/* Copyright 2002-2025 CS GROUP
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    * CS licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
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    *   http://www.apache.org/licenses/LICENSE-2.0
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   * Unless required by applicable law or agreed to in writing, software
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  package org.orekit.forces.gravity;
  
  
  import java.util.Collections;
  import java.util.List;
  
  import org.hipparchus.CalculusFieldElement;
  import org.hipparchus.analysis.differentiation.DerivativeStructure;
  import org.hipparchus.analysis.differentiation.Gradient;
  import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  import org.hipparchus.geometry.euclidean.threed.SphericalCoordinates;
  import org.hipparchus.geometry.euclidean.threed.Vector3D;
  import org.hipparchus.linear.Array2DRowRealMatrix;
  import org.hipparchus.linear.RealMatrix;
  import org.hipparchus.util.FastMath;
  import org.hipparchus.util.MathArrays;
  import org.orekit.forces.ForceModel;
  import org.orekit.forces.gravity.potential.NormalizedSphericalHarmonicsProvider;
  import org.orekit.forces.gravity.potential.NormalizedSphericalHarmonicsProvider.NormalizedSphericalHarmonics;
  import org.orekit.forces.gravity.potential.TideSystem;
  import org.orekit.forces.gravity.potential.TideSystemProvider;
  import org.orekit.frames.FieldStaticTransform;
  import org.orekit.frames.Frame;
  import org.orekit.frames.StaticTransform;
  import org.orekit.propagation.FieldSpacecraftState;
  import org.orekit.propagation.SpacecraftState;
  import org.orekit.time.AbsoluteDate;
  import org.orekit.time.FieldAbsoluteDate;
  import org.orekit.utils.FieldPVCoordinates;
  import org.orekit.utils.ParameterDriver;
  
  /** This class represents the gravitational field of a celestial body.
   * <p>
   * The algorithm implemented in this class has been designed by S. A. Holmes
   * and W. E. Featherstone from Department of Spatial Sciences, Curtin University
   * of Technology, Perth, Australia. It is described in their 2002 paper: <a
   * 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">
   * A unified approach to he Clenshaw summation and the recursive computation of
   * very high degree and order normalised associated Legendre functions</a>
   * (Journal of Geodesy (2002) 76: 279–299).
   * </p>
   * <p>
   * This model directly uses normalized coefficients and stable recursion algorithms
   * so it is more suited to high degree gravity fields than the classical Cunningham
   * Droziner models which use un-normalized coefficients.
   * </p>
   * <p>
   * Among the different algorithms presented in Holmes and Featherstone paper, this
   * class implements the <em>modified forward row method</em>. All recursion coefficients
   * are precomputed and stored for greater performance. This caching was suggested in the
   * paper but not used due to the large memory requirements. Since 2002, even low end
   * computers and mobile devices do have sufficient memory so this caching has become
   * feasible nowadays.
   * </p>
   * @author Luc Maisonobe
   * @since 6.0
   */
  
  public class HolmesFeatherstoneAttractionModel implements ForceModel, TideSystemProvider {
  
      /** Exponent scaling to avoid floating point overflow.
       * <p>The paper uses 10^280, we prefer a power of two to preserve accuracy thanks to
       * {@link FastMath#scalb(double, int)}, so we use 2^930 which has the same order of magnitude.
       */
      private static final int SCALING = 930;
  
      /** Central attraction scaling factor.
       * <p>
       * We use a power of 2 to avoid numeric noise introduction
       * in the multiplications/divisions sequences.
       * </p>
       */
      private static final double MU_SCALE = FastMath.scalb(1.0, 32);
  
      /** Driver for gravitational parameter. */
      private final ParameterDriver gmParameterDriver;
  
      /** Provider for the spherical harmonics. */
      private final NormalizedSphericalHarmonicsProvider provider;
  
      /** Rotating body. */
      private final Frame bodyFrame;
  
     /** Recursion coefficients g<sub>n,m</sub>/√j. */
     private final double[] gnmOj;
 
     /** Recursion coefficients h<sub>n,m</sub>/√j. */
     private final double[] hnmOj;
 
     /** Recursion coefficients e<sub>n,m</sub>. */
     private final double[] enm;
 
     /** Scaled sectorial Pbar<sub>m,m</sub>/u<sup>m</sup> &times; 2<sup>-SCALING</sup>. */
     private final double[] sectorial;
 
     /** Creates a new instance.
      * @param centralBodyFrame rotating body frame
      * @param provider provider for spherical harmonics
      * @since 6.0
      */
     public HolmesFeatherstoneAttractionModel(final Frame centralBodyFrame,
                                              final NormalizedSphericalHarmonicsProvider provider) {
 
         gmParameterDriver = new ParameterDriver(NewtonianAttraction.CENTRAL_ATTRACTION_COEFFICIENT,
                                                 provider.getMu(), MU_SCALE, 0.0, Double.POSITIVE_INFINITY);
 
         this.provider  = provider;
         this.bodyFrame = centralBodyFrame;
 
         // the pre-computed arrays hold coefficients from triangular arrays in a single
         // storing neither diagonal elements (n = m) nor the non-diagonal element n=1, m=0
         final int degree = provider.getMaxDegree();
         final int size = FastMath.max(0, degree * (degree + 1) / 2 - 1);
         gnmOj = new double[size];
         hnmOj = new double[size];
         enm   = new double[size];
 
         // pre-compute the recursion coefficients corresponding to equations 19 and 22
         // from Holmes and Featherstone paper
         // for cache efficiency, elements are stored in the same order they will be used
         // later on, i.e. from rightmost column to leftmost column
         int index = 0;
         for (int m = degree; m >= 0; --m) {
             final int j = (m == 0) ? 2 : 1;
             for (int n = FastMath.max(2, m + 1); n <= degree; ++n) {
                 final double f = (n - m) * (n + m + 1);
                 gnmOj[index] = 2 * (m + 1) / FastMath.sqrt(j * f);
                 hnmOj[index] = FastMath.sqrt((n + m + 2) * (n - m - 1) / (j * f));
                 enm[index]   = FastMath.sqrt(f / j);
                 ++index;
             }
         }
 
         // scaled sectorial terms corresponding to equation 28 in Holmes and Featherstone paper
         sectorial    = new double[degree + 1];
         sectorial[0] = FastMath.scalb(1.0, -SCALING);
         if (degree > 0) {
             sectorial[1] = FastMath.sqrt(3) * sectorial[0];
         }
         for (int m = 2; m < sectorial.length; ++m) {
             sectorial[m] = FastMath.sqrt((2 * m + 1) / (2.0 * m)) * sectorial[m - 1];
         }
 
     }
 
     /** {@inheritDoc} */
     @Override
     public boolean dependsOnPositionOnly() {
         return true;
     }
 
     /** {@inheritDoc} */
     public TideSystem getTideSystem() {
         return provider.getTideSystem();
     }
 
     /** Get the central attraction coefficient μ.
      * @return mu central attraction coefficient (m³/s²),
      * will throw an exception if gm PDriver has several
      * values driven (in this case the method
      * {@link #getMu(AbsoluteDate)} must be used.
      */
     public double getMu() {
         return gmParameterDriver.getValue();
     }
 
