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Exact Motion Planning for Tractor-Trailer Robots
 Petr Svestka Jules Vleugels

Department of Computer Science, Utrecht University P.O. Box 80.089, 3508 TB Utrecht, the Netherlands Email: fpetr,

A tractor-trailer robot consists of a carlike tractor towing a passive trailer. Due to its highly nonholonomic nature, the kinematics of this type of robot are complicated anddi cult to compute. We present exact closed-form solutions for the kinematic parameters of a tractortrailer robot and use them to construct an exact and e cient motion planner in the absence of obstacles. This local planner can be employed in a probabilistic global planner, which allows us to plan motions in the presence of obstacles.


1 Introduction
The motion planning problem iswell-known in the eld of robotics. Its objective is to nd collision-free paths for a robot moving amidst a set of obstacles. For free- ying robots, i.e., robots without restrictions on the motions they can perform, the motion planning problem is typically solved by computing a path in the free con guration space; such a path corresponds to a feasible free path in the workspace. This approach howeveris not possible for some motion planning problems, such as problems involving nonholonomic systems. In this case the robot can perform only restricted motions even in the absence of obstacles, which means that a path in the free con guration space is not necessarily feasible in the workspace. We refer to Laumond 5 for an introduction to nonholonomic motion planning; a good overview of the generalmotion planning problem was given by Latombe 4 . This paper investigates a particular and well-known nonholonomic system: the tractor-trailer robot. Informally, this robot consists of a carlike tractor towing a passive trailer. The tractor can perform motions that are similar to those of a car: it drives forwards or backwards while possibly steering left or right. The trailer follows the paththat is dictated by the motion of the tractor. In Section 2 we rst formally describe the tractor-trailer robot and discuss its nonholonomic constraints. We give closed-form solutions for its kinematic parameters and show how we can explicitly express its con guration at any given time instant. This makes it possible to construct an exact motion planning algorithm for the tractor-trailer robot in theabsence of obstacles, which is described in Section 3. Then in Section 4 we brie y discuss how this local planner can be employed in a probabilistic framework, which allows us to plan the motion of a tractor-trailer robot in the presence of obstacles. Finally, in Section 5 we give some conclusions and indicate open problems and directions of future work.

1.1 Related work

Due to the complexnature of the tractor-trailer robot, most work has focussed on obtaining approximations for its kinematics. The eld of Control Theory has produced some recent results 2, 9, 10, 11 on the n-trailer system a tractor towing n passive trailers of which the tractortrailer robot is a special case, but these approaches are not exact though a con guration can be

research was partiallysupported by ESPRIT Basic Research Action No. 6546 project PROMotion.


R2 L2 


F1 L1

R1 = F2

Figure 1: A model for the tractor-trailer robot. approximated arbitrarily close and or not e cient. Furthermore, they tend to have problems with incorporating possibly complex obstacles. Barraquand and Latombe 1 propose a heuristic brute-force approach to motion planning fortractor-trailer robots. It consists of heuristically searching a graph whose nodes are small axis-parallel cells in con guration space. Two such cells are connected in the graph if there exists a feasible path between two con gurations in the respective cells. The main drawback of their method is that when the heuristics fail it requires an exhaustive search in the discretized con guration space....
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