GLONASSAnalyticalPropagator.java

  1. /* Copyright 2002-2025 CS GROUP
  2.  * Licensed to CS GROUP (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.gnss;

  19. import org.hipparchus.analysis.differentiation.UnivariateDerivative2;
  20. import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  21. import org.hipparchus.geometry.euclidean.threed.Vector3D;
  22. import org.hipparchus.util.FastMath;
  23. import org.hipparchus.util.FieldSinCos;
  24. import org.hipparchus.util.MathArrays;
  25. import org.hipparchus.util.MathUtils;
  26. import org.hipparchus.util.Precision;
  27. import org.orekit.attitudes.Attitude;
  28. import org.orekit.attitudes.AttitudeProvider;
  29. import org.orekit.data.DataContext;
  30. import org.orekit.errors.OrekitException;
  31. import org.orekit.errors.OrekitMessages;
  32. import org.orekit.frames.Frame;
  33. import org.orekit.orbits.CartesianOrbit;
  34. import org.orekit.orbits.Orbit;
  35. import org.orekit.propagation.SpacecraftState;
  36. import org.orekit.propagation.analytical.AbstractAnalyticalPropagator;
  37. import org.orekit.propagation.analytical.gnss.data.GLONASSAlmanac;
  38. import org.orekit.propagation.analytical.gnss.data.GLONASSNavigationMessage;
  39. import org.orekit.propagation.analytical.gnss.data.GLONASSOrbitalElements;
  40. import org.orekit.propagation.analytical.gnss.data.GNSSConstants;
  41. import org.orekit.time.AbsoluteDate;
  42. import org.orekit.time.GLONASSDate;
  43. import org.orekit.time.TimeScale;
  44. import org.orekit.utils.PVCoordinates;

  46. /**
  47.  * This class aims at propagating a GLONASS orbit from {@link GLONASSOrbitalElements}.
  48.  * <p>
  49.  * <b>Caution:</b> The Glonass analytical propagator can only be used with {@link GLONASSAlmanac}.
  50.  * Using this propagator with a {@link GLONASSNavigationMessage} is prone to error.
  51.  * </p>
  52.  *
  53.  * @see <a href="http://russianspacesystems.ru/wp-content/uploads/2016/08/ICD-GLONASS-CDMA-General.-Edition-1.0-2016.pdf">
  54.  *       GLONASS Interface Control Document</a>
  55.  *
  56.  * @author Bryan Cazabonne
  57.  * @since 10.0
  58.  *
  59.  */
  60. public class GLONASSAnalyticalPropagator extends AbstractAnalyticalPropagator {

  62.     // Constants
  63.     /** Constant 7.0 / 3.0. */
  64.     private static final double SEVEN_THIRD = 7.0 / 3.0;

  66.     /** Constant 7.0 / 6.0. */
  67.     private static final double SEVEN_SIXTH = 7.0 / 6.0;

  69.     /** Constant 7.0 / 24.0. */
  70.     private static final double SEVEN_24TH = 7.0 / 24.0;

  72.     /** Constant 49.0 / 72.0. */
  73.     private static final double FN_72TH = 49.0 / 72.0;

  75.     /** Value of the earth's rotation rate in rad/s. */
  76.     private static final double GLONASS_AV = 7.2921150e-5;

  78.     /** Mean value of inclination for Glonass orbit is equal to 63°. */
  79.     private static final double GLONASS_MEAN_INCLINATION = 64.8;

  81.     /** Mean value of Draconian period for Glonass orbit is equal to 40544s : 11 hours 15 minutes 44 seconds. */
  82.     private static final double GLONASS_MEAN_DRACONIAN_PERIOD = 40544;

  84.     /** Second degree zonal coefficient of normal potential. */
  85.     private static final double GLONASS_J20 = 1.08262575e-3;

  87.     /** Equatorial radius of Earth (m). */
  88.     private static final double GLONASS_EARTH_EQUATORIAL_RADIUS = 6378136;

  90.     // Data used to solve Kepler's equation
  91.     /** First coefficient to compute Kepler equation solver starter. */
  92.     private static final double A;

  94.     /** Second coefficient to compute Kepler equation solver starter. */
  95.     private static final double B;

  97.     static {
  98.         final double k1 = 3 * FastMath.PI + 2;
  99.         final double k2 = FastMath.PI - 1;
  100.         final double k3 = 6 * FastMath.PI - 1;
  101.         A  = 3 * k2 * k2 / k1;
  102.         B  = k3 * k3 / (6 * k1);
  103.     }

  105.     /** The GLONASS orbital elements used. */
  106.     private final GLONASSOrbitalElements glonassOrbit;

  108.     /** The spacecraft mass (kg). */
  109.     private final double mass;

  111.     /** The ECI frame used for GLONASS propagation. */
  112.     private final Frame eci;

  114.     /** The ECEF frame used for GLONASS propagation. */
  115.     private final Frame ecef;

  117.     /** Data context for propagation. */
  118.     private final DataContext dataContext;

