HolmesFeatherstoneAttractionModel.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.forces.gravity;


  20. import java.util.stream.Stream;

  22. import org.hipparchus.Field;
  23. import org.hipparchus.RealFieldElement;
  24. import org.hipparchus.analysis.differentiation.DerivativeStructure;
  25. import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  26. import org.hipparchus.geometry.euclidean.threed.SphericalCoordinates;
  27. import org.hipparchus.geometry.euclidean.threed.Vector3D;
  28. import org.hipparchus.linear.Array2DRowRealMatrix;
  29. import org.hipparchus.linear.RealMatrix;
  30. import org.hipparchus.util.FastMath;
  31. import org.hipparchus.util.MathArrays;
  32. import org.orekit.errors.OrekitException;
  33. import org.orekit.errors.OrekitInternalError;
  34. import org.orekit.forces.AbstractForceModel;
  35. import org.orekit.forces.gravity.potential.NormalizedSphericalHarmonicsProvider;
  36. import org.orekit.forces.gravity.potential.NormalizedSphericalHarmonicsProvider.NormalizedSphericalHarmonics;
  37. import org.orekit.forces.gravity.potential.TideSystem;
  38. import org.orekit.forces.gravity.potential.TideSystemProvider;
  39. import org.orekit.frames.Frame;
  40. import org.orekit.frames.Transform;
  41. import org.orekit.propagation.FieldSpacecraftState;
  42. import org.orekit.propagation.SpacecraftState;
  43. import org.orekit.propagation.events.EventDetector;
  44. import org.orekit.propagation.events.FieldEventDetector;
  45. import org.orekit.time.AbsoluteDate;
  46. import org.orekit.time.FieldAbsoluteDate;
  47. import org.orekit.utils.FieldPVCoordinates;
  48. import org.orekit.utils.ParameterDriver;

  50. /** This class represents the gravitational field of a celestial body.
  51.  * <p>
  52.  * The algorithm implemented in this class has been designed by S. A. Holmes
  53.  * and W. E. Featherstone from Department of Spatial Sciences, Curtin University
  54.  * of Technology, Perth, Australia. It is described in their 2002 paper: <a
  55.  * href="http://cct.gfy.ku.dk/publ_others/ccta1870.pdf">A unified approach to
  56.  * the Clenshaw summation and the recursive computation of very high degree and
  57.  * order normalised associated Legendre functions</a> (Journal of Geodesy (2002)
  58.  * 76: 279–299).
  59.  * </p>
  60.  * <p>
  61.  * This model directly uses normalized coefficients and stable recursion algorithms
  62.  * so it is more suited to high degree gravity fields than the classical Cunningham
  63.  * Droziner models which use un-normalized coefficients.
  64.  * </p>
  65.  * <p>
  66.  * Among the different algorithms presented in Holmes and Featherstone paper, this
  67.  * class implements the <em>modified forward row method</em>. All recursion coefficients
  68.  * are precomputed and stored for greater performance. This caching was suggested in the
  69.  * paper but not used due to the large memory requirements. Since 2002, even low end
  70.  * computers and mobile devices do have sufficient memory so this caching has become
  71.  * feasible nowadays.
  72.  * <p>
  73.  * @author Luc Maisonobe
  74.  * @since 6.0
  75.  */

  77. public class HolmesFeatherstoneAttractionModel extends AbstractForceModel implements TideSystemProvider {

  79.     /** Exponent scaling to avoid floating point overflow.
  80.      * <p>The paper uses 10^280, we prefer a power of two to preserve accuracy thanks to
  81.      * {@link FastMath#scalb(double, int)}, so we use 2^930 which has the same order of magnitude.
  82.      */
  83.     private static final int SCALING = 930;

  85.     /** Central attraction scaling factor.
  86.      * <p>
  87.      * We use a power of 2 to avoid numeric noise introduction
  88.      * in the multiplications/divisions sequences.
  89.      * </p>
  90.      */
  91.     private static final double MU_SCALE = FastMath.scalb(1.0, 32);

  93.     /** Driver for gravitational parameter. */
  94.     private final ParameterDriver gmParameterDriver;

  96.     /** Provider for the spherical harmonics. */
  97.     private final NormalizedSphericalHarmonicsProvider provider;

  99.     /** Rotating body. */
  100.     private final Frame bodyFrame;

  102.     /** Recursion coefficients g<sub>n,m</sub>/√j. */
  103.     private final double[] gnmOj;

  105.     /** Recursion coefficients h<sub>n,m</sub>/√j. */
  106.     private final double[] hnmOj;

  108.     /** Recursion coefficients e<sub>n,m</sub>. */
  109.     private final double[] enm;

  111.     /** Scaled sectorial Pbar<sub>m,m</sub>/u<sup>m</sup> &times; 2<sup>-SCALING</sup>. */
  112.     private final double[] sectorial;

  114.     /** Creates a new instance.
  115.      * @param centralBodyFrame rotating body frame
  116.      * @param provider provider for spherical harmonics
  117.      * @since 6.0
  118.      */
  119.     public HolmesFeatherstoneAttractionModel(final Frame centralBodyFrame,
  120.                                              final NormalizedSphericalHarmonicsProvider provider) {

  122.         try {
  123.             gmParameterDriver = new ParameterDriver(NewtonianAttraction.CENTRAL_ATTRACTION_COEFFICIENT,
  124.                                                     provider.getMu(), MU_SCALE, 0.0, Double.POSITIVE_INFINITY);
  125.         } catch (OrekitException oe) {
  126.             // this should never occur as valueChanged above never throws an exception
  127.             throw new OrekitInternalError(oe);
  128.         }

  130.         this.provider  = provider;
  131.         this.bodyFrame = centralBodyFrame;

  133.         // the pre-computed arrays hold coefficients from triangular arrays in a single
  134.         // storing neither diagonal elements (n = m) nor the non-diagonal element n=1, m=0
  135.         final int degree = provider.getMaxDegree();
  136.         final int size = FastMath.max(0, degree * (degree + 1) / 2 - 1);
  137.         gnmOj = new double[size];
  138.         hnmOj = new double[size];
  139.         enm   = new double[size];

  141.         // pre-compute the recursion coefficients corresponding to equations 19 and 22
  142.         // from Holmes and Featherstone paper
  143.         // for cache efficiency, elements are stored in the same order they will be used
  144.         // later on, i.e. from rightmost column to leftmost column
  145.         int index = 0;
  146.         for (int m = degree; m >= 0; --m) {
  147.             final int j = (m == 0) ? 2 : 1;
  148.             for (int n = FastMath.max(2, m + 1); n <= degree; ++n) {
  149.                 final double f = (n - m) * (n + m + 1);
  150.                 gnmOj[index] = 2 * (m + 1) / FastMath.sqrt(j * f);
  151.                 hnmOj[index] = FastMath.sqrt((n + m + 2) * (n - m - 1) / (j * f));
  152.                 enm[index]   = FastMath.sqrt(f / j);
  153.                 ++index;
  154.             }
  155.         }

