FieldKeplerianOrbit.java

  1. /* Copyright 2002-2022 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.orbits;


  20. import java.util.List;
  21. import java.util.stream.Collectors;
  22. import java.util.stream.Stream;

  24. import org.hipparchus.CalculusFieldElement;
  25. import org.hipparchus.analysis.differentiation.FieldUnivariateDerivative1;
  26. import org.hipparchus.analysis.interpolation.FieldHermiteInterpolator;
  27. import org.hipparchus.exception.MathIllegalArgumentException;
  28. import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  29. import org.hipparchus.util.FastMath;
  30. import org.hipparchus.util.FieldSinCos;
  31. import org.hipparchus.util.MathArrays;
  32. import org.hipparchus.util.MathUtils;
  33. import org.hipparchus.util.Precision;
  34. import org.orekit.errors.OrekitException;
  35. import org.orekit.errors.OrekitIllegalArgumentException;
  36. import org.orekit.errors.OrekitInternalError;
  37. import org.orekit.errors.OrekitMessages;
  38. import org.orekit.frames.Frame;
  39. import org.orekit.time.FieldAbsoluteDate;
  40. import org.orekit.utils.FieldPVCoordinates;
  41. import org.orekit.utils.TimeStampedFieldPVCoordinates;


  44. /**
  45.  * This class handles traditional Keplerian orbital parameters.

  47.  * <p>
  48.  * The parameters used internally are the classical Keplerian elements:
  49.  *   <pre>
  50.  *     a
  51.  *     e
  52.  *     i
  53.  *     ω
  54.  *     Ω
  55.  *     v
  56.  *   </pre>
  57.  * where ω stands for the Perigee Argument, Ω stands for the
  58.  * Right Ascension of the Ascending Node and v stands for the true anomaly.
  59.  * <p>
  60.  * This class supports hyperbolic orbits, using the convention that semi major
  61.  * axis is negative for such orbits (and of course eccentricity is greater than 1).
  62.  * </p>
  63.  * <p>
  64.  * When orbit is either equatorial or circular, some Keplerian elements
  65.  * (more precisely ω and Ω) become ambiguous so this class should not
  66.  * be used for such orbits. For this reason, {@link EquinoctialOrbit equinoctial
  67.  * orbits} is the recommended way to represent orbits.
  68.  * </p>

  70.  * <p>
  71.  * The instance <code>KeplerianOrbit</code> is guaranteed to be immutable.
  72.  * </p>
  73.  * @see     Orbit
  74.  * @see    CircularOrbit
  75.  * @see    CartesianOrbit
  76.  * @see    EquinoctialOrbit
  77.  * @author Luc Maisonobe
  78.  * @author Guylaine Prat
  79.  * @author Fabien Maussion
  80.  * @author V&eacute;ronique Pommier-Maurussane
  81.  * @author Andrea Antolino
  82.  * @since 9.0
  83.  */
  84. public class FieldKeplerianOrbit<T extends CalculusFieldElement<T>> extends FieldOrbit<T> {

  86.     /** Name of the eccentricity parameter. */
  87.     private static final String ECCENTRICITY = "eccentricity";

  89.     /** First coefficient to compute Kepler equation solver starter. */
  90.     private static final double A;

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

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

  103.     /** Semi-major axis (m). */
  104.     private final T a;

  106.     /** Eccentricity. */
  107.     private final T e;

  109.     /** Inclination (rad). */
  110.     private final T i;

  112.     /** Perigee Argument (rad). */
  113.     private final T pa;

  115.     /** Right Ascension of Ascending Node (rad). */
  116.     private final T raan;

  118.     /** True anomaly (rad). */
  119.     private final T v;

  121.     /** Semi-major axis derivative (m/s). */
  122.     private final T aDot;

  124.     /** Eccentricity derivative. */
  125.     private final T eDot;

  127.     /** Inclination derivative (rad/s). */
  128.     private final T iDot;

  130.     /** Perigee Argument derivative (rad/s). */
  131.     private final T paDot;

  133.     /** Right Ascension of Ascending Node derivative (rad/s). */
  134.     private final T raanDot;

  136.     /** True anomaly derivative (rad/s). */
  137.     private final T vDot;

  139.     /** Partial Cartesian coordinates (position and velocity are valid, acceleration may be missing). */
  140.     private FieldPVCoordinates<T> partialPV;

  142.     /** Identity element. */
  143.     private final T one;

  145.     /** Zero element. */
  146.     private final T zero;

  148.     /** Third Canonical Vector. */
  149.     private final FieldVector3D<T> PLUS_K;

  151.     /** Creates a new instance.
  152.      * @param a  semi-major axis (m), negative for hyperbolic orbits
  153.      * @param e eccentricity (positive or equal to 0)
  154.      * @param i inclination (rad)
  155.      * @param pa perigee argument (ω, rad)
  156.      * @param raan right ascension of ascending node (Ω, rad)
  157.      * @param anomaly mean, eccentric or true anomaly (rad)
  158.      * @param type type of anomaly
  159.      * @param frame the frame in which the parameters are defined
  160.      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
  161.      * @param date date of the orbital parameters
  162.      * @param mu central attraction coefficient (m³/s²)
  163.      * @exception IllegalArgumentException if frame is not a {@link
  164.      * Frame#isPseudoInertial pseudo-inertial frame} or a and e don't match for hyperbolic orbits,
  165.      * or v is out of range for hyperbolic orbits
  166.      */
  167.     public FieldKeplerianOrbit(final T a, final T e, final T i,
  168.                                final T pa, final T raan,
  169.                                final T anomaly, final PositionAngle type,
  170.                                final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
  171.         throws IllegalArgumentException {
  172.         this(a, e, i, pa, raan, anomaly,
  173.              null, null, null, null, null, null,
  174.              type, frame, date, mu);
  175.     }

  177.     /** Creates a new instance.
  178.      * @param a  semi-major axis (m), negative for hyperbolic orbits
  179.      * @param e eccentricity (positive or equal to 0)
  180.      * @param i inclination (rad)
  181.      * @param pa perigee argument (ω, rad)
  182.      * @param raan right ascension of ascending node (Ω, rad)
  183.      * @param anomaly mean, eccentric or true anomaly (rad)
  184.      * @param aDot  semi-major axis derivative, null if unknown (m/s)
  185.      * @param eDot eccentricity derivative, null if unknown
  186.      * @param iDot inclination derivative, null if unknown (rad/s)
  187.      * @param paDot perigee argument derivative, null if unknown (rad/s)
  188.      * @param raanDot right ascension of ascending node derivative, null if unknown (rad/s)
  189.      * @param anomalyDot mean, eccentric or true anomaly derivative, null if unknown (rad/s)
  190.      * @param type type of anomaly
  191.      * @param frame the frame in which the parameters are defined
  192.      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
  193.      * @param date date of the orbital parameters
  194.      * @param mu central attraction coefficient (m³/s²)
  195.      * @exception IllegalArgumentException if frame is not a {@link
  196.      * Frame#isPseudoInertial pseudo-inertial frame} or a and e don't match for hyperbolic orbits,
  197.      * or v is out of range for hyperbolic orbits
  198.      */
  199.     public FieldKeplerianOrbit(final T a, final T e, final T i,
  200.                                final T pa, final T raan, final T anomaly,
  201.                                final T aDot, final T eDot, final T iDot,
  202.                                final T paDot, final T raanDot, final T anomalyDot,
  203.                                final PositionAngle type,
  204.                                final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
  205.         throws IllegalArgumentException {
  206.         super(frame, date, mu);
  207.         if (a.multiply(e.negate().add(1)).getReal() < 0) {
  208.             throw new OrekitIllegalArgumentException(OrekitMessages.ORBIT_A_E_MISMATCH_WITH_CONIC_TYPE, a.getReal(), e.getReal());
  209.         }

  211.         // Checking eccentricity range
  212.         checkParameterRangeInclusive(ECCENTRICITY, e.getReal(), 0.0, Double.POSITIVE_INFINITY);

  214.         this.a       =    a;
  215.         this.aDot    =    aDot;
  216.         this.e       =    e;
  217.         this.eDot    =    eDot;
  218.         this.i       =    i;
  219.         this.iDot    =    iDot;
  220.         this.pa      =    pa;
  221.         this.paDot   =    paDot;
  222.         this.raan    =    raan;
  223.         this.raanDot =    raanDot;

  225.         /** Identity element. */
  226.         this.one = a.getField().getOne();

  228.         /** Zero element. */
  229.         this.zero = a.getField().getZero();

  231.         /**Third canonical vector. */
  232.         this.PLUS_K = FieldVector3D.getPlusK(a.getField());

  234.         if (hasDerivatives()) {
  235.             final FieldUnivariateDerivative1<T> eUD        = new FieldUnivariateDerivative1<>(e, eDot);
  236.             final FieldUnivariateDerivative1<T> anomalyUD  = new FieldUnivariateDerivative1<>(anomaly,  anomalyDot);
  237.             final FieldUnivariateDerivative1<T> vUD;
  238.             switch (type) {
  239.                 case MEAN :
  240.                     vUD = (a.getReal() < 0) ?
  241.                           hyperbolicEccentricToTrue(meanToHyperbolicEccentric(anomalyUD, eUD), eUD) :
  242.                           ellipticEccentricToTrue(meanToEllipticEccentric(anomalyUD, eUD), eUD);
  243.                     break;
  244.                 case ECCENTRIC :
  245.                     vUD = (a.getReal() < 0) ?
  246.                           hyperbolicEccentricToTrue(anomalyUD, eUD) :
  247.                           ellipticEccentricToTrue(anomalyUD, eUD);
  248.                     break;
  249.                 case TRUE :
  250.                     vUD = anomalyUD;
  251.                     break;
  252.                 default : // this should never happen
  253.                     throw new OrekitInternalError(null);
  254.             }
  255.             this.v    = vUD.getValue();
  256.             this.vDot = vUD.getDerivative(1);
  257.         } else {
  258.             switch (type) {
  259.                 case MEAN :

  261.                     this.v = (a.getReal() < 0) ?
  262.                              hyperbolicEccentricToTrue(meanToHyperbolicEccentric(anomaly, e), e) :
  263.                              ellipticEccentricToTrue(meanToEllipticEccentric(anomaly, e), e);

  265.                     break;
  266.                 case ECCENTRIC :
  267.                     this.v = (a.getReal() < 0) ?
  268.                              hyperbolicEccentricToTrue(anomaly, e) :
  269.                              ellipticEccentricToTrue(anomaly, e);

  271.                     break;
  272.                 case TRUE :
  273.                     this.v = anomaly;
  274.                     break;
  275.                 default : // this should never happen
  276.                     throw new OrekitInternalError(null);
  277.             }
  278.             this.vDot = null;
  279.         }

