| /* |
| * Licensed to the Apache Software Foundation (ASF) under one or more |
| * contributor license agreements. See the NOTICE file distributed with |
| * this work for additional information regarding copyright ownership. |
| * The ASF 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.apache.commons.math4.geometry.euclidean.threed; |
| |
| import java.io.Serializable; |
| |
| import org.apache.commons.numbers.quaternion.Quaternion; |
| import org.apache.commons.geometry.euclidean.threed.Vector3D; |
| import org.apache.commons.geometry.euclidean.threed.rotation.QuaternionRotation; |
| import org.apache.commons.math4.Field; |
| import org.apache.commons.math4.RealFieldElement; |
| import org.apache.commons.math4.exception.MathArithmeticException; |
| import org.apache.commons.math4.exception.MathIllegalArgumentException; |
| import org.apache.commons.math4.exception.util.LocalizedFormats; |
| import org.apache.commons.math4.util.FastMath; |
| import org.apache.commons.math4.util.MathArrays; |
| |
| /** |
| * Implementation of rotation using {@link RealFieldElement}. |
| * <p>Instance of this class are guaranteed to be immutable.</p> |
| * |
| * @param <T> the type of the field elements |
| * @see FieldVector3D |
| * @see RotationOrder |
| * @since 3.2 |
| */ |
| |
| public class FieldRotation<T extends RealFieldElement<T>> implements Serializable { |
| |
| /** Serializable version identifier */ |
| private static final long serialVersionUID = 20130224l; |
| |
| /** Scalar coordinate of the quaternion. */ |
| private final T q0; |
| |
| /** First coordinate of the vectorial part of the quaternion. */ |
| private final T q1; |
| |
| /** Second coordinate of the vectorial part of the quaternion. */ |
| private final T q2; |
| |
| /** Third coordinate of the vectorial part of the quaternion. */ |
| private final T q3; |
| |
| /** Build a rotation from the quaternion coordinates. |
| * <p>A rotation can be built from a <em>normalized</em> quaternion, |
| * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> + |
| * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> + |
| * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized, |
| * the constructor can normalize it in a preprocessing step.</p> |
| * <p>Note that some conventions put the scalar part of the quaternion |
| * as the 4<sup>th</sup> component and the vector part as the first three |
| * components. This is <em>not</em> our convention. We put the scalar part |
| * as the first component.</p> |
| * @param q0 scalar part of the quaternion |
| * @param q1 first coordinate of the vectorial part of the quaternion |
| * @param q2 second coordinate of the vectorial part of the quaternion |
| * @param q3 third coordinate of the vectorial part of the quaternion |
| * @param needsNormalization if true, the coordinates are considered |
| * not to be normalized, a normalization preprocessing step is performed |
| * before using them |
| */ |
| public FieldRotation(final T q0, final T q1, final T q2, final T q3, final boolean needsNormalization) { |
| |
| if (needsNormalization) { |
| // normalization preprocessing |
| final T inv = |
| q0.multiply(q0).add(q1.multiply(q1)).add(q2.multiply(q2)).add(q3.multiply(q3)).sqrt().reciprocal(); |
| this.q0 = inv.multiply(q0); |
| this.q1 = inv.multiply(q1); |
| this.q2 = inv.multiply(q2); |
| this.q3 = inv.multiply(q3); |
| } else { |
| this.q0 = q0; |
| this.q1 = q1; |
| this.q2 = q2; |
| this.q3 = q3; |
| } |
| |
| } |
| |
| /** Build a rotation from an axis and an angle. |
| * <p>We use the convention that angles are oriented according to |
| * the effect of the rotation on vectors around the axis. That means |
| * that if (i, j, k) is a direct frame and if we first provide +k as |
| * the axis and π/2 as the angle to this constructor, and then |
| * {@link #applyTo(FieldVector3D) apply} the instance to +i, we will get |
| * +j.</p> |
| * <p>Another way to represent our convention is to say that a rotation |
| * of angle θ about the unit vector (x, y, z) is the same as the |
| * rotation build from quaternion components { cos(-θ/2), |
| * x * sin(-θ/2), y * sin(-θ/2), z * sin(-θ/2) }. |
| * Note the minus sign on the angle!</p> |
| * <p>On the one hand this convention is consistent with a vectorial |
| * perspective (moving vectors in fixed frames), on the other hand it |
| * is different from conventions with a frame perspective (fixed vectors |
| * viewed from different frames) like the ones used for example in spacecraft |
| * attitude community or in the graphics community.</p> |
| * @param axis axis around which to rotate |
| * @param angle rotation angle. |
| * @exception MathIllegalArgumentException if the axis norm is zero |
| * @deprecated as of 3.6, replaced with {@link |
| * #FieldRotation(FieldVector3D, RealFieldElement, RotationConvention)} |
| */ |
| @Deprecated |
| public FieldRotation(final FieldVector3D<T> axis, final T angle) |
| throws MathIllegalArgumentException { |
| this(axis, angle, RotationConvention.VECTOR_OPERATOR); |
| } |
| |
| /** Build a rotation from an axis and an angle. |
| * <p>We use the convention that angles are oriented according to |
| * the effect of the rotation on vectors around the axis. That means |
| * that if (i, j, k) is a direct frame and if we first provide +k as |
| * the axis and π/2 as the angle to this constructor, and then |
| * {@link #applyTo(FieldVector3D) apply} the instance to +i, we will get |
| * +j.</p> |
| * <p>Another way to represent our convention is to say that a rotation |
| * of angle θ about the unit vector (x, y, z) is the same as the |
| * rotation build from quaternion components { cos(-θ/2), |
| * x * sin(-θ/2), y * sin(-θ/2), z * sin(-θ/2) }. |
| * Note the minus sign on the angle!</p> |
| * <p>On the one hand this convention is consistent with a vectorial |
| * perspective (moving vectors in fixed frames), on the other hand it |
| * is different from conventions with a frame perspective (fixed vectors |
| * viewed from different frames) like the ones used for example in spacecraft |
| * attitude community or in the graphics community.</p> |
| * @param axis axis around which to rotate |
| * @param angle rotation angle. |
| * @param convention convention to use for the semantics of the angle |
| * @exception MathIllegalArgumentException if the axis norm is zero |
| * @since 3.6 |
| */ |
| public FieldRotation(final FieldVector3D<T> axis, final T angle, final RotationConvention convention) |
| throws MathIllegalArgumentException { |
| |
| final T norm = axis.getNorm(); |
| if (norm.getReal() == 0) { |
| throw new MathIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS); |
| } |
| |
| final T halfAngle = angle.multiply(convention == RotationConvention.VECTOR_OPERATOR ? -0.5 : 0.5); |
| final T coeff = halfAngle.sin().divide(norm); |
| |
| q0 = halfAngle.cos(); |
| q1 = coeff.multiply(axis.getX()); |
| q2 = coeff.multiply(axis.getY()); |
| q3 = coeff.multiply(axis.getZ()); |
| |
| } |
| |
| /** Build a rotation from a 3X3 matrix. |
| |
| * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices |
| * (which are matrices for which m.m<sup>T</sup> = I) with real |
| * coefficients. The module of the determinant of unit matrices is |
| * 1, among the orthogonal 3X3 matrices, only the ones having a |
| * positive determinant (+1) are rotation matrices.</p> |
| |
| * <p>When a rotation is defined by a matrix with truncated values |
| * (typically when it is extracted from a technical sheet where only |
| * four to five significant digits are available), the matrix is not |
| * orthogonal anymore. This constructor handles this case |
| * transparently by using a copy of the given matrix and applying a |
| * correction to the copy in order to perfect its orthogonality. If |
| * the Frobenius norm of the correction needed is above the given |
| * threshold, then the matrix is considered to be too far from a |
| * true rotation matrix and an exception is thrown.<p> |
| |
| * @param m rotation matrix |
| * @param threshold convergence threshold for the iterative |
| * orthogonality correction (convergence is reached when the |
| * difference between two steps of the Frobenius norm of the |
| * correction is below this threshold) |
| |
| * @exception NotARotationMatrixException if the matrix is not a 3X3 |
| * matrix, or if it cannot be transformed into an orthogonal matrix |
| * with the given threshold, or if the determinant of the resulting |
| * orthogonal matrix is negative |
| |
| */ |
| public FieldRotation(final T[][] m, final double threshold) |
| throws NotARotationMatrixException { |
| |
| // dimension check |
| if ((m.length != 3) || (m[0].length != 3) || |
| (m[1].length != 3) || (m[2].length != 3)) { |
| throw new NotARotationMatrixException( |
| LocalizedFormats.ROTATION_MATRIX_DIMENSIONS, |
| m.length, m[0].length); |
| } |
| |
| // compute a "close" orthogonal matrix |
| final T[][] ort = orthogonalizeMatrix(m, threshold); |
| |
| // check the sign of the determinant |
| final T d0 = ort[1][1].multiply(ort[2][2]).subtract(ort[2][1].multiply(ort[1][2])); |
| final T d1 = ort[0][1].multiply(ort[2][2]).subtract(ort[2][1].multiply(ort[0][2])); |
| final T d2 = ort[0][1].multiply(ort[1][2]).subtract(ort[1][1].multiply(ort[0][2])); |
| final T det = |
| ort[0][0].multiply(d0).subtract(ort[1][0].multiply(d1)).add(ort[2][0].multiply(d2)); |