  157.         // scaled sectorial terms corresponding to equation 28 in Holmes and Featherstone paper
  158.         sectorial    = new double[degree + 1];
  159.         sectorial[0] = FastMath.scalb(1.0, -SCALING);
  160.         sectorial[1] = FastMath.sqrt(3) * sectorial[0];
  161.         for (int m = 2; m < sectorial.length; ++m) {
  162.             sectorial[m] = FastMath.sqrt((2 * m + 1) / (2.0 * m)) * sectorial[m - 1];
  163.         }

  165.     }

  167.     /** {@inheritDoc} */
  168.     @Override
  169.     public boolean dependsOnPositionOnly() {
  170.         return true;
  171.     }

  173.     /** {@inheritDoc} */
  174.     public TideSystem getTideSystem() {
  175.         return provider.getTideSystem();
  176.     }

  178.     /** Get the central attraction coefficient μ.
  179.      * @return mu central attraction coefficient (m³/s²)
  180.      */
  181.     public double getMu() {
  182.         return gmParameterDriver.getValue();
  183.     }

  185.     /** Compute the value of the gravity field.
  186.      * @param date current date
  187.      * @param position position at which gravity field is desired in body frame
  188.      * @param mu central attraction coefficient to use
  189.      * @return value of the gravity field (central and non-central parts summed together)
  190.      * @exception OrekitException if position cannot be converted to central body frame
  191.      */
  192.     public double value(final AbsoluteDate date, final Vector3D position,
  193.                         final double mu)
  194.         throws OrekitException {
  195.         return mu / position.getNorm() + nonCentralPart(date, position, mu);
  196.     }

  198.     /** Compute the non-central part of the gravity field.
  199.      * @param date current date
  200.      * @param position position at which gravity field is desired in body frame
  201.      * @param mu central attraction coefficient to use
  202.      * @return value of the non-central part of the gravity field
  203.      * @exception OrekitException if position cannot be converted to central body frame
  204.      */
  205.     public double nonCentralPart(final AbsoluteDate date, final Vector3D position, final double mu)
  206.         throws OrekitException {

  208.         final int degree = provider.getMaxDegree();
  209.         final int order  = provider.getMaxOrder();
  210.         final NormalizedSphericalHarmonics harmonics = provider.onDate(date);

  212.         // allocate the columns for recursion
  213.         double[] pnm0Plus2 = new double[degree + 1];
  214.         double[] pnm0Plus1 = new double[degree + 1];
  215.         double[] pnm0      = new double[degree + 1];

  217.         // compute polar coordinates
  218.         final double x   = position.getX();
  219.         final double y   = position.getY();
  220.         final double z   = position.getZ();
  221.         final double x2  = x * x;
  222.         final double y2  = y * y;
  223.         final double z2  = z * z;
  224.         final double r2  = x2 + y2 + z2;
  225.         final double r   = FastMath.sqrt (r2);
  226.         final double rho = FastMath.sqrt(x2 + y2);
  227.         final double t   = z / r;   // cos(theta), where theta is the polar angle
  228.         final double u   = rho / r; // sin(theta), where theta is the polar angle
  229.         final double tOu = z / rho;

  231.         // compute distance powers
  232.         final double[] aOrN = createDistancePowersArray(provider.getAe() / r);

  234.         // compute longitude cosines/sines
  235.         final double[][] cosSinLambda = createCosSinArrays(position.getX() / rho, position.getY() / rho);

  237.         // outer summation over order
  238.         int    index = 0;
  239.         double value = 0;
  240.         for (int m = degree; m >= 0; --m) {

  242.             // compute tesseral terms without derivatives
  243.             index = computeTesseral(m, degree, index, t, u, tOu,
  244.                                     pnm0Plus2, pnm0Plus1, null, pnm0, null, null);

  246.             if (m <= order) {
  247.                 // compute contribution of current order to field (equation 5 of the paper)

  249.                 // inner summation over degree, for fixed order
  250.                 double sumDegreeS        = 0;
  251.                 double sumDegreeC        = 0;
  252.                 for (int n = FastMath.max(2, m); n <= degree; ++n) {
  253.                     sumDegreeS += pnm0[n] * aOrN[n] * harmonics.getNormalizedSnm(n, m);
  254.                     sumDegreeC += pnm0[n] * aOrN[n] * harmonics.getNormalizedCnm(n, m);
  255.                 }

  257.                 // contribution to outer summation over order
  258.                 value = value * u + cosSinLambda[1][m] * sumDegreeS + cosSinLambda[0][m] * sumDegreeC;

  260.             }

  262.             // rotate the recursion arrays
  263.             final double[] tmp = pnm0Plus2;
  264.             pnm0Plus2 = pnm0Plus1;
  265.             pnm0Plus1 = pnm0;
  266.             pnm0      = tmp;

  268.         }

  270.         // scale back
  271.         value = FastMath.scalb(value, SCALING);

  273.         // apply the global mu/r factor
  274.         return mu * value / r;

  276.     }

  278.     /** Compute the gradient of the non-central part of the gravity field.
  279.      * @param date current date
  280.      * @param position position at which gravity field is desired in body frame
  281.      * @param mu central attraction coefficient to use
  282.      * @return gradient of the non-central part of the gravity field
  283.      * @exception OrekitException if position cannot be converted to central body frame
  284.      */
  285.     public double[] gradient(final AbsoluteDate date, final Vector3D position, final double mu)
  286.         throws OrekitException {

  288.         final int degree = provider.getMaxDegree();
  289.         final int order  = provider.getMaxOrder();
  290.         final NormalizedSphericalHarmonics harmonics = provider.onDate(date);

  292.         // allocate the columns for recursion
  293.         double[] pnm0Plus2  = new double[degree + 1];
  294.         double[] pnm0Plus1  = new double[degree + 1];
  295.         double[] pnm0       = new double[degree + 1];
  296.         final double[] pnm1 = new double[degree + 1];

  298.         // compute polar coordinates
  299.         final double x    = position.getX();
  300.         final double y    = position.getY();
  301.         final double z    = position.getZ();
  302.         final double x2   = x * x;
  303.         final double y2   = y * y;
  304.         final double z2   = z * z;
  305.         final double r2   = x2 + y2 + z2;
  306.         final double r    = FastMath.sqrt (r2);
  307.         final double rho2 = x2 + y2;
  308.         final double rho  = FastMath.sqrt(rho2);
  309.         final double t    = z / r;   // cos(theta), where theta is the polar angle
  310.         final double u    = rho / r; // sin(theta), where theta is the polar angle
  311.         final double tOu  = z / rho;

  313.         // compute distance powers
  314.         final double[] aOrN = createDistancePowersArray(provider.getAe() / r);

  316.         // compute longitude cosines/sines
  317.         final double[][] cosSinLambda = createCosSinArrays(position.getX() / rho, position.getY() / rho);

  319.         // outer summation over order
  320.         int    index = 0;
  321.         double value = 0;
  322.         final double[] gradient = new double[3];
  323.         for (int m = degree; m >= 0; --m) {

  325.             // compute tesseral terms with derivatives
  326.             index = computeTesseral(m, degree, index, t, u, tOu,
  327.                                     pnm0Plus2, pnm0Plus1, null, pnm0, pnm1, null);

  329.             if (m <= order) {
  330.                 // compute contribution of current order to field (equation 5 of the paper)

