lamberthub: a hub of Lambert's problem solvers

A Python library designed to provide solutions to Lambert's problem, a classical problem in astrodynamics that involves determining the orbit of a spacecraft given two points in space and the time of flight between them. The problem is essential for trajectory planning, particularly for interplanetary missions.

This library implements multiple algorithms, each named after its author and publication year, for solving different variations of Lambert's problem. These algorithms can handle different types of orbits, including multi-revolution paths and direct transfers.

Installation

Multiple installation methods are supported:

Logo	Platform	Command
	PyPI	python -m pip install lamberthub
	GitHub	python -m pip install https://github.com/jorgepiloto/lamberthub/archive/main.zip

Available solvers

Algorithm	Reference
gauss1809	C. F. Gauss, Theoria motus corporum coelestium in sectionibus conicis solem ambientium. 1809.
battin1984	R. H. Battin and R. M. Vaughan, “An elegant lambert algorithm,” Journal of Guidance, Control, and Dynamics, vol. 7, no. 6, pp. 662–670, 1984.
gooding1990	R. Gooding, “A procedure for the solution of lambert’s orbital boundary-value problem,” Celestial Mechanics and Dynamical Astronomy, vol. 48, no. 2, pp. 145–165, 1990.
der2011	G. J. Der, “The superior Lambert algorithm,” AMOS, Maui, Hawaii, 2011.
thorne2004	J. D. Thorne, “Lambert’s theorem: a complete series solution,” The Journal of the Astronautical Sciences, vol. 52, no. 4, pp. 441–454, 2004.
avanzini2008	G. Avanzini, “A simple lambert algorithm,” Journal of Guidance, Control, and Dynamics, vol. 31, no. 6, pp. 1587–1594, 2008.
arora2013	N. Arora and R. P. Russell, “A fast and robust multiple revolution lambert algorithm using a cosine transformation,” Paper AAS, vol. 13, p. 728, 2013.
vallado2013	D. A. Vallado, Fundamentals of astrodynamics and applications. Springer Science & Business Media, 2013, vol. 12.
izzo2015	D. Izzo, “Revisiting lambert’s problem,” Celestial Mechanics and Dynamical Astronomy, vol. 121, no. 1, pp. 1–15, 2015.
jiang2016	R. Jiang, T. Chao, S. Wang, and M. Yang, “Improved semi-major Axis iterated method for Lambert's problem,” 2016 IEEE Chinese Guidance, Navigation and Control Conference, pp. 1423–1428, 2016.
pan2016	B. Pan and Y. Ma, “Lambert’s problem and solution by non-rational Bézier functions,” Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, first published online Nov. 16, 2016.
delatorre2018	D. De La Torre, R. Flores, and E. Fantino, “On the solution of Lambert's problem by regularization,” Acta Astronautica, vol. 153, pp. 26–38, 2018.
negrete2024	A. Negrete and O. Abdelkhalik, “An Exact Solution to Lambert's Problem Using Contour Integrals.”
mcelreath2025	J. McElreath, I. M. Down, and M. Majji, “A universal approach for solving the multi-revolution Lambert's problem,” Celestial Mechanics and Dynamical Astronomy, vol. 137, 22, 2025.

Using a solver

Any Lambert's problem algorithm implemented in lamberthub is a Python function which accepts the following parameters:

from lamberthub import authorYYYY


v1, v2 = authorYYYY(
    mu, r1, r2, tof, M=0, is_prograde=True, is_low_path=True,  # Type of solution
    maxiter=35, atol=1e-5, rtol=1e-7, full_output=False  # Iteration config
)

where author is the name of the author which developed the solver and YYYY the year of publication. Any of the solvers hosted by the ALL_SOLVERS list.

