HolmesFeatherstoneAttractionModel.java

  1. /* Copyright 2002-2020 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.stream.Stream;

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

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

  76. public class HolmesFeatherstoneAttractionModel extends AbstractForceModel implements TideSystemProvider {

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

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

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

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

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

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

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

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

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

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

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

  124.         this.provider  = provider;
  125.         this.bodyFrame = centralBodyFrame;

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

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

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

  159.     }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  250.             }

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

  258.         }

  260.         // scale back
  261.         value = FastMath.scalb(value, SCALING);

  263.         // apply the global mu/r factor
  264.         return mu * value / r;

  266.     }

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

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

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

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

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

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

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

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

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

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

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

  354.             }

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

  362.         }

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

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

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

  380.     }

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

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

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

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

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

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

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

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

  474.         }
  475.         // scale back
  476.         value       = value.scalb(SCALING);
  477.         gradient[0] = gradient[0].scalb(SCALING);
  478.         gradient[1] = gradient[1].scalb(SCALING);
  479.         gradient[2] = gradient[2].scalb(SCALING);

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

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

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

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

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

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

  522.     }

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

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

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

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

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

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

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

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

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

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

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

  631.             }

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

  642.         }

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


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

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


  673.     }

  675.     /** Container for gradient and Hessian. */
  676.     private static class GradientHessian {

  678.         /** Gradient. */
  679.         private final double[] gradient;

  681.         /** Hessian. */
  682.         private final double[][] hessian;

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

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

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

  710.     }

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

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

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

  730.         return aOrN;

  732.     }
  733.     /** Compute a/r powers array.
  734.      * @param aOr a/r
  735.      * @param <T> type of field used
  736.      * @return array containing (a/r)<sup>n</sup>
  737.      */
  738.     private <T extends RealFieldElement<T>> T[] createDistancePowersArray(final T aOr) {

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

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

  752.         return aOrN;

  754.     }

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

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

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

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

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

  788.         return cosSin;

  790.     }

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

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

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

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

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

  825.             }
  826.         }

  828.         return cosSin;

  830.     }

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

  858.         final double u2 = u * u;

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

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

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

  874.             ++localIndex;
  875.             ++n;

  877.         }

  879.         if (pnm1 != null) {

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

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

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

  895.                 ++localIndex;
  896.                 ++n;

  898.             }

  900.             if (pnm2 != null) {

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

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

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

  916.                     ++localIndex;
  917.                     ++n;

  919.                 }

  921.             }

  923.         }

  925.         return localIndex;

  927.     }

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

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

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

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

  974.         }
  975.         if (pnm1 != null) {

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

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

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

  991.                 ++localIndex;
  992.                 ++n;

  994.             }

  996.             if (pnm2 != null) {

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

  1005.                 // compute tesseral values and derivatives with respect to polar angle
  1006.                 localIndex = index;
  1007.                 while (n <= degree) {

  1009.                     // second derivative (differential of equation 30 with respect to theta)
  1010.                     pnm2[n] = tOu.multiply(pnm1[n]).subtract(pnm0[n].divide(u2)).multiply(m).subtract(u.multiply(pnm1Plus1[n]).multiply(enm[localIndex]));
  1011.                     ++localIndex;
  1012.                     ++n;

  1014.                 }

  1016.             }

  1018.         }
  1019.         return localIndex;

  1021.     }

  1023.     /** {@inheritDoc} */
  1024.     @Override
  1025.     public Vector3D acceleration(final SpacecraftState s, final double[] parameters) {

  1027.         final double mu = parameters[0];

  1029.         // get the position in body frame
  1030.         final AbsoluteDate date       = s.getDate();
  1031.         final Transform fromBodyFrame = bodyFrame.getTransformTo(s.getFrame(), date);
  1032.         final Transform toBodyFrame   = fromBodyFrame.getInverse();
  1033.         final Vector3D position       = toBodyFrame.transformPosition(s.getPVCoordinates().getPosition());

  1035.         // gradient of the non-central part of the gravity field
  1036.         return fromBodyFrame.transformVector(new Vector3D(gradient(date, position, mu)));

  1038.     }

  1040.     /** {@inheritDoc} */
  1041.     public <T extends RealFieldElement<T>> FieldVector3D<T> acceleration(final FieldSpacecraftState<T> s,
  1042.                                                                          final T[] parameters) {

  1044.         final T mu = parameters[0];

