7]: TABLE IV

PENTAPOD GAIT PHASES Figure 1. Numbering of legs for a hexapod robot with symmetrical distribution. STAGE

SUPPORT

TRANSFER

1 2 3 4 5 6

𝑙1 -𝑙2 -𝑙3 -𝑙4 -𝑙6 𝑙1 -𝑙2 -𝑙4 -𝑙5 -𝑙6 𝑙2 -𝑙3 -𝑙4 -𝑙5 -𝑙6 𝑙1 -𝑙2 -𝑙3 -𝑙4 -𝑙5 𝑙1 -𝑙2 -𝑙3 -𝑙5 -𝑙6 𝑙1 -𝑙3 -𝑙4 -𝑙5 -𝑙6

𝑙5 𝑙3 𝑙1 𝑙6 𝑙4 𝑙2

A. Tripod Gait: It is characterized by have 3 legs in every support stage [4]: TABLE I TRIPOD GAIT PHASES STAGE 1 2

SUPPORT 𝑙1 -𝑙4 -𝑙5 𝑙2 -𝑙3 -𝑙6

TRANSFER 𝑙2 -𝑙3 -𝑙6 𝑙1 -𝑙4 -𝑙5

Features: - 𝛽 = 1⁄2 in each of the legs - The stroke length is ½ of the stride length - Present a triangle such a support polygon - The gait is periodic, symmetric and regular. B. Quadruped gait: It is characterized by have 4 legs in every support stage [5]: TABLE II QUADRUPED GAIT PHASES STAGE 1 2 3

SUPPORT 𝑙2 -𝑙3 -𝑙5 -𝑙6 𝑙1 -𝑙2 -𝑙4 -𝑙5 𝑙1 -𝑙3 -𝑙4 -𝑙6

TRANSFER 𝑙1 -𝑙4 𝑙3 -𝑙6 𝑙2 -𝑙5

Features: - 𝛽 = 5⁄6 in each of the legs - The stroke length is ½ of the stride length - Present a pentagon such a support polygon - The gait is periodic, non-symmetric and regular.

V.

ANALYSIS FORMULATION

Due to the generality of parameters synthesis is possible that the implementation may be adaptable to many types of gaits, this will depend on the number of degrees of freedom and their distribution in the body. A. End-Effector Trajectory: To the displacement in the four types proposed, is presented in figure 2 a trajectory for the end-effector:


𝑍 = −[

8(𝐻 − 𝐻0 𝑆

)2

169

𝑆 − 𝑧0 − ] 2

(4)

On the other hand, the horizontal axis shows a linear equation: 𝐻 = 𝐴 + 𝐵𝑖 Where: 𝐻 : Coordinate on the horizontal axis 𝐴 : Starting point on the horizontal axis 𝐵 :𝑖

Figure 2. Transfer and support phases on three dimensional plane.

Where: 𝑥0 𝑦0 𝑧0 𝐻0 ℎ 𝑆 𝑖 𝛼

: Origin offset at 𝑋 axis : Origin offset at 𝑌 axis : Origin offset at 𝑍 axis : Midpoint in each phase : Maximum height in the transfer stage : Stride length [cm] : Iteration interval : Crab angle [0, 180]°.

The phases of the end-effector trajectory (support and transfer) have a distance of 𝑆/2, therefore the operating range for both phases is 𝐻 ∈ [𝐻0 − 𝑆⁄4 , 𝐻0 + 𝑆⁄4], where 𝑍 is dependent de 𝐻. Transfer Phase The vertical axis shows a parabolic equation, such as: (𝐻 − 𝑎)2 = 4𝑝(𝑉 − 𝑏) (3) Where: 𝐻 : Coordinate on the horizontal axis 𝑉 : Coordinate on the vertical axis (𝑎, 𝑏) : Vertex 𝑝 : Focal distance. The parameters are defined considering that the horizontal axis is formed by the XY plane and that the orientation depends exclusively of the crab angle. Vertical Axis :𝑍 Horizontal Axis : 𝐻 Vertex : (𝐻0 , ℎ + 𝑧0 ) Points on Parabola: (𝐻0 − 𝑆⁄4 , 𝑧0 ) & (𝐻0 + 𝑆⁄4 , 𝑧0 ) By replacing the vertex and one of the points on parabola in equation (3) it obtain the focus, and solving for Z in parabolic equation, is obtained: 16ℎ(𝐻 − 𝐻0 )2 𝑍 = −[ − 𝑧0 − ℎ] 𝑆2 If in transfer phase, the maximum height is equal to the distance walked, will be fulfilled that: “Maximum height = maximum distance”, and then replacing ℎ = 𝑆/2 in previous equation, obtain:

As 𝐻 is in the 𝑋𝑌 plane, the axes which are compounds are: 𝑋 = 𝑥0 + 𝐻 sin(±𝛼) 𝑌 = 𝑦0 + 𝐻 cos(±𝛼) It should to note that the start point and iteration interval are obtained by solving a system of two equations. Support Phase The horizontal axis shows a linear equation, such as: 𝐻 = 𝐴 + 𝐵𝑖 Where: 𝐻 : Coordinate on the horizontal axis 𝐴 : Starting point on the horizontal axis 𝐵 :𝑖 The equations for X and Y axes are: 𝑋 = 𝑥0 + 𝐻 sin(±𝛼) 𝑌 = 𝑦0 + 𝐻 cos(±𝛼) And the vertical axis is now a constant value: 𝑧0 Finally, the formulation of analysis can be summarized in the following table: TABLE V MOTION EQUATIONS FOR STAGES

TRANSFER 8(𝐻 − 𝐻0 )2 𝑆 𝑍 = −[ − 𝑧0 − ] 𝑆 2 𝐻 = 𝐴 ± 𝐵𝑖 𝑋 = 𝑥0 + 𝐻 sin(±𝛼) 𝑌 = 𝑦0 + 𝐻 cos(±𝛼)

VI.

SUPPORT 𝑍 = 𝑧0 𝐻 = 𝐴 ± 𝐵𝑖 𝑋 = 𝑥0 + 𝐻 sin(±𝛼) 𝑌 = 𝑦0 + 𝐻 cos(±𝛼)

APPLICATION MODEL

The analysis formulation must be validated by applying this to one type of gaits (this is called application model); now, will be performed the formulation of quadruped gait based on the analysis above, in order to obtain the position equations of each end-effector. A. Features of end-effector trajectory These features are general and can therefore be applied to any of the four gaits:  

𝑥0 = 12.38𝑐𝑚 (lateral length of the leg) 𝑦0 = 3𝑐𝑚  𝑙1,2


 

𝑦0 = −3𝑐𝑚  𝑙5,6 𝑧0 = −10.51𝑐𝑚 (robot height).

The origin offset parameters refer to the spatial position of the end-effector (in walking position) relative to the origin of the first system on the leg; considering that in the first system of the leg, the x-axis is pointing to the next system and y-axis is pointing to x-axis of universal system (this depends on the allocation of reference systems and degrees of freedom), see figure 3.

B. Quadruped Gait Formulation Using table V and considering the characteristics of the end-effector trajectory, is possible to obtain the position equations per stages: 

𝑆 = 7𝑐𝑚 Working area: The figure 4 and figure 5 show a surface generated for an 𝛼 [0,90]°.

First Stage: 0 ≤ 𝑖 ≤ 𝑆/3 𝑙1  𝐻 = [0: 𝑆/2] 𝑋1 = 12.38 + 1.5𝑖 sin(−𝛼) 𝑌1 = 𝑦0 + 1.5𝑖 cos(−𝛼) 𝑍1 = −[8(1.5𝑖 − 𝑆/4)2 ⁄𝑆 + 10.51 − 𝑆/2] 𝑙4  𝐻 = [0: 𝑆/2] 𝑋4 = 12.38 + 1.5𝑖 sin 𝛼 𝑌4 = 𝑦0 + 1.5𝑖 cos 𝛼 𝑍4 = −[8(1.5𝑖 − 𝑆/4)2 ⁄𝑆 + 10.51 − 𝑆/2] 𝑙3,5  [0: −𝑆/4] 𝑋3,5 = 12.38 − 0.75𝑖 sin(−𝛼) 𝑌3,5 = 𝑦0 − 0.75𝑖 cos(−𝛼) 𝑍3,5 = −10.51 𝑙2,6  [0: −𝑆/4] 𝑋2,6 = 12.38 − 0.75𝑖 sin(𝛼) 𝑌2,6 = 𝑦0 − 0.75𝑖 cos(𝛼) 𝑍2,6 = −10.51

Figure 3. Assignment of reference systems and spatial position of the endeffector in leg 2.

