HolmesFeatherstoneAttractionModel.java

  1. /* Copyright 2002-2022 CS GROUP
  2.  * Licensed to CS GROUP (CS) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * CS licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *   http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.orekit.forces.gravity;


  20. import java.util.Collections;
  21. import java.util.List;
  22. import java.util.stream.Stream;

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

  51. /** This class represents the gravitational field of a celestial body.
  52.  * <p>
  53.  * The algorithm implemented in this class has been designed by S. A. Holmes
  54.  * and W. E. Featherstone from Department of Spatial Sciences, Curtin University
  55.  * of Technology, Perth, Australia. It is described in their 2002 paper: <a
  56.  * href="https://www.researchgate.net/publication/226460594_A_unified_approach_to_the_Clenshaw_summation_and_the_recursive_computation_of_very_high_degree_and_order_normalised_associated_Legendre_functions">
  57.  * A unified approach to he Clenshaw summation and the recursive computation of
  58.  * very high degree and order normalised associated Legendre functions</a>
  59.  * (Journal of Geodesy (2002) 76: 279–299).
  60.  * </p>
  61.  * <p>
  62.  * This model directly uses normalized coefficients and stable recursion algorithms
  63.  * so it is more suited to high degree gravity fields than the classical Cunningham
  64.  * Droziner models which use un-normalized coefficients.
  65.  * </p>
  66.  * <p>
  67.  * Among the different algorithms presented in Holmes and Featherstone paper, this
  68.  * class implements the <em>modified forward row method</em>. All recursion coefficients
  69.  * are precomputed and stored for greater performance. This caching was suggested in the
  70.  * paper but not used due to the large memory requirements. Since 2002, even low end
  71.  * computers and mobile devices do have sufficient memory so this caching has become
  72.  * feasible nowadays.
  73.  * <p>
  74.  * @author Luc Maisonobe
  75.  * @since 6.0
  76.  */

  78. public class HolmesFeatherstoneAttractionModel extends AbstractForceModel implements TideSystemProvider {

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

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

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

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

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

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

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

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

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

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

  123.         gmParameterDriver = new ParameterDriver(NewtonianAttraction.CENTRAL_ATTRACTION_COEFFICIENT,
  124.                                                 provider.getMu(), MU_SCALE, 0.0, Double.POSITIVE_INFINITY);

  126.         this.provider  = provider;
  127.         this.bodyFrame = centralBodyFrame;

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

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

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

  161.     }

  163.     /** {@inheritDoc} */
  164.     @Override
  165.     public boolean dependsOnPositionOnly() {
  166.         return true;
  167.     }

  169.     /** {@inheritDoc} */
  170.     public TideSystem getTideSystem() {
  171.         return provider.getTideSystem();
  172.     }

  174.     /** Get the central attraction coefficient μ.
  175.      * @return mu central attraction coefficient (m³/s²)
  176.      */
  177.     public double getMu() {
  178.         return gmParameterDriver.getValue();
  179.     }

  181.     /** Compute the value of the gravity field.
  182.      * @param date current date
  183.      * @param position position at which gravity field is desired in body frame
  184.      * @param mu central attraction coefficient to use
  185.      * @return value of the gravity field (central and non-central parts summed together)
  186.      */
  187.     public double value(final AbsoluteDate date, final Vector3D position,
  188.                         final double mu) {
  189.         return mu / position.getNorm() + nonCentralPart(date, position, mu);
  190.     }

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

  200.         final int degree = provider.getMaxDegree();
  201.         final int order  = provider.getMaxOrder();
  202.         final NormalizedSphericalHarmonics harmonics = provider.onDate(date);

  204.         // allocate the columns for recursion
  205.         double[] pnm0Plus2 = new double[degree + 1];
  206.         double[] pnm0Plus1 = new double[degree + 1];
  207.         double[] pnm0      = new double[degree + 1];

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

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

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

  229.         // outer summation over order
  230.         int    index = 0;
  231.         double value = 0;
  232.         for (int m = degree; m >= 0; --m) {

