J. Braz. Soc. Mech. Sci. & Eng. vol.25 número4

S. da S. Fernandes

Departamento de Matemática Instituto Tecnológico de Aeronáutica 12228-900 São José dos Campos, SP. Brazil sandro@ief.ita.cta.br

Universal Closed-Form of Lagrangian

Multipliers for Coast-Arcs of Optimum

Space Trajectories

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 transclosed-formation (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).1

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 θ , measured from any convenient reference line through the center of attraction and the characteristic velocity C defined by

∫

= tf

t dt C

0

γ , (1)

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

max

γ γ≤ ≤

0 . (2)

Paper accepted October, 2003. Technical Editor: Atila P. Silva Freire.

The optimal trajectory problem is formulated as: It is proposed to transfer the space vehicle M from the initial state (r₀,u₀,v₀,θ₀,0)

at the time t₀ to the final state (r_f,u_f,v_f,θ_f,C_f) at the time f

t , 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:

u dt dr ₌

R r r v dt

du _{= − +}

2

2 _µ

S r uv dt

dv_{=− +}

r v dt d

=

θ

γ

=

dt dC

, (3)

whereµ 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 (

π

r,

π

u,

π

v,

π

θ,

π

C) are introduced

and the Hamiltonian function H is formed using Eqns (3),

C v

u r

r v S r uv R

r r v u

H π µ π ⎟π + π_θ +γπ

⎠ ⎞ ⎜

⎝ ⎛_{− +} + ⎟ ⎟ ⎠ ⎞ ⎜

⎜ ⎝ ⎛

+ − + =

2 2

. (4)

The optimal thrust acceleration γ is selected from the class of admissible controls, at each time, so that the Hamiltonian function H is a maximum. Therefore, taking R=γcosφ and S=γsinφ, where

φ defines the thrust direction from the local vertical, it follows that:

(

_{2 2}

)

1/2

* sin

v u

v

π π

π φ

+

= ,

(

_{2 2}

)

1/2

* cos

v u

u

π π

π φ

+

= . (5)

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

(

)

⎟

⎠ ⎞ ⎜

⎝

⎛ _{+ +}

+ + − ⎟ ⎟ ⎠ ⎞ ⎜

⎜ ⎝ ⎛

− +

= r u v u v C

r v r uv

r r v u

H π µ π π π_θ γ π 2 π 21 2 π

2

2 _/

,

(6)

(i) Θ<0→ γ*=0: null thrust arc;

(ii) Θ>0→ γ*=γmax: maximum thrust arc;

(iii) 0Θ≡ , during a finite interval of time → γ*=γ_intermediate:

singular arc. (7)

The function Θ is called switching function and is defined by

(

πu +πv

)

+πC

=

Θ 2 21/2

. (8)

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*,

) , , (

* _{r u v max u v C}

r v r uv

r r v u

H π µ _⎟⎟π − π + π_θ +γ Vπ π π

⎠ ⎞ ⎜

⎜ ⎝ ⎛

− + =

2 2

,

(9)

where

(

)

⎪ ⎪ ⎩ ⎪⎪ ⎨ ⎧

> Θ + +

< Θ =

0 0 0

2 1 2

2 _,

, )

, , (

/ C v

u C v u

π π

π π π π

V (10)

It is seen that the Lagrangian multipliers πu and πv 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 - π_u and π_v are the radial and circumferential components of the “primer” vector pv introduced by Lawden (1954) in the analysis of optimal space trajectories,

s r

v e e

p =πu +πv

,

where er and es 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:

θ

π π π µ π

r v r uv

r r v u

H r _⎟⎟ u− v+

⎠ ⎞ ⎜

⎜ ⎝ ⎛

− +

= 2 ₂ . (11)

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)

f e

p r

cos 1+

=

f e p u= µ sin

(

e f

)

p v= µ 1+ cos

f

+ =ω θ

(

π πω

)

π π

πr= p+ + e− f −

re f r

e f r

p cos sin

2

(

π πω

)

