CartesianOrbit.java

  1. /* Copyright 2002-2018 CS Systèmes d'Information
  2.  * Licensed to CS Systèmes d'Information (CS) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * CS licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *   http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.orekit.orbits;

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

  22. import org.hipparchus.analysis.differentiation.DSFactory;
  23. import org.hipparchus.analysis.differentiation.DerivativeStructure;
  24. import org.hipparchus.exception.LocalizedCoreFormats;
  25. import org.hipparchus.exception.MathIllegalStateException;
  26. import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  27. import org.hipparchus.geometry.euclidean.threed.Rotation;
  28. import org.hipparchus.geometry.euclidean.threed.RotationConvention;
  29. import org.hipparchus.geometry.euclidean.threed.Vector3D;
  30. import org.hipparchus.util.FastMath;
  31. import org.orekit.errors.OrekitMessages;
  32. import org.orekit.frames.Frame;
  33. import org.orekit.time.AbsoluteDate;
  34. import org.orekit.utils.CartesianDerivativesFilter;
  35. import org.orekit.utils.PVCoordinates;
  36. import org.orekit.utils.TimeStampedPVCoordinates;


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

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

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

  75.     /** Serializable UID. */
  76.     private static final long serialVersionUID = 20170414L;

  78.     /** Factory for first time derivatives. */
  79.     private static final DSFactory FACTORY = new DSFactory(1, 1);

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

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

  87.     /** Constructor from Cartesian parameters.
  88.      *
  89.      * <p> The acceleration provided in {@code pvCoordinates} is accessible using
  90.      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
  91.      * use {@code mu} and the position to compute the acceleration, including
  92.      * {@link #shiftedBy(double)} and {@link #getPVCoordinates(AbsoluteDate, Frame)}.
  93.      *
  94.      * @param pvaCoordinates the position, velocity and acceleration of the satellite.
  95.      * @param frame the frame in which the {@link PVCoordinates} are defined
  96.      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
  97.      * @param mu central attraction coefficient (m³/s²)
  98.      * @exception IllegalArgumentException if frame is not a {@link
  99.      * Frame#isPseudoInertial pseudo-inertial frame}
  100.      */
  101.     public CartesianOrbit(final TimeStampedPVCoordinates pvaCoordinates,
  102.                           final Frame frame, final double mu)
  103.         throws IllegalArgumentException {
  104.         super(pvaCoordinates, frame, mu);
  105.         hasNonKeplerianAcceleration = hasNonKeplerianAcceleration(pvaCoordinates, mu);
  106.         equinoctial = null;
  107.     }

  109.     /** Constructor from Cartesian parameters.
  110.      *
  111.      * <p> The acceleration provided in {@code pvCoordinates} is accessible using
  112.      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
  113.      * use {@code mu} and the position to compute the acceleration, including
  114.      * {@link #shiftedBy(double)} and {@link #getPVCoordinates(AbsoluteDate, Frame)}.
  115.      *
  116.      * @param pvaCoordinates the position and velocity of the satellite.
  117.      * @param frame the frame in which the {@link PVCoordinates} are defined
  118.      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
  119.      * @param date date of the orbital parameters
  120.      * @param mu central attraction coefficient (m³/s²)
  121.      * @exception IllegalArgumentException if frame is not a {@link
  122.      * Frame#isPseudoInertial pseudo-inertial frame}
  123.      */
  124.     public CartesianOrbit(final PVCoordinates pvaCoordinates, final Frame frame,
  125.                           final AbsoluteDate date, final double mu)
  126.         throws IllegalArgumentException {
  127.         this(new TimeStampedPVCoordinates(date, pvaCoordinates), frame, mu);
  128.     }

  130.     /** Constructor from any kind of orbital parameters.
  131.      * @param op orbital parameters to copy
  132.      */
  133.     public CartesianOrbit(final Orbit op) {
  134.         super(op.getPVCoordinates(), op.getFrame(), op.getMu());
  135.         hasNonKeplerianAcceleration = op.hasDerivatives();
  136.         if (op instanceof EquinoctialOrbit) {
  137.             equinoctial = (EquinoctialOrbit) op;
  138.         } else if (op instanceof CartesianOrbit) {
  139.             equinoctial = ((CartesianOrbit) op).equinoctial;
  140.         } else {
  141.             equinoctial = null;
  142.         }
  143.     }

  145.     /** {@inheritDoc} */
  146.     public OrbitType getType() {
  147.         return OrbitType.CARTESIAN;
  148.     }