  281.         // check true anomaly range
  282.         if (e.multiply(v.cos()).add(1).getReal() <= 0) {
  283.             final double vMax = e.reciprocal().negate().acos().getReal();
  284.             throw new OrekitIllegalArgumentException(OrekitMessages.ORBIT_ANOMALY_OUT_OF_HYPERBOLIC_RANGE,
  285.                                                      v.getReal(), e.getReal(), -vMax, vMax);
  286.         }

  288.         this.partialPV = null;

  290.     }

  292.     /** Constructor from Cartesian parameters.
  293.      *
  294.      * <p> The acceleration provided in {@code FieldPVCoordinates} is accessible using
  295.      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
  296.      * use {@code mu} and the position to compute the acceleration, including
  297.      * {@link #shiftedBy(CalculusFieldElement)} and {@link #getPVCoordinates(FieldAbsoluteDate, Frame)}.
  298.      *
  299.      * @param pvCoordinates the PVCoordinates of the satellite
  300.      * @param frame the frame in which are defined the {@link FieldPVCoordinates}
  301.      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
  302.      * @param mu central attraction coefficient (m³/s²)
  303.      * @exception IllegalArgumentException if frame is not a {@link
  304.      * Frame#isPseudoInertial pseudo-inertial frame}
  305.      */
  306.     public FieldKeplerianOrbit(final TimeStampedFieldPVCoordinates<T> pvCoordinates,
  307.                                final Frame frame, final T mu)
  308.         throws IllegalArgumentException {
  309.         this(pvCoordinates, frame, mu, hasNonKeplerianAcceleration(pvCoordinates, mu));
  310.     }

  312.     /** Constructor from Cartesian parameters.
  313.      *
  314.      * <p> The acceleration provided in {@code FieldPVCoordinates} is accessible using
  315.      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
  316.      * use {@code mu} and the position to compute the acceleration, including
  317.      * {@link #shiftedBy(CalculusFieldElement)} and {@link #getPVCoordinates(FieldAbsoluteDate, Frame)}.
  318.      *
  319.      * @param pvCoordinates the PVCoordinates of the satellite
  320.      * @param frame the frame in which are defined the {@link FieldPVCoordinates}
  321.      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
  322.      * @param mu central attraction coefficient (m³/s²)
  323.      * @param reliableAcceleration if true, the acceleration is considered to be reliable
  324.      * @exception IllegalArgumentException if frame is not a {@link
  325.      * Frame#isPseudoInertial pseudo-inertial frame}
  326.      */
  327.     private FieldKeplerianOrbit(final TimeStampedFieldPVCoordinates<T> pvCoordinates,
  328.                                 final Frame frame, final T mu,
  329.                                 final boolean reliableAcceleration)
  330.         throws IllegalArgumentException {

  332.         super(pvCoordinates, frame, mu);

  334.         // identity element
  335.         this.one = pvCoordinates.getPosition().getX().getField().getOne();

  337.         // zero element
  338.         this.zero = one.getField().getZero();

  340.         // third canonical vector
  341.         this.PLUS_K = FieldVector3D.getPlusK(one.getField());

  343.         // compute inclination
  344.         final FieldVector3D<T> momentum = pvCoordinates.getMomentum();
  345.         final T m2 = momentum.getNormSq();

  347.         i = FieldVector3D.angle(momentum, PLUS_K);
  348.         // compute right ascension of ascending node
  349.         raan = FieldVector3D.crossProduct(PLUS_K, momentum).getAlpha();
  350.         // preliminary computations for parameters depending on orbit shape (elliptic or hyperbolic)
  351.         final FieldVector3D<T> pvP     = pvCoordinates.getPosition();
  352.         final FieldVector3D<T> pvV     = pvCoordinates.getVelocity();
  353.         final FieldVector3D<T> pvA     = pvCoordinates.getAcceleration();

  355.         final T   r2      = pvP.getNormSq();
  356.         final T   r       = r2.sqrt();
  357.         final T   V2      = pvV.getNormSq();
  358.         final T   rV2OnMu = r.multiply(V2).divide(mu);

  360.         // compute semi-major axis (will be negative for hyperbolic orbits)
  361.         a = r.divide(rV2OnMu.negate().add(2.0));
  362.         final T muA = a.multiply(mu);

  364.         // compute true anomaly
  365.         if (a.getReal() > 0) {
  366.             // elliptic or circular orbit
  367.             final T eSE = FieldVector3D.dotProduct(pvP, pvV).divide(muA.sqrt());
  368.             final T eCE = rV2OnMu.subtract(1);
  369.             e = (eSE.multiply(eSE).add(eCE.multiply(eCE))).sqrt();
  370.             v = ellipticEccentricToTrue(eSE.atan2(eCE), e);
  371.         } else {
  372.             // hyperbolic orbit
  373.             final T eSH = FieldVector3D.dotProduct(pvP, pvV).divide(muA.negate().sqrt());
  374.             final T eCH = rV2OnMu.subtract(1);
  375.             e = (m2.negate().divide(muA).add(1)).sqrt();
  376.             v = hyperbolicEccentricToTrue((eCH.add(eSH)).divide(eCH.subtract(eSH)).log().divide(2), e);
  377.         }

  379.         // Checking eccentricity range
  380.         checkParameterRangeInclusive(ECCENTRICITY, e.getReal(), 0.0, Double.POSITIVE_INFINITY);

  382.         // compute perigee argument
  383.         final FieldVector3D<T> node = new FieldVector3D<>(raan, zero);
  384.         final T px = FieldVector3D.dotProduct(pvP, node);
  385.         final T py = FieldVector3D.dotProduct(pvP, FieldVector3D.crossProduct(momentum, node)).divide(m2.sqrt());
  386.         pa = py.atan2(px).subtract(v);

  388.         partialPV = pvCoordinates;

  390.         if (reliableAcceleration) {
  391.             // we have a relevant acceleration, we can compute derivatives

  393.             final T[][] jacobian = MathArrays.buildArray(a.getField(), 6, 6);
  394.             getJacobianWrtCartesian(PositionAngle.MEAN, jacobian);

  396.             final FieldVector3D<T> keplerianAcceleration    = new FieldVector3D<>(r.multiply(r2).reciprocal().multiply(mu.negate()), pvP);
  397.             final FieldVector3D<T> nonKeplerianAcceleration = pvA.subtract(keplerianAcceleration);
  398.             final T   aX                       = nonKeplerianAcceleration.getX();
  399.             final T   aY                       = nonKeplerianAcceleration.getY();
  400.             final T   aZ                       = nonKeplerianAcceleration.getZ();
  401.             aDot    = jacobian[0][3].multiply(aX).add(jacobian[0][4].multiply(aY)).add(jacobian[0][5].multiply(aZ));
  402.             eDot    = jacobian[1][3].multiply(aX).add(jacobian[1][4].multiply(aY)).add(jacobian[1][5].multiply(aZ));
  403.             iDot    = jacobian[2][3].multiply(aX).add(jacobian[2][4].multiply(aY)).add(jacobian[2][5].multiply(aZ));
  404.             paDot   = jacobian[3][3].multiply(aX).add(jacobian[3][4].multiply(aY)).add(jacobian[3][5].multiply(aZ));
  405.             raanDot = jacobian[4][3].multiply(aX).add(jacobian[4][4].multiply(aY)).add(jacobian[4][5].multiply(aZ));

  407.             // in order to compute true anomaly derivative, we must compute
  408.             // mean anomaly derivative including Keplerian motion and convert to true anomaly
  409.             final T MDot = getKeplerianMeanMotion().
  410.                            add(jacobian[5][3].multiply(aX)).add(jacobian[5][4].multiply(aY)).add(jacobian[5][5].multiply(aZ));
  411.             final FieldUnivariateDerivative1<T> eUD = new FieldUnivariateDerivative1<>(e, eDot);
  412.             final FieldUnivariateDerivative1<T> MUD = new FieldUnivariateDerivative1<>(getMeanAnomaly(), MDot);
  413.             final FieldUnivariateDerivative1<T> vUD = (a.getReal() < 0) ?
  414.                                             FieldKeplerianOrbit.hyperbolicEccentricToTrue(FieldKeplerianOrbit.meanToHyperbolicEccentric(MUD, eUD), eUD) :
  415.                                             FieldKeplerianOrbit.ellipticEccentricToTrue(FieldKeplerianOrbit.meanToEllipticEccentric(MUD, eUD), eUD);
  416.             vDot = vUD.getDerivative(1);

  418.         } else {
  419.             // acceleration is either almost zero or NaN,
  420.             // we assume acceleration was not known
  421.             // we don't set up derivatives
  422.             aDot    = null;
  423.             eDot    = null;
  424.             iDot    = null;
  425.             paDot   = null;
  426.             raanDot = null;
  427.             vDot    = null;
  428.         }

  430.     }

  432.     /** Constructor from Cartesian parameters.
  433.      *
  434.      * <p> The acceleration provided in {@code FieldPVCoordinates} is accessible using
  435.      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
  436.      * use {@code mu} and the position to compute the acceleration, including
  437.      * {@link #shiftedBy(CalculusFieldElement)} and {@link #getPVCoordinates(FieldAbsoluteDate, Frame)}.
  438.      *
  439.      * @param FieldPVCoordinates the PVCoordinates of the satellite
  440.      * @param frame the frame in which are defined the {@link FieldPVCoordinates}
  441.      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
  442.      * @param date date of the orbital parameters
  443.      * @param mu central attraction coefficient (m³/s²)
  444.      * @exception IllegalArgumentException if frame is not a {@link
  445.      * Frame#isPseudoInertial pseudo-inertial frame}
  446.      */
  447.     public FieldKeplerianOrbit(final FieldPVCoordinates<T> FieldPVCoordinates,
  448.                                final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
  449.         throws IllegalArgumentException {
  450.         this(new TimeStampedFieldPVCoordinates<>(date, FieldPVCoordinates), frame, mu);
  451.     }

  453.     /** Constructor from any kind of orbital parameters.
  454.      * @param op orbital parameters to copy
  455.      */
  456.     public FieldKeplerianOrbit(final FieldOrbit<T> op) {
  457.         this(op.getPVCoordinates(), op.getFrame(), op.getMu(), op.hasDerivatives());
  458.     }

  460.     /** {@inheritDoc} */
  461.     public OrbitType getType() {
  462.         return OrbitType.KEPLERIAN;
  463.     }

  465.     /** {@inheritDoc} */
  466.     public T getA() {
  467.         return a;
  468.     }

  470.     /** {@inheritDoc} */
  471.     public T getADot() {
  472.         return aDot;
  473.     }

  475.     /** {@inheritDoc} */
  476.     public T getE() {
  477.         return e;
  478.     }

  480.     /** {@inheritDoc} */
  481.     public T getEDot() {
  482.         return eDot;
  483.     }

  485.     /** {@inheritDoc} */
  486.     public T getI() {
  487.         return i;
  488.     }

  490.     /** {@inheritDoc} */
  491.     public T getIDot() {
  492.         return iDot;
  493.     }

  495.     /** Get the perigee argument.
  496.      * @return perigee argument (rad)
  497.      */
  498.     public T getPerigeeArgument() {
  499.         return pa;
  500.     }

