/* Copyright 2002-2016 CS Systèmes d'Information
    * Licensed to CS Systèmes d'Information (CS) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * CS licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
    * the License.  You may obtain a copy of the License at
    *
    *   http://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   * See the License for the specific language governing permissions and
   * limitations under the License.
   */
  package org.orekit.utils;
  
  import java.io.Serializable;
  
  import org.hipparchus.analysis.differentiation.DerivativeStructure;
  import org.hipparchus.exception.LocalizedCoreFormats;
  import org.hipparchus.exception.MathIllegalArgumentException;
  import org.hipparchus.exception.MathRuntimeException;
  import org.hipparchus.geometry.euclidean.threed.FieldRotation;
  import org.hipparchus.geometry.euclidean.threed.Rotation;
  import org.hipparchus.geometry.euclidean.threed.RotationConvention;
  import org.hipparchus.geometry.euclidean.threed.Vector3D;
  import org.hipparchus.linear.DecompositionSolver;
  import org.hipparchus.linear.MatrixUtils;
  import org.hipparchus.linear.QRDecomposition;
  import org.hipparchus.linear.RealMatrix;
  import org.hipparchus.linear.RealVector;
  import org.hipparchus.util.FastMath;
  import org.hipparchus.util.MathArrays;
  import org.orekit.errors.OrekitException;
  import org.orekit.errors.OrekitMessages;
  import org.orekit.time.TimeShiftable;
  
  /** Simple container for rotation/rotation rate/rotation acceleration triplets.
   * <p>
   * The state can be slightly shifted to close dates. This shift is based on
   * an approximate solution of the fixed acceleration motion. It is <em>not</em>
   * intended as a replacement for proper attitude propagation but should be
   * sufficient for either small time shifts or coarse accuracy.
   * </p>
   * <p>
   * This class is the angular counterpart to {@link PVCoordinates}.
   * </p>
   * <p>Instances of this class are guaranteed to be immutable.</p>
   * @author Luc Maisonobe
   */
  public class AngularCoordinates implements TimeShiftable<AngularCoordinates>, Serializable {
  
      /** Fixed orientation parallel with reference frame
       * (identity rotation, zero rotation rate and acceleration).
       */
      public static final AngularCoordinates IDENTITY =
              new AngularCoordinates(Rotation.IDENTITY, Vector3D.ZERO, Vector3D.ZERO);
  
      /** Serializable UID. */
      private static final long serialVersionUID = 20140414L;
  
      /** Rotation. */
      private final Rotation rotation;
  
      /** Rotation rate. */
      private final Vector3D rotationRate;
  
      /** Rotation acceleration. */
      private final Vector3D rotationAcceleration;
  
      /** Simple constructor.
       * <p> Sets the Coordinates to default : Identity, Ω = (0 0 0), dΩ/dt = (0 0 0).</p>
       */
      public AngularCoordinates() {
          this(Rotation.IDENTITY, Vector3D.ZERO, Vector3D.ZERO);
      }
  
      /** Builds a rotation/rotation rate pair.
       * @param rotation rotation
       * @param rotationRate rotation rate Ω (rad/s)
       */
      public AngularCoordinates(final Rotation rotation, final Vector3D rotationRate) {
          this(rotation, rotationRate, Vector3D.ZERO);
      }
  
      /** Builds a rotation/rotation rate/rotation acceleration triplet.
       * @param rotation rotation
       * @param rotationRate rotation rate Ω (rad/s)
       * @param rotationAcceleration rotation acceleration dΩ/dt (rad²/s²)
       */
      public AngularCoordinates(final Rotation rotation,
                                final Vector3D rotationRate, final Vector3D rotationAcceleration) {
          this.rotation             = rotation;
          this.rotationRate         = rotationRate;
          this.rotationAcceleration = rotationAcceleration;
      }
  
     /** Build the rotation that transforms a pair of pv coordinates into another one.
 
