CartesianOrbit.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;

  19. import java.io.Serializable;
  20. import java.util.stream.Stream;

  22. import org.hipparchus.analysis.differentiation.UnivariateDerivative2;
  23. import org.hipparchus.exception.LocalizedCoreFormats;
  24. import org.hipparchus.exception.MathIllegalStateException;
  25. import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  26. import org.hipparchus.geometry.euclidean.threed.Rotation;
  27. import org.hipparchus.geometry.euclidean.threed.RotationConvention;
  28. import org.hipparchus.geometry.euclidean.threed.Vector3D;
  29. import org.hipparchus.util.FastMath;
  30. import org.hipparchus.util.SinCos;
  31. import org.orekit.annotation.DefaultDataContext;
  32. import org.orekit.data.DataContext;
  33. import org.orekit.frames.Frame;
  34. import org.orekit.time.AbsoluteDate;
  35. import org.orekit.utils.CartesianDerivativesFilter;
  36. import org.orekit.utils.FieldPVCoordinates;
  37. import org.orekit.utils.PVCoordinates;
  38. import org.orekit.utils.TimeStampedPVCoordinates;


  41. /** This class holds Cartesian orbital parameters.

  43.  * <p>
  44.  * The parameters used internally are the Cartesian coordinates:
  45.  *   <ul>
  46.  *     <li>x</li>
  47.  *     <li>y</li>
  48.  *     <li>z</li>
  49.  *     <li>xDot</li>
  50.  *     <li>yDot</li>
  51.  *     <li>zDot</li>
  52.  *   </ul>
  53.  * contained in {@link PVCoordinates}.
  54.  *

  56.  * <p>
  57.  * Note that the implementation of this class delegates all non-Cartesian related
  58.  * computations ({@link #getA()}, {@link #getEquinoctialEx()}, ...) to an underlying
  59.  * instance of the {@link EquinoctialOrbit} class. This implies that using this class
  60.  * only for analytical computations which are always based on non-Cartesian
  61.  * parameters is perfectly possible but somewhat sub-optimal.
  62.  * </p>
  63.  * <p>
  64.  * The instance <code>CartesianOrbit</code> is guaranteed to be immutable.
  65.  * </p>
  66.  * @see    Orbit
  67.  * @see    KeplerianOrbit
  68.  * @see    CircularOrbit
  69.  * @see    EquinoctialOrbit
  70.  * @author Luc Maisonobe
  71.  * @author Guylaine Prat
  72.  * @author Fabien Maussion
  73.  * @author V&eacute;ronique Pommier-Maurussane
  74.  */
  75. public class CartesianOrbit extends Orbit {

  77.     /** Serializable UID. */
  78.     private static final long serialVersionUID = 20170414L;

  80.     /** Indicator for non-Keplerian derivatives. */
  81.     private final transient boolean hasNonKeplerianAcceleration;

  83.     /** Underlying equinoctial orbit to which high-level methods are delegated. */
  84.     private transient EquinoctialOrbit equinoctial;

  86.     /** Constructor from Cartesian parameters.
  87.      *
  88.      * <p> The acceleration provided in {@code pvCoordinates} is accessible using
  89.      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
  90.      * use {@code mu} and the position to compute the acceleration, including
  91.      * {@link #shiftedBy(double)} and {@link #getPVCoordinates(AbsoluteDate, Frame)}.
  92.      *
  93.      * @param pvaCoordinates the position, velocity and acceleration of the satellite.
  94.      * @param frame the frame in which the {@link PVCoordinates} are defined
  95.      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
  96.      * @param mu central attraction coefficient (m³/s²)
  97.      * @exception IllegalArgumentException if frame is not a {@link
  98.      * Frame#isPseudoInertial pseudo-inertial frame}
  99.      */
  100.     public CartesianOrbit(final TimeStampedPVCoordinates pvaCoordinates,
  101.                           final Frame frame, final double mu)
  102.         throws IllegalArgumentException {
  103.         super(pvaCoordinates, frame, mu);
  104.         hasNonKeplerianAcceleration = hasNonKeplerianAcceleration(pvaCoordinates, mu);
  105.         equinoctial = null;
  106.     }

