RichardsonExpansion.java
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package org.orekit.orbits;
import org.hipparchus.geometry.euclidean.threed.Vector3D;
import org.hipparchus.util.FastMath;
import org.hipparchus.util.SinCos;
import org.orekit.bodies.CR3BPSystem;
import org.orekit.errors.OrekitException;
import org.orekit.errors.OrekitMessages;
import org.orekit.utils.LagrangianPoints;
import org.orekit.utils.PVCoordinates;
/** Class implementing the Third-Order Richardson Expansion.
* @see "Dynamical systems, the three-body problem, and space mission design, Koon, Lo, Marsden, Ross"
* @author Vincent Mouraux
* @since 10.2
*/
public class RichardsonExpansion {
/** Distance between a Lagrangian Point and its closest primary body, meters. */
private final double gamma;
/** Mass ratio of the CR3BP System. */
private final double mu;
/** Distance between the two primary bodies, meters. */
private final double dDim;
/** Halo Orbit frequency. */
private final double wp;
/** Simple Halo Orbit coefficient. */
private final double k;
/** Simple Richardson coefficient. */
private final double a21;
/** Simple Richardson coefficient. */
private final double a22;
/** Simple Richardson coefficient. */
private final double a23;
/** Simple Richardson coefficient. */
private final double a24;
/** Simple Richardson coefficient. */
private final double b21;
/** Simple Richardson coefficient. */
private final double b22;
/** Simple Richardson coefficient. */
private final double d21;
/** Simple Richardson coefficient. */
private final double a31;
/** Simple Richardson coefficient. */
private final double a32;
/** Simple Richardson coefficient. */
private final double b31;
/** Simple Richardson coefficient. */
private final double b32;
/** Simple Richardson coefficient. */
private final double d31;
/** Simple Richardson coefficient. */
private final double d32;
/** Simple Richardson coefficient. */
private final double s1;
/** Simple Richardson coefficient. */
private final double s2;
/** Simple Richardson coefficient. */
private final double l1;
/** Simple Richardson coefficient. */
private final double l2;
/** Simple Halo Orbit coefficient. */
private final double delta;
/** Orbital Period of the Halo Orbit, seconds. */
private double orbitalPeriod;
/** Lagrangian Point considered. */
private LagrangianPoints point;
/** CR3BP System considered. */
private CR3BPSystem cr3bpSystem;
/** Simple Constructor.
* @param cr3bpSystem CR3BP System considered
* @param point Lagrangian Point considered
*/
public RichardsonExpansion(final CR3BPSystem cr3bpSystem,
final LagrangianPoints point) {
this.point = point;
this.cr3bpSystem = cr3bpSystem;
this.mu = cr3bpSystem.getMassRatio();
this.dDim = cr3bpSystem.getDdim();
this.gamma = cr3bpSystem.getGamma(point);
final double c2 = getCn(2.0);
final double c3 = getCn(3.0);
final double c4 = getCn(4.0);
wp = FastMath.sqrt(0.5 * (2.0 - c2 + FastMath.sqrt(9.0 * c2 * c2 - 8.0 * c2)));
final double ld = FastMath.sqrt(0.5 * (2.0 - c2 + FastMath.sqrt(9.0 * c2 * c2 - 8.0 * c2)));
k = (wp * wp + 1.0 + 2.0 * c2) / (2.0 * wp);
final double d1 = 3.0 * ld * ld * (k * (6.0 * ld * ld - 1.0) - 2.0 * ld) / k;
final double d2 = 8.0 * ld * ld * (k * (11.0 * ld * ld - 1.0) - 2.0 * ld) / k;
a21 = 3.0 * c3 * (k * k - 2.0) / (4.0 * (1.0 + 2.0 * c2));
a22 = 3.0 * c3 / (4.0 * (1.0 + 2.0 * c2));
a23 = -3.0 * c3 * ld * (3.0 * k * k * k * ld - 6.0 * k * (k - ld) + 4.0) / (4.0 * k * d1);
a24 = -3.0 * c3 * ld * (2.0 + 3.0 * k * ld) / (4.0 * k * d1);
b21 = -3.0 * c3 * ld * (3.0 * k * ld - 4.0) / (2.0 * d1);
