Elon Rimon

Exact robot navigation using artificial potential functions

A methodology for exact robot motion planning and control that unifies the purely kinematic path planning problem with the lower level feedback controller design is presented. Complete information about a freespace and goal is encoded in the form of a special artificial potential function, called a navigation function, that connects the kinematic planning problem with the dynamic execution problem in a provably correct fashion. The navigation function automatically gives rise to a bounded-torque feedback controller for the robots actuators that guarantees collision-free motion and convergence to the destination from almost all initial free configurations. A formula for navigation functions that guide a point-mass robot in a generalized sphere world is developed. The simplest member of this family is a space obtained by puncturing a disk by an arbitrary number of smaller disjoint disks representing obstacles. The other spaces are obtained from this model by a suitable coordinate transformation. Simulation results for planar scenarios are provided. >

Robot navigation functions on manifolds with boundary

This paper concerns the construction of a class of scalar valued analytic maps on analytic manifolds with boundary. These maps, which we term navigation functions, are constructed on an arbitrary sphere world-a compact connected subset of Euclidean n-space whose boundary is formed from the disjoint union of a finite number of (n - l)-spheres. We show that this class is invariant under composition with analytic diffeomorphisms: our sphere world construction immediately generates a navigation function on all manifolds into which a sphere world is deformable. On the other hand, certain well known results of S. Smale guarantee the existence of smooth navigation functions on any smooth manifold. This suggests that analytic navigation functions exist, as well, on more general analytic manifolds than the deformed sphere worlds we presently consider.

Spanning-tree based coverage of continuous areas by a mobile robot

This paper considers the problem of covering a continuous planar area by a square-shaped tool attached to a mobile robot. Using a tool-based approximation of the work-area, we present an algorithm that covers every point of the approximate area for tasks such as floor cleaning, lawn mowing, and field demining. The algorithm, called Spanning Tree Covering (STC), subdivides the work-area into disjoint cells corresponding to the square-shaped tool, then follows a spanning tree of the graph induced by the cells, while covering every point precisely once. We present and analyze three versions of the STC algorithm. The first version is off-line, where the robot has perfect apriori knowledge of its environment. The off-line STC algorithm computes an optimal covering path in linear time O(N), where N is the number of cells comprising the approximate area. The second version of STC is on-line, where the robot uses its sensors to detect obstacles and construct a spanning tree of the environment while covering the work-area. The on-line STC algorithm completes an optimal covering path in time O(N), but requires O(N) memory for its implementation. The third version of STC is “ant”-like. In this version, too, the robot has no apriori knowledge of the environment, but it may leave pheromone-like markers during the coverage process. The ant-like STC algorithm runs in time O(N), and requires only O(1) memory. Finally we present simulation results of the three STC algorithms, demonstrating their effectiveness in cases where the tool size is significantly smaller than the work-area characteristic dimension.

Mobility of bodies in contact. I. A new 2/sup nd/ order mobility index for multiple-finger grasps

Using a configuration-space approach, the paper develops a 2nd-order mobility theory for rigid bodies in contact. A major component of this theory is a coordinate invariant 2nd-order mobility index for a body, B, in frictionless contact with finger bodies A/sub 1/,...A/sub k/. The index is an integer that captures the inherent mobility of B in an equilibrium grasp due to second order, or surface curvature, effects. It differentiates between grasps which are deemed equivalent by classical 1st-order theories, but are physically different. We further show that 2nd-order effects can be used to lower the effective mobility of a grasped object, and discuss implications of this result for achieving new lower bounds on the number of contacting finger bodies needed to immobilize an object. Physical interpretation and stability analysis of 2nd-order effects are taken up in the companion paper.

TangentBug: A Range-Sensor-Based Navigation Algorithm

The Bug family algorithms navigate a 2-DOF mobile robot in a completely unknown environment using sensors. TangentBug is a new algorithm in this family, specifically designed for using a range sensor. TangentBug uses the range data to compute a locally short est path, based on a novel structure termed the local tangent graph (LTG). The robot uses the LTG for choosing the locally optimal di rection while moving toward the target, and for making local short- cuts and testing a leaving condition while moving along an obstacle boundary. The transition between these two modes of motion is governed by a globally convergent criterion, which is based on the distance of the robot from the target. We analyze the properties of TangentBug, and present simulation results that show that Tangent Bug consistently performs better than the classical Bug algorithms. The simulation results also show that TangentBug produces paths that in simple environments approach the globally optimal path, as the sensors maximal detection-range increases. The algorithm can be readily implemented on a mobile robot, and we discuss one such implementation.