  502.     /** Get the perigee argument derivative.
  503.      * <p>
  504.      * If the orbit was created without derivatives, the value returned is null.
  505.      * </p>
  506.      * @return perigee argument derivative (rad/s)
  507.      */
  508.     public T getPerigeeArgumentDot() {
  509.         return paDot;
  510.     }

  512.     /** Get the right ascension of the ascending node.
  513.      * @return right ascension of the ascending node (rad)
  514.      */
  515.     public T getRightAscensionOfAscendingNode() {
  516.         return raan;
  517.     }

  519.     /** Get the right ascension of the ascending node derivative.
  520.      * <p>
  521.      * If the orbit was created without derivatives, the value returned is null.
  522.      * </p>
  523.      * @return right ascension of the ascending node derivative (rad/s)
  524.      */
  525.     public T getRightAscensionOfAscendingNodeDot() {
  526.         return raanDot;
  527.     }

  529.     /** Get the true anomaly.
  530.      * @return true anomaly (rad)
  531.      */
  532.     public T getTrueAnomaly() {
  533.         return v;
  534.     }

  536.     /** Get the true anomaly derivative.
  537.      * <p>
  538.      * If the orbit was created without derivatives, the value returned is null.
  539.      * </p>
  540.      * @return true anomaly derivative (rad/s)
  541.      */
  542.     public T getTrueAnomalyDot() {
  543.         return vDot;
  544.     }

  546.     /** Get the eccentric anomaly.
  547.      * @return eccentric anomaly (rad)
  548.      */
  549.     public T getEccentricAnomaly() {
  550.         return (a.getReal() < 0) ? trueToHyperbolicEccentric(v, e) : trueToEllipticEccentric(v, e);
  551.     }

  553.     /** Get the eccentric anomaly derivative.
  554.      * <p>
  555.      * If the orbit was created without derivatives, the value returned is null.
  556.      * </p>
  557.      * @return eccentric anomaly derivative (rad/s)
  558.      */
  559.     public T getEccentricAnomalyDot() {

  561.         if (!hasDerivatives()) {
  562.             return null;
  563.         }

  565.         final FieldUnivariateDerivative1<T> eUD = new FieldUnivariateDerivative1<>(e, eDot);
  566.         final FieldUnivariateDerivative1<T> vUD = new FieldUnivariateDerivative1<>(v, vDot);
  567.         final FieldUnivariateDerivative1<T> EUD = (a.getReal() < 0) ?
  568.                                                 trueToHyperbolicEccentric(vUD, eUD) :
  569.                                                 trueToEllipticEccentric(vUD, eUD);
  570.         return EUD.getDerivative(1);

  572.     }

  574.     /** Get the mean anomaly.
  575.      * @return mean anomaly (rad)
  576.      */
  577.     public T getMeanAnomaly() {
  578.         return (a.getReal() < 0) ?
  579.                hyperbolicEccentricToMean(trueToHyperbolicEccentric(v, e), e) :
  580.                ellipticEccentricToMean(trueToEllipticEccentric(v, e), e);
  581.     }

  583.     /** Get the mean anomaly derivative.
  584.      * <p>
  585.      * If the orbit was created without derivatives, the value returned is null.
  586.      * </p>
  587.      * @return mean anomaly derivative (rad/s)
  588.      */
  589.     public T getMeanAnomalyDot() {

  591.         if (!hasDerivatives()) {
  592.             return null;
  593.         }

  595.         final FieldUnivariateDerivative1<T> eUD = new FieldUnivariateDerivative1<>(e, eDot);
  596.         final FieldUnivariateDerivative1<T> vUD = new FieldUnivariateDerivative1<>(v, vDot);
  597.         final FieldUnivariateDerivative1<T> MUD = (a.getReal() < 0) ?
  598.                                                 hyperbolicEccentricToMean(trueToHyperbolicEccentric(vUD, eUD), eUD) :
  599.                                                 ellipticEccentricToMean(trueToEllipticEccentric(vUD, eUD), eUD);
  600.         return MUD.getDerivative(1);

  602.     }

  604.     /** Get the anomaly.
  605.      * @param type type of the angle
  606.      * @return anomaly (rad)
  607.      */
  608.     public T getAnomaly(final PositionAngle type) {
  609.         return (type == PositionAngle.MEAN) ? getMeanAnomaly() :
  610.                                               ((type == PositionAngle.ECCENTRIC) ? getEccentricAnomaly() :
  611.                                                                                    getTrueAnomaly());
  612.     }

  614.     /** Get the anomaly derivative.
  615.      * <p>
  616.      * If the orbit was created without derivatives, the value returned is null.
  617.      * </p>
  618.      * @param type type of the angle
  619.      * @return anomaly derivative (rad/s)
  620.      */
  621.     public T getAnomalyDot(final PositionAngle type) {
  622.         return (type == PositionAngle.MEAN) ? getMeanAnomalyDot() :
  623.                                               ((type == PositionAngle.ECCENTRIC) ? getEccentricAnomalyDot() :
  624.                                                                                    getTrueAnomalyDot());
  625.     }

  627.     /** {@inheritDoc} */
  628.     @Override
  629.     public boolean hasDerivatives() {
  630.         return aDot != null;
  631.     }

  633.     /** Computes the true anomaly from the elliptic eccentric anomaly.
  634.      * @param E eccentric anomaly (rad)
  635.      * @param e eccentricity
  636.      * @param <T> type of the field elements
  637.      * @return v the true anomaly
  638.      */
  639.     public static <T extends CalculusFieldElement<T>> T ellipticEccentricToTrue(final T E, final T e) {
  640.         final T beta = e.divide(e.multiply(e).negate().add(1).sqrt().add(1));
  641.         final FieldSinCos<T> scE = FastMath.sinCos(E);
  642.         return E.add(beta.multiply(scE.sin()).divide(beta.multiply(scE.cos()).subtract(1).negate()).atan().multiply(2));
  643.     }

  645.     /** Computes the elliptic eccentric anomaly from the true anomaly.
  646.      * @param v true anomaly (rad)
  647.      * @param e eccentricity
  648.      * @param <T> type of the field elements
  649.      * @return E the elliptic eccentric anomaly
  650.      */
  651.     public static <T extends CalculusFieldElement<T>> T trueToEllipticEccentric(final T v, final T e) {
  652.         final T beta = e.divide(e.multiply(e).negate().add(1).sqrt().add(1));
  653.         final FieldSinCos<T> scv = FastMath.sinCos(v);
  654.         return v.subtract((beta.multiply(scv.sin()).divide(beta.multiply(scv.cos()).add(1))).atan().multiply(2));
  655.     }

  657.     /** Computes the true anomaly from the hyperbolic eccentric anomaly.
  658.      * @param H hyperbolic eccentric anomaly (rad)
  659.      * @param e eccentricity
  660.      * @param <T> type of the field elements
  661.      * @return v the true anomaly
  662.      */
  663.     public static <T extends CalculusFieldElement<T>> T hyperbolicEccentricToTrue(final T H, final T e) {
  664.         final T s    = e.add(1).divide(e.subtract(1)).sqrt();
  665.         final T tanH = H.multiply(0.5).tanh();
  666.         return s.multiply(tanH).atan().multiply(2);
  667.     }

  669.     /** Computes the hyperbolic eccentric anomaly from the true anomaly.
  670.      * @param v true anomaly (rad)
  671.      * @param e eccentricity
  672.      * @param <T> type of the field elements
  673.      * @return H the hyperbolic eccentric anomaly
  674.      */
  675.     public static <T extends CalculusFieldElement<T>> T trueToHyperbolicEccentric(final T v, final T e) {
  676.         final FieldSinCos<T> scv = FastMath.sinCos(v);
  677.         final T sinhH = e.multiply(e).subtract(1).sqrt().multiply(scv.sin()).divide(e.multiply(scv.cos()).add(1));
  678.         return sinhH.asinh();
  679.     }

  681.     /** Computes the mean anomaly from the hyperbolic eccentric anomaly.
  682.      * @param H hyperbolic eccentric anomaly (rad)
  683.      * @param e eccentricity
  684.      * @param <T> type of the field elements
  685.      * @return M the mean anomaly
  686.      */
  687.     public static <T extends CalculusFieldElement<T>> T hyperbolicEccentricToMean(final T H, final T e) {
  688.         return e.multiply(H.sinh()).subtract(H);
  689.     }

  691.     /** Computes the elliptic eccentric anomaly from the mean anomaly.
  692.      * <p>
  693.      * The algorithm used here for solving Kepler equation has been published
  694.      * in: "Procedures for  solving Kepler's Equation", A. W. Odell and
  695.      * R. H. Gooding, Celestial Mechanics 38 (1986) 307-334
  696.      * </p>
  697.      * @param M mean anomaly (rad)
  698.      * @param e eccentricity
  699.      * @param <T> type of the field elements
  700.      * @return E the eccentric anomaly
  701.      */
  702.     public static <T extends CalculusFieldElement<T>> T meanToEllipticEccentric(final T M, final T e) {
  703.         // reduce M to [-PI PI) interval
  704.         final T reducedM = MathUtils.normalizeAngle(M, M.getField().getZero());

  706.         // compute start value according to A. W. Odell and R. H. Gooding S12 starter
  707.         T E;
  708.         if (reducedM.abs().getReal() < 1.0 / 6.0) {
  709.             if (FastMath.abs(reducedM.getReal()) < Precision.SAFE_MIN) {
  710.                 // this is an Orekit change to the S12 starter, mainly used when T is some kind of derivative structure.
  711.                 // If reducedM is 0.0, the derivative of cbrt is infinite which induces NaN appearing later in
  712.                 // the computation. As in this case E and M are almost equal, we initialize E with reducedM
  713.                 E = reducedM;
  714.             } else {
  715.                 // this is the standard S12 starter
  716.                 E = reducedM.add(e.multiply( (reducedM.multiply(6).cbrt()).subtract(reducedM)));
  717.             }
  718.         } else {
  719.             final T pi = e.getPi();
  720.             if (reducedM.getReal() < 0) {
  721.                 final T w = reducedM.add(pi);
  722.                 E = reducedM.add(e.multiply(w.multiply(A).divide(w.negate().add(B)).subtract(pi).subtract(reducedM)));
  723.             } else {
  724.                 final T w = reducedM.negate().add(pi);
  725.                 E = reducedM.add(e.multiply(w.multiply(A).divide(w.negate().add(B)).negate().subtract(reducedM).add(pi)));
  726.             }
  727.         }

  729.         final T e1 = e.negate().add(1);
  730.         final boolean noCancellationRisk = (e1.getReal() + E.getReal() * E.getReal() / 6) >= 0.1;

  732.         // perform two iterations, each consisting of one Halley step and one Newton-Raphson step
  733.         for (int j = 0; j < 2; ++j) {

