Screw theory and higher order kinematic analysis of open serial and closed chains

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Abstract

In a pair of recent contributions by the authors of this paper, it has been shown that screw theory can be successfully employed in the acceleration analysis of open and closed spatial chains. In this contribution, a novel method for the velocity, acceleration, and jerk analysis of spatial chains is introduced. The method is based in obtaining recursive expressions for the velocity, acceleration, and jerk of the end effector of a serial manipulator. This new method facilitates greatly the derivation of some well known results concerning the velocity and acceleration analyses of spatial chains. Furthermore, the method provides proofs of several new results for the jerk analysis which were not possible using methods developed previously. In addition, the contribution shows how the results of open serial chains can be applied to closed chains and parallel manipulators. Finally, the method can be easily generalized to higher order analyses.

Introduction

Screw theory is without doubt an efficient mathematical tool for the study of spatial kinematics. The pioneering work by Ball[1], the outstanding treatises by Hunt[2], and Phillips3, 4and the multitude of contributions appearing in the literature are evidence of this. Furthermore, the isomorphism between screw theory and the Lie algebra, e(3), of the Euclidean group, E(3),[5]provide kinematicians with a wealth of results and techniques from modern differential geometry and Lie group theory.

Nevertheless, until recent years, screw theory had been confined to first-order or linear analysis; namely, the first-order velocity analysis of spatial chains, and the first-order static analysis of spatial chains. The original contributions by Sugimoto6, 7, and recent contributions by the authors8, 9have made some progress in extending screw theory beyond the first-order analyses. A recent paper by Narasimham and Kumar[10]also provides interesting results.

The aim of this present contribution is twofold. Firstly, to introduce a recursive method for the velocity, acceleration, and jerk analyses of spatial chains. This method provides far simpler proofs of many results, related to velocities and accelerations, that were obtained in previous contributions8, 9. Secondly, this recursive method allows the extension of screw theory to the jerk analysis of spatial chains. This extension appeared to be a formidable task using the methods developed by Rico and Duffy8, 9.

The extension of screw theory to the acceleration and jerk analyses of spatial chains is more than an interesting academic pursuit. The authors firmly believe that this extension provides important clues for a correct characterization of singularities of closed chains, some initial results along this direction have been reported by Rico and Gallardo[11].

A brief overview of the paper is now given. Section 2introduces results for the velocity and the acceleration of a rigid body which are already known, and novel results for the jerk of a rigid body as observed from a pair of different reference frames.

In Section 3a new method is introduced for expressing the velocity state, the reduced acceleration state, and a novel reduced jerk state of a rigid body as recursive expressions involving two different reference frames.

In Section 4new relationships are derived between the velocity, reduced acceleration, and reduced jerk states as observed from two different reference frames that nevertheless satisfy some conditions. It happens that these conditions are precisely those that are satisfied by the two serial chains that join two distinct lines in closed chains or in parallel manipulators.

The results of 2 Velocity, acceleration and jerk of a rigid body, 3 Recursive expressions for the velocity, reduced acceleration, and reduced jerk state of a rigid body, 4 Relationships between the velocity, acceleration and jerk of a rigid body as observed from a pair of different, but related, reference framesdo not require that the reference frames be connected at all. In contrast, the results in Section 5require that the links of an open serial chain be connected by helical pairs. There, the velocity, reduced acceleration, and reduced jerk states are expressed in terms of the screws of the open serial chain. The advantage of the recursive approach, developed in Section 2, is evident in the computation of the velocity and reduced acceleration state, those results were originally published by Rico and Duffy8, 9. Further, Section 5also extends the results to the reduced jerk state. This extension seemed impossible using the direct techniques employed previously by the authors8, 9. Finally, Section 6shows that under additional assumptions, the results obtained previously can be also applied to closed chains and parallel manipulators.

Section snippets

Velocity, acceleration and jerk of a rigid body

In this Section, it will be shown how to obtain relationships between the velocity, acceleration and jerk of a rigid body as observed from a pair of different reference frames. These relationships provide recursive equations for the velocity, acceleration, and jerk of the end effector of a serial manipulator. Although the results for velocities and accelerations are known, the results for the jerk are original.

The basis for the analysis is the well known result that relates the derivatives of a

Recursive expressions for the velocity, reduced acceleration, and reduced jerk state of a rigid body

In this section, the recursive expressions obtained in Section 2will be extended to recursive expressions for the velocity, reduced acceleration, and reduced jerk states of a rigid body; most of the results in this section are thought to be original.

Relationships between the velocity, acceleration and jerk of a rigid body as observed from a pair of different, but related, reference frames

In this section, several interesting relationships between the velocity, acceleration and jerk states of an arbitrary rigid body as observed from a pair of different reference frames that satisfy certain conditions will be obtained. It should be noted that the results in this section do not require the existence of any physical connection between the reference frames.

Expressions of the velocity, acceleration, and jerk states of the end effector of a serial chain in screw form

It should be noted that 3 Recursive expressions for the velocity, reduced acceleration, and reduced jerk state of a rigid body, 4 Relationships between the velocity, acceleration and jerk of a rigid body as observed from a pair of different, but related, reference framescontain results that are applicable to rigid bodies, or reference frames, that do not need to be connected in any form. In this section the rigid bodies form an open serial chain with adjacent bodies connected by a helical pair,

The velocity, acceleration and jerk analysis of closed chains and parallel manipulators

In this final section, it will be shown that after a few simplifications the results concerning with open serial chains, obtained in 4 Relationships between the velocity, acceleration and jerk of a rigid body as observed from a pair of different, but related, reference frames, 5 Expressions of the velocity, acceleration, and jerk states of the end effector of a serial chain in screw form, can be also applied to closed chains, and therefore to parallel manipulators.

Conclusions

The present contribution has introduced a new recursive technique for relating the velocity, acceleration and jerk states of a rigid body as observed from a pair of different reference frames. This recursive technique has allowed to prove, by a far simpler procedure, several results already proven, by the authors, using a direct technique. Further, the paper has extended the celebrated result by Coriolis from acceleration to jerk. Moreover, the necessary equations for solving the jerk analysis

Acknowledgements

Part of this work was completed during a summer visit of the first author to the Center for Intelligent Machines and Robotics, University of Florida. part of this work was carried out by the second author as part of the doctoral requirements in a graduate program at the Instituto Tecnológico de la Laguna (México). The authors thank Center of Excellence Fund of the State of Florida for its continuous support. This work was partially supported by the Consejo del Sistema Nacional de Educación

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