/* Copyright 2002-2025 CS GROUP
    * Licensed to CS GROUP (CS) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * CS licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
    * the License.  You may obtain a copy of the License at
    *
    *   http://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
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   * See the License for the specific language governing permissions and
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   */
  package org.orekit.orbits;
  
  
  import java.lang.reflect.Array;
  
  import org.hipparchus.CalculusFieldElement;
  import org.hipparchus.Field;
  import org.hipparchus.analysis.differentiation.FieldUnivariateDerivative1;
  import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  import org.hipparchus.util.FastMath;
  import org.hipparchus.util.FieldSinCos;
  import org.hipparchus.util.MathArrays;
  import org.orekit.errors.OrekitException;
  import org.orekit.errors.OrekitIllegalArgumentException;
  import org.orekit.errors.OrekitInternalError;
  import org.orekit.errors.OrekitMessages;
  import org.orekit.frames.FieldKinematicTransform;
  import org.orekit.frames.Frame;
  import org.orekit.time.FieldAbsoluteDate;
  import org.orekit.utils.FieldPVCoordinates;
  import org.orekit.utils.TimeStampedFieldPVCoordinates;
  
  
  /**
   * This class handles traditional Keplerian orbital parameters.
  
   * <p>
   * The parameters used internally are the classical Keplerian elements:
   *   <pre>
   *     a
   *     e
   *     i
   *     ω
   *     Ω
   *     v
   *   </pre>
   * where ω stands for the Perigee Argument, Ω stands for the
   * Right Ascension of the Ascending Node and v stands for the true anomaly.
   * <p>
   * This class supports hyperbolic orbits, using the convention that semi major
   * axis is negative for such orbits (and of course eccentricity is greater than 1).
   * </p>
   * <p>
   * When orbit is either equatorial or circular, some Keplerian elements
   * (more precisely ω and Ω) become ambiguous so this class should not
   * be used for such orbits. For this reason, {@link EquinoctialOrbit equinoctial
   * orbits} is the recommended way to represent orbits.
   * </p>
  
   * <p>
   * The instance <code>KeplerianOrbit</code> is guaranteed to be immutable.
   * </p>
   * @see     Orbit
   * @see    CircularOrbit
   * @see    CartesianOrbit
   * @see    EquinoctialOrbit
   * @author Luc Maisonobe
   * @author Guylaine Prat
   * @author Fabien Maussion
   * @author V&eacute;ronique Pommier-Maurussane
   * @author Andrea Antolino
   * @since 9.0
   * @param <T> type of the field elements
   */
  public class FieldKeplerianOrbit<T extends CalculusFieldElement<T>> extends FieldOrbit<T>
          implements PositionAngleBased<FieldKeplerianOrbit<T>> {
  
      /** Name of the eccentricity parameter. */
      private static final String ECCENTRICITY = "eccentricity";
  
      /** Semi-major axis (m). */
      private final T a;
  
      /** Eccentricity. */
      private final T e;
  
      /** Inclination (rad). */
      private final T i;
  
      /** Perigee Argument (rad). */
      private final T pa;
  
      /** Right Ascension of Ascending Node (rad). */
     private final T raan;
 
     /** Cached anomaly (rad). */
     private final T cachedAnomaly;
 
     /** Semi-major axis derivative (m/s). */
     private final T aDot;
 
     /** Eccentricity derivative. */
     private final T eDot;
 
     /** Inclination derivative (rad/s). */
     private final T iDot;
 
     /** Perigee Argument derivative (rad/s). */
     private final T paDot;
 
     /** Right Ascension of Ascending Node derivative (rad/s). */
     private final T raanDot;
 
     /** Derivative of cached anomaly (rad/s). */
     private final T cachedAnomalyDot;
 
     /** Cached type of position angle. */
     private final PositionAngleType cachedPositionAngleType;
 
     /** Partial Cartesian coordinates (position and velocity are valid, acceleration may be missing). */
     private FieldPVCoordinates<T> partialPV;
 
     /** PThird Canonical Vector. */
     private final FieldVector3D<T> PLUS_K;
 
     /** Creates a new instance.
      * @param a  semi-major axis (m), negative for hyperbolic orbits
      * @param e eccentricity (positive or equal to 0)
      * @param i inclination (rad)
      * @param pa perigee argument (ω, rad)
      * @param raan right ascension of ascending node (Ω, rad)
      * @param anomaly mean, eccentric or true anomaly (rad)
      * @param type type of anomaly
      * @param cachedPositionAngleType type of cached anomaly
      * @param frame the frame in which the parameters are defined
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param date date of the orbital parameters
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if frame is not a {@link
      * Frame#isPseudoInertial pseudo-inertial frame} or a and e don't match for hyperbolic orbits,
      * or v is out of range for hyperbolic orbits
      * @since 12.1
      */
     public FieldKeplerianOrbit(final T a, final T e, final T i,
                                final T pa, final T raan,
                                final T anomaly, final PositionAngleType type,
                                final PositionAngleType cachedPositionAngleType,
                                final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
         throws IllegalArgumentException {
         this(a, e, i, pa, raan, anomaly,
              a.getField().getZero(), a.getField().getZero(), a.getField().getZero(), a.getField().getZero(), a.getField().getZero(),
              computeKeplerianAnomalyDot(type, a, e, mu, anomaly, type), type, cachedPositionAngleType, frame, date, mu);
     }
 
     /** Creates a new instance.
      * @param a  semi-major axis (m), negative for hyperbolic orbits
      * @param e eccentricity (positive or equal to 0)
      * @param i inclination (rad)
      * @param pa perigee argument (ω, rad)
      * @param raan right ascension of ascending node (Ω, rad)
      * @param anomaly mean, eccentric or true anomaly (rad)
      * @param type type of anomaly
      * @param frame the frame in which the parameters are defined
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param date date of the orbital parameters
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if frame is not a {@link
      * Frame#isPseudoInertial pseudo-inertial frame} or a and e don't match for hyperbolic orbits,
      * or v is out of range for hyperbolic orbits
      * @since 12.1
      */
     public FieldKeplerianOrbit(final T a, final T e, final T i,
                                final T pa, final T raan,
                                final T anomaly, final PositionAngleType type,
                                final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
             throws IllegalArgumentException {
         this(a, e, i, pa, raan, anomaly, type, type, frame, date, mu);
     }
 
     /** Creates a new instance.
      * @param a  semi-major axis (m), negative for hyperbolic orbits
      * @param e eccentricity (positive or equal to 0)
      * @param i inclination (rad)
      * @param pa perigee argument (ω, rad)
      * @param raan right ascension of ascending node (Ω, rad)
      * @param anomaly mean, eccentric or true anomaly (rad)
      * @param aDot  semi-major axis derivative, null if unknown (m/s)
      * @param eDot eccentricity derivative, null if unknown
      * @param iDot inclination derivative, null if unknown (rad/s)
      * @param paDot perigee argument derivative, null if unknown (rad/s)
      * @param raanDot right ascension of ascending node derivative, null if unknown (rad/s)
      * @param anomalyDot mean, eccentric or true anomaly derivative, null if unknown (rad/s)
      * @param type type of anomaly
      * @param cachedPositionAngleType type of cached anomaly
      * @param frame the frame in which the parameters are defined
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param date date of the orbital parameters
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if frame is not a {@link
      * Frame#isPseudoInertial pseudo-inertial frame} or a and e don't match for hyperbolic orbits,
      * or v is out of range for hyperbolic orbits
      * @since 12.1
      */
     public FieldKeplerianOrbit(final T a, final T e, final T i,
                                final T pa, final T raan, final T anomaly,
                                final T aDot, final T eDot, final T iDot,
                                final T paDot, final T raanDot, final T anomalyDot,
                                final PositionAngleType type, final PositionAngleType cachedPositionAngleType,
                                final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
         throws IllegalArgumentException {
         super(frame, date, mu);
         this.cachedPositionAngleType = cachedPositionAngleType;
 
         if (a.multiply(e.negate().add(1)).getReal() < 0) {
             throw new OrekitIllegalArgumentException(OrekitMessages.ORBIT_A_E_MISMATCH_WITH_CONIC_TYPE, a.getReal(), e.getReal());
         }
 
         // Checking eccentricity range
         checkParameterRangeInclusive(ECCENTRICITY, e.getReal(), 0.0, Double.POSITIVE_INFINITY);
 
         this.a       =    a;
         this.aDot    =    aDot;
         this.e       =    e;
         this.eDot    =    eDot;
         this.i       =    i;
         this.iDot    =    iDot;
         this.pa      =    pa;
         this.paDot   =    paDot;
         this.raan    =    raan;
         this.raanDot =    raanDot;
 
         this.PLUS_K = FieldVector3D.getPlusK(a.getField());  // third canonical vector
 
         final FieldUnivariateDerivative1<T> cachedAnomalyUD = initializeCachedAnomaly(anomaly, anomalyDot, type);
         this.cachedAnomaly = cachedAnomalyUD.getValue();
         this.cachedAnomalyDot = cachedAnomalyUD.getFirstDerivative();
 
         // check true anomaly range
         if (!isElliptical()) {
             final T trueAnomaly = getTrueAnomaly();
             if (e.multiply(trueAnomaly.cos()).add(1).getReal() <= 0) {
                 final double vMax = e.reciprocal().negate().acos().getReal();
                 throw new OrekitIllegalArgumentException(OrekitMessages.ORBIT_ANOMALY_OUT_OF_HYPERBOLIC_RANGE,
                         trueAnomaly.getReal(), e.getReal(), -vMax, vMax);
             }
         }
 
         this.partialPV = null;
 
