AngularCoordinates.java
/* Copyright 2002-2022 CS GROUP
* Licensed to CS GROUP (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
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package org.orekit.utils;
import java.io.Serializable;
import org.hipparchus.CalculusFieldElement;
import org.hipparchus.analysis.differentiation.DSFactory;
import org.hipparchus.analysis.differentiation.Derivative;
import org.hipparchus.analysis.differentiation.DerivativeStructure;
import org.hipparchus.analysis.differentiation.UnivariateDerivative1;
import org.hipparchus.analysis.differentiation.UnivariateDerivative2;
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.FieldVector3D;
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 (rad/s²)
*/
public AngularCoordinates(final Rotation rotation,
final Vector3D rotationRate, final Vector3D rotationAcceleration) {
this.rotation = rotation;
this.rotationRate = rotationRate;
this.rotationAcceleration = rotationAcceleration;
}
/**
* Builds angular coordinates with the given rotation, zero angular
* velocity, and zero angular acceleration.
*
* @param rotation rotation
*/
public AngularCoordinates(final Rotation rotation) {
this(rotation, Vector3D.ZERO);
}
/** 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
*/
public AngularCoordinates(final PVCoordinates u1, final PVCoordinates u2,
final PVCoordinates v1, final PVCoordinates v2,
final double tolerance) {
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
*/
public AngularCoordinates(final PVCoordinates u, final PVCoordinates v) {
this(new FieldRotation<>(u.toDerivativeStructureVector(2),
v.toDerivativeStructureVector(2)));
}
/** Builds a AngularCoordinates from a {@link FieldRotation}<{@link Derivative}>.
* <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
* @param <U> type of the derivative
*/
public <U extends Derivative<U>> AngularCoordinates(final FieldRotation<U> 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}<{@link DerivativeStructure}>.
* <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
*/
public FieldRotation<DerivativeStructure> toDerivativeStructureRotation(final int order) {
// 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 DSFactory factory;
final DerivativeStructure q0DS;
final DerivativeStructure q1DS;
final DerivativeStructure q2DS;
final DerivativeStructure q3DS;
switch (order) {
case 0 :
factory = new DSFactory(1, order);
q0DS = factory.build(q0);
q1DS = factory.build(q1);
q2DS = factory.build(q2);
q3DS = factory.build(q3);
break;
case 1 :
factory = new DSFactory(1, order);
q0DS = factory.build(q0, q0Dot);
q1DS = factory.build(q1, q1Dot);
q2DS = factory.build(q2, q2Dot);
q3DS = factory.build(q3, q3Dot);
break;
case 2 :
factory = new DSFactory(1, order);
q0DS = factory.build(q0, q0Dot, q0DotDot);
q1DS = factory.build(q1, q1Dot, q1DotDot);
q2DS = factory.build(q2, q2Dot, q2DotDot);
q3DS = factory.build(q3, q3Dot, q3DotDot);
break;
default :
throw new OrekitException(OrekitMessages.OUT_OF_RANGE_DERIVATION_ORDER, order);
}
return new FieldRotation<>(q0DS, q1DS, q2DS, q3DS, false);
}
/** Transform the instance to a {@link FieldRotation}<{@link UnivariateDerivative1}>.
* <p>
* The {@link UnivariateDerivative1} coordinates correspond to time-derivatives up
* to the order 1.
* </p>
* @return rotation with time-derivatives embedded within the coordinates
*/
public FieldRotation<UnivariateDerivative1> toUnivariateDerivative1Rotation() {
// 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);
final UnivariateDerivative1 q0UD = new UnivariateDerivative1(q0, q0Dot);
final UnivariateDerivative1 q1UD = new UnivariateDerivative1(q1, q1Dot);
final UnivariateDerivative1 q2UD = new UnivariateDerivative1(q2, q2Dot);
final UnivariateDerivative1 q3UD = new UnivariateDerivative1(q3, q3Dot);
return new FieldRotation<>(q0UD, q1UD, q2UD, q3UD, false);
}
/** Transform the instance to a {@link FieldRotation}<{@link UnivariateDerivative2}>.
