Actuator Constrained Optimal Control of Formations Near the Libration Points
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The document discusses actuator constrained optimal control for spacecraft formations near libration points, focusing on dynamic sensitivities, control limitations, and method formulations. It presents a direct collocation method for managing highly constrained problems, enabling precise control and optimization despite actuator limitations. Conclusions highlight the applicability of the methodology for various formation configurations and requirements.
![university-log
Background
Transcription Formulations
Applications
Conclusions
Direct Collocation
Multiple Segment Formulations
Switching Segments and Time
Optimization via Direct Transcription
Define a parameter vector consisting of state and control values
at nodes (discrete points in time)
x = [· · · yT
(tj ) · · · · · · uT
(tj ) · · · t0 tf ]T
Convert the Optimal Control Problem into a Parameter
Optimization Problem
Minimize
J = φ(t0, y0, tf , yf ) +
Z tf
t0
L(t, y, u) dt ⇒
Minimize
F (x)
subject to
˙y = f(t, y, u)
0 = ψ0(t0, y0)
0 = ψf (tf , yf )
0 = β(t, y, u)
⇒
subject to
c(x) =
h
cT
ψ0
(x) cT
ψf
(x) cT
β (x) cT
˙y (x)
iT
= 0
Solve the resulting optimization problem with a standard
Nonlinear Programming (NLP) algorithm
Stanton, Marchand Actuator Constrained Optimal Control 7](https://image.slidesharecdn.com/astro-2008-140321090702-phpapp01/85/Actuator-Constrained-Optimal-Control-of-Formations-Near-the-Libration-Points-7-320.jpg)
![university-log
Background
Transcription Formulations
Applications
Conclusions
Costs and Constraints
Initial Guess
Sample Solution
Costs and Constraints
Constraints
Initial time and states specified
Final time and formation size and plane specified
r∗
cd = 1 km distance between chief and deputy, r∗
dd = 1.73 km
distance between deputies
Specified pointing r∗I
cs = [1 0 0]
State continuity (differential constraints) by segment
State equality across segments (at knots)
Weighted Costs
Minimize thrust
Minimize formation size deviations along trajectory
Minimize formation plane deviations along trajectory
J = w1J1 + w2J2 + w3J3
F(x) = w1F1(x) + w2F2(x) + w3F3(x)
Stanton, Marchand Actuator Constrained Optimal Control 11](https://image.slidesharecdn.com/astro-2008-140321090702-phpapp01/85/Actuator-Constrained-Optimal-Control-of-Formations-Near-the-Libration-Points-11-320.jpg)
Actuator Constrained Optimal Control of Formations Near the Libration Points
- 1. university-log Background Transcription Formulations Applications Conclusions Actuator Constrained Optimal Control of Formations Near the Libration Points Capt Stuart A. Stanton Dr. Belinda G. Marchand Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin AIAA/AAS Astrodynamics Specialist Conference, Aug 2008 Stanton, Marchand Actuator Constrained Optimal Control 1
- 2. university-log Background Transcription Formulations Applications Conclusions Outline Background Dynamic Sensitivities and Control Limitations Transcription Formulations Direct Collocation Multiple Segment Formulations Switching Segments and Time Applications Costs and Constraints Initial Guess Sample Solution Stanton, Marchand Actuator Constrained Optimal Control 2
- 3. university-log Background Transcription Formulations Applications Conclusions Dynamic Sensitivities and Control Limitations Formation Limitations for Deep-Space Imaging Formations Deputy Chief r∗ dd r∗ cd → r∗ cs Figure: Formation Pointing Fixed size, shape, and orientation of the formation Fixed orientation of each member of the formation (deputy spacecraft) Stanton, Marchand Actuator Constrained Optimal Control 3
