Distributed Coordination of Dynamic Rigid Bodies Nima Moshtagh, Ali Jadbabaie, Kostas Daniilidis Abstract— This paper provides a design methodology to construct a set of distributed control laws for a group of rigid bodies moving in 3D space. The motion of the rigid bodies is restricted by a nonholonomic constraint that prohibits the agents from spinning around their velocity vectors. The con- nections between two seemingly different flocking control laws presented by Tanner et al. [1] and by Justh and Krishnaprasad [2] are studied, and it is shown that they are actually the same laws expressed in different coordinate systems. The former is expressed in the fixed world frame, whereas the latter is expressed in the moving body frame. I. I NTRODUCTION The problem of distributed coordination of mobile agents has been studied in details in the literature [1]–[12]. There are generally two design methodologies for generating control laws that drive the multi-agent system into a formation. On the one hand, there is collection of literature [1], [8], [13] in which the control input, such as the acceleration vector of each agent, is a parameter that is expressed in some fixed global frame. On the other hand, there is a body of literature [5], [6], [14], [15] that develops distributed control laws for inputs, such as the rates of change of roll, pitch and yaw, which are expressed in the body frame of each agent. In this paper, the connections between two seemingly different formation control laws are studied. In particular, it is shown that the results presented by Tanner et al. [1] and by Justh and Krishnaprasad [2] are actually the same laws, but expressed in two different coordinate systems. In [2] the dynamics of each agent was described by the motion of its body frame. The body frame of each agent was represented by a natural Frenet frame that moved on the trajectory curve of that agent. They considered unit speed particles under the effect of gyroscopic forces that were per- pendicular to the velocity vector. In [2] the authors provided a detailed analysis of the relative motion of two particles in the three dimensional space, and derived curvature controllers that achieved the desired rectilinear and circular formations. Their control laws depended only on relative positions and orientations (i.e. shape of the formation). They extended their results to the multi-agent case, but only for the case of all-to- all communication network. Also, the proof of convergence for the multi-agent case was not presented in [2]. However, a detailed analysis of the distributed coordina- tion of a group of dynamic agent, for both cases that the The authors are with the GRASP Laboratory, University of Pennsylvania, Philadelphia, PA, USA {nima, jadbabai, kostas}@grasp.upenn.edu This work is supported in part by ONR Grant N0001406104 and ONR DURIP Grant N00014-07-1-0829 and ARO-MURI Grant W911NF-05-1- 0381. communication network is fixed and switching, is given by Tanner et al. [1]. They modeled each agent as point-mass particles and designed a control law for formation control of a group of mobile agents that drove the agents to some desired formation, while avoiding collisions among each other. The control input for each particle was its acceleration vector, which was expressed in the world frame. In this paper, we show that the flocking and coordination algorithm presented by Tanner et al. [1] for dynamic particles is actually the same as the one designed by Justh and Krishnaprasad in [2] with the difference that in the former the controller is expressed in a fixed world frame, whereas the controllers in the latter are expressed in the moving body frame of each agent. This paper is organized as follows. In Section II we present some known results on distributed coordination following the work of Tanner et al. [1], which gives us a set of control laws in the global frame. Then in Section III we develop the rigid body model that is used in this paper to describe the motion of each dynamic agent in space. In Section IV, a set of local controllers are derived by combining the results of the previous sections, and it turns out these local controllers are equivalent to the ones given by Justh and Krishnaprasad [2]. We also look into the circling formations of rigid bodies in Section V, and present both the body-frame inputs and the global-frame inputs that achieve the desired circular motion. Simulations and conclusions are presented in Sections VI and VII, respectively. II. DISTRIBUTED COORDINATION IN GLOBAL FRAME To study the problem of coordinated motion in a group of dynamical agents, we first express the dynamics in the fixed world frame {A}. By expressing the velocity and acceleration vectors in {A}, we can write the dynamics of each agent as a double integrator: ˙ r i = v i ˙ v i = a i , i =1,...,N . (1) Now, consider a system of N agents with dynamics (1) in R d ,d ∈{2, 3} moving with different velocities. Assume agents can communicate some information, say their veloci- ties, with their neighbors. We can represent the neighboring relations among agents by a weighted graph. Definition 2.1 (Proximity Graph): The proximity graph G = {V , E , W} is a weighted graph consisting of: • a set of vertices V indexed by the set of mobile agents; • a set of edges E = {(i, j ) | i, j ∈V , and i ∼ j }; • a set of weights W, over the set of edges E . Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007 WePI20.16 1-4244-1498-9/07/$25.00 ©2007 IEEE. 1480