D. Chablat, J. Angeles

1 The Computation of All 4R Serial Spherical Wrists With an Isotropic Architecture Damien Chablat* and Jorge Angeles** *Institut de Recherche en Communications et Cybernétique de Nantes1 1, rue de la Noë, 44321 Nantes, France **Department of Mechanical Engineering & Centre for Intelligent Machines, McGill University 817 Sherbrooke Street West, Montreal, Canada H3A 2K6 Damien.Chablat@irccyn.ec-nantes.fr, angeles@cim.mcgill.ca Phone: 33 2 40 37 69 54 - Fax: 33 2 40 37 69 30 1 Abstract A spherical wrist of the serial type with n revolute (R) joints is said to be isotropic if it can attain a posture whereby the singular values of its Jacobian matrix are all equal to 3 / n . What isotropy brings about is robustness to manufacturing, assembly, and measurement errors, thereby guaranteeing a maximum orientation accuracy. In this paper we investigate the existence of redundant isotropic architectures, which should add to the dexterity of the wrist under design by virtue of its extra degree of freedom. The problem formulation, for 4

n , leads to a system of eight quadratic equations with eight unknowns. The Bezout number of this system is thus 256 28 = , its BKK bound being 192. However, the actual number of solutions is shown to be 32. We list all solutions of the foregoing algebraic problem. All these solutions are real, but distinct solutions do not necessarily lead to distinct manipulators. Upon discarding those algebraic solutions that yield no new wrists, we end up with exactly eight distinct architectures, the eight corresponding manipulators being displayed at their isotropic postures.
2 Introduction The kinematic design of redundant spherical wrists under isotropy conditions is the subject of this paper. A manipulator is called isotropic if its Jacobian matrix can attain isotropic values at certain postures [1,6]. A matrix, in turn, is called isotropic if its singular values are all identical and nonzero. Furthermore, the matrix condition number can be defined as the ratio of its greatest to its smallest singular values [2]. Thus, isotropic matrices have a minimum condition number of unity. The kinematic structure of industrial manipulators are frequently decoupled into a positioning and an orientation submanipulator. The latter is designed with revolute joints whose axes intersect. However, when these three joints are coplanar, the manipulator becomes singular [3]. As a means to cope with singularities, redundant wrists have been suggested [4]. An extensive bibliography on the design of spherical wrists can be found in [5-6].

1 IRCCyN: UMR n°6597 CNRS, École Centrale de Nantes, Université de Nantes, École des Mines de Nantes To appear in Journal of Mechanical Design

D. Chablat, J. Angeles

2 An isotropic Jacobian matrix is of interest because its condition number is unity, and hence, a minimum [2,6]. It is noteworthy that the condition number gives an upper bound for the relative roundoff-error amplification, upon solving a system of linear equations, with respect to the relative roundoff-error in the data. The latter are contained in the numerical entries of the Jacobian matrix and in the twist [7]. In fact, machining tolerances and assembly errors bring about additional errors in the Jacobian entries. Prior to our analysis leading to the architectures sought, we recall a few geometric concepts in the subsection below. 2.1 Isotropic Sets of Points on the Unit Sphere Consider the set { }n k P S 1 ≡ of n ≥ 3 points on the unit sphere, of position vectors { }n k 1 e . Apparently, all the vectors of the foregoing set are of unit Euclidean norm. The second-moment tensor H of S is defined as

∑

n T k k 1 e e H

(1) The set S is said to be isotropic if its second-moment tensor is isotropic. Since H is symmetric and positive- definite, it is isotropic if it is proportional to the 3 3× identity matrix 1, the proportionality factor, denoted here with 2 σ , being the square of the triple singular value of H. In our case, apparently, the singular values of H coincide with its eigenvalues. We note that, if S is the set of vertices of a Platonic solid, then H is isotropic. Table 1 records the values of n and σ for each Platonic solid.

Tetrahedron Cube Octahedron Dodecahedron Icosahedron
n 4 8 6 20 12 σ
4/3 8/3 2 20/3 4 Table 1: The values of n and σ for the Platonic solids Remark 1: It is apparent that, if a point k P of an arbitrary set S of points on the unit sphere is replaced by its antipodal k Q , of position vector k k e q −

, then the second-moment tensor H of S is preserved. The replacement of a point on the unit sphere by its antipodal will be termed, henceforth, antipodal exchange. As a consequence of Remark 1, then, the isotropy of a set of points on the unit sphere is preserved under

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