EcksteinHechlerPropagator.java

  1. /* Copyright 2002-2015 CS Systèmes d'Information
  2.  * Licensed to CS Systèmes d'Information (CS) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * CS licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *   http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.orekit.propagation.analytical;

  19. import org.apache.commons.math3.analysis.differentiation.DerivativeStructure;
  20. import org.apache.commons.math3.geometry.euclidean.threed.FieldVector3D;
  21. import org.apache.commons.math3.geometry.euclidean.threed.Vector3D;
  22. import org.apache.commons.math3.util.FastMath;
  23. import org.apache.commons.math3.util.MathUtils;
  24. import org.orekit.attitudes.AttitudeProvider;
  25. import org.orekit.errors.OrekitException;
  26. import org.orekit.errors.OrekitMessages;
  27. import org.orekit.errors.PropagationException;
  28. import org.orekit.forces.gravity.potential.UnnormalizedSphericalHarmonicsProvider;
  29. import org.orekit.forces.gravity.potential.UnnormalizedSphericalHarmonicsProvider.UnnormalizedSphericalHarmonics;
  30. import org.orekit.orbits.CartesianOrbit;
  31. import org.orekit.orbits.CircularOrbit;
  32. import org.orekit.orbits.Orbit;
  33. import org.orekit.orbits.OrbitType;
  34. import org.orekit.orbits.PositionAngle;
  35. import org.orekit.propagation.SpacecraftState;
  36. import org.orekit.time.AbsoluteDate;
  37. import org.orekit.utils.TimeStampedPVCoordinates;

  39. /** This class propagates a {@link org.orekit.propagation.SpacecraftState}
  40.  *  using the analytical Eckstein-Hechler model.
  41.  * <p>The Eckstein-Hechler model is suited for near circular orbits
  42.  * (e < 0.1, with poor accuracy between 0.005 and 0.1) and inclination
  43.  * neither equatorial (direct or retrograde) nor critical (direct or
  44.  * retrograde).</p>
  45.  * <p>
  46.  * Note that before version 7.0, there was a large inconsistency in the generated
  47.  * orbits, and it was fixed as of version 7.0 of Orekit, with a visible side effect.
  48.  * The problems is that if the circular parameters produced by the Eckstein-Hechler
  49.  * model are used to build an orbit considered to be osculating, the velocity deduced
  50.  * from this orbit was <em>inconsistent with the position evolution</em>! The reason is
  51.  * that the model includes non-Keplerian effects but it does not include a corresponding
  52.  * circular/Cartesian conversion. As a consequence, all subsequent computation involving
  53.  * velocity were wrong. This includes attitude modes like yaw compensation and Doppler
  54.  * effect. As this effect was considered serious enough and as accurate velocities were
  55.  * considered important, the propagator now generates {@link CartesianOrbit Cartesian
  56.  * orbits} which are built in a special way to ensure consistency throughout propagation.
  57.  * A side effect is that if circular parameters are rebuilt by user from these propagated
  58.  * Cartesian orbit, the circular parameters will generally <em>not</em> match the initial
  59.  * orbit (differences in semi-major axis can exceed 120 m). The position however <em>will</em>
  60.  * match to sub-micrometer level, and this position will be identical to the positions
  61.  * that were generated by previous versions (in other words, the internals of the models
  62.  * have not been changed, only the output parameters have been changed). The correctness
  63.  * of the initialization has been assessed and is good, as it allows the subsequent orbit
  64.  * to remain close to a numerical reference orbit.
  65.  * </p>
  66.  * <p>
  67.  * If users need a more definitive initialization of an Eckstein-Hechler propagator, they
  68.  * should consider using a {@link org.orekit.propagation.conversion.PropagatorConverter
  69.  * propagator converter} to initialize their Eckstein-Hechler propagator using a complete
  70.  * sample instead of just a single initial orbit.
  71.  * </p>
  72.  * @see Orbit
  73.  * @author Guylaine Prat
  74.  */
  75. public class EcksteinHechlerPropagator extends AbstractAnalyticalPropagator {

  77.     /** Eckstein-Hechler model. */
  78.     private EHModel model;

  80.     /** Current mass. */
  81.     private double mass;

  83.     /** Reference radius of the central body attraction model (m). */
  84.     private double referenceRadius;

  86.     /** Central attraction coefficient (m³/s²). */
  87.     private double mu;

  89.     /** Un-normalized zonal coefficients. */
  90.     private double[] ck0;

  92.     /** Build a propagator from orbit and potential provider.
  93.      * <p>Mass and attitude provider are set to unspecified non-null arbitrary values.</p>
  94.      * @param initialOrbit initial orbit
  95.      * @param provider for un-normalized zonal coefficients
  96.      * @exception OrekitException if the zonal coefficients cannot be retrieved
  97.      * @exception PropagationException if the mean parameters cannot be computed
  98.      */
  99.     public EcksteinHechlerPropagator(final Orbit initialOrbit,
  100.                                      final UnnormalizedSphericalHarmonicsProvider provider)
  101.         throws PropagationException , OrekitException {
  102.         this(initialOrbit, DEFAULT_LAW, DEFAULT_MASS, provider,
  103.                 provider.onDate(initialOrbit.getDate()));
  104.     }

