MULTIPLE REVOLUTION LAMBERT´S TARGETING PROBLEM: AN ANALYTICALAPPROXIMATION

Claudio Bombardelli, Juan Luis Gonzalo, and Javier Roa

Space Dynamics Group,School of Aeronautical Engineering,

Technical University of Madrid

ABSTRACTSolving a Lambert’s Targeting Problem (LTP) consists of ob-taining the minimum required single-impulse delta-V to mod-ify the trajectory of a spacecraft in order to transfer it to a se-lected orbital position in a fixed time of flight. We present anapproximate analytical method for rapidly solving a genericLTP with limited loss of accuracy. The solution is built on anoptimum single-impulse time-free transfer with a D-matrixphasing correction. Because it exhibits good accuracy (bothin terms of delta-V and time of flight) near locally optimumtransfer conditions the method is useful for rapidly obtaininglow delta-V solutions for interplanetary trajectory optimiza-tion. In addition, the method can be employed to provide afirst guess solution for enhancing the convergence speed ofan accurate numerical Lambert solver.

Index Terms— Lambert’s Problem, D Matrix, QuarticEquation, Targeting

1. INTRODUCTION

Lambert’s problem (LP), i.e. the determination of the fi-nite set of orbits linking two position vectors in a two-bodygravitational field with a specified transfer time, is of funda-mental importance in orbital mechanics and has been studiedfor more than two centuries. If one accounts for multiplerevolutions and both prograde and retrograde transfers Lam-bert’s problem provides 2

(NDub +NR

ub + 1)

solutions whereNDub,N

Rub are, respectively, the maximum number of revo-

lutions that a direct (D) and retrograde (R) orbits can havebefore reaching the target location[1]. That means that iflong-duration transfer arcs are sought the problem becomesmore difficult to solve, the longer so the transfer time consid-ered.

The solutions of a Lambert’s problem are to be obtainednumerically and are essential for various tasks in astrodynam-ics, including the design of interplanetary trajectories. Mostefforts in the literature have been devoted to the improvementof the convergence speed of its numerical solution using dif-ferent choices for the iteration variable and ways to improve

its initial guess. A quite detailed historical review on the dif-ferent contributions to the method can be found in the intro-duction of reference[1].

When focusing on the design and optimization of spacetrajectories one encounters a subclass of Lambert’s Problemreferred to as Lambert’s Targeting Problem (LTP). It consistsof obtaining the minimum required single-impulse delta-V tomodify the trajectory of a spacecraft in order to transfer it toa selected orbital position in a fixed time of flight. If the so-lutions of a Lambert’s Problem have been obtained the LTPis solved right away by picking the one providing minimumdelta-V magnitude within the maximum number of revolu-tions physically allowed by the transfer time constraint. Thereason for distinguishing between LP and LTP is mostly re-lated to the method proposed in the present work, which isconcerned with the solution of an LTP but not the general so-lution of an LP.

Solving an LTP with the highest possible computationalefficiency is key to the design of optimized interplanetary tra-jectories, and, in particular, the ones including multiple grav-ity assists and deep space maneuvers (DSMs). For these typesof problems, the complete sampling (grid search) of the multi-dimensional solution space implies the solution of a num-ber of LTPs often exceeding the capability of even the mostadvanced computational means available today. One exam-ple is the design and optimization of multiple gravity assisttrajectories with multiple DSMs like the Messenger trajec-tory to Mercury, which employed seven trajectory arcs withfive deterministic DSM with delta-V larger than 70 m/s[2].The sheer number of LTPs to be solved in this case makesthe problem computationally intractable if a brute-force, grid-sampling approach is to be followed (see Conway’s book[3],page 202). Another relevant application for a high-efficiencyLTP solver is the design of “inter-satellitary” trajectories atthe giant planets where multi-revolutions transfers are com-mon and the number of possible flyby sequences make thedesign and optimization task “computationally gigantic”[4].It is clear that even a modest increase of efficiency in an LTPsolution algorithm would be extremely beneficial when fac-

ing these computational challenges. In addition, any intrinsicpruning capability that the algorithm could offer in order todiscard unintresting transfer arcs a priory is highly welcome.

Recent advances towards a highly efficient LTP solverhave been made by these authors[5], who proposed an ap-proximate analytical solution of the LTP based on Battin’soptimum single-impulse transfer solution and a linear phas-ing correction through the use of a novel Keplerian error statetransition matrix[6],[7]. The solution provides a useful toolfor preliminary trajectory design when high accuracy is nota must as long as the main features of the solution are pre-served.

