# American Institute of Mathematical Sciences

June 2019, 9(2): 225-256. doi: 10.3934/naco.2019016

## Indirect methods for fuel-minimal rendezvous with a large population of temporarily captured orbiters

 1 Department of Mathematics, University of Hawaii at Manoa, 2565 McCarthy Mall, Honolulu, Hawaii 96822, USA 2 Department of Computational Medicine and Bioinformatics, University of Michigan, 1600 Huron Parkway, Ann Arbor, MI 48105, USA

* Corresponding author: Monique Chyba

The paper is handled by Leong Wah June as the guest editor.

Received  December 2015 Revised  October 2018 Published  January 2019

Fund Project: This research is partially supported by the National Science Foundation (NSF) Division of Mathematical Sciences, award DMS-1109937, NSF Division of of Graduate Education, award DGE-0841223 and by the NASA, proposal Institute for the Science of Exploration Targets from the program Solar System Exploration Research Virtual Institute

A main objective of this work is to assess the feasibility of space missions to a new population of near Earth asteroids which temporarily orbit Earth, called temporarily captured orbiter. We design rendezvous missions to a large random sample from a database of over 16,000 simulated temporarily captured orbiters using an indirect method based on the maximum principle. The main contribution of this paper is the development of techniques to overcome the difficulty in initializing the algorithm with the construction of the so-called cloud of extremals.

Citation: Monique Chyba, Geoff Patterson. Indirect methods for fuel-minimal rendezvous with a large population of temporarily captured orbiters. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 225-256. doi: 10.3934/naco.2019016
##### References:
 [1] EL. Allgower and K. Georg, Numerical Continuation Methods, An Introduction, Springer, Berlin, 1990. doi: 10.1007/978-3-642-61257-2. [2] B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Springer-Verlag Berlin, 2003. [3] B. Bolin, R. Jedicke, M. Granvik, P. Brown, E. Howell, M. Nolan, M. Chyba, G. Patterson and R. Wainscoat, Detecting earth's temporarily captured natural irregular satellites - minimoons, Icarus, 241 (2014), 280-297. doi: 10.1016/j.icarus.2014.05.026. [4] B. J. Caillau, O. Cots and J. Gergaud, Differential continuation for regular optimal control problems, Optimization Methods and Software, 27 (2012), 177-196. doi: 10.1080/10556788.2011.593625. [5] J. B. Caillau, O. Cots and J. Gergaud, HAMPATH: on solving optimal control problems by indirect and path following methods, , http://apo.enseeiht.fr/hampath. [6] B. J. Caillau, B. Daoud and J. Gergaud, Minimum fuel control of the planar circular restricted three-body problem, Celestial Mech. Dynam. Astronom., 114 (2012), 137-150. doi: 10.1007/s10569-012-9443-x. [7] M. Chyba, T. Haberkorn and R. Jedicke, Minimum fuel round trip from a L2 Earth-Moon Halo orbit to Asteroid 2006RH120, Recent Advances in Celestial and Space Mechanics, Mathematics for Industry, Springer-Verlag, Japan, To Appear, 2016. doi: 10.1007/978-3-319-27464-5_4. [8] M. Chyba, G. Patterson, G. Picot, M. Granvik, R. Jedicke and J. Vaubaillon, Designing rendezvous missions with mini-moons using geometric optimal control, Journal of Industrial and Management Optimization, 10 (2014), 477-501. doi: 10.3934/jimo.2014.10.477. [9] M. Chyba, G. Patterson, G. Picot, M. Granvik, R. Jedicke and J. Vaubaillon, Time-minimal orbital transfers to temporarily-captured natural Earth satellites, Springer Verlag: Advances in Optimization and Control with Applications, 86 (2014), 213-235. doi: 10.1007/978-3-662-43404-8_12. [10] AV. Dmitruk and AM. Kaganovich, The hybrid maximum principle is a consequence of the pontryagin maximum principle, Systems Control Lett., 57 (2008), 964-970. doi: 10.1016/j.sysconle.2008.05.006. [11] M. Granvik, J. Vaubaillon and R. Jedicke, The population of natural Earth satellites, Icarus, 218 (2011), 262-277. doi: 10.1016/j.icarus.2011.12.003. [12] G. Mingotti, F. Topputo and F. Bernelli-Zazzera, A Method to Design Sun-Perturbed Earth-to-Moon Low-thrust Transfers with Ballistic Capture, AIDAA, 2007. [13] G. Patterson, Asteroid Rendezvous Missions Using Indirect Methods of Optimal Control, University of Hawaii at Manoa, dissertation, 2015. [14] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, John Wiley & Sons, New York, 1962. [15] C. Russell and V. Angelopoulos, The ARTEMIS Mission (Google eBook), Springer Science & Business Media, Nov 18, 2013 - Science - 112 pages. [16] D. Scheeres, The restricted hill four-body problem with applications to the earth-moon-sun system, Celestial Mechanics and Dynamical Astronomy, 70 (1998), 75-98. doi: 10.1023/A:1026498608950. [17] F. Topputo, On optimal two-impulse Earth-Moon transfers in a four-body model, Celest. Mech. Dyn. Astr., 117 (2013), 279-313. doi: 10.1007/s10569-013-9513-8. [18] Hodei Urrutxua, Daniel J. Scheeres, Claudio Bombardelli, Juan L. Gonzalo and Jesús Peláez", What Does it Take to Capture an Asteroid? A Case Study on Capturing Asteroid 2006 RH120, Advances in the Astronautical Sciences, (2014), AAS 14-276. [19] T. H. Sweetser, S. B. Broschart, V. Angelopoulos, G. J. Whiffen, D. C. Folta, M-K. Chung, S. J. Hatch and M. A. Woodard, ARTEMIS mission design, Space Science Reviews, 165 (2011), 27-57. [20] V. Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies, Academic Press, 1967.

