Zvi Shiller

Motion Planning in Dynamic Environments Using Velocity Obstacles

This paper presents a method for robot motion planning in dynamic environments. It consists of selecting avoidance maneuvers to avoid static and moving obstacles in the velocity space, based on the cur rent positions and velocities of the robot and obstacles. It is a first- order method, since it does not integrate velocities to yield positions as functions of time. The avoidance maneuvers are generated by selecting robot ve locities outside of the velocity obstacles, which represent the set of robot velocities that would result in a collision with a given obstacle that moves at a given velocity, at some future time. To ensure that the avoidance maneuver is dynamically feasible, the set of avoidance velocities is intersected with the set of admissible velocities, defined by the robots acceleration constraints. Computing new avoidance maneuvers at regular time intervals accounts for general obstacle trajectories. The trajectory from start to goal is computed by searching a tree of feasible avoidance maneuvers, computed at discrete time intervals. An exhaustive search of the tree yields near-optimal trajectories that either minimize distance or motion time. A heuristic search of the tree is applicable to on-line planning. The method is demonstrated for point and disk robots among static and moving obstacles, and for an automated vehicle in an intelligent vehicle highway system scenario.

Dynamic motion planning of autonomous vehicles

A method for planning the motions of autonomous vehicles moving on general terrains is presented that obtains the geometric path and vehicle speeds that minimize motion time considering vehicle dynamics, terrain topography, obstacles, and surface mobility. The terrain is represented by a smooth cubic B patch, and the geometric path consists of a B spline curve mapped to the surface. The time-optimal motions are computed by first obtaining the best obstacle-free path from all paths represented by a uniform grid. This path is further optimized using a local optimization procedure, using the optimal motion time along the path as the cost function and the control points of a B spline as the optimizing parameters. Examples are presented that demonstrate the method for a simple dynamic model of a vehicle moving on mountainous terrain. >

On computing the global time-optimal motions of robotic manipulators in the presence of obstacles

A method for computing the time-optimal motions of robotic manipulators is presented that considers the nonlinear manipulator dynamics, actuator constraints, joint limits, and obstacles. The optimization problem is reduced to a search for the time-optimal path in the n-dimensional position space. A small set of near-optimal paths is first efficiently selected for a grid, using a branch and bound search and a series of lower bound estimates on the traveling time along a given path. These paths are further optimized with a local path optimization to yield the global optimal solution. Obstacles are considered by eliminating the collision points from the tessellated space and by adding a penalty function to the motion time in the local optimization. The computational efficiency of the method stems from the reduced dimensionality of the searched spaced and from combining the grid search with a local optimization. The method is demonstrated in several examples for two- and six-degree-of-freedom manipulators with obstacles. >

Robot Path Planning with Obstacles, Actuator, Gripper, and Payload Constraints

A method is presented to obtain the time-optimal motions for robotic manipulators. It considers the full nonlinear dy namics of the manipulator, its actuator saturation limits, and gripper and payload constraints. It also accounts for both the presence of obstacles in the work space and restrictions on the motion of the manipulators joints. The method is com putationally practical and has been implemented for the optimal trajectory planning of general six degree-of-freedom manipulators. Examples are presented that demonstrate the substantial improvement in manipulator performance that can be achieved using this method.

Motion planning in dynamic environments using the relative velocity paradigm

A simple and efficient approach to the computation of avoidance maneuvers among moving obstacles is presented. The method is discussed for the case of a single maneuvering object avoiding several obstacles moving on known linear trajectories. The original dynamic problem is transformed into several static problems using the relative velocity between the maneuvering object and each obstacle. The static problems are converted into a single problem by means of a vector transformation, and the set of velocity vectors guaranteeing the avoidance of all the obstacles is computed. Within this set, the best maneuver for the particular approach can be selected. The geometric background of this approach is developed for both 2-D and 3-D cases, and the method is applied to an example of a 3-D avoidance maneuver.<<ETX>>

Motion planning in dynamic environments: obstacles moving along arbitrary trajectories

