# Fuel-Optimal Continuous-Thrust Orbital Rendezvous under Collision Avoidance Constraint

Topic: Optimal control | All

Séance du jeudi 9 octobre 2014, Salle V 213, 14h.

Richard EPENOY, Centre National d'Etudes Spatiales, Toulouse

1) Fuel-Optimal Continuous-Thrust Orbital Rendezvous under Collision Avoidance Constraint

This talk focuses on the design of a fuel-optimal maneuver strategy for the rendezvous between an active chaser satellite with continuous-thrust capability and a passive target satellite. The problem is formalized as an optimal control problem subject to a collision avoidance constraint. A new method for dealing with this state constraint is developed by building a sequence of unconstrained optimal control problems whose solutions converge toward the solution of the original problem. The efficiency of this method is demonstrated through numerical results obtained in the case of a rendezvous in Highly Elliptical Orbit.

2) An Optimal Control Approach for the Design of Low-energy Low-thrust Trajectories Between Libration Point Orbits

In this talk, we investigate the numerical computation of minimum-energy trajectories between Libration point orbits in the circular restricted three-body problem [1]. We will consider a low-thrust spacecraft and will focus our attention on its transfer between Lyapunov orbits around the Lagrange points L1 and L2 of the Earth-Moon three-body problem. These departure and arrival periodic orbits will be computed using Lindstedt-Poincaré techniques [2]. It is known from dynamical system theory [3-5] that almost zero-cost transfers exist for particular values of the transfer duration when the two orbits have the same Jacobi constant. These so-called heteroclinic connections follow in part the invariant manifolds of the departure and arrival orbits and require small impulsive thrusts that are not achievable by means of a low-thrust propulsion system. However, trying to determine low-energy low-thrust trajectories by solving the minimum-energy optimal control problem appears to be very difficult or even impossible from a medium value of the transfer duration. In particular, indirect shooting methods fail mainly due to the hypersensitivity of the state and costate equations. In this talk, we develop a three-step methodology for solving the minimum-energy optimal control problem without using information from the invariant manifolds of the initial and final orbits.

[1] V. G. Szebehely.

Theory of Orbits - The Restricted Problem of Three Bodies. Academic Press Inc., Harcourt Brace Jovanovich Publishers, Orlando, Florida, 1967, pp. 8-100.

[2] J. Masdemont.

High Order Expansions of Invariant Manifolds of Libration Point Orbits with Applications to Mission Design. Dynamical Systems, 20:1, 2005, pp. 59-113.

[3] W. S. Koon, M. W. Lo, J. E. Marsden, and S. D. Ross.

Heteroclinic Connections Between Periodic Orbits and Resonance Transitions in Celestial Mechanics. Chaos, 10:4, 2000, pp. 427-469.

[4} G. Gomez, W. S. Koon, J. E. Marsden, J. Masdemont, and S. D. Ross.

Connecting Orbits and Invariant Manifolds in the Spatial Restricted Three-body Problem. Nonlinearity, 17:5, 2004, pp. 1571-1606.

[5] E. Canalias and J. Masdemont.

Homoclinic and Heteroclinic Transfer Trajectories Between Lyapunov Orbits in the Sun-Earth and Earth-Moon Systems. Discrete and Continuous Dynamical Systems - Series A, 14:2, 2006, pp. 261-279.

Richard EPENOY, Centre National d'Etudes Spatiales, Toulouse

1) Fuel-Optimal Continuous-Thrust Orbital Rendezvous under Collision Avoidance Constraint

This talk focuses on the design of a fuel-optimal maneuver strategy for the rendezvous between an active chaser satellite with continuous-thrust capability and a passive target satellite. The problem is formalized as an optimal control problem subject to a collision avoidance constraint. A new method for dealing with this state constraint is developed by building a sequence of unconstrained optimal control problems whose solutions converge toward the solution of the original problem. The efficiency of this method is demonstrated through numerical results obtained in the case of a rendezvous in Highly Elliptical Orbit.

2) An Optimal Control Approach for the Design of Low-energy Low-thrust Trajectories Between Libration Point Orbits

In this talk, we investigate the numerical computation of minimum-energy trajectories between Libration point orbits in the circular restricted three-body problem [1]. We will consider a low-thrust spacecraft and will focus our attention on its transfer between Lyapunov orbits around the Lagrange points L1 and L2 of the Earth-Moon three-body problem. These departure and arrival periodic orbits will be computed using Lindstedt-Poincaré techniques [2]. It is known from dynamical system theory [3-5] that almost zero-cost transfers exist for particular values of the transfer duration when the two orbits have the same Jacobi constant. These so-called heteroclinic connections follow in part the invariant manifolds of the departure and arrival orbits and require small impulsive thrusts that are not achievable by means of a low-thrust propulsion system. However, trying to determine low-energy low-thrust trajectories by solving the minimum-energy optimal control problem appears to be very difficult or even impossible from a medium value of the transfer duration. In particular, indirect shooting methods fail mainly due to the hypersensitivity of the state and costate equations. In this talk, we develop a three-step methodology for solving the minimum-energy optimal control problem without using information from the invariant manifolds of the initial and final orbits.

[1] V. G. Szebehely.

Theory of Orbits - The Restricted Problem of Three Bodies. Academic Press Inc., Harcourt Brace Jovanovich Publishers, Orlando, Florida, 1967, pp. 8-100.

[2] J. Masdemont.

High Order Expansions of Invariant Manifolds of Libration Point Orbits with Applications to Mission Design. Dynamical Systems, 20:1, 2005, pp. 59-113.

[3] W. S. Koon, M. W. Lo, J. E. Marsden, and S. D. Ross.

Heteroclinic Connections Between Periodic Orbits and Resonance Transitions in Celestial Mechanics. Chaos, 10:4, 2000, pp. 427-469.

[4} G. Gomez, W. S. Koon, J. E. Marsden, J. Masdemont, and S. D. Ross.

Connecting Orbits and Invariant Manifolds in the Spatial Restricted Three-body Problem. Nonlinearity, 17:5, 2004, pp. 1571-1606.

[5] E. Canalias and J. Masdemont.

Homoclinic and Heteroclinic Transfer Trajectories Between Lyapunov Orbits in the Sun-Earth and Earth-Moon Systems. Discrete and Continuous Dynamical Systems - Series A, 14:2, 2006, pp. 261-279.