Competitive on-line coverage of grid environments by a mobile robot

We describe in this paper two on-line algorithms for covering planar areas by a square-shaped tool attached to a mobile robot. Let D be the tool size. The algorithms, called Spanning Tree Covering (STC) algorithms, incrementally subdivide the planar area into a grid of D-size cells, while following a spanning tree of a grid graph whose nodes are 2D-size cells. The two STC algorithms cover general planar grids. The first, Spiral-STC, employs uniform weights on the grid-graph edges and generates spiral-like covering patterns. The second, Scan-STC , assigns lower weights to edges aligned with a particular direction and generates scan-like covering patterns along this direction. Both algorithms cover any planar grid using a path whose length is at most (n + m)D, where n is the total number of D-size cells and m ≤ n is the number of boundary cells, defined as cells that share at least one point with the grid boundary. We also demonstrate that any on-line coverage algorithm generates a covering path whose length is at least (2 - e)lopt in worst case, where lopt is the length of the optimal off-line covering path. Since (n + m)D ≤ 2lopt, the bound is tight and the STC algorithms are worst-case optimal. Moreover, in practical environments m || n, and the STC algorithms generate close-to-optimal covering paths in such environments.

Caging Planar Bodies by One-Parameter Two-Fingered Gripping Systems

This paper is concerned with a caging problem: we wish to surround an object B by a multifingered hand such that B has some freedom to move, but still cannot escape the “cage” formed by the fingers. We introduce a new notion of the caging set, which is based on the configuration-space representation of the free motions of the hand system with respect to the object. Using stratified Morse theory, we show that the hand’s configuration at which the cage is broken corresponds to a frictionless equilibrium grasp. This allows us to formulate a technique for computing the caging set of a two-fingered hand whose opening is controlled by a single parameter. The tech-nique generalizes to one-parameter grip ping systems having a higher number of fingers.

Mobility of bodies in contact. II. How forces are generated by curvature effects

For part I, see ibid., p.696-708. The paper considers how forces are produced by compliance and surface curvature effects in systems where an object a is kinematically immobilized to second-order by finger bodies A/sub l/,...,A/sub k/. A class of configuration-space based elastic deformation models is introduced. Using these elastic deformation models, it is shown that any object which is kinematically immobilized to first or second-order is also dynamically locally asymptotically stable with respect to perturbations. Moreover, it is shown that for preloaded grasps kinematic immobility implies that the stiffness matrix of the grasp is positive definite. The stability result provides physical justification for using second-order effects for purposes of immobilization in practical applications. Simulations illustrate the concepts.

Obstacle Collision Detection Using Best Ellipsoid Fit

This paper describes a method for estimating the distance between a robot and its surrounding environment using best ellipsoid fit. The method consists of the following two stages. First we approximate the detailed geometry of the robot and its environment by minimum-volume enclosing ellipsoids. The computation of these ellipsoids is a convex optimization problem, for which efficient algorithms are known. Then we compute a conservative distance estimate using an important but little known formula for the distance of a point from and n-dimensional ellipse. The computation of the distance estimate (and its gradient vector) is shown to be an eigenvalue problem, whose solution can be rapidly found using standard techniques. We also present an incremental version of the distance computation, which takes place along a continuous trajectory taken by the robot. We have implemented the proposed approach and present some preliminary results.

Exact robot navigation using cost functions: the case of distinct spherical boundaries in E/sup n/

The utility of artificial potential functions is explored as a means of translating automatically a robot task description into a feedback control law to drive the robot actuators. A class of functions is sought which will guide a point robot amid any finite number of spherically bounded obstacles in Euclidean n-space toward an arbitrary destination point. By introducing a set of additional constraints, the subclass of navigation functions is defined. This class is dynamically sound in the sense that the actual mechanical system will inherit the essential aspects of the qualitative behavior of the gradient lines of the cost function. An existence proof is given by constructing a one parameter family of such functions; the parameter is used to guarantee the absence of local minima.<<ETX>>