  234.             // compute tesseral terms without derivatives
  235.             index = computeTesseral(m, degree, index, t, u, tOu,
  236.                                     pnm0Plus2, pnm0Plus1, null, pnm0, null, null);

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

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

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

  252.             }

  254.             // rotate the recursion arrays
  255.             final double[] tmp = pnm0Plus2;
  256.             pnm0Plus2 = pnm0Plus1;
  257.             pnm0Plus1 = pnm0;
  258.             pnm0      = tmp;

  260.         }

  262.         // scale back
  263.         value = FastMath.scalb(value, SCALING);

  265.         // apply the global mu/r factor
  266.         return mu * value / r;

  268.     }

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

  278.         final int degree = provider.getMaxDegree();
  279.         final int order  = provider.getMaxOrder();
  280.         final NormalizedSphericalHarmonics harmonics = provider.onDate(date);

  282.         // allocate the columns for recursion
  283.         double[] pnm0Plus2  = new double[degree + 1];
  284.         double[] pnm0Plus1  = new double[degree + 1];
  285.         double[] pnm0       = new double[degree + 1];
  286.         final double[] pnm1 = new double[degree + 1];

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

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

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

  309.         // outer summation over order
  310.         int    index = 0;
  311.         double value = 0;
  312.         final double[] gradient = new double[3];
  313.         for (int m = degree; m >= 0; --m) {

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

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

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

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

  356.             }

  358.             // rotate the recursion arrays
  359.             final double[] tmp = pnm0Plus2;
  360.             pnm0Plus2 = pnm0Plus1;
  361.             pnm0Plus1 = pnm0;
  362.             pnm0      = tmp;

  364.         }

  366.         // scale back
  367.         value       = FastMath.scalb(value,       SCALING);
  368.         gradient[0] = FastMath.scalb(gradient[0], SCALING);
  369.         gradient[1] = FastMath.scalb(gradient[1], SCALING);
  370.         gradient[2] = FastMath.scalb(gradient[2], SCALING);

  372.         // apply the global mu/r factor
  373.         final double muOr = mu / r;
  374.         value            *= muOr;
  375.         gradient[0]       = muOr * gradient[0] - value / r;
  376.         gradient[1]      *= muOr;
  377.         gradient[2]      *= muOr;

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

  382.     }

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

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

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

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

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

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

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

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

  476.         }
  477.         // scale back
  478.         value       = value.scalb(SCALING);
  479.         gradient[0] = gradient[0].scalb(SCALING);
  480.         gradient[1] = gradient[1].scalb(SCALING);
  481.         gradient[2] = gradient[2].scalb(SCALING);

  483.         // apply the global mu/r factor
  484.         final T muOr = r.reciprocal().multiply(mu);
  485.         value            = value.multiply(muOr);
  486.         gradient[0]       = muOr.multiply(gradient[0]).subtract(value.divide(r));
  487.         gradient[1]      = gradient[1].multiply(muOr);
  488.         gradient[2]      = gradient[2].multiply(muOr);

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

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

  504.         // row representing the gradient of r
  505.         jacobianPos[0][0] = xPos.divide(rPos);
  506.         jacobianPos[0][1] = yPos.divide(rPos);
  507.         jacobianPos[0][2] = zPos.divide(rPos);

  509.         // row representing the gradient of theta
  510.         jacobianPos[1][0] =  yPos.negate().divide(rho2Pos);
  511.         jacobianPos[1][1] =  xPos.divide(rho2Pos);
  512.         // jacobian[1][2] is already set to 0 at allocation time

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

  524.     }

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

  534.         final int degree = provider.getMaxDegree();
  535.         final int order  = provider.getMaxOrder();
  536.         final NormalizedSphericalHarmonics harmonics = provider.onDate(date);