This paper generalizes the concept of velocity obstacles given by Fiorini et al. (1998) to obstacles moving along arbitrary trajectories. We introduce the nonlinear velocity obstacle, which takes into account the shape, velocity and path curvature of the moving obstacle. The nonlinear v-obstacle allows selecting a single avoidance maneuver (if one exists) that avoids any number of obstacles moving on any known trajectories. For unknown trajectories, the nonlinear v-obstacles can be used to generate local avoidance maneuvers based on the current velocity and path curvature of the moving obstacle. This elevates the planning strategy to a second order method, compared to the first order avoidance using the linear v-obstacle, and zero order avoidance using only position information. Analytic expressions for the nonlinear v-obstacle are derived for general trajectories in the plane. The nonlinear v-obstacles are demonstrated in a complex traffic example.

Optimal obstacle avoidance based on the Hamilton-Jacobi-Bellman equation

This paper presents a method for generating shortest paths in cluttered environments, based on the Hamilton-Jacobi-Bellman (HJB) equation. Formulating the shortest obstacle avoidance problem as a time optimal control problem, the shortest paths are generated by following the negative gradient of the return function, which satisfies the HJB equation. A method to generate near-optimal paths is also presented, based on a psuedo return function. Paths generated by this method are guaranteed to reach the goal, at which the psuedo return function is shown to have a unique minimum. The computation required to generate the near-optimal paths is substantially lower than those of traditional potential field methods, making it applicable to on-line obstacle avoidance. Examples with circular obstacles demonstrate close correlation between the near-optimal and optimal paths, and the advantages of the proposed approach over traditional potential field methods. >

Time-Energy Optimal Control of Articulated Systems With Geometric Path Constraints

The motions of articulated systems along specified paths are optimized to minimize a time-energy cost function. The optimization problem is formulated in a reduced two-dimensional state space and solved using the Pontryagin maximum principle. The optimal control is shown to be smooth, as opposed to the typically discontinuous time optimal control. The numerical solution is obtained with a gradient search that iterates over the initial value of one co-state. Optimal trajectories are demonstrated numerically for a two-link planar manipulator and experimentally for the UCLA Direct Drive Arm. The smoother time-energy optimal trajectory is shown to result in smaller tracking errors than the time optimal trajectory.

Robust computation of path constrained time optimal motions

An algorithm is presented for the computation of path-constrained time-optimal motions of robotic manipulators exploring the nature of so-called critical points and critical arcs. At critical points the reflected inertia at one of the joints is zero, and the feasible acceleration range at the velocity limit is not unique. Time-optimal motions along critical arcs may be singular in the sense that the acceleration is neither maximum nor minimum. This is in contrast to existing motion optimization methods along specified paths which assume maximum acceleration or deceleration at all times. The consideration of critical arcs makes this algorithm robust near the switching points, which are potential points of failure in the other methods. Critical points can be anticipated along the path by mapping the locus of critical arcs to the position space. Examples are presented to demonstrate the algorithm and the existence of singular critical arcs.<<ETX>>

Navigation Among Moving Obstacles Using the NLVO: Principles and Applications to Intelligent Vehicles

Vehicle navigation in dynamic environments is a challenging task, especially when the motion of the obstacles populating the environment is unknown beforehand and is updated at runtime. Traditional motion planning approaches are too slow to be applied in real-time to this problem, whereas reactive navigation methods have generally a too short look-ahead horizon. Recently, iterative planning has emerged as a promising approach, however, it does not explicitly take into account the movements of the obstacles.This paper presents a real-time motion planning approach, based on the concept of the Non-LinearVobst (NLVO) (Shiller et al., 2001). Given a predicted environment, the NLVO models the set of velocities which lead to collisions with static and moving obstacles, and an estimation of the times-to-collision. At each controller iteration, an iterative A* motion planner evaluates the potential moves of the robot, based on the computed NLVO and the traveling time. Previous search results are reused to both minimize computation and maintain the global coherence of the solutions.We first review the concept of the NLVO, and then present the iterative planner. The planner is then applied to vehicle navigation and demonstrated in a complex traffic scenario.