this paper is as follows: in Section 2, we collect some mathematical preliminaries from the literature on controllability of nonlinear systems and on classification of free Lie algebras. These are drawn from classical references in control theory [4, 17, 18, 36, 40] and Lie algebras [15, 43]. In Section 3, using some outstanding results of Brockett on optimal steering of certain classes of systems as motivation [5], we discuss the use of sinusoidal inputs for steering systems of first order, i.e., systems where controllability is achieved after just one level of Lie brackets of the input vector fields. Section 4 attempts to expand the domain of applicability of these results to more complex systems, where several orders of Lie brackets are needed to obtain the full Lie algebra associated with the input distribution. The 4 MURRAY AND SASTRY

This paper describes a practical path planner for nonholonomic robots in environments with obstacles. The planner is based on building a one-dimensional, maximal clearance skeleton through the configuration space of the robot. However rather than using the Euclidean metric to determine clearance, a special metric which captures information about the nonholonomy of the robot is used. The robot navigates from start to goal states by loosely following the skeleton; the resulting paths taken by the robot are of low "complexity." We describe how much of the computation can be done off-line once and for all for a given robot, making for an efficient planner. The focus is on path planning for mobile robots, particularly the planar two-axle car, but the underlying ideas are quite general and may be applied to planners for other nonholonomic robots.

This paper aims at studying the trajectory planning for a car -- i.e. a non holonomic vehicle whose turning radius is lower bounded -- in a static and structured world. As for the structure of the world, we assume the existence of natural lanes within which the vehicle is able to move. The contribution of this paper is a smooth trajectory planner which, when given the polygonal line S representing the spine of a lane generates a trajectory C avoiding the obstacles of the world and which is smooth -- i.e. without backing up maneuvers -- and executable by the vehicle according to its own kinematic constraints. Besides C is topologically equivalent to S -- i.e. C must remain in the lane defined by S. C is made up of straight segments and circular arcs.

In this paper, we will compare different types of control, like nonlinear control, motion planning, fuzzy control, neural control, rule-based incremental control; we take robot motion as a comparative field, and more specifically motion of car-like robots. We will compare the different approaches on two main points: their theoretical basis (controllability, stability, robustness for a given application) and their conviviality (easy and user-friendly implementation of the control, application of artificial intelligence methods to improve control when faced with unknown situations). Very often, control techniques of the first type are called "classical control", while methods of the second type are called "intelligent control". We do not find these appellations are very relevant, as they tend to classify the first methods as guaranteed to work but impracticable and the second methods as easy to implement but magical; we will try to see in each method the advantages and the drawbacks rat...

The probabilistic path planner (PPP) is a general planning scheme that yields fast robot path planners for a wide variety of problems, involving, e.g., high degree of freedom articulated robots, nonholonomic robots, and multiple robots. In this paper we go into theoretical aspects regarding the behaviour of PPP. We formulate general properties that guarantee probabilistic completeness of PPP, and we show how these apply to various robot types. Furthermore, we present results which, under certain assumptions on the free configuration space, give estimates of the expected running times. For example, under one such assumption, we show that the expected running time of PPP grows only logarithmically with the complexity of the problem that it solves. 1 Introduction Robot path planning, which asks for the computation of collision free paths in environments containing obstacles, has received a great deal of attention in the last decades [Lat91, HA92]. We consider the basic problem, where the...

In this paper we discuss a special family of control laws, rule-based incremental control, and we address the problem of the parking maneuver for a car, first in simulation and then with a small mobile robot we have built. 1 INTRODUCTION Many systems lack a mathematical model or are illmodeled; however human experts can control them rather well. The knowledge generated by these experts seems to be stored in a declarative (if-then rules), symbolic way (no exact values on parameters are considered, but rather ranges) and the control actions are basically incremental (at each time, an input is either incremented, decremented by a given amount or left untouched), they are based upon the study of the tendencies of the system (variation trends are considered). Starting from these considerations, we have discussed in [6] a new family of control laws which can be defined informally as a finite set of rules: " if condition on the outputs then action on the inputs ". The actions on the inputs...

In this paper we discuss a special family of control laws, rule-based incremental control, and present two approaches (a time-varying state feedback stabilization and a motion planner) to the control of discrete nonholonomic systems with an application to a car-like robot in simulation and real-world. We compare both approaches and see that despite their very different appearance they induce similar behaviors on a car-like robot. 1 Rule-based incremental control Let us consider a discrete system with inputs u k and outputs y k sampled in time. For sake of clarity the next definition deals only with single input-single output systems but the generalization to higher dimensions is straightforward. A rule-based incremental control law [7] is defined by: ( u k+1 = u k + ffl k \Delta ffl k = '(y k ; y k\Gamma1 ; \Delta \Delta \Delta ; y k\Gammar ) with: \Delta 2 R + ; ' computable; ffl k 2 f\Gammam; \Gammam + 1; \Delta \Delta \Delta ; \Gamma1; 0; +1; \Delta \Delta \Delta ; m \G...

In this paper we introduce rule-based incremental controllers which have been used efficiently in realworld applications like controlling a laser cutting robot. The key point of these controllers is that the control laws have a very simple expression: an input between two time steps can remain constant or be increased or decreased by some amount. Although this may seem very crude, a large class of systems (linear and non linear) can be controlled in this way. We give some general results concerning incremental control in order to understand how this type of control works for linear systems. Then we address more complex problems as can be encountered in robot motion: controllability of carlike robots and parking maneuvers. The interesting point in this new type of control is furthermore its flexibility and relationship with learning, as we will see in the last section. 1 Introduction Many systems lack a mathematical model or are illmodeled; however human experts can control them rather...

This paper presents a sensor based docking strategy for a non holonomic mobile platform used to support material transportation operations in industrial like environments. A low cost infrared sensor system was designed and implemented, aiming at locating the mobile platform relative to the docking station where passive reflectors are installed. With this information, trajectories are generated and followed, docking the platform with a good accuracy (around \Sigma0:5cm in x and y, and \Sigma1 ffi in `), considering that the system was designed based on low cost sensors. The paper presents relevant experimental results. I. INTRODUCTION Material transportation and handling is a major component in many industrial processes. The commonly used Automated Guided Vehicles (AGVs) do not yield flexible transportation solutions, given that the vehicles follow fixed and pre-defined sets of paths. Mobile robots, on the contrary, have the ability to travel freely on all empty space of an environ...

This paper deals with the problem of planning and controlling the motion of a car like mobile moving in a dynamic and roadway like environment. The contribution presented here is a motion controller which executes in a reactive way a given nominal motion plan. Such a plan is made up of a smooth trajectory C and of time constraints of the type "reach location l at time t l ". Data concerning the actual environment of the vehicle considered are assumed to be obtained through perception. In order to get the required reactivity, we have developed a motion controller with two main components: the pilot which analyses the current situation and adapts the nominal plan accordingly, and the executor which generates the required motion commands. The pilot operates at a symbolic level using a set of behavioural rules. The executor makes use of a potential field approach to generate the motion commands. Keywords --- mobile-robot, non-holonomic-system, motion-execution, potentialfield. Acknowled...