Using R-Functions to Control the Shape of Soft Robots

Abstract

In this letter, we introduce a new approach for soft robot shape formation and morphing using approximate distance fields. The method uses concepts from constructive solid geometry, R-functions, to construct an approximate distance function to the boundary of a domain in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Re ^{d}$</tex-math></inline-formula> . The gradients of the R-functions can then be used to generate control algorithms for shape formation tasks for soft robots. By construction, R-functions are smooth and convex everywhere, possess precise differential properties, and easily extend from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Re ^{2}$</tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Re ^{3}$</tex-math></inline-formula> if needed. Furthermore, R-function theory provides a straightforward method to creating composite distance functions for any desired shape by combining subsets of distance functions. The process is highly efficient since the shape description is an analytical expression, and in this sense, it is better than competing control algorithms such as those based on potential fields. Although the method could also apply to swarm robots, in this letter it is applied to soft robots to demonstrate shape formation and morphing in 2-D (simulation and experimentation) and 3-D (simulation).