Page 271 - Autonomous Mobile Robots

Adaptive Neural-Fuzzy Control of Mobile Robots 257

model can be expressed in the matrix form (6.1) with

m 0 mL sin θ
 
0 m −mL cos θ
 
M(q) = 
mL sin θ −mL cos θ I

0 0 mL ˙ θ cos θ cos θ cos θ
   
C(q, ˙q) = 0 0 mL ˙ θ sin θ  , G(q) = 0, B(q) = 1  sin θ 
sin θ



R 1
0 0 0 R 2 −R 2
(6.87)
T
3
where q =[x c y c θ] ∈ R is the generalized coordinate with (x c , y c ) being
the coordinates of the center of mass of the vehicle, and θ being the orientation
T
2
angle of the vehicle with respect to the X-axis, τ =[τ r τ l ] ∈ R is the input
vector with τ r and τ l being the torques provided by the motors mounted on the
right and left respectively, m is the mass of the vehicle, I is its inertial moment
around the vertical axis at the center of mass, L denotes the distance between
the mid-distance of the rear wheels to the center of mass, 2R 1 denotes the radius
of the rear wheels, and 2R 2 is the distance between the two rear wheels. The
T
constraint forces are f = J (q)λ.
The nonholonomic constraints confine the vehicle to move only in the direc-
tion normal to the axis of the driving wheels, that is, the mobile bases satisfying
the conditions of pure rolling and nonslipping

˙ x c sin θ −˙y c cos θ + L ˙ θ = 0 (6.88)

From (6.88), it is known that J(q) and R(q) are in the form

   
sin θ cos θ −L sin θ
T
J (q) = − cos θ , R(q) = sin θ L cos θ  (6.89)




L 0 1
T
Thus, the constraint forces can be written as f = J (q)λ with

λ = m¨x c sin θ − m¨y c cos θ + mL ¨ θ (6.90)

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 257 — #29