- 4. university-log Background Transcription Formulations Applications Conclusions Dynamic Sensitivities and Control Limitations Dynamical Sensitivities Near the Libration Points Previous investigations have focused on unconstrained continuous control solutions Linear and nonlinear; feasible and optimal solutions Non-natural formations require extremely precise control (< nm/s2 accelerations) These controls are impossible to implement with existing actuator technology Continuous Control Finite Burn Implementation Thrust Limitations Min Thrust Mag. Figure: Implementing a Continuous Control Solution Cannot reproduce the fidelity of continuous control Continuous control may even be smaller than minimum thrust bound Stanton, Marchand Actuator Constrained Optimal Control 4
- 5. university-log Background Transcription Formulations Applications Conclusions Dynamic Sensitivities and Control Limitations Control Limitations for Deep-Space Imaging Formations Fixed thruster location on each spacecraft body Specified thrust acceleration magnitude Based on actuator performance capability T∗→ ↓ T∗ ↑ T∗ ←T∗ Figure: Spacecraft Body Stanton, Marchand Actuator Constrained Optimal Control 5
- 6. university-log Background Transcription Formulations Applications Conclusions Direct Collocation Multiple Segment Formulations Switching Segments and Time Transcription Methods for Highly Constrained Problems The libration point formation problem motivates a unique solution method Direct optimization methods serve as the foundation Modifications allow for creative treatment of difficult constraints Solution methods are generalized for any number of dynamical models. Stanton, Marchand Actuator Constrained Optimal Control 6
- 7. university-log Background Transcription Formulations Applications Conclusions Direct Collocation Multiple Segment Formulations Switching Segments and Time Optimization via Direct Transcription Define a parameter vector consisting of state and control values at nodes (discrete points in time) x = [· · · yT (tj ) · · · · · · uT (tj ) · · · t0 tf ]T Convert the Optimal Control Problem into a Parameter Optimization Problem Minimize J = φ(t0, y0, tf , yf ) + Z tf t0 L(t, y, u) dt ⇒ Minimize F (x) subject to ˙y = f(t, y, u) 0 = ψ0(t0, y0) 0 = ψf (tf , yf ) 0 = β(t, y, u) ⇒ subject to c(x) = h cT ψ0 (x) cT ψf (x) cT β (x) cT ˙y (x) iT = 0 Solve the resulting optimization problem with a standard Nonlinear Programming (NLP) algorithm Stanton, Marchand Actuator Constrained Optimal Control 7
- 8. university-log Background Transcription Formulations Applications Conclusions Direct Collocation Multiple Segment Formulations Switching Segments and Time Multiple Segment Formulations Account for state or control discontinuities by dividing the problem into segments Ideal treatment for finite burn control solutions Enforce appropriate constraints at the knots (segment boundaries) Include knot times or segment durations in parameter vector Time StateorControl Iteration p ↓ t0 ↓ Knot 1 ↑ ↓ Knot 2 tf ↑ ns = 3 segments nk = 2 interior knots nn = 5 nodes/segment Segment 1 Segment 2 Segment 3 Time StateorControl Iteration p + 1 ↓ t0 ↓ Knot 1 ↑ ↓ Knot 2 tf ↑ Segment 1 Segment 2 Segment 3 Figure: An Example of Segments and Knots Stanton, Marchand Actuator Constrained Optimal Control 8
- 9. university-log Background Transcription Formulations Applications Conclusions Direct Collocation Multiple Segment Formulations Switching Segments and Time Impacts of Fixed Spacecraft Orientation A traditional finite burn formulation specifies thrust (acceleration) magnitude, but not direction Assumes spacecraft can re-orient to deliver required thrust vector Control space U1: uT u = (T∗ )2 If spacecraft orientation is predetermined (according to other mission requirements) Actuator configuration must provide 3-axis maneuverability Assume thrusters are located on principal axes of body frame B ≡ {ˆxB, ˆyB, ˆzB} Control space U2: ui(ui − T∗ )(ui + T∗ ) = 0, i = ˆxB, . . . , ˆzB Fixed spacecraft orientation leads to discrete optimization, which gradient-type NLP algorithms cannot support. U1 U2 Figure: Control Spaces (a) U1 (Orientation Free), and (b) U2 (Orientation Fixed) Stanton, Marchand Actuator Constrained Optimal Control 9
