Introduction

    For more than a decade there has been a great interest in controlling large-scale dynamical systems composed of multiple mobile agents and a surge of results in cooperative and formation control is being witnessed. The framework of networked multi-agent systems has a variety of applications such as air traffic control [1], intelligent highways [2], multiple robots carrying out cooperative tasks [3], coordinated control of satellites for earth observation [4], RoboCup Soccer [5], to cite but a few examples and references. The main reasons for this growth is due to the fact that formations can implement instruments which are infeasible to implement monolithically.

    A large share of the topics on formation control focus on networks of agents being modeled as particles [6-9]. Modelling agents as particles gives a relatively simple control framework. Specifically, the control laws given in [6,7] constitute emulation of a mass-spring-damper system so that the agents eventually converge to each other. This entire behavior can be explained by employing the energy function defined in terms of the inertial frame and by showing that it decreases until the agents travel with asymptotic zero relative velocity. Furthermore, the Krasovskii-LaSalle's invariance principle is invoked to show convergence of relative positions as well.

    However, this approach is not sufficient enough when the problem of attitude synchronization, which is a vital part of formation control, is considered. This is due to the fact that if the mass moment of inertia is not spatially isotropic, the Euler equation describing the rotational motion becomes inherently nonlinear. This nonlinearity makes it difficult to show that attitude synchronization is attained using invariance principle through the energy function in the inertial frame.

    Applications of attitude synchronization include a coordinated cluster of satellites carrying telescopes for astronomical interferometry and enhancing resolution compared to a single satellite. Some of the real world applications include:

1. ST3 (Space Technology 3), developed by JPL, combines the images from two small telescopes flying in formation to produce images almost as good as those of a very expensive giant telescope [10].
2. LISA, a giant interferometer built using spacecraft separated by distance by ESA, will look for ripples in space called gravity waves, which are produced when black holes collapse [11].
3. TechSat 21, a set of three micro satellites flying in formation to operate as a "virtual satellite," is being developed by Air Force Research Laboratory (AFRL) [12].

    The application of attitude synchronization also extends to a fleet of sensor-equipped underwater vehicles that move together in an organized pattern to identify and track features in ocean [13].

Research abstract

   There have been a number of researches (e.g., [14-19]) within the framework of attitude synchronization. However, most of these works either make the agents follow an externally given trajectory or take a leader-follower approach. In [17] the authors showed that attitude consensus can be achieved using only the relative states between the spacecraft. Still they considered only the kinematic equation and the Euler equation describing the system dynamics is disregarded.

    However [18] and [19] explicitly consider the Euler equation while aiming for attitude synchronization in SO(3). Our research not only considers the system dynamics but also, to the best of our knowledge, produces the first result in attitude synchronization that uses a simple and intuitive control framework which emulates torques due to springs and dampers between the agents. We also further show that the angular velocities of each agent can be made to asymptotically converge towards a constant value, i.e. the agents can be made to rotate around a fixed rotational axis, by applying the virtual springs and dampers in different timings.

Reference

1. C. Tomlin, G. Pappas, and S. Sastry, "Conflict resolution for air traffic management: a study in multiagent hybrid systems," IEEE Trans. Autom. Contr., vol. 43, no. 4, pp. 509-521, 1998.
2. J. Z. Hernandez, S. Ossowski, and A. Garcia-Serrano, "Multiagent architectures for intelligent traffic management systems," Transportation Research Part C: Emerging Technologies, vol. 10, no. 5-6, pp. 473-506, 2002.
3. E. Pagello, A. D'Angeloc, F. Monteselloa, F. Garellia, and C. Ferraria, "Cooperative behaviors in multi-robot systems through implicit communication," Robotics and Autonomous Systems, vol. 29, no. 1, pp. 65-77, 1999.
4. P. Silvestrin, "Control and navigation aspects of the new earth observation missions of the European Space Agency," Ann. Rev. Contr., vol. 29, no. 2, pp. 247-260, 2005.
5. RoboCup Official Site, http://www.robocup.org/
6. H. G. Tanner, A. Jadbabaie, and G. J. Pappas, "Stable flocking of mobile agents, part I: Fixed topology," in Proc. IEEE Conf. Dec. Contr., (Maui, HI), pp. 2010-2015, December 2003.
7. H. G. Tanner, A. Jadbabaie, and G. J. Pappas, "Stable flocking of mobile agents, part II: Dynamic topology," in Proc. IEEE Conf. Dec. Contr., (Maui, HI), pp. 2016-2020, December 2003.
8. J. A. Fax and R. M. Murray, "Information flow and cooperative control of vehicle formations," IEEE Trans. Autom. Contr., vol. 49, no. 9, pp. 1495-1476, 2005.
9. T. Hayakawa, T. Matsuzawa, and S. Hara, "Formation control of multi-agent systems with sampled information," in Proc. IEEE Conf. Dec. Contr., (San Diego, CA), pp. 4333-4338, December 2006.
10. P. Ferguson, F. Busse, B. Engberg, J. How, M. Tillerson, N. Pohlman, A. Richards, and R. Twiggs, "Formation flying experiments on the Orion-Emerald mission," AIAA Space 2001 Conference and Exposition, (Albuquerque, NM), August 2001.
11. http://www.esa.int/esaCP/index.html
12. M. Martin, P. Klupar, S. Kilberg, J. Winter, "TechSat 21 and revolutionizing space missions using microsatellites," Proc. USU Small Satellite Conference, (Logan, UT), 2001.
13. S. Nair and N. E. Leonard, "Stable synchronization of rigid body networks," Networks and Heterogeneous Media, vol. 2, no. 4, pp. 595-624, 2007.
14. J. R. Lawton and R. W. Beard, "Synchronized multiple spacecraft rotations", Automatica, vol. 38, pp. 1359-1364.
15. A. K. Bondhus, K. Y. Pettersen, and J. T. Gravdahl, "Leader/follower synchronization of satellite attitude without angular velocity measurements, " in Proc. IEEE Conf. Dec. Contr. and the Eur. Contr. Conf., (Seville, Spain), pp. 7270-7277, December 2005.
16. W. Kang and H. H. Yeh, "Co-ordinated attitude control of multi-satellite systems," Int. J. Robust Non-linear Control, vol. 12, pp. 185-205, 2002.
17. Y. Igarashi, T. Hatanaka, M. Fujita, and M. W. Spong, "Passivity-based 3D attitude coordination: Convergence and connectivity," in Proc. IEEE Conf. Dec. Contr., (New Orleans, LA), pp. 2558-2565, December 2007.
18. A. Sarlette, R Sepulchre, and N. E. Leonard, "Autonomous rigid body attitude synchronization," in Proc. IEEE Conf. Dec. Contr., (New Orleans, LA), pp. 2566-2571, December 2007.
19. H. Bai, M. Arcak, and J. T. Wen, "A decentralized design for group alignment and synchronous rotation without inertial frame information," in Proc. IEEE Conf. Dec. Contr., (New Orleans, LA), pp. 2566-2571, December 2007.