1Ph.D. Student, Aerospace Research Institute, Ministry of Science, Research and Technology
2Assistant Professor, Aerospace Research Institute, Ministry of Science, Research and Technology
3Assistant Professor, Aerospace Department, Sharif University of Technology
Linear state bisection is introduced as a new method to find time-invariant state feedback control laws for a special form of underactuated nonlinear systems. The specialty of the systems considered is that every unactuated state should be coupled with at least two directly actuated states. The basic idea is based on bisecting actuated states and using linear combinations with adjustable parameters to stabilize the unactuated states. These linear combinations make the underactuated system virtually fullyactuated, making it suitable to be stabilized with well-known nonlinear control methods, like feedback linearization. In addition to its simplicity, one of the main contributions of this method is that it can be applied to systems with more than one unactuated state. Three underactuated systems are considered: an asymmetric rigid body, a planar rigid body with an unactuated internal degree of freedom and a system with two degrees of underactuation. It is shown through simulations that the proposed control laws can be effectively used to stabilize the special form of underactuated systems considered.
 Choukchu-Braham, A.; Cherki, B.; Djemai, M. and Busawon, K.; “Analysis and Control of Underactuated Mechanical Systems,” Springer Science and Business Media, 2014.
 Olfati-Saber, R.; “Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles,” Ph.D. Thesis, MIT University, 2001.
 Spong, M. W.; “Underactuated Mechanical Systems,” Control Problems in Robotics and Automation, Lecture Notes in Control and Information Sciences, Vol. 230, pp. 135-150, 1998.
 Liu, Y. and Yu, H.; “A Survey of Underactuated Mechanical Systems,” IET Control Theory and Applications, Vol. 7, No. 7, pp. 921-935, 2013.
 Voytsekhovsky, D. A. and Hirschorn, R. M.; “Stabilization of Single-Input Nonlinear Systems Using Higher Order Term Compensating Sliding Mode Control,” International Journal of Robust and Nonlinear Control, Vol. 18, No. 4-5, pp. 468-480, 2008.
 Dixon, W. E.; Behal, A.; Dawson, D. M. and Nagarkatti, S. P.; “Nonlinear Control of Engineering Systems: A Lyapunov-Based Approach,” Birkhäuser Basel, 2003.
 Rahman, E. A. A.; Nayfeh, A. H. and Masoud, Z. N.; “Dynamics and Control of Cranes: A Review,” Journal of Vibration and Control, Vol. 9, No 7, pp. 863-908, 2003.
 Block, D. J.; Astrom, K. J. and Spong, M. W.; “The Reaction Wheel Pendulum,” Synthesis Lectures on Controls and Mechatronics, Vol. 1, No. 1, pp. 1-105, 2007.
 Ghommam, J.; Mnif, F.; Benali, A. and Derbel, N.; “Asymptotic Backstepping Stabilization of an Underactuated Surface Vessel,” IEEE Transactions on Control Systems Technology, Vol. 14, No. 6, pp. 1150-1157, 2006.
 Huang, J.; Chuan-Jiang, L. I.; Guang-Fu, M. A. and Gang, L.; “Generalized Inversion Based Attitude Control for Underactuated Spacecraft,” Acta Automatica Sinica, Vol. 39, No. 3, pp. 285-292, 2013.
 Brockett, R. W.; “Asymptotic Stability and Feedback Stabilization,” Defense Technical Information Center, Harvard University, 1983.
 Reyhanoglu, M.; Cho, S.; Harris, N. and McClamroch, N. H.; “Discontinuous Feedback Control of a Special Class of Underactuated Mechanical Systems,” International Journal of Robust and Nonlinear Control, Vol. 10, No. 4, pp. 265-281, 2000.
 Morin, P. and Samson, C.; “Time-Varying Exponential Stabilization of the Attitude of a Rigid Spacecraft with Two Controls,” IEEE Conference on Decision and Control, Vol. 4, pp. 3938-3993, 1995.
 Acosta, J.; Ortega, R.; Astolfi, A. and Mahindrakur, A. D.; “Interconnection and Damping Assignment Passivity-Based Control of Mechanical Systems with Underactuation Degree One,” IEEE Transactions on Automatic Control, Vol. 50, No. 12, pp. 1936-1955, 2005.
 Sidi, M. J.; “Spacecraft Dynamics and Control, A Practical Engineering Approach,” Cambridge Aerospace Series, 2000.
 Reyhanoglu, M.; “Discontinuous Feedback Stabilization of the Angular Velocity of a Rigid Body with Two Control Torques,” 35th IEEE Conference on Decision and Control, Vol. 3, pp. 2692-2694, 1996.
 Wang, D.; Jia, Y.; Jin, L. and Xu, S.; “Control Analysis of an Underactuated Spacecraft under Disturbance,” Acta Astronautica, Vol. 83, pp. 44-53, 2013.
 Reyhanoglu, M.; Cho, S.; Harris, N. and McClamroch, N. H. and Kolmanovsky, I.; “Discontinuous Feedback Control of a Planar Rigid Body with an Unactuated Degree of Freedom,” IEEE Conference on Decision and Control, Vol. 1, pp. 433- 438, 1998.