HolmesFeatherstoneAttractionModel.java

  1. /* Copyright 2002-2019 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.      */
  191.     public double value(final AbsoluteDate date, final Vector3D position,
  192.                         final double mu) {
  193.         return mu / position.getNorm() + nonCentralPart(date, position, mu);
  194.     }

  196.     /** Compute the non-central part of the gravity field.
  197.      * @param date current date
  198.      * @param position position at which gravity field is desired in body frame
  199.      * @param mu central attraction coefficient to use
  200.      * @return value of the non-central part of the gravity field
  201.      */
  202.     public double nonCentralPart(final AbsoluteDate date, final Vector3D position, final double mu) {

  204.         final int degree = provider.getMaxDegree();
  205.         final int order  = provider.getMaxOrder();
  206.         final NormalizedSphericalHarmonics harmonics = provider.onDate(date);

  208.         // allocate the columns for recursion
  209.         double[] pnm0Plus2 = new double[degree + 1];
  210.         double[] pnm0Plus1 = new double[degree + 1];
  211.         double[] pnm0      = new double[degree + 1];

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

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

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

  233.         // outer summation over order
  234.         int    index = 0;
  235.         double value = 0;
  236.         for (int m = degree; m >= 0; --m) {

  238.             // compute tesseral terms without derivatives
  239.             index = computeTesseral(m, degree, index, t, u, tOu,
  240.                                     pnm0Plus2, pnm0Plus1, null, pnm0, null, null);

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

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

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

  256.             }

  258.             // rotate the recursion arrays
  259.             final double[] tmp = pnm0Plus2;
  260.             pnm0Plus2 = pnm0Plus1;
  261.             pnm0Plus1 = pnm0;
  262.             pnm0      = tmp;

  264.         }

  266.         // scale back
  267.         value = FastMath.scalb(value, SCALING);

  269.         // apply the global mu/r factor
  270.         return mu * value / r;

  272.     }

  274.     /** Compute the gradient of the non-central part of the gravity field.
  275.      * @param date current date
  276.      * @param position position at which gravity field is desired in body frame
  277.      * @param mu central attraction coefficient to use
  278.      * @return gradient of the non-central part of the gravity field
  279.      */
  280.     public double[] gradient(final AbsoluteDate date, final Vector3D position, final double mu) {

  282.         final int degree = provider.getMaxDegree();
  283.         final int order  = provider.getMaxOrder();
  284.         final NormalizedSphericalHarmonics harmonics = provider.onDate(date);

  286.         // allocate the columns for recursion
  287.         double[] pnm0Plus2  = new double[degree + 1];
  288.         double[] pnm0Plus1  = new double[degree + 1];
  289.         double[] pnm0       = new double[degree + 1];
  290.         final double[] pnm1 = new double[degree + 1];

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

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

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

  313.         // outer summation over order
  314.         int    index = 0;
  315.         double value = 0;
  316.         final double[] gradient = new double[3];
  317.         for (int m = degree; m >= 0; --m) {

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

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

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

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

  360.             }

  362.             // rotate the recursion arrays
  363.             final double[] tmp = pnm0Plus2;
  364.             pnm0Plus2 = pnm0Plus1;
  365.             pnm0Plus1 = pnm0;
  366.             pnm0      = tmp;

  368.         }

  370.         // scale back
  371.         value       = FastMath.scalb(value,       SCALING);
  372.         gradient[0] = FastMath.scalb(gradient[0], SCALING);
  373.         gradient[1] = FastMath.scalb(gradient[1], SCALING);
  374.         gradient[2] = FastMath.scalb(gradient[2], SCALING);

  376.         // apply the global mu/r factor
  377.         final double muOr = mu / r;
  378.         value            *= muOr;
  379.         gradient[0]       = muOr * gradient[0] - value / r;
  380.         gradient[1]      *= muOr;
  381.         gradient[2]      *= muOr;

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

  386.     }

  388.     /** Compute the gradient of the non-central part of the gravity field.
  389.      * @param date current date
  390.      * @param position position at which gravity field is desired in body frame
  391.      * @param mu central attraction coefficient to use
  392.      * @param <T> type of field used
  393.      * @return gradient of the non-central part of the gravity field
  394.      */
  395.     public <T extends RealFieldElement<T>> T[] gradient(final FieldAbsoluteDate<T> date, final FieldVector3D<T> position,
  396.                                                         final T mu) {

