Semidefinite Relaxations for Stochastic Optimal Control Policies

Recent results in the study of the Hamilton Jacobi Bellman (HJB) equation have led to the discovery of a formulation of the value function as a linear Partial Differential Equation (PDE) for stochastic nonlinear systems with a mild constraint on their disturbances. This has yielded promising directions for research in the planning and control of nonlinear systems. This work proposes a new method obtaining approximate solutions to these linear stochastic optimal control (SOC) problems. A candidate polynomial with variable coefficients is proposed as the solution to the SOC problem. A Sum of Squares (SOS) relaxation is then taken to the partial differential constraints, leading to a hierarchy of semidefinite relaxations with improving sub-optimality gap. The resulting approximate solutions are shown to be guaranteed over- and under-approximations for the optimal value function.Comment: Preprint. Accepted to American Controls Conference (ACC) 2014 in Portland, Oregon. 7 pages, colo

Optimal Navigation Functions for Nonlinear Stochastic Systems

This paper presents a new methodology to craft navigation functions for nonlinear systems with stochastic uncertainty. The method relies on the transformation of the Hamilton-Jacobi-Bellman (HJB) equation into a linear partial differential equation. This approach allows for optimality criteria to be incorporated into the navigation function, and generalizes several existing results in navigation functions. It is shown that the HJB and that existing navigation functions in the literature sit on ends of a spectrum of optimization problems, upon which tradeoffs may be made in problem complexity. In particular, it is shown that under certain criteria the optimal navigation function is related to Laplace's equation, previously used in the literature, through an exponential transform. Further, analytical solutions to the HJB are available in simplified domains, yielding guidance towards optimality for approximation schemes. Examples are used to illustrate the role that noise, and optimality can potentially play in navigation system design.Comment: Accepted to IROS 2014. 8 Page

Domain Decomposition for Stochastic Optimal Control

This work proposes a method for solving linear stochastic optimal control (SOC) problems using sum of squares and semidefinite programming. Previous work had used polynomial optimization to approximate the value function, requiring a high polynomial degree to capture local phenomena. To improve the scalability of the method to problems of interest, a domain decomposition scheme is presented. By using local approximations, lower degree polynomials become sufficient, and both local and global properties of the value function are captured. The domain of the problem is split into a non-overlapping partition, with added constraints ensuring $C^1$ continuity. The Alternating Direction Method of Multipliers (ADMM) is used to optimize over each domain in parallel and ensure convergence on the boundaries of the partitions. This results in improved conditioning of the problem and allows for much larger and more complex problems to be addressed with improved performance.Comment: 8 pages. Accepted to CDC 201

Convex Relaxations of SE(2) and SE(3) for Visual Pose Estimation

This paper proposes a new method for rigid body pose estimation based on spectrahedral representations of the tautological orbitopes of $SE(2)$ and $SE(3)$. The approach can use dense point cloud data from stereo vision or an RGB-D sensor (such as the Microsoft Kinect), as well as visual appearance data. The method is a convex relaxation of the classical pose estimation problem, and is based on explicit linear matrix inequality (LMI) representations for the convex hulls of $SE(2)$ and $SE(3)$. Given these representations, the relaxed pose estimation problem can be framed as a robust least squares problem with the optimization variable constrained to these convex sets. Although this formulation is a relaxation of the original problem, numerical experiments indicate that it is indeed exact - i.e. its solution is a member of $SE(2)$ or $SE(3)$ - in many interesting settings. We additionally show that this method is guaranteed to be exact for a large class of pose estimation problems.Comment: ICRA 2014 Preprin

Linear Hamilton Jacobi Bellman Equations in High Dimensions

The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal solution to large classes of control problems. Unfortunately, this generality comes at a price, the calculation of such solutions is typically intractible for systems with more than moderate state space size due to the curse of dimensionality. This work combines recent results in the structure of the HJB, and its reduction to a linear Partial Differential Equation (PDE), with methods based on low rank tensor representations, known as a separated representations, to address the curse of dimensionality. The result is an algorithm to solve optimal control problems which scales linearly with the number of states in a system, and is applicable to systems that are nonlinear with stochastic forcing in finite-horizon, average cost, and first-exit settings. The method is demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with system dimension two, six, and twelve respectively.Comment: 8 pages. Accepted to CDC 201

