Stability Analysis and Control of Rigid-Body Systems With Impacts and Friction

Abstract

Abstract-Many critical tasks in robotics, such as locomotion or manipulation, involve collisions between a rigid body and the environment or between multiple bodies. Methods based on sums-of-squares (SOS) for numerical computation of Lyapunov certificates are a powerful tool for analyzing the stability of continuous nonlinear systems, and can additionally be used to automatically synthesize stabilizing feedback controllers. 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 safety 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, a perching aircraft, and control design of a balancing robot.

Balancing and Step Recovery Capturability via Sums-of-Squares Optimization

“…Stepping events will be assumed to occur after a fixed period, T , and result in a discrete event with the post-step state given by a reset map x + = r(x − , s, Λ) for the foot location s ∈ [−1, 1] and impact impulse Λ ∈ R. Here, we take r(x − , s, Λ) to be affine in s. Note that some models will follow the traditional LIPM approach and assume zero impulse during stepping, and others will include impulsive impact forces transmitted through a massless leg. While, for models with impacts, it might be possible to explicitly define the impulse Λ, we instead will exploit an implicit definition of inelastic impacts to reduce overall problem complexity, similar to the approach taken in [28]. In this formulation, valid impulses must satisfy an implicit constraint of the form…”

“…See [27] and [18] for an overview of SOS programming. Recent applications have included verification and control design of robotic systems, [22,34], hybrid systems [25], and of mechanical systems undergoing contact [28].…”

“…With the Sprocedure multipliers, the above optimization program will be bilinear in the unknown polynomials (V appears both in the antecedent and consequent of the implications). To solve it, we adopt a two-stage technique of bilinear alternations, similar to the approaches used in [37,21,28]. While these approaches offer no guarantee of optimality, they are practically effective and relatively straightforward to implement.…”

“…Here, we present a unified approach to capturability computations, applicable to a significantly broader class of models and controllers. To enable computation for more complex models, we base our approach on recent developments in sums-of-squares (SOS) programming [27,18], which has been effectively used to compute control policies and regions of attraction [34,25,35,22,28] and to approximate reachable sets [7].…”

A Sums-of-Squares extension of policy iterations

“…The stability analysis of continuous-time hybrid systems with SOS certificates was investigated in [PP09]. [PTT16] have recently applied analogous techniques to perform stability analysis and controller synthesis in the context of robotics. Other studies rely on SOS reinforcement and moment relaxations to obtain hierarchies of approximations converging to sets of interest such as the DoA in continuous-time, either from outside in [HK14] or from inside in [KHJ12].…”