  538.         // allocate the columns for recursion
  539.         double[] pnm0Plus2  = new double[degree + 1];
  540.         double[] pnm0Plus1  = new double[degree + 1];
  541.         double[] pnm0       = new double[degree + 1];
  542.         double[] pnm1Plus1  = new double[degree + 1];
  543.         double[] pnm1       = new double[degree + 1];
  544.         final double[] pnm2 = new double[degree + 1];

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

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

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

  567.         // outer summation over order
  568.         int    index = 0;
  569.         double value = 0;
  570.         final double[]   gradient = new double[3];
  571.         final double[][] hessian  = new double[3][3];
  572.         for (int m = degree; m >= 0; --m) {

  574.             // compute tesseral terms
  575.             index = computeTesseral(m, degree, index, t, u, tOu,
  576.                                     pnm0Plus2, pnm0Plus1, pnm1Plus1, pnm0, pnm1, pnm2);

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

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

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

  633.             }

  635.             // rotate the recursion arrays
  636.             final double[] tmp0 = pnm0Plus2;
  637.             pnm0Plus2 = pnm0Plus1;
  638.             pnm0Plus1 = pnm0;
  639.             pnm0      = tmp0;
  640.             final double[] tmp1 = pnm1Plus1;
  641.             pnm1Plus1 = pnm1;
  642.             pnm1      = tmp1;

  644.         }

  646.         // scale back
  647.         value = FastMath.scalb(value, SCALING);
  648.         for (int i = 0; i < 3; ++i) {
  649.             gradient[i] = FastMath.scalb(gradient[i], SCALING);
  650.             for (int j = 0; j <= i; ++j) {
  651.                 hessian[i][j] = FastMath.scalb(hessian[i][j], SCALING);
  652.             }
  653.         }


  656.         // apply the global mu/r factor
  657.         final double muOr = mu / r;
  658.         value         *= muOr;
  659.         gradient[0]    = muOr * gradient[0] - value / r;
  660.         gradient[1]   *= muOr;
  661.         gradient[2]   *= muOr;
  662.         hessian[0][0]  = muOr * hessian[0][0] - 2 * gradient[0] / r;
  663.         hessian[1][0]  = muOr * hessian[1][0] -     gradient[1] / r;
  664.         hessian[2][0]  = muOr * hessian[2][0] -     gradient[2] / r;
  665.         hessian[1][1] *= muOr;
  666.         hessian[2][1] *= muOr;
  667.         hessian[2][2] *= muOr;

  669.         // convert gradient and Hessian from spherical to Cartesian
  670.         final SphericalCoordinates sc = new SphericalCoordinates(position);
  671.         return new GradientHessian(sc.toCartesianGradient(gradient),
  672.                                    sc.toCartesianHessian(hessian, gradient));


  675.     }

  677.     /** Container for gradient and Hessian. */
  678.     private static class GradientHessian {

  680.         /** Gradient. */
  681.         private final double[] gradient;

  683.         /** Hessian. */
  684.         private final double[][] hessian;

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

  698.         /** Get a reference to the gradient.
  699.          * @return gradient (a reference to the internal array is returned)
  700.          */
  701.         public double[] getGradient() {
  702.             return gradient;
  703.         }

  705.         /** Get a reference to the Hessian.
  706.          * @return Hessian (a reference to the internal array is returned)
  707.          */
  708.         public double[][] getHessian() {
  709.             return hessian;
  710.         }

  712.     }

  714.     /** Compute a/r powers array.
  715.      * @param aOr a/r
  716.      * @return array containing (a/r)<sup>n</sup>
  717.      */
  718.     private double[] createDistancePowersArray(final double aOr) {

  720.         // initialize array
  721.         final double[] aOrN = new double[provider.getMaxDegree() + 1];
  722.         aOrN[0] = 1;
  723.         aOrN[1] = aOr;

  725.         // fill up array
  726.         for (int n = 2; n < aOrN.length; ++n) {
  727.             final int p = n / 2;
  728.             final int q = n - p;
  729.             aOrN[n] = aOrN[p] * aOrN[q];
  730.         }

  732.         return aOrN;