  398.         final int degree = provider.getMaxDegree();
  399.         final int order  = provider.getMaxOrder();
  400.         final NormalizedSphericalHarmonics harmonics = provider.onDate(date.toAbsoluteDate());
  401.         final T zero = date.getField().getZero();
  402.         // allocate the columns for recursion
  403.         T[] pnm0Plus2  = MathArrays.buildArray(date.getField(), degree + 1);
  404.         T[] pnm0Plus1  = MathArrays.buildArray(date.getField(), degree + 1);
  405.         T[] pnm0       = MathArrays.buildArray(date.getField(), degree + 1);
  406.         final T[] pnm1 = MathArrays.buildArray(date.getField(), degree + 1);

  408.         // compute polar coordinates
  409.         final T x    = position.getX();
  410.         final T y    = position.getY();
  411.         final T z    = position.getZ();
  412.         final T x2   = x.multiply(x);
  413.         final T y2   = y.multiply(y);
  414.         final T z2   = z.multiply(z);
  415.         final T r2   = x2.add(y2).add(z2);
  416.         final T r    = r2.sqrt();
  417.         final T rho2 = x2.add(y2);
  418.         final T rho  = rho2.sqrt();
  419.         final T t    = z.divide(r);   // cos(theta), where theta is the polar angle
  420.         final T u    = rho.divide(r); // sin(theta), where theta is the polar angle
  421.         final T tOu  = z.divide(rho);

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

  426.         // compute longitude cosines/sines
  427.         final T[][] cosSinLambda = createCosSinArrays(rho.reciprocal().multiply(position.getX()), rho.reciprocal().multiply(position.getY()));
  428.         // outer summation over order
  429.         int    index = 0;
  430.         T value = zero;
  431.         final T[] gradient = MathArrays.buildArray(zero.getField(), 3);
  432.         for (int m = degree; m >= 0; --m) {

  434.             // compute tesseral terms with derivatives
  435.             index = computeTesseral(m, degree, index, t, u, tOu,
  436.                                     pnm0Plus2, pnm0Plus1, null, pnm0, pnm1, null);
  437.             if (m <= order) {
  438.                 // compute contribution of current order to field (equation 5 of the paper)

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

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

  480.         }
  481.         // scale back
  482.         value       = value.scalb(SCALING);
  483.         gradient[0] = gradient[0].scalb(SCALING);
  484.         gradient[1] = gradient[1].scalb(SCALING);
  485.         gradient[2] = gradient[2].scalb(SCALING);

  487.         // apply the global mu/r factor
  488.         final T muOr = r.reciprocal().multiply(mu);
  489.         value            = value.multiply(muOr);
  490.         gradient[0]       = muOr.multiply(gradient[0]).subtract(value.divide(r));
  491.         gradient[1]      = gradient[1].multiply(muOr);
  492.         gradient[2]      = gradient[2].multiply(muOr);

  494.         // convert gradient from spherical to Cartesian
  495.         // Cartesian coordinates
  496.         // remaining spherical coordinates
  497.         final T rPos     = position.getNorm();
  498.         // intermediate variables
  499.         final T xPos    = position.getX();
  500.         final T yPos    = position.getY();
  501.         final T zPos    = position.getZ();
  502.         final T rho2Pos = x.multiply(x).add(y.multiply(y));
  503.         final T rhoPos  = rho2.sqrt();
  504.         final T r2Pos   = rho2.add(z.multiply(z));

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

  508.         // row representing the gradient of r
  509.         jacobianPos[0][0] = xPos.divide(rPos);
  510.         jacobianPos[0][1] = yPos.divide(rPos);
  511.         jacobianPos[0][2] = zPos.divide(rPos);

  513.         // row representing the gradient of theta
  514.         jacobianPos[1][0] =  yPos.negate().divide(rho2Pos);
  515.         jacobianPos[1][1] =  xPos.divide(rho2Pos);
  516.         // jacobian[1][2] is already set to 0 at allocation time