  120.     /**
  121.      * Private constructor.
  122.      * @param glonassOrbit Glonass orbital elements
  123.      * @param eci Earth Centered Inertial frame
  124.      * @param ecef Earth Centered Earth Fixed frame
  125.      * @param provider Attitude provider
  126.      * @param mass Satellite mass (kg)
  127.      * @param context Data context
  128.      */
  129.     GLONASSAnalyticalPropagator(final GLONASSOrbitalElements glonassOrbit, final Frame eci,
  130.                                 final Frame ecef, final AttitudeProvider provider,
  131.                                 final double mass, final DataContext context) {
  132.         super(provider);
  133.         this.dataContext = context;
  134.         // Stores the GLONASS orbital elements
  135.         this.glonassOrbit = glonassOrbit;
  136.         // Sets the start date as the date of the orbital elements
  137.         setStartDate(glonassOrbit.getDate());
  138.         // Sets the mass
  139.         this.mass = mass;
  140.         // Sets the Earth Centered Inertial frame
  141.         this.eci  = eci;
  142.         // Sets the Earth Centered Earth Fixed frame
  143.         this.ecef = ecef;
  144.         // Sets initial state
  145.         final Orbit orbit = propagateOrbit(getStartDate());
  146.         final Attitude attitude = provider.getAttitude(orbit, orbit.getDate(), orbit.getFrame());
  147.         super.resetInitialState(new SpacecraftState(orbit, attitude).withMass(mass));
  148.     }

  150.     /**
  151.      * Gets the PVCoordinates of the GLONASS SV in {@link #getECEF() ECEF frame}.
  152.      *
  153.      * <p>The algorithm is defined at Appendix M.1 from GLONASS Interface Control Document,
  154.      * with automatic differentiation added to compute velocity and
  155.      * acceleration.</p>
  156.      *
  157.      * @param date the computation date
  158.      * @return the GLONASS SV PVCoordinates in {@link #getECEF() ECEF frame}
  159.      */
  160.     public PVCoordinates propagateInEcef(final AbsoluteDate date) {

  162.         // Interval of prediction dTpr
  163.         final UnivariateDerivative2 dTpr = getdTpr(date);

  165.         // Zero
  166.         final UnivariateDerivative2 zero = dTpr.getField().getZero();

  168.         // The number of whole orbits "w" on a prediction interval
  169.         final UnivariateDerivative2 w = FastMath.floor(dTpr.divide(GLONASS_MEAN_DRACONIAN_PERIOD + glonassOrbit.getDeltaT()));

  171.         // Current inclination
  172.         final UnivariateDerivative2 i = zero.newInstance(GLONASS_MEAN_INCLINATION / 180 * GNSSConstants.GLONASS_PI + glonassOrbit.getDeltaI());

  174.         // Eccentricity
  175.         final UnivariateDerivative2 e = zero.newInstance(glonassOrbit.getE());

  177.         // Mean draconique period in orbite w+1 and mean motion
  178.         final UnivariateDerivative2 tDR = w.multiply(2.0).add(1.0).multiply(glonassOrbit.getDeltaTDot()).
  179.                                           add(glonassOrbit.getDeltaT()).
  180.                                           add(GLONASS_MEAN_DRACONIAN_PERIOD);
  181.         final UnivariateDerivative2 n = tDR.divide(2.0 * GNSSConstants.GLONASS_PI).reciprocal();

  183.         // Semi-major axis : computed by successive approximation
  184.         final UnivariateDerivative2 sma = computeSma(tDR, i, e);

  186.         // (ae / p)^2 term
  187.         final UnivariateDerivative2 p     = sma.multiply(e.multiply(e).negate().add(1.0));
  188.         final UnivariateDerivative2 aeop  = p.divide(GLONASS_EARTH_EQUATORIAL_RADIUS).reciprocal();
  189.         final UnivariateDerivative2 aeop2 = aeop.multiply(aeop);

  191.         // Current longitude of the ascending node
  192.         final UnivariateDerivative2 lambda = computeLambda(dTpr, n, aeop2, i);

  194.         // Current argument of perigee
  195.         final UnivariateDerivative2 pa = computePA(dTpr, n, aeop2, i);

  197.         // Mean longitude at the instant the spacecraft passes the current ascending node
  198.         final UnivariateDerivative2 tanPAo2 = FastMath.tan(pa.divide(2.0));
  199.         final UnivariateDerivative2 coef    = tanPAo2.multiply(FastMath.sqrt(e.negate().add(1.0).divide(e.add(1.0))));
  200.         final UnivariateDerivative2 e0      = FastMath.atan(coef).multiply(2.0).negate();
  201.         final UnivariateDerivative2 m1      = pa.add(e0).subtract(FastMath.sin(e0).multiply(e));

  203.         // Current mean longitude
  204.         final UnivariateDerivative2 correction = dTpr.
  205.                                                  subtract(w.multiply(GLONASS_MEAN_DRACONIAN_PERIOD + glonassOrbit.getDeltaT())).
  206.                                                  subtract(w.square().multiply(glonassOrbit.getDeltaTDot()));
  207.         final UnivariateDerivative2 m = m1.add(n.multiply(correction));

