TimeStampedFieldAngularCoordinates.java
/* 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
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*/
package org.orekit.utils;
import java.util.Collection;
import org.hipparchus.Field;
import org.hipparchus.RealFieldElement;
import org.hipparchus.analysis.interpolation.FieldHermiteInterpolator;
import org.hipparchus.geometry.euclidean.threed.FieldRotation;
import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
import org.hipparchus.geometry.euclidean.threed.RotationConvention;
import org.hipparchus.util.FastMath;
import org.hipparchus.util.MathArrays;
import org.orekit.errors.OrekitException;
import org.orekit.errors.OrekitInternalError;
import org.orekit.errors.OrekitMessages;
import org.orekit.time.AbsoluteDate;
import org.orekit.time.TimeStamped;
/** {@link TimeStamped time-stamped} version of {@link FieldAngularCoordinates}.
* <p>Instances of this class are guaranteed to be immutable.</p>
* @param <T> the type of the field elements
* @author Luc Maisonobe
* @since 7.0
*/
public class TimeStampedFieldAngularCoordinates<T extends RealFieldElement<T>>
extends FieldAngularCoordinates<T> implements TimeStamped {
/** Serializable UID. */
private static final long serialVersionUID = 20140723L;
/** The date. */
private final AbsoluteDate date;
/** Builds a rotation/rotation rate pair.
* @param date coordinates date
* @param rotation rotation
* @param rotationRate rotation rate Ω (rad/s)
* @param rotationAcceleration rotation acceleration dΩ/dt (rad²/s²)
*/
public TimeStampedFieldAngularCoordinates(final AbsoluteDate date,
final FieldRotation<T> rotation,
final FieldVector3D<T> rotationRate,
final FieldVector3D<T> rotationAcceleration) {
super(rotation, rotationRate, rotationAcceleration);
this.date = date;
}
/** {@inheritDoc} */
public AbsoluteDate getDate() {
return date;
}
/** Revert a rotation/rotation rate pair.
* Build a pair which reverse the effect of another pair.
* @return a new pair whose effect is the reverse of the effect
* of the instance
*/
public TimeStampedFieldAngularCoordinates<T> revert() {
return new TimeStampedFieldAngularCoordinates<T>(date,
getRotation().revert(),
getRotation().applyInverseTo(getRotationRate().negate()),
getRotation().applyInverseTo(getRotationAcceleration().negate()));
}
/** Get a time-shifted state.
* <p>
* The state can be slightly shifted to close dates. This shift is based on
* a simple linear model. It is <em>not</em> intended as a replacement for
* proper attitude propagation but should be sufficient for either small
* time shifts or coarse accuracy.
* </p>
* @param dt time shift in seconds
* @return a new state, shifted with respect to the instance (which is immutable)
*/
public TimeStampedFieldAngularCoordinates<T> shiftedBy(final double dt) {
final FieldAngularCoordinates<T> sac = super.shiftedBy(dt);
return new TimeStampedFieldAngularCoordinates<T>(date.shiftedBy(dt),
sac.getRotation(), sac.getRotationRate(), sac.getRotationAcceleration());
}
/** Add an offset from the instance.
* <p>
* We consider here that the offset rotation is applied first and the
* instance is applied afterward. Note that angular coordinates do <em>not</em>
* commute under this operation, i.e. {@code a.addOffset(b)} and {@code
* b.addOffset(a)} lead to <em>different</em> results in most cases.
* </p>
* <p>
* The two methods {@link #addOffset(FieldAngularCoordinates) addOffset} and
* {@link #subtractOffset(FieldAngularCoordinates) subtractOffset} are designed
* so that round trip applications are possible. This means that both {@code
* ac1.subtractOffset(ac2).addOffset(ac2)} and {@code
* ac1.addOffset(ac2).subtractOffset(ac2)} return angular coordinates equal to ac1.
* </p>
* @param offset offset to subtract
* @return new instance, with offset subtracted
* @see #subtractOffset(FieldAngularCoordinates)
*/
public TimeStampedFieldAngularCoordinates<T> addOffset(final FieldAngularCoordinates<T> offset) {
final FieldVector3D<T> rOmega = getRotation().applyTo(offset.getRotationRate());
final FieldVector3D<T> rOmegaDot = getRotation().applyTo(offset.getRotationAcceleration());
return new TimeStampedFieldAngularCoordinates<T>(date,
getRotation().compose(offset.getRotation(), RotationConvention.VECTOR_OPERATOR),
getRotationRate().add(rOmega),
new FieldVector3D<T>( 1.0, getRotationAcceleration(),
1.0, rOmegaDot,
-1.0, FieldVector3D.crossProduct(getRotationRate(), rOmega)));
}
/** Subtract an offset from the instance.
