# American Institute of Mathematical Sciences

December  2007, 2(4): 597-626. doi: 10.3934/nhm.2007.2.597

## Stable synchronization of rigid body networks

 1 Control and Dynamical Systems, 107-81, California Institute of Technology, Pasadena, CA 91125, United States 2 Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, United States

Received  June 2007 Revised  September 2007 Published  September 2007

We address stable synchronization of a network of rotating and translating rigid bodies in three-dimensional space. Motivated by applications that require coordinated spinning spacecraft or diving underwater vehicles, we prove control laws that stably couple and coordinate the dynamics of multiple rigid bodies. We design decentralized, energy shaping control laws for each individual rigid body that depend on the relative orientation and relative position of its neighbors. Energy methods are used to prove stability of the coordinated multi-body dynamical system. To prove exponential stability, we break symmetry and consider a controlled dissipation term that requires each individual to measure its own velocity. The control laws are illustrated in simulation for a network of spinning rigid bodies.
Citation: Sujit Nair, Naomi Ehrich Leonard. Stable synchronization of rigid body networks. Networks & Heterogeneous Media, 2007, 2 (4) : 597-626. doi: 10.3934/nhm.2007.2.597
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