Quaternion Solution to the Problem of Optimal Control for Spacecraft’S Spatial Reorientation With Use of a Combined Quality Criterion Taking Into Account Energy Costs
Mikhail Levskii1*
1Research Institute of Space Systems, Khrunichev State Space Research-and-production Center, Korolev
*Correspondence to: Mikhail Levskii, Research Institute of Space Systems, Khrunichev State Space Research-and-production Center, Tikhonravova Street, 27, Korolev, Russia; E‑mail: levskii1966@mail.ru
Abstract
Objective: This research focuses on economically controlling spacecraft motion to minimize energy costs during reorientation. It constructs an optimal control method that considers both control forces and rotational kinetic energy, aiming for efficient spatial reorientation.
Methods: Utilizing the quaternion method and Pontryagin's maximum principle, the study devises a restricted control for optimal spacecraft turns. Analytical solutions are derived from differential equations relating orientation quaternion and angular velocity, with numerical simulations used for verification.
Results: The study presents a solution to the optimal control synthesis problem for spacecraft reorientation, optimizing for minimal energy costs. Key properties of optimal solutions are formulated analytically, aiding in determining optimal control algorithm parameters. Mathematical modeling illustrates the process and practical feasibility of the designed method for attitude control.
Conclusion: This research provides a comprehensive solution to spacecraft orientation optimal control, particularly beneficial for spacecraft equipped with electric-jet engines. The explicit control law improves efficiency and economizes spacecraft motion during orbit flight, contributing to smoother, continuous functions of time for both control functions and phase variables. The study's relevance lies in addressing the cost-effectiveness of spacecraft motion control, with potential for further research on optimal control with additional restrictions.
Keywords: spacecraft’s spatial attitude, quaternion, angular velocity, criterion of quality, maximum principle
1 INTRODUCTION
Problem of spacecraft reorientation into given angular position was solved. The solution method and the formalized description of spacecraft’s rotational motion kinematics are based on the quaternion models for description of rotational motion of solid body[1]. Spatial reorientation is a moving the spacecraft axes from one known attitude into another given angular position in finite time T. Angular attitude of spacecraft’s coordinate system is determined relative to a chosen reference basis. We considered the version of spatial turn when inertial coordinate system is reference system as often encountered case.
Much of papers have been dedicated to investigating a controlled rotation of solid body and problem of optimal control for spacecraft attitude in different statements and using different methods of solving[1-26]. For example, some authors propose the synthesis of optimal control based on the method of analytical design of optimal controllers[2], others use the concept of inverse problems of dynamics to obtain smooth controls for the implementation of the spatial rotation of the spacecraft, when the program trajectory is sought in the class of polynomials of a given degree, the coefficients of which are determined by the known values of the phase variables at the boundary points of the trajectory[3]. Special attention was paid to the problems of optimal control[2,4-24]. Optimization methods can also be different. In particular, solutions to the problem of reorientation of solid body of various configurations, based on the Pontryagin’s maximum principle, were considered in papers[9-24]. The time-optimal control is very importantly, therefore this maneuver is popular and interesting[5-12]. Other classical criteria for the quality of the control process were used previously (minimum fuel consumption[17], minimum energy consumption[12,17], etc.). Kinematic problems of rotation were considered in more detail[13-16]. Dynamic problems of optimal control are of particular interest and, at the same time, certain difficulties in solving the boundary value problem of a rotation; in some particular cases of control over a fixed time, the boundary two-point rotation problem is solved by the method of separation of variables[17]. Special regime of control for spacecraft rotation was examined also[18]. Specific features of attitude control for a spacecraft with inertial actuators (in particular, the gyrodins) have been researched earlier[25,26]. The patented method can be used in control system of a spacecraft, controlled by the gyrodins (or other inertial actuators)[27].
Earlier, planar rotations[5,8], relay controls for a turn[1,5-9,12], or the algorithms without optimization[3] for finding the smooth control functions were investigated. It is necessary to select out especially problems of the time-optimal turn[1,5-12], control problems for an axi-symmetric rigid body[10-13,18,20-22], and also kinematical problems of optimal turn[13-16]. Here, the dynamic problem of optimal control of a turn with the restricted control and the combined criterion of quality reflecting total energy costs (energy contribution of the control torques and integral costs of kinetic energy) is considered and solved in analytic form.
Analytical solution to an optimal turn problem in a closed form is of great practical interest because such solution allows the finished laws of the programmed control and variation of an optimal trajectory of spacecraft’s motion to be applied onboard. However, it is extremely difficult to obtain them for bodies (spacecrafts) with an arbitrary dynamic configuration. Some solutions (including analytical ones) were obtained for spherical[1,19] and dynamically symmetric bodies[10-13,20-22]. But analytical solution to the problem of three-dimensional turn with arbitrary boundary conditions (for angular position of a spacecraft) was not found for solid body with arbitrary distribution of mass. We know certain particular cases when general problem of a turn is solved[1,12,19]. Consequently, we can use only numerical methods (for approximate solution to this problem).
