Attitude Stabilization Using Modified Rodrigues Parameters without Angular Velocity Measurements

ABSTRACT

The optimal stabilization of a rigid body motion without angular velocity measurements is considered with the help of three internal rotors that effected by internal frictions. In this paper, the orientation of the body will be described in terms of the Modified Rodrigues parameters (MRPs). The optimal control law which stabilizes asymptotically this motion and minimizes the require like-energy cost is obtained in terms of the MRPs. Numerical study and simulation are introduced.

The optimal stabilization of a rigid body motion without angular velocity measurements is considered with the help of three internal rotors that effected by internal frictions. In this paper, the orientation of the body will be described in terms of the Modified Rodrigues parameters (MRPs). The optimal control law which stabilizes asymptotically this motion and minimizes the require like-energy cost is obtained in terms of the MRPs. Numerical study and simulation are introduced.

Cite this paper

nullA. El-Gohary and T. Tawfik, "Attitude Stabilization Using Modified Rodrigues Parameters without Angular Velocity Measurements,"*World Journal of Mechanics*, Vol. 1 No. 2, 2011, pp. 57-63. doi: 10.4236/wjm.2011.12008.

nullA. El-Gohary and T. Tawfik, "Attitude Stabilization Using Modified Rodrigues Parameters without Angular Velocity Measurements,"

References

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[2] B. T. Costic, D. M. Dawson, M. S. De Queiroz and V. Kapila, “A Quaternion-Based Adaptive Attitude Tracking Controller without Velocity Measurements,” Proceedings of the 39th IEEE Conference on Decision and Control, Sydeny, Vol. 3, 12-15 December 2000, pp. 2424- 2429.

[3] A. El-Gohary, “Optimal Control of the Rotational Motion of a Rigid Body Using Euler Parameters With the Help of Rotor System,” European Journal of Mechanics-A/Solids, Vol. 24, No. 1, 2005, pp. 111-125. doi:10.1016/j.euromechsol.2004.10.007

[4] A. El-Gohary, “New Optimal Control Law for Attitude of a Rigid Body without Angular Velocity Measurements,” Chaos, Solitons & Fractals, Vol. 25, No. 3, 2005, pp. 557-571. doi:10.1016/j.chaos.2004.12.008

[5] A. I. El-Gohary and T. S. Tawfik, “Optimal Control of a Rigid Body Motion Using Euler Parameters Without Angular Velocity Measurements,” Mechanics Research Communications, Vol. 37, No. 3, 2010, pp. 354-359. doi:10.1016/j.mechrescom.2010.02.004

[6] H. Schaub and J. L. Junkins, “Stereographic Orientation Parameters For Attitude Dynamics: A Generalization of the Rodrigues Parameters,” Journal of the Astronautical Sciences, Vol. 44, No. 1, 1996, pp. 1-19.

[7] G. R. Izzo and L. Pettazzi, “Command Shaping for a Flexible Satellite Platform Controlled by Advanced Fly- Wheels Systems,” Acta Astronautica, Vol. 60, No. 10-11, 2007, pp. 820-827.

[8] J. Li, M. Xu, Z. L.Wang and S. M. Wang, “Minimum-Torque Earth off-Nadir Pointing Control of Gravity-Gradient Stabilized Small Satellites,” Tsinghua Science and Technology, Vol. 5, No. 1, 2000, pp. 31-33.

[9] N. Krasovskii, “Problem of Stabilization of Controlled Motion,” In: I. G. Malkin, Ed., The Theory of Motion Stability, Nauka, Moscow, 1966, pp. 475-514.

[10] F. Lizarralde and J. T. Wen, “Attitude Control without Angular Velocity Measurement: A Passivity Approach,” IEEE transactions on Automatic Control, Vol. 41, No. 3, 1996, pp. 468-472. doi:10.1109/9.486654

[11] M. Lovera and A. Astolfib, “Spacecraft Attitude Control Using Magnetic Actuators,” Automatica, Vol. 40, No. 8, 2004, pp. 1405-1414. doi:10.1016/j.automatica.2004.02.022

[12] V. M. Matrosov, “On Stability of Motion,” Journal of Applied Mathematics and Mechanics, Vol. 26, No. 5, 1962, pp. 992-1002.

[13] A. Tayebi, “Unit Quaternion Observer Based Attitude Stabilization of a Rigid Spacecraft Without Velocity Measurement,” Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, 13-15 December 2006, pp. 1557-1561.

