The computation of all 4R serial spherical wrists with an isotropic architecture

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 rootn/3. 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 n=4, leads to a system of eight quadratic equations with eight unknowns. The Bezout number of this system is thus 2(8)=256, 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.