| if (det.getReal() < 0.0) { |
| throw new NotARotationMatrixException( |
| LocalizedFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT, |
| det); |
| } |
| |
| final T[] quat = mat2quat(ort); |
| q0 = quat[0]; |
| q1 = quat[1]; |
| q2 = quat[2]; |
| q3 = quat[3]; |
| |
| } |
| |
| /** Build the rotation that transforms a pair of vectors into another pair. |
| |
| * <p>Except for possible scale factors, if the instance were applied to |
| * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair |
| * (v<sub>1</sub>, v<sub>2</sub>).</p> |
| |
| * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is |
| * not the same as the angular separation between v<sub>1</sub> and |
| * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than |
| * v<sub>2</sub>, the corrected vector will be in the (±v<sub>1</sub>, |
| * +v<sub>2</sub>) half-plane.</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 |
| * @exception MathArithmeticException if the norm of one of the vectors is zero, |
| * or if one of the pair is degenerated (i.e. the vectors of the pair are collinear) |
| */ |
| public FieldRotation(FieldVector3D<T> u1, FieldVector3D<T> u2, FieldVector3D<T> v1, FieldVector3D<T> v2) |
| throws MathArithmeticException { |
| |
| // build orthonormalized base from u1, u2 |
| // this fails when vectors are null or collinear, which is forbidden to define a rotation |
| final FieldVector3D<T> u3 = FieldVector3D.crossProduct(u1, u2).normalize(); |
| u2 = FieldVector3D.crossProduct(u3, u1).normalize(); |
| u1 = u1.normalize(); |
| |
| // build an orthonormalized base from v1, v2 |
| // this fails when vectors are null or collinear, which is forbidden to define a rotation |
| final FieldVector3D<T> v3 = FieldVector3D.crossProduct(v1, v2).normalize(); |
| v2 = FieldVector3D.crossProduct(v3, v1).normalize(); |
| v1 = v1.normalize(); |
| |
| // buid a matrix transforming the first base into the second one |
| final T[][] array = MathArrays.buildArray(u1.getX().getField(), 3, 3); |
| array[0][0] = u1.getX().multiply(v1.getX()).add(u2.getX().multiply(v2.getX())).add(u3.getX().multiply(v3.getX())); |
| array[0][1] = u1.getY().multiply(v1.getX()).add(u2.getY().multiply(v2.getX())).add(u3.getY().multiply(v3.getX())); |
| array[0][2] = u1.getZ().multiply(v1.getX()).add(u2.getZ().multiply(v2.getX())).add(u3.getZ().multiply(v3.getX())); |
| array[1][0] = u1.getX().multiply(v1.getY()).add(u2.getX().multiply(v2.getY())).add(u3.getX().multiply(v3.getY())); |
| array[1][1] = u1.getY().multiply(v1.getY()).add(u2.getY().multiply(v2.getY())).add(u3.getY().multiply(v3.getY())); |
| array[1][2] = u1.getZ().multiply(v1.getY()).add(u2.getZ().multiply(v2.getY())).add(u3.getZ().multiply(v3.getY())); |
| array[2][0] = u1.getX().multiply(v1.getZ()).add(u2.getX().multiply(v2.getZ())).add(u3.getX().multiply(v3.getZ())); |
| array[2][1] = u1.getY().multiply(v1.getZ()).add(u2.getY().multiply(v2.getZ())).add(u3.getY().multiply(v3.getZ())); |
| array[2][2] = u1.getZ().multiply(v1.getZ()).add(u2.getZ().multiply(v2.getZ())).add(u3.getZ().multiply(v3.getZ())); |
| |
| T[] quat = mat2quat(array); |
| q0 = quat[0]; |
| q1 = quat[1]; |
| q2 = quat[2]; |
| q3 = quat[3]; |
| |
| } |
| |
| /** Build one of the rotations that transform one vector into another one. |
| |
| * <p>Except for a possible scale factor, if the instance were |
| * applied to the vector u it will produce the vector v. There is an |
| * infinite number of such rotations, this constructor choose the |
| * one with the smallest associated angle (i.e. the one whose axis |
| * is orthogonal to the (u, v) plane). If u and v are collinear, an |
| * arbitrary rotation axis is chosen.</p> |
| |
| * @param u origin vector |
| * @param v desired image of u by the rotation |
| * @exception MathArithmeticException if the norm of one of the vectors is zero |
| */ |
| public FieldRotation(final FieldVector3D<T> u, final FieldVector3D<T> v) throws MathArithmeticException { |
| |
| final T normProduct = u.getNorm().multiply(v.getNorm()); |
| if (normProduct.getReal() == 0) { |
| throw new MathArithmeticException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR); |
| } |
| |
| final T dot = FieldVector3D.dotProduct(u, v); |
| |
| if (dot.getReal() < ((2.0e-15 - 1.0) * normProduct.getReal())) { |
| // special case u = -v: we select a PI angle rotation around |
| // an arbitrary vector orthogonal to u |
| final FieldVector3D<T> w = u.orthogonal(); |
| q0 = normProduct.getField().getZero(); |
| q1 = w.getX().negate(); |
| q2 = w.getY().negate(); |
| q3 = w.getZ().negate(); |
| } else { |
| // general case: (u, v) defines a plane, we select |
| // the shortest possible rotation: axis orthogonal to this plane |
| q0 = dot.divide(normProduct).add(1.0).multiply(0.5).sqrt(); |
| final T coeff = q0.multiply(normProduct).multiply(2.0).reciprocal(); |
| final FieldVector3D<T> q = FieldVector3D.crossProduct(v, u); |
| q1 = coeff.multiply(q.getX()); |
| q2 = coeff.multiply(q.getY()); |
| q3 = coeff.multiply(q.getZ()); |
| } |
| |
| } |
| |
| /** Build a rotation from three Cardan or Euler elementary rotations. |
| |
| * <p>Cardan rotations are three successive rotations around the |
| * canonical axes X, Y and Z, each axis being used once. There are |
| * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler |
| * rotations are three successive rotations around the canonical |
| * axes X, Y and Z, the first and last rotations being around the |
| * same axis. There are 6 such sets of rotations (XYX, XZX, YXY, |
| * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p> |
| * <p>Beware that many people routinely use the term Euler angles even |
| * for what really are Cardan angles (this confusion is especially |
| * widespread in the aerospace business where Roll, Pitch and Yaw angles |
| * are often wrongly tagged as Euler angles).</p> |
| |
| * @param order order of rotations to use |
| * @param alpha1 angle of the first elementary rotation |
| * @param alpha2 angle of the second elementary rotation |
| * @param alpha3 angle of the third elementary rotation |
| * @deprecated as of 3.6, replaced with {@link |
| * #FieldRotation(RotationOrder, RotationConvention, |
| * RealFieldElement, RealFieldElement, RealFieldElement)} |
| */ |
| @Deprecated |
| public FieldRotation(final RotationOrder order, final T alpha1, final T alpha2, final T alpha3) { |
| this(order, RotationConvention.VECTOR_OPERATOR, alpha1, alpha2, alpha3); |
| } |
| |
| /** Build a rotation from three Cardan or Euler elementary rotations. |
| |
| * <p>Cardan rotations are three successive rotations around the |
| * canonical axes X, Y and Z, each axis being used once. There are |
| * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler |
| * rotations are three successive rotations around the canonical |
| * axes X, Y and Z, the first and last rotations being around the |
| * same axis. There are 6 such sets of rotations (XYX, XZX, YXY, |
| * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p> |
| * <p>Beware that many people routinely use the term Euler angles even |
| * for what really are Cardan angles (this confusion is especially |
| * widespread in the aerospace business where Roll, Pitch and Yaw angles |
| * are often wrongly tagged as Euler angles).</p> |
| |
| * @param order order of rotations to compose, from left to right |
| * (i.e. we will use {@code r1.compose(r2.compose(r3, convention), convention)}) |
| * @param convention convention to use for the semantics of the angle |
| * @param alpha1 angle of the first elementary rotation |
| * @param alpha2 angle of the second elementary rotation |
| * @param alpha3 angle of the third elementary rotation |
| * @since 3.6 |
| */ |
| public FieldRotation(final RotationOrder order, final RotationConvention convention, |
| final T alpha1, final T alpha2, final T alpha3) { |
| final T one = alpha1.getField().getOne(); |
| final FieldRotation<T> r1 = new FieldRotation<>(new FieldVector3D<>(one, order.getA1()), alpha1, convention); |
| final FieldRotation<T> r2 = new FieldRotation<>(new FieldVector3D<>(one, order.getA2()), alpha2, convention); |
| final FieldRotation<T> r3 = new FieldRotation<>(new FieldVector3D<>(one, order.getA3()), alpha3, convention); |
| final FieldRotation<T> composed = r1.compose(r2.compose(r3, convention), convention); |
| q0 = composed.q0; |
| q1 = composed.q1; |
| q2 = composed.q2; |
| q3 = composed.q3; |
| } |
| |
| /** Convert an orthogonal rotation matrix to a quaternion. |
| * @param ort orthogonal rotation matrix |
| * @return quaternion corresponding to the matrix |
| */ |
| private T[] mat2quat(final T[][] ort) { |
| |
| final T[] quat = MathArrays.buildArray(ort[0][0].getField(), 4); |
| |
| // There are different ways to compute the quaternions elements |
| // from the matrix. They all involve computing one element from |
| // the diagonal of the matrix, and computing the three other ones |
| // using a formula involving a division by the first element, |
| // which unfortunately can be zero. Since the norm of the |
| // quaternion is 1, we know at least one element has an absolute |
| // value greater or equal to 0.5, so it is always possible to |
| // select the right formula and avoid division by zero and even |
| // numerical inaccuracy. Checking the elements in turn and using |