µ π µ

π_u = _e+ _f −

e f p f

p cos

sin

(

)

(

π πω

)

µ

π µ

π µ π

− ⎥ ⎦ ⎤ ⎢ ⎣ ⎡

+ −

+ +

+ =

f

e p

v

p r e

f p

p r e f e f p r

p

1 sin

cos cos 2

2 2

ω

θ π

π = , (12)

where p is the semi-latus rectum, e is the eccentricity, ω is the pericenter argument and f is the true anomaly (fast phase) and

) , , ,

(π_p π_e π_f π_ω are adjoint variables to (p,e,f,ω). Along a coast-arc C and πC 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

f r

p_π

µ

2

=

H . (13)

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

2 r

p dt

df µ

= , (14)

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),

⎪ ⎪ ⎩ ⎪⎪ ⎨ ⎧ =

bdF bdE dD p

r

df 1 , (15)

where b=a 1−e2 .

( )

(

)

χ

µ rd

F a d

E a d

dD r

dt =

⎪ ⎩ ⎪ ⎨ ⎧

−

= , (16)

where χ is the generalized anomaly. The transformation defined by

r d

dt

=

χ

µ , (17)

is known as Sundman transformation (Battin, 1987).

Taking t₀ equal to the time of pericenter passage τ and 0

) (τ =

χ , it follows that

⎪ ⎩ ⎪ ⎨ ⎧

− =

F a

E a D

χ . (18)

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

(

)

( )

(

)

(

)

(

)

⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧

− −

− + =

α χ α

α χ α

χ

cosh 1 1

cos 1 1

2

1 2

e e

p

r , (19)

(

)

( )

(

)

⎪ ⎪ ⎪ ⎪ ⎪

⎩ ⎪⎪ ⎪ ⎪ ⎪

⎨ ⎧

− −

− −

⎟ ⎟ ⎠ ⎞ ⎜

⎜ ⎝ ⎛

+ =

−

α χ α α α χ

α χ α α α χ

χ χ

τ µ

sinh sin 6 2

3

e e p

t , (20)

where α is the reciprocal of a, i.e.

µ α 1 2 v2

r a= −

= , (21)

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

) , ( 1 ) ,

( 0

2 χ α U χ α

e p U

r

+ +

= , (22)

( )

( , )

1 ) ,

( ₁

3 χ α χ α

τ

µ U

e p U

t

+ + =

− . (23)

where

( )

(

)

⎪ ⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎨ ⎧

− =

α χ

α χ α

χ

cosh cos

1

) , (

0

U ,

( )

(

)

⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧

− − =

α α χ

α α χ

χ

α χ

sin sin ) , (

1

U , (24)

( )

(

)

⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧

− −

− =

α α χ α

α χ χ

α χ

cosh 1

cos 1

2

) , (

2

2

U ,

( )

(

)

⎪ ⎪ ⎪ ⎪ ⎪

⎩ ⎪⎪ ⎪ ⎪ ⎪

⎨ ⎧

− − −

− − =

α α

α χ α χ

α α

α χ α χ

χ

α χ

sinh sin

6

) , (

3

3

U ,

(25)

The variables α and χ 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 αand χ. For simplicity, the elliptic case will be taken to perform the transformation of variables. Thus,

(

)

(

χ α

)

α 1 cos

1 e

r = −

(

)

α µ 2

1

1 e

r

v = −

( )

χ α α

µ

sin 1

e r

u= θ=ω+ f. (26)

According to the properties of generalized canonical systems (Da Silva Fernandes, 1994a), Eqns (26) involve a new set of arbitrary parameters of integration - α, e,

ω

and χ - such that an intrinsic transformation of variables between the old set of parameters - p, e, ω and f - and the new one is defined

α

2

1 e

p= − ′ ω=ω′ e

e= ′ _⎟⎟

⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ −

′ + =

2 tan 1 1 2

tan χ α

e e f

. (27)

