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Extra info for Technique of my musical language, vol.1: text
Ur´ızar et al. Fig. 1 3-RPR parallel manipulator. tensively studied by several authors [7,9,10]. In , a new class of 3-RPR analytic manipulator was found that contrary to the four type of analytic manipulators studied in , this aforementioned manipulator is able to perform non-singular transitions. In , the authors obtained analytically the conditions for the existence of cusp points in the joint space for the RPR-2PRR planar parallel manipulator. In this paper, the locus of cusp points for the general 3-RPR planar parallel manipulator will be obtained in a numeric way in the 3-dimensional joint space basing on the conditions developed in .
In ), which is simply called the Cayley–Menger determinant of the involved points. P3 In terms of Cayley–Menger determinants, the squared distance between Pi and Pj can be expressed as D(i, j ) and the signed area1 of the triangle Pi Pj Pk , h √ p3 θ as ± 12 D(i, j, k). For a brief review P2 of the properties of Cayley–Menger deP P1 terminants, see . p y p1 p2 The bilateration problem in R2 consists of finding the feasible locations of x a point, say P3 , given its distances to two other points, say P1 and P2 , whose locations are known.
For three-degree-offreedom (DoF) parallel manipulators, those special positions appear as cusp points in the DKP singular curves represented in the joint space for a constant value of one input. Performing non-singular transitions by surrounding cusp points has been ex- J. M. V. 2010 45 46 M. Ur´ızar et al. Fig. 1 3-RPR parallel manipulator. tensively studied by several authors [7,9,10]. In , a new class of 3-RPR analytic manipulator was found that contrary to the four type of analytic manipulators studied in , this aforementioned manipulator is able to perform non-singular transitions.