| // the first one greater than 0.45 is safe (this leads to a simple |
| // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19) |
| T s = ort[0][0].add(ort[1][1]).add(ort[2][2]); |
| if (s.getReal() > -0.19) { |
| // compute q0 and deduce q1, q2 and q3 |
| quat[0] = s.add(1.0).sqrt().multiply(0.5); |
| T inv = quat[0].reciprocal().multiply(0.25); |
| quat[1] = inv.multiply(ort[1][2].subtract(ort[2][1])); |
| quat[2] = inv.multiply(ort[2][0].subtract(ort[0][2])); |
| quat[3] = inv.multiply(ort[0][1].subtract(ort[1][0])); |
| } else { |
| s = ort[0][0].subtract(ort[1][1]).subtract(ort[2][2]); |
| if (s.getReal() > -0.19) { |
| // compute q1 and deduce q0, q2 and q3 |
| quat[1] = s.add(1.0).sqrt().multiply(0.5); |
| T inv = quat[1].reciprocal().multiply(0.25); |
| quat[0] = inv.multiply(ort[1][2].subtract(ort[2][1])); |
| quat[2] = inv.multiply(ort[0][1].add(ort[1][0])); |
| quat[3] = inv.multiply(ort[0][2].add(ort[2][0])); |
| } else { |
| s = ort[1][1].subtract(ort[0][0]).subtract(ort[2][2]); |
| if (s.getReal() > -0.19) { |
| // compute q2 and deduce q0, q1 and q3 |
| quat[2] = s.add(1.0).sqrt().multiply(0.5); |
| T inv = quat[2].reciprocal().multiply(0.25); |
| quat[0] = inv.multiply(ort[2][0].subtract(ort[0][2])); |
| quat[1] = inv.multiply(ort[0][1].add(ort[1][0])); |
| quat[3] = inv.multiply(ort[2][1].add(ort[1][2])); |
| } else { |
| // compute q3 and deduce q0, q1 and q2 |
| s = ort[2][2].subtract(ort[0][0]).subtract(ort[1][1]); |
| quat[3] = s.add(1.0).sqrt().multiply(0.5); |
| T inv = quat[3].reciprocal().multiply(0.25); |
| quat[0] = inv.multiply(ort[0][1].subtract(ort[1][0])); |
| quat[1] = inv.multiply(ort[0][2].add(ort[2][0])); |
| quat[2] = inv.multiply(ort[2][1].add(ort[1][2])); |
| } |
| } |
| } |
| |
| return quat; |
| |
| } |
| |
| /** Revert a rotation. |
| * Build a rotation which reverse the effect of another |
| * rotation. This means that if r(u) = v, then r.revert(v) = u. The |
| * instance is not changed. |
| * @return a new rotation whose effect is the reverse of the effect |
| * of the instance |
| */ |
| public FieldRotation<T> revert() { |
| return new FieldRotation<>(q0.negate(), q1, q2, q3, false); |
| } |
| |
| /** Get the scalar coordinate of the quaternion. |
| * @return scalar coordinate of the quaternion |
| */ |
| public T getQ0() { |
| return q0; |
| } |
| |
| /** Get the first coordinate of the vectorial part of the quaternion. |
| * @return first coordinate of the vectorial part of the quaternion |
| */ |
| public T getQ1() { |
| return q1; |
| } |
| |
| /** Get the second coordinate of the vectorial part of the quaternion. |
| * @return second coordinate of the vectorial part of the quaternion |
| */ |
| public T getQ2() { |
| return q2; |
| } |
| |
| /** Get the third coordinate of the vectorial part of the quaternion. |
| * @return third coordinate of the vectorial part of the quaternion |
| */ |
| public T getQ3() { |
| return q3; |
| } |
| |
| /** Get the normalized axis of the rotation. |
| * @return normalized axis of the rotation |
| * @see #FieldRotation(FieldVector3D, RealFieldElement) |
| * @deprecated as of 3.6, replaced with {@link #getAxis(RotationConvention)} |
| */ |
| @Deprecated |
| public FieldVector3D<T> getAxis() { |
| return getAxis(RotationConvention.VECTOR_OPERATOR); |
| } |
| |
| /** Get the normalized axis of the rotation. |
| * <p> |
| * Note that as {@link #getAngle()} always returns an angle |
| * between 0 and π, changing the convention changes the |
| * direction of the axis, not the sign of the angle. |
| * </p> |
| * @param convention convention to use for the semantics of the angle |
| * @return normalized axis of the rotation |
| * @see #FieldRotation(FieldVector3D, RealFieldElement) |
| * @since 3.6 |
| */ |
| public FieldVector3D<T> getAxis(final RotationConvention convention) { |
| final T squaredSine = q1.multiply(q1).add(q2.multiply(q2)).add(q3.multiply(q3)); |
| if (squaredSine.getReal() == 0) { |
| final Field<T> field = squaredSine.getField(); |
| return new FieldVector3D<>(convention == RotationConvention.VECTOR_OPERATOR ? field.getOne(): field.getOne().negate(), |
| field.getZero(), |
| field.getZero()); |
| } else { |
| final double sgn = convention == RotationConvention.VECTOR_OPERATOR ? +1 : -1; |
| if (q0.getReal() < 0) { |
| T inverse = squaredSine.sqrt().reciprocal().multiply(sgn); |
| return new FieldVector3D<>(q1.multiply(inverse), q2.multiply(inverse), q3.multiply(inverse)); |
| } |
| final T inverse = squaredSine.sqrt().reciprocal().negate().multiply(sgn); |
| return new FieldVector3D<>(q1.multiply(inverse), q2.multiply(inverse), q3.multiply(inverse)); |
| } |
| } |
| |
| /** Get the angle of the rotation. |
| * @return angle of the rotation (between 0 and π) |
| * @see #FieldRotation(FieldVector3D, RealFieldElement) |
| */ |
| public T getAngle() { |
| if ((q0.getReal() < -0.1) || (q0.getReal() > 0.1)) { |
| return q1.multiply(q1).add(q2.multiply(q2)).add(q3.multiply(q3)).sqrt().asin().multiply(2); |
| } else if (q0.getReal() < 0) { |
| return q0.negate().acos().multiply(2); |
| } |
| return q0.acos().multiply(2); |
| } |
| |
| /** Get the Cardan or Euler angles corresponding to the instance. |
| |
| * <p>The equations show that each rotation can be defined by two |
| * different values of the Cardan or Euler angles set. For example |
| * if Cardan angles are used, the rotation defined by the angles |
| * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as |
| * the rotation defined by the angles π + a<sub>1</sub>, π |
| * - a<sub>2</sub> and π + a<sub>3</sub>. This method implements |
| * the following arbitrary choices:</p> |
| * <ul> |
| * <li>for Cardan angles, the chosen set is the one for which the |
| * second angle is between -π/2 and π/2 (i.e its cosine is |
| * positive),</li> |
| * <li>for Euler angles, the chosen set is the one for which the |
| * second angle is between 0 and π (i.e its sine is positive).</li> |
| * </ul> |
| |
| * <p>Cardan and Euler angle have a very disappointing drawback: all |
| * of them have singularities. This means that if the instance is |
| * too close to the singularities corresponding to the given |
| * rotation order, it will be impossible to retrieve the angles. For |
| * Cardan angles, this is often called gimbal lock. There is |
| * <em>nothing</em> to do to prevent this, it is an intrinsic problem |
| * with Cardan and Euler representation (but not a problem with the |
| * rotation itself, which is perfectly well defined). For Cardan |
| * angles, singularities occur when the second angle is close to |
| * -π/2 or +π/2, for Euler angle singularities occur when the |
| * second angle is close to 0 or π, this implies that the identity |
| * rotation is always singular for Euler angles!</p> |
| |
| * @param order rotation order to use |
| * @return an array of three angles, in the order specified by the set |
| * @exception CardanEulerSingularityException if the rotation is |
| * singular with respect to the angles set specified |
| * @deprecated as of 3.6, replaced with {@link #getAngles(RotationOrder, RotationConvention)} |
| */ |
| @Deprecated |
| public T[] getAngles(final RotationOrder order) |
| throws CardanEulerSingularityException { |
| return getAngles(order, RotationConvention.VECTOR_OPERATOR); |
| } |
| |
| /** Get the Cardan or Euler angles corresponding to the instance. |
| |
| * <p>The equations show that each rotation can be defined by two |
| * different values of the Cardan or Euler angles set. For example |
| * if Cardan angles are used, the rotation defined by the angles |
| * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as |
| * the rotation defined by the angles π + a<sub>1</sub>, π |
| * - a<sub>2</sub> and π + a<sub>3</sub>. This method implements |
| * the following arbitrary choices:</p> |
| * <ul> |
| * <li>for Cardan angles, the chosen set is the one for which the |
| * second angle is between -π/2 and π/2 (i.e its cosine is |
| * positive),</li> |
| * <li>for Euler angles, the chosen set is the one for which the |
| * second angle is between 0 and π (i.e its sine is positive).</li> |
| * </ul> |
| |
| * <p>Cardan and Euler angle have a very disappointing drawback: all |
| * of them have singularities. This means that if the instance is |
| * too close to the singularities corresponding to the given |
| * rotation order, it will be impossible to retrieve the angles. For |
| * Cardan angles, this is often called gimbal lock. There is |
| * <em>nothing</em> to do to prevent this, it is an intrinsic problem |
| * with Cardan and Euler representation (but not a problem with the |
| * rotation itself, which is perfectly well defined). For Cardan |
| * angles, singularities occur when the second angle is close to |
| * -π/2 or +π/2, for Euler angle singularities occur when the |
| * second angle is close to 0 or π, this implies that the identity |
| * rotation is always singular for Euler angles!</p> |
| |
| * @param order rotation order to use |
| * @param convention convention to use for the semantics of the angle |
| * @return an array of three angles, in the order specified by the set |
| * @exception CardanEulerSingularityException if the rotation is |
| * singular with respect to the angles set specified |
| * @since 3.6 |
| */ |
| public T[] getAngles(final RotationOrder order, RotationConvention convention) |