 



Second Stage: 𝑆/3 < 𝑖 ≤ 2𝑆/3 𝑙3  𝐻 = [−𝑆/4: 𝑆/4] 𝑋3 = 12.38 + (1.5𝑖 − 0.75𝑆) sin(−𝛼) 𝑌3 = 𝑦0 + (1.5𝑖 − 0.75𝑆) cos(−𝛼) 𝑍3 = −[8(1.5𝑖 − 0.75𝑆)2 ⁄𝑆 + 10.51 − 𝑆/2] 𝑙6  𝐻 = [−𝑆/4: 𝑆/4] 𝑋6 = 12.38 + (1.5𝑖 − 0.75𝑆) sin 𝛼 𝑌6 = 𝑦0 + (1.5𝑖 − 0.75𝑆) cos 𝛼 𝑍6 = −[8(1.5𝑖 − 0.75𝑆)2 ⁄𝑆 + 10.51 − 𝑆/2] 𝑙1  𝐻 = [𝑆/2: 𝑆/4] 𝑋1 = 12.38 + 0.75(𝑆 − 𝑖) sin(−𝛼) 𝑌1 = 𝑦0 + 0.75(𝑆 − 𝑖) cos(−𝛼) 𝑍1 = −10.51 𝑙4  𝐻 = [𝑆/2: 𝑆/4] 𝑋4 = 12.38 + 0.75(𝑆 − 𝑖) sin(𝛼) 𝑌4 = 𝑦0 + 0.75(𝑆 − 𝑖) cos(𝛼) 𝑍4 = −10.51 𝑙2  𝐻 = [−𝑆/4: −𝑆/2] 𝑋2 = 12.38 − 0.75𝑖 sin(−𝛼) 𝑌2 = 𝑦0 − 0.75𝑖 cos(−𝛼) 𝑍2 = −10.51 𝑙5  𝐻 = [−𝑆/4: −𝑆/2] 𝑋5 = 12.38 − 0.75𝑖 sin(𝛼) 𝑌5 = 𝑦0 − 0.75𝑖 cos(𝛼) 𝑍5 = −10.51

Figure 4. Working area for end-effector of the leg 3.



Third Stage: 2𝑆/3 < 𝑖 ≤ 𝑆 𝑙2  𝐻 = [−𝑆/2: 0] 𝑋2 = 12.38 + 1.5(𝑖 − 𝑆) sin 𝛼 𝑌2 = 𝑦0 + 1.5(𝑖 − 𝑆) cos 𝛼 𝑍2 = −[8(1.5(𝑖 − 𝑆) + 𝑆/4)2 ⁄𝑆 + 10.51 − 𝑆/2] 𝑙5  𝐻 = [−𝑆/2: 0] 𝑋5 = 12.38 + 1.5(𝑖 − 𝑆) sin(−𝛼) 𝑌5 = 𝑦0 + 1.5(𝑖 − 𝑆)𝑖 cos(−𝛼) 𝑍5 = −[8(1.5(𝑖 − 𝑆) + 𝑆/4)2 ⁄𝑆 + 10.51 − 𝑆/2] 𝑙1,3  𝐻 = [𝑆/4: 0] 𝑋1,3 = 12.38 + 0.75(𝑆 − 𝑖) sin(−𝛼)

Figure 5. Working area for end-effector of the leg 4.

170


171

𝑌1,3 = 𝑦0 + 0.75(𝑆 − 𝑖) cos(−𝛼) 𝑍1,3 = −10.51 𝑙4,6  𝐻 = [𝑆/4: 0] 𝑋4,6 = 12.38 + 0.75(𝑆 − 𝑖) sin(𝛼) 𝑌4,6 = 𝑦0 + 0.75(𝑆 − 𝑖) cos(𝛼) 𝑍4,6 = −10.51

VII.

RESULTS

In this section, the application model is extended to the four gaits, the purpose is get the position of the end-effector for each leg in the step trajectory. A. Tripod Gait It was generated using 300 iterations a gait period of 3 seconds, this has a stroke length of 2.5cm and an iteration delay of 10ms. The behavior is showing in the figure 6 and is due to the following parameters:  𝑆 = 5𝑐𝑚  𝑥0 = 12.38𝑐𝑚  𝑦0 = 5𝑐𝑚  𝑙1,2  𝑦0 = −5𝑐𝑚  𝑙5,6  𝑧0 = −10.51𝑐𝑚  𝛼 = 0°

Figure 7. Quadruped gait locomotion.

C. 4+2 Quadruped Gait It was generated using 110 iterations a gait period of 1.1 seconds, this has a stroke length of 3cm and an iteration delay of 10ms. The behavior is showing in the figure 8 and is due to the following parameters:  𝑆 = 6𝑐𝑚  𝑥0 = 12.38𝑐𝑚  𝑦0 = 4𝑐𝑚  𝑙1,2  𝑦0 = −4𝑐𝑚  𝑙5,6  𝑧0 = −10.51𝑐𝑚  𝛼 = −90°

Figure 6. Tripod gait locomotion.