     /** Get the central attraction coefficient μ.
      * @param date date at which mu wants to be known
      * @return mu central attraction coefficient (m³/s²)
      */
     public double getMu(final AbsoluteDate date) {
         return gmParameterDriver.getValue(date);
     }
 
     /** Compute the value of the gravity field.
      * @param date current date
      * @param position position at which gravity field is desired in body frame
      * @param mu central attraction coefficient to use
      * @return value of the gravity field (central and non-central parts summed together)
      */
     public double value(final AbsoluteDate date, final Vector3D position,
                         final double mu) {
         return mu / position.getNorm() + nonCentralPart(date, position, mu);
     }
 
     /** Compute the non-central part of the gravity field.
      * @param date current date
      * @param position position at which gravity field is desired in body frame
      * @param mu central attraction coefficient to use
      * @return value of the non-central part of the gravity field
      */
     public double nonCentralPart(final AbsoluteDate date, final Vector3D position, final double mu) {
 
         final int degree = provider.getMaxDegree();
         final int order  = provider.getMaxOrder();
         final NormalizedSphericalHarmonics harmonics = provider.onDate(date);
 
         // allocate the columns for recursion
         double[] pnm0Plus2 = new double[degree + 1];
         double[] pnm0Plus1 = new double[degree + 1];
         double[] pnm0      = new double[degree + 1];
 
         // compute polar coordinates
         final double x    = position.getX();
         final double y    = position.getY();
         final double z    = position.getZ();
         final double x2   = x * x;
         final double y2   = y * y;
         final double z2   = z * z;
         final double rho2 = x2 + y2;
         final double r2   = rho2 + z2;
         final double r    = FastMath.sqrt(r2);
         final double rho  = FastMath.sqrt(rho2);
         final double t    = z / r;   // cos(theta), where theta is the polar angle
         final double u    = rho / r; // sin(theta), where theta is the polar angle
         final double tOu  = z / rho;
 
         // compute distance powers
         final double[] aOrN = createDistancePowersArray(provider.getAe() / r);
 
         // compute longitude cosines/sines
         final double[][] cosSinLambda = createCosSinArrays(x / rho, y / rho);
 
         // outer summation over order
         int    index = 0;
         double value = 0;
         for (int m = degree; m >= 0; --m) {
 
             // compute tesseral terms without derivatives
             index = computeTesseral(m, degree, index, t, u, tOu,
                                     pnm0Plus2, pnm0Plus1, null, pnm0, null, null);
 
             if (m <= order) {
                 // compute contribution of current order to field (equation 5 of the paper)
 
                 // inner summation over degree, for fixed order
                 double sumDegreeS        = 0;
                 double sumDegreeC        = 0;
                 for (int n = FastMath.max(2, m); n <= degree; ++n) {
                     sumDegreeS += pnm0[n] * aOrN[n] * harmonics.getNormalizedSnm(n, m);
                     sumDegreeC += pnm0[n] * aOrN[n] * harmonics.getNormalizedCnm(n, m);
                 }
 
                 // contribution to outer summation over order
                 value = value * u + cosSinLambda[1][m] * sumDegreeS + cosSinLambda[0][m] * sumDegreeC;
 
             }
 
             // rotate the recursion arrays
             final double[] tmp = pnm0Plus2;
             pnm0Plus2 = pnm0Plus1;
             pnm0Plus1 = pnm0;
             pnm0      = tmp;
 
         }
 
         // scale back
         value = FastMath.scalb(value, SCALING);
 
         // apply the global mu/r factor
         return mu * value / r;
 
     }
 
     /** Compute the gradient of the non-central part of the gravity field.
      * <p>
      * If U represents the non-central part of the gravity field,
      * this method returns the negative gradient of U (-grad(U)).
      * </p>
      * @param date current date
      * @param position position at which gravity field is desired in body frame
      * @param mu central attraction coefficient to use
      * @return gradient of the non-central part of the gravity field
      */
     public double[] gradient(final AbsoluteDate date, final Vector3D position, final double mu) {
 
         final int degree = provider.getMaxDegree();
         final int order  = provider.getMaxOrder();
         final NormalizedSphericalHarmonics harmonics = provider.onDate(date);
 
         // allocate the columns for recursion
         double[] pnm0Plus2  = new double[degree + 1];
         double[] pnm0Plus1  = new double[degree + 1];
         double[] pnm0       = new double[degree + 1];
         final double[] pnm1 = new double[degree + 1];
 
         // compute polar coordinates
         final double x    = position.getX();
         final double y    = position.getY();
         final double z    = position.getZ();
         final double x2   = x * x;
         final double y2   = y * y;
         final double z2   = z * z;
         final double r2   = x2 + y2 + z2;
         final double r    = FastMath.sqrt (r2);
         final double rho2 = x2 + y2;
         final double rho  = FastMath.sqrt(rho2);
         final double t    = z / r;   // cos(theta), where theta is the polar angle
         final double u    = rho / r; // sin(theta), where theta is the polar angle
         final double tOu  = z / rho;
 
         // compute distance powers
         final double[] aOrN = createDistancePowersArray(provider.getAe() / r);
 
         // compute longitude cosines/sines
         final double[][] cosSinLambda = createCosSinArrays(x / rho, y / rho);
 
         // outer summation over order
         int    index = 0;
         double value = 0;
         final double[] gradient = new double[3];
         for (int m = degree; m >= 0; --m) {
 
             // compute tesseral terms with derivatives
             index = computeTesseral(m, degree, index, t, u, tOu,
                                     pnm0Plus2, pnm0Plus1, null, pnm0, pnm1, null);
 
             if (m <= order) {
                 // compute contribution of current order to field (equation 5 of the paper)
 
                 // inner summation over degree, for fixed order
                 double sumDegreeS        = 0;
                 double sumDegreeC        = 0;
                 double dSumDegreeSdR     = 0;
                 double dSumDegreeCdR     = 0;
                 double dSumDegreeSdTheta = 0;
                 double dSumDegreeCdTheta = 0;
                 for (int n = FastMath.max(2, m); n <= degree; ++n) {
                     final double qSnm  = aOrN[n] * harmonics.getNormalizedSnm(n, m);
                     final double qCnm  = aOrN[n] * harmonics.getNormalizedCnm(n, m);
                     final double nOr   = n / r;
                     final double s0    = pnm0[n] * qSnm;
                     final double c0    = pnm0[n] * qCnm;
                     final double s1    = pnm1[n] * qSnm;
                     final double c1    = pnm1[n] * qCnm;
                     sumDegreeS        += s0;
                     sumDegreeC        += c0;
                     dSumDegreeSdR     -= nOr * s0;
                     dSumDegreeCdR     -= nOr * c0;
                     dSumDegreeSdTheta += s1;
                     dSumDegreeCdTheta += c1;
                 }
 
