/* Copyright 2002-2024 CS GROUP
    * Licensed to CS GROUP (CS) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * 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
    * the License.  You may obtain a copy of the License at
    *
    *   http://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   * See the License for the specific language governing permissions and
   * limitations under the License.
   */
  package org.orekit.utils;
  
  import org.hipparchus.Field;
  import org.hipparchus.CalculusFieldElement;
  import org.hipparchus.analysis.differentiation.FDSFactory;
  import org.hipparchus.analysis.differentiation.FieldDerivative;
  import org.hipparchus.analysis.differentiation.FieldDerivativeStructure;
  import org.hipparchus.analysis.differentiation.FieldUnivariateDerivative1;
  import org.hipparchus.analysis.differentiation.FieldUnivariateDerivative2;
  import org.hipparchus.analysis.differentiation.UnivariateDerivative1;
  import org.hipparchus.analysis.differentiation.UnivariateDerivative2;
  import org.hipparchus.exception.LocalizedCoreFormats;
  import org.hipparchus.exception.MathIllegalArgumentException;
  import org.hipparchus.geometry.euclidean.threed.FieldRotation;
  import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  import org.hipparchus.geometry.euclidean.threed.RotationConvention;
  import org.hipparchus.linear.FieldDecompositionSolver;
  import org.hipparchus.linear.FieldMatrix;
  import org.hipparchus.linear.FieldQRDecomposition;
  import org.hipparchus.linear.FieldVector;
  import org.hipparchus.linear.MatrixUtils;
  import org.hipparchus.util.MathArrays;
  import org.orekit.errors.OrekitException;
  import org.orekit.errors.OrekitMessages;
  import org.orekit.time.FieldTimeShiftable;
  
  /** Simple container for rotation / rotation rate pairs, using {@link
   * CalculusFieldElement}.
   * <p>
   * The state can be slightly shifted to close dates. This shift is based on
   * a simple quadratic model. It is <em>not</em> intended as a replacement for
   * proper attitude propagation but should be sufficient for either small
   * time shifts or coarse accuracy.
   * </p>
   * <p>
   * This class is the angular counterpart to {@link FieldPVCoordinates}.
   * </p>
   * <p>Instances of this class are guaranteed to be immutable.</p>
   * @param <T> the type of the field elements
   * @author Luc Maisonobe
   * @since 6.0
   * @see AngularCoordinates
   */
  public class FieldAngularCoordinates<T extends CalculusFieldElement<T>>
          implements FieldTimeShiftable<FieldAngularCoordinates<T>, T> {
  
      /** rotation. */
      private final FieldRotation<T> rotation;
  
      /** rotation rate. */
      private final FieldVector3D<T> rotationRate;
  
      /** rotation acceleration. */
      private final FieldVector3D<T> rotationAcceleration;
  
      /** Builds a rotation/rotation rate pair.
       * @param rotation rotation
       * @param rotationRate rotation rate Ω (rad/s)
       */
      public FieldAngularCoordinates(final FieldRotation<T> rotation,
                                     final FieldVector3D<T> rotationRate) {
          this(rotation, rotationRate, FieldVector3D.getZero(rotation.getQ0().getField()));
      }
  
      /** Builds a rotation / rotation rate / rotation acceleration triplet.
       * @param rotation i.e. the orientation of the vehicle
       * @param rotationRate rotation rate Ω, i.e. the spin vector (rad/s)
       * @param rotationAcceleration angular acceleration vector dΩ/dt (rad/s²)
       */
      public FieldAngularCoordinates(final FieldRotation<T> rotation,
                                     final FieldVector3D<T> rotationRate,
                                     final FieldVector3D<T> rotationAcceleration) {
          this.rotation             = rotation;
          this.rotationRate         = rotationRate;
          this.rotationAcceleration = rotationAcceleration;
      }
  
      /** Build the rotation that transforms a pair of pv coordinates into another one.
  
       * <p><em>WARNING</em>! This method requires much more stringent assumptions on
       * its parameters than the similar {@link FieldRotation#FieldRotation(FieldVector3D, FieldVector3D,
       * FieldVector3D, FieldVector3D) constructor} from the {@link FieldRotation FieldRotation} class.
       * As far as the FieldRotation constructor is concerned, the {@code v₂} vector from
      * the second pair can be slightly misaligned. The FieldRotation constructor will
      * compensate for this misalignment and create a rotation that ensure {@code
      * v₁ = r(u₁)} and {@code v₂ ∈ plane (r(u₁), r(u₂))}. <em>THIS IS NOT
      * TRUE ANYMORE IN THIS CLASS</em>! As derivatives are involved and must be
      * preserved, this constructor works <em>only</em> if the two pairs are fully
      * consistent, i.e. if a rotation exists that fulfill all the requirements: {@code
      * v₁ = r(u₁)}, {@code v₂ = r(u₂)}, {@code dv₁/dt = dr(u₁)/dt}, {@code dv₂/dt
      * = dr(u₂)/dt}, {@code d²v₁/dt² = d²r(u₁)/dt²}, {@code d²v₂/dt² = d²r(u₂)/dt²}.</p>
      * @param u1 first vector of the origin pair
      * @param u2 second vector of the origin pair
      * @param v1 desired image of u1 by the rotation
      * @param v2 desired image of u2 by the rotation
      * @param tolerance relative tolerance factor used to check singularities
      */
     public FieldAngularCoordinates(final FieldPVCoordinates<T> u1, final FieldPVCoordinates<T> u2,
                                    final FieldPVCoordinates<T> v1, final FieldPVCoordinates<T> v2,
                                    final double tolerance) {
 
         try {
             // find the initial fixed rotation
             rotation = new FieldRotation<>(u1.getPosition(), u2.getPosition(),
                                            v1.getPosition(), v2.getPosition());
 
             // find rotation rate Ω such that
             //  Ω ⨯ v₁ = r(dot(u₁)) - dot(v₁)
             //  Ω ⨯ v₂ = r(dot(u₂)) - dot(v₂)
             final FieldVector3D<T> ru1Dot = rotation.applyTo(u1.getVelocity());
             final FieldVector3D<T> ru2Dot = rotation.applyTo(u2.getVelocity());
 
 
             rotationRate = inverseCrossProducts(v1.getPosition(), ru1Dot.subtract(v1.getVelocity()),
                                                 v2.getPosition(), ru2Dot.subtract(v2.getVelocity()),
                                                 tolerance);
 