Parameters

Parameters	Type	Description
mu	float	The gravitational parameter, that is, the mass of the attracting body times the gravitational constant.
r1	np.array	Initial position vector.
r2	np.array	Final position vector.
tof	float	Time of flight between initial and final vectors.
M	int	The number of revolutions. If zero (default), direct transfer is assumed.
is_prograde	bool	Controls the inclination of the final orbit. If True, inclination between 0 and 90 degrees. If False, inclination between 90 and 180 degrees.
is_low_path	bool	Selects the type of path when more than two solutions are available. No specific advantage unless there are mission constraints.
maxiter	int	Maximum number of iterations allowed when computing the solution.
atol	float	Absolute tolerance for the iterative method.
rtol	float	Relative tolerance for the iterative method.
full_output	bool	If True, returns additional information such as the number of iterations.

Returns

Returns	Type	Description
v1	np.array	Initial velocity vector.
v2	np.array	Final velocity vector.
numiter	int	Number of iterations (only if full_output is True).
tpi	float	Time per iteration (only if full_output is True).

Examples

Example: solving for a direct and prograde transfer orbit

Problem statement

Suppose you want to solve for the orbit of an interplanetary vehicle (that is Sun is the main attractor) form which you know that the initial and final positions are given by:

$$\vec{r_1} = \begin{bmatrix} 0.159321004 \\ 0.579266185 \\ 0.052359607 \end{bmatrix} \text{ [AU]} \quad \quad \vec{r_2} = \begin{bmatrix} 0.057594337 \\ 0.605750797 \\ 0.068345246 \end{bmatrix} \text{ [AU]} \quad \quad$$

The time of flight is $\Delta t = 0.010794065$ years. The orbit is prograde and direct, thus $M=0$. Remember that when $M=0$, there is only one possible solution, so the is_low_path flag does not play any role in this problem.

Solution

For this problem, gooding1990 is used. Any other solver would work too. Next, the parameters of the problem are instantiated. Finally, the initial and final velocity vectors are computed.

from lamberthub import gooding1990
import numpy as np


mu_sun = 39.47692641
r1 = np.array([0.159321004, 0.579266185, 0.052359607])
r2 = np.array([0.057594337, 0.605750797, 0.068345246])
tof = 0.010794065

v1, v2 = gooding1990(mu_sun, r1, r2, tof, M=0, is_prograde=True)
print(f"Initial velocity: {v1} [AU / years]")
print(f"Final velocity:   {v2} [AU / years]")

Result

Initial velocity: [-9.303608  3.01862016  1.53636008] [AU / years]
Final velocity:   [-9.511186  1.88884006  1.42137810] [AU / years]

Directly taken from An Introduction to the Mathematics and Methods of Astrodynamics, revised edition, by R.H. Battin, problem 7-12.

Performance comparison

Benchmarks run with pytest-benchmark. All Numba JIT functions are pre-compiled once before timing begins, compilation cost is never included in the measurements. Six test groups cover the main solver regimes:

Case group	What it tests
Zero-revolution nominal cases	Classic textbook cases, all solvers
Zero-revolution general cases	General elliptic, zero-revolution solvers
Zero-revolution hyperbolic cases	Hyperbolic transfers
One-revolution branch cases	First multi-revolution branch across prograde/retrograde and low/high paths
Near-minimum-energy multi-revolution cases	Near-minimum-energy cases with one and two revolutions
Near-tangent robustness cases	Near-tangent transfers that stress robustness

This section is auto-generated from CI benchmark artifacts.

• Generated: 2026-08-03 18:53 UTC
• Commit: 20a8135dd7d8
• Environment: Linux, Python 3.12.13, AMD EPYC 9V74 80-Core Processor

Times are in microseconds (lower is better). Speedup is relative to the slowest solver for each case. All solvers are JIT-warmed before timing begins.

