Abstract
Convex optimization is a class of mathematical programming problems with polynomial complexity for which state-of-the-art, highly efficient numerical algorithms with predeterminable computational bounds exist. Computational efficiency and tractability in aerospace engineering, especially in guidance, navigation, and control (GN&C), are of paramount importance. With theoretical guarantees on solutions and computational efficiency, convex optimization lends itself as a very appealing tool. Coinciding the strong drive toward autonomous operations of aerospace vehicles, convex optimization has seen rapidly increasing utility in solving aerospace GN&C problems with the potential for onboard real-time applications. This paper attempts to provide an overview on the problems to date in aerospace guidance, path planning, and control where convex optimization has been applied. Various convexification techniques are reviewed that have been used to convexify the originally nonconvex aerospace problems. Discussions on how to ensure the validity of the convexification process are provided. Some related implementation issues will be introduced as well.
Similar content being viewed by others
Change history
11 February 2022
A Correction to this paper has been published: https://doi.org/10.1007/s42064-022-0136-2
References
Betts, J. T. Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2nd end. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics, 2010: 247–255.
Nesterov, Y., Nemirovski, A. Interior-Point Polynomial Algorithms in Convex Programming. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics, 1994.
Boyd, S., Vandenberghe, L. Convex Optimization. New York: Cambridge University Press, 2004.
Mueller, M. W., D’Andrea, R. A model predictive controller for quadrocopter state interception. In: Proceedings of the 2013 European Control Conference, 2013: 1383–1389.
Augugliaro, F., Schoellig, A. P., D’Andrea, R. Generation of collision-free trajectories for a quadrocopter fleet: A sequential convex programming approach. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, 2012: 1917–1922.
Chen, Y., Cutler, M., How, J. P. Decoupled multiagent path planning via incremental sequential convex programming. In: Proceedings of the IEEE International Conference on Robotics and Automation, 2015: 5954–5961.
Alonso-Mora, J., Baker, S., Rus, D. Multi-robot navigation information via sequential convex programming. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, 2015: 4634–4641.
Alonso-Mora, J., Montijano, E., Schwager, M., Rus, D. Distributed multi-robot formation control among obstacles: A geometric and optimization approach with consensus. In: Proceedings of the IEEE International Conference on Robotics and Automation, 2016.
Alonso-Mora, J., Naegeli, T., Siegwart, R., Beardsley, P. Collision avoidance for aerial vehicles in multi-agent scenarios. Autonomous Robots, 2015, 39(1): 101–121.
Acikmese, B., Ploen, S. R. Convex programming approach to powered descent guidance for mars landing. Journal of Guidance, Control, and Dynamics, 2007, 30(5): 1353–1366.
Acikmese, B., Scharf, D., Blackmore, L., Wolf, A. Enhancements on the convex programming based powered descent guidance algorithm for mars landing. In: Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Guidance, Navigation, and Control and Co-located Conferences, 2008: AIAA 2008–6426.
Blackmore, L., Acikmese, B., Scharf, D. P. Minimum landing error powered descent guidance for mars landing using convex optimization. Journal of Guidance, Control, and Dynamics, 2010, 33(4): 1161–1171.
Acikmese, B., Carson, J., Blackmore, L. Lossless convexification of nonconvex control bound and pointing constraints of the soft landing optimal control problem. IEEE Transactions on Control Systems Technology, 2013, 21(6): 2104–2113.
Harris, M. W., Acikmese, B. Maximum divert for planetary landing using convex optimization. Journal of Optimization Theory and Applications, 2013, 162(3): 975–995.
Scharf, D. P., Ploen, S. R., Acikmese, B. A. Interpolation-enhanced powered descent guidance for onboard nominal, off-nominal, and multi-x scenarios. In: Proceedings of the AIAA Guidance, Navigation, and Control Conference, AIAA SciTech Forum, 2015: AIAA 2015–0850.
Pinson, R., Lu, P. Rapid generation of optimal asteroid powered descent trajectories. In: Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, 2015: AAS 15–616.