  518.         // row representing the gradient of phi
  519.         jacobianPos[2][0] = xPos.multiply(zPos).divide(rhoPos.multiply(r2Pos));
  520.         jacobianPos[2][1] = yPos.multiply(zPos).divide(rhoPos.multiply(r2Pos));
  521.         jacobianPos[2][2] = rhoPos.negate().divide(r2Pos);
  522.         final T[] cartGradPos = MathArrays.buildArray(zero.getField(), 3);
  523.         cartGradPos[0] = gradient[0].multiply(jacobianPos[0][0]).add(gradient[1].multiply(jacobianPos[1][0])).add(gradient[2].multiply(jacobianPos[2][0]));
  524.         cartGradPos[1] = gradient[0].multiply(jacobianPos[0][1]).add(gradient[1].multiply(jacobianPos[1][1])).add(gradient[2].multiply(jacobianPos[2][1]));
  525.         cartGradPos[2] = gradient[0].multiply(jacobianPos[0][2])                                      .add(gradient[2].multiply(jacobianPos[2][2]));
  526.         return cartGradPos;

  528.     }

  530.     /** Compute both the gradient and the hessian of the non-central part of the gravity field.
  531.      * @param date current date
  532.      * @param position position at which gravity field is desired in body frame
  533.      * @param mu central attraction coefficient to use
  534.      * @return gradient and hessian of the non-central part of the gravity field
  535.      */
  536.     private GradientHessian gradientHessian(final AbsoluteDate date, final Vector3D position, final double mu) {

  538.         final int degree = provider.getMaxDegree();
  539.         final int order  = provider.getMaxOrder();
  540.         final NormalizedSphericalHarmonics harmonics = provider.onDate(date);

  542.         // allocate the columns for recursion
  543.         double[] pnm0Plus2  = new double[degree + 1];
  544.         double[] pnm0Plus1  = new double[degree + 1];
  545.         double[] pnm0       = new double[degree + 1];
  546.         double[] pnm1Plus1  = new double[degree + 1];
  547.         double[] pnm1       = new double[degree + 1];
  548.         final double[] pnm2 = new double[degree + 1];

  550.         // compute polar coordinates
  551.         final double x    = position.getX();
  552.         final double y    = position.getY();
  553.         final double z    = position.getZ();
  554.         final double x2   = x * x;
  555.         final double y2   = y * y;
  556.         final double z2   = z * z;
  557.         final double r2   = x2 + y2 + z2;
  558.         final double r    = FastMath.sqrt (r2);
  559.         final double rho2 = x2 + y2;
  560.         final double rho  = FastMath.sqrt(rho2);
  561.         final double t    = z / r;   // cos(theta), where theta is the polar angle
  562.         final double u    = rho / r; // sin(theta), where theta is the polar angle
  563.         final double tOu  = z / rho;

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

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

  571.         // outer summation over order
  572.         int    index = 0;
  573.         double value = 0;
  574.         final double[]   gradient = new double[3];
  575.         final double[][] hessian  = new double[3][3];
  576.         for (int m = degree; m >= 0; --m) {

  578.             // compute tesseral terms
  579.             index = computeTesseral(m, degree, index, t, u, tOu,
  580.                                     pnm0Plus2, pnm0Plus1, pnm1Plus1, pnm0, pnm1, pnm2);

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

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

  623.                 // contribution to outer summation over order
  624.                 final double sML = cosSinLambda[1][m];
  625.                 final double cML = cosSinLambda[0][m];
  626.                 value            = value         * u + sML * sumDegreeS + cML * sumDegreeC;
  627.                 gradient[0]      = gradient[0]   * u + sML * dSumDegreeSdR + cML * dSumDegreeCdR;
  628.                 gradient[1]      = gradient[1]   * u + m * (cML * sumDegreeS - sML * sumDegreeC);
  629.                 gradient[2]      = gradient[2]   * u + sML * dSumDegreeSdTheta + cML * dSumDegreeCdTheta;
  630.                 hessian[0][0]    = hessian[0][0] * u + sML * d2SumDegreeSdRdR + cML * d2SumDegreeCdRdR;
  631.                 hessian[1][0]    = hessian[1][0] * u + m * (cML * dSumDegreeSdR - sML * dSumDegreeCdR);
  632.                 hessian[2][0]    = hessian[2][0] * u + sML * d2SumDegreeSdRdTheta + cML * d2SumDegreeCdRdTheta;
  633.                 hessian[1][1]    = hessian[1][1] * u - m * m * (sML * sumDegreeS + cML * sumDegreeC);
  634.                 hessian[2][1]    = hessian[2][1] * u + m * (cML * dSumDegreeSdTheta - sML * dSumDegreeCdTheta);
  635.                 hessian[2][2]    = hessian[2][2] * u + sML * d2SumDegreeSdThetadTheta + cML * d2SumDegreeCdThetadTheta;

