UC Berkeley

Discrete elastic rods (DER) is a recent formulation of a rod theory by Bergou et al. The material curve of the rod is approximated by a discrete set of lines connected at vertices. The formulation originated in the field of computer graphics and uses concepts from the nascent field of discrete differential geometry to characterize bending energies and torsional strains. Specifically, the discrete curvature vector associated with a vertex is used as a measure of bending strain and the length of the edges are used to account for stretching. Additionally, each edge is associated with a reference frame and a material frame, where the angle difference of the latter frame between adjacent edges is a measure of twist. Space- and time- parallel transport operators are introduced to update these frames in space and time respectively, so the torsion of the rod can be efficiently computed.

While DER is an elegant formulation, it is challenging to comprehend. In this dissertation, complete derivations for the expressions for the variations, gradients, and Hessians of kinematic variables induced by changes to the vertices are presented. These gradients are needed to numerically solve the governing equations of motion. The method by which a component of the rotation of the cross section is computed in the discrete elastic rod formulation is exceptional and exploits a phenomenon in differential geometry known as a holonomy. Relevant background from differential geometry and spherical geometry are presented to understand how the reference twist in the rod can be related to a solid angle enclosed by the trace of a unit tangent vector on a sphere and several examples are presented to illuminate the calculation of twist.

The second part of the dissertation is devoted to using the DER formulation to examine the dynamics of soft robots. To this end, a planar formulation of DER (PDER) is derived. Our work allows the governing equations of a discrete rod to be expressed in a canonical using Lagrange’s equations. This in turn allows us to use PDER with folded and branched elastic structures which feature in the designs of soft robots. To illustrate our developments, PDER is used to formulate and analyze the equations of motion needed to simulate the locomotion of a caterpillar-inspired soft robot on a rough surface.