FieldAngularCoordinates.java

  1. /* Copyright 2002-2019 CS Systèmes d'Information
  2.  * Licensed to CS Systèmes d'Information (CS) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * CS licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *   http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.orekit.utils;

  19. import org.hipparchus.Field;
  20. import org.hipparchus.RealFieldElement;
  21. import org.hipparchus.analysis.differentiation.DerivativeStructure;
  22. import org.hipparchus.analysis.differentiation.FDSFactory;
  23. import org.hipparchus.analysis.differentiation.FieldDerivativeStructure;
  24. import org.hipparchus.exception.LocalizedCoreFormats;
  25. import org.hipparchus.exception.MathIllegalArgumentException;
  26. import org.hipparchus.geometry.euclidean.threed.FieldRotation;
  27. import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  28. import org.hipparchus.geometry.euclidean.threed.RotationConvention;
  29. import org.hipparchus.linear.FieldDecompositionSolver;
  30. import org.hipparchus.linear.FieldMatrix;
  31. import org.hipparchus.linear.FieldQRDecomposition;
  32. import org.hipparchus.linear.FieldVector;
  33. import org.hipparchus.linear.MatrixUtils;
  34. import org.hipparchus.util.MathArrays;
  35. import org.orekit.errors.OrekitException;
  36. import org.orekit.errors.OrekitMessages;

  38. /** Simple container for rotation / rotation rate pairs, using {@link
  39.  * RealFieldElement}.
  40.  * <p>
  41.  * The state can be slightly shifted to close dates. This shift is based on
  42.  * a simple quadratic model. It is <em>not</em> intended as a replacement for
  43.  * proper attitude propagation but should be sufficient for either small
  44.  * time shifts or coarse accuracy.
  45.  * </p>
  46.  * <p>
  47.  * This class is the angular counterpart to {@link FieldPVCoordinates}.
  48.  * </p>
  49.  * <p>Instances of this class are guaranteed to be immutable.</p>
  50.  * @param <T> the type of the field elements
  51.  * @author Luc Maisonobe
  52.  * @since 6.0
  53.  * @see AngularCoordinates
  54.  */
  55. public class FieldAngularCoordinates<T extends RealFieldElement<T>> {


  58.     /** rotation. */
  59.     private final FieldRotation<T> rotation;

  61.     /** rotation rate. */
  62.     private final FieldVector3D<T> rotationRate;

  64.     /** rotation acceleration. */
  65.     private final FieldVector3D<T> rotationAcceleration;

  67.     /** Builds a rotation/rotation rate pair.
  68.      * @param rotation rotation
  69.      * @param rotationRate rotation rate Ω (rad/s)
  70.      */
  71.     public FieldAngularCoordinates(final FieldRotation<T> rotation,
  72.                                    final FieldVector3D<T> rotationRate) {
  73.         this(rotation, rotationRate,
  74.              new FieldVector3D<>(rotation.getQ0().getField().getZero(),
  75.                                  rotation.getQ0().getField().getZero(),
  76.                                  rotation.getQ0().getField().getZero()));
  77.     }

  79.     /** Builds a rotation / rotation rate / rotation acceleration triplet.
  80.      * @param rotation i.e. the orientation of the vehicle
  81.      * @param rotationRate rotation rate rate Ω, i.e. the spin vector (rad/s)
  82.      * @param rotationAcceleration angular acceleration vector dΩ/dt (rad²/s²)
  83.      */
  84.     public FieldAngularCoordinates(final FieldRotation<T> rotation,
  85.                                    final FieldVector3D<T> rotationRate,
  86.                                    final FieldVector3D<T> rotationAcceleration) {
  87.         this.rotation             = rotation;
  88.         this.rotationRate         = rotationRate;
  89.         this.rotationAcceleration = rotationAcceleration;
  90.     }

  92.     /** Build the rotation that transforms a pair of pv coordinates into another one.

  94.      * <p><em>WARNING</em>! This method requires much more stringent assumptions on
  95.      * its parameters than the similar {@link FieldRotation#FieldRotation(FieldVector3D, FieldVector3D,
  96.      * FieldVector3D, FieldVector3D) constructor} from the {@link FieldRotation FieldRotation} class.
  97.      * As far as the FieldRotation constructor is concerned, the {@code v₂} vector from
  98.      * the second pair can be slightly misaligned. The FieldRotation constructor will
  99.      * compensate for this misalignment and create a rotation that ensure {@code
  100.      * v₁ = r(u₁)} and {@code v₂ ∈ plane (r(u₁), r(u₂))}. <em>THIS IS NOT
  101.      * TRUE ANYMORE IN THIS CLASS</em>! As derivatives are involved and must be
  102.      * preserved, this constructor works <em>only</em> if the two pairs are fully
  103.      * consistent, i.e. if a rotation exists that fulfill all the requirements: {@code
  104.      * v₁ = r(u₁)}, {@code v₂ = r(u₂)}, {@code dv₁/dt = dr(u₁)/dt}, {@code dv₂/dt
  105.      * = dr(u₂)/dt}, {@code d²v₁/dt² = d²r(u₁)/dt²}, {@code d²v₂/dt² = d²r(u₂)/dt²}.</p>
  106.      * @param u1 first vector of the origin pair
  107.      * @param u2 second vector of the origin pair
  108.      * @param v1 desired image of u1 by the rotation
  109.      * @param v2 desired image of u2 by the rotation
  110.      * @param tolerance relative tolerance factor used to check singularities
  111.      */
  112.     public FieldAngularCoordinates(final FieldPVCoordinates<T> u1, final FieldPVCoordinates<T> u2,
  113.                                    final FieldPVCoordinates<T> v1, final FieldPVCoordinates<T> v2,
  114.                                    final double tolerance) {

  116.         try {
  117.             // find the initial fixed rotation
  118.             rotation = new FieldRotation<>(u1.getPosition(), u2.getPosition(),
  119.                                            v1.getPosition(), v2.getPosition());

  121.             // find rotation rate Ω such that
  122.             //  Ω ⨯ v₁ = r(dot(u₁)) - dot(v₁)
  123.             //  Ω ⨯ v₂ = r(dot(u₂)) - dot(v₂)
  124.             final FieldVector3D<T> ru1Dot = rotation.applyTo(u1.getVelocity());
  125.             final FieldVector3D<T> ru2Dot = rotation.applyTo(u2.getVelocity());


