AngularCoordinates.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 java.io.Serializable;

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

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

  58.     /** Fixed orientation parallel with reference frame
  59.      * (identity rotation, zero rotation rate and acceleration).
  60.      */
  61.     public static final AngularCoordinates IDENTITY =
  62.             new AngularCoordinates(Rotation.IDENTITY, Vector3D.ZERO, Vector3D.ZERO);

  64.     /** Serializable UID. */
  65.     private static final long serialVersionUID = 20140414L;

  67.     /** Rotation. */
  68.     private final Rotation rotation;

  70.     /** Rotation rate. */
  71.     private final Vector3D rotationRate;

  73.     /** Rotation acceleration. */
  74.     private final Vector3D rotationAcceleration;

  76.     /** Simple constructor.
  77.      * <p> Sets the Coordinates to default : Identity, Ω = (0 0 0), dΩ/dt = (0 0 0).</p>
  78.      */
  79.     public AngularCoordinates() {
  80.         this(Rotation.IDENTITY, Vector3D.ZERO, Vector3D.ZERO);
  81.     }

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

  91.     /** Builds a rotation/rotation rate/rotation acceleration triplet.
  92.      * @param rotation rotation
  93.      * @param rotationRate rotation rate Ω (rad/s)
  94.      * @param rotationAcceleration rotation acceleration dΩ/dt (rad²/s²)
  95.      */
  96.     public AngularCoordinates(final Rotation rotation,
  97.                               final Vector3D rotationRate, final Vector3D rotationAcceleration) {
  98.         this.rotation             = rotation;
  99.         this.rotationRate         = rotationRate;
  100.         this.rotationAcceleration = rotationAcceleration;
  101.     }

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

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

  127.         try {
  128.             // find the initial fixed rotation
  129.             rotation = new Rotation(u1.getPosition(), u2.getPosition(),
  130.                                     v1.getPosition(), v2.getPosition());

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

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

  154.         } catch (MathRuntimeException mrte) {
  155.             throw new OrekitException(mrte);
  156.         }

  158.     }

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

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

  169.      * @param u origin vector
  170.      * @param v desired image of u by the rotation
  171.      */
  172.     public AngularCoordinates(final PVCoordinates u, final PVCoordinates v) {
  173.         this(new FieldRotation<>(u.toDerivativeStructureVector(2),
  174.                                  v.toDerivativeStructureVector(2)));
  175.     }

  177.     /** Builds a AngularCoordinates from  a {@link FieldRotation}&lt;{@link DerivativeStructure}&gt;.
  178.      * <p>
  179.      * The rotation components must have time as their only derivation parameter and
  180.      * have consistent derivation orders.
  181.      * </p>
  182.      * @param r rotation with time-derivatives embedded within the coordinates
  183.      */
  184.     public AngularCoordinates(final FieldRotation<DerivativeStructure> r) {

  186.         final double q0       = r.getQ0().getReal();
  187.         final double q1       = r.getQ1().getReal();
  188.         final double q2       = r.getQ2().getReal();
  189.         final double q3       = r.getQ3().getReal();

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

  218.     }

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

  242.         final double v12 = v1.getNormSq();
  243.         final double v1n = FastMath.sqrt(v12);
  244.         final double v22 = v2.getNormSq();
  245.         final double v2n = FastMath.sqrt(v22);
  246.         final double threshold = tolerance * FastMath.max(v1n, v2n);

  248.         Vector3D omega;

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

  266.             final RealVector rhs = MatrixUtils.createRealVector(new double[] {
  267.                 c1.getX(), c1.getY(), c1.getZ(),
  268.                 c2.getX(), c2.getY(), c2.getZ()
  269.             });

  271.             // find the best solution we can
  272.             final DecompositionSolver solver = new QRDecomposition(m, threshold).getSolver();
  273.             final RealVector v = solver.solve(rhs);
  274.             omega = new Vector3D(v.getEntry(0), v.getEntry(1), v.getEntry(2));

  276.         } catch (MathIllegalArgumentException miae) {
  277.             if (miae.getSpecifier() == LocalizedCoreFormats.SINGULAR_MATRIX) {

  279.                 // handle some special cases for which we can compute a solution
  280.                 final double c12 = c1.getNormSq();
  281.                 final double c1n = FastMath.sqrt(c12);
  282.                 final double c22 = c2.getNormSq();
  283.                 final double c2n = FastMath.sqrt(c22);

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

  305.         }

  307.         // check results
  308.         final double d1 = Vector3D.distance(Vector3D.crossProduct(omega, v1), c1);
  309.         if (d1 > threshold) {
  310.             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, d1, 0, true);
  311.         }

  313.         final double d2 = Vector3D.distance(Vector3D.crossProduct(omega, v2), c2);
  314.         if (d2 > threshold) {
  315.             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, d2, 0, true);
  316.         }

  318.         return omega;

  320.     }

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

  332.         // quaternion components
  333.         final double q0 = rotation.getQ0();
  334.         final double q1 = rotation.getQ1();
  335.         final double q2 = rotation.getQ2();
  336.         final double q3 = rotation.getQ3();

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

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

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

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

  405.     }

  407.     /** Estimate rotation rate between two orientations.
  408.      * <p>Estimation is based on a simple fixed rate rotation
  409.      * during the time interval between the two orientations.</p>
  410.      * @param start start orientation
  411.      * @param end end orientation
  412.      * @param dt time elapsed between the dates of the two orientations
  413.      * @return rotation rate allowing to go from start to end orientations
  414.      */
  415.     public static Vector3D estimateRate(final Rotation start, final Rotation end, final double dt) {
  416.         final Rotation evolution = start.compose(end.revert(), RotationConvention.VECTOR_OPERATOR);
  417.         return new Vector3D(evolution.getAngle() / dt, evolution.getAxis(RotationConvention.VECTOR_OPERATOR));
  418.     }

  420.     /** Revert a rotation/rotation rate/ rotation acceleration triplet.
  421.      * Build a triplet which reverse the effect of another triplet.
  422.      * @return a new triplet whose effect is the reverse of the effect
  423.      * of the instance
  424.      */
  425.     public AngularCoordinates revert() {
  426.         return new AngularCoordinates(rotation.revert(),
  427.                                       rotation.applyInverseTo(rotationRate).negate(),
  428.                                       rotation.applyInverseTo(rotationAcceleration).negate());
  429.     }

  431.     /** Get a time-shifted state.
  432.      * <p>
  433.      * The state can be slightly shifted to close dates. This shift is based on
  434.      * an approximate solution of the fixed acceleration motion. It is <em>not</em>
  435.      * intended as a replacement for proper attitude propagation but should be
  436.      * sufficient for either small time shifts or coarse accuracy.
  437.      * </p>
  438.      * @param dt time shift in seconds
  439.      * @return a new state, shifted with respect to the instance (which is immutable)
  440.      */
  441.     public AngularCoordinates shiftedBy(final double dt) {

  443.         // the shiftedBy method is based on a local approximation.
  444.         // It considers separately the contribution of the constant
  445.         // rotation, the linear contribution or the rate and the
  446.         // quadratic contribution of the acceleration. The rate
  447.         // and acceleration contributions are small rotations as long
  448.         // as the time shift is small, which is the crux of the algorithm.
  449.         // Small rotations are almost commutative, so we append these small
  450.         // contributions one after the other, as if they really occurred
  451.         // successively, despite this is not what really happens.

  453.         // compute the linear contribution first, ignoring acceleration
  454.         // BEWARE: there is really a minus sign here, because if
  455.         // the target frame rotates in one direction, the vectors in the origin
  456.         // frame seem to rotate in the opposite direction
  457.         final double rate = rotationRate.getNorm();
  458.         final Rotation rateContribution = (rate == 0.0) ?
  459.                                           Rotation.IDENTITY :
  460.                                           new Rotation(rotationRate, rate * dt, RotationConvention.FRAME_TRANSFORM);

  462.         // append rotation and rate contribution
  463.         final AngularCoordinates linearPart =
  464.                 new AngularCoordinates(rateContribution.compose(rotation, RotationConvention.VECTOR_OPERATOR), rotationRate);

  466.         final double acc  = rotationAcceleration.getNorm();
  467.         if (acc == 0.0) {
  468.             // no acceleration, the linear part is sufficient
  469.             return linearPart;
  470.         }

  472.         // compute the quadratic contribution, ignoring initial rotation and rotation rate
  473.         // BEWARE: there is really a minus sign here, because if
  474.         // the target frame rotates in one direction, the vectors in the origin
  475.         // frame seem to rotate in the opposite direction
  476.         final AngularCoordinates quadraticContribution =
  477.                 new AngularCoordinates(new Rotation(rotationAcceleration,
  478.                                                     0.5 * acc * dt * dt,
  479.                                                     RotationConvention.FRAME_TRANSFORM),
  480.                                        new Vector3D(dt, rotationAcceleration),
  481.                                        rotationAcceleration);

  483.         // the quadratic contribution is a small rotation:
  484.         // its initial angle and angular rate are both zero.
  485.         // small rotations are almost commutative, so we append the small
  486.         // quadratic part after the linear part as a simple offset
  487.         return quadraticContribution.addOffset(linearPart);

  489.     }

  491.     /** Get the rotation.
  492.      * @return the rotation.
  493.      */
  494.     public Rotation getRotation() {
  495.         return rotation;
  496.     }

  498.     /** Get the rotation rate.
  499.      * @return the rotation rate vector Ω (rad/s).
  500.      */
  501.     public Vector3D getRotationRate() {
  502.         return rotationRate;
  503.     }

  505.     /** Get the rotation acceleration.
  506.      * @return the rotation acceleration vector dΩ/dt (rad²/s²).
  507.      */
  508.     public Vector3D getRotationAcceleration() {
  509.         return rotationAcceleration;
  510.     }

  512.     /** Add an offset from the instance.
  513.      * <p>
  514.      * We consider here that the offset rotation is applied first and the
  515.      * instance is applied afterward. Note that angular coordinates do <em>not</em>
  516.      * commute under this operation, i.e. {@code a.addOffset(b)} and {@code
  517.      * b.addOffset(a)} lead to <em>different</em> results in most cases.
  518.      * </p>
  519.      * <p>
  520.      * The two methods {@link #addOffset(AngularCoordinates) addOffset} and
  521.      * {@link #subtractOffset(AngularCoordinates) subtractOffset} are designed
  522.      * so that round trip applications are possible. This means that both {@code
  523.      * ac1.subtractOffset(ac2).addOffset(ac2)} and {@code
  524.      * ac1.addOffset(ac2).subtractOffset(ac2)} return angular coordinates equal to ac1.
  525.      * </p>
  526.      * @param offset offset to subtract
  527.      * @return new instance, with offset subtracted
  528.      * @see #subtractOffset(AngularCoordinates)
  529.      */
  530.     public AngularCoordinates addOffset(final AngularCoordinates offset) {
  531.         final Vector3D rOmega    = rotation.applyTo(offset.rotationRate);
  532.         final Vector3D rOmegaDot = rotation.applyTo(offset.rotationAcceleration);
  533.         return new AngularCoordinates(rotation.compose(offset.rotation, RotationConvention.VECTOR_OPERATOR),
  534.                                       rotationRate.add(rOmega),
  535.                                       new Vector3D( 1.0, rotationAcceleration,
  536.                                                     1.0, rOmegaDot,
  537.                                                    -1.0, Vector3D.crossProduct(rotationRate, rOmega)));
  538.     }

  540.     /** Subtract an offset from the instance.
  541.      * <p>
  542.      * We consider here that the offset rotation is applied first and the
  543.      * instance is applied afterward. Note that angular coordinates do <em>not</em>
  544.      * commute under this operation, i.e. {@code a.subtractOffset(b)} and {@code
  545.      * b.subtractOffset(a)} lead to <em>different</em> results in most cases.
  546.      * </p>
  547.      * <p>
  548.      * The two methods {@link #addOffset(AngularCoordinates) addOffset} and
  549.      * {@link #subtractOffset(AngularCoordinates) subtractOffset} are designed
  550.      * so that round trip applications are possible. This means that both {@code
  551.      * ac1.subtractOffset(ac2).addOffset(ac2)} and {@code
  552.      * ac1.addOffset(ac2).subtractOffset(ac2)} return angular coordinates equal to ac1.
  553.      * </p>
  554.      * @param offset offset to subtract
  555.      * @return new instance, with offset subtracted
  556.      * @see #addOffset(AngularCoordinates)
  557.      */
  558.     public AngularCoordinates subtractOffset(final AngularCoordinates offset) {
  559.         return addOffset(offset.revert());
  560.     }

  562.     /** Apply the rotation to a pv coordinates.
  563.      * @param pv vector to apply the rotation to
  564.      * @return a new pv coordinates which is the image of u by the rotation
  565.      */
  566.     public PVCoordinates applyTo(final PVCoordinates pv) {

  568.         final Vector3D transformedP = rotation.applyTo(pv.getPosition());
  569.         final Vector3D crossP       = Vector3D.crossProduct(rotationRate, transformedP);
  570.         final Vector3D transformedV = rotation.applyTo(pv.getVelocity()).subtract(crossP);
  571.         final Vector3D crossV       = Vector3D.crossProduct(rotationRate, transformedV);
  572.         final Vector3D crossCrossP  = Vector3D.crossProduct(rotationRate, crossP);
  573.         final Vector3D crossDotP    = Vector3D.crossProduct(rotationAcceleration, transformedP);
  574.         final Vector3D transformedA = new Vector3D( 1, rotation.applyTo(pv.getAcceleration()),
  575.                                                    -2, crossV,
  576.                                                    -1, crossCrossP,
  577.                                                    -1, crossDotP);

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

  581.     }

  583.     /** Apply the rotation to a pv coordinates.
  584.      * @param pv vector to apply the rotation to
  585.      * @return a new pv coordinates which is the image of u by the rotation
  586.      */
  587.     public TimeStampedPVCoordinates applyTo(final TimeStampedPVCoordinates pv) {

  589.         final Vector3D transformedP = getRotation().applyTo(pv.getPosition());
  590.         final Vector3D crossP       = Vector3D.crossProduct(getRotationRate(), transformedP);
  591.         final Vector3D transformedV = getRotation().applyTo(pv.getVelocity()).subtract(crossP);
  592.         final Vector3D crossV       = Vector3D.crossProduct(getRotationRate(), transformedV);
  593.         final Vector3D crossCrossP  = Vector3D.crossProduct(getRotationRate(), crossP);
  594.         final Vector3D crossDotP    = Vector3D.crossProduct(getRotationAcceleration(), transformedP);
  595.         final Vector3D transformedA = new Vector3D( 1, getRotation().applyTo(pv.getAcceleration()),
  596.                                                    -2, crossV,
  597.                                                    -1, crossCrossP,
  598.                                                    -1, crossDotP);

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

  602.     }

  604.     /** Apply the rotation to a pv coordinates.
  605.      * @param pv vector to apply the rotation to
  606.      * @param <T> type of the field elements
  607.      * @return a new pv coordinates which is the image of u by the rotation
  608.      * @since 9.0
  609.      */
  610.     public <T extends RealFieldElement<T>> FieldPVCoordinates<T> applyTo(final FieldPVCoordinates<T> pv) {

  612.         final FieldVector3D<T> transformedP = FieldRotation.applyTo(rotation, pv.getPosition());
  613.         final FieldVector3D<T> crossP       = FieldVector3D.crossProduct(rotationRate, transformedP);
  614.         final FieldVector3D<T> transformedV = FieldRotation.applyTo(rotation, pv.getVelocity()).subtract(crossP);
  615.         final FieldVector3D<T> crossV       = FieldVector3D.crossProduct(rotationRate, transformedV);
  616.         final FieldVector3D<T> crossCrossP  = FieldVector3D.crossProduct(rotationRate, crossP);
  617.         final FieldVector3D<T> crossDotP    = FieldVector3D.crossProduct(rotationAcceleration, transformedP);
  618.         final FieldVector3D<T> transformedA = new FieldVector3D<>( 1, FieldRotation.applyTo(rotation, pv.getAcceleration()),
  619.                                                                   -2, crossV,
  620.                                                                   -1, crossCrossP,
  621.                                                                   -1, crossDotP);