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##### References:
 [1] EL. Allgower and K. Georg, Numerical Continuation Methods, An Introduction, Springer, Berlin, 1990. doi: 10.1007/978-3-642-61257-2. [2] B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Springer-Verlag Berlin, 2003. [3] B. Bolin, R. Jedicke, M. Granvik, P. Brown, E. Howell, M. Nolan, M. Chyba, G. Patterson and R. Wainscoat, Detecting earth's temporarily captured natural irregular satellites - minimoons, Icarus, 241 (2014), 280-297. doi: 10.1016/j.icarus.2014.05.026. [4] B. J. Caillau, O. Cots and J. Gergaud, Differential continuation for regular optimal control problems, Optimization Methods and Software, 27 (2012), 177-196. doi: 10.1080/10556788.2011.593625. [5] J. B. Caillau, O. Cots and J. Gergaud, HAMPATH: on solving optimal control problems by indirect and path following methods, , http://apo.enseeiht.fr/hampath. [6] B. J. Caillau, B. Daoud and J. Gergaud, Minimum fuel control of the planar circular restricted three-body problem, Celestial Mech. Dynam. Astronom., 114 (2012), 137-150. doi: 10.1007/s10569-012-9443-x. [7] M. Chyba, T. Haberkorn and R. Jedicke, Minimum fuel round trip from a L2 Earth-Moon Halo orbit to Asteroid 2006RH120, Recent Advances in Celestial and Space Mechanics, Mathematics for Industry, Springer-Verlag, Japan, To Appear, 2016. doi: 10.1007/978-3-319-27464-5_4. [8] M. Chyba, G. Patterson, G. Picot, M. Granvik, R. Jedicke and J. Vaubaillon, Designing rendezvous missions with mini-moons using geometric optimal control, Journal of Industrial and Management Optimization, 10 (2014), 477-501. doi: 10.3934/jimo.2014.10.477. [9] M. Chyba, G. Patterson, G. Picot, M. Granvik, R. Jedicke and J. Vaubaillon, Time-minimal orbital transfers to temporarily-captured natural Earth satellites, Springer Verlag: Advances in Optimization and Control with Applications, 86 (2014), 213-235. doi: 10.1007/978-3-662-43404-8_12. [10] AV. Dmitruk and AM. Kaganovich, The hybrid maximum principle is a consequence of the pontryagin maximum principle, Systems Control Lett., 57 (2008), 964-970. doi: 10.1016/j.sysconle.2008.05.006. [11] M. Granvik, J. Vaubaillon and R. Jedicke, The population of natural Earth satellites, Icarus, 218 (2011), 262-277. doi: 10.1016/j.icarus.2011.12.003. [12] G. Mingotti, F. Topputo and F. Bernelli-Zazzera, A Method to Design Sun-Perturbed Earth-to-Moon Low-thrust Transfers with Ballistic Capture, AIDAA, 2007. [13] G. Patterson, Asteroid Rendezvous Missions Using Indirect Methods of Optimal Control, University of Hawaii at Manoa, dissertation, 2015. [14] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, John Wiley & Sons, New York, 1962. [15] C. Russell and V. Angelopoulos, The ARTEMIS Mission (Google eBook), Springer Science & Business Media, Nov 18, 2013 - Science - 112 