Universal closed-form of lagrangian multipliers for coast-arcs of optimum space trajectories

Universal closed-form of lagrangian multipliers for coast-arcs of optimum space trajectories

S. da S. Fernandes

Departamento de Matemática Instituto Tecnológico de Aeronáutica 12228-900 São José dos Campos, SP. Brazil [e-mail]

ABSTRACT

An universal closed-form solution of Lagrangian multipliers for the coast-arcs of optimum space trajectories in a Newtonian central force field is obtained by means of properties of generalized canonical systems and Sundman transformation. This closed-form solution, valid for all conics, is given as a function of a generalized anomaly.

Keywords: Optimal space trajectories, coast-arc problem, Lagrangian multipliers, universal variables, Sundman transformation

Introduction

The coast-arc problem is fundamental in the theory of optimal space trajectories of a constant exhaust velocity space vehicle with bounded or unbounded thrust magnitude (Marec, 1979); its solution defines the evolution of the Lawden’s primer vector (Lawden, 1954) on a coast-arc and describes the connection between the cut-off and restart conditions of the thruster. This important problem of Astrodynamics has been solved by several researchers through different formulations and methods involving integrations (Lawden, 1954; Eckenwiller, 1965; Hempel, 1966), canonical transformation theory (Powers and Tapley, 1969; Popescu, 1997), set of first integrals of motion (Marec, 1979), or properties of generalized canonical systems (Da Silva Fernandes, 1994a, 1994b, 1999a, 1999b, 2001). Expressions for the primer vector have been obtained for circular, elliptic, hyperbolic and parabolic motions, considering suitable sets of orbital elements. In this work, an universal closed-form solution is derived by using the Sundman transformation (Battin, 1987) and the generalized canonical approach. The coast-arc is formulated as proposed by Powers and Tapley (1969) through a two-dimensional formulation of the equations of motion and a closed-form solution, valid for all conics, is obtained as function of a generalized anomaly. Simplifications for near parabolic orbits are also presented.

Coast-Arc Problem

For completeness, previous results about the coast-arc problem are presented in this section (Da Silva Fernandes, 1999b).

Let us consider the two-dimensional motion of a space vehicle M powered by a constant exhaust velocity engine in a Newtonian central force field with the center of attraction at O. At time t, the state of the vehicle is defined by the radial distance r from the center of attraction; the radial and circumferential components of the velocity, u and v, respectively; the polar angle q , measured from any convenient reference line through the center of attraction and the characteristic velocity C defined by

where g denotes the magnitude of the thrust acceleration g, used as control variable and subject to the constraint

The optimal trajectory problem is formulated as: It is proposed to transfer the space vehicle M from the initial state (r₀,u₀,v₀,q₀,0) at the time to t₀ the final state (r_f,u_f,v_f,q _f,C_f) at the time t_f , such that the final characteristic velocity C_f is a minimum.

In the two-dimensional formulation (Powers and Tapley,1969), the well-known equations of motion in polar coordinates are:

where m is the gravitational parameter, R and S are the radial and circumferential components of the thrust acceleration, respectively.

Following the Pontryagin Maximum Principle (Pontryagin et al.,1962), the adjoint variables (p_r, p_u, p_v, p_q,p_C) are introduced and the Hamiltonian function H is formed using Eqns (3),

The optimal thrust acceleration g is selected from the class of admissible controls, at each time, so that the Hamiltonian function H is a maximum. Therefore, taking R= gcosf and S= gsinf, where f defines the thrust direction from the local vertical, it follows that:

From Eqns (4) and (5), the Hamiltonian function reduces to

which is maximum with respect to g , subject to the constraint defined by Eqn (2), for:

The function Q is called switching function and is defined by

The optimal control law is "bang-bang", i.e. alternating maximum thrust arcs - MT and null thrust (coast) arcs - NT, except in the singular case (Marec, 1979).

The optimal trajectories are governed by the maximized Hamiltonian function H^*,

where

It is seen that the Lagrangian multipliers p_u and p_v define the optimal thrust acceleration. The evolution of these multipliers on a null (coast) arc is fundamental because they define the connection between the cut-off and restart conditions of the engine as defined by Eqn (7). We note that the adjoint variables - Lagrangian multipliers - p_u and p_v are the radial and circumferential components of the "primer" vector p_v introduced by Lawden (1954) in the analysis of optimal space trajectories,

where e_r and e_s are the unit vectors along the radial and circumferential directions, respectively, of a moving frame of reference.

