Eric W. Justh

Equilibria and steering laws for planar formations

Abstract This paper presents a Lie group setting for the problem of control of formations, as a natural outcome of the analysis of a planar two-vehicle formation control law. The vehicle trajectories are described using the planar Frenet–Serret equations of motion, which capture the evolution of both the vehicle position and orientation for unit-speed motion subject to curvature (steering) control. The set of all possible (relative) equilibria for arbitrary G -invariant curvature controls is described (where G = SE (2) is a symmetry group for the control law), and a global convergence result for the two-vehicle control law is proved. An n -vehicle generalization of the two-vehicle control law is also presented, and the corresponding (relative) equilibria for the n -vehicle problem are characterized. Work is on-going to discover stability and convergence results for the n -vehicle problem.

Natural frames and interacting particles in three dimensions

Motivated by the problem of formation control for vehicles moving at unit speed in three-dimensional space, we are led to models of gyroscopically interacting particles, which require the machinery of curves and frames to describe and analyze. A Lie group formulation arises naturally, and we discuss the general problem of determining (relative) equilibria for arbitrary G-invariant controls (where G = SE(3) is a symmetry group for the control law). We then present global convergence (and non-collision) results for specific two-vehicle interaction laws in three dimensions, which lead to specific formations (i.e., relative equilibria). Generalizations of the interaction laws to n vehicles is also discussed, and simulation results presented.

Steering laws for motion camouflage

Motion camouflage is a stealth strategy observed in nature. We formulate the problem as a feedback system for particles moving at constant speed, and define what it means for the system to be in a state of motion camouflage. (Here, we focus on the planar setting, although the results can be generalized to three-dimensional motion.) We propose a biologically plausible feedback law, and use a high-gain limit to prove the accessibility of a motion-camouflage state in finite time. We discuss connections to work in missile guidance. We also present simulation results to explore the performance of the motion-camouflage feedback law for a variety of settings.

Steering laws and continuum models for planar formations

We consider a Lie group formulation for the problem of control of formations. Vehicle trajectories are described using the planar Frenet-Serret equations of motion, which capture the evolution of both vehicle position and orientation for unit-speed motion subject to curvature (steering) control. The Lie group structure can be exploited to determine the set of all possible (relative) equilibria for arbitrary G-invariant curvature controls, where G=SE(2) is a symmetry group for the control law. The main result is a convergence result for n vehicles (for finite n), using a Lyapunov function which for n=2, has been previously shown to yield global convergence. A continuum formulation of the basic equations is also presented.

Boundary following using gyroscopic control

Recent work in the study of interacting particles has demonstrated the effectiveness of gyroscopic interactions in producing desired stable spatial patterns (formations) of motion of a collective of particles. In this paper, we discuss the problem of how a single particle might interact with a fixed structure in space by exploiting gyroscopic feedback laws. We derive a gyroscopic feedback law modeling the interaction of a particle in the plane with an image particle representing the closest point on a simple closed curve bounding an obstacle and show that this law produces boundary-following behavior. We also provide a preliminary discussion of the three-dimensional case.

Adaptive optics with advanced phase-contrast techniques. II. High-resolution wave-front control

A wave-front control paradigm based on gradient-flow optimization is analyzed. In adaptive systems with gradient-flow dynamics, the output of the wave-front sensor is used to directly control high-resolution wavefront correctors without the need for wave-front phase reconstruction (direct-control systems). Here, adaptive direct-control systems with advanced phase-contrast wave-front sensors are analyzed theoretically, through numerical simulations, and experimentally. Adaptive system performance is studied for atmospheric-turbulence-induced phase distortions in the presence of input field intensity scintillations. The results demonstrate the effectiveness of this approach for high-resolution adaptive optics.

Motion camouflage in three dimensions

We formulate and analyze a three-dimensional model of motion camouflage, a stealth strategy observed in nature. The pursuer and evader trajectories are described using natural Frenet frames (or relatively parallel adapted frames), and the corresponding natural curvatures serve as controls. A high-gain feedback control law is derived. The biological plausibility of the feedback law is discussed, as is its connection to missile guidance. Simulations illustrating motion camouflage are also presented. This paper builds on recent work on motion camouflage in the planar setting

Pursuit and an evolutionary game

Pursuit is a familiar mechanical activity that humans and animals engage in—athletes chasing balls, predators seeking prey and insects manoeuvring in aerial territorial battles. In this paper, we discuss and compare strategies for pursuit, the occurrence in nature of a strategy known as motion camouflage, and some evolutionary arguments to support claims of prevalence of this strategy, as opposed to alternatives. We discuss feedback laws for a pursuer to realize motion camouflage, as well as two alternative strategies. We then set up a discrete-time evolutionary game to model competition among these strategies. This leads to a dynamics in the probability simplex in three dimensions, which captures the mean-field aspects of the evolutionary game. The analysis of this dynamics as an ascent equation solving a linear programming problem is consistent with observed behaviour in Monte Carlo experiments, and lends support to an evolutionary basis for prevalence of motion camouflage.

Analog CMOS implementation of high frequency least-mean square error learning circuit

A continuous-time analog CMOS circuit implementing the least mean square (LMS) adaptive learning algorithm demonstrates a frequency of operation of 80MHz, an adaptivity of 60dB, a minimum notch width of 25KHz, a minimum adapt time constant of 20/spl mu/s, and the simultaneous cancellation of two CW interferers. This frequency of operation is more than an order-of-magnitude greater than that reported for previous integrated analog learning processors. This paper as also describes the first use of an auto-zero circuit to cancel offset voltages of both the integrator and multiplier circuits used for weight learning. The use of the auto zero circuit results in a 20dB higher adaptivity than obtained by previous analog adaptive processors. The high frequency learning circuitry has a number of potential applications for both conventional and neural network signal processing, especially for communication applications.

Geometry of cyclic pursuit

Pursuit strategies (formulated using constant-speed particle models) provide a means for achieving cohesive behavior in systems of multiple mobile agents. In the present paper, we explore an n-agent cyclic pursuit scheme (i.e. agent i pursues agent i+1, modulo n) in which each agent employs a constant bearing pursuit strategy. We demonstrate the existence of an invariant submanifold, and state necessary and sufficient conditions for the existence of rectilinear and circling relative equilibria on that submanifold. We present a full analysis of steady-state solutions and stability characteristics for two-particle “mutual CB pursuit” and then outline steps to extend the nonlinear stability analysis to the many particle case.