[14] A. Tayebi and S. McGilvray, “Attitude Stabilization of a VTOL Quadrotor Aircraft,” IEEE Transactions on Control Systems Technology, Vol. 14, No. 3, 2006, pp. 562- 571. doi:10.1109/TCST.2006.872519

[15] P. Tsiotras, “A Passivity Approach to Attitude Stabilization Using Nonredundant Kinematic Parameterizations,” Proceedings of the 34th Conference on Decision and Control, New Orleans, 13-15 December 1995, pp. 13-15.

[16] P. Tsiotras, H. Shen and C. Hall, “Satellite Attitude Control and Power Tracking With Energy/Momentum Wheels,” Journal of Guidance, Control and Dynamics, Vol. 24, No. 1, 2001, pp. 23-34. doi:10.2514/2.4705

[1] M. R. Akella, “Rigid Body Attitude Tracking without Angular Velocity Feed Back,” System & Control Letters, Vol. 42, No. 4, 2001, pp. 321-326. doi:10.1016/S0167-6911(00)00102-X

[2] B. T. Costic, D. M. Dawson, M. S. De Queiroz and V. Kapila, “A Quaternion-Based Adaptive Attitude Tracking Controller without Velocity Measurements,” Proceedings of the 39th IEEE Conference on Decision and Control, Sydeny, Vol. 3, 12-15 December 2000, pp. 2424- 2429.

[3] A. El-Gohary, “Optimal Control of the Rotational Motion of a Rigid Body Using Euler Parameters With the Help of Rotor System,” European Journal of Mechanics-A/Solids, Vol. 24, No. 1, 2005, pp. 111-125. doi:10.1016/j.euromechsol.2004.10.007

[4] A. El-Gohary, “New Optimal Control Law for Attitude of a Rigid Body without Angular Velocity Measurements,” Chaos, Solitons & Fractals, Vol. 25, No. 3, 2005, pp. 557-571. doi:10.1016/j.chaos.2004.12.008

[5] A. I. El-Gohary and T. S. Tawfik, “Optimal Control of a Rigid Body Motion Using Euler Parameters Without Angular Velocity Measurements,” Mechanics Research Communications, Vol. 37, No. 3, 2010, pp. 354-359. doi:10.1016/j.mechrescom.2010.02.004

[6] H. Schaub and J. L. Junkins, “Stereographic Orientation Parameters For Attitude Dynamics: A Generalization of the Rodrigues Parameters,” Journal of the Astronautical Sciences, Vol. 44, No. 1, 1996, pp. 1-19.

[7] G. R. Izzo and L. Pettazzi, “Command Shaping for a Flexible Satellite Platform Controlled by Advanced Fly- Wheels Systems,” Acta Astronautica, Vol. 60, No. 10-11, 2007, pp. 820-827.

[8] J. Li, M. Xu, Z. L.Wang and S. M. Wang, “Minimum-Torque Earth off-Nadir Pointing Control of Gravity-Gradient Stabilized Small Satellites,” Tsinghua Science and Technology, Vol. 5, No. 1, 2000, pp. 31-33.

[9] N. Krasovskii, “Problem of Stabilization of Controlled Motion,” In: I. G. Malkin, Ed., The Theory of Motion Stability, Nauka, Moscow, 1966, pp. 475-514.

[10] F. Lizarralde and J. T. Wen, “Attitude Control without Angular Velocity Measurement: A Passivity Approach,” IEEE transactions on Automatic Control, Vol. 41, No. 3, 1996, pp. 468-472. doi:10.1109/9.486654

[11] M. Lovera and A. Astolfib, “Spacecraft Attitude Control Using Magnetic Actuators,” Automatica, Vol. 40, No. 8, 2004, pp. 1405-1414. doi:10.1016/j.automatica.2004.02.022

[12] V. M. Matrosov, “On Stability of Motion,” Journal of Applied Mathematics and Mechanics, Vol. 26, No. 5, 1962, pp. 992-1002.

[13] A. Tayebi, “Unit Quaternion Observer Based Attitude Stabilization of a Rigid Spacecraft Without Velocity Measurement,” Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, 13-15 December 2006, pp. 1557-1561.

[14] A. Tayebi and S. McGilvray, “Attitude Stabilization of a VTOL Quadrotor Aircraft,” IEEE Transactions on Control Systems Technology, Vol. 14, No. 3, 2006, pp. 562- 571. doi:10.1109/TCST.2006.872519

[15] P. Tsiotras, “A Passivity Approach to Attitude Stabilization Using Nonredundant Kinematic Parameterizations,” Proceedings of the 34th Conference on Decision and Control, New Orleans, 13-15 December 1995, pp. 13-15.

[16] P. Tsiotras, H. Shen and C. Hall, “Satellite Attitude Control and Power Tracking With Energy/Momentum Wheels,” Journal of Guidance, Control and Dynamics, Vol. 24, No. 1, 2001, pp. 23-34. doi:10.2514/2.4705