Convex Optimization for Rendezvous and Proximity Operation via Birkhoff Pseudospectral Method
Abstract
:1. Introduction
2. Problem Formulation
2.1. Relative Equations of Motion
 the Target is in a circular orbit near the Earth, and $\dot{\mathit{\omega}}=0$;
 the orbit radius of the Target ${r}_{T}$. is much larger than the distance between the two spacecraft $r$;
 both spacecraft are regarded as mass points, and all the orbit perturbations are ignored.
2.2. Rendezvous Trajectory Optimization
2.3. Convexification Techniques Applied to RTO
2.3.1. Equivalent Transformation of the Objective Function
2.3.2. Change of Variables for Dynamic Constraints
2.3.3. Linearization of the NoFly Zone
3. Discretization Based on Classical PS Method
3.1. TimeDomain Transformation
3.2. Classical PS Method for RTO
4. WellConditioned Birkhoff PS Methods for Multiple Dynamic Systems
4.1. Basic Properties of Birkhoff Interpolation
4.2. The FirstOrder Birkhoff PS Method for RTO
4.3. The SecondOrder Birkhoff PS Method for RTO
4.4. Sequential Convex Algorithm for RTO
Algorithm 1 The iteration process of the SC algorithm 
SC Algorithm for RTO 
Input: The initial state x_{0} = [r_{0},v_{0}] and the terminal state x_{f} = [r_{f},v_{f}]; the initial mass of the Chaser m_{0}; the initial and terminal time [t_{0},t_{f}] .

Step 1: Calculate the initial trajectory r^{0} without the nofly constraint and set it as the reference trajectory r^{ref} of the first iteration. Go to step 2. 
Step 2: Start the loop until meeting the trustregion constraint. 
1. For k = 1:Maxiter; 
2. Calculate the current trajectory {r^{k},u^{k}}; 
3. Calculate the maximum distance Δd_{max} between r^{k} and r^{ref}; 
4. Check the trustregion constraint: 
$$\Delta {d}_{\mathrm{max}}=\Vert {\mathit{r}}^{k}{\mathit{r}}^{ref}\Vert \le \epsilon .$$

If Δd_{max} ≤ ε established, exit the loop and go to step 3; otherwise, set r^{ref} = r^{k} and continue.

5. End. 
Step 3: Evaluate the accuracy of the solution e. Go to step 4. 
Step 4: Output the result. The optimal trajectory and optimal control are {r*,u*} ; the optimal objective value is j*. 
5. Numerical Simulation
5.1. Case 1: Excellent Computational Efficiency and Solution Accuracy under Different Grid Points
5.2. Case 2: Sensitivity Analysis under Different Conditions
6. Conclusions
Author Contributions
Funding
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Dimensionless Unit  Value 

Length ${L}_{u}$  ${R}_{e}$ 
Speed ${V}_{u}$  $\sqrt{{G}_{M}/{R}_{e}}$ 
Time ${T}_{u}$  $\sqrt{{({R}_{e})}^{3}/{G}_{M}}$ 
Acceleration ${a}_{u}$  ${{G}_{M}/({R}_{e})}^{2}$ 
Mass ${M}_{u}$  ${M}_{0}$ 
Force ${F}_{u}$  ${{M}_{0}{G}_{M}/({R}_{e})}^{2}$ 
Initialization Parameters  Value 

Initial Mass  ${m}_{0}=1000/{M}_{u}$ 
Initial and Terminal Time  ${t}_{0}=0,{t}_{f}=500/{T}_{u}$ 
Initial and Terminal Position  ${\mathit{r}}_{0}=[0,100,0]/{L}_{s};{\mathit{r}}_{f}=[0,20,0]/{L}_{u}$ 
Initial and Terminal Speed  ${\mathit{v}}_{0}=[0,0,0]/{V}_{u};{\mathit{v}}_{f}=[0,0,0]/{V}_{u}$ 
Nofly Zone Radius  $\begin{array}{c}{R}_{s}=10/{L}_{u}(Sphe\mathit{r}ical);\\ {R}_{x}=10/{L}_{u};{R}_{y}=15/{L}_{u};{R}_{z}=10/{L}_{u}(Ellipsoid)\end{array}$ 
