E. Aranda-Bricaire

Linearization of Discrete-Time Systems

The algebraic formalism developed in this paper unifies the study of the accessibility problem and various notions of feedback linearizability for discrete-time nonlinear systems. The accessibility problem for nonlinear discrete-time systems is shown to be easy to tackle by means of standard linear algebraic tools, whereas this is not the case for nonlinear continuous-time systems, in which case the most suitable approach is provided by differential geometry. The feedback linearization problem for discrete-time systems is recasted through the language of differential forms. In the event that a system is not feedback linearizable, the largest feedback linearizable subsystem is characterized within the same formalism using the notion of derived flag of a Pfaffian system. A discrete-time system may be linearizable by dynamic state feedback, though it is not linearizable by static state feedback. Necessary and sufficient conditions are given for the existence of a so-called linearizing output, which in turn is a sufficient condition for dynamic state feedback linearizability.

A linear algebraic framework for dynamic feedback linearization

To any accessible nonlinear system we associate a so-called infinitesimal Brunovsky form. This gives an algebraic criterion for strong accessibility as well as a generalization of Kronecker controllability indices. An output function which defines a right-invertible system without zero dynamics is shown to exist if and only if the basis of the Brunovsky form can be transformed into a system of exact differential forms. This is equivalent to the system being differentially flat and hence constitutes a necessary and sufficient condition for dynamic feedback linearizability. >

Constructive nonsmooth stabilization of triangular systems

The problem of local asymptotic continuous feedback stabilization of single-input nonlinear systems is considered here. The aim is to explicitly construct a continuous asymptotically stabilizing feedback for a class of nonlinear systems that is known to be continuous feedback asymptotically stabilizable, but for which the known results do not provide an effective method to compute the stabilizer. Computer simulations are included to show practical applicability of our approach.

Convergence and Collision Avoidance in Formation Control: A Survey of the Artificial Potential Functions Approach

Multi-agent Robots Systems (MARS) can be defined as sets of autonomous robots coordinated through a communication system to achieve cooperative tasks. During the last 20 years, MARS have found a wide range of applications in terrestrial, spatial and oceanic explorations emerging as a new research area (Cao et al. (1997)). Some advantages can be obtained from the collective behavior of MARS. For instance, the kind of tasks that can be accomplished are inherently more complex than those a single robot can accomplish. Also, the system becomes more flexible and fault-tolerant (Yamaguchi (2003)). The range of applications includes toxic residues cleaning, transportation and manipulation of large objects, alertness and exploration, searching and rescue tasks and simulation of biological entities behaviors (Arai et al. (2002)). The study ofMARS extends the classical problems of single robotswith new issues likemotion coordination, task decomposition and task assignment, network communications, searching and mapping, etc. Therefore, the study of MARS encompass distributed systems, artificial intelligence, game theory, biology, ethology, economics, control theory, etc. Motion coordination is an important research area of MARS, specifically formation control (Chen & Wang (2005)). The main goal is to coordinate a group of mobile agents or robots to achieve a desired formation pattern avoiding inter-agent collisions at the same time. The formation strategies are decentralized because it is assumed that every agent measures the position of a certain subset of agents and, eventually, it detects the position of other agents when a minimal allowed distance is violated and collision danger appears. Thus, the main intention is to achieve desired global behaviors through local interactions (Francis et al. (2004)). Also, the decentralized approaches offer greater autonomy for the robots, less computational load in control implementations and its applicability to large scale groups (Do (2007)). According to Desai (2002); Muhammad & Egerstedt (2004), the possible inter-agent communications and the desired relative position of every agent with respect to the others can be represented by a FormationGraph (FG). The application of different FG’s to the same group of robots produces different dynamics on the team behavior. In the literature, some special FG topologies are chosen and the convergence to the desired formation and non-collision is analyzed for any number of robots. A decentralized formation strategy must comply with two fundamental requirements: Global convergence to the desired formation and inter-agent collision avoidance (Cao et al. (1997)). 6

Discrete-time sliding mode path-tracking control for a wheeled mobile robot

In this work, it is presented a discrete time control strategy for the solution of the path-tracking problem for a wheeled mobile robot of the type (2,0). It is assumed that the mobile robot is remotely controlled over a communication network that induces a transport delay. The exact discrete-time model of the mobile robot including the induced delay is developed. It is presented a discrete-time strategy control based on a sliding mode approach that allows to solve the path-tracking problem. The closed loop stability of the overall system is clearly stated. The proposed control strategy is evaluated by mean of computer simulation

GLOBAL PATH-TRACKING FOR A MULTI-STEERED GENERAL N-TRAILER

Abstract A multi-steered general n-trailer is considered in this work. A global path-tracking control strategy for the n-trailer is proposed. Three nonlinear control laws are used to solve the problem by means of a commutation scheme. This commutation scheme allows to avoid the singularities inherent to a feedback linearization scheme. Simulation results show that the proposed scheme has good performance.

Non-collision conditions in multi-agent robots formation using local potential functions

An analysis of the convergence and non-collision conditions of a formation control strategy for multi-agent robots based on local potential functions is presented. The goal is to coordinate a group of agents, considered as points in plane, to achieve a particular formation. The control law is designed using local attractive forces only where every agent knows the position of another two agents reducing the requirements of the control law implementation. The control law guarantees the convergence to the desired formation but does not avoid inter-agent collisions. A set of necessary and sufficient non- collision conditions based on the explicit solution of the closed- loop system is derived. The conditions allow to conclude from the initial conditions whether or not the agents will collide. The formal proof is presented for the case of three agents. The result is extended to the case of formations of three unicycles.

Equivalence of Nonlinear Systems to Prime Systems under Generalized Output Transformations

Within a linear algebraic framework, we present a new characterization of the class of nonlinear systems which are equivalent to a prime system. We then introduce a class of generalized output transformations that can be thought of as a generalization to the nonlinear setting of a unimodular transformation in the output space. Our main result gives necessary and sufficient conditions for equivalence to a prime system under a certain group of transformations that includes generalized output transformations.

Trajectory tracking for groups of unicycles with convergence of the orientation angles

This paper presents a trajectory tracking control strategy for unicycles based on local potential functions and the leader-followers scheme. The leader robot must follow a prescribed trajectory and the rest of the robots follows the leader whereas achieves a desired formation at the same time. Every follower robot does not possess a global information about the leader path and the goals of the other followers. The control strategy does not influence directly the orientation angles, however a sufficient condition is established for their convergence to the same value. Finally, the performance of the proposed strategy is evaluated over an experimental set-up consisting of three unicycle-type robots and a computer vision system.

Linearization of discrete-time systems by exogenous dynamic feedback

It is shown that a discrete-time system may be linearizable by exogenous dynamic feedback, even if it cannot be linearized by endogenous feedback. This property is completely unexpected and constitutes a fundamental difference with respect to the continuous-time case. The notion of exogenous linearizing output is introduced. It is shown that existence of an exogenous linearizing output is a sufficient condition for dynamic linearizability. Necessary and sufficient conditions for the existence of an exogenous linearizing output are provided. The results of the paper are obtained using transformal operator matrices. The properties of such operators are studied. The theory is applied to the exact discrete-time model of a mobile robot, showing that the above-mentioned property concerns not only academic examples, but also physical systems.