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

In these equations, and are the position on the contact point in the plane, and and are the position of the contact point in the ball frame and are expressed using spherical coordinates. The radius of the ball is . Finally, expresses the orientation of the ball with respect to the contact point.

Steven M LaValle 2020-08-14