American Institute of Mathematical Sciences

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June  2016, 21(4): 1347-1388. doi: 10.3934/dcdsb.2016.21.1347

Planar tilting maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering

Jiamin Zhu 1, , Emmanuel Trélat 2, and  Max Cerf 1,

1. 

Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France, France
2. 

Sorbonne Universités, UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, F-75005, Paris

Received  April 2015 Published  March 2016

In this paper, we study the minimum time planar tilting maneuver of a spacecraft, from the theoretical as well as from the numerical point of view, with a particular focus on the chattering phenomenon. We prove that there exist optimal chattering arcs when a singular junction occurs. Our study is based on the Pontryagin Maximum Principle and on results by M.I. Zelikin and V.F. Borisov. We give sufficient conditions on the terminal values under which the optimal solutions do not contain any singular arc, and are bang-bang with a finite number of switchings. Moreover, we implement sub-optimal strategies by replacing the chattering control with a fixed number of piecewise constant controls. Numerical simulations illustrate our results.
Keywords: singular control, Pontryagin Maximum Principle, Spacecraft planar tilting maneuver, chattering arcs, sub-optimal strategy., minimum time control.
Mathematics Subject Classification: Primary: 49K15, 49M37, 65H2.
Citation: Jiamin Zhu, Emmanuel Trélat, Max Cerf. Planar tilting maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1347-1388. doi: 10.3934/dcdsb.2016.21.1347
References:
[1]

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint,, Springer, (2004).  doi: 10.1007/978-3-662-06404-7.  Google Scholar

[2]

E. Bakolas and Tsiotras, Optimal synthesis of the Zermelo-Markov-Dubins problem in a constant drift field,, Journal of Optimization Theory and Applications, 156 (2013), 469.  doi: 10.1007/s10957-012-0128-0.  Google Scholar

[3]

J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming,, Second edition, (2010).  doi: 10.1137/1.9780898718577.  Google Scholar

[4]

K. D. Bilimoria and B. B. Wie, Time-optimal three-axis reorientation of rigid spacecraft,, Journal of Guidance, 16 (1993), 446.  doi: 10.2514/3.21030.  Google Scholar

[5]

J. H. Blakelock, Automatic control of aircraft and missiles,, John Wiley and Sons, (1991), 251.   Google Scholar

[6]

J. F. Bonnans and A. Hermant, Well-posedness of the shooting algorithm for state constrained optimal control problems with a single constraint and control,, SIAM Journal on Control and Optimization, 46 (2007), 1398.  doi: 10.1137/06065756X.  Google Scholar

[7]

B. Bonnard and M. Chyba, The Role of Singular Trajectories in Control Theory,, Springer Verlag, (2003).   Google Scholar

[8]

B. Bonnard, J. B. Caillau and E. Trélat, Geometric optimal control of elliptic Keplerian orbits,, Discrete and Continuous Dynamical Systems, 5 (2005), 929.  doi: 10.3934/dcdsb.2005.5.929.  Google Scholar

[9]

B. Bonnard, J. B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control,, ESAIM: Control, 13 (2007), 207.  doi: 10.1051/cocv:2007012.  Google Scholar

[10]

B. Bonnard, L. Faubourg and E. Trélat, Mécanique Céleste et Contrôle de Systèmes Spatiaux,, Mathématiques and Applications, (2006).  doi: 10.1007/3-540-37640-2.  Google Scholar

[11]

B. Bonnard and I. Kupka, Generic properties of singular trajectories,, Annales de l'Institut Henri Poincaré, 14 (1997), 167.  doi: 10.1016/S0294-1449(97)80143-6.  Google Scholar

[12]

L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations,, Applications of Mathematics 17, (1983).  doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[13]

Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems,, SIAM Journal on Control and Optimization, 47 (2008), 1078.  doi: 10.1137/060663003.  Google Scholar

[14]

L. E. Dubins, On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents,, American Journal of mathematics, 79 (1957), 497.  doi: 10.2307/2372560.  Google Scholar

[15]

A. Fleming and I. M. Ross, Optimal control of spinning axisymmetric spacecraft: A pseudospectral approach,, AIAA Guidance, (2008), 7164.  doi: 10.2514/6.2008-7164.  Google Scholar

[16]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Mathematical Programming Language,, Murray Hill, (1987).  doi: 10.1007/978-3-642-83724-1_12.  Google Scholar