- 10. university-log Background Transcription Formulations Applications Conclusions Direct Collocation Multiple Segment Formulations Switching Segments and Time Managing Fixed Spacecraft Orientation Instead of optimizing control values (i.e. −T ∗ , 0, T∗ ), . . . Prespecify control values by segment and optimize switching times Knots are used to designate switching times in each control axis Segments are bounded by switches in any control The chronological ordering of knots changes at each iteration of the optimization ux t1,1 t1,2 t1,3 Iteration p uy t2,1 t2,2 t2,3 uz t3,1 t3,2 t3,3 Segments Time 1 2 3 4 5 6 7 8 9 10 ux t1,1 t1,2 t1,3 Iteration p + 1 uy t2,1 t2,2 t2,3 uz t3,1 t3,2 t3,3Segments Time 1 2 3 4 5 6 7 8 9 10 Figure: Conceptual Control Profile with Segment Divisions Stanton, Marchand Actuator Constrained Optimal Control 10
- 11. university-log Background Transcription Formulations Applications Conclusions Costs and Constraints Initial Guess Sample Solution Costs and Constraints Constraints Initial time and states specified Final time and formation size and plane specified r∗ cd = 1 km distance between chief and deputy, r∗ dd = 1.73 km distance between deputies Specified pointing r∗I cs = [1 0 0] State continuity (differential constraints) by segment State equality across segments (at knots) Weighted Costs Minimize thrust Minimize formation size deviations along trajectory Minimize formation plane deviations along trajectory J = w1J1 + w2J2 + w3J3 F(x) = w1F1(x) + w2F2(x) + w3F3(x) Stanton, Marchand Actuator Constrained Optimal Control 11
- 12. university-log Background Transcription Formulations Applications Conclusions Costs and Constraints Initial Guess Sample Solution Baseline Initial Guess Trajectory Legend Deputy 1 Trajectory Deputy 2 Trajectory Deputy 3 Trajectory t0 tf Control Legend Axis 1 Control (ux) Axis 2 Control (uy) Axis 3 Control (uz) −10 0 10 −10 −5 0 5 10 x (100 m) y(100m) Spacecraft Positions - −10 0 10 −10 −5 0 5 10 x (100 m) z(100m) −10 0 10 −10 −5 0 5 10 y (100 m) z(100m) −10 0 10 −10 0 10 −10 0 10 x Inertial Frame y z 0 1 2 3 4 5 −2 0 2 x 10 −9 Deputy 1 m/s 2 Control Accelerations - Body Frame Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1 0 1 2 3 4 5 −2 0 2 x 10 −9 Deputy 2 m/s 2 Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2 0 1 2 3 4 5 −2 0 2 x 10 −9 Deputy 3 m/s 2 Time (106 sec) Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3 Figure: Baseline Initial GuessStanton, Marchand Actuator Constrained Optimal Control 12
- 13. university-log Background Transcription Formulations Applications Conclusions Costs and Constraints Initial Guess Sample Solution Baseline Solution Trajectory Legend Deputy 1 Trajectory Deputy 2 Trajectory Deputy 3 Trajectory t0 tf Control Legend Axis 1 Control (ux) Axis 2 Control (uy) Axis 3 Control (uz) −10 0 10 −10 −5 0 5 10 x (100 m) y(100m) Spacecraft Positions - −10 0 10 −10 −5 0 5 10 x (100 m) z(100m) −10 0 10 −10 −5 0 5 10 y (100 m) z(100m) −10 0 10 −10 0 10 −10 0 10 x Inertial Frame y z 0 1 2 3 4 5 −2 0 2 x 10 −9 Deputy 1 m/s 2 Control Accelerations - Body Frame Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1Deputy 1 0 1 2 3 4 5 −2 0 2 x 10 −9 Deputy 2 m/s 2 Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2Deputy 2 0 1 2 3 4 5 −2 0 2 x 10 −9 Deputy 3 m/s 2 Time (106 sec) Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3Deputy 3 Figure: Baseline SolutionStanton, Marchand Actuator Constrained Optimal Control 13