     }
 
     /** Creates a new instance.
      * @param a  semi-major axis (m), negative for hyperbolic orbits
      * @param e eccentricity (positive or equal to 0)
      * @param i inclination (rad)
      * @param pa perigee argument (ω, rad)
      * @param raan right ascension of ascending node (Ω, rad)
      * @param anomaly mean, eccentric or true anomaly (rad)
      * @param aDot  semi-major axis derivative, null if unknown (m/s)
      * @param eDot eccentricity derivative, null if unknown
      * @param iDot inclination derivative, null if unknown (rad/s)
      * @param paDot perigee argument derivative, null if unknown (rad/s)
      * @param raanDot right ascension of ascending node derivative, null if unknown (rad/s)
      * @param anomalyDot mean, eccentric or true anomaly derivative, null if unknown (rad/s)
      * @param type type of anomaly
      * @param frame the frame in which the parameters are defined
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param date date of the orbital parameters
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if frame is not a {@link
      * Frame#isPseudoInertial pseudo-inertial frame} or a and e don't match for hyperbolic orbits,
      * or v is out of range for hyperbolic orbits
      */
     public FieldKeplerianOrbit(final T a, final T e, final T i,
                                final T pa, final T raan, final T anomaly,
                                final T aDot, final T eDot, final T iDot,
                                final T paDot, final T raanDot, final T anomalyDot,
                                final PositionAngleType type,
                                final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
             throws IllegalArgumentException {
         this(a, e, i, pa, raan, anomaly, aDot, eDot, iDot, paDot, raanDot, anomalyDot,
                 type, type, frame, date, mu);
     }
 
     /** Constructor from Cartesian parameters.
      *
      * <p> The acceleration provided in {@code FieldPVCoordinates} is accessible using
      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
      * use {@code mu} and the position to compute the acceleration, including
      * {@link #shiftedBy(CalculusFieldElement)} and {@link #getPVCoordinates(FieldAbsoluteDate, Frame)}.
      *
      * @param pvCoordinates the PVCoordinates of the satellite
      * @param frame the frame in which are defined the {@link FieldPVCoordinates}
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if frame is not a {@link
      * Frame#isPseudoInertial pseudo-inertial frame}
      */
     public FieldKeplerianOrbit(final TimeStampedFieldPVCoordinates<T> pvCoordinates,
                                final Frame frame, final T mu)
         throws IllegalArgumentException {
         this(pvCoordinates, frame, mu, hasNonKeplerianAcceleration(pvCoordinates, mu));
     }
 
     /** Constructor from Cartesian parameters.
      *
      * <p> The acceleration provided in {@code FieldPVCoordinates} is accessible using
      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
      * use {@code mu} and the position to compute the acceleration, including
      * {@link #shiftedBy(CalculusFieldElement)} and {@link #getPVCoordinates(FieldAbsoluteDate, Frame)}.
      *
      * @param pvCoordinates the PVCoordinates of the satellite
      * @param frame the frame in which are defined the {@link FieldPVCoordinates}
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param mu central attraction coefficient (m³/s²)
      * @param reliableAcceleration if true, the acceleration is considered to be reliable
      * @exception IllegalArgumentException if frame is not a {@link
      * Frame#isPseudoInertial pseudo-inertial frame}
      */
     private FieldKeplerianOrbit(final TimeStampedFieldPVCoordinates<T> pvCoordinates,
                                 final Frame frame, final T mu,
                                 final boolean reliableAcceleration)
         throws IllegalArgumentException {
 
         super(pvCoordinates, frame, mu);
 
         // third canonical vector
         this.PLUS_K = FieldVector3D.getPlusK(getOne().getField());
 
         // compute inclination
         final FieldVector3D<T> momentum = pvCoordinates.getMomentum();
         final T m2 = momentum.getNormSq();
 
         i = FieldVector3D.angle(momentum, PLUS_K);
         // compute right ascension of ascending node
         raan = FieldVector3D.crossProduct(PLUS_K, momentum).getAlpha();
         // preliminary computations for parameters depending on orbit shape (elliptic or hyperbolic)
         final FieldVector3D<T> pvP     = pvCoordinates.getPosition();
         final FieldVector3D<T> pvV     = pvCoordinates.getVelocity();
         final FieldVector3D<T> pvA     = pvCoordinates.getAcceleration();
 
         final T   r2      = pvP.getNormSq();
         final T   r       = r2.sqrt();
         final T   V2      = pvV.getNormSq();
         final T   rV2OnMu = r.multiply(V2).divide(mu);
 
         // compute semi-major axis (will be negative for hyperbolic orbits)
         a = r.divide(rV2OnMu.negate().add(2.0));
         final T muA = a.multiply(mu);
 
         // compute true anomaly
         if (isElliptical()) {
             // elliptic or circular orbit
             final T eSE = FieldVector3D.dotProduct(pvP, pvV).divide(muA.sqrt());
             final T eCE = rV2OnMu.subtract(1);
             e = (eSE.multiply(eSE).add(eCE.multiply(eCE))).sqrt();
             this.cachedPositionAngleType = PositionAngleType.ECCENTRIC;
             cachedAnomaly = eSE.atan2(eCE);
         } else {
             // hyperbolic orbit
             final T eSH = FieldVector3D.dotProduct(pvP, pvV).divide(muA.negate().sqrt());
             final T eCH = rV2OnMu.subtract(1);
             e = (m2.negate().divide(muA).add(1)).sqrt();
             this.cachedPositionAngleType = PositionAngleType.TRUE;
             cachedAnomaly = FieldKeplerianAnomalyUtility.hyperbolicEccentricToTrue(e, (eCH.add(eSH)).divide(eCH.subtract(eSH)).log().divide(2));
         }
 
         // Checking eccentricity range
         checkParameterRangeInclusive(ECCENTRICITY, e.getReal(), 0.0, Double.POSITIVE_INFINITY);
 
         // compute perigee argument
         final FieldVector3D<T> node = new FieldVector3D<>(raan, getZero());
         final T px = FieldVector3D.dotProduct(pvP, node);
         final T py = FieldVector3D.dotProduct(pvP, FieldVector3D.crossProduct(momentum, node)).divide(m2.sqrt());
         pa = py.atan2(px).subtract(getTrueAnomaly());
 
         partialPV = pvCoordinates;
 
         if (reliableAcceleration) {
             // we have a relevant acceleration, we can compute derivatives
 
             final T[][] jacobian = MathArrays.buildArray(a.getField(), 6, 6);
             getJacobianWrtCartesian(PositionAngleType.MEAN, jacobian);
 
             final FieldVector3D<T> keplerianAcceleration    = new FieldVector3D<>(r.multiply(r2).reciprocal().multiply(mu.negate()), pvP);
             final FieldVector3D<T> nonKeplerianAcceleration = pvA.subtract(keplerianAcceleration);
             final T   aX                       = nonKeplerianAcceleration.getX();
             final T   aY                       = nonKeplerianAcceleration.getY();
             final T   aZ                       = nonKeplerianAcceleration.getZ();
             aDot    = jacobian[0][3].multiply(aX).add(jacobian[0][4].multiply(aY)).add(jacobian[0][5].multiply(aZ));
             eDot    = jacobian[1][3].multiply(aX).add(jacobian[1][4].multiply(aY)).add(jacobian[1][5].multiply(aZ));
             iDot    = jacobian[2][3].multiply(aX).add(jacobian[2][4].multiply(aY)).add(jacobian[2][5].multiply(aZ));
             paDot   = jacobian[3][3].multiply(aX).add(jacobian[3][4].multiply(aY)).add(jacobian[3][5].multiply(aZ));
             raanDot = jacobian[4][3].multiply(aX).add(jacobian[4][4].multiply(aY)).add(jacobian[4][5].multiply(aZ));
 
             // in order to compute true anomaly derivative, we must compute
             // mean anomaly derivative including Keplerian motion and convert to true anomaly
             final T MDot = getKeplerianMeanMotion().
                            add(jacobian[5][3].multiply(aX)).add(jacobian[5][4].multiply(aY)).add(jacobian[5][5].multiply(aZ));
             final FieldUnivariateDerivative1<T> eUD = new FieldUnivariateDerivative1<>(e, eDot);
             final FieldUnivariateDerivative1<T> MUD = new FieldUnivariateDerivative1<>(getMeanAnomaly(), MDot);
             if (cachedPositionAngleType == PositionAngleType.ECCENTRIC) {
                 final FieldUnivariateDerivative1<T> EUD = (a.getReal() < 0) ?
                          FieldKeplerianAnomalyUtility.hyperbolicMeanToEccentric(eUD, MUD) :
                          FieldKeplerianAnomalyUtility.ellipticMeanToEccentric(eUD, MUD);
                 cachedAnomalyDot = EUD.getFirstDerivative();
             } else { // TRUE
                 final FieldUnivariateDerivative1<T> vUD = (a.getReal() < 0) ?
                          FieldKeplerianAnomalyUtility.hyperbolicMeanToTrue(eUD, MUD) :
                          FieldKeplerianAnomalyUtility.ellipticMeanToTrue(eUD, MUD);
                 cachedAnomalyDot = vUD.getFirstDerivative();
             }
 
         } else {
             // acceleration is either almost zero or NaN,
             // we assume acceleration was not known
             // we don't set up derivatives
             aDot    = getZero();
             eDot    = getZero();
             iDot    = getZero();
             paDot   = getZero();
             raanDot = getZero();
             cachedAnomalyDot    = computeKeplerianAnomalyDot(cachedPositionAngleType, a, e, mu, cachedAnomaly, cachedPositionAngleType);
         }
 
