AngularCoordinates.java

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

  19. import java.io.Serializable;

  21. import org.hipparchus.CalculusFieldElement;
  22. import org.hipparchus.analysis.differentiation.DSFactory;
  23. import org.hipparchus.analysis.differentiation.Derivative;
  24. import org.hipparchus.analysis.differentiation.DerivativeStructure;
  25. import org.hipparchus.analysis.differentiation.UnivariateDerivative1;
  26. import org.hipparchus.analysis.differentiation.UnivariateDerivative2;
  27. import org.hipparchus.exception.LocalizedCoreFormats;
  28. import org.hipparchus.exception.MathIllegalArgumentException;
  29. import org.hipparchus.exception.MathRuntimeException;
  30. import org.hipparchus.geometry.euclidean.threed.FieldRotation;
  31. import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  32. import org.hipparchus.geometry.euclidean.threed.Rotation;
  33. import org.hipparchus.geometry.euclidean.threed.RotationConvention;
  34. import org.hipparchus.geometry.euclidean.threed.Vector3D;
  35. import org.hipparchus.linear.DecompositionSolver;
  36. import org.hipparchus.linear.MatrixUtils;
  37. import org.hipparchus.linear.QRDecomposition;
  38. import org.hipparchus.linear.RealMatrix;
  39. import org.hipparchus.linear.RealVector;
  40. import org.hipparchus.util.FastMath;
  41. import org.hipparchus.util.MathArrays;
  42. import org.orekit.errors.OrekitException;
  43. import org.orekit.errors.OrekitMessages;
  44. import org.orekit.time.TimeShiftable;

  46. /** Simple container for rotation/rotation rate/rotation acceleration triplets.
  47.  * <p>
  48.  * The state can be slightly shifted to close dates. This shift is based on
  49.  * an approximate solution of the fixed acceleration motion. It is <em>not</em>
  50.  * intended as a replacement for proper attitude propagation but should be
  51.  * sufficient for either small time shifts or coarse accuracy.
  52.  * </p>
  53.  * <p>
  54.  * This class is the angular counterpart to {@link PVCoordinates}.
  55.  * </p>
  56.  * <p>Instances of this class are guaranteed to be immutable.</p>
  57.  * @author Luc Maisonobe
  58.  */
  59. public class AngularCoordinates implements TimeShiftable<AngularCoordinates>, Serializable {

  61.     /** Fixed orientation parallel with reference frame
  62.      * (identity rotation, zero rotation rate and acceleration).
  63.      */
  64.     public static final AngularCoordinates IDENTITY =
  65.             new AngularCoordinates(Rotation.IDENTITY, Vector3D.ZERO, Vector3D.ZERO);

  67.     /** Serializable UID. */
  68.     private static final long serialVersionUID = 20140414L;

  70.     /** Rotation. */
  71.     private final Rotation rotation;

  73.     /** Rotation rate. */
  74.     private final Vector3D rotationRate;

  76.     /** Rotation acceleration. */
  77.     private final Vector3D rotationAcceleration;

  79.     /** Simple constructor.
  80.      * <p> Sets the Coordinates to default : Identity, Ω = (0 0 0), dΩ/dt = (0 0 0).</p>
  81.      */
  82.     public AngularCoordinates() {
  83.         this(Rotation.IDENTITY, Vector3D.ZERO, Vector3D.ZERO);
  84.     }

  86.     /** Builds a rotation/rotation rate pair.
  87.      * @param rotation rotation
  88.      * @param rotationRate rotation rate Ω (rad/s)
  89.      */
  90.     public AngularCoordinates(final Rotation rotation, final Vector3D rotationRate) {
  91.         this(rotation, rotationRate, Vector3D.ZERO);
  92.     }

  94.     /** Builds a rotation/rotation rate/rotation acceleration triplet.
  95.      * @param rotation rotation
  96.      * @param rotationRate rotation rate Ω (rad/s)
  97.      * @param rotationAcceleration rotation acceleration dΩ/dt (rad/s²)
  98.      */
  99.     public AngularCoordinates(final Rotation rotation,
  100.                               final Vector3D rotationRate, final Vector3D rotationAcceleration) {
  101.         this.rotation             = rotation;
  102.         this.rotationRate         = rotationRate;
  103.         this.rotationAcceleration = rotationAcceleration;
  104.     }

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

  108.      * <p><em>WARNING</em>! This method requires much more stringent assumptions on
  109.      * its parameters than the similar {@link Rotation#Rotation(Vector3D, Vector3D,
  110.      * Vector3D, Vector3D) constructor} from the {@link Rotation Rotation} class.
  111.      * As far as the Rotation constructor is concerned, the {@code v₂} vector from
  112.      * the second pair can be slightly misaligned. The Rotation constructor will
  113.      * compensate for this misalignment and create a rotation that ensure {@code
  114.      * v₁ = r(u₁)} and {@code v₂ ∈ plane (r(u₁), r(u₂))}. <em>THIS IS NOT
  115.      * TRUE ANYMORE IN THIS CLASS</em>! As derivatives are involved and must be
  116.      * preserved, this constructor works <em>only</em> if the two pairs are fully
  117.      * consistent, i.e. if a rotation exists that fulfill all the requirements: {@code
  118.      * v₁ = r(u₁)}, {@code v₂ = r(u₂)}, {@code dv₁/dt = dr(u₁)/dt}, {@code dv₂/dt
  119.      * = dr(u₂)/dt}, {@code d²v₁/dt² = d²r(u₁)/dt²}, {@code d²v₂/dt² = d²r(u₂)/dt²}.</p>
  120.      * @param u1 first vector of the origin pair
  121.      * @param u2 second vector of the origin pair
  122.      * @param v1 desired image of u1 by the rotation
  123.      * @param v2 desired image of u2 by the rotation
  124.      * @param tolerance relative tolerance factor used to check singularities
  125.      */
  126.     public AngularCoordinates(final PVCoordinates u1, final PVCoordinates u2,
  127.                               final PVCoordinates v1, final PVCoordinates v2,
  128.                               final double tolerance) {

