FieldAngularCoordinates.java
/* Copyright 2002-2024 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
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* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
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package org.orekit.utils;
import org.hipparchus.Field;
import org.hipparchus.CalculusFieldElement;
import org.hipparchus.analysis.differentiation.FDSFactory;
import org.hipparchus.analysis.differentiation.FieldDerivative;
import org.hipparchus.analysis.differentiation.FieldDerivativeStructure;
import org.hipparchus.analysis.differentiation.FieldUnivariateDerivative1;
import org.hipparchus.analysis.differentiation.FieldUnivariateDerivative2;
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.geometry.euclidean.threed.FieldRotation;
import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
import org.hipparchus.geometry.euclidean.threed.RotationConvention;
import org.hipparchus.linear.FieldDecompositionSolver;
import org.hipparchus.linear.FieldMatrix;
import org.hipparchus.linear.FieldQRDecomposition;
import org.hipparchus.linear.FieldVector;
import org.hipparchus.linear.MatrixUtils;
import org.hipparchus.util.MathArrays;
import org.orekit.errors.OrekitException;
import org.orekit.errors.OrekitMessages;
import org.orekit.time.FieldTimeShiftable;
/** Simple container for rotation / rotation rate pairs, using {@link
* CalculusFieldElement}.
* <p>
* The state can be slightly shifted to close dates. This shift is based on
* a simple quadratic 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>
* <p>
* This class is the angular counterpart to {@link FieldPVCoordinates}.
* </p>
* <p>Instances of this class are guaranteed to be immutable.</p>
* @param <T> the type of the field elements
* @author Luc Maisonobe
* @since 6.0
* @see AngularCoordinates
*/
public class FieldAngularCoordinates<T extends CalculusFieldElement<T>>
implements FieldTimeShiftable<FieldAngularCoordinates<T>, T> {
/** rotation. */
private final FieldRotation<T> rotation;
/** rotation rate. */
private final FieldVector3D<T> rotationRate;
/** rotation acceleration. */
private final FieldVector3D<T> rotationAcceleration;
/** Builds a rotation/rotation rate pair.
* @param rotation rotation
* @param rotationRate rotation rate Ω (rad/s)
*/
public FieldAngularCoordinates(final FieldRotation<T> rotation,
final FieldVector3D<T> rotationRate) {
this(rotation, rotationRate,
new FieldVector3D<>(rotation.getQ0().getField().getZero(),
rotation.getQ0().getField().getZero(),
rotation.getQ0().getField().getZero()));
}
/** Builds a rotation / rotation rate / rotation acceleration triplet.
* @param rotation i.e. the orientation of the vehicle
* @param rotationRate rotation rate Ω, i.e. the spin vector (rad/s)
* @param rotationAcceleration angular acceleration vector dΩ/dt (rad/s²)
*/
public FieldAngularCoordinates(final FieldRotation<T> rotation,
final FieldVector3D<T> rotationRate,
final FieldVector3D<T> rotationAcceleration) {
this.rotation = rotation;
this.rotationRate = rotationRate;
this.rotationAcceleration = rotationAcceleration;
}
/** Build the rotation that transforms a pair of pv coordinates into another one.
* <p><em>WARNING</em>! This method requires much more stringent assumptions on
* its parameters than the similar {@link FieldRotation#FieldRotation(FieldVector3D, FieldVector3D,
* FieldVector3D, FieldVector3D) constructor} from the {@link FieldRotation FieldRotation} class.
