Simplex-Tree Based Kinematics of Foldable Objects as Multi-body Systems Involving Loops

Many practical multi-body systems involve loops. Studying the kinematics of such systems has been challenging, partly because of the requirement of maintaining loop closure constraints, which have conventionally been formulated as highly nonlinear equations in joint parameters. Recently, novel parameters defined by trees of triangles have been introduced for a broad class of linkage systems involving loops (e.g., spatial loops with spherical joints and planar loops with revolute joints); these parameters greatly simplify kinematics related computations and endow system configuration spaces with highly tractable piecewise convex geometries. In this paper, we describe a more general approach for multi-body systems, with loops, that allow construction trees of simplices. We illustrate the applicability and efficiency of our simplex-tree based approach to kinematics by a study of foldable objects. We present two sets of new parameters for single-vertex rigid fold kinematics; like the parameters in the triangle-tree prototype, each has a geometrically meaningful and computationally tractable constraint formulation, and each endows the configuration space with a nice geometry.