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

(

)

⎟

⎠ ⎞ ⎜

⎝ ⎛_{− +} ′

−

= _α π_χ

α χ π α π

2

1 2

2

e p

(

)

α

(

)

( )

πχ

α α χ χ π

α π

π

⎪⎭ ⎪ ⎬ ⎫ ⎪⎩

⎪ ⎨ ⎧

− ′ ′ − + ′ −

′ −

= ′ sin

1 1

1 2

2

2 _e e

π

_ω

=

π

_ω′

(

)

πχ

α α χ π 2 2 1 2 cos 2 1 e e e f ′ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ − ′ +

= . (28)

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

(

)

(

)

(

α ω χ

)

ω θ π π π π χ ω α π π π π ω π π π π θ , , , ; , , , , , , ; , , , , , , ; , , , e MATHIEU f e p MATHIEU v u r e f e p v u r ↓ ⎯ ⎯ ⎯ ⎯ → ⎯

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

( ) ( )

( )

( ) ( )

[

]

χ

ω α π α χ α α π α χ α π α χ α π π eM e e r er e r e r e r − − − − + − + − = sin 1 1 sin 1 cos 1 2 2 2 2 2 2 2 2

( ) ( ) ( )

( )

(

)

{

( )

( ) ( )} }

χ

ω α π α χ α χ α χ α α π α χ α π α χ α π α χ µα π cos 1 sin 2 1 cos 1 sin 1 sin 2 1 2 2 2 2 e e e e e e e e e r e u + + + − + − − − ⎪⎩ ⎪ ⎨ ⎧ ₋ + − =

( )

(

)

( ) ( )

(

)

( )

(

)

⎭ ⎬ ⎫ ⎥⎦ ⎤ ⎢⎣ ⎡ _{− +} + ⎥⎦ ⎤ ⎢⎣ ⎡ _{− −} − + ⎩ ⎨ ⎧ ⎥⎦ ⎤ − ⎢⎣ ⎡_{− +} + − − = χ ω α π α χ α χ α χ α π α χ α χ π α χ α χ απ µα π 2 sin 2 1 sin 2 1 2 sin 2 1 sin 2 1 1 2 cos 2 1 cos 2 2 3 2 1 1 2 2 3 2 e e e e e e e e e r e e v ω θ π

π = . (29)

Here M = µα3(t−τ) and µ(t−τ) 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 χ. 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

χ

π µ

r

=

H . (30)

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

0 α α= 0 e e= 0 ω ω=

(

₀

)

( ) ( ₀)

3 _{χ χ}

µα t−t =I −I , (31)

where

( )

χ α α

χ

χ) sin

( = −e

I . (32)

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

(

)

[

]

0

0 _{2(1 )} 3

1

χ α

α π µ τ χ π

π t r

e − − +

− + =

( )

0 0 ) 1 ( sin χ π α α χ π π e e e − − = 0 ω ω π π = 0 ) 1 ( χ

χ α π

π

e r

−

= . (33)

Note that t₀ =τ and χ(τ)=0.

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

(

)

_⎪⎭⎪⎬ ⎫ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + − + − + ⎪⎩ ⎪ ⎨ ⎧ + + − = χ ω α π π π π π er e e e p U U e U e p pU r e r ) 1 ( 1 2 3 1 1 2 1 1 3 1 0 2

{

( )

(

)

₍

₎

⎪⎭ ⎪ ⎬ ⎫ ⎥ ⎥ ⎦ ⎤ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − + + − + ⎢ ⎣ ⎡ ⎥⎦ ⎤ ⎢⎣ ⎡ + + − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − − − + − = χ ω α π α α α α π π α π α µα π 2 1 2 0 2 1 3 1 2 2 1 1 2 1 3 1 3 1 1 3 1 1 1 1 2 1 U e U e e U e p U U e e e p U e e U p U e r e u

(

)

( )

[

]

( )

)