[17]

A. T. Fuller, Relay control systems optimized for various performance criteria,, In Proceedings of the 1st World Congress IFAC, (1960), 510.   Google Scholar

[18]

A. T. Fuller, Study of an optimum non-linear control system,, International Journal of Electronics, 15 (1963), 63.  doi: 10.1080/00207216308937555.  Google Scholar

[19]

S. Gong, H. Baoyin and J. Li, Coupled attitude-orbit dynamics and control for displaced solar orbits,, Acta Astronautica, 65 (2009), 730.  doi: 10.1016/j.actaastro.2009.03.006.  Google Scholar

[20]

T. Haberkorn and E. Trélat, Convergence results for smooth regularizations of hybrid nonlinear optimal control problems,, SIAM Journal on Control and Optimization, 49 (2011), 1498.  doi: 10.1137/100809209.  Google Scholar

[21]

H. J. Kelley, R. E. Kopp, H. G. Moyer and H. Gardner, Singular extremals,, in Topics in Optimization (G. Leitmann, (1967), 63.   Google Scholar

[22]

D. Kim and J. D. Turner, Near-minimum-time control of asymmetric rigid spacecraft using two controls,, Automatica, 50 (2014), 2084.  doi: 10.1016/j.automatica.2014.05.038.  Google Scholar

[23]

A. J. Knutson and K. C. Howell, Coupled orbit and attitude dynamics for spacecraft comprised of multiple bodies in Earth-Moon Halo orbits,, In Proceedings of 63rd International Astronautical Congress, (2012), 5951.   Google Scholar

[24]

A. J. Krener, The high order maximal principle and its application to singular extremals,, SIAM Journal on Control and Optimization, 15 (1977), 256.  doi: 10.1137/0315019.  Google Scholar

[25]

I. A. K. Kupka, The ubiquity of Fuller's phenomenon,, Nonlinear controllability and optimal control, 133 (1990), 313.   Google Scholar

[26]

J. P. Laumond, Robot Motion Planning and Control,, Lecture Notes in Control and Information Sciences, (1998).  doi: 10.1007/BFb0036069.  Google Scholar

[27]

C. Marchal, Chattering arcs and chattering controls,, Journal of Optimization Theory and Applications, 11 (1973), 441.  doi: 10.1007/BF00935659.  Google Scholar

[28]

A. A. Markov, Some examples of the solution of a special kind of problem in greatest and least quantities,, (in Russian) Soobshch. Karkovsk. Mat. Obshch. 1, (1887), 250.   Google Scholar

[29]

J. P. Mcdanell and W. F. Powers, Necessary conditions joining optimal singular and nonsingular subarcs,, SIAM Journal on Control, 9 (1971), 161.  doi: 10.1137/0309014.  Google Scholar

[30]

T. G. McGee and J. K. Hedrick, Optimal path planning with a kinematic airplane model,, Journal of Guidance, 30 (2007), 629.  doi: 10.2514/1.25042.  Google Scholar

[31]

V. Y. Glizer, Optimal planar interception with fixed end conditions: Approximate solutions,, Journal of Optimization Theory and Applications, 93 (1997), 1.  doi: 10.1023/A:1022675631937.  Google Scholar

[32]

L. S. Pontryagin, Mathematical Theory of Optimal Processes,, CRC Press, (1987).   Google Scholar

[33]

R. Proulx and I. M. Ross, Time-optimal reorientation of asymmetric rigid bodies,, Advances in the Astronautical Sciences, 109 (2001), 1207.   Google Scholar

[34]

J. A. Reeds and L. A. Shepp, Optimal paths for a car that goes both forwards and backwards,, Pacific journal of mathematics, 145 (1990), 367.  doi: 10.2140/pjm.1990.145.367.  Google Scholar

[35]

H. Schättler and U. Ledzewicz, Synthesis of optimal controlled trajectories with chattering arcs,, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 161.   Google Scholar

[36]

H. Shen and Tsiotras, Time-optimal control of axisymmetric rigid spacecraft using two controls,, Journal of Guidance, 22 (1999), 682.  doi: 10.2514/2.4436.  Google Scholar

[37]

C. J. Silva and E. Trélat, Smooth regularization of bang-bang optimal control problems,, IEEE Trans. Automatic Control, 55 (2010), 2488.  doi: 10.1109/TAC.2010.2047742.  Google Scholar

