There are increasingly many researches in the field of underactuated systems, i.e., systems with fewer actuators than the system’s degrees of freedom. Several methods have been successful to derive control laws for certain classes of underactuated systems, e.g., [1–6].

The first general procedure for design and analysis of attitude control systems came in early 70’s [7]. Based on Meyer [7] and Lie algebraic methods in Hermes [8], Crouch [9] derived necessary and sufficient conditions for controllability of the underactuated spacecraft and constructed an algorithm which locally asymptotically stabilizes the spacecraft around an equilibrium point. The algorithm yields a piecewise constant discontinuous control inputs. In [10], a new discontinuous stabilizing feedback control strategy, which is simpler than the one in [9], was constructed.

An article by Samson [11] has shown that continuous, time-varying feedback can stabilize many classes of systems which cannot be stabilized by continuous state feedback. This result has triggered many researches on time-varying feedback to solve underactuated spacecraft control problem, e.g. [12–20]. In addition to time-varying feedback, several specific methods also have been used to solve the attitude control problem for underactuated case, e.g., center manifold theory [12], differentially flat systems [15, 19], backstepping technique [16, 18], hybrid control [20] and a smooth function, i.e., transverse function approach [17].

While all the papers mentioned above focus on the stabilization of the attitude, there are several papers that were focus on the stabilization of the velocities, e.g., [21–25]. In [22], Aeyels and Szafranski showed that a single control aligned with a principal axis cannot asymptotically stabilize the angular velocity of a rigid body. The extension of this research can be found in [23], where they showed that if a rigid body has an axis of symmetry then a single torque control can stabilize the angular velocity. An explicit polynomial feedback control for stabilizing the angular velocity of a rigid body for the symmetric case was given by Outbib and Sallet [24]. The results in [21, 25] showed that the Lie group theory can be used to achieve stabilization of the angular velocity for underactuated case.

A recent paper by Casagrande et al. [26] proposed a control strategy to stabilize both the attitude and the angular velocities of an underactuated spacecraft simultaneously. They used a switching control strategy combined with non-monotonic Lyapunov function to show global asymptotic stability of both the attitude and the angular velocities of an underactuated spacecraft. Although this non-monotonic Lyapunov function is not usual, a recent result by Ahmadi and Parrilo [27] showed that one can relax the monotonicity requirement of Lyapunov’s theorem to show the stability for larger class of systems.

In the last decade, a number of researches about spacecraft synchronization has increased, e.g. [28–30]. Lawton and Beard [28] designed two control strategies, velocity feedback approach and passivity-based damping approach, for maintaining attitude alignment among a group of spacecraft. Bondhus et al. [29] proposed a controller to synchronize two satellites when angular velocity measurements are not available by using nonlinear observers to estimate the angular velocities. Sarlette et al. [30] designed control laws to synchronize identical rigid bodies when the available information between the rigid bodies is restricted to relative orientations and angular velocities. To the best of our knowledge, there is one result by Hutagalung et al. [31] that has been successful to solve configuration consensus of multiple underactuated rigid bodies problem. However, they only show the convergence, not the stability, of the system.

In this research, we adopt a method in [26] to synchronize the attitude and the angular velocities of two underactuated spacecraft without referring to any leader or external reference. We propose a switching control strategy combined with multiple Lyapunov functions to show global asymptotic stabilization of both the attitude and the angular velocities errors simultaneously.

**References**

[1] M. Reyhanoglu, A. van der Schaft, N. H. McClamroch, and I. Kolmanovsky, “Dynamics and control of a class of underactuated mechanical systems,” *IEEE Trans. Autom. Contr.*, vol. 44, no. 9, pp. 1663–1671, 1999.

[2] F. Bullo, N. E. Leonard, and A. D. Lewis, “Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups,” *IEEE Trans. Autom. Contr.*, vol. 45, no. 8, pp. 1437–1454, 2000.

[3] R. Ortega, M. W. Spong, F. Gómez-Estern, and G. Blankenstein, “Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment,” *IEEE Trans. Autom. Contr.*, vol. 47, no. 8, pp. 1218–1233, 2002.

[4] S. Martínez and J. Cortés, “Motion control algorithms for simple mechanical systems with symmetry,” *Acta Appl. Math.*, vol. 76, no. 3, pp. 221–264, 2003.

[5] P. Morin and C. Samson, “Practical stablization of driftless systems on Lie groups: the transverse function approach,” *IEEE Trans. Autom. Contr.*, vol. 48, no. 9, pp. 1496–1508, 2003.

[6] J. Shen, N. H. McClamroch, and A. M. Bloch, “Local equilibrium controllability of multibody systems controlled via shape change,” *IEEE Trans. Autom. Contr.*, vol. 49, no. 4, pp. 506–520, 2004.

[7] G. Meyer, “Design and global analysis of spacecraft attitude control systems,” Tech. Rep. R-361, NASA, 1971.