  332.                 // inner summation over degree, for fixed order
  333.                 double sumDegreeS        = 0;
  334.                 double sumDegreeC        = 0;
  335.                 double dSumDegreeSdR     = 0;
  336.                 double dSumDegreeCdR     = 0;
  337.                 double dSumDegreeSdTheta = 0;
  338.                 double dSumDegreeCdTheta = 0;
  339.                 for (int n = FastMath.max(2, m); n <= degree; ++n) {
  340.                     final double qSnm  = aOrN[n] * harmonics.getNormalizedSnm(n, m);
  341.                     final double qCnm  = aOrN[n] * harmonics.getNormalizedCnm(n, m);
  342.                     final double nOr   = n / r;
  343.                     final double s0    = pnm0[n] * qSnm;
  344.                     final double c0    = pnm0[n] * qCnm;
  345.                     final double s1    = pnm1[n] * qSnm;
  346.                     final double c1    = pnm1[n] * qCnm;
  347.                     sumDegreeS        += s0;
  348.                     sumDegreeC        += c0;
  349.                     dSumDegreeSdR     -= nOr * s0;
  350.                     dSumDegreeCdR     -= nOr * c0;
  351.                     dSumDegreeSdTheta += s1;
  352.                     dSumDegreeCdTheta += c1;
  353.                 }

  355.                 // contribution to outer summation over order
  356.                 // beware that we need to order gradient using the mathematical conventions
  357.                 // compliant with the SphericalCoordinates class, so our lambda is its theta
  358.                 // (and hence at index 1) and our theta is its phi (and hence at index 2)
  359.                 final double sML = cosSinLambda[1][m];
  360.                 final double cML = cosSinLambda[0][m];
  361.                 value            = value       * u + sML * sumDegreeS        + cML * sumDegreeC;
  362.                 gradient[0]      = gradient[0] * u + sML * dSumDegreeSdR     + cML * dSumDegreeCdR;
  363.                 gradient[1]      = gradient[1] * u + m * (cML * sumDegreeS - sML * sumDegreeC);
  364.                 gradient[2]      = gradient[2] * u + sML * dSumDegreeSdTheta + cML * dSumDegreeCdTheta;

  366.             }

  368.             // rotate the recursion arrays
  369.             final double[] tmp = pnm0Plus2;
  370.             pnm0Plus2 = pnm0Plus1;
  371.             pnm0Plus1 = pnm0;
  372.             pnm0      = tmp;

  374.         }

  376.         // scale back
  377.         value       = FastMath.scalb(value,       SCALING);
  378.         gradient[0] = FastMath.scalb(gradient[0], SCALING);
  379.         gradient[1] = FastMath.scalb(gradient[1], SCALING);
  380.         gradient[2] = FastMath.scalb(gradient[2], SCALING);

  382.         // apply the global mu/r factor
  383.         final double muOr = mu / r;
  384.         value            *= muOr;
  385.         gradient[0]       = muOr * gradient[0] - value / r;
  386.         gradient[1]      *= muOr;
  387.         gradient[2]      *= muOr;

  389.         // convert gradient from spherical to Cartesian
  390.         return new SphericalCoordinates(position).toCartesianGradient(gradient);

  392.     }

  394.     /** Compute the gradient of the non-central part of the gravity field.
  395.      * @param date current date
  396.      * @param position position at which gravity field is desired in body frame
  397.      * @param mu central attraction coefficient to use
  398.      * @param <T> type of field used
  399.      * @return gradient of the non-central part of the gravity field
  400.      * @exception OrekitException if position cannot be converted to central body frame
  401.      */
  402.     public <T extends RealFieldElement<T>> T[] gradient(final FieldAbsoluteDate<T> date, final FieldVector3D<T> position,
  403.                                                         final T mu)
  404.         throws OrekitException {

  406.         final int degree = provider.getMaxDegree();
  407.         final int order  = provider.getMaxOrder();
  408.         final NormalizedSphericalHarmonics harmonics = provider.onDate(date.toAbsoluteDate());
  409.         final T zero = date.getField().getZero();
  410.         // allocate the columns for recursion
  411.         T[] pnm0Plus2  = MathArrays.buildArray(date.getField(), degree + 1);
  412.         T[] pnm0Plus1  = MathArrays.buildArray(date.getField(), degree + 1);
  413.         T[] pnm0       = MathArrays.buildArray(date.getField(), degree + 1);
  414.         final T[] pnm1 = MathArrays.buildArray(date.getField(), degree + 1);

  416.         // compute polar coordinates
  417.         final T x    = position.getX();
  418.         final T y    = position.getY();
  419.         final T z    = position.getZ();
  420.         final T x2   = x.multiply(x);
  421.         final T y2   = y.multiply(y);
  422.         final T z2   = z.multiply(z);
  423.         final T r2   = x2.add(y2).add(z2);
  424.         final T r    = r2.sqrt();
  425.         final T rho2 = x2.add(y2);
  426.         final T rho  = rho2.sqrt();
  427.         final T t    = z.divide(r);   // cos(theta), where theta is the polar angle
  428.         final T u    = rho.divide(r); // sin(theta), where theta is the polar angle
  429.         final T tOu  = z.divide(rho);

  431.         // compute distance powers
  432.         final T[] aOrN = createDistancePowersArray(r.reciprocal().multiply(provider.getAe()));

  434.         // compute longitude cosines/sines
  435.         final T[][] cosSinLambda = createCosSinArrays(rho.reciprocal().multiply(position.getX()), rho.reciprocal().multiply(position.getY()));
  436.         // outer summation over order
  437.         int    index = 0;
  438.         T value = zero;
  439.         final T[] gradient = MathArrays.buildArray(zero.getField(), 3);
  440.         for (int m = degree; m >= 0; --m) {

  442.             // compute tesseral terms with derivatives
  443.             index = computeTesseral(m, degree, index, t, u, tOu,
  444.                                     pnm0Plus2, pnm0Plus1, null, pnm0, pnm1, null);
  445.             if (m <= order) {
  446.                 // compute contribution of current order to field (equation 5 of the paper)

  448.                 // inner summation over degree, for fixed order
  449.                 T sumDegreeS        = zero;
  450.                 T sumDegreeC        = zero;
  451.                 T dSumDegreeSdR     = zero;
  452.                 T dSumDegreeCdR     = zero;
  453.                 T dSumDegreeSdTheta = zero;
  454.                 T dSumDegreeCdTheta = zero;
  455.                 for (int n = FastMath.max(2, m); n <= degree; ++n) {
  456.                     final T qSnm  = aOrN[n].multiply(harmonics.getNormalizedSnm(n, m));
  457.                     final T qCnm  = aOrN[n].multiply(harmonics.getNormalizedCnm(n, m));
  458.                     final T nOr   = r.reciprocal().multiply(n);
  459.                     final T s0    = pnm0[n].multiply(qSnm);
  460.                     final T c0    = pnm0[n].multiply(qCnm);
  461.                     final T s1    = pnm1[n].multiply(qSnm);
  462.                     final T c1    = pnm1[n].multiply(qCnm);
  463.                     sumDegreeS        = sumDegreeS       .add(s0);
  464.                     sumDegreeC        = sumDegreeC       .add(c0);
  465.                     dSumDegreeSdR     = dSumDegreeSdR    .subtract(nOr.multiply(s0));
  466.                     dSumDegreeCdR     = dSumDegreeCdR    .subtract(nOr.multiply(c0));
  467.                     dSumDegreeSdTheta = dSumDegreeSdTheta.add(s1);
  468.                     dSumDegreeCdTheta = dSumDegreeCdTheta.add(c1);
  469.                 }

