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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2023 Google Inc. All rights reserved.
// http://ceres-solver.org/
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
// * Neither the name of Google Inc. nor the names of its contributors may be
// used to endorse or promote products derived from this software without
// specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
//
// Author: sameeragarwal@google.com (Sameer Agarwal)
#include "ceres/rotation.h"
#include <algorithm>
#include <array>
#include <cmath>
#include <limits>
#include <random>
#include <string>
#include <utility>
#include "ceres/constants.h"
#include "ceres/internal/eigen.h"
#include "ceres/internal/euler_angles.h"
#include "ceres/internal/export.h"
#include "ceres/is_close.h"
#include "ceres/jet.h"
#include "ceres/stringprintf.h"
#include "ceres/test_util.h"
#include "glog/logging.h"
#include "gmock/gmock.h"
#include "gtest/gtest.h"
namespace ceres {
namespace internal {
inline constexpr double kPi = constants::pi_v<double>;
const double kHalfSqrt2 = 0.707106781186547524401;
// A tolerance value for floating-point comparisons.
static double const kTolerance = std::numeric_limits<double>::epsilon() * 10;
// Looser tolerance used for numerically unstable conversions.
static double const kLooseTolerance = 1e-9;
// Use as:
// double quaternion[4];
// EXPECT_THAT(quaternion, IsNormalizedQuaternion());
MATCHER(IsNormalizedQuaternion, "") {
double norm2 =
arg[0] * arg[0] + arg[1] * arg[1] + arg[2] * arg[2] + arg[3] * arg[3];
if (fabs(norm2 - 1.0) > kTolerance) {
*result_listener << "squared norm is " << norm2;
return false;
}
return true;
}
// Use as:
// double expected_quaternion[4];
// double actual_quaternion[4];
// EXPECT_THAT(actual_quaternion, IsNearQuaternion(expected_quaternion));
MATCHER_P(IsNearQuaternion, expected, "") {
// Quaternions are equivalent upto a sign change. So we will compare
// both signs before declaring failure.
bool is_near = true;
// NOTE: near (and far) can be defined as macros on the Windows platform (for
// ancient pascal calling convention). Do not use these identifiers.
for (int i = 0; i < 4; i++) {
if (fabs(arg[i] - expected[i]) > kTolerance) {
is_near = false;
break;
}
}
if (is_near) {
return true;
}
is_near = true;
for (int i = 0; i < 4; i++) {
if (fabs(arg[i] + expected[i]) > kTolerance) {
is_near = false;
break;
}
}
if (is_near) {
return true;
}
// clang-format off
*result_listener << "expected : "
<< expected[0] << " "
<< expected[1] << " "
<< expected[2] << " "
<< expected[3] << " "
<< "actual : "
<< arg[0] << " "
<< arg[1] << " "
<< arg[2] << " "
<< arg[3];
// clang-format on
return false;
}
// Use as:
// double expected_axis_angle[3];
// double actual_axis_angle[3];
// EXPECT_THAT(actual_axis_angle, IsNearAngleAxis(expected_axis_angle));
MATCHER_P(IsNearAngleAxis, expected, "") {
Eigen::Vector3d a(arg[0], arg[1], arg[2]);
Eigen::Vector3d e(expected[0], expected[1], expected[2]);
const double e_norm = e.norm();
double delta_norm = std::numeric_limits<double>::max();
if (e_norm > 0) {
// Deal with the sign ambiguity near PI. Since the sign can flip,
// we take the smaller of the two differences.
if (fabs(e_norm - kPi) < kLooseTolerance) {
delta_norm = std::min((a - e).norm(), (a + e).norm()) / e_norm;
} else {
delta_norm = (a - e).norm() / e_norm;
}
} else {
delta_norm = a.norm();
}
if (delta_norm <= kLooseTolerance) {
return true;
}
// clang-format off
*result_listener << " arg:"
<< " " << arg[0]
<< " " << arg[1]
<< " " << arg[2]
<< " was expected to be:"
<< " " << expected[0]
<< " " << expected[1]
<< " " << expected[2];
// clang-format on
return false;
}
// Use as:
// double matrix[9];
// EXPECT_THAT(matrix, IsOrthonormal());
MATCHER(IsOrthonormal, "") {
for (int c1 = 0; c1 < 3; c1++) {
for (int c2 = 0; c2 < 3; c2++) {
double v = 0;
for (int i = 0; i < 3; i++) {
v += arg[i + 3 * c1] * arg[i + 3 * c2];
}
double expected = (c1 == c2) ? 1 : 0;
if (fabs(expected - v) > kTolerance) {
*result_listener << "Columns " << c1 << " and " << c2
<< " should have dot product " << expected
<< " but have " << v;
return false;
}
}
}
return true;
}
// Use as:
// double matrix1[9];
// double matrix2[9];
// EXPECT_THAT(matrix1, IsNear3x3Matrix(matrix2));
MATCHER_P(IsNear3x3Matrix, expected, "") {
for (int i = 0; i < 9; i++) {
if (fabs(arg[i] - expected[i]) > kTolerance) {
*result_listener << "component " << i << " should be " << expected[i];
return false;
}
}
return true;
}
// Transforms a zero axis/angle to a quaternion.
TEST(Rotation, ZeroAngleAxisToQuaternion) {
double axis_angle[3] = {0, 0, 0};
double quaternion[4];
double expected[4] = {1, 0, 0, 0};
AngleAxisToQuaternion(axis_angle, quaternion);
EXPECT_THAT(quaternion, IsNormalizedQuaternion());
EXPECT_THAT(quaternion, IsNearQuaternion(expected));
}
// Test that exact conversion works for small angles.