  735.             final T f;
  736.             T fd;
  737.             final FieldSinCos<T> scE = FastMath.sinCos(E);
  738.             final T fdd  = e.multiply(scE.sin());
  739.             final T fddd = e.multiply(scE.cos());

  741.             if (noCancellationRisk) {

  743.                 f  = (E.subtract(fdd)).subtract(reducedM);
  744.                 fd = fddd.negate().add(1);
  745.             } else {


  748.                 f  = eMeSinE(E, e).subtract(reducedM);
  749.                 final T s = E.multiply(0.5).sin();
  750.                 fd = e1.add(e.multiply(s).multiply(s).multiply(2));
  751.             }
  752.             final T dee = f.multiply(fd).divide(f.multiply(fdd).multiply(0.5).subtract(fd.multiply(fd)));

  754.             // update eccentric anomaly, using expressions that limit underflow problems
  755.             final T w = fd.add(dee.multiply(fdd.add(dee.multiply(fddd.divide(3)))).multiply(0.5));
  756.             fd = fd.add(dee.multiply(fdd.add(dee.multiply(fddd).multiply(0.5))));
  757.             E = E.subtract(f.subtract(dee.multiply(fd.subtract(w))).divide(fd));

  759.         }

  761.         // expand the result back to original range
  762.         E = E.add(M).subtract(reducedM);
  763.         return E;
  764.     }

  766.     /** Accurate computation of E - e sin(E).
  767.      * <p>
  768.      * This method is used when E is close to 0 and e close to 1,
  769.      * i.e. near the perigee of almost parabolic orbits
  770.      * </p>
  771.      * @param E eccentric anomaly
  772.      * @param e eccentricity
  773.      * @param <T> Type of the field elements
  774.      * @return E - e sin(E)
  775.      */
  776.     private static <T extends CalculusFieldElement<T>> T eMeSinE(final T E, final T e) {

  778.         T x = (e.negate().add(1)).multiply(E.sin());
  779.         final T mE2 = E.negate().multiply(E);
  780.         T term = E;
  781.         double d    = 0;
  782.         // the inequality test below IS intentional and should NOT be replaced by a check with a small tolerance
  783.         for (T x0 = E.getField().getZero().add(Double.NaN); !Double.valueOf(x.getReal()).equals(Double.valueOf(x0.getReal()));) {
  784.             d += 2;
  785.             term = term.multiply(mE2.divide(d * (d + 1)));
  786.             x0 = x;
  787.             x = x.subtract(term);
  788.         }
  789.         return x;
  790.     }

  792.     /** Computes the hyperbolic eccentric anomaly from the mean anomaly.
  793.      * <p>
  794.      * The algorithm used here for solving hyperbolic Kepler equation is
  795.      * Danby's iterative method (3rd order) with Vallado's initial guess.
  796.      * </p>
  797.      * @param M mean anomaly (rad)
  798.      * @param e eccentricity
  799.      * @param <T> Type of the field elements
  800.      * @return H the hyperbolic eccentric anomaly
  801.      */
  802.     public static <T extends CalculusFieldElement<T>> T meanToHyperbolicEccentric(final T M, final T e) {

  804.         // Resolution of hyperbolic Kepler equation for Keplerian parameters

  806.         // Field value of pi
  807.         final T pi = e.getPi();

  809.         // Initial guess
  810.         T H;
  811.         if (e.getReal() < 1.6) {
  812.             if (-pi.getReal() < M.getReal() && M.getReal() < 0. || M.getReal() > pi.getReal()) {
  813.                 H = M.subtract(e);
  814.             } else {
  815.                 H = M.add(e);
  816.             }
  817.         } else {
  818.             if (e.getReal() < 3.6 && M.abs().getReal() > pi.getReal()) {
  819.                 H = M.subtract(e.copySign(M));
  820.             } else {
  821.                 H = M.divide(e.subtract(1));
  822.             }
  823.         }

  825.         // Iterative computation
  826.         int iter = 0;
  827.         do {
  828.             final T f3  = e.multiply(H.cosh());
  829.             final T f2  = e.multiply(H.sinh());
  830.             final T f1  = f3.subtract(1);
  831.             final T f0  = f2.subtract(H).subtract(M);
  832.             final T f12 = f1.multiply(2);
  833.             final T d   = f0.divide(f12);
  834.             final T fdf = f1.subtract(d.multiply(f2));
  835.             final T ds  = f0.divide(fdf);

  837.             final T shift = f0.divide(fdf.add(ds.multiply(ds.multiply(f3.divide(6)))));

  839.             H = H.subtract(shift);

  841.             if ((shift.abs().getReal()) <= 1.0e-12) {
  842.                 return H;
  843.             }

  845.         } while (++iter < 50);

  847.         throw new MathIllegalArgumentException(OrekitMessages.UNABLE_TO_COMPUTE_HYPERBOLIC_ECCENTRIC_ANOMALY,
  848.                                                iter);
  849.     }

  851.     /** Computes the mean anomaly from the elliptic eccentric anomaly.
  852.      * @param E eccentric anomaly (rad)
  853.      * @param e eccentricity
  854.      * @param <T> type of the field elements
  855.      * @return M the mean anomaly
  856.      */
  857.     public static <T extends CalculusFieldElement<T>> T ellipticEccentricToMean(final T E, final T e) {
  858.         return E.subtract(e.multiply(E.sin()));
  859.     }

  861.     /** {@inheritDoc} */
  862.     public T getEquinoctialEx() {
  863.         return e.multiply(pa.add(raan).cos());
  864.     }

  866.     /** {@inheritDoc} */
  867.     public T getEquinoctialExDot() {

  869.         if (!hasDerivatives()) {
  870.             return null;
  871.         }

  873.         final FieldUnivariateDerivative1<T> eUD    = new FieldUnivariateDerivative1<>(e,    eDot);
  874.         final FieldUnivariateDerivative1<T> paUD   = new FieldUnivariateDerivative1<>(pa,   paDot);
  875.         final FieldUnivariateDerivative1<T> raanUD = new FieldUnivariateDerivative1<>(raan, raanDot);
  876.         return eUD.multiply(paUD.add(raanUD).cos()).getDerivative(1);

  878.     }

  880.     /** {@inheritDoc} */
  881.     public T getEquinoctialEy() {
  882.         return  e.multiply((pa.add(raan)).sin());
  883.     }

  885.     /** {@inheritDoc} */
  886.     public T getEquinoctialEyDot() {

  888.         if (!hasDerivatives()) {
  889.             return null;
  890.         }

  892.         final FieldUnivariateDerivative1<T> eUD    = new FieldUnivariateDerivative1<>(e,    eDot);
  893.         final FieldUnivariateDerivative1<T> paUD   = new FieldUnivariateDerivative1<>(pa,   paDot);
  894.         final FieldUnivariateDerivative1<T> raanUD = new FieldUnivariateDerivative1<>(raan, raanDot);
  895.         return eUD.multiply(paUD.add(raanUD).sin()).getDerivative(1);

  897.     }

  899.     /** {@inheritDoc} */
  900.     public T getHx() {
  901.         // Check for equatorial retrograde orbit
  902.         if (FastMath.abs(i.subtract(i.getPi()).getReal()) < 1.0e-10) {
  903.             return this.zero.add(Double.NaN);
  904.         }
  905.         return  raan.cos().multiply(i.divide(2).tan());
  906.     }

  908.     /** {@inheritDoc} */
  909.     public T getHxDot() {

  911.         if (!hasDerivatives()) {
  912.             return null;
  913.         }

  915.         // Check for equatorial retrograde orbit
  916.         if (FastMath.abs(i.subtract(i.getPi()).getReal()) < 1.0e-10) {
  917.             return this.zero.add(Double.NaN);
  918.         }

  920.         final FieldUnivariateDerivative1<T> iUD    = new FieldUnivariateDerivative1<>(i,    iDot);
  921.         final FieldUnivariateDerivative1<T> raanUD = new FieldUnivariateDerivative1<>(raan, raanDot);
  922.         return raanUD.cos().multiply(iUD.multiply(0.5).tan()).getDerivative(1);

  924.     }

  926.     /** {@inheritDoc} */
  927.     public T getHy() {
  928.         // Check for equatorial retrograde orbit
  929.         if (FastMath.abs(i.subtract(i.getPi()).getReal()) < 1.0e-10) {
  930.             return this.zero.add(Double.NaN);
  931.         }
  932.         return  raan.sin().multiply(i.divide(2).tan());
  933.     }

  935.     /** {@inheritDoc} */
  936.     public T getHyDot() {

  938.         if (!hasDerivatives()) {
  939.             return null;
  940.         }

  942.         // Check for equatorial retrograde orbit
  943.         if (FastMath.abs(i.subtract(i.getPi()).getReal()) < 1.0e-10) {
  944.             return this.zero.add(Double.NaN);
  945.         }

  947.         final FieldUnivariateDerivative1<T> iUD    = new FieldUnivariateDerivative1<>(i,    iDot);
  948.         final FieldUnivariateDerivative1<T> raanUD = new FieldUnivariateDerivative1<>(raan, raanDot);
  949.         return raanUD.sin().multiply(iUD.multiply(0.5).tan()).getDerivative(1);

  951.     }

  953.     /** {@inheritDoc} */
  954.     public T getLv() {
  955.         return pa.add(raan).add(v);
  956.     }

  958.     /** {@inheritDoc} */
  959.     public T getLvDot() {
  960.         return hasDerivatives() ?
  961.                paDot.add(raanDot).add(vDot) :
  962.                null;
  963.     }

  965.     /** {@inheritDoc} */
  966.     public T getLE() {
  967.         return pa.add(raan).add(getEccentricAnomaly());
  968.     }

  970.     /** {@inheritDoc} */
  971.     public T getLEDot() {
  972.         return hasDerivatives() ?
  973.                paDot.add(raanDot).add(getEccentricAnomalyDot()) :
  974.                null;
  975.     }

  977.     /** {@inheritDoc} */
  978.     public T getLM() {
  979.         return pa.add(raan).add(getMeanAnomaly());
  980.     }

  982.     /** {@inheritDoc} */
  983.     public T getLMDot() {
  984.         return hasDerivatives() ?
  985.                paDot.add(raanDot).add(getMeanAnomalyDot()) :
  986.                null;
  987.     }

  989.     /** Compute position and velocity but not acceleration.
  990.      */
  991.     private void computePVWithoutA() {

  993.         if (partialPV != null) {
  994.             // already computed
  995.             return;
  996.         }

  998.         // preliminary variables
  999.         final FieldSinCos<T> scRaan = FastMath.sinCos(raan);
  1000.         final FieldSinCos<T> scPa   = FastMath.sinCos(pa);
  1001.         final FieldSinCos<T> scI    = FastMath.sinCos(i);
  1002.         final T cosRaan = scRaan.cos();
  1003.         final T sinRaan = scRaan.sin();
  1004.         final T cosPa   = scPa.cos();
  1005.         final T sinPa   = scPa.sin();
  1006.         final T cosI    = scI.cos();
  1007.         final T sinI    = scI.sin();
  1008.         final T crcp    = cosRaan.multiply(cosPa);
  1009.         final T crsp    = cosRaan.multiply(sinPa);
  1010.         final T srcp    = sinRaan.multiply(cosPa);
  1011.         final T srsp    = sinRaan.multiply(sinPa);

