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

Abstract

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 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.

1. Introduction

This paper presents numerical and analytical results for optimal low-thrust limited-power transfers between orbits with small eccentricities in a noncentral gravity field that includes the second zonal harmonic in the gravitational potential. This study has been motivated by the renewed interest in the use of low-thrust propulsion systems in space missions verified in the last two decades due to the development and the successes of two space mission powered by ionic propulsion: Deep Space One and SMART 1. Several researchers have obtained numerical and sometimes analytical solutions for a number of specific initial orbits and specific thrust profiles in central or noncentral gravity field [1–26].

It is assumed that the thrust direction is free and the thrust magnitude is unbounded [5, 27], that is, there exist no constraints on control variables. It is supposed that the orbital changes caused by the thrust and the perturbations due to the oblateness of the central body have the same order of magnitude. The optimization problem is formulated as a Mayer problem of optimal control with position and velocity vectors—Cartesian elements—as state variables. After applying the Pontryagin Maximum Principle [28], successive Mathieu transformations are performed and a suitable set of orbital elements is introduced. Hori method—a perturbation technique based on Lie series [29]—is applied in solving the canonical system of differential equations that governs the optimal trajectories. For transfers between close quasicircular orbits, new set of nonsingular orbital elements is introduced and the new Hamiltonian and the generating function, determined through the algorithm of Hori method, are transformed to these new elements through Mathieu transformations. First-order analytical solutions are obtained for transfers between close quasicircular orbits, considering maneuvers between near-equatorial orbits or nonequatorial orbits. These solutions are given in terms of systems of algebraic equations involving the imposed variations of the orbital elements, the duration of the maneuvers, the effects of the oblateness of the central body, and the initial values of the adjoint variables. The study of these transfers is particularly interesting because the orbits found in practice often have a small eccentricity, and the problem of slight modifications (corrections) of these orbits is frequently met [5]. For transfers between arbitrary orbits, a slightly different set of nonsingular elements is introduced, and the two-point boundary value problem described by averaged maximum Hamiltonian, expressed in nonsingular elements, is solved through a shooting method. A comparison between analytical and numerical results is presented for some maneuvers.

2. Optimal Space Trajectories

A low-thrust limited-power propulsion system, or LP system, is characterized by low-thrust acceleration level and high specific impulse [5]. The ratio between the maximum thrust acceleration and the gravity acceleration on the ground, , is between and . For such system, the fuel consumption is described by the variable defined as where γ is the magnitude of the thrust acceleration vector , used as control variable. The consumption variable is a monotonic decreasing function of the mass of space vehicle: where is the maximum power, and is the initial mass. The minimization of the final value of the fuel consumption is equivalent to the maximization of .

The general optimization problem concerned with low-thrust limited-power transfers (no rendezvous) will be formulated as a Mayer problem of optimal control by using Cartesian elements as state variables. Consider the motion of a space vehicle powered by a limited-power engine in a general gravity field. At time , the state of the vehicle is defined by the position vector , the velocity vector , and the consumption variable . The geometry of the transfer problem is illustrated in Figure 1. The control is unconstrained, that is, the thrust direction is free, and the thrust magnitude is unbounded.

Figure 1 

Geometry of the transfer problem.

The optimization problem is formulated as follows: it is proposed to transfer the space vehicle from the initial state at the initial time to the final state at the specified final time , such that the final consumption variable is a minimum. The state equations are where is the gravity acceleration. For transfers between orbits with small eccentricities, the initial and final conditions will be specified in terms of singular orbital elements introduced in next sections. It is also assumed that and the position of the vehicle in the initial orbit are specified.

According to the Pontryagin Maximum Principle [28], the optimal thrust acceleration must be selected from the admissible controls such that the Hamiltonian function reaches its maximum. The Hamiltonian function is formed using (2.3), where , and are the adjoint variables, and dot denotes the dot product. Since the optimization problem is unconstrained, is given by The optimal thrust acceleration is modulated [5], and the optimal trajectories are governed by the maximum Hamiltonian function , obtained from (2.4) and (2.5):

The consumption variable is ignorable, and is a first integral. From the transversality conditions, ; thus, Equation (2.6) reduces to The consumption variable is determined by simple quadrature.