 
             // find rotation acceleration dot(Ω) such that
             // dot(Ω) ⨯ v₁ = r(dotdot(u₁)) - 2 Ω ⨯ dot(v₁) - Ω ⨯  (Ω ⨯ v₁) - dotdot(v₁)
             // dot(Ω) ⨯ v₂ = r(dotdot(u₂)) - 2 Ω ⨯ dot(v₂) - Ω ⨯  (Ω ⨯ v₂) - dotdot(v₂)
             final FieldVector3D<T> ru1DotDot = rotation.applyTo(u1.getAcceleration());
             final FieldVector3D<T> oDotv1    = FieldVector3D.crossProduct(rotationRate, v1.getVelocity());
             final FieldVector3D<T> oov1      = FieldVector3D.crossProduct(rotationRate, rotationRate.crossProduct(v1.getPosition()));
             final FieldVector3D<T> c1        = new FieldVector3D<>(1, ru1DotDot, -2, oDotv1, -1, oov1, -1, v1.getAcceleration());
             final FieldVector3D<T> ru2DotDot = rotation.applyTo(u2.getAcceleration());
             final FieldVector3D<T> oDotv2    = FieldVector3D.crossProduct(rotationRate, v2.getVelocity());
             final FieldVector3D<T> oov2      = FieldVector3D.crossProduct(rotationRate, rotationRate.crossProduct( v2.getPosition()));
             final FieldVector3D<T> c2        = new FieldVector3D<>(1, ru2DotDot, -2, oDotv2, -1, oov2, -1, v2.getAcceleration());
             rotationAcceleration     = inverseCrossProducts(v1.getPosition(), c1, v2.getPosition(), c2, tolerance);
 
         } catch (MathIllegalArgumentException miae) {
             throw new OrekitException(miae);
         }
 
     }
 
     /** Builds a FieldAngularCoordinates from a field and a regular AngularCoordinates.
      * @param field field for the components
      * @param ang AngularCoordinates to convert
      */
     public FieldAngularCoordinates(final Field<T> field, final AngularCoordinates ang) {
         this.rotation             = new FieldRotation<>(field, ang.getRotation());
         this.rotationRate         = new FieldVector3D<>(field, ang.getRotationRate());
         this.rotationAcceleration = new FieldVector3D<>(field, ang.getRotationAcceleration());
     }
 
     /** Builds a FieldAngularCoordinates from  a {@link FieldRotation}&lt;{@link FieldDerivativeStructure}&gt;.
      * <p>
      * The rotation components must have time as their only derivation parameter and
      * have consistent derivation orders.
      * </p>
      * @param r rotation with time-derivatives embedded within the coordinates
      * @param <U> type of the derivative
      * @since 9.2
      */
     public <U extends FieldDerivative<T, U>> FieldAngularCoordinates(final FieldRotation<U> r) {
 
         final T q0       = r.getQ0().getValue();
         final T q1       = r.getQ1().getValue();
         final T q2       = r.getQ2().getValue();
         final T q3       = r.getQ3().getValue();
 
         rotation     = new FieldRotation<>(q0, q1, q2, q3, false);
         if (r.getQ0().getOrder() >= 1) {
             final T q0Dot    = r.getQ0().getPartialDerivative(1);
             final T q1Dot    = r.getQ1().getPartialDerivative(1);
             final T q2Dot    = r.getQ2().getPartialDerivative(1);
             final T q3Dot    = r.getQ3().getPartialDerivative(1);
             rotationRate =
                     new FieldVector3D<>(q0.linearCombination(q1.negate(), q0Dot, q0,          q1Dot,
                                                              q3,          q2Dot, q2.negate(), q3Dot).multiply(2),
                                         q0.linearCombination(q2.negate(), q0Dot, q3.negate(), q1Dot,
                                                              q0,          q2Dot, q1,          q3Dot).multiply(2),
                                         q0.linearCombination(q3.negate(), q0Dot, q2,          q1Dot,
                                                              q1.negate(), q2Dot, q0,          q3Dot).multiply(2));
             if (r.getQ0().getOrder() >= 2) {
                 final T q0DotDot = r.getQ0().getPartialDerivative(2);
                 final T q1DotDot = r.getQ1().getPartialDerivative(2);
                 final T q2DotDot = r.getQ2().getPartialDerivative(2);
                 final T q3DotDot = r.getQ3().getPartialDerivative(2);
                 rotationAcceleration =
                         new FieldVector3D<>(q0.linearCombination(q1.negate(), q0DotDot, q0,          q1DotDot,
                                                                  q3,          q2DotDot, q2.negate(), q3DotDot).multiply(2),
                                             q0.linearCombination(q2.negate(), q0DotDot, q3.negate(), q1DotDot,
                                                                  q0,          q2DotDot, q1,          q3DotDot).multiply(2),
                                             q0.linearCombination(q3.negate(), q0DotDot, q2,          q1DotDot,
                                                                  q1.negate(), q2DotDot, q0,          q3DotDot).multiply(2));
             } else {
                 rotationAcceleration = FieldVector3D.getZero(q0.getField());
             }
         } else {
             rotationRate         = FieldVector3D.getZero(q0.getField());
             rotationAcceleration = FieldVector3D.getZero(q0.getField());
         }
 
     }
 
     /** Fixed orientation parallel with reference frame
      * (identity rotation, zero rotation rate and acceleration).
      * @param field field for the components
      * @param <T> the type of the field elements
      * @return a new fixed orientation parallel with reference frame
      */
     public static <T extends CalculusFieldElement<T>> FieldAngularCoordinates<T> getIdentity(final Field<T> field) {
         return new FieldAngularCoordinates<>(field, AngularCoordinates.IDENTITY);
     }
 
     /** Find a vector from two known cross products.
      * <p>
      * We want to find Ω such that: Ω ⨯ v₁ = c₁ and Ω ⨯ v₂ = c₂
      * </p>
      * <p>
      * The first equation (Ω ⨯ v₁ = c₁) will always be fulfilled exactly,
      * and the second one will be fulfilled if possible.
      * </p>
      * @param v1 vector forming the first known cross product
      * @param c1 know vector for cross product Ω ⨯ v₁
      * @param v2 vector forming the second known cross product
      * @param c2 know vector for cross product Ω ⨯ v₂
      * @param tolerance relative tolerance factor used to check singularities
      * @param <T> the type of the field elements
      * @return vector Ω such that: Ω ⨯ v₁ = c₁ and Ω ⨯ v₂ = c₂
      * @exception MathIllegalArgumentException if vectors are inconsistent and
      * no solution can be found
      */
     private static <T extends CalculusFieldElement<T>> FieldVector3D<T> inverseCrossProducts(final FieldVector3D<T> v1, final FieldVector3D<T> c1,
                                                                                          final FieldVector3D<T> v2, final FieldVector3D<T> c2,
                                                                                          final double tolerance)
         throws MathIllegalArgumentException {
 