  209.         // Take into consideration the periodic perturbations
  210.         final FieldSinCos<UnivariateDerivative2> scPa = FastMath.sinCos(pa);
  211.         final UnivariateDerivative2 h = e.multiply(scPa.sin());
  212.         final UnivariateDerivative2 l = e.multiply(scPa.cos());
  213.         // δa1
  214.         final UnivariateDerivative2[] d1 = getParameterDifferentials(sma, i, h, l, m1);
  215.         // δa2
  216.         final UnivariateDerivative2[] d2 = getParameterDifferentials(sma, i, h, l, m);
  217.         // Apply corrections
  218.         final UnivariateDerivative2 smaCorr    = sma.add(d2[0]).subtract(d1[0]);
  219.         final UnivariateDerivative2 hCorr      = h.add(d2[1]).subtract(d1[1]);
  220.         final UnivariateDerivative2 lCorr      = l.add(d2[2]).subtract(d1[2]);
  221.         final UnivariateDerivative2 lambdaCorr = lambda.add(d2[3]).subtract(d1[3]);
  222.         final UnivariateDerivative2 iCorr      = i.add(d2[4]).subtract(d1[4]);
  223.         final UnivariateDerivative2 mCorr      = m.add(d2[5]).subtract(d1[5]);
  224.         final UnivariateDerivative2 eCorr      = FastMath.sqrt(hCorr.multiply(hCorr).add(lCorr.multiply(lCorr)));
  225.         final UnivariateDerivative2 paCorr;
  226.         if (eCorr.getValue() == 0.) {
  227.             paCorr = zero;
  228.         } else {
  229.             if (lCorr.getValue() == eCorr.getValue()) {
  230.                 paCorr = zero.newInstance(0.5 * GNSSConstants.GLONASS_PI);
  231.             } else if (lCorr.getValue() == -eCorr.getValue()) {
  232.                 paCorr = zero.newInstance(-0.5 * GNSSConstants.GLONASS_PI);
  233.             } else {
  234.                 paCorr = FastMath.atan2(hCorr, lCorr);
  235.             }
  236.         }

  238.         // Eccentric Anomaly
  239.         final UnivariateDerivative2 mk = mCorr.subtract(paCorr);
  240.         final UnivariateDerivative2 ek = getEccentricAnomaly(mk, eCorr);

  242.         // True Anomaly
  243.         final UnivariateDerivative2 vk =  getTrueAnomaly(ek, eCorr);

  245.         // Argument of Latitude
  246.         final UnivariateDerivative2 phik = vk.add(paCorr);

  248.         // Corrected Radius
  249.         final UnivariateDerivative2 pCorr = smaCorr.multiply(eCorr.multiply(eCorr).negate().add(1.0));
  250.         final UnivariateDerivative2 rk    = pCorr.divide(eCorr.multiply(FastMath.cos(vk)).add(1.0));

  252.         // Positions in orbital plane
  253.         final FieldSinCos<UnivariateDerivative2> scPhik = FastMath.sinCos(phik);
  254.         final UnivariateDerivative2 xk = scPhik.cos().multiply(rk);
  255.         final UnivariateDerivative2 yk = scPhik.sin().multiply(rk);

  257.         // Coordinates of position
  258.         final FieldSinCos<UnivariateDerivative2> scL = FastMath.sinCos(lambdaCorr);
  259.         final FieldSinCos<UnivariateDerivative2> scI = FastMath.sinCos(iCorr);
  260.         final FieldVector3D<UnivariateDerivative2> positionwithDerivatives =
  261.                         new FieldVector3D<>(xk.multiply(scL.cos()).subtract(yk.multiply(scL.sin()).multiply(scI.cos())),
  262.                                             xk.multiply(scL.sin()).add(yk.multiply(scL.cos()).multiply(scI.cos())),
  263.                                             yk.multiply(scI.sin()));

  265.         return new PVCoordinates(new Vector3D(positionwithDerivatives.getX().getValue(),
  266.                                               positionwithDerivatives.getY().getValue(),
  267.                                               positionwithDerivatives.getZ().getValue()),
  268.                                  new Vector3D(positionwithDerivatives.getX().getFirstDerivative(),
  269.                                               positionwithDerivatives.getY().getFirstDerivative(),
  270.                                               positionwithDerivatives.getZ().getFirstDerivative()),
  271.                                  new Vector3D(positionwithDerivatives.getX().getSecondDerivative(),
  272.                                               positionwithDerivatives.getY().getSecondDerivative(),
  273.                                               positionwithDerivatives.getZ().getSecondDerivative()));
  274.     }

  276.     /**
  277.      * Gets eccentric anomaly from mean anomaly.
  278.      * <p>The algorithm used to solve the Kepler equation has been published in:
  279.      * "Procedures for  solving Kepler's Equation", A. W. Odell and R. H. Gooding,
  280.      * Celestial Mechanics 38 (1986) 307-334</p>
  281.      * <p>It has been copied from the OREKIT library (KeplerianOrbit class).</p>
  282.      *
  283.      * @param mk the mean anomaly (rad)
  284.      * @param e the eccentricity
  285.      * @return the eccentric anomaly (rad)
  286.      */
  287.     private UnivariateDerivative2 getEccentricAnomaly(final UnivariateDerivative2 mk, final UnivariateDerivative2 e) {

