Abstract
In this paper we consider the problem of slewing a rigid spacecraft from rest to rest between prescribed attitudes while avoiding a forbidden direction during the slew. Using methods of differential geometric control theory, we derive a control law which minimises a cost functional penalizing both high angular velocities and proximity to the forbidden direction. The system of differential equations for the optimal angular velocities, arrived at by an application of Pontryagin's principle, turns out to be completely integrable. The constants of integration (which depend on the target attitude) can be determined analytically for a suitable choice of the cost functional; in the general case, a numerical scheme based on the integration of the variational equations is proposed.
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References
R.W. Brockett,System theory on group manifolds and coset spaces. SIAM J. Control Optim. 10 (1972),265–284.
______, Lie theory and control systems defined on spheres. SIAM J. Appl. Math. 25 (1973),213–225.
R.W. Brockett, R. S. Milman, and H. J. Sussmann. Differential geometric control theory. Birkhäuser-Verlag, 1983.
D. Folta, L. K. Newman, and D. Quinn, Design and implementation of satellite formations and constellations. Advances in the Astronautical Sci. 100, No. 1, 57–70.
P. C. Hughes, Spacecraft attitude dynamics. John Wiley & Sons, 1986.
B. Jakubczyk and W. Respondek, Geometry of feedback and optimal control. Marcel Dekker,1998.
J. L. Junkins and J.D. Turner, Optimal spacecraft rotational maneuvers. In: Elsevier Science Publishers,1986.
V. Jurdjevic, Non-Euclidean elastica. Amer. J. Math. 117 (1995),N o. 1, 93–124.
______, The geometry of the plate–ball problem. Arch. Ration. Mech. Anal. 124 (1993),305–328.
21-2, Geometric control theory. Cambridge University Press,1996.
21-3,In tegrable Hamiltonian systems on Lie groups: Kowalewski type. Ann. of Math. 150 (1999),605–644.
V. Jurdjevic and H. J. Sussmann, Control systems on Lie groups. J. Differential Equations 12 (1972), 313–329.
A.M. Kovalev, Nonlinear problems of control and observation in the theory of dynamical systems. (Russian), Kiev, Naukova Dumka,1980.
N. E. Leonard and J.E. Marsden, Stability and drift of underwater vehicle dynamics: mechanical systems with rigid motion symmetry. Phys. D 105 (1997),130–162.
N. E. Leonard and P. S. Krishnaprasad, Motion control of drift-free, left-invariant systems on Lie groups. IEEE Trans. Automat. Control 40 (1995),N o. 9, 1539–1554.
A. M. Letov, Engineering philosophy of optimization in the problem of analytical design of optimal controllers. In: Automatic and remote control III (Proc. Third Congr. IFAC, London) (1966), vol. 1, 22–28, Inst. Mech. Engrs., London (1967). 22 K. SPINDLER
A. D. Lewis and R.M. Murray, Configuration controllability of simple mechanical control systems. SIAM J. Control Optim. 35 (1997), No. 3, 766–790.
D. Mittenhuber, Dubins' problem in the hyperbolic plane using the open-disc model and in hyperbolic space. In: Proc. Conf. geometric control and non-holonomic mechanics, Mexico-City (1996), 101–152.
______,Dubins' problem is intrinsically three-dimensional. ESAIM Control Optim. Calc. Var. 3 (1998),1–22.
F. Monroy-Pérez, Three-dimensional non-Euclidean Dubins' problem. In: Proc. Conf. Geometric Control and Non-holonomic Mechanics, Mexico-City (1996),153–181.
______, Non-Euclidean Dubins' problem. J. Dynam. Control Systems 4 (1998), No. 2, 249–272.
K. Spindler, Optimal control on Lie groups with applications to attitude control. Math. Control Signals and Systems 11 (1998), No. 3, 197–219.
______, Attitude control of underactuated spacecraft. European J. Control 6 (2000), No. 3, 229–242.
G. C. Walsh, R. Montgomery, and S. S. Sastry, Optimal path planning on matrix Lie groups. In: Proc. 33rd Conf. on Decision and Control, Lake Buena Vista, Dec. (1994).
J. Wertz, Spacecraf t attitude determination and control. Kluwer Academic Publishers,1990.
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Spindler, K. Attitude Maneuvers Which Avoid a Forbidden Direction. Journal of Dynamical and Control Systems 8, 1–22 (2002). https://doi.org/10.1023/A:1013907732365
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DOI: https://doi.org/10.1023/A:1013907732365