  106.     /**
  107.      * Private helper constructor.
  108.      * @param initialOrbit initial orbit
  109.      * @param attitude attitude provider
  110.      * @param mass spacecraft mass
  111.      * @param provider for un-normalized zonal coefficients
  112.      * @param harmonics {@code provider.onDate(initialOrbit.getDate())}
  113.      * @exception OrekitException if the zonal coefficients cannot be retrieved
  114.      */
  115.     public EcksteinHechlerPropagator(final Orbit initialOrbit,
  116.                                      final AttitudeProvider attitude,
  117.                                      final double mass,
  118.                                      final UnnormalizedSphericalHarmonicsProvider provider,
  119.                                      final UnnormalizedSphericalHarmonics harmonics)
  120.         throws OrekitException {
  121.         this(initialOrbit, attitude, mass, provider.getAe(), provider.getMu(),
  122.                 harmonics.getUnnormalizedCnm(2, 0),
  123.                 harmonics.getUnnormalizedCnm(3, 0),
  124.                 harmonics.getUnnormalizedCnm(4, 0),
  125.                 harmonics.getUnnormalizedCnm(5, 0),
  126.                 harmonics.getUnnormalizedCnm(6, 0));
  127.     }

  129.     /** Build a propagator from orbit and potential.
  130.      * <p>Mass and attitude provider are set to unspecified non-null arbitrary values.</p>
  131.      * <p>The C<sub>n,0</sub> coefficients are the denormalized zonal coefficients, they
  132.      * are related to both the normalized coefficients
  133.      * <span style="text-decoration: overline">C</span><sub>n,0</sub>
  134.      *  and the J<sub>n</sub> one as follows:</p>
  135.      * <pre>
  136.      *   C<sub>n,0</sub> = [(2-δ<sub>0,m</sub>)(2n+1)(n-m)!/(n+m)!]<sup>½</sup>
  137.      *                      <span style="text-decoration: overline">C</span><sub>n,0</sub>
  138.      *   C<sub>n,0</sub> = -J<sub>n</sub>
  139.      * </pre>
  140.      * @param initialOrbit initial orbit
  141.      * @param referenceRadius reference radius of the Earth for the potential model (m)
  142.      * @param mu central attraction coefficient (m³/s²)
  143.      * @param c20 un-normalized zonal coefficient (about -1.08e-3 for Earth)
  144.      * @param c30 un-normalized zonal coefficient (about +2.53e-6 for Earth)
  145.      * @param c40 un-normalized zonal coefficient (about +1.62e-6 for Earth)
  146.      * @param c50 un-normalized zonal coefficient (about +2.28e-7 for Earth)
  147.      * @param c60 un-normalized zonal coefficient (about -5.41e-7 for Earth)
  148.      * @exception PropagationException if the mean parameters cannot be computed
  149.      * @see org.orekit.utils.Constants
  150.      */
  151.     public EcksteinHechlerPropagator(final Orbit initialOrbit,
  152.                                      final double referenceRadius, final double mu,
  153.                                      final double c20, final double c30, final double c40,
  154.                                      final double c50, final double c60)
  155.         throws PropagationException {
  156.         this(initialOrbit, DEFAULT_LAW, DEFAULT_MASS, referenceRadius, mu, c20, c30, c40, c50, c60);
  157.     }

  159.     /** Build a propagator from orbit, mass and potential provider.
  160.      * <p>Attitude law is set to an unspecified non-null arbitrary value.</p>
  161.      * @param initialOrbit initial orbit
  162.      * @param mass spacecraft mass
  163.      * @param provider for un-normalized zonal coefficients
  164.      * @exception OrekitException if the zonal coefficients cannot be retrieved
  165.      * @exception PropagationException if the mean parameters cannot be computed
  166.      */
  167.     public EcksteinHechlerPropagator(final Orbit initialOrbit, final double mass,
  168.                                      final UnnormalizedSphericalHarmonicsProvider provider)
  169.         throws PropagationException , OrekitException {
  170.         this(initialOrbit, DEFAULT_LAW, mass, provider, provider.onDate(initialOrbit.getDate()));
  171.     }

  173.     /** Build a propagator from orbit, mass and potential.
  174.      * <p>Attitude law is set to an unspecified non-null arbitrary value.</p>
  175.      * <p>The C<sub>n,0</sub> coefficients are the denormalized zonal coefficients, they
  176.      * are related to both the normalized coefficients
  177.      * <span style="text-decoration: overline">C</span><sub>n,0</sub>
  178.      *  and the J<sub>n</sub> one as follows:</p>
  179.      * <pre>
  180.      *   C<sub>n,0</sub> = [(2-δ<sub>0,m</sub>)(2n+1)(n-m)!/(n+m)!]<sup>½</sup>
  181.      *                      <span style="text-decoration: overline">C</span><sub>n,0</sub>
  182.      *   C<sub>n,0</sub> = -J<sub>n</sub>
  183.      * </pre>
  184.      * @param initialOrbit initial orbit
  185.      * @param mass spacecraft mass
  186.      * @param referenceRadius reference radius of the Earth for the potential model (m)
  187.      * @param mu central attraction coefficient (m³/s²)
  188.      * @param c20 un-normalized zonal coefficient (about -1.08e-3 for Earth)
  189.      * @param c30 un-normalized zonal coefficient (about +2.53e-6 for Earth)
  190.      * @param c40 un-normalized zonal coefficient (about +1.62e-6 for Earth)
  191.      * @param c50 un-normalized zonal coefficient (about +2.28e-7 for Earth)
  192.      * @param c60 un-normalized zonal coefficient (about -5.41e-7 for Earth)
  193.      * @exception PropagationException if the mean parameters cannot be computed
  194.      */
  195.     public EcksteinHechlerPropagator(final Orbit initialOrbit, final double mass,
  196.                                      final double referenceRadius, final double mu,
  197.                                      final double c20, final double c30, final double c40,
  198.                                      final double c50, final double c60)
  199.         throws PropagationException {
  200.         this(initialOrbit, DEFAULT_LAW, mass, referenceRadius, mu, c20, c30, c40, c50, c60);
  201.     }

