Sandro da Silva Fernandes

The Orbital Dynamics of Synchronous Satellites: Irregular Motions in the 2 : 1 Resonance

The orbital dynamics of synchronous satellites is studied. The 2 : 1 resonance is considered; in other words, the satellite completes two revolutions while the Earth completes one. In the development of the geopotential, the zonal harmonics 𝐽20 and 𝐽40 and the tesseral harmonics 𝐽22 and 𝐽42 are considered. The order of the dynamical system is reduced through successive Mathieu transformations, and the final system is solved by numerical integration. The Lyapunov exponents are used as tool to analyze the chaotic orbits.

A First-Order Analytical Theory for Optimal Low-Thrust Limited-Power Transfers between Arbitrary Elliptical Coplanar Orbits

A complete first-order analytical solution, which includes the short periodic terms, for the problem of optimal low-thrust limited-power transfers between arbitrary elliptic coplanar orbits in a Newtonian central gravity field is obtained through canonical transformation theory. The optimization problem is formulated as a Mayer problem of optimal control theory with Cartesian elements—position and velocity vectors—as state variables. After applying the Pontryagin maximum principle and determining the maximum Hamiltonian, classical orbital elements are introduced through a Mathieu transformation. The short periodic terms are then eliminated from the maximum Hamiltonian through an infinitesimal canonical transformation built through Hori method. Closed-form analytical solutions are obtained for the average canonical system by solving the Hamilton-Jacobi equation through separation of variables technique. For transfers between close orbits a simplified solution is straightforwardly derived by linearizing the new Hamiltonian and the generating function obtained through Hori method.

Optimal Two-Impulse Trajectories with Moderate Flight Time for Earth-Moon Missions

A study of optimal two-impulse trajectories with moderate flight time for Earth-Moon missions is presented. The optimization criterion is the total characteristic velocity. Three dynamical models are used to describe the motion of the space vehicle: the well-known patched-conic approximation and two versions of the planar circular restricted three-body problem (PCR3BP). In the patched-conic approximation model, the parameters to be optimized are two: initial phase angle of space vehicle and the first velocity impulse. In the PCR3BP models, the parameters to be optimized are four: initial phase angle of space vehicle, flight time, and the first and the second velocity impulses. In all cases, the optimization problem has one degree of freedom and can be solved by means of an algorithm based on gradient method in conjunction with Newton-Raphson method.

Generalized canonical systems—I: General properties

Abstract In this paper some properties of dynamical systems governed by a Hamiltonian function linear in the momenta (adjoint variables in optimal control theory) are presented. These properties are defined by the general solution of the dynamical system governed by any integrable part (if exists) of the Hamiltonian function. A family of Mathieu transformations is defined between the original variables of the system and the arbitrary parameters of integration of this general solution and, also, between two different sets of arbitrary parameters. These properties may be very useful in Space Dynamics.

Optimization of Low-Thrust Limited-Power Trajectories in a Noncentral Gravity Field—Transfers between Orbits with Small Eccentricities

Numerical and first-order analytical results are presented for optimal low-thrust limited-power trajectories in a gravity field that includes the second zonal harmonic 𝐽2 in the gravitational potential. Only transfers between orbits with small eccentricities are considered. The optimization problem is formulated as a Mayer problem of optimal control with Cartesian elements—position and velocity vectors—as state variables. After applying the Pontryagin Maximum Principle, successive canonical transformations are performed and a suitable set of orbital elements is introduced. Hori method—a perturbation technique based on Lie series—is applied in solving the canonical system of differential equations that governs the optimal trajectories. First-order analytical solutions are presented for transfers between close orbits, and a numerical solution is obtained for transfers between arbitrary orbits by solving the two-point boundary value problem described by averaged maximum Hamiltonian, expressed in nonsingular elements, through a shooting method. A comparison between analytical and numerical results is presented for some maneuvers.

Generalized canonical systems—III: Space dynamics applications: Solution of the coast-arc problem

Abstract Some properties of “generalized canonical systems”—canonical systems governed by a Hamiltonian function linear in the adjoint variables (momenta)—recently discussed, are applied in the study of optimal space trajectories. A systematical integration of the canonical system which governs the null thrust arcs, in a Newtonian central force field, is performed using these properties. Expressions for the primer vector are obtained for circular, elliptical, parabolic and hyperbolic motions.

Numerical and Analytical Study of Optimal Low-Thrust Limited-Power Transfers between Close Circular Coplanar Orbits

A numerical and analytical study of optimal low-thrust limited-power trajectories for simple transfer (no rendezvous) between close circular coplanar orbits in an inverse-square force field is presented. The numerical study is carried out by means of an indirect approach of the optimization problem in which the two-point boundary value problem, obtained from the set of necessary conditions describing the optimal solutions, is solved through a neighboring extremal algorithm based on the solution of the linearized two-point boundary value problem through Riccati transformation. The analytical study is provided by a linear theory which is expressed in terms of nonsingular elements and is determined through the canonical transformation theory. The fuel consumption is taken as the performance criterion and the analysis is carried out considering various radius ratios and transfer durations. The results are compared to the ones provided by a numerical method based on gradient techniques.

A NOTE ON THE SOLUTION OF THE COAST-ARC PROBLEM

Abstract In this note the solution of the coast-arc problem in Newtonian central field derived by means of properties of generalized canonical systems is revised. A different set of orbital elements is taken as arbitrary constants of integration that provides an unified approach for elliptical, hyperbolic and parabolic motions.

OPTIMUM LOW-THRUST LIMITED POWER TRANSFERS BETWEEN NEIGHBOURING ELLIPTIC NON-EQUATORIAL ORBITS IN A NON-CENTRAL GRAVITY FIELD

A complete first-order analytical solution is developed for the problem of optimum low-thrust limited power transfers between neighbouring elliptic non-equatorial orbits in a non-central gravity field. The optimization problem is formulated as a Mayer problem of optimal control with Cartesian elements as state variables. After applying the Pontryagin maximum principle and determining the optimal thrust acceleration, an intrinsic canonical transformation is performed: the Cartesian elements are changed by suitable orbital elements. Horis method is applied in determining a first-order analytical solution. Simple analytical solutions are obtained explicitly for long-time transfers.

Optimal low-thrust transfer between neighbouring quasi-circular orbits around an oblate planet

Abstract The problem of optimal low-thrust, limited power transfer between quasi-circular orbits ( e ⋍ 0 ) around an oblate planet is analysed. It is assumed that the orbital changes due to thrust acceleration and Earth oblateness are of the same order. A first order solution to the problem is obtained by application of Pontryagins Maximum Principle. Subsequently, by application of Horis method for generalized canonical systems, a first order solution in a small parameter ϵ is derived. Finally, three particular cases of long-time transfer and the orbit maintenance manoeuvre are considered. The results obtained are in agreement and represent an extension of the work done by Marec.