TEST(Rotation, SmallAngleAxisToQuaternion) {
// Small, finite value to test.
double theta = 1.0e-2;
double axis_angle[3] = {theta, 0, 0};
double quaternion[4];
double expected[4] = {cos(theta / 2), sin(theta / 2.0), 0, 0};
AngleAxisToQuaternion(axis_angle, quaternion);
EXPECT_THAT(quaternion, IsNormalizedQuaternion());
EXPECT_THAT(quaternion, IsNearQuaternion(expected));
}
// Test that approximate conversion works for very small angles.
TEST(Rotation, TinyAngleAxisToQuaternion) {
// Very small value that could potentially cause underflow.
double theta = pow(std::numeric_limits<double>::min(), 0.75);
double axis_angle[3] = {theta, 0, 0};
double quaternion[4];
double expected[4] = {cos(theta / 2), sin(theta / 2.0), 0, 0};
AngleAxisToQuaternion(axis_angle, quaternion);
EXPECT_THAT(quaternion, IsNormalizedQuaternion());
EXPECT_THAT(quaternion, IsNearQuaternion(expected));
}
// Transforms a rotation by pi/2 around X to a quaternion.
TEST(Rotation, XRotationToQuaternion) {
double axis_angle[3] = {kPi / 2, 0, 0};
double quaternion[4];
double expected[4] = {kHalfSqrt2, kHalfSqrt2, 0, 0};
AngleAxisToQuaternion(axis_angle, quaternion);
EXPECT_THAT(quaternion, IsNormalizedQuaternion());
EXPECT_THAT(quaternion, IsNearQuaternion(expected));
}
// Transforms a unit quaternion to an axis angle.
TEST(Rotation, UnitQuaternionToAngleAxis) {
double quaternion[4] = {1, 0, 0, 0};
double axis_angle[3];
double expected[3] = {0, 0, 0};
QuaternionToAngleAxis(quaternion, axis_angle);
EXPECT_THAT(axis_angle, IsNearAngleAxis(expected));
}
// Transforms a quaternion that rotates by pi about the Y axis to an axis angle.
TEST(Rotation, YRotationQuaternionToAngleAxis) {
double quaternion[4] = {0, 0, 1, 0};
double axis_angle[3];
double expected[3] = {0, kPi, 0};
QuaternionToAngleAxis(quaternion, axis_angle);
EXPECT_THAT(axis_angle, IsNearAngleAxis(expected));
}
// Transforms a quaternion that rotates by pi/3 about the Z axis to an axis
// angle.
TEST(Rotation, ZRotationQuaternionToAngleAxis) {
double quaternion[4] = {sqrt(3) / 2, 0, 0, 0.5};
double axis_angle[3];
double expected[3] = {0, 0, kPi / 3};
QuaternionToAngleAxis(quaternion, axis_angle);
EXPECT_THAT(axis_angle, IsNearAngleAxis(expected));
}
// Test that exact conversion works for small angles.
TEST(Rotation, SmallQuaternionToAngleAxis) {
// Small, finite value to test.
double theta = 1.0e-2;
double quaternion[4] = {cos(theta / 2), sin(theta / 2.0), 0, 0};
double axis_angle[3];
double expected[3] = {theta, 0, 0};
QuaternionToAngleAxis(quaternion, axis_angle);
EXPECT_THAT(axis_angle, IsNearAngleAxis(expected));
}
// Test that approximate conversion works for very small angles.
TEST(Rotation, TinyQuaternionToAngleAxis) {
// Very small value that could potentially cause underflow.
double theta = pow(std::numeric_limits<double>::min(), 0.75);
double quaternion[4] = {cos(theta / 2), sin(theta / 2.0), 0, 0};
double axis_angle[3];
double expected[3] = {theta, 0, 0};
QuaternionToAngleAxis(quaternion, axis_angle);
EXPECT_THAT(axis_angle, IsNearAngleAxis(expected));
}
TEST(Rotation, QuaternionToAngleAxisAngleIsLessThanPi) {
double quaternion[4];
double angle_axis[3];
const double half_theta = 0.75 * kPi;
quaternion[0] = cos(half_theta);
quaternion[1] = 1.0 * sin(half_theta);
quaternion[2] = 0.0;
quaternion[3] = 0.0;
QuaternionToAngleAxis(quaternion, angle_axis);
const double angle = std::hypot(angle_axis[0], angle_axis[1], angle_axis[2]);
EXPECT_LE(angle, kPi);
}
static constexpr int kNumTrials = 10000;
// Takes a bunch of random axis/angle values, converts them to quaternions,
// and back again.
TEST(Rotation, AngleAxisToQuaterionAndBack) {
std::mt19937 prng;
std::uniform_real_distribution<double> uniform_distribution{-1.0, 1.0};
for (int i = 0; i < kNumTrials; i++) {
double axis_angle[3];
// Make an axis by choosing three random numbers in [-1, 1) and
// normalizing.
double norm = 0;
for (double& coeff : axis_angle) {
coeff = uniform_distribution(prng);
norm += coeff * coeff;
}
norm = sqrt(norm);
// Angle in [-pi, pi).
double theta = uniform_distribution(
prng, std::uniform_real_distribution<double>::param_type{-kPi, kPi});
for (double& coeff : axis_angle) {
coeff = coeff * theta / norm;
}
double quaternion[4];
double round_trip[3];
// We use ASSERTs here because if there's one failure, there are
// probably many and spewing a million failures doesn't make anyone's
// day.