b22 = 3.0 * c3 * ld / d1;
d21 = -c3 / (2.0 * ld * ld);
a31 =
-9.0 * ld * (4.0 * c3 * (k * a23 - b21) + k * c4 * (4.0 + k * k)) / (4.0 * d2) +
(9.0 * ld * ld + 1.0 - c2) / (2.0 * d2) * (2.0 * c3 * (2.0 * a23 - k * b21) + c4 * (2.0 + 3.0 * k * k));
a32 =
-9.0 * ld * (4.0 * c3 * (k * a24 - b22) + k * c4) / (4.0 * d2) -
3 * (9.0 * ld * ld + 1.0 - c2) * (c3 * (k * b22 + d21 - 2.0 * a24) - c4) / (2.0 * d2);
b31 =
3.0 * 8.0 * ld * (3.0 * c3 * (k * b21 - 2.0 * a23) - c4 * (2.0 + 3.0 * k * k)) / (8.0 * d2) +
3.0 * ((9.0 * ld * ld + 1.0 + 2.0 * c2) * (4.0 * c3 * (k * a23 - b21) + k * c4 * (4.0 + k * k))) / (8.0 * d2);
b32 =
9.0 * ld * (c3 * (k * b22 + d21 - 2.0 * a24) - c4) / d2 +
3.0 * ((9.0 * ld * ld + 1.0 + 2.0 * c2) * (4.0 * c3 * (k * a24 - b22) + k * c4)) / (8.0 * d2);
d31 = 3.0 / (64.0 * ld * ld) * (4.0 * c3 * a24 + c4);
d32 = 3.0 / (64.0 * ld * ld) * (4.0 * c3 * (a23 - d21) + c4 * (4.0 + k * k));
s1 =
(3.0 * c3 * (2.0 * a21 * (k * k - 2.0) - a23 * (k * k + 2.0) - 2.0 * k * b21) / 2.0 -
3.0 * c4 * (3.0 * k * k * k * k - 8.0 * k * k + 8.0) / 8.0) / (2.0 * ld * (ld * (1.0 + k * k) - 2.0 * k));
s2 =
(3.0 * c3 * (2.0 * a22 * (k * k - 2.0) + a24 * (k * k + 2.0) + 2.0 * k * b22 + 5.0 * d21) / 2.0 +
3.0 * c4 * (12.0 - k * k) / 8.0) / (2.0 * ld * (ld * (1.0 + k * k) - 2.0 * k));
l1 = -3.0 * c3 * (2.0 * a21 + a23 + 5.0 * d21) / 2.0 - 3.0 * c4 * (12.0 - k * k) / 8.0 + 2.0 * ld * ld * s1;
l2 = 3.0 * c3 * (a24 - 2.0 * a22) / 2.0 + (9.0 * c4 / 8.0) + (2.0 * ld * ld * s2);
delta = wp * wp - c2;
}
/** Calculate Cn Richardson Coefficient.
* @param order Order 'n' of the coefficient needed.
* @return cn Cn Richardson Coefficient
*/
private double getCn(final double order) {
final double gamma3 = gamma * gamma * gamma;
final double cn;
switch (point) {
case L1:
cn = (1.0 / gamma3) * (mu + FastMath.pow(-1, order) * (1 - mu) * FastMath.pow(gamma, order + 1) / FastMath.pow(1 - gamma, order + 1));
break;
case L2:
cn = (1.0 / gamma3) * (FastMath.pow(-1, order) * mu + FastMath.pow(-1, order) * (1 - mu) * FastMath.pow(gamma, order + 1) / FastMath.pow(1 + gamma, order + 1));
break;
default:
throw new OrekitException(OrekitMessages.CANNOT_COMPUTE_LAGRANGIAN, point);
}
return cn;
}
/** Calculate first Guess.
* @param azr z-axis Amplitude of the required Halo Orbit, meters
* @param type type of the Halo Orbit ("Northern" or "Southern")
* @param t Orbit time, seconds (must be greater than 0)
* @param phi Orbit phase, rad
* @return firstGuess PVCoordinates of the first guess
*/
public PVCoordinates computeHaloFirstGuess(final double azr, final LibrationOrbitFamily type,
final double t, final double phi) {
// Z-Axis Halo Orbit Amplitude
final double az = azr / (gamma * dDim);
// X-Axis Halo Orbit Amplitude
final double ax = FastMath.sqrt((delta + l2 * az * az) / -l1);
final double nu = 1.0 + s1 * ax * ax + s2 * az * az;
final double tau = nu * t;
final double tau1 = wp * tau + phi;
final double m = (type == LibrationOrbitFamily.NORTHERN) ? 1 : 3;
final double dm = 2 - m;
final SinCos scTau1 = FastMath.sinCos(tau1);
final SinCos sc2Tau1 = SinCos.sum(scTau1, scTau1);
final SinCos sc3Tau1 = SinCos.sum(scTau1, sc2Tau1);
// First guess position relative to its Lagrangian point
final double firstx =
a21 * ax * ax +
a22 * az * az - ax * scTau1.cos() +
(a23 * ax * ax - a24 * az * az) * sc2Tau1.cos() +
(a31 * ax * ax * ax - a32 * ax * az * az) * sc3Tau1.cos();
final double firsty =
k * ax * scTau1.sin() +
(b21 * ax * ax - b22 * az * az) * sc2Tau1.sin() +
(b31 * ax * ax * ax - b32 * ax * az * az) * sc3Tau1.sin();
