Vectorized Formulation of Newton-Euler Dynamics for Efficiently Computing Three-Dimensional Folding Chains

Abstract

Within the wide field of self-assembly, the self-folding chain has unique potential for reliable and repeatable assembly of three-dimensional structures as demonstrated by protein biosynthesis. This potential could be translated to self-reconfiguring robots by utilizing magnetic forces between the chain components as a driving force for the folding process. Due to the constraints introduced by the joints between the chain components, simulation of the dynamics of longer chains is computationally intensive and challenging. This article presents a novel analytical approach to formulate the Newton–Euler dynamics of a self-reconfiguring chain in a single vectorized differential equation. The vectorized differential equation allows for a convenient implementation of a parallel processing architecture using single instruction multiple data (SIMD) or graphical processing unit (GPU) computation and as a result can improve simulation time of rigid body chains. Properties of existing interpretations of the Newton–Euler and Euler–Lagrange algorithms are discussed in their efficiency to compute the dynamics of rigid body chains. Finally, GPU and SIMD-supported simulation, based on the vectorized Newton–Euler equations described in this article, are compared, showing a significant improvement in computation time using GPU architecture for long chains with certain chain geometry.