| throws CardanEulerSingularityException { |
| |
| if (convention == RotationConvention.VECTOR_OPERATOR) { |
| if (order == RotationOrder.XYZ) { |
| |
| // r (+K) coordinates are : |
| // sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi) |
| // (-r) (+I) coordinates are : |
| // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta) |
| final // and we can choose to have theta in the interval [-PI/2 ; +PI/2] |
| FieldVector3D<T> v1 = applyTo(vector(0, 0, 1)); |
| final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0)); |
| if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(true); |
| } |
| return buildArray(v1.getY().negate().atan2(v1.getZ()), |
| v2.getZ().asin(), |
| v2.getY().negate().atan2(v2.getX())); |
| |
| } else if (order == RotationOrder.XZY) { |
| |
| // r (+J) coordinates are : |
| // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi) |
| // (-r) (+I) coordinates are : |
| // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi) |
| // and we can choose to have psi in the interval [-PI/2 ; +PI/2] |
| final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0)); |
| final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0)); |
| if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(true); |
| } |
| return buildArray(v1.getZ().atan2(v1.getY()), |
| v2.getY().asin().negate(), |
| v2.getZ().atan2(v2.getX())); |
| |
| } else if (order == RotationOrder.YXZ) { |
| |
| // r (+K) coordinates are : |
| // cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta) |
| // (-r) (+J) coordinates are : |
| // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi) |
| // and we can choose to have phi in the interval [-PI/2 ; +PI/2] |
| final FieldVector3D<T> v1 = applyTo(vector(0, 0, 1)); |
| final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0)); |
| if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(true); |
| } |
| return buildArray(v1.getX().atan2(v1.getZ()), |
| v2.getZ().asin().negate(), |
| v2.getX().atan2(v2.getY())); |
| |
| } else if (order == RotationOrder.YZX) { |
| |
| // r (+I) coordinates are : |
| // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta) |
| // (-r) (+J) coordinates are : |
| // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi) |
| // and we can choose to have psi in the interval [-PI/2 ; +PI/2] |
| final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0)); |
| final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0)); |
| if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(true); |
| } |
| return buildArray(v1.getZ().negate().atan2(v1.getX()), |
| v2.getX().asin(), |
| v2.getZ().negate().atan2(v2.getY())); |
| |
| } else if (order == RotationOrder.ZXY) { |
| |
| // r (+J) coordinates are : |
| // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi) |
| // (-r) (+K) coordinates are : |
| // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi) |
| // and we can choose to have phi in the interval [-PI/2 ; +PI/2] |
| final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0)); |
| final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1)); |
| if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(true); |
| } |
| return buildArray(v1.getX().negate().atan2(v1.getY()), |
| v2.getY().asin(), |
| v2.getX().negate().atan2(v2.getZ())); |
| |
| } else if (order == RotationOrder.ZYX) { |
| |
| // r (+I) coordinates are : |
| // cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta) |
| // (-r) (+K) coordinates are : |
| // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta) |
| // and we can choose to have theta in the interval [-PI/2 ; +PI/2] |
| final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0)); |
| final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1)); |
| if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(true); |
| } |
| return buildArray(v1.getY().atan2(v1.getX()), |
| v2.getX().asin().negate(), |
| v2.getY().atan2(v2.getZ())); |
| |
| } else if (order == RotationOrder.XYX) { |
| |
| // r (+I) coordinates are : |
| // cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta) |
| // (-r) (+I) coordinates are : |
| // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2) |
| // and we can choose to have theta in the interval [0 ; PI] |
| final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0)); |
| final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0)); |
| if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(false); |
| } |
| return buildArray(v1.getY().atan2(v1.getZ().negate()), |
| v2.getX().acos(), |
| v2.getY().atan2(v2.getZ())); |
| |
| } else if (order == RotationOrder.XZX) { |
| |
| // r (+I) coordinates are : |
| // cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi) |
| // (-r) (+I) coordinates are : |
| // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2) |
| // and we can choose to have psi in the interval [0 ; PI] |
| final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0)); |
| final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0)); |
| if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(false); |
| } |
| return buildArray(v1.getZ().atan2(v1.getY()), |
| v2.getX().acos(), |
| v2.getZ().atan2(v2.getY().negate())); |
| |
| } else if (order == RotationOrder.YXY) { |
| |
| // r (+J) coordinates are : |
| // sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi) |
| // (-r) (+J) coordinates are : |
| // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2) |
| // and we can choose to have phi in the interval [0 ; PI] |
| final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0)); |
| final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0)); |
| if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(false); |
| } |
| return buildArray(v1.getX().atan2(v1.getZ()), |
| v2.getY().acos(), |
| v2.getX().atan2(v2.getZ().negate())); |
| |
| } else if (order == RotationOrder.YZY) { |
| |
| // r (+J) coordinates are : |
| // -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi) |
| // (-r) (+J) coordinates are : |
| // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2) |
| // and we can choose to have psi in the interval [0 ; PI] |
| final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0)); |
| final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0)); |
| if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(false); |
| } |
| return buildArray(v1.getZ().atan2(v1.getX().negate()), |
| v2.getY().acos(), |
| v2.getZ().atan2(v2.getX())); |
| |
| } else if (order == RotationOrder.ZXZ) { |
| |
| // r (+K) coordinates are : |
| // sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi) |
| // (-r) (+K) coordinates are : |
| // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi) |
| // and we can choose to have phi in the interval [0 ; PI] |
| final FieldVector3D<T> v1 = applyTo(vector(0, 0, 1)); |
| final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1)); |
| if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(false); |
| } |
| return buildArray(v1.getX().atan2(v1.getY().negate()), |
| v2.getZ().acos(), |
| v2.getX().atan2(v2.getY())); |
| |
| } else { // last possibility is ZYZ |
| |
| // r (+K) coordinates are : |
| // cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta) |
| // (-r) (+K) coordinates are : |
| // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta) |
| // and we can choose to have theta in the interval [0 ; PI] |
| final FieldVector3D<T> v1 = applyTo(vector(0, 0, 1)); |
| final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1)); |
| if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(false); |
| } |
| return buildArray(v1.getY().atan2(v1.getX()), |
| v2.getZ().acos(), |
| v2.getY().atan2(v2.getX().negate())); |
| |
| } |
| } else { |
| if (order == RotationOrder.XYZ) { |
| |
| // r (Cartesian3D.plusI) coordinates are : |
| // cos (theta) cos (psi), -cos (theta) sin (psi), sin (theta) |
| // (-r) (Cartesian3D.plusK) coordinates are : |
| // sin (theta), -sin (phi) cos (theta), cos (phi) cos (theta) |
| // and we can choose to have theta in the interval [-PI/2 ; +PI/2] |
| FieldVector3D<T> v1 = applyTo(Vector3D.Unit.PLUS_X); |
| FieldVector3D<T> v2 = applyInverseTo(Vector3D.Unit.PLUS_Z); |
| if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(true); |
| } |
| return buildArray(v2.getY().negate().atan2(v2.getZ()), |
| v2.getX().asin(), |
| v1.getY().negate().atan2(v1.getX())); |
| |
| } else if (order == RotationOrder.XZY) { |
| |
| // r (Cartesian3D.plusI) coordinates are : |
| // cos (psi) cos (theta), -sin (psi), cos (psi) sin (theta) |
| // (-r) (Cartesian3D.plusJ) coordinates are : |
| // -sin (psi), cos (phi) cos (psi), sin (phi) cos (psi) |
| // and we can choose to have psi in the interval [-PI/2 ; +PI/2] |
| FieldVector3D<T> v1 = applyTo(Vector3D.Unit.PLUS_X); |
| FieldVector3D<T> v2 = applyInverseTo(Vector3D.Unit.PLUS_Y); |
| if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(true); |
| } |
| return buildArray(v2.getZ().atan2(v2.getY()), |
| v2.getX().asin().negate(), |
| v1.getZ().atan2(v1.getX())); |
| |
| } else if (order == RotationOrder.YXZ) { |
| |
| // r (Cartesian3D.plusJ) coordinates are : |
| // cos (phi) sin (psi), cos (phi) cos (psi), -sin (phi) |
| // (-r) (Cartesian3D.plusK) coordinates are : |
| // sin (theta) cos (phi), -sin (phi), cos (theta) cos (phi) |
| // and we can choose to have phi in the interval [-PI/2 ; +PI/2] |