                 // contribution to outer summation over order
                 // beware that we need to order gradient using the mathematical conventions
                 // compliant with the SphericalCoordinates class, so our lambda is its theta
                 // (and hence at index 1) and our theta is its phi (and hence at index 2)
                 final double sML = cosSinLambda[1][m];
                 final double cML = cosSinLambda[0][m];
                 value            = value       * u + sML * sumDegreeS        + cML * sumDegreeC;
                 gradient[0]      = gradient[0] * u + sML * dSumDegreeSdR     + cML * dSumDegreeCdR;
                 gradient[1]      = gradient[1] * u + m * (cML * sumDegreeS - sML * sumDegreeC);
                 gradient[2]      = gradient[2] * u + sML * dSumDegreeSdTheta + cML * dSumDegreeCdTheta;
 
             }
 
             // rotate the recursion arrays
             final double[] tmp = pnm0Plus2;
             pnm0Plus2 = pnm0Plus1;
             pnm0Plus1 = pnm0;
             pnm0      = tmp;
 
         }
 
         // scale back
         value       = FastMath.scalb(value,       SCALING);
         gradient[0] = FastMath.scalb(gradient[0], SCALING);
         gradient[1] = FastMath.scalb(gradient[1], SCALING);
         gradient[2] = FastMath.scalb(gradient[2], SCALING);
 
         // apply the global mu/r factor
         final double muOr = mu / r;
         value            *= muOr;
         gradient[0]       = muOr * gradient[0] - value / r;
         gradient[1]      *= muOr;
         gradient[2]      *= muOr;
 
         // convert gradient from spherical to Cartesian
         return new SphericalCoordinates(position).toCartesianGradient(gradient);
 
     }
 
     /** Compute the gradient of the non-central part of the gravity field.
      * <p>
      * If U represents the non-central part of the gravity field,
      * this method returns the negative gradient of U (-grad(U)).
      * </p>
      * @param date current date
      * @param position position at which gravity field is desired in body frame
      * @param mu central attraction coefficient to use
      * @param <T> type of field used
      * @return gradient of the non-central part of the gravity field
      */
     public <T extends CalculusFieldElement<T>> T[] gradient(final FieldAbsoluteDate<T> date, final FieldVector3D<T> position,
                                                         final T mu) {
 
         final int degree = provider.getMaxDegree();
         final int order  = provider.getMaxOrder();
         final NormalizedSphericalHarmonics harmonics = provider.onDate(date.toAbsoluteDate());
         final T zero = date.getField().getZero();
         // allocate the columns for recursion
         T[] pnm0Plus2  = MathArrays.buildArray(date.getField(), degree + 1);
         T[] pnm0Plus1  = MathArrays.buildArray(date.getField(), degree + 1);
         T[] pnm0       = MathArrays.buildArray(date.getField(), degree + 1);
         final T[] pnm1 = MathArrays.buildArray(date.getField(), degree + 1);
 
         // compute polar coordinates
         final T x    = position.getX();
         final T y    = position.getY();
         final T z    = position.getZ();
         final T x2   = x.square();
         final T y2   = y.square();
         final T rho2 = x2.add(y2);
         final T rho  = rho2.sqrt();
         final T z2   = z.square();
         final T r2   = rho2.add(z2);
         final T r    = r2.sqrt();
         final T t    = z.divide(r);   // cos(theta), where theta is the polar angle
         final T u    = rho.divide(r); // sin(theta), where theta is the polar angle
         final T tOu  = z.divide(rho);
 
         // compute distance powers
         final T[] aOrN = createDistancePowersArray(r.reciprocal().multiply(provider.getAe()));
 
         // compute longitude cosines/sines
         final T[][] cosSinLambda = createCosSinArrays(x.divide(rho), y.divide(rho));
         // outer summation over order
         int    index = 0;
         T value = zero;
         final T[] gradient = MathArrays.buildArray(zero.getField(), 3);
         for (int m = degree; m >= 0; --m) {
 
             // compute tesseral terms with derivatives
             index = computeTesseral(m, degree, index, t, u, tOu,
                                     pnm0Plus2, pnm0Plus1, null, pnm0, pnm1, null);
             if (m <= order) {
                 // compute contribution of current order to field (equation 5 of the paper)
 
                 // inner summation over degree, for fixed order
                 T sumDegreeS        = zero;
                 T sumDegreeC        = zero;
                 T dSumDegreeSdR     = zero;
                 T dSumDegreeCdR     = zero;
                 T dSumDegreeSdTheta = zero;
                 T dSumDegreeCdTheta = zero;
                 for (int n = FastMath.max(2, m); n <= degree; ++n) {
                     final T qSnm  = aOrN[n].multiply(harmonics.getNormalizedSnm(n, m));
                     final T qCnm  = aOrN[n].multiply(harmonics.getNormalizedCnm(n, m));
                     final T nOr   = r.reciprocal().multiply(n);
                     final T s0    = pnm0[n].multiply(qSnm);
                     final T c0    = pnm0[n].multiply(qCnm);
                     final T s1    = pnm1[n].multiply(qSnm);
                     final T c1    = pnm1[n].multiply(qCnm);
                     sumDegreeS        = sumDegreeS       .add(s0);
                     sumDegreeC        = sumDegreeC       .add(c0);
                     dSumDegreeSdR     = dSumDegreeSdR    .subtract(nOr.multiply(s0));
                     dSumDegreeCdR     = dSumDegreeCdR    .subtract(nOr.multiply(c0));
                     dSumDegreeSdTheta = dSumDegreeSdTheta.add(s1);
                     dSumDegreeCdTheta = dSumDegreeCdTheta.add(c1);
                 }
 
                 // contribution to outer summation over order
                 // beware that we need to order gradient using the mathematical conventions
                 // compliant with the SphericalCoordinates class, so our lambda is its theta
                 // (and hence at index 1) and our theta is its phi (and hence at index 2)
                 final T sML = cosSinLambda[1][m];
                 final T cML = cosSinLambda[0][m];
                 value            = value      .multiply(u).add(sML.multiply(sumDegreeS   )).add(cML.multiply(sumDegreeC));
                 gradient[0]      = gradient[0].multiply(u).add(sML.multiply(dSumDegreeSdR)).add(cML.multiply(dSumDegreeCdR));
                 gradient[1]      = gradient[1].multiply(u).add(cML.multiply(sumDegreeS).subtract(sML.multiply(sumDegreeC)).multiply(m));
                 gradient[2]      = gradient[2].multiply(u).add(sML.multiply(dSumDegreeSdTheta)).add(cML.multiply(dSumDegreeCdTheta));
             }
             // rotate the recursion arrays
             final T[] tmp = pnm0Plus2;
             pnm0Plus2 = pnm0Plus1;
             pnm0Plus1 = pnm0;
             pnm0      = tmp;
 
         }
         // scale back
         value       = value.scalb(SCALING);
         gradient[0] = gradient[0].scalb(SCALING);
         gradient[1] = gradient[1].scalb(SCALING);
         gradient[2] = gradient[2].scalb(SCALING);
 
         // apply the global mu/r factor
         final T muOr = r.reciprocal().multiply(mu);
         value            = value.multiply(muOr);
         gradient[0]      = muOr.multiply(gradient[0]).subtract(value.divide(r));
         gradient[1]      = gradient[1].multiply(muOr);
         gradient[2]      = gradient[2].multiply(muOr);
 
         // convert gradient from spherical to Cartesian
         // Cartesian coordinates
         // remaining spherical coordinates
 
