Trajectory tracking for nonholonomic systems. Theoretical background and applications

The problem of stabilizing reference trajectories for nonholonomic systems, often referred to as the trajectory tracking problem in the literature on mobile robots, is addressed. The first sections of this report set the theoretical background of the problem, with a focus on controllable driftless systems which are invariant on a Lie group. The interest of the differential geometry framework here adopted comes from the possibility of taking advantage of ubiquitous symmetry properties involved in the motion of mechanical bodies. Theoretical difficulties and impossibilities which set inevitable limits to what is achievable with feedback control are surveyed, and basic control design tools and techniques are recast within the approach here considered. A general method based on the so-called Transverse Function approach --developed by the authors--, yielding feedback controls which unconditionnally achieve the {\em practical} stabilization of arbitrary reference trajectories, including fixed points and non-admissible trajectories, is recalled. This property singles the proposed solution out of the abundant literature devoted to the subject. It is here complemented with novel results showing how the more common property of asymptotic stabilization of persistently exciting admissible trajectories can also be granted with this type of control. The last section of the report concerns the application of the approach to unicycle-type and car-like vehicles. The versatility and potentialities of the Transverse Function (TF) control approach are illustrated via simulations involving various reference trajectory properties, and a few complementary control issues are addressed. One of them concerns the possiblity of using control degrees of freedom to limit the vehicle's velocity inputs and the number of transient maneuvers associated with the reduction of initially large tracking errors. Another issue, illustrated by the car example, is related to possible extensions of the approach to systems which are not invariant on a Lie group.