  1013.         // reference axes defining the orbital plane
  1014.         final FieldVector3D<T> p = new FieldVector3D<>(crcp.subtract(cosI.multiply(srsp)),  srcp.add(cosI.multiply(crsp)), sinI.multiply(sinPa));
  1015.         final FieldVector3D<T> q = new FieldVector3D<>(crsp.add(cosI.multiply(srcp)).negate(), cosI.multiply(crcp).subtract(srsp), sinI.multiply(cosPa));

  1017.         if (a.getReal() > 0) {

  1019.             // elliptical case

  1021.             // elliptic eccentric anomaly
  1022.             final T uME2             = e.negate().add(1).multiply(e.add(1));
  1023.             final T s1Me2            = uME2.sqrt();
  1024.             final FieldSinCos<T> scE = FastMath.sinCos(getEccentricAnomaly());
  1025.             final T cosE             = scE.cos();
  1026.             final T sinE             = scE.sin();

  1028.             // coordinates of position and velocity in the orbital plane
  1029.             final T x      = a.multiply(cosE.subtract(e));
  1030.             final T y      = a.multiply(sinE).multiply(s1Me2);
  1031.             final T factor = FastMath.sqrt(getMu().divide(a)).divide(e.negate().multiply(cosE).add(1));
  1032.             final T xDot   = sinE.negate().multiply(factor);
  1033.             final T yDot   = cosE.multiply(s1Me2).multiply(factor);

  1035.             final FieldVector3D<T> position = new FieldVector3D<>(x, p, y, q);
  1036.             final FieldVector3D<T> velocity = new FieldVector3D<>(xDot, p, yDot, q);
  1037.             partialPV = new FieldPVCoordinates<>(position, velocity);

  1039.         } else {

  1041.             // hyperbolic case

  1043.             // compute position and velocity factors
  1044.             final FieldSinCos<T> scV = FastMath.sinCos(v);
  1045.             final T sinV             = scV.sin();
  1046.             final T cosV             = scV.cos();
  1047.             final T f                = a.multiply(e.multiply(e).negate().add(1));
  1048.             final T posFactor        = f.divide(e.multiply(cosV).add(1));
  1049.             final T velFactor        = FastMath.sqrt(getMu().divide(f));

  1051.             final FieldVector3D<T> position     = new FieldVector3D<>(posFactor.multiply(cosV), p, posFactor.multiply(sinV), q);
  1052.             final FieldVector3D<T> velocity     = new FieldVector3D<>(velFactor.multiply(sinV).negate(), p, velFactor.multiply(e.add(cosV)), q);
  1053.             partialPV = new FieldPVCoordinates<>(position, velocity);

  1055.         }

  1057.     }

  1059.     /** Compute non-Keplerian part of the acceleration from first time derivatives.
  1060.      * <p>
  1061.      * This method should be called only when {@link #hasDerivatives()} returns true.
  1062.      * </p>
  1063.      * @return non-Keplerian part of the acceleration
  1064.      */
  1065.     private FieldVector3D<T> nonKeplerianAcceleration() {

  1067.         final T[][] dCdP = MathArrays.buildArray(a.getField(), 6, 6);
  1068.         getJacobianWrtParameters(PositionAngle.MEAN, dCdP);

  1070.         final T nonKeplerianMeanMotion = getMeanAnomalyDot().subtract(getKeplerianMeanMotion());
  1071.         final T nonKeplerianAx =     dCdP[3][0].multiply(aDot).
  1072.                                  add(dCdP[3][1].multiply(eDot)).
  1073.                                  add(dCdP[3][2].multiply(iDot)).
  1074.                                  add(dCdP[3][3].multiply(paDot)).
  1075.                                  add(dCdP[3][4].multiply(raanDot)).
  1076.                                  add(dCdP[3][5].multiply(nonKeplerianMeanMotion));
  1077.         final T nonKeplerianAy =     dCdP[4][0].multiply(aDot).
  1078.                                  add(dCdP[4][1].multiply(eDot)).
  1079.                                  add(dCdP[4][2].multiply(iDot)).
  1080.                                  add(dCdP[4][3].multiply(paDot)).
  1081.                                  add(dCdP[4][4].multiply(raanDot)).
  1082.                                  add(dCdP[4][5].multiply(nonKeplerianMeanMotion));
  1083.         final T nonKeplerianAz =     dCdP[5][0].multiply(aDot).
  1084.                                  add(dCdP[5][1].multiply(eDot)).
  1085.                                  add(dCdP[5][2].multiply(iDot)).
  1086.                                  add(dCdP[5][3].multiply(paDot)).
  1087.                                  add(dCdP[5][4].multiply(raanDot)).
  1088.                                  add(dCdP[5][5].multiply(nonKeplerianMeanMotion));

  1090.         return new FieldVector3D<>(nonKeplerianAx, nonKeplerianAy, nonKeplerianAz);

  1092.     }

  1094.     /** {@inheritDoc} */
  1095.     protected TimeStampedFieldPVCoordinates<T> initPVCoordinates() {

  1097.         // position and velocity
  1098.         computePVWithoutA();

  1100.         // acceleration
  1101.         final T r2 = partialPV.getPosition().getNormSq();
  1102.         final FieldVector3D<T> keplerianAcceleration = new FieldVector3D<>(r2.multiply(FastMath.sqrt(r2)).reciprocal().multiply(getMu().negate()),
  1103.                                                                            partialPV.getPosition());
  1104.         final FieldVector3D<T> acceleration = hasDerivatives() ?
  1105.                                               keplerianAcceleration.add(nonKeplerianAcceleration()) :
  1106.                                               keplerianAcceleration;

  1108.         return new TimeStampedFieldPVCoordinates<>(getDate(), partialPV.getPosition(), partialPV.getVelocity(), acceleration);

  1110.     }

  1112.     /** {@inheritDoc} */
  1113.     public FieldKeplerianOrbit<T> shiftedBy(final double dt) {
  1114.         return shiftedBy(getDate().getField().getZero().add(dt));
  1115.     }

  1117.     /** {@inheritDoc} */
  1118.     public FieldKeplerianOrbit<T> shiftedBy(final T dt) {

  1120.         // use Keplerian-only motion
  1121.         final FieldKeplerianOrbit<T> keplerianShifted = new FieldKeplerianOrbit<>(a, e, i, pa, raan,
  1122.                                                                                   getKeplerianMeanMotion().multiply(dt).add(getMeanAnomaly()),
  1123.                                                                                   PositionAngle.MEAN, getFrame(), getDate().shiftedBy(dt), getMu());

  1125.         if (hasDerivatives()) {

  1127.             // extract non-Keplerian acceleration from first time derivatives
  1128.             final FieldVector3D<T> nonKeplerianAcceleration = nonKeplerianAcceleration();

  1130.             // add quadratic effect of non-Keplerian acceleration to Keplerian-only shift
  1131.             keplerianShifted.computePVWithoutA();
  1132.             final FieldVector3D<T> fixedP   = new FieldVector3D<>(one, keplerianShifted.partialPV.getPosition(),
  1133.                                                                   dt.multiply(dt).multiply(0.5), nonKeplerianAcceleration);
  1134.             final T   fixedR2 = fixedP.getNormSq();
  1135.             final T   fixedR  = fixedR2.sqrt();
  1136.             final FieldVector3D<T> fixedV  = new FieldVector3D<>(one, keplerianShifted.partialPV.getVelocity(),
  1137.                                                                  dt, nonKeplerianAcceleration);
  1138.             final FieldVector3D<T> fixedA  = new FieldVector3D<>(fixedR2.multiply(fixedR).reciprocal().multiply(getMu().negate()),
  1139.                                                                  keplerianShifted.partialPV.getPosition(),
  1140.                                                                  one, nonKeplerianAcceleration);

  1142.             // build a new orbit, taking non-Keplerian acceleration into account
  1143.             return new FieldKeplerianOrbit<>(new TimeStampedFieldPVCoordinates<>(keplerianShifted.getDate(),
  1144.                                                                                  fixedP, fixedV, fixedA),
  1145.                                              keplerianShifted.getFrame(), keplerianShifted.getMu());

  1147.         } else {
  1148.             // Keplerian-only motion is all we can do
  1149.             return keplerianShifted;
  1150.         }

  1152.     }

  1154.     /** {@inheritDoc}
  1155.      * <p>
  1156.      * The interpolated instance is created by polynomial Hermite interpolation
  1157.      * on Keplerian elements, without derivatives (which means the interpolation
  1158.      * falls back to Lagrange interpolation only).
  1159.      * </p>
  1160.      * <p>
  1161.      * As this implementation of interpolation is polynomial, it should be used only
  1162.      * with small samples (about 10-20 points) in order to avoid <a
  1163.      * href="http://en.wikipedia.org/wiki/Runge%27s_phenomenon">Runge's phenomenon</a>
  1164.      * and numerical problems (including NaN appearing).
  1165.      * </p>
  1166.      * <p>
  1167.      * If orbit interpolation on large samples is needed, using the {@link
  1168.      * org.orekit.propagation.analytical.Ephemeris} class is a better way than using this
  1169.      * low-level interpolation. The Ephemeris class automatically handles selection of
  1170.      * a neighboring sub-sample with a predefined number of point from a large global sample
  1171.      * in a thread-safe way.
  1172.      * </p>
  1173.      */
  1174.     public FieldKeplerianOrbit<T> interpolate(final FieldAbsoluteDate<T> date, final Stream<FieldOrbit<T>> sample) {

  1176.         // first pass to check if derivatives are available throughout the sample
  1177.         final List<FieldOrbit<T>> list = sample.collect(Collectors.toList());
  1178.         boolean useDerivatives = true;
  1179.         for (final FieldOrbit<T> orbit : list) {
  1180.             useDerivatives = useDerivatives && orbit.hasDerivatives();
  1181.         }

  1183.         // set up an interpolator
  1184.         final FieldHermiteInterpolator<T> interpolator = new FieldHermiteInterpolator<>();