         final T v12 = v1.getNormSq();
         final T v1n = v12.sqrt();
         final T v22 = v2.getNormSq();
         final T v2n = v22.sqrt();
         final T threshold;
         if (v1n.getReal() >= v2n.getReal()) {
             threshold = v1n.multiply(tolerance);
         }
         else {
             threshold = v2n.multiply(tolerance);
         }
         FieldVector3D<T> omega = null;
 
         try {
             // create the over-determined linear system representing the two cross products
             final FieldMatrix<T> m = MatrixUtils.createFieldMatrix(v12.getField(), 6, 3);
             m.setEntry(0, 1, v1.getZ());
             m.setEntry(0, 2, v1.getY().negate());
             m.setEntry(1, 0, v1.getZ().negate());
             m.setEntry(1, 2, v1.getX());
             m.setEntry(2, 0, v1.getY());
             m.setEntry(2, 1, v1.getX().negate());
             m.setEntry(3, 1, v2.getZ());
             m.setEntry(3, 2, v2.getY().negate());
             m.setEntry(4, 0, v2.getZ().negate());
             m.setEntry(4, 2, v2.getX());
             m.setEntry(5, 0, v2.getY());
             m.setEntry(5, 1, v2.getX().negate());
 
             final T[] kk = MathArrays.buildArray(v2n.getField(), 6);
             kk[0] = c1.getX();
             kk[1] = c1.getY();
             kk[2] = c1.getZ();
             kk[3] = c2.getX();
             kk[4] = c2.getY();
             kk[5] = c2.getZ();
             final FieldVector<T> rhs = MatrixUtils.createFieldVector(kk);
 
             // find the best solution we can
             final FieldDecompositionSolver<T> solver = new FieldQRDecomposition<>(m).getSolver();
             final FieldVector<T> v = solver.solve(rhs);
             omega = new FieldVector3D<>(v.getEntry(0), v.getEntry(1), v.getEntry(2));
 
         } catch (MathIllegalArgumentException miae) {
             if (miae.getSpecifier() == LocalizedCoreFormats.SINGULAR_MATRIX) {
 
                 // handle some special cases for which we can compute a solution
                 final T c12 = c1.getNormSq();
                 final T c1n = c12.sqrt();
                 final T c22 = c2.getNormSq();
                 final T c2n = c22.sqrt();
                 if (c1n.getReal() <= threshold.getReal() && c2n.getReal() <= threshold.getReal()) {
                     // simple special case, velocities are cancelled
                     return new FieldVector3D<>(v12.getField().getZero(), v12.getField().getZero(), v12.getField().getZero());
                 } else if (v1n.getReal() <= threshold.getReal() && c1n.getReal() >= threshold.getReal()) {
                     // this is inconsistent, if v₁ is zero, c₁ must be 0 too
                     throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE,
                                                            c1n.getReal(), 0, true);
                 } else if (v2n.getReal() <= threshold.getReal() && c2n.getReal() >= threshold.getReal()) {
                     // this is inconsistent, if v₂ is zero, c₂ must be 0 too
                     throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE,
                                                            c2n.getReal(), 0, true);
                 } else if (v1.crossProduct(v1).getNorm().getReal() <= threshold.getReal() && v12.getReal() > threshold.getReal()) {
                     // simple special case, v₂ is redundant with v₁, we just ignore it
                     // use the simplest Ω: orthogonal to both v₁ and c₁
                     omega = new FieldVector3D<>(v12.reciprocal(), v1.crossProduct(c1));
                 } else {
                     throw miae;
                 }
             } else {
                 throw miae;
             }
         }
         // check results
         final T d1 = FieldVector3D.distance(omega.crossProduct(v1), c1);
         if (d1.getReal() > threshold.getReal()) {
             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, 0, true);
         }
 
         final T d2 = FieldVector3D.distance(omega.crossProduct(v2), c2);
         if (d2.getReal() > threshold.getReal()) {
             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, 0, true);
         }
 
         return omega;
 
     }
 
     /** Transform the instance to a {@link FieldRotation}&lt;{@link FieldDerivativeStructure}&gt;.
      * <p>
      * The {@link FieldDerivativeStructure} coordinates correspond to time-derivatives up
      * to the user-specified order.
      * </p>
      * @param order derivation order for the vector components
      * @return rotation with time-derivatives embedded within the coordinates
           * @since 9.2
      */
     public FieldRotation<FieldDerivativeStructure<T>> toDerivativeStructureRotation(final int order) {
 
         // quaternion components
         final T q0 = rotation.getQ0();
         final T q1 = rotation.getQ1();
         final T q2 = rotation.getQ2();
         final T q3 = rotation.getQ3();
 
         // first time-derivatives of the quaternion
         final T oX    = rotationRate.getX();
         final T oY    = rotationRate.getY();
         final T oZ    = rotationRate.getZ();
         final T q0Dot = q0.linearCombination(q1.negate(), oX, q2.negate(), oY, q3.negate(), oZ).multiply(0.5);
         final T q1Dot = q0.linearCombination(q0,          oX, q3.negate(), oY, q2,          oZ).multiply(0.5);
         final T q2Dot = q0.linearCombination(q3,          oX, q0,          oY, q1.negate(), oZ).multiply(0.5);
         final T q3Dot = q0.linearCombination(q2.negate(), oX, q1,          oY, q0,          oZ).multiply(0.5);
 
         // second time-derivatives of the quaternion
         final T oXDot = rotationAcceleration.getX();
         final T oYDot = rotationAcceleration.getY();
         final T oZDot = rotationAcceleration.getZ();
         final T q0DotDot = q0.linearCombination(array6(q1, q2,  q3, q1Dot, q2Dot,  q3Dot),
                                                 array6(oXDot, oYDot, oZDot, oX, oY, oZ)).multiply(-0.5);
         final T q1DotDot = q0.linearCombination(array6(q0, q2, q3.negate(), q0Dot, q2Dot, q3Dot.negate()),
                                                 array6(oXDot, oZDot, oYDot, oX, oZ, oY)).multiply(0.5);
         final T q2DotDot = q0.linearCombination(array6(q0, q3, q1.negate(), q0Dot, q3Dot, q1Dot.negate()),
                                                 array6(oYDot, oXDot, oZDot, oY, oX, oZ)).multiply(0.5);
         final T q3DotDot = q0.linearCombination(array6(q0, q1, q2.negate(), q0Dot, q1Dot, q2Dot.negate()),
                                                 array6(oZDot, oYDot, oXDot, oZ, oY, oX)).multiply(0.5);
 