For noncentral field, the gravity acceleration is given by [30] with the disturbing function defined by where is the gravitational parameter, is the mean equatorial radius of the central body, is the coefficient for the second zonal harmonic of the potential, is the latitude, and is the Legendre polynomial of order 2. is a row vector of partial derivatives, and superscript denotes its transpose.

Using (2.8) and (2.9), the maximum Hamiltonian function can be put in the following form: where is the undisturbed Hamiltonian function, and are disturbing functions concerning the oblateness of the central body and the optimal thrust acceleration, respectively. and are assumed to be of the same order of magnitude.

3. Transformation from Cartesian Elements to a Set of Orbital Elements

Consider the canonical system of differential equations governed by the undisturbed Hamiltonian : where is the unit vector pointing radially outward of the moving frame of reference (Figure 2). The general solution of the state equations is well known in Astrodynamics [31], and the general solution of the adjoint equations is obtained through properties of generalized canonical systems [32], as described in the appendix. Thus, where and are unit vectors along circumferential and normal directions of the moving frame of reference, respectively; is the semimajor axis, is the eccentricity, is the inclination of orbital plane, is the longitude of the ascending node, is the argument of pericenter, is the true anomaly, is the eccentric anomaly, is the mean anomaly, is the mean motion, and , and so forth are functions of the elliptic motion which can be expressed explicitly in terms of the eccentricity and the mean anomaly through Lagrange series [31]. The true, eccentric and mean anomalies are related through the equations: The unit vectors , and of the moving frame of reference are written in the fixed frame of reference as

Figure 2 

Frames of reference.

Equations (3.2)–(3.5) define a Mathieu transformation between the Cartesian elements and the orbital ones . Since the Hamiltonian function is invariant with respect to this canonical transformation, it follows, from (2.11) through (3.5), that where with the disturbing function given by

The new Hamiltonian function , defined by (3.8)–(3.10), describes the optimal low-thrust limited-power trajectories in a noncentral gravity field which includes the perturbative effects of the second zonal harmonic of the gravitational potential.

4. Averaged Maximum Hamiltonian for Optimal Transfers

In order to eliminate the short periodic terms from the maximum Hamiltonian function , Hori method [29] is applied. It is assumed that is of zero-order and is of the first-order in a small parameter associated to the magnitude of the thrust acceleration, and has the same order of .

Consider an infinitesimal canonical transformation: The new variables are designated by the prime. According to the algorithm of Hori method [29], at order 0, denotes the new undisturbed Hamiltonian. Now, consider the canonical system described by : general solution of which is given by The subscript 0 denotes the constants of integration.

Introducing the general solution defined above into the equation of order 1 of the algorithm of Hori method, it reduces to According to [29], the mean value of must be calculated, and, is obtained through integration of the remaining part. and are given by the following equations: Terms factored by have been omitted in equations above, since only transfers (no rendez-vous) are considered [2].

It should be noted that the averaged maximum Hamiltonian and the generating function become singular for circular and/or equatorial orbits. In order to avoid these singularities, suitable sets of nonsingular elements will be introduced in the next sections.

5. Optimal Transfers between Close Quasicircular Orbits

In this section, approximate first-order analytical solutions will be obtained for transfers between close quasicircular orbits.

5.1. Transfers between Close Quasicircular Nonequatorial Orbits

Consider the Mathieu transformation [33] defined by The new set of canonical variables, designated by the double prime, is nonsingular for circular nonequatorial orbits.

Introducing (5.2) into (4.6) and using the expansions of the elliptic motion in terms of eccentricity and mean anomaly [31], one finds that the new averaged maximum Hamiltonian and the generating function are written up to the zeroth order in eccentricity as Double prime is omitted to simplify the notation.

For transfers between close orbits, and can be linearized around a suitable reference orbit, and an approximate first-order analytical solution can be determined. This solution is given by where is the vector of initial value of the adjoint variables, is a symmetric matrix concerning the optimal thrust acceleration and is a vector containing the perturbative effects of the oblateness of the central body. The variables and are introduced to make the adjoint and the state vectors dimensionless, and the variables and are introduced to simplify the matrix . The adjoint vector is constant. The matrix and the vector are given by where with is the initial time, is the final time, and . The overbar denotes the orbital elements of the reference orbit.