  203.     /** Build a propagator from orbit, attitude provider and potential provider.
  204.      * <p>Mass is set to an unspecified non-null arbitrary value.</p>
  205.      * @param initialOrbit initial orbit
  206.      * @param attitudeProv attitude provider
  207.      * @param provider for un-normalized zonal coefficients
  208.      * @exception OrekitException if the zonal coefficients cannot be retrieved
  209.      * @exception PropagationException if the mean parameters cannot be computed
  210.      */
  211.     public EcksteinHechlerPropagator(final Orbit initialOrbit,
  212.                                      final AttitudeProvider attitudeProv,
  213.                                      final UnnormalizedSphericalHarmonicsProvider provider)
  214.         throws PropagationException , OrekitException {
  215.         this(initialOrbit, attitudeProv, DEFAULT_MASS, provider,
  216.                 provider.onDate(initialOrbit.getDate()));
  217.     }

  219.     /** Build a propagator from orbit, attitude provider and potential.
  220.      * <p>Mass is set to an unspecified non-null arbitrary value.</p>
  221.      * <p>The C<sub>n,0</sub> coefficients are the denormalized zonal coefficients, they
  222.      * are related to both the normalized coefficients
  223.      * <span style="text-decoration: overline">C</span><sub>n,0</sub>
  224.      *  and the J<sub>n</sub> one as follows:</p>
  225.      * <pre>
  226.      *   C<sub>n,0</sub> = [(2-δ<sub>0,m</sub>)(2n+1)(n-m)!/(n+m)!]<sup>½</sup>
  227.      *                     <span style="text-decoration: overline">C</span><sub>n,0</sub>
  228.      *   C<sub>n,0</sub> = -J<sub>n</sub>
  229.      * </pre>
  230.      * @param initialOrbit initial orbit
  231.      * @param attitudeProv attitude provider
  232.      * @param referenceRadius reference radius of the Earth for the potential model (m)
  233.      * @param mu central attraction coefficient (m³/s²)
  234.      * @param c20 un-normalized zonal coefficient (about -1.08e-3 for Earth)
  235.      * @param c30 un-normalized zonal coefficient (about +2.53e-6 for Earth)
  236.      * @param c40 un-normalized zonal coefficient (about +1.62e-6 for Earth)
  237.      * @param c50 un-normalized zonal coefficient (about +2.28e-7 for Earth)
  238.      * @param c60 un-normalized zonal coefficient (about -5.41e-7 for Earth)
  239.      * @exception PropagationException if the mean parameters cannot be computed
  240.      */
  241.     public EcksteinHechlerPropagator(final Orbit initialOrbit,
  242.                                      final AttitudeProvider attitudeProv,
  243.                                      final double referenceRadius, final double mu,
  244.                                      final double c20, final double c30, final double c40,
  245.                                      final double c50, final double c60)
  246.         throws PropagationException {
  247.         this(initialOrbit, attitudeProv, DEFAULT_MASS, referenceRadius, mu, c20, c30, c40, c50, c60);
  248.     }

  250.     /** Build a propagator from orbit, attitude provider, mass and potential provider.
  251.      * @param initialOrbit initial orbit
  252.      * @param attitudeProv attitude provider
  253.      * @param mass spacecraft mass
  254.      * @param provider for un-normalized zonal coefficients
  255.      * @exception OrekitException if the zonal coefficients cannot be retrieved
  256.      * @exception PropagationException if the mean parameters cannot be computed
  257.      */
  258.     public EcksteinHechlerPropagator(final Orbit initialOrbit,
  259.                                      final AttitudeProvider attitudeProv,
  260.                                      final double mass,
  261.                                      final UnnormalizedSphericalHarmonicsProvider provider)
  262.         throws PropagationException , OrekitException {
  263.         this(initialOrbit, attitudeProv, mass, provider,
  264.                 provider.onDate(initialOrbit.getDate()));
  265.     }

  267.     /** Build a propagator from orbit, attitude provider, mass and potential.
  268.      * <p>The C<sub>n,0</sub> coefficients are the denormalized zonal coefficients, they
  269.      * are related to both the normalized coefficients
  270.      * <span style="text-decoration: overline">C</span><sub>n,0</sub>
  271.      *  and the J<sub>n</sub> one as follows:</p>
  272.      * <pre>
  273.      *   C<sub>n,0</sub> = [(2-δ<sub>0,m</sub>)(2n+1)(n-m)!/(n+m)!]<sup>½</sup>
  274.      *                      <span style="text-decoration: overline">C</span><sub>n,0</sub>
  275.      *   C<sub>n,0</sub> = -J<sub>n</sub>
  276.      * </pre>
  277.      * @param initialOrbit initial orbit
  278.      * @param attitudeProv attitude provider
  279.      * @param mass spacecraft mass
  280.      * @param referenceRadius reference radius of the Earth for the potential model (m)
  281.      * @param mu central attraction coefficient (m³/s²)
  282.      * @param c20 un-normalized zonal coefficient (about -1.08e-3 for Earth)
  283.      * @param c30 un-normalized zonal coefficient (about +2.53e-6 for Earth)
  284.      * @param c40 un-normalized zonal coefficient (about +1.62e-6 for Earth)
  285.      * @param c50 un-normalized zonal coefficient (about +2.28e-7 for Earth)
  286.      * @param c60 un-normalized zonal coefficient (about -5.41e-7 for Earth)
  287.      * @exception PropagationException if the mean parameters cannot be computed
  288.      */
  289.     public EcksteinHechlerPropagator(final Orbit initialOrbit,
  290.                                      final AttitudeProvider attitudeProv,
  291.                                      final double mass,
  292.                                      final double referenceRadius, final double mu,
  293.                                      final double c20, final double c30, final double c40,
  294.                                      final double c50, final double c60)
  295.         throws PropagationException {

