Page 82 - Introduction to Autonomous Mobile Robots

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Mobile Robot Kinematics
The sliding constraints comprising C β() can be used to go even further, enabling us
1 s
to evaluate the maneuverability and workspace of the robot rather than just its predicted
motion. Next, we examine methods for using the sliding constraints, sometimes in conjunc-
tion with rolling constraints, to generate powerful analyses of the maneuverability of a
robot chassis.

3.3 Mobile Robot Maneuverability

The kinematic mobility of a robot chassis is its ability to directly move in the environment.
The basic constraint limiting mobility is the rule that every wheel must satisfy its sliding
constraint. Therefore, we can formally derive robot mobility by starting from equation
(3.26).
In addition to instantaneous kinematic motion, a mobile robot is able to further manip-
ulate its position, over time, by steering steerable wheels. As we will see in section 3.3.3,
the overall maneuverability of a robot is thus a combination of the mobility available based
on the kinematic sliding constraints of the standard wheels, plus the additional freedom
contributed by steering and spinning the steerable standard wheels.

3.3.1 Degree of mobility
Equation (3.26) imposes the constraint that every wheel must avoid any lateral slip. Of
course, this holds separately for each and every wheel, and so it is possible to specify this
constraint separately for fixed and for steerable standard wheels:

·
C R θ()ξ = 0 (3.35)
I
1f
·
β
C ()R θ()ξ = 0 (3.36)
1s s I
For both of these constraints to be satisfied, the motion vector R θ()ξ · I must belong to
the null space of the projection matrix C β() , which is simply a combination of C and
1 s 1f
C . Mathematically, the null space of C β() is the space N such that for any vector n in
1s 1 s
N, C β()n = 0 . If the kinematic constraints are to be honored, then the motion of the
s
1
robot must always be within this space . The kinematic constraints [equations (3.35) and
N
(3.36)] can also be demonstrated geometrically using the concept of a robot’s instantaneous
center of rotation (ICR ).
Consider a single standard wheel. It is forced by the sliding constraint to have zero lat-
eral motion. This can be shown geometrically by drawing a zero motion line through its
horizontal axis, perpendicular to the wheel plane (figure 3.12). At any given instant, wheel
motion along the zero motion line must be zero. In other words, the wheel must be moving