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In general, optimal control problems rely on numerically rather than analytically solving methods, due to their nonlinearities. The direct method, one of the numerically solving methods, is mainly to transform the optimal control problem into a nonlinear optimization problem with finite dimensions, via discretizing the objective functional and the forced dynamical equations directly. However, in the procedure of the direct method, the classical discretizations of the forced equations will reduce or affect the accuracy of the resulting optimization problem as well as the discrete optimal control. In view of this fact, more accurate and efficient numerical algorithms should be employed to approximate the forced dynamical equations. As verified, the discrete variational difference schemes for forced Birkhoffian systems exhibit excellent numerical behaviors in terms of high accuracy, long-time stability and precise energy prediction. Thus, the forced dynamical equations in optimal control problems, after being represented as forced Birkhoffian equations, can be discretized according to the discrete variational difference schemes for forced Birkhoffian systems. Compared with the method of employing traditional difference schemes to discretize the forced dynamical equations, this way yields faithful nonlinear optimization problems and consequently gives accurate and efficient discrete optimal control. Subsequently, in the paper we are to apply the proposed method of numerically solving optimal control problems to the rendezvous and docking problem of spacecrafts. First, we make a reasonable simplification, i.e., the rendezvous and docking process of two spacecrafts is reduced to the problem of optimally transferring the chaser spacecraft with a continuously acting force from one circular orbit around the Earth to another one. During this transfer, the goal is to minimize the control effort. Second, the dynamical equations of the chaser spacecraft are represented as the form of the forced Birkhoffian equation. Then in this case, the discrete variational difference scheme for forced Birkhoffian system can be employed to discretize the chaser spacecraft's equations of motion. With further discretizing the control effort and the boundary conditions, the resulting nonlinear optimization problem is obtained. Finally, the optimization problem is solved directly by the nonlinear programming method and then the discrete optimal control is achieved. The obtained optimal control is efficient enough to realize the rendezvous and docking process, even though it is only an approximation of the continuous one. Simulation results fully verify the efficiency of the proposed method for numerically solving optimal control problems, if the fact that the time step is chosen to be very large to limit the dimension of the optimization problem is noted.
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Keywords:
- Birkhoffian system /
- optimal control /
- nonlinear programming /
- rendezvous and docking
[1] Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoff System (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔, 史荣昌, 张永发, 吴惠彬 1996 Birkhoff 系统动力学 (北京: 北京理工大学出版社)]
[2] Cui J C, Song D, Guo Y X 2012 Acta Phys. Sin. 61 244501 (in Chinese) [崔金超, 宋端, 郭永新 2012 61 244501]
[3] Cui J C, Zhao Z, Guo Y X 2013 Acta Phys. Sin. 62 090205 (in Chinese) [崔金超, 赵喆, 郭永新 2013 62 090205]
[4] Zhang Y 2010 Commun. Theor. Phys. 53 166
[5] Zhai X H, Zhang Y 2014 Nonlinear Dyn. 77 73
[6] Zhang Y 2010 Chin. Phys. B 19 080301
[7] Kong X L, Wu H B, Mei F X 2012 J. Geom. Phys. 62 1157
[8] Liu S X, Liu C, Guo Y X 2011 Acta Phys. Sin. 60 064501 (in Chinese) [刘世兴, 刘畅, 郭永新 2011 60 064501]
[9] Liu S X, Hua W, Guo Y X 2014 Chin. Phys. B 23 064501
[10] Mei F X, Wu H B 2015 Chin. Phys. B 24 104502
[11] Mei F X, Wu H B 2015 Chin. Phys. B 24 054501
[12] Sun Y J, Shang Z J 2005 Phys. Lett. A 336 358
[13] Su H L, Sun Y J, Qin M Z, Scherer R 2007 Int. J. Pure Appl. Math. 40 341
[14] Kong X L, Wu H B, Mei F X 2016 Chin. Phys. B 25 010203
[15] Liu C 2012 Ph. D. Dissertation (Beijing: Beijing Institue of Technology) (in Chinese) [刘畅 2012 博士学位论文 (北京: 北京理工大学)]
[16] Kong X L, Wu H B, Mei F X 2013 Appl. Math. Comp. 225 326
[17] Kong X L, Wu H B, Mei F X 2013 Nonlinear Dyn. 74 711
[18] Gill P E, Jay L O, Leonard M W, Petzold L R, Sharma V 2000 J. Comput. Appl. Math. 120 197
[19] Zhang Y 2008 Chin. Phys. B 17 4365
[20] Marsden J E, Ratiu T S 1999 Introduction to Mechanics and Symmetry (New York: Springer)
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[1] Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoff System (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔, 史荣昌, 张永发, 吴惠彬 1996 Birkhoff 系统动力学 (北京: 北京理工大学出版社)]
[2] Cui J C, Song D, Guo Y X 2012 Acta Phys. Sin. 61 244501 (in Chinese) [崔金超, 宋端, 郭永新 2012 61 244501]
[3] Cui J C, Zhao Z, Guo Y X 2013 Acta Phys. Sin. 62 090205 (in Chinese) [崔金超, 赵喆, 郭永新 2013 62 090205]
[4] Zhang Y 2010 Commun. Theor. Phys. 53 166
[5] Zhai X H, Zhang Y 2014 Nonlinear Dyn. 77 73
[6] Zhang Y 2010 Chin. Phys. B 19 080301
[7] Kong X L, Wu H B, Mei F X 2012 J. Geom. Phys. 62 1157
[8] Liu S X, Liu C, Guo Y X 2011 Acta Phys. Sin. 60 064501 (in Chinese) [刘世兴, 刘畅, 郭永新 2011 60 064501]
[9] Liu S X, Hua W, Guo Y X 2014 Chin. Phys. B 23 064501
[10] Mei F X, Wu H B 2015 Chin. Phys. B 24 104502
[11] Mei F X, Wu H B 2015 Chin. Phys. B 24 054501
[12] Sun Y J, Shang Z J 2005 Phys. Lett. A 336 358
[13] Su H L, Sun Y J, Qin M Z, Scherer R 2007 Int. J. Pure Appl. Math. 40 341
[14] Kong X L, Wu H B, Mei F X 2016 Chin. Phys. B 25 010203
[15] Liu C 2012 Ph. D. Dissertation (Beijing: Beijing Institue of Technology) (in Chinese) [刘畅 2012 博士学位论文 (北京: 北京理工大学)]
[16] Kong X L, Wu H B, Mei F X 2013 Appl. Math. Comp. 225 326
[17] Kong X L, Wu H B, Mei F X 2013 Nonlinear Dyn. 74 711
[18] Gill P E, Jay L O, Leonard M W, Petzold L R, Sharma V 2000 J. Comput. Appl. Math. 120 197
[19] Zhang Y 2008 Chin. Phys. B 17 4365
[20] Marsden J E, Ratiu T S 1999 Introduction to Mechanics and Symmetry (New York: Springer)
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