  297.         super(attitudeProv);
  298.         this.mass = mass;

  300.         try {

  302.             // store model coefficients
  303.             this.referenceRadius = referenceRadius;
  304.             this.mu  = mu;
  305.             this.ck0 = new double[] {
  306.                 0.0, 0.0, c20, c30, c40, c50, c60
  307.             };

  309.             // compute mean parameters
  310.             // transform into circular adapted parameters used by the Eckstein-Hechler model
  311.             resetInitialState(new SpacecraftState(initialOrbit,
  312.                                                   attitudeProv.getAttitude(initialOrbit,
  313.                                                                            initialOrbit.getDate(),
  314.                                                                            initialOrbit.getFrame()),
  315.                                                   mass));

  317.         } catch (OrekitException oe) {
  318.             throw new PropagationException(oe);
  319.         }
  320.     }

  322.     /** {@inheritDoc} */
  323.     public void resetInitialState(final SpacecraftState state)
  324.         throws PropagationException {
  325.         super.resetInitialState(state);
  326.         this.mass = state.getMass();
  327.         computeMeanParameters((CircularOrbit) OrbitType.CIRCULAR.convertType(state.getOrbit()));
  328.     }

  330.     /** Compute mean parameters according to the Eckstein-Hechler analytical model.
  331.      * @param osculating osculating orbit
  332.      * @exception PropagationException if orbit goes outside of supported range
  333.      * (trajectory inside the Brillouin sphere, too eccentric, equatorial, critical
  334.      * inclination) or if convergence cannot be reached
  335.      */
  336.     private void computeMeanParameters(final CircularOrbit osculating)
  337.         throws PropagationException {

  339.         // sanity check
  340.         if (osculating.getA() < referenceRadius) {
  341.             throw new PropagationException(OrekitMessages.TRAJECTORY_INSIDE_BRILLOUIN_SPHERE,
  342.                                            osculating.getA());
  343.         }

  345.         // rough initialization of the mean parameters
  346.         EHModel current = new EHModel(osculating);

  348.         // threshold for each parameter
  349.         final double epsilon         = 1.0e-13;
  350.         final double thresholdA      = epsilon * (1 + FastMath.abs(current.mean.getA()));
  351.         final double thresholdE      = epsilon * (1 + current.mean.getE());
  352.         final double thresholdAngles = epsilon * FastMath.PI;

  354.         int i = 0;
  355.         while (i++ < 100) {

  357.             // recompute the osculating parameters from the current mean parameters
  358.             final DerivativeStructure[] parameters = current.propagateParameters(current.mean.getDate());

  360.             // adapted parameters residuals
  361.             final double deltaA      = osculating.getA()          - parameters[0].getValue();
  362.             final double deltaEx     = osculating.getCircularEx() - parameters[1].getValue();
  363.             final double deltaEy     = osculating.getCircularEy() - parameters[2].getValue();
  364.             final double deltaI      = osculating.getI()          - parameters[3].getValue();
  365.             final double deltaRAAN   = MathUtils.normalizeAngle(osculating.getRightAscensionOfAscendingNode() -
  366.                                                                 parameters[4].getValue(),
  367.                                                                 0.0);
  368.             final double deltaAlphaM = MathUtils.normalizeAngle(osculating.getAlphaM() - parameters[5].getValue(), 0.0);

  370.             // update mean parameters
  371.             current = new EHModel(new CircularOrbit(current.mean.getA()          + deltaA,
  372.                                                     current.mean.getCircularEx() + deltaEx,
  373.                                                     current.mean.getCircularEy() + deltaEy,
  374.                                                     current.mean.getI()          + deltaI,
  375.                                                     current.mean.getRightAscensionOfAscendingNode() + deltaRAAN,
  376.                                                     current.mean.getAlphaM()     + deltaAlphaM,
  377.                                                     PositionAngle.MEAN,
  378.                                                     current.mean.getFrame(),
  379.                                                     current.mean.getDate(), mu));

  381.             // check convergence
  382.             if ((FastMath.abs(deltaA)      < thresholdA) &&
  383.                 (FastMath.abs(deltaEx)     < thresholdE) &&
  384.                 (FastMath.abs(deltaEy)     < thresholdE) &&
  385.                 (FastMath.abs(deltaI)      < thresholdAngles) &&
  386.                 (FastMath.abs(deltaRAAN)   < thresholdAngles) &&
  387.                 (FastMath.abs(deltaAlphaM) < thresholdAngles)) {
  388.                 model = current;
  389.                 return;
  390.             }

  392.         }

  394.         throw new PropagationException(OrekitMessages.UNABLE_TO_COMPUTE_ECKSTEIN_HECHLER_MEAN_PARAMETERS, i);

  396.     }

  398.     /** {@inheritDoc} */
  399.     public CartesianOrbit propagateOrbit(final AbsoluteDate date)
  400.         throws PropagationException {
  401.         // compute Cartesian parameters, taking derivatives into account
  402.         // to make sure velocity and acceleration are consistent
  403.         return new CartesianOrbit(toCartesian(date, model.propagateParameters(date)),
  404.                                   model.mean.getFrame(), mu);
  405.     }

  407.     /** Local class for Eckstein-Hechler model, with fixed mean parameters. */
  408.     private class EHModel {

  410.         /** Mean orbit. */
  411.         private final CircularOrbit mean;