* <p>
* We consider here that the offset Rotation is applied first and the
* instance is applied afterward. Note that angular coordinates do <em>not</em>
* commute under this operation, i.e. {@code a.subtractOffset(b)} and {@code
* b.subtractOffset(a)} lead to <em>different</em> results in most cases.
* </p>
* <p>
* The two methods {@link #addOffset(FieldAngularCoordinates) addOffset} and
* {@link #subtractOffset(FieldAngularCoordinates) subtractOffset} are designed
* so that round trip applications are possible. This means that both {@code
* ac1.subtractOffset(ac2).addOffset(ac2)} and {@code
* ac1.addOffset(ac2).subtractOffset(ac2)} return angular coordinates equal to ac1.
* </p>
* @param offset offset to subtract
* @return new instance, with offset subtracted
* @see #addOffset(FieldAngularCoordinates)
*/
public TimeStampedFieldAngularCoordinates<T> subtractOffset(final FieldAngularCoordinates<T> offset) {
return addOffset(offset.revert());
}
/** Interpolate angular coordinates.
* <p>
* The interpolated instance is created by polynomial Hermite interpolation
* on Rodrigues vector ensuring rotation rate remains the exact derivative of rotation.
* </p>
* <p>
* This method is based on Sergei Tanygin's paper <a
* href="http://www.agi.com/downloads/resources/white-papers/Attitude-interpolation.pdf">Attitude
* Interpolation</a>, changing the norm of the vector to match the modified Rodrigues
* vector as described in Malcolm D. Shuster's paper <a
* href="http://www.ladispe.polito.it/corsi/Meccatronica/02JHCOR/2011-12/Slides/Shuster_Pub_1993h_J_Repsurv_scan.pdf">A
* Survey of Attitude Representations</a>. This change avoids the singularity at π.
* There is still a singularity at 2π, which is handled by slightly offsetting all rotations
* when this singularity is detected.
* </p>
* <p>
* Note that even if first time derivatives (rotation rates)
* from sample can be ignored, the interpolated instance always includes
* interpolated derivatives. This feature can be used explicitly to
* compute these derivatives when it would be too complex to compute them
* from an analytical formula: just compute a few sample points from the
* explicit formula and set the derivatives to zero in these sample points,
* then use interpolation to add derivatives consistent with the rotations.
* </p>
* @param date interpolation date
* @param filter filter for derivatives from the sample to use in interpolation
* @param sample sample points on which interpolation should be done
* @param <T> the type of the field elements
* @return a new position-velocity, interpolated at specified date
* @exception OrekitException if the number of point is too small for interpolating
*/
public static <T extends RealFieldElement<T>>
TimeStampedFieldAngularCoordinates<T> interpolate(final AbsoluteDate date,
final AngularDerivativesFilter filter,
final Collection<TimeStampedFieldAngularCoordinates<T>> sample)
throws OrekitException {
// get field properties
final Field<T> field = sample.iterator().next().getRotation().getQ0().getField();
final T zero = field.getZero();
final T one = field.getOne();
// set up safety elements for 2π singularity avoidance
final double epsilon = 2 * FastMath.PI / sample.size();
final double threshold = FastMath.min(-(1.0 - 1.0e-4), -FastMath.cos(epsilon / 4));
// set up a linear model canceling mean rotation rate
final FieldVector3D<T> meanRate;
if (filter != AngularDerivativesFilter.USE_R) {
FieldVector3D<T> sum = new FieldVector3D<T>(zero, zero, zero);
for (final TimeStampedFieldAngularCoordinates<T> datedAC : sample) {