         final FDSFactory<T> factory;
         final FieldDerivativeStructure<T> q0DS;
         final FieldDerivativeStructure<T> q1DS;
         final FieldDerivativeStructure<T> q2DS;
         final FieldDerivativeStructure<T> q3DS;
         switch (order) {
             case 0 :
                 factory = new FDSFactory<>(q0.getField(), 1, order);
                 q0DS = factory.build(q0);
                 q1DS = factory.build(q1);
                 q2DS = factory.build(q2);
                 q3DS = factory.build(q3);
                 break;
             case 1 :
                 factory = new FDSFactory<>(q0.getField(), 1, order);
                 q0DS = factory.build(q0, q0Dot);
                 q1DS = factory.build(q1, q1Dot);
                 q2DS = factory.build(q2, q2Dot);
                 q3DS = factory.build(q3, q3Dot);
                 break;
             case 2 :
                 factory = new FDSFactory<>(q0.getField(), 1, order);
                 q0DS = factory.build(q0, q0Dot, q0DotDot);
                 q1DS = factory.build(q1, q1Dot, q1DotDot);
                 q2DS = factory.build(q2, q2Dot, q2DotDot);
                 q3DS = factory.build(q3, q3Dot, q3DotDot);
                 break;
             default :
                 throw new OrekitException(OrekitMessages.OUT_OF_RANGE_DERIVATION_ORDER, order);
         }
 
         return new FieldRotation<>(q0DS, q1DS, q2DS, q3DS, false);
 
     }
 
     /** Transform the instance to a {@link FieldRotation}&lt;{@link UnivariateDerivative1}&gt;.
      * <p>
      * The {@link UnivariateDerivative1} coordinates correspond to time-derivatives up
      * to the order 1.
      * </p>
      * @return rotation with time-derivatives embedded within the coordinates
      */
     public FieldRotation<FieldUnivariateDerivative1<T>> toUnivariateDerivative1Rotation() {
 
         // quaternion components
         final T q0 = rotation.getQ0();
         final T q1 = rotation.getQ1();
         final T q2 = rotation.getQ2();
         final T q3 = rotation.getQ3();
 
         // first time-derivatives of the quaternion
         final T oX    = rotationRate.getX();
         final T oY    = rotationRate.getY();
         final T oZ    = rotationRate.getZ();
         final T q0Dot = q0.linearCombination(q1.negate(), oX, q2.negate(), oY, q3.negate(), oZ).multiply(0.5);
         final T q1Dot = q0.linearCombination(q0,          oX, q3.negate(), oY, q2,          oZ).multiply(0.5);
         final T q2Dot = q0.linearCombination(q3,          oX, q0,          oY, q1.negate(), oZ).multiply(0.5);
         final T q3Dot = q0.linearCombination(q2.negate(), oX, q1,          oY, q0,          oZ).multiply(0.5);
 
         final FieldUnivariateDerivative1<T> q0UD = new FieldUnivariateDerivative1<>(q0, q0Dot);
         final FieldUnivariateDerivative1<T> q1UD = new FieldUnivariateDerivative1<>(q1, q1Dot);
         final FieldUnivariateDerivative1<T> q2UD = new FieldUnivariateDerivative1<>(q2, q2Dot);
         final FieldUnivariateDerivative1<T> q3UD = new FieldUnivariateDerivative1<>(q3, q3Dot);
 
         return new FieldRotation<>(q0UD, q1UD, q2UD, q3UD, false);
 
     }
 
     /** Transform the instance to a {@link FieldRotation}&lt;{@link UnivariateDerivative2}&gt;.
      * <p>
      * The {@link UnivariateDerivative2} coordinates correspond to time-derivatives up
      * to the order 2.
      * </p>
      * @return rotation with time-derivatives embedded within the coordinates
      */
     public FieldRotation<FieldUnivariateDerivative2<T>> toUnivariateDerivative2Rotation() {
 
         // quaternion components
         final T q0 = rotation.getQ0();
         final T q1 = rotation.getQ1();
         final T q2 = rotation.getQ2();
         final T q3 = rotation.getQ3();
 
         // first time-derivatives of the quaternion
         final T oX    = rotationRate.getX();
         final T oY    = rotationRate.getY();
         final T oZ    = rotationRate.getZ();
         final T q0Dot = q0.linearCombination(q1.negate(), oX, q2.negate(), oY, q3.negate(), oZ).multiply(0.5);
         final T q1Dot = q0.linearCombination(q0,          oX, q3.negate(), oY, q2,          oZ).multiply(0.5);
         final T q2Dot = q0.linearCombination(q3,          oX, q0,          oY, q1.negate(), oZ).multiply(0.5);
         final T q3Dot = q0.linearCombination(q2.negate(), oX, q1,          oY, q0,          oZ).multiply(0.5);
 
         // second time-derivatives of the quaternion
         final T oXDot = rotationAcceleration.getX();
         final T oYDot = rotationAcceleration.getY();
         final T oZDot = rotationAcceleration.getZ();
         final T q0DotDot = q0.linearCombination(array6(q1, q2,  q3, q1Dot, q2Dot,  q3Dot),
                                                 array6(oXDot, oYDot, oZDot, oX, oY, oZ)).multiply(-0.5);
         final T q1DotDot = q0.linearCombination(array6(q0, q2, q3.negate(), q0Dot, q2Dot, q3Dot.negate()),
                                                 array6(oXDot, oZDot, oYDot, oX, oZ, oY)).multiply(0.5);
         final T q2DotDot = q0.linearCombination(array6(q0, q3, q1.negate(), q0Dot, q3Dot, q1Dot.negate()),
                                                 array6(oYDot, oXDot, oZDot, oY, oX, oZ)).multiply(0.5);
         final T q3DotDot = q0.linearCombination(array6(q0, q1, q2.negate(), q0Dot, q1Dot, q2Dot.negate()),
                                                 array6(oZDot, oYDot, oXDot, oZ, oY, oX)).multiply(0.5);
 
         final FieldUnivariateDerivative2<T> q0UD = new FieldUnivariateDerivative2<>(q0, q0Dot, q0DotDot);
         final FieldUnivariateDerivative2<T> q1UD = new FieldUnivariateDerivative2<>(q1, q1Dot, q1DotDot);
         final FieldUnivariateDerivative2<T> q2UD = new FieldUnivariateDerivative2<>(q2, q2Dot, q2DotDot);
         final FieldUnivariateDerivative2<T> q3UD = new FieldUnivariateDerivative2<>(q3, q3Dot, q3DotDot);
 
         return new FieldRotation<>(q0UD, q1UD, q2UD, q3UD, false);
 
     }
 
     /** Build an arry of 6 elements.
      * @param e1 first element
      * @param e2 second element
      * @param e3 third element
      * @param e4 fourth element
      * @param e5 fifth element
      * @param e6 sixth element
      * @return a new array
      * @since 9.2
      */
     private T[] array6(final T e1, final T e2, final T e3, final T e4, final T e5, final T e6) {
         final T[] array = MathArrays.buildArray(e1.getField(), 6);
         array[0] = e1;
         array[1] = e2;
         array[2] = e3;
         array[3] = e4;
         array[4] = e5;
         array[5] = e6;
         return array;
     }
 