Near-minimum-energy multi-revolution cases

Near-minimum-energy

Rank	Solver	Revolutions	prograde	Path	Median (µs)	Mean (µs)	IQR (µs)	Speedup	Rounds
1	mcelreath2025	1	Yes	high	47.2	48.0	0.6	13.9x	4323
2	der2011	1	Yes	high	53.4	54.3	0.8	12.3x	3842
3	izzo2015	1	Yes	high	60.1	61.0	0.9	10.9x	3431
4	gooding1990	1	Yes	high	94.6	96.7	1.8	6.9x	2191
5	negrete2024	1	Yes	high	192.7	195.0	3.0	3.4x	1046
6	arora2013	1	Yes	high	211.1	214.5	8.6	3.1x	975
7	delatorre2018	1	Yes	high	655.1	658.2	9.4	1.0x	317
1	mcelreath2025	1	Yes	low	47.2	48.0	0.6	15.0x	4326
2	izzo2015	1	Yes	low	61.4	62.5	1.0	11.5x	3361
3	der2011	1	Yes	low	79.2	80.6	1.3	8.9x	2614
4	gooding1990	1	Yes	low	94.2	96.3	1.9	7.5x	2164
5	negrete2024	1	Yes	low	192.4	194.9	3.0	3.7x	1048
6	arora2013	1	Yes	low	231.9	235.2	9.0	3.1x	878
7	delatorre2018	1	Yes	low	707.8	709.1	8.4	1.0x	289
1	mcelreath2025	2	Yes	high	47.6	48.7	0.7	13.8x	4263
2	der2011	2	Yes	high	49.2	50.0	0.8	13.4x	4177
3	izzo2015	2	Yes	high	61.0	62.0	0.9	10.8x	3368
4	gooding1990	2	Yes	high	97.0	99.0	1.7	6.8x	2142
5	negrete2024	2	Yes	high	193.8	196.5	3.5	3.4x	1038
6	arora2013	2	Yes	high	214.7	218.1	7.6	3.1x	952
7	delatorre2018	2	Yes	high	657.6	658.6	9.4	1.0x	315
1	mcelreath2025	2	Yes	low	47.2	48.1	0.6	14.5x	4335
2	der2011	2	Yes	low	50.1	62.5	15.9	13.7x	4163
3	izzo2015	2	Yes	low	60.3	61.2	0.9	11.4x	3382
4	gooding1990	2	Yes	low	96.7	98.7	1.8	7.1x	2108
5	negrete2024	2	Yes	low	194.1	196.2	2.5	3.5x	1038
6	arora2013	2	Yes	low	226.7	229.3	6.9	3.0x	902
7	delatorre2018	2	Yes	low	686.3	687.1	8.2	1.0x	300

One-revolution branch cases

Der article II

Rank	Solver	Revolutions	prograde	Path	Median (µs)	Mean (µs)	IQR (µs)	Speedup	Rounds
1	mcelreath2025	1	No	high	47.7	48.8	0.7	13.2x	4238
2	der2011	1	No	high	49.8	50.7	0.7	12.7x	4089
3	izzo2015	1	No	high	61.6	63.9	1.1	10.2x	3332
4	gooding1990	1	No	high	97.1	99.8	2.2	6.5x	2110
5	arora2013	1	No	high	147.5	150.3	4.1	4.3x	1393
6	negrete2024	1	No	high	194.3	197.3	3.9	3.2x	1039
7	delatorre2018	1	No	high	630.6	633.5	10.7	1.0x	324
1	mcelreath2025	1	No	low	48.0	49.1	0.7	12.9x	4248
2	der2011	1	No	low	50.1	51.0	0.7	12.3x	4097
3	izzo2015	1	No	low	62.1	63.3	1.1	10.0x	3305
4	gooding1990	1	No	low	97.0	99.5	2.3	6.4x	2132
5	arora2013	1	No	low	149.3	152.1	4.5	4.1x	1372
6	negrete2024	1	No	low	194.5	197.3	2.7	3.2x	1036
7	delatorre2018	1	No	low	618.5	620.4	10.2	1.0x	331
1	mcelreath2025	1	Yes	high	47.9	49.0	0.7	13.5x	4249
2	der2011	1	Yes	high	49.7	50.5	0.7	13.0x	4134
3	izzo2015	1	Yes	high	61.2	62.2	1.0	10.6x	3369
4	gooding1990	1	Yes	high	96.4	99.7	2.6	6.7x	2120
5	arora2013	1	Yes	high	157.2	160.0	4.2	4.1x	1303
6	negrete2024	1	Yes	high	196.8	201.5	8.9	3.3x	1026
7	delatorre2018	1	Yes	high	646.4	647.2	10.2	1.0x	317
1	mcelreath2025	1	Yes	low	47.8	48.8	0.7	13.0x	4285
2	der2011	1	Yes	low	49.9	50.8	0.7	12.5x	4087
3	izzo2015	1	Yes	low	61.6	62.7	1.0	10.1x	3310
4	gooding1990	1	Yes	low	97.7	100.1	2.2	6.4x	2067
5	arora2013	1	Yes	low	162.2	165.5	4.7	3.8x	1263
6	negrete2024	1	Yes	low	196.5	199.4	3.8	3.2x	1025
7	delatorre2018	1	Yes	low	623.1	625.2	9.3	1.0x	329

