Lagrangian multipliers for coast-arcs of optimum space trajectories

Lagrangian Multipliers for Coast-Arcs of Optimum Space Trajectories

Sandro da Silva Fernandes

Departamento de Matemática

Instituto Tecnológico de Aeronáutica

12228-900 São José dos Campos, SP. Brazil

[e-mail]

Some properties of generalized canonical systems - special dynamical systems described by a Hamiltonian function linear in the adjoint variables - are applied in determining the solution of the two-dimensional coast-arc problem in an inverse-square gravity field. A complete closed-form solution for Lagrangian multipliers - adjoint variables - is obtained by means of such properties for elliptic, circular, parabolic and hyperbolic motions. Classic orbital elements are taken as constants of integration of this solution in the case of elliptic, parabolic and hyperbolic motions. For circular motion, a set of nonsingular orbital elements is introduced as constants of integration in order to eliminate the singularity of the solution.

Keywords: Optimal space trajetories, coast-arc problem, Lagrangian multipliers

Introduction

The coast-arc problem (Lawden, 1954) that defines the optimum trajectory of a constant exhaust velocity space vehicle with bounded or unbounded thrust magnitude is described by a special class of system of differential equations, termed "generalized canonical systems" (Da Silva Fernandes, 1994a). Such systems described by a Hamiltonian function linear in the momenta (adjoint variables in optimal control theory) have intrinsic properties concerning the Mathieu transformations defined by the general solution of the system of differential equations governed by the integrable kernel

_o of the Hamiltonian. The main feature of this general solution is that the solution of the differential equations for the momenta involves the calculation of the Jacobian matrix of the point transformation defined by the general solution of the differential equations for the generalized coordinates.

The coast-arc problem is an old problem in Astrodynamics and was solved by Lawden (1954, 1963) by means of direct integration of the adjoint differential equations; by Eckenwiller (1965) and Hempel (1966) through methods involving numerous integrations; by Marec (1979) through the set of first integrals of motion derived by Pines (1964), Edelbaum and Pines (1970), and Moyer (1969); and, by Powers and Tapley (1969) and Popescu (1997) through canonical transformation theory. However, all of these methods involves more algebraic calculations than the approach presented in this paper.

By using the properties of generalized canonical systems, integration of the system of differential equations describing the coast-arc in a Newtonian central force field - inverse-square gravity field - is performed. As will be shown, the generalized canonical approach requires the evaluation of only one integral, closely related to the classic Kepler’s equation.

A complete closed-form solution will be obtained for elliptic, circular, parabolic and hyperbolic motions. Classic orbital elements are taken as constants of integration of this solution in the case of elliptic, parabolic and hyperbolic motions. For circular motion, the solution given in terms of classic orbital elements becomes singular, and a new set of orbital elements - nonsingular elements - is introduced as constants of integration of the general solution of the system of differential equations.

The coast-arc problem will be formulated as proposed by Powers and Tapley(1969), considering a two-dimensional fligth in a inverse-square gravity field with the radial distance, the radial and circumferential components of the velocity and a polar angle as state variables. The three-dimensional case was previously discussed, considering a vector formulation of the problem with position and velocity vectors as state variables, as well as a different set of orbital elements as constants of integration(Da Silva Fernandes, 1994b, 1994c, 1999a).

In contrast with the most other methods cited above, the generalized canonical approach presented in the paper can be applied in the study of the coast-arc problem considering the perturbative effects of additional forces - drag perturbations, effects of oblateness of the central body,...etc. -, by using the canonical perturbation methods of Celestial Mechanics, for which the solution in an inverse-square gravity field plays a fundamental role.

For completeness, statements of three important propositions about generalized canonical systems are presented in the Appendix.

Coast-Arc Problem

Let us consider the two-dimensional motion of a space vehicle powered by a constant exhaust velocity engine in a Newtonian central force field with the center of attraction at O (Figure 1). 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, known as Lawden problem, is formulated as follows.

It is proposed to transfer the space vehicle from the initial state at the time t₀ to the final state at the time t_f , such that the final characteristic velocity 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_n ,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 = g cosf and S = g sinf, 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 fuction 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). In this case, the Maximum Principle does not allow one to obtain the optimal thrust acceleration.

The optimal trajectories are then 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.

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

with the state equations defined by Eqn (3) and the adjoint equations given by

From the last differential equation in Eqn (12) and the transversality condition, it follows that p_C = - 1. On the other hand, C = constant during a coast-arc. Accordingly, the order of the system of differential equations defined by the Hamiltonian H, Eqn (11), is reduced. The general solution of this reduced system will be presented in what follows.

