Sampling-based Polytopic Trees for Approximate Optimal Control of Piecewise Affine Systems
This work addresses control problems for robotics applications with contact dynamics, offering an incremental improvement over existing techniques like LQR-trees.
The paper tackles the computational challenges in controlling piecewise affine systems, such as those in robot locomotion, by introducing a method that combines trajectory optimization and feedback control into a single step using mixed-integer convex programming, enabling handling of hard constraints.
Piecewise affine (PWA) systems are widely used to model highly nonlinear behaviors such as contact dynamics in robot locomotion and manipulation. Existing control techniques for PWA systems have computational drawbacks, both in offline design and online implementation. In this paper, we introduce a method to obtain feedback control policies and a corresponding set of admissible initial conditions for discrete-time PWA systems such that all the closed-loop trajectories reach a goal polytope, while a cost function is optimized. The idea is conceptually similar to LQR-trees \cite{tedrake2010lqr}, which consists of 3 steps: (1) open-loop trajectory optimization, (2) feedback control for computation of "funnels" of states around trajectories, and (3) repeating (1) and (2) in a way that the funnels are grown backward from the goal in a tree fashion and fill the state-space as much as possible. We show PWA dynamics can be exploited to combine step (1) and (2) into a single step that is tackled using mixed-integer convex programming, which makes the method suitable for dealing with hard constraints. Illustrative examples on contact-based dynamics are presented.