* As far as the FieldRotation constructor is concerned, the {@code v₂} vector from
* the second pair can be slightly misaligned. The FieldRotation 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 FieldAngularCoordinates(final FieldPVCoordinates<T> u1, final FieldPVCoordinates<T> u2,
final FieldPVCoordinates<T> v1, final FieldPVCoordinates<T> v2,
final double tolerance) {
try {
// find the initial fixed rotation
rotation = new FieldRotation<>(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 FieldVector3D<T> ru1Dot = rotation.applyTo(u1.getVelocity());
final FieldVector3D<T> 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 FieldVector3D<T> ru1DotDot = rotation.applyTo(u1.getAcceleration());
final FieldVector3D<T> oDotv1 = FieldVector3D.crossProduct(rotationRate, v1.getVelocity());
final FieldVector3D<T> oov1 = FieldVector3D.crossProduct(rotationRate, rotationRate.crossProduct(v1.getPosition()));
final FieldVector3D<T> c1 = new FieldVector3D<>(1, ru1DotDot, -2, oDotv1, -1, oov1, -1, v1.getAcceleration());
final FieldVector3D<T> ru2DotDot = rotation.applyTo(u2.getAcceleration());
final FieldVector3D<T> oDotv2 = FieldVector3D.crossProduct(rotationRate, v2.getVelocity());
final FieldVector3D<T> oov2 = FieldVector3D.crossProduct(rotationRate, rotationRate.crossProduct( v2.getPosition()));
final FieldVector3D<T> c2 = new FieldVector3D<>(1, ru2DotDot, -2, oDotv2, -1, oov2, -1, v2.getAcceleration());
rotationAcceleration = inverseCrossProducts(v1.getPosition(), c1, v2.getPosition(), c2, tolerance);
} catch (MathIllegalArgumentException miae) {
throw new OrekitException(miae);
}
}
/** Builds a FieldAngularCoordinates from a field and a regular AngularCoordinates.
* @param field field for the components
* @param ang AngularCoordinates to convert
*/
public FieldAngularCoordinates(final Field<T> field, final AngularCoordinates ang) {
this.rotation = new FieldRotation<>(field, ang.getRotation());
this.rotationRate = new FieldVector3D<>(field, ang.getRotationRate());
this.rotationAcceleration = new FieldVector3D<>(field, ang.getRotationAcceleration());
}
/** Builds a FieldAngularCoordinates from a {@link FieldRotation}<{@link FieldDerivativeStructure}>.
* <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
* @since 9.2
*/
public <U extends FieldDerivative<T, U>> FieldAngularCoordinates(final FieldRotation<U> r) {
final T q0 = r.getQ0().getValue();
final T q1 = r.getQ1().getValue();
final T q2 = r.getQ2().getValue();
final T q3 = r.getQ3().getValue();
rotation = new FieldRotation<>(q0, q1, q2, q3, false);
if (r.getQ0().getOrder() >= 1) {
final T q0Dot = r.getQ0().getPartialDerivative(1);
final T q1Dot = r.getQ1().getPartialDerivative(1);
final T q2Dot = r.getQ2().getPartialDerivative(1);
final T q3Dot = r.getQ3().getPartialDerivative(1);
rotationRate =
new FieldVector3D<>(q0.linearCombination(q1.negate(), q0Dot, q0, q1Dot,
q3, q2Dot, q2.negate(), q3Dot).multiply(2),
q0.linearCombination(q2.negate(), q0Dot, q3.negate(), q1Dot,
q0, q2Dot, q1, q3Dot).multiply(2),
q0.linearCombination(q3.negate(), q0Dot, q2, q1Dot,
q1.negate(), q2Dot, q0, q3Dot).multiply(2));
if (r.getQ0().getOrder() >= 2) {
final T q0DotDot = r.getQ0().getPartialDerivative(2);
final T q1DotDot = r.getQ1().getPartialDerivative(2);
final T q2DotDot = r.getQ2().getPartialDerivative(2);
final T q3DotDot = r.getQ3().getPartialDerivative(2);
rotationAcceleration =
new FieldVector3D<>(q0.linearCombination(q1.negate(), q0DotDot, q0, q1DotDot,
q3, q2DotDot, q2.negate(), q3DotDot).multiply(2),
q0.linearCombination(q2.negate(), q0DotDot, q3.negate(), q1DotDot,
q0, q2DotDot, q1, q3DotDot).multiply(2),
q0.linearCombination(q3.negate(), q0DotDot, q2, q1DotDot,
q1.negate(), q2DotDot, q0, q3DotDot).multiply(2));
} else {
rotationAcceleration = FieldVector3D.getZero(q0.getField());
}
} else {
rotationRate = FieldVector3D.getZero(q0.getField());
rotationAcceleration = FieldVector3D.getZero(q0.getField());
}
}
/** Fixed orientation parallel with reference frame
* (identity rotation, zero rotation rate and acceleration).
