Time of Flight
4.6 Lambert’s Problem
anomalies from true anomalies and the TOF. The real advantage of theΔE formulation is the case where flight time betweenr0andr is known. This scenario is Lambert’s problem, which we describe in the next section.
shows that two very different orbits provide a transfer fromr1tor2with the same TOF t2−t1. The short-way orbit has less energy (smaller semimajor axis) than the long-way orbit (dashed line) because a satellite on the short-way path travels a shorter distance in the same flight time. In some scenarios, the long-way path may become a hyperbolic trajectory in order to complete the transfer in the same flight time as the short-way path.
The reader should also note that the short-way path is not always in the counter-clockwise direction (as shown in Figures 4.14a and 4.15); the directions of the short-way and long-short-way transfers depend on the position vectorsr1andr2.
We can compute the transfer angle or difference in true anomaly from the dot product of the two position vectors:
cosΔθ =r1 r2
r1r2 (4.85)
The“calculator” inverse cosine operation always places the angle in the first or second quadrant (i.e., a short-way transfer). If the long-way path is desired, then the inverse-cosine operation of Eq. (4.85) must be subtracted from 2π. The reader should also note that the angular momentum vectorh for the short-way path in Figure 4.15 is directed out of the page while the angular momentum vector of the long-way path is into the page.
Many different techniques for solving Lambert’s problem have been developed, includ-ing Lambert’s original formulation, Gauss’ method, and Battin’s method. Vallado [2;
pp. 467–498] presents a very good overview of these various methods including algo-rithms for solving Lambert’s problem. Here we will formulate Lambert’s problem and show one iterative technique for obtaining the solution. Using the orbit-propagation
2 1
r1
r2
TOF1-2= t2– t1
TOF1-2= t2– t1
“Long-way” orbit
∆θ > 180°
“Short-way” orbit
∆θ < 180°
Figure 4.15 Short-way and long-way transfers between r1andr2with the same time of flight.
equations (4.84a) and (4.84b), we can write expressions for the unknown initial and ter-minal velocity vectors in terms ofr1andr2
v1=1
g r2−f r1 (4.86)
v2= fr1+g
g r2−f r1 (4.87)
Clearly, determining the Lagrangian coefficients is the key to obtainingv1andv2. Next, equate the two equation sets that define the Lagrangian coefficients in terms ofΔθ and ΔE; that is, Eqs. (4.70)–(4.73) and Eqs. (4.80)–(4.83) with a change in subscript notation to indicate positions 1 and 2:
f = 1−r2
p 1−cosΔθ = 1−a r1
1−cosΔE (4.88)
g=r1r2sinΔθ
pμ = t2−t1 − a3
μ ΔE −sinΔE (4.89)
f = μ p
1−cos Δθ
p −1
r1−1 r2
tanΔθ
2 = − μasinΔE r1r2
(4.90) g= 1−r1
p 1−cosΔθ = 1−a
r2 1−cosΔE (4.91)
Equations (4.88)–(4.91) are composed of seven variables: r1, r2, a, p,Δθ, ΔE, and t2– t1. Of these seven variables, four are known: r1, r2,Δθ, and TOF t2– t1. Therefore, we have three unknown values: semimajor axis a, parameter p, and difference in eccentric anom-alyΔE. At first glance, it appears that we have an overdetermined system of four equa-tions and three unknowns. However, recall that the four Lagrangian coefficients are not independent because of the condition f g−f g = 1 that was obtained by computing the angular momentum vector of the propagated state (r, v) [see Eqs. (4.56)–(4.59) for details]. In truth, we have three independent Lagrangian-coefficient equations and three unknowns. The difficulty is that these equations are transcendental functions of the unknown quantities. Therefore, a closed-form solution cannot be determined. We require an iterative search method for solution.
A basic iterative algorithm for solving Lambert’s problem follows:
1) Given position vectorsr1andr2(and direction of travel, i.e. short-way or long-way path), determine the radius magnitudes r1 and r2 and transfer angle Δθ using Eq. (4.85).
2) Guess a trial value of one of the three unknown quantities (a, p, orΔE).
3) Use the f and f equations (4.88) and (4.90) to determine the other two remaining unknown values.
4) Use the g equation (4.89) to determine the TOF for the trial value.
5) Adjust the iteration parameter (go back to step 3) until the computed flight time in step 4 matches the actual TOF.
As previously stated, we will present one method for solving Lambert’s problem.