      * <p><em>WARNING</em>! This method requires much more stringent assumptions on
      * its parameters than the similar {@link Rotation#Rotation(Vector3D, Vector3D,
      * Vector3D, Vector3D) constructor} from the {@link Rotation Rotation} class.
      * As far as the Rotation constructor is concerned, the {@code v₂} vector from
      * the second pair can be slightly misaligned. The Rotation constructor will
      * compensate for this misalignment and create a rotation that ensure {@code
      * v₁ = r(u₁)} and {@code v₂ ∈ plane (r(u₁), r(u₂))}. <em>THIS IS NOT
      * TRUE ANYMORE IN THIS CLASS</em>! As derivatives are involved and must be
      * preserved, this constructor works <em>only</em> if the two pairs are fully
      * consistent, i.e. if a rotation exists that fulfill all the requirements: {@code
      * v₁ = r(u₁)}, {@code v₂ = r(u₂)}, {@code dv₁/dt = dr(u₁)/dt}, {@code dv₂/dt
      * = dr(u₂)/dt}, {@code d²v₁/dt² = d²r(u₁)/dt²}, {@code d²v₂/dt² = d²r(u₂)/dt²}.</p>
      * @param u1 first vector of the origin pair
      * @param u2 second vector of the origin pair
      * @param v1 desired image of u1 by the rotation
      * @param v2 desired image of u2 by the rotation
      * @param tolerance relative tolerance factor used to check singularities
      * @exception OrekitException if the vectors are inconsistent for the
      * rotation to be found (null, aligned, ...)
      */
     public AngularCoordinates(final PVCoordinates u1, final PVCoordinates u2,
                               final PVCoordinates v1, final PVCoordinates v2,
                               final double tolerance)
         throws OrekitException {
 
         try {
             // find the initial fixed rotation
             rotation = new Rotation(u1.getPosition(), u2.getPosition(),
                                     v1.getPosition(), v2.getPosition());
 
             // find rotation rate Ω such that
             //  Ω ⨯ v₁ = r(dot(u₁)) - dot(v₁)
             //  Ω ⨯ v₂ = r(dot(u₂)) - dot(v₂)
             final Vector3D ru1Dot = rotation.applyTo(u1.getVelocity());
             final Vector3D ru2Dot = rotation.applyTo(u2.getVelocity());
             rotationRate = inverseCrossProducts(v1.getPosition(), ru1Dot.subtract(v1.getVelocity()),
                                                 v2.getPosition(), ru2Dot.subtract(v2.getVelocity()),
                                                 tolerance);
 
             // find rotation acceleration dot(Ω) such that
             // dot(Ω) ⨯ v₁ = r(dotdot(u₁)) - 2 Ω ⨯ dot(v₁) - Ω ⨯  (Ω ⨯ v₁) - dotdot(v₁)
             // dot(Ω) ⨯ v₂ = r(dotdot(u₂)) - 2 Ω ⨯ dot(v₂) - Ω ⨯  (Ω ⨯ v₂) - dotdot(v₂)
             final Vector3D ru1DotDot = rotation.applyTo(u1.getAcceleration());
             final Vector3D oDotv1    = Vector3D.crossProduct(rotationRate, v1.getVelocity());
             final Vector3D oov1      = Vector3D.crossProduct(rotationRate, Vector3D.crossProduct(rotationRate, v1.getPosition()));
             final Vector3D c1        = new Vector3D(1, ru1DotDot, -2, oDotv1, -1, oov1, -1, v1.getAcceleration());
             final Vector3D ru2DotDot = rotation.applyTo(u2.getAcceleration());
             final Vector3D oDotv2    = Vector3D.crossProduct(rotationRate, v2.getVelocity());
             final Vector3D oov2      = Vector3D.crossProduct(rotationRate, Vector3D.crossProduct(rotationRate, v2.getPosition()));
             final Vector3D c2        = new Vector3D(1, ru2DotDot, -2, oDotv2, -1, oov2, -1, v2.getAcceleration());
             rotationAcceleration     = inverseCrossProducts(v1.getPosition(), c1, v2.getPosition(), c2, tolerance);
 
         } catch (MathRuntimeException mrte) {
             throw new OrekitException(mrte);
         }
 
     }
 
     /** Build one of the rotations that transform one pv coordinates into another one.
 
      * <p>Except for a possible scale factor, if the instance were
      * applied to the vector u it will produce the vector v. There is an
      * infinite number of such rotations, this constructor choose the
      * one with the smallest associated angle (i.e. the one whose axis
      * is orthogonal to the (u, v) plane). If u and v are collinear, an
      * arbitrary rotation axis is chosen.</p>
 
      * @param u origin vector
      * @param v desired image of u by the rotation
      * @exception OrekitException if the vectors components cannot be converted to
      * {@link DerivativeStructure} with proper order
      */
     public AngularCoordinates(final PVCoordinates u, final PVCoordinates v) throws OrekitException {
         this(new FieldRotation<DerivativeStructure>(u.toDerivativeStructureVector(2),
                                                     v.toDerivativeStructureVector(2)));
     }
 
     /** Builds a AngularCoordinates from  a {@link FieldRotation}&lt;{@link DerivativeStructure}&gt;.
      * <p>
      * The rotation components must have time as their only derivation parameter and
      * have consistent derivation orders.
      * </p>
      * @param r rotation with time-derivatives embedded within the coordinates
      */
     public AngularCoordinates(final FieldRotation<DerivativeStructure> r) {
 
         final double q0       = r.getQ0().getReal();
         final double q1       = r.getQ1().getReal();
         final double q2       = r.getQ2().getReal();
         final double q3       = r.getQ3().getReal();
 
         rotation     = new Rotation(q0, q1, q2, q3, false);
         if (r.getQ0().getOrder() >= 1) {
             final double q0Dot    = r.getQ0().getPartialDerivative(1);
             final double q1Dot    = r.getQ1().getPartialDerivative(1);
             final double q2Dot    = r.getQ2().getPartialDerivative(1);
             final double q3Dot    = r.getQ3().getPartialDerivative(1);
             rotationRate =
                     new Vector3D(2 * MathArrays.linearCombination(-q1, q0Dot,  q0, q1Dot,  q3, q2Dot, -q2, q3Dot),
                                  2 * MathArrays.linearCombination(-q2, q0Dot, -q3, q1Dot,  q0, q2Dot,  q1, q3Dot),
                                  2 * MathArrays.linearCombination(-q3, q0Dot,  q2, q1Dot, -q1, q2Dot,  q0, q3Dot));
             if (r.getQ0().getOrder() >= 2) {
                 final double q0DotDot = r.getQ0().getPartialDerivative(2);
                 final double q1DotDot = r.getQ1().getPartialDerivative(2);
                 final double q2DotDot = r.getQ2().getPartialDerivative(2);
                 final double q3DotDot = r.getQ3().getPartialDerivative(2);
                 rotationAcceleration =
                         new Vector3D(2 * MathArrays.linearCombination(-q1, q0DotDot,  q0, q1DotDot,  q3, q2DotDot, -q2, q3DotDot),
                                      2 * MathArrays.linearCombination(-q2, q0DotDot, -q3, q1DotDot,  q0, q2DotDot,  q1, q3DotDot),
                                      2 * MathArrays.linearCombination(-q3, q0DotDot,  q2, q1DotDot, -q1, q2DotDot,  q0, q3DotDot));
             } else {
                 rotationAcceleration = Vector3D.ZERO;
             }
         } else {
             rotationRate         = Vector3D.ZERO;
             rotationAcceleration = Vector3D.ZERO;
         }
 
     }
 
     /** Find a vector from two known cross products.
      * <p>
      * We want to find Ω such that: Ω ⨯ v₁ = c₁ and Ω ⨯ v₂ = c₂
      * </p>
      * <p>
      * The first equation (Ω ⨯ v₁ = c₁) will always be fulfilled exactly,
      * and the second one will be fulfilled if possible.
      * </p>
      * @param v1 vector forming the first known cross product
      * @param c1 know vector for cross product Ω ⨯ v₁
      * @param v2 vector forming the second known cross product
      * @param c2 know vector for cross product Ω ⨯ v₂
      * @param tolerance relative tolerance factor used to check singularities
      * @return vector Ω such that: Ω ⨯ v₁ = c₁ and Ω ⨯ v₂ = c₂
      * @exception MathIllegalArgumentException if vectors are inconsistent and
      * no solution can be found
      */
     private static Vector3D inverseCrossProducts(final Vector3D v1, final Vector3D c1,
                                                  final Vector3D v2, final Vector3D c2,
                                                  final double tolerance)
         throws MathIllegalArgumentException {
 
         final double v12 = v1.getNormSq();
         final double v1n = FastMath.sqrt(v12);
         final double v22 = v2.getNormSq();
         final double v2n = FastMath.sqrt(v22);
         final double threshold = tolerance * FastMath.max(v1n, v2n);
 
         Vector3D omega;
 
         try {
             // create the over-determined linear system representing the two cross products
             final RealMatrix m = MatrixUtils.createRealMatrix(6, 3);
             m.setEntry(0, 1,  v1.getZ());
             m.setEntry(0, 2, -v1.getY());
             m.setEntry(1, 0, -v1.getZ());
             m.setEntry(1, 2,  v1.getX());
             m.setEntry(2, 0,  v1.getY());
             m.setEntry(2, 1, -v1.getX());
             m.setEntry(3, 1,  v2.getZ());
             m.setEntry(3, 2, -v2.getY());
             m.setEntry(4, 0, -v2.getZ());
             m.setEntry(4, 2,  v2.getX());
             m.setEntry(5, 0,  v2.getY());
             m.setEntry(5, 1, -v2.getX());
 
             final RealVector rhs = MatrixUtils.createRealVector(new double[] {
                 c1.getX(), c1.getY(), c1.getZ(),
                 c2.getX(), c2.getY(), c2.getZ()
             });
 
             // find the best solution we can
             final DecompositionSolver solver = new QRDecomposition(m, threshold).getSolver();
             final RealVector v = solver.solve(rhs);
             omega = new Vector3D(v.getEntry(0), v.getEntry(1), v.getEntry(2));
 
         } catch (MathIllegalArgumentException miae) {
             if (miae.getSpecifier() == LocalizedCoreFormats.SINGULAR_MATRIX) {
 
                 // handle some special cases for which we can compute a solution
                 final double c12 = c1.getNormSq();
                 final double c1n = FastMath.sqrt(c12);
                 final double c22 = c2.getNormSq();
                 final double c2n = FastMath.sqrt(c22);
 
                 if (c1n <= threshold && c2n <= threshold) {
                     // simple special case, velocities are cancelled
                     return Vector3D.ZERO;
                 } else if (v1n <= threshold && c1n >= threshold) {
                     // this is inconsistent, if v₁ is zero, c₁ must be 0 too
                     throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, c1n, 0, true);
                 } else if (v2n <= threshold && c2n >= threshold) {
                     // this is inconsistent, if v₂ is zero, c₂ must be 0 too
                     throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, c2n, 0, true);
                 } else if (Vector3D.crossProduct(v1, v2).getNorm() <= threshold && v12 > threshold) {
                     // simple special case, v₂ is redundant with v₁, we just ignore it
                     // use the simplest Ω: orthogonal to both v₁ and c₁
                     omega = new Vector3D(1.0 / v12, Vector3D.crossProduct(v1, c1));
                 } else {
                     throw miae;
                 }
             } else {
                 throw miae;
             }
 
         }
 
         // check results
         final double d1 = Vector3D.distance(Vector3D.crossProduct(omega, v1), c1);
         if (d1 > threshold) {
             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, d1, 0, true);
         }
 
         final double d2 = Vector3D.distance(Vector3D.crossProduct(omega, v2), c2);
         if (d2 > threshold) {
             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, d2, 0, true);
         }
 
         return omega;
 
     }
 
     /** Transform the instance to a {@link FieldRotation}&lt;{@link DerivativeStructure}&gt;.
      * <p>
      * The {@link DerivativeStructure} coordinates correspond to time-derivatives up
      * to the user-specified order.
      * </p>
      * @param order derivation order for the vector components
      * @return rotation with time-derivatives embedded within the coordinates
      * @exception OrekitException if the user specified order is too large
      */
     public FieldRotation<DerivativeStructure> toDerivativeStructureRotation(final int order)
         throws OrekitException {
 
         // quaternion components
         final double q0 = rotation.getQ0();
         final double q1 = rotation.getQ1();
         final double q2 = rotation.getQ2();
         final double q3 = rotation.getQ3();
 
         // first time-derivatives of the quaternion
         final double oX    = rotationRate.getX();
         final double oY    = rotationRate.getY();
         final double oZ    = rotationRate.getZ();
         final double q0Dot = 0.5 * MathArrays.linearCombination(-q1, oX, -q2, oY, -q3, oZ);
         final double q1Dot = 0.5 * MathArrays.linearCombination( q0, oX, -q3, oY,  q2, oZ);
         final double q2Dot = 0.5 * MathArrays.linearCombination( q3, oX,  q0, oY, -q1, oZ);
         final double q3Dot = 0.5 * MathArrays.linearCombination(-q2, oX,  q1, oY,  q0, oZ);
 
         // second time-derivatives of the quaternion
         final double oXDot = rotationAcceleration.getX();
         final double oYDot = rotationAcceleration.getY();
         final double oZDot = rotationAcceleration.getZ();
         final double q0DotDot = -0.5 * MathArrays.linearCombination(new double[] {
             q1, q2,  q3, q1Dot, q2Dot,  q3Dot
         }, new double[] {
             oXDot, oYDot, oZDot, oX, oY, oZ
         });
         final double q1DotDot =  0.5 * MathArrays.linearCombination(new double[] {
             q0, q2, -q3, q0Dot, q2Dot, -q3Dot
         }, new double[] {
             oXDot, oZDot, oYDot, oX, oZ, oY
         });
         final double q2DotDot =  0.5 * MathArrays.linearCombination(new double[] {
             q0, q3, -q1, q0Dot, q3Dot, -q1Dot
         }, new double[] {
             oYDot, oXDot, oZDot, oY, oX, oZ
         });
         final double q3DotDot =  0.5 * MathArrays.linearCombination(new double[] {
             q0, q1, -q2, q0Dot, q1Dot, -q2Dot
         }, new double[] {
             oZDot, oYDot, oXDot, oZ, oY, oX
         });
 