| FieldVector3D<T> v1 = applyTo(Vector3D.Unit.PLUS_Y); |
| FieldVector3D<T> v2 = applyInverseTo(Vector3D.Unit.PLUS_Z); |
| if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(true); |
| } |
| return buildArray(v2.getX().atan2(v2.getZ()), |
| v2.getY().asin().negate(), |
| v1.getX().atan2(v1.getY())); |
| |
| } else if (order == RotationOrder.YZX) { |
| |
| // r (Cartesian3D.plusJ) coordinates are : |
| // sin (psi), cos (psi) cos (phi), -cos (psi) sin (phi) |
| // (-r) (Cartesian3D.plusI) coordinates are : |
| // cos (theta) cos (psi), sin (psi), -sin (theta) cos (psi) |
| // and we can choose to have psi in the interval [-PI/2 ; +PI/2] |
| FieldVector3D<T> v1 = applyTo(Vector3D.Unit.PLUS_Y); |
| FieldVector3D<T> v2 = applyInverseTo(Vector3D.Unit.PLUS_X); |
| if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(true); |
| } |
| return buildArray(v2.getZ().negate().atan2(v2.getX()), |
| v2.getY().asin(), |
| v1.getZ().negate().atan2(v1.getY())); |
| |
| } else if (order == RotationOrder.ZXY) { |
| |
| // r (Cartesian3D.plusK) coordinates are : |
| // -cos (phi) sin (theta), sin (phi), cos (phi) cos (theta) |
| // (-r) (Cartesian3D.plusJ) coordinates are : |
| // -sin (psi) cos (phi), cos (psi) cos (phi), sin (phi) |
| // and we can choose to have phi in the interval [-PI/2 ; +PI/2] |
| FieldVector3D<T> v1 = applyTo(Vector3D.Unit.PLUS_Z); |
| FieldVector3D<T> v2 = applyInverseTo(Vector3D.Unit.PLUS_Y); |
| if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(true); |
| } |
| return buildArray(v2.getX().negate().atan2(v2.getY()), |
| v2.getZ().asin(), |
| v1.getX().negate().atan2(v1.getZ())); |
| |
| } else if (order == RotationOrder.ZYX) { |
| |
| // r (Cartesian3D.plusK) coordinates are : |
| // -sin (theta), cos (theta) sin (phi), cos (theta) cos (phi) |
| // (-r) (Cartesian3D.plusI) coordinates are : |
| // cos (psi) cos (theta), sin (psi) cos (theta), -sin (theta) |
| // and we can choose to have theta in the interval [-PI/2 ; +PI/2] |
| FieldVector3D<T> v1 = applyTo(Vector3D.Unit.PLUS_Z); |
| FieldVector3D<T> v2 = applyInverseTo(Vector3D.Unit.PLUS_X); |
| if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(true); |
| } |
| return buildArray(v2.getY().atan2(v2.getX()), |
| v2.getZ().asin().negate(), |
| v1.getY().atan2(v1.getZ())); |
| |
| } else if (order == RotationOrder.XYX) { |
| |
| // r (Cartesian3D.plusI) coordinates are : |
| // cos (theta), sin (phi2) sin (theta), cos (phi2) sin (theta) |
| // (-r) (Cartesian3D.plusI) coordinates are : |
| // cos (theta), sin (theta) sin (phi1), -sin (theta) cos (phi1) |
| // and we can choose to have theta in the interval [0 ; PI] |
| FieldVector3D<T> v1 = applyTo(Vector3D.Unit.PLUS_X); |
| FieldVector3D<T> v2 = applyInverseTo(Vector3D.Unit.PLUS_X); |
| if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(false); |
| } |
| return buildArray(v2.getY().atan2(v2.getZ().negate()), |
| v2.getX().acos(), |
| v1.getY().atan2(v1.getZ())); |
| |
| } else if (order == RotationOrder.XZX) { |
| |
| // r (Cartesian3D.plusI) coordinates are : |
| // cos (psi), -cos (phi2) sin (psi), sin (phi2) sin (psi) |
| // (-r) (Cartesian3D.plusI) coordinates are : |
| // cos (psi), sin (psi) cos (phi1), sin (psi) sin (phi1) |
| // and we can choose to have psi in the interval [0 ; PI] |
| FieldVector3D<T> v1 = applyTo(Vector3D.Unit.PLUS_X); |
| FieldVector3D<T> v2 = applyInverseTo(Vector3D.Unit.PLUS_X); |
| if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(false); |
| } |
| return buildArray(v2.getZ().atan2(v2.getY()), |
| v2.getX().acos(), |
| v1.getZ().atan2(v1.getY().negate())); |
| |
| } else if (order == RotationOrder.YXY) { |
| |
| // r (Cartesian3D.plusJ) coordinates are : |
| // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2) |
| // (-r) (Cartesian3D.plusJ) coordinates are : |
| // sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi) |
| // and we can choose to have phi in the interval [0 ; PI] |
| FieldVector3D<T> v1 = applyTo(Vector3D.Unit.PLUS_Y); |
| FieldVector3D<T> v2 = applyInverseTo(Vector3D.Unit.PLUS_Y); |
| if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(false); |
| } |
| return buildArray(v2.getX().atan2(v2.getZ()), |
| v2.getY().acos(), |
| v1.getX().atan2(v1.getZ().negate())); |
| |
| } else if (order == RotationOrder.YZY) { |
| |
| // r (Cartesian3D.plusJ) coordinates are : |
| // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2) |
| // (-r) (Cartesian3D.plusJ) coordinates are : |
| // -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi) |
| // and we can choose to have psi in the interval [0 ; PI] |
| FieldVector3D<T> v1 = applyTo(Vector3D.Unit.PLUS_Y); |
| FieldVector3D<T> v2 = applyInverseTo(Vector3D.Unit.PLUS_Y); |
| if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(false); |
| } |
| return buildArray(v2.getZ().atan2(v2.getX().negate()), |
| v2.getY().acos(), |
| v1.getZ().atan2(v1.getX())); |
| |
| } else if (order == RotationOrder.ZXZ) { |
| |
| // r (Cartesian3D.plusK) coordinates are : |
| // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi) |
| // (-r) (Cartesian3D.plusK) coordinates are : |
| // sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi) |
| // and we can choose to have phi in the interval [0 ; PI] |
| FieldVector3D<T> v1 = applyTo(Vector3D.Unit.PLUS_Z); |
| FieldVector3D<T> v2 = applyInverseTo(Vector3D.Unit.PLUS_Z); |
| if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(false); |
| } |
| return buildArray(v2.getX().atan2(v2.getY().negate()), |
| v2.getZ().acos(), |
| v1.getX().atan2(v1.getY())); |
| |
| } else { // last possibility is ZYZ |
| |
| // r (Cartesian3D.plusK) coordinates are : |
| // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta) |
| // (-r) (Cartesian3D.plusK) coordinates are : |
| // cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta) |
| // and we can choose to have theta in the interval [0 ; PI] |
| FieldVector3D<T> v1 = applyTo(Vector3D.Unit.PLUS_Z); |
| FieldVector3D<T> v2 = applyInverseTo(Vector3D.Unit.PLUS_Z); |
| if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) { |
| throw new CardanEulerSingularityException(false); |
| } |
| return buildArray(v2.getY().atan2(v2.getX()), |
| v2.getZ().acos(), |
| v1.getY().atan2(v1.getX().negate())); |
| |
| } |
| } |
| |
| } |
| |
| /** Create a dimension 3 array. |
| * @param a0 first array element |
| * @param a1 second array element |
| * @param a2 third array element |
| * @return new array |
| */ |
| private T[] buildArray(final T a0, final T a1, final T a2) { |
| final T[] array = MathArrays.buildArray(a0.getField(), 3); |
| array[0] = a0; |
| array[1] = a1; |
| array[2] = a2; |
| return array; |
| } |
| |
| /** Create a constant vector. |
| * @param x abscissa |
| * @param y ordinate |
| * @param z height |
| * @return a constant vector |
| */ |
| private FieldVector3D<T> vector(final double x, final double y, final double z) { |
| final T zero = q0.getField().getZero(); |
| return new FieldVector3D<>(zero.add(x), zero.add(y), zero.add(z)); |
| } |
| |
| /** Get the 3X3 matrix corresponding to the instance |
| * @return the matrix corresponding to the instance |
| */ |
| public T[][] getMatrix() { |
| |
| // products |
| final T q0q0 = q0.multiply(q0); |
| final T q0q1 = q0.multiply(q1); |
| final T q0q2 = q0.multiply(q2); |
| final T q0q3 = q0.multiply(q3); |
| final T q1q1 = q1.multiply(q1); |
| final T q1q2 = q1.multiply(q2); |
| final T q1q3 = q1.multiply(q3); |
| final T q2q2 = q2.multiply(q2); |
| final T q2q3 = q2.multiply(q3); |
| final T q3q3 = q3.multiply(q3); |
| |
| // create the matrix |
| final T[][] m = MathArrays.buildArray(q0.getField(), 3, 3); |
| |
| m [0][0] = q0q0.add(q1q1).multiply(2).subtract(1); |
| m [1][0] = q1q2.subtract(q0q3).multiply(2); |
| m [2][0] = q1q3.add(q0q2).multiply(2); |
| |
| m [0][1] = q1q2.add(q0q3).multiply(2); |
| m [1][1] = q0q0.add(q2q2).multiply(2).subtract(1); |
| m [2][1] = q2q3.subtract(q0q1).multiply(2); |
| |
| m [0][2] = q1q3.subtract(q0q2).multiply(2); |
| m [1][2] = q2q3.add(q0q1).multiply(2); |
| m [2][2] = q0q0.add(q3q3).multiply(2).subtract(1); |
| |
| return m; |
| |
| } |
| |
| /** Convert to a constant vector without derivatives. |
| * @return a constant vector |
| */ |
| public QuaternionRotation toRotation() { |
| return QuaternionRotation.of(q0.getReal(), q1.getReal(), q2.getReal(), q3.getReal()); |
| } |
| |
| /** Apply the rotation to a vector. |
| * @param u vector to apply the rotation to |
| * @return a new vector which is the image of u by the rotation |
| */ |
| public FieldVector3D<T> applyTo(final FieldVector3D<T> u) { |
| |
| final T x = u.getX(); |
| final T y = u.getY(); |
| final T z = u.getZ(); |
| |
| final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z)); |
| |
| return new FieldVector3D<>(q0.multiply(x.multiply(q0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x), |
| q0.multiply(y.multiply(q0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y), |
| q0.multiply(z.multiply(q0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z)); |
| |
| } |
| |
| /** Apply the rotation to a vector. |
| * @param u vector to apply the rotation to |
| * @return a new vector which is the image of u by the rotation |
| */ |
| public FieldVector3D<T> applyTo(final Vector3D u) { |
| |
| final double x = u.getX(); |
| final double y = u.getY(); |
| final double z = u.getZ(); |