  128.             rotationRate = inverseCrossProducts(v1.getPosition(), ru1Dot.subtract(v1.getVelocity()),
  129.                                                 v2.getPosition(), ru2Dot.subtract(v2.getVelocity()),
  130.                                                 tolerance);


  133.             // find rotation acceleration dot(Ω) such that
  134.             // dot(Ω) ⨯ v₁ = r(dotdot(u₁)) - 2 Ω ⨯ dot(v₁) - Ω ⨯  (Ω ⨯ v₁) - dotdot(v₁)
  135.             // dot(Ω) ⨯ v₂ = r(dotdot(u₂)) - 2 Ω ⨯ dot(v₂) - Ω ⨯  (Ω ⨯ v₂) - dotdot(v₂)
  136.             final FieldVector3D<T> ru1DotDot = rotation.applyTo(u1.getAcceleration());
  137.             final FieldVector3D<T> oDotv1    = FieldVector3D.crossProduct(rotationRate, v1.getVelocity());
  138.             final FieldVector3D<T> oov1      = FieldVector3D.crossProduct(rotationRate, rotationRate.crossProduct(v1.getPosition()));
  139.             final FieldVector3D<T> c1        = new FieldVector3D<>(1, ru1DotDot, -2, oDotv1, -1, oov1, -1, v1.getAcceleration());
  140.             final FieldVector3D<T> ru2DotDot = rotation.applyTo(u2.getAcceleration());
  141.             final FieldVector3D<T> oDotv2    = FieldVector3D.crossProduct(rotationRate, v2.getVelocity());
  142.             final FieldVector3D<T> oov2      = FieldVector3D.crossProduct(rotationRate, rotationRate.crossProduct( v2.getPosition()));
  143.             final FieldVector3D<T> c2        = new FieldVector3D<>(1, ru2DotDot, -2, oDotv2, -1, oov2, -1, v2.getAcceleration());
  144.             rotationAcceleration     = inverseCrossProducts(v1.getPosition(), c1, v2.getPosition(), c2, tolerance);

  146.         } catch (MathIllegalArgumentException miae) {
  147.             throw new OrekitException(miae);
  148.         }

  150.     }

  152.     /** Builds a FieldAngularCoordinates from a field and a regular AngularCoordinates.
  153.      * @param field field for the components
  154.      * @param ang AngularCoordinates to convert
  155.      */
  156.     public FieldAngularCoordinates(final Field<T> field, final AngularCoordinates ang) {
  157.         this.rotation             = new FieldRotation<>(field, ang.getRotation());
  158.         this.rotationRate         = new FieldVector3D<>(field, ang.getRotationRate());
  159.         this.rotationAcceleration = new FieldVector3D<>(field, ang.getRotationAcceleration());
  160.     }

  162.     /** Builds a FieldAngularCoordinates from  a {@link FieldRotation}&lt;{@link FieldDerivativeStructure}&gt;.
  163.      * <p>
  164.      * The rotation components must have time as their only derivation parameter and
  165.      * have consistent derivation orders.
  166.      * </p>
  167.      * @param r rotation with time-derivatives embedded within the coordinates
  168.      * @since 9.2
  169.      */
  170.     public FieldAngularCoordinates(final FieldRotation<FieldDerivativeStructure<T>> r) {

  172.         final T q0       = r.getQ0().getValue();
  173.         final T q1       = r.getQ1().getValue();
  174.         final T q2       = r.getQ2().getValue();
  175.         final T q3       = r.getQ3().getValue();

  177.         rotation     = new FieldRotation<>(q0, q1, q2, q3, false);
  178.         if (r.getQ0().getOrder() >= 1) {
  179.             final T q0Dot    = r.getQ0().getPartialDerivative(1);
  180.             final T q1Dot    = r.getQ1().getPartialDerivative(1);
  181.             final T q2Dot    = r.getQ2().getPartialDerivative(1);
  182.             final T q3Dot    = r.getQ3().getPartialDerivative(1);
  183.             rotationRate =
  184.                     new FieldVector3D<>(q0.linearCombination(q1.negate(), q0Dot, q0,          q1Dot,
  185.                                                              q3,          q2Dot, q2.negate(), q3Dot).multiply(2),
  186.                                         q0.linearCombination(q2.negate(), q0Dot, q3.negate(), q1Dot,
  187.                                                              q0,          q2Dot, q1,          q3Dot).multiply(2),
  188.                                         q0.linearCombination(q3.negate(), q0Dot, q2,          q1Dot,
  189.                                                              q1.negate(), q2Dot, q0,          q3Dot).multiply(2));
  190.             if (r.getQ0().getOrder() >= 2) {
  191.                 final T q0DotDot = r.getQ0().getPartialDerivative(2);
  192.                 final T q1DotDot = r.getQ1().getPartialDerivative(2);
  193.                 final T q2DotDot = r.getQ2().getPartialDerivative(2);
  194.                 final T q3DotDot = r.getQ3().getPartialDerivative(2);
  195.                 rotationAcceleration =
  196.                         new FieldVector3D<>(q0.linearCombination(q1.negate(), q0DotDot, q0,          q1DotDot,
  197.                                                                  q3,          q2DotDot, q2.negate(), q3DotDot).multiply(2),
  198.                                             q0.linearCombination(q2.negate(), q0DotDot, q3.negate(), q1DotDot,
  199.                                                                  q0,          q2DotDot, q1,          q3DotDot).multiply(2),
  200.                                             q0.linearCombination(q3.negate(), q0DotDot, q2,          q1DotDot,
  201.                                                                  q1.negate(), q2DotDot, q0,          q3DotDot).multiply(2));
  202.             } else {
  203.                 rotationAcceleration = FieldVector3D.getZero(q0.getField());
  204.             }
  205.         } else {
  206.             rotationRate         = FieldVector3D.getZero(q0.getField());
  207.             rotationAcceleration = FieldVector3D.getZero(q0.getField());
  208.         }

  210.     }

  212.     /** Fixed orientation parallel with reference frame
  213.      * (identity rotation, zero rotation rate and acceleration).
  214.      * @param field field for the components
  215.      * @param <T> the type of the field elements
  216.      * @return a new fixed orientation parallel with reference frame
  217.      */
  218.     public static <T extends RealFieldElement<T>> FieldAngularCoordinates<T> getIdentity(final Field<T> field) {
  219.         return new FieldAngularCoordinates<>(field, AngularCoordinates.IDENTITY);
  220.     }