  734.     }
  735.     /** Compute a/r powers array.
  736.      * @param aOr a/r
  737.      * @param <T> type of field used
  738.      * @return array containing (a/r)<sup>n</sup>
  739.      */
  740.     private <T extends CalculusFieldElement<T>> T[] createDistancePowersArray(final T aOr) {

  742.         // initialize array
  743.         final T[] aOrN = MathArrays.buildArray(aOr.getField(), provider.getMaxDegree() + 1);
  744.         aOrN[0] = aOr.getField().getOne();
  745.         aOrN[1] = aOr;

  747.         // fill up array
  748.         for (int n = 2; n < aOrN.length; ++n) {
  749.             final int p = n / 2;
  750.             final int q = n - p;
  751.             aOrN[n] = aOrN[p].multiply(aOrN[q]);
  752.         }

  754.         return aOrN;

  756.     }

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

  766.         // initialize arrays
  767.         final double[][] cosSin = new double[2][provider.getMaxOrder() + 1];
  768.         cosSin[0][0] = 1;
  769.         cosSin[1][0] = 0;
  770.         if (provider.getMaxOrder() > 0) {
  771.             cosSin[0][1] = cosLambda;
  772.             cosSin[1][1] = sinLambda;

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

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

  785.                 cosSin[0][m] = cosSin[0][p] * cosSin[0][q] - cosSin[1][p] * cosSin[1][q];
  786.                 cosSin[1][m] = cosSin[1][p] * cosSin[0][q] + cosSin[0][p] * cosSin[1][q];
  787.             }
  788.         }

  790.         return cosSin;

  792.     }

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

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

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

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

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

  827.             }
  828.         }

  830.         return cosSin;

  832.     }

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

  860.         final double u2 = u * u;

  862.         // initialize recursion from sectorial terms
  863.         int n = FastMath.max(2, m);
  864.         if (n == m) {
  865.             pnm0[n] = sectorial[n];
  866.             ++n;
  867.         }

  869.         // compute tesseral values
  870.         int localIndex = index;
  871.         while (n <= degree) {

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

  876.             ++localIndex;
  877.             ++n;

  879.         }

  881.         if (pnm1 != null) {

  883.             // initialize recursion from sectorial terms
  884.             n = FastMath.max(2, m);
  885.             if (n == m) {
  886.                 pnm1[n] = m * tOu * pnm0[n];
  887.                 ++n;
  888.             }

  890.             // compute tesseral values and derivatives with respect to polar angle
  891.             localIndex = index;
  892.             while (n <= degree) {

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

  897.                 ++localIndex;
  898.                 ++n;

  900.             }

  902.             if (pnm2 != null) {

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

  911.                 // compute tesseral values and derivatives with respect to polar angle
  912.                 localIndex = index;
  913.                 while (n <= degree) {

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

  918.                     ++localIndex;
  919.                     ++n;

  921.                 }

  923.             }

  925.         }

  927.         return localIndex;

  929.     }

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

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

  967.         // compute tesseral values
  968.         int localIndex = index;
  969.         while (n <= degree) {

  971.             // value (equation 27 of the paper)
  972.             pnm0[n] = t.multiply(gnmOj[localIndex]).multiply(pnm0Plus1[n]).subtract(u2.multiply(pnm0Plus2[n]).multiply(hnmOj[localIndex]));
  973.             ++localIndex;
  974.             ++n;

  976.         }
  977.         if (pnm1 != null) {

  979.             // initialize recursion from sectorial terms
  980.             n = FastMath.max(2, m);
  981.             if (n == m) {
  982.                 pnm1[n] = tOu.multiply(m).multiply(pnm0[n]);
  983.                 ++n;
  984.             }

  986.             // compute tesseral values and derivatives with respect to polar angle
  987.             localIndex = index;
  988.             while (n <= degree) {

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

  993.                 ++localIndex;
  994.                 ++n;