Below, we solve the problem of reorientation with new index of quality and the restricted control. In this article, the adopted functional of quality characterizes the energy costs as combination of costs of control resources and rotation energy. Its minimization is very important task in practice of spacecraft flight. For the time being, the issues of cost-effectiveness remain relevant for spacecraft motion control. A finding and investigating the optimal control of spacecraft reorientation from initial angular position into given spatial position (with respect to the chosen combined indicator) is the purpose of our research.
Optimal control problems for spacecraft orientation with use of the combined criterions of optimality were investigated earlier[21-23]. However, in contrast to the published papers, the quality criterion used by us provides smooth control functions (it is firstly) and spacecraft motion with kinetic energy of rotation which is minimum for given time of a turn (it is secondly). Additionally, in our problem of optimal reorientation, the restricted control torque acts to a spacecraft for its rotation. Also, it should be noted that solution obtained in the presented research can be applied for any type of a spacecraft (but not only for spacecrafts with inertial actuators[25,26]), in contrast to the previous researches.
4 CONCLUSION
Optimal control program of spacecraft’s reorientation with minimal costs of energy has been found; it is demonstrated that the control when angular momentum is parallel to a controlling torque within the entire interval of spatial turn is optimum. The issues of profitability and economical control of spacecraft rotations is relevant in present time, therefore the studied problem is very important. The solved problem differs from other problems with a combined functional in index form, in the presence of restrictions on the control and does not concern to an axi-symmetric body[21-23]. The obtained results demonstrate that the designed control method of spacecraft’s three-dimensional reorientation is feasible in practice.
Acknowledgements
The author would like to express greatest thanks to colleagues and specialists from Korolev Rocket Space Corporation (in Russia), and to all participants of the seminars and International conferences on optimization problems.
Conflicts of Interest
The author declared that there are no conflict of interest.
Author Contribution
Levskii M was responsible for the original draft, including methodology, validation, mathematical formulating and analysis of formulas, writing, conceptualization, supervision, numerical modeling and design of the figures that illustrated example of mathematical simulation of optimal rotation.
Abbreviation List
ERJ, Electric-rocket engines
References
[1] Branets VN, Shmyglevskii IP. Use of quaternions in problems of orientation of solid bodies. Nauka: Moscow, USSR, 1973.
[2] Sinitsin LI, Kramlikh AV. Synthesis of the optimal control law for the reorientation of a nanosatellite using the procedure of analytical construction of optimal regulators. J Phys: Conf Ser, 2021; 1745: 012053.[DOI]
[3] Velishchanskii MA, Krishchenko AP, Tkachev SB. Synthesis of spacecraft reorientation algorithms using the concept of the inverse dynamic problem. J Comput Syst Sci Int, 2003; 42: 811‑818.[DOI]
[4] Junkins JL, Turner JD. Optimal spacecraft rotational maneuvers. Elsevier: USA, 1986.
[5] Reshmin SA. The threshold absolute value of a relay control bringing a satellite to a gravitationally stable position in optimal time. Dokl Phys, 2018; 63: 257-261.[DOI]
[6] Scrivener S, Thompson R. Survey of time-optimal attitude maneuvers. J Guid Contr Dyn, 1994; 17: 225-233.[DOI]
[7] Zhou H, Wang D, Wu B et al. Time-optimal reorientation for rigid satellite with reaction wheels. Int J Contr, 2012; 85: 1452-1463.[DOI]
[8] Reshmin SA. Threshold absolute value of a relay control when time-optimally bringing a satellite to a gravitationally stable position. J Comput Syst Sci Int, 2018; 57: 713‑722.[DOI]
[9] Levskii MV. Pontryagin’s maximum principle in optimal control problems of orientation of a spacecraft. J Comput Syst Sci Int, 2008; 47: 974‑986.[DOI]
[10] Shen H, Tsiotras P. Time-optimal control of axi-symmetric rigid spacecraft with two controls. AIAA J Guid Control Dyn, 1999; 22: 682-694.[DOI]
[11] Molodenkov AV, Sapunkov YG. Analytical solution of the minimum time slew maneuver problem for an axially symmetric spacecraft in the class of conical motions. J Comput Syst Sci Int, 2018; 57: 302-318.[DOI]
[12] Branets VN, Chertok MB, Kaznacheev YV. Optimal turn of a rigid body with a single axis of symmetry. Kosm Issled, 1984, 22: 352-360.[DOI]
[13] Zelepukina OV, Chelnokov YN. Construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric rigid body. Mech Solids, 2011; 46: 519‑533.[DOI]
[14] Levskii MV. Optimal spacecraft terminal attitude control synthesis by the quaternion method. Mech. Solids, 2009; 44: 169-183.[DOI]
[15] Levskii MV. Kinematically optimal spacecraft attitude control. J Comput Syst Sci Int, 2015; 54: 116-132.[DOI]
[16] Biryukov VG, Chelnokov YN. Construction of optimal laws of variation of the angular momentum vector of a rigid body. Mech Solids, 2014; 49: 479‑494.[DOI]
[17] Aipanov SA, Zhakypov AT. The method of separation of variables and its application to the problem of a spacecrafts optimal turn. Cosmic Res, 2020; 58: 53-63.[DOI]
[18] Molodenkov AV, Sapunkov YG. Special control regime in the problem of optimal turn of an axially symmetric spacecraft. J. Comput Syst Sci Int, 2010; 49: 891-899.[DOI]
[19] Levskii MV. About method for solving the optimal control problems of spacecraft spatial orientation. Проблемы нелинейного анализа в инженерных системах, 2015; 21: 61-75.