  222.     /** Find a vector from two known cross products.
  223.      * <p>
  224.      * We want to find Ω such that: Ω ⨯ v₁ = c₁ and Ω ⨯ v₂ = c₂
  225.      * </p>
  226.      * <p>
  227.      * The first equation (Ω ⨯ v₁ = c₁) will always be fulfilled exactly,
  228.      * and the second one will be fulfilled if possible.
  229.      * </p>
  230.      * @param v1 vector forming the first known cross product
  231.      * @param c1 know vector for cross product Ω ⨯ v₁
  232.      * @param v2 vector forming the second known cross product
  233.      * @param c2 know vector for cross product Ω ⨯ v₂
  234.      * @param tolerance relative tolerance factor used to check singularities
  235.      * @param <T> the type of the field elements
  236.      * @return vector Ω such that: Ω ⨯ v₁ = c₁ and Ω ⨯ v₂ = c₂
  237.      * @exception MathIllegalArgumentException if vectors are inconsistent and
  238.      * no solution can be found
  239.      */
  240.     private static <T extends RealFieldElement<T>> FieldVector3D<T> inverseCrossProducts(final FieldVector3D<T> v1, final FieldVector3D<T> c1,
  241.                                                                                          final FieldVector3D<T> v2, final FieldVector3D<T> c2,
  242.                                                                                          final double tolerance)
  243.         throws MathIllegalArgumentException {

  245.         final T v12 = v1.getNormSq();
  246.         final T v1n = v12.sqrt();
  247.         final T v22 = v2.getNormSq();
  248.         final T v2n = v22.sqrt();
  249.         final T threshold;
  250.         if (v1n.getReal() >= v2n.getReal()) {
  251.             threshold = v1n.multiply(tolerance);
  252.         }
  253.         else {
  254.             threshold = v2n.multiply(tolerance);
  255.         }
  256.         FieldVector3D<T> omega = null;

  258.         try {
  259.             // create the over-determined linear system representing the two cross products
  260.             final FieldMatrix<T> m = MatrixUtils.createFieldMatrix(v12.getField(), 6, 3);
  261.             m.setEntry(0, 1, v1.getZ());
  262.             m.setEntry(0, 2, v1.getY().negate());
  263.             m.setEntry(1, 0, v1.getZ().negate());
  264.             m.setEntry(1, 2, v1.getX());
  265.             m.setEntry(2, 0, v1.getY());
  266.             m.setEntry(2, 1, v1.getX().negate());
  267.             m.setEntry(3, 1, v2.getZ());
  268.             m.setEntry(3, 2, v2.getY().negate());
  269.             m.setEntry(4, 0, v2.getZ().negate());
  270.             m.setEntry(4, 2, v2.getX());
  271.             m.setEntry(5, 0, v2.getY());
  272.             m.setEntry(5, 1, v2.getX().negate());

  274.             final T[] kk = MathArrays.buildArray(v2n.getField(), 6);
  275.             kk[0] = c1.getX();
  276.             kk[1] = c1.getY();
  277.             kk[2] = c1.getZ();
  278.             kk[3] = c2.getX();
  279.             kk[4] = c2.getY();
  280.             kk[5] = c2.getZ();
  281.             final FieldVector<T> rhs = MatrixUtils.createFieldVector(kk);

  283.             // find the best solution we can
  284.             final FieldDecompositionSolver<T> solver = new FieldQRDecomposition<>(m).getSolver();
  285.             final FieldVector<T> v = solver.solve(rhs);
  286.             omega = new FieldVector3D<>(v.getEntry(0), v.getEntry(1), v.getEntry(2));

  288.         } catch (MathIllegalArgumentException miae) {
  289.             if (miae.getSpecifier() == LocalizedCoreFormats.SINGULAR_MATRIX) {

  291.                 // handle some special cases for which we can compute a solution
  292.                 final T c12 = c1.getNormSq();
  293.                 final T c1n = c12.sqrt();
  294.                 final T c22 = c2.getNormSq();
  295.                 final T c2n = c22.sqrt();
  296.                 if (c1n.getReal() <= threshold.getReal() && c2n.getReal() <= threshold.getReal()) {
  297.                     // simple special case, velocities are cancelled
  298.                     return new FieldVector3D<>(v12.getField().getZero(), v12.getField().getZero(), v12.getField().getZero());
  299.                 } else if (v1n.getReal() <= threshold.getReal() && c1n.getReal() >= threshold.getReal()) {
  300.                     // this is inconsistent, if v₁ is zero, c₁ must be 0 too
  301.                     throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE,
  302.                                                            c1n.getReal(), 0, true);
  303.                 } else if (v2n.getReal() <= threshold.getReal() && c2n.getReal() >= threshold.getReal()) {
  304.                     // this is inconsistent, if v₂ is zero, c₂ must be 0 too
  305.                     throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE,
  306.                                                            c2n.getReal(), 0, true);
  307.                 } else if (v1.crossProduct(v1).getNorm().getReal() <= threshold.getReal() && v12.getReal() > threshold.getReal()) {
  308.                     // simple special case, v₂ is redundant with v₁, we just ignore it
  309.                     // use the simplest Ω: orthogonal to both v₁ and c₁
  310.                     omega = new FieldVector3D<>(v12.reciprocal(), v1.crossProduct(c1));
  311.                 }
  312.             } else {
  313.                 throw miae;
  314.             }
  315.         }
  316.         // check results
  317.         final T d1 = FieldVector3D.distance(omega.crossProduct(v1), c1);
  318.         if (d1.getReal() > threshold.getReal()) {
  319.             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, 0, true);
  320.         }

  322.         final T d2 = FieldVector3D.distance(omega.crossProduct(v2), c2);
  323.         if (d2.getReal() > threshold.getReal()) {
  324.             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, 0, true);
  325.         }

  327.         return omega;

  329.     }

  331.     /** Transform the instance to a {@link FieldRotation}&lt;{@link FieldDerivativeStructure}&gt;.
  332.      * <p>
  333.      * The {@link DerivativeStructure} coordinates correspond to time-derivatives up
  334.      * to the user-specified order.
  335.      * </p>
  336.      * @param order derivation order for the vector components
  337.      * @return rotation with time-derivatives embedded within the coordinates
  338.           * @since 9.2
  339.      */
  340.     public FieldRotation<FieldDerivativeStructure<T>> toDerivativeStructureRotation(final int order) {