Before demonstrating one solution technique, it is very important to note that the
Lagrangian-coefficient expressions (4.80)–(4.83) hold for elliptical orbits only (note the existence of eccentric anomalyΔE which only pertains to ellipses). In order to accom-modate hyperbolic orbits, we require equations for the Lagrangian coefficients in terms of change in hyperbolic anomalyΔF [the expressions in terms of Δθ, Eqs. (4.70)–(4.73), hold for all orbits]. It is for this reason that a universal-variable formulation is often employed in order to develop Lagrangian-coefficient equations that are valid for ellipti-cal, parabolic, and hyperbolic orbits. We will not pursue universal variables here; the interested reader may consult References [1–3] for methods that solve Lambert’s prob-lem using universal variables.
One technique for solving Lambert’s problem is the p-iteration method. As the name implies, we guess a trial value of parameter and iterate on p until the computed TOF matches the actual flight time. Although a complete p-iteration algorithm must accom-modate hyperbolic orbits (and hence express the Lagrangian coefficients in terms ofΔF), our discussion here will only consider elliptical orbits. The p-iteration method is selected because it is somewhat more intuitive than other methods and therefore relatively easy to comprehend. Again, the objective here is to present an introduction to Lambert’s solu-tion. The formulation of the p-iteration method is from Bate et al. [1; pp. 241–251].
Because p is our iteration variable, we must determine the two other unknown vari-ables, a andΔE, from the Lagrangian-coefficient equations. To do so requires manipu-lation of the f and f equations (4.88) and (4.90). These steps are not shown here (see Bate et al. [1] for details). Semimajor axis a can be determined from a trial value of p using
a= mkp
2m−l2 p2+ 2klp−k2 (4.92)
where the auxiliary variables k, l, and m are constants that are functions of the known values
k= r1r2 1−cosΔθ (4.93)
l= r1+ r2 (4.94)
m= r1r2 1 + cosΔθ (4.95)
Clearly, the constants k, l, and m can be computed directly fromr1andr2, that is, the given information for Lambert’s problem. Equation (4.92) shows that semimajor axis becomes infinite (i.e., a parabolic orbit) when the denominator term is zero. The two roots of the quadratic denominator in Eq. (4.92) are
pmin= k
l+ 2m (4.96)
pmax= k
l− 2m (4.97)
Here we use the“min” and “max” subscripts to denote the parameter limits for elliptical transfers. Therefore, if pmin< p < pmax, semimajor axis (4.92) is positive and the orbit between r1and r2is an ellipse. As p approaches the limits pminor pmax, the transfer approaches a parabolic trajectory. For p > pmax, the transfer is hyperbolic (however, we will not consider hyperbolic transfers here). Because we are only considering elliptical transfers, we can restrict our p-iteration search between pminand pmax(of course, a gen-eral p-iteration method must consider elliptical and hyperbolic transfers).
Next, we compute the f, g, and f coefficients using Eqs. (4.88)–(4.90) for the trial value of p and the known difference in true anomalyΔθ
f = 1−r2
p 1−cosΔθ g=r1r2sinΔθ
pμ f = μ
p
1−cosΔθ
p −1
r1−1 r2
tanΔθ 2
We can determine cosΔE and sin ΔE using the right-hand sides of Eqs. (4.88) and (4.90) with the numerical values of the f and f coefficients (computed above) and the trial semi-major axis a that has been computed using Eq. (4.92)
cosΔE = 1−r1
a 1−f (4.98)
sinΔE = −r1r2f
μa (4.99)
Both sine and cosine of ΔE are required to resolve the correct quadrant (e.g., using MATLAB’s atan2 function). The reader should ensure that ΔE is always positive, 0 <ΔE < 2π, so that the TOF calculation is correct. The trial TOF is computed from the g coefficient and Eq. (4.89)
t2−t1= g + a3
μ ΔE −sinΔE (4.100)
If the trial flight time computed using Eq. (4.100) matches the actual TOF, then the trial value of p is correct and we have determined the correct orbit fromr1tor2. If there is any error in flight time, then parameter p must be adjusted until the flight-time error is driven to a negligible value. One way to adjust p between iterations is to use the secant search method:
pi+ 1= pi−τi
pi−pi−1
τi−τi−1 (4.101)
where piis the parameter for the ith iteration, andτiis the difference between the flight time computed using Eq. (4.100) and the actual (or desired) flight time. Note that the fraction term on the right-hand side of Eq. (4.101) is the inverse of the finite-difference approximation of the derivative dτ/dp. Therefore, the secant method is essentially New-ton’s root-solving algorithm where the derivative term is replaced by a finite difference.
After p is updated using Eq. (4.101), the p-iteration algorithm must update a [Eq. (4.92)], ΔE [Eqs. (4.98) and (4.99)], and TOF [Eq. (4.100)]. Convergence to the correct orbit occurs when τi <ε where ε is an acceptable (small) TOF error. When the converged solution is obtained, we can compute the two velocity vectors using the Lagrangian coef-ficients and Eqs, (4.86) and (4.87). The following example illustrates the solution to Lam-bert’s problem using the p-iteration method.