sum = sum.add(datedAC.getRotationRate());
}
meanRate = new FieldVector3D<T>(1.0 / sample.size(), sum);
} else {
if (sample.size() < 2) {
throw new OrekitException(OrekitMessages.NOT_ENOUGH_DATA_FOR_INTERPOLATION,
sample.size());
}
FieldVector3D<T> sum = new FieldVector3D<T>(zero, zero, zero);
TimeStampedFieldAngularCoordinates<T> previous = null;
for (final TimeStampedFieldAngularCoordinates<T> datedAC : sample) {
if (previous != null) {
sum = sum.add(estimateRate(previous.getRotation(), datedAC.getRotation(),
datedAC.date.durationFrom(previous.getDate())));
}
previous = datedAC;
}
meanRate = new FieldVector3D<T>(1.0 / (sample.size() - 1), sum);
}
TimeStampedFieldAngularCoordinates<T> offset =
new TimeStampedFieldAngularCoordinates<T>(date, new FieldRotation<T>(one, zero, zero, zero, false),
meanRate, new FieldVector3D<T>(zero, zero, zero));
boolean restart = true;
for (int i = 0; restart && i < sample.size() + 2; ++i) {
// offset adaptation parameters
restart = false;
// set up an interpolator taking derivatives into account
final FieldHermiteInterpolator<T> interpolator = new FieldHermiteInterpolator<T>();
// add sample points
final double[] previous = new double[] {
1.0, 0.0, 0.0, 0.0
};
for (final TimeStampedFieldAngularCoordinates<T> ac : sample) {
// remove linear offset from the current coordinates
final T dt = zero.add(ac.date.durationFrom(date));
final TimeStampedFieldAngularCoordinates<T> fixed = ac.subtractOffset(offset.shiftedBy(dt.getReal()));
final T[][] rodrigues = getModifiedRodrigues(fixed, previous, threshold);
if (rodrigues == null) {
// the sample point is close to a modified Rodrigues vector singularity
// we need to change the linear offset model to avoid this
restart = true;
break;
}
switch (filter) {
case USE_RRA:
// populate sample with rotation, rotation rate and acceleration data
interpolator.addSamplePoint(dt, rodrigues[0], rodrigues[1], rodrigues[2]);
break;
case USE_RR:
// populate sample with rotation and rotation rate data
interpolator.addSamplePoint(dt, rodrigues[0], rodrigues[1]);
break;
case USE_R:
// populate sample with rotation data only
interpolator.addSamplePoint(dt, rodrigues[0]);
break;
default :
// this should never happen
throw new OrekitInternalError(null);
}
}
if (restart) {
// interpolation failed, some intermediate rotation was too close to 2π
// we need to offset all rotations to avoid the singularity
offset = offset.addOffset(new FieldAngularCoordinates<T>(new FieldRotation<T>(new FieldVector3D<T>(one, zero, zero),
zero.add(epsilon),
RotationConvention.VECTOR_OPERATOR),
new FieldVector3D<T>(zero, zero, zero),
new FieldVector3D<T>(zero, zero, zero)));
} else {
// interpolation succeeded with the current offset
final T[][] p = interpolator.derivatives(field.getZero(), 2);
return createFromModifiedRodrigues(p, offset);
}
}
// this should never happen
throw new OrekitInternalError(null);
}
/** Create a 6 elements array.
* @param field field to which coordinates belong
* @param a0 first element
* @param a1 second element
* @param a2 third element
* @param a3 fourth element
* @param a4 fifth element
* @param a5 sixth element
* @param <T> the type of the field elements
* @return array containing a0, a1, a2, a3, a4, a5
*/
private static <T extends RealFieldElement<T>> T[] array6(final Field<T> field,
final T a0, final T a1, final T a2,
final T a3, final T a4, final T a5) {
final T[] array = MathArrays.buildArray(field, 6);
array[0] = a0;
array[1] = a1;
array[2] = a2;
array[3] = a3;
array[4] = a4;
array[5] = a5;
return array;
}
/** Create a 3x3 matrix.
