Chapter 3. Algorithm for Lambert's Problem

Transcription

1 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

2 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) (ii) (iii) Singularity when the radial distances at two positions are equal Formulations are different for different types of conics 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 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 (t 2 - t 1 ) depend only upon the semi major axis (a ) and the sum of the distances (r 1 + r 2 ) of the initial and final points from the center of the force and the length of the chord (c) joining these points, i.e. (t 2 -tj).[j;, =F(a,r 1 +r 2,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 8 t is called transfer angle. Figure 3.1 Geometry of Lambert's problem 28

4 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" r 1 + r 2 " 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 r o ', is the distance between center of force and Po. 1 -Crt + r 2 ) 2 B, 2 1 -c ~---:.~-+--;-ro---""1ipo F o 1 -(r + r) 2 [ 2 1 -c 2 E' 2 Figure 3.2 Geometry of the transformation 29

5 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 (r o )' 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 =IE 2 -Ell where E"E 2 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, + r 2 IJ and " a IJ are also invariants 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 LP o F o P 2 = E = (E; - E;). 2 The time 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

6 It is well known that the periapsis can be expressed as (3.3) which implies that (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], if ellipse ifhyperbola (3.5) and from the properties of parabola, (3.6) where 8/ is the transfer angle. From Figure 3.2, it is clear that 2v =8/ 8 _/ =V. 2 Substitution of (3.5) in (3.4) gives that rop 2 1 l-e =-sec -E o a 2 (3.7) Also, in orbital mechanics, we know that 1 l+e?1 tan 2 _ 2 1- eo 2 V = 0 tan - - E (3.8) After adding tan 2 ~ E both sides, it can be shown that 2 (3.9) 31

7 From the equations (3.7) and (3.9) we get, 4tan 2..!.. E 2rop _ 2 a -? (1+tan--E)(tan -v+tan -E) (3.10) Use of equation (3.7) in the equation (3.2), the transfer time equation takes the form as 1 Ri 4tan3-E ~ 2-3(t 2 -t 1 ) 3 8r 1 1? 1 - Op [(1 + tan 2 - E)(tan 2 - V + tan - - E)] tan 3 -E =E - sin E +? 1?2 1? 1 (1 + tan - - E)(tan- - v + tan - - E) (3.11 ) Define some auxiliary quantities in the following way, m y2= _ (l+x)(1+x) (3.12) The transfer time equation takes the form 3 2 E -sine y - y =m 1 4tan 3 - E 2 (3.13) 32

8 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 if ellipse if parabola - 1 ~ x ~ 00 if hyperbola (3.15) where E = (E; - E;) as defined earlier. 2 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_J 4 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

9 E-sinE=! F(..!-, 1; l;-x) 3.17) 31E dx tan - 2 The hypergeometric function F is expanded in terms of continued fraction as given below x :---- 4x x 7+. (3.18) Hence, the unified transfer time equation, which is applicable for all types of conics, is (3.19) From equations (3.16) and (3.17), df _ 1(1 F) dx 2x l+x (3.20) When x vanishes, the equation (3.20) becomes in determinate. The indeterminacy can be eliminated by defining F as F = 1 where l+xg and we get 1 G= x x 5+ 16x x df dx = (l-g)f 2(1+x) (3.21 ) Further simplification is attempted by defining G as G = 1 where 4x 3+- ~ 34

10 9x q= x x x 13+. (3.22) resulting in F = 4x+3q [4x+q(3+x)] and df (2x+3q) =---- dx (l+x)[4x+q(3+x)] (3.23) The transfer time equation in generalized form is 3 2 m(2x+ 3q) y - y = (l + x)[4x +q(3 + x)] (3.24) Solution process 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

11 (3.26) where (v 2 - 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.)2 a, = 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./ 8xy2 a =a =--'------,,---..:.- I (3.29) But the semi-parameter of the original orbit is found by 36