* <p>
* The {@link UnivariateDerivative2} coordinates correspond to time-derivatives up
* to the order 2.
* </p>
* @return rotation with time-derivatives embedded within the coordinates
*/
public FieldRotation<UnivariateDerivative2> toUnivariateDerivative2Rotation() {
// 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 UnivariateDerivative2 q0UD = new UnivariateDerivative2(q0, q0Dot, q0DotDot);
final UnivariateDerivative2 q1UD = new UnivariateDerivative2(q1, q1Dot, q1DotDot);
final UnivariateDerivative2 q2UD = new UnivariateDerivative2(q2, q2Dot, q2DotDot);
final UnivariateDerivative2 q3UD = new UnivariateDerivative2(q3, q3Dot, q3DotDot);
return new FieldRotation<>(q0UD, q1UD, q2UD, q3UD, 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 rotation. Same as {@link #shiftedBy(double)} except
* only the shifted rotation is computed.
* <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)
* @see #shiftedBy(double)
*/
public Rotation rotationShiftedBy(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 Rotation linearPart =
rateContribution.compose(rotation, RotationConvention.VECTOR_OPERATOR);
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 Rotation quadraticContribution =
new Rotation(rotationAcceleration,
0.5 * acc * dt * dt,
RotationConvention.FRAME_TRANSFORM);
// 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
.compose(linearPart, RotationConvention.VECTOR_OPERATOR);
}
/** 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 (rad/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);
}
/** Apply the rotation to a pv coordinates.
* @param pv vector to apply the rotation to
* @param <T> type of the field elements
* @return a new pv coordinates which is the image of u by the rotation
* @since 9.0
*/
public <T extends CalculusFieldElement<T>> FieldPVCoordinates<T> applyTo(final FieldPVCoordinates<T> pv) {
final FieldVector3D<T> transformedP = FieldRotation.applyTo(rotation, pv.getPosition());
final FieldVector3D<T> crossP = FieldVector3D.crossProduct(rotationRate, transformedP);
final FieldVector3D<T> transformedV = FieldRotation.applyTo(rotation, pv.getVelocity()).subtract(crossP);
final FieldVector3D<T> crossV = FieldVector3D.crossProduct(rotationRate, transformedV);
final FieldVector3D<T> crossCrossP = FieldVector3D.crossProduct(rotationRate, crossP);
final FieldVector3D<T> crossDotP = FieldVector3D.crossProduct(rotationAcceleration, transformedP);
final FieldVector3D<T> transformedA = new FieldVector3D<>( 1, FieldRotation.applyTo(rotation, pv.getAcceleration()),
-2, crossV,
-1, crossCrossP,
-1, crossDotP);
return new FieldPVCoordinates<>(transformedP, transformedV, transformedA);
}
/** Apply the rotation to a pv coordinates.
* @param pv vector to apply the rotation to
* @param <T> type of the field elements
* @return a new pv coordinates which is the image of u by the rotation
* @since 9.0
*/
public <T extends CalculusFieldElement<T>> TimeStampedFieldPVCoordinates<T> applyTo(final TimeStampedFieldPVCoordinates<T> pv) {
final FieldVector3D<T> transformedP = FieldRotation.applyTo(rotation, pv.getPosition());
final FieldVector3D<T> crossP = FieldVector3D.crossProduct(rotationRate, transformedP);
final FieldVector3D<T> transformedV = FieldRotation.applyTo(rotation, pv.getVelocity()).subtract(crossP);
final FieldVector3D<T> crossV = FieldVector3D.crossProduct(rotationRate, transformedV);
final FieldVector3D<T> crossCrossP = FieldVector3D.crossProduct(rotationRate, crossP);
final FieldVector3D<T> crossDotP = FieldVector3D.crossProduct(rotationAcceleration, transformedP);
final FieldVector3D<T> transformedA = new FieldVector3D<>( 1, FieldRotation.applyTo(rotation, pv.getAcceleration()),
-2, crossV,
-1, crossCrossP,
-1, crossDotP);
return new TimeStampedFieldPVCoordinates<>(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));
}
}