  413.         // CHECKSTYLE: stop JavadocVariable check

  415.         // preprocessed values
  416.         private final double xnotDot;
  417.         private final double rdpom;
  418.         private final double rdpomp;
  419.         private final double eps1;
  420.         private final double eps2;
  421.         private final double xim;
  422.         private final double ommD;
  423.         private final double rdl;
  424.         private final double aMD;

  426.         private final double kh;
  427.         private final double kl;

  429.         private final double ax1;
  430.         private final double ay1;
  431.         private final double as1;
  432.         private final double ac2;
  433.         private final double axy3;
  434.         private final double as3;
  435.         private final double ac4;
  436.         private final double as5;
  437.         private final double ac6;

  439.         private final double ex1;
  440.         private final double exx2;
  441.         private final double exy2;
  442.         private final double ex3;
  443.         private final double ex4;

  445.         private final double ey1;
  446.         private final double eyx2;
  447.         private final double eyy2;
  448.         private final double ey3;
  449.         private final double ey4;

  451.         private final double rx1;
  452.         private final double ry1;
  453.         private final double r2;
  454.         private final double r3;
  455.         private final double rl;

  457.         private final double iy1;
  458.         private final double ix1;
  459.         private final double i2;
  460.         private final double i3;
  461.         private final double ih;

  463.         private final double lx1;
  464.         private final double ly1;
  465.         private final double l2;
  466.         private final double l3;
  467.         private final double ll;

  469.         // CHECKSTYLE: resume JavadocVariable check

  471.         /** Create a model for specified mean orbit.
  472.          * @param mean mean orbit
  473.          * @exception PropagationException if mean orbit is not within model supported domain
  474.          */
  475.         public EHModel(final CircularOrbit mean) throws PropagationException {

  477.             this.mean = mean;

  479.             // preliminary processing
  480.             double q = referenceRadius / mean.getA();
  481.             double ql = q * q;
  482.             final double g2 = ck0[2] * ql;
  483.             ql *= q;
  484.             final double g3 = ck0[3] * ql;
  485.             ql *= q;
  486.             final double g4 = ck0[4] * ql;
  487.             ql *= q;
  488.             final double g5 = ck0[5] * ql;
  489.             ql *= q;
  490.             final double g6 = ck0[6] * ql;

  492.             final double cosI1 = FastMath.cos(mean.getI());
  493.             final double sinI1 = FastMath.sin(mean.getI());
  494.             final double sinI2 = sinI1 * sinI1;
  495.             final double sinI4 = sinI2 * sinI2;
  496.             final double sinI6 = sinI2 * sinI4;

  498.             if (sinI2 < 1.0e-10) {
  499.                 throw new PropagationException(OrekitMessages.ALMOST_EQUATORIAL_ORBIT,
  500.                                                FastMath.toDegrees(mean.getI()));
  501.             }

  503.             if (FastMath.abs(sinI2 - 4.0 / 5.0) < 1.0e-3) {
  504.                 throw new PropagationException(OrekitMessages.ALMOST_CRITICALLY_INCLINED_ORBIT,
  505.                                                FastMath.toDegrees(mean.getI()));
  506.             }

  508.             if (mean.getE() > 0.1) {
  509.                 // if 0.005 < e < 0.1 no error is triggered, but accuracy is poor
  510.                 throw new PropagationException(OrekitMessages.TOO_LARGE_ECCENTRICITY_FOR_PROPAGATION_MODEL,
  511.                                                mean.getE());
  512.             }

  514.             xnotDot = FastMath.sqrt(mu / mean.getA()) / mean.getA();

  516.             rdpom = -0.75 * g2 * (4.0 - 5.0 * sinI2);
  517.             rdpomp = 7.5 * g4 * (1.0 - 31.0 / 8.0 * sinI2 + 49.0 / 16.0 * sinI4) -
  518.                     13.125 * g6 * (1.0 - 8.0 * sinI2 + 129.0 / 8.0 * sinI4 - 297.0 / 32.0 * sinI6);

  520.             q = 3.0 / (32.0 * rdpom);
  521.             eps1 = q * g4 * sinI2 * (30.0 - 35.0 * sinI2) -
  522.                     175.0 * q * g6 * sinI2 * (1.0 - 3.0 * sinI2 + 2.0625 * sinI4);
  523.             q = 3.0 * sinI1 / (8.0 * rdpom);
  524.             eps2 = q * g3 * (4.0 - 5.0 * sinI2) - q * g5 * (10.0 - 35.0 * sinI2 + 26.25 * sinI4);

  526.             xim = mean.getI();
  527.             ommD = cosI1 * (1.50    * g2 - 2.25 * g2 * g2 * (2.5 - 19.0 / 6.0 * sinI2) +
  528.                             0.9375  * g4 * (7.0 * sinI2 - 4.0) +
  529.                             3.28125 * g6 * (2.0 - 9.0 * sinI2 + 8.25 * sinI4));

  531.             rdl = 1.0 - 1.50 * g2 * (3.0 - 4.0 * sinI2);
  532.             aMD = rdl +
  533.                     2.25 * g2 * g2 * (9.0 - 263.0 / 12.0 * sinI2 + 341.0 / 24.0 * sinI4) +
  534.                     15.0 / 16.0 * g4 * (8.0 - 31.0 * sinI2 + 24.5 * sinI4) +
  535.                     105.0 / 32.0 * g6 * (-10.0 / 3.0 + 25.0 * sinI2 - 48.75 * sinI4 + 27.5 * sinI6);