  108.     /** Constructor from Cartesian parameters.
  109.      *
  110.      * <p> The acceleration provided in {@code pvCoordinates} is accessible using
  111.      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
  112.      * use {@code mu} and the position to compute the acceleration, including
  113.      * {@link #shiftedBy(double)} and {@link #getPVCoordinates(AbsoluteDate, Frame)}.
  114.      *
  115.      * @param pvaCoordinates the position and velocity of the satellite.
  116.      * @param frame the frame in which the {@link PVCoordinates} are defined
  117.      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
  118.      * @param date date of the orbital parameters
  119.      * @param mu central attraction coefficient (m³/s²)
  120.      * @exception IllegalArgumentException if frame is not a {@link
  121.      * Frame#isPseudoInertial pseudo-inertial frame}
  122.      */
  123.     public CartesianOrbit(final PVCoordinates pvaCoordinates, final Frame frame,
  124.                           final AbsoluteDate date, final double mu)
  125.         throws IllegalArgumentException {
  126.         this(new TimeStampedPVCoordinates(date, pvaCoordinates), frame, mu);
  127.     }

  129.     /** Constructor from any kind of orbital parameters.
  130.      * @param op orbital parameters to copy
  131.      */
  132.     public CartesianOrbit(final Orbit op) {
  133.         super(op.getPVCoordinates(), op.getFrame(), op.getMu());
  134.         hasNonKeplerianAcceleration = op.hasDerivatives();
  135.         if (op instanceof EquinoctialOrbit) {
  136.             equinoctial = (EquinoctialOrbit) op;
  137.         } else if (op instanceof CartesianOrbit) {
  138.             equinoctial = ((CartesianOrbit) op).equinoctial;
  139.         } else {
  140.             equinoctial = null;
  141.         }
  142.     }

  144.     /** {@inheritDoc} */
  145.     public OrbitType getType() {
  146.         return OrbitType.CARTESIAN;
  147.     }

  149.     /** Lazy evaluation of equinoctial parameters. */
  150.     private void initEquinoctial() {
  151.         if (equinoctial == null) {
  152.             if (hasDerivatives()) {
  153.                 // getPVCoordinates includes accelerations that will be interpreted as derivatives
  154.                 equinoctial = new EquinoctialOrbit(getPVCoordinates(), getFrame(), getDate(), getMu());
  155.             } else {
  156.                 // get rid of Keplerian acceleration so we don't assume
  157.                 // we have derivatives when in fact we don't have them
  158.                 equinoctial = new EquinoctialOrbit(new PVCoordinates(getPVCoordinates().getPosition(),
  159.                                                                      getPVCoordinates().getVelocity()),
  160.                                                    getFrame(), getDate(), getMu());
  161.             }
  162.         }
  163.     }

  165.     /** Get the position/velocity with derivatives.
  166.      * @return position/velocity with derivatives
  167.      * @since 10.2
  168.      */
  169.     private FieldPVCoordinates<UnivariateDerivative2> getPVDerivatives() {
  170.         // PVA coordinates
  171.         final PVCoordinates pva = getPVCoordinates();
  172.         final Vector3D      p   = pva.getPosition();
  173.         final Vector3D      v   = pva.getVelocity();
  174.         final Vector3D      a   = pva.getAcceleration();
  175.         // Field coordinates
  176.         final FieldVector3D<UnivariateDerivative2> pG = new FieldVector3D<>(new UnivariateDerivative2(p.getX(), v.getX(), a.getX()),
  177.                                                                new UnivariateDerivative2(p.getY(), v.getY(), a.getY()),
  178.                                                                new UnivariateDerivative2(p.getZ(), v.getZ(), a.getZ()));
  179.         final FieldVector3D<UnivariateDerivative2> vG = new FieldVector3D<>(new UnivariateDerivative2(v.getX(), a.getX(), 0.0),
  180.                                                                new UnivariateDerivative2(v.getY(), a.getY(), 0.0),
  181.                                                                new UnivariateDerivative2(v.getZ(), a.getZ(), 0.0));
  182.         return new FieldPVCoordinates<>(pG, vG);
  183.     }

  185.     /** {@inheritDoc} */
  186.     public double getA() {
  187.         final double r  = getPVCoordinates().getPosition().getNorm();
  188.         final double V2 = getPVCoordinates().getVelocity().getNormSq();
  189.         return r / (2 - r * V2 / getMu());
  190.     }

