Many critical tasks in robotics, such as locomotion or manipulation, involve collisions between a rigid body and the environment or between multiple bodies. Sums-of-squares (SOS) based methods for numerical computation of Lyapunov certificates are a powerful tool for analyzing the stability of continuous nonlinear systems, which can play a powerful role in motion planning and control design. Here, we present a method for applying sums-of-squares verification to rigid bodies with Coulomb friction undergoing discontinuous, inelastic impact events. The proposed algorithm explicitly generates Lyapunov certificates for stability, positive invariance, and reachability over admissible (non-penetrating) states and contact forces. We leverage the complementarity formulation of contact, which naturally generates the semialgebraic constraints that define this admissible region. The approach is demonstrated on multiple robotics examples, including simple models of a walking robot and a perching aircraft.

Dynamic perching maneuvers for fixed-wing aircraft are becoming increasingly plausible due to recent progress in perching using `micro-spines' mounted on tuned suspensions and, separately, on feedback motion planning techniques for post-stall maneuvering. In this paper, we bring these complementary techniques together by efficiently estimating the mechanical stability of the plane when it makes contact with a vertical surface; the resulting landing funnel can then be used in a feedback motion planning algorithm for the flight controller. We consider a simplified model of the perching dynamics and report an extension of the region of attraction techniques, using sums-of-squares optimization, which combines polynomial approximations of barrier constraints with the traditional Lyapunov methods to achieve tight estimation of the true region of attraction for the model. We demonstrate the new method on a variety of design parameters for the perching system, suggesting a potential use as a mechanical system or controller design tool.

Gel is a hardware description language that enables quick scripting of high level designs and can be easily extended to new design patterns. It is expression oriented and extremely succinct. Modules are described as functions and composed through function calls. Types and bit widths are inferred automatically to guarantee correctness. Together these features reduce hardware development time, allowing complex designs to be scripted quickly. A simulator and logic analyzer are available to help in the development process. A compiler has been developed that translates Gel to Verilog, and a number of applications have been demonstrated. This paper introduces the core language, demonstrates its extensibility, and shows how design patterns can easily be created. Finally, we compare a few applications written in Gel against equivalents written in Verilog.