Example 4.9 A ground station determines two position vectors for a weather-forecasting satellite operated by the National Oceanic and Atmospheric Administration (NOAA). The two position vectors in the ECI frame are
r1=
−5,655 144
−3,697 284
−2,426 687
km, r2=
5, 891 286 2, 874 322
−2,958 454 km
The flight time between position vectors is 63 min. Use the p-iteration method to deter-mine the NOAA satellite’s orbit.
The difference in true anomaly is computed using Eq. (4.85) cosΔθ =r1 r2
r1r2 =−0 712062
Therefore, the two possible values areΔθ = 135.40 (short way) or Δθ= 224.60 (long way). Because the flight time is relatively“large” (63 min) and both arcs are significant, it is not apparent which path is correct (as a counter example, it would be easy to select the short-way path if the flight time was 6 min and we had to choose betweenΔθ = 20 or Δθ = 340 ). Let us begin with the short-way path, Δθ = 135.40 . The auxiliary constants are computed using Eqs. (4.93)–(4.95):
k= r1r2 1−cosΔθ = 8 839433 107 km2 l= r1+ r2= 14, 370 85 km
m= r1r2 1 + cosΔθ = 1 486631 107 km2
We see that k, l, and m are the same constants, whether we use the short-way or long-way path. Equations (4.96) and (4.97) provide the lower and upper bounds on p for an ellip-tical transfer:
pmin= k
l+ 2m= 4, 459 042 km pmax= k
l− 2m= 9, 911 799 km We can select the first trial value of parameter closer to pmin
p1= 0 7pmin+ 0 3pmax= 6, 094 869 km Using Eq. (4.92), the corresponding trial semimajor axis is
a= mkp
2m−l2 p2+ 2klp−k2= 7, 255 803 km
The three independent Lagrangian coefficients are computed using r1, r2,Δθ, and the trial value of p
f= 1−r2
p 1−cosΔθ = – 1 020182 g=r1r2sinΔθ
pμ = 735 466534 s
f= μ p
1−cosΔθ
p −1
r1−1
r2 tanΔθ
2 = 5 049967 10−5 s−1
Coefficients f, f , and the trial value of a are used to solve for change in eccentric anomaly:
cosΔE = 1−r1
a 1−f = – 0 998824 and sinΔE =−r1r2f
μa = – 0 048482 Usingatan2 and ensuring a positive value, we obtain ΔE = 3.190094 rad. The trial TOF is computed using Eq. (4.100) and g,ΔE, and a
t2−t1= g + a3
μ ΔE −sinΔE = 3,905 86 s = 65 0977 min
Hence the TOF error isτ1= 2.0977 min for the current iterate. We cannot use the secant method for the second iteration (we do not yet have past-iteration data), so we select a second trial value of p that is closer to pmax:
p2= 0 3pmin+ 0 7pmax= 8, 275 972 km
The p-iteration algorithm recalculates a, f, g, f ,ΔE, and the trial flight time using the above sequence of equations. The trial flight time is 27.52 min and the corresponding error is τ2=–35.48 min which is significantly worse than the first trial. However, the two trial values of p were chosen arbitrarily. The third trial value of p is computed using the secant search (4.101)
p3= p2−τ2
p2−p1
τ2−τ1
= 6, 216 625 km
This procedure repeats until the flight-time error is less than 10−4min (0.006 s). Table 4.2 summarizes the p-iteration scheme for the short-way transfer withΔθ = 135.40 . The converged value for parameter is p = 6,144.013 km and the corresponding semimajor axis is a = 7,193.6 km. We can compute the eccentricity of the short-way solution:
Table 4.2 p-iteration trials for the short-way transfer (Example 4.9).
Iteration Trial p
(km) Trial t2– t1
(min)
Flight-time error (min)
1 6,094.869 65.0977 2.0977
2 8,275.972 27.5198 –35.4802
3 6,216.626 60.1459 –2.8541
4 6,036.473 67.7868 4.7868
5 6,149.333 62.7814 –0.2186
6 6,144.404 62.9840 –0.0160
7 6,144.013 63.0001 5.82(10−5)
e= 1−p
a= 0 3820
Hence, the way orbit is not circular. Furthermore, the perigee radius of the short-way orbit is rp= p/(1 + e) = 4,445.8 km which is less than the radius of the Earth. Clearly, the short-way orbit is not feasible.
We can repeat the p-iteration steps for the long-way path whereΔθ= 224.60 . As pre-viously stated, the k, l, and m constants and limits pminand pmaxremain the same as those computed for the short-way path. The iterations corresponding to the long-way path are summarized in Table 4.3. The semimajor axis and eccentricity of the converged orbit solution are a = 7,184.60 km and e = 0.001, respectively, which indicate a near-circular orbit. Hence, the NOAA satellite follows the long-way path between positionsr1and r2. Figure 4.16 shows the long-way transfer on the nearly circular orbit.