  192.     /** {@inheritDoc} */
  193.     public double getADot() {
  194.         if (hasDerivatives()) {
  195.             final FieldPVCoordinates<UnivariateDerivative2> pv = getPVDerivatives();
  196.             final UnivariateDerivative2 r  = pv.getPosition().getNorm();
  197.             final UnivariateDerivative2 V2 = pv.getVelocity().getNormSq();
  198.             final UnivariateDerivative2 a  = r.divide(r.multiply(V2).divide(getMu()).subtract(2).negate());
  199.             return a.getDerivative(1);
  200.         } else {
  201.             return Double.NaN;
  202.         }
  203.     }

  205.     /** {@inheritDoc} */
  206.     public double getE() {
  207.         final double muA = getMu() * getA();
  208.         if (muA > 0) {
  209.             // elliptic or circular orbit
  210.             final Vector3D pvP   = getPVCoordinates().getPosition();
  211.             final Vector3D pvV   = getPVCoordinates().getVelocity();
  212.             final double rV2OnMu = pvP.getNorm() * pvV.getNormSq() / getMu();
  213.             final double eSE     = Vector3D.dotProduct(pvP, pvV) / FastMath.sqrt(muA);
  214.             final double eCE     = rV2OnMu - 1;
  215.             return FastMath.sqrt(eCE * eCE + eSE * eSE);
  216.         } else {
  217.             // hyperbolic orbit
  218.             final Vector3D pvM = getPVCoordinates().getMomentum();
  219.             return FastMath.sqrt(1 - pvM.getNormSq() / muA);
  220.         }
  221.     }

  223.     /** {@inheritDoc} */
  224.     public double getEDot() {
  225.         if (hasDerivatives()) {
  226.             final FieldPVCoordinates<UnivariateDerivative2> pv = getPVDerivatives();
  227.             final FieldVector3D<UnivariateDerivative2> pvP   = pv.getPosition();
  228.             final FieldVector3D<UnivariateDerivative2> pvV   = pv.getVelocity();
  229.             final UnivariateDerivative2 r       = pvP.getNorm();
  230.             final UnivariateDerivative2 V2      = pvV.getNormSq();
  231.             final UnivariateDerivative2 rV2OnMu = r.multiply(V2).divide(getMu());
  232.             final UnivariateDerivative2 a       = r.divide(rV2OnMu.negate().add(2));
  233.             final UnivariateDerivative2 eSE     = FieldVector3D.dotProduct(pvP, pvV).divide(a.multiply(getMu()).sqrt());
  234.             final UnivariateDerivative2 eCE     = rV2OnMu.subtract(1);
  235.             final UnivariateDerivative2 e       = eCE.multiply(eCE).add(eSE.multiply(eSE)).sqrt();
  236.             return e.getDerivative(1);
  237.         } else {
  238.             return Double.NaN;
  239.         }
  240.     }

  242.     /** {@inheritDoc} */
  243.     public double getI() {
  244.         return Vector3D.angle(Vector3D.PLUS_K, getPVCoordinates().getMomentum());
  245.     }

  247.     /** {@inheritDoc} */
  248.     public double getIDot() {
  249.         if (hasDerivatives()) {
  250.             final FieldPVCoordinates<UnivariateDerivative2> pv = getPVDerivatives();
  251.             final FieldVector3D<UnivariateDerivative2> momentum =
  252.                             FieldVector3D.crossProduct(pv.getPosition(), pv.getVelocity());
  253.             final UnivariateDerivative2 i = FieldVector3D.angle(Vector3D.PLUS_K, momentum);
  254.             return i.getDerivative(1);
  255.         } else {
  256.             return Double.NaN;
  257.         }
  258.     }

  260.     /** {@inheritDoc} */
  261.     public double getEquinoctialEx() {
  262.         initEquinoctial();
  263.         return equinoctial.getEquinoctialEx();
  264.     }

  266.     /** {@inheritDoc} */
  267.     public double getEquinoctialExDot() {
  268.         initEquinoctial();
  269.         return equinoctial.getEquinoctialExDot();
  270.     }

  272.     /** {@inheritDoc} */
  273.     public double getEquinoctialEy() {
  274.         initEquinoctial();
  275.         return equinoctial.getEquinoctialEy();
  276.     }

  278.     /** {@inheritDoc} */
  279.     public double getEquinoctialEyDot() {
  280.         initEquinoctial();
  281.         return equinoctial.getEquinoctialEyDot();
  282.     }

  284.     /** {@inheritDoc} */
  285.     public double getHx() {
  286.         final Vector3D w = getPVCoordinates().getMomentum().normalize();
  287.         // Check for equatorial retrograde orbit
  288.         if ((w.getX() * w.getX() + w.getY() * w.getY()) == 0 && w.getZ() < 0) {
  289.             return Double.NaN;
  290.         }
  291.         return -w.getY() / (1 + w.getZ());
  292.     }