| |
| final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z)); |
| |
| return new FieldVector3D<>(q0.multiply(q0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x), |
| q0.multiply(q0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y), |
| q0.multiply(q0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z)); |
| |
| } |
| |
| /** Apply the rotation to a vector stored in an array. |
| * @param in an array with three items which stores vector to rotate |
| * @param out an array with three items to put result to (it can be the same |
| * array as in) |
| */ |
| public void applyTo(final T[] in, final T[] out) { |
| |
| final T x = in[0]; |
| final T y = in[1]; |
| final T z = in[2]; |
| |
| final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z)); |
| |
| out[0] = q0.multiply(x.multiply(q0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x); |
| out[1] = q0.multiply(y.multiply(q0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y); |
| out[2] = q0.multiply(z.multiply(q0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z); |
| |
| } |
| |
| /** Apply the rotation to a vector stored in an array. |
| * @param in an array with three items which stores vector to rotate |
| * @param out an array with three items to put result to |
| */ |
| public void applyTo(final double[] in, final T[] out) { |
| |
| final double x = in[0]; |
| final double y = in[1]; |
| final double z = in[2]; |
| |
| final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z)); |
| |
| out[0] = q0.multiply(q0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x); |
| out[1] = q0.multiply(q0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y); |
| out[2] = q0.multiply(q0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z); |
| |
| } |
| |
| /** Apply a rotation to a vector. |
| * @param rot rotation to apply |
| * @param u vector to apply the rotation to |
| * @param <T> the type of the field elements |
| * @return a new vector which is the image of u by the rotation |
| */ |
| public static <T extends RealFieldElement<T>> FieldVector3D<T> applyTo(final QuaternionRotation rot, final FieldVector3D<T> u) { |
| final Quaternion r = rot.getQuaternion(); |
| final T x = u.getX(); |
| final T y = u.getY(); |
| final T z = u.getZ(); |
| |
| final T s = x.multiply(r.getX()).add(y.multiply(r.getY())).add(z.multiply(r.getZ())); |
| |
| return new FieldVector3D<>(x.multiply(r.getW()).subtract(z.multiply(r.getY()).subtract(y.multiply(r.getZ()))).multiply(r.getW()).add(s.multiply(r.getX())).multiply(2).subtract(x), |
| y.multiply(r.getW()).subtract(x.multiply(r.getZ()).subtract(z.multiply(r.getX()))).multiply(r.getW()).add(s.multiply(r.getY())).multiply(2).subtract(y), |
| z.multiply(r.getW()).subtract(y.multiply(r.getX()).subtract(x.multiply(r.getY()))).multiply(r.getW()).add(s.multiply(r.getZ())).multiply(2).subtract(z)); |
| |
| } |
| |
| /** Apply the inverse of the rotation to a vector. |
| * @param u vector to apply the inverse of the rotation to |
| * @return a new vector which such that u is its image by the rotation |
| */ |
| public FieldVector3D<T> applyInverseTo(final FieldVector3D<T> u) { |
| |
| final T x = u.getX(); |
| final T y = u.getY(); |
| final T z = u.getZ(); |
| |
| final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z)); |
| final T m0 = q0.negate(); |
| |
| return new FieldVector3D<>(m0.multiply(x.multiply(m0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x), |
| m0.multiply(y.multiply(m0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y), |
| m0.multiply(z.multiply(m0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z)); |
| |
| } |
| |
| /** Apply the inverse of the rotation to a vector. |
| * @param u vector to apply the inverse of the rotation to |
| * @return a new vector which such that u is its image by the rotation |
| */ |
| public FieldVector3D<T> applyInverseTo(final Vector3D u) { |
| |
| final double x = u.getX(); |
| final double y = u.getY(); |
| final double z = u.getZ(); |
| |
| final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z)); |
| final T m0 = q0.negate(); |
| |
| return new FieldVector3D<>(m0.multiply(m0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x), |
| m0.multiply(m0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y), |
| m0.multiply(m0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z)); |
| |
| } |
| |
| /** Apply the inverse of the rotation to a vector stored in an array. |
| * @param in an array with three items which stores vector to rotate |
| * @param out an array with three items to put result to (it can be the same |
| * array as in) |
| */ |
| public void applyInverseTo(final T[] in, final T[] out) { |
| |
| final T x = in[0]; |
| final T y = in[1]; |
| final T z = in[2]; |
| |
| final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z)); |
| final T m0 = q0.negate(); |
| |
| out[0] = m0.multiply(x.multiply(m0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x); |
| out[1] = m0.multiply(y.multiply(m0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y); |
| out[2] = m0.multiply(z.multiply(m0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z); |
| |
| } |
| |
| /** Apply the inverse of the rotation to a vector stored in an array. |
| * @param in an array with three items which stores vector to rotate |
| * @param out an array with three items to put result to |
| */ |
| public void applyInverseTo(final double[] in, final T[] out) { |
| |
| final double x = in[0]; |
| final double y = in[1]; |
| final double z = in[2]; |
| |
| final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z)); |
| final T m0 = q0.negate(); |
| |
| out[0] = m0.multiply(m0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x); |
| out[1] = m0.multiply(m0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y); |
| out[2] = m0.multiply(m0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z); |
| |
| } |
| |
| /** Apply the inverse of a rotation to a vector. |
| * @param rot rotation to apply |
| * @param u vector to apply the inverse of the rotation to |
| * @param <T> the type of the field elements |
| * @return a new vector which such that u is its image by the rotation |
| */ |
| public static <T extends RealFieldElement<T>> FieldVector3D<T> applyInverseTo(final QuaternionRotation rot, final FieldVector3D<T> u) { |
| final Quaternion r = rot.getQuaternion(); |
| final T x = u.getX(); |
| final T y = u.getY(); |
| final T z = u.getZ(); |
| |
| final T s = x.multiply(r.getX()).add(y.multiply(r.getY())).add(z.multiply(r.getZ())); |
| final double m0 = -r.getW(); |
| |
| return new FieldVector3D<>(x.multiply(m0).subtract(z.multiply(r.getY()).subtract(y.multiply(r.getZ()))).multiply(m0).add(s.multiply(r.getX())).multiply(2).subtract(x), |
| y.multiply(m0).subtract(x.multiply(r.getZ()).subtract(z.multiply(r.getX()))).multiply(m0).add(s.multiply(r.getY())).multiply(2).subtract(y), |
| z.multiply(m0).subtract(y.multiply(r.getX()).subtract(x.multiply(r.getY()))).multiply(m0).add(s.multiply(r.getZ())).multiply(2).subtract(z)); |
| |
| } |
| |
| /** Apply the instance to another rotation. |
| * <p> |
| * Calling this method is equivalent to call |
| * {@link #compose(FieldRotation, RotationConvention) |
| * compose(r, RotationConvention.VECTOR_OPERATOR)}. |
| * </p> |
| * @param r rotation to apply the rotation to |
| * @return a new rotation which is the composition of r by the instance |
| */ |
| public FieldRotation<T> applyTo(final FieldRotation<T> r) { |
| return compose(r, RotationConvention.VECTOR_OPERATOR); |
| } |
| |
| /** Compose the instance with another rotation. |
| * <p> |
| * If the semantics of the rotations composition corresponds to a |
| * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention, |
| * applying the instance to a rotation is computing the composition |
| * in an order compliant with the following rule : let {@code u} be any |
| * vector and {@code v} its image by {@code r1} (i.e. |
| * {@code r1.applyTo(u) = v}). Let {@code w} be the image of {@code v} by |
| * rotation {@code r2} (i.e. {@code r2.applyTo(v) = w}). Then |
| * {@code w = comp.applyTo(u)}, where |
| * {@code comp = r2.compose(r1, RotationConvention.VECTOR_OPERATOR)}. |
| * </p> |
| * <p> |
| * If the semantics of the rotations composition corresponds to a |
| * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention, |
| * the application order will be reversed. So keeping the exact same |
| * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w} |
| * and {@code comp} as above, {@code comp} could also be computed as |
| * {@code comp = r1.compose(r2, RotationConvention.FRAME_TRANSFORM)}. |
| * </p> |
| * @param r rotation to apply the rotation to |
| * @param convention convention to use for the semantics of the angle |
| * @return a new rotation which is the composition of r by the instance |
| */ |
| public FieldRotation<T> compose(final FieldRotation<T> r, final RotationConvention convention) { |
| return convention == RotationConvention.VECTOR_OPERATOR ? |
| composeInternal(r) : r.composeInternal(this); |
| } |
| |
| /** Compose the instance with another rotation using vector operator convention. |
| * @param r rotation to apply the rotation to |
| * @return a new rotation which is the composition of r by the instance |
| * using vector operator convention |
| */ |
