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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

this paper, which presents randomized, algorithmic techniques for path planning that are particular suited for problems that involve dierential constraints.

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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.

the author and should not be interpreted as representing the o cial policies, either expressed or A good model of the mechanics of a task is a resource for a robot, just as actuators and sensors are resources. The e ective use of frictional, gravitational, and dynamic forces can substitute for extra actuators; the expectation derived from a good model can minimize sensing requirements. Despite this, most robot systems attempt to dominate or nullify task mechanics, rather than exploit them. There has been little e ort to understand the manipulation capabilities of even the simplest robots under more complete mechanics models. This thesis addresses that knowledge de cit by studying graspless or nonprehensile manipulation. Nonprehensile manipulation exploits task mechanics to achieve a goal state without grasping, allowing simple mechanisms to accomplish complex tasks. With nonprehensile manipulation, a robot can manipulate objects too large or heavy to be grasped and lifted, and a low-degree-of-freedom robot can control more degrees-of-freedom of an object by allowing relative motion between the object and the manipulator. Two key problems are determining controllability of and motion planning for

This paper aims at studying the trajectory planning for a car -- i.e. a non holonomic vehicle whose turning radius is lower bounded -- in a static and structured world. As for the structure of the world, we assume the existence of natural lanes within which the vehicle is able to move. The contribution of this paper is a smooth trajectory planner which, when given the polygonal line S representing the spine of a lane generates a trajectory C avoiding the obstacles of the world and which is smooth -- i.e. without backing up maneuvers -- and executable by the vehicle according to its own kinematic constraints. Besides C is topologically equivalent to S -- i.e. C must remain in the lane defined by S. C is made up of straight segments and circular arcs.

Motivated by applications in robotics, we formulate the problem of minimizing the total angle cost of a TSP tour for a set of points in Euclidean space, where the angle cost of a tour is the sum of the direction changes at the points. We establish the NP-hardness of both this problem and its relaxation to the cycle cover problem. We then consider the issue of designing approximation algorithms for these problems and show that both problems can be approximated to within a ratio of O(log n) in polynomial time. We also consider the problem of simultaneously approximating both the angle and the length measure for a TSP tour. In studying the resulting tradeoff, we choose to focus on the sum of the two performance ratios and provide tight bounds on the sum. Finally, we consider the extremal value of the angle measure and obtain essentially tight bounds for it. In this extended abstract we restrict our attention to the planar setting, but all our results are easily extended to higher dimensio...

Abstract. Although routing is a well-studied problem in various contexts, there remain unsolved problems in routing edges for graph layouts. In contrast with techniques from other domains such as VLSI CAD and robotics, where physical constraints play a major role, aesthetics play the more important role in graph layout. For graphs, we seek paths that are easy to follow and add meaning to the layout. We describe a collection of aesthetic attributes applicable to drawing edges in graphs, and present a general approach for routing individual edges subject to these principles. We also give implementation details and survey di culties that arise in an implementation. 1

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