  294.     /** {@inheritDoc} */
  295.     public double getHxDot() {
  296.         if (hasDerivatives()) {
  297.             final FieldPVCoordinates<UnivariateDerivative2> pv = getPVDerivatives();
  298.             final FieldVector3D<UnivariateDerivative2> w =
  299.                             FieldVector3D.crossProduct(pv.getPosition(), pv.getVelocity()).normalize();
  300.             // Check for equatorial retrograde orbit
  301.             final double x = w.getX().getValue();
  302.             final double y = w.getY().getValue();
  303.             final double z = w.getZ().getValue();
  304.             if ((x * x + y * y) == 0 && z < 0) {
  305.                 return Double.NaN;
  306.             }
  307.             final UnivariateDerivative2 hx = w.getY().negate().divide(w.getZ().add(1));
  308.             return hx.getDerivative(1);
  309.         } else {
  310.             return Double.NaN;
  311.         }
  312.     }

  314.     /** {@inheritDoc} */
  315.     public double getHy() {
  316.         final Vector3D w = getPVCoordinates().getMomentum().normalize();
  317.         // Check for equatorial retrograde orbit
  318.         if ((w.getX() * w.getX() + w.getY() * w.getY()) == 0 && w.getZ() < 0) {
  319.             return Double.NaN;
  320.         }
  321.         return  w.getX() / (1 + w.getZ());
  322.     }

  324.     /** {@inheritDoc} */
  325.     public double getHyDot() {
  326.         if (hasDerivatives()) {
  327.             final FieldPVCoordinates<UnivariateDerivative2> pv = getPVDerivatives();
  328.             final FieldVector3D<UnivariateDerivative2> w =
  329.                             FieldVector3D.crossProduct(pv.getPosition(), pv.getVelocity()).normalize();
  330.             // Check for equatorial retrograde orbit
  331.             final double x = w.getX().getValue();
  332.             final double y = w.getY().getValue();
  333.             final double z = w.getZ().getValue();
  334.             if ((x * x + y * y) == 0 && z < 0) {
  335.                 return Double.NaN;
  336.             }
  337.             final UnivariateDerivative2 hy = w.getX().divide(w.getZ().add(1));
  338.             return hy.getDerivative(1);
  339.         } else {
  340.             return Double.NaN;
  341.         }
  342.     }

  344.     /** {@inheritDoc} */
  345.     public double getLv() {
  346.         initEquinoctial();
  347.         return equinoctial.getLv();
  348.     }

  350.     /** {@inheritDoc} */
  351.     public double getLvDot() {
  352.         initEquinoctial();
  353.         return equinoctial.getLvDot();
  354.     }

  356.     /** {@inheritDoc} */
  357.     public double getLE() {
  358.         initEquinoctial();
  359.         return equinoctial.getLE();
  360.     }

  362.     /** {@inheritDoc} */
  363.     public double getLEDot() {
  364.         initEquinoctial();
  365.         return equinoctial.getLEDot();
  366.     }

  368.     /** {@inheritDoc} */
  369.     public double getLM() {
  370.         initEquinoctial();
  371.         return equinoctial.getLM();
  372.     }

  374.     /** {@inheritDoc} */
  375.     public double getLMDot() {
  376.         initEquinoctial();
  377.         return equinoctial.getLMDot();
  378.     }

  380.     /** {@inheritDoc} */
  381.     public boolean hasDerivatives() {
  382.         return hasNonKeplerianAcceleration;
  383.     }

  385.     /** {@inheritDoc} */
  386.     protected TimeStampedPVCoordinates initPVCoordinates() {
  387.         // nothing to do here, as the canonical elements are already the Cartesian ones
  388.         return getPVCoordinates();
  389.     }

  391.     /** {@inheritDoc} */
  392.     public CartesianOrbit shiftedBy(final double dt) {
  393.         final PVCoordinates shiftedPV = (getA() < 0) ? shiftPVHyperbolic(dt) : shiftPVElliptic(dt);
  394.         return new CartesianOrbit(shiftedPV, getFrame(), getDate().shiftedBy(dt), getMu());
  395.     }