AngleAxisToQuaternion(axis_angle, quaternion);
ASSERT_THAT(quaternion, IsNormalizedQuaternion());
QuaternionToAngleAxis(quaternion, round_trip);
ASSERT_THAT(round_trip, IsNearAngleAxis(axis_angle));
}
}
// Takes a bunch of random quaternions, converts them to axis/angle,
// and back again.
TEST(Rotation, QuaterionToAngleAxisAndBack) {
std::mt19937 prng;
std::uniform_real_distribution<double> uniform_distribution{-1.0, 1.0};
for (int i = 0; i < kNumTrials; i++) {
double quaternion[4];
// Choose four random numbers in [-1, 1) and normalize.
double norm = 0;
for (double& coeff : quaternion) {
coeff = uniform_distribution(prng);
norm += coeff * coeff;
}
norm = sqrt(norm);
for (double& coeff : quaternion) {
coeff = coeff / norm;
}
double axis_angle[3];
double round_trip[4];
QuaternionToAngleAxis(quaternion, axis_angle);
AngleAxisToQuaternion(axis_angle, round_trip);
ASSERT_THAT(round_trip, IsNormalizedQuaternion());
ASSERT_THAT(round_trip, IsNearQuaternion(quaternion));
}
}
// Transforms a zero axis/angle to a rotation matrix.
TEST(Rotation, ZeroAngleAxisToRotationMatrix) {
double axis_angle[3] = {0, 0, 0};
double matrix[9];
double expected[9] = {1, 0, 0, 0, 1, 0, 0, 0, 1};
AngleAxisToRotationMatrix(axis_angle, matrix);
EXPECT_THAT(matrix, IsOrthonormal());
EXPECT_THAT(matrix, IsNear3x3Matrix(expected));
}
TEST(Rotation, NearZeroAngleAxisToRotationMatrix) {
double axis_angle[3] = {1e-24, 2e-24, 3e-24};
double matrix[9];
double expected[9] = {1, 0, 0, 0, 1, 0, 0, 0, 1};
AngleAxisToRotationMatrix(axis_angle, matrix);
EXPECT_THAT(matrix, IsOrthonormal());
EXPECT_THAT(matrix, IsNear3x3Matrix(expected));
}
// Transforms a rotation by pi/2 around X to a rotation matrix and back.
TEST(Rotation, XRotationToRotationMatrix) {
double axis_angle[3] = {kPi / 2, 0, 0};
double matrix[9];
// The rotation matrices are stored column-major.
double expected[9] = {1, 0, 0, 0, 0, 1, 0, -1, 0};
AngleAxisToRotationMatrix(axis_angle, matrix);
EXPECT_THAT(matrix, IsOrthonormal());
EXPECT_THAT(matrix, IsNear3x3Matrix(expected));
double round_trip[3];
RotationMatrixToAngleAxis(matrix, round_trip);
EXPECT_THAT(round_trip, IsNearAngleAxis(axis_angle));
}
// Transforms an axis angle that rotates by pi about the Y axis to a
// rotation matrix and back.
TEST(Rotation, YRotationToRotationMatrix) {
double axis_angle[3] = {0, kPi, 0};
double matrix[9];
double expected[9] = {-1, 0, 0, 0, 1, 0, 0, 0, -1};
AngleAxisToRotationMatrix(axis_angle, matrix);
EXPECT_THAT(matrix, IsOrthonormal());
EXPECT_THAT(matrix, IsNear3x3Matrix(expected));
double round_trip[3];
RotationMatrixToAngleAxis(matrix, round_trip);
EXPECT_THAT(round_trip, IsNearAngleAxis(axis_angle));
}
TEST(Rotation, NearPiAngleAxisRoundTrip) {
double in_axis_angle[3];
double matrix[9];
double out_axis_angle[3];
std::mt19937 prng;
std::uniform_real_distribution<double> uniform_distribution{-1.0, 1.0};
for (int i = 0; i < kNumTrials; i++) {
// Make an axis by choosing three random numbers in [-1, 1) and
// normalizing.
double norm = 0;
for (double& coeff : in_axis_angle) {
coeff = uniform_distribution(prng);
norm += coeff * coeff;
}
norm = sqrt(norm);
// Angle in [pi - kMaxSmallAngle, pi).
constexpr double kMaxSmallAngle = 1e-8;
double theta =
uniform_distribution(prng,
std::uniform_real_distribution<double>::param_type{
kPi - kMaxSmallAngle, kPi});
for (double& coeff : in_axis_angle) {
coeff *= (theta / norm);
}
AngleAxisToRotationMatrix(in_axis_angle, matrix);
RotationMatrixToAngleAxis(matrix, out_axis_angle);
EXPECT_THAT(in_axis_angle, IsNearAngleAxis(out_axis_angle));
}
}
TEST(Rotation, AtPiAngleAxisRoundTrip) {
// A rotation of kPi about the X axis;
// clang-format off
static constexpr double kMatrix[3][3] = {
{1.0, 0.0, 0.0},
{0.0, -1.0, 0.0},
{0.0, 0.0, -1.0}
};
// clang-format on
double in_matrix[9];
// Fill it from kMatrix in col-major order.
for (int j = 0, k = 0; j < 3; ++j) {
for (int i = 0; i < 3; ++i, ++k) {
in_matrix[k] = kMatrix[i][j];
}
}
const double expected_axis_angle[3] = {kPi, 0, 0};
double out_matrix[9];
double axis_angle[3];
RotationMatrixToAngleAxis(in_matrix, axis_angle);
AngleAxisToRotationMatrix(axis_angle, out_matrix);
LOG(INFO) << "AngleAxis = " << axis_angle[0] << " " << axis_angle[1] << " "
<< axis_angle[2];
LOG(INFO) << "Expected AngleAxis = " << kPi << " 0 0";
double out_rowmajor[3][3];
for (int j = 0, k = 0; j < 3; ++j) {
for (int i = 0; i < 3; ++i, ++k) {
out_rowmajor[i][j] = out_matrix[k];
}
}
LOG(INFO) << "Rotation:";
LOG(INFO) << "EXPECTED | ACTUAL";
for (int i = 0; i < 3; ++i) {
std::string line;
for (int j = 0; j < 3; ++j) {
StringAppendF(&line, "%g ", kMatrix[i][j]);
}
line += " | ";
for (int j = 0; j < 3; ++j) {
StringAppendF(&line, "%g ", out_rowmajor[i][j]);
}
LOG(INFO) << line;
}
EXPECT_THAT(axis_angle, IsNearAngleAxis(expected_axis_angle));
EXPECT_THAT(out_matrix, IsNear3x3Matrix(in_matrix));
}
// Transforms an axis angle that rotates by pi/3 about the Z axis to a
// rotation matrix.
TEST(Rotation, ZRotationToRotationMatrix) {
double axis_angle[3] = {0, 0, kPi / 3};
double matrix[9];
// This is laid-out row-major on the screen but is actually stored
// column-major.
// clang-format off
double expected[9] = { 0.5, sqrt(3) / 2, 0, // Column 1
-sqrt(3) / 2, 0.5, 0, // Column 2
0, 0, 1 }; // Column 3
// clang-format on
AngleAxisToRotationMatrix(axis_angle, matrix);
EXPECT_THAT(matrix, IsOrthonormal());
EXPECT_THAT(matrix, IsNear3x3Matrix(expected));
double round_trip[3];
RotationMatrixToAngleAxis(matrix, round_trip);
EXPECT_THAT(round_trip, IsNearAngleAxis(axis_angle));
}
// Takes a bunch of random axis/angle values, converts them to rotation
// matrices, and back again.
TEST(Rotation, AngleAxisToRotationMatrixAndBack) {
std::mt19937 prng;
std::uniform_real_distribution<double> uniform_distribution{-1.0, 1.0};
for (int i = 0; i < kNumTrials; i++) {
double axis_angle[3];
// Make an axis by choosing three random numbers in [-1, 1) and
// normalizing.
double norm = 0;
for (double& i : axis_angle) {
i = uniform_distribution(prng);
norm += i * i;
}
norm = sqrt(norm);
// Angle in [-pi, pi).
double theta = uniform_distribution(
prng, std::uniform_real_distribution<double>::param_type{-kPi, kPi});
for (double& i : axis_angle) {
i = i * theta / norm;
}
double matrix[9];
double round_trip[3];
AngleAxisToRotationMatrix(axis_angle, matrix);
ASSERT_THAT(matrix, IsOrthonormal());
RotationMatrixToAngleAxis(matrix, round_trip);
for (int i = 0; i < 3; ++i) {
EXPECT_NEAR(round_trip[i], axis_angle[i], kLooseTolerance);
}
}
}
// Takes a bunch of random axis/angle values near zero, converts them
// to rotation matrices, and back again.
TEST(Rotation, AngleAxisToRotationMatrixAndBackNearZero) {
std::mt19937 prng;
std::uniform_real_distribution<double> uniform_distribution{-1.0, 1.0};
for (int i = 0; i < kNumTrials; i++) {
double axis_angle[3];
// Make an axis by choosing three random numbers in [-1, 1) and
// normalizing.
double norm = 0;
for (double& i : axis_angle) {
i = uniform_distribution(prng);
norm += i * i;
}
norm = sqrt(norm);
// Tiny theta.
constexpr double kScale = 1e-16;
double theta =
uniform_distribution(prng,
std::uniform_real_distribution<double>::param_type{
-kScale * kPi, kScale * kPi});
for (double& i : axis_angle) {
i = i * theta / norm;
}
double matrix[9];
double round_trip[3];
AngleAxisToRotationMatrix(axis_angle, matrix);
ASSERT_THAT(matrix, IsOrthonormal());
RotationMatrixToAngleAxis(matrix, round_trip);
for (int i = 0; i < 3; ++i) {
EXPECT_NEAR(
round_trip[i], axis_angle[i], std::numeric_limits<double>::epsilon());
}
}
}
// Transposes a 3x3 matrix.
static void Transpose3x3(double m[9]) {
std::swap(m[1], m[3]);
std::swap(m[2], m[6]);
std::swap(m[5], m[7]);
}
// Convert Euler angles from radians to degrees.
static void ToDegrees(double euler_angles[3]) {
for (int i = 0; i < 3; ++i) {
euler_angles[i] *= 180.0 / kPi;
}
}
// Compare the 3x3 rotation matrices produced by the axis-angle
// rotation 'aa' and the Euler angle rotation 'ea' (in radians).
static void CompareEulerToAngleAxis(double aa[3], double ea[3]) {
double aa_matrix[9];
AngleAxisToRotationMatrix(aa, aa_matrix);
Transpose3x3(aa_matrix); // Column to row major order.
double ea_matrix[9];
ToDegrees(ea); // Radians to degrees.
const int kRowStride = 3;
EulerAnglesToRotationMatrix(ea, kRowStride, ea_matrix);
EXPECT_THAT(aa_matrix, IsOrthonormal());
EXPECT_THAT(ea_matrix, IsOrthonormal());
EXPECT_THAT(ea_matrix, IsNear3x3Matrix(aa_matrix));
}
// Test with rotation axis along the x/y/z axes.
// Also test zero rotation.