* @param field field for the components
* @param <T> the type of the field elements
* @return a new fixed orientation parallel with reference frame
*/
public static <T extends CalculusFieldElement<T>> FieldAngularCoordinates<T> getIdentity(final Field<T> field) {
return new FieldAngularCoordinates<>(field, AngularCoordinates.IDENTITY);
}
/** 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
* @param <T> the type of the field elements
* @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 <T extends CalculusFieldElement<T>> FieldVector3D<T> inverseCrossProducts(final FieldVector3D<T> v1, final FieldVector3D<T> c1,
final FieldVector3D<T> v2, final FieldVector3D<T> c2,
final double tolerance)
throws MathIllegalArgumentException {
final T v12 = v1.getNormSq();
final T v1n = v12.sqrt();
final T v22 = v2.getNormSq();
final T v2n = v22.sqrt();
final T threshold;
if (v1n.getReal() >= v2n.getReal()) {
threshold = v1n.multiply(tolerance);
}
else {
threshold = v2n.multiply(tolerance);
}
FieldVector3D<T> omega = null;
try {
// create the over-determined linear system representing the two cross products
final FieldMatrix<T> m = MatrixUtils.createFieldMatrix(v12.getField(), 6, 3);
m.setEntry(0, 1, v1.getZ());
m.setEntry(0, 2, v1.getY().negate());
m.setEntry(1, 0, v1.getZ().negate());
m.setEntry(1, 2, v1.getX());
m.setEntry(2, 0, v1.getY());
m.setEntry(2, 1, v1.getX().negate());
m.setEntry(3, 1, v2.getZ());
m.setEntry(3, 2, v2.getY().negate());
m.setEntry(4, 0, v2.getZ().negate());
m.setEntry(4, 2, v2.getX());
m.setEntry(5, 0, v2.getY());
m.setEntry(5, 1, v2.getX().negate());
final T[] kk = MathArrays.buildArray(v2n.getField(), 6);
kk[0] = c1.getX();
kk[1] = c1.getY();
kk[2] = c1.getZ();
kk[3] = c2.getX();
kk[4] = c2.getY();
kk[5] = c2.getZ();
final FieldVector<T> rhs = MatrixUtils.createFieldVector(kk);
// find the best solution we can
final FieldDecompositionSolver<T> solver = new FieldQRDecomposition<>(m).getSolver();
final FieldVector<T> v = solver.solve(rhs);
omega = new FieldVector3D<>(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 T c12 = c1.getNormSq();
final T c1n = c12.sqrt();
final T c22 = c2.getNormSq();
final T c2n = c22.sqrt();
if (c1n.getReal() <= threshold.getReal() && c2n.getReal() <= threshold.getReal()) {
// simple special case, velocities are cancelled
return new FieldVector3D<>(v12.getField().getZero(), v12.getField().getZero(), v12.getField().getZero());
} else if (v1n.getReal() <= threshold.getReal() && c1n.getReal() >= threshold.getReal()) {
// this is inconsistent, if v₁ is zero, c₁ must be 0 too
throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE,
c1n.getReal(), 0, true);
} else if (v2n.getReal() <= threshold.getReal() && c2n.getReal() >= threshold.getReal()) {
// this is inconsistent, if v₂ is zero, c₂ must be 0 too
throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE,
c2n.getReal(), 0, true);
} else if (v1.crossProduct(v1).getNorm().getReal() <= threshold.getReal() && v12.getReal() > threshold.getReal()) {
// simple special case, v₂ is redundant with v₁, we just ignore it
// use the simplest Ω: orthogonal to both v₁ and c₁
omega = new FieldVector3D<>(v12.reciprocal(), v1.crossProduct(c1));
} else {
throw miae;
}
} else {
throw miae;
}
}
// check results
final T d1 = FieldVector3D.distance(omega.crossProduct(v1), c1);
if (d1.getReal() > threshold.getReal()) {
throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, 0, true);
}
final T d2 = FieldVector3D.distance(omega.crossProduct(v2), c2);
if (d2.getReal() > threshold.getReal()) {
throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, 0, true);
}
return omega;
}
/** Transform the instance to a {@link FieldRotation}<{@link FieldDerivativeStructure}>.