  1046.         // check for faster computation dedicated to derivatives with respect to state
  1047.         if (isGradientStateDerivative(s)) {
  1048.             @SuppressWarnings("unchecked")
  1049.             final FieldVector3D<Gradient> p = (FieldVector3D<Gradient>) s.getPVCoordinates().getPosition();
  1050.             @SuppressWarnings("unchecked")
  1051.             final FieldVector3D<T> a = (FieldVector3D<T>) accelerationWrtState(s.getDate().toAbsoluteDate(),
  1052.                                                                                s.getFrame(), p,
  1053.                                                                                (Gradient) mu);
  1054.             return a;
  1055.         } else if (isDSStateDerivative(s)) {
  1056.             @SuppressWarnings("unchecked")
  1057.             final FieldVector3D<DerivativeStructure> p = (FieldVector3D<DerivativeStructure>) s.getPVCoordinates().getPosition();
  1058.             @SuppressWarnings("unchecked")
  1059.             final FieldVector3D<T> a = (FieldVector3D<T>) accelerationWrtState(s.getDate().toAbsoluteDate(),
  1060.                                                                                s.getFrame(), p,
  1061.                                                                                (DerivativeStructure) mu);
  1062.             return a;
  1063.         }

  1065.         // get the position in body frame
  1066.         final FieldAbsoluteDate<T> date          = s.getDate();
  1067.         final Transform            fromBodyFrame = bodyFrame.getTransformTo(s.getFrame(), date.toAbsoluteDate());
  1068.         final Transform            toBodyFrame   = fromBodyFrame.getInverse();
  1069.         final FieldVector3D<T>     position      = toBodyFrame.transformPosition(s.getPVCoordinates().getPosition());

  1071.         // gradient of the non-central part of the gravity field
  1072.         return fromBodyFrame.transformVector(new FieldVector3D<>(gradient(date, position, mu)));

  1074.     }

  1076.     /** {@inheritDoc} */
  1077.     public Stream<EventDetector> getEventsDetectors() {
  1078.         return Stream.empty();
  1079.     }

  1081.     @Override
  1082.     /** {@inheritDoc} */
  1083.     public <T extends RealFieldElement<T>> Stream<FieldEventDetector<T>> getFieldEventsDetectors(final Field<T> field) {
  1084.         return Stream.empty();
  1085.     }

  1087.     /** Check if a field state corresponds to derivatives with respect to state.
  1088.      * @param state state to check
  1089.      * @param <T> type of the filed elements
  1090.      * @return true if state corresponds to derivatives with respect to state
  1091.      * @since 9.0
  1092.      */
  1093.     private <T extends RealFieldElement<T>> boolean isDSStateDerivative(final FieldSpacecraftState<T> state) {
  1094.         try {
  1095.             final DerivativeStructure dsMass = (DerivativeStructure) state.getMass();
  1096.             final int o = dsMass.getOrder();
  1097.             final int p = dsMass.getFreeParameters();
  1098.             if (o != 1 || p < 3) {
  1099.                 return false;
  1100.             }
  1101.             @SuppressWarnings("unchecked")
  1102.             final FieldPVCoordinates<DerivativeStructure> pv = (FieldPVCoordinates<DerivativeStructure>) state.getPVCoordinates();
  1103.             return isVariable(pv.getPosition().getX(), 0) &&
  1104.                    isVariable(pv.getPosition().getY(), 1) &&
  1105.                    isVariable(pv.getPosition().getZ(), 2);
  1106.         } catch (ClassCastException cce) {
  1107.             return false;
  1108.         }
  1109.     }

  1111.     /** Check if a field state corresponds to derivatives with respect to state.
  1112.      * @param state state to check
  1113.      * @param <T> type of the filed elements
  1114.      * @return true if state corresponds to derivatives with respect to state
  1115.      * @since 10.2
  1116.      */
  1117.     private <T extends RealFieldElement<T>> boolean isGradientStateDerivative(final FieldSpacecraftState<T> state) {
  1118.         try {
  1119.             final Gradient gMass = (Gradient) state.getMass();
  1120.             final int p = gMass.getFreeParameters();
  1121.             if (p < 3) {
  1122.                 return false;
  1123.             }
  1124.             @SuppressWarnings("unchecked")
  1125.             final FieldPVCoordinates<Gradient> pv = (FieldPVCoordinates<Gradient>) state.getPVCoordinates();
  1126.             return isVariable(pv.getPosition().getX(), 0) &&
  1127.                    isVariable(pv.getPosition().getY(), 1) &&
  1128.                    isVariable(pv.getPosition().getZ(), 2);
  1129.         } catch (ClassCastException cce) {
  1130.             return false;
  1131.         }
  1132.     }

