An algebraic formulation of kinematic isotropy and design of isotropic 6-6 Stewart platform manipulators

In this paper, we present a new method to study and design isotropic spatial parallel manipulators. We construct the Jacobian matrices of linear and angular velocities symbolically and analyse them. The isotropy of the angular velocity and that of the linear velocity are treated in a decoupled manner using an algebraic formulation that lead to eigenproblems of certain symmetric matrices. The criteria for isotropy in the individual cases, as well as their combination are expressed in closed-form in terms of the minimum number of algebraic equations. The proposed method is applied to the design of isotropic 6-6 Stewart platforms having semi-regular hexagonal top and bottom platforms. Symbolic expressions for two different families of isotropic configurations are obtained. Several mechanically feasible configurations are presented demonstrating orientation, position, and combined kinematic isotropy. Methods are presented for designing a manipulator for combined isotropy with partially specified position and orientation, and for determining the isotropic configurations of a manipulator with a given geometry. The sensitivity of the isotropy conditions with respect to variation in the configuration parameters is studied numerically using the example of an existing manipulator.