pages. [16] D. Scheeres, The restricted hill four-body problem with applications to the earth-moon-sun system, Celestial Mechanics and Dynamical Astronomy, 70 (1998), 75-98. doi: 10.1023/A:1026498608950. [17] F. Topputo, On optimal two-impulse Earth-Moon transfers in a four-body model, Celest. Mech. Dyn. Astr., 117 (2013), 279-313. doi: 10.1007/s10569-013-9513-8. [18] Hodei Urrutxua, Daniel J. Scheeres, Claudio Bombardelli, Juan L. Gonzalo and Jesús Peláez", What Does it Take to Capture an Asteroid? A Case Study on Capturing Asteroid 2006 RH120, Advances in the Astronautical Sciences, (2014), AAS 14-276. [19] T. H. Sweetser, S. B. Broschart, V. Angelopoulos, G. J. Whiffen, D. C. Folta, M-K. Chung, S. J. Hatch and M. A. Woodard, ARTEMIS mission design, Space Science Reviews, 165 (2011), 27-57. [20] V. Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies, Academic Press, 1967.
Simulated TCOs orbits from the database calculated in [11], plotted in the inertial Earth-Moon (EM)-barycentric reference frame
The distribution of TCOs capture times less than 3 years
Distributions of the periapsis distances for all 16,923 temporarily captured orbiter, with respect to the Earth (perigee), Moon (perilune), $L_1$ (peri-$L_1$), and $L_2$ (peri-$L_2$)
Fixed control structure
$\Delta v$ (m/s) vs. total thrusting time $\tau$ (hours)
Rendezvous points (small white dots) for TCO #1. Also shown are vectors indicating the direction of the velocity of the spacecraft and direction of the Sun for each rendezvous point
Rendezvous points (in white) for TCOs #1-100 (right). These are still only a small portion of all the rendezvous points $\mathbf{q}_{rdvz}$ we consider for the 16,923 minimoons
Example of cloud extremal construction. In plain, our constructed trajectory. In dashed, the resulting cloud extremal. The Earth (left) and Moon (right) are shown as black circles, and the CR3BP equilibrium points $L_i$ are plotted for reference as x's
Control structure for constructing cloud extremals
The Halo orbit discretized into 100 equally spaced (in terms of time) points
Delta-v distribution for the 12,558 cloud elements
Transfer time distribution for the 12,558 cloud elements
Initial Sun true anomaly distribution for the 12,558 cloud elements
Positions $\mathbf{s}_{cloud}$ of cloud points $\mathbf{q}_{cloud}$ for the 12,558 cloud elements, shown at different zoom levels
Distribution of TCO cloud distances (LD, position only). Left: all TCOs, Right: truncated for distances less than 0.5 LD only
Distribution of TCO cloud distances (non-dimensional, position and velocity)
Initial results: Distribution of Δv for 1,338 computed transfers
