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Research Papers

# Control Design for a Class of Nonholonomic Systems Via Reference Vector Fields and Output Regulation

[+] Author and Article Information
Dimitra Panagou

Assistant Professor
Department of Aerospace Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: dpanagou@umich.edu

Herbert G. Tanner

Associate Professor
Mechanical Engineering Department,
University of Delaware,
Newark, DE 19716
e-mail: btanner@udel.edu

Kostas J. Kyriakopoulos

Professor
School of Mechanical Engineering,
National Technical University of Athens,
Zografou,
Athens 15780, Greece
e-mail: kkyria@mail.ntua.gr

An isolated critical point is called a rose if it has elliptic type of sectors only, i.e., if in a neighborhood around it, all integral curves begin and end at the critical point; an example is the dipole [31].

Note that our characterization of the system states into “leafwise” and “transverse” applies when n0 = 0 as well, i.e., when A(q) has no zero columns. In this case, one trivially takes $x≜q$, i.e. all system coordinates qi are thought as leafwise, while the leaf space $L$ coincides with the configuration space $C$.

Note, furthermore, that any vector field which has a single critical point x = 0 of either elliptic or parabolic sectors [31] may serve as a valid choice for F, since in both cases all integral curves converge to the critical point.

Note that the selection of the consistency errors (or outputs) sμ(⋅) depends on the analytical form of F, and it is not necessarily unique. This implies that for different choices of sμ(⋅), one may end up with different control laws.

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 4, 2013; final manuscript received March 31, 2015; published online May 26, 2015. Assoc. Editor: Evangelos Papadopoulos.

J. Dyn. Sys., Meas., Control 137(8), 081011 (Aug 01, 2015) (9 pages) Paper No: DS-13-1007; doi: 10.1115/1.4030335 History: Received January 04, 2013; Revised March 31, 2015; Online May 26, 2015

## Abstract

This paper presents procedural guidelines for the construction of discontinuous state feedback controllers for driftless, kinematic nonholonomic systems, with extensions to a class of dynamic nonholonomic systems with drift. Given an n-dimensional kinematic nonholonomic system subject to κ Pfaffian constraints, system states are partitioned into “leafwise” and “transverse,” based on the structure of the Pfaffian constraint matrix. A reference vector field F is defined as a function of the leafwise states only in a way that it is nonsingular everywhere except for a submanifold containing the origin. The induced decomposition of the configuration space, together with requiring the system vector field to be aligned with F, suggests choices for Lyapunov-like functions. The proposed approach recasts the original nonholonomic control problem as an output regulation problem, which although nontrivial, may admit solutions based on standard tools.

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## Figures

Fig. 1

The dipolar vector field (5), for λ = 2 and p = [1 0]

Fig. 2

The dipolar vector field (5), for λ = 2 and p=(1/2)[11]⊤

Fig. 3

The system trajectories x(t) under the control laws (23), (20), and (26)

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