Maximum Thrust Limit  $\left[\begin{array}{cc}{T}_{\mathrm{min}}& {T}_{\mathrm{max}}\end{array}\right]=\left[\begin{array}{cc}0& 10/{F}_{u}\end{array}\right]$ 
Reciprocal of effective exhaust velocity  $\mu ={V}_{u}/2000$ 
Appendix B
References
 Fehse, W. Automated Rendezvous and Docking of Spacecraft; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar] [CrossRef]
 Chamberlin, J.A.; Rose, J.T. Gemini rendezvous program. J. Spacecr. Rocket. 1964, 1, 13–18. [Google Scholar] [CrossRef]
 Young, K.A.; Alexander, J.D. Apollo lunar rendezvous. J. Spacecr. Rocket. 1970, 7, 1083–1086. [Google Scholar] [CrossRef]
 Murtazin, R.F.; Budylov, S.G. Short rendezvous missions for advanced Russian human spacecraft. Acta Astronaut. 2010, 67, 900–909. [Google Scholar] [CrossRef]
 Goodman, J.L. History of space shuttle rendezvous and proximity operations. J. Spacecr. Rocket. 2006, 43, 944–959. [Google Scholar] [CrossRef]
 Woffinden, D.C.; Geller, D.K. Navigating the road to autonomous orbital rendezvous. J. Spacecr. Rocket. 2007, 44, 898–909. [Google Scholar] [CrossRef]
 Luo, Y.; Zhang, J.; Tang, G. Survey of orbital dynamics and control of space rendezvous. Chin. J. Aeronaut. 2014, 27, 1–11. [Google Scholar] [CrossRef]
 Morante, D.; Sanjurjo Rivo, M.; Soler, M. A Survey on LowThrust Trajectory Optimization Approaches. Aerospace 2021, 8, 88. [Google Scholar] [CrossRef]
 Bucchioni, G.; Innocenti, M. Rendezvous in CisLunar Space near Rectilinear Halo Orbit: Dynamics and Control Issues. Aerospace 2021, 8, 68. [Google Scholar] [CrossRef]
 Kawano, I.; Mokuno, M.; Kasai, T.; Suzuki, T. Result and evaluation of autonomous rendezvous docking experiment of ETSVII. Guid. Navig. Control Conf. Exhib. 1999, 38, 105–111. [Google Scholar] [CrossRef]
 Thomas, M.D.; David, M. XSS10 microsatellite flight demonstration program results. In Spacecraft Platforms and Infrastructure; SPIE: Orlando, FL, USA, 2004; Volume 5419, pp. 16–25. [Google Scholar]
 Timothy, E.R. Demonstration of autonomous rendezvous technology (DART) project summary. In Space Systems Technology and Operations; SPIE: Orlando, FL, USA, 2003; Volume 5088, pp. 10–19. [Google Scholar]
 Manny, R.L.; ChihTsai, C.; Michael, W.B.; Thomas, P.W.; David, L.C.; William, B.G.; Peter, W.S.; Peter, A.S.; Mark, A.L. Orbital express autonomous rendezvous and capture sensor system (ARCSS) flight test results. In Sensors and Systems for Space Applications II; SPIE: Orlando, FL, USA, 2008. [Google Scholar]
 Zhou, J. Tiangong1/Shenzhou8 rendezvous and docking process. In Space Rendezvous and Docking Technology; National Defense Industry Press: Beijing, China, 2013; pp. 56–62. [Google Scholar]
 Yang Cheng, Z.X. Tianzhou3 Completes Autonomous Rapid Rendezvous and Docking. Available online: https://www.chinadefenseobservation.com/?p=8682 (accessed on 20 September 2021).
 Trent, J.; Perrotto, J.B. Spacex Dragon Attached to Space Station in Spaceflight First. Available online: https://www.nasa.gov/home/hqnews/2012/may/HQ_12172_SpaceX_Dragon_Berth.html (accessed on 25 May 2012).
