Constructive Proof for Complete Controllability of a Rolling Pseudo-Hyperbolic Space

We present a constructive proof for complete controllability of the rolling motion of a 2-dimensional pseudo-hyperbolic space, with index zero, over the affine space associated with the tangent space at a point. This motion is assumed to have the constraints of no-twist and no-slip (pure rolling). It is already known that, under mild conditions, this system is controllable, which means that any admissible configuration of the rolling manifold can be obtained from any other by rolling without violating the constraints. This result is obtained via some algebraic theoretical arguments that do not specify how to reach a configuration from another. But from a practical point of view an important issue is to know exactly how this can be achieved. The main objective of the present paper is to address this problem. We will describe a procedure that shows precisely how to move the 2-dimensional pseudo-hyperbolic space between two admissible configurations. This is accomplished by generating the forbidden motions of twist and slip using only pure rolling motions.