IEEE Int. Conf. on Robotics and Automation May 16{21, 1998. Leuven (BE). Vol. 1, pp. 27{32
Path Planning with Uncertainty for Car-Like Robots Th. Fraichard and R. Mermond a b
Inria Rh^one-Alpes & Gravir Zirst. 655 av. de l'Europe, 38330 Montbonnot Saint Martin, France
[email protected] June 16, 1998
Abstract | this paper presents the rst path planner taking into account both non-holonomic and uncertainty constraints. The case of a car-like robot subject to cumulative and unbounded con guration uncertainty related to bounded control and sensing errors is considered. Assuming the existence of landmarks allowing the robot to relocalize itself in particular places, we present an algorithm that computes paths that are both feasible and robust. In other words, they respect the non-holonomic constraints of a car-like robot and the robot is assured to reach its goal by following them as long as its control and sensing errors remain within bounded sets. Keywords | mobile-robot, non-holonomic-system, path-planning, uncer-
tainty, landmark.
Acknowledgements | this work was partially supported by the InriaInretsc Praxitele programme on individual urban public transports [1994-1997], and the Inco-Copernicus ERB-IC15-CT96-0702 project \Multi-agent robot systems for industrial applications in the transport domain" [1997-1999]. a b c
Institut National de Recherche en Informatique et en Automatique. Lab. d'Informatique GRAphique, VIsion et Robotique. Institut National de REcherche sur les Transports et leur Securite.
Path Planning with Uncertainty for Car-Like Robots Th. Fraichard and R. Mermond
InriaRh^one-Alpes & Graviry Inria. Zirst. 655 av. de l'Europe. 38330 Montbonnot Saint Martin, France
Abstract | this paper presents the rst path planner taking into account both non-holonomic and uncertainty constraints. The case of a car-like robot subject to cumulative and unbounded con guration uncertainty related to bounded control and sensing errors is considered. Assuming the existence of landmarks allowing the robot to relocalize itself in particular places, we present an algorithm that computes paths that are both feasible and robust. In other words, they respect the non-holonomic constraints of a car-like robot and the robot is assured to reach its goal by following them as long as its control and sensing errors remain within bounded sets.
wheeled mobile robots, whereas uncertainty has been of primary concern in assembly tasks (the `peg-in-hole' problem). Uncertainty is also a challenging problem in the context of path planning for mobile robots. Yet, despite the fact that most wheeled mobile robots are subject to nonholonomic constraints, previous research works on path planning with uncertainty for mobile robots never considered non-holonomic constraints. It is therefore our purpose to address robust non-holonomic path planning, i.e. path planning with both uncertainty and non-holonomic constraints. The case of a car-like robot moving in a stationary workspace is considered. A car-like robot is the typical non-holonomic robot: at each time instant, it can only move forward and backward in a direction tangent to its main axis; besides its turning radius is lower bounded. The robot is subject to cumulative and unbounded con guration uncertainty related to bounded control and sensing errors. To overcome the cumulative nature of the con guration uncertainty, the existence of landmarks has to be assumed (otherwise, it is impossible to reach distant goals). Broadly speaking, a landmark is any device that allows the robot to reduce its con guration uncertainty, e.g. GPS, triangulation systems, beacons, etc. In this framework, this paper presents a robust nonholonomic path planner. To the best of our knowledge, it is the rst time that the combined complexity of both non-holonomic and uncertainty constraints are considered simultaneously. The contribution of this paper is the following: a model of the evolution of the con guration uncertainty for a car-like robot is given. This model is then used to develop a `local' robust non-holonomic path planner; this planner is local in the sense that it does not aim at being complete (for the sake of eciency, the search for a solution path is restricted to a nite and discrete subset of the whole set of solution paths). The local planner does not take landmarks into account. However it is shown how to embed it in a global path planning scheme such as the Ariadne's Clew Algorithm [3], or the Probabilistic Path Planner [10] in order to obtain a general and complete robust non-holonomic path planner.
1 Introduction
Path planning for robots is a fundamental issue in Robotics (cf. [12]). The purpose of a path planner is to compute a path, i.e. a continuous sequence of con gurations,1 that leads the robot to its goal. The primary concern of path planning is to compute collision-free paths. Several extensions to the basic path planning problem have been considered. One of them regards the non-holonomy problem. This problem arises when a robot is subject to non-holonomic kinematic constraints that restrict the set of its admissible directions of motion (a wheel is a non-holonomic system; it can only move in a direction perpendicular to its axle). In this case, the planned paths must be feasible also, i.e. respect these non-holonomic constraints. Another key issue in path planning is the uncertainty problem. It is related to the various sources of uncertainty a ecting the actual robot (control and sensing errors, inaccurate models of the environment, etc.). These uncertainties may lead to failures during the execution of a path preventing the robot from ever reaching its goal. A solution to this problem is to design a path planner that explicitly takes into account uncertainty so as to compute robust paths, i.e. paths that guarantee that the goal will be reached in spite of these uncertainties. As shown in the literature review of x2, non-holonomy and uncertainty have been addressed separately to a large extent. Non-holonomy is a key issue in path planning for
Inst. Nat. de Recherche en Informatique et en Automatique. y Lab. d'Informatique GRAphique, VIsion et Robotique. 1 The con guration of a robot is a set of independent parametres that uniquely describes the position and orientation of every part of the robot.
2 Background
Uncertainty appeared in path planning as early as 1976 [20]. Ever since, it has motivated a lot of research
1
works, especially for assembly tasks. Non-holonomy was introduced later for mobile robots [13], and it also formed the basis of a large body of research. The reader is referred to [12] for more details on this two topics. As for non-holonomic path planning, following a result established in [6], most of the existing path planners compute paths made up of straight segments connected with tangent circular arcs of minimum radius, e.g. [2, 9, 14]. As for uncertain path planning, the seminal preimage concept introduced in [16] has been applied to mobile robots, e.g. [1, 15]. Other approaches have been developed also, e.g. [4, 5, 19]. However, the computational issues involved have restricted these works to simple cases (point robots performing straight motions) and, to the best of our knowledge, the combined complexity of both uncertain and non-holonomic constraints has not been fully addressed before.
(q , u )
x
q d
x
q
d
y y Fig. 2: (q , u ) with u = (d; ), represented in C (left) and in W (right).
A non-deterministic (or bounded-set) representation was chosen to model the uncertainty [7]: the con guration uncertainty of A is characterized by two parametres that respectively represent the uncertainty on the position of A and on its orientation, namely u = (d; ) 2 IR S1 . Accordingly (q , u ), i.e. the set of possible con gurations for A when its nominal con guration is q and its uncertainty u , is a cylindrical region of C de ned as (Fig. 2): (q; u ) = fqp j dist(q; qp )