     }
 
     /** Constructor from Cartesian parameters.
      *
      * <p> The acceleration provided in {@code FieldPVCoordinates} is accessible using
      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
      * use {@code mu} and the position to compute the acceleration, including
      * {@link #shiftedBy(CalculusFieldElement)} and {@link #getPVCoordinates(FieldAbsoluteDate, Frame)}.
      *
      * @param FieldPVCoordinates the PVCoordinates of the satellite
      * @param frame the frame in which are defined the {@link FieldPVCoordinates}
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param date date of the orbital parameters
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if frame is not a {@link
      * Frame#isPseudoInertial pseudo-inertial frame}
      */
     public FieldKeplerianOrbit(final FieldPVCoordinates<T> FieldPVCoordinates,
                                final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
         throws IllegalArgumentException {
         this(new TimeStampedFieldPVCoordinates<>(date, FieldPVCoordinates), frame, mu);
     }
 
     /** Constructor from any kind of orbital parameters.
      * @param op orbital parameters to copy
      */
     public FieldKeplerianOrbit(final FieldOrbit<T> op) {
         this(op.getPVCoordinates(), op.getFrame(), op.getMu());
     }
 
     /** Constructor from Field and KeplerianOrbit.
      * <p>Build a FieldKeplerianOrbit from non-Field KeplerianOrbit.</p>
      * @param field CalculusField to base object on
      * @param op non-field orbit with only "constant" terms
      * @since 12.0
      */
     public FieldKeplerianOrbit(final Field<T> field, final KeplerianOrbit op) {
         this(field.getZero().newInstance(op.getA()), field.getZero().newInstance(op.getE()), field.getZero().newInstance(op.getI()),
                 field.getZero().newInstance(op.getPerigeeArgument()), field.getZero().newInstance(op.getRightAscensionOfAscendingNode()),
                 field.getZero().newInstance(op.getAnomaly(op.getCachedPositionAngleType())),
                 field.getZero().newInstance(op.getADot()),
                 field.getZero().newInstance(op.getEDot()),
                 field.getZero().newInstance(op.getIDot()),
                 field.getZero().newInstance(op.getPerigeeArgumentDot()),
                 field.getZero().newInstance(op.getRightAscensionOfAscendingNodeDot()),
                 field.getZero().newInstance(op.getAnomalyDot(op.getCachedPositionAngleType())),
                 op.getCachedPositionAngleType(), op.getFrame(),
                 new FieldAbsoluteDate<>(field, op.getDate()), field.getZero().newInstance(op.getMu()));
     }
 
     /** Constructor from Field and Orbit.
      * <p>Build a FieldKeplerianOrbit from any non-Field Orbit.</p>
      * @param field CalculusField to base object on
      * @param op non-field orbit with only "constant" terms
      * @since 12.0
      */
     public FieldKeplerianOrbit(final Field<T> field, final Orbit op) {
         this(field, (KeplerianOrbit) OrbitType.KEPLERIAN.convertType(op));
     }
 
     /** {@inheritDoc} */
     @Override
     public OrbitType getType() {
         return OrbitType.KEPLERIAN;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getA() {
         return a;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getADot() {
         return aDot;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getE() {
         return e;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getEDot() {
         return eDot;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getI() {
         return i;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getIDot() {
         return iDot;
     }
 
     /** Get the perigee argument.
      * @return perigee argument (rad)
      */
     public T getPerigeeArgument() {
         return pa;
     }
 
     /** Get the perigee argument derivative.
      * @return perigee argument derivative (rad/s)
      */
     public T getPerigeeArgumentDot() {
         return paDot;
     }
 
     /** Get the right ascension of the ascending node.
      * @return right ascension of the ascending node (rad)
      */
     public T getRightAscensionOfAscendingNode() {
         return raan;
     }
 
     /** Get the right ascension of the ascending node derivative.
      * @return right ascension of the ascending node derivative (rad/s)
      */
     public T getRightAscensionOfAscendingNodeDot() {
         return raanDot;
     }
 
     /** Get the true anomaly.
      * @return true anomaly (rad)
      */
     public T getTrueAnomaly() {
         switch (cachedPositionAngleType) {
             case MEAN: return (a.getReal() < 0) ? FieldKeplerianAnomalyUtility.hyperbolicMeanToTrue(e, cachedAnomaly) :
                     FieldKeplerianAnomalyUtility.ellipticMeanToTrue(e, cachedAnomaly);
 
             case TRUE: return cachedAnomaly;
 
             case ECCENTRIC: return (a.getReal() < 0) ? FieldKeplerianAnomalyUtility.hyperbolicEccentricToTrue(e, cachedAnomaly) :
                     FieldKeplerianAnomalyUtility.ellipticEccentricToTrue(e, cachedAnomaly);
 
             default: throw new OrekitInternalError(null);
         }
     }
 
     /** Get the true anomaly derivative.
      * @return true anomaly derivative (rad/s)
      */
     public T getTrueAnomalyDot() {
         switch (cachedPositionAngleType) {
             case MEAN:
                 final FieldUnivariateDerivative1<T> eUD = new FieldUnivariateDerivative1<>(e, eDot);
                 final FieldUnivariateDerivative1<T> MUD = new FieldUnivariateDerivative1<>(cachedAnomaly, cachedAnomalyDot);
                 final FieldUnivariateDerivative1<T> vUD = (a.getReal() < 0) ?
                          FieldKeplerianAnomalyUtility.hyperbolicMeanToTrue(eUD, MUD) :
                          FieldKeplerianAnomalyUtility.ellipticMeanToTrue(eUD, MUD);
                 return vUD.getFirstDerivative();
 
             case TRUE:
                 return cachedAnomalyDot;
 
             case ECCENTRIC:
                 final FieldUnivariateDerivative1<T> eUD2 = new FieldUnivariateDerivative1<>(e, eDot);
                 final FieldUnivariateDerivative1<T> EUD = new FieldUnivariateDerivative1<>(cachedAnomaly, cachedAnomalyDot);
                 final FieldUnivariateDerivative1<T> vUD2 = (a.getReal() < 0) ?
                          FieldKeplerianAnomalyUtility.hyperbolicEccentricToTrue(eUD2, EUD) :
                          FieldKeplerianAnomalyUtility.ellipticEccentricToTrue(eUD2, EUD);
                 return vUD2.getFirstDerivative();
 
             default:
                 throw new OrekitInternalError(null);
         }
     }
 
     /** Get the eccentric anomaly.
      * @return eccentric anomaly (rad)
      */
     public T getEccentricAnomaly() {
         switch (cachedPositionAngleType) {
             case MEAN:
                 return (a.getReal() < 0) ? FieldKeplerianAnomalyUtility.hyperbolicMeanToEccentric(e, cachedAnomaly) :
                     FieldKeplerianAnomalyUtility.ellipticMeanToEccentric(e, cachedAnomaly);
 
             case ECCENTRIC:
                 return cachedAnomaly;
 
             case TRUE:
                 return (a.getReal() < 0) ? FieldKeplerianAnomalyUtility.hyperbolicTrueToEccentric(e, cachedAnomaly) :
                     FieldKeplerianAnomalyUtility.ellipticTrueToEccentric(e, cachedAnomaly);
 
             default:
                 throw new OrekitInternalError(null);
         }
     }
 
     /** Get the eccentric anomaly derivative.
      * @return eccentric anomaly derivative (rad/s)
      */
     public T getEccentricAnomalyDot() {
         switch (cachedPositionAngleType) {
             case ECCENTRIC:
                 return cachedAnomalyDot;
 
             case TRUE:
                 final FieldUnivariateDerivative1<T> eUD = new FieldUnivariateDerivative1<>(e, eDot);
                 final FieldUnivariateDerivative1<T> vUD = new FieldUnivariateDerivative1<>(cachedAnomaly, cachedAnomalyDot);
                 final FieldUnivariateDerivative1<T> EUD = (a.getReal() < 0) ?
                          FieldKeplerianAnomalyUtility.hyperbolicTrueToEccentric(eUD, vUD) :
                          FieldKeplerianAnomalyUtility.ellipticTrueToEccentric(eUD, vUD);
                 return EUD.getFirstDerivative();
 
             case MEAN:
                 final FieldUnivariateDerivative1<T> eUD2 = new FieldUnivariateDerivative1<>(e, eDot);
                 final FieldUnivariateDerivative1<T> MUD = new FieldUnivariateDerivative1<>(cachedAnomaly, cachedAnomalyDot);
                 final FieldUnivariateDerivative1<T> EUD2 = (a.getReal() < 0) ?
                          FieldKeplerianAnomalyUtility.hyperbolicMeanToEccentric(eUD2, MUD) :
                          FieldKeplerianAnomalyUtility.ellipticMeanToEccentric(eUD2, MUD);
                 return EUD2.getFirstDerivative();
 
             default:
                 throw new OrekitInternalError(null);
         }
     }
 
     /** Get the mean anomaly.
      * @return mean anomaly (rad)
      */
     public T getMeanAnomaly() {
         switch (cachedPositionAngleType) {
             case ECCENTRIC: return (a.getReal() < 0) ? FieldKeplerianAnomalyUtility.hyperbolicEccentricToMean(e, cachedAnomaly) :
                     FieldKeplerianAnomalyUtility.ellipticEccentricToMean(e, cachedAnomaly);
 
             case MEAN: return cachedAnomaly;
 
             case TRUE: return (a.getReal() < 0) ? FieldKeplerianAnomalyUtility.hyperbolicTrueToMean(e, cachedAnomaly) :
                     FieldKeplerianAnomalyUtility.ellipticTrueToMean(e, cachedAnomaly);
 
             default: throw new OrekitInternalError(null);
         }
     }
 
     /** Get the mean anomaly derivative.
 