 Betts, J.T. Survey of numerical methods for trajectory optimization. J. Guid. Control Dyn. 1998, 21, 193–207. [Google Scholar] [CrossRef]
 Rao, A.V. A survey of numerical methods for optimal control. Adv. Astronaut. Sci. 2010, 135, 1–32. [Google Scholar]
 Trélat, E. Optimal control and applications to aerospace: Some results and challenges. J. Optim. Theory Appl. 2012, 154, 713–758. [Google Scholar] [CrossRef]
 Tang, G.; Jiang, F.; Li, J. FuelOptimal LowThrust Trajectory Optimization Using Indirect Method and Successive Convex Programming. IEEE Trans. Aerosp. Electron. Syst. 2018, 54, 2053–2066. [Google Scholar] [CrossRef]
 Andersen, E.D.; Roos, C.; Terlaky, T. On implementing a primaldual interiorpoint method for conic quadratic optimization. Math. Program. 2003, 95, 249–277. [Google Scholar] [CrossRef]
 Guo, X.; Zhu, M. Direct trajectory optimization based on a mapped Chebyshev pseudospectral method. Chin. J. Aeronaut 2013, 26, 401–412. [Google Scholar] [CrossRef]
 Benson, D.A.; Huntington, G.T.; Thorvaldsen, T.P.; Rao, A.V. Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method. J. Guid. Control Dyn. 2006, 29, 1435–1440. [Google Scholar] [CrossRef]
 Boyd, S.; Boyd, S.P.; Vandenberghe, L. Convex Optimization; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
 Oumer, A.M.; Kim, D.K. RealTime Fuel Optimization and Guidance for Spacecraft Rendezvous and Docking. Aerospace 2022, 9, 276. [Google Scholar] [CrossRef]
 Malyuta, D.; Reynolds, T.; Szmuk, M.; Mesbahi, M.; Acikmese, B.; Carson, J.M. Discretization Performance and Accuracy Analysis for the Rocket Powered Descent Guidance Problem. In Proceedings of the AIAA Scitech 2019 Forum, San Diego, CA, USA, 7–11 January 2019. [Google Scholar] [CrossRef]
 Wächter, A.; Biegler, L.T. On the implementation of an interiorpoint filter linesearch algorithm for largescale nonlinear programming. Math. Program. 2006, 106, 25–57. [Google Scholar] [CrossRef]
 Gill, P.E.; Murray, W.; Saunders, M.A. SNOPT: An SQP Algorithm for LargeScale Constrained Optimization. SIAM Rev. 2005, 47, 99–131. [Google Scholar] [CrossRef]
 Betts, J.T. Practical Methods for Optimal Control and Estimation Using Nonlinear Programming; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
 Betts, J.T.; Huffman, W.P. Mesh refinement in direct transcription methods for optimal control. Optim. Control Appl. Methods 1998, 19, 1–21. [Google Scholar] [CrossRef]
 Elnagar, G.; Kazemi, M.A.; Razzaghi, M. The pseudospectral Legendre method for discretizing optimal control problems. IEEE Trans. Autom. Control 1995, 40, 1793–1796. [Google Scholar] [CrossRef]
 Elnagar, G.; Kazemi, M.A. Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems. Comput. Optim. Appl. 1998, 11, 195–217. [Google Scholar] [CrossRef]
 Fahroo, F.; Ross, I.M. Costate Estimation by a Legendre Pseudospectral Method. J. Guid. Control Dyn. 2001, 24, 270–277. [Google Scholar] [CrossRef]
 Benson, D. A Gauss Pseudospectral Transcription for Optimal Control. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 2005. [Google Scholar]
 Fahroo, F.; Ross, I.M. Advances in pseudospectral methods for optimal control. In Proceedings of the AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, HI, USA, 18 August–21 August 2008. [Google Scholar] [CrossRef]