  537.             final double qq = -1.5 * g2 / rdl;
  538.             final double qA   = 0.75 * g2 * g2 * sinI2;
  539.             final double qB   = 0.25 * g4 * sinI2;
  540.             final double qC   = 105.0 / 16.0 * g6 * sinI2;
  541.             final double qD   = -0.75 * g3 * sinI1;
  542.             final double qE   = 3.75 * g5 * sinI1;
  543.             kh = 0.375 / rdpom;
  544.             kl = kh / sinI1;

  546.             ax1 = qq * (2.0 - 3.5 * sinI2);
  547.             ay1 = qq * (2.0 - 2.5 * sinI2);
  548.             as1 = qD * (4.0 - 5.0 * sinI2) +
  549.                   qE * (2.625 * sinI4 - 3.5 * sinI2 + 1.0);
  550.             ac2 = qq * sinI2 +
  551.                   qA * 7.0 * (2.0 - 3.0 * sinI2) +
  552.                   qB * (15.0 - 17.5 * sinI2) +
  553.                   qC * (3.0 * sinI2 - 1.0 - 33.0 / 16.0 * sinI4);
  554.             axy3 = qq * 3.5 * sinI2;
  555.             as3 = qD * 5.0 / 3.0 * sinI2 +
  556.                   qE * 7.0 / 6.0 * sinI2 * (1.0 - 1.125 * sinI2);
  557.             ac4 = qA * sinI2 +
  558.                   qB * 4.375 * sinI2 +
  559.                   qC * 0.75 * (1.1 * sinI4 - sinI2);

  561.             as5 = qE * 21.0 / 80.0 * sinI4;

  563.             ac6 = qC * -11.0 / 80.0 * sinI4;

  565.             ex1 = qq * (1.0 - 1.25 * sinI2);
  566.             exx2 = qq * 0.5 * (3.0 - 5.0 * sinI2);
  567.             exy2 = qq * (2.0 - 1.5 * sinI2);
  568.             ex3 = qq * 7.0 / 12.0 * sinI2;
  569.             ex4 = qq * 17.0 / 8.0 * sinI2;

  571.             ey1 = qq * (1.0 - 1.75 * sinI2);
  572.             eyx2 = qq * (1.0 - 3.0 * sinI2);
  573.             eyy2 = qq * (2.0 * sinI2 - 1.5);
  574.             ey3 = qq * 7.0 / 12.0 * sinI2;
  575.             ey4 = qq * 17.0 / 8.0 * sinI2;

  577.             q  = -qq * cosI1;
  578.             rx1 =  3.5 * q;
  579.             ry1 = -2.5 * q;
  580.             r2 = -0.5 * q;
  581.             r3 =  7.0 / 6.0 * q;
  582.             rl = g3 * cosI1 * (4.0 - 15.0 * sinI2) -
  583.                  2.5 * g5 * cosI1 * (4.0 - 42.0 * sinI2 + 52.5 * sinI4);

  585.             q = 0.5 * qq * sinI1 * cosI1;
  586.             iy1 =  q;
  587.             ix1 = -q;
  588.             i2 =  q;
  589.             i3 =  q * 7.0 / 3.0;
  590.             ih = -g3 * cosI1 * (4.0 - 5.0 * sinI2) +
  591.                  2.5 * g5 * cosI1 * (4.0 - 14.0 * sinI2 + 10.5 * sinI4);

  593.             lx1 = qq * (7.0 - 77.0 / 8.0 * sinI2);
  594.             ly1 = qq * (55.0 / 8.0 * sinI2 - 7.50);
  595.             l2 = qq * (1.25 * sinI2 - 0.5);
  596.             l3 = qq * (77.0 / 24.0 * sinI2 - 7.0 / 6.0);
  597.             ll = g3 * (53.0 * sinI2 - 4.0 - 57.5 * sinI4) +
  598.                  2.5 * g5 * (4.0 - 96.0 * sinI2 + 269.5 * sinI4 - 183.75 * sinI6);

  600.         }

  602.         /** Extrapolate an orbit up to a specific target date.
  603.          * @param date target date for the orbit
  604.          * @return propagated parameters
  605.          * @exception PropagationException if some parameters are out of bounds
  606.          */
  607.         public DerivativeStructure[] propagateParameters(final AbsoluteDate date)
  608.             throws PropagationException {

  610.             // keplerian evolution
  611.             final DerivativeStructure dt =
  612.                     new DerivativeStructure(1, 2, 0, date.durationFrom(mean.getDate()));
  613.             final DerivativeStructure xnot = dt.multiply(xnotDot);

  615.             // secular effects

  617.             // eccentricity
  618.             final DerivativeStructure x   = xnot.multiply(rdpom + rdpomp);
  619.             final DerivativeStructure cx  = x.cos();
  620.             final DerivativeStructure sx  = x.sin();
  621.             final DerivativeStructure exm = cx.multiply(mean.getCircularEx()).
  622.                                             add(sx.multiply(eps2 - (1.0 - eps1) * mean.getCircularEy()));
  623.             final DerivativeStructure eym = sx.multiply((1.0 + eps1) * mean.getCircularEx()).
  624.                                             add(cx.multiply(mean.getCircularEy() - eps2)).
  625.                                             add(eps2);

  627.             // no secular effect on inclination

  629.             // right ascension of ascending node
  630.             final DerivativeStructure omm =
  631.                     new DerivativeStructure(1, 2,
  632.                                             MathUtils.normalizeAngle(mean.getRightAscensionOfAscendingNode() + ommD * xnot.getValue(),
  633.                                                                      FastMath.PI),
  634.                                             ommD * xnotDot,
  635.                                             0.0);

