chapter 3 algorithm for lambert's problem -...

Chapter 3

Algorithm for Lambert's Problem

Abstract The solution process of Lambert problem, which is used in all analytical techniques

that generate lunar transfer trajectories, is described. Algorithms based on the formulation of

Battin and Vaughan for determining the in-plane characteristics of the conic and the

formulation of Der for determining out-of-plane characteristics are used for the solution of the

Lambert problem. These algorithms are uniformly valid for determining all types of conics:

circular, elliptical and hyperbolic and avoids mathematical singularities existing in other

formulations. The superiority of the combined algorithm, referred to as BVD algorithm, in

various scenarios encountered in the transfer trajectory design is established.

3.1 Introduction

. The determination of an orbit connecting two position vectors in a specified flight time

under a central force field is referred to as Lambert's problem in the literature. When we try to

solve transfer trajectory design problem analytically, the trajectory is split into many phases.

Lunar transfer consists of two phases: (i) a geocentric conic from a point on the parking orbit

to a point on the boundary of MSI of the moon (ii) a selenocentric conic along which the

spacecraft travels from the point on the boundary of the MSI to the target point. Each phase

is posed as a two-body Lambert's problem. But these phases are synchronized at the

boundary of the MSI using large number of iterations. So, the solution of the Lambert's

problem must be obtained numerous times before synchronization.

26

A variety of methods of dealing with this classical problem have been discussed over

the years by many authors. Escobal [7] gives algorithms based on six iterative methods and

recommends a method based on 'true anomaly iteration'. But this method has the following

shortcomings:

(i) Singularity when the radial distances at two positions are equal

(ii) Formulations are different for different types of conics

(iii) Slow convergence to the required solution when the transfer angle is large

A method must be more reliable, robust and fast converging for use in transfer trajectory

design problem. Battin and Vaughan [10] present a new formulation that overcomes the

shortcomings observed in other methods. They exploit a new principle discovered by Battin,

which is a fundamental property of two body orbits. The new principle is the invariance of the

mean point under a certain geometric transformation, which brings the mean point radius into

coincidence with an orbital apse. However, these formulations determine only in-plane

characteristics of the conic that is semi-major axis, eccentricity and true anomaly etc. To

define an orbit completely, the out-of-plane parameters such as inclination and right ascension

of ascending node must also be determined. For this, the velocity vector at the initial point

must be known. In general, the velocity vector is computed using f and g series

approximation. Again, the approach is different for different conics. For the analytical

techniques, a unified approach is desirable. Der [11] provides a unified formulation to get the

velocity vector at the initial point based on f and g series approximations. This fonnulation is

used herein to compute out-of-plane parameters of the conic determined using the Battin's

formulation. The superiority of this combination of Battin and Der's algorithms is demonstrated

with some case studies.

27

3.2 Battin Universal Formulation

The formulation presented in this section follows the approach of Battin and Vaughan

[10J. In general, the solution of the Lambert's problem is obtained by formulating the transfer

time equation in an appropriate form and by solving it. Lambert's theorem provides a

functional relationship between transfer time and some parameters of the conic connecting

two points in space. We state Lambert's theorem without proof, which is the basis of the

transfer time equation.

Lambert's Theorem: The orbital transfer time (t2 - t1) depend only upon the semi major axis

(a ) and the sum of the distances (r1 + r2 ) of the initial and final points from the center of the

force and the length of the chord (c) joining these points, i.e.

(t2 -tj).[j;, =F(a,r1 +r2 ,c)

where j.1 is gravitational constant of the central body.

This equation is known as transfer time equation. The geometry of the Lambert's problem is

illustrated in Figure 3.1. The angle 8t

is called transfer angle.

Figure 3.1 Geometry of Lambert's problem

28

3.2.1 Geometrical transformation

Before formulating the transfer time equation, Battin and Vaughan effect a

geometrical transformation. This transformation is based on a clever interpretation of

Lambert's theorem itself and helps avoid the singularity when the transfer angle is 180 deg.