12 (3.30) Complete orbital characteristics using Der's formulation 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 t 1 ). 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= l-cos.fi z 1 2 cosh.fi-1 -z if ellipse if parabola if hyperbola s= 1 6 sinhh-h (h/ if ellipse if parabola if hyperbola (iii) Express the f and g functions as functions of x, t, C and S as follows 37

13 df fi1 2 f =-=--x(l-ax S) dt r l r 2 (iv) In f and g series expansion, the radial direction vector (;2) at t 2 is expressed as follows: We get the velocity vector VI at ti' VI = '2 - f ~. g Now, with (~, VI) known at t 1 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 x o )' (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 a value 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 , compute the velocity vector at t l and then complete information of the orbit. 38

14 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

15 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

16 Table 3.1 Determination efficiency for circular orbits Sample problem 1 Position at tl Xl = km, Yl = km, Zl = km Position at 12 X2 = km, Y2 = km, Z2 = km Time of Flight sec Orbit a (km) e i (deg) Q (deg) (j) (deg) v (deg) (j) + v (deg) Expected Determined

17 Table 3.2 Determination efficiency for elliptical orbits Sample problem 2 Position at tl Xl = km, Yl = km, Zl = km Position at ~ X2 = km, Y2 = km, Z2= km Time of Flight sec Orbit a (km) e i (deg) Q (deg) ill (deg) v (deg) ill+ v (deg) Expected Determined E

18 Table 3.3 Determination efficiency for hyperbolic orbits Sample problem 3 Position at tl Xl = km, Yl = km, Zl = km Position at ~ X2 = km, Y2 = km, Z2 = km Time of Flight sec Orbit a (km) e i (deg) Q (deg) m(deg) v (deg) m+ V (deg) Expected Determined E

19 Table 3.4 Determination efficiency for heliocentric orbits Sample problem 4 Position at t1 X1 = km, Y1 = km, Z1 = km Position at ~ X2 = km, Y2= km, Z2 = km Time of Flight days Orbit a (km) e i (deg) n (deg) OJ (deg) v (deg) Expected Determined

20 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 E E E E E E E E E E E E E E E E E E E E E E E E BVD - Battin, Vaughan and Der algorithm 2TAI - True Anomaly Iteration V_Diff a _Diff : Absolute difference between determined and actual velocities : 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, v 2 = 85 deg, Or = 60 deg 45

21 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 E E E E E E E E E E E E E E E E E E E E E E E E E E BVO - Battin, Vaughan and Oer algorithm 2TAI - True Anomaly Iteration V_0iff a _Oiff : Absolute difference between determined and actual velocities : Absolute difference between determined and actual semi major axes Other Orbit Parameters: a = km, i =-45 deg, Q = 120 deg, (j) = 150 deg VI =25 deg, v 2 = 85 deg, (J/ = 60 deg 46

22 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 E E E E E E E E E E E E E E E E BVO - Battin, Vaughan and Oer algorithm 2TAI - True Anomaly Iteration V_0iff a _Oiff : Absolute difference between determined and actual velocities : Absolute difference between determined and actual semi major axes Other Orbit Parameters: a = km, e = 0.5, Q = 120 deg, CtJ = 150 deg VI =25 deg, v 2 = 85 deg, Or = 60 deg 47

23 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 E E E E E E E E E E E E E E E E E E E E E E E-15 NC 0.0 NC3 5 NC E : E E BVD - Battin, Vaughan and Der algorithm 2TAI - True Anomaly Iteration V_Diff a _Diff : Absolute difference between determined and actual velocities : Absolute difference between determined and actual semi major axes 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

24 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 E E E E E E E E E-14 NC3 3.6E-12 NC 5 NC E-14 NC 3.6E-12 NC 5 NC E-15 NC 0.0 NC 5 NC E E E E E E E E BVD - Battin, Vaughan and Der algorithm 2TAI True Anomaly Iteration V_Diff a _Diff : Absolute difference between determined and actual velocities : Absolute difference between determined and actual semi major axes 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