     /** Estimate rotation rate between two orientations.
      * <p>Estimation is based on a simple fixed rate rotation
      * during the time interval between the two orientations.</p>
      * @param start start orientation
      * @param end end orientation
      * @param dt time elapsed between the dates of the two orientations
      * @param <T> the type of the field elements
      * @return rotation rate allowing to go from start to end orientations
      */
     public static <T extends CalculusFieldElement<T>>
         FieldVector3D<T> estimateRate(final FieldRotation<T> start,
                                       final FieldRotation<T> end,
                                       final double dt) {
         return estimateRate(start, end, start.getQ0().getField().getZero().newInstance(dt));
     }
 
     /** Estimate rotation rate between two orientations.
      * <p>Estimation is based on a simple fixed rate rotation
      * during the time interval between the two orientations.</p>
      * @param start start orientation
      * @param end end orientation
      * @param dt time elapsed between the dates of the two orientations
      * @param <T> the type of the field elements
      * @return rotation rate allowing to go from start to end orientations
      */
     public static <T extends CalculusFieldElement<T>>
         FieldVector3D<T> estimateRate(final FieldRotation<T> start,
                                       final FieldRotation<T> end,
                                       final T dt) {
         final FieldRotation<T> evolution = start.compose(end.revert(), RotationConvention.VECTOR_OPERATOR);
         return new FieldVector3D<>(evolution.getAngle().divide(dt),
                                    evolution.getAxis(RotationConvention.VECTOR_OPERATOR));
     }
 
     /**
      * Revert a rotation / rotation rate / rotation acceleration triplet.
      *
      * <p> Build a triplet which reverse the effect of another triplet.
      *
      * @return a new triplet whose effect is the reverse of the effect
      * of the instance
      */
     public FieldAngularCoordinates<T> revert() {
         return new FieldAngularCoordinates<>(rotation.revert(),
                                              rotation.applyInverseTo(rotationRate.negate()),
                                              rotation.applyInverseTo(rotationAcceleration.negate()));
     }
 
     /** Get a time-shifted rotation. Same as {@link #shiftedBy(double)} except
      * only the shifted rotation is computed.
      * <p>
      * The state can be slightly shifted to close dates. This shift is based on
      * an approximate solution of the fixed acceleration motion. It is <em>not</em>
      * intended as a replacement for proper attitude propagation but should be
      * sufficient for either small time shifts or coarse accuracy.
      * </p>
      * @param dt time shift in seconds
      * @return a new state, shifted with respect to the instance (which is immutable)
      * @see  #shiftedBy(CalculusFieldElement)
      * @since 11.2
      */
     public FieldRotation<T> rotationShiftedBy(final T dt) {
 
         // the shiftedBy method is based on a local approximation.
         // It considers separately the contribution of the constant
         // rotation, the linear contribution or the rate and the
         // quadratic contribution of the acceleration. The rate
         // and acceleration contributions are small rotations as long
         // as the time shift is small, which is the crux of the algorithm.
         // Small rotations are almost commutative, so we append these small
         // contributions one after the other, as if they really occurred
         // successively, despite this is not what really happens.
 
         // compute the linear contribution first, ignoring acceleration
         // BEWARE: there is really a minus sign here, because if
         // the target frame rotates in one direction, the vectors in the origin
         // frame seem to rotate in the opposite direction
         final T rate = rotationRate.getNorm();
         final FieldRotation<T> rateContribution = (rate.getReal() == 0.0) ?
                 FieldRotation.getIdentity(dt.getField()) :
                 new FieldRotation<>(rotationRate, rate.multiply(dt), RotationConvention.FRAME_TRANSFORM);
 
         // append rotation and rate contribution
         final FieldRotation<T> linearPart =
                 rateContribution.compose(rotation, RotationConvention.VECTOR_OPERATOR);
 
         final T acc  = rotationAcceleration.getNorm();
         if (acc.getReal() == 0.0) {
             // no acceleration, the linear part is sufficient
             return linearPart;
         }
 
         // compute the quadratic contribution, ignoring initial rotation and rotation rate
         // BEWARE: there is really a minus sign here, because if
         // the target frame rotates in one direction, the vectors in the origin
         // frame seem to rotate in the opposite direction
         final FieldRotation<T> quadraticContribution =
                 new FieldRotation<>(rotationAcceleration,
                         acc.multiply(dt.square()).multiply(0.5),
                         RotationConvention.FRAME_TRANSFORM);
 
         // the quadratic contribution is a small rotation:
         // its initial angle and angular rate are both zero.
         // small rotations are almost commutative, so we append the small
         // quadratic part after the linear part as a simple offset
         return quadraticContribution
                 .compose(linearPart, RotationConvention.VECTOR_OPERATOR);
 
     }
 
     /** Get a time-shifted state.
      * <p>
      * The state can be slightly shifted to close dates. This shift is based on
      * a simple quadratic model. It is <em>not</em> intended as a replacement for
      * proper attitude propagation but should be sufficient for either small
      * time shifts or coarse accuracy.
      * </p>
      * @param dt time shift in seconds
      * @return a new state, shifted with respect to the instance (which is immutable)
      */
     @Override
     public FieldAngularCoordinates<T> shiftedBy(final double dt) {
         return shiftedBy(rotation.getQ0().getField().getZero().newInstance(dt));
     }
 
     /** Get a time-shifted state.
      * <p>
      * The state can be slightly shifted to close dates. This shift is based on
      * a simple quadratic model. It is <em>not</em> intended as a replacement for
      * proper attitude propagation but should be sufficient for either small
      * time shifts or coarse accuracy.
      * </p>
      * @param dt time shift in seconds
      * @return a new state, shifted with respect to the instance (which is immutable)
      */
     @Override
     public FieldAngularCoordinates<T> shiftedBy(final T dt) {
 
         // the shiftedBy method is based on a local approximation.
         // It considers separately the contribution of the constant
         // rotation, the linear contribution or the rate and the
         // quadratic contribution of the acceleration. The rate
         // and acceleration contributions are small rotations as long
         // as the time shift is small, which is the crux of the algorithm.
         // Small rotations are almost commutative, so we append these small
         // contributions one after the other, as if they really occurred
         // successively, despite this is not what really happens.
 
