Designing Modular Lattice Systems with Chiral Space Groups

Abstract

We propose to use the concept of chiral space groups used by crystallography science to define and design lattice robots. Chiral space groups are of great interest because they give all possible sets of discrete displacements having a group structure and a translational symmetry. We explain the analogy between lattice robot kinematics and crystal symmetry, and identify three fundamental properties of lattice robots: (1) discreteness; (2) translational symmetry; and (3) composition. Then we give the possible connector symmetries and orientations into a chiral space group, and the possible sliding and hinge joint locations and orientations compatible with the displacements in the groups. We present a framework for the design of lattice robots by assembling compatible joints and connectors into a chiral space group. Several two-dimensional and three-dimensional examples of designs are given to illustrate the framework. Moreover, we list the symmetries of the two chiral space groups P432 and P622 because they contain the symmetries of all the 65 chiral space groups and allow the design of any lattice system.