  397.     /** {@inheritDoc}
  398.      * <p>
  399.      * The interpolated instance is created by polynomial Hermite interpolation
  400.      * ensuring velocity remains the exact derivative of position.
  401.      * </p>
  402.      * <p>
  403.      * As this implementation of interpolation is polynomial, it should be used only
  404.      * with small samples (about 10-20 points) in order to avoid <a
  405.      * href="http://en.wikipedia.org/wiki/Runge%27s_phenomenon">Runge's phenomenon</a>
  406.      * and numerical problems (including NaN appearing).
  407.      * </p>
  408.      * <p>
  409.      * If orbit interpolation on large samples is needed, using the {@link
  410.      * org.orekit.propagation.analytical.Ephemeris} class is a better way than using this
  411.      * low-level interpolation. The Ephemeris class automatically handles selection of
  412.      * a neighboring sub-sample with a predefined number of point from a large global sample
  413.      * in a thread-safe way.
  414.      * </p>
  415.      */
  416.     public CartesianOrbit interpolate(final AbsoluteDate date, final Stream<Orbit> sample) {
  417.         final TimeStampedPVCoordinates interpolated =
  418.                 TimeStampedPVCoordinates.interpolate(date, CartesianDerivativesFilter.USE_PVA,
  419.                                                      sample.map(orbit -> orbit.getPVCoordinates()));
  420.         return new CartesianOrbit(interpolated, getFrame(), date, getMu());
  421.     }

  423.     /** Compute shifted position and velocity in elliptic case.
  424.      * @param dt time shift
  425.      * @return shifted position and velocity
  426.      */
  427.     private PVCoordinates shiftPVElliptic(final double dt) {

  429.         // preliminary computation
  430.         final Vector3D pvP   = getPVCoordinates().getPosition();
  431.         final Vector3D pvV   = getPVCoordinates().getVelocity();
  432.         final double r2      = pvP.getNormSq();
  433.         final double r       = FastMath.sqrt(r2);
  434.         final double rV2OnMu = r * pvV.getNormSq() / getMu();
  435.         final double a       = getA();
  436.         final double eSE     = Vector3D.dotProduct(pvP, pvV) / FastMath.sqrt(getMu() * a);
  437.         final double eCE     = rV2OnMu - 1;
  438.         final double e2      = eCE * eCE + eSE * eSE;

  440.         // we can use any arbitrary reference 2D frame in the orbital plane
  441.         // in order to simplify some equations below, we use the current position as the u axis
  442.         final Vector3D u     = pvP.normalize();
  443.         final Vector3D v     = Vector3D.crossProduct(getPVCoordinates().getMomentum(), u).normalize();

  445.         // the following equations rely on the specific choice of u explained above,
  446.         // some coefficients that vanish to 0 in this case have already been removed here
  447.         final double ex        = (eCE - e2) * a / r;
  448.         final double ey        = -FastMath.sqrt(1 - e2) * eSE * a / r;
  449.         final double beta      = 1 / (1 + FastMath.sqrt(1 - e2));
  450.         final double thetaE0   = FastMath.atan2(ey + eSE * beta * ex, r / a + ex - eSE * beta * ey);
  451.         final SinCos scThetaE0 = FastMath.sinCos(thetaE0);
  452.         final double thetaM0   = thetaE0 - ex * scThetaE0.sin() + ey * scThetaE0.cos();

  454.         // compute in-plane shifted eccentric argument
  455.         final double thetaM1 = thetaM0 + getKeplerianMeanMotion() * dt;
  456.         final double thetaE1 = meanToEccentric(thetaM1, ex, ey);
  457.         final SinCos scTE    = FastMath.sinCos(thetaE1);
  458.         final double cTE     = scTE.cos();
  459.         final double sTE     = scTE.sin();

  461.         // compute shifted in-plane Cartesian coordinates
  462.         final double exey   = ex * ey;
  463.         final double exCeyS = ex * cTE + ey * sTE;
  464.         final double x      = a * ((1 - beta * ey * ey) * cTE + beta * exey * sTE - ex);
  465.         final double y      = a * ((1 - beta * ex * ex) * sTE + beta * exey * cTE - ey);
  466.         final double factor = FastMath.sqrt(getMu() / a) / (1 - exCeyS);
  467.         final double xDot   = factor * (-sTE + beta * ey * exCeyS);
  468.         final double yDot   = factor * ( cTE - beta * ex * exCeyS);

  470.         final Vector3D shiftedP = new Vector3D(x, u, y, v);
  471.         final Vector3D shiftedV = new Vector3D(xDot, u, yDot, v);
  472.         if (hasNonKeplerianAcceleration) {