         // compute the linear contribution first, ignoring acceleration
         // BEWARE: there is really a minus sign here, because if
         // the target frame rotates in one direction, the vectors in the origin
         // frame seem to rotate in the opposite direction
         final T rate = rotationRate.getNorm();
         final T zero = rate.getField().getZero();
         final T one  = rate.getField().getOne();
         final FieldRotation<T> rateContribution = (rate.getReal() == 0.0) ?
                                                   new FieldRotation<>(one, zero, zero, zero, false) :
                                                   new FieldRotation<>(rotationRate,
                                                                       rate.multiply(dt),
                                                                       RotationConvention.FRAME_TRANSFORM);
 
         // append rotation and rate contribution
         final FieldAngularCoordinates<T> linearPart =
                 new FieldAngularCoordinates<>(rateContribution.compose(rotation, RotationConvention.VECTOR_OPERATOR),
                                               rotationRate);
 
         final T acc  = rotationAcceleration.getNorm();
         if (acc.getReal() == 0.0) {
             // no acceleration, the linear part is sufficient
             return linearPart;
         }
 
         // compute the quadratic contribution, ignoring initial rotation and rotation rate
         // BEWARE: there is really a minus sign here, because if
         // the target frame rotates in one direction, the vectors in the origin
         // frame seem to rotate in the opposite direction
         final FieldAngularCoordinates<T> quadraticContribution =
                 new FieldAngularCoordinates<>(new FieldRotation<>(rotationAcceleration,
                                                                   acc.multiply(dt.square().multiply(0.5)),
                                                                   RotationConvention.FRAME_TRANSFORM),
                                               new FieldVector3D<>(dt, rotationAcceleration),
                                               rotationAcceleration);
 
         // the quadratic contribution is a small rotation:
         // its initial angle and angular rate are both zero.
         // small rotations are almost commutative, so we append the small
         // quadratic part after the linear part as a simple offset
         return quadraticContribution.addOffset(linearPart);
 
     }
 
     /** Get the rotation.
      * @return the rotation.
      */
     public FieldRotation<T> getRotation() {
         return rotation;
     }
 
     /** Get the rotation rate.
      * @return the rotation rate vector (rad/s).
      */
     public FieldVector3D<T> getRotationRate() {
         return rotationRate;
     }
 
     /** Get the rotation acceleration.
      * @return the rotation acceleration vector dΩ/dt (rad/s²).
      */
     public FieldVector3D<T> getRotationAcceleration() {
         return rotationAcceleration;
     }
 
     /** Add an offset from the instance.
      * <p>
      * We consider here that the offset rotation is applied first and the
      * instance is applied afterward. Note that angular coordinates do <em>not</em>
      * commute under this operation, i.e. {@code a.addOffset(b)} and {@code
      * b.addOffset(a)} lead to <em>different</em> results in most cases.
      * </p>
      * <p>
      * The two methods {@link #addOffset(FieldAngularCoordinates) addOffset} and
      * {@link #subtractOffset(FieldAngularCoordinates) subtractOffset} are designed
      * so that round trip applications are possible. This means that both {@code
      * ac1.subtractOffset(ac2).addOffset(ac2)} and {@code
      * ac1.addOffset(ac2).subtractOffset(ac2)} return angular coordinates equal to ac1.
      * </p>
      * @param offset offset to subtract
      * @return new instance, with offset subtracted
      * @see #subtractOffset(FieldAngularCoordinates)
      */
     public FieldAngularCoordinates<T> addOffset(final FieldAngularCoordinates<T> offset) {
         final FieldVector3D<T> rOmega    = rotation.applyTo(offset.rotationRate);
         final FieldVector3D<T> rOmegaDot = rotation.applyTo(offset.rotationAcceleration);
         return new FieldAngularCoordinates<>(rotation.compose(offset.rotation, RotationConvention.VECTOR_OPERATOR),
                                              rotationRate.add(rOmega),
                                              new FieldVector3D<>( 1.0, rotationAcceleration,
                                                                   1.0, rOmegaDot,
                                                                  -1.0, FieldVector3D.crossProduct(rotationRate, rOmega)));
     }
 
     /** Subtract an offset from the instance.
      * <p>
      * We consider here that the offset Rotation is applied first and the
      * instance is applied afterward. Note that angular coordinates do <em>not</em>
      * commute under this operation, i.e. {@code a.subtractOffset(b)} and {@code
      * b.subtractOffset(a)} lead to <em>different</em> results in most cases.
      * </p>
      * <p>
      * The two methods {@link #addOffset(FieldAngularCoordinates) addOffset} and
      * {@link #subtractOffset(FieldAngularCoordinates) subtractOffset} are designed
      * so that round trip applications are possible. This means that both {@code
      * ac1.subtractOffset(ac2).addOffset(ac2)} and {@code
      * ac1.addOffset(ac2).subtractOffset(ac2)} return angular coordinates equal to ac1.
      * </p>
      * @param offset offset to subtract
      * @return new instance, with offset subtracted
      * @see #addOffset(FieldAngularCoordinates)
      */
     public FieldAngularCoordinates<T> subtractOffset(final FieldAngularCoordinates<T> offset) {
         return addOffset(offset.revert());
     }
 
     /** Convert to a regular angular coordinates.
      * @return a regular angular coordinates
      */
     public AngularCoordinates toAngularCoordinates() {
         return new AngularCoordinates(rotation.toRotation(), rotationRate.toVector3D(),
                                       rotationAcceleration.toVector3D());
     }
 
     /** Apply the rotation to a pv coordinates.
      * @param pv vector to apply the rotation to
      * @return a new pv coordinates which is the image of pv by the rotation
      */
     public FieldPVCoordinates<T> applyTo(final PVCoordinates pv) {
 
         final FieldVector3D<T> transformedP = rotation.applyTo(pv.getPosition());
         final FieldVector3D<T> crossP       = FieldVector3D.crossProduct(rotationRate, transformedP);
         final FieldVector3D<T> transformedV = rotation.applyTo(pv.getVelocity()).subtract(crossP);
         final FieldVector3D<T> crossV       = FieldVector3D.crossProduct(rotationRate, transformedV);
         final FieldVector3D<T> crossCrossP  = FieldVector3D.crossProduct(rotationRate, crossP);
         final FieldVector3D<T> crossDotP    = FieldVector3D.crossProduct(rotationAcceleration, transformedP);
         final FieldVector3D<T> transformedA = new FieldVector3D<>( 1, rotation.applyTo(pv.getAcceleration()),
                                                                   -2, crossV,
                                                                   -1, crossCrossP,
                                                                   -1, crossDotP);
 
         return new FieldPVCoordinates<>(transformedP, transformedV, transformedA);
 