- 14. university-log Background Transcription Formulations Applications Conclusions Conclusions A modified collocation method with a segment-time switching algorithm leads to highly constrained control solutions Generalized formulation allows users to input formation configuration, size, orientation, and rotation rate thruster capability and placement dynamic model and reference trajectory initial and terminal conditions Suited to aid in establishing requirements and capabilities for highly constrained formations Stanton, Marchand Actuator Constrained Optimal Control 14
- 15. university-log Appendix References I Howell, K.C. and Keeter, T.M. “Station-Keeping Strategies for Libration Point Orbits: Target Point and Floquet Mode Approaches.” Advances in the Astronautical Sciences, 89(2), 1995, pp. 1377-1396. Barden, B.T. and Howell, K.C. “Formation Flying in the Vicinity of Libration Point Orbits.” Advances in the Astronautical Sciences, 99(2), 1998, pp. 969-988. Luquette, R.J. and Sanner, R.M. “A non-linear approach to spacecraft formation control in the vicinity of a collinear libration point.” AAS/AIAA Astrodynamics Conference, 30 Jul-2 Aug 2001 (Quebec, Canada: AAS), AAS Paper 01-330. Scheeres, D.J. and Vinh, N.X. “Dynamics and Control of Relative Motion in an Unstable Orbit,” AIAA Paper 2000-4235, Aug 2000. Scheeres, D.J., Hsiao, F.-Y., and Vinh, N.X. “Stabilizing Motion Relative to an Unstable Orbit: Applications to Spacecraft Formation Flight.” Journal of Guidance, Control, and Dynamics, 26(1), Jan-Feb 2003, pp. 62-73. Gurfil, P., Idan, M., and Kasdin, N.J. “Adaptive Neural Control of Deep-Space Formation Flying.” American Control Conference, 8-10 May 2002 (Anchorage, AK: ACC), pp. 2842-2847. Gurfil, P., Idan, M., and Kasdin, N.J. “Adaptive Neural Control of Deep-Space Formation Flying.” Journal of Guidance, Control, and Dynamics, 26(3), May-Jun 2003, pp.491-501. Gurfil, P. and Kasdin, N.J. “Stability and control of spacecraft formation flying in trajectories of the restricted three-body problem.” Acta Astronautica, 54, 2004, pp. 433-453. Stanton, Marchand Actuator Constrained Optimal Control 15
- 16. university-log Appendix References II Marchand, B.G. and Howell, K.C. “Formation Flight Near L1 and L2 in the Sun-Earth/Moon Ephemeris System Including Solar Radiation Pressure.” In Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, Big Sky, MT, Aug, 2003. AAS Paper 03-596. Howell, K.C. and Marchand, B.G. “Design and Control of Formations Near the Libration Points of the Sun-Earth/Moon Ephemeris System.” In Proceedings of the Space Flight Mechanics Symposium - Goddard Space Flight Center, Greenbelt, MD, Oct 2003. Marchand, B.G. and Howell, K.C. “Aspherical Formations Near the Libration Points in the Sun-Earth/Moon Ephemeris System.” AAS/AIAA Space Flight Mechanics Meeting, 7-12 Februray 2004 (Maui, HI: AAS), AAS Paper 04-157. Howell, K.C. and Marchand, B.G. “Formations Near the Libration Points: Design Strategies Using Natural and Non-Natural Arcs.” In Proceedings of GSFC 2nd International Symposium on Formation Flying Missions and Technologies, Greenbelt, MD, Sep 2004. Howell, K.C. and Marchand, B.G. “Natural and Non-Natural Spacecraft Formations Near the L1 and L2 Libration Points in the Sun-Earth/Moon Ephemeris System.” Dynamical Systems: An International Journal, Special Issue: Dynamical Systems in Dynamical Astronomy and Space Mission Design, 20(1), Mar 2005, pp. 149-173. Marchand, B.G. and Howell, K.C. “Control Strategies for Formation Flight in the Vicinity of the Libration Points.” Journal of Guidance, Control, and Dynamics. 28(6), Nov-Dec 2005, pp. 1210-1219. Marchand, B.G. Spacecraft Formation Keeping Near the Libration Points of the Sun-Earth/Moon System. PhD Dissertation. Purdue University. August 2004. Stanton, Marchand Actuator Constrained Optimal Control 16
