Geometric control methods for nonlinear systems and robotic applications

by Altafini, Claudio

This thesis is a collection of seven independent papers dealing with different topics in the analysis and control of nonlinear systems, mainly discussed using differential geometric methods and mainly inspired by applications to Robotics. Paper A proposes a geometric framework for the study of certain redundant robotic chains. Interpreting the forward kinematic map from joint space to the workspace of the end-effector as a Riemannian submersion allows to give clear geometric characterizations of several properties of redundant robots, for example of the Moore-Penrose pseudoinverse as the horizontal lift of the Riemannian submersion. Furthermore, it enables to pull back to joint space the motion control algorithms designed in workspace, all respecting the different structures of the two model spaces. The generation of motion in a geometric setting continues in Paper B, where the reduction by groups invariance of first and second order variational problems is discussed for a configuration space which is a semidirect product of a Lie group and a vector space, endowed with the Riemannian connection of a positive definite metric tensor instead of the natural affine connection. Paper C treats motion on Lie groups in presence of constraints that are not invariant: for a kinematic control system on the Lie group, the combination of inputs that satisfies the constraints is computed in coordinates via the Wei- Norman formula and in a coordinate-free setting by finding the annihilator of the coadjoint orbit of the constraint one form at the point of interest. For a class of linear switching systems with controllable logic, an interpretation is proposed in Paper D in terms of bilinear control systems. The main consequence is the characterization of the reachable set of the switching system as having only the structure of a semigroup since, in general, the logic inputs cannot reverse the direction of the flow. Paper E considers the nilpotent, filiform Lie group of transformations corresponding to a control system in chained form and shows how to obtain an abelian left coset out of it by factoring out the characteristic line field. The control theoretic interpretation is the arclength reparameterization normally used in differential flatness methods. Paper F investigates the so-called general n-trailer i.e. a variant of the multibody wheeled vehicle discussed in the literature. Properties like controllability, singular locus and existence of canonical forms are analyzed. The last paper presents practical experiments on backward driving for a particular multibody vehicle in the class of general n-trailers. For the situation under investigation, the system behaves like an unstable, saturated nonlinear system. The proposed hybrid control scheme is able to avoid jack-knife saturations on line by driving forward and realigning the bodies of the system when needed. iv