  637.             // latitude argument
  638.             final DerivativeStructure xlm =
  639.                     new DerivativeStructure(1, 2,
  640.                                             MathUtils.normalizeAngle(mean.getAlphaM() + aMD * xnot.getValue(), FastMath.PI),
  641.                                             aMD * xnotDot,
  642.                                             0.0);

  644.             // periodical terms
  645.             final DerivativeStructure cl1 = xlm.cos();
  646.             final DerivativeStructure sl1 = xlm.sin();
  647.             final DerivativeStructure cl2 = cl1.multiply(cl1).subtract(sl1.multiply(sl1));
  648.             final DerivativeStructure sl2 = cl1.multiply(sl1).add(sl1.multiply(cl1));
  649.             final DerivativeStructure cl3 = cl2.multiply(cl1).subtract(sl2.multiply(sl1));
  650.             final DerivativeStructure sl3 = cl2.multiply(sl1).add(sl2.multiply(cl1));
  651.             final DerivativeStructure cl4 = cl3.multiply(cl1).subtract(sl3.multiply(sl1));
  652.             final DerivativeStructure sl4 = cl3.multiply(sl1).add(sl3.multiply(cl1));
  653.             final DerivativeStructure cl5 = cl4.multiply(cl1).subtract(sl4.multiply(sl1));
  654.             final DerivativeStructure sl5 = cl4.multiply(sl1).add(sl4.multiply(cl1));
  655.             final DerivativeStructure cl6 = cl5.multiply(cl1).subtract(sl5.multiply(sl1));

  657.             final DerivativeStructure qh  = eym.subtract(eps2).multiply(kh);
  658.             final DerivativeStructure ql  = exm.multiply(kl);

  660.             final DerivativeStructure exmCl1 = exm.multiply(cl1);
  661.             final DerivativeStructure exmSl1 = exm.multiply(sl1);
  662.             final DerivativeStructure eymCl1 = eym.multiply(cl1);
  663.             final DerivativeStructure eymSl1 = eym.multiply(sl1);
  664.             final DerivativeStructure exmCl2 = exm.multiply(cl2);
  665.             final DerivativeStructure exmSl2 = exm.multiply(sl2);
  666.             final DerivativeStructure eymCl2 = eym.multiply(cl2);
  667.             final DerivativeStructure eymSl2 = eym.multiply(sl2);
  668.             final DerivativeStructure exmCl3 = exm.multiply(cl3);
  669.             final DerivativeStructure exmSl3 = exm.multiply(sl3);
  670.             final DerivativeStructure eymCl3 = eym.multiply(cl3);
  671.             final DerivativeStructure eymSl3 = eym.multiply(sl3);
  672.             final DerivativeStructure exmCl4 = exm.multiply(cl4);
  673.             final DerivativeStructure exmSl4 = exm.multiply(sl4);
  674.             final DerivativeStructure eymCl4 = eym.multiply(cl4);
  675.             final DerivativeStructure eymSl4 = eym.multiply(sl4);

  677.             // semi major axis
  678.             final DerivativeStructure rda = exmCl1.multiply(ax1).
  679.                                             add(eymSl1.multiply(ay1)).
  680.                                             add(sl1.multiply(as1)).
  681.                                             add(cl2.multiply(ac2)).
  682.                                             add(exmCl3.add(eymSl3).multiply(axy3)).
  683.                                             add(sl3.multiply(as3)).
  684.                                             add(cl4.multiply(ac4)).
  685.                                             add(sl5.multiply(as5)).
  686.                                             add(cl6.multiply(ac6));

  688.             // eccentricity
  689.             final DerivativeStructure rdex = cl1.multiply(ex1).
  690.                                              add(exmCl2.multiply(exx2)).
  691.                                              add(eymSl2.multiply(exy2)).
  692.                                              add(cl3.multiply(ex3)).
  693.                                              add(exmCl4.add(eymSl4).multiply(ex4));
  694.             final DerivativeStructure rdey = sl1.multiply(ey1).
  695.                                              add(exmSl2.multiply(eyx2)).
  696.                                              add(eymCl2.multiply(eyy2)).
  697.                                              add(sl3.multiply(ey3)).
  698.                                              add(exmSl4.subtract(eymCl4).multiply(ey4));

  700.             // ascending node
  701.             final DerivativeStructure rdom = exmSl1.multiply(rx1).
  702.                                              add(eymCl1.multiply(ry1)).
  703.                                              add(sl2.multiply(r2)).
  704.                                              add(eymCl3.subtract(exmSl3).multiply(r3)).
  705.                                              add(ql.multiply(rl));

  707.             // inclination
  708.             final DerivativeStructure rdxi = eymSl1.multiply(iy1).
  709.                                              add(exmCl1.multiply(ix1)).
  710.                                              add(cl2.multiply(i2)).
  711.                                              add(exmCl3.add(eymSl3).multiply(i3)).
  712.                                              add(qh.multiply(ih));

  714.             // latitude argument
  715.             final DerivativeStructure rdxl = exmSl1.multiply(lx1).
  716.                                              add(eymCl1.multiply(ly1)).
  717.                                              add(sl2.multiply(l2)).
  718.                                              add(exmSl3.subtract(eymCl3).multiply(l3)).
  719.                                              add(ql.multiply(ll));

  721.             // osculating parameters
  722.             return new DerivativeStructure[] {
  723.                 rda.add(1.0).multiply(mean.getA()),
  724.                 rdex.add(exm),
  725.                 rdey.add(eym),
  726.                 rdxi.add(xim),
  727.                 rdom.add(omm),
  728.                 rdxl.add(xlm)
  729.             };

