this paper is as follows: in Section 2, we collect some mathematical preliminaries from the literature on controllability of nonlinear systems and on classification of free Lie algebras. These are drawn from classical references in control theory [4, 17, 18, 36, 40] and Lie algebras [15, 43]. In Section 3, using some outstanding results of Brockett on optimal steering of certain classes of systems as motivation [5], we discuss the use of sinusoidal inputs for steering systems of first order, i.e., systems where controllability is achieved after just one level of Lie brackets of the input vector fields. Section 4 attempts to expand the domain of applicability of these results to more complex systems, where several orders of Lie brackets are needed to obtain the full Lie algebra associated with the input distribution. The 4 MURRAY AND SASTRY

The problem of synthesizing and analyzing collective autonomous agents has only recently begun to be practically studied by the robotics community. This paper overviews the most prominent directions of research, defines key terms, and summarizes the main issues. Finally, it briefly describes our approach to controlling group behavior and its relation to the field as a whole.

This paper describes a practical path planner for nonholonomic robots in environments with obstacles. The planner is based on building a one-dimensional, maximal clearance skeleton through the configuration space of the robot. However rather than using the Euclidean metric to determine clearance, a special metric which captures information about the nonholonomy of the robot is used. The robot navigates from start to goal states by loosely following the skeleton; the resulting paths taken by the robot are of low "complexity." We describe how much of the computation can be done off-line once and for all for a given robot, making for an efficient planner. The focus is on path planning for mobile robots, particularly the planar two-axle car, but the underlying ideas are quite general and may be applied to planners for other nonholonomic robots.

In the first part of this paper, we dene approximate polynomial gcds (greatest common divisors) and extended gcds provided that approximations to the zeros of the input polynomials are available. We relate our novel definition to the older and weaker ones, based on perturbation of the coefficients of the input polynomials, we demonstrate some deficiency of the latter definitions (which our denition avoids), and we propose new effective sequential and parallel (RNC and NC) algorithms for computing approximate gcds and extended gcds. Our stronger results are obtained with no increase of the asymptotic bounds on the computational cost. This is partly due to application of our recent nearly optimal algorithms for approximating polynomial zeros. In the second part of our paper, working under the older and more customary definition of approximate gcds, we modify and develop an alternative approach, which was previously based on the computation of the Singular Value Decomposition (SVD) of the associat...

This work considers the path planning problem for planar revolute manipulators operating in a workspace of polygonal obstacles. This problem is solved by determining the topological characteristics of obstacles in configuration space, thereby determining where feasible paths can be found. A collision-free path is then calculated by using the mathematical description of the boundaries of only those configuration space obstacles with which collisions are possible. The key to this technique is a simple test for determining whether two disjoint obstacles are connected in configuration space. This test allows the path planner to restrict its calculations to regions in which collisionfree paths are guaranteed a priori, thus avoiding unnecessary computations and resulting in an efficient implementation. Typical timing results for environments consisting of four polyhedral obstacles comprised of a total of 27 vertices are on the order of 22 ms on a SPARC-IPC workstation. I. Introduction The p...

Exact computation is an important paradigm for the implementation of geometric algorithms. In this paper, we consider for the first time the practically important problem of collision detection under this aspect. The task is to decide whether a polyhedral object can perform a prescribed sequence of translations and rotations in the presence of stationary polyhedral obstacles. We present an exact decision method for this problem which is purely based on integer arithmetic. Our approach guarantees that the required binary length of intermediate numbers is bounded by 14L+ 22, where L denotes the maximal bit-size of any input value. 1 Introduction Exact computation is widely recognized as one of the key issues in the design of geometric algorithms in the near future. Recent work on exact algorithms focuses on traditional problems of computational geometry, such as the construction of Voronoi diagrams [9, 1, 6]. However, there are reasons for believing that exact computation will also be a...

We present an algebraic algorithm to generate the exact general sweep boundary of a 2D curved object which changes its shape dynamically while moving along a parametric curve trajectory.

Demining and unexploded ordnance (UXO) clearance are extremely tedious and dangerous tasks. The use of robots bypasses the hazards and potentially increases the efficiency of both tasks. A first crucial step towards robotic mine/UXO clearance is to locate all the targets. This requires a path planner that generates a path to pass a detector over all points of a mine/UXO field, i.e., a planner that is complete. The current state of the art in path planning for mine/UXO clearance is to move a robot randomly or use simple heuristics. These methods do not possess completeness guarantees which are vital for locating all of the mines/UXOs. Using such random approaches is akin to intentionally using imperfect detectors. In this paper, we first overview our prior complete coverage algorithm and compare it with randomized approaches. In addition to the provable guarantees, we demonstrate that complete coverage achieves coverage in shorter time than random coverage. We also show that the use of complete approaches enables the creation of a filter to reject bad sensor readings, which is necessary for successful deployment of robots. We propose a new approach to handle sensor uncertainty that uses geometrical and topological features rather than sensor uncertainty models. We have verified our results by performing experiments in unstructured indoor environments. Finally, for scenarios where some a priori information about a minefield is available, we expedite the demining process by introducing a probabilistic method so that a demining robot does not have to perform exhaustive coverage.