[8] H. Hermes, “On the synthesis of a stabilizing feedback control via Lie algebraic methods,” *SIAM J. Contr. Optimiz.*, vol. 18, no. 4, pp. 352–361, 1980.

[9] P. E. Crouch, “Spacecraft attitude control and stabilization: Applications of geometric control theory to rigid body models,” *IEEE Trans. Autom. Contr.*, vol. AC-29, no. 4, pp. 321–331, 1984.

[10] H. Krishnan, M. Reyhanoglu, and H. McClamroch, “Attitude stabilization of a rigid spacecraft using two control torques: A nonlinear control approach based on the spacecraft attitude dynamics,” *Automatica*, vol. 30, no. 6, pp. 1023–1027, 1994.

[11] C. Samson, “Velocity and torque feedback control of a nonholonomic cart,” in *Advanced Robot Control*, (New York), Springer-Verlag, 1991.

[12] P. Morin, C. Samson, J.-B. Pomet, and Z.-P. Jiang, “Time-varying feedback stabilization of the attitude of a rigid spacecraft with two controls,” *Syst. Contr. Lett.*, vol. 25, no. 5, pp. 375–385, 1995.

[13] J.-M. Coron and E.-Y. Keraï, “Explicit feedbacks stabilizing the attitude of a rigid spacecraft with two control torques,” *Automatica*, vol. 32, no. 5, pp. 669–677, 1996.

[14] P. Morin and C. Samson, “Time-varying exponential stabilization of a rigid spacecraft with two control torques,” *IEEE Trans. Autom. Contr.*, vol. 42, no. 4, pp. 528–534, 1997.

[15] P. Tsiotras and V. Doumtchenko, “Control of spacecraft subject to actuator failures: Stateof-the-art and open problems,” *J. Astro. Sci.*, vol. 48, no. 2/3, pp. 337–358, 2000.

[16] A. Behal, D. Dawson, E. Zergeroglu, and Y. Fang, “Nonlinear tracking control of an underactuated spacecraft,” in *Proc. Amer. Contr. Conf.*, (Anchorage, AK), pp. 4684–4699, 2002.

[17] P. Morin and C. Samson, “Control of underactuated mechanical systems by the transverse function approach,” in *Proc. IEEE Conf. Dec. Contr.*, (Seville, Spain), pp. 7508–7513, 2005.

[18] M. A. Karami and F. Sassani, “Nonlinear attitude control of an underactuated spacecraft subject to disturbance torques,” in *Proc. Amer. Contr. Conf.*, (New York, NY), pp. 3150–3155, 2007.

[19] P. Tsiotras and J. Luo, “Control of underactuated spacecraft with bounded inputs,” *Automatica*, vol. 36, pp. 1153–1169, 2000.

[20] A. R. Teel and R. G. Sanfelice, “On robust, global stabilization of the attitude of an underactuated rigid body using hybrid feedback,” in *Proc. Amer. Contr. Conf.*, (Seattle, WA), pp. 2909–2914, 2008.

[21] F. Bullo, “Stabilization of relative equilibria for underactuated systems on riemannian manifolds,” *Automatica*, vol. 36, pp. 1819–1834, 2000.

[22] D. Aeyels and M. Szafranski, “Comments on the stabilizability of the angular velocity of a rigid body,” *Syst. Contr. Lett.*, vol. 10, pp. 35–39, 1988.

[23] E. D. Sontag and H. J. Sussmann, “Further comments on the stabilizability of the angular velocity of a rigid body,” *Syst. Contr. Lett.*, vol. 12, pp. 213–217, 1989.

[24] R. Outbib and G. Sallet, “Stabilizability of the angular velocity of a rigid body revisited,” *Syst. Contr. Lett.*, vol. 18, pp. 93–98, 1992.

[25] N. Nordkvist and F. Bullo, “Control algorithms along relative equilibria of underactuated Lagrangian systems on Lie groups,” *IEEE Trans. Autom. Contr.*, vol. 53, no. 11, pp. 2651–2658, 2008.

[26] D. Casagrande, A. Astolfi, and T. Parisini, “Global asymptotic stabilization of the attitude and the angular rates of an underactuated non-symmetric rigid body,” *Automatica*, vol. 44, pp. 1781–1789, 2008.

[27] A. A. Ahmadi and P. A. Parrilo, “Non-monotonic Lyapunov functions for stability of discrete time nonlinear and switched systems,” in *Proc. IEEE Conf. Dec. Contr.*, (Cancun, MX), pp. 614–621, 2008.

[28] J. R. Lawton and R. W. Beard, “Synchronized multiple spacecraft rotations,” *Automatica*, vol. 38, pp. 1359–1364, 2002.

[29] 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.*, (Seville, Spain), pp. 7270–7277, 2005.

[30] 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, 2007.

[31] M. Hutagalung, T. Hayakawa, and T. Urakubo, “Configuration consensus of two underactuated planar rigid bodies,” in *Proc. IEEE Conf. Dec. Contr.*, (Cancun, MX), pp. 5016–5021, 2008.