Abstract
Space maneuverings of the space vehicle require the capability of onboard trajectory planning. Convex programming based optimization strategy gets much attention in the design of trajectory planning methods with deterministic convergence properties. Due to the nonlinear dynamics, space-trajectory planning problems are always non-convex and difficult to be solved by the convex programming approach directly. This paper presents a Newton–Kantorovich/convex programming (N–K/CP) approach, based on the combination of the convex programming and the Newton–Kantorovich (N–K) method, to solve the nonlinear and non-convex space-trajectory planning problem. This trajectory planning problem is formulated as a nonlinear optimal control problem. By linearization and relaxation techniques, the nonlinear optimal control problem is convexified as a convex programming problem, which can be solved efficiently with convex programming solvers. For the linearized convex optimization problem, N–K method is introduced to design an iterative solving algorithm, the solution of which approximates the original trajectory planning problem with high accuracy. The convergence of the proposed N–K/CP approach is proved, and the effectiveness is demonstrated by numerical experiments and comparisons with other state-of-the-art methods.
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Zhao, J., Zhou, R.: Distributed time-constrained guidance using nonlinear model predictive control. Nonlinear Dyn. (2015). doi:10.1007/s11071-015-2578-z
Lu, P., Liu, X.: Autonomous trajectory planning for rendezvous and proximity operations by conic optimization. J. Guid. Control Dyn. 36(2), 375–389 (2013)
Zong, Q., Wang, F., Tian, B., Rui, R.: Robust adaptive dynamic surface control design for a flexible air-breathing hypersonic vehicle with input constraints and uncertainty. Nonlinear Dyn. 78(1), 289–315 (2014)
Anand, P., Rao, M.B.: Venkateswarlu Ch. Dynamic optimization of a copolymerization reactor using tabu search. ISA Trans. 55, 13–26 (2015)
Zhao, J., Zhou, R.: Pigeon-inspired optimization applied to constrained gliding trajectories. Nonlinear Dyn. 82(4), 1781–1795 (2015)
Smith, I. E.: General formulation of the iterative guidance mode, NASA TM X-53414 (1966)
Chandler, D.C., Smith, I.E.: Development of the iterative guidance mode with its application to various vehicles and missions. J. Space Rockets 4(7), 898–903 (1967)
McHenry, R.L., Brand, T.J., Long, A.D., Cockrell, B.F., Thibodeau III, J.R.: Space shuttle ascent guidance, navigation, and control. J. Astronaut. Sci. 27(1), 1–38 (1979)
Lawden, D.F.: Optimal trajectory for space Navigation. Butterworth, London (1963)
Gath, P.F., Calise, A.J.: Optimal of launch vehicle ascent trajectories with path constraints and coast arcs. J. Guid. Control Dyn. 24(2), 296–304 (2001)
Lu, P., Griffin, B., Dukeman, G. A., Chavez, F. R.: Rapid optimal multi-burn ascent planning and guidance.In: AIAA Guidance, Navigation, and Control Conference, Hilton Head, SC (2007)
Pan, B., Chen, Z., Lu, P., Gao, B.: Reduced transversality conditions in optimal space trajectories. J. Guid. Control Dyn. 36(5), 1289–1300 (2013)
Brusch, R.G.: Constrained impulsive trajectory optimization for orbit-to-orbit transfer. J. Guid. Control Dyn. 2(3), 204–212 (1979)
Bock, H. G., Plitt, K. J.: A multiple shooting algorithm for direct solution of optimal control problems. In: Proceedings of the 9th IFAC World Congress, Budapest, Hungary (1984)
Darby, C.L., Hager, W.W., Rao, A.V.: An hp-adaptive pseudospectral method for solving optimal control problems. Optim. Control Appl. Methods 32(4), 476–502 (2011)
Vlassenbroeck, J., Dooren, R.V.: A Chebyshev technique for solving nonlinear optimal control problems. IEEE Trans Autom. Control 33(4), 333–340 (1988)
Yu, W., Chen, W.: Guidance law with circular no-fly zone constraint. Nonlinear Dyn. 78, 1953–1971 (2014)
Betts, J.T.: Survey of numerical methods for trajectory optimization. J. Guid. Control Dyn. 21(2), 193–207 (1998)
Acikmese, B., Ploen, S.R.: Convex programming approach to powered descent guidance for Mars landing. J. Guid. Control Dyn. 30(5), 1353–1366 (2007)
Blackmore, L., Acikmese, B., Scharf, D.P.: Minimum-landing-error powered descent guidance for Mars landing using convex optimization. J. Guid. Control Dyn. 33(4), 1161–1171 (2010)
Acikmese, B., Blackmore, L.: Lossless convexification for a class of optimal problems with nonconvex control constraints. Automatica 47(2), 341–347 (2011)
Berkovitz, L.D.: Convexity and Optimization in \({\mathbb{R}}^{n}\). Wiley, New York (2002)
Nesterov, Y., Nemirovsky, A.: Interior-Point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics, Philadelphia (1994)
Wang, Y., Boyd, S.: Fast model predictive control using online optimization. IEEE Trans. Control Syst. Technol. 18(2), 267–278 (2010)
Zhang, S., Acikmese, B., Swei, S., Prabhu, D.: Convex programming approach to real-time trajectory optimization for Mars aerocapture. in: IEEE Aerospace Conference (2015)
Vaddi, S.: Convex optimization formulation of robust nonlinear model-predictive-controller design. In: AIAA Guidance, Navigation, and Control Conference (2010)
Liu, X., Lu, P.: Solving nonconvex optimal control problems by convex optimization. J. Guid. Control Dyn. 37(3), 750–765 (2014)
Boyd, J.P.: An analytical and numerical study of the two-dimensional Bratu Equation. J. Sci. Comput. 1(2), 183–206 (1986)
Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover Inc., New York (2000)
Chen, Q., Zhang, Y., Liao, S., Wan, F.: Newton–Kantorovich/pseudospectral solution to perturbed astrodynamic two-point boundary-value problems. J. Guid. Control Dyn. 36(2), 485–498 (2013)
Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math Prog. 95(1), 3–51 (2003)
Acikmese, B., Carson, J.M., Blackmore, L.: Lossless convexification of non-convex control bound and pointing constraints of the soft landing optimal control problem. IEEE Trans. Control Syst. Technol. 21(6), 2104–2113 (2013)
Ye, Y.: Interior Point Algorithms. Wiley, New York (1997)
Grant, M., Boyd, S., Ye, Y.: CVX: matlab software for disciplined convex programming. [online] http://www.stanford.edu/~boyd/cvx/. (2009)
Acknowledgements
This research has been funded in part by the National Natural Science Foundation of China under Grant 61174221/11272062 and the National Key R&D Program of China under grant No. 2016YFB 1200100. The authors would like to thank all the reviewers for their valuable comments and helpful suggestions to improve the quality of the paper.
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Cheng, X., Li, H. & Zhang, R. Autonomous trajectory planning for space vehicles with a Newton–Kantorovich/convex programming approach. Nonlinear Dyn 89, 2795–2814 (2017). https://doi.org/10.1007/s11071-017-3626-7
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DOI: https://doi.org/10.1007/s11071-017-3626-7