Probabilistic roadmap planning methods have been shown to perform well in a number of practical situations, but their performance degrades when paths are required to pass through narrow passages in the free space. We propose a new method of sampling the configuration space in which randomly generated configurations, free or not, are retracted onto the medial axis of the free space. We give algorithms that perform this retraction while avoiding explicit computation of the medial axis, and we show that sampling and retracting in this manner increases the number of nodes found in small volume corridors in a way that is independent of the volume of the corridor and depends only on the characteristics of the obstacles bounding it. Theoretical and experimental results are given to show that this improves performance on problems requiring traversal of narrow passages. 1

Several motion planning methods using networks of randomly generated nodes in the free space have been shown to perform well in a number of cases, however their performance degrades when paths are required topass through narrow passages in the free space. In [16] weproposed MAPRM, a method of sampling the con guration space in which randomly generated con gurations, free or not, are retracted onto the medial axis of the free space without having to rst compute the medial axis; this was shown to increase sampling in narrow passages. In this paper we give details of the MAPRM algorithm for the case of a free- ying rigid body moving in three dimensions, and show that the retraction may be carried out without explicitly computing the Cobstacles or the medial axis. We give theoretical arguments to show that this improves sampling in narrow corridors, and present preliminary experimental results comparing the performance to uniform random sampling from the free space.

We propose a new approach to solving the point-topoint inverse kinematics problem for highly redundant manipulators. It is inspired by recent motion planning research and explicitly takes into account constraints due to joint limits and self-collisions. Central to our approach is the novel notion of kinematic roadmap for a manipulator. The kinematic roadmap captures the connectivity of the configuration space of a manipulator in a finite graph like structure. The standard formulation of inverse kinematics problem is then solved using this roadmap. Our current implementation, based on Ariadne's Clew Algorithm [BA 93], is composed of two sub-algorithms: EXPLORE, an appealingly simple algorithm that builds the kinematic roadmap by placing landmarks in the configuration space

This paper addresses the problem of generating smooth trajectories between an initial and a final position and orientation in space. The main idea is to define a functional depending on velocity or its derivatives that measures the smoothness of a trajectory and find trajectories that minimize this functional. In order to ensure that the computed trajectories are independent of the parameterization of positions and orientations, we use the notions of Riemannian metric and covariant derivative from differential geometry and formulate the problem as a variational problem on the Lie group of spatial rigid body displacements, SE(3). We show that by choosing an appropriate measure of smoothness, the trajectories can be made to satisfy boundary conditions on the velocities or higher order derivatives. Dynamically smooth trajectories can be obtained by incorporating the inertia of the system into the definition of the Riemannian metric. We state the necessary conditions for the shortest dista...

Nonlinear control of mechanical systems is a challenging discipline that lies at the intersection between control theory and geometric mechanics. This thesis sheds new light on this interplay while investigating motion control problems for Lagrangian systems. Both stability and motion planning aspects are treated within a unified framework that accounts for a large class of devices such as robotic manipulators, autonomous vehicles and locomotion systems. One distinguishing feature of mechanical systems is the number of control forces. For systems with as many input forces as degrees of freedom, many control problems are tractable. One contribution of this thesis is a set of trajectory tracking controllers designed via the notions of configuration and velocity error. The proposed approach includes as special cases a variety of results on joint and workspace control of manipulators as well as on attitude and position control of vehicles. Whenever fewer input forces are available than deg...

Abstract not found

This thesis is about force controlled compliant robot motion, with the emphasis on: 1) mod- elling of arbitrary and time-varying contact situations between a rigid manipulated object and rigid objects in its environment, 2) motion specification in terms of allowed velocities and ac- celerations for the manipulated object, maintaining the contact with the physical constraints but without generating too large contact forces, and 3) on-line identification of uncertainties in the instantaneous geometric parameters of the motion constraint model, i.e., the position of the contact points, the direction of the contact normal, and the local curvature parameters. Requirements for generality, simplicity and robustness have guided the research work.

The set of spatial rigid body motions forms a Lie group known as the special Euclidean group in three dimensions, SE(3). Chasles's theorem states that there exists a screw motion between two arbitrary elements of SE(3). In this paper we investigate whether there exist a Riemannian metric whose geodesics are screw motions. We prove that no Riemannian metric with such geodesics exists and we show that the metrics whose geodesicsare screwmotions form a two-parameter family of semi-Riemannian metrics. 1 INTRODUCTION The set of all three-dimensional rigid body displacements forms a Lie group [4, 5]. This group is generally referred to as SE(3), the special Euclidean group in three dimensions. The tangent space at the identity endowed with the Lie bracket operation has the structure of a Lie algebra and is denoted by se(3). It is isomorphic to the set of all twists and the Lie bracket of two twists corresponds to the motor product of the respective motors. A good discussion on the geometr...

We develop a geometric framework for the stability analysis of multifingered grasps and propose a measure of grasp stability for arbitrary perturbations and loading conditions. The measure requires a choice of metric on the group of rigid body displacements. We show that although the stability of a grasp itself does not depend on the choice of metric, comparison of the stability of different grasps depends on the metric. Finally, we provide some insight into the choice of metrics for stability analysis. 1

The set of rigid body motions forms the Lie group SE(3), the special Euclidean group in three dimensions. In this paper we investigate Riemannian metrics and affine connections on SE(3) that are suited for kinematic analysis and robot trajectory planning. In the first part of the paper, we study metrics whose geodesics are screw motions. We prove that no Riemannian metric can have such geodesics and we show that metrics whose geodesics are screw motions form a two-parameter family of semi-Riemannian metrics. In the second part of the paper we investigate affine connections which through the covariant derivative give the expression for the acceleration of a rigid body that agrees with the expression used in kinematics. We prove that there is a unique symmetric connection with this property. Further, we show that there is a family of Riemannian metrics that are compatible with such a connection. These metrics are products of the bi-invariant metric on the group of rotations and a positiv...