Abstract
The bounded quasi-periodic relative trajectories are investigated in this paper for on-orbit surveillance, inspection or repair, which requires rapid changes in formation configuration for full three-dimensional imaging and unpredictable evolutions of relative trajectories for non-allied spacecraft. A linearized differential equation for modeling J 2 perturbed relative dynamics is derived without any simplified treatment of full short-period effects. The equation serves as a nominal reference model for stationkeeping controller to generate the quasi-periodic trajectories near the equilibrium, i.e., the location of the chief. The developed model exhibits good numerical accuracy and is applicable to an elliptic orbit with small eccentricity inheriting from the osculating conversion of orbital elements. A Hamiltonian structure-preserving controller is derived for the three-dimensional time-periodic system that models the J 2-perturbed relative dynamics on a mean circular orbit. The equilibrium of the system has time-varying topological types and no fixed-dimensional unstable/stable/center manifolds, which are quite different from the two-dimensional time-independent system with a permanent pair of hyperbolic eigenvalues and fixed-dimensions of unstable/stable/ center manifolds. The unstable and stable manifolds are employed to change the hyperbolic equilibrium to elliptic one with the poles assigned on the imaginary axis. The detailed investigations are conducted on the critical controller gain for Floquet stability and the optimal gain for the fuel cost, respectively. Any initial relative position and velocity leads to a bounded trajectory around the controlled elliptic equilibrium. The numerical simulation indicates that the controller effectively stabilizes motions relative to the perturbed elliptic orbit with small eccentricity and unperturbed elliptic orbit with arbitrary eccentricity. The developed controller stabilizes the quasi-periodic relative trajectories involved in six foundational motions with different frequencies generated by the eigenvectors of the Floquet multipliers, rather than to track a reference relative configuration. Only the relative positions are employed for the feedback without the information from the direct measurement or the filter estimation of relative velocity. So the current controller has potential applications in formation flying for its less computation overload for on-board computer, less constraint on the measurements, and easily-achievable quasi-periodic relative trajectories.
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References
Alfriend, K.T., Yan, H.: An orbital elements approach to the nonlinear formation flying problem. In: 1st International Formation Flying Symposium, Toulouse (2002)
Alfriend, K.T., Vadali, S., Gurfil, P., How, J., Breger, L.: Spacecraft Formation Flying: Dynamics, Control and Navigation. Elsevier Astrodynamics Series (2010)
Badawya A., McInnes C.: Small spacecraft formation using potential functions. Acta Astronaut. 65(11–12), 1783–1788 (2009)
Born G.H., Goldstein D.B., Thompson B.: An analytical theory for orbit determination. J. Astronaut. Sci. 49(2), 345–361 (2001)
Breger, L., How, P.: Partial J 2-invariance for Spacecraft Formations. In: AIAA/AAS Astrodynamics Conference, Keystone, AIAA 2006-6585 (2006)
Breger L.S., How J.P.: Gauss’s variational equation-based dynamics and control for formation flying spacecraft. J. Guid. Control Dyn. 30(2), 437–448 (2007)
Clohessy W., Wiltshire R.: Terminal guidance system for satellite rendezvous. J. Aerosp. Sci. 27(9), 653–658 (1960)
D’Amico S., Montenbruck O.: Proximity operations of formation flying spacecraft using an eccentricity/inclination vector separation. J. Guid. Control Dyn. 29(3), 554–563 (2006)
Duan, X., Bainum, P.M.: Design of spacecraft formation flying orbits. In: AAS, pp. 503–588 (2003)
Gim D.W., Alfriend K.T.: State transition matrix of relative motion for the perturbed noncircular reference orbit. J. Guid. Control Dyn. 26(6), 956–971 (2003)
Gurfil P., Mishne D.: Cyclic spacecraft formations:relative motion control using line-of-sight measurements only. J. Guid. Control Dyn. 30(1), 214–226 (2007)
Hamel J., de Lafontaine J.: Linearized dynamics of formation flying spacecraft on a J 2- perturbed elliptical orbit. J. Guid. Control Dyn. 30(6), 1649–1658 (2007)
Izzo, D., Sabatini, M., Valente, C.: A new linear model describing formation flying dynamics under J 2 effects. In: Proceedings of 17th AIDAA National Congress, vol. 1, pp. 493–500. Rome (2003)
Kasdin N.J., Gurfil P., Kolemen E.: Canonical modeling of relative spacecraft motion via epicyclic orbital elements. Celest. Mech. Dyn. Astron. 92(4), 337–370 (2005)
Kechichian J.A.: Motion in general elliptic orbits with respect to a dragging and precessing coordinate frame. J. Astronaut. Sci. 46(1), 25–46 (1998)
