Using Motion Planning for Knot Untangling

In this paper we investigate the application of motion planning techniques to the untangling of mathematical knots. Knot untangling can be viewed as a high-dimensional planning problem in reparametrizable configuration spaces. In the past, simulated annealing and other energy minimization methods have been used to find knot untangling paths. We have developed a probabilistic planner that is capable of untangling knots described by over 400 variables. We have tested on known difficult benchmarks in this area and untangled them more quickly than has been achieved with minimization in the literature. In this work, the use of motion planning techniques is critical for the untangling. Our planner defines local goals and makes combined use of energy minimization and randomized tree-based planning. We also show how to produce candidates with a minimal number of segments for a given knot. The planner developed in this work is novel in that it is used to study issues arising in practical motion planning for high-dimensional and reparametrizable geometry. The use of energy methods, local goals and tree-based expansion is also novel and may suggest solutions in other planning applications. Finally, we discuss some possible applications of our untangling planner in computational topology, in the study of DNA rings and protein folding and for planning with flexible robots. KEY WORDS—motion planning, knots, flexible objects, probabilistic methods