Linearly Solvable Stochastic Control Lyapunov Functions

This paper presents a new method for synthesizing stochastic control Lyapunov functions for a class of nonlinear stochastic control systems. The technique relies on a transformation of the classical nonlinear Hamilton-Jacobi-Bellman partial differential equation to a linear partial differential equation for a class of problems with a particular constraint on the stochastic forcing. This linear partial differential equation can then be relaxed to a linear differential inclusion, allowing for relaxed solutions to be generated using sum of squares programming. The resulting relaxed solutions are in fact viscosity super/subsolutions, and by the maximum principle are pointwise upper and lower bounds to the underlying value function, even for coarse polynomial approximations. Furthermore, the pointwise upper bound is shown to be a stochastic control Lyapunov function, yielding a method for generating nonlinear controllers with pointwise bounded distance from the optimal cost when using the optimal controller. These approximate solutions may be computed with non-increasing error via a hierarchy of semidefinite optimization problems. Finally, this paper develops a-priori bounds on trajectory suboptimality when using these approximate value functions, as well as demonstrates that these methods, and bounds, can be applied to a more general class of nonlinear systems not obeying the constraint on stochastic forcing. Simulated examples illustrate the methodology.Comment: Published in SIAM Journal of Control and Optimizatio

Omnidirectional Sensory and Motor Volumes in Electric Fish

Active sensing organisms, such as bats, dolphins, and weakly electric fish, generate a 3-D space for active sensation by emitting self-generated energy into the environment. For a weakly electric fish, we demonstrate that the electrosensory space for prey detection has an unusual, omnidirectional shape. We compare this sensory volume with the animal's motor volume—the volume swept out by the body over selected time intervals and over the time it takes to come to a stop from typical hunting velocities. We find that the motor volume has a similar omnidirectional shape, which can be attributed to the fish's backward-swimming capabilities and body dynamics. We assessed the electrosensory space for prey detection by analyzing simulated changes in spiking activity of primary electrosensory afferents during empirically measured and synthetic prey capture trials. The animal's motor volume was reconstructed from video recordings of body motion during prey capture behavior. Our results suggest that in weakly electric fish, there is a close connection between the shape of the sensory and motor volumes. We consider three general spatial relationships between 3-D sensory and motor volumes in active and passive-sensing animals, and we examine hypotheses about these relationships in the context of the volumes we quantify for weakly electric fish. We propose that the ratio of the sensory volume to the motor volume provides insight into behavioral control strategies across all animals

Suboptimal stabilizing controllers for linearly solvable system

This paper presents a novel method to synthesize stochastic control Lyapunov functions for a class of nonlinear, stochastic control systems. In this work, the classical nonlinear Hamilton-Jacobi-Bellman partial differential equation is transformed into a linear partial differential equation for a class of systems with a particular constraint on the stochastic disturbance. It is shown that this linear partial differential equation can be relaxed to a linear differential inclusion, allowing for approximating polynomial solutions to be generated using sum of squares programming. It is shown that the resulting solutions are stochastic control Lyapunov functions with a number of compelling properties. In particular, a-priori bounds on trajectory suboptimality are shown for these approximate value functions. The result is a technique whereby approximate solutions may be computed with non-increasing error via a hierarchy of semidefinite optimization problems

Convex Model Predictive Control for Vehicular Systems

In this work, we present a method to perform Model Predictive Control (MPC) over systems whose state is an element of $SO(n)$ for $n=2,3$. This is done without charts or any local linearization, and instead is performed by operating over the orbitope of rotation matrices. This results in a novel MPC scheme without the drawbacks associated with conventional linearization techniques. Instead, second order cone- or semidefinite-constraints on state variables are the only requirement beyond those of a QP-scheme typical for MPC of linear systems. Of particular emphasis is the application to aeronautical and vehicular systems, wherein the method removes many of the transcendental trigonometric terms associated with these systems' state space equations. Furthermore, the method is shown to be compatible with many existing variants of MPC, including obstacle avoidance via Mixed Integer Linear Programming (MILP)

Novel nano-composite biomaterials that respond to light

Composites of nanoparticles and polymers are finding wide applications to alter material properties, conductivity, and utility. Here, we show that nano-composites can be designed to heat in the presence of near infrared light. This process is useful in transitioning materials through a transition temperature for a range of applications. For example, shape-memory materials (including polymers, metals, and ceramics) are those that are processed into a temporary shape and respond to some external stimuli (e.g., temperature) to undergo a transition back to a permanent shape and may be useful in a range of applications from aerospace to fabrics, to biomedical devices and microsystem components. In this work, we formulated composites of gold nanorods (\u3c1% by volume) and biodegradable networks, where exposure to infrared light induced heating and consequently, shape transitions. The heating is repeatable and tunable based on nanorod concentration and light intensity