# Geometric control methods for nonlinear systems and robotic applications

Abstract (Summary)

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
Bibliographical Information:

Advisor:

School:Kungliga Tekniska högskolan

School Location:Sweden

Source Type:Doctoral Dissertation

Keywords:Geometric control; Differential geometric methods; Nonlinear control systems; Control of mechanical systems; Robot dynamics and control; Lie groups; Variational problems; Group symmetry; Reachability; Switching systems

ISBN:91-7283-094-8

Date of Publication:01/01/2001