  342.         // quaternion components
  343.         final T q0 = rotation.getQ0();
  344.         final T q1 = rotation.getQ1();
  345.         final T q2 = rotation.getQ2();
  346.         final T q3 = rotation.getQ3();

  348.         // first time-derivatives of the quaternion
  349.         final T oX    = rotationRate.getX();
  350.         final T oY    = rotationRate.getY();
  351.         final T oZ    = rotationRate.getZ();
  352.         final T q0Dot = q0.linearCombination(q1.negate(), oX, q2.negate(), oY, q3.negate(), oZ).multiply(0.5);
  353.         final T q1Dot = q0.linearCombination(q0,          oX, q3.negate(), oY, q2,          oZ).multiply(0.5);
  354.         final T q2Dot = q0.linearCombination(q3,          oX, q0,          oY, q1.negate(), oZ).multiply(0.5);
  355.         final T q3Dot = q0.linearCombination(q2.negate(), oX, q1,          oY, q0,          oZ).multiply(0.5);

  357.         // second time-derivatives of the quaternion
  358.         final T oXDot = rotationAcceleration.getX();
  359.         final T oYDot = rotationAcceleration.getY();
  360.         final T oZDot = rotationAcceleration.getZ();
  361.         final T q0DotDot = q0.linearCombination(array6(q1, q2,  q3, q1Dot, q2Dot,  q3Dot),
  362.                                                 array6(oXDot, oYDot, oZDot, oX, oY, oZ)).
  363.                            multiply(-0.5);
  364.         final T q1DotDot = q0.linearCombination(array6(q0, q2, q3.negate(), q0Dot, q2Dot, q3Dot.negate()),
  365.                                                 array6(oXDot, oZDot, oYDot, oX, oZ, oY)).multiply(0.5);
  366.         final T q2DotDot =  q0.linearCombination(array6(q0, q3, q1.negate(), q0Dot, q3Dot, q1Dot.negate()),
  367.                                                  array6(oYDot, oXDot, oZDot, oY, oX, oZ)).multiply(0.5);
  368.         final T q3DotDot =  q0.linearCombination(array6(q0, q1, q2.negate(), q0Dot, q1Dot, q2Dot.negate()),
  369.                                                  array6(oZDot, oYDot, oXDot, oZ, oY, oX)).multiply(0.5);

  371.         final FDSFactory<T> factory;
  372.         final FieldDerivativeStructure<T> q0DS;
  373.         final FieldDerivativeStructure<T> q1DS;
  374.         final FieldDerivativeStructure<T> q2DS;
  375.         final FieldDerivativeStructure<T> q3DS;
  376.         switch(order) {
  377.             case 0 :
  378.                 factory = new FDSFactory<>(q0.getField(), 1, order);
  379.                 q0DS = factory.build(q0);
  380.                 q1DS = factory.build(q1);
  381.                 q2DS = factory.build(q2);
  382.                 q3DS = factory.build(q3);
  383.                 break;
  384.             case 1 :
  385.                 factory = new FDSFactory<>(q0.getField(), 1, order);
  386.                 q0DS = factory.build(q0, q0Dot);
  387.                 q1DS = factory.build(q1, q1Dot);
  388.                 q2DS = factory.build(q2, q2Dot);
  389.                 q3DS = factory.build(q3, q3Dot);
  390.                 break;
  391.             case 2 :
  392.                 factory = new FDSFactory<>(q0.getField(), 1, order);
  393.                 q0DS = factory.build(q0, q0Dot, q0DotDot);
  394.                 q1DS = factory.build(q1, q1Dot, q1DotDot);
  395.                 q2DS = factory.build(q2, q2Dot, q2DotDot);
  396.                 q3DS = factory.build(q3, q3Dot, q3DotDot);
  397.                 break;
  398.             default :
  399.                 throw new OrekitException(OrekitMessages.OUT_OF_RANGE_DERIVATION_ORDER, order);
  400.         }

  402.         return new FieldRotation<>(q0DS, q1DS, q2DS, q3DS, false);

  404.     }

  406.     /** Build an arry of 6 elements.
  407.      * @param e1 first element
  408.      * @param e2 second element
  409.      * @param e3 third element
  410.      * @param e4 fourth element
  411.      * @param e5 fifth element
  412.      * @param e6 sixth element
  413.      * @return a new array
  414.      * @since 9.2
  415.      */
  416.     private T[] array6(final T e1, final T e2, final T e3, final T e4, final T e5, final T e6) {
  417.         final T[] array = MathArrays.buildArray(e1.getField(), 6);
  418.         array[0] = e1;
  419.         array[1] = e2;
  420.         array[2] = e3;
  421.         array[3] = e4;
  422.         array[4] = e5;
  423.         array[5] = e6;
  424.         return array;
  425.     }

  427.     /** Estimate rotation rate between two orientations.
  428.      * <p>Estimation is based on a simple fixed rate rotation
  429.      * during the time interval between the two orientations.</p>
  430.      * @param start start orientation
  431.      * @param end end orientation
  432.      * @param dt time elapsed between the dates of the two orientations
  433.      * @param <T> the type of the field elements
  434.      * @return rotation rate allowing to go from start to end orientations
  435.      */
  436.     public static <T extends RealFieldElement<T>>
  437.         FieldVector3D<T> estimateRate(final FieldRotation<T> start,
  438.                                       final FieldRotation<T> end,
  439.                                       final double dt) {
  440.         return estimateRate(start, end, start.getQ0().getField().getZero().add(dt));
  441.     }

  443.     /** Estimate rotation rate between two orientations.
  444.      * <p>Estimation is based on a simple fixed rate rotation
  445.      * during the time interval between the two orientations.</p>
  446.      * @param start start orientation
  447.      * @param end end orientation
  448.      * @param dt time elapsed between the dates of the two orientations
  449.      * @param <T> the type of the field elements
  450.      * @return rotation rate allowing to go from start to end orientations
  451.      */
  452.     public static <T extends RealFieldElement<T>>
  453.         FieldVector3D<T> estimateRate(final FieldRotation<T> start,
  454.                                       final FieldRotation<T> end,
  455.                                       final T dt) {
  456.         final FieldRotation<T> evolution = start.compose(end.revert(), RotationConvention.VECTOR_OPERATOR);
  457.         return new FieldVector3D<>(evolution.getAngle().divide(dt),
  458.                                    evolution.getAxis(RotationConvention.VECTOR_OPERATOR));
  459.     }