 Rea, J. Launch Vehicle Trajectory Optimization Using a Legendre Pseudospectral Method. In Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, Austin, TX, USA, 11–14 August 2003. [Google Scholar] [CrossRef]
 Bedrossian, N.; Kang, W. Pseudospectral Optimal Control Theory Makes Debut Flight, Saves NASA $1M in Under Three Hours. SIAM News 2007, 40, 1–3. [Google Scholar]
 Sagliano, M. Generalized hp Pseudospectral Convex Programming for Powered Descent and Landing. J. Guid. Control. Dyn. 2018, 42, 1562–1570. [Google Scholar] [CrossRef]
 Koeppen, N.; Ross, I.M.; Wilcox, L.C.; Proulx, R.J. Fast Mesh Refinement in Pseudospectral Optimal Control. J. Guid. Control Dyn. 2019, 42, 711–722. [Google Scholar] [CrossRef]
 Gong, Q.; Fahroo, F.; Ross, I.M. Spectral Algorithm for Pseudospectral Methods in Optimal Control. J. Guid. Control Dyn. 2008, 31, 460–471. [Google Scholar] [CrossRef]
 Hesthaven, J.S. Integration Preconditioning of Pseudospectral Operators. I. Basic Linear Operators. SIAM J. Numer. Anal. 1998, 35, 1571–1593. [Google Scholar] [CrossRef]
 Elbarbary, E.M.E. Integration Preconditioning Matrix for Ultraspherical Pseudospectral Operators. SIAM J. Sci. Comput. 2006, 28, 1186–1201. [Google Scholar] [CrossRef]
 Ross, I.M.; Proulx, R.J. Further Results on Fast Birkhoff Pseudospectral Optimal Control Programming. J. Guid. Control Dyn. 2019, 42, 2086–2092. [Google Scholar] [CrossRef]
 Wang, L.L.; Samson, M.D.; Zhao, X. A wellconditioned collocation method using a pseudospectral integration matrix. SIAM J. Sci. Comput. 2014, 36, A907–A929. [Google Scholar] [CrossRef]
 Du, K. On WellConditioned Spectral Collocation and Spectral Methods by the Integral Reformulation. SIAM J. Sci. Comput. 2016, 38, A3247–A3263. [Google Scholar] [CrossRef]
 McCoid, C.; Trummer, M.R. Preconditioning of spectral methods via Birkhoff interpolation. Numer. Algorithm 2017, 79, 555–573. [Google Scholar] [CrossRef]
 Clohessy, W.H.; Wiltshire, R.S. Terminal guidance system for satellite rendezvous. J. Aerosp. Sci. 1960, 27, 653–658. [Google Scholar] [CrossRef]
 Chobotov, V.A. (Ed.) Orbital Mechanics; AIAA, Inc.: Reston, WV, USA, 2002. [Google Scholar]
 Luo, Y.Z.; Tang, G.J.; Lei, Y.J. Optimal multiobjective linearized impulsive rendezvous. J. Guid. Control Dyn. 2007, 30, 383–389. [Google Scholar] [CrossRef]
 Acikmese, B.; Ploen, S.R. Convex programming approach to powered descent guidance for Mars landing. J. Guid Control Dyn. 2007, 30, 1353–1366. [Google Scholar] [CrossRef]
 Açıkmeşe, B.; Carson, J.M.; Blackmore, L. Lossless convexification of nonconvex control bound and pointing constraints of the soft landing optimal control problem. IEEE Trans. Control Syst. Technol. 2013, 21, 2104–2113. [Google Scholar] [CrossRef]
 Liu, X.; Lu, P.; Pan, B. Survey of convex optimization for aerospace applications. Astrodynamics 2017, 1, 23–40. [Google Scholar] [CrossRef]
 Lu, P.; Liu, X. Autonomous Trajectory Planning for Rendezvous and Proximity Operations by Conic Optimization. J. Guid. Control Dyn. 2013, 36, 375–390. [Google Scholar] [CrossRef]
 Gong, Q.; Ross, I.M.; Fahroo, F. Spectral and Pseudospectral Optimal Control Over Arbitrary Grids. J. Optim. Theory Appl. 2016, 169, 759–783. [Google Scholar] [CrossRef]