The general solution of the state equations, Eqns (3), is well-known from the classic Two-Body Problem (Bate et al., 1971),

where p is the semi-latus rectum, e is the eccentricity, w is the pericenter argument and f is the true anomaly. According to Propositon 3 (Appendix) f is the fast phase of the system.

The general solution of the adjoint equations is obtained through the evaluation of the inverse of the Jacobian matrix of the point transformation defined by Eqns (13), or equivalently, the Jacobian matrix of the inverse transformation (Appendix, Proposition 2).

Let us consider the inverse of the point transformation defined by Eqn (13),

Therefore, the partial derivatives are:

the other partial derivatives are null.

From Eqns (5) and (15), it follows that

where (p_p, p_e ,p_f ,p_w) are adjoint variables to _(p,e,f,w).

Following Proposition 2, the new Hamiltonian function resulting from the Mathieu (extended point) transformation defined by Eqns (14) and (16) is

The general solution of the new state equations defined by is very simple,

where p₀, e₀, f₀ and w₀ are arbitrary constants of integration, t₀ is the initial time and J(f) is defined by

For t₀ equal to the time of pericenter passage t, the time equation in Eqn (18) reduces to the classic Kepler’s equation for elliptic orbits, if we introduce the eccentric anomaly E and the mean anomaly M,

where n is the mean motion. Similar results are obtained for parabolic and hyperbolic motions.

Following Proposition 1, the general solution of the new adjoint equations are obtained through the evaluation of the partial derivatives of the point transformation defined by Eqns (18). Therefore,

where

with

F is the hyperbolic eccentric anomaly and D is the parabolic eccentric anomaly.

Thus, taking t₀ = t , we get

Therefore, from Eqns (16), (18) and (28), it follows that

Note that the subscript ‘0’ denoting the constants of integration in the right-hand side of the equations above is omitted for simplicity.

We note that the adjoint variables p_u and pv are, respectively, the radial and circumferential components of the "primer" vector p_vintroduced by Lawden (1954) in the analysis of optimal space trajectories,

where e_rand e_sare the unit vectors along the radial and circumferential directions, respectively, of a moving frame of reference (Fig.1). Introducing Lawden’s constants and the functions I₁ and I₂ , the "primer" vector can be put in the form

with

with I(f) defined by Eqn (23)-(25) according to the type of motion: elliptic, hyperbolic or parabolic. For simplicity, the subscript denoting the constants of integration are omitted in Eqns (34) - (40). Lawden’s constants and functions I₁ and I₂ can be rewritten in a particular form for each type of motion if suitable sets of orbital elements are introduced (Da Silva Fernandes, 1999a).

It should be noted that Eqns (29) - (32) have singularities for circular motion, e = 0. In order to avoid this singularity, a new set of orbital elements - nonsingular elements - will be introduced. Following Propositon 3, new expressions for the adjoint variables ( p_r ,p_u ,p_n ,p_q) will be derived.

Elimination of Singularity

In this section, a set of nonsingular orbital elements is introduced to eliminate the singularity of the circular case (Da Silva Fernandes, 1999b).

Let us consider the following set of orbital elements:

The prime denotes the new variable. These elements are usually known as equitoctial elements (Broucke and Cefola, 1972).

According to Propositon 3, (p', h, k, ) define a new set of arbitrary parameters of integration of the general solution of state equations; that is,

To obtain the adjoint variables ( p_r ,p_u ,p_n ,p_q) in terms of the new set of orbital elements and adjoint variables, we proceed as described below. First, we express the old adjoint variables (p_p, p_e ,p_f ,p_w) in terms of the new set of canonical variables (p', h, k, ,p_p',p_h ,p_k , ) .

From Eqns (41), we evaluate the partial derivatives:

the other partial derivatives are null.

Therefore, from Eqns (43) and Proposition 3 (Appendix), we get

For simplicity, the right-hand side of Eqns (44) is written in terms of the old state variables.

Now, from Eqns (16), (42) and (44), it results:

Equations (42) and (45) define a time independent Mathieu (extended point) transformation between the canonical variables (r ,u ,v , q, p_r ,p_u ,p_n ,p_q) and the canonical set of nonsingular elements (p', h, k, ,p_p',p_h ,p_k , ),

The new Hamiltonian function resulting from this canonical transformation is

For simplicity, the prime denoting the new variable p' will be omitted in the following paragraphs.

The general solution of the canonical system defined by the Hamiltonian function is obtained using the same approach presented in the preceeding section. This general solution is given by:

for the state equations, and,

for adjoint equations, where is the true longitude of pericenter.