  150.     /** Lazy evaluation of equinoctial parameters. */
  151.     private void initEquinoctial() {
  152.         if (equinoctial == null) {
  153.             if (hasDerivatives()) {
  154.                 // getPVCoordinates includes accelerations that will be interpreted as derivatives
  155.                 equinoctial = new EquinoctialOrbit(getPVCoordinates(), getFrame(), getDate(), getMu());
  156.             } else {
  157.                 // get rid of Keplerian acceleration so we don't assume
  158.                 // we have derivatives when in fact we don't have them
  159.                 equinoctial = new EquinoctialOrbit(new PVCoordinates(getPVCoordinates().getPosition(),
  160.                                                                      getPVCoordinates().getVelocity()),
  161.                                                    getFrame(), getDate(), getMu());
  162.             }
  163.         }
  164.     }

  166.     /** Get position with derivatives.
  167.      * @return position with derivatives
  168.      */
  169.     private FieldVector3D<DerivativeStructure> getPositionDS() {
  170.         final Vector3D p = getPVCoordinates().getPosition();
  171.         final Vector3D v = getPVCoordinates().getVelocity();
  172.         return new FieldVector3D<>(FACTORY.build(p.getX(), v.getX()),
  173.                                    FACTORY.build(p.getY(), v.getY()),
  174.                                    FACTORY.build(p.getZ(), v.getZ()));
  175.     }

  177.     /** Get velocity with derivatives.
  178.      * @return velocity with derivatives
  179.      */
  180.     private FieldVector3D<DerivativeStructure> getVelocityDS() {
  181.         final Vector3D v = getPVCoordinates().getVelocity();
  182.         final Vector3D a = getPVCoordinates().getAcceleration();
  183.         return new FieldVector3D<>(FACTORY.build(v.getX(), a.getX()),
  184.                                    FACTORY.build(v.getY(), a.getY()),
  185.                                    FACTORY.build(v.getZ(), a.getZ()));
  186.     }

  188.     /** {@inheritDoc} */
  189.     public double getA() {
  190.         final double r  = getPVCoordinates().getPosition().getNorm();
  191.         final double V2 = getPVCoordinates().getVelocity().getNormSq();
  192.         return r / (2 - r * V2 / getMu());
  193.     }

  195.     /** {@inheritDoc} */
  196.     public double getADot() {
  197.         if (hasDerivatives()) {
  198.             final DerivativeStructure r  = getPositionDS().getNorm();
  199.             final DerivativeStructure V2 = getVelocityDS().getNormSq();
  200.             final DerivativeStructure a  = r.divide(r.multiply(V2).divide(getMu()).subtract(2).negate());
  201.             return a.getPartialDerivative(1);
  202.         } else {
  203.             return Double.NaN;
  204.         }
  205.     }

  207.     /** {@inheritDoc} */
  208.     public double getE() {
  209.         final double muA = getMu() * getA();
  210.         if (muA > 0) {
  211.             // elliptic or circular orbit
  212.             final Vector3D pvP   = getPVCoordinates().getPosition();
  213.             final Vector3D pvV   = getPVCoordinates().getVelocity();
  214.             final double rV2OnMu = pvP.getNorm() * pvV.getNormSq() / getMu();
  215.             final double eSE     = Vector3D.dotProduct(pvP, pvV) / FastMath.sqrt(muA);
  216.             final double eCE     = rV2OnMu - 1;
  217.             return FastMath.sqrt(eCE * eCE + eSE * eSE);
  218.         } else {
  219.             // hyperbolic orbit
  220.             final Vector3D pvM = getPVCoordinates().getMomentum();
  221.             return FastMath.sqrt(1 - pvM.getNormSq() / muA);
  222.         }
  223.     }

  225.     /** {@inheritDoc} */
  226.     public double getEDot() {
  227.         if (hasDerivatives()) {
  228.             final FieldVector3D<DerivativeStructure> pvP   = getPositionDS();
  229.             final FieldVector3D<DerivativeStructure> pvV   = getVelocityDS();
  230.             final DerivativeStructure r       = getPositionDS().getNorm();
  231.             final DerivativeStructure V2      = getVelocityDS().getNormSq();
  232.             final DerivativeStructure rV2OnMu = r.multiply(V2).divide(getMu());
  233.             final DerivativeStructure a       = r.divide(rV2OnMu.negate().add(2));
  234.             final DerivativeStructure eSE     = FieldVector3D.dotProduct(pvP, pvV).divide(a.multiply(getMu()).sqrt());
  235.             final DerivativeStructure eCE     = rV2OnMu.subtract(1);
  236.             final DerivativeStructure e       = eCE.multiply(eCE).add(eSE.multiply(eSE)).sqrt();
  237.             return e.getPartialDerivative(1);
  238.         } else {
  239.             return Double.NaN;
  240.         }
  241.     }

