Speaker
Description
Designing mechanisms and machines that execute periodic motions typically requires considerations of speed and energy efficiency. This is a challenging task due to the deep interdependence created by the mechanical dynamics between the hardware (morphology) and the control (motion). The co-design of such mechatronic systems involves the simultaneous and integrated development of both the physical components (such as actuators and mechanical structures) and the control algorithms that govern the system's behavior.
To address this in a design assistant, we formulate an extended trajectory optimization problem to simultaneously identify optimal designs that generate optimal periodic motions. Such problems are inherently non-convex due to nonlinear dynamics and implicit boundary constraints coupling initial and final conditions.
To cope with this challenge, we explore the use of the Koopman operator framework to tackle trajectory optimization problems with a partially convex approach. While the Koopman operator has been successfully applied in model predictive control, handling mixed boundary constraints within this framework remains an open challenge. This work makes two key contributions: (1) we demonstrate why full convexification of the problem is fundamentally infeasible within the Koopman operator framework, and (2) we propose a bilevel optimization approach that convexifies the high-dimensional lower-level problem, resulting in a simplified, low-dimensional upper-level problem. This decomposition not only facilitates efficient computation but also supports global optimization strategies.
The bi-level structure is particularly suited for co-design, as it allows the morphology design (upper-level) and motion planning (lower-level) to be addressed independently yet collaboratively. To validate the method, we present case studies on two systems: the mathematical pendulum and the compass-gait walker, showcasing the approach's effectiveness in optimizing periodic motion.