[38]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis,, Translated from the German by R. Bartels, (1993).  doi: 10.1007/978-1-4757-2272-7.  Google Scholar

[39]

H. J. Sussmann and G. Tang, Shortest paths for the reeds-shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control,, Rutgers Center for Systems and Control Technical Report, 10 (1991), 1.   Google Scholar

[40]

H. J. Sussmann, The Markov-Dubins problem with angular acceleration control,, In Proccedings of the 36th IEEE Conference on Decision and Control, 3 (1997), 2639.  doi: 10.1109/CDC.1997.657778.  Google Scholar

[41]

L. Techy and C. A. Woolsey, Minimum-time path-planning for unmanned aerial vehicles in steady uniform winds,, Journal of Guidance, 32 (2009), 1736.  doi: 10.2514/1.44580.  Google Scholar

[42]

J. D. Thorne and C. D. Hall, Minimum-time continuous-thrust orbit transfers using the Kustaanheimo-Stiefel transformation,, Journal of Guidance, 20 (1997), 836.   Google Scholar

[43]

E. Trélat, Optimal control and applications to aerospace: Some results and challenges,, Journal of Optimization Theory and Applications, 154 (2012), 713.  doi: 10.1007/s10957-012-0050-5.  Google Scholar

[44]

E. Trélat, Contrôle Optimal: Théorie & Applications,, Vuibert, (2005).   Google Scholar

[45]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Mathematical Programming, 106 (2006), 25.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

[46]

P. K. C. Wang and F. Hadaegh, Coordination and Control of Multiple Microspacecraft Moving in Formation,, Journal of the Astronautical Sciences, 44 (1996), 315.   Google Scholar

[47]

W. M. Wonham, Note on a problem in optimal non-linear control,, Journal of Electronics and Control, 15 (1963), 59.  doi: 10.1080/00207216308937554.  Google Scholar

[48]

X. Yue, Y. Yang and Z. Geng, Indirect optimization for finite-thrust time-optimal orbital maneuver,, Journal of Guidance, 33 (2010), 628.  doi: 10.2514/1.44885.  Google Scholar

[49]

M. I. Zelikin and V. F. Borisov, Theory of Chattering Control, with Applications to Astronautics, Robotics, Economics and Engineering,, Systems & Control: Foundations & Applications, (1994).  doi: 10.1007/978-1-4612-2702-1.  Google Scholar

[50]

M. I. Zelikin and V. F. Borisov, Optimal chattering feedback control,, Journal of Mathematical Sciences, 114 (2003), 1227.  doi: 10.1023/A:1022082011808.  Google Scholar

[51]

J. Zhu, E. Trélat and M. Cerf, Minimum time control of the rocket attitude reorientation associated with orbit dynamics,, SIAM J. Control Optim., 54 (2016), 391.  doi: 10.1137/15M1028716.  Google Scholar

show all references

References:
[1]

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint,, Springer, (2004).  doi: 10.1007/978-3-662-06404-7.  Google Scholar

[2]

E. Bakolas and Tsiotras, Optimal synthesis of the Zermelo-Markov-Dubins problem in a constant drift field,, Journal of Optimization Theory and Applications, 156 (2013), 469.  doi: 10.1007/s10957-012-0128-0.  Google Scholar

[3]

J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming,, Second edition, (2010).  doi: 10.1137/1.9780898718577.  Google Scholar

[4]

K. D. Bilimoria and B. B. Wie, Time-optimal three-axis reorientation of rigid spacecraft,, Journal of Guidance, 16 (1993), 446.  doi: 10.2514/3.21030.  Google Scholar

[5]

J. H. Blakelock, Automatic control of aircraft and missiles,, John Wiley and Sons, (1991), 251.   Google Scholar

[6]

J. F. Bonnans and A. Hermant, Well-posedness of the shooting algorithm for state constrained optimal control problems with a single constraint and control,, SIAM Journal on Control and Optimization, 46 (2007), 1398.  doi: 10.1137/06065756X.  Google Scholar

[7]

B. Bonnard and M. Chyba, The Role of Singular Trajectories in Control Theory,, Springer Verlag, (2003).   Google Scholar

[8]

B. Bonnard, J. B. Caillau and E. Trélat, Geometric optimal control of elliptic Keplerian orbits,, Discrete and Continuous Dynamical Systems, 5 (2005), 929.  doi: 10.3934/dcdsb.2005.5.929.  Google Scholar