* <p>
* The {@link FieldDerivativeStructure} 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
* @since 9.2
*/
public FieldRotation<FieldDerivativeStructure<T>> toDerivativeStructureRotation(final int order) {
// quaternion components
final T q0 = rotation.getQ0();
final T q1 = rotation.getQ1();
final T q2 = rotation.getQ2();
final T q3 = rotation.getQ3();
// first time-derivatives of the quaternion
final T oX = rotationRate.getX();
final T oY = rotationRate.getY();
final T oZ = rotationRate.getZ();
final T q0Dot = q0.linearCombination(q1.negate(), oX, q2.negate(), oY, q3.negate(), oZ).multiply(0.5);
final T q1Dot = q0.linearCombination(q0, oX, q3.negate(), oY, q2, oZ).multiply(0.5);
final T q2Dot = q0.linearCombination(q3, oX, q0, oY, q1.negate(), oZ).multiply(0.5);
final T q3Dot = q0.linearCombination(q2.negate(), oX, q1, oY, q0, oZ).multiply(0.5);
// second time-derivatives of the quaternion
final T oXDot = rotationAcceleration.getX();
final T oYDot = rotationAcceleration.getY();
final T oZDot = rotationAcceleration.getZ();
final T q0DotDot = q0.linearCombination(array6(q1, q2, q3, q1Dot, q2Dot, q3Dot),
array6(oXDot, oYDot, oZDot, oX, oY, oZ)).multiply(-0.5);
final T q1DotDot = q0.linearCombination(array6(q0, q2, q3.negate(), q0Dot, q2Dot, q3Dot.negate()),
array6(oXDot, oZDot, oYDot, oX, oZ, oY)).multiply(0.5);
final T q2DotDot = q0.linearCombination(array6(q0, q3, q1.negate(), q0Dot, q3Dot, q1Dot.negate()),
array6(oYDot, oXDot, oZDot, oY, oX, oZ)).multiply(0.5);
final T q3DotDot = q0.linearCombination(array6(q0, q1, q2.negate(), q0Dot, q1Dot, q2Dot.negate()),
array6(oZDot, oYDot, oXDot, oZ, oY, oX)).multiply(0.5);
final FDSFactory<T> factory;
final FieldDerivativeStructure<T> q0DS;
final FieldDerivativeStructure<T> q1DS;
final FieldDerivativeStructure<T> q2DS;
final FieldDerivativeStructure<T> q3DS;
switch (order) {
case 0 :
factory = new FDSFactory<>(q0.getField(), 1, order);
q0DS = factory.build(q0);
q1DS = factory.build(q1);
q2DS = factory.build(q2);
q3DS = factory.build(q3);
break;
case 1 :
factory = new FDSFactory<>(q0.getField(), 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 FDSFactory<>(q0.getField(), 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<FieldUnivariateDerivative1<T>> toUnivariateDerivative1Rotation() {
// quaternion components
final T q0 = rotation.getQ0();
final T q1 = rotation.getQ1();
final T q2 = rotation.getQ2();
final T q3 = rotation.getQ3();
// first time-derivatives of the quaternion
final T oX = rotationRate.getX();
final T oY = rotationRate.getY();
final T oZ = rotationRate.getZ();
final T q0Dot = q0.linearCombination(q1.negate(), oX, q2.negate(), oY, q3.negate(), oZ).multiply(0.5);
final T q1Dot = q0.linearCombination(q0, oX, q3.negate(), oY, q2, oZ).multiply(0.5);
final T q2Dot = q0.linearCombination(q3, oX, q0, oY, q1.negate(), oZ).multiply(0.5);
final T q3Dot = q0.linearCombination(q2.negate(), oX, q1, oY, q0, oZ).multiply(0.5);
final FieldUnivariateDerivative1<T> q0UD = new FieldUnivariateDerivative1<>(q0, q0Dot);
final FieldUnivariateDerivative1<T> q1UD = new FieldUnivariateDerivative1<>(q1, q1Dot);
final FieldUnivariateDerivative1<T> q2UD = new FieldUnivariateDerivative1<>(q2, q2Dot);
final FieldUnivariateDerivative1<T> q3UD = new FieldUnivariateDerivative1<>(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<FieldUnivariateDerivative2<T>> toUnivariateDerivative2Rotation() {