         // intermediate variables
         final T xPos    = position.getX();
         final T yPos    = position.getY();
         final T zPos    = position.getZ();
         final T rho2Pos = x.square().add(y.square());
         final T rhoPos  = rho2.sqrt();
         final T r2Pos   = rho2.add(z.square());
         final T rPos    = r2Pos.sqrt();
 
         final T[][] jacobianPos = MathArrays.buildArray(zero.getField(), 3, 3);
 
         // row representing the gradient of r
         jacobianPos[0][0] = xPos.divide(rPos);
         jacobianPos[0][1] = yPos.divide(rPos);
         jacobianPos[0][2] = zPos.divide(rPos);
 
         // row representing the gradient of theta
         jacobianPos[1][0] =  yPos.negate().divide(rho2Pos);
         jacobianPos[1][1] =  xPos.divide(rho2Pos);
         // jacobian[1][2] is already set to 0 at allocation time
 
         // row representing the gradient of phi
         final T rhoPosTimesR2Pos = rhoPos.multiply(r2Pos);
         jacobianPos[2][0] = xPos.multiply(zPos).divide(rhoPosTimesR2Pos);
         jacobianPos[2][1] = yPos.multiply(zPos).divide(rhoPosTimesR2Pos);
         jacobianPos[2][2] = rhoPos.negate().divide(r2Pos);
         final T[] cartGradPos = MathArrays.buildArray(zero.getField(), 3);
         cartGradPos[0] = gradient[0].multiply(jacobianPos[0][0]).add(gradient[1].multiply(jacobianPos[1][0])).add(gradient[2].multiply(jacobianPos[2][0]));
         cartGradPos[1] = gradient[0].multiply(jacobianPos[0][1]).add(gradient[1].multiply(jacobianPos[1][1])).add(gradient[2].multiply(jacobianPos[2][1]));
         cartGradPos[2] = gradient[0].multiply(jacobianPos[0][2])                                      .add(gradient[2].multiply(jacobianPos[2][2]));
         return cartGradPos;
 
     }
 
     /** Compute both the gradient and the hessian of the non-central part of the gravity field.
      * <p>
      * If U represents the non-central part of the gravity field,
      * this method returns the negative gradient and hessian of U (-grad(U) and -hessian(U)).
      * </p>
      * @param date current date
      * @param position position at which gravity field is desired in body frame
      * @param mu central attraction coefficient to use
      * @return gradient and hessian of the non-central part of the gravity field
      */
     private GradientHessian gradientHessian(final AbsoluteDate date, final Vector3D position, final double mu) {
 
         final int degree = provider.getMaxDegree();
         final int order  = provider.getMaxOrder();
         final NormalizedSphericalHarmonics harmonics = provider.onDate(date);
 
         // allocate the columns for recursion
         double[] pnm0Plus2  = new double[degree + 1];
         double[] pnm0Plus1  = new double[degree + 1];
         double[] pnm0       = new double[degree + 1];
         double[] pnm1Plus1  = new double[degree + 1];
         double[] pnm1       = new double[degree + 1];
         final double[] pnm2 = new double[degree + 1];
 
         // compute polar coordinates
         final double x    = position.getX();
         final double y    = position.getY();
         final double z    = position.getZ();
         final double x2   = x * x;
         final double y2   = y * y;
         final double z2   = z * z;
         final double rho2 = x2 + y2;
         final double rho  = FastMath.sqrt(rho2);
         final double r2   = rho2 + z2;
         final double r    = FastMath.sqrt(r2);
         final double t    = z / r;   // cos(theta), where theta is the polar angle
         final double u    = rho / r; // sin(theta), where theta is the polar angle
         final double tOu  = z / rho;
 
         // compute distance powers
         final double[] aOrN = createDistancePowersArray(provider.getAe() / r);
 
         // compute longitude cosines/sines
         final double[][] cosSinLambda = createCosSinArrays(x / rho, y / rho);
 
         // outer summation over order
         int    index = 0;
         double value = 0;
         final double[]   gradient = new double[3];
         final double[][] hessian  = new double[3][3];
         for (int m = degree; m >= 0; --m) {
 
             // compute tesseral terms
             index = computeTesseral(m, degree, index, t, u, tOu,
                                     pnm0Plus2, pnm0Plus1, pnm1Plus1, pnm0, pnm1, pnm2);
 
             if (m <= order) {
                 // compute contribution of current order to field (equation 5 of the paper)
 
                 // inner summation over degree, for fixed order
                 double sumDegreeS               = 0;
                 double sumDegreeC               = 0;
                 double dSumDegreeSdR            = 0;
                 double dSumDegreeCdR            = 0;
                 double dSumDegreeSdTheta        = 0;
                 double dSumDegreeCdTheta        = 0;
                 double d2SumDegreeSdRdR         = 0;
                 double d2SumDegreeSdRdTheta     = 0;
                 double d2SumDegreeSdThetadTheta = 0;
                 double d2SumDegreeCdRdR         = 0;
                 double d2SumDegreeCdRdTheta     = 0;
                 double d2SumDegreeCdThetadTheta = 0;
                 for (int n = FastMath.max(2, m); n <= degree; ++n) {
                     final double qSnm         = aOrN[n] * harmonics.getNormalizedSnm(n, m);
                     final double qCnm         = aOrN[n] * harmonics.getNormalizedCnm(n, m);
                     final double nOr          = n / r;
                     final double nnP1Or2      = nOr * (n + 1) / r;
                     final double s0           = pnm0[n] * qSnm;
                     final double c0           = pnm0[n] * qCnm;
                     final double s1           = pnm1[n] * qSnm;
                     final double c1           = pnm1[n] * qCnm;
                     final double s2           = pnm2[n] * qSnm;
                     final double c2           = pnm2[n] * qCnm;
                     sumDegreeS               += s0;
                     sumDegreeC               += c0;
                     dSumDegreeSdR            -= nOr * s0;
                     dSumDegreeCdR            -= nOr * c0;
                     dSumDegreeSdTheta        += s1;
                     dSumDegreeCdTheta        += c1;
                     d2SumDegreeSdRdR         += nnP1Or2 * s0;
                     d2SumDegreeSdRdTheta     -= nOr * s1;
                     d2SumDegreeSdThetadTheta += s2;
                     d2SumDegreeCdRdR         += nnP1Or2 * c0;
                     d2SumDegreeCdRdTheta     -= nOr * c1;
                     d2SumDegreeCdThetadTheta += c2;
                 }
 