Near-tangent robustness cases

Near tangent

Rank	Solver	Revolutions	prograde	Path	Median (µs)	Mean (µs)	IQR (µs)	Speedup	Rounds
1	mcelreath2025	0	Yes	high	47.4	48.5	0.7	6.7x	4332
2	arora2013	0	Yes	high	54.6	55.4	0.7	5.8x	3753
3	izzo2015	0	Yes	high	60.0	61.0	1.0	5.3x	3438
4	der2011	0	Yes	high	65.7	66.9	1.0	4.8x	3176
5	gooding1990	0	Yes	high	79.5	81.0	1.2	4.0x	2591
6	negrete2024	0	Yes	high	128.4	130.2	1.2	2.5x	1572
7	delatorre2018	0	Yes	high	316.1	318.9	8.8	1.0x	644

Zero-revolution general cases

Curtiss book

Rank	Solver	Revolutions	prograde	Path	Median (µs)	Mean (µs)	IQR (µs)	Speedup	Rounds
1	battin1984	0	Yes	low	33.1	33.6	0.3	990.6x	6196
2	der2011	0	Yes	low	39.3	40.1	0.5	834.2x	5166
3	jiang2016	0	Yes	low	40.9	41.7	0.5	802.9x	5028
4	mcelreath2025	0	Yes	low	47.1	51.8	1.7	696.7x	4342
5	arora2013	0	Yes	low	55.7	56.8	0.7	588.8x	3672
6	izzo2015	0	Yes	low	60.1	61.0	1.0	546.3x	3411
7	gooding1990	0	Yes	low	79.1	80.5	1.2	415.0x	2581
8	thorne2004	0	Yes	low	97.5	99.2	1.9	336.5x	2099
9	avanzini2008	0	Yes	low	118.5	121.5	2.7	276.9x	1719
10	negrete2024	0	Yes	low	125.8	128.3	1.4	260.7x	1602
11	delatorre2018	0	Yes	low	315.2	318.3	9.2	104.1x	646
12	pan2016	0	Yes	low	32808.5	32679.4	483.2	1.0x	7

Der article I

Rank	Solver	Revolutions	prograde	Path	Median (µs)	Mean (µs)	IQR (µs)	Speedup	Rounds
1	battin1984	0	No	high	35.3	35.9	0.4	885.7x	5749
2	jiang2016	0	No	high	42.5	43.2	0.5	736.3x	4803
3	mcelreath2025	0	No	high	47.2	48.0	0.6	662.1x	4361
4	arora2013	0	No	high	56.6	57.6	0.7	552.5x	3599
5	der2011	0	No	high	57.2	58.3	1.0	546.4x	3614
6	izzo2015	0	No	high	60.8	61.7	0.9	514.3x	3375
7	gooding1990	0	No	high	80.8	82.5	1.4	387.1x	2500
8	thorne2004	0	No	high	101.2	103.1	2.1	309.0x	2020
9	avanzini2008	0	No	high	124.8	127.5	2.8	250.5x	1631
10	negrete2024	0	No	high	126.8	128.4	1.2	246.4x	1588
11	delatorre2018	0	No	high	314.5	318.4	9.5	99.4x	648
12	pan2016	0	No	high	31257.4	31297.2	134.9	1.0x	7
1	battin1984	0	Yes	low	36.6	37.1	0.3	857.8x	5568
2	der2011	0	Yes	low	39.9	40.6	0.5	785.5x	5099
3	jiang2016	0	Yes	low	42.0	42.9	0.5	747.1x	4864
4	mcelreath2025	0	Yes	low	45.5	46.2	0.6	689.1x	4319
5	arora2013	0	Yes	low	53.6	54.4	0.6	585.6x	3801
6	izzo2015	0	Yes	low	59.8	60.7	0.9	524.5x	3428
7	gooding1990	0	Yes	low	79.5	82.4	1.3	394.5x	2548
8	thorne2004	0	Yes	low	105.6	107.5	2.2	296.9x	1952
9	avanzini2008	0	Yes	low	125.4	127.8	2.7	250.2x	1635
10	negrete2024	0	Yes	low	127.4	128.8	1.1	246.2x	1581
11	delatorre2018	0	Yes	low	288.7	289.2	9.3	108.6x	721
12	pan2016	0	Yes	low	31364.5	31397.0	69.0	1.0x	7

