Patrick G. Xavier

Kinodynamic motion planning

Kinodynamic planning attempts to solve a robot motion problem subject to simultaneous kinematic and dynamics constraints. In the general problem, given a robot system, we must find a minimal-time trajectory that goes from a start position and velocity to a goal position and velocity while avoiding obstacles by a safety margin and respecting constraints on velocity and acceleration. We consider the simplified case of a point mass under Newtonian mechanics, together with velocity and acceleration bounds. The point must be flown from a start to a goal, amidst polyhedral obstacles in 2D or 3D. Although exact solutions to this problem are not known, we provide the first provably good approximation algorithm, and show that it runs in polynomial time

On the complexity of kinodynamic planning

The following problem, is considered: given a robot system find a minimal-time trajectory from a start position and velocity to a goal position and velocity, while avoiding obstacles and respecting dynamic constraints on velocity and acceleration. The simplified case of a point mass under Newtonian mechanics together with velocity and acceleration bounds is considered. The point must be flown from a start to a goal, amid 2-D or 3-D polyhedral obstacles. While exact solutions to this problem are not known, the first provably good approximation algorithm is given and shown to run in polynomial time.

Fast swept-volume distance for robust collision detection

The need for collision detection arises in several robotics areas, including motion-planning, online collision avoidance, and simulation. At the heart of most current methods are algorithms for interference detection and/or distance computation. A few recent algorithms and implementations are very fast, but to use them for accurate collision detection, very small step sizes can be necessary, reducing their effective efficiency. We present a fast, implemented technique for doing exact distance computation and interference detection for translationally-swept bodies. For rotationally swept bodies, we adapt this technique to improve accuracy, for any given step size, in distance computation and interference detection. We present preliminary experiments which show that the combination of basic and swept-body calculations holds much promise for faster accurate collision detection.

Provably good approximation algorithms for optimal kinodynamic planning for Cartesian robots and open-chain manipulators

Inoptimal kinodynamic planning, given a robot system, we must find a minimal-time trajectory that goes from a start state to a goal state while avoiding obstacles by a speed-dependent safety margin and respecting dynamics bounds. With Canny and Reif [1], we approached this problem from anɛ-approximation standpoint and introduced a provably good approximation algorithm for optimal kinodynamic planning for a robot obeying particle dynamics. If a solution exists, this algorithm returns a trajectoryɛ-close to optimal in time polynomial in both (1/ɛ) and the geometric complexity.We extend [1] and [2] tod-link three-dimensional robots with full rigid-body dynamics amidst obstacles. Specifically, we describe polynomial-time approximation algorithms for Cartesian robots obeyingL2 dynamic bounds and for open-kinematic-chain manipulators with revolute and prismatic joints. The latter class includes many industrial manipulators. The correctness and complexity of these algorithms rely on new trajectory tracking lemmas for robots with coupled dynamics bounds.

A provably good approximation algorithm for optimal-time trajectory planning

The authors describe the first implementation of a good polynomial-time approximation algorithm for kinodynamic planning. Attention is given to the following problem: given a robot system, find a minimal-time trajectory from a start position and velocity to a goal position and velocity, while avoiding obstacles and respecting dynamic constraints on velocity and acceleration. From the class of approximate minimal-time trajectories for a given problem instance that the theoretical algorithm would find, the proposed implementation will find a trajectory with minimal useless chattering. In addition, the authors present an improved analysis of the accuracy of the approximation strength of this approach. This analysis reveals that the algorithm produces approximations good to a small additive error in state space and exact in time while only sacrificing the epsilon -approximation factor in safety, where epsilon is an error term. In addition, the analysis halves the polynomial complexity of the algorithm in (1/ epsilon ), and it provides a simple characterization of when the algorithm will find a trajectory that is exact at the start and goal.<<ETX>>

Provably good approximation algorithms for optimal kinodynamic planning: Robots with decoupled dynamics bounds