TEST(EulerAnglesToRotationMatrix, OnAxis) {
int n_tests = 0;
for (double x = -1.0; x <= 1.0; x += 1.0) {
for (double y = -1.0; y <= 1.0; y += 1.0) {
for (double z = -1.0; z <= 1.0; z += 1.0) {
if ((x != 0) + (y != 0) + (z != 0) > 1) continue;
double axis_angle[3] = {x, y, z};
double euler_angles[3] = {x, y, z};
CompareEulerToAngleAxis(axis_angle, euler_angles);
++n_tests;
}
}
}
CHECK_EQ(7, n_tests);
}
// Test that a random rotation produces an orthonormal rotation
// matrix.
TEST(EulerAnglesToRotationMatrix, IsOrthonormal) {
std::mt19937 prng;
std::uniform_real_distribution<double> uniform_distribution{-180.0, 180.0};
for (int trial = 0; trial < kNumTrials; ++trial) {
double euler_angles_degrees[3];
for (double& euler_angles_degree : euler_angles_degrees) {
euler_angles_degree = uniform_distribution(prng);
}
double rotation_matrix[9];
EulerAnglesToRotationMatrix(euler_angles_degrees, 3, rotation_matrix);
EXPECT_THAT(rotation_matrix, IsOrthonormal());
}
}
static double sample_euler[][3] = {{0.5235988, 1.047198, 0.7853982},
{0.5235988, 1.047198, 0.5235988},
{0.7853982, 0.5235988, 1.047198}};
// ZXY Intrinsic Euler Angle to rotation matrix conversion test from
// scipy/spatial/transform/test/test_rotation.py
TEST(EulerAngles, IntrinsicEulerSequence312ToRotationMatrixCanned) {
// clang-format off
double const expected[][9] =
{{0.306186083320088, -0.249999816228639, 0.918558748402491,
0.883883627842492, 0.433012359189203, -0.176776777947208,
-0.353553128699351, 0.866025628186053, 0.353553102817459},
{ 0.533493553519713, -0.249999816228639, 0.808012821828067,
0.808012821828067, 0.433012359189203, -0.399519181705765,
-0.249999816228639, 0.866025628186053, 0.433012359189203},
{ 0.047366781483451, -0.612372449482883, 0.789149143778432,
0.659739427618959, 0.612372404654096, 0.435596057905909,
-0.750000183771249, 0.500000021132493, 0.433012359189203}};
// clang-format on
for (int i = 0; i < 3; ++i) {
double results[9];
EulerAnglesToRotation<IntrinsicZXY>(sample_euler[i], results);
ASSERT_THAT(results, IsNear3x3Matrix(expected[i]));
}
}
// ZXY Extrinsic Euler Angle to rotation matrix conversion test from
// scipy/spatial/transform/test/test_rotation.py
TEST(EulerAngles, ExtrinsicEulerSequence312ToRotationMatrix) {
// clang-format off
double const expected[][9] =
{{0.918558725988105, 0.176776842651999, 0.353553128699352,
0.249999816228639, 0.433012359189203, -0.866025628186053,
-0.306186150563275, 0.883883614901527, 0.353553102817459},
{ 0.966506404215301, -0.058012606358071, 0.249999816228639,
0.249999816228639, 0.433012359189203, -0.866025628186053,
-0.058012606358071, 0.899519223970752, 0.433012359189203},
{ 0.659739424151467, -0.047366829779744, 0.750000183771249,
0.612372449482883, 0.612372404654096, -0.500000021132493,
-0.435596000136163, 0.789149175666285, 0.433012359189203}};
// clang-format on
for (int i = 0; i < 3; ++i) {
double results[9];
EulerAnglesToRotation<ExtrinsicZXY>(sample_euler[i], results);
ASSERT_THAT(results, IsNear3x3Matrix(expected[i]));
}
}
// ZXZ Intrinsic Euler Angle to rotation matrix conversion test from
// scipy/spatial/transform/test/test_rotation.py
TEST(EulerAngles, IntrinsicEulerSequence313ToRotationMatrix) {
// clang-format off
double expected[][9] =
{{0.435595832832961, -0.789149008363071, 0.433012832394307,
0.659739379322704, -0.047367454164077, -0.750000183771249,
0.612372616786097, 0.612372571957297, 0.499999611324802},
{ 0.625000065470068, -0.649518902838302, 0.433012832394307,
0.649518902838302, 0.124999676794869, -0.750000183771249,
0.433012832394307, 0.750000183771249, 0.499999611324802},
{-0.176777132429787, -0.918558558684756, 0.353553418477159,
0.883883325123719, -0.306186652473014, -0.353553392595246,
0.433012832394307, 0.249999816228639, 0.866025391583588}};
// clang-format on
for (int i = 0; i < 3; ++i) {
double results[9];
EulerAnglesToRotation<IntrinsicZXZ>(sample_euler[i], results);
ASSERT_THAT(results, IsNear3x3Matrix(expected[i]));
}
}
// ZXZ Extrinsic Euler Angle to rotation matrix conversion test from
// scipy/spatial/transform/test/test_rotation.py
TEST(EulerAngles, ExtrinsicEulerSequence313ToRotationMatrix) {
// clang-format off
double expected[][9] =
{{0.435595832832961, -0.659739379322704, 0.612372616786097,
0.789149008363071, -0.047367454164077, -0.612372571957297,
0.433012832394307, 0.750000183771249, 0.499999611324802},