                 // contribution to outer summation over order
                 final double sML = cosSinLambda[1][m];
                 final double cML = cosSinLambda[0][m];
                 value            = value         * u + sML * sumDegreeS + cML * sumDegreeC;
                 gradient[0]      = gradient[0]   * u + sML * dSumDegreeSdR + cML * dSumDegreeCdR;
                 gradient[1]      = gradient[1]   * u + m * (cML * sumDegreeS - sML * sumDegreeC);
                 gradient[2]      = gradient[2]   * u + sML * dSumDegreeSdTheta + cML * dSumDegreeCdTheta;
                 hessian[0][0]    = hessian[0][0] * u + sML * d2SumDegreeSdRdR + cML * d2SumDegreeCdRdR;
                 hessian[1][0]    = hessian[1][0] * u + m * (cML * dSumDegreeSdR - sML * dSumDegreeCdR);
                 hessian[2][0]    = hessian[2][0] * u + sML * d2SumDegreeSdRdTheta + cML * d2SumDegreeCdRdTheta;
                 hessian[1][1]    = hessian[1][1] * u - m * m * (sML * sumDegreeS + cML * sumDegreeC);
                 hessian[2][1]    = hessian[2][1] * u + m * (cML * dSumDegreeSdTheta - sML * dSumDegreeCdTheta);
                 hessian[2][2]    = hessian[2][2] * u + sML * d2SumDegreeSdThetadTheta + cML * d2SumDegreeCdThetadTheta;
 
             }
 
             // rotate the recursion arrays
             final double[] tmp0 = pnm0Plus2;
             pnm0Plus2 = pnm0Plus1;
             pnm0Plus1 = pnm0;
             pnm0      = tmp0;
             final double[] tmp1 = pnm1Plus1;
             pnm1Plus1 = pnm1;
             pnm1      = tmp1;
 
         }
 
         // scale back
         value = FastMath.scalb(value, SCALING);
         for (int i = 0; i < 3; ++i) {
             gradient[i] = FastMath.scalb(gradient[i], SCALING);
             for (int j = 0; j <= i; ++j) {
                 hessian[i][j] = FastMath.scalb(hessian[i][j], SCALING);
             }
         }
 
 
         // apply the global mu/r factor
         final double muOr = mu / r;
         value         *= muOr;
         gradient[0]    = muOr * gradient[0] - value / r;
         gradient[1]   *= muOr;
         gradient[2]   *= muOr;
         hessian[0][0]  = muOr * hessian[0][0] - 2 * gradient[0] / r;
         hessian[1][0]  = muOr * hessian[1][0] -     gradient[1] / r;
         hessian[2][0]  = muOr * hessian[2][0] -     gradient[2] / r;
         hessian[1][1] *= muOr;
         hessian[2][1] *= muOr;
         hessian[2][2] *= muOr;
 
         // convert gradient and Hessian from spherical to Cartesian
         final SphericalCoordinates sc = new SphericalCoordinates(position);
         return new GradientHessian(sc.toCartesianGradient(gradient),
                                    sc.toCartesianHessian(hessian, gradient));
 
 
     }
 
     /** Container for gradient and Hessian. */
     private static class GradientHessian {
 
         /** Gradient. */
         private final double[] gradient;
 
         /** Hessian. */
         private final double[][] hessian;
 
         /** Simple constructor.
          * <p>
          * A reference to the arrays is stored, they are <strong>not</strong> cloned.
          * </p>
          * @param gradient gradient
          * @param hessian hessian
          */
         GradientHessian(final double[] gradient, final double[][] hessian) {
             this.gradient = gradient;
             this.hessian  = hessian;
         }
 
         /** Get a reference to the gradient.
          * @return gradient (a reference to the internal array is returned)
          */
         public double[] getGradient() {
             return gradient;
         }
 
         /** Get a reference to the Hessian.
          * @return Hessian (a reference to the internal array is returned)
          */
         public double[][] getHessian() {
             return hessian;
         }
 
     }
 
     /** Compute a/r powers array.
      * @param aOr a/r
      * @return array containing (a/r)<sup>n</sup>
      */
     private double[] createDistancePowersArray(final double aOr) {
 
         // initialize array
         final double[] aOrN = new double[provider.getMaxDegree() + 1];
         aOrN[0] = 1;
         if (provider.getMaxDegree() > 0) {
             aOrN[1] = aOr;
         }
 
         // fill up array
         for (int n = 2; n < aOrN.length; ++n) {
             final int p = n / 2;
             final int q = n - p;
             aOrN[n] = aOrN[p] * aOrN[q];
         }
 
         return aOrN;
 
     }
     /** Compute a/r powers array.
      * @param aOr a/r
      * @param <T> type of field used
      * @return array containing (a/r)<sup>n</sup>
      */
     private <T extends CalculusFieldElement<T>> T[] createDistancePowersArray(final T aOr) {
 
         // initialize array
         final T[] aOrN = MathArrays.buildArray(aOr.getField(), provider.getMaxDegree() + 1);
         aOrN[0] = aOr.getField().getOne();
         if (provider.getMaxDegree() > 0) {
             aOrN[1] = aOr;
         }
 
         // fill up array
         for (int n = 2; n < aOrN.length; ++n) {
             final int p = n / 2;
             final int q = n - p;
             aOrN[n] = aOrN[p].multiply(aOrN[q]);
         }
 
         return aOrN;
 
     }
 
     /** Compute longitude cosines and sines.
      * @param cosLambda cos(λ)
      * @param sinLambda sin(λ)
      * @return array containing cos(m &times; λ) in row 0
      * and sin(m &times; λ) in row 1
      */
     private double[][] createCosSinArrays(final double cosLambda, final double sinLambda) {
 
         // initialize arrays
         final double[][] cosSin = new double[2][provider.getMaxOrder() + 1];
         cosSin[0][0] = 1;
         cosSin[1][0] = 0;
         if (provider.getMaxOrder() > 0) {
             cosSin[0][1] = cosLambda;
             cosSin[1][1] = sinLambda;
 
             // fill up array
             for (int m = 2; m < cosSin[0].length; ++m) {
 
                 // m * lambda is split as p * lambda + q * lambda, trying to avoid
                 // p or q being much larger than the other. This reduces the number of
                 // intermediate results reused to compute each value, and hence should limit
                 // as much as possible roundoff error accumulation
                 // (this does not change the number of floating point operations)
                 final int p = m / 2;
                 final int q = m - p;
 
                 cosSin[0][m] = cosSin[0][p] * cosSin[0][q] - cosSin[1][p] * cosSin[1][q];
                 cosSin[1][m] = cosSin[1][p] * cosSin[0][q] + cosSin[0][p] * cosSin[1][q];
             }
         }
 
         return cosSin;
 
     }
 
     /** Compute longitude cosines and sines.
      * @param cosLambda cos(λ)
      * @param sinLambda sin(λ)
      * @param <T> type of field used
      * @return array containing cos(m &times; λ) in row 0
      * and sin(m &times; λ) in row 1
      */
     private <T extends CalculusFieldElement<T>> T[][] createCosSinArrays(final T cosLambda, final T sinLambda) {
 
         final T one = cosLambda.getField().getOne();
         final T zero = cosLambda.getField().getZero();
         // initialize arrays
         final T[][] cosSin = MathArrays.buildArray(one.getField(), 2, provider.getMaxOrder() + 1);
         cosSin[0][0] = one;
         cosSin[1][0] = zero;
         if (provider.getMaxOrder() > 0) {
             cosSin[0][1] = cosLambda;
             cosSin[1][1] = sinLambda;
 