(

₍

₎

⎭ ⎬ ⎫ − ⎢ ⎣ ⎡ ⎥⎦ ⎤ + − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + + − − + ⎩ ⎨ ⎧ ⎥⎦ ⎤ − − ⎢⎣ ⎡_{− +} + − = e U eU U e e U e p U U eU U e p e U e U e r p e v 1 2 1 1 3 2 1 2 1 2 1 2 2 3 2 1 1 0 1 2 1 3 1 0 1 2 2 1 0 χ ω α π α π π α απ α µ π ω θ π

π = . (34)

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

⎪⎭ ⎪ ⎬ ⎫ ⎪⎩

⎪ ⎨ ⎧

⎟⎟ ⎠ ⎞ ⎜⎜

⎝ ⎛ ₊ + +

+ −

= e D

r D

p D p D p r

π π

π π

π _{α ω} 3 5

2 ₅

1 2

1

(

)

(

⎪⎭ ⎪ ⎬ ⎫ ⎟⎟ ⎠ ⎞ − −

− + ⎪⎩

⎪ ⎨ ⎧

− +

+ − =

D e u

D p D

pD p p

D p pD D r

π

π π

π µ

π _{α ω}

6 4

2 2 2

5 1

3 4 1 2

2 1

(

)

⎪⎭ ⎪ ⎬ ⎫ ⎟ ⎟ ⎠ ⎞ ⎜

⎜ ⎝ ⎛

+ − + ⎪⎩

⎪ ⎨ ⎧

+ +

+ −

= _{e D}

v

p D pD D

p p D p r

p _{π π π π}

µ

π _{α ω}

5 3

2 2

1 2 5

ω

θ π

π = . (35)

As described in the Section 2, the Lagrangian multipliers πu and πv define the “primer” vector pv. Thus,

{

(

)

(

)

₍

₎

(

)

( )

[

]

( )

)

(

₍

₎

,

1 2

1 1

3

2 1

2 1 2 1 2 2 3 2

1

2 1 3

1 3 1

1 3

1 1 1

1

2 1

1 0 1 2 1 3

1 0 1 2

2 1 0

2 1 2 0 2

1 3

1 2

2

1 1

s e r e

v

e e U eU U e e U e p U

U eU U e p e

U e U e r

p

e U

e U e

e

U e p U U e e e

p U e

e

U p U e r

⎭ ⎬ ⎫ − ⎢

⎣ ⎡

⎥⎦ ⎤ + − + ⎟ ⎠ ⎞ ⎜

⎝ ⎛

+ + +

− − +

⎩ ⎨ ⎧

⎥⎦ ⎤ − − ⎢⎣

⎡_{− +} + − +

⎪⎭ ⎪ ⎬ ⎫ ⎥ ⎥ ⎦ ⎤ ⎟ ⎟ ⎠ ⎞ ⎜

⎜ ⎝ ⎛

− − +

+ − +

⎢ ⎣ ⎡

⎥⎦ ⎤ ⎢⎣

⎡ + + −

+ ⎟ ⎠ ⎞ ⎜

⎝ ⎛

+ − −

−

+ −

=

χ ω

α

χ ω

α

π α

π

π α απ

α µ

π α α

α

α π

π α π α µα

p

(36)

for all conics, and

(

)

(

(

)

, 5

3 2 2

1 5

1

3 4 1 2

2 1

5

2 6

4

2 2 2

s D

e r

D e v

e p D pD

D p p D p r

p e D p D

pD p p

D p pD D r

⎪⎭ ⎪ ⎬ ⎫ ⎟ ⎟ ⎠ ⎞ ⎜

⎜ ⎝ ⎛

+ − +

⎪⎩ ⎪ ⎨ ⎧

+ +

+ − + ⎪⎭ ⎪ ⎬ ⎫ ⎟⎟ ⎠ ⎞ − −

− + ⎪⎩

⎪ ⎨ ⎧

− +

+ − =

π

π π

π µ π

π π

π µ

ω α

ω α

p

(37)

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.

References

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