Der article II

Rank	Solver	Revolutions	prograde	Path	Median (µs)	Mean (µs)	IQR (µs)	Speedup	Rounds
1	battin1984	0	No	high	35.5	36.0	0.3	1016.3x	5705
2	der2011	0	No	high	40.3	41.0	0.5	894.3x	5042
3	jiang2016	0	No	high	40.7	41.4	0.5	885.5x	5017
4	mcelreath2025	0	No	high	47.2	48.0	0.6	763.6x	4324
5	izzo2015	0	No	high	60.2	61.2	1.0	599.0x	3415
6	arora2013	0	No	high	63.3	64.5	0.8	569.4x	3219
7	gooding1990	0	No	high	81.8	83.3	1.2	440.4x	2486
8	thorne2004	0	No	high	90.2	92.2	2.0	399.5x	2274
9	avanzini2008	0	No	high	121.3	125.0	3.1	297.0x	1710
10	negrete2024	0	No	high	128.2	130.3	1.4	281.1x	1572
11	delatorre2018	0	No	high	296.7	299.8	8.9	121.5x	685
12	pan2016	0	No	high	36042.0	36043.5	146.6	1.0x	6
1	battin1984	0	Yes	high	37.2	37.8	0.4	976.3x	5472
2	der2011	0	Yes	high	39.7	40.4	0.5	912.6x	5130
3	jiang2016	0	Yes	high	40.5	41.2	0.5	896.1x	5024
4	mcelreath2025	0	Yes	high	47.2	48.2	0.6	768.2x	4392
5	arora2013	0	Yes	high	53.9	54.9	0.6	672.5x	3771
6	izzo2015	0	Yes	high	60.3	61.2	0.9	602.0x	3415
7	gooding1990	0	Yes	high	81.1	82.7	1.3	447.6x	2498
8	thorne2004	0	Yes	high	89.0	90.7	1.9	407.5x	2291
9	avanzini2008	0	Yes	high	128.3	131.7	3.6	282.8x	1588
10	negrete2024	0	Yes	high	128.3	129.8	1.2	282.8x	1570
11	delatorre2018	0	Yes	high	306.9	321.9	9.7	118.2x	659
12	pan2016	0	Yes	high	36276.9	36228.8	305.5	1.0x	6