      * @return mean anomaly derivative (rad/s)
      */
     public T getMeanAnomalyDot() {
         switch (cachedPositionAngleType) {
             case MEAN:
                 return cachedAnomalyDot;
 
             case ECCENTRIC:
                 final FieldUnivariateDerivative1<T> eUD = new FieldUnivariateDerivative1<>(e, eDot);
                 final FieldUnivariateDerivative1<T> EUD = new FieldUnivariateDerivative1<>(cachedAnomaly, cachedAnomalyDot);
                 final FieldUnivariateDerivative1<T> MUD = (a.getReal() < 0) ?
                          FieldKeplerianAnomalyUtility.hyperbolicEccentricToMean(eUD, EUD) :
                          FieldKeplerianAnomalyUtility.ellipticEccentricToMean(eUD, EUD);
                 return MUD.getFirstDerivative();
 
             case TRUE:
                 final FieldUnivariateDerivative1<T> eUD2 = new FieldUnivariateDerivative1<>(e, eDot);
                 final FieldUnivariateDerivative1<T> vUD = new FieldUnivariateDerivative1<>(cachedAnomaly, cachedAnomalyDot);
                 final FieldUnivariateDerivative1<T> MUD2 = (a.getReal() < 0) ?
                          FieldKeplerianAnomalyUtility.hyperbolicTrueToMean(eUD2, vUD) :
                          FieldKeplerianAnomalyUtility.ellipticTrueToMean(eUD2, vUD);
                 return MUD2.getFirstDerivative();
 
             default:
                 throw new OrekitInternalError(null);
         }
     }
 
     /** Get the anomaly.
      * @param type type of the angle
      * @return anomaly (rad)
      */
     public T getAnomaly(final PositionAngleType type) {
         return (type == PositionAngleType.MEAN) ? getMeanAnomaly() :
                                               ((type == PositionAngleType.ECCENTRIC) ? getEccentricAnomaly() :
                                                                                    getTrueAnomaly());
     }
 
     /** Get the anomaly derivative.
      * @param type type of the angle
      * @return anomaly derivative (rad/s)
      */
     public T getAnomalyDot(final PositionAngleType type) {
         return (type == PositionAngleType.MEAN) ? getMeanAnomalyDot() :
                                               ((type == PositionAngleType.ECCENTRIC) ? getEccentricAnomalyDot() :
                                                                                    getTrueAnomalyDot());
     }
 
     /** {@inheritDoc} */
     @Override
     public boolean hasNonKeplerianAcceleration() {
         return aDot.getReal() != 0. || eDot.getReal() != 0 || iDot.getReal() != 0. || raanDot.getReal() != 0. || paDot.getReal() != 0. ||
                 FastMath.abs(cachedAnomalyDot.subtract(computeKeplerianAnomalyDot(cachedPositionAngleType, a, e, getMu(), cachedAnomaly, cachedPositionAngleType)).getReal()) > TOLERANCE_POSITION_ANGLE_RATE;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getEquinoctialEx() {
         return e.multiply(pa.add(raan).cos());
     }
 
     /** {@inheritDoc} */
     @Override
     public T getEquinoctialExDot() {
         if (!hasNonKeplerianRates()) {
             return getZero();
         }
         final FieldUnivariateDerivative1<T> eUD    = new FieldUnivariateDerivative1<>(e,    eDot);
         final FieldUnivariateDerivative1<T> paUD   = new FieldUnivariateDerivative1<>(pa,   paDot);
         final FieldUnivariateDerivative1<T> raanUD = new FieldUnivariateDerivative1<>(raan, raanDot);
         return eUD.multiply(paUD.add(raanUD).cos()).getFirstDerivative();
 
     }
 
     /** {@inheritDoc} */
     @Override
     public T getEquinoctialEy() {
         return e.multiply((pa.add(raan)).sin());
     }
 
     /** {@inheritDoc} */
     @Override
     public T getEquinoctialEyDot() {
         if (!hasNonKeplerianRates()) {
             return getZero();
         }
         final FieldUnivariateDerivative1<T> eUD    = new FieldUnivariateDerivative1<>(e,    eDot);
         final FieldUnivariateDerivative1<T> paUD   = new FieldUnivariateDerivative1<>(pa,   paDot);
         final FieldUnivariateDerivative1<T> raanUD = new FieldUnivariateDerivative1<>(raan, raanDot);
         return eUD.multiply(paUD.add(raanUD).sin()).getFirstDerivative();
 
     }
 
     /** {@inheritDoc} */
     @Override
     public T getHx() {
         // Check for equatorial retrograde orbit
         if (FastMath.abs(i.subtract(i.getPi()).getReal()) < 1.0e-10) {
             return getZero().add(Double.NaN);
         }
         return raan.cos().multiply(i.divide(2).tan());
     }
 
     /** {@inheritDoc} */
     @Override
     public T getHxDot() {
 
         // Check for equatorial retrograde orbit
         if (FastMath.abs(i.subtract(i.getPi()).getReal()) < 1.0e-10) {
             return getZero().add(Double.NaN);
         }
         if (!hasNonKeplerianRates()) {
             return getZero();
         }
 
         final FieldUnivariateDerivative1<T> iUD    = new FieldUnivariateDerivative1<>(i,    iDot);
         final FieldUnivariateDerivative1<T> raanUD = new FieldUnivariateDerivative1<>(raan, raanDot);
         return raanUD.cos().multiply(iUD.multiply(0.5).tan()).getFirstDerivative();
 
     }
 
     /** {@inheritDoc} */
     @Override
     public T getHy() {
         // Check for equatorial retrograde orbit
         if (FastMath.abs(i.subtract(i.getPi()).getReal()) < 1.0e-10) {
             return getZero().add(Double.NaN);
         }
         return raan.sin().multiply(i.divide(2).tan());
     }
 
     /** {@inheritDoc} */
     @Override
     public T getHyDot() {
 
         // Check for equatorial retrograde orbit
         if (FastMath.abs(i.subtract(i.getPi()).getReal()) < 1.0e-10) {
             return getZero().add(Double.NaN);
         }
         if (!hasNonKeplerianRates()) {
             return getZero();
         }
 
         final FieldUnivariateDerivative1<T> iUD    = new FieldUnivariateDerivative1<>(i,    iDot);
         final FieldUnivariateDerivative1<T> raanUD = new FieldUnivariateDerivative1<>(raan, raanDot);
         return raanUD.sin().multiply(iUD.multiply(0.5).tan()).getFirstDerivative();
 
     }
 
     /** {@inheritDoc} */
     @Override
     public T getLv() {
         return pa.add(raan).add(getTrueAnomaly());
     }
 
     /** {@inheritDoc} */
     @Override
     public T getLvDot() {
         return paDot.add(raanDot).add(getTrueAnomalyDot());
     }
 
     /** {@inheritDoc} */
     @Override
     public T getLE() {
         return pa.add(raan).add(getEccentricAnomaly());
     }
 
     /** {@inheritDoc} */
     @Override
     public T getLEDot() {
         return paDot.add(raanDot).add(getEccentricAnomalyDot());
     }
 
     /** {@inheritDoc} */
     @Override
     public T getLM() {
         return pa.add(raan).add(getMeanAnomaly());
     }
 
     /** {@inheritDoc} */
     @Override
     public T getLMDot() {
         return paDot.add(raanDot).add(getMeanAnomalyDot());
     }
 
     /** Initialize cached anomaly with rate.
      * @param anomaly input anomaly
      * @param anomalyDot rate of input anomaly
      * @param inputType position angle type passed as input
      * @return anomaly to cache with rate
      * @since 12.1
      */
     private FieldUnivariateDerivative1<T> initializeCachedAnomaly(final T anomaly, final T anomalyDot,
                                                                   final PositionAngleType inputType) {
         if (cachedPositionAngleType == inputType) {
             return new FieldUnivariateDerivative1<>(anomaly, anomalyDot);
 