  130.         try {
  131.             // find the initial fixed rotation
  132.             rotation = new Rotation(u1.getPosition(), u2.getPosition(),
  133.                                     v1.getPosition(), v2.getPosition());

  135.             // find rotation rate Ω such that
  136.             //  Ω ⨯ v₁ = r(dot(u₁)) - dot(v₁)
  137.             //  Ω ⨯ v₂ = r(dot(u₂)) - dot(v₂)
  138.             final Vector3D ru1Dot = rotation.applyTo(u1.getVelocity());
  139.             final Vector3D ru2Dot = rotation.applyTo(u2.getVelocity());
  140.             rotationRate = inverseCrossProducts(v1.getPosition(), ru1Dot.subtract(v1.getVelocity()),
  141.                                                 v2.getPosition(), ru2Dot.subtract(v2.getVelocity()),
  142.                                                 tolerance);

  144.             // find rotation acceleration dot(Ω) such that
  145.             // dot(Ω) ⨯ v₁ = r(dotdot(u₁)) - 2 Ω ⨯ dot(v₁) - Ω ⨯  (Ω ⨯ v₁) - dotdot(v₁)
  146.             // dot(Ω) ⨯ v₂ = r(dotdot(u₂)) - 2 Ω ⨯ dot(v₂) - Ω ⨯  (Ω ⨯ v₂) - dotdot(v₂)
  147.             final Vector3D ru1DotDot = rotation.applyTo(u1.getAcceleration());
  148.             final Vector3D oDotv1    = Vector3D.crossProduct(rotationRate, v1.getVelocity());
  149.             final Vector3D oov1      = Vector3D.crossProduct(rotationRate, Vector3D.crossProduct(rotationRate, v1.getPosition()));
  150.             final Vector3D c1        = new Vector3D(1, ru1DotDot, -2, oDotv1, -1, oov1, -1, v1.getAcceleration());
  151.             final Vector3D ru2DotDot = rotation.applyTo(u2.getAcceleration());
  152.             final Vector3D oDotv2    = Vector3D.crossProduct(rotationRate, v2.getVelocity());
  153.             final Vector3D oov2      = Vector3D.crossProduct(rotationRate, Vector3D.crossProduct(rotationRate, v2.getPosition()));
  154.             final Vector3D c2        = new Vector3D(1, ru2DotDot, -2, oDotv2, -1, oov2, -1, v2.getAcceleration());
  155.             rotationAcceleration     = inverseCrossProducts(v1.getPosition(), c1, v2.getPosition(), c2, tolerance);

  157.         } catch (MathRuntimeException mrte) {
  158.             throw new OrekitException(mrte);
  159.         }

  161.     }

  163.     /** Build one of the rotations that transform one pv coordinates into another one.

  165.      * <p>Except for a possible scale factor, if the instance were
  166.      * applied to the vector u it will produce the vector v. There is an
  167.      * infinite number of such rotations, this constructor choose the
  168.      * one with the smallest associated angle (i.e. the one whose axis
  169.      * is orthogonal to the (u, v) plane). If u and v are collinear, an
  170.      * arbitrary rotation axis is chosen.</p>

  172.      * @param u origin vector
  173.      * @param v desired image of u by the rotation
  174.      */
  175.     public AngularCoordinates(final PVCoordinates u, final PVCoordinates v) {
  176.         this(new FieldRotation<>(u.toDerivativeStructureVector(2),
  177.                                  v.toDerivativeStructureVector(2)));
  178.     }

  180.     /** Builds a AngularCoordinates from  a {@link FieldRotation}&lt;{@link Derivative}&gt;.
  181.      * <p>
  182.      * The rotation components must have time as their only derivation parameter and
  183.      * have consistent derivation orders.
  184.      * </p>
  185.      * @param r rotation with time-derivatives embedded within the coordinates
  186.      * @param <U> type of the derivative
  187.      */
  188.     public <U extends Derivative<U>> AngularCoordinates(final FieldRotation<U> r) {

  190.         final double q0       = r.getQ0().getReal();
  191.         final double q1       = r.getQ1().getReal();
  192.         final double q2       = r.getQ2().getReal();
  193.         final double q3       = r.getQ3().getReal();

  195.         rotation     = new Rotation(q0, q1, q2, q3, false);
  196.         if (r.getQ0().getOrder() >= 1) {
  197.             final double q0Dot    = r.getQ0().getPartialDerivative(1);
  198.             final double q1Dot    = r.getQ1().getPartialDerivative(1);
  199.             final double q2Dot    = r.getQ2().getPartialDerivative(1);
  200.             final double q3Dot    = r.getQ3().getPartialDerivative(1);
  201.             rotationRate =
  202.                     new Vector3D(2 * MathArrays.linearCombination(-q1, q0Dot,  q0, q1Dot,  q3, q2Dot, -q2, q3Dot),
  203.                                  2 * MathArrays.linearCombination(-q2, q0Dot, -q3, q1Dot,  q0, q2Dot,  q1, q3Dot),
  204.                                  2 * MathArrays.linearCombination(-q3, q0Dot,  q2, q1Dot, -q1, q2Dot,  q0, q3Dot));
  205.             if (r.getQ0().getOrder() >= 2) {
  206.                 final double q0DotDot = r.getQ0().getPartialDerivative(2);
  207.                 final double q1DotDot = r.getQ1().getPartialDerivative(2);
  208.                 final double q2DotDot = r.getQ2().getPartialDerivative(2);
  209.                 final double q3DotDot = r.getQ3().getPartialDerivative(2);
  210.                 rotationAcceleration =
  211.                         new Vector3D(2 * MathArrays.linearCombination(-q1, q0DotDot,  q0, q1DotDot,  q3, q2DotDot, -q2, q3DotDot),
  212.                                      2 * MathArrays.linearCombination(-q2, q0DotDot, -q3, q1DotDot,  q0, q2DotDot,  q1, q3DotDot),
  213.                                      2 * MathArrays.linearCombination(-q3, q0DotDot,  q2, q1DotDot, -q1, q2DotDot,  q0, q3DotDot));
  214.             } else {
  215.                 rotationAcceleration = Vector3D.ZERO;
  216.             }
  217.         } else {
  218.             rotationRate         = Vector3D.ZERO;
  219.             rotationAcceleration = Vector3D.ZERO;
  220.         }