  996.             }

  998.             if (pnm2 != null) {

  1000.                 // initialize recursion from sectorial terms
  1001.                 n = FastMath.max(2, m);
  1002.                 if (n == m) {
  1003.                     pnm2[n] =   tOu.multiply(pnm1[n]).subtract(pnm0[n].divide(u2)).multiply(m);
  1004.                     ++n;
  1005.                 }

  1007.                 // compute tesseral values and derivatives with respect to polar angle
  1008.                 localIndex = index;
  1009.                 while (n <= degree) {

  1011.                     // second derivative (differential of equation 30 with respect to theta)
  1012.                     pnm2[n] = tOu.multiply(pnm1[n]).subtract(pnm0[n].divide(u2)).multiply(m).subtract(u.multiply(pnm1Plus1[n]).multiply(enm[localIndex]));
  1013.                     ++localIndex;
  1014.                     ++n;

  1016.                 }

  1018.             }

  1020.         }
  1021.         return localIndex;

  1023.     }

  1025.     /** {@inheritDoc} */
  1026.     @Override
  1027.     public Vector3D acceleration(final SpacecraftState s, final double[] parameters) {

  1029.         final double mu = parameters[0];

  1031.         // get the position in body frame
  1032.         final AbsoluteDate date       = s.getDate();
  1033.         final Transform fromBodyFrame = bodyFrame.getTransformTo(s.getFrame(), date);
  1034.         final Transform toBodyFrame   = fromBodyFrame.getInverse();
  1035.         final Vector3D position       = toBodyFrame.transformPosition(s.getPVCoordinates().getPosition());

  1037.         // gradient of the non-central part of the gravity field
  1038.         return fromBodyFrame.transformVector(new Vector3D(gradient(date, position, mu)));

  1040.     }

  1042.     /** {@inheritDoc} */
  1043.     public <T extends CalculusFieldElement<T>> FieldVector3D<T> acceleration(final FieldSpacecraftState<T> s,
  1044.                                                                          final T[] parameters) {

  1046.         final T mu = parameters[0];

  1048.         // check for faster computation dedicated to derivatives with respect to state
  1049.         if (isGradientStateDerivative(s)) {
  1050.             @SuppressWarnings("unchecked")
  1051.             final FieldVector3D<Gradient> p = (FieldVector3D<Gradient>) s.getPVCoordinates().getPosition();
  1052.             @SuppressWarnings("unchecked")
  1053.             final FieldVector3D<T> a = (FieldVector3D<T>) accelerationWrtState(s.getDate().toAbsoluteDate(),
  1054.                                                                                s.getFrame(), p,
  1055.                                                                                (Gradient) mu);
  1056.             return a;
  1057.         } else if (isDSStateDerivative(s)) {
  1058.             @SuppressWarnings("unchecked")
  1059.             final FieldVector3D<DerivativeStructure> p = (FieldVector3D<DerivativeStructure>) s.getPVCoordinates().getPosition();
  1060.             @SuppressWarnings("unchecked")
  1061.             final FieldVector3D<T> a = (FieldVector3D<T>) accelerationWrtState(s.getDate().toAbsoluteDate(),
  1062.                                                                                s.getFrame(), p,
  1063.                                                                                (DerivativeStructure) mu);
  1064.             return a;
  1065.         }

  1067.         // get the position in body frame
  1068.         final FieldAbsoluteDate<T> date          = s.getDate();
  1069.         final Transform            fromBodyFrame = bodyFrame.getTransformTo(s.getFrame(), date.toAbsoluteDate());
  1070.         final Transform            toBodyFrame   = fromBodyFrame.getInverse();
  1071.         final FieldVector3D<T>     position      = toBodyFrame.transformPosition(s.getPVCoordinates().getPosition());

  1073.         // gradient of the non-central part of the gravity field
  1074.         return fromBodyFrame.transformVector(new FieldVector3D<>(gradient(date, position, mu)));

  1076.     }

  1078.     /** {@inheritDoc} */
  1079.     public Stream<EventDetector> getEventsDetectors() {
  1080.         return Stream.empty();
  1081.     }

