Abstract
The problem of energy-optimal turn maneuver of a solid having arbitrary dynamic configuration is considered within the framework of the quaternion formulation with arbitrary boundary conditions and without restrictions imposed on the control function. This turn maneuver is performed in fixed time. In the class of generalized conical motions, the optimal turn maneuver problem has been modified. As a result, it is possible to obtain its analytical solution. The analytical solution of the modified problem can be considered as an approximate (quasi-optimal) solution of the traditional optimal turn maneuver problem. An algorithm for the quasi-optimal turn maneuver of a solid is given. Given numerical examples illustrate that the solution of the modified problem is a close approximation for the solution of the traditional problem on the optimal turn maneuver of a solid. In addition, the examples show that the kinematic characteristics of the optimal motion in the traditional problem (attitude quaternion and angular velocity vector) depend weakly on the dynamic configuration of a solid and are mainly determined by the boundary conditions of the problem (anyway, for ϕ≥ π/2 where ϕ is the Euler angle). The weak dependence of the kinematic characteristics of the optimal motion of a solid on its dynamic configuration guarantees the closeness of the solutions of the modified and traditional problems of optimal turn maneuver of a body with an arbitrary dynamic configuration.
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References
V. N. Branets and I. P. Shmyglevskii, Application of Quaternions to Rigid Body Attitude Problems (Nauka, Moscow, 1973) [in Russian].
S. L. Scrivener and R. C. Thompson, “Survey of Time-Optimal Attitude Maneuvers,” J. Guid. Cont. Dyn. 17 (2), 225–233 (1994).
Yu. N. Chelnokov, “Quaternion Solution of Kinematic Problems of Control of Orientation of Rigid Bodies: Equations of Motion, Problem Statements, Programmed Motion and Control,” Izv. Akad. Nauk. Mekh. Tv. Tela, No. 4, 7–14 (1993) [Mech. Sol. (Engl. Transl.) 33 (5), 4–10 (1993)].
V. N. Branets, M. B. Chertok, and Yu. V. Kaznacheev, “Optimal Turn of a Solid Body with one Symmetry Axis,” Kosm. Iss. 22 (3) 352–360 (1984) [Cosm. Res. (Engl. Transl.)].
A. N. Sirotin, “Optimal Control of Reorientation of Symmetric Solid Body from a Rest Position to a Rest Position,” Izv. Akad. Nauk SSSR, Mekh. Tv. Tela, No. 1, 36–47 (1989).
A. N. Sirotin, “On Time-Optimal Spatial Reorientation in a Rest Position of a rotating Spherically Symmetric Body,” Izv. Akad. Nauk SSSR, Mekh. Tv. Tela, No. 3, 18–27 (1997) [Mech. Sol. (Engl. Transl.) 32 (3), 14–21 (1997)].
A. V. Molodenkov and Ya. G. Sapunkov, “A New Class of Analytic Solutions in the Optimal Turn Problem for a Spherically Symmetric Body,” Izv. Ros. Akad. Nauk. Mekh. Tv. Tela, No. 2, 16–27 (2012). [Mech. Sol. (Engl. Trans.) 47 (2), 167–177 (2012)].
A. V. Molodenkov, “Quaternion Solution of the Problem of Optimal Turn of a Rigid Body with Spherical Distribution of Mass,” in Problems of Mechanics and Control. Collection of Scientific Papers (PGU, Perm, 1995), pp. 122–131 [in Russian].
Ya. G. Sapunkov and A. V. Molodenkov, “The Investigation of Characteristics of Distant Sounding System of the Earth with the Help of Cosmic Device,” Supp. Mekhatron. Avtomatiz. Upravl.: Avtomatich. Avtomatizir. Upr. Let. App. No. 6, 10–15 (2008).
A. V. Molodenkov and Ya. G. Sapunkov, “Analytical Solution of the Optimal Attitude Maneuver Problem with a Combined Objective Functional for a Rigid Body in the Class of Conical Motions,” Izv. Ros. Akad. Nauk. Mekh. Tv. Tela, No. 2, 3–16 (2016). [Mech. Sol. (Engl. Trans.) 51 (2), 135–147 (2016)].
A. V. Molodenkov and Ya. G. Sapunkov, “Analytical Approximate Solution of the Problem of a Spacecraft’s Optimal Turn with Arbitrary Boundary Conditions,” Izv. Akad. Nauk. Teor. Sys. Upr., No. 3, 131–141 (2015) [J. Comp. Sys. Sci. Int. (Engl. Trans.) 54 (3), 458–468 (2015)].
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Nauka, Moscow, 1961; Gordon and Breach, New York, 1986)
A. V. Molodenkov, “On the Solution of the Darboux Problem,” Izv. Ros. Akad. Nauk. Mekh. Tv. Tela, No. 2, 3–13 (2007) [Mech. Sol. (Engl. Transl.) 42 (2), 167–176 (2007)].
Yu. R. Banit, M. Yu. Belyaev, T. A. Dobrinskaya, et al., Estimating the inertia tensor of the International Space Station on the base of the telemetry information, Preprint KIAM No. 57 (Keldysh Institute of Applied Mathematics, Moscow, 2002) [in Russian].
F. Li and P. M. Bainum, “Numerical Approach for Solving Rigid Spacecraft Minimum Time Attitude Maneuvers,” J. Guid. Cont. Dyn. 13 (1), 38–45 (1990).
G. J. Lastman, “A Shooting Method for Solving Two-Point Boundary-Value Problems Arising from Non-Singular Bang-Bang Optimal Control Problems,” Int. J. Cont. 27 (4), 513–524 (1978).
N. Bedrossian, S. Bhatt, W. Kang, and M. Ross, “Zero-Propellant Maneuver Guidance,” IEEE Cont. Sys. Mag. 29 (5), 53–73 (2009).
Description of the experiment “Flight testing of the ISS turn control process using jet engines of the Russian Segment of ISS with minimum fuel consumption and taking into account restrictions on the ISS design loads,” http://knts.tsniimash.ru/ru/site/Experiment_q.aspx?idE=328&id=5.
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Russian Text © Author(s), 2019, published in Izvestiya Akademii Nauk, Mekhanika Tverdogo Tela, 2019, No. 2, pp. 140–154.
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Molodenkov, A.V., Sapunkov, Y.G. Analytical Quasi-Optimal Solution for the Problem on Turn Maneuver of an Arbitrary Solid with Arbitrary Doundary Conditions. Mech. Solids 54, 474–485 (2019). https://doi.org/10.3103/S0025654419020110
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DOI: https://doi.org/10.3103/S0025654419020110