Pinson, R., Lu, P. Trajectory design employing convex optimization for landing on irregularly shaped asteroids. In: Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, 2016: AIAA 2016–5378.
Acikmese, B., Blackmore, L. Lossless convexification for a class of optimal problems with nonconvex control constraints. Automatica, 2011, 47(2): 341–347.
Harris, M. W., Acikmese, B. Lossless convexification of non-convex optimal control problems for state constrained linear systems. Automatica, 2014, 50(9): 2304–2311.
Lu, P., Liu, X. Autonomous trajectory planning for rendezvous and proximity operations by conic optimization. Journal of Guidance, Control, and Dynamics, 2013, 36(2): 375–389.
Liu, X., Lu, P. Robust trajectory optimization for highly constrained rendezvous and proximity operations. In: Proceedings of the AIAA Guidance, Navigation, and Control (GNC) Conference, Guidance, Navigation, and Control and Co-located Conferences, 2013: AIAA 2013–4720.
Liu, X., Lu, P. Solving nonconvex optimal control problems by convex optimization. Journal of Guidance, Control, and Dynamics, 2014, 37(3): 750–765.
Liu, X. Autonomous trajectory planning by convex optimization. Ph.D. Thesis. Iowa State University, 2013.
Grzymisch, J., Fichter, W. Optimal rendezvous guidance with enhanced bearings-only observability. Journal of Guidance, Control, and Dynamics, 2015, 38(6): 1131–1140.
Louembet, C., Arzelier, D., Deaconu, G. Robust rendezvous planning under maneuver execution errors. Journal of Guidance, Control, and Dynamics, 2015, 38(1): 76–93.
Mueller, J. B. Onboard planning of collision avoidance maneuvers using robust optimization. In: Proceedings of the AIAA Infotech@Aerospace Conference, 2009: AIAA 2009–2051.
Mueller, J. B., Griesemer, P. R., Thomas, S. Avoidance maneuver planning incorporating station-keeping constraints and automatic relaxation. Journal of Aerospace Information Systems, 2013, 10(6): 306–322.
Carson, J. M., Acikmese, B. A model-predictive control technique with guaranteed resolvability and required thruster. In: Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, Guidance, Navigation, and Control and Co-located Conferences, 2006: AIAA 2006–6780.
Tillerson, M., Inalhan, G., How, J. P. Co-ordination and control of distributed spacecraft systems using convex optimization techniques. International Journal of Robust and Nonlinear Control, 2002, 12(2-3): 207–242.
Deaconu, G., Louembet, C., Theron, A. Designing continuously constrained spacecraft relative trajectories for proximity operations. Journal of Guidance, Control, and Dynamics, 2015, 38(7): 1208–1217.
Dai, R., Sun, C. Path planning of spatial rigid motion with constrained attitude. Journal of Guidance, Control, and Dynamics, 2015, 38(8): 1356–1365.
Wu, Y., Cao, X., Xing, Y., Zheng, P., Zhang, S. Relative motion coupled control for formation flying spacecraft via convex optimization. Aerospace Science and Technology, 2010, 14(6): 415–428.
De Bruijn, F., Theil, S., Choukroun, D., Gill, E. Geostationary satellite station-keeping using convex optimization. Journal of Guidance, Control, and Dynamics, 2016, 39(3): 605–616.
De Bruijn, F., Theil, S., Choukroun, D., Gill, E. Collocation of geostationary satellites using convex optimization. Journal of Guidance, Control, and Dynamics, 2016, 39(6): 1303–1313.
Morgan, D., Chung, S. J., Hadaegh, F. Y. Model predictive control of swarms of spacecraft using sequential convex optimization. Journal of Guidance, Control, and Dynamics, 2014, 37(6): 1725–1740.
Liu, X., Shen, Z., Lu, P. Entry trajectory optimization by second-order cone programming. Journal of Guidance, Control, and Dynamics, 2016, 39(2): 227–241.
Liu, X., Shen, Z., Lu, P. Solving the maximum-crossrange problem via successive second-order cone programming with a line search. Aerospace Science and Technology, 2015, 47: 10–20.