For elliptic motion, Kepler’s equation can be put in the form

that, in view of Eqns (41), can be rewritten as

Therefore, from Eqns (45), (48) and (49), it follows that:

Except for the adjoint variables to the nonsingular orbital elements and the true longitude of pericenter, the subscript denoting the constants of integration is omitted. The function of the elliptic motion is given by

For circular motion, Eqns (50) - (53) simplify,

For circular motion, means the initial position of the space vehicle in orbit.

Introducing new constants, , and functions F, G, I₃ and I₄, the "primer" vector can be written in nonsingular elements as follows:

with

For circular motion, Eqns (59)-(67) simplify and the "primer" vector can be put in the form:

where a is the radius of circular orbit, and

Appendix

In what follows, properties of generalized canonical systems are stated. For details see previous work (Da Silva Fernandes, 1994a). Only integrable systems will be considered in following discussions.

Let us consider the generalized canonical system,

governed by the Hamiltonian function H,

where x is the n-dimensional state (generalized coordinates) vector and l is the n-dimensional adjoint (momenta) vector. Dot denotes the scalar vector product and T denotes the transposed matrix. The integration of this system of differential equations splits up into two parts: firstly, one performs the integration of the first set of differential equations - state equations - and, with the known solution of these equations, one performs the integration of the second set of differential equations - adjoint equations.

Since, by hypothesis, the system is completely integrable, follows from the Hamilton-Jacobi theory (Lanczos, 1970; Arnold, 1987; Goldstein, 1980) that

where S is the generating function of a canonical transformation that reduces the Hamiltonian H to a constant Hamiltonian equal to zero. A complete solution S(c₀, l, t) of Eqn (A.3) is given by

where c₀ is n-dimensional vector of arbitrary constants of integration corresponding to the new state variables.

The transformation of variables defined by the generating function S(c₀, l , t) is given by (Goldstein, 1980)

where is the Jacobian matrix,

and b₀ is n-dimensional vector of arbitrary constants of integration corresponding to the new adjoint variables.

The general solution of Eqn (A.1) is then given by

This general solution can also be interpreted in a different way. Eqns (A.7) define a time dependent Mathieu (extended point) transformation between the variables (x;l) and the new ones (c₀; b₀). This new result can be proved by applying Delaunay-Poincaré condition (Lanczos, 1970; Arnold, 1987; Goldstein, 1980),

For periodic system, the general solution can be expressed by

where q is the fast phase, (c₀; b₀) are arbitrary constants of integration, and B is a n-dimensional vector involving n arbitrary constants of integration and the fast phase q. D_Y is the Jacobian matrix

Eqns (A.8) are obtained in the same way described in the preceding paragraphs, considering that in the periodic case, the Hamilton-Jacobi equation assumes the form

H = ,

with the new Hamiltonian = WB₀, where W is the frequency of the fast phase q. The generating function is then given by S(C,l) = -l^Ty(C), where C = [C₁C₂C_n-1 q]^T. Note that in this case, the integration is not complete.

If the general solution of the system defined by Eqn (A.1) is expressed in terms of another set of arbitrary constants of integration b₁ and c₁, that is

then there exists the following relation between the two sets of arbitrary constants of integration,

where is the Jacobian matrix

which is given in terms of Jacobian matrices and by

Similar result holds for periodic systems. These results are consequence of group property of canonical transformations (Lanczos, 1970).

It should be noted that, in all cases above, the general solution of adjoint equations is obtained from the general solution of the state equations through the evaluation of the Jacobian matrix D_j, or more precisely, of its inverse.

Accordingly, the following propositions can be stated from the previous results:

Proposition 1: A time dependent Mathieu (extended point) transformation between the variables (x;l) of the system and the arbitrary constants of integration (c₀; b₀) is defined by Eqn (A.5),

This canonical transformation reduces the Hamiltonian function H to a constant Hamiltonian equal to zero.

Proposition 2: For periodic system, a time independent Mathieu (extended point) transformation between the variables (x;l) of the system and the arbitrary parameters (constants and fast phase) of integration (C₁, C₂, ...,C_n-1, q; B₁, B₂, ..., B_n-1, B_q) is defined by Eqn (A.8),

The new Hamiltonian function resulting from this canonical transformation only involves the frequency of the fast phase q and the adjoint variable to q,

The frequency can be a function of the arbitrary constants C₁, C₂, ...,C_n-1.

Proposition 3: A time independent Mathieu (extended point) transformation between two different sets of arbitrary constants of integration (c₀; b₀) and (c₁; b₁) is defined by Eqn (A.11),

Similar result holds for periodic systems with general solution involving a fast phase.