// quaternion components
final T q0 = rotation.getQ0();
final T q1 = rotation.getQ1();
final T q2 = rotation.getQ2();
final T q3 = rotation.getQ3();
// first time-derivatives of the quaternion
final T oX = rotationRate.getX();
final T oY = rotationRate.getY();
final T oZ = rotationRate.getZ();
final T q0Dot = q0.linearCombination(q1.negate(), oX, q2.negate(), oY, q3.negate(), oZ).multiply(0.5);
final T q1Dot = q0.linearCombination(q0, oX, q3.negate(), oY, q2, oZ).multiply(0.5);
final T q2Dot = q0.linearCombination(q3, oX, q0, oY, q1.negate(), oZ).multiply(0.5);
final T q3Dot = q0.linearCombination(q2.negate(), oX, q1, oY, q0, oZ).multiply(0.5);
// second time-derivatives of the quaternion
final T oXDot = rotationAcceleration.getX();
final T oYDot = rotationAcceleration.getY();
final T oZDot = rotationAcceleration.getZ();
final T q0DotDot = q0.linearCombination(array6(q1, q2, q3, q1Dot, q2Dot, q3Dot),
array6(oXDot, oYDot, oZDot, oX, oY, oZ)).multiply(-0.5);
final T q1DotDot = q0.linearCombination(array6(q0, q2, q3.negate(), q0Dot, q2Dot, q3Dot.negate()),
array6(oXDot, oZDot, oYDot, oX, oZ, oY)).multiply(0.5);
final T q2DotDot = q0.linearCombination(array6(q0, q3, q1.negate(), q0Dot, q3Dot, q1Dot.negate()),
array6(oYDot, oXDot, oZDot, oY, oX, oZ)).multiply(0.5);
final T q3DotDot = q0.linearCombination(array6(q0, q1, q2.negate(), q0Dot, q1Dot, q2Dot.negate()),
array6(oZDot, oYDot, oXDot, oZ, oY, oX)).multiply(0.5);
final FieldUnivariateDerivative2<T> q0UD = new FieldUnivariateDerivative2<>(q0, q0Dot, q0DotDot);
final FieldUnivariateDerivative2<T> q1UD = new FieldUnivariateDerivative2<>(q1, q1Dot, q1DotDot);
final FieldUnivariateDerivative2<T> q2UD = new FieldUnivariateDerivative2<>(q2, q2Dot, q2DotDot);
final FieldUnivariateDerivative2<T> q3UD = new FieldUnivariateDerivative2<>(q3, q3Dot, q3DotDot);
return new FieldRotation<>(q0UD, q1UD, q2UD, q3UD, false);
}
/** Build an arry of 6 elements.
* @param e1 first element
* @param e2 second element
* @param e3 third element
* @param e4 fourth element
* @param e5 fifth element
* @param e6 sixth element
* @return a new array
* @since 9.2
*/
private T[] array6(final T e1, final T e2, final T e3, final T e4, final T e5, final T e6) {
final T[] array = MathArrays.buildArray(e1.getField(), 6);
array[0] = e1;
array[1] = e2;
array[2] = e3;
array[3] = e4;
array[4] = e5;
array[5] = e6;
return array;
}
/** 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
* @param <T> the type of the field elements
* @return rotation rate allowing to go from start to end orientations
*/
public static <T extends CalculusFieldElement<T>>
FieldVector3D<T> estimateRate(final FieldRotation<T> start,
final FieldRotation<T> end,
final double dt) {
return estimateRate(start, end, start.getQ0().getField().getZero().newInstance(dt));
}
/** 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
* @param <T> the type of the field elements
* @return rotation rate allowing to go from start to end orientations
*/
public static <T extends CalculusFieldElement<T>>
FieldVector3D<T> estimateRate(final FieldRotation<T> start,
final FieldRotation<T> end,
final T dt) {
final FieldRotation<T> evolution = start.compose(end.revert(), RotationConvention.VECTOR_OPERATOR);
return new FieldVector3D<>(evolution.getAngle().divide(dt),
evolution.getAxis(RotationConvention.VECTOR_OPERATOR));
}
/**
* Revert a rotation / rotation rate / rotation acceleration triplet.
