Robot kinematics and coordinate transformations

Abstract

This paper introduces a class of linearizing coordinate transformations for mechanical systems whose moment of inertia matrix has a square root which is a jacobian. The transformations, when they exist, define a local isometry from joint space to euclidean space, hence, may afford further insight into the transient behavior of robot motion. It remains to be seen whether any appreciably large class of robots admit such linearizing isometries. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. NOTE: At the time of publication, author Daniel Koditschek was affiliated with Yale University. Currently, he is a faculty member in the Department of Electrical and Systems Engineering at the University of Pennsylvania.This conference paper is available at ScholarlyCommons: http://repository.upenn.edu/ese_papers/417 AbstractThis paper introduces a class of linearizing coordinate transformations for mechanical systems whose moment of inertia matrix has a square root which is a jacobian. The transformations. when they exist. clcfine a local isometry from joint space to euclidean spacc. hence. may afford further insight into the transient behavior of robot motion. It remains to be seen whether any appreciably large class of robots admit such linearizing isometries.

Modeling and Control of Mechanical Systems in Terms of Quasi-Velocities

“…Since the Jacobian is an invertible matrix, ξ(q) must be an invertible function meaning that there is a one-to-one correspondence between ξ and q. Under this circumstance, ξ and v are indeed alternative possibilities for generalized coordinates and generalized velocities and that can fundamentally simplify the equations of motion Bedrossian (1992); Gu and Loh (1987); Kodischeck (1985);Spong (1992). It can be also seen from (7b) that if ξ(q) exists and it is a smooth function, then the expression in the parenthesis of the right-hand side of (14) vanishes, i.e.,…”

“…As a result, the corresponding dynamic formulation in not invariant and a solution depends on measure units or a weighting matrix selected Aghili (2005); Angeles (2003); Lipkin and Duffy (1988); Luca and Manes (1994); Manes (1992). There also exist other techniques to describe the equations of motion in terms of quasi-velocities, i.e., a vector whose Euclidean norm is proportional to the square root of the system's kinetic energy, which can lead to simplification of these equations Aghili (2008;2007); Bedrossian (1992); Gu (2000); Gu and Loh (1987); Herman (2005); Herman and Kozlowski (2006); Jain and Rodriguez (1995); Junkins and Schaub (1997); Kodischeck (1985); Kozlowski (1998); Loduha and Ravani (1995); Papastavridis (1998) ;Rodriguez and Kertutz-Delgado (1992); Sinclair et al (2006);Spong (1992). A recent survey on some of these techniques can be found in Herman and Kozlowski (2006).…”

“…In short, the square-root factorization of mass matrix is used as a transformation to obtain the quasi-velocities, which are a linear combination of the velocity and the generalized coordinates Herman and Kozlowski (2006); Papastavridis (1998). It was shown by Kodistchek Kodischeck (1985) that if the square-root factorization of the inertia matrix is integrable, then the robot dynamics can be significantly simplified. In such a case, transforming the generalized coordinates to quasi-coordinates by making use of the integrable factorization modifies the robot dynamics to a system of double integrator.…”

A survey of equations of motion in terms of inertial quasi-velocities for serial manipulators

“…[23,33,58]. Factorization of the inertia matrix (which is always positive definite) of a N -link robot manipulator as multiplication of a matrix and its transposition, and definition of a canonical transformation lead to robot dynamic equations which are particularly simple.…”

A Sufficient Condition for the Feedback Quasilinearization of Control Mechanical Systems

“…There has been active research on quasilinearization [2,5,7,8], but the results were obtained by the zero curvature condition or by some complicated PDE conditions, producing restrictive outcomes. Then, very strong results were finally obtained in [4] where easily verifiable quasilinearizability conditions were derived.…”