Joint Search of Optimal Topology and Trajectory for Planar Linkages

TL;DR

This paper introduces a semi-automatic, optimization-based approach to design planar linkages that realize user-specified trajectories, improving efficiency and optimality over traditional trial-and-error methods.

Contribution

It formulates the linkage design as a non-smooth MIQCQP and compares three algorithms, proposing a hybrid method that efficiently finds optimal solutions.

Findings

Hybrid algorithm outperforms simulated annealing in speed and optimality.

Effective design of planar linkages for complex trajectories within a few hours.

Validated linkages successfully used as robot legs.

Abstract

We present an algorithm to compute planar linkage topology and geometry, given a user-specified end-effector trajectory. Planar linkage structures convert rotational or prismatic motions of a single actuator into an arbitrarily complex periodic motion, \refined{which is an important component when building low-cost, modular robots, mechanical toys, and foldable structures in our daily lives (chairs, bikes, and shelves). The design of such structures require trial and error even for experienced engineers. Our research provides semi-automatic methods for exploring novel designs given high-level specifications and constraints.} We formulate this problem as a non-smooth numerical optimization with quadratic objective functions and non-convex quadratic constraints involving mixed-integer decision variables (MIQCQP). We propose and compare three approximate algorithms to solve this problem: mixed-integer conic-programming (MICP), mixed-integer nonlinear programming (MINLP), and simulated annealing (SA). We evaluated these algorithms searching for planar linkages involving $10-14$ rigid links. Our results show that the best performance can be achieved by combining MICP and MINLP, leading to a hybrid algorithm capable of finding the planar linkages within a couple of hours on a desktop machine, which significantly outperforms the SA baseline in terms of optimality. We highlight the effectiveness of our optimized planar linkages by using them as legs of a walking robot.