  289.         // reduce M to [-PI PI] interval
  290.         final UnivariateDerivative2 reducedM = new UnivariateDerivative2(MathUtils.normalizeAngle(mk.getValue(), 0.0),
  291.                                                                          mk.getFirstDerivative(),
  292.                                                                          mk.getSecondDerivative());

  294.         // compute start value according to A. W. Odell and R. H. Gooding S12 starter
  295.         UnivariateDerivative2 ek;
  296.         if (FastMath.abs(reducedM.getValue()) < 1.0 / 6.0) {
  297.             if (FastMath.abs(reducedM.getValue()) < Precision.SAFE_MIN) {
  298.                 // this is an Orekit change to the S12 starter.
  299.                 // If reducedM is 0.0, the derivative of cbrt is infinite which induces NaN appearing later in
  300.                 // the computation. As in this case E and M are almost equal, we initialize ek with reducedM
  301.                 ek = reducedM;
  302.             } else {
  303.                 // this is the standard S12 starter
  304.                 ek = reducedM.add(reducedM.multiply(6).cbrt().subtract(reducedM).multiply(e));
  305.             }
  306.         } else {
  307.             if (reducedM.getValue() < 0) {
  308.                 final UnivariateDerivative2 w = reducedM.add(FastMath.PI);
  309.                 ek = reducedM.add(w.multiply(-A).divide(w.subtract(B)).subtract(FastMath.PI).subtract(reducedM).multiply(e));
  310.             } else {
  311.                 final UnivariateDerivative2 minusW = reducedM.subtract(FastMath.PI);
  312.                 ek = reducedM.add(minusW.multiply(A).divide(minusW.add(B)).add(FastMath.PI).subtract(reducedM).multiply(e));
  313.             }
  314.         }

  316.         final UnivariateDerivative2 e1 = e.negate().add(1.0);
  317.         final boolean noCancellationRisk = (e1.getValue() + ek.getValue() * ek.getValue() / 6) >= 0.1;

  319.         // perform two iterations, each consisting of one Halley step and one Newton-Raphson step
  320.         for (int j = 0; j < 2; ++j) {
  321.             final UnivariateDerivative2 f;
  322.             UnivariateDerivative2 fd;
  323.             final UnivariateDerivative2 fdd  = ek.sin().multiply(e);
  324.             final UnivariateDerivative2 fddd = ek.cos().multiply(e);
  325.             if (noCancellationRisk) {
  326.                 f  = ek.subtract(fdd).subtract(reducedM);
  327.                 fd = fddd.subtract(1).negate();
  328.             } else {
  329.                 f  = eMeSinE(ek, e).subtract(reducedM);
  330.                 final UnivariateDerivative2 s = ek.multiply(0.5).sin();
  331.                 fd = s.multiply(s).multiply(e.multiply(2.0)).add(e1);
  332.             }
  333.             final UnivariateDerivative2 dee = f.multiply(fd).divide(f.multiply(0.5).multiply(fdd).subtract(fd.multiply(fd)));

  335.             // update eccentric anomaly, using expressions that limit underflow problems
  336.             final UnivariateDerivative2 w = fd.add(dee.multiply(0.5).multiply(fdd.add(dee.multiply(fdd).divide(3))));
  337.             fd = fd.add(dee.multiply(fdd.add(dee.multiply(0.5).multiply(fdd))));
  338.             ek = ek.subtract(f.subtract(dee.multiply(fd.subtract(w))).divide(fd));
  339.         }

  341.         // expand the result back to original range
  342.         ek = ek.add(mk.getValue() - reducedM.getValue());

  344.         // Returns the eccentric anomaly
  345.         return ek;
  346.     }

  348.     /**
  349.      * Accurate computation of E - e sin(E).
  350.      *
  351.      * @param E eccentric anomaly
  352.      * @param ecc the eccentricity
  353.      * @return E - e sin(E)
  354.      */
  355.     private UnivariateDerivative2 eMeSinE(final UnivariateDerivative2 E, final UnivariateDerivative2 ecc) {
  356.         UnivariateDerivative2 x = E.sin().multiply(ecc.negate().add(1.0));
  357.         final UnivariateDerivative2 mE2 = E.negate().multiply(E);
  358.         UnivariateDerivative2 term = E;
  359.         UnivariateDerivative2 d    = E.getField().getZero();
  360.         // the inequality test below IS intentional and should NOT be replaced by a check with a small tolerance
  361.         for (UnivariateDerivative2 x0 = d.add(Double.NaN); !Double.valueOf(x.getValue()).equals(x0.getValue());) {
  362.             d = d.add(2);
  363.             term = term.multiply(mE2.divide(d.multiply(d.add(1))));
  364.             x0 = x;
  365.             x = x.subtract(term);
  366.         }
  367.         return x;
  368.     }

