The subject of this paper is a special class of three-degree-of-freedom parallel manipulators. The singular configurations of the two Jacobian matrices are first studied. The isotropic configurations are then found based on the characteristic length of this manipulator. The isoconditioning loci for the Jacobian matrices are plotted to define a global performance index allowing the comparison of the different working modes. The index thus resulting is compared with the Cartesian workspace surface and the average of the condition number.
Various performance indices have been devised to assess the kinetostatic performances of serial and parallel 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 references cited here. A dimensionless quality index was recently introduced in [1] 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 of the end-effector, because the Jacobian determinant is independent of this location. The proof of the foregoing result is available in [2], 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 [3]. It is thus natural that this concept found its way in this context. Indeed, the condition number of the Jacobian matrix was proposed in [4] 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 [3] 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 [5] to cope with the above-mentioned inhomogeneity.
In this paper we use the characteristic length to normalize the Jacobian matrix of a three-dof planar manipulator and to calculate the isoconditioning loci for all its working modes.
A planar three-dof manipulator with three parallel PRR chains, the object of this paper, is shown in Fig. 1. This manipulator has been frequently studied, in particular in [6][7]. The actuated joint variables are the displacements of the three prismatic joints, the Cartesian variables being the position vector p of the operation point P and the orientation θ of the platform.
The trajectories of the points A i define an equilateral triangle
whose geometric center is the point O, while the points B 1 , B 2 and B 3 , whose geometric center is the point P, lie at the corners of an equilateral triangle. We thus have α i = π + (i -1)(2π/3), for i = 1, 2, 3. Moreover, l = l 1 = l 2 = l 3 , with l i denoting the length of A i B i and r = r 1 = r 2 = r 3 , with r i denoting the length of B i P, in units of length that need not be specified in the paper. The layout of the trajectories of points A i is defined by the radius R of the circle inscribed in the associated triangle.
The velocity ṗ of point P can be obtained in three different forms, depending on which leg is traversed, namely,
with matrix E defined as
The velocity ȧi of points A i is given by ȧi
where e i is a unit vector in the direction of the ith prismatic joint.
We would like to eliminate the three idle joint rates η1 , η2 and η3 from eqs.(1a-c), which we do upon dot-multiplying the former by (b ia i ) T , thus obtaining
Equations (2a-c) can now be cast in vector form, namely,
with ρ ρ ρ thus being the vector of actuated joint rates.
Moreover, A and B are, respectively, the direct-kinematics and the inverse-kinematics matrices of the manipulator, defined as
When A and B are nonsingular, we obtain the relations
with K denoting the inverse of J.
Parallel singularities occur when the determinant of matrix A vanishes [8][9]. At these configurations, it is possible to move locally the operation point P with the actuators locked, the structure thus resulting cannot resist arbitrary forces, and control is lost. To avoid any performance deterioration, it is necessary to have a Cartesian workspace free of parallel singularities. For the planar manipulator studied, such configurations are reached whenever the axes A 1 B 1 , A 2 B 2 and A 3 B 3 intersect (possibly at infinity), as depicted in Fig. 2.
In the presence of such configurations, moreover, the manipulator cannot resist a force applied at the intersection point. These configurations are located inside the Cartesian workspace and form the boundaries of the joint workspace [8].
Serial singularities occur when det(B) = 0. In the presence of theses singularities, there is a direction along which no Cartesian velocity can be produced. Serial singularities define the boundary of the Cartesian workspace. For the topology under study, the serial singularities occur whenever (b ia i ) T e i = 0 for at least one value of i, as depicted in Fig. 3 for i = 2.
We derive below the loci of equal condition number of the matrices A, B and K. To do this, we first recall the definition of condition number of an m × n matrix M, with m ≤ n, κ(M). This number can be defined in various ways; for our purposes, we define κ(M) as the ratio
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