{ 0.625000065470068, -0.649518902838302, 0.433012832394307,
0.649518902838302, 0.124999676794869, -0.750000183771249,
0.433012832394307, 0.750000183771249, 0.499999611324802},
{-0.176777132429787, -0.883883325123719, 0.433012832394307,
0.918558558684756, -0.306186652473014, -0.249999816228639,
0.353553418477159, 0.353553392595246, 0.866025391583588}};
// clang-format on
for (int i = 0; i < 3; ++i) {
double results[9];
EulerAnglesToRotation<ExtrinsicZXZ>(sample_euler[i], results);
ASSERT_THAT(results, IsNear3x3Matrix(expected[i]));
}
}
template <typename T>
struct GeneralEulerAngles : public ::testing::Test {
public:
static constexpr bool kIsParityOdd = T::kIsParityOdd;
static constexpr bool kIsProperEuler = T::kIsProperEuler;
static constexpr bool kIsIntrinsic = T::kIsIntrinsic;
template <typename URBG>
static void RandomEulerAngles(double* euler, URBG& prng) {
using ParamType = std::uniform_real_distribution<double>::param_type;
std::uniform_real_distribution<double> uniform_distribution{-kPi, kPi};
// Euler angles should be in
// [-pi,pi) x [0,pi) x [-pi,pi])
// if the outer axes are repeated and
// [-pi,pi) x [-pi/2,pi/2) x [-pi,pi])
// otherwise
euler[0] = uniform_distribution(prng);
euler[2] = uniform_distribution(prng);
if constexpr (kIsProperEuler) {
euler[1] = uniform_distribution(prng, ParamType{0, kPi});
} else {
euler[1] = uniform_distribution(prng, ParamType{-kPi / 2, kPi / 2});
}
}
static void CheckPrincipalRotationMatrixProduct(double angles[3]) {
// Convert Shoemake's Euler angle convention into 'apparent' rotation axes
// sequences, i.e. the alphabetic code (ZYX, ZYZ, etc.) indicates in what
// sequence rotations about different axes are applied
constexpr int i = T::kAxes[0];
constexpr int j = (3 + (kIsParityOdd ? (i - 1) % 3 : (i + 1) % 3)) % 3;
constexpr int k = kIsProperEuler ? i : 3 ^ i ^ j;
constexpr auto kSeq =
kIsIntrinsic ? std::array{k, j, i} : std::array{i, j, k};
double aa_matrix[9];
Eigen::Map<Eigen::Matrix3d, 0, Eigen::Stride<1, 3>> aa(aa_matrix);
aa.setIdentity();
for (int i = 0; i < 3; ++i) {
Eigen::Vector3d angle_axis;
if constexpr (kIsIntrinsic) {
angle_axis = -angles[i] * Eigen::Vector3d::Unit(kSeq[i]);
} else {
angle_axis = angles[i] * Eigen::Vector3d::Unit(kSeq[i]);
}
Eigen::Matrix3d m;
AngleAxisToRotationMatrix(angle_axis.data(), m.data());
aa = m * aa;
}
if constexpr (kIsIntrinsic) {
aa.transposeInPlace();
}
double ea_matrix[9];
EulerAnglesToRotation<T>(angles, ea_matrix);
EXPECT_THAT(aa_matrix, IsOrthonormal());
EXPECT_THAT(ea_matrix, IsOrthonormal());
EXPECT_THAT(ea_matrix, IsNear3x3Matrix(aa_matrix));
}
};
using EulerSystemList = ::testing::Types<ExtrinsicXYZ,
ExtrinsicXYX,
ExtrinsicXZY,
ExtrinsicXZX,
ExtrinsicYZX,
ExtrinsicYZY,
ExtrinsicYXZ,
ExtrinsicYXY,
ExtrinsicZXY,
ExtrinsicZXZ,
ExtrinsicZYX,
ExtrinsicZYZ,
IntrinsicZYX,
IntrinsicXYX,
IntrinsicYZX,
IntrinsicXZX,
IntrinsicXZY,
IntrinsicYZY,
IntrinsicZXY,
IntrinsicYXY,
IntrinsicYXZ,
IntrinsicZXZ,
IntrinsicXYZ,
IntrinsicZYZ>;
TYPED_TEST_SUITE(GeneralEulerAngles, EulerSystemList);
TYPED_TEST(GeneralEulerAngles, EulerAnglesToRotationMatrixAndBack) {
std::mt19937 prng;
std::uniform_real_distribution<double> uniform_distribution{-1.0, 1.0};
for (int i = 0; i < kNumTrials; ++i) {
double euler[3];
TestFixture::RandomEulerAngles(euler, prng);
double matrix[9];
double round_trip[3];
EulerAnglesToRotation<TypeParam>(euler, matrix);
ASSERT_THAT(matrix, IsOrthonormal());
RotationMatrixToEulerAngles<TypeParam>(matrix, round_trip);
for (int j = 0; j < 3; ++j)
ASSERT_NEAR(euler[j], round_trip[j], 128.0 * kLooseTolerance);
}
}
// Check that the rotation matrix converted from euler angles is equivalent to
// product of three principal axis rotation matrices
// R_euler = R_a2(euler_2) * R_a1(euler_1) * R_a0(euler_0)
TYPED_TEST(GeneralEulerAngles, PrincipalRotationMatrixProduct) {
std::mt19937 prng;
double euler[3];
for (int i = 0; i < kNumTrials; ++i) {
TestFixture::RandomEulerAngles(euler, prng);
TestFixture::CheckPrincipalRotationMatrixProduct(euler);
}
}
// Gimbal lock (euler[1] == +/-pi) handling test. If a rotation matrix
// represents a gimbal-locked configuration, then converting this rotation
// matrix to euler angles and back must produce the same rotation matrix.
//
// From scipy/spatial/transform/test/test_rotation.py, but additionally covers
// gimbal lock handling for proper euler angles, which scipy appears to fail to
// do properly.