         } else {
             final FieldUnivariateDerivative1<T> eUD = new FieldUnivariateDerivative1<>(e, eDot);
             final FieldUnivariateDerivative1<T> anomalyUD = new FieldUnivariateDerivative1<>(anomaly, anomalyDot);
 
             if (a.getReal() < 0) {
                 switch (cachedPositionAngleType) {
                     case MEAN:
                         if (inputType == PositionAngleType.ECCENTRIC) {
                             return FieldKeplerianAnomalyUtility.hyperbolicEccentricToMean(eUD, anomalyUD);
                         } else {
                             return FieldKeplerianAnomalyUtility.hyperbolicTrueToMean(eUD, anomalyUD);
                         }
 
                     case ECCENTRIC:
                         if (inputType == PositionAngleType.MEAN) {
                             return FieldKeplerianAnomalyUtility.hyperbolicMeanToEccentric(eUD, anomalyUD);
                         } else {
                             return FieldKeplerianAnomalyUtility.hyperbolicTrueToEccentric(eUD, anomalyUD);
                         }
 
                     case TRUE:
                         if (inputType == PositionAngleType.MEAN) {
                             return FieldKeplerianAnomalyUtility.hyperbolicMeanToTrue(eUD, anomalyUD);
                         } else {
                             return FieldKeplerianAnomalyUtility.hyperbolicEccentricToTrue(eUD, anomalyUD);
                         }
 
                     default:
                         break;
                 }
 
             } else {
                 switch (cachedPositionAngleType) {
                     case MEAN:
                         if (inputType == PositionAngleType.ECCENTRIC) {
                             return FieldKeplerianAnomalyUtility.ellipticEccentricToMean(eUD, anomalyUD);
                         } else {
                             return FieldKeplerianAnomalyUtility.ellipticTrueToMean(eUD, anomalyUD);
                         }
 
                     case ECCENTRIC:
                         if (inputType == PositionAngleType.MEAN) {
                             return FieldKeplerianAnomalyUtility.ellipticMeanToEccentric(eUD, anomalyUD);
                         } else {
                             return FieldKeplerianAnomalyUtility.ellipticTrueToEccentric(eUD, anomalyUD);
                         }
 
                     case TRUE:
                         if (inputType == PositionAngleType.MEAN) {
                             return FieldKeplerianAnomalyUtility.ellipticMeanToTrue(eUD, anomalyUD);
                         } else {
                             return FieldKeplerianAnomalyUtility.ellipticEccentricToTrue(eUD, anomalyUD);
                         }
 
                     default:
                         break;
                 }
 
             }
             throw new OrekitInternalError(null);
         }
 
     }
 
     /** Compute reference axes.
      * @return reference axes
      * @since 12.0
      */
     private FieldVector3D<T>[] referenceAxes() {
 
         // preliminary variables
         final FieldSinCos<T> scRaan = FastMath.sinCos(raan);
         final FieldSinCos<T> scPa   = FastMath.sinCos(pa);
         final FieldSinCos<T> scI    = FastMath.sinCos(i);
         final T cosRaan = scRaan.cos();
         final T sinRaan = scRaan.sin();
         final T cosPa   = scPa.cos();
         final T sinPa   = scPa.sin();
         final T cosI    = scI.cos();
         final T sinI    = scI.sin();
         final T crcp    = cosRaan.multiply(cosPa);
         final T crsp    = cosRaan.multiply(sinPa);
         final T srcp    = sinRaan.multiply(cosPa);
         final T srsp    = sinRaan.multiply(sinPa);
 
         // reference axes defining the orbital plane
         @SuppressWarnings("unchecked")
         final FieldVector3D<T>[] axes = (FieldVector3D<T>[]) Array.newInstance(FieldVector3D.class, 2);
         axes[0] = new FieldVector3D<>(crcp.subtract(cosI.multiply(srsp)),  srcp.add(cosI.multiply(crsp)), sinI.multiply(sinPa));
         axes[1] = new FieldVector3D<>(crsp.add(cosI.multiply(srcp)).negate(), cosI.multiply(crcp).subtract(srsp), sinI.multiply(cosPa));
 
         return axes;
 
     }
 
     /** Compute position and velocity but not acceleration.
      */
     private void computePVWithoutA() {
 
         if (partialPV != null) {
             // already computed
             return;
         }
 
         final FieldVector3D<T>[] axes = referenceAxes();
 
         if (isElliptical()) {
 
             // elliptical case
 
             // elliptic eccentric anomaly
             final T uME2             = e.negate().add(1).multiply(e.add(1));
             final T s1Me2            = uME2.sqrt();
             final FieldSinCos<T> scE = FastMath.sinCos(getEccentricAnomaly());
             final T cosE             = scE.cos();
             final T sinE             = scE.sin();
 
             // coordinates of position and velocity in the orbital plane
             final T x      = a.multiply(cosE.subtract(e));
             final T y      = a.multiply(sinE).multiply(s1Me2);
             final T factor = FastMath.sqrt(getMu().divide(a)).divide(e.negate().multiply(cosE).add(1));
             final T xDot   = sinE.negate().multiply(factor);
             final T yDot   = cosE.multiply(s1Me2).multiply(factor);
 
             final FieldVector3D<T> position = new FieldVector3D<>(x, axes[0], y, axes[1]);
             final FieldVector3D<T> velocity = new FieldVector3D<>(xDot, axes[0], yDot, axes[1]);
             partialPV = new FieldPVCoordinates<>(position, velocity);
 
         } else {
 
             // hyperbolic case
 
             // compute position and velocity factors
             final FieldSinCos<T> scV = FastMath.sinCos(getTrueAnomaly());
             final T sinV             = scV.sin();
             final T cosV             = scV.cos();
             final T f                = a.multiply(e.square().negate().add(1));
             final T posFactor        = f.divide(e.multiply(cosV).add(1));
             final T velFactor        = FastMath.sqrt(getMu().divide(f));
 
             final FieldVector3D<T> position     = new FieldVector3D<>(posFactor.multiply(cosV), axes[0], posFactor.multiply(sinV), axes[1]);
             final FieldVector3D<T> velocity     = new FieldVector3D<>(velFactor.multiply(sinV).negate(), axes[0], velFactor.multiply(e.add(cosV)), axes[1]);
             partialPV = new FieldPVCoordinates<>(position, velocity);
 
         }
 
     }
 
     /** {@inheritDoc} */
     @Override
     protected FieldVector3D<T> initPosition() {
         final FieldVector3D<T>[] axes = referenceAxes();
 
         if (isElliptical()) {
 
             // elliptical case
 
             // elliptic eccentric anomaly
             final T uME2             = e.negate().add(1).multiply(e.add(1));
             final T s1Me2            = uME2.sqrt();
             final FieldSinCos<T> scE = FastMath.sinCos(getEccentricAnomaly());
             final T cosE             = scE.cos();
             final T sinE             = scE.sin();
 
             return new FieldVector3D<>(a.multiply(cosE.subtract(e)), axes[0], a.multiply(sinE).multiply(s1Me2), axes[1]);
 
         } else {
 
             // hyperbolic case
 
             // compute position and velocity factors
             final FieldSinCos<T> scV = FastMath.sinCos(getTrueAnomaly());
             final T sinV             = scV.sin();
             final T cosV             = scV.cos();
             final T f                = a.multiply(e.square().negate().add(1));
             final T posFactor        = f.divide(e.multiply(cosV).add(1));
 
             return new FieldVector3D<>(posFactor.multiply(cosV), axes[0], posFactor.multiply(sinV), axes[1]);
 
         }
 
     }
 
     /** {@inheritDoc} */
     @Override
     protected TimeStampedFieldPVCoordinates<T> initPVCoordinates() {
 
         // position and velocity
         computePVWithoutA();
 
         // acceleration
         final T r2 = partialPV.getPosition().getNormSq();
         final FieldVector3D<T> keplerianAcceleration = new FieldVector3D<>(r2.multiply(FastMath.sqrt(r2)).reciprocal().multiply(getMu().negate()),
                                                                            partialPV.getPosition());
         final FieldVector3D<T> acceleration = hasNonKeplerianRates() ?
                                               keplerianAcceleration.add(nonKeplerianAcceleration()) :
                                               keplerianAcceleration;
 
         return new TimeStampedFieldPVCoordinates<>(getDate(), partialPV.getPosition(), partialPV.getVelocity(), acceleration);
 
     }
 
     /** {@inheritDoc} */
     @Override
     public FieldKeplerianOrbit<T> inFrame(final Frame inertialFrame) {
         final FieldPVCoordinates<T> fieldPVCoordinates;
         if (hasNonKeplerianAcceleration()) {
             fieldPVCoordinates = getPVCoordinates(inertialFrame);
         } else {
             final FieldKinematicTransform<T> transform = getFrame().getKinematicTransformTo(inertialFrame, getDate());
             fieldPVCoordinates = transform.transformOnlyPV(getPVCoordinates());
         }
         final FieldKeplerianOrbit<T> fieldOrbit = new FieldKeplerianOrbit<>(fieldPVCoordinates, inertialFrame, getDate(), getMu());
         if (fieldOrbit.getCachedPositionAngleType() == getCachedPositionAngleType()) {
             return fieldOrbit;
         } else {
             return fieldOrbit.withCachedPositionAngleType(getCachedPositionAngleType());
         }
     }
 
     /** {@inheritDoc} */
     @Override
     public FieldKeplerianOrbit<T> withCachedPositionAngleType(final PositionAngleType positionAngleType) {
         return new FieldKeplerianOrbit<>(a, e, i, pa, raan, getAnomaly(positionAngleType), aDot, eDot, iDot, paDot, raanDot,
                 getAnomalyDot(positionAngleType), positionAngleType, getFrame(), getDate(), getMu());
     }
 
     /** {@inheritDoc} */
     @Override
     public FieldKeplerianOrbit<T> shiftedBy(final double dt) {
         return shiftedBy(getZero().newInstance(dt));
     }
 
     /** {@inheritDoc} */
     @Override
     public FieldKeplerianOrbit<T> shiftedBy(final T dt) {
 
         // use Keplerian-only motion
         final FieldKeplerianOrbit<T> keplerianShifted = new FieldKeplerianOrbit<>(a, e, i, pa, raan,
                                                                                   getKeplerianMeanMotion().multiply(dt).add(getMeanAnomaly()),
                                                                                   PositionAngleType.MEAN, cachedPositionAngleType, getFrame(), getDate().shiftedBy(dt), getMu());
 
         if (!dt.isZero() && hasNonKeplerianRates()) {
 
             // extract non-Keplerian acceleration from first time derivatives
             final FieldVector3D<T> nonKeplerianAcceleration = nonKeplerianAcceleration();
 