  1186.         // second pass to feed interpolator
  1187.         FieldAbsoluteDate<T> previousDate = null;
  1188.         T                    previousPA   = zero.add(Double.NaN);
  1189.         T                    previousRAAN = zero.add(Double.NaN);
  1190.         T                    previousM    = zero.add(Double.NaN);
  1191.         for (final FieldOrbit<T> orbit : list) {
  1192.             final FieldKeplerianOrbit<T> kep = (FieldKeplerianOrbit<T>) OrbitType.KEPLERIAN.convertType(orbit);
  1193.             final T continuousPA;
  1194.             final T continuousRAAN;
  1195.             final T continuousM;
  1196.             if (previousDate == null) {
  1197.                 continuousPA   = kep.getPerigeeArgument();
  1198.                 continuousRAAN = kep.getRightAscensionOfAscendingNode();
  1199.                 continuousM    = kep.getMeanAnomaly();
  1200.             } else {
  1201.                 final T dt      = kep.getDate().durationFrom(previousDate);
  1202.                 final T keplerM = previousM.add(kep.getKeplerianMeanMotion().multiply(dt));
  1203.                 continuousPA   = MathUtils.normalizeAngle(kep.getPerigeeArgument(), previousPA);
  1204.                 continuousRAAN = MathUtils.normalizeAngle(kep.getRightAscensionOfAscendingNode(), previousRAAN);
  1205.                 continuousM    = MathUtils.normalizeAngle(kep.getMeanAnomaly(), keplerM);
  1206.             }
  1207.             previousDate = kep.getDate();
  1208.             previousPA   = continuousPA;
  1209.             previousRAAN = continuousRAAN;
  1210.             previousM    = continuousM;
  1211.             final T[] toAdd = MathArrays.buildArray(getA().getField(), 6);
  1212.             toAdd[0] = kep.getA();
  1213.             toAdd[1] = kep.getE();
  1214.             toAdd[2] = kep.getI();
  1215.             toAdd[3] = continuousPA;
  1216.             toAdd[4] = continuousRAAN;
  1217.             toAdd[5] = continuousM;
  1218.             if (useDerivatives) {
  1219.                 final T[] toAddDot = MathArrays.buildArray(one.getField(), 6);
  1220.                 toAddDot[0] = kep.getADot();
  1221.                 toAddDot[1] = kep.getEDot();
  1222.                 toAddDot[2] = kep.getIDot();
  1223.                 toAddDot[3] = kep.getPerigeeArgumentDot();
  1224.                 toAddDot[4] = kep.getRightAscensionOfAscendingNodeDot();
  1225.                 toAddDot[5] = kep.getMeanAnomalyDot();
  1226.                 interpolator.addSamplePoint(kep.getDate().durationFrom(date),
  1227.                                             toAdd, toAddDot);
  1228.             } else {
  1229.                 interpolator.addSamplePoint(this.zero.add(kep.getDate().durationFrom(date)),
  1230.                                             toAdd);
  1231.             }
  1232.         }

  1234.         // interpolate
  1235.         final T[][] interpolated = interpolator.derivatives(zero, 1);

  1237.         // build a new interpolated instance
  1238.         return new FieldKeplerianOrbit<>(interpolated[0][0], interpolated[0][1], interpolated[0][2],
  1239.                                          interpolated[0][3], interpolated[0][4], interpolated[0][5],
  1240.                                          interpolated[1][0], interpolated[1][1], interpolated[1][2],
  1241.                                          interpolated[1][3], interpolated[1][4], interpolated[1][5],
  1242.                                          PositionAngle.MEAN, getFrame(), date, getMu());

  1244.     }

  1246.     /** {@inheritDoc} */
  1247.     protected T[][] computeJacobianMeanWrtCartesian() {
  1248.         if (a.getReal() > 0) {
  1249.             return computeJacobianMeanWrtCartesianElliptical();
  1250.         } else {
  1251.             return computeJacobianMeanWrtCartesianHyperbolic();
  1252.         }
  1253.     }

  1255.     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
  1256.      * <p>
  1257.      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
  1258.      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
  1259.      * yDot for j=4, zDot for j=5).
  1260.      * </p>
  1261.      * @return 6x6 Jacobian matrix
  1262.      */
  1263.     private T[][] computeJacobianMeanWrtCartesianElliptical() {

  1265.         final T[][] jacobian = MathArrays.buildArray(getA().getField(), 6, 6);

  1267.         // compute various intermediate parameters
  1268.         computePVWithoutA();
  1269.         final FieldVector3D<T> position = partialPV.getPosition();
  1270.         final FieldVector3D<T> velocity = partialPV.getVelocity();
  1271.         final FieldVector3D<T> momentum = partialPV.getMomentum();
  1272.         final T v2         = velocity.getNormSq();
  1273.         final T r2         = position.getNormSq();
  1274.         final T r          = r2.sqrt();
  1275.         final T r3         = r.multiply(r2);

  1277.         final T px         = position.getX();
  1278.         final T py         = position.getY();
  1279.         final T pz         = position.getZ();
  1280.         final T vx         = velocity.getX();
  1281.         final T vy         = velocity.getY();
  1282.         final T vz         = velocity.getZ();
  1283.         final T mx         = momentum.getX();
  1284.         final T my         = momentum.getY();
  1285.         final T mz         = momentum.getZ();

  1287.         final T mu         = getMu();
  1288.         final T sqrtMuA    = FastMath.sqrt(a.multiply(mu));
  1289.         final T sqrtAoMu   = FastMath.sqrt(a.divide(mu));
  1290.         final T a2         = a.multiply(a);
  1291.         final T twoA       = a.multiply(2);
  1292.         final T rOnA       = r.divide(a);

  1294.         final T oMe2       = e.multiply(e).negate().add(1);
  1295.         final T epsilon    = oMe2.sqrt();
  1296.         final T sqrtRec    = epsilon.reciprocal();

  1298.         final FieldSinCos<T> scI  = FastMath.sinCos(i);
  1299.         final FieldSinCos<T> scPA = FastMath.sinCos(pa);
  1300.         final T cosI       = scI.cos();
  1301.         final T sinI       = scI.sin();
  1302.         final T cosPA      = scPA.cos();
  1303.         final T sinPA      = scPA.sin();

  1305.         final T pv         = FieldVector3D.dotProduct(position, velocity);
  1306.         final T cosE       = a.subtract(r).divide(a.multiply(e));
  1307.         final T sinE       = pv.divide(e.multiply(sqrtMuA));

  1309.         // da
  1310.         final FieldVector3D<T> vectorAR = new FieldVector3D<>(a2.multiply(2).divide(r3), position);
  1311.         final FieldVector3D<T> vectorARDot = velocity.scalarMultiply(a2.multiply(mu.divide(2.).reciprocal()));
  1312.         fillHalfRow(this.one, vectorAR,    jacobian[0], 0);
  1313.         fillHalfRow(this.one, vectorARDot, jacobian[0], 3);

  1315.         // de
  1316.         final T factorER3 = pv.divide(twoA);
  1317.         final FieldVector3D<T> vectorER   = new FieldVector3D<>(cosE.multiply(v2).divide(r.multiply(mu)), position,
  1318.                                                                 sinE.divide(sqrtMuA), velocity,
  1319.                                                                 factorER3.negate().multiply(sinE).divide(sqrtMuA), vectorAR);
  1320.         final FieldVector3D<T> vectorERDot = new FieldVector3D<>(sinE.divide(sqrtMuA), position,
  1321.                                                                  cosE.multiply(mu.divide(2.).reciprocal()).multiply(r), velocity,
  1322.                                                                  factorER3.negate().multiply(sinE).divide(sqrtMuA), vectorARDot);
  1323.         fillHalfRow(this.one, vectorER,    jacobian[1], 0);
  1324.         fillHalfRow(this.one, vectorERDot, jacobian[1], 3);

  1326.         // dE / dr (Eccentric anomaly)
  1327.         final T coefE = cosE.divide(e.multiply(sqrtMuA));
  1328.         final FieldVector3D<T>  vectorEAnR =
  1329.             new FieldVector3D<>(sinE.negate().multiply(v2).divide(e.multiply(r).multiply(mu)), position, coefE, velocity,
  1330.                                 factorER3.negate().multiply(coefE), vectorAR);

  1332.         // dE / drDot
  1333.         final FieldVector3D<T>  vectorEAnRDot =
  1334.             new FieldVector3D<>(sinE.multiply(-2).multiply(r).divide(e.multiply(mu)), velocity, coefE, position,
  1335.                                 factorER3.negate().multiply(coefE), vectorARDot);

  1337.         // precomputing some more factors
  1338.         final T s1 = sinE.negate().multiply(pz).divide(r).subtract(cosE.multiply(vz).multiply(sqrtAoMu));
  1339.         final T s2 = cosE.negate().multiply(pz).divide(r3);
  1340.         final T s3 = sinE.multiply(vz).divide(sqrtMuA.multiply(-2));
  1341.         final T t1 = sqrtRec.multiply(cosE.multiply(pz).divide(r).subtract(sinE.multiply(vz).multiply(sqrtAoMu)));
  1342.         final T t2 = sqrtRec.multiply(sinE.negate().multiply(pz).divide(r3));
  1343.         final T t3 = sqrtRec.multiply(cosE.subtract(e)).multiply(vz).divide(sqrtMuA.multiply(2));
  1344.         final T t4 = sqrtRec.multiply(e.multiply(sinI).multiply(cosPA).multiply(sqrtRec).subtract(vz.multiply(sqrtAoMu)));
  1345.         final FieldVector3D<T> s = new FieldVector3D<>(cosE.divide(r), this.PLUS_K,
  1346.                                                        s1,       vectorEAnR,
  1347.                                                        s2,       position,
  1348.                                                        s3,       vectorAR);
  1349.         final FieldVector3D<T> sDot = new FieldVector3D<>(sinE.negate().multiply(sqrtAoMu), this.PLUS_K,
  1350.                                                           s1,               vectorEAnRDot,
  1351.                                                           s3,               vectorARDot);
  1352.         final FieldVector3D<T> t =
  1353.             new FieldVector3D<>(sqrtRec.multiply(sinE).divide(r), this.PLUS_K).add(new FieldVector3D<>(t1, vectorEAnR,
  1354.                                                                                                        t2, position,
  1355.                                                                                                        t3, vectorAR,
  1356.                                                                                                        t4, vectorER));
  1357.         final FieldVector3D<T> tDot = new FieldVector3D<>(sqrtRec.multiply(cosE.subtract(e)).multiply(sqrtAoMu), this.PLUS_K,
  1358.                                                           t1,                                                    vectorEAnRDot,
  1359.                                                           t3,                                                    vectorARDot,
  1360.                                                           t4,                                                    vectorERDot);

  1362.         // di
  1363.         final T factorI1 = sinI.negate().multiply(sqrtRec).divide(sqrtMuA);
  1364.         final T i1 =  factorI1;
  1365.         final T i2 =  factorI1.negate().multiply(mz).divide(twoA);
  1366.         final T i3 =  factorI1.multiply(mz).multiply(e).divide(oMe2);
  1367.         final T i4 = cosI.multiply(sinPA);
  1368.         final T i5 = cosI.multiply(cosPA);
  1369.         fillHalfRow(i1, new FieldVector3D<>(vy, vx.negate(), this.zero), i2, vectorAR, i3, vectorER, i4, s, i5, t,
  1370.                     jacobian[2], 0);
  1371.         fillHalfRow(i1, new FieldVector3D<>(py.negate(), px, this.zero), i2, vectorARDot, i3, vectorERDot, i4, sDot, i5, tDot,
  1372.                     jacobian[2], 3);