  637.             }

  639.             // rotate the recursion arrays
  640.             final double[] tmp0 = pnm0Plus2;
  641.             pnm0Plus2 = pnm0Plus1;
  642.             pnm0Plus1 = pnm0;
  643.             pnm0      = tmp0;
  644.             final double[] tmp1 = pnm1Plus1;
  645.             pnm1Plus1 = pnm1;
  646.             pnm1      = tmp1;

  648.         }

  650.         // scale back
  651.         value = FastMath.scalb(value, SCALING);
  652.         for (int i = 0; i < 3; ++i) {
  653.             gradient[i] = FastMath.scalb(gradient[i], SCALING);
  654.             for (int j = 0; j <= i; ++j) {
  655.                 hessian[i][j] = FastMath.scalb(hessian[i][j], SCALING);
  656.             }
  657.         }


  660.         // apply the global mu/r factor
  661.         final double muOr = mu / r;
  662.         value         *= muOr;
  663.         gradient[0]    = muOr * gradient[0] - value / r;
  664.         gradient[1]   *= muOr;
  665.         gradient[2]   *= muOr;
  666.         hessian[0][0]  = muOr * hessian[0][0] - 2 * gradient[0] / r;
  667.         hessian[1][0]  = muOr * hessian[1][0] -     gradient[1] / r;
  668.         hessian[2][0]  = muOr * hessian[2][0] -     gradient[2] / r;
  669.         hessian[1][1] *= muOr;
  670.         hessian[2][1] *= muOr;
  671.         hessian[2][2] *= muOr;

  673.         // convert gradient and Hessian from spherical to Cartesian
  674.         final SphericalCoordinates sc = new SphericalCoordinates(position);
  675.         return new GradientHessian(sc.toCartesianGradient(gradient),
  676.                                    sc.toCartesianHessian(hessian, gradient));


  679.     }

  681.     /** Container for gradient and Hessian. */
  682.     private static class GradientHessian {

  684.         /** Gradient. */
  685.         private final double[] gradient;

  687.         /** Hessian. */
  688.         private final double[][] hessian;

  690.         /** Simple constructor.
  691.          * <p>
  692.          * A reference to the arrays is stored, they are <strong>not</strong> cloned.
  693.          * </p>
  694.          * @param gradient gradient
  695.          * @param hessian hessian
  696.          */
  697.         GradientHessian(final double[] gradient, final double[][] hessian) {
  698.             this.gradient = gradient;
  699.             this.hessian  = hessian;
  700.         }

  702.         /** Get a reference to the gradient.
  703.          * @return gradient (a reference to the internal array is returned)
  704.          */
  705.         public double[] getGradient() {
  706.             return gradient;
  707.         }

  709.         /** Get a reference to the Hessian.
  710.          * @return Hessian (a reference to the internal array is returned)
  711.          */
  712.         public double[][] getHessian() {
  713.             return hessian;
  714.         }

  716.     }

  718.     /** Compute a/r powers array.
  719.      * @param aOr a/r
  720.      * @return array containing (a/r)<sup>n</sup>
  721.      */
  722.     private double[] createDistancePowersArray(final double aOr) {

  724.         // initialize array
  725.         final double[] aOrN = new double[provider.getMaxDegree() + 1];
  726.         aOrN[0] = 1;
  727.         aOrN[1] = aOr;