[20] Molodenkov AV, Sapunkov YG. Analytical solution of the optimal slew problem for an axisymmetric spacecraft in the class of conical motions. J Comput Syst Sci Int, 2016; 55: 969‑985.[DOI]
[21] Molodenkov AV, Sapunkov YG. Analytical quasi-optimal solution of the slew problem for an axially symmetric rigid body with a combined performance index. J Comput Syst Sci Int, 2020; 59: 347–357.[DOI]
[22] Sapunkov YG, Molodenkov AV. Analytical solution of the problem on an axisymmetric spacecraft attitude maneuver optimal with respect to a combined functional. Autom Remote Contr, 2021; 82: 1183-1200.[DOI]
[23] Molodenkov AV, Sapunkov YG. Analytical solution of the optimal attitude maneuver problem with a combined objective functional for a rigid body in the class of conical motions. Mech Solids, 2016; 51: 135-147.[DOI]
[24] Levskii MV. Optimal control of the angular momentum for a solid (spacecraft) performing a spatial turn. Mech Solids, 2023; 58: 63‑77.[DOI]
[25] Lam QM. Robust and adaptive reconfigurable control for satellite attitude control subject to under-actuated control condition of reaction wheel assembly. Math Eng Sci Aerosp, 2018; 9: 47‑63.[DOI]
[26] Levskii MV. Special aspects in attitude control of a spacecraft, equipped with inertial actuators. J Comp Sci Appl Inform Technol, 2017; 2: 1-9.[DOI]
[27] Levskii MV. Method of controlling a spacecraft turn. RF Patent No. 2093433, 1997.
[28] Gorshkov OA, Muravev VA, Shagayda AA. Holl's and ionic plasma engines for spacecrafts. Mashinostroenie: Moscow, 2008, Russia.
[29] Pontryagin LS, Boltyanskii VG, Gamkrelidze RV et al. The mathematical theory of optimal processes. Gordon and Breach: New York, 1986, USA.
[30] Chanyal BC. Quaternionic approach on the Dirac–Maxwell, Bernoulli and Navier–Stokes equations for dyonic fluid plasma. Int J Mod Ph, 2019; 34: 1950202.[DOI]
[31] Chanyal BC, Pathak M. Quaternionic approach to dual magnetohydrodynamics of dyonic cold plasma. Adv High En Ph, 2018; 2018: 1-13.[DOI]
[32] Chanyal BC, Sandhya. A comparative study of quaternionic rotational Dirac equation and its interpretation. In. J Geom Meth Mod Ph, 2020; 17: 2050018.[DOI]
[33] Levskii MV. Device for regular rigid body precession parameters formation. RF Patent No. 2146638, 2000.
[34] Levskii MV. Method of controlling a spacecraft turn and system for its realization. RF Patent No. 2114771, 1998.
[35] Levskii MV. System for control of spacecraft turn. RF Patent No. 2006431, 1994.
[36] Kumar S, Kanwar V, Singh S. Modified efficient families of two and three-step predictor-corrector iterative methods for solving nonlinear equations. Appl Math, 2010; 1: 153-158.[DOI]
[37] Kulkov VM, Obukhov VA, Egorov YuG et al. Comparative estimation of efficiency of applying the perspective types of electric-rocket engines on small spacecrafts. Vestn Samar Gos Aerokosm Univ, 2012; 3: 187‑195.
Copyright © 2024 The Author(s). This open-access article is licensed under a Creative Commons Attribution 4.0 International License ( https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, sharing, adaptation, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright ©