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
- * 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.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));
- }
- }