On the kinetostatic optimization of revolute-coupled planar manipulators

doi:10.1016/S0094-114X(01)00081-7

Mechanism and Machine Theory

Volume 37, Issue 4, April 2002, Pages 351-374

https://doi.org/10.1016/S0094-114X(01)00081-7 Get rights and content

Abstract

Proposed in this paper is a kinetostatic performance index for the optimum dimensioning of planar manipulators of the serial type. The index is based on the concept of distance of the underlying Jacobian matrix to a given isotropic matrix that is used as a reference model for the purpose of performance evaluation. Applications of the index fall in the realm of design, but control applications are outlined. The paper focuses on planar manipulators, the basic concepts being currently extended to their three-dimensional counterparts.

Introduction

Various performance indices have been devised to assess the kinetostatic performance of serial manipulators. Among these, the concepts of service angle [1], dexterous workspace [2] and manipulability [3] are worth mentioning. All these different concepts allow the definition of the kinetostatic performance of a manipulator from correspondingly different viewpoints. However, with the exception of Yoshikawa's manipulability index [3], none of these considers the invertibility of the Jacobian matrix. A dimensionless quality index was recently introduced by Lee et al. [4] based on the ratio of the Jacobian determinant to its maximum absolute value, as applicable to parallel manipulators. This index does not take into account the location of the operation point in the end-effector, for the Jacobian determinant is independent of this location. The proof of the foregoing fact is available in [5], as pertaining to serial manipulators, its extension to their parallel counterparts being straightforward. The condition number of a given matrix is well known to provide a measure of invertibility of the matrix [6]. It is thus natural that this concept found its way in this context. Indeed, the condition number of the Jacobian matrix was proposed by Salisbury and Craig [7] as a figure of merit to minimize when designing manipulators for maximum accuracy. In fact, the condition number gives, for a square matrix, a measure of the relative roundoff-error amplification of the computed results [6] with respect to the data roundoff-error. As is well known, however, the dimensional inhomogeneity of the entries of the Jacobian matrix prevents the straightforward application of the condition number as a measure of Jacobian invertibility. The characteristic length was introduced in [8] to cope with the above-mentioned inhomogeneity. Apparently, nevertheless, this concept has found strong opposition within some circles, mainly because of the lack of a direct geometric interpretation of the concept. It is the aim of this paper to shed more light on this debate, while proposing a novel performance index that lends itself to a straightforward manipulation and leads to sound geometric relations. Briefly stated, the performance index proposed here is based on the concept of distance in the space of m×n matrices, which is based, in turn, on the concept of inner product of this space. The performance index underlying this paper thus measures the distance of a given Jacobian matrix from an isotropic matrix of the same gestalt. With the purpose of rendering the Jacobian matrix dimensionally homogeneous, we resort to the concept of posture-dependent conditioning length. Thus, given at an arbitrary serial manipulator in an arbitrary posture, it is possible to define a unique length that renders this matrix dimensionally homogeneous and of minimum distance to isotropy. The characteristic length of the manipulator is then defined as the conditioning length corresponding to the posture that renders the above-mentioned distance a minimum over all possible manipulator postures. This paper is devoted to planar manipulators, the concepts being currently extended to spatial ones.

Section snippets

Algebraic background

Given two arbitrary m×n matrices A and B of real entries, their inner product, represented by $(A, B)$ , is defined as $(A, B)≡ tr AWB^{T},$ where $W$ is a positive-definite n×n weighting matrix that is introduced to allow for suitable normalization. The entries of $W$ need not be dimensionally homogeneous, and, in fact, they should not if $A$ and $B$ are not. However, the product $AWB^{T}$ must be dimensionally homogeneous; else, its trace is meaningless. The norm of the space of m×n matrices induced by the above inner

Isotropic sets of points

Consider the set $S ≡{P_{k}}_{1}^{n}$ of n points in the plane, of position vectors ${p_{k}}_{1}^{n}$ and centroid C, of position vector c, i.e., $c ≡ 1 n ∑_{1}^{n} p_{k} .$ The summation appearing in the right-hand side of the above expression is known as the first moment of $S$ with respect to the origin O from which the position vectors stem. The second moment of $S$ with respect to C is defined as a tensor M, namely, $M ≡∑_{1}^{n} (p_{k} − c)(p_{k} − c)^{T} .$ It is now apparent that the root-mean-square value of the distances {d_k}ⁿ₁ of $S$ , d_rms, to the centroid

An outline of kinematic chains

The connection between sets of points and planar manipulators of the serial type is the concept of simple kinematic chain. For completeness, we recall here some basic definitions pertaining to this concept.

The posture-dependent conditioning length of planar n-revolute manipulators

Under the assumption that the manipulator finds itself at a posture $P$ that is given by its set of joint angles, {θ_k}₁ⁿ, we start by dividing the last n rows of the Jacobian by a length $l_{P}$ , as yet to be determined. This length will be found so as to minimize the distance of the normalized Jacobian to a corresponding isotropic matrix K, subscript $P$ reminding us that, as the manipulator changes its posture, so does the length $l_{P}$ . This length will be termed the conditioning length of the

Applications to design and control

Manipulators are designed for a family of tasks, more so than for a specific task – manipulator design for a specific task defeats the purpose of using a manipulator, in the first place! The first step in designing a manipulator, moreover, is to dimension its links. It is apparent that from a purely geometric viewpoint, the link lengths are not as important as the link-length ratios. Once these ratios are optimally determined, the link lengths can be obtained based on requirements such as

Conclusions

The conditioning length $l_{P}$ was defined for a given posture of a planar manipulator. This concept allows us to normalize the Jacobian matrix so as to render it in nondimensional form. We base the definition of the characteristic length on an objective function z that gives a geometric significance to the conditioning length. Moreover, the objective function introduced here is defined as a measure of the distance of the normalized – nondimensional – Jacobian matrix to an isotropic reference

Acknowledgements

The first author acknowledges support from France's Institut National dc Recherche en Informatique et en Automatique. The second author acknowledges support from the Natural Sciences and Engineering Research Council, of Canada, and of Singapore's Nanyang Technological University, where he completed the research work reported here, while on sabbatical from McGill University.

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¹: URLS: (D.Chablat), www.cim.megill.ca (J. Angeles). IRCCyN: UMR 6597 CNRS, Ecole Centrale de Nantes, Universite de Nantes, Ecole des Mines de Nantes.

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On the kinetostatic optimization of revolute-coupled planar manipulators

Abstract

Introduction

Section snippets

Algebraic background

Isotropic sets of points

An outline of kinematic chains

The posture-dependent conditioning length of planar n-revolute manipulators

Applications to design and control

Conclusions

Acknowledgements

Details of kinematics of manipulators with the method of volumes

Mekh. Mashin

The workspace of a mechanical manipulator

ASME J. Mech. Design

Manipulability of robotic mechanisms

Int. J. Robotics Res.

A practical quality index based on the octahedral manipulator

Int. J. Robotic Res.

Fundamentals of Robotic Mechanical Systems

Matrix Computations