Lecture 3B.Supplemental_4
Supplementary Material - Linearisation of Quadrotor Equations of Motion Presented by Lucy Tang, PhD student. In lecture, we claimed that in the linearised-equations of motion for the quadrotor, the second-derivative of position is proportional to u , and the fourth-derivative of position is proportional to u!. In this segment, we"ll derive these relationships e#plicitly. $ecall the quadrotor%s equations-of-motion, which come from the linear and angular momentum balances&
'e can use u to represent the total thrust applied and u ! to represent the moment vector. 'e could then rewrite the equations-of-motion in terms of u and the components of u !&
In the equilibrium hover configuration, the position and yaw angle of the quadrotor can be at some arbitrary value r ( and ( respectively. )owever, the angles and , as are all *ero. 'e want to derive e#pressions for the equations , , and well as r of motion when the quadrotor is near this equilibrium-configuration.
+irst, let"s consider the value of cos cos near the equilibrium-configuration when (. /round (, the function cos cos can be appro#imated using the Taylor e#pansion which is shown here& cos, - cos, -
(
d cos, d
(
'e can consider all the terms after the first two terms in this series to be negligibly small. The value of cos cos at ( is . The derivative of cos cos is -sin . 0ince sin , at ( is (. The appro#imation of cos cos near ( is simply .
'e can confirm this qualitatively by loo1ing at the value of the cosine function, near (. Loo1ing at the plot of the cosine function for angles between -2( to 2( 2(, we see that the cosine values are indeed close to . 'e can repeat this process to appro#imate the function sin near (. /gain, we use the Taylor series. +or sin sin we arrive at the following e#pression& sin, - sin, -
(
d sin, d
(
0ince sin sin, at ( is (, and the derivative of sin is cos cos, we can simplify the e#pression, and since the value of cos cos at ( is , the value of sin sin near ( is appro#imately &
This suggests that around (, we e#pect the sin function to loo1 appro#imately linear. 3#amining the values of sin for angles from -2( -2( to 2( 2(, we see that the sin function indeed loo1s linear. 'e will use these two appro#imations to linearise the equations-of-motion of the quadrotor. /gain, we want to use these appro#imations to help direct simplified versions of the equations-of-motion that apply when the quadrotor is near the equilibrium hover configuration. +irst, consider the linear momentum equation. 'e can e#plicitly write the rotation matri# in terms of the 3uler angles&
'e can then perform the matri#-multiplication to arrive at the following second order differential equations&
/t equilibrium, the pitch-angle and the roll-angle are both appro#imately (. Therefore we can use the equation we derived earlier to appro#imate the sines and cosines of these angles. 0in 0in , sin sin , cos cos sin sin
0ubstituting in these appro#imations reduces the differential equations to the ones shown below&
cos sin u m x sin cos u m y mg u m z 'e can clearly see that the second-derivative of position is proportional to the input, u. 4ow consider the relationship between the angular-velocity components p, q, q, r and the first derivatives of the 3uler angles. /gain, we carry out the matri#-multiplication & to arrive at three equations to relating p, q and r to , and
'e can then substitute in the appro#imations for the sines and cosines of and , giving us the equations shown here&
p q r 4e#t, we can appro#imate all terms that are the product of an angle and an angles derivative as (. 4ear )over, and and all angular derivatives are close to (. The product of two terms near ( will be very, very small, and as a result we can appro#imate these terms to (. 0ubstituting these appro#imations into the equations, tells us that around hover, the angular velocity components are appro#imately the time derivatives of the 3uler angles. P q
r +inally consider the angular-momentum equation&
+irst, we appro#imate the off-diagonal inertia terms as close to (.
I#y Iy# I#* I*# Iy# I*y ( This allows us to simplify the inertia matri#&
Performing the matri#-multiplication gives us the following set of equations& I xx p u! x I yy qr I zz qr I yy q u! y I xx pr I zz pr
u! z I xx pq I yy pq I zz r 'e saw earlier that around hover, p, q and r are appro#imately , and respectively and are therefore also close to (. The pattern of any two angular-velocitycomponents can then be appro#imated as (. This gives us the following set of equations&
5sing again the appro#imation of the angular-velocity components as the 3uler angle derivatives, we arrive at this set of second-order differential equations&
u! x I xx u! y I yy u! z I zz
4ow let"s go bac1 to the linear-momentum equation in the the #-direction&
'e differentiate this equation twice&
and substitute the appro#imations of the angular-momentum equations-of-motion into the e#pression for the fourth derivative of #. 6mitting terms that don%t contain the second-derivative of an 3uler angle, we arrive at the following e#pression for the fourth-derivative of #&
7arrying out this procedure in the y- and *-directions yields similar equations. 'e can see that the fourth-derivative of position is indeed proportional to u !.