* @param field field to which coordinates belong
* @param a00 first element, first row
* @param a01 second element, first row
* @param a02 third element, first row
* @param a10 first element, second row
* @param a11 second element, second row
* @param a12 third element, second row
* @param a20 first element, third row
* @param a21 second element, third row
* @param a22 third element, third row
* @param <T> the type of the field elements
* @return array containing a0, a1, a2
*/
private static <T extends RealFieldElement<T>> T[][] matrix33(final Field<T> field,
final T a00, final T a01, final T a02,
final T a10, final T a11, final T a12,
final T a20, final T a21, final T a22) {
final T[][] matrix = MathArrays.buildArray(field, 3, 3);
matrix[0][0] = a00;
matrix[0][1] = a01;
matrix[0][2] = a02;
matrix[1][0] = a10;
matrix[1][1] = a11;
matrix[1][2] = a12;
matrix[2][0] = a20;
matrix[2][1] = a21;
matrix[2][2] = a22;
return matrix;
}
/** 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 fixed coordinates to convert, with offset already fixed
* @param previous previous quaternion used
* @param threshold threshold for rotations too close to 2π
* @param <T> the type of the field elements
* @return modified Rodrigues vector and derivative, or null if rotation is too close to 2π
*/
private static <T extends RealFieldElement<T>> T[][] getModifiedRodrigues(final TimeStampedFieldAngularCoordinates<T> fixed,
final double[] previous, final double threshold) {
// make sure all interpolated points will be on the same branch
T q0 = fixed.getRotation().getQ0();
T q1 = fixed.getRotation().getQ1();
T q2 = fixed.getRotation().getQ2();
T q3 = fixed.getRotation().getQ3();
if (MathArrays.linearCombination(q0.getReal(), previous[0],
q1.getReal(), previous[1],
q2.getReal(), previous[2],
q3.getReal(), previous[3]) < 0) {
q0 = q0.negate();
q1 = q1.negate();
q2 = q2.negate();
q3 = q3.negate();
}
previous[0] = q0.getReal();
previous[1] = q1.getReal();
previous[2] = q2.getReal();
previous[3] = q3.getReal();
// check modified Rodrigues vector singularity
if (q0.getReal() < threshold) {
// this is an intermediate point that happens to be 2PI away from reference
// we need to change the linear offset model to avoid this point
return null;
}
final Field<T> field = q0.getField();
final T oX = fixed.getRotationRate().getX();
final T oY = fixed.getRotationRate().getY();
final T oZ = fixed.getRotationRate().getZ();
final T oXDot = fixed.getRotationAcceleration().getX();
final T oYDot = fixed.getRotationAcceleration().getY();
final T oZDot = fixed.getRotationAcceleration().getZ();
// first time-derivatives of the quaternion
final T q0Dot = q0.linearCombination(q1.negate(), oX, q2.negate(), oY, q3.negate(), oZ).multiply(0.5);
final T q1Dot = q1.linearCombination(q0, oX, q3.negate(), oY, q2, oZ).multiply(0.5);
final T q2Dot = q2.linearCombination(q3, oX, q0, oY, q1.negate(), oZ).multiply(0.5);
final T q3Dot = q3.linearCombination(q2.negate(), oX, q1, oY, q0, oZ).multiply(0.5);
// second time-derivatives of the quaternion
final T q0DotDot = q0.linearCombination(array6(field, q1, q2, q3, q1Dot, q2Dot, q3Dot),
array6(field, oXDot, oYDot, oZDot, oX, oY, oZ)).multiply(-0.5);
final T q1DotDot = q1.linearCombination(array6(field, q0, q2, q3.negate(), q0Dot, q2Dot, q3Dot.negate()),
array6(field, oXDot, oZDot, oYDot, oX, oZ, oY)).multiply(0.5);
final T q2DotDot = q2.linearCombination(array6(field, q0, q3, q1.negate(), q0Dot, q3Dot, q1Dot.negate()),
array6(field, oYDot, oXDot, oZDot, oY, oX, oZ)).multiply(0.5);
final T q3DotDot = q3.linearCombination(array6(field, q0, q1, q2.negate(), q0Dot, q1Dot, q2Dot.negate()),
array6(field, oZDot, oYDot, oXDot, oZ, oY, oX)).multiply(0.5);