  370.     /** Gets true anomaly from eccentric anomaly.
  371.     *
  372.     * @param ek the eccentric anomaly (rad)
  373.     * @param ecc the eccentricity
  374.     * @return the true anomaly (rad)
  375.     */
  376.     private UnivariateDerivative2 getTrueAnomaly(final UnivariateDerivative2 ek, final UnivariateDerivative2 ecc) {
  377.         final UnivariateDerivative2 svk = ek.sin().multiply(FastMath.sqrt( ecc.square().negate().add(1.0)));
  378.         final UnivariateDerivative2 cvk = ek.cos().subtract(ecc);
  379.         return svk.atan2(cvk);
  380.     }

  382.     /**
  383.      * Get the interval of prediction.
  384.      *
  385.      * @param date the considered date
  386.      * @return the duration from GLONASS orbit Reference epoch (s)
  387.      */
  388.     private UnivariateDerivative2 getdTpr(final AbsoluteDate date) {
  389.         final TimeScale glonass = dataContext.getTimeScales().getGLONASS();
  390.         final GLONASSDate tEnd = new GLONASSDate(date, glonass);
  391.         final GLONASSDate tSta = new GLONASSDate(glonassOrbit.getDate(), glonass);
  392.         final int n  = tEnd.getDayNumber();
  393.         final int na = tSta.getDayNumber();
  394.         final int deltaN;
  395.         if (na == 27) {
  396.             deltaN = n - na - FastMath.round((float) (n - na) / 1460) * 1460;
  397.         } else {
  398.             deltaN = n - na - FastMath.round((float) (n - na) / 1461) * 1461;
  399.         }
  400.         final UnivariateDerivative2 ti = new UnivariateDerivative2(tEnd.getSecInDay(), 1.0, 0.0);

  402.         return ti.subtract(glonassOrbit.getTime()).add(86400 * deltaN);
  403.     }

  405.     /**
  406.      * Computes the semi-major axis of orbit using technique of successive approximations.
  407.      * @param tDR mean draconique period (s)
  408.      * @param i current inclination (rad)
  409.      * @param e eccentricity
  410.      * @return the semi-major axis (m).
  411.      */
  412.     private UnivariateDerivative2 computeSma(final UnivariateDerivative2 tDR,
  413.                                              final UnivariateDerivative2 i,
  414.                                              final UnivariateDerivative2 e) {

  416.         // Zero
  417.         final UnivariateDerivative2 zero = tDR.getField().getZero();

  419.         // If one of the input parameter is equal to Double.NaN, an infinite loop can occur.
  420.         // In that case, we do not compute the value of the semi major axis.
  421.         // We decided to return a Double.NaN value instead.
  422.         if (Double.isNaN(tDR.getValue()) || Double.isNaN(i.getValue()) || Double.isNaN(e.getValue())) {
  423.             return zero.add(Double.NaN);
  424.         }

  426.         // Common parameters
  427.         final UnivariateDerivative2 sinI         = FastMath.sin(i);
  428.         final UnivariateDerivative2 sin2I        = sinI.multiply(sinI);
  429.         final UnivariateDerivative2 ome2         = e.multiply(e).negate().add(1.0);
  430.         final UnivariateDerivative2 ome2Pow3o2   = FastMath.sqrt(ome2).multiply(ome2);
  431.         final UnivariateDerivative2 pa           = zero.newInstance(glonassOrbit.getPa());
  432.         final UnivariateDerivative2 cosPA        = FastMath.cos(pa);
  433.         final UnivariateDerivative2 opecosPA     = e.multiply(cosPA).add(1.0);
  434.         final UnivariateDerivative2 opecosPAPow2 = opecosPA.multiply(opecosPA);
  435.         final UnivariateDerivative2 opecosPAPow3 = opecosPAPow2.multiply(opecosPA);

  437.         // Initial approximation
  438.         UnivariateDerivative2 tOCK = tDR;

  440.         // Successive approximations
  441.         // The process of approximation ends when fulfilling the following condition: |a(n+1) - a(n)| < 1cm
  442.         UnivariateDerivative2 an   = zero;
  443.         UnivariateDerivative2 anp1 = zero;
  444.         boolean isLastStep = false;
  445.         while (!isLastStep) {

  447.             // a(n+1) computation
  448.             final UnivariateDerivative2 tOCKo2p     = tOCK.divide(2.0 * GNSSConstants.GLONASS_PI);
  449.             final UnivariateDerivative2 tOCKo2pPow2 = tOCKo2p.multiply(tOCKo2p);
  450.             anp1 = FastMath.cbrt(tOCKo2pPow2.multiply(GNSSConstants.GLONASS_MU));

  452.             // p(n+1) computation
  453.             final UnivariateDerivative2 p = anp1.multiply(ome2);