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

  625.     }

  627.     /** Apply the rotation to a pv coordinates.
  628.      * @param pv vector to apply the rotation to
  629.      * @param <T> type of the field elements
  630.      * @return a new pv coordinates which is the image of u by the rotation
  631.      * @since 9.0
  632.      */
  633.     public <T extends RealFieldElement<T>> TimeStampedFieldPVCoordinates<T> applyTo(final TimeStampedFieldPVCoordinates<T> pv) {

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

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

  648.     }

  650.     /** Convert rotation, rate and acceleration to modified Rodrigues vector and derivatives.
  651.      * <p>
  652.      * The modified Rodrigues vector is tan(θ/4) u where θ and u are the
  653.      * rotation angle and axis respectively.
  654.      * </p>
  655.      * @param sign multiplicative sign for quaternion components
  656.      * @return modified Rodrigues vector and derivatives (vector on row 0, first derivative
  657.      * on row 1, second derivative on row 2)
  658.      * @see #createFromModifiedRodrigues(double[][])
  659.      */
  660.     public double[][] getModifiedRodrigues(final double sign) {

  662.         final double q0    = sign * getRotation().getQ0();
  663.         final double q1    = sign * getRotation().getQ1();
  664.         final double q2    = sign * getRotation().getQ2();
  665.         final double q3    = sign * getRotation().getQ3();
  666.         final double oX    = getRotationRate().getX();
  667.         final double oY    = getRotationRate().getY();
  668.         final double oZ    = getRotationRate().getZ();
  669.         final double oXDot = getRotationAcceleration().getX();
  670.         final double oYDot = getRotationAcceleration().getY();
  671.         final double oZDot = getRotationAcceleration().getZ();

  673.         // first time-derivatives of the quaternion
  674.         final double q0Dot = 0.5 * MathArrays.linearCombination(-q1, oX, -q2, oY, -q3, oZ);
  675.         final double q1Dot = 0.5 * MathArrays.linearCombination( q0, oX, -q3, oY,  q2, oZ);
  676.         final double q2Dot = 0.5 * MathArrays.linearCombination( q3, oX,  q0, oY, -q1, oZ);
  677.         final double q3Dot = 0.5 * MathArrays.linearCombination(-q2, oX,  q1, oY,  q0, oZ);

  679.         // second time-derivatives of the quaternion
  680.         final double q0DotDot = -0.5 * MathArrays.linearCombination(new double[] {
  681.             q1, q2,  q3, q1Dot, q2Dot,  q3Dot
  682.         }, new double[] {
  683.             oXDot, oYDot, oZDot, oX, oY, oZ
  684.         });
  685.         final double q1DotDot =  0.5 * MathArrays.linearCombination(new double[] {
  686.             q0, q2, -q3, q0Dot, q2Dot, -q3Dot
  687.         }, new double[] {
  688.             oXDot, oZDot, oYDot, oX, oZ, oY
  689.         });
  690.         final double q2DotDot =  0.5 * MathArrays.linearCombination(new double[] {
  691.             q0, q3, -q1, q0Dot, q3Dot, -q1Dot
  692.         }, new double[] {
  693.             oYDot, oXDot, oZDot, oY, oX, oZ
  694.         });
  695.         final double q3DotDot =  0.5 * MathArrays.linearCombination(new double[] {
  696.             q0, q1, -q2, q0Dot, q1Dot, -q2Dot
  697.         }, new double[] {
  698.             oZDot, oYDot, oXDot, oZ, oY, oX
  699.         });

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

  708.         final double r1       = inv * q1;
  709.         final double r2       = inv * q2;
  710.         final double r3       = inv * q3;

  712.         final double mInvR1   = -inv * r1;
  713.         final double mInvR2   = -inv * r2;
  714.         final double mInvR3   = -inv * r3;

