Pursuit formations of unicycles

Abstract

In this paper, the stability of equilibrium formations for multiple unicycle systems in cyclic pursuit is studied in detail. The cyclic pursuit setup is particularly simple in that each unicycle $i$ pursues only one other unicycle, unicycle $i+1$ (modulo $n$), where $n$ is the number of unicycles. This research is principally motivated by the historical development of pursuit problems found in the mathematics and science literature, which dates as far back as 1732 and yet continues to be of current interest. On the other hand, it is anticipated that the analytical techniques and solutions pertaining to these problems will prove relevant to the study of multiagent systems and in cooperative control engineering.

Introduction

Problems based on the notion of pursuit have appealed to the curiosity of mathematicians and scientists over a period spanning centuries. These ideas apparently originated in the mathematics of pursuit curves (c. 1732), first studied by French scientist Pierre Bouguer (Bernhart, 1959). Simply put, if a point $a$ in space moves along a known curve, then another point $p$ describes a pursuit curve if the motion of $p$ is always directed towards $a$ and the two points move with equal speeds. More than a century later, in 1877, Edouard Lucas asked, what trajectories would be generated if three dogs, initially placed at the vertices of an equilateral triangle, were to run one after the other? In 1880, Henri Brocard replied with the answer that each dog's pursuit curve would be that of a logarithmic spiral and that the dogs would meet at a common point, known now as the Brocard point of a triangle (Bernhart, 1959). As a consequence of these old ideas, contemporary researchers have shown notable interest in problems based on the latter concept of cyclic pursuit, on which this paper is based.

Herein, we generalize the notion of cyclic pursuit to systems of $n$ ordered and identical planar agents, where each individual agent $i$ pursues the next, $i+1$ modulo $n$. In particular, we study the stability of equilibrium formations when the agents are modelled as unicycles. Multiagent systems and cooperative control have become topics of growing popularity within the systems engineering research community. Certainly, the possible applications for multiple cooperating agents are numerous, and include: terrestrial, space, and oceanic exploration; military surveillance and rescue; or even automated transportation systems. Therefore, from an engineering perspective, the challenging problem of how to employ only local interactions (e.g., pursuit) to generate global behaviors for the collective is of distinct interest. For a sampling and review of some recent research in multiagent and cooperative control, see Gazi and Passino (2002), Jadbabaie, Lin, and Morse (2003), Justh and Krishnaprasad (2003), Marshall, Brouke, and Francis (2004), Ögren, Fiorelli, and Leonard (2004).

We take as inspiration for our study the historical development of cyclic pursuit problems found in the mathematics and science literature. In one of his several Scripta Mathematica articles on the subject, Bernhart (1959) reveals an intriguing history of cyclic pursuit, beginning with Brocard's response to Lucas in 1880. Among his findings, Bernhart reported on a Pi Mu Epsilon talk given by a man named Peterson, who apparently extended the original three dogs problem to $n$ ordered “bugs” that start at the vertices of a regular $n$-polygon. He is said to have illustrated his results for the square using four “cannibalistic spiders.” Thus, if each bug pursues the next modulo $n$ (i.e., cyclic pursuit) at fixed speed, the bugs will trace out logarithmic spirals and eventually meet at the polygon's center. Watton and Kydon (1969) provide a solution to this regular $n$-bugs problem, also noting that the constant speed assumption is not necessary.

Suppose the bugs do not start at the vertices of a regular $n$-polygon. Klamkin and Newman (1971) show that, for three bugs, so long as the bugs are not initially arranged so that they are all collinear, they will meet at a common point and this meeting will be mutual. The $n$-bug problem was later examined by the authors Behroozi and Gagnon (1979), who proved that “a bug cannot capture another bug which is not capturing another bug (i.e., mutual capture), except by head-on collision.” They used their result to show that, specifically for the 4-bugs problem, the terminal capture is indeed mutual. Very recently, Richardson (2001a) resolved this issue for $n$-bugs, showing “it is possible for bugs to capture their own prey without all bugs simultaneously doing so, even for non-collinear initial positions.” However, he also proved that if these initial positions are chosen randomly, then the probability that a non-mutual capture will occur is zero. Other variations on the traditional cyclic pursuit problem have also been considered. For example, Bruckstein, Cohen, and Efrat (1991)z studied both continuous and discrete pursuit problems, as well as both constant and varying speed scenarios. For a more complete review, see Bernhart (1959), Richardson (2001a).

Suppose we now imagine that each “bug” is instead an autonomous agent in the plane. In what follows, we generalize the cyclic pursuit concept to autonomous agents and discuss its properties as a possible coordination framework for multiagent systems. In particular, we consider the case when each agent is subject to a single nonholonomic motion constraint, or equivalently, modelled as a unicycle. Therefore, depending on the allowed control energy, each agent will require some finite time to steer itself towards its prey. What global motions can be generated? We first asked this question in Marshall, Broucke, and Francis (2003), where preliminary results appeared. Recently, the author Richardson (2001b) posed a similar question for a particular constant speed version of the $n$-bugs problem. He showed that the system's limiting behavior exponentially resembles a regular$n$-polygon, but only when $n⩾7$.

Thus, our primary motivation is to follow historical development and study the achievable formations for unicycles under cyclic pursuit. Then again, practically speaking, the study of cyclic pursuit may result in a feasible strategy for multiple vehicle systems since it is distributed (i.e., decentralized and there is no leader) and rather simple in that each agent is required to sense information from only one other agent. Our study begins by classifying all possible equilibrium formations for unicycles in cyclic pursuit. We first state the results of a global stability analysis for the case when $n=2$, which originally appeared in Marshall et al. (2003), followed by a complete local stability analysis for the general case when $n⩾2$. Moreover, in each case it is exposed how the multiple unicycle system's global behavior can be changed by appropriate controller gain assignments.