  222.     }

  224.     /** Find a vector from two known cross products.
  225.      * <p>
  226.      * We want to find Ω such that: Ω ⨯ v₁ = c₁ and Ω ⨯ v₂ = c₂
  227.      * </p>
  228.      * <p>
  229.      * The first equation (Ω ⨯ v₁ = c₁) will always be fulfilled exactly,
  230.      * and the second one will be fulfilled if possible.
  231.      * </p>
  232.      * @param v1 vector forming the first known cross product
  233.      * @param c1 know vector for cross product Ω ⨯ v₁
  234.      * @param v2 vector forming the second known cross product
  235.      * @param c2 know vector for cross product Ω ⨯ v₂
  236.      * @param tolerance relative tolerance factor used to check singularities
  237.      * @return vector Ω such that: Ω ⨯ v₁ = c₁ and Ω ⨯ v₂ = c₂
  238.      * @exception MathIllegalArgumentException if vectors are inconsistent and
  239.      * no solution can be found
  240.      */
  241.     private static Vector3D inverseCrossProducts(final Vector3D v1, final Vector3D c1,
  242.                                                  final Vector3D v2, final Vector3D c2,
  243.                                                  final double tolerance)
  244.         throws MathIllegalArgumentException {

  246.         final double v12 = v1.getNormSq();
  247.         final double v1n = FastMath.sqrt(v12);
  248.         final double v22 = v2.getNormSq();
  249.         final double v2n = FastMath.sqrt(v22);
  250.         final double threshold = tolerance * FastMath.max(v1n, v2n);

  252.         Vector3D omega;

  254.         try {
  255.             // create the over-determined linear system representing the two cross products
  256.             final RealMatrix m = MatrixUtils.createRealMatrix(6, 3);
  257.             m.setEntry(0, 1,  v1.getZ());
  258.             m.setEntry(0, 2, -v1.getY());
  259.             m.setEntry(1, 0, -v1.getZ());
  260.             m.setEntry(1, 2,  v1.getX());
  261.             m.setEntry(2, 0,  v1.getY());
  262.             m.setEntry(2, 1, -v1.getX());
  263.             m.setEntry(3, 1,  v2.getZ());
  264.             m.setEntry(3, 2, -v2.getY());
  265.             m.setEntry(4, 0, -v2.getZ());
  266.             m.setEntry(4, 2,  v2.getX());
  267.             m.setEntry(5, 0,  v2.getY());
  268.             m.setEntry(5, 1, -v2.getX());

  270.             final RealVector rhs = MatrixUtils.createRealVector(new double[] {
  271.                 c1.getX(), c1.getY(), c1.getZ(),
  272.                 c2.getX(), c2.getY(), c2.getZ()
  273.             });

  275.             // find the best solution we can
  276.             final DecompositionSolver solver = new QRDecomposition(m, threshold).getSolver();
  277.             final RealVector v = solver.solve(rhs);
  278.             omega = new Vector3D(v.getEntry(0), v.getEntry(1), v.getEntry(2));

  280.         } catch (MathIllegalArgumentException miae) {
  281.             if (miae.getSpecifier() == LocalizedCoreFormats.SINGULAR_MATRIX) {

  283.                 // handle some special cases for which we can compute a solution
  284.                 final double c12 = c1.getNormSq();
  285.                 final double c1n = FastMath.sqrt(c12);
  286.                 final double c22 = c2.getNormSq();
  287.                 final double c2n = FastMath.sqrt(c22);

  289.                 if (c1n <= threshold && c2n <= threshold) {
  290.                     // simple special case, velocities are cancelled
  291.                     return Vector3D.ZERO;
  292.                 } else if (v1n <= threshold && c1n >= threshold) {
  293.                     // this is inconsistent, if v₁ is zero, c₁ must be 0 too
  294.                     throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, c1n, 0, true);
  295.                 } else if (v2n <= threshold && c2n >= threshold) {
  296.                     // this is inconsistent, if v₂ is zero, c₂ must be 0 too
  297.                     throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, c2n, 0, true);
  298.                 } else if (Vector3D.crossProduct(v1, v2).getNorm() <= threshold && v12 > threshold) {
  299.                     // simple special case, v₂ is redundant with v₁, we just ignore it
  300.                     // use the simplest Ω: orthogonal to both v₁ and c₁
  301.                     omega = new Vector3D(1.0 / v12, Vector3D.crossProduct(v1, c1));
  302.                 } else {
  303.                     throw miae;
  304.                 }
  305.             } else {
  306.                 throw miae;
  307.             }

  309.         }

  311.         // check results
  312.         final double d1 = Vector3D.distance(Vector3D.crossProduct(omega, v1), c1);
  313.         if (d1 > threshold) {
  314.             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, d1, 0, true);
  315.         }

  317.         final double d2 = Vector3D.distance(Vector3D.crossProduct(omega, v2), c2);
  318.         if (d2 > threshold) {
  319.             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, d2, 0, true);
  320.         }

  322.         return omega;

  324.     }

  326.     /** Transform the instance to a {@link FieldRotation}&lt;{@link DerivativeStructure}&gt;.
  327.      * <p>
  328.      * The {@link DerivativeStructure} coordinates correspond to time-derivatives up
  329.      * to the user-specified order.
  330.      * </p>
  331.      * @param order derivation order for the vector components
  332.      * @return rotation with time-derivatives embedded within the coordinates
  333.      */
  334.     public FieldRotation<DerivativeStructure> toDerivativeStructureRotation(final int order) {

  336.         // quaternion components
  337.         final double q0 = rotation.getQ0();
  338.         final double q1 = rotation.getQ1();
  339.         final double q2 = rotation.getQ2();
  340.         final double q3 = rotation.getQ3();

  342.         // first time-derivatives of the quaternion
  343.         final double oX    = rotationRate.getX();
  344.         final double oY    = rotationRate.getY();
  345.         final double oZ    = rotationRate.getZ();
  346.         final double q0Dot = 0.5 * MathArrays.linearCombination(-q1, oX, -q2, oY, -q3, oZ);
  347.         final double q1Dot = 0.5 * MathArrays.linearCombination( q0, oX, -q3, oY,  q2, oZ);
  348.         final double q2Dot = 0.5 * MathArrays.linearCombination( q3, oX,  q0, oY, -q1, oZ);
  349.         final double q3Dot = 0.5 * MathArrays.linearCombination(-q2, oX,  q1, oY,  q0, oZ);

  351.         // second time-derivatives of the quaternion
  352.         final double oXDot = rotationAcceleration.getX();
  353.         final double oYDot = rotationAcceleration.getY();
  354.         final double oZDot = rotationAcceleration.getZ();
  355.         final double q0DotDot = -0.5 * MathArrays.linearCombination(new double[] {
  356.             q1, q2,  q3, q1Dot, q2Dot,  q3Dot
  357.         }, new double[] {
  358.             oXDot, oYDot, oZDot, oX, oY, oZ
  359.         });
  360.         final double q1DotDot =  0.5 * MathArrays.linearCombination(new double[] {
  361.             q0, q2, -q3, q0Dot, q2Dot, -q3Dot
  362.         }, new double[] {
  363.             oXDot, oZDot, oYDot, oX, oZ, oY
  364.         });
  365.         final double q2DotDot =  0.5 * MathArrays.linearCombination(new double[] {
  366.             q0, q3, -q1, q0Dot, q3Dot, -q1Dot
  367.         }, new double[] {
  368.             oYDot, oXDot, oZDot, oY, oX, oZ
  369.         });
  370.         final double q3DotDot =  0.5 * MathArrays.linearCombination(new double[] {
  371.             q0, q1, -q2, q0Dot, q1Dot, -q2Dot
  372.         }, new double[] {
  373.             oZDot, oYDot, oXDot, oZ, oY, oX
  374.         });

  376.         final DSFactory factory;
  377.         final DerivativeStructure q0DS;
  378.         final DerivativeStructure q1DS;
  379.         final DerivativeStructure q2DS;
  380.         final DerivativeStructure q3DS;
  381.         switch(order) {
  382.             case 0 :
  383.                 factory = new DSFactory(1, order);
  384.                 q0DS = factory.build(q0);
  385.                 q1DS = factory.build(q1);
  386.                 q2DS = factory.build(q2);
  387.                 q3DS = factory.build(q3);
  388.                 break;
  389.             case 1 :
  390.                 factory = new DSFactory(1, order);
  391.                 q0DS = factory.build(q0, q0Dot);
  392.                 q1DS = factory.build(q1, q1Dot);
  393.                 q2DS = factory.build(q2, q2Dot);
  394.                 q3DS = factory.build(q3, q3Dot);
  395.                 break;
  396.             case 2 :
  397.                 factory = new DSFactory(1, order);
  398.                 q0DS = factory.build(q0, q0Dot, q0DotDot);
  399.                 q1DS = factory.build(q1, q1Dot, q1DotDot);
  400.                 q2DS = factory.build(q2, q2Dot, q2DotDot);
  401.                 q3DS = factory.build(q3, q3Dot, q3DotDot);
  402.                 break;
  403.             default :
  404.                 throw new OrekitException(OrekitMessages.OUT_OF_RANGE_DERIVATION_ORDER, order);
  405.         }

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

  409.     }

  411.     /** Transform the instance to a {@link FieldRotation}&lt;{@link UnivariateDerivative1}&gt;.
  412.      * <p>
  413.      * The {@link UnivariateDerivative1} coordinates correspond to time-derivatives up
  414.      * to the order 1.
  415.      * </p>
  416.      * @return rotation with time-derivatives embedded within the coordinates
  417.      */
  418.     public FieldRotation<UnivariateDerivative1> toUnivariateDerivative1Rotation() {

  420.         // quaternion components
  421.         final double q0 = rotation.getQ0();
  422.         final double q1 = rotation.getQ1();
  423.         final double q2 = rotation.getQ2();
  424.         final double q3 = rotation.getQ3();

  426.         // first time-derivatives of the quaternion
  427.         final double oX    = rotationRate.getX();
  428.         final double oY    = rotationRate.getY();
  429.         final double oZ    = rotationRate.getZ();
  430.         final double q0Dot = 0.5 * MathArrays.linearCombination(-q1, oX, -q2, oY, -q3, oZ);
  431.         final double q1Dot = 0.5 * MathArrays.linearCombination( q0, oX, -q3, oY,  q2, oZ);
  432.         final double q2Dot = 0.5 * MathArrays.linearCombination( q3, oX,  q0, oY, -q1, oZ);
  433.         final double q3Dot = 0.5 * MathArrays.linearCombination(-q2, oX,  q1, oY,  q0, oZ);

  435.         final UnivariateDerivative1 q0UD = new UnivariateDerivative1(q0, q0Dot);
  436.         final UnivariateDerivative1 q1UD = new UnivariateDerivative1(q1, q1Dot);
  437.         final UnivariateDerivative1 q2UD = new UnivariateDerivative1(q2, q2Dot);
  438.         final UnivariateDerivative1 q3UD = new UnivariateDerivative1(q3, q3Dot);

  440.         return new FieldRotation<>(q0UD, q1UD, q2UD, q3UD, false);

  442.     }

  444.     /** Transform the instance to a {@link FieldRotation}&lt;{@link UnivariateDerivative2}&gt;.
  445.      * <p>
  446.      * The {@link UnivariateDerivative2} coordinates correspond to time-derivatives up
  447.      * to the order 2.
  448.      * </p>
  449.      * @return rotation with time-derivatives embedded within the coordinates
  450.      */
  451.     public FieldRotation<UnivariateDerivative2> toUnivariateDerivative2Rotation() {

  453.         // quaternion components
  454.         final double q0 = rotation.getQ0();
  455.         final double q1 = rotation.getQ1();
  456.         final double q2 = rotation.getQ2();
  457.         final double q3 = rotation.getQ3();

  459.         // first time-derivatives of the quaternion
  460.         final double oX    = rotationRate.getX();
  461.         final double oY    = rotationRate.getY();
  462.         final double oZ    = rotationRate.getZ();
  463.         final double q0Dot = 0.5 * MathArrays.linearCombination(-q1, oX, -q2, oY, -q3, oZ);
  464.         final double q1Dot = 0.5 * MathArrays.linearCombination( q0, oX, -q3, oY,  q2, oZ);
  465.         final double q2Dot = 0.5 * MathArrays.linearCombination( q3, oX,  q0, oY, -q1, oZ);
  466.         final double q3Dot = 0.5 * MathArrays.linearCombination(-q2, oX,  q1, oY,  q0, oZ);

  468.         // second time-derivatives of the quaternion
  469.         final double oXDot = rotationAcceleration.getX();
  470.         final double oYDot = rotationAcceleration.getY();
  471.         final double oZDot = rotationAcceleration.getZ();
  472.         final double q0DotDot = -0.5 * MathArrays.linearCombination(new double[] {
  473.             q1, q2,  q3, q1Dot, q2Dot,  q3Dot
  474.         }, new double[] {
  475.             oXDot, oYDot, oZDot, oX, oY, oZ
  476.         });
  477.         final double q1DotDot =  0.5 * MathArrays.linearCombination(new double[] {
  478.             q0, q2, -q3, q0Dot, q2Dot, -q3Dot
  479.         }, new double[] {
  480.             oXDot, oZDot, oYDot, oX, oZ, oY
  481.         });
  482.         final double q2DotDot =  0.5 * MathArrays.linearCombination(new double[] {
  483.             q0, q3, -q1, q0Dot, q3Dot, -q1Dot
  484.         }, new double[] {
  485.             oYDot, oXDot, oZDot, oY, oX, oZ
  486.         });
  487.         final double q3DotDot =  0.5 * MathArrays.linearCombination(new double[] {
  488.             q0, q1, -q2, q0Dot, q1Dot, -q2Dot
  489.         }, new double[] {
  490.             oZDot, oYDot, oXDot, oZ, oY, oX
  491.         });

  493.         final UnivariateDerivative2 q0UD = new UnivariateDerivative2(q0, q0Dot, q0DotDot);
  494.         final UnivariateDerivative2 q1UD = new UnivariateDerivative2(q1, q1Dot, q1DotDot);
  495.         final UnivariateDerivative2 q2UD = new UnivariateDerivative2(q2, q2Dot, q2DotDot);
  496.         final UnivariateDerivative2 q3UD = new UnivariateDerivative2(q3, q3Dot, q3DotDot);

  498.         return new FieldRotation<>(q0UD, q1UD, q2UD, q3UD, false);

  500.     }

  502.     /** Estimate rotation rate between two orientations.
  503.      * <p>Estimation is based on a simple fixed rate rotation
  504.      * during the time interval between the two orientations.</p>
  505.      * @param start start orientation
  506.      * @param end end orientation
  507.      * @param dt time elapsed between the dates of the two orientations
  508.      * @return rotation rate allowing to go from start to end orientations
  509.      */
  510.     public static Vector3D estimateRate(final Rotation start, final Rotation end, final double dt) {
  511.         final Rotation evolution = start.compose(end.revert(), RotationConvention.VECTOR_OPERATOR);
  512.         return new Vector3D(evolution.getAngle() / dt, evolution.getAxis(RotationConvention.VECTOR_OPERATOR));
  513.     }

  515.     /** Revert a rotation/rotation rate/ rotation acceleration triplet.
  516.      * Build a triplet which reverse the effect of another triplet.
  517.      * @return a new triplet whose effect is the reverse of the effect
  518.      * of the instance
  519.      */
  520.     public AngularCoordinates revert() {
  521.         return new AngularCoordinates(rotation.revert(),
  522.                                       rotation.applyInverseTo(rotationRate).negate(),
  523.                                       rotation.applyInverseTo(rotationAcceleration).negate());
  524.     }

  526.     /** Get a time-shifted state.
  527.      * <p>
  528.      * The state can be slightly shifted to close dates. This shift is based on
  529.      * an approximate solution of the fixed acceleration motion. It is <em>not</em>
  530.      * intended as a replacement for proper attitude propagation but should be
  531.      * sufficient for either small time shifts or coarse accuracy.
  532.      * </p>
  533.      * @param dt time shift in seconds
  534.      * @return a new state, shifted with respect to the instance (which is immutable)
  535.      */
  536.     public AngularCoordinates shiftedBy(final double dt) {

  538.         // the shiftedBy method is based on a local approximation.
  539.         // It considers separately the contribution of the constant
  540.         // rotation, the linear contribution or the rate and the
  541.         // quadratic contribution of the acceleration. The rate
  542.         // and acceleration contributions are small rotations as long
  543.         // as the time shift is small, which is the crux of the algorithm.
  544.         // Small rotations are almost commutative, so we append these small
  545.         // contributions one after the other, as if they really occurred
  546.         // successively, despite this is not what really happens.

  548.         // compute the linear contribution first, ignoring acceleration
  549.         // BEWARE: there is really a minus sign here, because if
  550.         // the target frame rotates in one direction, the vectors in the origin
  551.         // frame seem to rotate in the opposite direction
  552.         final double rate = rotationRate.getNorm();
  553.         final Rotation rateContribution = (rate == 0.0) ?
  554.                                           Rotation.IDENTITY :
  555.                                           new Rotation(rotationRate, rate * dt, RotationConvention.FRAME_TRANSFORM);

  557.         // append rotation and rate contribution
  558.         final AngularCoordinates linearPart =
  559.                 new AngularCoordinates(rateContribution.compose(rotation, RotationConvention.VECTOR_OPERATOR), rotationRate);

  561.         final double acc  = rotationAcceleration.getNorm();
  562.         if (acc == 0.0) {
  563.             // no acceleration, the linear part is sufficient
  564.             return linearPart;
  565.         }

  567.         // compute the quadratic contribution, ignoring initial rotation and rotation rate
  568.         // BEWARE: there is really a minus sign here, because if
  569.         // the target frame rotates in one direction, the vectors in the origin
  570.         // frame seem to rotate in the opposite direction
  571.         final AngularCoordinates quadraticContribution =
  572.                 new AngularCoordinates(new Rotation(rotationAcceleration,
  573.                                                     0.5 * acc * dt * dt,
  574.                                                     RotationConvention.FRAME_TRANSFORM),
  575.                                        new Vector3D(dt, rotationAcceleration),
  576.                                        rotationAcceleration);

  578.         // the quadratic contribution is a small rotation:
  579.         // its initial angle and angular rate are both zero.
  580.         // small rotations are almost commutative, so we append the small
  581.         // quadratic part after the linear part as a simple offset
  582.         return quadraticContribution.addOffset(linearPart);

  584.     }

  586.     /** Get the rotation.
  587.      * @return the rotation.
  588.      */
  589.     public Rotation getRotation() {
  590.         return rotation;
  591.     }

  593.     /** Get the rotation rate.
  594.      * @return the rotation rate vector Ω (rad/s).
  595.      */
  596.     public Vector3D getRotationRate() {
  597.         return rotationRate;
  598.     }

  600.     /** Get the rotation acceleration.
  601.      * @return the rotation acceleration vector dΩ/dt (rad/s²).
  602.      */
  603.     public Vector3D getRotationAcceleration() {
  604.         return rotationAcceleration;
  605.     }

  607.     /** Add an offset from the instance.
  608.      * <p>
  609.      * We consider here that the offset rotation is applied first and the
  610.      * instance is applied afterward. Note that angular coordinates do <em>not</em>
  611.      * commute under this operation, i.e. {@code a.addOffset(b)} and {@code
  612.      * b.addOffset(a)} lead to <em>different</em> results in most cases.
  613.      * </p>
  614.      * <p>
  615.      * The two methods {@link #addOffset(AngularCoordinates) addOffset} and
  616.      * {@link #subtractOffset(AngularCoordinates) subtractOffset} are designed
  617.      * so that round trip applications are possible. This means that both {@code
  618.      * ac1.subtractOffset(ac2).addOffset(ac2)} and {@code
  619.      * ac1.addOffset(ac2).subtractOffset(ac2)} return angular coordinates equal to ac1.
  620.      * </p>
  621.      * @param offset offset to subtract
  622.      * @return new instance, with offset subtracted
  623.      * @see #subtractOffset(AngularCoordinates)
  624.      */
  625.     public AngularCoordinates addOffset(final AngularCoordinates offset) {
  626.         final Vector3D rOmega    = rotation.applyTo(offset.rotationRate);
  627.         final Vector3D rOmegaDot = rotation.applyTo(offset.rotationAcceleration);
  628.         return new AngularCoordinates(rotation.compose(offset.rotation, RotationConvention.VECTOR_OPERATOR),
  629.                                       rotationRate.add(rOmega),
  630.                                       new Vector3D( 1.0, rotationAcceleration,
  631.                                                     1.0, rOmegaDot,
  632.                                                    -1.0, Vector3D.crossProduct(rotationRate, rOmega)));
  633.     }

  635.     /** Subtract an offset from the instance.
  636.      * <p>
  637.      * We consider here that the offset rotation is applied first and the
  638.      * instance is applied afterward. Note that angular coordinates do <em>not</em>
  639.      * commute under this operation, i.e. {@code a.subtractOffset(b)} and {@code
  640.      * b.subtractOffset(a)} lead to <em>different</em> results in most cases.
  641.      * </p>
  642.      * <p>
  643.      * The two methods {@link #addOffset(AngularCoordinates) addOffset} and
  644.      * {@link #subtractOffset(AngularCoordinates) subtractOffset} are designed
  645.      * so that round trip applications are possible. This means that both {@code
  646.      * ac1.subtractOffset(ac2).addOffset(ac2)} and {@code
  647.      * ac1.addOffset(ac2).subtractOffset(ac2)} return angular coordinates equal to ac1.
  648.      * </p>
  649.      * @param offset offset to subtract
  650.      * @return new instance, with offset subtracted
  651.      * @see #addOffset(AngularCoordinates)
  652.      */
  653.     public AngularCoordinates subtractOffset(final AngularCoordinates offset) {
  654.         return addOffset(offset.revert());
  655.     }

  657.     /** Apply the rotation to a pv coordinates.
  658.      * @param pv vector to apply the rotation to
  659.      * @return a new pv coordinates which is the image of u by the rotation
  660.      */
  661.     public PVCoordinates applyTo(final PVCoordinates pv) {

  663.         final Vector3D transformedP = rotation.applyTo(pv.getPosition());
  664.         final Vector3D crossP       = Vector3D.crossProduct(rotationRate, transformedP);
  665.         final Vector3D transformedV = rotation.applyTo(pv.getVelocity()).subtract(crossP);
  666.         final Vector3D crossV       = Vector3D.crossProduct(rotationRate, transformedV);
  667.         final Vector3D crossCrossP  = Vector3D.crossProduct(rotationRate, crossP);
  668.         final Vector3D crossDotP    = Vector3D.crossProduct(rotationAcceleration, transformedP);
  669.         final Vector3D transformedA = new Vector3D( 1, rotation.applyTo(pv.getAcceleration()),
  670.                                                    -2, crossV,
  671.                                                    -1, crossCrossP,
  672.                                                    -1, crossDotP);

  674.         return new PVCoordinates(transformedP, transformedV, transformedA);

  676.     }

  678.     /** Apply the rotation to a pv coordinates.
  679.      * @param pv vector to apply the rotation to
  680.      * @return a new pv coordinates which is the image of u by the rotation
  681.      */
  682.     public TimeStampedPVCoordinates applyTo(final TimeStampedPVCoordinates pv) {