  474.             // extract non-Keplerian part of the initial acceleration
  475.             final Vector3D nonKeplerianAcceleration = new Vector3D(1, getPVCoordinates().getAcceleration(),
  476.                                                                    getMu() / (r2 * r), pvP);

  478.             // add the quadratic motion due to the non-Keplerian acceleration to the Keplerian motion
  479.             final Vector3D fixedP   = new Vector3D(1, shiftedP,
  480.                                                    0.5 * dt * dt, nonKeplerianAcceleration);
  481.             final double   fixedR2 = fixedP.getNormSq();
  482.             final double   fixedR  = FastMath.sqrt(fixedR2);
  483.             final Vector3D fixedV  = new Vector3D(1, shiftedV,
  484.                                                   dt, nonKeplerianAcceleration);
  485.             final Vector3D fixedA  = new Vector3D(-getMu() / (fixedR2 * fixedR), shiftedP,
  486.                                                   1, nonKeplerianAcceleration);

  488.             return new PVCoordinates(fixedP, fixedV, fixedA);

  490.         } else {
  491.             // don't include acceleration,
  492.             // so the shifted orbit is not considered to have derivatives
  493.             return new PVCoordinates(shiftedP, shiftedV);
  494.         }

  496.     }

  498.     /** Compute shifted position and velocity in hyperbolic case.
  499.      * @param dt time shift
  500.      * @return shifted position and velocity
  501.      */
  502.     private PVCoordinates shiftPVHyperbolic(final double dt) {

  504.         final PVCoordinates pv = getPVCoordinates();
  505.         final Vector3D pvP   = pv.getPosition();
  506.         final Vector3D pvV   = pv.getVelocity();
  507.         final Vector3D pvM   = pv.getMomentum();
  508.         final double r2      = pvP.getNormSq();
  509.         final double r       = FastMath.sqrt(r2);
  510.         final double rV2OnMu = r * pvV.getNormSq() / getMu();
  511.         final double a       = getA();
  512.         final double muA     = getMu() * a;
  513.         final double e       = FastMath.sqrt(1 - Vector3D.dotProduct(pvM, pvM) / muA);
  514.         final double sqrt    = FastMath.sqrt((e + 1) / (e - 1));

  516.         // compute mean anomaly
  517.         final double eSH     = Vector3D.dotProduct(pvP, pvV) / FastMath.sqrt(-muA);
  518.         final double eCH     = rV2OnMu - 1;
  519.         final double H0      = FastMath.log((eCH + eSH) / (eCH - eSH)) / 2;
  520.         final double M0      = e * FastMath.sinh(H0) - H0;

  522.         // find canonical 2D frame with p pointing to perigee
  523.         final double v0      = 2 * FastMath.atan(sqrt * FastMath.tanh(H0 / 2));
  524.         final Vector3D p     = new Rotation(pvM, v0, RotationConvention.FRAME_TRANSFORM).applyTo(pvP).normalize();
  525.         final Vector3D q     = Vector3D.crossProduct(pvM, p).normalize();

  527.         // compute shifted eccentric anomaly
  528.         final double M1      = M0 + getKeplerianMeanMotion() * dt;
  529.         final double H1      = KeplerianAnomalyUtility.hyperbolicMeanToEccentric(e, M1);

  531.         // compute shifted in-plane Cartesian coordinates
  532.         final double cH     = FastMath.cosh(H1);
  533.         final double sH     = FastMath.sinh(H1);
  534.         final double sE2m1  = FastMath.sqrt((e - 1) * (e + 1));

  536.         // coordinates of position and velocity in the orbital plane
  537.         final double x      = a * (cH - e);
  538.         final double y      = -a * sE2m1 * sH;
  539.         final double factor = FastMath.sqrt(getMu() / -a) / (e * cH - 1);
  540.         final double xDot   = -factor * sH;
  541.         final double yDot   =  factor * sE2m1 * cH;

  543.         final Vector3D shiftedP = new Vector3D(x, p, y, q);
  544.         final Vector3D shiftedV = new Vector3D(xDot, p, yDot, q);
  545.         if (hasNonKeplerianAcceleration) {

  547.             // extract non-Keplerian part of the initial acceleration
  548.             final Vector3D nonKeplerianAcceleration = new Vector3D(1, getPVCoordinates().getAcceleration(),
  549.                                                                    getMu() / (r2 * r), pvP);