*
* <p> 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 FieldAngularCoordinates<T> revert() {
return new FieldAngularCoordinates<>(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(CalculusFieldElement)
* @since 11.2
*/
public FieldRotation<T> rotationShiftedBy(final T 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 T rate = rotationRate.getNorm();
final FieldRotation<T> rateContribution = (rate.getReal() == 0.0) ?
FieldRotation.getIdentity(dt.getField()) :
new FieldRotation<>(rotationRate, rate.multiply(dt), RotationConvention.FRAME_TRANSFORM);
// append rotation and rate contribution
final FieldRotation<T> linearPart =
rateContribution.compose(rotation, RotationConvention.VECTOR_OPERATOR);
final T acc = rotationAcceleration.getNorm();
if (acc.getReal() == 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 FieldRotation<T> quadraticContribution =
new FieldRotation<>(rotationAcceleration,
acc.multiply(dt.square()).multiply(0.5),
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
* a simple quadratic 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)
*/
@Override
public FieldAngularCoordinates<T> shiftedBy(final double dt) {
return shiftedBy(rotation.getQ0().getField().getZero().newInstance(dt));
}
/** Get a time-shifted state.
* <p>
* The state can be slightly shifted to close dates. This shift is based on
* a simple quadratic 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)
*/
@Override
public FieldAngularCoordinates<T> shiftedBy(final T 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 T rate = rotationRate.getNorm();
final T zero = rate.getField().getZero();
final T one = rate.getField().getOne();
final FieldRotation<T> rateContribution = (rate.getReal() == 0.0) ?
new FieldRotation<>(one, zero, zero, zero, false) :
new FieldRotation<>(rotationRate,
rate.multiply(dt),
RotationConvention.FRAME_TRANSFORM);
// append rotation and rate contribution
final FieldAngularCoordinates<T> linearPart =
new FieldAngularCoordinates<>(rateContribution.compose(rotation, RotationConvention.VECTOR_OPERATOR),
rotationRate);
final T acc = rotationAcceleration.getNorm();
if (acc.getReal() == 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 FieldAngularCoordinates<T> quadraticContribution =
new FieldAngularCoordinates<>(new FieldRotation<>(rotationAcceleration,
acc.multiply(dt.square().multiply(0.5)),
RotationConvention.FRAME_TRANSFORM),
new FieldVector3D<>(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 FieldRotation<T> getRotation() {
return rotation;
}
/** Get the rotation rate.
* @return the rotation rate vector (rad/s).
*/
public FieldVector3D<T> getRotationRate() {
return rotationRate;
}
/** Get the rotation acceleration.
* @return the rotation acceleration vector dΩ/dt (rad/s²).
*/
public FieldVector3D<T> 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(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 FieldAngularCoordinates<T> addOffset(final FieldAngularCoordinates<T> offset) {
final FieldVector3D<T> rOmega = rotation.applyTo(offset.rotationRate);
final FieldVector3D<T> rOmegaDot = rotation.applyTo(offset.rotationAcceleration);
return new FieldAngularCoordinates<>(rotation.compose(offset.rotation, RotationConvention.VECTOR_OPERATOR),
rotationRate.add(rOmega),
new FieldVector3D<>( 1.0, rotationAcceleration,
1.0, rOmegaDot,
-1.0, FieldVector3D.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(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 FieldAngularCoordinates<T> subtractOffset(final FieldAngularCoordinates<T> offset) {
return addOffset(offset.revert());
}
/** Convert to a regular angular coordinates.
