Page 298 - Autonomous Mobile Robots

Adaptive Control of Mobile Robots 285

7.4 SIMULATION
Consider a wheeled mobile robot moving on a horizontal plane, as shown in
Figure 7.1, which has three wheels (two are differential drive wheels, one
T
is a caster wheel), and is characterized by the configuration q =[x, y, θ] .
We assume that the robot does not contain flexible parts, all steering axes are
perpendicular to the ground, the contact between wheels and the ground satisfies
the condition of pure rolling and nonslipping.
The complete nonholonomic dynamic model of the wheeled mobile robot
is given by

J(q)˙q = 0 (7.59)
T
M(q)¨q + C(q, ˙q)˙q + G(q) = B(q)K N I + J (q)λ (7.60)
LI + RI + K a ω = ν (7.61)
˙
The constraint of the nonslipping condition can be written as

˙ x cos θ +˙y sin θ = 0

From the constraint, we have

J(q) =[cos θ, sin θ,0]

which leads to
 
− sin θ 0
S(q) =  cos θ 0 
0 1
Lagrange formulation can be used to derive the dynamic equations of the
wheeled mobile robot. Because the mobile base is constrained to the horizontal
plane, its potential energy remain constant, and accordingly G(q) = 0. The
kinematic energy K is given by [37]

1 T
K = ˙q M(q)˙q
2
where
 
m 0 0 0
M(q) =  0 m 0 0 
0 0 I 0
with m 0 being the mass of the wheeled mobile robot, and I 0 being its inertia
moment around the vertical axis at point Q. As a consequence, we obtain
∂K
˙
C(q, ˙q)˙q = M(q)˙q − = 0
∂q

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c007” — 2006/3/31 — 16:43 — page 285 — #19