  461.     /**
  462.      * Revert a rotation / rotation rate / rotation acceleration triplet.
  463.      *
  464.      * <p> Build a triplet which reverse the effect of another triplet.
  465.      *
  466.      * @return a new triplet whose effect is the reverse of the effect
  467.      * of the instance
  468.      */
  469.     public FieldAngularCoordinates<T> revert() {
  470.         return new FieldAngularCoordinates<>(rotation.revert(),
  471.                                              rotation.applyInverseTo(rotationRate.negate()),
  472.                                              rotation.applyInverseTo(rotationAcceleration.negate()));
  473.     }

  475.     /** Get a time-shifted state.
  476.      * <p>
  477.      * The state can be slightly shifted to close dates. This shift is based on
  478.      * a simple quadratic model. It is <em>not</em> intended as a replacement for
  479.      * proper attitude propagation but should be sufficient for either small
  480.      * time shifts or coarse accuracy.
  481.      * </p>
  482.      * @param dt time shift in seconds
  483.      * @return a new state, shifted with respect to the instance (which is immutable)
  484.      */
  485.     public FieldAngularCoordinates<T> shiftedBy(final double dt) {
  486.         return shiftedBy(rotation.getQ0().getField().getZero().add(dt));
  487.     }

  489.     /** Get a time-shifted state.
  490.      * <p>
  491.      * The state can be slightly shifted to close dates. This shift is based on
  492.      * a simple quadratic model. It is <em>not</em> intended as a replacement for
  493.      * proper attitude propagation but should be sufficient for either small
  494.      * time shifts or coarse accuracy.
  495.      * </p>
  496.      * @param dt time shift in seconds
  497.      * @return a new state, shifted with respect to the instance (which is immutable)
  498.      */
  499.     public FieldAngularCoordinates<T> shiftedBy(final T dt) {

  501.         // the shiftedBy method is based on a local approximation.
  502.         // It considers separately the contribution of the constant
  503.         // rotation, the linear contribution or the rate and the
  504.         // quadratic contribution of the acceleration. The rate
  505.         // and acceleration contributions are small rotations as long
  506.         // as the time shift is small, which is the crux of the algorithm.
  507.         // Small rotations are almost commutative, so we append these small
  508.         // contributions one after the other, as if they really occurred
  509.         // successively, despite this is not what really happens.

  511.         // compute the linear contribution first, ignoring acceleration
  512.         // BEWARE: there is really a minus sign here, because if
  513.         // the target frame rotates in one direction, the vectors in the origin
  514.         // frame seem to rotate in the opposite direction
  515.         final T rate = rotationRate.getNorm();
  516.         final T zero = rate.getField().getZero();
  517.         final T one  = rate.getField().getOne();
  518.         final FieldRotation<T> rateContribution = (rate.getReal() == 0.0) ?
  519.                                                   new FieldRotation<>(one, zero, zero, zero, false) :
  520.                                                   new FieldRotation<>(rotationRate,
  521.                                                                       rate.multiply(dt),
  522.                                                                       RotationConvention.FRAME_TRANSFORM);

  524.         // append rotation and rate contribution
  525.         final FieldAngularCoordinates<T> linearPart =
  526.                 new FieldAngularCoordinates<>(rateContribution.compose(rotation, RotationConvention.VECTOR_OPERATOR),
  527.                                               rotationRate);

  529.         final T acc  = rotationAcceleration.getNorm();
  530.         if (acc.getReal() == 0.0) {
  531.             // no acceleration, the linear part is sufficient
  532.             return linearPart;
  533.         }

  535.         // compute the quadratic contribution, ignoring initial rotation and rotation rate
  536.         // BEWARE: there is really a minus sign here, because if
  537.         // the target frame rotates in one direction, the vectors in the origin
  538.         // frame seem to rotate in the opposite direction
  539.         final FieldAngularCoordinates<T> quadraticContribution =
  540.                 new FieldAngularCoordinates<>(new FieldRotation<>(rotationAcceleration,
  541.                                                                   acc.multiply(dt.multiply(0.5).multiply(dt)),
  542.                                                                   RotationConvention.FRAME_TRANSFORM),
  543.                                               new FieldVector3D<>(dt, rotationAcceleration),
  544.                                               rotationAcceleration);

  546.         // the quadratic contribution is a small rotation:
  547.         // its initial angle and angular rate are both zero.
  548.         // small rotations are almost commutative, so we append the small
  549.         // quadratic part after the linear part as a simple offset
  550.         return quadraticContribution.addOffset(linearPart);

  552.     }

  554.     /** Get the rotation.
  555.      * @return the rotation.
  556.      */
  557.     public FieldRotation<T> getRotation() {
  558.         return rotation;
  559.     }

  561.     /** Get the rotation rate.
  562.      * @return the rotation rate vector (rad/s).
  563.      */
  564.     public FieldVector3D<T> getRotationRate() {
  565.         return rotationRate;
  566.     }

  568.     /** Get the rotation acceleration.
  569.      * @return the rotation acceleration vector dΩ/dt (rad²/s²).
  570.      */
  571.     public FieldVector3D<T> getRotationAcceleration() {
  572.         return rotationAcceleration;
  573.     }

  575.     /** Add an offset from the instance.
  576.      * <p>
  577.      * We consider here that the offset rotation is applied first and the
  578.      * instance is applied afterward. Note that angular coordinates do <em>not</em>
  579.      * commute under this operation, i.e. {@code a.addOffset(b)} and {@code
  580.      * b.addOffset(a)} lead to <em>different</em> results in most cases.
  581.      * </p>
  582.      * <p>
  583.      * The two methods {@link #addOffset(FieldAngularCoordinates) addOffset} and
  584.      * {@link #subtractOffset(FieldAngularCoordinates) subtractOffset} are designed
  585.      * so that round trip applications are possible. This means that both {@code
  586.      * ac1.subtractOffset(ac2).addOffset(ac2)} and {@code
  587.      * ac1.addOffset(ac2).subtractOffset(ac2)} return angular coordinates equal to ac1.
  588.      * </p>
  589.      * @param offset offset to subtract
  590.      * @return new instance, with offset subtracted
  591.      * @see #subtractOffset(FieldAngularCoordinates)
  592.      */
  593.     public FieldAngularCoordinates<T> addOffset(final FieldAngularCoordinates<T> offset) {
  594.         final FieldVector3D<T> rOmega    = rotation.applyTo(offset.rotationRate);
  595.         final FieldVector3D<T> rOmegaDot = rotation.applyTo(offset.rotationAcceleration);
  596.         return new FieldAngularCoordinates<>(rotation.compose(offset.rotation, RotationConvention.VECTOR_OPERATOR),
  597.                                              rotationRate.add(rOmega),
  598.                                              new FieldVector3D<>( 1.0, rotationAcceleration,
  599.                                                                   1.0, rOmegaDot,
  600.                                                                  -1.0, FieldVector3D.crossProduct(rotationRate, rOmega)));
  601.     }