Zero-revolution hyperbolic cases

GMAT hyperbolic

Rank	Solver	Revolutions	prograde	Path	Median (µs)	Mean (µs)	IQR (µs)	Speedup	Rounds
1	battin1984	0	No	low	35.4	35.9	0.3	1027.1x	5726
2	der2011	0	No	low	39.7	40.4	0.4	917.5x	5084
3	jiang2016	0	No	low	40.8	41.5	0.5	893.1x	4998
4	mcelreath2025	0	No	low	48.5	49.3	0.6	751.1x	4195
5	izzo2015	0	No	low	60.3	61.1	0.9	604.1x	3392
6	arora2013	0	No	low	62.6	63.5	1.0	581.2x	3284
7	gooding1990	0	No	low	80.8	82.2	1.0	450.3x	2552
8	thorne2004	0	No	low	101.2	103.2	2.1	359.6x	2030
9	avanzini2008	0	No	low	126.8	130.3	3.1	287.0x	1613
10	negrete2024	0	No	low	132.5	134.1	1.2	274.6x	1521
11	delatorre2018	0	No	low	329.7	331.9	8.9	110.4x	617
12	pan2016	0	No	low	36395.9	36396.0	49.4	1.0x	6
1	battin1984	0	Yes	low	34.5	35.0	0.3	1059.8x	5902
2	der2011	0	Yes	low	39.7	40.5	0.5	920.5x	5128
3	jiang2016	0	Yes	low	40.1	40.8	0.4	912.5x	5079
4	mcelreath2025	0	Yes	low	48.0	48.8	0.6	761.9x	4242
5	arora2013	0	Yes	low	53.4	54.2	0.9	685.1x	3912
6	izzo2015	0	Yes	low	59.8	60.7	0.9	611.6x	3426
7	gooding1990	0	Yes	low	80.7	82.0	1.1	453.2x	2544
8	thorne2004	0	Yes	low	99.6	101.7	2.0	367.1x	2065
9	avanzini2008	0	Yes	low	125.0	127.8	2.8	292.5x	1636
10	negrete2024	0	Yes	low	132.7	135.4	1.5	275.5x	1524
11	delatorre2018	0	Yes	low	335.0	334.9	13.2	109.2x	619
12	pan2016	0	Yes	low	36563.6	36566.5	135.8	1.0x	6

Zero-revolution nominal cases

Battin book

Rank	Solver	Revolutions	prograde	Path	Median (µs)	Mean (µs)	IQR (µs)	Speedup	Rounds
1	gauss1809	0	Yes	low	33.7	34.7	0.5	1059.5x	6087
2	battin1984	0	Yes	low	37.0	38.0	0.4	965.9x	5494
3	der2011	0	Yes	low	50.5	52.2	1.2	707.6x	4030
4	jiang2016	0	Yes	low	51.0	52.7	1.1	700.2x	4028
5	mcelreath2025	0	Yes	low	60.4	62.5	1.3	591.2x	3436
6	arora2013	0	Yes	low	78.9	81.2	1.9	452.7x	2634
7	izzo2015	0	Yes	low	83.0	85.5	1.7	430.4x	2439
8	gooding1990	0	Yes	low	104.1	114.2	2.9	342.9x	1980
9	thorne2004	0	Yes	low	126.8	130.7	5.1	281.7x	1628
10	negrete2024	0	Yes	low	127.4	130.8	4.2	280.2x	1583
11	vallado2013	0	Yes	low	131.6	134.5	4.4	271.4x	1530
12	avanzini2008	0	Yes	low	156.1	161.6	5.1	228.8x	1315
13	delatorre2018	0	Yes	low	309.0	312.3	14.9	115.6x	666
14	pan2016	0	Yes	low	35707.4	35730.6	209.5	1.0x	6

Vallado book

Rank	Solver	Revolutions	prograde	Path	Median (µs)	Mean (µs)	IQR (µs)	Speedup	Rounds
1	battin1984	0	Yes	low	39.1	40.6	0.6	8.8x	5273
2	gauss1809	0	Yes	low	52.6	54.2	1.0	6.5x	3860
3	jiang2016	0	Yes	low	56.1	58.2	1.4	6.1x	3686
4	mcelreath2025	0	Yes	low	61.5	63.9	1.4	5.6x	3402
5	vallado2013	0	Yes	low	67.6	70.3	1.0	5.1x	2990
6	der2011	0	Yes	low	69.7	72.2	1.5	4.9x	2919
7	arora2013	0	Yes	low	78.6	80.9	2.0	4.4x	2617
8	izzo2015	0	Yes	low	82.3	84.8	1.7	4.2x	2508
9	gooding1990	0	Yes	low	106.7	110.3	3.1	3.2x	1894
10	pan2016	0	Yes	low	115.9	119.4	3.7	3.0x	1775
11	thorne2004	0	Yes	low	125.1	129.0	5.0	2.7x	1641
12	negrete2024	0	Yes	low	128.4	131.9	4.4	2.7x	1569
13	avanzini2008	0	Yes	low	164.9	172.2	7.4	2.1x	1246
14	delatorre2018	0	Yes	low	343.7	347.3	14.1	1.0x	595