             // add quadratic effect of non-Keplerian acceleration to Keplerian-only shift
             keplerianShifted.computePVWithoutA();
             final FieldVector3D<T> fixedP   = new FieldVector3D<>(getOne(), keplerianShifted.partialPV.getPosition(),
                                                                   dt.square().multiply(0.5), nonKeplerianAcceleration);
             final T   fixedR2 = fixedP.getNormSq();
             final T   fixedR  = fixedR2.sqrt();
             final FieldVector3D<T> fixedV  = new FieldVector3D<>(getOne(), keplerianShifted.partialPV.getVelocity(),
                                                                  dt, nonKeplerianAcceleration);
             final FieldVector3D<T> fixedA  = new FieldVector3D<>(fixedR2.multiply(fixedR).reciprocal().multiply(getMu().negate()),
                                                                  keplerianShifted.partialPV.getPosition(),
                                                                  getOne(), nonKeplerianAcceleration);
 
             // build a new orbit, taking non-Keplerian acceleration into account
             return new FieldKeplerianOrbit<>(new TimeStampedFieldPVCoordinates<>(keplerianShifted.getDate(),
                                                                                  fixedP, fixedV, fixedA),
                                              keplerianShifted.getFrame(), keplerianShifted.getMu());
 
         } else {
             // Keplerian-only motion is all we can do
             return keplerianShifted;
         }
 
     }
 
     /** {@inheritDoc} */
     @Override
     protected T[][] computeJacobianMeanWrtCartesian() {
         if (isElliptical()) {
             return computeJacobianMeanWrtCartesianElliptical();
         } else {
             return computeJacobianMeanWrtCartesianHyperbolic();
         }
     }
 
     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
      * <p>
      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
      * yDot for j=4, zDot for j=5).
      * </p>
      * @return 6x6 Jacobian matrix
      */
     private T[][] computeJacobianMeanWrtCartesianElliptical() {
 
         final T[][] jacobian = MathArrays.buildArray(getA().getField(), 6, 6);
 
         // compute various intermediate parameters
         computePVWithoutA();
         final FieldVector3D<T> position = partialPV.getPosition();
         final FieldVector3D<T> velocity = partialPV.getVelocity();
         final FieldVector3D<T> momentum = partialPV.getMomentum();
         final T v2         = velocity.getNormSq();
         final T r2         = position.getNormSq();
         final T r          = r2.sqrt();
         final T r3         = r.multiply(r2);
 
         final T px         = position.getX();
         final T py         = position.getY();
         final T pz         = position.getZ();
         final T vx         = velocity.getX();
         final T vy         = velocity.getY();
         final T vz         = velocity.getZ();
         final T mx         = momentum.getX();
         final T my         = momentum.getY();
         final T mz         = momentum.getZ();
 
         final T mu         = getMu();
         final T sqrtMuA    = FastMath.sqrt(a.multiply(mu));
         final T sqrtAoMu   = FastMath.sqrt(a.divide(mu));
         final T a2         = a.square();
         final T twoA       = a.multiply(2);
         final T rOnA       = r.divide(a);
 
         final T oMe2       = e.square().negate().add(1);
         final T epsilon    = oMe2.sqrt();
         final T sqrtRec    = epsilon.reciprocal();
 
         final FieldSinCos<T> scI  = FastMath.sinCos(i);
         final FieldSinCos<T> scPA = FastMath.sinCos(pa);
         final T cosI       = scI.cos();
         final T sinI       = scI.sin();
         final T cosPA      = scPA.cos();
         final T sinPA      = scPA.sin();
 
         final T pv         = FieldVector3D.dotProduct(position, velocity);
         final T cosE       = a.subtract(r).divide(a.multiply(e));
         final T sinE       = pv.divide(e.multiply(sqrtMuA));
 
         // da
         final FieldVector3D<T> vectorAR = new FieldVector3D<>(a2.multiply(2).divide(r3), position);
         final FieldVector3D<T> vectorARDot = velocity.scalarMultiply(a2.multiply(mu.divide(2.).reciprocal()));
         fillHalfRow(getOne(), vectorAR,    jacobian[0], 0);
         fillHalfRow(getOne(), vectorARDot, jacobian[0], 3);
 
         // de
         final T factorER3 = pv.divide(twoA);
         final FieldVector3D<T> vectorER   = new FieldVector3D<>(cosE.multiply(v2).divide(r.multiply(mu)), position,
                                                                 sinE.divide(sqrtMuA), velocity,
                                                                 factorER3.negate().multiply(sinE).divide(sqrtMuA), vectorAR);
         final FieldVector3D<T> vectorERDot = new FieldVector3D<>(sinE.divide(sqrtMuA), position,
                                                                  cosE.multiply(mu.divide(2.).reciprocal()).multiply(r), velocity,
                                                                  factorER3.negate().multiply(sinE).divide(sqrtMuA), vectorARDot);
         fillHalfRow(getOne(), vectorER,    jacobian[1], 0);
         fillHalfRow(getOne(), vectorERDot, jacobian[1], 3);
 
         // dE / dr (Eccentric anomaly)
         final T coefE = cosE.divide(e.multiply(sqrtMuA));
         final FieldVector3D<T>  vectorEAnR =
             new FieldVector3D<>(sinE.negate().multiply(v2).divide(e.multiply(r).multiply(mu)), position, coefE, velocity,
                                 factorER3.negate().multiply(coefE), vectorAR);
 
         // dE / drDot
         final FieldVector3D<T>  vectorEAnRDot =
             new FieldVector3D<>(sinE.multiply(-2).multiply(r).divide(e.multiply(mu)), velocity, coefE, position,
                                 factorER3.negate().multiply(coefE), vectorARDot);
 
         // precomputing some more factors
         final T s1 = sinE.negate().multiply(pz).divide(r).subtract(cosE.multiply(vz).multiply(sqrtAoMu));
         final T s2 = cosE.negate().multiply(pz).divide(r3);
         final T s3 = sinE.multiply(vz).divide(sqrtMuA.multiply(-2));
         final T t1 = sqrtRec.multiply(cosE.multiply(pz).divide(r).subtract(sinE.multiply(vz).multiply(sqrtAoMu)));
         final T t2 = sqrtRec.multiply(sinE.negate().multiply(pz).divide(r3));
         final T t3 = sqrtRec.multiply(cosE.subtract(e)).multiply(vz).divide(sqrtMuA.multiply(2));
         final T t4 = sqrtRec.multiply(e.multiply(sinI).multiply(cosPA).multiply(sqrtRec).subtract(vz.multiply(sqrtAoMu)));
         final FieldVector3D<T> s = new FieldVector3D<>(cosE.divide(r), this.PLUS_K,
                                                        s1,       vectorEAnR,
                                                        s2,       position,
                                                        s3,       vectorAR);
         final FieldVector3D<T> sDot = new FieldVector3D<>(sinE.negate().multiply(sqrtAoMu), this.PLUS_K,
                                                           s1,               vectorEAnRDot,
                                                           s3,               vectorARDot);
         final FieldVector3D<T> t =
             new FieldVector3D<>(sqrtRec.multiply(sinE).divide(r), this.PLUS_K).add(new FieldVector3D<>(t1, vectorEAnR,
                                                                                                        t2, position,
                                                                                                        t3, vectorAR,
                                                                                                        t4, vectorER));
         final FieldVector3D<T> tDot = new FieldVector3D<>(sqrtRec.multiply(cosE.subtract(e)).multiply(sqrtAoMu), this.PLUS_K,
                                                           t1,                                                    vectorEAnRDot,
                                                           t3,                                                    vectorARDot,
                                                           t4,                                                    vectorERDot);
 
         // di
         final T factorI1 = sinI.negate().multiply(sqrtRec).divide(sqrtMuA);
         final T i1 =  factorI1;
         final T i2 =  factorI1.negate().multiply(mz).divide(twoA);
         final T i3 =  factorI1.multiply(mz).multiply(e).divide(oMe2);
         final T i4 = cosI.multiply(sinPA);
         final T i5 = cosI.multiply(cosPA);
         fillHalfRow(i1, new FieldVector3D<>(vy, vx.negate(), getZero()), i2, vectorAR, i3, vectorER, i4, s, i5, t,
                     jacobian[2], 0);
         fillHalfRow(i1, new FieldVector3D<>(py.negate(), px, getZero()), i2, vectorARDot, i3, vectorERDot, i4, sDot, i5, tDot,
                     jacobian[2], 3);
 
         // dpa
         fillHalfRow(cosPA.divide(sinI), s,    sinPA.negate().divide(sinI), t,    jacobian[3], 0);
         fillHalfRow(cosPA.divide(sinI), sDot, sinPA.negate().divide(sinI), tDot, jacobian[3], 3);
 
         // dRaan
         final T factorRaanR = (a.multiply(mu).multiply(oMe2).multiply(sinI).multiply(sinI)).reciprocal();
         fillHalfRow( factorRaanR.negate().multiply(my), new FieldVector3D<>(getZero(), vz, vy.negate()),
                      factorRaanR.multiply(mx), new FieldVector3D<>(vz.negate(), getZero(),  vx),
                      jacobian[4], 0);
         fillHalfRow(factorRaanR.negate().multiply(my), new FieldVector3D<>(getZero(), pz.negate(),  py),
                      factorRaanR.multiply(mx), new FieldVector3D<>(pz, getZero(), px.negate()),
                      jacobian[4], 3);
 