  551.             // add the quadratic motion due to the non-Keplerian acceleration to the Keplerian motion
  552.             final Vector3D fixedP   = new Vector3D(1, shiftedP,
  553.                                                    0.5 * dt * dt, nonKeplerianAcceleration);
  554.             final double   fixedR2 = fixedP.getNormSq();
  555.             final double   fixedR  = FastMath.sqrt(fixedR2);
  556.             final Vector3D fixedV  = new Vector3D(1, shiftedV,
  557.                                                   dt, nonKeplerianAcceleration);
  558.             final Vector3D fixedA  = new Vector3D(-getMu() / (fixedR2 * fixedR), shiftedP,
  559.                                                   1, nonKeplerianAcceleration);

  561.             return new PVCoordinates(fixedP, fixedV, fixedA);

  563.         } else {
  564.             // don't include acceleration,
  565.             // so the shifted orbit is not considered to have derivatives
  566.             return new PVCoordinates(shiftedP, shiftedV);
  567.         }

  569.     }

  571.     /** Computes the eccentric in-plane argument from the mean in-plane argument.
  572.      * @param thetaM = mean in-plane argument (rad)
  573.      * @param ex first component of eccentricity vector
  574.      * @param ey second component of eccentricity vector
  575.      * @return the eccentric in-plane argument.
  576.      */
  577.     private double meanToEccentric(final double thetaM, final double ex, final double ey) {
  578.         // Generalization of Kepler equation to in-plane parameters
  579.         // with thetaE = eta + E and
  580.         //      thetaM = eta + M = thetaE - ex.sin(thetaE) + ey.cos(thetaE)
  581.         // and eta being counted from an arbitrary reference in the orbital plane
  582.         double thetaE        = thetaM;
  583.         double thetaEMthetaM = 0.0;
  584.         int    iter          = 0;
  585.         do {
  586.             final SinCos scThetaE = FastMath.sinCos(thetaE);

  588.             final double f2 = ex * scThetaE.sin() - ey * scThetaE.cos();
  589.             final double f1 = 1.0 - ex * scThetaE.cos() - ey * scThetaE.sin();
  590.             final double f0 = thetaEMthetaM - f2;

  592.             final double f12 = 2.0 * f1;
  593.             final double shift = f0 * f12 / (f1 * f12 - f0 * f2);

  595.             thetaEMthetaM -= shift;
  596.             thetaE         = thetaM + thetaEMthetaM;

  598.             if (FastMath.abs(shift) <= 1.0e-12) {
  599.                 return thetaE;
  600.             }

  602.         } while (++iter < 50);

  604.         throw new MathIllegalStateException(LocalizedCoreFormats.CONVERGENCE_FAILED);

  606.     }

  608.     /** Create a 6x6 identity matrix.
  609.      * @return 6x6 identity matrix
  610.      */
  611.     private double[][] create6x6Identity() {
  612.         // this is the fastest way to set the 6x6 identity matrix
  613.         final double[][] identity = new double[6][6];
  614.         for (int i = 0; i < 6; i++) {
  615.             identity[i][i] = 1.0;
  616.         }
  617.         return identity;
  618.     }

  620.     @Override
  621.     protected double[][] computeJacobianMeanWrtCartesian() {
  622.         return create6x6Identity();
  623.     }

  625.     @Override
  626.     protected double[][] computeJacobianEccentricWrtCartesian() {
  627.         return create6x6Identity();
  628.     }

  630.     @Override
  631.     protected double[][] computeJacobianTrueWrtCartesian() {
  632.         return create6x6Identity();
  633.     }

  635.     /** {@inheritDoc} */
  636.     public void addKeplerContribution(final PositionAngle type, final double gm,
  637.                                       final double[] pDot) {

  639.         final PVCoordinates pv = getPVCoordinates();

  641.         // position derivative is velocity
  642.         final Vector3D velocity = pv.getVelocity();
  643.         pDot[0] += velocity.getX();
  644.         pDot[1] += velocity.getY();
  645.         pDot[2] += velocity.getZ();

  647.         // velocity derivative is Newtonian acceleration
  648.         final Vector3D position = pv.getPosition();
  649.         final double r2         = position.getNormSq();
  650.         final double coeff      = -gm / (r2 * FastMath.sqrt(r2));
  651.         pDot[3] += coeff * position.getX();
  652.         pDot[4] += coeff * position.getY();
  653.         pDot[5] += coeff * position.getZ();