         final DerivativeStructure q0DS;
         final DerivativeStructure q1DS;
         final DerivativeStructure q2DS;
         final DerivativeStructure q3DS;
         switch(order) {
             case 0 :
                 q0DS = new DerivativeStructure(1, 0, q0);
                 q1DS = new DerivativeStructure(1, 0, q1);
                 q2DS = new DerivativeStructure(1, 0, q2);
                 q3DS = new DerivativeStructure(1, 0, q3);
                 break;
             case 1 :
                 q0DS = new DerivativeStructure(1, 1, q0, q0Dot);
                 q1DS = new DerivativeStructure(1, 1, q1, q1Dot);
                 q2DS = new DerivativeStructure(1, 1, q2, q2Dot);
                 q3DS = new DerivativeStructure(1, 1, q3, q3Dot);
                 break;
             case 2 :
                 q0DS = new DerivativeStructure(1, 2, q0, q0Dot, q0DotDot);
                 q1DS = new DerivativeStructure(1, 2, q1, q1Dot, q1DotDot);
                 q2DS = new DerivativeStructure(1, 2, q2, q2Dot, q2DotDot);
                 q3DS = new DerivativeStructure(1, 2, q3, q3Dot, q3DotDot);
                 break;
             default :
                 throw new OrekitException(OrekitMessages.OUT_OF_RANGE_DERIVATION_ORDER, order);
         }
 
         return new FieldRotation<DerivativeStructure>(q0DS, q1DS, q2DS, q3DS, false);
 
     }
 
     /** Estimate rotation rate between two orientations.
      * <p>Estimation is based on a simple fixed rate rotation
      * during the time interval between the two orientations.</p>
      * @param start start orientation
      * @param end end orientation
      * @param dt time elapsed between the dates of the two orientations
      * @return rotation rate allowing to go from start to end orientations
      */
     public static Vector3D estimateRate(final Rotation start, final Rotation end, final double dt) {
         final Rotation evolution = start.compose(end.revert(), RotationConvention.VECTOR_OPERATOR);
         return new Vector3D(evolution.getAngle() / dt, evolution.getAxis(RotationConvention.VECTOR_OPERATOR));
     }
 
     /** Revert a rotation/rotation rate/ rotation acceleration triplet.
      * Build a triplet which reverse the effect of another triplet.
      * @return a new triplet whose effect is the reverse of the effect
      * of the instance
      */
     public AngularCoordinates revert() {
         return new AngularCoordinates(rotation.revert(),
                                       rotation.applyInverseTo(rotationRate).negate(),
                                       rotation.applyInverseTo(rotationAcceleration).negate());
     }
 
     /** Get a time-shifted state.
      * <p>
      * The state can be slightly shifted to close dates. This shift is based on
      * an approximate solution of the fixed acceleration motion. It is <em>not</em>
      * intended as a replacement for proper attitude propagation but should be
      * sufficient for either small time shifts or coarse accuracy.
      * </p>
      * @param dt time shift in seconds
      * @return a new state, shifted with respect to the instance (which is immutable)
      */
     public AngularCoordinates shiftedBy(final double dt) {
 
         // the shiftedBy method is based on a local approximation.
         // It considers separately the contribution of the constant
         // rotation, the linear contribution or the rate and the
         // quadratic contribution of the acceleration. The rate
         // and acceleration contributions are small rotations as long
         // as the time shift is small, which is the crux of the algorithm.
         // Small rotations are almost commutative, so we append these small
         // contributions one after the other, as if they really occurred
         // successively, despite this is not what really happens.
 
         // compute the linear contribution first, ignoring acceleration
         // BEWARE: there is really a minus sign here, because if
         // the target frame rotates in one direction, the vectors in the origin
         // frame seem to rotate in the opposite direction
         final double rate = rotationRate.getNorm();
         final Rotation rateContribution = (rate == 0.0) ?
                                           Rotation.IDENTITY :
                                           new Rotation(rotationRate, rate * dt, RotationConvention.FRAME_TRANSFORM);
 
         // append rotation and rate contribution
         final AngularCoordinates linearPart =
                 new AngularCoordinates(rateContribution.compose(rotation, RotationConvention.VECTOR_OPERATOR), rotationRate);
 
         final double acc  = rotationAcceleration.getNorm();
         if (acc == 0.0) {
             // no acceleration, the linear part is sufficient
             return linearPart;
         }
 