  455.             // Tock(n+1) computation
  456.             final UnivariateDerivative2 aeop  = p.divide(GLONASS_EARTH_EQUATORIAL_RADIUS).reciprocal();
  457.             final UnivariateDerivative2 aeop2 = aeop.multiply(aeop);
  458.             final UnivariateDerivative2 term1 = aeop2.multiply(GLONASS_J20).multiply(1.5);
  459.             final UnivariateDerivative2 term2 = sin2I.multiply(2.5).negate().add(2.0);
  460.             final UnivariateDerivative2 term3 = ome2Pow3o2.divide(opecosPAPow2);
  461.             final UnivariateDerivative2 term4 = opecosPAPow3.divide(ome2);
  462.             tOCK = tDR.divide(term1.multiply(term2.multiply(term3).add(term4)).negate().add(1.0));

  464.             // Check convergence
  465.             if (FastMath.abs(anp1.subtract(an).getReal()) <= 0.01) {
  466.                 isLastStep = true;
  467.             }

  469.             an = anp1;
  470.         }

  472.         return an;

  474.     }

  476.     /**
  477.      * Computes the current longitude of the ascending node.
  478.      * @param dTpr interval of prediction (s)
  479.      * @param n mean motion (rad/s)
  480.      * @param aeop2 square of the ratio between the radius of the ellipsoid and p, with p = sma * (1 - ecc²)
  481.      * @param i inclination (rad)
  482.      * @return the current longitude of the ascending node (rad)
  483.      */
  484.     private UnivariateDerivative2 computeLambda(final UnivariateDerivative2 dTpr,
  485.                                                 final UnivariateDerivative2 n,
  486.                                                 final UnivariateDerivative2 aeop2,
  487.                                                 final UnivariateDerivative2 i) {
  488.         final UnivariateDerivative2 cosI = FastMath.cos(i);
  489.         final UnivariateDerivative2 precession = aeop2.multiply(n).multiply(cosI).multiply(1.5 * GLONASS_J20);
  490.         return dTpr.multiply(precession.add(GLONASS_AV)).negate().add(glonassOrbit.getLambda());
  491.     }

  493.     /**
  494.      * Computes the current argument of perigee.
  495.      * @param dTpr interval of prediction (s)
  496.      * @param n mean motion (rad/s)
  497.      * @param aeop2 square of the ratio between the radius of the ellipsoid and p, with p = sma * (1 - ecc²)
  498.      * @param i inclination (rad)
  499.      * @return the current argument of perigee (rad)
  500.      */
  501.     private UnivariateDerivative2 computePA(final UnivariateDerivative2 dTpr,
  502.                                             final UnivariateDerivative2 n,
  503.                                             final UnivariateDerivative2 aeop2,
  504.                                             final UnivariateDerivative2 i) {
  505.         final UnivariateDerivative2 cosI  = FastMath.cos(i);
  506.         final UnivariateDerivative2 cos2I = cosI.multiply(cosI);
  507.         final UnivariateDerivative2 precession = aeop2.multiply(n).multiply(cos2I.multiply(5.0).negate().add(1.0)).multiply(0.75 * GLONASS_J20);
  508.         return dTpr.multiply(precession).negate().add(glonassOrbit.getPa());
  509.     }

  511.     /**
  512.      * Computes the differentials δa<sub>i</sub>.
  513.      * <p>
  514.      * The value of i depends of the type of longitude (i = 2 for the current mean longitude;
  515.      * i = 1 for the mean longitude at the instant the spacecraft passes the current ascending node)
  516.      * </p>
  517.      * @param a semi-major axis (m)
  518.      * @param i inclination (rad)
  519.      * @param h x component of the eccentricity (rad)
  520.      * @param l y component of the eccentricity (rad)
  521.      * @param m longitude (current or at the ascending node instant)
  522.      * @return the differentials of the orbital parameters
  523.      */
  524.     private UnivariateDerivative2[] getParameterDifferentials(final UnivariateDerivative2 a, final UnivariateDerivative2 i,
  525.                                                               final UnivariateDerivative2 h, final UnivariateDerivative2 l,
  526.                                                               final UnivariateDerivative2 m) {

  528.         // B constant
  529.         final UnivariateDerivative2 aeoa  = a.divide(GLONASS_EARTH_EQUATORIAL_RADIUS).reciprocal();
  530.         final UnivariateDerivative2 aeoa2 = aeoa.multiply(aeoa);
  531.         final UnivariateDerivative2 b     = aeoa2.multiply(1.5 * GLONASS_J20);

