Page 100 - Introduction to Autonomous Mobile Robots

85
Mobile Robot Kinematics
θ denotes the angle between the X axis of the robot reference frame, and the X axis asso-
R I
ω
ciated with the final position and are the tangent and the angular velocity respectively.
v
On the other hand, if α ∈ I , where
2
,
,
⁄
⁄
I = – ( π – π 2 ] ∪ ( π 2 π] (3.54)
2
redefining the forward direction of the robot by setting v = v – , we obtain a system
described by a matrix equation of the form

ρ · cos α 0
· sin α v
α = – ----------- 1 ω (3.55)
ρ
β · sin α
----------- 0
ρ

3.6.2.3 Remarks on the kinematic model in polar coordinates [eq. (3.53) and (3.55)]
• The coordinate transformation is not defined at x = y = 0 ; as in such a point the deter-
minant of the Jacobian matrix of the transformation is not defined, that is unbounded.
• For α ∈ I the forward direction of the robot points toward the goal, for α ∈ I it is the
1 2
reverse direction.
• By properly defining the forward direction of the robot at its initial configuration, it is
α
always possible to have α ∈ I at t = 0 . However, this does not mean that remains
1
in for all time . Hence, to avoid that the robot changes direction during approaching
I
t
I
the goal, it is necessary to determine, if possible, the controller in such a way that α ∈ I 1
for all , whenever α 0() ∈ I 1 . The same applies for the reverse direction (see stability
t
issues below).
3.6.2.4 The control law
ω
v
The control signals and must now be designed to drive the robot from its actual con-
figuration, say ρ α β,( , ) , to the goal position. It is obvious that equation (3.53) presents
0 0 0
a discontinuity at ρ = 0 ; thus the theorem of Brockett does not obstruct smooth stabiliz-
ability.
If we consider now the linear control law
v = k ρ (3.56)
ρ

ω = k α + k β (3.57)
α
β