         // compute the quadratic contribution, ignoring initial rotation and rotation rate
         // BEWARE: there is really a minus sign here, because if
         // the target frame rotates in one direction, the vectors in the origin
         // frame seem to rotate in the opposite direction
         final AngularCoordinates quadraticContribution =
                 new AngularCoordinates(new Rotation(rotationAcceleration,
                                                     0.5 * acc * dt * dt,
                                                     RotationConvention.FRAME_TRANSFORM),
                                        new Vector3D(dt, rotationAcceleration),
                                        rotationAcceleration);
 
         // the quadratic contribution is a small rotation:
         // its initial angle and angular rate are both zero.
         // small rotations are almost commutative, so we append the small
         // quadratic part after the linear part as a simple offset
         return quadraticContribution.addOffset(linearPart);
 
     }
 
     /** Get the rotation.
      * @return the rotation.
      */
     public Rotation getRotation() {
         return rotation;
     }
 
     /** Get the rotation rate.
      * @return the rotation rate vector Ω (rad/s).
      */
     public Vector3D getRotationRate() {
         return rotationRate;
     }
 
     /** Get the rotation acceleration.
      * @return the rotation acceleration vector dΩ/dt (rad²/s²).
      */
     public Vector3D getRotationAcceleration() {
         return rotationAcceleration;
     }
 
     /** Add an offset from the instance.
      * <p>
      * We consider here that the offset rotation is applied first and the
      * instance is applied afterward. Note that angular coordinates do <em>not</em>
      * commute under this operation, i.e. {@code a.addOffset(b)} and {@code
      * b.addOffset(a)} lead to <em>different</em> results in most cases.
      * </p>
      * <p>
      * The two methods {@link #addOffset(AngularCoordinates) addOffset} and
      * {@link #subtractOffset(AngularCoordinates) subtractOffset} are designed
      * so that round trip applications are possible. This means that both {@code
      * ac1.subtractOffset(ac2).addOffset(ac2)} and {@code
      * ac1.addOffset(ac2).subtractOffset(ac2)} return angular coordinates equal to ac1.
      * </p>
      * @param offset offset to subtract
      * @return new instance, with offset subtracted
      * @see #subtractOffset(AngularCoordinates)
      */
     public AngularCoordinates addOffset(final AngularCoordinates offset) {
         final Vector3D rOmega    = rotation.applyTo(offset.rotationRate);
         final Vector3D rOmegaDot = rotation.applyTo(offset.rotationAcceleration);
         return new AngularCoordinates(rotation.compose(offset.rotation, RotationConvention.VECTOR_OPERATOR),
                                       rotationRate.add(rOmega),
                                       new Vector3D( 1.0, rotationAcceleration,
                                                     1.0, rOmegaDot,
                                                    -1.0, Vector3D.crossProduct(rotationRate, rOmega)));
     }
 
     /** Subtract an offset from the instance.
      * <p>
      * We consider here that the offset rotation is applied first and the
      * instance is applied afterward. Note that angular coordinates do <em>not</em>
      * commute under this operation, i.e. {@code a.subtractOffset(b)} and {@code
      * b.subtractOffset(a)} lead to <em>different</em> results in most cases.
      * </p>
      * <p>
      * The two methods {@link #addOffset(AngularCoordinates) addOffset} and
      * {@link #subtractOffset(AngularCoordinates) subtractOffset} are designed
      * so that round trip applications are possible. This means that both {@code
      * ac1.subtractOffset(ac2).addOffset(ac2)} and {@code
      * ac1.addOffset(ac2).subtractOffset(ac2)} return angular coordinates equal to ac1.
      * </p>
      * @param offset offset to subtract
      * @return new instance, with offset subtracted
      * @see #addOffset(AngularCoordinates)
      */
     public AngularCoordinates subtractOffset(final AngularCoordinates offset) {
         return addOffset(offset.revert());
     }
 
     /** Apply the rotation to a pv coordinates.
      * @param pv vector to apply the rotation to
      * @return a new pv coordinates which is the image of u by the rotation
      */
     public PVCoordinates applyTo(final PVCoordinates pv) {
 
         final Vector3D transformedP = rotation.applyTo(pv.getPosition());
         final Vector3D crossP       = Vector3D.crossProduct(rotationRate, transformedP);
         final Vector3D transformedV = rotation.applyTo(pv.getVelocity()).subtract(crossP);
         final Vector3D crossV       = Vector3D.crossProduct(rotationRate, transformedV);
         final Vector3D crossCrossP  = Vector3D.crossProduct(rotationRate, crossP);
         final Vector3D crossDotP    = Vector3D.crossProduct(rotationAcceleration, transformedP);
         final Vector3D transformedA = new Vector3D( 1, rotation.applyTo(pv.getAcceleration()),
                                                    -2, crossV,
                                                    -1, crossCrossP,
                                                    -1, crossDotP);
 
         return new PVCoordinates(transformedP, transformedV, transformedA);
 