             // fill up array
             for (int m = 2; m < cosSin[0].length; ++m) {
 
                 // m * lambda is split as p * lambda + q * lambda, trying to avoid
                 // p or q being much larger than the other. This reduces the number of
                 // intermediate results reused to compute each value, and hence should limit
                 // as much as possible roundoff error accumulation
                 // (this does not change the number of floating point operations)
                 final int p = m / 2;
                 final int q = m - p;
 
                 cosSin[0][m] = cosSin[0][p].multiply(cosSin[0][q]).subtract(cosSin[1][p].multiply(cosSin[1][q]));
                 cosSin[1][m] = cosSin[1][p].multiply(cosSin[0][q]).add(cosSin[0][p].multiply(cosSin[1][q]));
 
             }
         }
 
         return cosSin;
 
     }
 
     /** Compute one order of tesseral terms.
      * <p>
      * This corresponds to equations 27 and 30 of the paper.
      * </p>
      * @param m current order
      * @param degree max degree
      * @param index index in the flattened array
      * @param t cos(θ), where θ is the polar angle
      * @param u sin(θ), where θ is the polar angle
      * @param tOu t/u
      * @param pnm0Plus2 array containing scaled P<sub>n,m+2</sub>/u<sup>m+2</sup>
      * @param pnm0Plus1 array containing scaled P<sub>n,m+1</sub>/u<sup>m+1</sup>
      * @param pnm1Plus1 array containing scaled dP<sub>n,m+1</sub>/u<sup>m+1</sup>
      * (may be null if second derivatives are not needed)
      * @param pnm0 array to fill with scaled P<sub>n,m</sub>/u<sup>m</sup>
      * @param pnm1 array to fill with scaled dP<sub>n,m</sub>/u<sup>m</sup>
      * (may be null if first derivatives are not needed)
      * @param pnm2 array to fill with scaled d²P<sub>n,m</sub>/u<sup>m</sup>
      * (may be null if second derivatives are not needed)
      * @return new value for index
      */
     private int computeTesseral(final int m, final int degree, final int index,
                                 final double t, final double u, final double tOu,
                                 final double[] pnm0Plus2, final double[] pnm0Plus1, final double[] pnm1Plus1,
                                 final double[] pnm0, final double[] pnm1, final double[] pnm2) {
 
         final double u2 = u * u;
 
         // initialize recursion from sectorial terms
         int n = FastMath.max(2, m);
         if (n == m) {
             pnm0[n] = sectorial[n];
             ++n;
         }
 
         // compute tesseral values
         int localIndex = index;
         while (n <= degree) {
 
             // value (equation 27 of the paper)
             pnm0[n] = gnmOj[localIndex] * t * pnm0Plus1[n] - hnmOj[localIndex] * u2 * pnm0Plus2[n];
 
             ++localIndex;
             ++n;
 
         }
 
         if (pnm1 != null) {
 
             // initialize recursion from sectorial terms
             n = FastMath.max(2, m);
             if (n == m) {
                 pnm1[n] = m * tOu * pnm0[n];
                 ++n;
             }
 
             // compute tesseral values and derivatives with respect to polar angle
             localIndex = index;
             while (n <= degree) {
 
                 // first derivative (equation 30 of the paper)
                 pnm1[n] = m * tOu * pnm0[n] - enm[localIndex] * u * pnm0Plus1[n];
 
                 ++localIndex;
                 ++n;
 
             }
 
             if (pnm2 != null) {
 
                 // initialize recursion from sectorial terms
                 n = FastMath.max(2, m);
                 if (n == m) {
                     pnm2[n] = m * (tOu * pnm1[n] - pnm0[n] / u2);
                     ++n;
                 }
 
                 // compute tesseral values and derivatives with respect to polar angle
                 localIndex = index;
                 while (n <= degree) {
 
                     // second derivative (differential of equation 30 with respect to theta)
                     pnm2[n] = m * (tOu * pnm1[n] - pnm0[n] / u2) - enm[localIndex] * u * pnm1Plus1[n];
 
                     ++localIndex;
                     ++n;
 
                 }
 
             }
 
         }
 
         return localIndex;
 
     }
 
     /** Compute one order of tesseral terms.
      * <p>
      * This corresponds to equations 27 and 30 of the paper.
      * </p>
      * @param m current order
      * @param degree max degree
      * @param index index in the flattened array
      * @param t cos(θ), where θ is the polar angle
      * @param u sin(θ), where θ is the polar angle
      * @param tOu t/u
      * @param pnm0Plus2 array containing scaled P<sub>n,m+2</sub>/u<sup>m+2</sup>
      * @param pnm0Plus1 array containing scaled P<sub>n,m+1</sub>/u<sup>m+1</sup>
      * @param pnm1Plus1 array containing scaled dP<sub>n,m+1</sub>/u<sup>m+1</sup>
      * (may be null if second derivatives are not needed)
      * @param pnm0 array to fill with scaled P<sub>n,m</sub>/u<sup>m</sup>
      * @param pnm1 array to fill with scaled dP<sub>n,m</sub>/u<sup>m</sup>
      * (may be null if first derivatives are not needed)
      * @param pnm2 array to fill with scaled d²P<sub>n,m</sub>/u<sup>m</sup>
      * (may be null if second derivatives are not needed)
      * @param <T> instance of field element
      * @return new value for index
      */
     private <T extends CalculusFieldElement<T>> int computeTesseral(final int m, final int degree, final int index,
                                                                 final T t, final T u, final T tOu,
                                                                 final T[] pnm0Plus2, final T[] pnm0Plus1, final T[] pnm1Plus1,
                                                                 final T[] pnm0, final T[] pnm1, final T[] pnm2) {
 
         final T u2 = u.square();
         final T zero = u.getField().getZero();
         // initialize recursion from sectorial terms
         int n = FastMath.max(2, m);
         if (n == m) {
             pnm0[n] = zero.newInstance(sectorial[n]);
             ++n;
         }
 
         // compute tesseral values
         int localIndex = index;
         while (n <= degree) {
 
             // value (equation 27 of the paper)
             pnm0[n] = t.multiply(gnmOj[localIndex]).multiply(pnm0Plus1[n]).subtract(u2.multiply(pnm0Plus2[n]).multiply(hnmOj[localIndex]));
             ++localIndex;
             ++n;
 
         }
         if (pnm1 != null) {
 
             // initialize recursion from sectorial terms
             n = FastMath.max(2, m);
             if (n == m) {
                 pnm1[n] = tOu.multiply(m).multiply(pnm0[n]);
                 ++n;
             }
 
             // compute tesseral values and derivatives with respect to polar angle
             localIndex = index;
             while (n <= degree) {
 
                 // first derivative (equation 30 of the paper)
                 pnm1[n] = tOu.multiply(m).multiply(pnm0[n]).subtract(u.multiply(enm[localIndex]).multiply(pnm0Plus1[n]));
 
                 ++localIndex;
                 ++n;
 
             }
 
             if (pnm2 != null) {
 
                 // initialize recursion from sectorial terms
                 n = FastMath.max(2, m);
                 if (n == m) {
                     pnm2[n] =   tOu.multiply(pnm1[n]).subtract(pnm0[n].divide(u2)).multiply(m);
                     ++n;
                 }
 
                 // compute tesseral values and derivatives with respect to polar angle
                 localIndex = index;
                 while (n <= degree) {
 