Liu, X., Shen, Z. Rapid smooth entry trajectory planning for high lift/drag hypersonic glide vehicles. Journal of Optimization Theory and Applications, 2016, 168(3): 917–943.
Liu, X., Shen, Z., Lu, P. Exact convex relaxation for optimal flight of aerodynamically controlled missiles. IEEE Transactions on Aerospace and Electronic Systems, 2016, 52(4): 1881–1892.
Liu, X. Fuel-optimal rocket landing with aerodynamic controls. In: Proceedings of the AIAA Guidance, Navigation, and Control Conference, AIAA SciTech Forum, 2017: AIAA 2017–1732.
Szmuk, M., Acikmese, B., Berning, A. W. Successive convexification for fuel-optimal powered landing with aerodynamic drag and non-convex constraints. In: Proceedings of the AIAA Guidance, Navigation, and Control Conference, AIAA SciTech Forum, 2016: AIAA 2016–0378.
Tam, M., Lightsey, E. G. Constrained spacecraft reorientation using mixed integer convex programming. Acta Astronautica, 2016, 127: 31–40.
Richards, A., Schouwenaars, T., How, J. P., Feron, E. Spacecraft trajectory planning with avoidance constraints using mixed-integer linear programming. Journal of Guidance, Control, and Dynamics, 2002, 25(4): 755–764.
Tillerson, M. J. Coordination and control of multiple spacecraft using convex optimization techniques. Master Thesis. Massachusetts Institute of Technology, 2002.
Culligan, K., Valenti, M., Kuwata, Y., How, J. P. Three-dimensional flight experiments using on-line mixed-integer linear programming trajectory optimization. In: Proceedings of the 2007 American Control Conference, 2007: 5322–5327.
Kamal, W. A., Gu, D.-W., Postlethwaite, I. Real time trajectory planning for UAVs Using MILP. In: Proceedings of the 44th IEEE Conference on Decision and Control and the European Control Conference, 2005: 3381–3386.
Forsmo, E. J. Optimal path planning for unmanned aerial systems. Master Thesis. Norwegian University of Science and Technology, 2012.
Culligan, K. F. Online trajectory planning for UAVs using mixed integer linear programming. Master Thesis. Massachusetts Institute of Technology, 2006.
Mao, Y., Szmuk, M., Ackimese, B. Successive convexification of non-convex optimal control problems and its convergence properties. In: Proceedings of the 2016 IEEE 55th Control and Decision Conference, 2016: 3636–3641.
Wang, Z., Grant, M. J. Constrained trajectory optimization for planetary entry via sequential convex programming. In: Proceedings of the AIAA Atmospheric Flight Mechanics Conference, AIAA AVIATION Forum, 2016: AIAA 2016–3241.
Banks, S., Dinesh, K. Approximate optimal control and stability of nonlinear finite-and infinite-dimensional systems. Annals of Operations Research, 2000, 98(1): 19–44.
Mueller, J. B., Larsson, R. Collision avoidance maneuver planning with robust optimization. In: Proceedings of the 7th International ESA Conference on Guidance, Navigation and Control Systems, 2008.
Harris, M. W., Acikmese, B. Minimum time rendezvous of multiple spacecraft using differential drag. Journal of Guidance, Control, and Dynamics, 2014, 37(2): 365–373.
Sun, C., Dai, R., Lu, P. Solving polynomial optimal control problems via iterative convex optimization. In: Proceedings of the AIAA Guidance, Navigation, and Control Conference, AIAA SciTech Forum, 2016: AIAA 2016–0371.
Nocedal, J., Wright, S. J. Numerical Optimization, 2nd end. New York: Springer-Verlag, 2006.
Andersen, E. D., Roos, C., Terlaky, T. On implementing a primal-dual interior-point method for conic quadratic optimization. Mathematical Programming, 2003, 95(2): 249–277.
Toh, K. C., Todd, M. J., Tutuncu, R. H. SDPT3-A Matlab software package for semidefinite programming, Version 1.3. Optimization Methods and Software, 1999, 11(1-4): 545–581.
