Concurrent Optimization of Morphology Design and PID Control for Running Robots on Unstructured Terrains

Abstract Designing legged robots for running in complex, unstructured environments requires systematic formulations of heterogeneous, multi-domain constraints and variables, including component choices, geometric choices, controller profiles, and touchdown conditions, which remains underexplored for small-scale (10-500 g) legged robots. At these scales, identifying feasible and robust running solutions becomes increasingly difficult—especially on rough terrain—due to limitations in physical parameters, reduced sensing capabilities, lack of a priori terrain knowledge, and limited onboard computing. This paper introduces an optimization formulation and solution framework that concurrently designs the physical system (morphology) and controllers for small-scale robots that are abstracted as torque-driven spring-loaded inverted pendulums (TD-SLIP). A set of constraints is defined to promote a symmetric gait, bounding touchdown conditions and vertical/horizontal displacements, which, along with the objective function, promote a greater number of gait cycles. Motion control is achieved with a combination of optimally tuned PID control of the stance phase and optimizing the motor-actuation-period for the flight phase. Optimization is performed using a standard mixed-discrete Particle Swarm Optimization algorithm. The optimized design on flat terrain achieved 19 stable gait cycles. Initializing with the flat terrain optimized design is found to be uniquely helpful in driving the optimization search over the design space for rough terrain, while achieving at least 14% better performance compared to the flat terrain design. Evaluated over a large set of unseen rough terrains, the optimized designs demonstrate stable gaits with 8 or more cycles in most cases.