  1374.         // dpa
  1375.         fillHalfRow(cosPA.divide(sinI), s,    sinPA.negate().divide(sinI), t,    jacobian[3], 0);
  1376.         fillHalfRow(cosPA.divide(sinI), sDot, sinPA.negate().divide(sinI), tDot, jacobian[3], 3);

  1378.         // dRaan
  1379.         final T factorRaanR = (a.multiply(mu).multiply(oMe2).multiply(sinI).multiply(sinI)).reciprocal();
  1380.         fillHalfRow( factorRaanR.negate().multiply(my), new FieldVector3D<>(zero, vz, vy.negate()),
  1381.                      factorRaanR.multiply(mx), new FieldVector3D<>(vz.negate(), zero,  vx),
  1382.                      jacobian[4], 0);
  1383.         fillHalfRow(factorRaanR.negate().multiply(my), new FieldVector3D<>(zero, pz.negate(),  py),
  1384.                      factorRaanR.multiply(mx), new FieldVector3D<>(pz, zero, px.negate()),
  1385.                      jacobian[4], 3);

  1387.         // dM
  1388.         fillHalfRow(rOnA, vectorEAnR,    sinE.negate(), vectorER,    jacobian[5], 0);
  1389.         fillHalfRow(rOnA, vectorEAnRDot, sinE.negate(), vectorERDot, jacobian[5], 3);

  1391.         return jacobian;

  1393.     }

  1395.     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
  1396.      * <p>
  1397.      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
  1398.      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
  1399.      * yDot for j=4, zDot for j=5).
  1400.      * </p>
  1401.      * @return 6x6 Jacobian matrix
  1402.      */
  1403.     private T[][] computeJacobianMeanWrtCartesianHyperbolic() {

  1405.         final T[][] jacobian = MathArrays.buildArray(getA().getField(), 6, 6);

  1407.         // compute various intermediate parameters
  1408.         computePVWithoutA();
  1409.         final FieldVector3D<T> position = partialPV.getPosition();
  1410.         final FieldVector3D<T> velocity = partialPV.getVelocity();
  1411.         final FieldVector3D<T> momentum = partialPV.getMomentum();
  1412.         final T r2         = position.getNormSq();
  1413.         final T r          = r2.sqrt();
  1414.         final T r3         = r.multiply(r2);

  1416.         final T x          = position.getX();
  1417.         final T y          = position.getY();
  1418.         final T z          = position.getZ();
  1419.         final T vx         = velocity.getX();
  1420.         final T vy         = velocity.getY();
  1421.         final T vz         = velocity.getZ();
  1422.         final T mx         = momentum.getX();
  1423.         final T my         = momentum.getY();
  1424.         final T mz         = momentum.getZ();

  1426.         final T mu         = getMu();
  1427.         final T absA       = a.negate();
  1428.         final T sqrtMuA    = absA.multiply(mu).sqrt();
  1429.         final T a2         = a.multiply(a);
  1430.         final T rOa        = r.divide(absA);

  1432.         final FieldSinCos<T> scI = FastMath.sinCos(i);
  1433.         final T cosI       = scI.cos();
  1434.         final T sinI       = scI.sin();

  1436.         final T pv         = FieldVector3D.dotProduct(position, velocity);

  1438.         // da
  1439.         final FieldVector3D<T> vectorAR = new FieldVector3D<>(a2.multiply(-2).divide(r3), position);
  1440.         final FieldVector3D<T> vectorARDot = velocity.scalarMultiply(a2.multiply(-2).divide(mu));
  1441.         fillHalfRow(this.one.negate(), vectorAR,    jacobian[0], 0);
  1442.         fillHalfRow(this.one.negate(), vectorARDot, jacobian[0], 3);

  1444.         // differentials of the momentum
  1445.         final T m      = momentum.getNorm();
  1446.         final T oOm    = m.reciprocal();
  1447.         final FieldVector3D<T> dcXP = new FieldVector3D<>(this.zero, vz, vy.negate());
  1448.         final FieldVector3D<T> dcYP = new FieldVector3D<>(vz.negate(),   this.zero,  vx);
  1449.         final FieldVector3D<T> dcZP = new FieldVector3D<>( vy, vx.negate(),   this.zero);
  1450.         final FieldVector3D<T> dcXV = new FieldVector3D<>(  this.zero,  z.negate(),   y);
  1451.         final FieldVector3D<T> dcYV = new FieldVector3D<>(  z,   this.zero,  x.negate());
  1452.         final FieldVector3D<T> dcZV = new FieldVector3D<>( y.negate(),   x,   this.zero);
  1453.         final FieldVector3D<T> dCP  = new FieldVector3D<>(mx.multiply(oOm), dcXP, my.multiply(oOm), dcYP, mz.multiply(oOm), dcZP);
  1454.         final FieldVector3D<T> dCV  = new FieldVector3D<>(mx.multiply(oOm), dcXV, my.multiply(oOm), dcYV, mz.multiply(oOm), dcZV);

  1456.         // dp
  1457.         final T mOMu   = m.divide(mu);
  1458.         final FieldVector3D<T> dpP  = new FieldVector3D<>(mOMu.multiply(2), dCP);
  1459.         final FieldVector3D<T> dpV  = new FieldVector3D<>(mOMu.multiply(2), dCV);

  1461.         // de
  1462.         final T p      = m.multiply(mOMu);
  1463.         final T moO2ae = absA.multiply(2).multiply(e).reciprocal();
  1464.         final T m2OaMu = p.negate().divide(absA);
  1465.         fillHalfRow(moO2ae, dpP, m2OaMu.multiply(moO2ae), vectorAR,    jacobian[1], 0);
  1466.         fillHalfRow(moO2ae, dpV, m2OaMu.multiply(moO2ae), vectorARDot, jacobian[1], 3);

  1468.         // di
  1469.         final T cI1 = m.multiply(sinI).reciprocal();
  1470.         final T cI2 = cosI.multiply(cI1);
  1471.         fillHalfRow(cI2, dCP, cI1.negate(), dcZP, jacobian[2], 0);
  1472.         fillHalfRow(cI2, dCV, cI1.negate(), dcZV, jacobian[2], 3);


  1475.         // dPA
  1476.         final T cP1     =  y.multiply(oOm);
  1477.         final T cP2     =  x.negate().multiply(oOm);
  1478.         final T cP3     =  mx.multiply(cP1).add(my.multiply(cP2)).negate();
  1479.         final T cP4     =  cP3.multiply(oOm);
  1480.         final T cP5     =  r2.multiply(sinI).multiply(sinI).negate().reciprocal();
  1481.         final T cP6     = z.multiply(cP5);
  1482.         final T cP7     = cP3.multiply(cP5);
  1483.         final FieldVector3D<T> dacP  = new FieldVector3D<>(cP1, dcXP, cP2, dcYP, cP4, dCP, oOm, new FieldVector3D<>(my.negate(), mx, this.zero));
  1484.         final FieldVector3D<T> dacV  = new FieldVector3D<>(cP1, dcXV, cP2, dcYV, cP4, dCV);
  1485.         final FieldVector3D<T> dpoP  = new FieldVector3D<>(cP6, dacP, cP7, this.PLUS_K);
  1486.         final FieldVector3D<T> dpoV  = new FieldVector3D<>(cP6, dacV);

  1488.         final T re2     = r2.multiply(e).multiply(e);
  1489.         final T recOre2 = p.subtract(r).divide(re2);
  1490.         final T resOre2 = pv.multiply(mOMu).divide(re2);
  1491.         final FieldVector3D<T> dreP  = new FieldVector3D<>(mOMu, velocity, pv.divide(mu), dCP);
  1492.         final FieldVector3D<T> dreV  = new FieldVector3D<>(mOMu, position, pv.divide(mu), dCV);
  1493.         final FieldVector3D<T> davP  = new FieldVector3D<>(resOre2.negate(), dpP, recOre2, dreP, resOre2.divide(r), position);
  1494.         final FieldVector3D<T> davV  = new FieldVector3D<>(resOre2.negate(), dpV, recOre2, dreV);
  1495.         fillHalfRow(this.one, dpoP, this.one.negate(), davP, jacobian[3], 0);
  1496.         fillHalfRow(this.one, dpoV, this.one.negate(), davV, jacobian[3], 3);

  1498.         // dRAAN
  1499.         final T cO0 = cI1.multiply(cI1);
  1500.         final T cO1 =  mx.multiply(cO0);
  1501.         final T cO2 =  my.negate().multiply(cO0);
  1502.         fillHalfRow(cO1, dcYP, cO2, dcXP, jacobian[4], 0);
  1503.         fillHalfRow(cO1, dcYV, cO2, dcXV, jacobian[4], 3);

  1505.         // dM
  1506.         final T s2a    = pv.divide(absA.multiply(2));
  1507.         final T oObux  = m.multiply(m).add(absA.multiply(mu)).sqrt().reciprocal();
  1508.         final T scasbu = pv.multiply(oObux);
  1509.         final FieldVector3D<T> dauP = new FieldVector3D<>(sqrtMuA.reciprocal(), velocity, s2a.negate().divide(sqrtMuA), vectorAR);
  1510.         final FieldVector3D<T> dauV = new FieldVector3D<>(sqrtMuA.reciprocal(), position, s2a.negate().divide(sqrtMuA), vectorARDot);
  1511.         final FieldVector3D<T> dbuP = new FieldVector3D<>(oObux.multiply(mu.divide(2.)), vectorAR,    m.multiply(oObux), dCP);
  1512.         final FieldVector3D<T> dbuV = new FieldVector3D<>(oObux.multiply(mu.divide(2.)), vectorARDot, m.multiply(oObux), dCV);
  1513.         final FieldVector3D<T> dcuP = new FieldVector3D<>(oObux, velocity, scasbu.negate().multiply(oObux), dbuP);
  1514.         final FieldVector3D<T> dcuV = new FieldVector3D<>(oObux, position, scasbu.negate().multiply(oObux), dbuV);
  1515.         fillHalfRow(this.one, dauP, e.negate().divide(rOa.add(1)), dcuP, jacobian[5], 0);
  1516.         fillHalfRow(this.one, dauV, e.negate().divide(rOa.add(1)), dcuV, jacobian[5], 3);

  1518.         return jacobian;

  1520.     }

  1522.     /** {@inheritDoc} */
  1523.     protected T[][] computeJacobianEccentricWrtCartesian() {
  1524.         if (a.getReal() > 0) {
  1525.             return computeJacobianEccentricWrtCartesianElliptical();
  1526.         } else {
  1527.             return computeJacobianEccentricWrtCartesianHyperbolic();
  1528.         }
  1529.     }

  1531.     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
  1532.      * <p>
  1533.      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
  1534.      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
  1535.      * yDot for j=4, zDot for j=5).
  1536.      * </p>
  1537.      * @return 6x6 Jacobian matrix
  1538.      */
  1539.     private T[][] computeJacobianEccentricWrtCartesianElliptical() {