  731.         }

  733.     }

  735.     /** Convert circular parameters <em>with derivatives</em> to Cartesian coordinates.
  736.      * @param date date of the orbital parameters
  737.      * @param parameters circular parameters (a, ex, ey, i, raan, alphaM)
  738.      * @return Cartesian coordinates consistent with values and derivatives
  739.      */
  740.     private TimeStampedPVCoordinates toCartesian(final AbsoluteDate date, final DerivativeStructure[] parameters) {

  742.         // evaluate coordinates in the orbit canonical reference frame
  743.         final DerivativeStructure cosOmega = parameters[4].cos();
  744.         final DerivativeStructure sinOmega = parameters[4].sin();
  745.         final DerivativeStructure cosI     = parameters[3].cos();
  746.         final DerivativeStructure sinI     = parameters[3].sin();
  747.         final DerivativeStructure alphaE   = meanToEccentric(parameters[5], parameters[1], parameters[2]);
  748.         final DerivativeStructure cosAE    = alphaE.cos();
  749.         final DerivativeStructure sinAE    = alphaE.sin();
  750.         final DerivativeStructure ex2      = parameters[1].multiply(parameters[1]);
  751.         final DerivativeStructure ey2      = parameters[2].multiply(parameters[2]);
  752.         final DerivativeStructure exy      = parameters[1].multiply(parameters[2]);
  753.         final DerivativeStructure q        = ex2.add(ey2).subtract(1).negate().sqrt();
  754.         final DerivativeStructure beta     = q.add(1).reciprocal();
  755.         final DerivativeStructure bx2      = beta.multiply(ex2);
  756.         final DerivativeStructure by2      = beta.multiply(ey2);
  757.         final DerivativeStructure bxy      = beta.multiply(exy);
  758.         final DerivativeStructure u        = bxy.multiply(sinAE).subtract(parameters[1].add(by2.subtract(1).multiply(cosAE)));
  759.         final DerivativeStructure v        = bxy.multiply(cosAE).subtract(parameters[2].add(bx2.subtract(1).multiply(sinAE)));
  760.         final DerivativeStructure x        = parameters[0].multiply(u);
  761.         final DerivativeStructure y        = parameters[0].multiply(v);

  763.         // canonical orbit reference frame
  764.         final FieldVector3D<DerivativeStructure> p =
  765.                 new FieldVector3D<DerivativeStructure>(x.multiply(cosOmega).subtract(y.multiply(cosI.multiply(sinOmega))),
  766.                                                        x.multiply(sinOmega).add(y.multiply(cosI.multiply(cosOmega))),
  767.                                                        y.multiply(sinI));

  769.         // dispatch derivatives
  770.         final Vector3D p0 = new Vector3D(p.getX().getValue(),
  771.                                          p.getY().getValue(),
  772.                                          p.getZ().getValue());
  773.         final Vector3D p1 = new Vector3D(p.getX().getPartialDerivative(1),
  774.                                          p.getY().getPartialDerivative(1),
  775.                                          p.getZ().getPartialDerivative(1));
  776.         final Vector3D p2 = new Vector3D(p.getX().getPartialDerivative(2),
  777.                                          p.getY().getPartialDerivative(2),
  778.                                          p.getZ().getPartialDerivative(2));
  779.         return new TimeStampedPVCoordinates(date, p0, p1, p2);

  781.     }

  783.     /** Computes the eccentric latitude argument from the mean latitude argument.
  784.      * @param alphaM = M + Ω mean latitude argument (rad)
  785.      * @param ex e cos(Ω), first component of circular eccentricity vector
  786.      * @param ey e sin(Ω), second component of circular eccentricity vector
  787.      * @return the eccentric latitude argument.
  788.      */
  789.     private DerivativeStructure meanToEccentric(final DerivativeStructure alphaM,
  790.                                                 final DerivativeStructure ex,
  791.                                                 final DerivativeStructure ey) {
  792.         // Generalization of Kepler equation to circular parameters
  793.         // with alphaE = PA + E and
  794.         //      alphaM = PA + M = alphaE - ex.sin(alphaE) + ey.cos(alphaE)
  795.         DerivativeStructure alphaE        = alphaM;
  796.         DerivativeStructure shift         = alphaM.getField().getZero();
  797.         DerivativeStructure alphaEMalphaM = alphaM.getField().getZero();
  798.         DerivativeStructure cosAlphaE     = alphaE.cos();
  799.         DerivativeStructure sinAlphaE     = alphaE.sin();
  800.         int    iter          = 0;
  801.         do {
  802.             final DerivativeStructure f2 = ex.multiply(sinAlphaE).subtract(ey.multiply(cosAlphaE));
  803.             final DerivativeStructure f1 = alphaM.getField().getOne().subtract(ex.multiply(cosAlphaE)).subtract(ey.multiply(sinAlphaE));
  804.             final DerivativeStructure f0 = alphaEMalphaM.subtract(f2);

  806.             final DerivativeStructure f12 = f1.multiply(2);
  807.             shift = f0.multiply(f12).divide(f1.multiply(f12).subtract(f0.multiply(f2)));

  809.             alphaEMalphaM  = alphaEMalphaM.subtract(shift);
  810.             alphaE         = alphaM.add(alphaEMalphaM);
  811.             cosAlphaE      = alphaE.cos();
  812.             sinAlphaE      = alphaE.sin();

  814.         } while ((++iter < 50) && (FastMath.abs(shift.getValue()) > 1.0e-12));

  816.         return alphaE;

  818.     }

  820.     /** {@inheritDoc} */
  821.     protected double getMass(final AbsoluteDate date) {
  822.         return mass;
  823.     }

  825. }
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