  471.                 // contribution to outer summation over order
  472.                 // beware that we need to order gradient using the mathematical conventions
  473.                 // compliant with the SphericalCoordinates class, so our lambda is its theta
  474.                 // (and hence at index 1) and our theta is its phi (and hence at index 2)
  475.                 final T sML = cosSinLambda[1][m];
  476.                 final T cML = cosSinLambda[0][m];
  477.                 value            = value      .multiply(u).add(sML.multiply(sumDegreeS   )).add(cML.multiply(sumDegreeC));
  478.                 gradient[0]      = gradient[0].multiply(u).add(sML.multiply(dSumDegreeSdR)).add(cML.multiply(dSumDegreeCdR));
  479.                 gradient[1]      = gradient[1].multiply(u).add(cML.multiply(sumDegreeS).subtract(sML.multiply(sumDegreeC)).multiply(m));
  480.                 gradient[2]      = gradient[2].multiply(u).add(sML.multiply(dSumDegreeSdTheta)).add(cML.multiply(dSumDegreeCdTheta));
  481.             }
  482.             // rotate the recursion arrays
  483.             final T[] tmp = pnm0Plus2;
  484.             pnm0Plus2 = pnm0Plus1;
  485.             pnm0Plus1 = pnm0;
  486.             pnm0      = tmp;

  488.         }
  489.         // scale back
  490.         value       = value.scalb(SCALING);
  491.         gradient[0] = gradient[0].scalb(SCALING);
  492.         gradient[1] = gradient[1].scalb(SCALING);
  493.         gradient[2] = gradient[2].scalb(SCALING);

  495.         // apply the global mu/r factor
  496.         final T muOr = r.reciprocal().multiply(mu);
  497.         value            = value.multiply(muOr);
  498.         gradient[0]       = muOr.multiply(gradient[0]).subtract(value.divide(r));
  499.         gradient[1]      = gradient[1].multiply(muOr);
  500.         gradient[2]      = gradient[2].multiply(muOr);

  502.         // convert gradient from spherical to Cartesian
  503.         // Cartesian coordinates
  504.         // remaining spherical coordinates
  505.         final T rPos     = position.getNorm();
  506.         // intermediate variables
  507.         final T xPos    = position.getX();
  508.         final T yPos    = position.getY();
  509.         final T zPos    = position.getZ();
  510.         final T rho2Pos = x.multiply(x).add(y.multiply(y));
  511.         final T rhoPos  = rho2.sqrt();
  512.         final T r2Pos   = rho2.add(z.multiply(z));

  514.         final T[][] jacobianPos = MathArrays.buildArray(zero.getField(), 3, 3);

  516.         // row representing the gradient of r
  517.         jacobianPos[0][0] = xPos.divide(rPos);
  518.         jacobianPos[0][1] = yPos.divide(rPos);
  519.         jacobianPos[0][2] = zPos.divide(rPos);

  521.         // row representing the gradient of theta
  522.         jacobianPos[1][0] =  yPos.negate().divide(rho2Pos);
  523.         jacobianPos[1][1] =  xPos.divide(rho2Pos);
  524.         // jacobian[1][2] is already set to 0 at allocation time

  526.         // row representing the gradient of phi
  527.         jacobianPos[2][0] = xPos.multiply(zPos).divide(rhoPos.multiply(r2Pos));
  528.         jacobianPos[2][1] = yPos.multiply(zPos).divide(rhoPos.multiply(r2Pos));
  529.         jacobianPos[2][2] = rhoPos.negate().divide(r2Pos);
  530.         final T[] cartGradPos = MathArrays.buildArray(zero.getField(), 3);
  531.         cartGradPos[0] = gradient[0].multiply(jacobianPos[0][0]).add(gradient[1].multiply(jacobianPos[1][0])).add(gradient[2].multiply(jacobianPos[2][0]));
  532.         cartGradPos[1] = gradient[0].multiply(jacobianPos[0][1]).add(gradient[1].multiply(jacobianPos[1][1])).add(gradient[2].multiply(jacobianPos[2][1]));
  533.         cartGradPos[2] = gradient[0].multiply(jacobianPos[0][2])                                      .add(gradient[2].multiply(jacobianPos[2][2]));
  534.         return cartGradPos;

  536.     }

  538.     /** Compute both the gradient and the hessian of the non-central part of the gravity field.
  539.      * @param date current date
  540.      * @param position position at which gravity field is desired in body frame
  541.      * @param mu central attraction coefficient to use
  542.      * @return gradient and hessian of the non-central part of the gravity field
  543.      * @exception OrekitException if position cannot be converted to central body frame
  544.      */
  545.     private GradientHessian gradientHessian(final AbsoluteDate date, final Vector3D position, final double mu)
  546.         throws OrekitException {

  548.         final int degree = provider.getMaxDegree();
  549.         final int order  = provider.getMaxOrder();
  550.         final NormalizedSphericalHarmonics harmonics = provider.onDate(date);

  552.         // allocate the columns for recursion
  553.         double[] pnm0Plus2  = new double[degree + 1];
  554.         double[] pnm0Plus1  = new double[degree + 1];
  555.         double[] pnm0       = new double[degree + 1];
  556.         double[] pnm1Plus1  = new double[degree + 1];
  557.         double[] pnm1       = new double[degree + 1];
  558.         final double[] pnm2 = new double[degree + 1];

  560.         // compute polar coordinates
  561.         final double x    = position.getX();
  562.         final double y    = position.getY();
  563.         final double z    = position.getZ();
  564.         final double x2   = x * x;
  565.         final double y2   = y * y;
  566.         final double z2   = z * z;
  567.         final double r2   = x2 + y2 + z2;
  568.         final double r    = FastMath.sqrt (r2);
  569.         final double rho2 = x2 + y2;
  570.         final double rho  = FastMath.sqrt(rho2);
  571.         final double t    = z / r;   // cos(theta), where theta is the polar angle
  572.         final double u    = rho / r; // sin(theta), where theta is the polar angle
  573.         final double tOu  = z / rho;

  575.         // compute distance powers
  576.         final double[] aOrN = createDistancePowersArray(provider.getAe() / r);

  578.         // compute longitude cosines/sines
  579.         final double[][] cosSinLambda = createCosSinArrays(position.getX() / rho, position.getY() / rho);

  581.         // outer summation over order
  582.         int    index = 0;
  583.         double value = 0;
  584.         final double[]   gradient = new double[3];
  585.         final double[][] hessian  = new double[3][3];
  586.         for (int m = degree; m >= 0; --m) {

  588.             // compute tesseral terms
  589.             index = computeTesseral(m, degree, index, t, u, tOu,
  590.                                     pnm0Plus2, pnm0Plus1, pnm1Plus1, pnm0, pnm1, pnm2);

