Flocking for Multiple Ellipsoidal Agents with Limited Communication Ranges

This paper contributes a design of distributed controllers for flocking of mobile agents with an ellipsoidal shape and a limited communication range. A separation condition for ellipsoidal agents is first derived. Smooth step functions are then introduced. These functions and the separation condition between the ellipsoidal agents are embedded in novel pairwise potential functions to design flocking control algorithms. The proposed flocking design results in (1) smooth controllers despite of the agents’ limited communication ranges, (2) no collisions between any agents, (3) asymptotic convergence of each agent’s generalized velocity to a desired velocity, and (4) boundedness of the flock size, defined as the sum of all distances between the agents, by a constant. 1. Introduction Flocking, referred to as a collective motion of a large number of self-propelled entities, has attracted a lot of attention of researchers in biology, physics, and computer science [1–4]. Engineering applications of flocking include search, rescue, coverage, surveillance, sensor networks, and cooperative transportation [5–14]. In 1987, Reynolds [2] introduced three rules of flocking: (1) separation: avoid collision with nearby flock-mates; (2) alignment: attempt to match velocity with nearby flock-mates; (3) cohesion: attempt to stay close to nearby flock-mates. Since then, there has been a number of modifications and extensions of the above three rules and additional rules to result in many algorithms to realize these rules. The graph theory and the Lyapunov direct method were used to solve consensus problems in [15–17]. Local artificial potentials between neighboring agents were used to deal with separation (collision avoidance) and cohesion problems in [12, 13, 18–23]. The leader-follower approach to a target tracking problem was used in [24, 25]. Other related work includes geometric formation optimization [26, 27], pattern formation [28], and task allocation [29]. In all the above cited references, the agents are considered as a single point, a circular disk, or a sphere. In practice, many agents have a nonspherical, especially long and narrow, shape. If these agents are fitted to spheres, there is a problem of the large conservative volume. To illustrate this problem, we look at an example of fitting a cylindrical agent with a radius of and a length of to an ellipsoid with semiaxes of , , and , and a sphere with a radius of as shown Figure 1. By shrinking the space along the direction of the major axis of the ellipsoid, we can find , , and . Therefore, the conservative