Initial results: Rendezvous with TCO #9. The Moon's orbit is shown as the ellipse around the Earth (gray dot). The thin grey path is the orbit of the TCO, starting from its capture point (triangle) to its escape point (square). The circle on the TCO orbit marks where the TCO was when the craft departed, and the star is the point of rendezvous. The dark gray path is the spacecraft's trajectory. It's three boosts are marked as white dots (including the final rendezvous boost)
Initial results: Rendezvous with minimoon #14. The legend is the same as for Figure 18
Initial results: Distribution of $\Delta v$ for the best computed transfers for 93 minimoons
Initial results: Position components of $\mathbf{q}_{rdvz}$ versus $\Delta v$
Initial results: Velocity components of $\mathbf{q}_{rdvz}$ versus $\Delta v$
Initial results: CR3BP Energy at $\mathbf{q}_{rdvz}$ versus $\Delta v$
Initial results: CR3BP Energy at $\mathbf{q}_{cloud}$ versus resulting $\Delta v$
Initial results: Transfer times versus $\Delta v$
Initial results: Continuation distance versus $\Delta v$
Initial results: Continuation final Sun true anomaly difference versus $\Delta v$
Large scale results: Distribution of $\Delta v$ for the 3,000 TCOs
Large scale results: Rendezvous with minimoon #9,046
Large scale results: Rendezvous with minimoon #16,844
Original (left) and refined (right) rendezvous with minimoon #5
Refined results: distribution of $\Delta v$ improvement for 93 TCOs
Original (left) and refined (right) rendezvous with TCO #40
Original (left) and refined (right) rendezvous with TCO #9,046
Numerical values for the CR3BP and the CR4BP
 CR3BP parameters CR4BP parameters $\mu$ $1.2153\cdot 10^{-1}$ $\mu_S$ $3.289\cdot 10^5$ 1 norm. dist. (LD) $384400$ km $r_S$ $3.892\cdot 10^2$ 1 norm. time $104.379$ h $\omega_S$ $-0.925$ rad/norm. time
 CR3BP parameters CR4BP parameters $\mu$ $1.2153\cdot 10^{-1}$ $\mu_S$ $3.289\cdot 10^5$ 1 norm. dist. (LD) $384400$ km $r_S$ $3.892\cdot 10^2$ 1 norm. time $104.379$ h $\omega_S$ $-0.925$ rad/norm. time
Comparison of chosen values $t_i$ versus solution values $t_i^*$. Units are in hours
 $t_1$ $t_2$ $t_3$ $t_4$ 1.00 240.00 241.00 479.00 $t_1^*$ $t_2^*$ $t_3^*$ $t_4^*$ 0.42 265.64 266.43 478.62
 $t_1$ $t_2$ $t_3$ $t_4$ 1.00 240.00 241.00 479.00 $t_1^*$ $t_2^*$ $t_3^*$ $t_4^*$ 0.42 265.64 266.43 478.62
Comparison of chosen values $\mathbf{P}(0)$ versus solution values $\mathbf{P}(0)^*$
 $p_x(0)$ $p_y(0)$ $p_z(0)$ $p_{\dot{x}}(0)$ $p_{\dot{y}}(0)$ $p_{\dot{z}}(0)$ $p_m(0)$ 0.0059 -0.0022 -0.0075 -0.0026 0.0051 -0.0070 -0.0000089 $p_x^*(0)$ $p_y^*(0)$ $p_z^*(0)$ $p_{\dot{x}}^*(0)$ $p_{\dot{y}}^*(0)$ $p_{\dot{z}}^*(0)$ $p_m^*(0)$ 0.1029 -0.0144 -0.0323 0.0116 0.0301 0.0015 -0.000071
 $p_x(0)$ $p_y(0)$ $p_z(0)$ $p_{\dot{x}}(0)$ $p_{\dot{y}}(0)$ $p_{\dot{z}}(0)$ $p_m(0)$ 0.0059 -0.0022 -0.0075 -0.0026 0.0051 -0.0070 -0.0000089 $p_x^*(0)$ $p_y^*(0)$ $p_z^*(0)$ $p_{\dot{x}}^*(0)$ $p_{\dot{y}}^*(0)$ $p_{\dot{z}}^*(0)$ $p_m^*(0)$ 0.1029 -0.0144 -0.0323 0.0116 0.0301 0.0015 -0.000071
