Instead of a wheel, consider rolling a ball in the plane. Place a
ball on a table and try rolling it with your palm placed flat on top
of it. It should feel like there are two degrees of freedom: rolling
forward and rolling side to side. The ball should not be able to spin
in place. The directions can be considered as two action variables.
The total degrees of freedom of the ball is five, however, because it
can achieve any orientation in and any
position in the
plane; thus,
. Given that there are only two
action variables, is it possible to roll the ball into any
configuration? It is shown in [632,491] that this is
possible, even for the more general problem of one sphere rolling on
another (the plane is a special case of a sphere with infinite
radius). This problem can actually arise in robotic manipulation when
a spherical object come into contact (e.g., a robot hand may have
fingers with spherical tips); see
[103,676,725,729].
The resulting transition equation was shown in [716] (also see [725]) to be
Steven M LaValle 2020-08-14