  243.     /** {@inheritDoc} */
  244.     public double getI() {
  245.         return Vector3D.angle(Vector3D.PLUS_K, getPVCoordinates().getMomentum());
  246.     }

  248.     /** {@inheritDoc} */
  249.     public double getIDot() {
  250.         if (hasDerivatives()) {
  251.             final FieldVector3D<DerivativeStructure> momentum =
  252.                             FieldVector3D.crossProduct(getPositionDS(), getVelocityDS());
  253.             final DerivativeStructure i = FieldVector3D.angle(Vector3D.PLUS_K, momentum);
  254.             return i.getPartialDerivative(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 FieldVector3D<DerivativeStructure> w =
  298.                             FieldVector3D.crossProduct(getPositionDS(), getVelocityDS()).normalize();
  299.             // Check for equatorial retrograde orbit
  300.             final double x = w.getX().getValue();
  301.             final double y = w.getY().getValue();
  302.             final double z = w.getZ().getValue();
  303.             if (((x * x + y * y) == 0) && z < 0) {
  304.                 return Double.NaN;
  305.             }
  306.             final DerivativeStructure hx = w.getY().negate().divide(w.getZ().add(1));
  307.             return hx.getPartialDerivative(1);
  308.         } else {
  309.             return Double.NaN;
  310.         }
  311.     }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  451.         // compute in-plane shifted eccentric argument
  452.         final double thetaM1 = thetaM0 + getKeplerianMeanMotion() * dt;
  453.         final double thetaE1 = meanToEccentric(thetaM1, ex, ey);
  454.         final double cTE     = FastMath.cos(thetaE1);
  455.         final double sTE     = FastMath.sin(thetaE1);

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

  466.         final Vector3D shiftedP = new Vector3D(x, u, y, v);
  467.         final Vector3D shiftedV = new Vector3D(xDot, u, yDot, v);
  468.         if (hasNonKeplerianAcceleration) {

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

  474.             // add the quadratic motion due to the non-Keplerian acceleration to the Keplerian motion
  475.             final Vector3D fixedP   = new Vector3D(1, shiftedP,
  476.                                                    0.5 * dt * dt, nonKeplerianAcceleration);
  477.             final double   fixedR2 = fixedP.getNormSq();
  478.             final double   fixedR  = FastMath.sqrt(fixedR2);
  479.             final Vector3D fixedV  = new Vector3D(1, shiftedV,
  480.                                                   dt, nonKeplerianAcceleration);
  481.             final Vector3D fixedA  = new Vector3D(-getMu() / (fixedR2 * fixedR), shiftedP,
  482.                                                   1, nonKeplerianAcceleration);

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

  486.         } else {
  487.             // don't include acceleration,
  488.             // so the shifted orbit is not considered to have derivatives
  489.             return new PVCoordinates(shiftedP, shiftedV);
  490.         }

  492.     }

  494.     /** Compute shifted position and velocity in hyperbolic case.
  495.      * @param dt time shift
  496.      * @return shifted position and velocity
  497.      */
  498.     private PVCoordinates shiftPVHyperbolic(final double dt) {

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

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

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

  523.         // compute shifted eccentric anomaly
  524.         final double M1      = M0 + getKeplerianMeanMotion() * dt;
  525.         final double H1      = meanToHyperbolicEccentric(M1, e);

  527.         // compute shifted in-plane Cartesian coordinates
  528.         final double cH     = FastMath.cosh(H1);
  529.         final double sH     = FastMath.sinh(H1);
  530.         final double sE2m1  = FastMath.sqrt((e - 1) * (e + 1));

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

  539.         final Vector3D shiftedP = new Vector3D(x, p, y, q);
  540.         final Vector3D shiftedV = new Vector3D(xDot, p, yDot, q);
  541.         if (hasNonKeplerianAcceleration) {