Initial results: Winding number pairs and corresponding $\Delta v$ statistics for 1,338 transfers
 $w_e$ $w_m$ count mean $\Delta v$ median $\Delta v$ min $\Delta v$ max $\Delta v$ -1 -1 640 919.6 891.6 255.9 2171.2 -2 -2 241 1003.7 956.5 195.6 4850.1 0 0 78 977.0 889.9 428.9 2241.0 -3 -3 54 1044.9 857.5 354.1 3753.0 1 1 48 1030.3 864.2 782.5 2886.7 -1 0 39 781.4 788.4 269.5 1259.0 -5 -5 28 633.7 502.5 368.6 959.9 -2 -1 24 919.3 893.7 387.8 1468.1 -4 -4 22 1478.8 1000.8 397.1 2947.7 0 1 22 1891.7 2059.4 898.6 2480.2 -2 -3 21 1034.6 937.0 410.3 2033.3 -1 -2 18 1042.9 987.1 409.4 1855.2 -4 -3 15 1579.3 1460.0 775.5 2767.9 0 -1 13 1018.2 1019.4 462.9 1524.1 2 0 13 2788.0 3113.7 683.6 3160.3 -4 -5 10 2207.8 2201.8 2163.5 2264.7 -5 -4 6 851.0 795.7 740.2 1116.2 -3 -4 6 1361.4 1474.7 574.1 1695.3 -3 -2 5 1163.4 1361.2 388.0 1438.9 -2 2 4 906.0 900.5 846.6 976.4 -6 -4 3 886.8 883.0 632.3 1145.0 -3 -1 3 1924.5 1926.2 1909.6 1937.6 -1 1 3 898.1 753.7 641.6 1298.9 3 1 3 1363.8 1363.8 1363.8 1363.8 -18 2 2 3072.1 3072.1 3063.3 3081.0 -1 -3 2 1723.9 1723.9 1653.3 1794.5 2 -1 2 902.9 902.9 559.6 1246.2 2 1 2 1089.8 1089.8 1048.6 1131.0 3 0 2 1090.1 1090.1 813.0 1367.1 11 2 2 5735.4 5735.4 5015.0 6455.7 -5 -3 1 700.9 700.9 700.9 700.9 -4 -2 1 857.7 857.7 857.7 857.7 -2 0 1 710.7 710.7 710.7 710.7 4 3 1 1447.7 1447.7 1447.7 1447.7 6 1 1 1351.3 1351.3 1351.3 1351.3 7 1 1 1165.2 1165.2 1165.2 1165.2 8 1 1 1095.6 1095.6 1095.6 1095.6
 $w_e$ $w_m$ count mean $\Delta v$ median $\Delta v$ min $\Delta v$ max $\Delta v$ -1 -1 640 919.6 891.6 255.9 2171.2 -2 -2 241 1003.7 956.5 195.6 4850.1 0 0 78 977.0 889.9 428.9 2241.0 -3 -3 54 1044.9 857.5 354.1 3753.0 1 1 48 1030.3 864.2 782.5 2886.7 -1 0 39 781.4 788.4 269.5 1259.0 -5 -5 28 633.7 502.5 368.6 959.9 -2 -1 24 919.3 893.7 387.8 1468.1 -4 -4 22 1478.8 1000.8 397.1 2947.7 0 1 22 1891.7 2059.4 898.6 2480.2 -2 -3 21 1034.6 937.0 410.3 2033.3 -1 -2 18 1042.9 987.1 409.4 1855.2 -4 -3 15 1579.3 1460.0 775.5 2767.9 0 -1 13 1018.2 1019.4 462.9 1524.1 2 0 13 2788.0 3113.7 683.6 3160.3 -4 -5 10 2207.8 2201.8 2163.5 2264.7 -5 -4 6 851.0 795.7 740.2 1116.2 -3 -4 6 1361.4 1474.7 574.1 1695.3 -3 -2 5 1163.4 1361.2 388.0 1438.9 -2 2 4 906.0 900.5 846.6 976.4 -6 -4 3 886.8 883.0 632.3 1145.0 -3 -1 3 1924.5 1926.2 1909.6 1937.6 -1 1 3 898.1 753.7 641.6 1298.9 3 1 3 1363.8 1363.8 1363.8 1363.8 -18 2 2 3072.1 3072.1 3063.3 3081.0 -1 -3 2 1723.9 1723.9 1653.3 1794.5 2 -1 2 902.9 902.9 559.6 1246.2 2 1 2 1089.8 1089.8 1048.6 1131.0 3 0 2 1090.1 1090.1 813.0 1367.1 11 2 2 5735.4 5735.4 5015.0 6455.7 -5 -3 1 700.9 700.9 700.9 700.9 -4 -2 1 857.7 857.7 857.7 857.7 -2 0 1 710.7 710.7 710.7 710.7 4 3 1 1447.7 1447.7 1447.7 1447.7 6 1 1 1351.3 1351.3 1351.3 1351.3 7 1 1 1165.2 1165.2 1165.2 1165.2 8 1 1 1095.6 1095.6 1095.6 1095.6
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