 Fornberg, B. A Practical Guide to Pseudospectral Methods; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar] [CrossRef]
 Fahroo, F.; Ross, I.M. Pseudospectral methods for infinitehorizon optimal control problems. J. Guid Control Dyn. 2008, 31, 927–936. [Google Scholar] [CrossRef]
 Garg, D.; Hager, W.W.; Rao, A.V. Pseudospectral Methods for Solving InfniteHorizon Optimal Control Problems. Automatica 2011, 47, 829–837. [Google Scholar] [CrossRef]
 Sagliano, M. Pseudospectral Convex Optimization for Powered Descent and Landing. J. Guid. Control Dyn. 2017, 41, 320–334. [Google Scholar] [CrossRef]
 Fletcher, C.A.J. Computational Galerkin Methods. In Computational Galerkin Methods; Springer: Berlin/Heidelberg, Germany, 1984; pp. 72–85. [Google Scholar] [CrossRef]
 Lorentz, G.G.; Zeller, K.L. Birkhoff Interpolation. SIAM J. Numer. Anal. 1971, 8, 43–48. [Google Scholar] [CrossRef]
 Schoenberg, I.J. On HermiteBirkhoff interpolation. J. Math. Anal. Appl. 1966, 16, 538–543. [Google Scholar] [CrossRef]
 Zhao, D.J.; Song, Z.Y. Reentry trajectory optimization with waypoint and nofly zone constraints using multiphase convex programming. ACTA Astronaut. 2017, 137, 60–69. [Google Scholar] [CrossRef]
 Hu, Q.; Xie, J.; Liu, X. Trajectory optimization for accompanying satellite obstacle avoidance. Aerosp. Sci. Technol. 2018, 82, 220–233. [Google Scholar] [CrossRef]
 Löfberg, J. YALMIP: A Toolbox for Modeling and Optimization in MATLAB2004. In Proceedings of the 2004 IEEE International Conference on Robotics and Automation, CACSD Conference, Taipei, Taiwan, 2–4 September 2004. [Google Scholar]
N  ZOH  FDPSM ^{1}  FBPSM ^{2}  SBPSM ^{3}  

T_{total} (s)  e(m) ^{4}  J_{obj} (kg)  T_{total} (s)  e(m)  J_{obj} (kg)  T_{total} (s)  e(m)  J_{obj} (kg)  T_{total} (s)  e(m)  J_{obj} (kg)  
30  19.50  40.471  0.5547  5.35  0.296  0.5524  5.01  0.296  0.5524  4.92  0.296  0.5524 
60  24.88  20.196  0.5538  8.78  0.328  0.5536  9.69  0.328  0.5536  8.33  0.328  0.5536 
90  31.46  13.398  0.5525  10.39  0.177  0.5531  10.62  0.177  0.5531  10.42  0.177  0.5531 
120  40.82  10.038  0.5502  18.97  0.100  0.5532  17.19  0.100  0.5532  18.03  0.100  0.5532 
150  51.33  8.042  0.5523  64.11  0.131  0.5515  24.33  0.002  0.5529  20.98  0.002  0.5529 
180  61.63  6.819  0.5508  76.44  0.034  0.5524  26.85  0.021  0.5529  23.78  0.021  0.5529 
210  70.91  5.744  0.5522  94.61  0.326  0.5501  35.76  0.034  0.5529  29.19  0.034  0.5528 
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Zhang, Z.; Zhao, D.; Li, X.; Kong, C.; Su, M. Convex Optimization for Rendezvous and Proximity Operation via Birkhoff Pseudospectral Method. Aerospace 2022, 9, 505. https://doi.org/10.3390/aerospace9090505
Zhang Z, Zhao D, Li X, Kong C, Su M. Convex Optimization for Rendezvous and Proximity Operation via Birkhoff Pseudospectral Method. Aerospace. 2022; 9(9):505. https://doi.org/10.3390/aerospace9090505
Chicago/Turabian StyleZhang, Zhiwei, Dangjun Zhao, Xianbin Li, Chunyang Kong, and Ming Su. 2022. "Convex Optimization for Rendezvous and Proximity Operation via Birkhoff Pseudospectral Method" Aerospace 9, no. 9: 505. https://doi.org/10.3390/aerospace9090505