Page 91 - Introduction to Autonomous Mobile Robots

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Consider the fixed standard wheel sliding constraint: Chapter 3

·
cos ( α + β) sin ( α + β) lsin β R θ()ξ = 0 (3.43)
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·
ξ
ξ
This constraint must use robot motion rather than pose because the point is to con-
strain robot motion perpendicular to the wheel plane to be zero. The constraint is noninte-
grable, depending explicitly on robot motion. Therefore, the sliding constraint is a
nonholonomic constraint. Consider a bicycle configuration, with one fixed standard wheel
and one steerable standard wheel. Because the fixed wheel sliding constraint will be in
force for such a robot, we can conclude that the bicycle is a nonholonomic robot.
But suppose that one locks the bicycle steering system, so that it becomes two fixed stan-
dard wheels with separate but parallel axes. We know that δ = 1 for such a configura-
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tion. Is it nonholonomic? Although it may not appear so because of the sliding and rolling
constraints, the locked bicycle is actually holonomic. Consider the workspace of this
locked bicycle. It consists of a single infinite line along which the bicycle can move (assum-
ing the steering was frozen straight ahead). For formulaic simplicity, assume that this infi-
nite line is aligned with X I in the global reference frame and that
{ β 12 = π 2 α = 0 α = π} . In this case the sliding constraints of both wheels can be
,
⁄
,
,
1
2
,
replaced with an equally complete set of constraints on the robot pose: y ={ 0 θ = 0} .
This eliminates two nonholonomic constraints, corresponding to the sliding constraints of
the two wheels.
The only remaining nonholonomic kinematic constraints are the rolling constraints for
each wheel:
·
·
– sin ( α + β) cos ( α + β) lcos β R θ()ξ + rϕ = 0 (3.44)
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This constraint is required for each wheel to relate the speed of wheel spin to the speed
of motion projected along the wheel plane. But in the case of our locked bicycle, given the
initial rotational position of a wheel at the origin, ϕ o , we can replace this constraint with
one that directly relates position on the line, x, with wheel rotation angle, ϕ :
⁄
ϕ = ( x r) + ϕ o .
The locked bicycle is an example of the first type of holonomic robot – where constraints
do exist but are all holonomic kinematic constraints. This is the case for all holonomic
robots with δ < 3 . The second type of holonomic robot exists when there are no kinematic
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constraints, that is, N = 0 and N = 0 . Since there are no kinematic constraints, there are
f
s
also no nonholonomic kinematic constraints and so such a robot is always holonomic. This
is the case for all holonomic robots with δ = . 3
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