  1134.     /** Check if a derivative represents a specified variable.
  1135.      * @param ds derivative to check
  1136.      * @param index index of the variable
  1137.      * @return true if the derivative represents a specified variable
  1138.      * @since 9.0
  1139.      */
  1140.     private boolean isVariable(final DerivativeStructure ds, final int index) {
  1141.         final double[] derivatives = ds.getAllDerivatives();
  1142.         boolean check = true;
  1143.         for (int i = 1; i < derivatives.length; ++i) {
  1144.             check &= derivatives[i] == ((index + 1 == i) ? 1.0 : 0.0);
  1145.         }
  1146.         return check;
  1147.     }

  1149.     /** Check if a derivative represents a specified variable.
  1150.      * @param g derivative to check
  1151.      * @param index index of the variable
  1152.      * @return true if the derivative represents a specified variable
  1153.      * @since 10.2
  1154.      */
  1155.     private boolean isVariable(final Gradient g, final int index) {
  1156.         final double[] derivatives = g.getGradient();
  1157.         boolean check = true;
  1158.         for (int i = 0; i < derivatives.length; ++i) {
  1159.             check &= derivatives[i] == ((index == i) ? 1.0 : 0.0);
  1160.         }
  1161.         return check;
  1162.     }

  1164.     /** Compute acceleration derivatives with respect to state parameters.
  1165.      * <p>
  1166.      * From a theoretical point of view, this method computes the same values
  1167.      * as {@link #acceleration(FieldSpacecraftState, RealFieldElement[])} in the
  1168.      * specific case of {@link DerivativeStructure} with respect to state, so
  1169.      * it is less general. However, it is *much* faster in this important case.
  1170.      * <p>
  1171.      * <p>
  1172.      * The derivatives should be computed with respect to position. The input
  1173.      * parameters already take into account the free parameters (6 or 7 depending
  1174.      * on derivation with respect to mass being considered or not) and order
  1175.      * (always 1). Free parameters at indices 0, 1 and 2 correspond to derivatives
  1176.      * with respect to position. Free parameters at indices 3, 4 and 5 correspond
  1177.      * to derivatives with respect to velocity (these derivatives will remain zero
  1178.      * as acceleration due to gravity does not depend on velocity). Free parameter
  1179.      * at index 6 (if present) corresponds to to derivatives with respect to mass
  1180.      * (this derivative will remain zero as acceleration due to gravity does not
  1181.      * depend on mass).
  1182.      * </p>
  1183.      * @param date current date
  1184.      * @param frame inertial reference frame for state (both orbit and attitude)
  1185.      * @param position position of spacecraft in inertial frame
  1186.      * @param mu central attraction coefficient to use
  1187.      * @return acceleration with all derivatives specified by the input parameters
  1188.      * own derivatives
  1189.      * @since 6.0
  1190.      */
  1191.     private FieldVector3D<DerivativeStructure> accelerationWrtState(final AbsoluteDate date, final Frame frame,
  1192.                                                                     final FieldVector3D<DerivativeStructure> position,
  1193.                                                                     final DerivativeStructure mu) {

  1195.         // free parameters
  1196.         final int freeParameters = mu.getFreeParameters();

  1198.         // get the position in body frame
  1199.         final Transform fromBodyFrame = bodyFrame.getTransformTo(frame, date);
  1200.         final Transform toBodyFrame   = fromBodyFrame.getInverse();
  1201.         final Vector3D positionBody   = toBodyFrame.transformPosition(position.toVector3D());

  1203.         // compute gradient and Hessian
  1204.         final GradientHessian gh   = gradientHessian(date, positionBody, mu.getReal());

  1206.         // gradient of the non-central part of the gravity field
  1207.         final double[] gInertial = fromBodyFrame.transformVector(new Vector3D(gh.getGradient())).toArray();

  1209.         // Hessian of the non-central part of the gravity field
  1210.         final RealMatrix hBody     = new Array2DRowRealMatrix(gh.getHessian(), false);
  1211.         final RealMatrix rot       = new Array2DRowRealMatrix(toBodyFrame.getRotation().getMatrix());
  1212.         final RealMatrix hInertial = rot.transpose().multiply(hBody).multiply(rot);

  1214.         // distribute all partial derivatives in a compact acceleration vector
  1215.         final double[] derivatives = new double[freeParameters + 1];
  1216.         final DerivativeStructure[] accDer = new DerivativeStructure[3];
  1217.         for (int i = 0; i < 3; ++i) {

  1219.             // first element is value of acceleration (i.e. gradient of field)
  1220.             derivatives[0] = gInertial[i];

  1222.             // Jacobian of acceleration (i.e. Hessian of field)
  1223.             derivatives[1] = hInertial.getEntry(i, 0);
  1224.             derivatives[2] = hInertial.getEntry(i, 1);
  1225.             derivatives[3] = hInertial.getEntry(i, 2);

