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To recover from large perturbations, a legged robot must make and break contact with its environment at various locations. These contact switches make it natural to model the robot as a hybrid system. If we apply Model Predictive Control to the feedback design of this hybrid system, the on/off behavior of contacts can be directly encoded using binary variables in a Mixed Integer Programming problem, which scales badly with the number of time steps and is too slow for online computation. We propose novel techniques for the design of stabilizing controllers for such hybrid systems. We approximate the dynamics of the system as a discrete-time Piecewise Affine (PWA) system, and compute the state feedback controllers across the hybrid modes offline via Lyapunov theory. The Lyapunov stability conditions are translated into Linear Matrix Inequalities. A Piecewise Quadratic Lyapunov function together with a Piecewise Linear (PL) feedback controller can be obtained by Semidefinite Programming (SDP). We show that we can embed a quadratic objective in the SDP, designing a controller approximating the Piecewise-Affine Quadratic Regulator. Moreover, we observe that our formulation restricted to the linear system case appears to always produce exactly the unique stabilizing solution to the Discrete Algebraic Riccati Equation. In addition, we extend the search from the PL controller to the PWA controller via Bilinear Matrix Inequalities. Finally, we demonstrate and evaluate our methods on a few PWA systems, including a simplified humanoid robot model.

In this paper, we design nonlinear state feedback controllers for discrete-time polynomial dynamical systems via the occupation measure approach. We propose the discrete-time controlled Liouville equation, and use it to formulate the controller synthesis problem as an infinite-dimensional linear programming problem on measures, which is then relaxed as finitedimensional semidefinite programming problems on moments of measures and their duals on sums-of-squares polynomials. Nonlinear controllers can be extracted from the solutions to the relaxed problems. The advantage of the occupation measure approach is that we solve convex problems instead of generally non-convex problems, and the computational complexity is polynomial in the state and input dimensions, and hence the approach is more scalable. In addition, we show that the approach can be applied to over-approximating the backward reachable set of discrete-time autonomous polynomial systems and the controllable set of discrete-time polynomial systems under known state feedback control laws. We illustrate our approach on several dynamical systems.

In this paper, we address the trajectory planning problem in uncertain nonconvex static and dynamic environments that contain obstacles with probabilistic location, size, and geometry. To address this problem, we provide a risk bounded trajectory planning method that looks for continuous-time trajectories with guaranteed bounded risk over the planning time horizon. Risk is defined as the probability of collision with uncertain obstacles. Existing approaches to address risk bounded trajectory planning problems either are limited to Gaussian uncertainties and convex obstacles or rely on sampling-based methods that need uncertainty samples and time discretization. To address the risk bounded trajectory planning problem, we leverage the notion of risk contours to transform the risk bounded planning problem into a deterministic optimization problem. Risk contours are the set of all points in the uncertain environment with guaranteed bounded risk. The obtained deterministic optimization is, in general, nonlinear and nonconvex time-varying optimization. We provide convex methods based on sum-of-squares optimization to efficiently solve the obtained nonconvex time-varying optimization problem and obtain the continuous-time risk bounded trajectories without time discretization. The provided approach deals with arbitrary probabilistic uncertainties, nonconvex and nonlinear, static and dynamic obstacles, and is suitable for online trajectory planning problems.

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