     }
 
     /** Apply the rotation to a pv coordinates.
      * @param pv vector to apply the rotation to
      * @return a new pv coordinates which is the image of pv by the rotation
      */
     public TimeStampedFieldPVCoordinates<T> applyTo(final TimeStampedPVCoordinates pv) {
 
         final FieldVector3D<T> transformedP = rotation.applyTo(pv.getPosition());
         final FieldVector3D<T> crossP       = FieldVector3D.crossProduct(rotationRate, transformedP);
         final FieldVector3D<T> transformedV = rotation.applyTo(pv.getVelocity()).subtract(crossP);
         final FieldVector3D<T> crossV       = FieldVector3D.crossProduct(rotationRate, transformedV);
         final FieldVector3D<T> crossCrossP  = FieldVector3D.crossProduct(rotationRate, crossP);
         final FieldVector3D<T> crossDotP    = FieldVector3D.crossProduct(rotationAcceleration, transformedP);
         final FieldVector3D<T> transformedA = new FieldVector3D<>( 1, rotation.applyTo(pv.getAcceleration()),
                                                                   -2, crossV,
                                                                   -1, crossCrossP,
                                                                   -1, crossDotP);
 
         return new TimeStampedFieldPVCoordinates<>(pv.getDate(), transformedP, transformedV, transformedA);
 
     }
 
     /** Apply the rotation to a pv coordinates.
      * @param pv vector to apply the rotation to
      * @return a new pv coordinates which is the image of pv by the rotation
      * @since 9.0
      */
     public FieldPVCoordinates<T> applyTo(final FieldPVCoordinates<T> pv) {
 
         final FieldVector3D<T> transformedP = rotation.applyTo(pv.getPosition());
         final FieldVector3D<T> crossP       = FieldVector3D.crossProduct(rotationRate, transformedP);
         final FieldVector3D<T> transformedV = rotation.applyTo(pv.getVelocity()).subtract(crossP);
         final FieldVector3D<T> crossV       = FieldVector3D.crossProduct(rotationRate, transformedV);
         final FieldVector3D<T> crossCrossP  = FieldVector3D.crossProduct(rotationRate, crossP);
         final FieldVector3D<T> crossDotP    = FieldVector3D.crossProduct(rotationAcceleration, transformedP);
         final FieldVector3D<T> transformedA = new FieldVector3D<>( 1, rotation.applyTo(pv.getAcceleration()),
                                                                   -2, crossV,
                                                                   -1, crossCrossP,
                                                                   -1, crossDotP);
 
         return new FieldPVCoordinates<>(transformedP, transformedV, transformedA);
 
     }
 
     /** Apply the rotation to a pv coordinates.
      * @param pv vector to apply the rotation to
      * @return a new pv coordinates which is the image of pv by the rotation
      * @since 9.0
      */
     public TimeStampedFieldPVCoordinates<T> applyTo(final TimeStampedFieldPVCoordinates<T> pv) {
 
         final FieldVector3D<T> transformedP = rotation.applyTo(pv.getPosition());
         final FieldVector3D<T> crossP       = FieldVector3D.crossProduct(rotationRate, transformedP);
         final FieldVector3D<T> transformedV = rotation.applyTo(pv.getVelocity()).subtract(crossP);
         final FieldVector3D<T> crossV       = FieldVector3D.crossProduct(rotationRate, transformedV);
         final FieldVector3D<T> crossCrossP  = FieldVector3D.crossProduct(rotationRate, crossP);
         final FieldVector3D<T> crossDotP    = FieldVector3D.crossProduct(rotationAcceleration, transformedP);
         final FieldVector3D<T> transformedA = new FieldVector3D<>( 1, rotation.applyTo(pv.getAcceleration()),
                                                                   -2, crossV,
                                                                   -1, crossCrossP,
                                                                   -1, crossDotP);
 
         return new TimeStampedFieldPVCoordinates<>(pv.getDate(), transformedP, transformedV, transformedA);
 
     }
 
     /** Convert rotation, rate and acceleration to modified Rodrigues vector and derivatives.
      * <p>
      * The modified Rodrigues vector is tan(θ/4) u where θ and u are the
      * rotation angle and axis respectively.
      * </p>
      * @param sign multiplicative sign for quaternion components
      * @return modified Rodrigues vector and derivatives (vector on row 0, first derivative
      * on row 1, second derivative on row 2)
      * @see #createFromModifiedRodrigues(CalculusFieldElement[][])
      * @since 9.0
      */
     public T[][] getModifiedRodrigues(final double sign) {
 
         final T q0    = getRotation().getQ0().multiply(sign);
         final T q1    = getRotation().getQ1().multiply(sign);
         final T q2    = getRotation().getQ2().multiply(sign);
         final T q3    = getRotation().getQ3().multiply(sign);
         final T oX    = getRotationRate().getX();
         final T oY    = getRotationRate().getY();
         final T oZ    = getRotationRate().getZ();
         final T oXDot = getRotationAcceleration().getX();
         final T oYDot = getRotationAcceleration().getY();
         final T oZDot = getRotationAcceleration().getZ();
 
         // first time-derivatives of the quaternion
         final T q0Dot = q0.linearCombination(q1.negate(), oX, q2.negate(), oY, q3.negate(), oZ).multiply(0.5);
         final T q1Dot = q0.linearCombination( q0, oX, q3.negate(), oY,  q2, oZ).multiply(0.5);
         final T q2Dot = q0.linearCombination( q3, oX,  q0, oY, q1.negate(), oZ).multiply(0.5);
         final T q3Dot = q0.linearCombination(q2.negate(), oX,  q1, oY,  q0, oZ).multiply(0.5);
 
         // second time-derivatives of the quaternion
         final T q0DotDot = linearCombination(q1, oXDot, q2, oYDot, q3, oZDot,
                                              q1Dot, oX, q2Dot, oY, q3Dot, oZ).
                            multiply(-0.5);
         final T q1DotDot = linearCombination(q0, oXDot, q2, oZDot, q3.negate(), oYDot,
                                              q0Dot, oX, q2Dot, oZ, q3Dot.negate(), oY).
                            multiply(0.5);
         final T q2DotDot = linearCombination(q0, oYDot, q3, oXDot, q1.negate(), oZDot,
                                              q0Dot, oY, q3Dot, oX, q1Dot.negate(), oZ).
                            multiply(0.5);
         final T q3DotDot = linearCombination(q0, oZDot, q1, oYDot, q2.negate(), oXDot,
                                              q0Dot, oZ, q1Dot, oY, q2Dot.negate(), oX).
                            multiply(0.5);
 