     }
 
     /** Apply the rotation to a pv coordinates.
      * @param pv vector to apply the rotation to
      * @return a new pv coordinates which is the image of u by the rotation
      */
     public TimeStampedPVCoordinates applyTo(final TimeStampedPVCoordinates pv) {
 
         final Vector3D transformedP = getRotation().applyTo(pv.getPosition());
         final Vector3D crossP       = Vector3D.crossProduct(getRotationRate(), transformedP);
         final Vector3D transformedV = getRotation().applyTo(pv.getVelocity()).subtract(crossP);
         final Vector3D crossV       = Vector3D.crossProduct(getRotationRate(), transformedV);
         final Vector3D crossCrossP  = Vector3D.crossProduct(getRotationRate(), crossP);
         final Vector3D crossDotP    = Vector3D.crossProduct(getRotationAcceleration(), transformedP);
         final Vector3D transformedA = new Vector3D( 1, getRotation().applyTo(pv.getAcceleration()),
                                                    -2, crossV,
                                                    -1, crossCrossP,
                                                    -1, crossDotP);
 
         return new TimeStampedPVCoordinates(pv.getDate(), transformedP, transformedV, transformedA);
 
     }
 
     /** Convert rotation, rate and acceleration to modified Rodrigues vector and derivatives.
      * <p>
      * The modified Rodrigues vector is tan(θ/4) u where θ and u are the
      * rotation angle and axis respectively.
      * </p>
      * @param sign multiplicative sign for quaternion components
      * @return modified Rodrigues vector and derivatives (vector on row 0, first derivative
      * on row 1, second derivative on row 2)
      * @see #createFromModifiedRodrigues(double[][])
      */
     public double[][] getModifiedRodrigues(final double sign) {
 
         final double q0    = sign * getRotation().getQ0();
         final double q1    = sign * getRotation().getQ1();
         final double q2    = sign * getRotation().getQ2();
         final double q3    = sign * getRotation().getQ3();
         final double oX    = getRotationRate().getX();
         final double oY    = getRotationRate().getY();
         final double oZ    = getRotationRate().getZ();
         final double oXDot = getRotationAcceleration().getX();
         final double oYDot = getRotationAcceleration().getY();
         final double oZDot = getRotationAcceleration().getZ();
 
         // first time-derivatives of the quaternion
         final double q0Dot = 0.5 * MathArrays.linearCombination(-q1, oX, -q2, oY, -q3, oZ);
         final double q1Dot = 0.5 * MathArrays.linearCombination( q0, oX, -q3, oY,  q2, oZ);
         final double q2Dot = 0.5 * MathArrays.linearCombination( q3, oX,  q0, oY, -q1, oZ);
         final double q3Dot = 0.5 * MathArrays.linearCombination(-q2, oX,  q1, oY,  q0, oZ);
 
         // second time-derivatives of the quaternion
         final double q0DotDot = -0.5 * MathArrays.linearCombination(new double[] {
             q1, q2,  q3, q1Dot, q2Dot,  q3Dot
         }, new double[] {
             oXDot, oYDot, oZDot, oX, oY, oZ
         });
         final double q1DotDot =  0.5 * MathArrays.linearCombination(new double[] {
             q0, q2, -q3, q0Dot, q2Dot, -q3Dot
         }, new double[] {
             oXDot, oZDot, oYDot, oX, oZ, oY
         });
         final double q2DotDot =  0.5 * MathArrays.linearCombination(new double[] {
             q0, q3, -q1, q0Dot, q3Dot, -q1Dot
         }, new double[] {
             oYDot, oXDot, oZDot, oY, oX, oZ
         });
         final double q3DotDot =  0.5 * MathArrays.linearCombination(new double[] {
             q0, q1, -q2, q0Dot, q1Dot, -q2Dot
         }, new double[] {
             oZDot, oYDot, oXDot, oZ, oY, oX
         });
 