  729.         // fill up array
  730.         for (int n = 2; n < aOrN.length; ++n) {
  731.             final int p = n / 2;
  732.             final int q = n - p;
  733.             aOrN[n] = aOrN[p] * aOrN[q];
  734.         }

  736.         return aOrN;

  738.     }
  739.     /** Compute a/r powers array.
  740.      * @param aOr a/r
  741.      * @param <T> type of field used
  742.      * @return array containing (a/r)<sup>n</sup>
  743.      */
  744.     private <T extends RealFieldElement<T>> T[] createDistancePowersArray(final T aOr) {

  746.         // initialize array
  747.         final T[] aOrN = MathArrays.buildArray(aOr.getField(), provider.getMaxDegree() + 1);
  748.         aOrN[0] = aOr.getField().getOne();
  749.         aOrN[1] = aOr;

  751.         // fill up array
  752.         for (int n = 2; n < aOrN.length; ++n) {
  753.             final int p = n / 2;
  754.             final int q = n - p;
  755.             aOrN[n] = aOrN[p].multiply(aOrN[q]);
  756.         }

  758.         return aOrN;

  760.     }

  762.     /** Compute longitude cosines and sines.
  763.      * @param cosLambda cos(λ)
  764.      * @param sinLambda sin(λ)
  765.      * @return array containing cos(m &times; λ) in row 0
  766.      * and sin(m &times; λ) in row 1
  767.      */
  768.     private double[][] createCosSinArrays(final double cosLambda, final double sinLambda) {

  770.         // initialize arrays
  771.         final double[][] cosSin = new double[2][provider.getMaxOrder() + 1];
  772.         cosSin[0][0] = 1;
  773.         cosSin[1][0] = 0;
  774.         if (provider.getMaxOrder() > 0) {
  775.             cosSin[0][1] = cosLambda;
  776.             cosSin[1][1] = sinLambda;

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

  781.                 // m * lambda is split as p * lambda + q * lambda, trying to avoid
  782.                 // p or q being much larger than the other. This reduces the number of
  783.                 // intermediate results reused to compute each value, and hence should limit
  784.                 // as much as possible roundoff error accumulation
  785.                 // (this does not change the number of floating point operations)
  786.                 final int p = m / 2;
  787.                 final int q = m - p;

  789.                 cosSin[0][m] = cosSin[0][p] * cosSin[0][q] - cosSin[1][p] * cosSin[1][q];
  790.                 cosSin[1][m] = cosSin[1][p] * cosSin[0][q] + cosSin[0][p] * cosSin[1][q];
  791.             }
  792.         }

  794.         return cosSin;

  796.     }

  798.     /** Compute longitude cosines and sines.
  799.      * @param cosLambda cos(λ)
  800.      * @param sinLambda sin(λ)
  801.      * @param <T> type of field used
  802.      * @return array containing cos(m &times; λ) in row 0
  803.      * and sin(m &times; λ) in row 1
  804.      */
  805.     private <T extends RealFieldElement<T>> T[][] createCosSinArrays(final T cosLambda, final T sinLambda) {

  807.         final T one = cosLambda.getField().getOne();
  808.         final T zero = cosLambda.getField().getZero();
  809.         // initialize arrays
  810.         final T[][] cosSin = MathArrays.buildArray(one.getField(), 2, provider.getMaxOrder() + 1);
  811.         cosSin[0][0] = one;
  812.         cosSin[1][0] = zero;
  813.         if (provider.getMaxOrder() > 0) {
  814.             cosSin[0][1] = cosLambda;
  815.             cosSin[1][1] = sinLambda;

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

  820.                 // m * lambda is split as p * lambda + q * lambda, trying to avoid
  821.                 // p or q being much larger than the other. This reduces the number of
  822.                 // intermediate results reused to compute each value, and hence should limit
  823.                 // as much as possible roundoff error accumulation
  824.                 // (this does not change the number of floating point operations)
  825.                 final int p = m / 2;
  826.                 final int q = m - p;

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

  831.             }
  832.         }

  834.         return cosSin;

  836.     }

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

  864.         final double u2 = u * u;

  866.         // initialize recursion from sectorial terms
  867.         int n = FastMath.max(2, m);
  868.         if (n == m) {
  869.             pnm0[n] = sectorial[n];
  870.             ++n;
  871.         }

