Package org.orekit.control.heuristics.lambert

Class LambertSolver

java.lang.Object

org.orekit.control.heuristics.lambert.LambertSolver

public class LambertSolver
extends Object

Lambert solver, assuming Keplerian motion.

An orbit is determined from two position vectors.

References: Battin, R.H., An Introduction to the Mathematics and Methods of Astrodynamics, AIAA Education, 1999. Lancaster, E.R. and Blanchard, R.C., A Unified Form of Lambert’s Theorem, Goddard Space Flight Center, 1968.

Since:

13.1

Author:

Joris Olympio, Romain Serra

See Also:

LambertBoundaryConditions, LambertBoundaryVelocities

Constructor Summary

Constructors

Constructor	Description
LambertSolver(double mu)	Creator.

Method Summary

Modifier and Type	Method	Description
RealMatrix	computeJacobian(boolean posigrade, int nRev, LambertBoundaryConditions boundaryConditions)	Computes the Jacobian matrix of the Lambert solution.
LambertBoundaryVelocities	solve(boolean posigrade, int nRev, LambertBoundaryConditions boundaryConditions)	Solve for the corresponding velocity vectors given two position vectors and a duration.
static Vector2D	solveNormalized2D(double r1, double r2, double dth, double tau, int mRev)	Lambert's solver for the historical, planar problem.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

LambertSolver

public LambertSolver(double mu)

Creator.

Parameters:

mu - gravitational constant

Method Detail

solve

public LambertBoundaryVelocities solve(boolean posigrade,
                                       int nRev,
                                       LambertBoundaryConditions boundaryConditions)

Solve for the corresponding velocity vectors given two position vectors and a duration.

The logic for setting posigrade and nRev is that the sweep angle Δυ travelled by the object between t1 and t2 is 2π nRev +1 - α if posigrade is false and 2π nRev + α if posigrade is true, where α is the separation angle between p1 and p2, which is always computed between 0 and π (because in 3D without a normal reference, vector angles cannot go past π).

This implies that posigrade should be set to true if p2 is located in the half orbit starting at p1 and it should be set to false if p2 is located in the half orbit ending at p1, regardless of the number of periods between t1 and t2, and nRev should be set accordingly.

As an example, if t2 is less than half a period after t1, then posigrade should be true and nRev should be 0. If t2 is more than half a period after t1 but less than one period after t1, posigrade should be false and nRev should be 0.

If solving fails completely, null is returned. If only the computation of terminal velocity fails, a partial pair of velocities is returned (with some NaNs).

Parameters:

posigrade - flag indicating the direction of motion

nRev - number of revolutions

boundaryConditions - Lambert problem boundary conditions

Returns:

boundary velocity vectors

solveNormalized2D

public static Vector2D solveNormalized2D(double r1,
                                         double r2,
                                         double dth,
                                         double tau,
                                         int mRev)

Lambert's solver for the historical, planar problem. Assume mu=1.

Parameters:

r1 - radius 1

r2 - radius 2

dth - sweep angle

tau - time of flight

mRev - number of revs

Returns:

velocity at departure in (T, N) basis. Is Vector2D.NaN if solving fails

computeJacobian

public RealMatrix computeJacobian(boolean posigrade,
                                  int nRev,
                                  LambertBoundaryConditions boundaryConditions)

Computes the Jacobian matrix of the Lambert solution. The rows represent the initial and terminal velocity vectors. The columns represent the parameters: initial time, initial position, terminal time, terminal velocity.

Reference: Di Lizia, P., Armellin, R., Zazzera, F. B., and Berz, M. High Order Expansion of the Solution of Two-Point Boundary Value Problems using Differential Algebra: Applications to Spacecraft Dynamics.

Parameters:

posigrade - direction flag

nRev - number of revolutions

boundaryConditions - Lambert boundary conditions

Returns:

Jacobian matrix