We consider the general problem of steering an infinitesimal propelled and steerable particle with no rotational inertia traversing on a mathematically smooth (vs. frictionless) surface, where both the speed and body yaw rate serve as inputs to the system and lateral motion is not allowed, i.e. through a no sideslip condition. More specifically, focus is on derivation of relevant state equations in control input form, numerical and visual confirmation of state equation accuracy through specific simulations, and exploring interesting approaches to nonholonomic path planning with associated numerical simulation and visualization. Given the state equations (3rd order), in the interests of practical validation, they were exercised by considering motion on a number of smooth surfaces. The surfaces were selected for a variety of reasons, such as: the resulting qualitative trajectory is known a priori, there exists an opportunity to check numerical results with respect to previous results, or the surfaces are iconic and/or are geometrically rich.
Nonholonomic steering on the surface is a very interesting and challenging problem and several approaches are investigated: (1) steering using sinusoids (detailed), (2) steering on a trajectory, and (3) “drive-and-turn” (valid in this case). Prior to implementing the steering using sinusoids algorithm, it was necessary to transform the system into “one-chained” form. The first step entailed conversion to an approximate one-chained form model that possesses a certain structure, from which the process established by Murray and Sastry can be successfully launched, where two special smooth scalar functions of the states are sought that possess special relationships to Lie-related distribution spaces associated with the control input vectors. Inputs are then transformed as well via specialized Lie derivatives. It was demonstrated through simulation that steering to an arbitrary system state on a faceted surface can be accomplished with sinusoidal inputs in only one maneuver set (i.e. maneuver A & B). Using this fact, the work presented culminates with steering to an arbitrary system state on a smooth surface that can be accomplished by essentially iterating on a steering algorithm that assumes the particle is on a plane tangent to the smooth surface at the desired destination. In this regard, it is shown that a sequence of maneuver sets converges rather quickly in the example demonstrated. Applications of this work pertain to the fairly general situation of steering a vehicle on a smooth surface, a practical vehicle navigation and control problem.