The Lambert's theorem mentioned above is restated that if the terminal points P1 and P2 are

held fixed, the shape of the orbit may be altered by moving the foci without altering the flight

time, provided that" r1 + r2 " and" a" are unchanged in the process. Further, they use the

mean value theorem for the choice of apses line. The mean value theorem, in this context, is :

on any smooth arc of a curve joining the points P1 and P2 , there is at least one intermediate

point Po such that the tangent to the curve at Po is parallel to the chord joining P1 and P2. The

mean point radius I ro', is the distance between center of force and Po.

E'2

1-c2

1-c2

1-Crt + r2 )

2 B,2

-----------~---:.~-+--;-ro---""1IPoFo

1-(r + r)·2 [ 2

Figure 3.2 Geometry of the transformation

29

Based on the two theorems mentioned above, they transformed the geometry of the

orbit such that major axis is perpendicular to P1P2 and is the periapsis of the transformed

orbit. The transformed orbit is shown in Figure 3.2. They invented that there are several

invariants under this transformation. One such invariant is the mean point radius (ro)' Also,

the difference between eccentric anomalies (or equivalent anomalies in case of other conics)

is another invariant of this orbital transformation: (i.e) IE; -E;I =IE2 -Ell where E"E2

are eccentric anomalies of P1 and P2 respectively in pre-transformed orbit and E;, E; are

eccentric anomalies of P1 and P2 respectively in the transformed orbit. Evidently, "r, + r2 IJ and

" a IJ are also invariants.

3.2.2 Transfer time equation

The transfer time equation is formulated in the transformed orbit and the solution is

obtained. Later on, the solution is interpreted for the original geometry. From the geometrical

properties of the triangle P1FoP2, it is clear that LPoFoP2 = E = (E; - E;). The time2

required to traverse to P2 from Po (transfer time) is given by Kepler's equation for the ellipse

as

(3.1)

where a and eo are semi major axis and eccentricity of the transformed orbit. This equation

is referred to as transfer time equation. Equation (3.1) can be rewritten as,

(3.2)

30

It is well known that the periapsis can be expressed as

which implies that

(3.3)

(3.4)

Now, 'iJ is expressed in terms of rop , which is radius to the mean point of the parabola Pl

and P2 as [40],

and from the properties of parabola,

if ellipse

if hyperbola(3.5)

(3.6)

where 8/ is the transfer angle. From Figure 3.2, it is clear that 2v =8/

Substitution of (3.5) in (3.4) gives that

rop 2 1l-e =-sec -E

o a 2

Also, in orbital mechanics, we know that

1 l+e ?1tan 2 _ V =__0 tan - - E2 1- eo 2

After adding tan 2 ~ E both sides, it can be shown that2

31

8_/ =V.2

(3.7)

(3.8)

(3.9)

From the equations (3.7) and (3.9) we get,

24tan 2 ..!.. E

rop _ 2a - ?1 2 1 2 1

(1+tan--E)(tan -v+tan -E)2 2 2

Use of equation (3.7) in the equation (3.2), the transfer time equation takes the form as

1

Ri4tan3-E

~ 2-3(t2 -t1) 38r 1 1 ? 1 -

Op [(1 + tan 2 - E)(tan 2 - V + tan - - E)] 2

2 2 21

4tan 3-E

=E - sin E + ? 1 ?21 ? 1(1 + tan - - E)(tan - - v + tan - - E)

2 2 2

Define some auxiliary quantities in the following way,

(3.10)

(3.11 )

my2= _(l+x)(1+x)

The transfer time equation takes the form

3 2 E -sinEy - y =m

14tan 3- E

2

32

(3.12)

(3.13)

3.2.3 Generalization of transfer time equation

The transfer time equation (3.13) is applicable only when the conic is ellipse. Battin

and Vaughan using the concept of hypergeometric function derive a general form. The right

hand side of equation (3.13) can be rewritten as

(3.14)

The equation (3.14) can be generalized by introducing the definition of x as follows

2 I ' ,tan -(E -E)4 2 I

x= 0

tan 2 .l (H - H )4 2 q

where E = (E; - E;) as defined earlier.2

if ellipse

if parabola - 1~ x ~ 00

if hyperbola

(3.15)

