Ian R. Manchester

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.

Stable dynamic walking over uneven terrain

We propose a constructive control design for stabilization of non-periodic trajectories of underactuated robots. An important example of such a system is an underactuated “dynamic walking” biped robot traversing rough or uneven terrain. The stabilization problem is inherently challenging due to the nonlinearity, open-loop instability, hybrid (impact) dynamics, and target motions which are not known in advance. The proposed technique is to compute a transverse linearization about the desired motion: a linear impulsive system which locally represents “transversal” dynamics about a target trajectory. This system is then exponentially stabilized using a modified receding-horizon control design, providing exponential orbital stability of the target trajectory of the original nonlinear system. The proposed method is experimentally verified using a compass-gait walker: a two-degree-of-freedom biped with hip actuation but pointed stilt-like feet. The technique is, however, very general and can be applied to a wide variety of hybrid nonlinear systems.

Circular-Navigation-Guidance Law for Precision Missile/Target Engagements

A new precision guidance law with impact angle constraint for a two-dimensional planar intercept is presented. It is based on the principle of following a circular arc to the target, hence the name circular navigation guidance. The guidance law does not require range-to-target information. We prove that it attains a perfect intercept under certain ideal conditions. In a broader range of conditions, it is shown to perform favorably when compared to another law from the literature.

Can we make a robot ballerina perform a pirouette? Orbital stabilization of periodic motions of underactuated mechanical systems☆

This paper provides an introduction to several problems and techniques related to controlling periodic motions of dynamical systems. In particular, we consider planning periodic motions and designing feedback controllers for orbital stabilization. We review classical and recent design methods based on the Poincare first-return map and the transverse linearization. We begin with general nonlinear systems and then specialize to a class of underactuated mechanical systems for which a particularly rich structure allows many of the problems to be solved analytically.

Bounding on rough terrain with the LittleDog robot

A motion planning algorithm is described for bounding over rough terrain with the LittleDog robot. Unlike walking gaits, bounding is highly dynamic and cannot be planned with quasi-steady approximations. LittleDog is modeled as a planar five-link system, with a 16-dimensional state space; computing a plan over rough terrain in this high-dimensional state space that respects the kinodynamic constraints due to underactuation and motor limits is extremely challenging. Rapidly Exploring Random Trees (RRTs) are known for fast kinematic path planning in high-dimensional configuration spaces in the presence of obstacles, but search efficiency degrades rapidly with the addition of challenging dynamics. A computationally tractable planner for bounding was developed by modifying the RRT algorithm by using: (1) motion primitives to reduce the dimensionality of the problem; (2) Reachability Guidance, which dynamically changes the sampling distribution and distance metric to address differential constraints and discontinuous motion primitive dynamics; and (3) sampling with a Voronoi bias in a lower-dimensional “task space” for bounding. Short trajectories were demonstrated to work on the robot, however open-loop bounding is inherently unstable. A feedback controller based on transverse linearization was implemented, and shown in simulation to stabilize perturbations in the presence of noise and time delays.

Circular Navigation Missile Guidance with Incomplete Information and Uncertain Autopilot Model

4Goldberg, D., Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley Longman, Reading, MA, 1989, p. 22. 5Isaacs, R., Differential Games, Wiley, New York, 1965, p. 279. 6Horie, K., “Collocation with Nonlinear Programming for Two-Sided Flight Path Optimization,” Ph.D. Dissertation, Dept. of Aeronautical and Astronautical Engineering, Univ. of Illinois at Urbana–Champaign, Urbana, IL, Feb. 2002. 7Goldberg, D. E., and Voessner, S., “Optimizing Global-Local Search Hybrids,” Proceedings of the Genetic and Evolutionary Computation Conference 1999, Morgan Kaufmann, San Francisco, 1999.

Stability Analysis and Control Design for an Underactuated Walking Robot via Computation of a Transverse Linearization

The problem is to create a hybrid periodic motion, reminiscent of walking, for a model of an underactuated biped robot. We show how to construct a transverse linearization analytically and how to use it for stability analysis and for design of an exponentially orbitally stabilizing controller. In doing so, we extend a technique recently developed for continuous-time controlled mechanical systems with degree of underactuation one. All derivations are shown on an example of a three-link walking robot, modeled as a system with impulse effects.

Precision missile guidance using radar/multiple-video sensor fusion via communication channels with bit-rate constraints

We consider a precision missile guidance problem in which the objective is twofold: to come as close as possible to hitting the target, and also to do so from a particular direction. The effectiveness of a guidance law is strongly dependent on the quality of information available to it. In this work we construct a precision guidance law that combines information from an on-board video camera with data transmitted from ground-based radars and video cameras mounted on unmanned aerial vehicles. The communication channels are bit-rate limited, and recent results in control and estimation over finite-data-rate channels are used to construct a nonlinear state estimator. Simulations show the performance improvements which are possible compared to the use of the on-board sensor alone.

Transverse contraction criteria for existence, stability, and robustness of a limit cycle

This paper derives a differential contraction condition for the existence of an orbitally-stable limit cycle in an autonomous system. This transverse contraction condition can be represented as a pointwise linear matrix inequality (LMI), thus allowing convex optimization tools such as sum-of-squares programming to be used to search for certificates of the existence of a stable limit cycle. Many desirable properties of contracting dynamics are extended to this context, including preservation of contraction under a broad class of interconnections. In addition, by introducing the concepts of differential dissipativity and transverse differential dissipativity, contraction and transverse contraction can be established for large scale systems via LMI conditions on component subsystems.

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.