// the modified Rodrigues is tan(θ/4) u where θ and u are the rotation angle and axis respectively
// this can be rewritten using quaternion components:
// r (q₁ / (1+q₀), q₂ / (1+q₀), q₃ / (1+q₀))
// applying the derivation chain rule to previous expression gives rDot and rDotDot
final T inv = q0.add(1.0).reciprocal();
final T mTwoInvQ0Dot = inv.multiply(q0Dot).multiply(-2);
final T r1 = inv.multiply(q1);
final T r2 = inv.multiply(q2);
final T r3 = inv.multiply(q3);
final T mInvR1 = inv.multiply(r1).negate();
final T mInvR2 = inv.multiply(r2).negate();
final T mInvR3 = inv.multiply(r3).negate();
final T r1Dot = r1.linearCombination(inv, q1Dot, mInvR1, q0Dot);
final T r2Dot = r2.linearCombination(inv, q2Dot, mInvR2, q0Dot);
final T r3Dot = r3.linearCombination(inv, q3Dot, mInvR3, q0Dot);
final T r1DotDot = r1.linearCombination(inv, q1DotDot, mTwoInvQ0Dot, r1Dot, mInvR1, q0DotDot);
final T r2DotDot = r2.linearCombination(inv, q2DotDot, mTwoInvQ0Dot, r2Dot, mInvR2, q0DotDot);
final T r3DotDot = r3.linearCombination(inv, q3DotDot, mTwoInvQ0Dot, r3Dot, mInvR3, q0DotDot);
return matrix33(field,
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 derivatives)
* @param offset linear offset model to add (its date must be consistent with the modified Rodrigues vector)
* @param <T> the type of the field elements
* @return angular coordinates
*/
private static <T extends RealFieldElement<T>>
TimeStampedFieldAngularCoordinates<T> createFromModifiedRodrigues(final T[][] r,
final TimeStampedFieldAngularCoordinates<T> offset) {
// rotation
final T rSquared = r[0][0].multiply(r[0][0]).add(r[0][1].multiply(r[0][1])).add(r[0][2].multiply(r[0][2]));
final T oPQ0 = rSquared.add(1).reciprocal().multiply(2);
final T q0 = oPQ0.subtract(1);
final T q1 = oPQ0.multiply(r[0][0]);
final T q2 = oPQ0.multiply(r[0][1]);
final T q3 = oPQ0.multiply(r[0][2]);
// rotation rate
final T oPQ02 = oPQ0.multiply(oPQ0);
final T q0Dot = oPQ02.negate().multiply(q0.linearCombination(r[0][0], r[1][0], r[0][1], r[1][1], r[0][2], r[1][2]));
final T q1Dot = oPQ0.multiply(r[1][0]).add(r[0][0].multiply(q0Dot));
final T q2Dot = oPQ0.multiply(r[1][1]).add(r[0][1].multiply(q0Dot));
final T q3Dot = oPQ0.multiply(r[1][2]).add(r[0][2].multiply(q0Dot));
final T oX = q1.linearCombination(q1.negate(), q0Dot, q0, q1Dot, q3, q2Dot, q2.negate(), q3Dot).multiply(2);
final T oY = q2.linearCombination(q2.negate(), q0Dot, q3.negate(), q1Dot, q0, q2Dot, q1, q3Dot).multiply(2);
final T oZ = q3.linearCombination(q3.negate(), q0Dot, q2, q1Dot, q1.negate(), q2Dot, q0, q3Dot).multiply(2);
// rotation acceleration
final T q0DotDot = q0.getField().getOne().subtract(q0).divide(oPQ0).multiply(q0Dot).multiply(q0Dot).
subtract(oPQ02.multiply(q0.linearCombination(r[0][0], r[2][0], r[0][1], r[2][1], r[0][2], r[2][2]))).
subtract(q1Dot.multiply(q1Dot).add(q2Dot.multiply(q2Dot)).add(q3Dot.multiply(q3Dot)));
final T q1DotDot = q1.linearCombination(oPQ0, r[2][0], r[1][0].multiply(2), q0Dot, r[0][0], q0DotDot);
final T q2DotDot = q2.linearCombination(oPQ0, r[2][1], r[1][1].multiply(2), q0Dot, r[0][1], q0DotDot);
final T q3DotDot = q3.linearCombination(oPQ0, r[2][2], r[1][2].multiply(2), q0Dot, r[0][2], q0DotDot);
final T oXDot = q1.linearCombination(q1.negate(), q0DotDot, q0, q1DotDot, q3, q2DotDot, q2.negate(), q3DotDot).multiply(2);
final T oYDot = q2.linearCombination(q2.negate(), q0DotDot, q3.negate(), q1DotDot, q0, q2DotDot, q1, q3DotDot).multiply(2);
final T oZDot = q3.linearCombination(q3.negate(), q0DotDot, q2, q1DotDot, q1.negate(), q2DotDot, q0, q3DotDot).multiply(2);
return new TimeStampedFieldAngularCoordinates<T>(offset.getDate(),
new FieldRotation<T>(q0, q1, q2, q3, false),
new FieldVector3D<T>(oX, oY, oZ),
new FieldVector3D<T>(oXDot, oYDot, oZDot)).addOffset(offset);
}
}