  533.         // Commons Parameters
  534.         final FieldSinCos<UnivariateDerivative2> scI   = FastMath.sinCos(i);
  535.         final FieldSinCos<UnivariateDerivative2> scLk  = FastMath.sinCos(m);
  536.         final FieldSinCos<UnivariateDerivative2> sc2Lk = FieldSinCos.sum(scLk, scLk);
  537.         final FieldSinCos<UnivariateDerivative2> sc3Lk = FieldSinCos.sum(scLk, sc2Lk);
  538.         final FieldSinCos<UnivariateDerivative2> sc4Lk = FieldSinCos.sum(sc2Lk, sc2Lk);
  539.         final UnivariateDerivative2 cosI   = scI.cos();
  540.         final UnivariateDerivative2 sinI   = scI.sin();
  541.         final UnivariateDerivative2 cosI2  = cosI.multiply(cosI);
  542.         final UnivariateDerivative2 sinI2  = sinI.multiply(sinI);
  543.         final UnivariateDerivative2 cosLk  = scLk.cos();
  544.         final UnivariateDerivative2 sinLk  = scLk.sin();
  545.         final UnivariateDerivative2 cos2Lk = sc2Lk.cos();
  546.         final UnivariateDerivative2 sin2Lk = sc2Lk.sin();
  547.         final UnivariateDerivative2 cos3Lk = sc3Lk.cos();
  548.         final UnivariateDerivative2 sin3Lk = sc3Lk.sin();
  549.         final UnivariateDerivative2 cos4Lk = sc4Lk.cos();
  550.         final UnivariateDerivative2 sin4Lk = sc4Lk.sin();

  552.         // h*cos(nLk), l*cos(nLk), h*sin(nLk) and l*sin(nLk)
  553.         // n = 1
  554.         final UnivariateDerivative2 hCosLk = h.multiply(cosLk);
  555.         final UnivariateDerivative2 hSinLk = h.multiply(sinLk);
  556.         final UnivariateDerivative2 lCosLk = l.multiply(cosLk);
  557.         final UnivariateDerivative2 lSinLk = l.multiply(sinLk);
  558.         // n = 2
  559.         final UnivariateDerivative2 hCos2Lk = h.multiply(cos2Lk);
  560.         final UnivariateDerivative2 hSin2Lk = h.multiply(sin2Lk);
  561.         final UnivariateDerivative2 lCos2Lk = l.multiply(cos2Lk);
  562.         final UnivariateDerivative2 lSin2Lk = l.multiply(sin2Lk);
  563.         // n = 3
  564.         final UnivariateDerivative2 hCos3Lk = h.multiply(cos3Lk);
  565.         final UnivariateDerivative2 hSin3Lk = h.multiply(sin3Lk);
  566.         final UnivariateDerivative2 lCos3Lk = l.multiply(cos3Lk);
  567.         final UnivariateDerivative2 lSin3Lk = l.multiply(sin3Lk);
  568.         // n = 4
  569.         final UnivariateDerivative2 hCos4Lk = h.multiply(cos4Lk);
  570.         final UnivariateDerivative2 hSin4Lk = h.multiply(sin4Lk);
  571.         final UnivariateDerivative2 lCos4Lk = l.multiply(cos4Lk);
  572.         final UnivariateDerivative2 lSin4Lk = l.multiply(sin4Lk);

  574.         // 1 - (3 / 2)*sin²i
  575.         final UnivariateDerivative2 om3o2xSinI2 = sinI2.multiply(1.5).negate().add(1.0);

  577.         // Compute Differentials
  578.         // δa
  579.         final UnivariateDerivative2 dakT1 = b.multiply(2.0).multiply(om3o2xSinI2).multiply(lCosLk.add(hSinLk));
  580.         final UnivariateDerivative2 dakT2 = b.multiply(sinI2).multiply(hSinLk.multiply(0.5).subtract(lCosLk.multiply(0.5)).
  581.                                                                      add(cos2Lk).add(lCos3Lk.multiply(3.5)).add(hSin3Lk.multiply(3.5)));
  582.         final UnivariateDerivative2 dak = dakT1.add(dakT2);

  584.         // δh
  585.         final UnivariateDerivative2 dhkT1 = b.multiply(om3o2xSinI2).multiply(sinLk.add(lSin2Lk.multiply(1.5)).subtract(hCos2Lk.multiply(1.5)));
  586.         final UnivariateDerivative2 dhkT2 = b.multiply(sinI2).multiply(0.25).multiply(sinLk.subtract(sin3Lk.multiply(SEVEN_THIRD)).add(lSin2Lk.multiply(5.0)).
  587.                                                                                     subtract(lSin4Lk.multiply(8.5)).add(hCos4Lk.multiply(8.5)).add(hCos2Lk));
  588.         final UnivariateDerivative2 dhkT3 = lSin2Lk.multiply(cosI2).multiply(b).multiply(0.5).negate();
  589.         final UnivariateDerivative2 dhk = dhkT1.subtract(dhkT2).add(dhkT3);

  591.         // δl
  592.         final UnivariateDerivative2 dlkT1 = b.multiply(om3o2xSinI2).multiply(cosLk.add(lCos2Lk.multiply(1.5)).add(hSin2Lk.multiply(1.5)));
  593.         final UnivariateDerivative2 dlkT2 = b.multiply(sinI2).multiply(0.25).multiply(cosLk.negate().subtract(cos3Lk.multiply(SEVEN_THIRD)).subtract(hSin2Lk.multiply(5.0)).
  594.                                                                                     subtract(lCos4Lk.multiply(8.5)).subtract(hSin4Lk.multiply(8.5)).add(lCos2Lk));
  595.         final UnivariateDerivative2 dlkT3 = hSin2Lk.multiply(cosI2).multiply(b).multiply(0.5);
  596.         final UnivariateDerivative2 dlk = dlkT1.subtract(dlkT2).add(dlkT3);