This generalization can easily be verified by deriving transfer time equations for other types of

conics in similar lines. The general unified form of equation (3.14) becomes

E-sin E = _1 (tan-I E __I_J4 tan 3 .l E 2x E 1+ x

2

=_!£(tan-I EJ

dx E(3.16)

The hypergeometric function for tan;;,E is F(~, 1; %;- x). So, the equation (3.16)

becomes

33

E-sin E =!£F(..!-, 1; l;-x)4 31E dx 2 2tan -

2

3.17)

The hypergeometric function F is expanded in terms of continued fraction as given below

x1+----:----

4x3+ 9

5+ x7+ .

Hence, the unified transfer time equation, which is applicable for all types of conics, is

From equations (3.16) and (3.17),

dF _ 1(1 F)- ----dx 2x l+x

(3.18)

(3.19)

(3.20)

When x vanishes, the equation (3.20) becomes in determinate. The indeterminacy can be

eliminated by defining Fas F = 1 wherel+xG

1G=--------

4x3+------

9x5+ 16x

7 + 9+ 25x

11+ ......

and we get

dF (l-G)F=

dx 2(1+x)

Further simplification is attempted by defining Gas G = 14x

where3+-

~

34

(3.21 )

9xq=5+-----

16x7+------

9+ 25x

11 + 36x13+ .

resulting in

F = 4x+3q[4x+q(3+x)]

and

dF (2x+3q)- - ------=----dx (l+x)[4x+q(3+x)]

The transfer time equation in generalized form is

3 2 m(2x+ 3q)y - y =

(l + x)[4x +q(3 + x)]

3.2.4 Solution process

(3.22)

(3.23)

(3.24)

The solution of Lambert's problem is obtained by solving the cubic transfer time

equation (3.24). But since x is also an unknown parameter, the equation (3.12) relating

y and x is rewritten as quadtratic in x and solved to get,

x= [1_1]2 +!!!.-_ (1+1)2 / 2

(3.25)

The solution of the Lambert's problem is obtained by finding simultaneous solution of the

equations (3.24) and (3.25) by successive substitution method.

An initial guess for x must be made to initiate the solution process. We know that x

is parameter depending on the difference in the eccentric anomalies, so its value is set as

dependent on the difference in true anomalies. The initial guess for x is given by

35

(3.26)

where (v2 - VI) is the difference in true anomalies of the points P1 and P2. Substitution of

this value in equation (3.24) and solving the cubic equation, gives a value for y , say Yo' In

the next iteration, x = Xl is computed using y = Yo in equation (3.25) and used to solve

equation (3.24) to get Yl' This iterative process is continued till two successive values of X

and y do not differ by more than a prefixed tolerance level. In this process, because we are

solving a cubic equation, the choice for the root must be understood. Analysis of equation

(3.13) leads to the conclusion that the right hand side of equation (3.13) is always positive for

all types of conics. Hence the equation (3.13) can have exactly only one positive real root,

which is taken as the solution of the cubic equation.

After achieving convergence, the semi-major axis and the eccentricity of the conic

(transformed) obtained and then these parameters are computed for the original orbit. If

ai' e, and P, are semi major axis, eccentricity and the semi-parameter respectively, then

ms(l + ,.1.)2a, = 2

8xy(3.27)

(3.28)

Since the semi major axis is unchanged under the transformation, the semi major axis (a) of

the original orbit is given by

ms(l + ,.1./a =a =--'------,,---..:.-

I 8xy2

But the semi-parameter of the original orbit is found by

36

(3.29)

3.2.4.1 Complete orbital characteristics using Der's formulation

(3.30)

Having determined the size (a) and shape (e) of the orbit and x, to know the

complete information about the orbit connecting P1 (at t[) and P2 (att 2 ), it is now left to

compute the velocity vector VI at P1(at t1 ). For this, a formulation described by Der [11],

which is again a universal approach, is followed. This approach is based on f and g series

expansion ahd is valid for all types of orbits. The procedure that computes the velocity vector

from' a ' and' x ' is explained in the following steps:

(i) compute a =~ and z = QX2

a

(ii) compute

c=

s=

l-cos.fi

z1

2cosh.fi -1

-z

1

6sinhh-h

(h/

if ellipse

if parabola

if hyperbola

if ellipse

if parabola

if hyperbola

(iii) Express the f and g functions as functions of x, t, C and Sas follows

37

· df fi1 2f =-=--x(l-ax S)dt rl r2

(iv) In f and g series expansion, the radial direction vector (;2) at t2 is expressed as

follows:

We get the velocity vector VI at tI' VI = '2 - f ~ .g

Now, with (~, VI) known at t1 the complete information about the orbit is available.

3.3 Computational Algorithm

(1) Guess a value for x, which is a function of eccentric anomalies of the two points

using equation (3.26) (say xo)'

(2) Compute the auxiliary parameters l,m,~(x) (use equations (3.12) and (3.22)).

(3) Solve the cubic transfer time equation (3.24) to get avalue for y (say Yo)'

(4) Compute a new value for x using equation (3.25).

(5) Repeat steps (2), (3) and (4) until two consecutive values of x and y do not differ by

more than a pre-assigned small value.

(6) Using the converged parameters x and y, compute semi major axis and eccentricity

using equations (3.29) and (3.30)

(7) Use the procedure described in Section 3.2.4.1, compute the velocity vector at t l and

then complete information of the orbit.

38

The above algorithm combines the steps of Battin's and Der's formulations and is referred to

as 'BVD algorithm'.

3.4 Performance of BVO Algorithm

The efficiency of the algorithm is assessed with the help of four sample problems

representing different scenarios encountered in transfer trajectory design:

(1) Circular orbits (2) Elliptic orbits (3) Hyperbolic orbits (4) Heliocentric orbits.

Positions for the sample problems are generated in the following way: (i) Orbit characteristics:

semi major axis (a), eccentricity (e), inclination (i), right ascension of ascending node (Q),

argument of periapsis (OJ) are fixed for the above cases (ii) The positions at two different

timings corresponding to two different true anomalies were generated using the software

'GaPS' [35]. These positions are used to determine the complete characteristics using BVD

algorithm and compared with the actual orbit characteristics to test reproduction efficiency.

In all scenarios, the determination accuracy is same and the convergence pattern is also the

same. It is clear that even for determination of a hyperbolic orbit, this algorithm works. The

accuracy achieved in the case interplanetary heliocentric orbit is remarkable considering the

magnitude of the semi major axis. Performance efficiency is evaluated by comparison of this

algorithm with the 'true anomaly iteration method' (TAl) that was recommended by Escobal

[7]. Evaluation is based on velocity difference between the determined and the actual orbits at

t l , semi major axis difference and the number of iterations taken for convergence. The

performance of BVD algorithm is uniformly same for all semi major axes and better than TAl

accuracies (Table 3.5). This obseNation holds good for all eccentricities and inclinations

(Tables 3.6 and 3.7). Further, even for high eccentricity orbits (as high as 0.99) the number of

39

iterations is very small whereas TAl method requires more number of iterations. For various

transfer angles, the comparison is given in Table 3.8. The BVD method functions in all

situations and the performance is uniformly good whereas the TAl method requires more

iteration producing the results with less accuracy. Also, when the transfer angle is about 310

deg, there is no convergence with TAl method. This is because in this region the radial

distances of the two positions are nearty equal. This is illustrated in Table 3.9. In this region,

either the number of iterations is very high or there is no convergence. It is clear that in a

region of transfer angles where TAl method fails, the BVD algorithm succeeds. This is

desirable in transfer trajectory design problems because the transfer angle involved and the

radial distances involved are not known beforehand.

3.5 Conclusions

The solution process based on Battin and Vaughan's formulation for determining a

conic with its in-plane characteristics and Der's formulation for deriving the out-of-plane

characteristics of the determined conic is described for completeness sake. These

formulations are uniformly valid for all types of conics: circular, elliptical, parabolic and

hyperbolic. Its supremacy over 'true anomaly iteration method' is established. The

shortcomings of the 'true anomaly iteration method' such as singularity when the radial

distances of two positions are equal are overcome. This is desirable because in transfer

trajectory design problem in which these algorithms are used, different kinds of scenarios

involving different conic and singularities will occur and the occurrence is unknown a priori.

