Symmetries in Workspace Densities of Discretely Actuated Manipulators

Abstract

Discrete actuators have a finite number of states, e.g. stepper motors and bistable pneumatic cylinders. Given that the number of dis­tinct configurations attainable by discretely actuated manipulators grows exponentially in the number of actuated degrees of freedom, i.e. K ⁿ for n actuators or actuated modules each with K states, brute force representation of discrete manipulator workspaces is not feasible in the highly actuated case. Approximating the workspaces of segments of discrete manipulators as density functions on the Euclidean group (which describes workspace positions and orientations) the whole workspace can be approximated as an n-fold convolution of these functions. We have shown in the past that using convolution as a computational tool reduces an O(K ⁿ)problem to something that is linear in n.

When the workspace density function of a manipulator has certain discrete and/or continuous symmetries, the amount of data that must be stored can be reduced dramatically, and convolutions can be performed very efficiently. Here we explore various symmetries that result in practical manipulator designs.