The Lagrangian coefficients associated with the converged long-way orbit are
Table 4.3 p-iteration trials for the long-way transfer (Example 4.9).
Iteration Trial p
(km) Trial t2– t1
(min) Flight-time error
(min)
1 6,094.869 37.4175 –25.5825
2 8,275.972 134.6874 71.6874
3 6,668.512 48.3316 –14.6684
4 6,941.555 55.3307 –7.6693
5 7,240.742 65.0128 2.0128
6 7,178.545 62.7892 –0.2108
7 7,184.442 62.9948 –0.0052
8 7,184.592 63.0000 1.40(10−5)
1 2
r1
r2
TOF1-2= t2– t1= 63 min
∆θ = 224.6°
Perigee
Figure 4.16 Long-way transfer for a NOAA weather satellite (Example 4.9).
f=– 0 713771, g = – 677 398216 s, f = 7 273210 10−4 s−1, and g = – 0 710752 We can use Eqs. (4.86) and (4.87) to determine the initial and terminal velocity vectors in the ECI frame:
v1=1
g r2−f r1 =
−2 7381
−0 3474 6 9244
km/s
v2= fr1+g
g r2−f r1 =
−2 1670
−2 4422
−6 6865 km/s
Finally, we can use either state, (r1,v1) or (r2,v2), to determine the remaining orbital ele-ments. We find that the initial and terminal true anomalies areθ1=–40 and θ2= 184.6 as shown in Figure 4.16. The inclination of the NOAA satellite is i = 98.77 , which is a nearly polar orbit.
As a final note for this example, let us observe the trends in the flight time and orbital path at the limiting values of parameter. Figure 4.17 shows TOF error (trial flight time minus actual flight time) for short-way and long-way paths with pmin< p < pmax. For short-way paths with p pmin, the transfer becomes a very thin, long ellipse with a very large apogee distance as shown in Figure 4.18a. Because the satellite passes through apo-gee on the short-way path with p pmin(a“lofted” transfer; see Figure 4.18a), the flight
3000 4000 5000 6000 7000 8000 9000 10000 11000 Parameter, p, km
–100 0 100 200 300 400 500
Time-of-flight error, min
Short-way path
Long-way path
pmin pmax
Short-way
solution Long-way solution
Figure 4.17 Time-of-flight errors vs. parameter (Example 4.9).
time is very long and hence the TOF error is very large as seen in Figure 4.17. As p becomes larger and approaches pmax, the satellite passes through perigee on its short-way path and hence the flight time becomes smaller. When p = pmax, the transfer is a parabola and the short-way path passes through perigee as shown in Figure 4.18b.
For p > pmax, the short-way transfer becomes hyperbolic and the flight time continues to diminish. Figure 4.19 shows the long-way paths for p = pmin (parabola) and p pmax(elliptical transit through apogee). When using the limiting values for p, the apse directions and flight times of the long-way paths are essentially reversed when com-pared with the short-way paths.
Short-way p pmin
Short-way p = pmax
(a)
(b)
r1
r2 r2
r1
Figure 4.18 Short-way paths: (a) p pminand (b) p = pmax(Example 4.9).
Long -way p = pmin
(b)
r2
Long -way p pmax
r1
(a)
r2
r1
Figure 4.19 Long-way paths: (a) p = pminand (b) p pmax(Example 4.9).
Example 4.9 used a secant search to iterate on p; other root-solving methods (such as the bisection method or Brent’s method) may be used. As a practical matter, it may be useful to check the flight time at the limiting values of p (e.g., p = 1 001pmin and p= 0 999pmax) to ensure that the actual flight time is bracketed. Another search option is to employ a preliminary“brute-force” search where the flight-time error is computed for a small number of equally spaced values of p between pminand pmax. A good initial guess for p can then be determined by interpolating the TOF error data and the subse-quent secant search should rapidly converge to the solution.
It is important to restate that this section has presented an algorithm for solving Lam-bert’s problem when the orbit transfer is an ellipse. Extending these results to hyperbolic trajectories requires expressing the Lagrangian coefficients in terms of difference in hyperbolic anomalyΔF or using universal variables to accommodate all possible orbits with one set of equations. We will solve Lambert’s problem to obtain interplanetary transfers. As we shall see in Chapter 10, the transfer between two planetary orbits (such as an Earth–Mars transfer) is an ellipse with the sun as the primary gravitational body.
Therefore, we may use our “ellipse-only” formulation of Lambert’s problem (and the p-iteration method) to design trajectories for interplanetary missions.