  873.         // compute tesseral values
  874.         int localIndex = index;
  875.         while (n <= degree) {

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

  880.             ++localIndex;
  881.             ++n;

  883.         }

  885.         if (pnm1 != null) {

  887.             // initialize recursion from sectorial terms
  888.             n = FastMath.max(2, m);
  889.             if (n == m) {
  890.                 pnm1[n] = m * tOu * pnm0[n];
  891.                 ++n;
  892.             }

  894.             // compute tesseral values and derivatives with respect to polar angle
  895.             localIndex = index;
  896.             while (n <= degree) {

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

  901.                 ++localIndex;
  902.                 ++n;

  904.             }

  906.             if (pnm2 != null) {

  908.                 // initialize recursion from sectorial terms
  909.                 n = FastMath.max(2, m);
  910.                 if (n == m) {
  911.                     pnm2[n] = m * (tOu * pnm1[n] - pnm0[n] / u2);
  912.                     ++n;
  913.                 }

  915.                 // compute tesseral values and derivatives with respect to polar angle
  916.                 localIndex = index;
  917.                 while (n <= degree) {

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

  922.                     ++localIndex;
  923.                     ++n;

  925.                 }

  927.             }

  929.         }

  931.         return localIndex;

  933.     }

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

  962.         final T u2 = u.multiply(u);
  963.         final T zero = u.getField().getZero();
  964.         // initialize recursion from sectorial terms
  965.         int n = FastMath.max(2, m);
  966.         if (n == m) {
  967.             pnm0[n] = zero.add(sectorial[n]);
  968.             ++n;
  969.         }

  971.         // compute tesseral values
  972.         int localIndex = index;
  973.         while (n <= degree) {

  975.             // value (equation 27 of the paper)
  976.             pnm0[n] = t.multiply(gnmOj[localIndex]).multiply(pnm0Plus1[n]).subtract(u2.multiply(pnm0Plus2[n]).multiply(hnmOj[localIndex]));
  977.             ++localIndex;
  978.             ++n;

  980.         }
  981.         if (pnm1 != null) {

  983.             // initialize recursion from sectorial terms
  984.             n = FastMath.max(2, m);
  985.             if (n == m) {
  986.                 pnm1[n] = tOu.multiply(m).multiply(pnm0[n]);
  987.                 ++n;
  988.             }

  990.             // compute tesseral values and derivatives with respect to polar angle
  991.             localIndex = index;
  992.             while (n <= degree) {

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

  997.                 ++localIndex;
  998.                 ++n;

  1000.             }

  1002.             if (pnm2 != null) {

  1004.                 // initialize recursion from sectorial terms
  1005.                 n = FastMath.max(2, m);
  1006.                 if (n == m) {
  1007.                     pnm2[n] =   tOu.multiply(pnm1[n]).subtract(pnm0[n].divide(u2)).multiply(m);
  1008.                     ++n;
  1009.                 }

  1011.                 // compute tesseral values and derivatives with respect to polar angle
  1012.                 localIndex = index;
  1013.                 while (n <= degree) {

  1015.                     // second derivative (differential of equation 30 with respect to theta)
  1016.                     pnm2[n] = tOu.multiply(pnm1[n]).subtract(pnm0[n].divide(u2)).multiply(m).subtract(u.multiply(pnm1Plus1[n]).multiply(enm[localIndex]));
  1017.                     ++localIndex;
  1018.                     ++n;

  1020.                 }

  1022.             }

  1024.         }
  1025.         return localIndex;

  1027.     }

  1029.     /** {@inheritDoc} */
  1030.     @Override
  1031.     public Vector3D acceleration(final SpacecraftState s, final double[] parameters) {

  1033.         final double mu = parameters[0];

  1035.         // get the position in body frame
  1036.         final AbsoluteDate date       = s.getDate();
  1037.         final Transform fromBodyFrame = bodyFrame.getTransformTo(s.getFrame(), date);
  1038.         final Transform toBodyFrame   = fromBodyFrame.getInverse();
  1039.         final Vector3D position       = toBodyFrame.transformPosition(s.getPVCoordinates().getPosition());