         // the modified Rodrigues is tan(θ/4) u where θ and u are the rotation angle and axis respectively
         // this can be rewritten using quaternion components:
         //      r (q₁ / (1+q₀), q₂ / (1+q₀), q₃ / (1+q₀))
         // applying the derivation chain rule to previous expression gives rDot and rDotDot
         final double inv          = 1.0 / (1.0 + q0);
         final double mTwoInvQ0Dot = -2 * inv * q0Dot;
 
         final double r1       = inv * q1;
         final double r2       = inv * q2;
         final double r3       = inv * q3;
 
         final double mInvR1   = -inv * r1;
         final double mInvR2   = -inv * r2;
         final double mInvR3   = -inv * r3;
 
         final double r1Dot    = MathArrays.linearCombination(inv, q1Dot, mInvR1, q0Dot);
         final double r2Dot    = MathArrays.linearCombination(inv, q2Dot, mInvR2, q0Dot);
         final double r3Dot    = MathArrays.linearCombination(inv, q3Dot, mInvR3, q0Dot);
 
         final double r1DotDot = MathArrays.linearCombination(inv, q1DotDot, mTwoInvQ0Dot, r1Dot, mInvR1, q0DotDot);
         final double r2DotDot = MathArrays.linearCombination(inv, q2DotDot, mTwoInvQ0Dot, r2Dot, mInvR2, q0DotDot);
         final double r3DotDot = MathArrays.linearCombination(inv, q3DotDot, mTwoInvQ0Dot, r3Dot, mInvR3, q0DotDot);
 
         return new double[][] {
             {
                 r1,       r2,       r3
             }, {
                 r1Dot,    r2Dot,    r3Dot
             }, {
                 r1DotDot, r2DotDot, r3DotDot
             }
         };
 
     }
 
     /** Convert a modified Rodrigues vector and derivatives to angular coordinates.
      * @param r modified Rodrigues vector (with first and second times derivatives)
      * @return angular coordinates
      * @see #getModifiedRodrigues(double)
      */
     public static AngularCoordinates createFromModifiedRodrigues(final double[][] r) {
 
         // rotation
         final double rSquared = r[0][0] * r[0][0] + r[0][1] * r[0][1] + r[0][2] * r[0][2];
         final double oPQ0     = 2 / (1 + rSquared);
         final double q0       = oPQ0 - 1;
         final double q1       = oPQ0 * r[0][0];
         final double q2       = oPQ0 * r[0][1];
         final double q3       = oPQ0 * r[0][2];
 
         // rotation rate
         final double oPQ02    = oPQ0 * oPQ0;
         final double q0Dot    = -oPQ02 * MathArrays.linearCombination(r[0][0], r[1][0], r[0][1], r[1][1],  r[0][2], r[1][2]);
         final double q1Dot    = oPQ0 * r[1][0] + r[0][0] * q0Dot;
         final double q2Dot    = oPQ0 * r[1][1] + r[0][1] * q0Dot;
         final double q3Dot    = oPQ0 * r[1][2] + r[0][2] * q0Dot;
         final double oX       = 2 * MathArrays.linearCombination(-q1, q0Dot,  q0, q1Dot,  q3, q2Dot, -q2, q3Dot);
         final double oY       = 2 * MathArrays.linearCombination(-q2, q0Dot, -q3, q1Dot,  q0, q2Dot,  q1, q3Dot);
         final double oZ       = 2 * MathArrays.linearCombination(-q3, q0Dot,  q2, q1Dot, -q1, q2Dot,  q0, q3Dot);
 
         // rotation acceleration
         final double q0DotDot = (1 - q0) / oPQ0 * q0Dot * q0Dot -
                                 oPQ02 * MathArrays.linearCombination(r[0][0], r[2][0], r[0][1], r[2][1], r[0][2], r[2][2]) -
                                 (q1Dot * q1Dot + q2Dot * q2Dot + q3Dot * q3Dot);
         final double q1DotDot = MathArrays.linearCombination(oPQ0, r[2][0], 2 * r[1][0], q0Dot, r[0][0], q0DotDot);
         final double q2DotDot = MathArrays.linearCombination(oPQ0, r[2][1], 2 * r[1][1], q0Dot, r[0][1], q0DotDot);
         final double q3DotDot = MathArrays.linearCombination(oPQ0, r[2][2], 2 * r[1][2], q0Dot, r[0][2], q0DotDot);
         final double oXDot    = 2 * MathArrays.linearCombination(-q1, q0DotDot,  q0, q1DotDot,  q3, q2DotDot, -q2, q3DotDot);
         final double oYDot    = 2 * MathArrays.linearCombination(-q2, q0DotDot, -q3, q1DotDot,  q0, q2DotDot,  q1, q3DotDot);
         final double oZDot    = 2 * MathArrays.linearCombination(-q3, q0DotDot,  q2, q1DotDot, -q1, q2DotDot,  q0, q3DotDot);
 
         return new AngularCoordinates(new Rotation(q0, q1, q2, q3, false),
                                       new Vector3D(oX, oY, oZ),
                                       new Vector3D(oXDot, oYDot, oZDot));
 
     }
 
 }