TYPED_TEST(GeneralEulerAngles, GimbalLocked) {
constexpr auto kBoundaryAngles = TestFixture::kIsProperEuler
? std::array{0.0, kPi}
: std::array{-kPi / 2, kPi / 2};
constexpr double gimbal_locked_configurations[4][3] = {
{0.78539816, kBoundaryAngles[1], 0.61086524},
{0.61086524, kBoundaryAngles[0], 0.34906585},
{0.61086524, kBoundaryAngles[1], 0.43633231},
{0.43633231, kBoundaryAngles[0], 0.26179939}};
double angle_estimates[3];
double mat_expected[9];
double mat_estimated[9];
for (const auto& euler_angles : gimbal_locked_configurations) {
EulerAnglesToRotation<TypeParam>(euler_angles, mat_expected);
RotationMatrixToEulerAngles<TypeParam>(mat_expected, angle_estimates);
EulerAnglesToRotation<TypeParam>(angle_estimates, mat_estimated);
ASSERT_THAT(mat_expected, IsNear3x3Matrix(mat_estimated));
}
}
// Tests using Jets for specific behavior involving auto differentiation
// near singularity points.
using J3 = Jet<double, 3>;
using J4 = Jet<double, 4>;
namespace {
// Converts an array of N real numbers (doubles) to an array of jets
template <int N>
void ArrayToArrayOfJets(double const* const src, Jet<double, N>* dst) {
for (int i = 0; i < N; ++i) {
dst[i] = Jet<double, N>(src[i], i);
}
}
// Generically initializes a Jet with type T and a N-dimensional dual part
// N is explicitly given (instead of inferred from sizeof...(Ts)) so that the
// dual part can be initialized from Eigen expressions
template <int N, typename T, typename... Ts>
Jet<T, N> MakeJet(T a, const T& v0, Ts&&... v) {
Jet<T, N> j;
j.a = a; // Real part
((j.v << v0), ..., std::forward<Ts>(v)); // Fill dual part with N components
return j;
}
J3 MakeJ3(double a, double v0, double v1, double v2) {
J3 j;
j.a = a;
j.v[0] = v0;
j.v[1] = v1;
j.v[2] = v2;
return j;
}
J4 MakeJ4(double a, double v0, double v1, double v2, double v3) {
J4 j;
j.a = a;
j.v[0] = v0;
j.v[1] = v1;
j.v[2] = v2;
j.v[3] = v3;
return j;
}
} // namespace
// Use EXPECT_THAT(x, testing::PointWise(JetClose(prec), y); to achieve Jet
// array comparison
MATCHER_P(JetClose, relative_precision, "") {
using internal::IsClose;
using LHSJetType = std::remove_reference_t<std::tuple_element_t<0, arg_type>>;
using RHSJetType = std::remove_reference_t<std::tuple_element_t<1, arg_type>>;
constexpr int kDualPartDimension = LHSJetType::DIMENSION;
static_assert(
kDualPartDimension == RHSJetType::DIMENSION,
"Can only compare Jets with dual parts having equal dimensions");
auto&& [x, y] = arg;
double relative_error;
double absolute_error;
if (!IsClose(
x.a, y.a, relative_precision, &relative_error, &absolute_error)) {
*result_listener << "Real part mismatch: x.a = " << x.a
<< " and y.a = " << y.a
<< " where the relative error between them is "
<< relative_error
<< " and the absolute error between them is "
<< absolute_error;
return false;
}
for (int i = 0; i < kDualPartDimension; i++) {
if (!IsClose(x.v[i],
y.v[i],
relative_precision,
&relative_error,
&absolute_error)) {
*result_listener << "Dual part mismatch: x.v[" << i << "] = " << x.v[i]
<< " and y.v[" << i << "] = " << y.v[i]
<< " where the relative error between them is "
<< relative_error
<< " and the absolute error between them is "
<< absolute_error;
return false;
}
}
return true;
}
// Log-10 of a value well below machine precision.
static const int kSmallTinyCutoff = static_cast<int>(
2 * log(std::numeric_limits<double>::epsilon()) / log(10.0));
// Log-10 of a value just below values representable by double.
static const int kTinyZeroLimit =
static_cast<int>(1 + log(std::numeric_limits<double>::min()) / log(10.0));
// Test that exact conversion works for small angles when jets are used.
TEST(Rotation, SmallAngleAxisToQuaternionForJets) {
// Examine small x rotations that are still large enough
// to be well within the range represented by doubles.
for (int i = -2; i >= kSmallTinyCutoff; i--) {
double theta = pow(10.0, i);
J3 axis_angle[3] = {J3(theta, 0), J3(0, 1), J3(0, 2)};
J3 quaternion[4];
J3 expected[4] = {
MakeJ3(cos(theta / 2), -sin(theta / 2) / 2, 0, 0),
MakeJ3(sin(theta / 2), cos(theta / 2) / 2, 0, 0),
MakeJ3(0, 0, sin(theta / 2) / theta, 0),
MakeJ3(0, 0, 0, sin(theta / 2) / theta),
};
AngleAxisToQuaternion(axis_angle, quaternion);
EXPECT_THAT(quaternion, testing::Pointwise(JetClose(kTolerance), expected));
}
}
// Test that conversion works for very small angles when jets are used.
TEST(Rotation, TinyAngleAxisToQuaternionForJets) {
// Examine tiny x rotations that extend all the way to where
// underflow occurs.
for (int i = kSmallTinyCutoff; i >= kTinyZeroLimit; i--) {
double theta = pow(10.0, i);
J3 axis_angle[3] = {J3(theta, 0), J3(0, 1), J3(0, 2)};
J3 quaternion[4];
// To avoid loss of precision in the test itself,
// a finite expansion is used here, which will
// be exact up to machine precision for the test values used.