  1083.     @Override
  1084.     /** {@inheritDoc} */
  1085.     public <T extends CalculusFieldElement<T>> Stream<FieldEventDetector<T>> getFieldEventsDetectors(final Field<T> field) {
  1086.         return Stream.empty();
  1087.     }

  1089.     /** Check if a field state corresponds to derivatives with respect to state.
  1090.      * @param state state to check
  1091.      * @param <T> type of the filed elements
  1092.      * @return true if state corresponds to derivatives with respect to state
  1093.      * @since 9.0
  1094.      */
  1095.     private <T extends CalculusFieldElement<T>> boolean isDSStateDerivative(final FieldSpacecraftState<T> state) {
  1096.         try {
  1097.             final DerivativeStructure dsMass = (DerivativeStructure) state.getMass();
  1098.             final int o = dsMass.getOrder();
  1099.             final int p = dsMass.getFreeParameters();
  1100.             if (o != 1 || p < 3) {
  1101.                 return false;
  1102.             }
  1103.             @SuppressWarnings("unchecked")
  1104.             final FieldPVCoordinates<DerivativeStructure> pv = (FieldPVCoordinates<DerivativeStructure>) state.getPVCoordinates();
  1105.             return isVariable(pv.getPosition().getX(), 0) &&
  1106.                    isVariable(pv.getPosition().getY(), 1) &&
  1107.                    isVariable(pv.getPosition().getZ(), 2);
  1108.         } catch (ClassCastException cce) {
  1109.             return false;
  1110.         }
  1111.     }

  1113.     /** Check if a field state corresponds to derivatives with respect to state.
  1114.      * @param state state to check
  1115.      * @param <T> type of the filed elements
  1116.      * @return true if state corresponds to derivatives with respect to state
  1117.      * @since 10.2
  1118.      */
  1119.     private <T extends CalculusFieldElement<T>> boolean isGradientStateDerivative(final FieldSpacecraftState<T> state) {
  1120.         try {
  1121.             final Gradient gMass = (Gradient) state.getMass();
  1122.             final int p = gMass.getFreeParameters();
  1123.             if (p < 3) {
  1124.                 return false;
  1125.             }
  1126.             @SuppressWarnings("unchecked")
  1127.             final FieldPVCoordinates<Gradient> pv = (FieldPVCoordinates<Gradient>) state.getPVCoordinates();
  1128.             return isVariable(pv.getPosition().getX(), 0) &&
  1129.                    isVariable(pv.getPosition().getY(), 1) &&
  1130.                    isVariable(pv.getPosition().getZ(), 2);
  1131.         } catch (ClassCastException cce) {
  1132.             return false;
  1133.         }
  1134.     }

  1136.     /** Check if a derivative represents a specified variable.
  1137.      * @param ds derivative to check
  1138.      * @param index index of the variable
  1139.      * @return true if the derivative represents a specified variable
  1140.      * @since 9.0
  1141.      */
  1142.     private boolean isVariable(final DerivativeStructure ds, final int index) {
  1143.         final double[] derivatives = ds.getAllDerivatives();
  1144.         boolean check = true;
  1145.         for (int i = 1; i < derivatives.length; ++i) {
  1146.             check &= derivatives[i] == ((index + 1 == i) ? 1.0 : 0.0);
  1147.         }
  1148.         return check;
  1149.     }

  1151.     /** Check if a derivative represents a specified variable.
  1152.      * @param g derivative to check
  1153.      * @param index index of the variable
  1154.      * @return true if the derivative represents a specified variable
  1155.      * @since 10.2
  1156.      */
  1157.     private boolean isVariable(final Gradient g, final int index) {
  1158.         final double[] derivatives = g.getGradient();
  1159.         boolean check = true;
  1160.         for (int i = 0; i < derivatives.length; ++i) {
  1161.             check &= derivatives[i] == ((index == i) ? 1.0 : 0.0);
  1162.         }
  1163.         return check;
  1164.     }

