We introduce a semidefinite relaxation for optimal control of linear systems with time scaling. These problems are inherently nonconvex, since the system dynamics involves bilinear products between the discretization time step and the system state and controls. The proposed relaxation is closely related to the standard second-order semidefinite relaxation for quadratic constraints, but we carefully select a subset of the possible bilinear terms and apply a change of variables to achieve empirically tight relaxations while keeping the computational load light. We further extend our method to handle piecewise-affine (PWA) systems by formulating the PWA optimal-control problem as a shortest-path problem in a graph of convex sets (GCS). In this GCS, different paths represent different mode sequences for the PWA system, and the convex sets model the relaxed dynamics within each mode. By combining a tight convex relaxation of the GCS problem with our semidefinite relaxation with time scaling, we can solve PWA optimal-control problems through a single semidefinite program.

Learning-based neural-network (NN) control policies have shown impressive empirical performance in a wide range of tasks in robotics and control. However, formal (Lyapunov) stability guarantees over the region-of-attraction (ROA) for NN controllers with nonlinear dynamical systems are challenging to obtain, and most existing approaches rely on expensive solvers for sums-of-squares (SOS), mixed-integer programming (MIP), or satisfiability modulo theories (SMT). In this paper, we demonstrate a new framework for learning NN controllers together with Lyapunov certificates using fast empirical falsification and strategic regularizations. We propose a novel formulation that defines a larger verifiable region-of-attraction (ROA) than shown in the literature, and refines the conventional restrictive constraints on Lyapunov derivatives to focus only on certifiable ROAs. The Lyapunov condition is rigorously verified post-hoc using branch-and-bound with scalable linear bound propagation-based NN verification techniques. The approach is efficient and flexible, and the full training and verification procedure is accelerated on GPUs without relying on expensive solvers for SOS, MIP, nor SMT. The flexibility and efficiency of our framework allow us to demonstrate Lyapunov-stable output feedback control with synthesized NN-based controllers and NN-based observers with formal stability guarantees, for the first time in literature.

The empirical success of reinforcement learning (RL) in contact-rich manipulation leaves much to be understood from a model-based perspective, where the key difficulties are often attributed to 1) the explosion of contact modes, 2) stiff, nonsmooth contact dynamics and the resulting exploding/discontinuous gradients, and 3) the nonconvexity of the planning problem. The stochastic nature of RL addresses 1) and 2) by effectively sampling and averaging the contact modes. On the other hand, model-based methods have tackled the same challenges by smoothing contact dynamics analytically. Our first contribution is to establish the theoretical equivalence of the two smoothing schemes for simple systems, and provide qualitative and empirical equivalence on several complex examples. In order to further alleviate 2), our second contribution is a convex, differentiable, and quasi-dynamic formulation of contact dynamics, which is amenable to both smoothing schemes, and has proven to be highly effective for contact-rich planning. Our final contribution resolves 3), where we show that classical sampling-based motion planning algorithms can be effective in global planning when contact modes are abstracted via smoothing. Applying our method on several challenging contact-rich manipulation tasks, we demonstrate that efficient model-based motion planning can achieve results comparable to RL, but with dramatically less computation.

Connected and automated vehicles (CAVs) have the potential to improve the safety of automated driving by utilizing increased awareness about their surroundings in real time vehicle control. In this paper we propose a framework for a shared perception system suitable for CAVs and explain the algorithms used in the system. Finally, we experimentally demonstrate the benefit of our shared perception system for automated vehicles in uncertain environments.