  1227.             // next element is derivative with respect to parameter mu
  1228.             if (derivatives.length > 4 && isVariable(mu, 3)) {
  1229.                 derivatives[4] = gInertial[i] / mu.getReal();
  1230.             }

  1232.             accDer[i] = position.getX().getFactory().build(derivatives);

  1234.         }

  1236.         return new FieldVector3D<>(accDer);

  1238.     }

  1240.     /** Compute acceleration derivatives with respect to state parameters.
  1241.      * <p>
  1242.      * From a theoretical point of view, this method computes the same values
  1243.      * as {@link #acceleration(FieldSpacecraftState, RealFieldElement[])} in the
  1244.      * specific case of {@link DerivativeStructure} with respect to state, so
  1245.      * it is less general. However, it is *much* faster in this important case.
  1246.      * <p>
  1247.      * <p>
  1248.      * The derivatives should be computed with respect to position. The input
  1249.      * parameters already take into account the free parameters (6 or 7 depending
  1250.      * on derivation with respect to mass being considered or not) and order
  1251.      * (always 1). Free parameters at indices 0, 1 and 2 correspond to derivatives
  1252.      * with respect to position. Free parameters at indices 3, 4 and 5 correspond
  1253.      * to derivatives with respect to velocity (these derivatives will remain zero
  1254.      * as acceleration due to gravity does not depend on velocity). Free parameter
  1255.      * at index 6 (if present) corresponds to to derivatives with respect to mass
  1256.      * (this derivative will remain zero as acceleration due to gravity does not
  1257.      * depend on mass).
  1258.      * </p>
  1259.      * @param date current date
  1260.      * @param frame inertial reference frame for state (both orbit and attitude)
  1261.      * @param position position of spacecraft in inertial frame
  1262.      * @param mu central attraction coefficient to use
  1263.      * @return acceleration with all derivatives specified by the input parameters
  1264.      * own derivatives
  1265.      * @since 10.2
  1266.      */
  1267.     private FieldVector3D<Gradient> accelerationWrtState(final AbsoluteDate date, final Frame frame,
  1268.                                                          final FieldVector3D<Gradient> position,
  1269.                                                          final Gradient mu) {

  1271.         // free parameters
  1272.         final int freeParameters = mu.getFreeParameters();

  1274.         // get the position in body frame
  1275.         final Transform fromBodyFrame = bodyFrame.getTransformTo(frame, date);
  1276.         final Transform toBodyFrame   = fromBodyFrame.getInverse();
  1277.         final Vector3D positionBody   = toBodyFrame.transformPosition(position.toVector3D());

  1279.         // compute gradient and Hessian
  1280.         final GradientHessian gh   = gradientHessian(date, positionBody, mu.getReal());

  1282.         // gradient of the non-central part of the gravity field
  1283.         final double[] gInertial = fromBodyFrame.transformVector(new Vector3D(gh.getGradient())).toArray();

  1285.         // Hessian of the non-central part of the gravity field
  1286.         final RealMatrix hBody     = new Array2DRowRealMatrix(gh.getHessian(), false);
  1287.         final RealMatrix rot       = new Array2DRowRealMatrix(toBodyFrame.getRotation().getMatrix());
  1288.         final RealMatrix hInertial = rot.transpose().multiply(hBody).multiply(rot);

  1290.         // distribute all partial derivatives in a compact acceleration vector
  1291.         final double[] derivatives = new double[freeParameters];
  1292.         final Gradient[] accDer = new Gradient[3];
  1293.         for (int i = 0; i < 3; ++i) {

  1295.             // Jacobian of acceleration (i.e. Hessian of field)
  1296.             derivatives[0] = hInertial.getEntry(i, 0);
  1297.             derivatives[1] = hInertial.getEntry(i, 1);
  1298.             derivatives[2] = hInertial.getEntry(i, 2);

  1300.             // next element is derivative with respect to parameter mu
  1301.             if (derivatives.length > 3 && isVariable(mu, 3)) {
  1302.                 derivatives[3] = gInertial[i] / mu.getReal();
  1303.             }

  1305.             accDer[i] = new Gradient(gInertial[i], derivatives);

  1307.         }

  1309.         return new FieldVector3D<>(accDer);

  1311.     }

  1313.     /** {@inheritDoc} */
  1314.     public ParameterDriver[] getParametersDrivers() {
  1315.         return new ParameterDriver[] {
  1316.             gmParameterDriver
  1317.         };
  1318.     }

  1320. }
Created with JaCoCo 0.8.6.202009150832