We review the steps leading to the derivation of the analyt-ical LTP solver providing additional formulas and examplesnot previously described.

2. ANALYTICAL LTP SOLVER

Let us consider the Lambert’s Targeting Problem (LTP) of aspacecraft departing from a point r1 with initial velocity v0

and arriving at a point r2 in a fixed time of flight ∆tL whilemoving in a Keplerian gravitational field of gravitational pa-rameter µ. The required delta-V targeting impulse ∆vL canbe broken down into an optimum time-free impulsive delta-V(∆vB) and a phsing correction term (∆vC):

∆vL = ∆vB + ∆vC . (1)

Time-free delta-VThe optimum time-free impulsive delta-V (∆vB) can be ob-tained analytically following the method proposed by Battin,which reduces to the solution of the quartic equation[8]:

x4 − Px3 +Qx− 1 = 0 (2)

where x > 0 is the square of the ratio between the skew-radial and skew-chordal components of the optimum depar-ture velocity

x2 =vρ1vc1

, (3)

while:

P =

√2r1r2µc

cos∆θ

2(v0π · ur1) ,

Q =

√2r1r2µc

cos∆θ

2(v0π · uc) ,

with r1, r2,∆θ, c indicating, respectively, the magnitudesof the departure and arrival location, the angular width of thetransfer arc and its chordal length, while v0π,ur1,uc are, re-spectively, the transfer plane component of the spacecraft ini-tial velocity v0, the radial unit vector at departure and thechordal unit vector from r1 to r2.

Eq. (2) is a quartic algebraic equation that can be solvedanalytically using different methods (e.g. by Lagrange resol-vent) or numerically (e.g. with Newton’s or Halley’s method).Once solved, the optimal chordal and radial components of v1

are determined using Eq. (3) and the following relation:

1

vc1=

√2r1r2µc

x cos∆θ

2, (4)

Having determined the optimum v1 the total single-impulse transfer velocity vector is obtained as:

∆vB = v1 − v0, (5)

while the angular momentum and eccentricity vector of thetransfer orbit are obtained as:

h = huh = r1 × v1

e = eue =v1 × h

µ− ur1.

The true and eccentric anomalies at r1 and r2 can now bedetermined from the relations:

cos θ1,2 = ur1,2 · ue; sin θ1,2 = (ue × ur1,2) · uh,

cosE1,2 =e− cos θ1,2

1 + e cos θ1,2; sinE1,2 =

√1− e2 sin θ1,21 + e cos θ1,2

.

The semi-major axis and mean motion are readily com-puted as:

a =r1

1− e cosE1, n =

õ

a3

Finally the transfer time satisfies Kepler’s equation:

∆tB =E2 − E1 − e (sinE2 − sinE1)

n.

For hyperbolic orbits (e > 1), all preceding relations arestill valid after considering the relation between hyperbolicand elliptic eccentric anomaly:

H = iE. (6)

D matrix targetingWell-designed transfer arcs are characterized by high phasingefficiencies or, equivalently, small magnitude velocity cor-rections ∆vC . When this is the case one can obtain a suf-ficiently accurate estimation of ∆vC starting from the opti-mum single-impulse transfer trajectory. The sought maneu-ver correction will be the one adjusting the trajectory phasing

without changing the orbit radius at the arrival true anomalyθ2.

The phasing correction to be applied is best estimated as:

δt = ∆tL −∆tB −N TB , (7)

whereN is the total number of full revolutions of the clos-est optimum single-impulse transfer:

N =

[∆tL −∆tB

TB

],

with [] denoting the nearest integer function.The velocity correction ∆vC associated with δt can be

obtained from the linear relation linking the radial positionshift (δr) and time delay (δt) of a Keplerian orbit at trueanomaly θ to the applied delta-V maneuver along the radial(δvr,1) and transversal direction (δvθ,1) at true anomaly θ1[6],[7]: [

δr (θ)δt (θ)

]≈ D (θ, θ1)

[δvr,1δvθ,1

]. (8)

The D matrix is essentially a Keplerian error state tran-sition matrix where time (t) is a state variable together withthe radial distance (r) and the radial and transversal velocity(vr, vθ) while the independent variable is the angular positionθ on the reference orbit (conveniently measured from the ref-erence orbit eccentricity vector).

