Control Theorists Chart New Waters for

Call it the department chair’s dilemma: How do you get a swarm of independently minded minions moving in concert toward some useful goal? The problem may be inherently insoluble in a professorial setting, in part because the notion of academic motion is purely metaphoric. But encouraging signs of progress have emerged for a more literal analog. Researchers in control theory are perfecting a repertoire of micromanagerial algorithms necessary—or at least sufficient—for the task of steering groups of otherwise autonomous vehicles in the service of a common goal. In an invited presentation at the Sixth SIAM Conference on Control and its Applications, held in New Orleans in July, Naomi Ehrich Leonard of Princeton University described progress she and colleagues have made in the theory of collective motion. Leonard is an expert on geometric mechanics and control, with a particular interest in cooperative control and collective motion of natural and engineered systems, from birds and fish to autonomous underwater vehicles. She was one of the key participants in the Autonomous Ocean Sampling Network II project, which field-tested a fleet of underwater gliders in Monterey Bay, California, in 2003, and she is the principal investigator for the Adaptive Sampling and Prediction (ASAP) project, which will conduct further tests in Monterey Bay next summer. A central goal of these projects is to develop an autonomous system for observing and predicting ocean dynamics. The fundamental idea is for autonomous vehicles to act as mobile sensor platforms, coordinating their motion to collect as efficiently as possible the most useful data for assimilation into ocean-forecasting models. The oceanographic goal of the 2003 experiment was to study upwelling events. Control theoretically, the objective was to test algorithms that coordinate the movement of independent vehicles. The researchers found that they could keep three gliders at the vertices of a linearly translating equilateral triangle over a run of approximately 16 hours (see Figure 1). In one experiment the gliders stayed roughly three kilometers apart during the entire run. With an additional control term in the second half of the run, one edge of the triangle was kept perpendicular to the direction of the group’s motion—an impressive accomplishment, given that the gliders could communicate only when they surfaced, every couple of hours. In a second experiment, the researchers successfully reduced the distance between gliders, moving in a triangular formation, from six to three kilometers, despite currents that rivaled the effective speed of the gliders. The results have helped shape plans for the month-long ASAP trials in 2006. Meanwhile, the control theorists are developing an increasingly sophisticated theory of multi-agent control. In her talk, Leonard highlighted recent work with Rodolphe Sepulchre of l’Université de Liège in Belgium and her student Derek Paley in the department of mechanical and aerospace engineering at Princeton. The trio have studied an idealized problem concerning the stabilization of collective motion in the plane. Their approach provides a set of basic procedures, such as switching from linear to circular trajectories, that can be used in the design of more complicated maneuvers. The idealized problem is closely related to the theory of coupled phase oscillators. It posits point particles in the complex plane, moving with constant, unit speed. The use of complex variables is a notational convenience, allowing the velocity vector to be written in the form eiθk, where θk is the orientation of particle k. The control is a steering rate, dθk/dt = uk, which depends only on the relative positions and orientations of the particles, rjk = rj – rk and θjk = θj – θk. (Getting the group to move in a particular direction requires the inclusion of a reference heading; the simplest way to do so is to couple the reference heading to ust one particle—in effect, to pick a leader and give it a compass.) The key to the stability analysis begins with some simple algebra. The group’s center of mass is the average of the individual position vectors r = 1/NΣrk. Its linear momentum (giving each particle unit mass) is p = 1/N Σ eiθk; p can also be interpreted as the centroid of the phasors (headings) of the particles. The magnitude of p lies between 0 and 1: It’s at a maximum when the particles are traveling in synch, e.g., along parallel lines, and 0 when the particles are conspiring to be “anti-synchronized,” e.g., equispaced and moving in a circle. This suggests that the control be based on the gradient of the “phase potential,” K |p| 2, where K is a parameter that can be positive or negative. Doing so leads to the phase control uk = ω0 – K/N Σ sin θjk, where ω0 is a constant angular rate. If ω0 = 0, the control produces a steady state consisting of straight-line trajectories—all in the same direction if K < 0, and scattered but in a balanced distribution (so that the center of Control Theorists Chart New Waters for Synchronized Swimmers