Page 84 - Introduction to Autonomous Mobile Robots

Mobile Robot Kinematics

a) b) 69

β t()
s

Figure 3.13
(a) Differential drive robot with two individually motorized wheels and a castor wheel, e.g., the Pyg-
malion robot at EPFL. (b) Tricycle with two fixed standard wheels and one steered standard wheel,
e.g. Piaggio minitransporter.

The Ackerman vehicle of figure 3.12a demonstrates another way in which a wheel may
be unable to contribute an independent constraint to the robot kinematics. This vehicle has
two steerable standard wheels. Given the instantaneous position of just one of these steer-
able wheels and the position of the fixed rear wheels, there is only a single solution for the
ICR . The position of the second steerable wheel is absolutely constrained by the ICR .
Therefore, it offers no independent constraints to robot motion.
Robot chassis kinematics is therefore a function of the set of independent constraints
arising from all standard wheels. The mathematical interpretation of independence is
related to the rank of a matrix. Recall that the rank of a matrix is the smallest number of
independent rows or columns. Equation (3.26) represents all sliding constraints imposed by
the wheels of the mobile robot. Therefore rank C β() is the number of independent con-
1 s
straints.
The greater the number of independent constraints, and therefore the greater the rank of
C β() , the more constrained is the mobility of the robot. For example, consider a robot
1 s
with a single fixed standard wheel. Remember that we consider only standard wheels. This
robot may be a unicycle or it may have several Swedish wheels; however, it has exactly one
fixed standard wheel. The wheel is at a position specified by parameters αβ l,, relative
to the robot’s local reference frame. C β() is comprised of C and C . However, since
1 s 1f 1s
there are no steerable standard wheels C is empty and therefore C β() contains only
1s 1 s
C . Because there is one fixed standard wheel, this matrix has a rank of one and therefore
1f
this robot has a single independent constrain on mobility:

C β() = C = cos ( α + β) sin ( α + β) lsin β (3.37)
1 s 1f