  1166.     /** Compute acceleration derivatives with respect to state parameters.
  1167.      * <p>
  1168.      * From a theoretical point of view, this method computes the same values
  1169.      * as {@link #acceleration(FieldSpacecraftState, CalculusFieldElement[])} in the
  1170.      * specific case of {@link DerivativeStructure} with respect to state, so
  1171.      * it is less general. However, it is *much* faster in this important case.
  1172.      * <p>
  1173.      * <p>
  1174.      * The derivatives should be computed with respect to position. The input
  1175.      * parameters already take into account the free parameters (6 or 7 depending
  1176.      * on derivation with respect to mass being considered or not) and order
  1177.      * (always 1). Free parameters at indices 0, 1 and 2 correspond to derivatives
  1178.      * with respect to position. Free parameters at indices 3, 4 and 5 correspond
  1179.      * to derivatives with respect to velocity (these derivatives will remain zero
  1180.      * as acceleration due to gravity does not depend on velocity). Free parameter
  1181.      * at index 6 (if present) corresponds to to derivatives with respect to mass
  1182.      * (this derivative will remain zero as acceleration due to gravity does not
  1183.      * depend on mass).
  1184.      * </p>
  1185.      * @param date current date
  1186.      * @param frame inertial reference frame for state (both orbit and attitude)
  1187.      * @param position position of spacecraft in inertial frame
  1188.      * @param mu central attraction coefficient to use
  1189.      * @return acceleration with all derivatives specified by the input parameters
  1190.      * own derivatives
  1191.      * @since 6.0
  1192.      */
  1193.     private FieldVector3D<DerivativeStructure> accelerationWrtState(final AbsoluteDate date, final Frame frame,
  1194.                                                                     final FieldVector3D<DerivativeStructure> position,
  1195.                                                                     final DerivativeStructure mu) {

  1197.         // free parameters
  1198.         final int freeParameters = mu.getFreeParameters();

  1200.         // get the position in body frame
  1201.         final Transform fromBodyFrame = bodyFrame.getTransformTo(frame, date);
  1202.         final Transform toBodyFrame   = fromBodyFrame.getInverse();
  1203.         final Vector3D positionBody   = toBodyFrame.transformPosition(position.toVector3D());

  1205.         // compute gradient and Hessian
  1206.         final GradientHessian gh   = gradientHessian(date, positionBody, mu.getReal());

  1208.         // gradient of the non-central part of the gravity field
  1209.         final double[] gInertial = fromBodyFrame.transformVector(new Vector3D(gh.getGradient())).toArray();

  1211.         // Hessian of the non-central part of the gravity field
  1212.         final RealMatrix hBody     = new Array2DRowRealMatrix(gh.getHessian(), false);
  1213.         final RealMatrix rot       = new Array2DRowRealMatrix(toBodyFrame.getRotation().getMatrix());
  1214.         final RealMatrix hInertial = rot.transpose().multiply(hBody).multiply(rot);

  1216.         // distribute all partial derivatives in a compact acceleration vector
  1217.         final double[] derivatives = new double[freeParameters + 1];
  1218.         final DerivativeStructure[] accDer = new DerivativeStructure[3];
  1219.         for (int i = 0; i < 3; ++i) {

  1221.             // first element is value of acceleration (i.e. gradient of field)
  1222.             derivatives[0] = gInertial[i];

  1224.             // Jacobian of acceleration (i.e. Hessian of field)
  1225.             derivatives[1] = hInertial.getEntry(i, 0);
  1226.             derivatives[2] = hInertial.getEntry(i, 1);
  1227.             derivatives[3] = hInertial.getEntry(i, 2);

  1229.             // next element is derivative with respect to parameter mu
  1230.             if (derivatives.length > 4 && isVariable(mu, 3)) {
  1231.                 derivatives[4] = gInertial[i] / mu.getReal();
  1232.             }

  1234.             accDer[i] = position.getX().getFactory().build(derivatives);

  1236.         }

  1238.         return new FieldVector3D<>(accDer);