* @return a regular angular coordinates
*/
public AngularCoordinates toAngularCoordinates() {
return new AngularCoordinates(rotation.toRotation(), rotationRate.toVector3D(),
rotationAcceleration.toVector3D());
}
/** 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 pv by the rotation
*/
public FieldPVCoordinates<T> applyTo(final PVCoordinates pv) {
final FieldVector3D<T> transformedP = rotation.applyTo(pv.getPosition());
final FieldVector3D<T> crossP = FieldVector3D.crossProduct(rotationRate, transformedP);
final FieldVector3D<T> transformedV = rotation.applyTo(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, rotation.applyTo(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
* @return a new pv coordinates which is the image of pv by the rotation
*/
public TimeStampedFieldPVCoordinates<T> applyTo(final TimeStampedPVCoordinates pv) {
final FieldVector3D<T> transformedP = rotation.applyTo(pv.getPosition());
final FieldVector3D<T> crossP = FieldVector3D.crossProduct(rotationRate, transformedP);
final FieldVector3D<T> transformedV = rotation.applyTo(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, rotation.applyTo(pv.getAcceleration()),
-2, crossV,
-1, crossCrossP,
-1, crossDotP);
return new TimeStampedFieldPVCoordinates<>(pv.getDate(), 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 pv by the rotation
* @since 9.0
*/
public FieldPVCoordinates<T> applyTo(final FieldPVCoordinates<T> pv) {
final FieldVector3D<T> transformedP = rotation.applyTo(pv.getPosition());
final FieldVector3D<T> crossP = FieldVector3D.crossProduct(rotationRate, transformedP);
final FieldVector3D<T> transformedV = rotation.applyTo(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, rotation.applyTo(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
* @return a new pv coordinates which is the image of pv by the rotation
* @since 9.0
*/
public TimeStampedFieldPVCoordinates<T> applyTo(final TimeStampedFieldPVCoordinates<T> pv) {
final FieldVector3D<T> transformedP = rotation.applyTo(pv.getPosition());
final FieldVector3D<T> crossP = FieldVector3D.crossProduct(rotationRate, transformedP);
final FieldVector3D<T> transformedV = rotation.applyTo(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, rotation.applyTo(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(CalculusFieldElement[][])
* @since 9.0
*/
public T[][] getModifiedRodrigues(final double sign) {
final T q0 = getRotation().getQ0().multiply(sign);
final T q1 = getRotation().getQ1().multiply(sign);
final T q2 = getRotation().getQ2().multiply(sign);
final T q3 = getRotation().getQ3().multiply(sign);
final T oX = getRotationRate().getX();
final T oY = getRotationRate().getY();
final T oZ = getRotationRate().getZ();
final T oXDot = getRotationAcceleration().getX();
final T oYDot = getRotationAcceleration().getY();
final T oZDot = 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 = q0.linearCombination( q0, oX, q3.negate(), oY, q2, oZ).multiply(0.5);
final T q2Dot = q0.linearCombination( q3, oX, q0, oY, q1.negate(), oZ).multiply(0.5);
final T q3Dot = q0.linearCombination(q2.negate(), oX, q1, oY, q0, oZ).multiply(0.5);
// second time-derivatives of the quaternion
final T q0DotDot = linearCombination(q1, oXDot, q2, oYDot, q3, oZDot,
q1Dot, oX, q2Dot, oY, q3Dot, oZ).
multiply(-0.5);
final T q1DotDot = linearCombination(q0, oXDot, q2, oZDot, q3.negate(), oYDot,
q0Dot, oX, q2Dot, oZ, q3Dot.negate(), oY).
multiply(0.5);
final T q2DotDot = linearCombination(q0, oYDot, q3, oXDot, q1.negate(), oZDot,
q0Dot, oY, q3Dot, oX, q1Dot.negate(), oZ).
multiply(0.5);
final T q3DotDot = linearCombination(q0, oZDot, q1, oYDot, q2.negate(), oXDot,
q0Dot, oZ, q1Dot, oY, q2Dot.negate(), 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).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 = q0.linearCombination(inv, q1Dot, mInvR1, q0Dot);
final T r2Dot = q0.linearCombination(inv, q2Dot, mInvR2, q0Dot);
final T r3Dot = q0.linearCombination(inv, q3Dot, mInvR3, q0Dot);
final T r1DotDot = q0.linearCombination(inv, q1DotDot, mTwoInvQ0Dot, r1Dot, mInvR1, q0DotDot);
final T r2DotDot = q0.linearCombination(inv, q2DotDot, mTwoInvQ0Dot, r2Dot, mInvR2, q0DotDot);
final T r3DotDot = q0.linearCombination(inv, q3DotDot, mTwoInvQ0Dot, r3Dot, mInvR3, q0DotDot);
final T[][] rodrigues = MathArrays.buildArray(q0.getField(), 3, 3);
rodrigues[0][0] = r1;
rodrigues[0][1] = r2;
rodrigues[0][2] = r3;
rodrigues[1][0] = r1Dot;
rodrigues[1][1] = r2Dot;
rodrigues[1][2] = r3Dot;
rodrigues[2][0] = r1DotDot;
rodrigues[2][1] = r2DotDot;
rodrigues[2][2] = r3DotDot;
return rodrigues;
}
/**
* Compute a linear combination.