  1041.         // gradient of the non-central part of the gravity field
  1042.         return fromBodyFrame.transformVector(new Vector3D(gradient(date, position, mu)));

  1044.     }

  1046.     /** {@inheritDoc} */
  1047.     public <T extends RealFieldElement<T>> FieldVector3D<T> acceleration(final FieldSpacecraftState<T> s,
  1048.                                                                          final T[] parameters) {

  1050.         final T mu = parameters[0];

  1052.         // check for faster computation dedicated to derivatives with respect to state
  1053.         if (isStateDerivative(s)) {
  1054.             @SuppressWarnings("unchecked")
  1055.             final FieldVector3D<DerivativeStructure> p = (FieldVector3D<DerivativeStructure>) s.getPVCoordinates().getPosition();
  1056.             @SuppressWarnings("unchecked")
  1057.             final FieldVector3D<T> a = (FieldVector3D<T>) accelerationWrtState(s.getDate().toAbsoluteDate(),
  1058.                                                                                s.getFrame(), p,
  1059.                                                                                (DerivativeStructure) mu);
  1060.             return a;
  1061.         }

  1063.         // get the position in body frame
  1064.         final FieldAbsoluteDate<T> date          = s.getDate();
  1065.         final Transform            fromBodyFrame = bodyFrame.getTransformTo(s.getFrame(), date.toAbsoluteDate());
  1066.         final Transform            toBodyFrame   = fromBodyFrame.getInverse();
  1067.         final FieldVector3D<T>     position      = toBodyFrame.transformPosition(s.getPVCoordinates().getPosition());

  1069.         // gradient of the non-central part of the gravity field
  1070.         return fromBodyFrame.transformVector(new FieldVector3D<>(gradient(date, position, mu)));

  1072.     }

  1074.     /** {@inheritDoc} */
  1075.     public Stream<EventDetector> getEventsDetectors() {
  1076.         return Stream.empty();
  1077.     }

  1079.     @Override
  1080.     /** {@inheritDoc} */
  1081.     public <T extends RealFieldElement<T>> Stream<FieldEventDetector<T>> getFieldEventsDetectors(final Field<T> field) {
  1082.         return Stream.empty();
  1083.     }

  1085.     /** Check if a field state corresponds to derivatives with respect to state.
  1086.      * @param state state to check
  1087.      * @param <T> type of the filed elements
  1088.      * @return true if state corresponds to derivatives with respect to state
  1089.      * @since 9.0
  1090.      */
  1091.     private <T extends RealFieldElement<T>> boolean isStateDerivative(final FieldSpacecraftState<T> state) {
  1092.         try {
  1093.             final DerivativeStructure dsMass = (DerivativeStructure) state.getMass();
  1094.             final int o = dsMass.getOrder();
  1095.             final int p = dsMass.getFreeParameters();
  1096.             if (o != 1 || (p < 3)) {
  1097.                 return false;
  1098.             }
  1099.             @SuppressWarnings("unchecked")
  1100.             final FieldPVCoordinates<DerivativeStructure> pv = (FieldPVCoordinates<DerivativeStructure>) state.getPVCoordinates();
  1101.             return isVariable(pv.getPosition().getX(), 0) &&
  1102.                    isVariable(pv.getPosition().getY(), 1) &&
  1103.                    isVariable(pv.getPosition().getZ(), 2);
  1104.         } catch (ClassCastException cce) {
  1105.             return false;
  1106.         }
  1107.     }

  1109.     /** Check if a derivative represents a specified variable.
  1110.      * @param ds derivative to check
  1111.      * @param index index of the variable
  1112.      * @return true if the derivative represents a specified variable
  1113.      * @since 9.0
  1114.      */
  1115.     private boolean isVariable(final DerivativeStructure ds, final int index) {
  1116.         final double[] derivatives = ds.getAllDerivatives();
  1117.         boolean check = true;
  1118.         for (int i = 1; i < derivatives.length; ++i) {
  1119.             check &= derivatives[i] == ((index + 1 == i) ? 1.0 : 0.0);
  1120.         }
  1121.         return check;
  1122.     }