The D matrix terms were first obtained in reference[6] andare here rewritten in a more compact way in terms of angularmomentum, radial distance and classical orbital elements ofthe reference orbit:

D1,1 =r2

hsin (θ − θ1) ,

D1,2 =r2r1h× 2− 2 cos (θ − θ1)− e sin θ1 sin (θ − θ1)

a (1− e2).

D2,1 =a4(1-e2

)2r1h2

[− (4∆CE − e∆C2E) (cosE1 − e)

+(6e∆E − 4

(1 + e2

)∆SE + e∆S2E

)sinE1

],

D2,2 =a4√

1− e24r1h2

[12(1− e2

)∆E − 3e2∆S2E

+6e3∆SE +(2(2− e2

)sinE1 − esin2E1

)×

(4∆CE − e∆C2E ) + (4 cosE1 − e cos 2E1)×(e∆S2E − 2∆SE

(2 − e2

))],

where:

∆E = E − E1

∆SE = sinE − sinE1

∆CE = cosE − cosE1

∆S2E = sin 2E − sin 2E1

∆C2E = cos 2E − cos 2E1.

The necessary targeting delta-V conditions are obtainedby inversion of Eq.(8) after setting zero radial error et en-counters and the required targeting time delay in Eq.(7). Inthis manner: [

δvr,1δvθ,1

]= D∗

[0δt

].

where we have introduced the “guidance matrix” D∗ = D−1.Finally, the sought velocity correction yields:

∆vC ≈ δvr,1ur1 + δvθ,1uθ. (9)

Eq(9), together with (5), provide the approximate solutionof the LTP (Eq.(1)).

Performance of the method

The performance of the method, in terms of accuracyand speed, has been tested by producing pork-chop plots ofdirect transfer trajectories to Mars and to the asteroid 65803Didymos[9]. In all cases, contour plots were produced forboth the departure energy (C3) and arrival excess velocity atthe target celestial body (∆V ). The plots were obtained bya 1000×1000 discretization of the departure epoch × timeof flight coordinates leading to a one million Lambert solvercalls. The analytical formulas derived in this article werecoded in fortran using a GNU gfortran compiler (version5.3.1) in simple precision with an Intel Core processor i7-4790@3.6GHz. The use of simple precision fits well withinthe scope of a preliminary optimization tool and its futureGPU implementation.

In order to assess the accuracy of the method and its com-putational speed three very fast Lambert solver algorithmswere considered: the classical Gooding’s method[10], therecently published Izzo’s method[11] and the Arora-Russellmethod[1]. The double-precision fortran versions of thetwo former methods were kindly made available by Jacob

Table 1. Simplified ephemerides for the considered celestial bodiesEarth Mars 65803 Didymos

reference epoch (MJD) 58849 58849 57200

semi-major axis (AU) 1.00 1.52 1.64

eccentricity 0.0167 0.0934 0.384

inclination (deg) 0.00280 1.85 3.41

argument of pericenter (deg) 287 285 319

longitude of ascending node (deg) 176 49.5 73.2

mean anomaly (deg) 357 247 190

Williams through the Fortran Astrodynamics Toolkit git-hub repository[12]. A double-precision fortran code for theArora-Russell method was kindly provided to the authors byNitin Arora and Ryan Russell.

Simplified ephemerides, reported in Table 1, were used inall cases in order to ease the reproducibility of the results.

It is important to underline that in order to obtain suf-ficiently accurate approximations the analytical LTP solverneeds to be run starting from two different optimum singleimpulse problems. For the departure C3 estimation problemthe targeting is clearly done over an optimum single impulseproblem of a trajectory starting from Earth. Conversely, forthe arrival excess velocity estimation problem a reversed tra-jectory starting from the target body and headed towards theEarth is employed (meaning that the sign of all velocity vectorcomponents has to be switched).

Mars trajectoriesFigures 1 and 2 show the analytically and numerically com-puted pork-chop plots for an Earth-Mars transfer in the years2025-2030. Contour plots are given for both departure energy(fig. 1) and arrival excess velocity at Mars (2). The qualityof the analytical pork-chop plot is remarkable and all mainfeatures appear to be well reproduced. In addition the delta-Vand launch date error for all main local optimum solutions arenegligible.

As for the computing time the method, provides a reduc-tion of a factor of 4.3, 5.1, 10.0 with respect to the methodof Arora-Russell, Izzo and Gooding, respectively. The maxi-mum number of revolutions for the previous methods (whichmildly affects Arora-Russell and Izzo’s method but does af-fect Gooding’s one considerably) was set to 2 in this case.

