Geometric decompositions of collective motion.

Collective motion in nature is a captivating phenomenon. Revealing the underlying mechanisms, which are of biological and theoretical interest, will require empirical data, modelling and analysis techniques. Here, we contribute a geometric viewpoint, yielding a novel method of analysing movement. Snapshots of collective motion are portrayed as tangent vectors on configuration space, with length determined by the total kinetic energy. Using the geometry of fibre bundles and connections, this portrait is split into orthogonal components each tangential to a lower dimensional manifold derived from configuration space. The resulting decomposition, when interleaved with classical shape space construction, is categorized into a family of kinematic modes-including rigid translations, rigid rotations, inertia tensor transformations, expansions and compressions. Snapshots of empirical data from natural collectives can be allocated to these modes and weighted by fractions of total kinetic energy. Such quantitative measures can provide insight into the variation of the driving goals of a collective, as illustrated by applying these methods to a publicly available dataset of pigeon flocking. The geometric framework may also be profitably employed in the control of artificial systems of interacting agents such as robots.

Symmetry and reduction in collectives: low-dimensional cyclic pursuit.

We investigate low-dimensional examples of cyclic pursuit in a collective, wherein each agent employs a constant bearing (CB) steering law relative to exactly one other agent. For the case of three agents in the plane, we characterize relative equilibria and pure shape equilibria of associated closed-loop dynamics. Re-scaling time yields a reduction of phase space to two dimensions and effective tools for stability analysis. Study of bifurcation of a family of collinear equilibria dependent on a single CB control parameter reveals the presence of a rich collection of trajectories that are periodic in shape and undergo precession in physical space. For collectives in three dimensions, with an appropriate notion of CB pursuit strategy and corresponding steering law, the two-agent case proves to be explicitly integrable. These results suggest control schemes for small teams of mobile robotic agents engaged in area coverage tasks such as search and rescue, and raise interesting possibilities for behaviour in biological contexts.

Optimality, reduction and collective motion.

The planar self-steering particle model of agents in a collective gives rise to dynamics on the N-fold direct product of SE(2), the rigid motion group in the plane. Assuming a connected, undirected graph of interaction between agents, we pose a family of symmetric optimal control problems with a coupling parameter capturing the strength of interactions. The Hamiltonian system associated with the necessary conditions for optimality is reducible to a Lie-Poisson dynamical system possessing interesting structure. In particular, the strong coupling limit reveals additional (hidden) symmetry, beyond the manifest one used in reduction: this enables explicit integration of the dynamics, and demonstrates the presence of a 'master clock' that governs all agents to steer identically. For finite coupling strength, we show that special solutions exist with steering controls proportional across the collective. These results suggest that optimality principles may provide a framework for understanding imitative behaviours observed in certain animal aggregations.

Echolocating bats use a nearly time-optimal strategy to intercept prey.

Acquisition of food in many animal species depends on the pursuit and capture of moving prey. Among modern humans, the pursuit and interception of moving targets plays a central role in a variety of sports, such as tennis, football, Frisbee, and baseball. Studies of target pursuit in animals, ranging from dragonflies to fish and dogs to humans, have suggested that they all use a constant bearing (CB) strategy to pursue prey or other moving targets. CB is best known as the interception strategy employed by baseball outfielders to catch ballistic fly balls. CB is a time-optimal solution to catch targets moving along a straight line, or in a predictable fashion--such as a ballistic baseball, or a piece of food sinking in water. Many animals, however, have to capture prey that may make evasive and unpredictable maneuvers. Is CB an optimum solution to pursuing erratically moving targets? Do animals faced with such erratic prey also use CB? In this paper, we address these questions by studying prey capture in an insectivorous echolocating bat. Echolocating bats rely on sonar to pursue and capture flying insects. The bat's prey may emerge from foliage for a brief time, fly in erratic three-dimensional paths before returning to cover. Bats typically take less than one second to detect, localize and capture such insects. We used high speed stereo infra-red videography to study the three dimensional flight paths of the big brown bat, Eptesicus fuscus, as it chased erratically moving insects in a dark laboratory flight room. We quantified the bat's complex pursuit trajectories using a simple delay differential equation. Our analysis of the pursuit trajectories suggests that bats use a constant absolute target direction strategy during pursuit. We show mathematically that, unlike CB, this approach minimizes the time it takes for a pursuer to intercept an unpredictably moving target. Interestingly, the bat's behavior is similar to the interception strategy implemented in some guided missiles. We suggest that the time-optimal strategy adopted by the bat is in response to the evolutionary pressures of having to capture erratic and fast moving insects.

Coordinated orbit transfer for satellite clusters.

We propose a control law that allows a satellite formation to achieve orbit transfer. During the transfer, the formation can be either maintained or modified to a desired formation. Based on the orbit transfer control law proposed by Chang, Chichka, and Marsden for single satellite, we add coupling terms to the summation of Lyapunov functions for single satellites. These terms are functions of the difference between the mean anomalies (or perigee passage times) of formation members. The asymptotic stability of the desired formation in desired orbits is proved.

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.

Analysis and synthesis of feedforward neural networks using discrete affine wavelet transformations.

A representation of a class of feedforward neural networks in terms of discrete affine wavelet transforms is developed. It is shown that by appropriate grouping of terms, feedforward neural networks with sigmoidal activation functions can be viewed as architectures which implement affine wavelet decompositions of mappings. It is shown that the wavelet transform formalism provides a mathematical framework within which it is possible to perform both analysis and synthesis of feedforward networks. For the purpose of analysis, the wavelet formulation characterizes a class of mappings which can be implemented by feedforward networks as well as reveals an exact implementation of a given mapping in this class. Spatio-spectral localization properties of wavelets can be exploited in synthesizing a feedforward network to perform a given approximation task. Two synthesis procedures based on spatio-spectral localization that reduce the training problem to one of convex optimization are outlined.

The self-tuning controller: comparison with human performance in the control of arterial pressure.

A self-tuning controller was implemented for the automated infusion of sodium nitroprusside to lower mean arterial pressure in anesthetized dogs. The system incorporated a recursive least-squares parameter identifier and a modified minimum-variance controller. The onset delay was estimated on-line, the performance criterion included the cost of control, and requested step-changes were automatically translated into five successive smaller steps to reduce overshoot. The performance of the system in lowering mean arterial pressure was quantitatively compared with that of a well-trained anesthesiologist. In 10 runs in four animals, the automated system performed as well as the physician who devoted 100% of his attention to the task. Since the stability of the self-tuning controller cannot be guaranteed, such a system should be operated only in the presence of appropriate supervisory algorithms.