  716.         final double r1Dot    = MathArrays.linearCombination(inv, q1Dot, mInvR1, q0Dot);
  717.         final double r2Dot    = MathArrays.linearCombination(inv, q2Dot, mInvR2, q0Dot);
  718.         final double r3Dot    = MathArrays.linearCombination(inv, q3Dot, mInvR3, q0Dot);

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

  724.         return new double[][] {
  725.             {
  726.                 r1,       r2,       r3
  727.             }, {
  728.                 r1Dot,    r2Dot,    r3Dot
  729.             }, {
  730.                 r1DotDot, r2DotDot, r3DotDot
  731.             }
  732.         };

  734.     }

  736.     /** Convert a modified Rodrigues vector and derivatives to angular coordinates.
  737.      * @param r modified Rodrigues vector (with first and second times derivatives)
  738.      * @return angular coordinates
  739.      * @see #getModifiedRodrigues(double)
  740.      */
  741.     public static AngularCoordinates createFromModifiedRodrigues(final double[][] r) {

  743.         // rotation
  744.         final double rSquared = r[0][0] * r[0][0] + r[0][1] * r[0][1] + r[0][2] * r[0][2];
  745.         final double oPQ0     = 2 / (1 + rSquared);
  746.         final double q0       = oPQ0 - 1;
  747.         final double q1       = oPQ0 * r[0][0];
  748.         final double q2       = oPQ0 * r[0][1];
  749.         final double q3       = oPQ0 * r[0][2];

  751.         // rotation rate
  752.         final double oPQ02    = oPQ0 * oPQ0;
  753.         final double q0Dot    = -oPQ02 * MathArrays.linearCombination(r[0][0], r[1][0], r[0][1], r[1][1],  r[0][2], r[1][2]);
  754.         final double q1Dot    = oPQ0 * r[1][0] + r[0][0] * q0Dot;
  755.         final double q2Dot    = oPQ0 * r[1][1] + r[0][1] * q0Dot;
  756.         final double q3Dot    = oPQ0 * r[1][2] + r[0][2] * q0Dot;
  757.         final double oX       = 2 * MathArrays.linearCombination(-q1, q0Dot,  q0, q1Dot,  q3, q2Dot, -q2, q3Dot);
  758.         final double oY       = 2 * MathArrays.linearCombination(-q2, q0Dot, -q3, q1Dot,  q0, q2Dot,  q1, q3Dot);
  759.         final double oZ       = 2 * MathArrays.linearCombination(-q3, q0Dot,  q2, q1Dot, -q1, q2Dot,  q0, q3Dot);

  761.         // rotation acceleration
  762.         final double q0DotDot = (1 - q0) / oPQ0 * q0Dot * q0Dot -
  763.                                 oPQ02 * MathArrays.linearCombination(r[0][0], r[2][0], r[0][1], r[2][1], r[0][2], r[2][2]) -
  764.                                 (q1Dot * q1Dot + q2Dot * q2Dot + q3Dot * q3Dot);
  765.         final double q1DotDot = MathArrays.linearCombination(oPQ0, r[2][0], 2 * r[1][0], q0Dot, r[0][0], q0DotDot);
  766.         final double q2DotDot = MathArrays.linearCombination(oPQ0, r[2][1], 2 * r[1][1], q0Dot, r[0][1], q0DotDot);
  767.         final double q3DotDot = MathArrays.linearCombination(oPQ0, r[2][2], 2 * r[1][2], q0Dot, r[0][2], q0DotDot);
  768.         final double oXDot    = 2 * MathArrays.linearCombination(-q1, q0DotDot,  q0, q1DotDot,  q3, q2DotDot, -q2, q3DotDot);
  769.         final double oYDot    = 2 * MathArrays.linearCombination(-q2, q0DotDot, -q3, q1DotDot,  q0, q2DotDot,  q1, q3DotDot);
  770.         final double oZDot    = 2 * MathArrays.linearCombination(-q3, q0DotDot,  q2, q1DotDot, -q1, q2DotDot,  q0, q3DotDot);

  772.         return new AngularCoordinates(new Rotation(q0, q1, q2, q3, false),
  773.                                       new Vector3D(oX, oY, oZ),
  774.                                       new Vector3D(oXDot, oYDot, oZDot));

  776.     }

  778. }
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