We consider the following problem: given a robot system, find a minimal-time trajectory that goes from a start state to a goal state while avoiding obstacles by a speed-dependent safety margin and respecting dynamics bounds. In [1] we developed a provably good approximation algorithm for the minimum-time trajectory problem for a robot system with decoupled dynamics bounds (e.g., a point robot in ℝ3). This algorithm differs from previous work in three ways. It is possible (1) to bound the goodness of the approximation by an error termɛ; (2) to bound the computational complexity of our algorithm polynomially; and (3) to express the complexity as a polynomial function of the error term. Hence, given the geometric obstacles, dynamics bounds, and the error termɛ, the algorithm returns a solution that isɛ-close to optimal and requires only a polynomial (in (1/ɛ)) amount of time.We extend the results of [1] in two ways. First, we modify it to halve the exponent in the polynomial bounds from 6d to 3d, so that the new algorithm isO(cdN 1/ɛ)3d), whereN is the geometric complexity of the obstacles andc is a robot-dependent constant. Second, the new algorithm finds a trajectory that matches the optimal in time with anɛ factor sacrificed in the obstacle-avoidance safety margin. Similar results hold for polyhedral Cartesian manipulators in polyhedral environments.The new results indicate that an implementation of the algorithm could be reasonable, and a preliminary implementation has been done for the planar case.

Provably good approximation algorithms for optimal kinodynamic planning for Cartesian robots and open chain manipulators

We consider the following problem: given a robot system, find a minimal-time trajectory from a start state to a goal state, while avoiding obstacles by a speed-dependent safety margin and respecting dynamics bounds. In [CDRX] we developed a provably good approximation algorithm for the minimum-time trajectory problem for a robot system with decoupled dynamics bounds. This algorithm differed from previous work in three ways: it is possible (1) to bound the goodness of the approximation by an error term ε (2) to polynomially bound the running time (complexity) of our algorithm; and (3) to express the complexity as a polynomial function of the error term. We extend these results to <italic>d</italic>-link, revolute-joint 3D robots will full rigid body dynamics. Specifically, we first prove a generalized trajectory-tracking lemma for robots with coupled dynamics bounds. Using this result we describe polynomial-time approximation algorithms for Cartesian robots obeying <italic>L</italic><subscrpt>2</subscrpt> dynamics bounds and open kinematic chain manipulators with revolute and prismatic joints; the latter class includes most industrial manipulators. We obtain a general <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>2</supscrpt> (log <italic>n</italic>)(1/ε<supscrpt>6<italic>d</italic>-1</supscrpt>) algorithm, where <italic>n</italic> is the geometric complexity. The algorithm is simple, but the new game-theoretic proof techniques we introduce are subtle, and employ tools from disparate parts of computational geometry, robotics, and dynamical systems.

A generic algorithm for constructing hierarchical representations of geometric objects

For a number of years, robotics researchers have exploited hierarchical representations of geometrical objects and scenes in motion-planning, collision-avoidance, and simulation. However, few general techniques exist for automatically constructing them. We present a generic, bottom-up algorithm that uses a heuristic clustering technique to produce balanced, coherent hierarchies. Its worst-case running time is O(N/sup 2/logN), but for non-pathological cases it is O(NlogN), where N is the number of input primitives. We have completed a preliminary C++ implementation for input collections of 3D convex polygons and 3D convex polyhedra and conducted simple experiments with scenes of up to 12,000 polygons, which take only a few minutes to process. We present examples using spheres and convex hulls as hierarchy primitives.

Shortest path planning for a tethered robot or an anchored cable

We consider the problem of planning shortest paths for a tethered robot with a finite length tether in a 2D environment with polygonal obstacles. We present an algorithm that runs in time O((k/sub l/+1)/sup 2/n/sup 4/) and finds the shortest path or correctly determines that none exists that obeys the constraints, where n is the number obstacle vertices, and k/sub l/ is the number loops in the initial configuration of the tether. The robot may cross its tether but nothing can cross obstacles, which cause the tether to bend. The algorithm can also be applied to planning a shortest path for the free end of an anchored cable.

The Umbra Simulation Framework

Umbra is a new Sandia-developed modeling and simulation framework. The Umbra framework allows users to quickly build models and simulations for intelligent system development, analysis, experimentation, and control and supports tradeoff analyses of complex robotic systems, device, and component concepts. Umbra links together heterogeneous collections of modeling tools. The models in Umbra include 3D geometry and physics models of robots, devices and their environments. Model components can be built with varying levels of fidelity and readily switched to allow models built with low fidelity for conceptual analysis to be gradually converted to high fidelity models for later phase detailed analysis. Within control environments, the models can be readily replaced with actual control elements. This paper describes Umbra at a functional level and describes issues that Sandia uses Umbra to address.