  684.         final Vector3D transformedP = getRotation().applyTo(pv.getPosition());
  685.         final Vector3D crossP       = Vector3D.crossProduct(getRotationRate(), transformedP);
  686.         final Vector3D transformedV = getRotation().applyTo(pv.getVelocity()).subtract(crossP);
  687.         final Vector3D crossV       = Vector3D.crossProduct(getRotationRate(), transformedV);
  688.         final Vector3D crossCrossP  = Vector3D.crossProduct(getRotationRate(), crossP);
  689.         final Vector3D crossDotP    = Vector3D.crossProduct(getRotationAcceleration(), transformedP);
  690.         final Vector3D transformedA = new Vector3D( 1, getRotation().applyTo(pv.getAcceleration()),
  691.                                                    -2, crossV,
  692.                                                    -1, crossCrossP,
  693.                                                    -1, crossDotP);

  695.         return new TimeStampedPVCoordinates(pv.getDate(), transformedP, transformedV, transformedA);

  697.     }

  699.     /** Apply the rotation to a pv coordinates.
  700.      * @param pv vector to apply the rotation to
  701.      * @param <T> type of the field elements
  702.      * @return a new pv coordinates which is the image of u by the rotation
  703.      * @since 9.0
  704.      */
  705.     public <T extends CalculusFieldElement<T>> FieldPVCoordinates<T> applyTo(final FieldPVCoordinates<T> pv) {

  707.         final FieldVector3D<T> transformedP = FieldRotation.applyTo(rotation, pv.getPosition());
  708.         final FieldVector3D<T> crossP       = FieldVector3D.crossProduct(rotationRate, transformedP);
  709.         final FieldVector3D<T> transformedV = FieldRotation.applyTo(rotation, pv.getVelocity()).subtract(crossP);
  710.         final FieldVector3D<T> crossV       = FieldVector3D.crossProduct(rotationRate, transformedV);
  711.         final FieldVector3D<T> crossCrossP  = FieldVector3D.crossProduct(rotationRate, crossP);
  712.         final FieldVector3D<T> crossDotP    = FieldVector3D.crossProduct(rotationAcceleration, transformedP);
  713.         final FieldVector3D<T> transformedA = new FieldVector3D<>( 1, FieldRotation.applyTo(rotation, pv.getAcceleration()),
  714.                                                                   -2, crossV,
  715.                                                                   -1, crossCrossP,
  716.                                                                   -1, crossDotP);

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

  720.     }

  722.     /** Apply the rotation to a pv coordinates.
  723.      * @param pv vector to apply the rotation to
  724.      * @param <T> type of the field elements
  725.      * @return a new pv coordinates which is the image of u by the rotation
  726.      * @since 9.0
  727.      */
  728.     public <T extends CalculusFieldElement<T>> TimeStampedFieldPVCoordinates<T> applyTo(final TimeStampedFieldPVCoordinates<T> pv) {

  730.         final FieldVector3D<T> transformedP = FieldRotation.applyTo(rotation, pv.getPosition());
  731.         final FieldVector3D<T> crossP       = FieldVector3D.crossProduct(rotationRate, transformedP);
  732.         final FieldVector3D<T> transformedV = FieldRotation.applyTo(rotation, pv.getVelocity()).subtract(crossP);
  733.         final FieldVector3D<T> crossV       = FieldVector3D.crossProduct(rotationRate, transformedV);
  734.         final FieldVector3D<T> crossCrossP  = FieldVector3D.crossProduct(rotationRate, crossP);
  735.         final FieldVector3D<T> crossDotP    = FieldVector3D.crossProduct(rotationAcceleration, transformedP);
  736.         final FieldVector3D<T> transformedA = new FieldVector3D<>( 1, FieldRotation.applyTo(rotation, pv.getAcceleration()),
  737.                                                                   -2, crossV,
  738.                                                                   -1, crossCrossP,
  739.                                                                   -1, crossDotP);

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

  743.     }

  745.     /** Convert rotation, rate and acceleration to modified Rodrigues vector and derivatives.
  746.      * <p>
  747.      * The modified Rodrigues vector is tan(θ/4) u where θ and u are the
  748.      * rotation angle and axis respectively.
  749.      * </p>
  750.      * @param sign multiplicative sign for quaternion components
  751.      * @return modified Rodrigues vector and derivatives (vector on row 0, first derivative
  752.      * on row 1, second derivative on row 2)
  753.      * @see #createFromModifiedRodrigues(double[][])
  754.      */
  755.     public double[][] getModifiedRodrigues(final double sign) {

  757.         final double q0    = sign * getRotation().getQ0();
  758.         final double q1    = sign * getRotation().getQ1();
  759.         final double q2    = sign * getRotation().getQ2();
  760.         final double q3    = sign * getRotation().getQ3();
  761.         final double oX    = getRotationRate().getX();
  762.         final double oY    = getRotationRate().getY();
  763.         final double oZ    = getRotationRate().getZ();
  764.         final double oXDot = getRotationAcceleration().getX();
  765.         final double oYDot = getRotationAcceleration().getY();
  766.         final double oZDot = getRotationAcceleration().getZ();

  768.         // first time-derivatives of the quaternion
  769.         final double q0Dot = 0.5 * MathArrays.linearCombination(-q1, oX, -q2, oY, -q3, oZ);
  770.         final double q1Dot = 0.5 * MathArrays.linearCombination( q0, oX, -q3, oY,  q2, oZ);
  771.         final double q2Dot = 0.5 * MathArrays.linearCombination( q3, oX,  q0, oY, -q1, oZ);
  772.         final double q3Dot = 0.5 * MathArrays.linearCombination(-q2, oX,  q1, oY,  q0, oZ);

  774.         // second time-derivatives of the quaternion
  775.         final double q0DotDot = -0.5 * MathArrays.linearCombination(new double[] {
  776.             q1, q2,  q3, q1Dot, q2Dot,  q3Dot
  777.         }, new double[] {
  778.             oXDot, oYDot, oZDot, oX, oY, oZ
  779.         });
  780.         final double q1DotDot =  0.5 * MathArrays.linearCombination(new double[] {
  781.             q0, q2, -q3, q0Dot, q2Dot, -q3Dot
  782.         }, new double[] {
  783.             oXDot, oZDot, oYDot, oX, oZ, oY
  784.         });
  785.         final double q2DotDot =  0.5 * MathArrays.linearCombination(new double[] {
  786.             q0, q3, -q1, q0Dot, q3Dot, -q1Dot
  787.         }, new double[] {
  788.             oYDot, oXDot, oZDot, oY, oX, oZ
  789.         });
  790.         final double q3DotDot =  0.5 * MathArrays.linearCombination(new double[] {
  791.             q0, q1, -q2, q0Dot, q1Dot, -q2Dot
  792.         }, new double[] {
  793.             oZDot, oYDot, oXDot, oZ, oY, oX
  794.         });