| private FieldRotation<T> composeInternal(final FieldRotation<T> r) { |
| return new FieldRotation<>(r.q0.multiply(q0).subtract(r.q1.multiply(q1).add(r.q2.multiply(q2)).add(r.q3.multiply(q3))), |
| r.q1.multiply(q0).add(r.q0.multiply(q1)).add(r.q2.multiply(q3).subtract(r.q3.multiply(q2))), |
| r.q2.multiply(q0).add(r.q0.multiply(q2)).add(r.q3.multiply(q1).subtract(r.q1.multiply(q3))), |
| r.q3.multiply(q0).add(r.q0.multiply(q3)).add(r.q1.multiply(q2).subtract(r.q2.multiply(q1))), |
| false); |
| } |
| |
| /** Apply the instance to another rotation. |
| * <p> |
| * Calling this method is equivalent to call |
| * {@link #compose(QuaternionRotation, RotationConvention) |
| * compose(r, RotationConvention.VECTOR_OPERATOR)}. |
| * </p> |
| * @param r rotation to apply the rotation to |
| * @return a new rotation which is the composition of r by the instance |
| */ |
| public FieldRotation<T> applyTo(final QuaternionRotation r) { |
| return compose(r, RotationConvention.VECTOR_OPERATOR); |
| } |
| |
| /** Compose the instance with another rotation. |
| * <p> |
| * If the semantics of the rotations composition corresponds to a |
| * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention, |
| * applying the instance to a rotation is computing the composition |
| * in an order compliant with the following rule : let {@code u} be any |
| * vector and {@code v} its image by {@code r1} (i.e. |
| * {@code r1.applyTo(u) = v}). Let {@code w} be the image of {@code v} by |
| * rotation {@code r2} (i.e. {@code r2.applyTo(v) = w}). Then |
| * {@code w = comp.applyTo(u)}, where |
| * {@code comp = r2.compose(r1, RotationConvention.VECTOR_OPERATOR)}. |
| * </p> |
| * <p> |
| * If the semantics of the rotations composition corresponds to a |
| * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention, |
| * the application order will be reversed. So keeping the exact same |
| * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w} |
| * and {@code comp} as above, {@code comp} could also be computed as |
| * {@code comp = r1.compose(r2, RotationConvention.FRAME_TRANSFORM)}. |
| * </p> |
| * @param r rotation to apply the rotation to |
| * @param convention convention to use for the semantics of the angle |
| * @return a new rotation which is the composition of r by the instance |
| */ |
| public FieldRotation<T> compose(final QuaternionRotation r, final RotationConvention convention) { |
| return convention == RotationConvention.VECTOR_OPERATOR ? |
| composeInternal(r) : applyTo(r, this); |
| } |
| |
| /** Compose the instance with another rotation using vector operator convention. |
| * @param rot rotation to apply the rotation to |
| * @return a new rotation which is the composition of r by the instance |
| * using vector operator convention |
| */ |
| private FieldRotation<T> composeInternal(final QuaternionRotation rot) { |
| final Quaternion r = rot.getQuaternion(); |
| return new FieldRotation<>(q0.multiply(r.getW()).subtract(q1.multiply(r.getX()).add(q2.multiply(r.getY())).add(q3.multiply(r.getZ()))), |
| q0.multiply(r.getX()).add(q1.multiply(r.getW())).add(q3.multiply(r.getY()).subtract(q2.multiply(r.getZ()))), |
| q0.multiply(r.getY()).add(q2.multiply(r.getW())).add(q1.multiply(r.getZ()).subtract(q3.multiply(r.getX()))), |
| q0.multiply(r.getZ()).add(q3.multiply(r.getW())).add(q2.multiply(r.getX()).subtract(q1.multiply(r.getY()))), |
| false); |
| } |
| |
| /** Apply a rotation to another rotation. |
| * Applying a rotation to another rotation is computing the composition |
| * in an order compliant with the following rule : let u be any |
| * vector and v its image by rInner (i.e. rInner.applyTo(u) = v), let w be the image |
| * of v by rOuter (i.e. rOuter.applyTo(v) = w), then w = comp.applyTo(u), |
| * where comp = applyTo(rOuter, rInner). |
| * @param rot1 rotation to apply |
| * @param rInner rotation to apply the rotation to |
| * @param <T> the type of the field elements |
| * @return a new rotation which is the composition of r by the instance |
| */ |
| public static <T extends RealFieldElement<T>> FieldRotation<T> applyTo(final QuaternionRotation rot1, final FieldRotation<T> rInner) { |
| final Quaternion r1 = rot1.getQuaternion(); |
| return new FieldRotation<>(rInner.q0.multiply(r1.getW()).subtract(rInner.q1.multiply(r1.getX()).add(rInner.q2.multiply(r1.getY())).add(rInner.q3.multiply(r1.getZ()))), |
| rInner.q1.multiply(r1.getW()).add(rInner.q0.multiply(r1.getX())).add(rInner.q2.multiply(r1.getZ()).subtract(rInner.q3.multiply(r1.getY()))), |
| rInner.q2.multiply(r1.getW()).add(rInner.q0.multiply(r1.getY())).add(rInner.q3.multiply(r1.getX()).subtract(rInner.q1.multiply(r1.getZ()))), |
| rInner.q3.multiply(r1.getW()).add(rInner.q0.multiply(r1.getZ())).add(rInner.q1.multiply(r1.getY()).subtract(rInner.q2.multiply(r1.getX()))), |
| false); |
| } |
| |
| /** Apply the inverse of the instance to another rotation. |
| * <p> |
| * Calling this method is equivalent to call |
| * {@link #composeInverse(FieldRotation, RotationConvention) |
| * composeInverse(r, RotationConvention.VECTOR_OPERATOR)}. |
| * </p> |
| * @param r rotation to apply the rotation to |
| * @return a new rotation which is the composition of r by the inverse |
| * of the instance |
| */ |
| public FieldRotation<T> applyInverseTo(final FieldRotation<T> r) { |
| return composeInverse(r, RotationConvention.VECTOR_OPERATOR); |
| } |
| |
| /** Compose the inverse of the instance with another rotation. |
| * <p> |
| * If the semantics of the rotations composition corresponds to a |
| * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention, |
| * applying the inverse of the instance to a rotation is computing |
| * the composition in an order compliant with the following rule : |
| * let {@code u} be any vector and {@code v} its image by {@code r1} |
| * (i.e. {@code r1.applyTo(u) = v}). Let {@code w} be the inverse image |
| * of {@code v} by {@code r2} (i.e. {@code r2.applyInverseTo(v) = w}). |
| * Then {@code w = comp.applyTo(u)}, where |
| * {@code comp = r2.composeInverse(r1)}. |
| * </p> |
| * <p> |
| * If the semantics of the rotations composition corresponds to a |
| * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention, |
| * the application order will be reversed, which means it is the |
| * <em>innermost</em> rotation that will be reversed. So keeping the exact same |
| * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w} |
| * and {@code comp} as above, {@code comp} could also be computed as |
| * {@code comp = r1.revert().composeInverse(r2.revert(), RotationConvention.FRAME_TRANSFORM)}. |
| * </p> |
| * @param r rotation to apply the rotation to |
| * @param convention convention to use for the semantics of the angle |
| * @return a new rotation which is the composition of r by the inverse |
| * of the instance |
| */ |
| public FieldRotation<T> composeInverse(final FieldRotation<T> r, final RotationConvention convention) { |
| return convention == RotationConvention.VECTOR_OPERATOR ? |
| composeInverseInternal(r) : r.composeInternal(revert()); |
| } |
| |
| /** Compose the inverse of the instance with another rotation |
| * using vector operator convention. |
| * @param r rotation to apply the rotation to |
| * @return a new rotation which is the composition of r by the inverse |
| * of the instance using vector operator convention |
| */ |
| private FieldRotation<T> composeInverseInternal(FieldRotation<T> r) { |
| return new FieldRotation<>(r.q0.multiply(q0).add(r.q1.multiply(q1).add(r.q2.multiply(q2)).add(r.q3.multiply(q3))).negate(), |
| r.q0.multiply(q1).add(r.q2.multiply(q3).subtract(r.q3.multiply(q2))).subtract(r.q1.multiply(q0)), |
| r.q0.multiply(q2).add(r.q3.multiply(q1).subtract(r.q1.multiply(q3))).subtract(r.q2.multiply(q0)), |
| r.q0.multiply(q3).add(r.q1.multiply(q2).subtract(r.q2.multiply(q1))).subtract(r.q3.multiply(q0)), |
| false); |
| } |
| |
| /** Apply the inverse of the instance to another rotation. |
| * <p> |
| * Calling this method is equivalent to call |
| * {@link #composeInverse(QuaternionRotation, RotationConvention) |
| * composeInverse(r, RotationConvention.VECTOR_OPERATOR)}. |
| * </p> |
| * @param r rotation to apply the rotation to |
| * @return a new rotation which is the composition of r by the inverse |
| * of the instance |
| */ |
| public FieldRotation<T> applyInverseTo(final QuaternionRotation r) { |
| return composeInverse(r, RotationConvention.VECTOR_OPERATOR); |
| } |
| |
| /** Compose the inverse of the instance with another rotation. |
| * <p> |
| * If the semantics of the rotations composition corresponds to a |
| * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention, |
| * applying the inverse of the instance to a rotation is computing |
| * the composition in an order compliant with the following rule : |
| * let {@code u} be any vector and {@code v} its image by {@code r1} |
| * (i.e. {@code r1.applyTo(u) = v}). Let {@code w} be the inverse image |
| * of {@code v} by {@code r2} (i.e. {@code r2.applyInverseTo(v) = w}). |
| * Then {@code w = comp.applyTo(u)}, where |
| * {@code comp = r2.composeInverse(r1)}. |
| * </p> |
| * <p> |
| * If the semantics of the rotations composition corresponds to a |
| * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention, |
| * the application order will be reversed, which means it is the |