         // the modified Rodrigues is tan(θ/4) u where θ and u are the rotation angle and axis respectively
         // this can be rewritten using quaternion components:
         //      r (q₁ / (1+q₀), q₂ / (1+q₀), q₃ / (1+q₀))
         // applying the derivation chain rule to previous expression gives rDot and rDotDot
         final T inv          = q0.add(1).reciprocal();
         final T mTwoInvQ0Dot = inv.multiply(q0Dot).multiply(-2);
 
         final T r1       = inv.multiply(q1);
         final T r2       = inv.multiply(q2);
         final T r3       = inv.multiply(q3);
 
         final T mInvR1   = inv.multiply(r1).negate();
         final T mInvR2   = inv.multiply(r2).negate();
         final T mInvR3   = inv.multiply(r3).negate();
 
         final T r1Dot    = q0.linearCombination(inv, q1Dot, mInvR1, q0Dot);
         final T r2Dot    = q0.linearCombination(inv, q2Dot, mInvR2, q0Dot);
         final T r3Dot    = q0.linearCombination(inv, q3Dot, mInvR3, q0Dot);
 
         final T r1DotDot = q0.linearCombination(inv, q1DotDot, mTwoInvQ0Dot, r1Dot, mInvR1, q0DotDot);
         final T r2DotDot = q0.linearCombination(inv, q2DotDot, mTwoInvQ0Dot, r2Dot, mInvR2, q0DotDot);
         final T r3DotDot = q0.linearCombination(inv, q3DotDot, mTwoInvQ0Dot, r3Dot, mInvR3, q0DotDot);
 
         final T[][] rodrigues = MathArrays.buildArray(q0.getField(), 3, 3);
         rodrigues[0][0] = r1;
         rodrigues[0][1] = r2;
         rodrigues[0][2] = r3;
         rodrigues[1][0] = r1Dot;
         rodrigues[1][1] = r2Dot;
         rodrigues[1][2] = r3Dot;
         rodrigues[2][0] = r1DotDot;
         rodrigues[2][1] = r2DotDot;
         rodrigues[2][2] = r3DotDot;
         return rodrigues;
 
     }
 
     /**
      * Compute a linear combination.
      * @param a1 first factor of the first term
      * @param b1 second factor of the first term
      * @param a2 first factor of the second term
      * @param b2 second factor of the second term
      * @param a3 first factor of the third term
      * @param b3 second factor of the third term
      * @param a4 first factor of the fourth term
      * @param b4 second factor of the fourth term
      * @param a5 first factor of the fifth term
      * @param b5 second factor of the fifth term
      * @param a6 first factor of the sixth term
      * @param b6 second factor of the sicth term
      * @return a<sub>1</sub>&times;b<sub>1</sub> + a<sub>2</sub>&times;b<sub>2</sub> +
      * a<sub>3</sub>&times;b<sub>3</sub> + a<sub>4</sub>&times;b<sub>4</sub> +
      * a<sub>5</sub>&times;b<sub>5</sub> + a<sub>6</sub>&times;b<sub>6</sub>
      */
     private T linearCombination(final T a1, final T b1, final T a2, final T b2, final T a3, final T b3,
                                 final T a4, final T b4, final T a5, final T b5, final T a6, final T b6) {
 
         final T[] a = MathArrays.buildArray(a1.getField(), 6);
         a[0] = a1;
         a[1] = a2;
         a[2] = a3;
         a[3] = a4;
         a[4] = a5;
         a[5] = a6;
 
         final T[] b = MathArrays.buildArray(b1.getField(), 6);
         b[0] = b1;
         b[1] = b2;
         b[2] = b3;
         b[3] = b4;
         b[4] = b5;
         b[5] = b6;
 
         return a1.linearCombination(a, b);
 
     }
 
     /** Convert a modified Rodrigues vector and derivatives to angular coordinates.
      * @param r modified Rodrigues vector (with first and second times derivatives)
      * @param <T> the type of the field elements
      * @return angular coordinates
      * @see #getModifiedRodrigues(double)
      * @since 9.0
      */
     public static <T extends CalculusFieldElement<T>>  FieldAngularCoordinates<T> createFromModifiedRodrigues(final T[][] r) {
 
         // rotation
         final T rSquared = r[0][0].square().add(r[0][1].square()).add(r[0][2].square());
         final T oPQ0     = rSquared.add(1).reciprocal().multiply(2);
         final T q0       = oPQ0.subtract(1);
         final T q1       = oPQ0.multiply(r[0][0]);
         final T q2       = oPQ0.multiply(r[0][1]);
         final T q3       = oPQ0.multiply(r[0][2]);
 
         // rotation rate
         final T oPQ02    = oPQ0.multiply(oPQ0);
         final T q0Dot    = oPQ02.multiply(q0.linearCombination(r[0][0], r[1][0], r[0][1], r[1][1],  r[0][2], r[1][2])).negate();
         final T q1Dot    = oPQ0.multiply(r[1][0]).add(r[0][0].multiply(q0Dot));
         final T q2Dot    = oPQ0.multiply(r[1][1]).add(r[0][1].multiply(q0Dot));
         final T q3Dot    = oPQ0.multiply(r[1][2]).add(r[0][2].multiply(q0Dot));
         final T oX       = q0.linearCombination(q1.negate(), q0Dot,  q0, q1Dot,  q3, q2Dot, q2.negate(), q3Dot).multiply(2);
         final T oY       = q0.linearCombination(q2.negate(), q0Dot, q3.negate(), q1Dot,  q0, q2Dot,  q1, q3Dot).multiply(2);
         final T oZ       = q0.linearCombination(q3.negate(), q0Dot,  q2, q1Dot, q1.negate(), q2Dot,  q0, q3Dot).multiply(2);
 
         // rotation acceleration
         final T q0DotDot = q0.subtract(1).negate().divide(oPQ0).multiply(q0Dot.square()).
                            subtract(oPQ02.multiply(q0.linearCombination(r[0][0], r[2][0], r[0][1], r[2][1], r[0][2], r[2][2]))).
                            subtract(q1Dot.square().add(q2Dot.square()).add(q3Dot.square()));
         final T q1DotDot = q0.linearCombination(oPQ0, r[2][0], r[1][0].add(r[1][0]), q0Dot, r[0][0], q0DotDot);
         final T q2DotDot = q0.linearCombination(oPQ0, r[2][1], r[1][1].add(r[1][1]), q0Dot, r[0][1], q0DotDot);
         final T q3DotDot = q0.linearCombination(oPQ0, r[2][2], r[1][2].add(r[1][2]), q0Dot, r[0][2], q0DotDot);
         final T oXDot    = q0.linearCombination(q1.negate(), q0DotDot,  q0, q1DotDot,  q3, q2DotDot, q2.negate(), q3DotDot).multiply(2);
         final T oYDot    = q0.linearCombination(q2.negate(), q0DotDot, q3.negate(), q1DotDot,  q0, q2DotDot,  q1, q3DotDot).multiply(2);
         final T oZDot    = q0.linearCombination(q3.negate(), q0DotDot,  q2, q1DotDot, q1.negate(), q2DotDot,  q0, q3DotDot).multiply(2);
 
         return new FieldAngularCoordinates<>(new FieldRotation<>(q0, q1, q2, q3, false),
                                              new FieldVector3D<>(oX, oY, oZ),
                                              new FieldVector3D<>(oXDot, oYDot, oZDot));
 
     }
 
 }