  1124.     /** Compute acceleration derivatives with respect to state parameters.
  1125.      * <p>
  1126.      * From a theoretical point of view, this method computes the same values
  1127.      * as {@link #acceleration(FieldSpacecraftState, RealFieldElement[])} in the
  1128.      * specific case of {@link DerivativeStructure} with respect to state, so
  1129.      * it is less general. However, it is *much* faster in this important case.
  1130.      * <p>
  1131.      * <p>
  1132.      * The derivatives should be computed with respect to position. The input
  1133.      * parameters already take into account the free parameters (6 or 7 depending
  1134.      * on derivation with respect to mass being considered or not) and order
  1135.      * (always 1). Free parameters at indices 0, 1 and 2 correspond to derivatives
  1136.      * with respect to position. Free parameters at indices 3, 4 and 5 correspond
  1137.      * to derivatives with respect to velocity (these derivatives will remain zero
  1138.      * as acceleration due to gravity does not depend on velocity). Free parameter
  1139.      * at index 6 (if present) corresponds to to derivatives with respect to mass
  1140.      * (this derivative will remain zero as acceleration due to gravity does not
  1141.      * depend on mass).
  1142.      * </p>
  1143.      * @param date current date
  1144.     * @param frame inertial reference frame for state (both orbit and attitude)
  1145.      * @param position position of spacecraft in inertial frame
  1146.      * @param mu central attraction coefficient to use
  1147.      * @return acceleration with all derivatives specified by the input parameters
  1148.      * own derivatives
  1149.           * @since 6.0
  1150.      */
  1151.     private FieldVector3D<DerivativeStructure> accelerationWrtState(final AbsoluteDate date, final Frame frame,
  1152.                                                                     final FieldVector3D<DerivativeStructure> position,
  1153.                                                                     final DerivativeStructure mu) {

  1155.         // get the position in body frame
  1156.         final Transform fromBodyFrame = bodyFrame.getTransformTo(frame, date);
  1157.         final Transform toBodyFrame   = fromBodyFrame.getInverse();
  1158.         final Vector3D positionBody   = toBodyFrame.transformPosition(position.toVector3D());

  1160.         // compute gradient and Hessian
  1161.         final GradientHessian gh   = gradientHessian(date, positionBody, mu.getReal());

  1163.         // gradient of the non-central part of the gravity field
  1164.         final double[] gInertial = fromBodyFrame.transformVector(new Vector3D(gh.getGradient())).toArray();

  1166.         // Hessian of the non-central part of the gravity field
  1167.         final RealMatrix hBody     = new Array2DRowRealMatrix(gh.getHessian(), false);
  1168.         final RealMatrix rot       = new Array2DRowRealMatrix(toBodyFrame.getRotation().getMatrix());
  1169.         final RealMatrix hInertial = rot.transpose().multiply(hBody).multiply(rot);

  1171.         // distribute all partial derivatives in a compact acceleration vector
  1172.         final double[] derivatives = new double[1 + position.getX().getFreeParameters()];
  1173.         final DerivativeStructure[] accDer = new DerivativeStructure[3];
  1174.         for (int i = 0; i < 3; ++i) {

  1176.             // first element is value of acceleration (i.e. gradient of field)
  1177.             derivatives[0] = gInertial[i];

  1179.             // next three elements are one row of the Jacobian of acceleration (i.e. Hessian of field)
  1180.             derivatives[1] = hInertial.getEntry(i, 0);
  1181.             derivatives[2] = hInertial.getEntry(i, 1);
  1182.             derivatives[3] = hInertial.getEntry(i, 2);

  1184.             // next element is derivative with respect to parameter mu
  1185.             if (derivatives.length > 4 && isVariable(mu, 3)) {
  1186.                 derivatives[4] = gInertial[i] / mu.getReal();
  1187.             }

  1189.             accDer[i] = position.getX().getFactory().build(derivatives);

  1191.         }

  1193.         return new FieldVector3D<>(accDer);

  1195.     }

  1197.     /** {@inheritDoc} */
  1198.     public ParameterDriver[] getParametersDrivers() {
  1199.         return new ParameterDriver[] {
  1200.             gmParameterDriver
  1201.         };
  1202.     }

  1204. }
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