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

  547.             // add the quadratic motion due to the non-Keplerian acceleration to the Keplerian motion
  548.             final Vector3D fixedP   = new Vector3D(1, shiftedP,
  549.                                                    0.5 * dt * dt, nonKeplerianAcceleration);
  550.             final double   fixedR2 = fixedP.getNormSq();
  551.             final double   fixedR  = FastMath.sqrt(fixedR2);
  552.             final Vector3D fixedV  = new Vector3D(1, shiftedV,
  553.                                                   dt, nonKeplerianAcceleration);
  554.             final Vector3D fixedA  = new Vector3D(-getMu() / (fixedR2 * fixedR), shiftedP,
  555.                                                   1, nonKeplerianAcceleration);

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

  559.         } else {
  560.             // don't include acceleration,
  561.             // so the shifted orbit is not considered to have derivatives
  562.             return new PVCoordinates(shiftedP, shiftedV);
  563.         }

  565.     }

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

  585.             final double f2 = ex * sinThetaE - ey * cosThetaE;
  586.             final double f1 = 1.0 - ex * cosThetaE - ey * sinThetaE;
  587.             final double f0 = thetaEMthetaM - f2;

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

  592.             thetaEMthetaM -= shift;
  593.             thetaE         = thetaM + thetaEMthetaM;

  595.             if (FastMath.abs(shift) <= 1.0e-12) {
  596.                 return thetaE;
  597.             }

  599.         } while (++iter < 50);

  601.         throw new MathIllegalStateException(LocalizedCoreFormats.CONVERGENCE_FAILED);

  603.     }

  605.     /** Computes the hyperbolic eccentric anomaly from the mean anomaly.
  606.      * <p>
  607.      * The algorithm used here for solving hyperbolic Kepler equation is
  608.      * Danby's iterative method (3rd order) with Vallado's initial guess.
  609.      * </p>
  610.      * @param M mean anomaly (rad)
  611.      * @param ecc eccentricity
  612.      * @return the hyperbolic eccentric anomaly
  613.      */
  614.     private double meanToHyperbolicEccentric(final double M, final double ecc) {

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

  618.         // Initial guess
  619.         double H;
  620.         if (ecc < 1.6) {
  621.             if ((-FastMath.PI < M && M < 0.) || M > FastMath.PI) {
  622.                 H = M - ecc;
  623.             } else {
  624.                 H = M + ecc;
  625.             }
  626.         } else {
  627.             if (ecc < 3.6 && FastMath.abs(M) > FastMath.PI) {
  628.                 H = M - FastMath.copySign(ecc, M);
  629.             } else {
  630.                 H = M / (ecc - 1.);
  631.             }
  632.         }

  634.         // Iterative computation
  635.         int iter = 0;
  636.         do {
  637.             final double f3  = ecc * FastMath.cosh(H);
  638.             final double f2  = ecc * FastMath.sinh(H);
  639.             final double f1  = f3 - 1.;
  640.             final double f0  = f2 - H - M;
  641.             final double f12 = 2. * f1;
  642.             final double d   = f0 / f12;
  643.             final double fdf = f1 - d * f2;
  644.             final double ds  = f0 / fdf;

  646.             final double shift = f0 / (fdf + ds * ds * f3 / 6.);

  648.             H -= shift;

  650.             if (FastMath.abs(shift) <= 1.0e-12) {
  651.                 return H;
  652.             }

  654.         } while (++iter < 50);

  656.         throw new MathIllegalStateException(OrekitMessages.UNABLE_TO_COMPUTE_HYPERBOLIC_ECCENTRIC_ANOMALY,
  657.                                             iter);
  658.     }

  660.     /** Create a 6x6 identity matrix.
  661.      * @return 6x6 identity matrix
  662.      */
  663.     private double[][] create6x6Identity() {
  664.         // this is the fastest way to set the 6x6 identity matrix
  665.         final double[][] identity = new double[6][6];
  666.         for (int i = 0; i < 6; i++) {
  667.             identity[i][i] = 1.0;
  668.         }
  669.         return identity;
  670.     }

  672.     @Override
  673.     protected double[][] computeJacobianMeanWrtCartesian() {
  674.         return create6x6Identity();
  675.     }

  677.     @Override
  678.     protected double[][] computeJacobianEccentricWrtCartesian() {
  679.         return create6x6Identity();
  680.     }

  682.     @Override
  683.     protected double[][] computeJacobianTrueWrtCartesian() {
  684.         return create6x6Identity();
  685.     }