  655.     }

  657.     /**  Returns a string representation of this Orbit object.
  658.      * @return a string representation of this object
  659.      */
  660.     public String toString() {
  661.         // use only the six defining elements, like the other Orbit.toString() methods
  662.         final String comma = ", ";
  663.         final PVCoordinates pv = getPVCoordinates();
  664.         final Vector3D p = pv.getPosition();
  665.         final Vector3D v = pv.getVelocity();
  666.         return "Cartesian parameters: {P(" +
  667.                 p.getX() + comma +
  668.                 p.getY() + comma +
  669.                 p.getZ() + "), V(" +
  670.                 v.getX() + comma +
  671.                 v.getY() + comma +
  672.                 v.getZ() + ")}";
  673.     }

  675.     /** Replace the instance with a data transfer object for serialization.
  676.      * <p>
  677.      * This intermediate class serializes all needed parameters,
  678.      * including position-velocity which are <em>not</em> serialized by parent
  679.      * {@link Orbit} class.
  680.      * </p>
  681.      * @return data transfer object that will be serialized
  682.      */
  683.     @DefaultDataContext
  684.     private Object writeReplace() {
  685.         return new DTO(this);
  686.     }

  688.     /** Internal class used only for serialization. */
  689.     @DefaultDataContext
  690.     private static class DTO implements Serializable {

  692.         /** Serializable UID. */
  693.         private static final long serialVersionUID = 20170414L;

  695.         /** Double values. */
  696.         private double[] d;

  698.         /** Frame in which are defined the orbital parameters. */
  699.         private final Frame frame;

  701.         /** Simple constructor.
  702.          * @param orbit instance to serialize
  703.          */
  704.         private DTO(final CartesianOrbit orbit) {

  706.             final TimeStampedPVCoordinates pv = orbit.getPVCoordinates();

  708.             // decompose date
  709.             final AbsoluteDate j2000Epoch =
  710.                     DataContext.getDefault().getTimeScales().getJ2000Epoch();
  711.             final double epoch  = FastMath.floor(pv.getDate().durationFrom(j2000Epoch));
  712.             final double offset = pv.getDate().durationFrom(j2000Epoch.shiftedBy(epoch));

  714.             if (orbit.hasDerivatives()) {
  715.                 this.d = new double[] {
  716.                     epoch, offset, orbit.getMu(),
  717.                     pv.getPosition().getX(),     pv.getPosition().getY(),     pv.getPosition().getZ(),
  718.                     pv.getVelocity().getX(),     pv.getVelocity().getY(),     pv.getVelocity().getZ(),
  719.                     pv.getAcceleration().getX(), pv.getAcceleration().getY(), pv.getAcceleration().getZ()
  720.                 };
  721.             } else {
  722.                 this.d = new double[] {
  723.                     epoch, offset, orbit.getMu(),
  724.                     pv.getPosition().getX(),     pv.getPosition().getY(),     pv.getPosition().getZ(),
  725.                     pv.getVelocity().getX(),     pv.getVelocity().getY(),     pv.getVelocity().getZ()
  726.                 };
  727.             }

  729.             this.frame = orbit.getFrame();

  731.         }

  733.         /** Replace the deserialized data transfer object with a {@link CartesianOrbit}.
  734.          * @return replacement {@link CartesianOrbit}
  735.          */
  736.         private Object readResolve() {
  737.             final AbsoluteDate j2000Epoch =
  738.                     DataContext.getDefault().getTimeScales().getJ2000Epoch();
  739.             if (d.length >= 12) {
  740.                 // we have derivatives
  741.                 return new CartesianOrbit(new TimeStampedPVCoordinates(j2000Epoch.shiftedBy(d[0]).shiftedBy(d[1]),
  742.                                                                        new Vector3D(d[3], d[ 4], d[ 5]),
  743.                                                                        new Vector3D(d[6], d[ 7], d[ 8]),
  744.                                                                        new Vector3D(d[9], d[10], d[11])),
  745.                                           frame, d[2]);
  746.             } else {
  747.                 // we don't have derivatives
  748.                 return new CartesianOrbit(new TimeStampedPVCoordinates(j2000Epoch.shiftedBy(d[0]).shiftedBy(d[1]),
  749.                                                                        new Vector3D(d[3], d[ 4], d[ 5]),
  750.                                                                        new Vector3D(d[6], d[ 7], d[ 8])),
  751.                                           frame, d[2]);
  752.             }
  753.         }

  755.     }

  757. }
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