Various performance indices are used for the design of serial manipulators. One method of optimization relies on the condition number of the Jacobian matrix. The minimization of the condition number leads, under certain conditions, to isotropic configurations, for which the roundoff-error amplification is lowest. In this paper, the isotropy conditions, introduced elsewhere, are the motivation behind the introduction of isotropic sets of points. By connecting together these points, we define families of isotropic manipulators. This paper is devoted to planar manipulators, the concepts being currently extended to their spatial counterparts. Furthermore, only manipulators with revolute joints are considered here.
Various performance indices have been devised to assess the kinetostatic performance of serial manipulators. The literature on performance indices is extremely rich to fit in the limits of this paper, the interested reader being invited to look at it in the rather recent references cited here. A dimensionless quality index was recently introduced by Lee, Duffy, and Hunt (1998) 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 (Angeles, 1997), as pertaining to serial manipulators, its extension to their parallel counterparts being straightforward. The condition number of a given matrix, on the other hand is well known to provide a measure of invertibility of the matrix (Golub and Van Loan, 1989). 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 (1982) 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 (Golub and Van Loan, 1989) 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 (Angeles and López-Cajún, 1992) 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 in this debate, by resorting to the concept of isotropic sets of points. Briefly stated, the application of isotropic sets of points to the design of manipulator architectures relies on the concept of distance in the space of m × n matrices, which is based, in turn, on the Frobenius norm of matrices. With the purpose of rendering the Jacobian matrix dimensionally homogeneous, moreover, we introduce the concept of posture-dependent conditioning length. Thus, given 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.
It is noteworthy that isotropy comprising symmetry at its core, manipulators with only revolute joints are considered here. It should be apparent that mixing actuated revolutes with actuated prismatic joints would destroy symmetry, and hence, isotropy.
When comparing two dimensionless m × n matrices A and B, we can define the distance d(A, B) between them as the Frobenius norm of their difference, namely,
An m × n isotropic matrix, with m < n, is one with a singular value σ > 0 of multiplicity m, and hence, if the m × n matrix C is isotropic, then
where 1 is the m × m identity matrix. Note that the generalized inverse of C can be computed without roundoff-error, for it is proportional to C T , namely,
Furthermore, the condition number κ(A) of a square matrix A is defined as (Golub and Van Loan, 1989)
where any norm can be used. For purposes of the paper, we shall use the Frobenius norm for matrices and the Euclidean norm for vectors. Henceforth we assume, moreover, a planar n-revolute manipulator, as depicted in Fig. 1, with Jacobian matrix J given by (Angeles, 1997)
where r i is the vector directed from the center of the ith revolute to the operation point P of the end-effector, and matrix E represents a counterclockwise rotation of 90 • . It will prove convenient to partition J into
Therefore, while the entries of A are dimensionless, those of B have units of length. Thus, the sole singular value of A, i.e., the nonnegative square root of the scalar of AA T , is √ n, and hence, dimensionless, and pertains to the mapping from joint-rates into end-effector angular velocity. The singular values of B, which are the nonnegative square roots of the eigenvalues of BB T , have units of length, and account for the mapping from joint-rates into operation-point velocity. It is thus apparent that the singular values of J have different dimensions and hence, it is impossible to compute κ(J) as in eq.( 4), for the norm of J cannot be defined. The normalization of the Jacobian for purposes of rendering it dimensionless has been judged to be dependent on the normalizing length (Paden, and Sastry, 1988;Li, 1990).
As a means to avoid the
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