King L.B., Parker G.G., Deshmukh S., Chong J.H.: Study of interspacecraft coulomb forces and implications for formation flying. J. Propuls. Power 19(3), 497–505 (2003)
Kong E., Kwon D., Schweighart S., Elias L., Sedwick R., Miller D.: Electromagnetic formation flight for multi-satellite arrays. J. Spacecr. Rockets 41(4), 659–666 (2004)
Koon, W.S., Marsden, J.E., Murray, R.M., Masdemont, J.: J 2 dynamics and formation flight. In: AIAA Guidance, Navigation, and Control Conference and Exhibit, Montreal, AIAA 2001-4090 (2001)
Lawden D.: Fundamentals of space navigation. J. Br. Interplanet. Soc. 13(2), 87–101 (1954)
Lewis, J. Space weapons in the 2005 US defense budget request. In: Workshop on Outer Space and Security, paper 2375, Geneva (2004)
Martinusi V., Gurfil P.: Solutions and periodicity of satellite relative motion under even zonal harmonics perturbations. Celest. Mech. Dyn. Astron. 111, 387–414 (2011)
Montenbruck O., Kirschner M., D’Amico S., Bettadpur S.: E/I-vector separation for safe switching of the GRACE formation. Aerosp. Sci. Technol. 10(7), 628–635 (2006)
Ross I.M.: Linearized dynamic equations for spacecraft subject to J 2 perturbation. J. Guid. Control Dyn. 26(4), 657–659 (2003)
Rupp, T., Boge, T., Kiehling, R., Sellmaier, F.: Flight dynamics challenges of the german on-orbit servicing mission DEOS. In: 21st International Symposium on Space Flight Dynamics, ISSFD09f, Toulouse (2009)
Sabatini, M., Izzo, D., Palmerini, G.: Analysis and control of convenient orbital configuration for formation flying missions. In: 16th AAS/AIAA Space Flight Mechanics Conference, AAS 06-120, Tampa (2006)
Sabatini M., Izzo D., Palmerini G.: Minimum control for spacecraft formations in a J 2 perturbed environment. Celest. Mech. Dyn. Astron. 105, 141–157 (2009)
Schaub H., Alfriend K.T.: J 2 invariant relative orbits for spacecraft formations. Celest. Mech. Dyn. Astron. 79, 77–95 (2001)
Schaub H., Alfriend K.T.: Hybrid cartesian and orbit element feedback law for formation flying spacecraft. J. Guid. Control Dyn. 25(2), 387–393 (2002)
Scheeres D.J., Hsiao F.Y., Vinh N.X.: Stabilizing motion relative to an unstable orbit: applications to spacecraft formation flight. J. Guid. Control Dyn. 26(1), 62–73 (2003)
Schweighart S., Sedwick R.: High-fidelity linearized J 2 model for satellite formation flight. J. Guid. Control Dyn. 25(6), 1073–1080 (2002)
Sengupta P., Vadali S.R., Alfriend K.T.: Averaged relative motion and applications to formation flight near perturbed orbits. J. Guid. Control Dyn. 31(2), 258–272 (2008)
Vadali S.R.: Model for linearized satellite relative motion about a J 2-perturbed mean circular orbit. J. Guid. Control Dyn. 32(5), 1687–1691 (2009)
Vaddi S.S., Vadali S.R.: Linear and nonlinear control laws for formation flying. Adv. Astronaut. Sci. 114(1), 171–187 (2003)
Vadali, S.R., Schaub, H., Alfriend, K.T.: Initial conditions and fuel-optimal control for formation flying satellite. In: AIAA GNC Conference, Paper No. AIAA 99-426, Portland (1999)
Vadali S.R., Alfriend K.T., Vaddi S.S.: Hill’s equations, mean orbital elements, and formation flying of satellites. Adv. Astronaut. Scie. 106, 187–204 (2000)
Vadali S.R., Vaddi S.S., Alfriend K.T.: An intelligent control concept for formation flying satellites. Int. J. Robust Nonlinear Control 12, 97–115 (2002)
Vadali S.R., Sengupta P., Yan H., Alfriend K.T.: Fundamental frequencies of satellite relative motion and control of formations. J. Guid. Control Dyn. 31(5), 1239–1248 (2008)
Wang, P.K.C., Hadaegh, F.Y.: Formation flying of multiple spacecraft with autonomous rendezvous and docking capability. In: IET Control Theory Application, pp. 494–504 (2007)
Xu M., Xu S.J.: J 2 invariant relative orbits via differential correction algorithm. Acta Mech. Sin. 23(5), 585–595 (2007)
Xu M., Xu S.J.: Structure- preserving stabilization for Hamiltonian system and its applications in solar sail. J. Guid. Control Dyn. 32(3), 997–1004 (2009)
Xu M., Xu S.J.: Nonlinear dynamical analysis for displaced orbits above a planet. Celest. Mech. Dyn. Astron. 102(4), 327–353 (2008)
Xu M., Wang Y., Xu S.J.: On the existence of J 2 invariant relative orbits from the dynamical system point of view. Celest. Mech. Dyn. Astron. 112(4), 427–444 (2012)
Yan, H., Alfriend, K.T.: Numerical searches and optimal control of J 2 invariant orbit. In: 16th Annual AAS/AIAA Spaceflight Mechanics Meeting, AAS 06-163, Tampa (2006)
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Xu, M., Zhu, J., Tan, T. et al. Application of Hamiltonian structure-preserving control to formation flying on a J 2-perturbed mean circular orbit. Celest Mech Dyn Astr 113, 403–433 (2012). https://doi.org/10.1007/s10569-012-9430-2
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DOI: https://doi.org/10.1007/s10569-012-9430-2