Page 299 - Autonomous Mobile Robots

286 Autonomous Mobile Robots

From Figure 7.1, we have

 
− sin θ − sin θ
B(q) = 1/P  cos θ cos θ 


L −L
where P is the radius of the wheels and 2L is the length of the axis of the
two fixed differential drive wheels as shown in Figure 7.1. The matrices K N =
diag[K N1 , K N2 ], L = diag[L 1 , L 2 ], R = diag[R 1 , R 2 ], K a = diag[K a1 , K a2 ], and
ω is given by (7.10) and (7.11).
Following the description in Section 7.2, the dynamics of the wheeled
mobile robot can be written as

˙ x = v 1 cos θ
˙ y = v 1 sin θ

˙ θ = v 2
T
M(q)S(q)˙v + C 1 (q)v + G = BK N I + J λ (7.62)
dI
L + RI + K a µv = ν (7.63)
dt

where

 
−m 0 cos θ ˙ θ 0
T
C 1 = −m 0 sin θ ˙ θ 0 , v =[v 1 , v 2 ]


0 0
with v 1 , v 2 the linear and angular velocities of the robot.
Considering the coordinates transformation X = T 1 (q) and state feedback
u = T −1 (q)v given by [38]
2
     
x 1 0 0 1 x
x 2 =  cos θ sin θ 0 y
 

  
x 3 − sin θ cos θ 0 θ

u 1 = v 2
u 2 = v 1 − v 2 x 2

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c007” — 2006/3/31 — 16:43 — page 286 — #20