                     // second derivative (differential of equation 30 with respect to theta)
                     pnm2[n] = tOu.multiply(pnm1[n]).subtract(pnm0[n].divide(u2)).multiply(m).subtract(u.multiply(pnm1Plus1[n]).multiply(enm[localIndex]));
                     ++localIndex;
                     ++n;
 
                 }
 
             }
 
         }
         return localIndex;
 
     }
 
     /** {@inheritDoc} */
     @Override
     public Vector3D acceleration(final SpacecraftState s, final double[] parameters) {
 
         final double mu = parameters[0];
 
         // get the position in body frame
         final AbsoluteDate date       = s.getDate();
         final StaticTransform fromBodyFrame = bodyFrame.getStaticTransformTo(s.getFrame(), date);
         final StaticTransform toBodyFrame   = fromBodyFrame.getInverse();
         final Vector3D position       = toBodyFrame.transformPosition(s.getPosition());
 
         // gradient of the non-central part of the gravity field
         return fromBodyFrame.transformVector(new Vector3D(gradient(date, position, mu)));
 
     }
 
     /** {@inheritDoc} */
     public <T extends CalculusFieldElement<T>> FieldVector3D<T> acceleration(final FieldSpacecraftState<T> s,
                                                                          final T[] parameters) {
 
         final T mu = parameters[0];
 
         // check for faster computation dedicated to derivatives with respect to state and gravitational parameter
         if (s.getDate().hasZeroField()) {
             if (isGradientStateDerivative(s) && isGradientConstantOrMuDerivative(mu)) {
                 @SuppressWarnings("unchecked")
                 final FieldVector3D<Gradient> p = (FieldVector3D<Gradient>) s.getPosition();
                 @SuppressWarnings("unchecked")
                 final FieldVector3D<T> a = (FieldVector3D<T>) accelerationWrtState(s.getDate().toAbsoluteDate(),
                         s.getFrame(), p,
                         (Gradient) mu);
                 return a;
             } else if (isDSStateDerivative(s) && isDSConstantOrMuDerivative(mu)) {
                 @SuppressWarnings("unchecked")
                 final FieldVector3D<DerivativeStructure> p = (FieldVector3D<DerivativeStructure>) s.getPosition();
                 @SuppressWarnings("unchecked")
                 final FieldVector3D<T> a = (FieldVector3D<T>) accelerationWrtState(s.getDate().toAbsoluteDate(),
                         s.getFrame(), p,
                         (DerivativeStructure) mu);
                 return a;
             }
         }
 
         // get the position in body frame
         final FieldAbsoluteDate<T> date             = s.getDate();
         final FieldStaticTransform<T> fromBodyFrame = bodyFrame.getStaticTransformTo(s.getFrame(), date);
         final FieldStaticTransform<T> toBodyFrame   = fromBodyFrame.getInverse();
         final FieldVector3D<T> position             = toBodyFrame.transformPosition(s.getPosition());
 
         // gradient of the non-central part of the gravity field
         return fromBodyFrame.transformVector(new FieldVector3D<>(gradient(date, position, mu)));
 
     }
 
     /** Check if a field state corresponds to derivatives with respect to state.
      * @param state state to check
      * @param <T> type of the filed elements
      * @return true if state corresponds to derivatives with respect to state
      * @since 9.0
      */
     private <T extends CalculusFieldElement<T>> boolean isDSStateDerivative(final FieldSpacecraftState<T> state) {
         try {
             final DerivativeStructure dsMass = (DerivativeStructure) state.getMass();
             final int o = dsMass.getOrder();
             final int p = dsMass.getFreeParameters();
             if (o != 1 || p < 3) {
                 return false;
             }
             @SuppressWarnings("unchecked")
             final FieldPVCoordinates<DerivativeStructure> pv = (FieldPVCoordinates<DerivativeStructure>) state.getPVCoordinates();
             return isVariable(pv.getPosition().getX(), 0) &&
                    isVariable(pv.getPosition().getY(), 1) &&
                    isVariable(pv.getPosition().getZ(), 2);
         } catch (ClassCastException cce) {
             return false;
         }
     }
 
     /** Check if a field state corresponds to derivatives with respect to state.
      * @param state state to check
      * @param <T> type of the filed elements
      * @return true if state corresponds to derivatives with respect to state
      * @since 10.2
      */
     private <T extends CalculusFieldElement<T>> boolean isGradientStateDerivative(final FieldSpacecraftState<T> state) {
         try {
             final Gradient gMass = (Gradient) state.getMass();
             final int p = gMass.getFreeParameters();
             if (p < 3) {
                 return false;
             }
             @SuppressWarnings("unchecked")
             final FieldPVCoordinates<Gradient> pv = (FieldPVCoordinates<Gradient>) state.getPVCoordinates();
             return isVariable(pv.getPosition().getX(), 0) &&
                    isVariable(pv.getPosition().getY(), 1) &&
                    isVariable(pv.getPosition().getZ(), 2);
         } catch (ClassCastException cce) {
             return false;
         }
     }
 
     /**
      * Check if a field gravitational parameter is constant or represents an independent variable.
      * @param mu field gravitational parameter
      * @return true if the field gravitational parameter is constant or represents an independent variable
      * @param <T> type of field
      * @since 13.1.3
      */
     private <T extends CalculusFieldElement<T>> boolean isDSConstantOrMuDerivative(final T mu) {
         try {
             final double[] derivatives = ((DerivativeStructure) mu).getAllDerivatives();
             boolean check = true;
             for (int i = 1; i < derivatives.length; i++) {
                 if (i == 4) {
                     check &= derivatives[i] == 0.0 || derivatives[i] == 1.0;
                 } else {
                     check &= derivatives[i] == 0.0;
                 }
             }
             return check;
         } catch (ClassCastException cce) {
             return false;
         }
     }
 
     /** Check if a derivative represents a specified variable.
      * @param ds derivative to check
      * @param index index of the variable
      * @return true if the derivative represents a specified variable
      * @since 9.0
      */
     private boolean isVariable(final DerivativeStructure ds, final int index) {
         final double[] derivatives = ds.getAllDerivatives();
         boolean check = true;
         for (int i = 1; i < derivatives.length; ++i) {
             check &= derivatives[i] == ((index + 1 == i) ? 1.0 : 0.0);
         }
         return check;
     }
 
     /**
      * Check if a field gravitational parameter is constant or represents an independent variable.
      * @param mu field gravitational parameter
      * @return true if the field gravitational parameter is constant or represents an independent variable
      * @param <T> type of field
      * @since 13.1.3
      */
     private <T extends CalculusFieldElement<T>> boolean isGradientConstantOrMuDerivative(final T mu) {
         try {
             final double[] derivatives = ((Gradient) mu).getGradient();
             boolean check = true;
             for (int i = 0; i < derivatives.length; i++) {
                 if (i == 3) {
                     check &= derivatives[i] == 0.0 || derivatives[i] == 1.0;
                 } else {
                     check &= derivatives[i] == 0.0;
                 }
             }
             return check;
         } catch (ClassCastException cce) {
             return false;
         }
     }
 