[9]

B. Bonnard, J. B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control,, ESAIM: Control, 13 (2007), 207.  doi: 10.1051/cocv:2007012.  Google Scholar

[10]

B. Bonnard, L. Faubourg and E. Trélat, Mécanique Céleste et Contrôle de Systèmes Spatiaux,, Mathématiques and Applications, (2006).  doi: 10.1007/3-540-37640-2.  Google Scholar

[11]

B. Bonnard and I. Kupka, Generic properties of singular trajectories,, Annales de l'Institut Henri Poincaré, 14 (1997), 167.  doi: 10.1016/S0294-1449(97)80143-6.  Google Scholar

[12]

L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations,, Applications of Mathematics 17, (1983).  doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[13]

Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems,, SIAM Journal on Control and Optimization, 47 (2008), 1078.  doi: 10.1137/060663003.  Google Scholar

[14]

L. E. Dubins, On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents,, American Journal of mathematics, 79 (1957), 497.  doi: 10.2307/2372560.  Google Scholar

[15]

A. Fleming and I. M. Ross, Optimal control of spinning axisymmetric spacecraft: A pseudospectral approach,, AIAA Guidance, (2008), 7164.  doi: 10.2514/6.2008-7164.  Google Scholar

[16]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Mathematical Programming Language,, Murray Hill, (1987).  doi: 10.1007/978-3-642-83724-1_12.  Google Scholar

[17]

A. T. Fuller, Relay control systems optimized for various performance criteria,, In Proceedings of the 1st World Congress IFAC, (1960), 510.   Google Scholar

[18]

A. T. Fuller, Study of an optimum non-linear control system,, International Journal of Electronics, 15 (1963), 63.  doi: 10.1080/00207216308937555.  Google Scholar

[19]

S. Gong, H. Baoyin and J. Li, Coupled attitude-orbit dynamics and control for displaced solar orbits,, Acta Astronautica, 65 (2009), 730.  doi: 10.1016/j.actaastro.2009.03.006.  Google Scholar

[20]

T. Haberkorn and E. Trélat, Convergence results for smooth regularizations of hybrid nonlinear optimal control problems,, SIAM Journal on Control and Optimization, 49 (2011), 1498.  doi: 10.1137/100809209.  Google Scholar

[21]

H. J. Kelley, R. E. Kopp, H. G. Moyer and H. Gardner, Singular extremals,, in Topics in Optimization (G. Leitmann, (1967), 63.   Google Scholar

[22]

D. Kim and J. D. Turner, Near-minimum-time control of asymmetric rigid spacecraft using two controls,, Automatica, 50 (2014), 2084.  doi: 10.1016/j.automatica.2014.05.038.  Google Scholar

[23]

A. J. Knutson and K. C. Howell, Coupled orbit and attitude dynamics for spacecraft comprised of multiple bodies in Earth-Moon Halo orbits,, In Proceedings of 63rd International Astronautical Congress, (2012), 5951.   Google Scholar

[24]

A. J. Krener, The high order maximal principle and its application to singular extremals,, SIAM Journal on Control and Optimization, 15 (1977), 256.  doi: 10.1137/0315019.  Google Scholar

[25]

I. A. K. Kupka, The ubiquity of Fuller's phenomenon,, Nonlinear controllability and optimal control, 133 (1990), 313.   Google Scholar

[26]

J. P. Laumond, Robot Motion Planning and Control,, Lecture Notes in Control and Information Sciences, (1998).  doi: 10.1007/BFb0036069.  Google Scholar

[27]

C. Marchal, Chattering arcs and chattering controls,, Journal of Optimization Theory and Applications, 11 (1973), 441.  doi: 10.1007/BF00935659.  Google Scholar

[28]

A. A. Markov, Some examples of the solution of a special kind of problem in greatest and least quantities,, (in Russian) Soobshch. Karkovsk. Mat. Obshch. 1, (1887), 250.   Google Scholar

[29]

J. P. Mcdanell and W. F. Powers, Necessary conditions joining optimal singular and nonsingular subarcs,, SIAM Journal on Control, 9 (1971), 161.  doi: 10.1137/0309014.  Google Scholar

[30]

