The Mixed Approach for Motion Planning: Learning Global Strategies from a Local Planner

In this paper, we propose a mixed approach for motion planning that decomposes the problem into two levels. At the global level, we build a graph whose nodes represent relatively large cells of the Configuration Space of the robotic system. Adjacent cells are connected by edges weighted by the probability for the local planner to succeed in computing a trajectory from a point in one cell to a goal in the other. These probabilities are used by a minimum cost path finding algorithm to generate subgoals for the local planner. They are updated using a Bayesian rule from the results of the execution of planned trajectories at the local level. At the global level, no geometric information is stored, thus eliminating the expensive transformation of obstacles into the Configuration Space needed by usual global methods. We take advantage of the ability of our local planner to move close to obstacles so that only a crude discretization of the Configuration Space is needed. This makes it possible to apply this technique to robotic systems with a large number of degrees of freedom. In mobile robot applications, sensors being used by the local planner, this method achieves the learning of planning strategies in an unknown environment without building a complete geometric model of the world. 1 . I n t r o d u c t i o n 1.1. Position of the Problem The general problem of motion planning can be stated as follows : given an algebraic description of the boundary of the moving objects and of the obstacles in Cartesian Space, find a collision free path for the moving objects from a set of initial positions to a set of goals. This general problem can be decomposed in a variety of sub-problems, according to whether the description of the world is planar or three-dimensional, whether there is only one moving object or many of them, whether there are connected by revolute or prismatic joints to form articulated chains or not. Theoretical issues posed by this problem have been thoroughly examined by Schwartz and Sharir [Sc]. They propose an algorithm answering the general motion planning problem, based on Tarski's algorithm for deciding statements in the quantified elementary theory of real numbers. Their algorithm, which is not of direct practical application, is polynomial in the number of constraints describing the obstacles, but exponential in the number of degrees of freedom. Other works suggest the inherent exponential complexity of planning. An early result by Reif [Re] proves PSPACE hardness of a special instance of the planning problem. Other special cases where in turn examined by Hopcroft, Joseph and Whitesides [Ho], Spirakis and Yap [Sp] among others. Few practical algorithms have nevertheless been proposed for the case of highest interest, that of an articulated chain of solids in a three dimensional environment. Algorithms developed by Lozano-Perez [Lo] Faverjon [Fa84, Fa86]t are based on a representation of obstacles in the Configuration Space of the manipulator. The Configuration Space of a system is any set of independent parameters that enables to describe the position of all points bound to the moving bodies. As a straightforward translation of the description of obstacles from Cartesian Space to Configuration Space is not possible for manipulators, these algorithms rely on a subdivision of configuration parameters in small ranges. This provides a grid whose cells are either labelled as free, or intersecting Configuration Obstacles, the transform of obstacles in Configuration Space. A path is searched as a sequence of nodes in the graph describing the connectivity between free regions of Configuration Space. Such a description requires a memory space exponential in the number of configuration parameters, and again, planning of a path is exponential in the number of configuration parameters. This imposes a practical limitation on the number of degrees of freedom involved: known algorithms limit themselves to planning a global path for the first three joints of a manipulator, while a heuristic approach is used for motions of the hand. On the other hand, local methods have proved to be very powerful for computing motion of a manipulator. Local information on the environment only is used to compute displacements at any time, without keeping track of any landmarks. The so-called Potential Field Method relies on a minimization including a term attracting the manipulator towards the goal, and repulsive terms that push bodies of the manipulator away from the obstacles. In [FT87] we propose an alternative to the Potential Field Method for locally computing trajectories. A task is expressed by the minimization of the relevant measures of the problem written as a function of configuration parameters. Moving objects have a simplified local view of the environment as planes separating them from the obstacles, that are transformed into linear constraints in Configuration Space. 1.2. Overview of the Approach In this paper we propose to uncouple the general problem of path planning into a low complexity local planner and a higher complexity global planner working on a graph of cells representing relatively large sets of configuration parameters. The main idea underlying this approach Faverjon and Tournassoud 1131 is that we want to turn the power of local methods to profit so as to deal with the high complexity of global planning only at the relevant level of description. At the higher level no geometric description of the obstacles is used but only weigths indicating the probability for a trajectory computed locally not to lead to any Given initial and goal configurations, a classical minimum cost path finding algorithm in the graph yields a list of cells giving the global shape of the path joining the node containing the init ial configuration to the node containing the goal. Then the robot, starting from its initial position, describes a trajectory computed by the local planner, taking as subgoal a point located inside the next cell on the path. During the execution of the path, the weights are updated using a model of learning from results of motions generated by the local planner. The robot eventually reaches the final goal, or there is failure, meaning the robot is blocked while aiming at some cell. In this case, we put a higher weight to the corresponding transition and compute again a global path from current configuration, based on the updated weights. This produces a new path avoiding the problematic arc. Let us underline that this approach is relevant only in the case it is performed using a loose grid in Configuration Space, or we again deal with a space of small cells either occupied or free, which is not better than the global approach in terms of computational cost Dealing with a loose graph is only made possible because of the intrinsic power of the local planner that produces long pieces of collision free trajectory. Our approach presents other interesting features. First local computation can be based on a local model of the visible obstacles acquired through proximity sensors such as ultrasonic sensors, or a stereo vision system, as a substitute to a complete geometric model of the environment. Transitory mobile obstacles wi l l in particular be taken into account by the local planner. Figure ( la ) shows the trajectory of a mobile robot in an unkown environment using this approach. Figure ( l b ) shows the trajectory obtained with the same initial and goal positions when we take into account the knowledge obtained from the first execution. The following sections describe in more details the different parts of this approach, namely the Sute Graph, global planning, local planning, and the learning process. Figure 1. a) Before learning, b) After learning.