  1541.         // start by computing the Jacobian with mean angle
  1542.         final T[][] jacobian = computeJacobianMeanWrtCartesianElliptical();

  1544.         // Differentiating the Kepler equation M = E - e sin E leads to:
  1545.         // dM = (1 - e cos E) dE - sin E de
  1546.         // which is inverted and rewritten as:
  1547.         // dE = a/r dM + sin E a/r de
  1548.         final FieldSinCos<T> scE = FastMath.sinCos(getEccentricAnomaly());
  1549.         final T aOr              = e.negate().multiply(scE.cos()).add(1).reciprocal();

  1551.         // update anomaly row
  1552.         final T[] eRow           = jacobian[1];
  1553.         final T[] anomalyRow     = jacobian[5];
  1554.         for (int j = 0; j < anomalyRow.length; ++j) {
  1555.             anomalyRow[j] = aOr.multiply(anomalyRow[j].add(scE.sin().multiply(eRow[j])));
  1556.         }

  1558.         return jacobian;

  1560.     }

  1562.     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
  1563.      * <p>
  1564.      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
  1565.      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
  1566.      * yDot for j=4, zDot for j=5).
  1567.      * </p>
  1568.      * @return 6x6 Jacobian matrix
  1569.      */
  1570.     private T[][] computeJacobianEccentricWrtCartesianHyperbolic() {

  1572.         // start by computing the Jacobian with mean angle
  1573.         final T[][] jacobian = computeJacobianMeanWrtCartesianHyperbolic();

  1575.         // Differentiating the Kepler equation M = e sinh H - H leads to:
  1576.         // dM = (e cosh H - 1) dH + sinh H de
  1577.         // which is inverted and rewritten as:
  1578.         // dH = 1 / (e cosh H - 1) dM - sinh H / (e cosh H - 1) de
  1579.         final T H      = getEccentricAnomaly();
  1580.         final T coshH  = H.cosh();
  1581.         final T sinhH  = H.sinh();
  1582.         final T absaOr = e.multiply(coshH).subtract(1).reciprocal();
  1583.         // update anomaly row
  1584.         final T[] eRow       = jacobian[1];
  1585.         final T[] anomalyRow = jacobian[5];

  1587.         for (int j = 0; j < anomalyRow.length; ++j) {
  1588.             anomalyRow[j] = absaOr.multiply(anomalyRow[j].subtract(sinhH.multiply(eRow[j])));

  1590.         }

  1592.         return jacobian;

  1594.     }

  1596.     /** {@inheritDoc} */
  1597.     protected T[][] computeJacobianTrueWrtCartesian() {
  1598.         if (a.getReal() > 0) {
  1599.             return computeJacobianTrueWrtCartesianElliptical();
  1600.         } else {
  1601.             return computeJacobianTrueWrtCartesianHyperbolic();
  1602.         }
  1603.     }

  1605.     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
  1606.      * <p>
  1607.      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
  1608.      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
  1609.      * yDot for j=4, zDot for j=5).
  1610.      * </p>
  1611.      * @return 6x6 Jacobian matrix
  1612.      */
  1613.     private T[][] computeJacobianTrueWrtCartesianElliptical() {

  1615.         // start by computing the Jacobian with eccentric angle
  1616.         final T[][] jacobian = computeJacobianEccentricWrtCartesianElliptical();
  1617.         // Differentiating the eccentric anomaly equation sin E = sqrt(1-e^2) sin v / (1 + e cos v)
  1618.         // and using cos E = (e + cos v) / (1 + e cos v) to get rid of cos E leads to:
  1619.         // dE = [sqrt (1 - e^2) / (1 + e cos v)] dv - [sin E / (1 - e^2)] de
  1620.         // which is inverted and rewritten as:
  1621.         // dv = sqrt (1 - e^2) a/r dE + [sin E / sqrt (1 - e^2)] a/r de
  1622.         final T e2               = e.multiply(e);
  1623.         final T oMe2             = e2.negate().add(1);
  1624.         final T epsilon          = oMe2.sqrt();
  1625.         final FieldSinCos<T> scE = FastMath.sinCos(getEccentricAnomaly());
  1626.         final T aOr              = e.multiply(scE.cos()).negate().add(1).reciprocal();
  1627.         final T aFactor          = epsilon.multiply(aOr);
  1628.         final T eFactor          = scE.sin().multiply(aOr).divide(epsilon);

  1630.         // update anomaly row
  1631.         final T[] eRow           = jacobian[1];
  1632.         final T[] anomalyRow     = jacobian[5];
  1633.         for (int j = 0; j < anomalyRow.length; ++j) {
  1634.             anomalyRow[j] = aFactor.multiply(anomalyRow[j]).add(eFactor.multiply(eRow[j]));
  1635.         }
  1636.         return jacobian;

  1638.     }

  1640.     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
  1641.      * <p>
  1642.      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
  1643.      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
  1644.      * yDot for j=4, zDot for j=5).
  1645.      * </p>
  1646.      * @return 6x6 Jacobian matrix
  1647.      */
  1648.     private T[][] computeJacobianTrueWrtCartesianHyperbolic() {

  1650.         // start by computing the Jacobian with eccentric angle
  1651.         final T[][] jacobian = computeJacobianEccentricWrtCartesianHyperbolic();

  1653.         // Differentiating the eccentric anomaly equation sinh H = sqrt(e^2-1) sin v / (1 + e cos v)
  1654.         // and using cosh H = (e + cos v) / (1 + e cos v) to get rid of cosh H leads to:
  1655.         // dH = [sqrt (e^2 - 1) / (1 + e cos v)] dv + [sinh H / (e^2 - 1)] de
  1656.         // which is inverted and rewritten as:
  1657.         // dv = sqrt (1 - e^2) a/r dH - [sinh H / sqrt (e^2 - 1)] a/r de
  1658.         final T e2       = e.multiply(e);
  1659.         final T e2Mo     = e2.subtract(1);
  1660.         final T epsilon  = e2Mo.sqrt();
  1661.         final T H        = getEccentricAnomaly();
  1662.         final T coshH    = H.cosh();
  1663.         final T sinhH    = H.sinh();
  1664.         final T aOr      = e.multiply(coshH).subtract(1).reciprocal();
  1665.         final T aFactor  = epsilon.multiply(aOr);
  1666.         final T eFactor  = sinhH .multiply(aOr).divide(epsilon);

  1668.         // update anomaly row
  1669.         final T[] eRow           = jacobian[1];
  1670.         final T[] anomalyRow     = jacobian[5];
  1671.         for (int j = 0; j < anomalyRow.length; ++j) {
  1672.             anomalyRow[j] = aFactor.multiply(anomalyRow[j]).subtract(eFactor.multiply(eRow[j]));
  1673.         }

  1675.         return jacobian;

  1677.     }

  1679.     /** {@inheritDoc} */
  1680.     public void addKeplerContribution(final PositionAngle type, final T gm,
  1681.                                       final T[] pDot) {
  1682.         final T oMe2;
  1683.         final T ksi;
  1684.         final T absA = a.abs();
  1685.         final T n    = absA.reciprocal().multiply(gm).sqrt().divide(absA);
  1686.         switch (type) {
  1687.             case MEAN :
  1688.                 pDot[5] = pDot[5].add(n);
  1689.                 break;
  1690.             case ECCENTRIC :
  1691.                 oMe2 = e.multiply(e).negate().add(1).abs();
  1692.                 ksi  = e.multiply(v.cos()).add(1);
  1693.                 pDot[5] = pDot[5].add( n.multiply(ksi).divide(oMe2));
  1694.                 break;
  1695.             case TRUE :
  1696.                 oMe2 = e.multiply(e).negate().add(1).abs();
  1697.                 ksi  = e.multiply(v.cos()).add(1);
  1698.                 pDot[5] = pDot[5].add(n.multiply(ksi).multiply(ksi).divide(oMe2.multiply(oMe2.sqrt())));
  1699.                 break;
  1700.             default :
  1701.                 throw new OrekitInternalError(null);
  1702.         }
  1703.     }

  1705.     /**  Returns a string representation of this Keplerian parameters object.
  1706.      * @return a string representation of this object
  1707.      */
  1708.     public String toString() {
  1709.         return new StringBuilder().append("Keplerian parameters: ").append('{').
  1710.                                   append("a: ").append(a.getReal()).
  1711.                                   append("; e: ").append(e.getReal()).
  1712.                                   append("; i: ").append(FastMath.toDegrees(i.getReal())).
  1713.                                   append("; pa: ").append(FastMath.toDegrees(pa.getReal())).
  1714.                                   append("; raan: ").append(FastMath.toDegrees(raan.getReal())).
  1715.                                   append("; v: ").append(FastMath.toDegrees(v.getReal())).
  1716.                                   append(";}").toString();
  1717.     }

  1719.     /** Check if the given parameter is within an acceptable range.
  1720.      * The bounds are inclusive: an exception is raised when either of those conditions are met:
  1721.      * <ul>
  1722.      *     <li>The parameter is strictly greater than upperBound</li>
  1723.      *     <li>The parameter is strictly lower than lowerBound</li>
  1724.      * </ul>
  1725.      * <p>
  1726.      * In either of these cases, an OrekitException is raised.
  1727.      * </p>
  1728.      * @param parameterName name of the parameter
  1729.      * @param parameter value of the parameter
  1730.      * @param lowerBound lower bound of the acceptable range (inclusive)
  1731.      * @param upperBound upper bound of the acceptable range (inclusive)
  1732.      */
  1733.     private void checkParameterRangeInclusive(final String parameterName, final double parameter,
  1734.                                               final double lowerBound, final double upperBound) {
  1735.         if (parameter < lowerBound || parameter > upperBound) {
  1736.             throw new OrekitException(OrekitMessages.INVALID_PARAMETER_RANGE, parameterName,
  1737.                                       parameter, lowerBound, upperBound);
  1738.         }
  1739.     }

  1741.     /** {@inheritDoc} */
  1742.     @Override
  1743.     public KeplerianOrbit toOrbit() {
  1744.         if (hasDerivatives()) {
  1745.             return new KeplerianOrbit(a.getReal(), e.getReal(), i.getReal(),
  1746.                                       pa.getReal(), raan.getReal(), v.getReal(),
  1747.                                       aDot.getReal(), eDot.getReal(), iDot.getReal(),
  1748.                                       paDot.getReal(), raanDot.getReal(), vDot.getReal(),
  1749.                                       PositionAngle.TRUE,
  1750.                                       getFrame(), getDate().toAbsoluteDate(), getMu().getReal());
  1751.         } else {
  1752.             return new KeplerianOrbit(a.getReal(), e.getReal(), i.getReal(),
  1753.                                       pa.getReal(), raan.getReal(), v.getReal(),
  1754.                                       PositionAngle.TRUE,
  1755.                                       getFrame(), getDate().toAbsoluteDate(), getMu().getReal());
  1756.         }
  1757.     }


  1760. }
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