In the last years the small satellites have played an important role in the technological development. The attractive short period of design and low cost of them and the capacity to solve problems that are usually considered as problems to big and expensive spacecrafts lead us to study the control problem of these satellites. Active three-axis magnetic attitude stabilization of a low Earth orbit satellite is considered in this work. The control is created by interaction between the magnetic moment generated by magnetorquers mounted on the satellite body and the geomagnetic field. This problem is quite complex and difficult to solve. To overcome this difficulty guidance control is considered, where we use ε-strategies introduced by Pontryagin in the frame of differential games theory. Qualitative analysis and results of numerical simulation are presented. 1. Introduction The problem of attitude control of spacecraft has been widely studied in the last years. If the spacecraft is equipped with three independent actuators, a complete solution to the set point and tracking control problems is available. This problem was solved by different ways (see, e.g., [1–4]). If only two independent actuators are available, as discussed in detail in [5], the problem of attitude regulation is not solvable by means of continuous time-invariant control laws, whereas a time-varying control law, achieving local asymptotic nonexponential stability, was proposed in [6]. Since magnetorquers are relatively reliable, lightweight, and energy efficient, they have become attractive for small and inexpensive satellites, but the above results are not directly applicable if the satellite is equipped with these magnetic coils as actuators. Such actuators operate on the basis of the interaction between the magnetic field generated by magnetic coils installed on the satellite body and the magnetic field of the Earth. This implies a spacecraft control significantly different. There are two classical uses of magnetic torque rods in attitude control. One is for momentum management of wheel-based systems [7]. The other is for angular momentum and nutation control of spinning [8], momentum-biased [9], and dual-spin spacecraft [10]. The present study is one of a growing number that considers active three-axis magnetic attitude stabilization of a low Earth orbit satellite considering a nontilted dipole model for the geomagnetic field. Magnetic coils have been extensively used since the early sixties as a simple and reliable technology to implement attitude control actuators in low Earth orbit
References
[1]
J. T.-Y. Wen and K. Kreutz-Delgado, “The attitude control problem,” IEEE Transactions on Automatic Control, vol. 36, no. 10, pp. 1148–1162, 1991.
[2]
O.-E. Fjellstad and T. I. Fossen, “Comments on “The attitude control problem”,” IEEE Transactions on Automatic Control, vol. 39, no. 3, pp. 699–700, 1994.
[3]
F. Caccavale and L. Villani, “Output feedback control for attitude tracking,” Systems and Control Letters, vol. 38, no. 2, pp. 91–98, 1999.
[4]
M. R. Akella, “Rigid body attitude tracking without angular velocity feedback,” Systems and Control Letters, vol. 42, no. 4, pp. 321–326, 2001.
[5]
C. I. Byrnes and A. Isidori, “On the attitude stabilization of rigid spacecraft,” Automatica, vol. 27, no. 1, pp. 87–95, 1991.
[6]
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,” Systems and Control Letters, vol. 25, no. 5, pp. 375–385, 1995.
[7]
H. Iida and K. Ninomiya, “A new approach to magnetic angular momentum management for large scientific satellites,” NEC Research and Development, vol. 37, no. 1, pp. 60–77, 1996.
[8]
L. D. D. Ferreira and J. J. Cruz, “Attitude and spin rate control of a spinning satellite using geomagnetic field,” Journal of Guidance, Control, and Dynamics, vol. 14, no. 1, pp. 216–218, 1991.
[9]
M. E. Pittelkau, “Optimal periodic control for spacecraft pointing and attitude determination,” Journal of Guidance, Control, and Dynamics, vol. 16, no. 6, pp. 1078–1084, 1993.
[10]
K. T. Alfriend, “Magnetic attitude control system for dual-spin satellites,” AIAA Journal, vol. 13, no. 6, pp. 817–822, 1975.
[11]
J. S. White, F. H. Shigemoto, and K. Bourquin, “Satellite attitude control utilizing the Earth’s magnetic field,” Tech. Rep. D-1068, NASA, Washington, DC, USA, 1961.
[12]
A. C. Stickler and K. T. Alfriend, “Elementary magnetic attitude control system,” Journal of Spacecraft and Rockets, vol. 13, no. 5, pp. 282–287, 1976.
[13]
F. Martel, P. K. Pal, and M. L. Psiaki, “Active magnetic control system for gravity gradient stabilized spacecraft,” in Proceedings of the 2nd Annual AIAA/USU Conference on Small Satellites, Logan, Utah, USA, September 1988.
[14]
K. L. Musser and W. L. Ebert, “Autonomous spacecraft attitude control using magnetic torquing only,” in Proceedings of the Flight Mechanics/Estimation Theory Symposium, pp. 23–38, NASA, May 1989.
[15]
R. Wi？niewski, “Nonlinear control for satellite detumbling based on magnetic torquing,” in Proceedings of the 22nd (JSDE) Joint Service Data Exchange for Guidance, Navigation and Control, Scottsdale, Ariz, USA, October-November 1994.
[16]
R. Wi？niewski and M. Blanke, “Three-axis satellite attitude control based on magnetic torquing,” in Proceedings of the 13th IFAC World Congress, San Francisco, Calif, USA, June 1996.
