Mark M. Tobenkin

LQR-trees: Feedback Motion Planning via Sums-of-Squares Verification

Advances in the direct computation of Lyapunov functions using convex optimization make it possible to efficiently evaluate regions of attraction for smooth non-linear systems. Here we present a feedback motion-planning algorithm which uses rigorously computed stability regions to build a sparse tree of LQR-stabilized trajectories. The region of attraction of this non-linear feedback policy “probabilistically covers” the entire controllable subset of state space, verifying that all initial conditions that are capable of reaching the goal will reach the goal. We numerically investigate the properties of this systematic non-linear feedback design algorithm on simple non-linear systems, prove the property of probabilistic coverage, and discuss extensions and implementation details of the basic algorithm.

Fast self-healing gradients

We present CRF-Gradient, a self-healing gradient algorithm that provably reconfigures in O(diameter) time. Self-healing gradients are a frequently used building block for distributed self-healing systems, but previous algorithms either have a healing rate limited by the shortest link in the network or must rebuild invalid regions from scratch. We have verified CRF-Gradient in simulation and on a network of Mica2 motes. Our approach can also be generalized and applied to create other self-healing calculations, such as cumulative probability fields.

Convex optimization of nonlinear feedback controllers via occupation measures

The construction of feedback control laws for underactuated nonlinear robotic systems with input saturation limits is crucial for dynamic robotic tasks such as walking, running, or flying. Existing techniques for feedback control design are either restricted to linear systems, rely on discretizations of the state space, or require solving a nonconvex optimization problem that requires feasible initialization. This paper presents a method for designing feedback controllers for polynomial systems that maximize the size of the time-limited backwards reachable set (BRS). In contrast to traditional approaches based on Lyapunov’s criteria for stability, we rely on the notion of occupation measures to pose this problem as an infinite-dimensional linear program which can then be approximated in finite dimension via semidefinite programs (SDPs). The solution to each SDP yields a polynomial control policy and an outer approximation of the largest achievable BRS which is well suited for use in a trajectory library or feedback motion planning algorithm. We demonstrate the efficacy and scalability of our approach on six nonlinear systems. Comparisons to an infinite-horizon linear quadratic regulator approach and an approach relying on Lyapunov’s criteria for stability are also included in order to illustrate the improved performance of the presented technique.

Invariant Funnels around Trajectories using Sum-of-Squares Programming

Abstract This paper presents numerical methods for computing regions of finite-time invariance (funnels) around solutions of polynomial differential equations. The methods are compared on stabilized trajectories of a six-state model of a satellite. First, we present a method which exactly certifies sufficient conditions for invariance despite relying on approximate trajectories from numerical integration. Our second method relaxes the constraints of the first by sampling in time. On the model system, this recovered almost identical funnels but was much faster to compute. In both cases, funnels are verified using Sum-of-Squares programming to search over time-varying quadratic Lyapunov functions. We examine both time-varying rescalings of quadratic forms computed from linearizations and searching directly over time-varying quadratic Lyapunov functions. On the model system the latter provided larger funnels at the cost of increased computation time.

Lyapunov analysis of rigid body systems with impacts and friction via sums-of-squares

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.

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

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.

Identification of nonlinear systems with stable oscillations

We propose a convex optimization procedure for identification of nonlinear systems that exhibit stable limit cycles. It extends the “robust identification error” framework in which a convex upper bound on simulation error is optimized to fit rational polynomial models with a strong stability guarantee. In this work, we relax the stability constraint using the concepts of transverse dynamics and orbital stability, thus allowing systems with autonomous oscillations to be identified. The resulting optimization problem is convex. The method is illustrated by identifying a high-fidelity model from experimental recordings of a live rat hippocampal neuron in culture.

Algebraic verification for parameterized motion planning libraries

Recent progress in algorithms for estimating regions of attraction and invariant sets of nonlinear systems has led to the application of these techniques to motion planning in complex environments. In most instances, the verification occurs offline as the algorithms are still too computationally demanding for realtime implementation; as a result any online planner is restricted to applying the finite set of motion plans that were verified offline. In this paper we attempt to present a partial remedy by algebraically verifying families of parameterized feedback controllers. We provide a specific example using LQR controllers parameterized by their goal or nominal motion. We formulate this verification using robust region of attraction techniques in sums-of-squares optimization, and show that perturbations of a Lyapunov or Riccati equation can be used to provide algebraically parameterized Lyapunov candidates. The resulting verified “funnels” then provide a parameterized motion library that can be used efficiently in online planning. We present a number of numerical examples to demonstrate the effectiveness of our approach.

Stable Nonlinear System Identification: Convexity, Model Class, and Consistency

Abstract Recently a new approach to black-box nonlinear system identification has been introduced which searches over a convex set of stable nonlinear models for the one which minimizes a convex upper bound of long-term simulation error. In this paper, we further study the properties of the proposed model set and identification algorithm and provide two theoretical results: (a) we show that the proposed model set includes all quadratically stable nonlinear systems, as well as some more complex systems; (b) we study the statistical consistency of the proposed identification method applied to a linear system with noisy measurements. It is shown a modification related to instrumental variables gives consistent parameter estimates.

Fast Self-stabilization for Gradients

Gradients are distributed distance estimates used as a building block in many sensor network applications. In large or long-lived deployments, it is important for the estimate to self-stabilize in response to changes in the network or ongoing computations, but existing algorithms may repair very slowly, produce distorted estimates, or suffer large transients. The CRF-Gradient algorithm[1] addresses these shortcomings, and in this paper we prove that it self-stabilizes in O (diameter ) time--more specifically, in 4 ·diameter /c + k seconds, where k is a small constant and c is the minimum speed of multi-hop message propagation.