B. Quadruped Gait It was generated using 210 iterations a gait period of 2.1 seconds, this has a stroke length of 3.5cm and an iteration delay of 10ms. The behavior is showing in the figure 7 and is due to the following parameters:  𝑆 = 7𝑐𝑚  𝑥0 = 12.38𝑐𝑚  𝑦0 = 3𝑐𝑚  𝑙1,2  𝑦0 = −3𝑐𝑚  𝑙5,6  𝑧0 = −10.51𝑐𝑚  𝛼 = 90°

Figure 8. Locomotion of the 4+2 quadruped gait.

D. Pentapod Gait It was generated using 130 iterations a gait period of 1.3 seconds, this has a stroke length of 2.5cm and an iteration delay of 10ms. The behavior is showing in the figure 9 and is due to the following parameters:  𝑆 = 5𝑐𝑚  𝑥0 = 12.38𝑐𝑚  𝑦0 = 4𝑐𝑚  𝑙1,2  𝑦0 = −4𝑐𝑚  𝑙5,6  𝑧0 = −10.51𝑐𝑚  𝛼 = 35°.


172

APPENDIX Software developed for the analysis of locomotion:

https://drive.google.com/folderview?id=0B_JXOKvBtg_ HVmF2QWp5YW93Ums&usp=sharing REFERENCES [1]

[2]

[3] Figure 9. Pentapod gait locomotion.

VIII.

DISCUSSIONS

The results obtained of the iterations during a complete cycle of locomotion, show that the legs have a behavior and characteristics identical to those described in the walking gaits, thus proving that yes could parameterize and formulate the motion equations in the four types of gaits and this through a single analysis. The crab angle influence can be observed in the behavior of x and y axis at different results, thus proving, that determines the direction of motion regardless of gait type; but if want to perform a backward movement, this can be generated by a down counter for 𝑖.

IX. 

 



CONCLUSIONS

Based on the motion parameters proposed, the description of the four gaits was performed without resorting a parameters or special characteristics, thus providing a generalized analysis model and independent to the selected type. To improve the static equilibrium of the robot (support polygon), should be apply an origin offset of positive value to the 𝑙1,2 legs and negative to 𝑙5,6 legs. The parameter “crab angle” solves the problem of independent equations in the horizontal plane, the orientation of the trajectory depends only on this parameter, which determines the equations per axis of dependent manner. If the analysis is implemented in a hexapod robot and want to test it in a non-uniform terrain, will require the use of sensors to detect surface; this will not alter the design of the analysis, only will make the 𝑍0 parameter becomes a variable.

ACKNOWLEDGMENT The authors would like to thank to Filiberto Azabache for his help and support in making this work possible. Also to RGEEUPAO for providing all the facilities that enabled this research possible.

[4]

[5] [6]

[7]

Xilun Ding, Zhiying Wang, Alberto Rovetta and J.M. Zhu, “Climbing and Walking Robots”, Beihang University, Politecnico di Milano, China, Italy, 2010, p.291 Shibendu Shekhar Roy, Ajay Kumar Singh and Dilip Kumar Pratihar, “Analysis of Six-legged Walking Robots”, 14th National Conference on Machines and Mechanisms, NIT, Durgapur, India, 2009, p3 Andrés Prieto-Moreno Torres, “Estudio de la locomoción de un robot cuadrúpedo mediante la generación de patrones biológicos”, Escuela Politécnica Superior, p.1 Tsu-Tian Lee, Ching-Ming Liao, and Ting-Kou Chen, “On the Stability Properties of Hexapod Tripod Gait”, IEEE Journal of Robotics and Automation, Vol. 4. No. 4. August, 1988, p.427 Nestor Martínez De Oraa, Pedro Fernández Gómez, “Robot Hexápodo”, Universitat Politècnica de Catalunya, 2003, pp.32-33 Lianqing Yu Yujin Wang, Weijun Tao, “Gait Analysis and Implementation of a Simple Quadruped Robot”, 2nd International Conference on Industrial Mechatronics and Automation, 2010, pp.431432 G. Clark Haynes, Alfred A. Rizzi, “Gaits and Gait Transitions for Legged Robots”, IEEE International Conference on Robotics and Automation Florida, 2006, p.1121

Guillermo Evangelista is an Electronic Engineer graduated from the Universidad Privada Antenor Orrego. He currently teach Electronic Engineering in the UPAO, where he is also a member of the Engineering Research Unit and advisor Engineering Research Group Electronics (RGEE). His research interest is the mathematical modeling for polyarticulated robots and UAV units, as well as the development of solutions based artificial vision for industrial applications and non-invasive monitoring. Denis Lázaro is student of Electronic Engineering at the Universidad Privada Antenor Orrego, and present is VicePresident of the RAS-IEEE / UPAO.

Translational Motion Analysis of a Hexapod Walking Robot - English

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