* @param a1 first factor of the first term
* @param b1 second factor of the first term
* @param a2 first factor of the second term
* @param b2 second factor of the second term
* @param a3 first factor of the third term
* @param b3 second factor of the third term
* @param a4 first factor of the fourth term
* @param b4 second factor of the fourth term
* @param a5 first factor of the fifth term
* @param b5 second factor of the fifth term
* @param a6 first factor of the sixth term
* @param b6 second factor of the sicth term
* @return a<sub>1</sub>×b<sub>1</sub> + a<sub>2</sub>×b<sub>2</sub> +
* a<sub>3</sub>×b<sub>3</sub> + a<sub>4</sub>×b<sub>4</sub> +
* a<sub>5</sub>×b<sub>5</sub> + a<sub>6</sub>×b<sub>6</sub>
*/
private T linearCombination(final T a1, final T b1, final T a2, final T b2, final T a3, final T b3,
final T a4, final T b4, final T a5, final T b5, final T a6, final T b6) {
final T[] a = MathArrays.buildArray(a1.getField(), 6);
a[0] = a1;
a[1] = a2;
a[2] = a3;
a[3] = a4;
a[4] = a5;
a[5] = a6;
final T[] b = MathArrays.buildArray(b1.getField(), 6);
b[0] = b1;
b[1] = b2;
b[2] = b3;
b[3] = b4;
b[4] = b5;
b[5] = b6;
return a1.linearCombination(a, b);
}
/** Convert a modified Rodrigues vector and derivatives to angular coordinates.
* @param r modified Rodrigues vector (with first and second times derivatives)
* @param <T> the type of the field elements
* @return angular coordinates
* @see #getModifiedRodrigues(double)
* @since 9.0
*/
public static <T extends CalculusFieldElement<T>> FieldAngularCoordinates<T> createFromModifiedRodrigues(final T[][] r) {
// rotation
final T rSquared = r[0][0].square().add(r[0][1].square()).add(r[0][2].square());
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.multiply(q0.linearCombination(r[0][0], r[1][0], r[0][1], r[1][1], r[0][2], r[1][2])).negate();
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 = q0.linearCombination(q1.negate(), q0Dot, q0, q1Dot, q3, q2Dot, q2.negate(), q3Dot).multiply(2);
final T oY = q0.linearCombination(q2.negate(), q0Dot, q3.negate(), q1Dot, q0, q2Dot, q1, q3Dot).multiply(2);
final T oZ = q0.linearCombination(q3.negate(), q0Dot, q2, q1Dot, q1.negate(), q2Dot, q0, q3Dot).multiply(2);
// rotation acceleration
final T q0DotDot = q0.subtract(1).negate().divide(oPQ0).multiply(q0Dot.square()).
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.square().add(q2Dot.square()).add(q3Dot.square()));
final T q1DotDot = q0.linearCombination(oPQ0, r[2][0], r[1][0].add(r[1][0]), q0Dot, r[0][0], q0DotDot);
final T q2DotDot = q0.linearCombination(oPQ0, r[2][1], r[1][1].add(r[1][1]), q0Dot, r[0][1], q0DotDot);
final T q3DotDot = q0.linearCombination(oPQ0, r[2][2], r[1][2].add(r[1][2]), q0Dot, r[0][2], q0DotDot);
final T oXDot = q0.linearCombination(q1.negate(), q0DotDot, q0, q1DotDot, q3, q2DotDot, q2.negate(), q3DotDot).multiply(2);
final T oYDot = q0.linearCombination(q2.negate(), q0DotDot, q3.negate(), q1DotDot, q0, q2DotDot, q1, q3DotDot).multiply(2);
final T oZDot = q0.linearCombination(q3.negate(), q0DotDot, q2, q1DotDot, q1.negate(), q2DotDot, q0, q3DotDot).multiply(2);
return new FieldAngularCoordinates<>(new FieldRotation<>(q0, q1, q2, q3, false),
new FieldVector3D<>(oX, oY, oZ),
new FieldVector3D<>(oXDot, oYDot, oZDot));
}
}