Anirudha Majumdar

Control design along trajectories with sums of squares programming

Motivated by the need for formal guarantees on the stability and safety of controllers for challenging robot control tasks, we present a control design procedure that explicitly seeks to maximize the size of an invariant “funnel” that leads to a predefined goal set. Our certificates of invariance are given in terms of sums of squares proofs of a set of appropriately defined Lyapunov inequalities. These certificates, together with our proposed polynomial controllers, can be efficiently obtained via semidefinite optimization. Our approach can handle time-varying dynamics resulting from tracking a given trajectory, input saturations (e.g. torque limits), and can be extended to deal with uncertainty in the dynamics and state. The resulting controllers can be used by space-filling feedback motion planning algorithms to fill up the space with significantly fewer trajectories. We demonstrate our approach on a severely torque limited underactuated double pendulum (Acrobot) and provide extensive simulation and hardware validation.

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.

Robust Online Motion Planning with Regions of Finite Time Invariance

In this paper we consider the problem of generating motion plans for a nonlinear dynamical system that are guaranteed to succeed despite uncertainty in the environment, parametric model uncertainty, disturbances, and/or errors in state estimation. Furthermore, we consider the case where these plans must be generated online, because constraints such as obstacles in the environment may not be known until they are perceived (with a noisy sensor) at runtime. Previous work on feedback motion planning for nonlinear systems was limited to offline planning due to the computational cost of safety verification. Here we take a trajectory library approach by designing controllers that stabilize the nominal trajectories in the library and precomputing regions of finite time invariance (”funnels”) for the resulting closed loop system. We leverage sums-of-squares programming in order to efficiently compute funnels which take into account bounded disturbances and uncertainty. The resulting funnel library is then used to sequentially compose motion plans at runtime while ensuring the safety of the robot. A major advantage of the work presented here is that by explicitly taking into account the effect of uncertainty, the robot can evaluate motion plans based on how vulnerable they are to disturbances.We demonstrate our method on a simulation of a plane flying through a two dimensional forest of polygonal trees with parametric uncertainty and disturbances in the form of a bounded ”cross-wind”.

DSOS and SDSOS optimization: LP and SOCP-based alternatives to sum of squares optimization

Sum of squares (SOS) optimization has been a powerful and influential addition to the theory of optimization in the past decade. Its reliance on relatively large-scale semidefinite programming, however, has seriously challenged its ability to scale in many practical applications. In this paper, we introduce DSOS and SDSOS optimization as more tractable alternatives to sum of squares optimization that rely instead on linear programming and second order cone programming. These are optimization problems over certain subsets of sum of squares polynomials and positive semidefinite matrices and can be of potential interest in general applications of semidefinite programming where scalability is a limitation.

Control and verification of high-dimensional systems with DSOS and SDSOS programming

In this paper, we consider linear programming (LP) and second order cone programming (SOCP) based alternatives to sum of squares (SOS) programming and apply this framework to high-dimensional problems arising in control applications. Despite the wide acceptance of SOS programming in the control and optimization communities, scalability has been a key challenge due to its reliance on semidefinite programming (SDP) as its main computational engine. While SDPs have many appealing features, current SDP solvers do not approach the scalability or numerical maturity of LP and SOCP solvers. Our approach is based on the recent work of Ahmadi and Majumdar [1], which replaces the positive semidefiniteness constraint inherent in the SOS approach with stronger conditions based on diagonal dominance and scaled diagonal dominance. This leads to the DSOS and SDSOS cones of polynomials, which can be optimized over using LP and SOCP respectively. We demonstrate this approach on four high dimensional control problems that are currently well beyond the reach of SOS programming: computing a region of attraction for a 22 dimensional system, analysis of a 50 node network of oscillators, searching for degree 3 controllers and degree 8 Lyapunov functions for an Acrobot system (with the resulting controller validated on a hardware platform), and a balancing controller for a 30 state and 14 control input model of the ATLAS humanoid robot. While there is additional conservatism introduced by our approach, extensive numerical experiments on smaller instances of our problems demonstrate that this conservatism can be small compared to SOS programming.

Flying between obstacles with an autonomous knife-edge maneuver

Avian flight far exceeds our best aircraft control systems. We have conducted a series of experiments at the Concord Field Station demonstrating the extraordinary maneuverability of the common pigeon, showing it darting through tight spaces and recovering from large disturbances with ease. Our goal is to understand how to make small fixed-wing aircraft achieve similar feats in equally challenging environments.

Complexity of ten decision problems in continuous time dynamical systems

We show that for continuous time dynamical systems described by polynomial differential equations of modest degree (typically equal to three), the following decision problems which arise in numerous areas of systems and control theory cannot have a polynomial time (or even pseudo-polynomial time) algorithm unless P=NP: local attractivity of an equilibrium point, stability of an equilibrium point in the sense of Lyapunov, boundedness of trajectories, convergence of all trajectories in a ball to a given equilibrium point, existence of a quadratic Lyapunov function, invariance of a ball, invariance of a quartic semialgebraic set under linear dynamics, local collision avoidance, and existence of a stabilizing control law. We also extend our earlier NP-hardness proof of testing local asymptotic stability for polynomial vector fields to the case of trigonometric differential equations of degree four.

Funnel Libraries for Real-Time Robust Feedback Motion Planning

We consider the problem of generating motion plans for a robot that are guaranteed to succeed despite uncertainty in the environment, parametric model uncertainty, and disturbances. Furthermore, we consider scenarios where these plans must be generated in real time, because constraints such as obstacles in the environment may not be known until they are perceived (with a noisy sensor) at runtime. Our approach is to pre-compute a library of “funnels” along different maneuvers of the system that the state is guaranteed to remain within (despite bounded disturbances) when the feedback controller corresponding to the maneuver is executed. We leverage powerful computational machinery from convex optimization (sums-of-squares programming in particular) to compute these funnels. The resulting funnel library is then used to sequentially compose motion plans at runtime while ensuring the safety of the robot. A major advantage of the work presented here is that by explicitly taking into account the effect of uncertainty, the robot can evaluate motion plans based on how vulnerable they are to disturbances. We demonstrate and validate our method using extensive hardware experiments on a small fixed-wing airplane avoiding obstacles at high speed (~12 mph), along with thorough simulation experiments of ground vehicle and quadrotor models navigating through cluttered environments. To our knowledge, these demonstrations constitute one of the first examples of provably safe and robust control for robotic systems with complex nonlinear dynamics that need to plan in real time in environments with complex geometric constraints.

Safety verification of reactive controllers for UAV flight in cluttered environments using barrier certificates

Unmanned aerial vehicles (UAVs) have a so-far untapped potential to operate at high speeds through cluttered environments. Many of these systems are limited by their adhoc reactive controllers using simple visual cues like optical flow. Here we consider the problem of formally verifying an output-feedback controller for an aircraft operating in an unknown environment. Using recent advances in sums-of-squares programming that allow for efficient computation of barrier functions, we search for global certificates of safety for the closed-loop system in a given environment. In contrast to previous work, we use rational functions to globally approximate non-smooth dynamics and use multiple barrier functions to guard against more than one obstacle. We expect that these formal verification techniques will allow for the comparison, and ultimately optimization, of reactive controllers for robustness to varying initial conditions and environments.

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.