These propositions provides the basis of the analysis of the coast-arc problem.

Conclusion

A closed-form solution of the coast-arc problem in Newtonian central force field is obtained applying properties of generalized canonical systems for elliptic, circular, parabolic and hyperbolic motions. This generalized canonical approach that involves a set of Mathieu transformations, requires the evaluation of only one integral related to the classic Kepler’s equation. Equations describing Lawden’s "primer" vector are presented in singular and non-singular orbital elements.

The generalized canonical approach presented in the paper can be applied to study more general problems that includes effects of additional forces - drag perturbations, effects of oblatenesse of the central body,.etc. - using the canonical perturbation methods of Celestial Mechanics.

Acknowledgments

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

Manuscript received: July 1999, Technical Editor: Hans Ingo Weber.

• Arnold, V.I., 1987, "Métodos Matemáticos da Mecânica Clássica", Editora Mir, Moscow, 479 p.
• Bate, R.R., Mueller, D.D. and White, J.E., 1971, "Fundamentals of Astrodynamics", Dover, New York, USA, 455 p.
• Broucke, R.A. and Cefola, P.J.,1972, "On the Equinoctial Orbit Elements", Celest. Mech., Vol. 5, pp 303-310.
• Da Silva Fernandes, S., 1994a, "Generalized Canonical Systems - I. General Properties", Acta Astronautica, Vol. 32, No.5, pp. 331-338.
• Da Silva Fernandes, S., 1994b, "Generalized Canonical Systems - III. Space Dynamics Applications: Solution of the Coast-Arc Problem", Acta Astronautica, Vol. 32, No.5, pp. 347-354.
• Da Silva Fernandes, S., 1994c, "Applications of Generalized Canonical Systems in the Study of Optimal Space Trajectories", RBCM - J. of Brazilian Soc. of Mechanical Sciences XVI - Special Issue, pp. 552-558.
• Da Silva Fernandes, S., 1999a, "A Note on the Solution of the Coast-Arc Problem", Acta Astronautica, Vol. 45, No.1, pp. 53-57.
• Da Silva Fernandes, S., 1999b, "Generalized Canonical System Applications to Optimal Trajectory Analysis", Journal of Guidance, Control and Dynamics, Vol. 22, No.6, pp. 918-921.
• Eckenwiler, M.W., 1965, "Closed-Form Lagrangian Multipliers for Coast Periods of Optimum Trajectories", AIAA Journal, Vol. 3, No.6, pp.1149-1151.
• Edelbaum, T.N. and Pines, S., 1970, "Fifth and Sixth Integrals for Optimum Rocket Trajectories in a Central Field", AIAA Journal, Vol. 8, No.7, pp.1201-1204.
• Goldstein, H., 1980, "Classical Mechanics", Addison-Wesley Publishing Company, Reading, 672 p.
• Hempel, P.R., 1966, "Representation of the Lagrangian Multipliers for Coast Periods of Optimum Trajectories", AIAA Journal, Vol. 4, No.4, pp. 729-730.
• Lanczos, C., 1970, "The Variational Principles of Mechanics", Dover, New York, 418 p.
• Lawden, D.F., 1954, "Fundamentals of Space Navigation", Journal of Britsh Interplanetary Society, Vol.13, pp. 87-101.
• Lawden, D.F., 1963, "Optimal Trajectories for Space Navigation", Butterworths, London.
• Marec,J.P., 1979, "Optimal Space Trajectories", Elsevier, Amsterdam, The Netherlands, 329 p.
• Moyer, H.G.,1969, "Integrals for Impulsive Orbit Transfer from Noether’s Theorem", AIAA Journal, Vol. 7, No.7, pp. 1232-1235.
• Pines, S., 1964, "Constants of Motion for Optimum Thrust Trajectories in a Central Gravity Field", AIAA Journal, Vol. 2, No.11, pp. 2010-2014.
• Pontryagin, L.S., Boltyanskii,V.G., Gamkrelidze, R.V., Mischenko, E.F. ,1962 , "The Mathematical Theory of Optimal Process", Interscience Publishers Wiley, New York, USA, 362 p.
• Popescu, M., 1997, "Applications of Canonical Transformations in Optimizing Orbital Transfers", Journal of Guidance, Control and Dynamics, Vol. 20, No.4, pp. 774-779.
• Powers, W.F., and Tapley, B.D., 1969, "Canonical Transformation Applications to Optimal Trajectory Analysis", AIAA Journal, Vol.7, No.3, pp. 394-399.

Publication Dates

• Publication in this collection
  24 Oct 2001
• Date of issue
  2001

History

• Received
  July 1999