  592.             if (m <= order) {
  593.                 // compute contribution of current order to field (equation 5 of the paper)

  595.                 // inner summation over degree, for fixed order
  596.                 double sumDegreeS               = 0;
  597.                 double sumDegreeC               = 0;
  598.                 double dSumDegreeSdR            = 0;
  599.                 double dSumDegreeCdR            = 0;
  600.                 double dSumDegreeSdTheta        = 0;
  601.                 double dSumDegreeCdTheta        = 0;
  602.                 double d2SumDegreeSdRdR         = 0;
  603.                 double d2SumDegreeSdRdTheta     = 0;
  604.                 double d2SumDegreeSdThetadTheta = 0;
  605.                 double d2SumDegreeCdRdR         = 0;
  606.                 double d2SumDegreeCdRdTheta     = 0;
  607.                 double d2SumDegreeCdThetadTheta = 0;
  608.                 for (int n = FastMath.max(2, m); n <= degree; ++n) {
  609.                     final double qSnm         = aOrN[n] * harmonics.getNormalizedSnm(n, m);
  610.                     final double qCnm         = aOrN[n] * harmonics.getNormalizedCnm(n, m);
  611.                     final double nOr          = n / r;
  612.                     final double nnP1Or2      = nOr * (n + 1) / r;
  613.                     final double s0           = pnm0[n] * qSnm;
  614.                     final double c0           = pnm0[n] * qCnm;
  615.                     final double s1           = pnm1[n] * qSnm;
  616.                     final double c1           = pnm1[n] * qCnm;
  617.                     final double s2           = pnm2[n] * qSnm;
  618.                     final double c2           = pnm2[n] * qCnm;
  619.                     sumDegreeS               += s0;
  620.                     sumDegreeC               += c0;
  621.                     dSumDegreeSdR            -= nOr * s0;
  622.                     dSumDegreeCdR            -= nOr * c0;
  623.                     dSumDegreeSdTheta        += s1;
  624.                     dSumDegreeCdTheta        += c1;
  625.                     d2SumDegreeSdRdR         += nnP1Or2 * s0;
  626.                     d2SumDegreeSdRdTheta     -= nOr * s1;
  627.                     d2SumDegreeSdThetadTheta += s2;
  628.                     d2SumDegreeCdRdR         += nnP1Or2 * c0;
  629.                     d2SumDegreeCdRdTheta     -= nOr * c1;
  630.                     d2SumDegreeCdThetadTheta += c2;
  631.                 }

  633.                 // contribution to outer summation over order
  634.                 final double sML = cosSinLambda[1][m];
  635.                 final double cML = cosSinLambda[0][m];
  636.                 value            = value         * u + sML * sumDegreeS + cML * sumDegreeC;
  637.                 gradient[0]      = gradient[0]   * u + sML * dSumDegreeSdR + cML * dSumDegreeCdR;
  638.                 gradient[1]      = gradient[1]   * u + m * (cML * sumDegreeS - sML * sumDegreeC);
  639.                 gradient[2]      = gradient[2]   * u + sML * dSumDegreeSdTheta + cML * dSumDegreeCdTheta;
  640.                 hessian[0][0]    = hessian[0][0] * u + sML * d2SumDegreeSdRdR + cML * d2SumDegreeCdRdR;
  641.                 hessian[1][0]    = hessian[1][0] * u + m * (cML * dSumDegreeSdR - sML * dSumDegreeCdR);
  642.                 hessian[2][0]    = hessian[2][0] * u + sML * d2SumDegreeSdRdTheta + cML * d2SumDegreeCdRdTheta;
  643.                 hessian[1][1]    = hessian[1][1] * u - m * m * (sML * sumDegreeS + cML * sumDegreeC);
  644.                 hessian[2][1]    = hessian[2][1] * u + m * (cML * dSumDegreeSdTheta - sML * dSumDegreeCdTheta);
  645.                 hessian[2][2]    = hessian[2][2] * u + sML * d2SumDegreeSdThetadTheta + cML * d2SumDegreeCdThetadTheta;

  647.             }

  649.             // rotate the recursion arrays
  650.             final double[] tmp0 = pnm0Plus2;
  651.             pnm0Plus2 = pnm0Plus1;
  652.             pnm0Plus1 = pnm0;
  653.             pnm0      = tmp0;
  654.             final double[] tmp1 = pnm1Plus1;
  655.             pnm1Plus1 = pnm1;
  656.             pnm1      = tmp1;

  658.         }

  660.         // scale back
  661.         value = FastMath.scalb(value, SCALING);
  662.         for (int i = 0; i < 3; ++i) {
  663.             gradient[i] = FastMath.scalb(gradient[i], SCALING);
  664.             for (int j = 0; j <= i; ++j) {
  665.                 hessian[i][j] = FastMath.scalb(hessian[i][j], SCALING);
  666.             }
  667.         }


  670.         // apply the global mu/r factor
  671.         final double muOr = mu / r;
  672.         value         *= muOr;
  673.         gradient[0]    = muOr * gradient[0] - value / r;
  674.         gradient[1]   *= muOr;
  675.         gradient[2]   *= muOr;
  676.         hessian[0][0]  = muOr * hessian[0][0] - 2 * gradient[0] / r;
  677.         hessian[1][0]  = muOr * hessian[1][0] -     gradient[1] / r;
  678.         hessian[2][0]  = muOr * hessian[2][0] -     gradient[2] / r;
  679.         hessian[1][1] *= muOr;
  680.         hessian[2][1] *= muOr;
  681.         hessian[2][2] *= muOr;

  683.         // convert gradient and Hessian from spherical to Cartesian
  684.         final SphericalCoordinates sc = new SphericalCoordinates(position);
  685.         return new GradientHessian(sc.toCartesianGradient(gradient),
  686.                                    sc.toCartesianHessian(hessian, gradient));


  689.     }

  691.     /** Container for gradient and Hessian. */
  692.     private static class GradientHessian {

  694.         /** Gradient. */
  695.         private final double[] gradient;

  697.         /** Hessian. */
  698.         private final double[][] hessian;

  700.         /** Simple constructor.
  701.          * <p>
  702.          * A reference to the arrays is stored, they are <strong>not</strong> cloned.
  703.          * </p>
  704.          * @param gradient gradient
  705.          * @param hessian hessian
  706.          */
  707.         GradientHessian(final double[] gradient, final double[][] hessian) {
  708.             this.gradient = gradient;
  709.             this.hessian  = hessian;
  710.         }

  712.         /** Get a reference to the gradient.
  713.          * @return gradient (a reference to the internal array is returned)
  714.          */
  715.         public double[] getGradient() {
  716.             return gradient;
  717.         }

  719.         /** Get a reference to the Hessian.
  720.          * @return Hessian (a reference to the internal array is returned)
  721.          */
  722.         public double[][] getHessian() {
  723.             return hessian;
  724.         }

  726.     }

  728.     /** Compute a/r powers array.
  729.      * @param aOr a/r
  730.      * @return array containing (a/r)<sup>n</sup>
  731.      */
  732.     private double[] createDistancePowersArray(final double aOr) {