Sturm, J. F. Using SeDuMi 1.02: A Matlab toolbox for optimization over symmetric cones. Optimization Methods and Software, 1999, 11(1-4): 625–653.
Domahidi, A., Chu, E., Boyd, S. ECOS: An SOCP solver for embedded system. In: Proceedings of the European Control Conference, 2013: 3071–3076.
Lofberg, J. YALMIP: A toolbox for modeling and optimization in MATLAB, In: Proceedings of the 2004 IEEE International Conference on Robotics and Automation, 2004: 284–289.
Grant, M., Boyd, S. CVX: Matlab software for disciplined convex programming, version 2.1. 2016. Available at http://cvxr.com/cvx.
Mattingley, J., Boyd, S. CVXGEN: A code generator for embedded convex optimization. Optimization and Engineering, 2010, 13(1): 1–27.
Chu, E., Parikh, N., Domahidi, A., Boyd, S. Code generation for embedded second-order cone programming. In: Proceedings of the European Control Conference, 2013.
Dueri, D., Acikmese, B., Scharf, D., Harris, M. W. Customized real-time interior-point methods for onboard powered descent guidance. Journal of Guidance, Control, and Dynamics, 2017, 40(2): 197–212.
Scharf, D. P., Regehr, M. W., Dueri, D., Acikmese, B., Vaughan, G. M., Benito, J. ADAPT: Demonstrations of onboard large-divert guidance with a reusable launch vehicle. In: Proceedings of the 2014 IEEE Aerospace Conference, 2014: 1–18.
Scharf, D., Acikmese, B., Dueri, D., Benito, J., Casoliva, J. Implementation and experimental demonstration of onboard powered-descent guidance. Journal of Guidance, Control, and Dynamics, 2017, 40(2): 213–229.
Liu, X., Shen, Z., Lu, P. Closed-loop optimization of guidance gain for constrained impact. Journal of Guidance, Control, and Dynamics, 2017, 40(2): 453–460.
Lu, P. Introducing computational guidance and control. Journal of Guidance, Control, and Dynamics, 2017, 40(2): 193.
Acknowledgements
The author at Beijing Institute of Technology gratefully acknowledges the support to this work by the National Natural Science Foundation of China (Grant No. 61603017).
Author information
Authors and Affiliations
Corresponding author
Additional information
Xinfu Liu received his B.E. degree in the Department of Automation, Central South University, Changsha, China, in 2008, and Ph.D. degree in aerospace engineering from Iowa State University, Ames, USA, in 2013. From 2013 to 2016 he was a postdoctoral research associate at Beihang University, Beijing, China. Since 2016, he has been an associate professor in the School of Aerospace Engineering, Beijing Institute of Technology, Beijing, China. His research interests are in optimal control, autonomous trajectory planning of flight vehicles, and convex optimization and its applications.
Binfeng Pan is an associate professor in the School of Astronautics, Northwestern Polytechnical University (NPU). He received his M.S. and Ph.D. degrees in aerospace engineering from NPU in 2007 and 2010 respectively. His research interests are in the trajectory optimization, onboard guidance and control algorithms design.
Ping Lu received his baccalaureate degree from the Beijing Institute of Aeronautics, and Ph.D. degree in aerospace engineering from the University of Michigan. He was on the faculty of aerospace engineering at Iowa State University from 1990 to 2016 where his last position was professor. He joined the San Diego State University in 2016 to be a professor and the Chair of the Aerospace Engineering Department. His research interests and expertise are in aerospace guidance, flight control, and autonomous trajectory planning and optimization. Prof. Lu was the recipient of the prestigious American Institute of Aeronautics and Astronautics (AIAA) Mechanics and Control of Flight Award in 2008, “for contributions in advanced guidance algorithms for entry and ascent fligh”. He is an AIAA Fellow, and the Editor-in-Chief of Journal of Guidance, Control, and Dynamics.
Rights and permissions
About this article
Cite this article
Liu, X., Lu, P. & Pan, B. Survey of convex optimization for aerospace applications. Astrodyn 1, 23–40 (2017). https://doi.org/10.1007/s42064-017-0003-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42064-017-0003-8