  603.     /** Subtract an offset from the instance.
  604.      * <p>
  605.      * We consider here that the offset Rotation is applied first and the
  606.      * instance is applied afterward. Note that angular coordinates do <em>not</em>
  607.      * commute under this operation, i.e. {@code a.subtractOffset(b)} and {@code
  608.      * b.subtractOffset(a)} lead to <em>different</em> results in most cases.
  609.      * </p>
  610.      * <p>
  611.      * The two methods {@link #addOffset(FieldAngularCoordinates) addOffset} and
  612.      * {@link #subtractOffset(FieldAngularCoordinates) subtractOffset} are designed
  613.      * so that round trip applications are possible. This means that both {@code
  614.      * ac1.subtractOffset(ac2).addOffset(ac2)} and {@code
  615.      * ac1.addOffset(ac2).subtractOffset(ac2)} return angular coordinates equal to ac1.
  616.      * </p>
  617.      * @param offset offset to subtract
  618.      * @return new instance, with offset subtracted
  619.      * @see #addOffset(FieldAngularCoordinates)
  620.      */
  621.     public FieldAngularCoordinates<T> subtractOffset(final FieldAngularCoordinates<T> offset) {
  622.         return addOffset(offset.revert());
  623.     }

  625.     /** Convert to a regular angular coordinates.
  626.      * @return a regular angular coordinates
  627.      */
  628.     public AngularCoordinates toAngularCoordinates() {
  629.         return new AngularCoordinates(rotation.toRotation(), rotationRate.toVector3D(),
  630.                                       rotationAcceleration.toVector3D());
  631.     }

  633.     /** Apply the rotation to a pv coordinates.
  634.      * @param pv vector to apply the rotation to
  635.      * @return a new pv coordinates which is the image of u by the rotation
  636.      */
  637.     public FieldPVCoordinates<T> applyTo(final PVCoordinates pv) {

  639.         final FieldVector3D<T> transformedP = rotation.applyTo(pv.getPosition());
  640.         final FieldVector3D<T> crossP       = FieldVector3D.crossProduct(rotationRate, transformedP);
  641.         final FieldVector3D<T> transformedV = rotation.applyTo(pv.getVelocity()).subtract(crossP);
  642.         final FieldVector3D<T> crossV       = FieldVector3D.crossProduct(rotationRate, transformedV);
  643.         final FieldVector3D<T> crossCrossP  = FieldVector3D.crossProduct(rotationRate, crossP);
  644.         final FieldVector3D<T> crossDotP    = FieldVector3D.crossProduct(rotationAcceleration, transformedP);
  645.         final FieldVector3D<T> transformedA = new FieldVector3D<>( 1, rotation.applyTo(pv.getAcceleration()),
  646.                                                                   -2, crossV,
  647.                                                                   -1, crossCrossP,
  648.                                                                   -1, crossDotP);

  650.         return new FieldPVCoordinates<>(transformedP, transformedV, transformedA);

  652.     }

  654.     /** Apply the rotation to a pv coordinates.
  655.      * @param pv vector to apply the rotation to
  656.      * @return a new pv coordinates which is the image of u by the rotation
  657.      */
  658.     public TimeStampedFieldPVCoordinates<T> applyTo(final TimeStampedPVCoordinates pv) {

  660.         final FieldVector3D<T> transformedP = rotation.applyTo(pv.getPosition());
  661.         final FieldVector3D<T> crossP       = FieldVector3D.crossProduct(rotationRate, transformedP);
  662.         final FieldVector3D<T> transformedV = rotation.applyTo(pv.getVelocity()).subtract(crossP);
  663.         final FieldVector3D<T> crossV       = FieldVector3D.crossProduct(rotationRate, transformedV);
  664.         final FieldVector3D<T> crossCrossP  = FieldVector3D.crossProduct(rotationRate, crossP);
  665.         final FieldVector3D<T> crossDotP    = FieldVector3D.crossProduct(rotationAcceleration, transformedP);
  666.         final FieldVector3D<T> transformedA = new FieldVector3D<>( 1, rotation.applyTo(pv.getAcceleration()),
  667.                                                                   -2, crossV,
  668.                                                                   -1, crossCrossP,
  669.                                                                   -1, crossDotP);

  671.         return new TimeStampedFieldPVCoordinates<>(pv.getDate(), transformedP, transformedV, transformedA);

  673.     }

  675.     /** Apply the rotation to a pv coordinates.
  676.      * @param pv vector to apply the rotation to
  677.      * @return a new pv coordinates which is the image of u by the rotation
  678.      * @since 9.0
  679.      */
  680.     public FieldPVCoordinates<T> applyTo(final FieldPVCoordinates<T> pv) {

  682.         final FieldVector3D<T> transformedP = rotation.applyTo(pv.getPosition());
  683.         final FieldVector3D<T> crossP       = FieldVector3D.crossProduct(rotationRate, transformedP);
  684.         final FieldVector3D<T> transformedV = rotation.applyTo(pv.getVelocity()).subtract(crossP);
  685.         final FieldVector3D<T> crossV       = FieldVector3D.crossProduct(rotationRate, transformedV);
  686.         final FieldVector3D<T> crossCrossP  = FieldVector3D.crossProduct(rotationRate, crossP);
  687.         final FieldVector3D<T> crossDotP    = FieldVector3D.crossProduct(rotationAcceleration, transformedP);
  688.         final FieldVector3D<T> transformedA = new FieldVector3D<>( 1, rotation.applyTo(pv.getAcceleration()),
  689.                                                                   -2, crossV,
  690.                                                                   -1, crossCrossP,
  691.                                                                   -1, crossDotP);