Didymos trajectoriesFigures 3 and 4 show the analytically and numerically com-puted pork-chop plots for a transfer to the binary asteroid65803 Didymos in the years 2019-2022. Contour plots aregiven for both departure energy (fig. 3) and arrival excess ve-locity (4). Similarly to the previous case the structure of thepork-chop plots is well reproduced by the analytical method

departure date [year]

transfe

r tim

e [days]

2025.5 2026 2026.5 2027 2027.5 2028 2028.5 2029 2029.5 2030100

200

300

400

500

600

700

800

900

1000

8

9

10

11

12

13

14

C3=7.8344 km

2/s

2

(tdep

05−08−2026)

(tdep

10−27−2026)

C3=9.1575 km

2/s

2

C3=8.8803 km

2/s

2

(tdep

11−27−2028)

C3=8.9610 km

2/s

2

(tdep

12−05−2028)

departure date [year]

transfe

r tim

e [days]

2025.5 2026 2026.5 2027 2027.5 2028 2028.5 2029 2029.5 2030100

200

300

400

500

600

700

800

900

1000

8

9

10

11

12

13

14

C3=7.8344 km

2/s

2

(tdep

05−08−2026)

C3=8.8803 km

2/s

2

(tdep

11−27−2028)

(tdep

10−27−2026)C

3=8.9610 km

2/s

2

(tdep

12−05−2028)

C3=9.1599 km

2/s

2

Fig. 1. Numerical (top) and analytical (bottom) pork-chopplot for the injection energy (C3) of an Earth-Mars transfer.

departure date [year]

transfe

r tim

e [days]

2025.5 2026 2026.5 2027 2027.5 2028 2028.5 2029 2029.5 2030100

200

300

400

500

600

700

800

900

1000

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

∆V=2.7444 km/s

(tdep

04−14−2026)

∆V=3.5716 km/s(t

dep 12−16−2025)

∆V=2.8267 km/s

∆V=2.3323 km/s

∆V=2.9493 km/s

∆V=2.5592 km/s

(tdep

04−22−2026)

∆V=2.3764 km/s

(tdep

07−15−2026)

(tdep

08−29−2028)

(tdep

09−30−2028)

(tdep

07−22−2028)

transfe

r tim

e [days]

departure date [year]2025.5 2026 2026.5 2027 2027.5 2028 2028.5 2029 2029.5 2030

100

200

300

400

500

600

700

800

900

1000

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

(tdep

09−30−2028)

(tdep

07−15−2026)

∆V=2.3764 km/s(t

dep 04−14−2026)

∆V=2.7444 km/s

(tdep

07−22−2028)∆V=2.3324 km/s

∆V=2.9465 km/s

(tdep

08−28−2028)∆V=2.5588 km/s

∆V=3.5650 km/s

∆V=2.9462 km/s

(tdep

12−14−2025) (tdep

04−20−2026)

Fig. 2. Numerical (top) and analytical (bottom) pork-chopplot for the arrival excess velocity (∆V ) of an Earth-Marstransfer.

although the accuracy of the delta-V and launch date nearminimum condition is not as remarkable as in the Mars case,especially for the C3 plot. This is due to the low phasing effi-ciency (as low as ∼67%) of some of the best solutions found.In any case, the error remains within the limit of a few per-

departure date [year]

tra

nsfe

r tim

e [

da

ys]

2019.5 2020 2020.5 2021 2021.5 2022100

200

300

400

500

600

700

800

900

5

10

15

20

25

30

35

(tdep

01−22−2021)

C3=1.8545 km

2/s

2

C3=2.1082 km

2/s

2

C3=3.1844 km

2/s

2

(tdep

08−16−2020)

(tdep

08−26−2021)

C3=24.948 km

2/s

2

(tdep

10−28−2020)

departure date [year]

tra

nsfe

r tim

e [

da

ys]

2019.5 2020 2020.5 2021 2021.5 2022100

200

300

400

500

600

700

800

900

5

10

15

20

25

30

35

(tdep

08−16−2020)

C3=1.9623 km

2/s

2

(tdep

08−25−2021)

C3=1.7534 km

2/s

2

C3=2.8281 km

2/s

2

(tdep

01−24−2021)

C3=25.150 km

2/s

2

(tdep

10−27−2020)