40

Table 3.1

Determination efficiency for circular orbits· Sample problem 1

Position at tl

Xl = 887.785388 km , Yl =-6487.436867 km, Zl = 2474.873734 km

Position at 12

X2 = 5174.392438 km , Y2 = -4012.563133 km, Z2 = -2474.873734 km

Time of Flight

971.419445 sec

Orbit a (km) e i (deg) Q (deg) (j) (deg) v (deg) (j) + v (deg)

Expected 7000.0 0.0 45.0 120.0 150.0 0.0 150.0

Determined 7000.0 0.0 45.0 120.0 341.248 168.752 150.0

41

Table 3.2

Determination efficiency for elliptical orbits· Sample problem 2

Position at tl

Xl =2273.316685 km , Yl =-2187.500000 km, Zl =1515.544457 km

Position at ~

X2 = 909.326674 km , Y2 = -3675.000000 km, Z2= -1818.653348 km

Time of Flight

318.644416 sec

Orbit a (km) e i (deg) Q (deg) ill (deg) v (deg) ill + v (deg)

Expected 7000.0 0.5 120.0 120.0 150.0 0.0 150.0

Determined 7000.0 0.5 120.0 120.0 150.0 0.9E-14 150.0

42

Table 3.3

Determination efficiency for hyperbolic orbits· Sample problem 3

Position at tl

Xl =1268.264840 km , Yl =-9267.766953 km, Zl =3535.533906 km

Position at ~

X2 =10559.984568 km , Y2 =-8188.904353 km, Z2 =-5050.762723 km

Time of Flight

1351.026057 sec

Orbit a (km) e i (deg) Q (deg) m(deg) v (deg) m+ V (deg)

Expected -20000.0 1.5 45.0 120.0 150.0 0.0 150.0

Determined -20000.0 1.5 45.0 120.0 150.0 0.22E-12 150.0

43

Table 3.4

Determination efficiency for heliocentric orbits· Sample problem 4

Position at t1

X1 = -165066454.034 km , Y1 = 144849637.245 km, Z1 = -3423998.596 km

Position at ~

X2 = -194482766.201 km , Y2= 118099855.449 km, Z2 = -2847890.309 km

Time of Flight

20.346053 days

Orbit a (km) e i (deg) n (deg) OJ (deg) v (deg)

