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RESEARCH PAPERS

# The Kinetostatic Optimization of Robotic Manipulators: The Inverse and the Direct Problems

[+] Author and Article Information
Waseem A. Khan

Department of Mechanical Engineering & Centre for Intelligent Machines, McGill University, Montreal, QC H3A 2A7, Canadawakhan@cim.mcgill.ca

Jorge Angeles

Department of Mechanical Engineering & Centre for Intelligent Machines, McGill University, Montreal, QC H3A 2A7, Canadaangeles@cim.mcgill.ca

Nonredundant manipulators with two to five axes require a coefficient of the trace in the definition of the Frobenius norm of $1∕2$ to $1∕5$, correspondingly.

Although $a1, we do not take $M=b1$ because $b1$ does not affect the manipulator condition number.

J. Mech. Des 128(1), 168-178 (Aug 19, 2005) (11 pages) doi:10.1115/1.2120808 History: Received May 02, 2005; Revised August 19, 2005

## Abstract

The design of a robotic manipulator begins with the dimensioning of its various links to meet performance specifications. However, a methodology for the determination of the manipulator architecture, i.e., the fundamental geometry of the links, regardless of their shapes, is still lacking. Attempts have been made to apply the classical paradigms of linkage synthesis for motion generation, as in the Burmester Theory. The problem with this approach is that it relies on a specific task, described in the form of a discrete set of end-effector poses, which kills the very purpose of using robots, namely, their adaptability to a family of tasks. Another approach relies on the minimization of a condition number of the Jacobian matrix over the architectural parameters and the posture variables of the manipulator. This approach is not trouble-free either, for the matrices involved can have entries that bear different units, the matrix singular values thus being of disparate dimensions, which prevents the evaluation of any version of the condition number. As a means to cope with dimensional inhomogeneity, the concept of characteristic length was put forth. However, this concept has been slow in finding acceptance within the robotics community, probably because it lacks a direct geometric interpretation. In this paper the concept is revisited and put forward from a different point of view. In this vein, the concept of homogeneous space is introduced in order to relieve the designer from the concept of characteristic length. Within this space the link lengths are obtained as ratios, their optimum values as well as those of all angles involved being obtained by minimizing a condition number of the dimensionally homogeneous Jacobian. Further, a comparison between the condition number based on the two-norm and that based on the Frobenius norm is provided, where it is shown that the use of the Frobenius norm is more suitable for design purposes. Formulation of the inverse problem—obtaining link lengths—and the direct problem—obtaining the characteristic length of a given manipulator—are described. Finally a geometric interpretation of the characteristic length is provided. The application of the concept to the design and kinetostatic performance evaluation of serial robots is illustrated with examples.

###### FIGURES IN THIS ARTICLE
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Copyright © 2006 by American Society of Mechanical Engineers
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## Figures

Figure 1

A homogeneous line in homogeneous space

Figure 2

The point O¯i′ on Hi, defined as the point of this line closest to the operation point P¯

Figure 3

The two singular values of the two-revolute manipulator with a2∕a1=2∕2 vs θ2

Figure 4

Inverse of condition number κ2(J) vs θ2

Figure 5

Inverse of condition number κF(J) vs θ2

Figure 6

Displays of the optimum manipulator at: (a) The isotropic posture; and (b) the maximum-reach posture

Figure 7

Displays of the second optimum manipulator at: (a) The isotropic posture; and (b) the maximum-reach posture

Figure 8

A planar manipulator

Figure 9

Isotropic 3R planar manipulator: (a) At the isotropic posture; and (b) a geometric interpretation of its characteristic length

Figure 10

Equilateral 3R planar manipulator: (a) At the isotropic posture; and (b) a geometric interpretation of its characteristic length

Figure 11

4R planar manipulator with equal link lengths at optimum posture

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