  687.     /** {@inheritDoc} */
  688.     public void addKeplerContribution(final PositionAngle type, final double gm,
  689.                                       final double[] pDot) {

  691.         final PVCoordinates pv = getPVCoordinates();

  693.         // position derivative is velocity
  694.         final Vector3D velocity = pv.getVelocity();
  695.         pDot[0] += velocity.getX();
  696.         pDot[1] += velocity.getY();
  697.         pDot[2] += velocity.getZ();

  699.         // velocity derivative is Newtonian acceleration
  700.         final Vector3D position = pv.getPosition();
  701.         final double r2         = position.getNormSq();
  702.         final double coeff      = -gm / (r2 * FastMath.sqrt(r2));
  703.         pDot[3] += coeff * position.getX();
  704.         pDot[4] += coeff * position.getY();
  705.         pDot[5] += coeff * position.getZ();

  707.     }

  709.     /**  Returns a string representation of this Orbit object.
  710.      * @return a string representation of this object
  711.      */
  712.     public String toString() {
  713.         return "Cartesian parameters: " + getPVCoordinates().toString();
  714.     }

  716.     /** Replace the instance with a data transfer object for serialization.
  717.      * <p>
  718.      * This intermediate class serializes all needed parameters,
  719.      * including position-velocity which are <em>not</em> serialized by parent
  720.      * {@link Orbit} class.
  721.      * </p>
  722.      * @return data transfer object that will be serialized
  723.      */
  724.     private Object writeReplace() {
  725.         return new DTO(this);
  726.     }

  728.     /** Internal class used only for serialization. */
  729.     private static class DTO implements Serializable {

  731.         /** Serializable UID. */
  732.         private static final long serialVersionUID = 20170414L;

  734.         /** Double values. */
  735.         private double[] d;

  737.         /** Frame in which are defined the orbital parameters. */
  738.         private final Frame frame;

  740.         /** Simple constructor.
  741.          * @param orbit instance to serialize
  742.          */
  743.         private DTO(final CartesianOrbit orbit) {

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

  747.             // decompose date
  748.             final double epoch  = FastMath.floor(pv.getDate().durationFrom(AbsoluteDate.J2000_EPOCH));
  749.             final double offset = pv.getDate().durationFrom(AbsoluteDate.J2000_EPOCH.shiftedBy(epoch));

  751.             if (orbit.hasDerivatives()) {
  752.                 this.d = new double[] {
  753.                     epoch, offset, orbit.getMu(),
  754.                     pv.getPosition().getX(),     pv.getPosition().getY(),     pv.getPosition().getZ(),
  755.                     pv.getVelocity().getX(),     pv.getVelocity().getY(),     pv.getVelocity().getZ(),
  756.                     pv.getAcceleration().getX(), pv.getAcceleration().getY(), pv.getAcceleration().getZ()
  757.                 };
  758.             } else {
  759.                 this.d = new double[] {
  760.                     epoch, offset, orbit.getMu(),
  761.                     pv.getPosition().getX(),     pv.getPosition().getY(),     pv.getPosition().getZ(),
  762.                     pv.getVelocity().getX(),     pv.getVelocity().getY(),     pv.getVelocity().getZ()
  763.                 };
  764.             }

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

  768.         }

  770.         /** Replace the deserialized data transfer object with a {@link CartesianOrbit}.
  771.          * @return replacement {@link CartesianOrbit}
  772.          */
  773.         private Object readResolve() {
  774.             if (d.length >= 12) {
  775.                 // we have derivatives
  776.                 return new CartesianOrbit(new TimeStampedPVCoordinates(AbsoluteDate.J2000_EPOCH.shiftedBy(d[0]).shiftedBy(d[1]),
  777.                                                                        new Vector3D(d[3], d[ 4], d[ 5]),
  778.                                                                        new Vector3D(d[6], d[ 7], d[ 8]),
  779.                                                                        new Vector3D(d[9], d[10], d[11])),
  780.                                           frame, d[2]);
  781.             } else {
  782.                 // we don't have derivatives
  783.                 return new CartesianOrbit(new TimeStampedPVCoordinates(AbsoluteDate.J2000_EPOCH.shiftedBy(d[0]).shiftedBy(d[1]),
  784.                                                                        new Vector3D(d[3], d[ 4], d[ 5]),
  785.                                                                        new Vector3D(d[6], d[ 7], d[ 8])),
  786.                                           frame, d[2]);
  787.             }
  788.         }

  790.     }

  792. }
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