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We present an algorithm to compute planar linkage topology and geometry,given a user-specified end-effector trajectory. Planar linkage structuresconvert rotational or prismatic motions of a single actuator into anarbitrarily complex periodic motion, \refined{which is an important componentwhen building low-cost, modular robots, mechanical toys, and foldablestructures in our daily lives (chairs, bikes, and shelves). The design of suchstructures require trial and error even for experienced engineers. Our researchprovides semi-automatic methods for exploring novel designs given high-levelspecifications and constraints.} We formulate this problem as a non-smoothnumerical optimization with quadratic objective functions and non-convexquadratic constraints involving mixed-integer decision variables (MIQCQP). Wepropose 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 searchingfor planar linkages involving $10-14$ rigid links. Our results show that thebest performance can be achieved by combining MICP and MINLP, leading to ahybrid algorithm capable of finding the planar linkages within a couple ofhours on a desktop machine, which significantly outperforms the SA baseline interms of optimality. We highlight the effectiveness of our optimized planarlinkages by using them as legs of a walking robot.

Joint Search of Optimal Topology and Trajectory for Planar Linkages