Fig. 3. Numerical (top) and analytical (bottom) pork-chopplot for the injection energy (C3) of an Earth-Didymos trans-fer.

departure date [year]

tra

nsfe

r tim

e [

da

ys]

2019.5 2020 2020.5 2021 2021.5 2022100

200

300

400

500

600

700

800

900

1

1.5

2

2.5

3

3.5

4

4.5

∆V=0.6552 km/s

∆V=0.6552 km/s

(tdep

11−27−2020)

(tdep

11−21−2020)

departure date [year]

tra

nsfe

r tim

e [

da

ys]

2019.5 2020 2020.5 2021 2021.5 2022100

200

300

400

500

600

700

800

900

1

1.5

2

2.5

3

3.5

4

4.5

(tdep

11−21−2020)

(tdep

11−27−2020)

∆V=0.6544 km/s

∆V=0.6815 km/s

Fig. 4. Numerical (top) and analytical (bottom) pork-chopplot for the arrival excess velocity (∆V ) of an Earth-Didymostransfer.

cent, which may still be acceptable at a preliminary missiondesign stage.

The computational speed reduction factor in this case is

4.0, 5.3, 11.1 with respect to the method of Arora-Russell,Izzo and Gooding, respectively. The difference when com-pared to the Mars case is due to the problem dependency ofthe numerical methods (the computational time of the ana-lytical method is virtually constant). For problems involvingmulti-rev solutions with a high number of revolutions the dif-ference in performance may be substantial.

3. CONCLUSIONS

A novel approximate analytical solution of the Lambert’s Tar-geting Problem (LTP) was presented and investigated. Al-though not yet extensively tested, the solution appears to beable to reproduce the most interesting parts of a generic pork-chop plot for interplanetary transfer trajectories. In particular,it offers remarkably good accuracy (considering its analyti-cal character) in both estimated delta-V and launch date atlocal minimum points. As for computational speed, the pro-posed solution is at least 4 times faster than the most efficientLambert solvers currently available and more than one orderof magnitude faster than Gooding’s method. A limited efforthas been done in order to increase the efficiency of its imple-mentation, which opens interesting possibilities for the future.The limited accuracy of the method, which is obviously itsmain drawback, should be traded off with the need of compu-tationally effective solutions when dealing with massive andhighly complex optimization problems. Many of these as-pects will need to be studied extensively in the future.

AcknowledgementsThis work has been supported by the Spanish Ministry ofEconomy and Competitiveness within the framework of theresearch project “Dynamical Analysis, Advanced OrbitalPropagation, and Simulation of Complex Space Systems”(ESP2013-41634-P). The authors also want to thank thefunding received from the European Union Seventh Frame-work Programme (FP7/2007-2013) under grant agreementN 607457 (LEOSWEEP). Last but not least, we thank RyanRussell (University of Texas at Austin), Nitin Arora (JetPropulsion Laboratory) and Jacob Williams (NASA Johnson)for kindly providing fortran implementations of differentLambert solvers.

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[6] Claudio Bombardelli, “Analytical formulation of im-pulsive collision avoidance dynamics,” Celestial Me-chanics and Dynamical Astronomy, vol. 118, no. 2, pp.99–114, 2014. 1, 2, 2

[7] Claudio Bombardelli and Javier Hernando-Ayuso, “Op-timal impulsive collision avoidance in low earth orbit,”Journal of Guidance, Control, and Dynamics, vol. 38,no. 2, pp. 217–225, 2015. 1, 2

[8] Richard H. Battin, An introduction to the mathematicsand methods of astrodynamics, Aiaa, 1999. 2

[9] A Galvez, I Carnelli, P Michel, AF Cheng, C Reed,S Ulamec, J Biele, P Zbrll, and R Landis, “Aida: The as-teroid impact & deflection assessment mission,” in Eu-ropean Planetary Science Congress, EPSC2013-1043,2013. 2

[10] R.H Gooding, “A procedure for the solution of lambert’sorbital boundary-value problem,” Celestial Mechanicsand Dynamical Astronomy, vol. 48, no. 2, pp. 145–165,1990. 2

[11] Dario Izzo, “Revisiting lambert’s problem,” CelestialMechanics and Dynamical Astronomy, vol. 121, no. 1,pp. 1–15, 2014. 2

[12] Jacob Williams, “Fortran-astrodynamics-toolkit,”https://github.com/jacobwilliams/Fortran-Astrodynamics-Toolkit, Accessed:2016-02-02. 2