According to the Eqns (9) and (10), the coast-arc problem is described by the following Hamiltonian function:

The general solution of the system of differential equations governed by the Hamiltonian H can be obtained through the properties of generalized canonical systems and is given by (Da Silva Fernandes, 1999b, 2001)

where p is the semi-latus rectum, e is the eccentricity, w is the pericenter argument and f is the true anomaly (fast phase) and (p_p, p_e, p_f, p_w) are adjoint variables to (p, e, f, w) . Along a coast-arc C and p_C are ignorable variables. It should be noted that Eqns (12) define a Mathieu transformation between the Cartesian elements and the orbital ones.

The new Hamiltonian function resulting from the Mathieu (extended point) transformation defined by Eqns (12) is

The general solution of the system of differential equations governed by the new Hamiltonian function (13) is very simple (Da Silva Fernandes, 1999b, 2001). Nevertheless, this solution has distinct forms according to the type of motion: elliptic, parabolic and hyperbolic. This trouble is intrinsically related to different time-of-flight equations for elliptic, parabolic and hyperbolic orbits. This drawback is overcome in a reformulation of the time-of-flight equation by the introduction of a generalized eccentric anomaly that allow to derive an universal solution for the coast-arc problem valid for all orbits. In next sections, the change of variable, known as Sundman transformation (Battin, 1987), is introduced and applied in deriving an universal solution.

Sundman Transformation

The time-of-flight equation for each kind of conic is obtained from the integration of the differential equation

by introducing the eccentric anomaly E for elliptic orbit, the hyperbolic eccentric anomaly F for hyperbolic orbit and the parabolic eccentric anomaly D for parabolic orbit through the differential relationships (Battin, 1987),

where .

Therefore, from Eqns (14) and (15), it follows that

where c is the generalized anomaly.

The transformation defined by

is known as Sundman transformation (Battin, 1987).

Taking t₀ equal to the time of pericenter passage t and c (t) = 0, it follows that

Accordingly, the radial distance and the time-of-flight are given by

where a is the reciprocal of a, i.e.

and may be positive, negative or zero, according to the kind of conic: ellipse, hyperbola or parabola, respectively. Introducing the universal functions U_n(c,a), n=0, 1, 2, 3, Eqns (19) and (20) can be put in a form valid for all conics,

where

The variables a and c are called universal variables and will be applied in the next section to get an universal solution of the coast-arc problem.

Universal Solution

The solution of the coast-arc problem in universal variables is obtained as described below. The general solution of the state equations is rewritten in terms of the universal variables a and c. For simplicity, the elliptic case will be taken to perform the transformation of variables. Thus,

According to the properties of generalized canonical systems (Da Silva Fernandes, 1994a), Eqns (26) involve a new set of arbitrary parameters of integration - a, e, w and c - such that an intrinsic transformation of variables between the old set of parameters - p, e, w and f - and the new one is defined

Prime is used to denote the new variables.

The transformation between the old and new adjoint variables is obtained through the evaluation of the inverse of the Jacobian matrix of the point transformation defined by Eqns (27), and is given by

Equations (12), (27) and (28) define two Mathieu (extended point) transformations,

Therefore, the general solution of the adjoint equations governed by the Hamiltonian function H, defined by Eqn (11), is given in terms of the universal variables by

Here and are given as functions of the new variables through Eqn (20). It should be noted that Eqns (26) and (29) are expressed in terms of the fast phase c. In order to obtain the complete integration of the system of differential equations governed by the Hamiltonian function H, we proceed as described below.

According to the group property of canonical transformations, Eqns (26) and (29) define a new canonical - Mathieu - transformation. The Hamiltonian function resulting from this new canonical transformation is given by

The general solution of the new state equations governed by the Hamiltonian H is

where

The general solution of the new adjoint equations is obtained through the evaluation of the inverse of the Jacobian matrix of the point transformation defined by Eqns (31), and is given by

Note that t₀=t and c(t)=0.

Introducing Eqns (31) and (33) into Eqns (29), and using the universal functions U_n (c,a), n = 0, 1, 2, 3, defined by Eqns (24) - (25), one gets:

These equations are valid for all conics. For simplicity, the subscript denoting the constants of integration is omitted.

For near parabolic orbits, a ® 0, Eqns (34) can be greatly simplified expanding in Taylor series the U functions and truncating the series at the first terms. Accordingly,

As described in the Section 2, the Lagrangian multipliers p_u and p_v define the "primer" vector p_v. Thus,

for all conics, and

for near parabolic orbits.

Conclusions

In this paper, an universal closed-form of Lagragian multipliers for coast-arcs of optimum space trajectories of a constant exhaust velocity space vehicle, valid for all conics, is presented. This closed-form of Lagragian multipliers has been obtained by means of properties of generalized canonical systems and the Sundman transformation. Expressions of Lawden’s "primer" vector are also presented, including simplifications for parabolic orbits.

Acknowledgements

This research has been supported by CNPq under contract 300081/1994-4.

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Publication Dates

• Publication in this collection
  18 Mar 2004
• Date of issue
  Dec 2003