Primitives for Smoothing Mobile zyx Robot Trajectories zy * zyx Sara Fleury, Philippe Soukres, Jean-Paul Laumond, Raja Chatila LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse, France e-mail : {Sara, soueres, jpl, raja}Blaas.fr Abstract: zyxwvutsrq Clothoids are very useful zyxwvuts for smooth- ing the motion of a mobile robot mowing along zyxwvuts a tra- jectory. This paper addresses the problem of smooth- ing the mobile robot motions when cusps, i e . , changes of motion direction along the trajectory, are imposed. We pinpoint some special curves (that we call uanti- clothoids”) and we discuss how they can be used to- gether with clothoids in order to smooth a predefined trajectory. 1 Introduction Clothoids (or Cornu spirals) are known as very use- ful curves to smooth trajectories. Their equation is n = k,s+no where K is the curvature, s the arc length, no the initial curvature and kc a constant characteriz- ing the shape of the clothoid. Clothoids allow to link curves of infinite radius of curvature (i.e., lines) and curves of finite radius of curvature, with a continuous change of the curvature. Clothoids have practical ap- plications in railway and highway design. They have been introduced in Robotics zyxwvuts ([5, 2, 91) for smoothing the motions of a mobile robot moving “forward” on a broken line (i.e., without a change of orientation along the trajectory). This paper pinpoints the involutes of circles’. The natural equation of an involute of a circle is p2 = 2k,s, p being the radius of curvature and k, the radius of the circle. Another general form of this equation is p = k,(B - Bo) + po where B is the direction of the tangent, Bo the initial direction of the curve2 and po the initial radius of curvature. k, is the characterlstic parameter of the involute. Such curves allow to link “curves” of infinite curvature (i.e., curves reduced to a point) and curves of finite curvature, with a contin- uous change of the curvature. This property of the circle involutes leads us to call them anticlothoids in the context of this paper. Like clothoids, anti- clothoids are the time-optimal trajectories of a two driving wheels mobile robot 161. Both types of curves are dual from the point of view of control. They are produced by applying respectively a same constant ac- celeration on both wheels (anticlothoids) or constant *This work is supported by the ECC Esprit 3 Project 6546 ‘The involute of a circle is the curve described by the end of 2As s is a measure of the length of the curve, (0 - 00) me.+ PROMotion. a thread as it is unwound from a stationary spool [q sues the angular variation of the me. and opposite accelerations (clothoids). The purpose of this paper is to show how to use anticlothoids in order to smooth the motions of a mobile robot when cusps are imposed (i.e., when the robot has to change the direction of its motion). We shall first overview some known results on tra- jectory smoothing, mainly using clothoids ($2). Then we introduce both types of curves from a control the- ory viewpoint, and we show how a mobile robot can execute motions supported by them (f3). Geometric properties are then proven (f4). Two connected oriented straight line segments being given as a reference trajectory, we show how to plan a mo- tion such that: 1. the velocities of the driving wheels are continuous and never simultaneously null (i.e the motion is smooth), 2. the trajectory never lies farther than some fixed threshold from the condition is r e quired for collision 3. the produced trajectory respects the imposed di- rections of motion. The trajectories will consist of sequences of clothoids and anticlothoids. In a last section (f5), we shall conclude by sketch- ing a method for smoothing a mobile robot’s trajec- tory consisting of a polygonal line, while accounting for non-collision and direction changes. 2 Related work Clothoids are extensively used for the design of highways 41 in order to smoothly join straight lines in Computer Aided Design [CAD) [12]. They were more recently introduced in Robotics. Indeed, most trajectory planners produce trajectories consisting of straight lines and turns that force the robot to stop, because of the discontinuity in the an- gular speed when it has to chan e direction. In order to eliminate these stops, smoot%ing of the reference trajectory to produce a swift motion has been a long- time objective, for which the use of clothoids is very popular. The smoothing problem was also addressed for pro- ducing directly a trajectory that joins a sequence of robot configurations (2, y, 0)i smoothly, i.e., without stopping. with circu I ar portions. The are also used as splines 832 1050-4729/93 $3.00 0 1993 IEEE