Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

Planning Algorithms: Introductory Material

A new motion planning method for robots in static workspaces is presented. This method proceeds in two phases: a learning phase and a query phase. In the learning phase, a probabilistic roadmap is constructed and stored as a graph whose nodes correspond to collision-free configurations and whose edges correspond to feasible paths between these configurations. These paths are computed using a simple and fast local planner. In the query phase, any given start and goal configurations of the robot are connected to two nodes of the roadmap; the roadmap is then searched for a path joining these two nodes. The method is general and easy to implement. It can be applied to virtually any type of holonomic robot. It requires selecting certain parameters (e.g., the duration of the learning phase) whose values depend on the scene, that is the robot and its workspace. But these values turn out to be relatively easy to choose, Increased efficiency can also be achieved by tailoring some components of the method (e.g., the local planner) to the considered robots. In this paper the method is applied to planar articulated robots with many degrees of freedom. Experimental results show that path planning can be done in a fraction of a second on a contemporary workstation (/spl ap/150 MIPS), after learning for relatively short periods of time (a few dozen seconds).

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Probabilistic roadmaps for path planning in high-dimensional configuration spaces

During the last decade, sampling-based path planning algorithms, such as probabilistic roadmaps (PRM) and rapidly exploring random trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g. as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g. showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e. such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.

Sampling-based algorithms for optimal motion planning

During the last decade, sampling-based path planning algorithms, such as Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g., as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g., showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e., such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.

Sampling-based Algorithms for Optimal Motion Planning

This paper presents algorithms for computing constraints on the position of an object due to the presence of ther objects. This problem arises in applications that require choosing how to arrange or how to move objects without collisions. The approach presented here is based on characterizing the position and orientation of an object as a single point in a configuration space, in which each coordinate represents a degree of freedom in the position or orientation of the object. The configurations forbidden to this object, due to the presence of other objects, can then be characterized as regions in the configuration space, called configuration space obstacles. The paper presents algorithms for computing these configuration space obstacles when the objects are polygons or polyhedra.

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Spatial Planning: A Configuration Space Approach

This paper describes a collision avoidance algorithm for planning a safe path for a polyhedral object moving among known polyhedral objects. The algorithm transforms the obstacles so that they represent the locus of forbidden positions for an arbitrary reference point on the moving object. A trajectory of this reference point which avoids all forbidden regions is free of collisions. Trajectories are found by searching a network which indicates, for each vertex in the transformed obstacles, which other vertices can be reached safely.

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An algorithm for planning collision-free paths among polyhedral obstacles

This paper describes an experimental very high level programming system for computer controlled mechanical assembly. AUTOPASS (AuTomated Parts Assembly System). The AUTOPASS language is oriented towards objects and assembly operations, rather than motions of mechanical assembly machines. It is intended to enable the user to concentrate on the overall assembly sequence and to program with English-like statements using names and terminology that are familiar to him. To relate assembly operations to manipulator motions, the AUTOPASS compiler uses an internal representation of the assembly world. This representation consists of a geometric data base generated prior to compilation and updated during compilation; it thus represents the state of the world at each assembly step. The level of the language has been chosen to provide a high degree of assistance to the user without the system's having to perform artificial intelligence type problem solving operations.

AUTOPASS: an automatic programming system for computer controlled mechanical assembly

Very high level languages for describing mechanical assembly require a representation of the geometric and physical properties of 3-D objects including parts, tools, and the assembler itself. This paper describes a geometric modeling system that generates a data base in which objects and assemblies are represented by nodes in a graph structure. The edges of the graph represent relationships among objects such as part-of, attachment, constraint, and assembly. The nodes also store positional relationships between objects and physical properties such as material type. The user designs objects by combining positive and negative parameterized primitive volumes, for example, cubes and cones, which are represented internally as polyhedra. The data base is built by invoking a procedural representation of the primitive volumes, which generates vertex, edge, and surface lists of instances of the volumes. Several applications in the automatic assembly domain have been implemented using the geometric modeling system as a basis.

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A geometric modeling system for automated mechanical assembly

In an earlier paper, the authors presented an algorithm for finding all polyhedral solid objects with a given set of vertices and straight line edges (its wire frame). This paper extends the Wire Frame algorithm to find all solid polyhedral objects with a given set of two dimensional projections. These projections may contain depth information in the form of dashed and solid lines, may represent cross sections, and may be overall or detail views. The choice of labeling conventions in the projections determines the difficulty of the problem. It is shown that with certain conventions and projections the problem of fleshing out projections essentially reduces to the problem of fleshing out wire frames. Even if no labeling is used, the Projections algorithm presented here finds all solutions even though it is possible to construct simple examples with a very large number of solutions. Such examples have a large amount of symmetry and various accidental coincidences which typically do not occur in objects of practical interest. Because of its generality, the algorithm can handle pathological cases if they arise. This Projections algorithm, which has applications in the conversion of engineering drawings in a Computer Aided Design, Computer Aided Manufacturing (CADCAM) system, has been implemented. The algorithm has successfully found solutions to problems that are rather complex in terms of either the number of possible solutions or the inherent complexity of projections of objects of engineering interest.

Fleshing out projections

Given a polyhedral object, its wire frame is the set of its edges and vertices. In this paper we present an algorithm which discovers all objects with a given wire frame. This algorithm, which has a number of applications to mechanical design besides being of mathematical interest, has been implemented and has performed well on complex objects.

Michael A. Wesley

Papers

An algorithm for planning collision-free paths among polyhedral obstacles

AUTOPASS: an automatic programming system for computer controlled mechanical assembly

A geometric modeling system for automated mechanical assembly

Fleshing out projections

Fleshing out wire frames