To find the optimal rendezvous time and order of each target in a multi-rendezvous mission, it’s nessassory to evaluate Δv between different targets with different departure times and transfer durations. To improve the efficiency of Δv evaluation, this paper proposes an analytical method for assessing the optimality of a given double-impulse trajectory by leveraging relative dynamic equations based on orbital differences. Furthermore, a gradient-based nonlinear programming algorithm is established for the rapid optimization of three-impulsive trajectory when the double-impulse trajectory is suboptimal. The methodology is validated through various case studies, encompassing both low-Earth-orbits and asteroid orbits. The results indicate that our optimality checks align with the prime vector theory while demanding fewer computations. Additionally, the three-impulsive solutions rapidly obtained via our nonlinear programming algorithm are consistent with those obtained from multi-impulse evolutionary optimization models. A multi-target rendezvous optimization case also proves its applicability to optimization of multi-rendezvous missions.
SukhanovAADurãoOLazzaroDLow-cost main-belt asteroid sample return. J Spacecraft Rockets2001; 38(5): 736–744.
2.
D’ArrigoPApiesSS. A mission for the exploration of the main asteroid belt using a swarm of microsatellites. Acta Astronaut2006; 59(8-11): 689–699.
3.
IzzoDGetznerIHennesD, et al.Evolving solutions to TSP variants for active space debris removal. Proceedings of the 2015 annual conference on genetic and evolutionary computation, Madrid, Spain, 11–15 July 2015.
4.
BaranovAAGrishkoDA. Review of path planning in prospective multi-target active debris removal missions in low Earth orbits. Prog Aero Sci2024; 145: 100982.
5.
ChaiRSavvarisATsourdosA, et al.A review of optimization techniques in spacecraft flight trajectory design. Prog Aero Sci2019; 109: 100543.
6.
FedericiLZavoliAColasurdoG. Evolutionary optimization of multirendezvous impulsive trajectories. International Journal of Aerospace Engineering2021; 2021: 1–19.
7.
ZhangZZhangNChenZ, et al.Global trajectory optimization of multispacecraft successive rendezvous using multitree search. J Guid Control Dynam2023; 47: 1–15.
8.
NelsonSLZarchanP. Alternative approach to the solution of lambert’s problem. J Guid Control Dynam1992; 15(4): 1003–1009.
9.
PrussingJE. A class of optimal two-impulse rendezvous using multiple-revolution lambert solutions. J Astronaut Sci2000; 48: 131–148.
10.
JifukuRIchikawaABandoM. Optimal pulse strategies for relative orbit transfer along a circular orbit. J Guid Control Dynam2011; 34(5): 1329–1341.
11.
SerraRArzelierDRondepierreA. Analytical solutions for impulsive elliptic out-of-plane rendezvous problem via primer vector theory. IEEE Trans Control Syst Technol2017; 26(1): 207–221.
12.
LandauD. Efficient maneuver placement for automated trajectory design. J Guid Control Dynam2018; 41(7): 1531–1541.
13.
AryaVŞaloğluKTaheriE, et al.Generation of multiple-revolution many-impulse optimal spacecraft maneuvers. J Spacecraft Rockets2023; 60(6): 1699–1711.
14.
HennesDIzzoDLandauD. Fast approximators for optimal low-thrust hops between main belt asteroids. 2016 IEEE symposium series on computational intelligence (SSCI), Athens, Greece, 10 October 2016.
15.
HuangAYLuoYZLiHN. Fast estimation of perturbed impulsive rendezvous via semi-analytical equality-constrained optimization. J Guid Control Dynam2020; 43(12): 2383–2390.
16.
ShenHX. Explicit approximation for J2-Perturbed low-thrust transfers between circular orbits. J Guid Control Dynam2021; 44(8): 1525–1531.
17.
GuoXRenDWuD, et al.Dnn estimation of low-thrust transfer time: focusing on fast transfers in multi-asteroid rendezvous missions. Acta Astronaut2023; 204: 518–530.
18.
RussellRP. Complete lambert solver including second-order sensitivities. J Guid Control Dynam2022; 45(2): 196–212.
19.
NegreteAAbdelkhalikOO. Exact solution to lambert’s problem using contour integrals. J Guid Control Dynam2024; 0(0): 1–10.
20.
LawdenDF. Optimal trajectories for space navigation. London: Butterworths, 1963.
21.
EdelbaumTN. How many impulses?Astronaut Aeronaut1967; 11(5): 64–69.
22.
LionPMHandelsmanM. Primer vector on fixed-time impulsive trajectories. AIAA J1968; 6(1): 127–132.
23.
PrussingJE. Optimal impulsive linear systems: sufficient conditions and maximum number of impulses. J Astronaut Sci1995; 43(2): 195–206.
24.
PrussingJE. Optimal four-impulse fixed-time rendezvous in the vicinity of a circular orbit. AIAA J1969; 7(5): 928–935.
25.
CarterTEAlvarezSA. Quadratic-based computation of four-impulse optimal rendezvous near circular orbit. J Guid Control Dynam2000; 23(1): 109–117.
26.
LuoYZZhangJLiH, et al.Interactive optimization approach for optimal impulsive rendezvous using primer vector and evolutionary algorithms. Acta Astronaut2010; 67(3-4): 396–405.
27.
HuangAHnLYzL. Analytical delta-v-based nonlinear programming of multitarget rendezvous and flyby trajectories. J Guid Control Dynam2024; 0(0): 1–9.
28.
HuangAYzLHnL. Global optimization of multiple-spacecraft rendezvous mission via decomposition and dynamics-guide evolution approach. J Guid Control Dynam2022; 45(1): 171–178.
29.
YangZLuoYZZhangJ, et al.Homotopic perturbed lambert algorithm for long-duration rendezvous optimization. J Guid Control Dynam2015; 38(11): 2215–2223.
30.
GimDWAlfriendKT. State transition matrix of relative motion for the perturbed noncircular reference orbit. J Guid Control Dynam2003; 26(6): 956–971.
31.
StefanoC. The equations of relative motion in the orbital reference frame. Celestial Mech Dyn Astron2016; 124(1): 215–234.
32.
GillPEMurrayWSaundersMA. Snopt: an sqp algorithm for large-scale constrained optimization. SIAM Rev2005; 47(1): 99–131.