  796.         // the modified Rodrigues is tan(θ/4) u where θ and u are the rotation angle and axis respectively
  797.         // this can be rewritten using quaternion components:
  798.         //      r (q₁ / (1+q₀), q₂ / (1+q₀), q₃ / (1+q₀))
  799.         // applying the derivation chain rule to previous expression gives rDot and rDotDot
  800.         final double inv          = 1.0 / (1.0 + q0);
  801.         final double mTwoInvQ0Dot = -2 * inv * q0Dot;

  803.         final double r1       = inv * q1;
  804.         final double r2       = inv * q2;
  805.         final double r3       = inv * q3;

  807.         final double mInvR1   = -inv * r1;
  808.         final double mInvR2   = -inv * r2;
  809.         final double mInvR3   = -inv * r3;

  811.         final double r1Dot    = MathArrays.linearCombination(inv, q1Dot, mInvR1, q0Dot);
  812.         final double r2Dot    = MathArrays.linearCombination(inv, q2Dot, mInvR2, q0Dot);
  813.         final double r3Dot    = MathArrays.linearCombination(inv, q3Dot, mInvR3, q0Dot);

  815.         final double r1DotDot = MathArrays.linearCombination(inv, q1DotDot, mTwoInvQ0Dot, r1Dot, mInvR1, q0DotDot);
  816.         final double r2DotDot = MathArrays.linearCombination(inv, q2DotDot, mTwoInvQ0Dot, r2Dot, mInvR2, q0DotDot);
  817.         final double r3DotDot = MathArrays.linearCombination(inv, q3DotDot, mTwoInvQ0Dot, r3Dot, mInvR3, q0DotDot);

  819.         return new double[][] {
  820.             {
  821.                 r1,       r2,       r3
  822.             }, {
  823.                 r1Dot,    r2Dot,    r3Dot
  824.             }, {
  825.                 r1DotDot, r2DotDot, r3DotDot
  826.             }
  827.         };

  829.     }

  831.     /** Convert a modified Rodrigues vector and derivatives to angular coordinates.
  832.      * @param r modified Rodrigues vector (with first and second times derivatives)
  833.      * @return angular coordinates
  834.      * @see #getModifiedRodrigues(double)
  835.      */
  836.     public static AngularCoordinates createFromModifiedRodrigues(final double[][] r) {

  838.         // rotation
  839.         final double rSquared = r[0][0] * r[0][0] + r[0][1] * r[0][1] + r[0][2] * r[0][2];
  840.         final double oPQ0     = 2 / (1 + rSquared);
  841.         final double q0       = oPQ0 - 1;
  842.         final double q1       = oPQ0 * r[0][0];
  843.         final double q2       = oPQ0 * r[0][1];
  844.         final double q3       = oPQ0 * r[0][2];

  846.         // rotation rate
  847.         final double oPQ02    = oPQ0 * oPQ0;
  848.         final double q0Dot    = -oPQ02 * MathArrays.linearCombination(r[0][0], r[1][0], r[0][1], r[1][1],  r[0][2], r[1][2]);
  849.         final double q1Dot    = oPQ0 * r[1][0] + r[0][0] * q0Dot;
  850.         final double q2Dot    = oPQ0 * r[1][1] + r[0][1] * q0Dot;
  851.         final double q3Dot    = oPQ0 * r[1][2] + r[0][2] * q0Dot;
  852.         final double oX       = 2 * MathArrays.linearCombination(-q1, q0Dot,  q0, q1Dot,  q3, q2Dot, -q2, q3Dot);
  853.         final double oY       = 2 * MathArrays.linearCombination(-q2, q0Dot, -q3, q1Dot,  q0, q2Dot,  q1, q3Dot);
  854.         final double oZ       = 2 * MathArrays.linearCombination(-q3, q0Dot,  q2, q1Dot, -q1, q2Dot,  q0, q3Dot);

  856.         // rotation acceleration
  857.         final double q0DotDot = (1 - q0) / oPQ0 * q0Dot * q0Dot -
  858.                                 oPQ02 * MathArrays.linearCombination(r[0][0], r[2][0], r[0][1], r[2][1], r[0][2], r[2][2]) -
  859.                                 (q1Dot * q1Dot + q2Dot * q2Dot + q3Dot * q3Dot);
  860.         final double q1DotDot = MathArrays.linearCombination(oPQ0, r[2][0], 2 * r[1][0], q0Dot, r[0][0], q0DotDot);
  861.         final double q2DotDot = MathArrays.linearCombination(oPQ0, r[2][1], 2 * r[1][1], q0Dot, r[0][1], q0DotDot);
  862.         final double q3DotDot = MathArrays.linearCombination(oPQ0, r[2][2], 2 * r[1][2], q0Dot, r[0][2], q0DotDot);
  863.         final double oXDot    = 2 * MathArrays.linearCombination(-q1, q0DotDot,  q0, q1DotDot,  q3, q2DotDot, -q2, q3DotDot);
  864.         final double oYDot    = 2 * MathArrays.linearCombination(-q2, q0DotDot, -q3, q1DotDot,  q0, q2DotDot,  q1, q3DotDot);
  865.         final double oZDot    = 2 * MathArrays.linearCombination(-q3, q0DotDot,  q2, q1DotDot, -q1, q2DotDot,  q0, q3DotDot);

  867.         return new AngularCoordinates(new Rotation(q0, q1, q2, q3, false),
  868.                                       new Vector3D(oX, oY, oZ),
  869.                                       new Vector3D(oXDot, oYDot, oZDot));

  871.     }

  873. }
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