  693.         return new FieldPVCoordinates<>(transformedP, transformedV, transformedA);

  695.     }

  697.     /** Apply the rotation to a pv coordinates.
  698.      * @param pv vector to apply the rotation to
  699.      * @return a new pv coordinates which is the image of u by the rotation
  700.      * @since 9.0
  701.      */
  702.     public TimeStampedFieldPVCoordinates<T> applyTo(final TimeStampedFieldPVCoordinates<T> pv) {

  704.         final FieldVector3D<T> transformedP = rotation.applyTo(pv.getPosition());
  705.         final FieldVector3D<T> crossP       = FieldVector3D.crossProduct(rotationRate, transformedP);
  706.         final FieldVector3D<T> transformedV = rotation.applyTo(pv.getVelocity()).subtract(crossP);
  707.         final FieldVector3D<T> crossV       = FieldVector3D.crossProduct(rotationRate, transformedV);
  708.         final FieldVector3D<T> crossCrossP  = FieldVector3D.crossProduct(rotationRate, crossP);
  709.         final FieldVector3D<T> crossDotP    = FieldVector3D.crossProduct(rotationAcceleration, transformedP);
  710.         final FieldVector3D<T> transformedA = new FieldVector3D<>( 1, rotation.applyTo(pv.getAcceleration()),
  711.                                                                   -2, crossV,
  712.                                                                   -1, crossCrossP,
  713.                                                                   -1, crossDotP);

  715.         return new TimeStampedFieldPVCoordinates<>(pv.getDate(), transformedP, transformedV, transformedA);

  717.     }

  719.     /** Convert rotation, rate and acceleration to modified Rodrigues vector and derivatives.
  720.      * <p>
  721.      * The modified Rodrigues vector is tan(θ/4) u where θ and u are the
  722.      * rotation angle and axis respectively.
  723.      * </p>
  724.      * @param sign multiplicative sign for quaternion components
  725.      * @return modified Rodrigues vector and derivatives (vector on row 0, first derivative
  726.      * on row 1, second derivative on row 2)
  727.      * @see #createFromModifiedRodrigues(RealFieldElement[][])
  728.      * @since 9.0
  729.      */
  730.     public T[][] getModifiedRodrigues(final double sign) {

  732.         final T q0    = getRotation().getQ0().multiply(sign);
  733.         final T q1    = getRotation().getQ1().multiply(sign);
  734.         final T q2    = getRotation().getQ2().multiply(sign);
  735.         final T q3    = getRotation().getQ3().multiply(sign);
  736.         final T oX    = getRotationRate().getX();
  737.         final T oY    = getRotationRate().getY();
  738.         final T oZ    = getRotationRate().getZ();
  739.         final T oXDot = getRotationAcceleration().getX();
  740.         final T oYDot = getRotationAcceleration().getY();
  741.         final T oZDot = getRotationAcceleration().getZ();

  743.         // first time-derivatives of the quaternion
  744.         final T q0Dot = q0.linearCombination(q1.negate(), oX, q2.negate(), oY, q3.negate(), oZ).multiply(0.5);
  745.         final T q1Dot = q0.linearCombination( q0, oX, q3.negate(), oY,  q2, oZ).multiply(0.5);
  746.         final T q2Dot = q0.linearCombination( q3, oX,  q0, oY, q1.negate(), oZ).multiply(0.5);
  747.         final T q3Dot = q0.linearCombination(q2.negate(), oX,  q1, oY,  q0, oZ).multiply(0.5);

  749.         // second time-derivatives of the quaternion
  750.         final T q0DotDot = linearCombination(q1, oXDot, q2, oYDot, q3, oZDot,
  751.                                              q1Dot, oX, q2Dot, oY, q3Dot, oZ).
  752.                            multiply(-0.5);
  753.         final T q1DotDot = linearCombination(q0, oXDot, q2, oZDot, q3.negate(), oYDot,
  754.                                              q0Dot, oX, q2Dot, oZ, q3Dot.negate(), oY).
  755.                            multiply(0.5);
  756.         final T q2DotDot = linearCombination(q0, oYDot, q3, oXDot, q1.negate(), oZDot,
  757.                                              q0Dot, oY, q3Dot, oX, q1Dot.negate(), oZ).
  758.                            multiply(0.5);
  759.         final T q3DotDot = linearCombination(q0, oZDot, q1, oYDot, q2.negate(), oXDot,
  760.                                              q0Dot, oZ, q1Dot, oY, q2Dot.negate(), oX).
  761.                            multiply(0.5);

  763.         // the modified Rodrigues is tan(θ/4) u where θ and u are the rotation angle and axis respectively
  764.         // this can be rewritten using quaternion components:
  765.         //      r (q₁ / (1+q₀), q₂ / (1+q₀), q₃ / (1+q₀))
  766.         // applying the derivation chain rule to previous expression gives rDot and rDotDot
  767.         final T inv          = q0.add(1).reciprocal();
  768.         final T mTwoInvQ0Dot = inv.multiply(q0Dot).multiply(-2);

  770.         final T r1       = inv.multiply(q1);
  771.         final T r2       = inv.multiply(q2);
  772.         final T r3       = inv.multiply(q3);

  774.         final T mInvR1   = inv.multiply(r1).negate();
  775.         final T mInvR2   = inv.multiply(r2).negate();
  776.         final T mInvR3   = inv.multiply(r3).negate();

  778.         final T r1Dot    = q0.linearCombination(inv, q1Dot, mInvR1, q0Dot);
  779.         final T r2Dot    = q0.linearCombination(inv, q2Dot, mInvR2, q0Dot);
  780.         final T r3Dot    = q0.linearCombination(inv, q3Dot, mInvR3, q0Dot);

  782.         final T r1DotDot = q0.linearCombination(inv, q1DotDot, mTwoInvQ0Dot, r1Dot, mInvR1, q0DotDot);
  783.         final T r2DotDot = q0.linearCombination(inv, q2DotDot, mTwoInvQ0Dot, r2Dot, mInvR2, q0DotDot);
  784.         final T r3DotDot = q0.linearCombination(inv, q3DotDot, mTwoInvQ0Dot, r3Dot, mInvR3, q0DotDot);