         // dM
         fillHalfRow(rOnA, vectorEAnR,    sinE.negate(), vectorER,    jacobian[5], 0);
         fillHalfRow(rOnA, vectorEAnRDot, sinE.negate(), vectorERDot, jacobian[5], 3);
 
         return jacobian;
 
     }
 
     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
      * <p>
      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
      * yDot for j=4, zDot for j=5).
      * </p>
      * @return 6x6 Jacobian matrix
      */
     private T[][] computeJacobianMeanWrtCartesianHyperbolic() {
 
         final T[][] jacobian = MathArrays.buildArray(getA().getField(), 6, 6);
 
         // compute various intermediate parameters
         computePVWithoutA();
         final FieldVector3D<T> position = partialPV.getPosition();
         final FieldVector3D<T> velocity = partialPV.getVelocity();
         final FieldVector3D<T> momentum = partialPV.getMomentum();
         final T r2         = position.getNormSq();
         final T r          = r2.sqrt();
         final T r3         = r.multiply(r2);
 
         final T x          = position.getX();
         final T y          = position.getY();
         final T z          = position.getZ();
         final T vx         = velocity.getX();
         final T vy         = velocity.getY();
         final T vz         = velocity.getZ();
         final T mx         = momentum.getX();
         final T my         = momentum.getY();
         final T mz         = momentum.getZ();
 
         final T mu         = getMu();
         final T absA       = a.negate();
         final T sqrtMuA    = absA.multiply(mu).sqrt();
         final T a2         = a.square();
         final T rOa        = r.divide(absA);
 
         final FieldSinCos<T> scI = FastMath.sinCos(i);
         final T cosI       = scI.cos();
         final T sinI       = scI.sin();
 
         final T pv         = FieldVector3D.dotProduct(position, velocity);
 
         // da
         final FieldVector3D<T> vectorAR = new FieldVector3D<>(a2.multiply(-2).divide(r3), position);
         final FieldVector3D<T> vectorARDot = velocity.scalarMultiply(a2.multiply(-2).divide(mu));
         fillHalfRow(getOne().negate(), vectorAR,    jacobian[0], 0);
         fillHalfRow(getOne().negate(), vectorARDot, jacobian[0], 3);
 
         // differentials of the momentum
         final T m      = momentum.getNorm();
         final T oOm    = m.reciprocal();
         final FieldVector3D<T> dcXP = new FieldVector3D<>(getZero(), vz, vy.negate());
         final FieldVector3D<T> dcYP = new FieldVector3D<>(vz.negate(),   getZero(),  vx);
         final FieldVector3D<T> dcZP = new FieldVector3D<>( vy, vx.negate(),   getZero());
         final FieldVector3D<T> dcXV = new FieldVector3D<>(  getZero(),  z.negate(),   y);
         final FieldVector3D<T> dcYV = new FieldVector3D<>(  z,   getZero(),  x.negate());
         final FieldVector3D<T> dcZV = new FieldVector3D<>( y.negate(),   x,   getZero());
         final FieldVector3D<T> dCP  = new FieldVector3D<>(mx.multiply(oOm), dcXP, my.multiply(oOm), dcYP, mz.multiply(oOm), dcZP);
         final FieldVector3D<T> dCV  = new FieldVector3D<>(mx.multiply(oOm), dcXV, my.multiply(oOm), dcYV, mz.multiply(oOm), dcZV);
 
         // dp
         final T mOMu   = m.divide(mu);
         final FieldVector3D<T> dpP  = new FieldVector3D<>(mOMu.multiply(2), dCP);
         final FieldVector3D<T> dpV  = new FieldVector3D<>(mOMu.multiply(2), dCV);
 
         // de
         final T p      = m.multiply(mOMu);
         final T moO2ae = absA.multiply(2).multiply(e).reciprocal();
         final T m2OaMu = p.negate().divide(absA);
         fillHalfRow(moO2ae, dpP, m2OaMu.multiply(moO2ae), vectorAR,    jacobian[1], 0);
         fillHalfRow(moO2ae, dpV, m2OaMu.multiply(moO2ae), vectorARDot, jacobian[1], 3);
 
         // di
         final T cI1 = m.multiply(sinI).reciprocal();
         final T cI2 = cosI.multiply(cI1);
         fillHalfRow(cI2, dCP, cI1.negate(), dcZP, jacobian[2], 0);
         fillHalfRow(cI2, dCV, cI1.negate(), dcZV, jacobian[2], 3);
 
 
         // dPA
         final T cP1     =  y.multiply(oOm);
         final T cP2     =  x.negate().multiply(oOm);
         final T cP3     =  mx.multiply(cP1).add(my.multiply(cP2)).negate();
         final T cP4     =  cP3.multiply(oOm);
         final T cP5     =  r2.multiply(sinI).multiply(sinI).negate().reciprocal();
         final T cP6     = z.multiply(cP5);
         final T cP7     = cP3.multiply(cP5);
         final FieldVector3D<T> dacP  = new FieldVector3D<>(cP1, dcXP, cP2, dcYP, cP4, dCP, oOm, new FieldVector3D<>(my.negate(), mx, getZero()));
         final FieldVector3D<T> dacV  = new FieldVector3D<>(cP1, dcXV, cP2, dcYV, cP4, dCV);
         final FieldVector3D<T> dpoP  = new FieldVector3D<>(cP6, dacP, cP7, this.PLUS_K);
         final FieldVector3D<T> dpoV  = new FieldVector3D<>(cP6, dacV);
 
         final T re2     = r2.multiply(e.square());
         final T recOre2 = p.subtract(r).divide(re2);
         final T resOre2 = pv.multiply(mOMu).divide(re2);
         final FieldVector3D<T> dreP  = new FieldVector3D<>(mOMu, velocity, pv.divide(mu), dCP);
         final FieldVector3D<T> dreV  = new FieldVector3D<>(mOMu, position, pv.divide(mu), dCV);
         final FieldVector3D<T> davP  = new FieldVector3D<>(resOre2.negate(), dpP, recOre2, dreP, resOre2.divide(r), position);
         final FieldVector3D<T> davV  = new FieldVector3D<>(resOre2.negate(), dpV, recOre2, dreV);
         fillHalfRow(getOne(), dpoP, getOne().negate(), davP, jacobian[3], 0);
         fillHalfRow(getOne(), dpoV, getOne().negate(), davV, jacobian[3], 3);
 
         // dRAAN
         final T cO0 = cI1.square();
         final T cO1 =  mx.multiply(cO0);
         final T cO2 =  my.negate().multiply(cO0);
         fillHalfRow(cO1, dcYP, cO2, dcXP, jacobian[4], 0);
         fillHalfRow(cO1, dcYV, cO2, dcXV, jacobian[4], 3);
 
         // dM
         final T s2a    = pv.divide(absA.multiply(2));
         final T oObux  = m.square().add(absA.multiply(mu)).sqrt().reciprocal();
         final T scasbu = pv.multiply(oObux);
         final FieldVector3D<T> dauP = new FieldVector3D<>(sqrtMuA.reciprocal(), velocity, s2a.negate().divide(sqrtMuA), vectorAR);
         final FieldVector3D<T> dauV = new FieldVector3D<>(sqrtMuA.reciprocal(), position, s2a.negate().divide(sqrtMuA), vectorARDot);
         final FieldVector3D<T> dbuP = new FieldVector3D<>(oObux.multiply(mu.divide(2.)), vectorAR,    m.multiply(oObux), dCP);
         final FieldVector3D<T> dbuV = new FieldVector3D<>(oObux.multiply(mu.divide(2.)), vectorARDot, m.multiply(oObux), dCV);
         final FieldVector3D<T> dcuP = new FieldVector3D<>(oObux, velocity, scasbu.negate().multiply(oObux), dbuP);
         final FieldVector3D<T> dcuV = new FieldVector3D<>(oObux, position, scasbu.negate().multiply(oObux), dbuV);
         fillHalfRow(getOne(), dauP, e.negate().divide(rOa.add(1)), dcuP, jacobian[5], 0);
         fillHalfRow(getOne(), dauV, e.negate().divide(rOa.add(1)), dcuV, jacobian[5], 3);
 
         return jacobian;
 
     }
 
     /** {@inheritDoc} */
     @Override
     protected T[][] computeJacobianEccentricWrtCartesian() {
         if (isElliptical()) {
             return computeJacobianEccentricWrtCartesianElliptical();
         } else {
             return computeJacobianEccentricWrtCartesianHyperbolic();
         }
     }
 
     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
      * <p>
      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
      * yDot for j=4, zDot for j=5).
      * </p>
      * @return 6x6 Jacobian matrix
      */
     private T[][] computeJacobianEccentricWrtCartesianElliptical() {
 
         // start by computing the Jacobian with mean angle
         final T[][] jacobian = computeJacobianMeanWrtCartesianElliptical();
 
         // Differentiating the Kepler equation M = E - e sin E leads to:
         // dM = (1 - e cos E) dE - sin E de
         // which is inverted and rewritten as:
         // dE = a/r dM + sin E a/r de
         final FieldSinCos<T> scE = FastMath.sinCos(getEccentricAnomaly());
         final T aOr              = e.negate().multiply(scE.cos()).add(1).reciprocal();
 
         // update anomaly row
         final T[] eRow           = jacobian[1];
         final T[] anomalyRow     = jacobian[5];
         for (int j = 0; j < anomalyRow.length; ++j) {
             anomalyRow[j] = aOr.multiply(anomalyRow[j].add(scE.sin().multiply(eRow[j])));
         }
 
         return jacobian;
 
