CHAPTER 2. COMPUTATIONAL SOLUTIONS TO LAMBERT’S PROB-
2.2 Solution Methods to Lambert’s Problem
2.2.2 Gooding’s Method
The second Lambert solution method considered is known as Gooding’s method [10, 20].
This method is an extension of a Lancaster and Blanchard’s unified form of Lambert’s theorem developed in the 1960s[9]. Gooding was able to formulate robust initial guesses as well as derivatives for a high order root finding routine. The final version of the algorithm developed for this study differs from the algorithms presented in Gooding’s papers [10, 20], but can provide approximately 100% convergence. This method will also be shown to be one of the most computationally efficient algorithms.
For this solution method, two necessary parameters, T and q, are defined in the same way as, T and λ, in Battin’s method. However, the choice of the independent universal variable differs. In this case the independent universal variable refers to a choice in variables in which
the resulting solution method applies to both elliptical and hyperbolic orbits. The sign of q is chosen the same as λ, as
q = λ =
√r1r2 s cosθ
2 (2.30)
T ≡ r8µ
s3∆t (2.31)
The independent variable for this method is defined as
x2 = 1 − s
2a (2.32)
Elliptical orbits cover the range from −1 to 1, with a value of 0 corresponding to the minimum energy solution. An x value of exactly 1 represents a parabolic orbit, while x values larger than 1 are hyperbolic orbits.
Two additional variables, K and E, that are functions of the problem geometry are defined as
K = q2= 1 − c
s (2.33)
E(x) = x2− 1 = − s
2a (2.34)
2.2.2.1 Formulation of the elliptical and hyperbolic time equation
The following time-of-flight equation formulation works for both elliptical and hyperbolic orbits. However, this formulation does not work well when the orbit is near parabolic. For near parabolic orbits a separate formulation is necessary. The following equations allow the time equation to be represented in terms of the independent variable, x, and the physical parameters of the problem.
y =p|E| (2.35)
z =√
1 + KE (2.36)
f = y (z − qx) (2.37)
g = xz − qE (2.38)
d =
atan(f /g), E < 0 atanh(f /g), E > 0
(2.39)
The inverse tangent function should be computed unambiguously, typically by the ATAN2 function. The time-of-flight function for elliptical and hyperbolic orbits for the root-finding routine is then defined as follows:
F (x) = 2 E
x − qz −d y
− T = 0 (2.40)
To solve for x for a corresponding required time parameter, T , a second order Halley iterative method is used [21]. Halley’s method requires second order derivative of the time equation with respect to the independent variable x. This high-order method requires more computations, when compared with the commonly used Newton-Raphson methods, but significantly increases the rate of convergence. In addition, initial guess conditions are much more relaxed when compared to low order root finding methods. Halley’s method for solving F (x) = 0 for x is given by
xn+1= xn− 2FnFn0
2 (Fn0)2− FnFn00 (2.41)
The first and second derivatives of F (x) with respect to x are given by
F0 = −
3x(F + T ) + 4q3x z − 4
E (2.42)
F00= −
where T is the specified time-of-flight.
2.2.2.2 Formulation of the time equation for near parabolic orbits
For near parabolic orbits, in the neighborhood of x ≈ 1, Eq. (2.40) suffers from a loss of significant digits. For near parabolic orbits the Gooding time-of-flight equation can be written as
Equation (2.45) can be represented by a series function, as done by Gooding’s original algorithms. However, series-based solutions are more difficult to implement, and there is no significant increase in the algorithm’s performance when the transcendental form is directly implemented, likely due to compiler optimization. For this reason, all the necessary derivatives for the algorithm are kept in the transcendental form. The first and second order derivatives for Eq. (2.45) are then given by
σ0 = − u
It is still necessary to calculate the derivatives of the near parabolic time equation, which are given as
F0 = −2x σ0(−E) − q5σ0(−KE)
(2.48)
F00= F0
x + 4x2 σ00(−E) − q7σ00(−KE)
(2.49)
2.2.2.3 Initial guess generation
With the time equation formulation and all necessary derivatives completed for all orbit forms, the last step is to determine initial guesses. By examining Fig. 2.2, it is clear that the time curves are strictly monotonically decreasing, except for the case when q is exactly 1. An initial guess generation scheme, based on a bi-linear approximation, was developed by Gooding.
This initial guess generation method is extremely robust and is outlined below.
0 5 10 15 20 25 30
-1 -0.5 0 0.5 1 1.5 2
Normalized Time-of-Flight T(x)
x
q=-1.0 q=-0.7 q=0.0 q=0.7 q=1.0
Figure 2.2 Plot of T (x) vs x for Gooding’s method.
The first step in determining the initial guess is to determine the sign of the solution, x.
The sign is determined by evaluating the time-of-flight from Eq. (2.40) at x = 0. If the desired time, T , is less than T0, the initial guess is obtained as
x0 = T0
T0− T
4T (2.50)
However, in the case that the desired time is greater than T0, the following equations are necessary for the initial guess:
x01= − T − T0
T − T0+ 4 (2.51)
W = x01+ c1 r
2 − θ
π (2.52)
x03=
x01 , W ≥ 0
x01+ (−W )161 (x02− x01) , W < 0
(2.53)
λ = 1 + c1x03
4 4 − T − T0
− c2x203√
1 + x01 (2.54)
x0 = λx03 (2.55)
This final equation for the initial guess it designed to give robust initial guesses even when the solution is close to -1. The following constants were determined through the numerical study:
c1 = 0.5, c2 = 0.03.
The basics of the final algorithm are provided below. The inputs to the algorithm are q and T . The algorithm developed in this section converges for approximately 100% of solutions and is also one of the most efficient of the four algorithms. Gooding’s method is outlined as
1. Evaluate T0 when x = 0 and determine the initial guess from Eqs. (2.50) or (2.55).
2. If x is close to 1.0, calculate E and K to evaluate F , F0, and F00from Eqs. (2.44), (2.48), and (2.49), respectively.
3. Otherwise calculate z, d, y, and E and evaluate F , F0, and F00 from Eqs. (2.40), (2.42), and (2.43), respectively.
4. Update x using Halley’s method.
5. Go to step 2 and repeat until the desired tolerance for the change in x is met or a maximum number of iterations is exceeded.
6. Output the converged semi-major axis, a, or the initial and final radius vectors.