A contact covariant approach to optimal control with applications to sub-Riemannian geometry

We discuss contact geometry naturally related with optimal control problems (and Pontryagin Maximum Principle). We explore and expand the observations of [Ohsawa, 2015], providing simple and elegant characterizations of normal and abnormal sub-Riemannian extremals.Comment: A small correction in the statement and proof of Thm 6.15. Watch our publication: https://youtu.be/V04N9X3NxYA and https://youtu.be/jghdRK2IaU

Nilpotentization of the kinematics of the n-trailer system at singular points and motion planning through the singular locus

We propose in this paper a constructive procedure that transforms locally, even at singular configurations, the kinematics of a car towing trailers into Kumpera-Ruiz normal form. This construction converts the nonholonomic motion planning problem into an algebraic problem (the resolution of a system of polynomial equations), which we illustrate by steering the two-trailer system in a neighborhood of singular configurations. We show also that the n-trailer system is a universal local model for all Goursat structures and that all Goursat structures are locally nilpotentizable.Comment: LaTeX2e, 23 pages, 4 figures, submitted to International journal of contro

Contact systems and corank one involutive subdistributions

We give necessary and sufficient geometric conditions for a distribution (or a Pfaffian system) to be locally equivalent to the canonical contact system on Jn(R,Rm), the space of n-jets of maps from R into Rm. We study the geometry of that class of systems, in particular, the existence of corank one involutive subdistributions. We also distinguish regular points, at which the system is equivalent to the canonical contact system, and singular points, at which we propose a new normal form that generalizes the canonical contact system on Jn(R,Rm) in a way analogous to that how Kumpera-Ruiz normal form generalizes the canonical contact system on Jn(R,R), which is also called Goursat normal form.Comment: LaTeX2e, 29 pages, submitted to Acta applicandae mathematica

Conic nonholonomic constraints on surfaces and control systems

This paper addresses the equivalence problem of conic submanifolds in the tangent bundle of a smooth 2-dimensional manifold. We deal with this problem under the prism of feedback equivalence of control systems, both control-affine and fully nonlinear. We also give equivalence results for a special class of conic submanifolds via the study of the Lie algebra of infinitesimal symmetries of the corresponding control systems. Then we consider the classification problem of conic submanifolds, which is achieved via feedback classification of nonlinear control system. Our results describe and completely characterise conic systems, and include several normal and canonical forms.Comment: 42 pages, 5 appendices, preprin

Input-output linearization and decoupling of mechanical control systems

In this work, we present a problem of simultaneous input-output feedback linearization and decoupling (non-interacting) for mechanical control systems with outputs. We show that the natural requirement of preserving mechanical structure of the system and of transformations imposes supplementary conditions when compared to the classical solution of the same problem for general control systems. These conditions can be expressed using objects on the configuration space only. We illustrate our results with several examples of mechanical control systems

From Morse triangular form of ODE control systems to feedback canonical form of DAE control systems

In this paper, we relate the feedback canonical form FBCF [24] of differential-algebraic control systems (DACSs) with the famous Morse canonical form MCF [28],[27] of ordinary differential equation control systems (ODECSs). First, a procedure called an explicitation (with driving variables) is proposed to connect the two above categories of control systems by attaching to a DACS a class of ODECSs with two kinds of inputs (the original control input u and a vector of driving variables v). Then, we show that any ODECS with two kinds of inputs can be transformed into its extended MCF via two intermediate forms: the extended Morse triangular form and the extended Morse normal form. Next, we illustrate that the FBCF of a DACS and the extended MCF of the explicitation system have a perfect one-to-one correspondence. At last, an algorithm is proposed to transform a given DACS into its FBCF via the explicitation procedure and a numerical example is given to show the efficiency of the proposed algorithm

Geometric analysis of nonlinear differential-algebraic equations via nonlinear control theory

For nonlinear differential-algebraic equations (DAEs), we define two kinds of equivalences, namely, the external and internal equivalence. Roughly speaking, the word "external" means that we consider a DAE (locally) everywhere and "internal" means that we consider the DAE on its (locally) maximal invariant submanifold (i.e., where its solutions exist) only. First, we revise the geometric reduction method in DAEs solution theory and formulate an implementable algorithm to realize that method. Then a procedure named explicitation with driving variables is proposed to connect nonlinear DAEs with nonlinear control systems and we show that the driving variables of an explicitation system can be reduced under some involutivity conditions. Finally, due to the explicitation, we will use some notions from nonlinear control theory to derive two nonlinear generalizations of the Weierstrass form.Comment: 38 page

Feedback Linearizability of Strict Feedforward Systems

For any strict feedforward system that is feedback linearizable we provide (following our earlier results) an algorithm, along with explicit transformations, that linearizes the system by change of coordinates and feedback in two steps: first, we bring the system to a newly introduced Nonlinear Brunovský canonical form (NBr) and then we go from (NBr) to a linear system. The whole linearization procedure includes diffeo-quadratures (differentiating, integrating, and composing functions) but not solving PDE’s. Application to feedback stabilization of strict feedforward systems is given