  1240.     }

  1242.     /** Compute acceleration derivatives with respect to state parameters.
  1243.      * <p>
  1244.      * From a theoretical point of view, this method computes the same values
  1245.      * as {@link #acceleration(FieldSpacecraftState, CalculusFieldElement[])} in the
  1246.      * specific case of {@link DerivativeStructure} with respect to state, so
  1247.      * it is less general. However, it is *much* faster in this important case.
  1248.      * <p>
  1249.      * <p>
  1250.      * The derivatives should be computed with respect to position. The input
  1251.      * parameters already take into account the free parameters (6 or 7 depending
  1252.      * on derivation with respect to mass being considered or not) and order
  1253.      * (always 1). Free parameters at indices 0, 1 and 2 correspond to derivatives
  1254.      * with respect to position. Free parameters at indices 3, 4 and 5 correspond
  1255.      * to derivatives with respect to velocity (these derivatives will remain zero
  1256.      * as acceleration due to gravity does not depend on velocity). Free parameter
  1257.      * at index 6 (if present) corresponds to to derivatives with respect to mass
  1258.      * (this derivative will remain zero as acceleration due to gravity does not
  1259.      * depend on mass).
  1260.      * </p>
  1261.      * @param date current date
  1262.      * @param frame inertial reference frame for state (both orbit and attitude)
  1263.      * @param position position of spacecraft in inertial frame
  1264.      * @param mu central attraction coefficient to use
  1265.      * @return acceleration with all derivatives specified by the input parameters
  1266.      * own derivatives
  1267.      * @since 10.2
  1268.      */
  1269.     private FieldVector3D<Gradient> accelerationWrtState(final AbsoluteDate date, final Frame frame,
  1270.                                                          final FieldVector3D<Gradient> position,
  1271.                                                          final Gradient mu) {

  1273.         // free parameters
  1274.         final int freeParameters = mu.getFreeParameters();

  1276.         // get the position in body frame
  1277.         final Transform fromBodyFrame = bodyFrame.getTransformTo(frame, date);
  1278.         final Transform toBodyFrame   = fromBodyFrame.getInverse();
  1279.         final Vector3D positionBody   = toBodyFrame.transformPosition(position.toVector3D());

  1281.         // compute gradient and Hessian
  1282.         final GradientHessian gh   = gradientHessian(date, positionBody, mu.getReal());

  1284.         // gradient of the non-central part of the gravity field
  1285.         final double[] gInertial = fromBodyFrame.transformVector(new Vector3D(gh.getGradient())).toArray();

  1287.         // Hessian of the non-central part of the gravity field
  1288.         final RealMatrix hBody     = new Array2DRowRealMatrix(gh.getHessian(), false);
  1289.         final RealMatrix rot       = new Array2DRowRealMatrix(toBodyFrame.getRotation().getMatrix());
  1290.         final RealMatrix hInertial = rot.transpose().multiply(hBody).multiply(rot);

  1292.         // distribute all partial derivatives in a compact acceleration vector
  1293.         final double[] derivatives = new double[freeParameters];
  1294.         final Gradient[] accDer = new Gradient[3];
  1295.         for (int i = 0; i < 3; ++i) {

  1297.             // Jacobian of acceleration (i.e. Hessian of field)
  1298.             derivatives[0] = hInertial.getEntry(i, 0);
  1299.             derivatives[1] = hInertial.getEntry(i, 1);
  1300.             derivatives[2] = hInertial.getEntry(i, 2);

  1302.             // next element is derivative with respect to parameter mu
  1303.             if (derivatives.length > 3 && isVariable(mu, 3)) {
  1304.                 derivatives[3] = gInertial[i] / mu.getReal();
  1305.             }

  1307.             accDer[i] = new Gradient(gInertial[i], derivatives);

  1309.         }

  1311.         return new FieldVector3D<>(accDer);

  1313.     }

  1315.     /** {@inheritDoc} */
  1316.     public List<ParameterDriver> getParametersDrivers() {
  1317.         return Collections.singletonList(gmParameterDriver);
  1318.     }

  1320. }
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