Scalable enforcement of geometric non-interference constraints for gradient-based optimization

Abstract

Many design optimization problems include constraints to prevent intersection of the geometric shape being optimized with other objects or with domain boundaries. When applying gradient-based optimization to such problems, the constraint function must provide an accurate representation of the domain boundary and be smooth, amenable to numerical differentiation, and fast-to-evaluate for a large number of points. We propose the use of tensor-product B-splines to construct an efficient-to-evaluate level set function that locally approximates the signed distance function for representing geometric non-interference constraints. Adapting ideas from the surface reconstruction methods, we formulate an energy minimization problem to compute the B-spline control points that define the level set function given an oriented point cloud sampled over a geometric shape. Unlike previous explicit non-interference constraint formulations, our method requires an initial setup operation, but results in a more efficient-to-evaluate and scalable representation of geometric non-interference constraints. This paper presents the results of accuracy and scaling studies performed on our formulation. We demonstrate our method by solving a medical robot design optimization problem with non-interference constraints. We achieve constraint evaluation times on the order of 10-6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-6}$$\\end{document} seconds per point on a modern desktop workstation, and a maximum on-surface error of less than 1.0% of the minimum bounding box diagonal for all examples studied. Overall, our method provides an effective formulation for non-interference constraint enforcement with high computational efficiency for gradient-based design optimization problems whose solutions require at least hundreds of evaluations of constraints and their derivatives.