final double firstz =
dm * az * scTau1.cos() +
dm * d21 * ax * az * (sc2Tau1.cos() - 3.0) +
dm * (d32 * az * ax * ax - d31 * az * az * az) * sc3Tau1.cos();
// First guess Velocity relative to its Lagrangian point
final double vx =
ax * wp * nu * scTau1.sin() -
2.0 * (a23 * ax * ax - a24 * az * az) * wp * nu * sc2Tau1.sin() -
3.0 * (a31 * ax * ax * ax - a32 * ax * az * az) * wp * nu * sc3Tau1.sin();
final double vy =
k * ax * wp * nu * scTau1.cos() +
2.0 * wp * nu * (b21 * ax * ax - b22 * az * az) * sc2Tau1.cos() +
3.0 * wp * nu * (b31 * ax * ax * ax - b32 * ax * az * az) * sc3Tau1.cos();
final double vz =
-dm * az * wp * nu * scTau1.sin() -
2.0 * dm * d21 * ax * az * wp * nu * sc2Tau1.sin() -
3.0 * dm * (d32 * az * ax * ax - d31 * az * az * az) * wp * nu * sc3Tau1.sin();
// Return PV Coordinates
return point == LagrangianPoints.L1 ?
new PVCoordinates(new Vector3D(firstx * gamma + 1.0 - mu - gamma, firsty * gamma, firstz * gamma), new Vector3D(vx * gamma, vy * gamma, vz * gamma)) :
new PVCoordinates(new Vector3D(firstx * gamma + 1.0 - mu + gamma, firsty * gamma, firstz * gamma), new Vector3D(vx * gamma, vy * gamma, vz * gamma));
}
/** Calculate first Guess.
* @param ayr x-axis Amplitude of the required Lyapunov Orbit, meters
* @param t time
* @param phi Orbit phase, rad
* @return firstGuess PVCoordinates of the first guess
*/
public PVCoordinates computeLyapunovFirstGuess(final double ayr, final double t, final double phi) {
// Z-Axis Lyapunov Orbit Amplitude
final double az = 0;
// y-Axis Lyapunov Orbit Amplitude
final double ay = ayr / (gamma * dDim);
final double ax = ay / k;
final double nu = 1.0 + s1 * ax * ax + s2 * az * az;
final double tau = nu * t;
final double tau1 = wp * tau + phi;
// First guess position relative to its Lagrangian point
final double firstx = -ax * FastMath.cos(tau1);
final double firsty = 0.0;
final double firstz = 0.0;
// First guess Velocity relative to its Lagrangian point
final double vx = 0.0;
final double vy = k * ax * wp * nu * FastMath.cos(tau1);
final double vz = 0.0;
// Return PV Coordinates
return point == LagrangianPoints.L1 ?
new PVCoordinates(new Vector3D(firstx * gamma + 1.0 - mu - gamma, firsty * gamma, firstz * gamma),
new Vector3D(vx * gamma, vy * gamma, vz * gamma)) :
new PVCoordinates(new Vector3D(firstx * gamma + 1.0 - mu + gamma, firsty * gamma, firstz * gamma),
new Vector3D(vx * gamma, vy * gamma, vz * gamma));
}
/** Return the orbital period of the Halo Orbit.
* @param azr z-axis Amplitude of the required Halo Orbit, meters
* @return the orbitalPeriod
*/
public double getHaloOrbitalPeriod(final double azr) {
final double az = azr / (gamma * dDim);
final double ax = FastMath.sqrt((delta + l2 * az * az) / -l1);
final double nu = 1.0 + s1 * ax * ax + s2 * az * az;
orbitalPeriod = FastMath.abs(2.0 * FastMath.PI / (wp * nu));
return orbitalPeriod;
}
/** Return the orbital period of the Halo Orbit.
* @param axr x-axis Amplitude of the required Lyapunov Orbit, meters
* @return the orbitalPeriod
*/
public double getLyapunovOrbitalPeriod(final double axr) {
final double az = 0;
final double ax = axr / (gamma * dDim);
final double nu = 1.0 + s1 * ax * ax + s2 * az * az;
orbitalPeriod = FastMath.abs(2.0 * FastMath.PI / (wp * nu));
return orbitalPeriod;
}
/** Get the considered CR3BP system.
* @return CRR3BP system
*/
public CR3BPSystem getCr3bpSystem() {
return cr3bpSystem;
}
/** Get the considered lagrangian point.
* @return lagrangian point
*/
public LagrangianPoints getLagrangianPoint() {
return point;
}
}