| * <em>innermost</em> rotation that will be reversed. So keeping the exact same |
| * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w} |
| * and {@code comp} as above, {@code comp} could also be computed as |
| * {@code comp = r1.revert().composeInverse(r2.revert(), RotationConvention.FRAME_TRANSFORM)}. |
| * </p> |
| * @param r rotation to apply the rotation to |
| * @param convention convention to use for the semantics of the angle |
| * @return a new rotation which is the composition of r by the inverse |
| * of the instance |
| */ |
| public FieldRotation<T> composeInverse(final QuaternionRotation r, final RotationConvention convention) { |
| return convention == RotationConvention.VECTOR_OPERATOR ? |
| composeInverseInternal(r) : applyTo(r, revert()); |
| } |
| |
| /** Compose the inverse of the instance with another rotation |
| * using vector operator convention. |
| * @param rot rotation to apply the rotation to |
| * @return a new rotation which is the composition of r by the inverse |
| * of the instance using vector operator convention |
| */ |
| private FieldRotation<T> composeInverseInternal(QuaternionRotation rot) { |
| final Quaternion r = rot.getQuaternion(); |
| return new FieldRotation<>(q0.multiply(r.getW()).add(q1.multiply(r.getX()).add(q2.multiply(r.getY())).add(q3.multiply(r.getZ()))).negate(), |
| q1.multiply(r.getW()).add(q3.multiply(r.getY()).subtract(q2.multiply(r.getZ()))).subtract(q0.multiply(r.getX())), |
| q2.multiply(r.getW()).add(q1.multiply(r.getZ()).subtract(q3.multiply(r.getX()))).subtract(q0.multiply(r.getY())), |
| q3.multiply(r.getW()).add(q2.multiply(r.getX()).subtract(q1.multiply(r.getY()))).subtract(q0.multiply(r.getZ())), |
| false); |
| } |
| |
| /** Apply the inverse of a rotation to another rotation. |
| * Applying the inverse of a rotation to another rotation is computing |
| * the composition in an order compliant with the following rule : |
| * let u be any vector and v its image by rInner (i.e. rInner.applyTo(u) = v), |
| * let w be the inverse image of v by rOuter |
| * (i.e. rOuter.applyInverseTo(v) = w), then w = comp.applyTo(u), where |
| * comp = applyInverseTo(rOuter, rInner). |
| * @param rotOuter rotation to apply the rotation to |
| * @param rInner rotation to apply the rotation to |
| * @param <T> the type of the field elements |
| * @return a new rotation which is the composition of r by the inverse |
| * of the instance |
| */ |
| public static <T extends RealFieldElement<T>> FieldRotation<T> applyInverseTo(final QuaternionRotation rotOuter, final FieldRotation<T> rInner) { |
| final Quaternion rOuter = rotOuter.getQuaternion(); |
| return new FieldRotation<>(rInner.q0.multiply(rOuter.getW()).add(rInner.q1.multiply(rOuter.getX()).add(rInner.q2.multiply(rOuter.getY())).add(rInner.q3.multiply(rOuter.getZ()))).negate(), |
| rInner.q0.multiply(rOuter.getX()).add(rInner.q2.multiply(rOuter.getZ()).subtract(rInner.q3.multiply(rOuter.getY()))).subtract(rInner.q1.multiply(rOuter.getW())), |
| rInner.q0.multiply(rOuter.getY()).add(rInner.q3.multiply(rOuter.getX()).subtract(rInner.q1.multiply(rOuter.getZ()))).subtract(rInner.q2.multiply(rOuter.getW())), |
| rInner.q0.multiply(rOuter.getZ()).add(rInner.q1.multiply(rOuter.getY()).subtract(rInner.q2.multiply(rOuter.getX()))).subtract(rInner.q3.multiply(rOuter.getW())), |
| false); |
| } |
| |
| /** Perfect orthogonality on a 3X3 matrix. |
| * @param m initial matrix (not exactly orthogonal) |
| * @param threshold convergence threshold for the iterative |
| * orthogonality correction (convergence is reached when the |
| * difference between two steps of the Frobenius norm of the |
| * correction is below this threshold) |
| * @return an orthogonal matrix close to m |
| * @exception NotARotationMatrixException if the matrix cannot be |
| * orthogonalized with the given threshold after 10 iterations |
| */ |
| private T[][] orthogonalizeMatrix(final T[][] m, final double threshold) |
| throws NotARotationMatrixException { |
| |
| T x00 = m[0][0]; |
| T x01 = m[0][1]; |
| T x02 = m[0][2]; |
| T x10 = m[1][0]; |
| T x11 = m[1][1]; |
| T x12 = m[1][2]; |
| T x20 = m[2][0]; |
| T x21 = m[2][1]; |
| T x22 = m[2][2]; |
| double fn = 0; |
| double fn1; |
| |
| final T[][] o = MathArrays.buildArray(m[0][0].getField(), 3, 3); |
| |
| // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M) |
| int i = 0; |
| while (++i < 11) { |
| |
| // Mt.Xn |
| final T mx00 = m[0][0].multiply(x00).add(m[1][0].multiply(x10)).add(m[2][0].multiply(x20)); |
| final T mx10 = m[0][1].multiply(x00).add(m[1][1].multiply(x10)).add(m[2][1].multiply(x20)); |
| final T mx20 = m[0][2].multiply(x00).add(m[1][2].multiply(x10)).add(m[2][2].multiply(x20)); |
| final T mx01 = m[0][0].multiply(x01).add(m[1][0].multiply(x11)).add(m[2][0].multiply(x21)); |
| final T mx11 = m[0][1].multiply(x01).add(m[1][1].multiply(x11)).add(m[2][1].multiply(x21)); |
| final T mx21 = m[0][2].multiply(x01).add(m[1][2].multiply(x11)).add(m[2][2].multiply(x21)); |
| final T mx02 = m[0][0].multiply(x02).add(m[1][0].multiply(x12)).add(m[2][0].multiply(x22)); |
| final T mx12 = m[0][1].multiply(x02).add(m[1][1].multiply(x12)).add(m[2][1].multiply(x22)); |
| final T mx22 = m[0][2].multiply(x02).add(m[1][2].multiply(x12)).add(m[2][2].multiply(x22)); |
| |
| // Xn+1 |
| o[0][0] = x00.subtract(x00.multiply(mx00).add(x01.multiply(mx10)).add(x02.multiply(mx20)).subtract(m[0][0]).multiply(0.5)); |
| o[0][1] = x01.subtract(x00.multiply(mx01).add(x01.multiply(mx11)).add(x02.multiply(mx21)).subtract(m[0][1]).multiply(0.5)); |
| o[0][2] = x02.subtract(x00.multiply(mx02).add(x01.multiply(mx12)).add(x02.multiply(mx22)).subtract(m[0][2]).multiply(0.5)); |
| o[1][0] = x10.subtract(x10.multiply(mx00).add(x11.multiply(mx10)).add(x12.multiply(mx20)).subtract(m[1][0]).multiply(0.5)); |
| o[1][1] = x11.subtract(x10.multiply(mx01).add(x11.multiply(mx11)).add(x12.multiply(mx21)).subtract(m[1][1]).multiply(0.5)); |
| o[1][2] = x12.subtract(x10.multiply(mx02).add(x11.multiply(mx12)).add(x12.multiply(mx22)).subtract(m[1][2]).multiply(0.5)); |
| o[2][0] = x20.subtract(x20.multiply(mx00).add(x21.multiply(mx10)).add(x22.multiply(mx20)).subtract(m[2][0]).multiply(0.5)); |
| o[2][1] = x21.subtract(x20.multiply(mx01).add(x21.multiply(mx11)).add(x22.multiply(mx21)).subtract(m[2][1]).multiply(0.5)); |
| o[2][2] = x22.subtract(x20.multiply(mx02).add(x21.multiply(mx12)).add(x22.multiply(mx22)).subtract(m[2][2]).multiply(0.5)); |
| |
| // correction on each elements |
| final double corr00 = o[0][0].getReal() - m[0][0].getReal(); |
| final double corr01 = o[0][1].getReal() - m[0][1].getReal(); |
| final double corr02 = o[0][2].getReal() - m[0][2].getReal(); |
| final double corr10 = o[1][0].getReal() - m[1][0].getReal(); |
| final double corr11 = o[1][1].getReal() - m[1][1].getReal(); |
| final double corr12 = o[1][2].getReal() - m[1][2].getReal(); |
| final double corr20 = o[2][0].getReal() - m[2][0].getReal(); |
| final double corr21 = o[2][1].getReal() - m[2][1].getReal(); |
| final double corr22 = o[2][2].getReal() - m[2][2].getReal(); |
| |
| // Frobenius norm of the correction |
| fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 + |
| corr10 * corr10 + corr11 * corr11 + corr12 * corr12 + |
| corr20 * corr20 + corr21 * corr21 + corr22 * corr22; |
| |
| // convergence test |
| if (FastMath.abs(fn1 - fn) <= threshold) { |
| return o; |
| } |
| |
| // prepare next iteration |
| x00 = o[0][0]; |
| x01 = o[0][1]; |
| x02 = o[0][2]; |
| x10 = o[1][0]; |
| x11 = o[1][1]; |
| x12 = o[1][2]; |
| x20 = o[2][0]; |
| x21 = o[2][1]; |
| x22 = o[2][2]; |
| fn = fn1; |
| |
| } |
| |
| // the algorithm did not converge after 10 iterations |
| throw new NotARotationMatrixException(LocalizedFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX, |
| i - 1); |
| |
| } |
| |
| /** Compute the <i>distance</i> between two rotations. |
| * <p>The <i>distance</i> is intended here as a way to check if two |
| * rotations are almost similar (i.e. they transform vectors the same way) |
| * or very different. It is mathematically defined as the angle of |
| * the rotation r that prepended to one of the rotations gives the other |
| * one:</p> |
| * <div style="white-space: pre"><code> |
| * r<sub>1</sub>(r) = r<sub>2</sub> |
| * </code></div> |
| * <p>This distance is an angle between 0 and π. Its value is the smallest |
| * possible upper bound of the angle in radians between r<sub>1</sub>(v) |
| * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is |
| * reached for some v. The distance is equal to 0 if and only if the two |
| * rotations are identical.</p> |
| * <p>Comparing two rotations should always be done using this value rather |
| * than for example comparing the components of the quaternions. It is much |
| * more stable, and has a geometric meaning. Also comparing quaternions |
| * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64) |
| * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite |
| * their components are different (they are exact opposites).</p> |
| * @param r1 first rotation |
| * @param r2 second rotation |
| * @param <T> the type of the field elements |
| * @return <i>distance</i> between r1 and r2 |
| */ |
| public static <T extends RealFieldElement<T>> T distance(final FieldRotation<T> r1, final FieldRotation<T> r2) { |
| return r1.composeInverseInternal(r2).getAngle(); |
| } |
| |
| } |