T. G. McGee and J. K. Hedrick, Optimal path planning with a kinematic airplane model,, Journal of Guidance, 30 (2007), 629.  doi: 10.2514/1.25042.  Google Scholar

[31]

V. Y. Glizer, Optimal planar interception with fixed end conditions: Approximate solutions,, Journal of Optimization Theory and Applications, 93 (1997), 1.  doi: 10.1023/A:1022675631937.  Google Scholar

[32]

L. S. Pontryagin, Mathematical Theory of Optimal Processes,, CRC Press, (1987).   Google Scholar

[33]

R. Proulx and I. M. Ross, Time-optimal reorientation of asymmetric rigid bodies,, Advances in the Astronautical Sciences, 109 (2001), 1207.   Google Scholar

[34]

J. A. Reeds and L. A. Shepp, Optimal paths for a car that goes both forwards and backwards,, Pacific journal of mathematics, 145 (1990), 367.  doi: 10.2140/pjm.1990.145.367.  Google Scholar

[35]

H. Schättler and U. Ledzewicz, Synthesis of optimal controlled trajectories with chattering arcs,, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 161.   Google Scholar

[36]

H. Shen and Tsiotras, Time-optimal control of axisymmetric rigid spacecraft using two controls,, Journal of Guidance, 22 (1999), 682.  doi: 10.2514/2.4436.  Google Scholar

[37]

C. J. Silva and E. Trélat, Smooth regularization of bang-bang optimal control problems,, IEEE Trans. Automatic Control, 55 (2010), 2488.  doi: 10.1109/TAC.2010.2047742.  Google Scholar

[38]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis,, Translated from the German by R. Bartels, (1993).  doi: 10.1007/978-1-4757-2272-7.  Google Scholar

[39]

H. J. Sussmann and G. Tang, Shortest paths for the reeds-shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control,, Rutgers Center for Systems and Control Technical Report, 10 (1991), 1.   Google Scholar

[40]

H. J. Sussmann, The Markov-Dubins problem with angular acceleration control,, In Proccedings of the 36th IEEE Conference on Decision and Control, 3 (1997), 2639.  doi: 10.1109/CDC.1997.657778.  Google Scholar

[41]

L. Techy and C. A. Woolsey, Minimum-time path-planning for unmanned aerial vehicles in steady uniform winds,, Journal of Guidance, 32 (2009), 1736.  doi: 10.2514/1.44580.  Google Scholar

[42]

J. D. Thorne and C. D. Hall, Minimum-time continuous-thrust orbit transfers using the Kustaanheimo-Stiefel transformation,, Journal of Guidance, 20 (1997), 836.   Google Scholar

[43]

E. Trélat, Optimal control and applications to aerospace: Some results and challenges,, Journal of Optimization Theory and Applications, 154 (2012), 713.  doi: 10.1007/s10957-012-0050-5.  Google Scholar

[44]

E. Trélat, Contrôle Optimal: Théorie & Applications,, Vuibert, (2005).   Google Scholar

[45]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Mathematical Programming, 106 (2006), 25.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

[46]

P. K. C. Wang and F. Hadaegh, Coordination and Control of Multiple Microspacecraft Moving in Formation,, Journal of the Astronautical Sciences, 44 (1996), 315.   Google Scholar

[47]

W. M. Wonham, Note on a problem in optimal non-linear control,, Journal of Electronics and Control, 15 (1963), 59.  doi: 10.1080/00207216308937554.  Google Scholar

[48]

X. Yue, Y. Yang and Z. Geng, Indirect optimization for finite-thrust time-optimal orbital maneuver,, Journal of Guidance, 33 (2010), 628.  doi: 10.2514/1.44885.  Google Scholar

[49]

M. I. Zelikin and V. F. Borisov, Theory of Chattering Control, with Applications to Astronautics, Robotics, Economics and Engineering,, Systems & Control: Foundations & Applications, (1994).  doi: 10.1007/978-1-4612-2702-1.  Google Scholar

[50]

M. I. Zelikin and V. F. Borisov, Optimal chattering feedback control,, Journal of Mathematical Sciences, 114 (2003), 1227.  doi: 10.1023/A:1022082011808.  Google Scholar

[51]

J. Zhu, E. Trélat and M. Cerf, Minimum time control of the rocket attitude reorientation associated with orbit dynamics,, SIAM J. Control Optim., 54 (2016), 391.  doi: 10.1137/15M1028716.  Google Scholar

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