Frank C. Park

In this article we develop a mathematical theory for optimizing the kinematic dexterity of robotic mechanisms and obtain a collection of analytical tools for robot design. The performance criteria we consider are workspace volume and dexterity; by the latter we mean the ability to move and apply forces in arbitrary directions as easily as possible. Clearly, dexterity and workspace volume are intrinsic to a mechanism, so that any mathematical formulation of these properties must necessarily be in

Abstract We provide a tutorial and review of the state-of-the-art in robot dynamics algorithms that rely on methods from differential geometry, particularly the theory of Lie groups. After reviewing the underlying Lie group structure of the rigid-body motions and the geometric formulation of the equations of motion for a single rigid body, we show how classical screw-theoretic concepts can be expressed in a reference frame-invariant way using Lie-theoretic concepts and derive recursive algorithm

We present a particle filtering algorithm for visual tracking, in which the state equations for the object motion evolve on the two-dimensional affine group. We first formulate, in a coordinate-invariant and geometrically meaningful way, particle filtering on the affine group that allows for combined state—covariance estimation. Measurement likelihoods are also calculated from the image covariance descriptors using incremental principal geodesic analysis, a generalization of principal component

Abstract This article addresses the problem of designing a robotic mechanism such that its end‐effector frame comes closest to reaching a set of desired goal frames. We formulate this as an optimization problem, in which the kinematic parameters are selected to minimize the total distance between the end‐effector frame and each goal frame. The objective function is defined in terms of a class of distance metrics on the rigid body motions that are invariant with respect to choice of fixed referen

Abstract This article presents a simulation‐based strategy for sizing the actuators of a redundantly actuated robotic mechanism. The class of robotic mechanisms we consider may contain one or more closed loops and possess an arbitrary number of active and passive joints, and the number of actuators may exceed the mechanism's kinematic degrees of freedom. Our approach relies on a series of dynamic simulations of the mechanism, by applying Taguchi's method to systematically perform the simulations

Abstract Motion planning for high-DOF multi-arm systems operating in complex environments remains a challenging problem, with many motion planning algorithms requiring evaluation of the minimum collision distance and its derivative. Because of the computational complexity of calculating the collision distance, recent methods have attempted to leverage data-driven machine learning methods to learn the collision distance. Because of the significant training dataset requirements for high-DOF robots

Abstract In this article we generalize the concept of Bézier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Bézier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their algebraic group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Bézier curves. Spatial dis