  598.         // δλ
  599.         final UnivariateDerivative2 dokT1 = b.negate().multiply(cosI);
  600.         final UnivariateDerivative2 dokT2 = lSinLk.multiply(3.5).subtract(hCosLk.multiply(2.5)).subtract(sin2Lk.multiply(0.5)).
  601.                                           subtract(lSin3Lk.multiply(SEVEN_SIXTH)).add(hCos3Lk.multiply(SEVEN_SIXTH));
  602.         final UnivariateDerivative2 dok = dokT1.multiply(dokT2);

  604.         // δi
  605.         final UnivariateDerivative2 dik = b.multiply(sinI).multiply(cosI).multiply(0.5).
  606.                         multiply(lCosLk.negate().add(hSinLk).add(cos2Lk).add(lCos3Lk.multiply(SEVEN_THIRD)).add(hSin3Lk.multiply(SEVEN_THIRD)));

  608.         // δL
  609.         final UnivariateDerivative2 dLkT1 = b.multiply(2.0).multiply(om3o2xSinI2).multiply(lSinLk.multiply(1.75).subtract(hCosLk.multiply(1.75)));
  610.         final UnivariateDerivative2 dLkT2 = b.multiply(sinI2).multiply(3.0).multiply(hCosLk.multiply(SEVEN_24TH).negate().subtract(lSinLk.multiply(SEVEN_24TH)).
  611.                                                                                    subtract(hCos3Lk.multiply(FN_72TH)).add(lSin3Lk.multiply(FN_72TH)).add(sin2Lk.multiply(0.25)));
  612.         final UnivariateDerivative2 dLkT3 = b.multiply(cosI2).multiply(lSinLk.multiply(3.5).subtract(hCosLk.multiply(2.5)).subtract(sin2Lk.multiply(0.5)).
  613.                                                                      subtract(lSin3Lk.multiply(SEVEN_SIXTH)).add(hCos3Lk.multiply(SEVEN_SIXTH)));
  614.         final UnivariateDerivative2 dLk = dLkT1.add(dLkT2).add(dLkT3);

  616.         // Final array
  617.         final UnivariateDerivative2[] differentials = MathArrays.buildArray(a.getField(), 6);
  618.         differentials[0] = dak.multiply(a);
  619.         differentials[1] = dhk;
  620.         differentials[2] = dlk;
  621.         differentials[3] = dok;
  622.         differentials[4] = dik;
  623.         differentials[5] = dLk;

  625.         return differentials;
  626.     }

  628.     /** {@inheritDoc} */
  629.     protected double getMass(final AbsoluteDate date) {
  630.         return mass;
  631.     }

  633.     /**
  634.      * Get the Earth gravity coefficient used for GLONASS propagation.
  635.      * @return the Earth gravity coefficient.
  636.      */
  637.     public static double getMU() {
  638.         return GNSSConstants.GLONASS_MU;
  639.     }

  641.     /**
  642.      * Gets the underlying GLONASS orbital elements.
  643.      *
  644.      * @return the underlying GLONASS orbital elements
  645.      */
  646.     public GLONASSOrbitalElements getGLONASSOrbitalElements() {
  647.         return glonassOrbit;
  648.     }

  650.     /**
  651.      * Gets the Earth Centered Inertial frame used to propagate the orbit.
  652.      * @return the ECI frame
  653.      */
  654.     public Frame getECI() {
  655.         return eci;
  656.     }

  658.     /**
  659.      * Gets the Earth Centered Earth Fixed frame used to propagate GLONASS orbits.
  660.      * @return the ECEF frame
  661.      */
  662.     public Frame getECEF() {
  663.         return ecef;
  664.     }

  666.     /** {@inheritDoc} */
  667.     public Frame getFrame() {
  668.         return eci;
  669.     }

  671.     /** {@inheritDoc} */
  672.     public void resetInitialState(final SpacecraftState state) {
  673.         throw new OrekitException(OrekitMessages.NON_RESETABLE_STATE);
  674.     }

  676.     /** {@inheritDoc} */
  677.     protected void resetIntermediateState(final SpacecraftState state, final boolean forward) {
  678.         throw new OrekitException(OrekitMessages.NON_RESETABLE_STATE);
  679.     }

  681.     /** {@inheritDoc} */
  682.     public Orbit propagateOrbit(final AbsoluteDate date) {
  683.         // Gets the PVCoordinates in ECEF frame
  684.         final PVCoordinates pvaInECEF = propagateInEcef(date);
  685.         // Transforms the PVCoordinates to ECI frame
  686.         final PVCoordinates pvaInECI = ecef.getTransformTo(eci, date).transformPVCoordinates(pvaInECEF);
  687.         // Returns the Cartesian orbit
  688.         return new CartesianOrbit(pvaInECI, eci, date, GNSSConstants.GLONASS_MU);
  689.     }

  691. }
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