  786.         final T[][] rodrigues = MathArrays.buildArray(q0.getField(), 3, 3);
  787.         rodrigues[0][0] = r1;
  788.         rodrigues[0][1] = r2;
  789.         rodrigues[0][2] = r3;
  790.         rodrigues[1][0] = r1Dot;
  791.         rodrigues[1][1] = r2Dot;
  792.         rodrigues[1][2] = r3Dot;
  793.         rodrigues[2][0] = r1DotDot;
  794.         rodrigues[2][1] = r2DotDot;
  795.         rodrigues[2][2] = r3DotDot;
  796.         return rodrigues;

  798.     }

  800.     /**
  801.      * Compute a linear combination.
  802.      * @param a1 first factor of the first term
  803.      * @param b1 second factor of the first term
  804.      * @param a2 first factor of the second term
  805.      * @param b2 second factor of the second term
  806.      * @param a3 first factor of the third term
  807.      * @param b3 second factor of the third term
  808.      * @param a4 first factor of the fourth term
  809.      * @param b4 second factor of the fourth term
  810.      * @param a5 first factor of the fifth term
  811.      * @param b5 second factor of the fifth term
  812.      * @param a6 first factor of the sixth term
  813.      * @param b6 second factor of the sicth term
  814.      * @return a<sub>1</sub>&times;b<sub>1</sub> + a<sub>2</sub>&times;b<sub>2</sub> +
  815.      * a<sub>3</sub>&times;b<sub>3</sub> + a<sub>4</sub>&times;b<sub>4</sub> +
  816.      * a<sub>5</sub>&times;b<sub>5</sub> + a<sub>6</sub>&times;b<sub>6</sub>
  817.      */
  818.     private T linearCombination(final T a1, final T b1, final T a2, final T b2, final T a3, final T b3,
  819.                                 final T a4, final T b4, final T a5, final T b5, final T a6, final T b6) {

  821.         final T[] a = MathArrays.buildArray(a1.getField(), 6);
  822.         a[0] = a1;
  823.         a[1] = a2;
  824.         a[2] = a3;
  825.         a[3] = a4;
  826.         a[4] = a5;
  827.         a[5] = a6;

  829.         final T[] b = MathArrays.buildArray(b1.getField(), 6);
  830.         b[0] = b1;
  831.         b[1] = b2;
  832.         b[2] = b3;
  833.         b[3] = b4;
  834.         b[4] = b5;
  835.         b[5] = b6;

  837.         return a1.linearCombination(a, b);

  839.     }

  841.     /** Convert a modified Rodrigues vector and derivatives to angular coordinates.
  842.      * @param r modified Rodrigues vector (with first and second times derivatives)
  843.      * @param <T> the type of the field elements
  844.      * @return angular coordinates
  845.      * @see #getModifiedRodrigues(double)
  846.      * @since 9.0
  847.      */
  848.     public static <T extends RealFieldElement<T>>  FieldAngularCoordinates<T> createFromModifiedRodrigues(final T[][] r) {

  850.         // rotation
  851.         final T rSquared = r[0][0].multiply(r[0][0]).add(r[0][1].multiply(r[0][1])).add(r[0][2].multiply(r[0][2]));
  852.         final T oPQ0     = rSquared.add(1).reciprocal().multiply(2);
  853.         final T q0       = oPQ0.subtract(1);
  854.         final T q1       = oPQ0.multiply(r[0][0]);
  855.         final T q2       = oPQ0.multiply(r[0][1]);
  856.         final T q3       = oPQ0.multiply(r[0][2]);

  858.         // rotation rate
  859.         final T oPQ02    = oPQ0.multiply(oPQ0);
  860.         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();
  861.         final T q1Dot    = oPQ0.multiply(r[1][0]).add(r[0][0].multiply(q0Dot));
  862.         final T q2Dot    = oPQ0.multiply(r[1][1]).add(r[0][1].multiply(q0Dot));
  863.         final T q3Dot    = oPQ0.multiply(r[1][2]).add(r[0][2].multiply(q0Dot));
  864.         final T oX       = q0.linearCombination(q1.negate(), q0Dot,  q0, q1Dot,  q3, q2Dot, q2.negate(), q3Dot).multiply(2);
  865.         final T oY       = q0.linearCombination(q2.negate(), q0Dot, q3.negate(), q1Dot,  q0, q2Dot,  q1, q3Dot).multiply(2);
  866.         final T oZ       = q0.linearCombination(q3.negate(), q0Dot,  q2, q1Dot, q1.negate(), q2Dot,  q0, q3Dot).multiply(2);

  868.         // rotation acceleration
  869.         final T q0DotDot = q0.subtract(1).negate().divide(oPQ0).multiply(q0Dot).multiply(q0Dot).
  870.                            subtract(oPQ02.multiply(q0.linearCombination(r[0][0], r[2][0], r[0][1], r[2][1], r[0][2], r[2][2]))).
  871.                            subtract(q1Dot.multiply(q1Dot).add(q2Dot.multiply(q2Dot)).add(q3Dot.multiply(q3Dot)));
  872.         final T q1DotDot = q0.linearCombination(oPQ0, r[2][0], r[1][0].add(r[1][0]), q0Dot, r[0][0], q0DotDot);
  873.         final T q2DotDot = q0.linearCombination(oPQ0, r[2][1], r[1][1].add(r[1][1]), q0Dot, r[0][1], q0DotDot);
  874.         final T q3DotDot = q0.linearCombination(oPQ0, r[2][2], r[1][2].add(r[1][2]), q0Dot, r[0][2], q0DotDot);
  875.         final T oXDot    = q0.linearCombination(q1.negate(), q0DotDot,  q0, q1DotDot,  q3, q2DotDot, q2.negate(), q3DotDot).multiply(2);
  876.         final T oYDot    = q0.linearCombination(q2.negate(), q0DotDot, q3.negate(), q1DotDot,  q0, q2DotDot,  q1, q3DotDot).multiply(2);
  877.         final T oZDot    = q0.linearCombination(q3.negate(), q0DotDot,  q2, q1DotDot, q1.negate(), q2DotDot,  q0, q3DotDot).multiply(2);

  879.         return new FieldAngularCoordinates<>(new FieldRotation<>(q0, q1, q2, q3, false),
  880.                                              new FieldVector3D<>(oX, oY, oZ),
  881.                                              new FieldVector3D<>(oXDot, oYDot, oZDot));

  883.     }

  885. }
Created with JaCoCo 0.8.2.201808211720