lambert: Solve Lambert's problem to determine a transfer orbit

In Rafael-Ayala/asteRisk: Computation of Satellite Position

Solve Lambert's problem to determine a transfer orbit

Description

Given 2 position vectors and a time difference between them, calculates the velocity that the object should have at the initial and final states to follow a transfer orbit in the target time (Lambert's problem). The transfer orbit can be elliptical, parabolic or hyperbolic, all of which are dealt with. In the case of elliptical orbits, multi-revolution orbits are possible. Additionally, low-path and high-path orbits are also possible. All possible solutions are returned for the case of elliptical transfers. The user must specify if a retrograde transfer is desired. By default, prograde transfers are calculated. Currently, the formulation by Dario Izzo is applied to solve Lambert's problem (https://link.springer.com/article/10.1007/s10569-014-9587-y).

Usage

lambert(initialPosition, finalPosition, initialTime, finalTime, retrogradeTransfer=FALSE, 
        centralBody="Earth", maxIterations=2000, atol=0.00001, rtol=0.000001)

Arguments

initialPosition	Vector with the 3 components of the initial position in Cartesian coordinates, in meters.
finalPosition	Vector with the 3 components of the initial position in Cartesian coordinates, in meters.
initialTime	Either date-time string in UTC indicating the time corresponding to the initial state vector of the satellite, or numeric value indicating the starting time in seconds since an arbitrary reference instant. If provided as a date-time string in UTC, finalTime can be provided either as another date-time string in UTC (in which case the transfer time will be determined as the difference between the 2 date-time strings), or as a numeric value indicating the seconds ellapsed since initialTime. If provided as a numeric value, finalTime can only be provided as another numeric value indicating the number of seconds since the same arbitrary reference for objects in deep space, and also for objects near Earth if targetTime is provided as a date-time string.
finalTime	Either date-time string in UTC indicating the time corresponding to the final state vector of the satellite, or numeric value indicating the final time in seconds. If initialTime was provided as a date-time string in UTC, then finalTime can be provided as either of the two (date-time string or numeric value in seconds). In this case, providing finalTime as a numeric value will be interpreted as the number of seconds since the instant specified for initialTime. If initialTime was provided as a numeric value (indicating the time in seconds since an arbitrary reference point), then finalTime can only be provided as another numeric value, which will be interpreted as the number of seconds since the same reference instant as that used for initialTime, and therefore the transfer time will be calculated as the difference between initialTime and finalTime.
retrogradeTransfer	Logical indicating if retrograde transfer orbits should be calculated, i.e., transfer orbits with an inclination higher than 180º. By default, retrogradeTransfer=FALSE, and prograde transfer orbits are calculated.
centralBody	String indicating the central body around which the satellite is orbiting. Can be one of c("Sun", "Mercury", "Venus", "Earth", "Moon", "Mars", "Jupiter", "Saturn", "Uranus", "Neptune") (case insensitive).
maxIterations	Maximum number of iterations to perform at the different root-finding steps when solving Lambert's problem.
atol	Absolute tolerance value used for the convergence criterion at the different root-finding steps when solving Lambert's problem.
rtol	Relative tolerance value used for the convergence criterion at the different root-finding steps when solving Lambert's problem.

Value

A list with a number of elements equal to the number of possible transfer orbits found. For hyperbolic and parabolic transfer orbits, there will always be a single possible transfer orbit. For elliptic transfer orbits, multi-revolution orbits may exist if the transfer time is large enough. If they are possible, they will be calculated together with the basic, single-revolution orbit. Each element of the top-level list is in itself a list with the following elements:

numberRevs	Number of complete revolutions that will be performed in this transfer orbit. Always equals 0 for parabolic and hyperbolic transfer orbits, as well as for the non-multirevolution orbit of elliptic transfers.
path	String indicating if the transfer orbit is of the high-path type (i.e., has its second focus located beyond the vector connecting the initial and final positions) or of the low-path type (second focus located between this vector and the first focus).
orbitType	String indicating the type of transfer orbit (elliptic, parabolic or hyperbolic).
v1	Velocity vector in m/s that the object should have at the start of the transfer orbit.
v2	Velocity vector in m/s that the object should have at the end of the transfer orbit.

References

https://link.springer.com/article/10.1007/s10569-014-9587-y

Examples

# Consider the following initial and final positions:
initialPosition <- c(15945.34, 0, 0) * 1000
finalPosition <- c(12214.83899, 10249.46731, 0) * 1000
# Given a time difference of 76 minutes between the 2 states, calculate the
# velocity that the spacecraft should have at the beginning and end of the
# transfer orbit
lambertSolution <- lambert(initialPosition, finalPosition, 0, 76*60)
length(lambertSolution)
lambertSolution[[1]]$orbitType
lambertSolution[[1]]$v1
lambertSolution[[1]]$v2

# A single transfer orbit is possible (elliptic, single-revolution orbit)

Rafael-Ayala/asteRisk documentation built on May 16, 2024, 5:24 p.m.