Expected 197163372.454 0.23885398 1.294 182.381 186.344 130.0

Determined 197163372.454 0.23885398 1.294 182.381 186.344 130.0

44

Table 3.5

Comparison of Lambert problem solutions with variation

in semi major axis

Semi major V_Diff (km/s) a_Diff (km) No. of Iterations

axis (km) BVD1 TAI2 BVD TAl BVD TAl

7000.0 1.78E-15 8.0E-10 9.1E-13 2.2E-04 4 12

10000.0 1.78E-15 6.7E-10 3.5E-11 3.2E-04 4 12

20000.0 1.78E-15 4.7E-10 8.0E-11 6.4E-04 4 12

40000.0 8.88E-15 3.4E-10 1.4E-10 1.2E-03 4 12

80000.0 8.88E-15 2.4E-10 3.2E-10 2.6E-03 4 12

160000.0 4.44E-15 1.7E-10 6.4E-10 5.1E-03 4 12

1BVD - Battin, Vaughan and Der algorithm

2TAI - True Anomaly Iteration

V_Diff : Absolute difference between determined and actual velocities

a _Diff : Absolute difference between determined and actual semi major axes

Other Orbit Parameters: e =0.5, i = 45 deg, Q. = 120 deg, OJ = 150 deg

VI =25 deg, v2 = 85 deg, Or = 60 deg

45

Table 3.6

Comparison of Lambert problem solutions with variation in eccentricity

Eccentricity V_Oiff (km/s) a_Oiff (km) No. of Iterations

BV01 TAI2 BVO TAl BVO TAl

0.0 0.0 2.8E-10 3.6E-13 6.5E-04 1 2

0.01 0.0 2.8E-10 7.3E-13 6.5E-04 3 7

0.10 1.78E-15 3.1 E-10 7.3E-13 6.5E-04 4 7

0.25 1.78E-15 3.6E-10 3.6E-13 6.5E-04 4 7

0.50 1.78E-15 4.7E-10 8.0E-11 6.4E-04 4 12

0.90 1.06E-14 9.9E-10 2.3E-10 6.1 E-04 4 27

0.99 4.01E-12 3.9E-09 5.4E-07 1.7E-04 4 27

1BVO - Battin, Vaughan and Oer algorithm


V_0iff : Absolute difference between determined and actual velocities

a _Oiff : Absolute difference between determined and actual semi major axes

Other Orbit Parameters: a =20000 km, i =-45 deg, Q = 120 deg, (j) = 150 deg

VI =25 deg, v2 = 85 deg, (J/ = 60 deg

46

Table 3.7

Comparison of Lambert problem solutions with variation in inclination

Inclination V_0iff (km/s) a_Oiff (km) No. of Iterations

(deg) BV01 TAI2 BVO TAl BVO TAl

0.0 8.8E-16 4.7E-10 8.0E-11 6.4E-04 4 12

30.0 8.8E-16 4.7E-10 8.0E-11 6.4E-04 4 12

63.4 8.8E-16 4.7E-10 8.0E-11 6.4E-04 4 12

100.0 8.8E-16 4.7E-10 8.0E-11 6.4E-04 4 12

1BVO - Battin, Vaughan and Oer algorithm


V_0iff : Absolute difference between determined and actual velocities

a _Oiff : Absolute difference between determined and actual semi major axes

Other Orbit Parameters: a = 20000 km, e = 0.5, Q = 120 deg, CtJ = 150 deg

VI =25 deg, v2 = 85 deg, Or = 60 deg

47

Table 3.8

Comparison of Lambert problem solutions with variation in transfer angle

Transfer V_Diff (km/s) a_Diff (km) No. of Iterations

Angle (deg) BVD1 TAI2 BVD TAl BVD TAl

20.0 0.0 4.7E-10 3.2E-10 6.4E04 3 20

40.0 0.0 4.7E-10 0.0 6.4E04 3 16

60.0 1.78E-15 4.5E-10 8.0E-11 6.4E04 4 12

90.0 8.8E-16 4.5E-10 4.8E-11 6.4E04 4 6

180.0 0.0 4.5E-10 0.0 6.4E04 5 8

210.0 3.5E-14 4.8E-10 0.0 6.4E04 5 14

295.0 9.4E-14 5.8E-09 3.6E-12 6.4E-04 5 862

310.0 8.8E-15 NC 0.0 NC3 5 NC

355.0 2.6E-13 7.41:-10 3.6E-12 6.4E-04 5 30





Other Orbit Parameters: a =20000 km, e =0.5 , i=45 deg, Q =120 deg, OJ =150 deg

3NC - No Convergence (after 4000 iterations)

48

Table 3.9

Comparison of Lambert problem solutions in the singularity region (r1=r2)

Transfer V_Diff (km/s) a_Diff (km) No. of Iterations

angle (deg) BVD1 TAi2 BVD TAl BVD TAl

290.0 5.2E-14 4.7E-10 3.6E-12 6.4E04 5 673

295.0 9.4E-14 5.8E-10 7.3E-12 6.4E04 5 872

296.0 5.4E-14 NC3 3.6E-12 NC 5 NC

300.0 6.5E-14 NC 3.6E-12 NC 5 NC

310.0 8.8E-15 NC 0.0 NC 5 NC

315.0 4.6E-14 4.7E-10 3.6E-12 6.4E04 5 27

320.0 1.78E-13 4.8E-09 3.6E-12 6.4E-04 5 33


2TAI • True Anomaly Iteration



Other Orbit Parameters: a =20000 km, e =0.5, i =45 deg, Q =120 deg, OJ =150 deg

3NC - No Convergence (after 4000 iterations)

49

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