  734.         // initialize array
  735.         final double[] aOrN = new double[provider.getMaxDegree() + 1];
  736.         aOrN[0] = 1;
  737.         aOrN[1] = aOr;

  739.         // fill up array
  740.         for (int n = 2; n < aOrN.length; ++n) {
  741.             final int p = n / 2;
  742.             final int q = n - p;
  743.             aOrN[n] = aOrN[p] * aOrN[q];
  744.         }

  746.         return aOrN;

  748.     }
  749.     /** Compute a/r powers array.
  750.      * @param aOr a/r
  751.      * @param <T> type of field used
  752.      * @return array containing (a/r)<sup>n</sup>
  753.      */
  754.     private <T extends RealFieldElement<T>> T[] createDistancePowersArray(final T aOr) {

  756.         // initialize array
  757.         final T[] aOrN = MathArrays.buildArray(aOr.getField(), provider.getMaxDegree() + 1);
  758.         aOrN[0] = aOr.getField().getOne();
  759.         aOrN[1] = aOr;

  761.         // fill up array
  762.         for (int n = 2; n < aOrN.length; ++n) {
  763.             final int p = n / 2;
  764.             final int q = n - p;
  765.             aOrN[n] = aOrN[p].multiply(aOrN[q]);
  766.         }

  768.         return aOrN;

  770.     }

  772.     /** Compute longitude cosines and sines.
  773.      * @param cosLambda cos(λ)
  774.      * @param sinLambda sin(λ)
  775.      * @return array containing cos(m &times; λ) in row 0
  776.      * and sin(m &times; λ) in row 1
  777.      */
  778.     private double[][] createCosSinArrays(final double cosLambda, final double sinLambda) {

  780.         // initialize arrays
  781.         final double[][] cosSin = new double[2][provider.getMaxOrder() + 1];
  782.         cosSin[0][0] = 1;
  783.         cosSin[1][0] = 0;
  784.         if (provider.getMaxOrder() > 0) {
  785.             cosSin[0][1] = cosLambda;
  786.             cosSin[1][1] = sinLambda;

  788.             // fill up array
  789.             for (int m = 2; m < cosSin[0].length; ++m) {

  791.                 // m * lambda is split as p * lambda + q * lambda, trying to avoid
  792.                 // p or q being much larger than the other. This reduces the number of
  793.                 // intermediate results reused to compute each value, and hence should limit
  794.                 // as much as possible roundoff error accumulation
  795.                 // (this does not change the number of floating point operations)
  796.                 final int p = m / 2;
  797.                 final int q = m - p;

  799.                 cosSin[0][m] = cosSin[0][p] * cosSin[0][q] - cosSin[1][p] * cosSin[1][q];
  800.                 cosSin[1][m] = cosSin[1][p] * cosSin[0][q] + cosSin[0][p] * cosSin[1][q];
  801.             }
  802.         }

  804.         return cosSin;

  806.     }

  808.     /** Compute longitude cosines and sines.
  809.      * @param cosLambda cos(λ)
  810.      * @param sinLambda sin(λ)
  811.      * @param <T> type of field used
  812.      * @return array containing cos(m &times; λ) in row 0
  813.      * and sin(m &times; λ) in row 1
  814.      */
  815.     private <T extends RealFieldElement<T>> T[][] createCosSinArrays(final T cosLambda, final T sinLambda) {

  817.         final T one = cosLambda.getField().getOne();
  818.         final T zero = cosLambda.getField().getZero();
  819.         // initialize arrays
  820.         final T[][] cosSin = MathArrays.buildArray(one.getField(), 2, provider.getMaxOrder() + 1);
  821.         cosSin[0][0] = one;
  822.         cosSin[1][0] = zero;
  823.         if (provider.getMaxOrder() > 0) {
  824.             cosSin[0][1] = cosLambda;
  825.             cosSin[1][1] = sinLambda;

  827.             // fill up array
  828.             for (int m = 2; m < cosSin[0].length; ++m) {

  830.                 // m * lambda is split as p * lambda + q * lambda, trying to avoid
  831.                 // p or q being much larger than the other. This reduces the number of
  832.                 // intermediate results reused to compute each value, and hence should limit
  833.                 // as much as possible roundoff error accumulation
  834.                 // (this does not change the number of floating point operations)
  835.                 final int p = m / 2;
  836.                 final int q = m - p;

  838.                 cosSin[0][m] = cosSin[0][p].multiply(cosSin[0][q]).subtract(cosSin[1][p].multiply(cosSin[1][q]));
  839.                 cosSin[1][m] = cosSin[1][p].multiply(cosSin[0][q]).add(cosSin[0][p].multiply(cosSin[1][q]));

  841.             }
  842.         }

  844.         return cosSin;

  846.     }

  848.     /** Compute one order of tesseral terms.
  849.      * <p>
  850.      * This corresponds to equations 27 and 30 of the paper.
  851.      * </p>
  852.      * @param m current order
  853.      * @param degree max degree
  854.      * @param index index in the flattened array
  855.      * @param t cos(θ), where θ is the polar angle
  856.      * @param u sin(θ), where θ is the polar angle
  857.      * @param tOu t/u
  858.      * @param pnm0Plus2 array containing scaled P<sub>n,m+2</sub>/u<sup>m+2</sup>
  859.      * @param pnm0Plus1 array containing scaled P<sub>n,m+1</sub>/u<sup>m+1</sup>
  860.      * @param pnm1Plus1 array containing scaled dP<sub>n,m+1</sub>/u<sup>m+1</sup>
  861.      * (may be null if second derivatives are not needed)
  862.      * @param pnm0 array to fill with scaled P<sub>n,m</sub>/u<sup>m</sup>
  863.      * @param pnm1 array to fill with scaled dP<sub>n,m</sub>/u<sup>m</sup>
  864.      * (may be null if first derivatives are not needed)
  865.      * @param pnm2 array to fill with scaled d²P<sub>n,m</sub>/u<sup>m</sup>
  866.      * (may be null if second derivatives are not needed)
  867.      * @return new value for index
  868.      */
  869.     private int computeTesseral(final int m, final int degree, final int index,
  870.                                 final double t, final double u, final double tOu,
  871.                                 final double[] pnm0Plus2, final double[] pnm0Plus1, final double[] pnm1Plus1,
  872.                                 final double[] pnm0, final double[] pnm1, final double[] pnm2) {

  874.         final double u2 = u * u;

  876.         // initialize recursion from sectorial terms
  877.         int n = FastMath.max(2, m);
  878.         if (n == m) {
  879.             pnm0[n] = sectorial[n];
  880.             ++n;
  881.         }

  883.         // compute tesseral values
  884.         int localIndex = index;
  885.         while (n <= degree) {

  887.             // value (equation 27 of the paper)
  888.             pnm0[n] = gnmOj[localIndex] * t * pnm0Plus1[n] - hnmOj[localIndex] * u2 * pnm0Plus2[n];

  890.             ++localIndex;
  891.             ++n;

  893.         }

  895.         if (pnm1 != null) {

  897.             // initialize recursion from sectorial terms
  898.             n = FastMath.max(2, m);
  899.             if (n == m) {
  900.                 pnm1[n] = m * tOu * pnm0[n];
  901.                 ++n;
  902.             }

