Hyperbolic pseudoinverses for kinematics in the Euclidean group

The kinematics of a robot manipulator are described in terms of the mapping connecting its joint space and the 6-dimensional Euclidean group of motions SE(3). The associated Jacobian matrices map into its Lie algebra se(3), the space of twists describing infinitesimal motion of a rigid body. Control methods generally require knowledge of an inverse for the Jacobian. However for an arm with fewer or greater than six actuated joints or at singularities of the kinematic mapping this breaks down. The Moore--Penrose pseudoinverse has frequently been used as a surrogate but is not invariant under change of coordinates. Since the Euclidean Lie algebra carries a pencil of invariant bilinear forms that are indefinite, a family of alternative hyperbolic pseudoinverses is available. Generalised Gram matrices and the classification of screw systems are used to determine conditions for their existence. The existence or otherwise of these pseudoinverses also relates to a classical problem addressed by Sylvester concerning the conditions for a system of lines to be in involution or, equivalently, the corresponding system of generalised forces to be in equilibrium

A CORDIC-Based Pipelined Architecture for Direct Kinematic Position Computation

The kinematic equation of an fi-link manipulator involves the chain product of n homogeneous link transformation matrices and reveals the requirement for computing a large set of elementary operations: multiplications, additions, and trigonometric functions. However, these elementary operations, in general, cannot be efficiently computed in general-purpose uniprocessor computers. The CORDIC (COordinate Rotation DIgital Computer) algorithms are the natural candidates for efficiently computing these elementary operations and the interconnection of these CORDIC processors to exploit the great potential of pipelining provides a better solution for computing the direct kinematics. This paper describes a novel CORDIC-based pipelined architecture for the computation of direct kinematic position solution based on the decomposition of the homogeneous link transformation matrix. It is found that a homogeneous link transformation matrix can be decomposed into a product of two matrices, each of which can be computed by two CORDIC processors arranged in parallel, forming a 2-stage cascade CORDIC computational module. Extending this idea to an n-link manipulator, n 2-stage CORDIC computational modules, consisting of 4n CORDIC processors, can be concatenated to form a pipelined architecture for computing the position and orientation of the end-effector of the manipulator. Since the initial delay time of the proposed pipelined architecture is 80n μs and the pipelined time is 40μs, the proposed CORDIC-based architecture requires a total computation time of (80n + 120)μs for computing the position and orientation of the end- effector of an n-link manipulator

Multi-loop position analysis via iterated linear programming

Robotics: Science and Systems Conference (RSS), 2006, Filadelfia (EE.UU.)This paper presents a numerical method able to isolate all configurations that an arbitrary loop linkage can adopt, within given ranges for its degrees of freedom. The procedure is general, in the sense that it can be applied to single or multiple intermingled loops of arbitrary topology, and complete, in the sense that all possible solutions get accurately bounded, irrespectively of whether the analyzed linkage is rigid or mobile. The problem is tackled by formulating a system of linear, parabolic, and hyperbolic equations, which is here solved by a new strategy exploiting its structure. The method is conceptually simple, geometric in nature, and easy to implement, yet it provides solutions at the desired accuracy in short computation times.This work was supported by the project 'Planificador de trayectorias para sistemas robotizados de arquitectura arbitraria' (J-00930).Peer Reviewe

Advanced Strategies for Robot Manipulators

Amongst the robotic systems, robot manipulators have proven themselves to be of increasing importance and are widely adopted to substitute for human in repetitive and/or hazardous tasks. Modern manipulators are designed complicatedly and need to do more precise, crucial and critical tasks. So, the simple traditional control methods cannot be efficient, and advanced control strategies with considering special constraints are needed to establish. In spite of the fact that groundbreaking researches have been carried out in this realm until now, there are still many novel aspects which have to be explored

Mathematical Analysis of the Joint Motion of Redundant Robots Under Pseudo-Inverse Control

Redundant robots that are kinematically controlled using Jacobian pseudo-inverses may not have repeatable joint motions, when the end-effector traces a closed path in the workspace. This phenomenon is known as joint drift. The joint drift problem was initially observed and analyzed by Klein and Huang[lO]. Shamir and Yomdin(l5J also analyzed this problem using differential geometric approach. Klein and Kee[ll.] observed through numerical experiments that the drift had predictable properties. In this paper we present a measure of the drift motion, we show this measure is useful for predicting the stability properties of drifts. We further show that this measure of drift does indeed exhibit the properties numerically observed by Klein and Kee