Sensor-based Trajectory Deformation: Application to Reactive Navigation of Nonholonomic Robots - Visual Servoing via Advanced Numerical Methods - page 316

Fig. 17.2 Mobile robot
Hilare 2 towing a trailer

A configuration of this robot is represented by q =( x , y , θ , ϕ ) where ( x , y ) is the

position of the center of the robot, is the

orientation of the trailer with respect to the robot. The control vector fields are

θ is the orientation of the robot and ϕ

⎛ ⎛

⎞
⎠

⎞
⎠

cos θ 0

0
1

⎝ ⎝

sin θ

X 1 = X 2 =

0

1
l t

l l t

− sin ϕ − 1 − cos ϕ

where l r (resp. l t ) is the distance between the center of the robot (resp. the trailer)
and the trailer connection. The inputs of the system are u 1 and u 2 the linear and
angular velocities of the robot. To get a basis of

4

R at each configuration q ,we

define two additional vector fields:

⎛ ⎞ ⎛ ⎞

− sin θ − sin( θ + ϕ )

⎝ ⎠ ⎝ ⎠ .

cos θ cos( θ + ϕ )

X 3 = X 4 =

0
0

− l t − l r cos ϕ

− l t

The linearized system is thus defined by the matrices

⎛ ⎞

00 − u 1 sin θ − u 3 cos θ − u 4 cos( θ + ϕ ) − u 4 cos( θ + ϕ )

⎝ ⎠

00 u 1 cos θ − u 3 sin θ − u 4 sin( θ + ϕ ) − u 4 sin( θ + ϕ )

A ( s )=

00 0 u 4 l r s ϕ

− u 1 c ϕ+ u 2 l r s ϕ
l t

00 0

⎛ ⎞

cos θ 0 − sin θ − sin( θ + ϕ )

⎝ ⎠ .

sin θ 0 cos θ cos( θ + ϕ )

B ( s )=

0 1 0 − l t − l r cos ϕ

1
l t

l r
l t

− sin ϕ − 1 − cos ϕ 0 − l t

The input perturbation is defined by truncated Fourier series over inputs u 1 and u 2 .
The configuration potential field is defined by a decreasing function of the distance
to obstacles in the workspace. We refer the reader to [8] for details.