  904.             // compute tesseral values and derivatives with respect to polar angle
  905.             localIndex = index;
  906.             while (n <= degree) {

  908.                 // first derivative (equation 30 of the paper)
  909.                 pnm1[n] = m * tOu * pnm0[n] - enm[localIndex] * u * pnm0Plus1[n];

  911.                 ++localIndex;
  912.                 ++n;

  914.             }

  916.             if (pnm2 != null) {

  918.                 // initialize recursion from sectorial terms
  919.                 n = FastMath.max(2, m);
  920.                 if (n == m) {
  921.                     pnm2[n] = m * (tOu * pnm1[n] - pnm0[n] / u2);
  922.                     ++n;
  923.                 }

  925.                 // compute tesseral values and derivatives with respect to polar angle
  926.                 localIndex = index;
  927.                 while (n <= degree) {

  929.                     // second derivative (differential of equation 30 with respect to theta)
  930.                     pnm2[n] = m * (tOu * pnm1[n] - pnm0[n] / u2) - enm[localIndex] * u * pnm1Plus1[n];

  932.                     ++localIndex;
  933.                     ++n;

  935.                 }

  937.             }

  939.         }

  941.         return localIndex;

  943.     }

  945.     /** Compute one order of tesseral terms.
  946.      * <p>
  947.      * This corresponds to equations 27 and 30 of the paper.
  948.      * </p>
  949.      * @param m current order
  950.      * @param degree max degree
  951.      * @param index index in the flattened array
  952.      * @param t cos(θ), where θ is the polar angle
  953.      * @param u sin(θ), where θ is the polar angle
  954.      * @param tOu t/u
  955.      * @param pnm0Plus2 array containing scaled P<sub>n,m+2</sub>/u<sup>m+2</sup>
  956.      * @param pnm0Plus1 array containing scaled P<sub>n,m+1</sub>/u<sup>m+1</sup>
  957.      * @param pnm1Plus1 array containing scaled dP<sub>n,m+1</sub>/u<sup>m+1</sup>
  958.      * (may be null if second derivatives are not needed)
  959.      * @param pnm0 array to fill with scaled P<sub>n,m</sub>/u<sup>m</sup>
  960.      * @param pnm1 array to fill with scaled dP<sub>n,m</sub>/u<sup>m</sup>
  961.      * (may be null if first derivatives are not needed)
  962.      * @param pnm2 array to fill with scaled d²P<sub>n,m</sub>/u<sup>m</sup>
  963.      * (may be null if second derivatives are not needed)
  964.      * @param <T> instance of field element
  965.      * @return new value for index
  966.      */
  967.     private <T extends RealFieldElement<T>> int computeTesseral(final int m, final int degree, final int index,
  968.                                                                 final T t, final T u, final T tOu,
  969.                                                                 final T[] pnm0Plus2, final T[] pnm0Plus1, final T[] pnm1Plus1,
  970.                                                                 final T[] pnm0, final T[] pnm1, final T[] pnm2) {

  972.         final T u2 = u.multiply(u);
  973.         final T zero = u.getField().getZero();
  974.         // initialize recursion from sectorial terms
  975.         int n = FastMath.max(2, m);
  976.         if (n == m) {
  977.             pnm0[n] = zero.add(sectorial[n]);
  978.             ++n;
  979.         }

  981.         // compute tesseral values
  982.         int localIndex = index;
  983.         while (n <= degree) {

  985.             // value (equation 27 of the paper)
  986.             pnm0[n] = t.multiply(gnmOj[localIndex]).multiply(pnm0Plus1[n]).subtract(u2.multiply(pnm0Plus2[n]).multiply(hnmOj[localIndex]));
  987.             ++localIndex;
  988.             ++n;

  990.         }
  991.         if (pnm1 != null) {

  993.             // initialize recursion from sectorial terms
  994.             n = FastMath.max(2, m);
  995.             if (n == m) {
  996.                 pnm1[n] = tOu.multiply(m).multiply(pnm0[n]);
  997.                 ++n;
  998.             }

  1000.             // compute tesseral values and derivatives with respect to polar angle
  1001.             localIndex = index;
  1002.             while (n <= degree) {

  1004.                 // first derivative (equation 30 of the paper)
  1005.                 pnm1[n] = tOu.multiply(m).multiply(pnm0[n]).subtract(u.multiply(enm[localIndex]).multiply(pnm0Plus1[n]));

  1007.                 ++localIndex;
  1008.                 ++n;

  1010.             }

  1012.             if (pnm2 != null) {

  1014.                 // initialize recursion from sectorial terms
  1015.                 n = FastMath.max(2, m);
  1016.                 if (n == m) {
  1017.                     pnm2[n] =   tOu.multiply(pnm1[n]).subtract(pnm0[n].divide(u2)).multiply(m);
  1018.                     ++n;
  1019.                 }

  1021.                 // compute tesseral values and derivatives with respect to polar angle
  1022.                 localIndex = index;
  1023.                 while (n <= degree) {

  1025.                     // second derivative (differential of equation 30 with respect to theta)
  1026.                     pnm2[n] = tOu.multiply(pnm1[n]).subtract(pnm0[n].divide(u2)).multiply(m).subtract(u.multiply(pnm1Plus1[n]).multiply(enm[localIndex]));
  1027.                     ++localIndex;
  1028.                     ++n;

  1030.                 }

  1032.             }

  1034.         }
  1035.         return localIndex;

  1037.     }

  1039.     /** {@inheritDoc} */
  1040.     @Override
  1041.     public Vector3D acceleration(final SpacecraftState s, final double[] parameters)
  1042.         throws OrekitException {

  1044.         final double mu = parameters[0];

  1046.         // get the position in body frame
  1047.         final AbsoluteDate date       = s.getDate();
  1048.         final Transform fromBodyFrame = bodyFrame.getTransformTo(s.getFrame(), date);
  1049.         final Transform toBodyFrame   = fromBodyFrame.getInverse();
  1050.         final Vector3D position       = toBodyFrame.transformPosition(s.getPVCoordinates().getPosition());

  1052.         // gradient of the non-central part of the gravity field
  1053.         return fromBodyFrame.transformVector(new Vector3D(gradient(date, position, mu)));

  1055.     }

  1057.     /** {@inheritDoc} */
  1058.     public <T extends RealFieldElement<T>> FieldVector3D<T> acceleration(final FieldSpacecraftState<T> s,
  1059.                                                                          final T[] parameters)
  1060.         throws OrekitException {

  1062.         final T mu = parameters[0];

  1064.         // check for faster computation dedicated to derivatives with respect to state
  1065.         if (isStateDerivative(s)) {
  1066.             @SuppressWarnings("unchecked")
  1067.             final FieldVector3D<DerivativeStructure> p = (FieldVector3D<DerivativeStructure>) s.getPVCoordinates().getPosition();
  1068.             @SuppressWarnings("unchecked")
  1069.             final FieldVector3D<T> a = (FieldVector3D<T>) accelerationWrtState(s.getDate().toAbsoluteDate(),
  1070.                                                                                s.getFrame(), p,
  1071.                                                                                (DerivativeStructure) mu);
  1072.             return a;
  1073.         }