     /** Check if a derivative represents a specified variable.
      * @param g derivative to check
      * @param index index of the variable
      * @return true if the derivative represents a specified variable
      * @since 10.2
      */
     private boolean isVariable(final Gradient g, final int index) {
         final double[] derivatives = g.getGradient();
         boolean check = true;
         for (int i = 0; i < derivatives.length; ++i) {
             check &= derivatives[i] == ((index == i) ? 1.0 : 0.0);
         }
         return check;
     }
 
     /** Compute acceleration derivatives with respect to state parameters.
      * <p>
      * From a theoretical point of view, this method computes the same values
      * as {@link #acceleration(FieldSpacecraftState, CalculusFieldElement[])} in the
      * specific case of {@link DerivativeStructure} with respect to state, so
      * it is less general. However, it is *much* faster in this important case.
      * <p>
      * <p>
      * The derivatives should be computed with respect to position. The input
      * parameters already take into account the free parameters (6 or 7 depending
      * on derivation with respect to mass being considered or not) and order
      * (always 1). Free parameters at indices 0, 1 and 2 correspond to derivatives
      * with respect to position. Free parameters at indices 3, 4 and 5 correspond
      * to derivatives with respect to velocity (these derivatives will remain zero
      * as acceleration due to gravity does not depend on velocity). Free parameter
      * at index 6 (if present) corresponds to to derivatives with respect to mass
      * (this derivative will remain zero as acceleration due to gravity does not
      * depend on mass).
      * </p>
      * @param date current date
      * @param frame inertial reference frame for state (both orbit and attitude)
      * @param position position of spacecraft in inertial frame
      * @param mu central attraction coefficient to use
      * @return acceleration with all derivatives specified by the input parameters
      * own derivatives
      * @since 6.0
      */
     private FieldVector3D<DerivativeStructure> accelerationWrtState(final AbsoluteDate date, final Frame frame,
                                                                     final FieldVector3D<DerivativeStructure> position,
                                                                     final DerivativeStructure mu) {
 
         // free parameters
         final int freeParameters = mu.getFreeParameters();
 
         // get the position in body frame
         final StaticTransform fromBodyFrame = bodyFrame.getStaticTransformTo(frame, date);
         final StaticTransform toBodyFrame   = fromBodyFrame.getInverse();
         final Vector3D positionBody   = toBodyFrame.transformPosition(position.toVector3D());
 
         // compute gradient and Hessian
         final GradientHessian gh   = gradientHessian(date, positionBody, mu.getReal());
 
         // gradient of the non-central part of the gravity field
         final double[] gInertial = fromBodyFrame.transformVector(new Vector3D(gh.getGradient())).toArray();
 
         // Hessian of the non-central part of the gravity field
         final RealMatrix hBody     = new Array2DRowRealMatrix(gh.getHessian(), false);
         final RealMatrix rot       = new Array2DRowRealMatrix(toBodyFrame.getRotation().getMatrix());
         final RealMatrix hInertial = rot.transposeMultiply(hBody).multiply(rot);
 
         // distribute all partial derivatives in a compact acceleration vector
         final double[] derivatives = new double[freeParameters + 1];
         final DerivativeStructure[] accDer = new DerivativeStructure[3];
         for (int i = 0; i < 3; ++i) {
 
             // first element is value of acceleration (i.e. gradient of field)
             derivatives[0] = gInertial[i];
 
             // Jacobian of acceleration (i.e. Hessian of field)
             derivatives[1] = hInertial.getEntry(i, 0);
             derivatives[2] = hInertial.getEntry(i, 1);
             derivatives[3] = hInertial.getEntry(i, 2);
 
             // next element is derivative with respect to parameter mu
             if (derivatives.length > 4 && isVariable(mu, 3)) {
                 derivatives[4] = gInertial[i] / mu.getReal();
             }
 
             accDer[i] = position.getX().getFactory().build(derivatives);
 
         }
 
         return new FieldVector3D<>(accDer);
 
     }
 
     /** Compute acceleration derivatives with respect to state parameters.
      * <p>
      * From a theoretical point of view, this method computes the same values
      * as {@link #acceleration(FieldSpacecraftState, CalculusFieldElement[])} in the
      * specific case of {@link DerivativeStructure} with respect to state, so
      * it is less general. However, it is *much* faster in this important case.
      * <p>
      * <p>
      * The derivatives should be computed with respect to position. The input
      * parameters already take into account the free parameters (6 or 7 depending
      * on derivation with respect to mass being considered or not) and order
      * (always 1). Free parameters at indices 0, 1 and 2 correspond to derivatives
      * with respect to position. Free parameters at indices 3, 4 and 5 correspond
      * to derivatives with respect to velocity (these derivatives will remain zero
      * as acceleration due to gravity does not depend on velocity). Free parameter
      * at index 6 (if present) corresponds to to derivatives with respect to mass
      * (this derivative will remain zero as acceleration due to gravity does not
      * depend on mass).
      * </p>
      * @param date current date
      * @param frame inertial reference frame for state (both orbit and attitude)
      * @param position position of spacecraft in inertial frame
      * @param mu central attraction coefficient to use
      * @return acceleration with all derivatives specified by the input parameters
      * own derivatives
      * @since 10.2
      */
     private FieldVector3D<Gradient> accelerationWrtState(final AbsoluteDate date, final Frame frame,
                                                          final FieldVector3D<Gradient> position,
                                                          final Gradient mu) {
 
         // free parameters
         final int freeParameters = mu.getFreeParameters();
 
         // get the position in body frame
         final StaticTransform fromBodyFrame = bodyFrame.getStaticTransformTo(frame, date);
         final StaticTransform toBodyFrame   = fromBodyFrame.getInverse();
         final Vector3D positionBody   = toBodyFrame.transformPosition(position.toVector3D());
 
         // compute gradient and Hessian
         final GradientHessian gh   = gradientHessian(date, positionBody, mu.getReal());
 
         // gradient of the non-central part of the gravity field
         final double[] gInertial = fromBodyFrame.transformVector(new Vector3D(gh.getGradient())).toArray();
 
         // Hessian of the non-central part of the gravity field
         final RealMatrix hBody     = new Array2DRowRealMatrix(gh.getHessian(), false);
         final RealMatrix rot       = new Array2DRowRealMatrix(toBodyFrame.getRotation().getMatrix());
         final RealMatrix hInertial = rot.transposeMultiply(hBody).multiply(rot);
 
         // distribute all partial derivatives in a compact acceleration vector
         final double[] derivatives = new double[freeParameters];
         final Gradient[] accDer = new Gradient[3];
         for (int i = 0; i < 3; ++i) {
 
             // Jacobian of acceleration (i.e. Hessian of field)
             derivatives[0] = hInertial.getEntry(i, 0);
             derivatives[1] = hInertial.getEntry(i, 1);
             derivatives[2] = hInertial.getEntry(i, 2);
 
             // next element is derivative with respect to parameter mu
             if (derivatives.length > 3 && isVariable(mu, 3)) {
                 derivatives[3] = gInertial[i] / mu.getReal();
             }
 
             accDer[i] = new Gradient(gInertial[i], derivatives);
 
         }
 
         return new FieldVector3D<>(accDer);
 
     }
 
     /** {@inheritDoc} */
     public List<ParameterDriver> getParametersDrivers() {
         return Collections.singletonList(gmParameterDriver);
     }
 
 }