[17]
E. Silani and M. Lovera, “Magnetic spacecraft attitude control: a survey and some new results,” Control Engineering Practice, vol. 13, no. 3, pp. 357–371, 2005.
[18]
R. Wi？niewski and M. Blanke, “Fully magnetic attitude control for spacecraft subject to gravity gradient,” Automatica, vol. 35, no. 7, pp. 1201–1214, 1999.
[19]
R. Wi？niewski, “Linear time-varying approach to satellite attitude control using only electromagnetic actuation,” Journal of Guidance, Control, and Dynamics, vol. 23, no. 4, pp. 640–647, 2000.
[20]
M. L. Psiaki, “Magnetic torquer attitude control via asymptotic periodic linear quadratic regulation,” Journal of Guidance, Control, and Dynamics, vol. 24, no. 2, pp. 386–394, 2001.
[21]
R. Wi？niewski and F. L. Markley, “Optimal magnetic attitude control,” in Proceedings of the 14th IFAC World Congress, Beijing, China, July 1999.
[22]
M. Lovera, E. De Marchi, and S. Bittanti, “Periodic attitude control techniques for small satellites with magnetic actuators,” IEEE Transactions on Control Systems Technology, vol. 10, no. 1, pp. 90–95, 2002.
[23]
H. Yan, I. M. Ross, and K. T. Alfriend, “Pseudospectral feedback control for three-axis magnetic attitude stabilization in elliptic orbits,” Journal of Guidance, Control, and Dynamics, vol. 30, no. 4, pp. 1107–1115, 2007.
[24]
M. Corno and M. Lovera, “Spacecraft attitude dynamics and control in the presence of large magnetic residuals,” Control Engineering Practice, vol. 17, no. 4, pp. 456–468, 2009.
[25]
T. Pulecchi, M. Lovera, and A. Varga, “Optimal discrete-time design of three-axis magnetic attitude control laws,” IEEE Transactions on Control Systems Technology, vol. 18, no. 3, pp. 714–722, 2010.
[26]
M. Abdelrahman, I. Chang, and S.-Y. Park, “Magnetic torque attitude control of a satellite using the state-dependent Riccati equation technique,” International Journal of Non-Linear Mechanics, vol. 46, no. 5, pp. 758–771, 2011.
[27]
C. Arduini and P. Baiocco, “Active magnetic damping attitude control for gravity gradient stabilized spacecraft,” Journal of Guidance, Control, and Dynamics, vol. 20, no. 1, pp. 117–122, 1997.
[28]
C. J. Damaren, “Comments on “Fully magnetic attitude control for spacecraft subject to gravity gradient”,” Automatica, vol. 38, no. 12, p. 2189, 2002.
[29]
M. Lovera and A. Astolfi, “Global attitude regulation using magnetic control,” in Proceedings of the 40th IEEE Conference on Decision and Control (CDC '01), pp. 4604–4609, Orlando, FL, USA, December 2001.
[30]
M. Lovera and A. Astolfi, “Spacecraft attitude control using magnetic actuators,” Automatica, vol. 40, no. 8, pp. 1405–1414, 2004.
[31]
M. Lovera and A. Astolfi, “Global magnetic attitude control of inertially pointing spacecraft,” Journal of Guidance, Control, and Dynamics, vol. 28, no. 5, pp. 1065–1067, 2005.
[32]
M. Lovera and A. Astolfi, “Global magnetic attitude control of spacecraft in the presence of gravity gradient,” IEEE Transactions on Aerospace and Electronic Systems, vol. 42, no. 3, pp. 796–805, 2006.
[33]
V. Bushenkov and G. Smirnov, Stabilization Problems with Constraints: Analysis and Computational Aspects, Gordon and Breach Science Publishers, Amsterdam, The Netherlands, 1997.
[34]
L. S. Pontryagin, “Linear differential games. I,” Soviet Mathematics. Doklady, vol. 8, pp. 769–771, 1967.
[35]
R. N. Izmailov, “The peak effect in stationary linear systems with scalar inputs and outputs,” Automation and Remote Control, vol. 48, no. 8, pp. 1018–1024, 1987.
[36]
H. J. Sussmann and P. V. Kokotovic, “The peaking phenomenon and the global stabilization of nonlinear systems,” IEEE Transactions on Automatic Control, vol. 36, no. 4, pp. 424–440, 1991.
[37]
G. Smirnov, V. Bushenkov, and F. Miranda, “Advances on the transient growth quantification in linear control systems,” International Journal of Applied Mathematics and Statistics, vol. 14, no. J09, pp. 82–92, 2009.
[38]
S. P. Bhat, “Controllability of nonlinear time-varying systems: applications to spacecraft attitude control using magnetic actuation,” IEEE Transactions on Automatic Control, vol. 50, no. 11, pp. 1725–1735, 2005.
[39]
G. Smirnov, Introduction to the Theory of Differential Inclusions, American Mathematical Society, Providence, RI, USA, 2002.
[40]
I. V. Burkov, “Asymptotic stabilization of the position of a rigid body with fixed point without velocity measurements,” Systems and Control Letters, vol. 25, no. 3, pp. 205–209, 1995.
[41]
S. Sastry, Nonlinear Systems: Analysis, Stability, and Control, Interdisciplinary Applied Mathematics, Vol. 10, Springer, Berlin, Germany, 1999.