  1075.         // get the position in body frame
  1076.         final FieldAbsoluteDate<T> date          = s.getDate();
  1077.         final Transform            fromBodyFrame = bodyFrame.getTransformTo(s.getFrame(), date.toAbsoluteDate());
  1078.         final Transform            toBodyFrame   = fromBodyFrame.getInverse();
  1079.         final FieldVector3D<T>     position      = toBodyFrame.transformPosition(s.getPVCoordinates().getPosition());

  1081.         // gradient of the non-central part of the gravity field
  1082.         return fromBodyFrame.transformVector(new FieldVector3D<>(gradient(date, position, mu)));

  1084.     }

  1086.     /** {@inheritDoc} */
  1087.     public Stream<EventDetector> getEventsDetectors() {
  1088.         return Stream.empty();
  1089.     }

  1091.     @Override
  1092.     /** {@inheritDoc} */
  1093.     public <T extends RealFieldElement<T>> Stream<FieldEventDetector<T>> getFieldEventsDetectors(final Field<T> field) {
  1094.         return Stream.empty();
  1095.     }

  1097.     /** Check if a field state corresponds to derivatives with respect to state.
  1098.      * @param state state to check
  1099.      * @param <T> type of the filed elements
  1100.      * @return true if state corresponds to derivatives with respect to state
  1101.      * @since 9.0
  1102.      */
  1103.     private <T extends RealFieldElement<T>> boolean isStateDerivative(final FieldSpacecraftState<T> state) {
  1104.         try {
  1105.             final DerivativeStructure dsMass = (DerivativeStructure) state.getMass();
  1106.             final int o = dsMass.getOrder();
  1107.             final int p = dsMass.getFreeParameters();
  1108.             if (o != 1 || (p < 3)) {
  1109.                 return false;
  1110.             }
  1111.             @SuppressWarnings("unchecked")
  1112.             final FieldPVCoordinates<DerivativeStructure> pv = (FieldPVCoordinates<DerivativeStructure>) state.getPVCoordinates();
  1113.             return isVariable(pv.getPosition().getX(), 0) &&
  1114.                    isVariable(pv.getPosition().getY(), 1) &&
  1115.                    isVariable(pv.getPosition().getZ(), 2);
  1116.         } catch (ClassCastException cce) {
  1117.             return false;
  1118.         }
  1119.     }

  1121.     /** Check if a derivative represents a specified variable.
  1122.      * @param ds derivative to check
  1123.      * @param index index of the variable
  1124.      * @return true if the derivative represents a specified variable
  1125.      * @since 9.0
  1126.      */
  1127.     private boolean isVariable(final DerivativeStructure ds, final int index) {
  1128.         final double[] derivatives = ds.getAllDerivatives();
  1129.         boolean check = true;
  1130.         for (int i = 1; i < derivatives.length; ++i) {
  1131.             check &= derivatives[i] == ((index + 1 == i) ? 1.0 : 0.0);
  1132.         }
  1133.         return check;
  1134.     }

  1136.     /** Compute acceleration derivatives with respect to state parameters.
  1137.      * <p>
  1138.      * From a theoretical point of view, this method computes the same values
  1139.      * as {@link #acceleration(FieldSpacecraftState, RealFieldElement[])} in the
  1140.      * specific case of {@link DerivativeStructure} with respect to state, so
  1141.      * it is less general. However, it is *much* faster in this important case.
  1142.      * <p>
  1143.      * <p>
  1144.      * The derivatives should be computed with respect to position. The input
  1145.      * parameters already take into account the free parameters (6 or 7 depending
  1146.      * on derivation with respect to mass being considered or not) and order
  1147.      * (always 1). Free parameters at indices 0, 1 and 2 correspond to derivatives
  1148.      * with respect to position. Free parameters at indices 3, 4 and 5 correspond
  1149.      * to derivatives with respect to velocity (these derivatives will remain zero
  1150.      * as acceleration due to gravity does not depend on velocity). Free parameter
  1151.      * at index 6 (if present) corresponds to to derivatives with respect to mass
  1152.      * (this derivative will remain zero as acceleration due to gravity does not
  1153.      * depend on mass).
  1154.      * </p>
  1155.      * @param date current date
  1156.     * @param frame inertial reference frame for state (both orbit and attitude)
  1157.      * @param position position of spacecraft in inertial frame
  1158.      * @param mu central attraction coefficient to use
  1159.      * @return acceleration with all derivatives specified by the input parameters
  1160.      * own derivatives
  1161.      * @exception OrekitException if derivatives cannot be computed
  1162.      * @since 6.0
  1163.      */
  1164.     private FieldVector3D<DerivativeStructure> accelerationWrtState(final AbsoluteDate date, final Frame frame,
  1165.                                                                     final FieldVector3D<DerivativeStructure> position,
  1166.                                                                     final DerivativeStructure mu)
  1167.         throws OrekitException {

  1169.         // get the position in body frame
  1170.         final Transform fromBodyFrame = bodyFrame.getTransformTo(frame, date);
  1171.         final Transform toBodyFrame   = fromBodyFrame.getInverse();
  1172.         final Vector3D positionBody   = toBodyFrame.transformPosition(position.toVector3D());

  1174.         // compute gradient and Hessian
  1175.         final GradientHessian gh   = gradientHessian(date, positionBody, mu.getReal());

  1177.         // gradient of the non-central part of the gravity field
  1178.         final double[] gInertial = fromBodyFrame.transformVector(new Vector3D(gh.getGradient())).toArray();

  1180.         // Hessian of the non-central part of the gravity field
  1181.         final RealMatrix hBody     = new Array2DRowRealMatrix(gh.getHessian(), false);
  1182.         final RealMatrix rot       = new Array2DRowRealMatrix(toBodyFrame.getRotation().getMatrix());
  1183.         final RealMatrix hInertial = rot.transpose().multiply(hBody).multiply(rot);

  1185.         // distribute all partial derivatives in a compact acceleration vector
  1186.         final double[] derivatives = new double[1 + position.getX().getFreeParameters()];
  1187.         final DerivativeStructure[] accDer = new DerivativeStructure[3];
  1188.         for (int i = 0; i < 3; ++i) {

  1190.             // first element is value of acceleration (i.e. gradient of field)
  1191.             derivatives[0] = gInertial[i];

  1193.             // next three elements are one row of the Jacobian of acceleration (i.e. Hessian of field)
  1194.             derivatives[1] = hInertial.getEntry(i, 0);
  1195.             derivatives[2] = hInertial.getEntry(i, 1);
  1196.             derivatives[3] = hInertial.getEntry(i, 2);

  1198.             // next element is derivative with respect to parameter mu
  1199.             if (derivatives.length > 4 && isVariable(mu, 3)) {
  1200.                 derivatives[4] = gInertial[i] / mu.getReal();
  1201.             }

  1203.             accDer[i] = position.getX().getFactory().build(derivatives);

  1205.         }

  1207.         return new FieldVector3D<>(accDer);

  1209.     }

  1211.     /** {@inheritDoc} */
  1212.     public ParameterDriver[] getParametersDrivers() {
  1213.         return new ParameterDriver[] {
  1214.             gmParameterDriver
  1215.         };
  1216.     }

  1218. }
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