- 17. university-log Appendix References III Marchand, B.G., Howell, K.C., and Betts, J.T. “Discrete Nonlinear Optimal Control of S/C Formations Near the L1 and L2 Points of the Sun-Earth/Moon System.” AAS/AIAA Astrodynamics Specialists Conference, Lake Tahoe, CA, August, 2005. Infeld, S.I., Josselyn, S.B., Murray, W., and Ross, I.M. “Design and Control of Libration Point Spacecraft Formations.” Journal of Guidance, Control, and Dynamics, 30(4), Jul-Aug 2007, pp. 899-909. Mueller, J. “Thruster Options for Microspacecraft: a Review and Evaluation of Existing Hardware and Emerging Technologies.” 33rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, 6-9 July 1997 (Seattle, WA: AIAA), AIAA Paper 97-3058. Gonzales, A.D. and Baker, R.P. “Microchip Laser Propulsion for Small Satellites.” 37th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, 8-11 July 2001 (Salt Lake City, UT: AIAA), AIAA Paper 2001-3789. Gonzales, D.A., and Baker, R.P. “Microchip Laser Propulsion for Small Satellites.” Proceedings of the 37th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Salt Lake City, Utah, Jul 2001. AIAA Paper 2001-3789. Gonzales, A.D. and Baker, R.P. “Micropropulsion using a Nd:YAG microchip laser.” Proceedings of SPIE - The International Society for Optical Engineering, 4760(ii), 2002, p 752-765. Phipps, C. and Luke, J. “Diode Laser-Driven Microthrusters–A New Departure for Micropropulsion. AIAA Journal, 40(2), 2002, pp. 310-318. Phipps, C., Luke, J., and Helgeson, W. “Laser space propulsion overview.” Proceedings of SPIE - The International Society for Optical Engineering, 6606, Advanced Laser Technologies 2006, 2007, p 660602. Stanton, Marchand Actuator Constrained Optimal Control 17
- 18. university-log Appendix References IV Coulter, D. “NASA’s Terrestrial Planet Finder Missions.” In Proceedings of SPIE Vol. 5487, Bellingham, WA, 2004, pp. 1207-1215. Lay, O.P. et al. “Architecture Trade Study for the Terrestrial Planet Finder Interferometer.” In Proceedings of SPIE Vol. 5905, Bellingham, WA, 2005, 590502. Cash, W. and Gendreau, K. “MAXIM Science and Technology.” In Proceedings of SPIE Vol. 5491, Bellingham, WA, 2004, pp. 199-211. Gill, Philip E., Murray, Walter, and Saunders, Michael A. “User’s Guide for SNOPT Version 7: Software for Large-Scale Nonlinear Programming.” February 12, 2006. Hull, David G. “Conversion of Optimal Control Problems into Parameter Optimization Problems.” Journal of Guidance, Control, and Dynamics, 20(1), Jan-Feb 1997, pp. 57-60. Stanton, Marchand Actuator Constrained Optimal Control 18
- 19. university-log Appendix Varying Parameters to Obtain Different Solutions Final time Number of nodes and knots Initial guess Thrust magnitude Table: Comparison of Solutions with Various Parameters Baseline tf nk, tf Feasible Guess Thrust nn 4 4 6 4 4 nk 10 10 20 10 10 tf (106 sec) 5.1183 10.2366 10.2366 5.1183 5.1183 Guess Baseline Baseline Baseline Feasible Baseline wThrust 1 400 1 400 1 400 1 400 1 1600 wDistance 0.1 0.1 0.1 0.1 0.1 wPlane 1 1 1 1 1 Thrust (km/s2) 2.0e-12 2.0e-12 2.0e-12 2.0e-12 4.0e-12 n 2333 2333 6779 2333 2333 # Iterations 45 157 194 95 272 Computational Time (sec) 253.91 956.41 3000.35 575.88 1570.04 Weighted Thrust Cost 1.6233 5.8133 6.6838 10.9247 1.1324 Weighted Formation Cost 16.1818 634.3255 67.7092 35.5675 6.9286 Weighted Plane Cost 0.8591 84.6949 4.6852 4.9035 1.0632 Total Cost 18.6643 724.8338 79.0782 51.3956 9.1242 Stanton, Marchand Actuator Constrained Optimal Control 19