     }
 
     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
      * <p>
      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
      * yDot for j=4, zDot for j=5).
      * </p>
      * @return 6x6 Jacobian matrix
      */
     private T[][] computeJacobianEccentricWrtCartesianHyperbolic() {
 
         // start by computing the Jacobian with mean angle
         final T[][] jacobian = computeJacobianMeanWrtCartesianHyperbolic();
 
         // Differentiating the Kepler equation M = e sinh H - H leads to:
         // dM = (e cosh H - 1) dH + sinh H de
         // which is inverted and rewritten as:
         // dH = 1 / (e cosh H - 1) dM - sinh H / (e cosh H - 1) de
         final T H      = getEccentricAnomaly();
         final T coshH  = H.cosh();
         final T sinhH  = H.sinh();
         final T absaOr = e.multiply(coshH).subtract(1).reciprocal();
         // update anomaly row
         final T[] eRow       = jacobian[1];
         final T[] anomalyRow = jacobian[5];
 
         for (int j = 0; j < anomalyRow.length; ++j) {
             anomalyRow[j] = absaOr.multiply(anomalyRow[j].subtract(sinhH.multiply(eRow[j])));
 
         }
 
         return jacobian;
 
     }
 
     /** {@inheritDoc} */
     @Override
     protected T[][] computeJacobianTrueWrtCartesian() {
         if (isElliptical()) {
             return computeJacobianTrueWrtCartesianElliptical();
         } else {
             return computeJacobianTrueWrtCartesianHyperbolic();
         }
     }
 
     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
      * <p>
      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
      * yDot for j=4, zDot for j=5).
      * </p>
      * @return 6x6 Jacobian matrix
      */
     private T[][] computeJacobianTrueWrtCartesianElliptical() {
 
         // start by computing the Jacobian with eccentric angle
         final T[][] jacobian = computeJacobianEccentricWrtCartesianElliptical();
         // Differentiating the eccentric anomaly equation sin E = sqrt(1-e^2) sin v / (1 + e cos v)
         // and using cos E = (e + cos v) / (1 + e cos v) to get rid of cos E leads to:
         // dE = [sqrt (1 - e^2) / (1 + e cos v)] dv - [sin E / (1 - e^2)] de
         // which is inverted and rewritten as:
         // dv = sqrt (1 - e^2) a/r dE + [sin E / sqrt (1 - e^2)] a/r de
         final T e2               = e.square();
         final T oMe2             = e2.negate().add(1);
         final T epsilon          = oMe2.sqrt();
         final FieldSinCos<T> scE = FastMath.sinCos(getEccentricAnomaly());
         final T aOr              = e.multiply(scE.cos()).negate().add(1).reciprocal();
         final T aFactor          = epsilon.multiply(aOr);
         final T eFactor          = scE.sin().multiply(aOr).divide(epsilon);
 
         // update anomaly row
         final T[] eRow           = jacobian[1];
         final T[] anomalyRow     = jacobian[5];
         for (int j = 0; j < anomalyRow.length; ++j) {
             anomalyRow[j] = aFactor.multiply(anomalyRow[j]).add(eFactor.multiply(eRow[j]));
         }
         return jacobian;
 
     }
 
     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
      * <p>
      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
      * yDot for j=4, zDot for j=5).
      * </p>
      * @return 6x6 Jacobian matrix
      */
     private T[][] computeJacobianTrueWrtCartesianHyperbolic() {
 
         // start by computing the Jacobian with eccentric angle
         final T[][] jacobian = computeJacobianEccentricWrtCartesianHyperbolic();
 
         // Differentiating the eccentric anomaly equation sinh H = sqrt(e^2-1) sin v / (1 + e cos v)
         // and using cosh H = (e + cos v) / (1 + e cos v) to get rid of cosh H leads to:
         // dH = [sqrt (e^2 - 1) / (1 + e cos v)] dv + [sinh H / (e^2 - 1)] de
         // which is inverted and rewritten as:
         // dv = sqrt (1 - e^2) a/r dH - [sinh H / sqrt (e^2 - 1)] a/r de
         final T e2       = e.square();
         final T e2Mo     = e2.subtract(1);
         final T epsilon  = e2Mo.sqrt();
         final T H        = getEccentricAnomaly();
         final T coshH    = H.cosh();
         final T sinhH    = H.sinh();
         final T aOr      = e.multiply(coshH).subtract(1).reciprocal();
         final T aFactor  = epsilon.multiply(aOr);
         final T eFactor  = sinhH.multiply(aOr).divide(epsilon);
 
         // update anomaly row
         final T[] eRow           = jacobian[1];
         final T[] anomalyRow     = jacobian[5];
         for (int j = 0; j < anomalyRow.length; ++j) {
             anomalyRow[j] = aFactor.multiply(anomalyRow[j]).subtract(eFactor.multiply(eRow[j]));
         }
 
         return jacobian;
 
     }
 
     /** {@inheritDoc} */
     @Override
     public void addKeplerContribution(final PositionAngleType type, final T gm,
                                       final T[] pDot) {
         pDot[5] = pDot[5].add(computeKeplerianAnomalyDot(type, a, e, gm, cachedAnomaly, cachedPositionAngleType));
     }
 
     /**
      * Compute rate of argument of latitude.
      * @param type position angle type of rate
      * @param a semi major axis
      * @param e eccentricity
      * @param mu mu
      * @param anomaly anomaly
      * @param cachedType position angle type of passed anomaly
      * @param <T> field type
      * @return first-order time derivative for anomaly
      * @since 12.2
      */
     private static <T extends CalculusFieldElement<T>> T computeKeplerianAnomalyDot(final PositionAngleType type, final T a, final T e,
                                                                                     final T mu, final T anomaly, final PositionAngleType cachedType) {
         final T absA = a.abs();
         final T n    = absA.reciprocal().multiply(mu).sqrt().divide(absA);
         if (type == PositionAngleType.MEAN) {
             return n;
         }
         final T ksi;
         final T oMe2;
         final T trueAnomaly = FieldKeplerianAnomalyUtility.convertAnomaly(cachedType, anomaly, e, PositionAngleType.TRUE);
         if (type == PositionAngleType.ECCENTRIC) {
             oMe2 = e.square().negate().add(1).abs();
             ksi = e.multiply(trueAnomaly.cos()).add(1);
             return n.multiply(ksi).divide(oMe2);
         } else {
             oMe2 = e.square().negate().add(1).abs();
             ksi  = e.multiply(trueAnomaly.cos()).add(1);
             return n.multiply(ksi).multiply(ksi).divide(oMe2.multiply(oMe2.sqrt()));
         }
     }
 
     /**  Returns a string representation of this Keplerian parameters object.
      * @return a string representation of this object
      */
     public String toString() {
         return new StringBuilder().append("Keplerian parameters: ").append('{').
                                   append("a: ").append(a.getReal()).
                                   append("; e: ").append(e.getReal()).
                                   append("; i: ").append(FastMath.toDegrees(i.getReal())).
                                   append("; pa: ").append(FastMath.toDegrees(pa.getReal())).
                                   append("; raan: ").append(FastMath.toDegrees(raan.getReal())).
                                   append("; v: ").append(FastMath.toDegrees(getTrueAnomaly().getReal())).
                                   append(";}").toString();
     }
 
     /** {@inheritDoc} */
     @Override
     public PositionAngleType getCachedPositionAngleType() {
         return cachedPositionAngleType;
     }
 
     /** {@inheritDoc} */
     @Override
     public boolean hasNonKeplerianRates() {
         return hasNonKeplerianAcceleration();
     }
 
     /** {@inheritDoc} */
     @Override
     public FieldKeplerianOrbit<T> withKeplerianRates() {
         return new FieldKeplerianOrbit<>(getA(), getE(), getI(), getPerigeeArgument(), getRightAscensionOfAscendingNode(),
                 cachedAnomaly, cachedPositionAngleType, getFrame(), getDate(), getMu());
     }
 
     /** Check if the given parameter is within an acceptable range.
      * The bounds are inclusive: an exception is raised when either of those conditions are met:
      * <ul>
      *     <li>The parameter is strictly greater than upperBound</li>
      *     <li>The parameter is strictly lower than lowerBound</li>
      * </ul>
      * <p>
      * In either of these cases, an OrekitException is raised.
      * </p>
      * @param parameterName name of the parameter
      * @param parameter value of the parameter
      * @param lowerBound lower bound of the acceptable range (inclusive)
      * @param upperBound upper bound of the acceptable range (inclusive)
      */
     private void checkParameterRangeInclusive(final String parameterName, final double parameter,
                                               final double lowerBound, final double upperBound) {
         if (parameter < lowerBound || parameter > upperBound) {
             throw new OrekitException(OrekitMessages.INVALID_PARAMETER_RANGE, parameterName,
                                       parameter, lowerBound, upperBound);
         }
     }
 
     /** {@inheritDoc} */
     @Override
     public KeplerianOrbit toOrbit() {
         final double cachedPositionAngle = cachedAnomaly.getReal();
         return new KeplerianOrbit(a.getReal(), e.getReal(), i.getReal(),
                                   pa.getReal(), raan.getReal(), cachedPositionAngle,
                                   aDot.getReal(), eDot.getReal(), iDot.getReal(),
                                   paDot.getReal(), raanDot.getReal(),
                                   cachedAnomalyDot.getReal(),
                                   cachedPositionAngleType, cachedPositionAngleType,
                                   getFrame(), getDate().toAbsoluteDate(), getMu().getReal());
     }
 
 
 }