J3 expected[4] = {
MakeJ3(1.0, 0, 0, 0),
MakeJ3(0, 0.5, 0, 0),
MakeJ3(0, 0, 0.5, 0),
MakeJ3(0, 0, 0, 0.5),
};
AngleAxisToQuaternion(axis_angle, quaternion);
EXPECT_THAT(quaternion, testing::Pointwise(JetClose(kTolerance), expected));
}
}
// Test that derivatives are correct for zero rotation.
TEST(Rotation, ZeroAngleAxisToQuaternionForJets) {
J3 axis_angle[3] = {J3(0, 0), J3(0, 1), J3(0, 2)};
J3 quaternion[4];
J3 expected[4] = {
MakeJ3(1.0, 0, 0, 0),
MakeJ3(0, 0.5, 0, 0),
MakeJ3(0, 0, 0.5, 0),
MakeJ3(0, 0, 0, 0.5),
};
AngleAxisToQuaternion(axis_angle, quaternion);
EXPECT_THAT(quaternion, testing::Pointwise(JetClose(kTolerance), expected));
}
// Test that exact conversion works for small angles.
TEST(Rotation, SmallQuaternionToAngleAxisForJets) {
// Examine small x rotations that are still large enough
// to be well within the range represented by doubles.
for (int i = -2; i >= kSmallTinyCutoff; i--) {
double theta = pow(10.0, i);
double s = sin(theta);
double c = cos(theta);
J4 quaternion[4] = {J4(c, 0), J4(s, 1), J4(0, 2), J4(0, 3)};
J4 axis_angle[3];
// clang-format off
J4 expected[3] = {
MakeJ4(2*theta, -2*s, 2*c, 0, 0),
MakeJ4(0, 0, 0, 2*theta/s, 0),
MakeJ4(0, 0, 0, 0, 2*theta/s),
};
// clang-format on
QuaternionToAngleAxis(quaternion, axis_angle);
EXPECT_THAT(axis_angle, testing::Pointwise(JetClose(kTolerance), expected));
}
}
// Test that conversion works for very small angles.
TEST(Rotation, TinyQuaternionToAngleAxisForJets) {
// Examine tiny x rotations that extend all the way to where
// underflow occurs.
for (int i = kSmallTinyCutoff; i >= kTinyZeroLimit; i--) {
double theta = pow(10.0, i);
double s = sin(theta);
double c = cos(theta);
J4 quaternion[4] = {J4(c, 0), J4(s, 1), J4(0, 2), J4(0, 3)};
J4 axis_angle[3];
// To avoid loss of precision in the test itself,
// a finite expansion is used here, which will
// be exact up to machine precision for the test values used.
// clang-format off
J4 expected[3] = {
MakeJ4(2*theta, -2*s, 2.0, 0, 0),
MakeJ4(0, 0, 0, 2.0, 0),
MakeJ4(0, 0, 0, 0, 2.0),
};
// clang-format on
QuaternionToAngleAxis(quaternion, axis_angle);
EXPECT_THAT(axis_angle, testing::Pointwise(JetClose(kTolerance), expected));
}
}
// Test that conversion works for no rotation.
TEST(Rotation, ZeroQuaternionToAngleAxisForJets) {
J4 quaternion[4] = {J4(1, 0), J4(0, 1), J4(0, 2), J4(0, 3)};
J4 axis_angle[3];
J4 expected[3] = {
MakeJ4(0, 0, 2.0, 0, 0),
MakeJ4(0, 0, 0, 2.0, 0),
MakeJ4(0, 0, 0, 0, 2.0),
};
QuaternionToAngleAxis(quaternion, axis_angle);
EXPECT_THAT(axis_angle, testing::Pointwise(JetClose(kTolerance), expected));
}
// The following 4 test cases cover the conversion of Euler Angles to rotation
// matrices for Jets
//
// The dual parts (with dimension 3) of the resultant matrix of Jets contain the
// derivative of each matrix element w.r.t. the input Euler Angles. In other
// words, for each element in R = EulerAnglesToRotationMatrix(angles), we have
// R_ij.v = jacobian(R_ij, angles)
//
// The test data (dual parts of the Jets) is generated by analytically
// differentiating the formulas for Euler Angle to Rotation Matrix conversion
// Test ZXY/312 Intrinsic Euler Angles to rotation matrix conversion using Jets
// The two ZXY test cases specifically cover handling of Tait-Bryan angles
// i.e. last axis of rotation is different from the first
TEST(EulerAngles, Intrinsic312EulerSequenceToRotationMatrixForJets) {
J3 euler_angles[3];
J3 rotation_matrix[9];
ArrayToArrayOfJets(sample_euler[0], euler_angles);
EulerAnglesToRotation<IntrinsicZXY>(euler_angles, rotation_matrix);
{
// clang-format off
const J3 expected[] = {
MakeJ3( 0.306186083320, -0.883883627842, -0.176776571821, -0.918558748402), // NOLINT
MakeJ3(-0.249999816229, -0.433012359189, 0.433012832394, 0.000000000000), // NOLINT
MakeJ3( 0.918558748402, 0.176776777947, 0.176776558880, 0.306186083320), // NOLINT
MakeJ3( 0.883883627842, 0.306186083320, 0.306185986727, 0.176776777947), // NOLINT
MakeJ3( 0.433012359189, -0.249999816229, -0.750000183771, 0.000000000000), // NOLINT
MakeJ3(-0.176776777947, 0.918558748402, -0.306185964313, 0.883883627842), // NOLINT
MakeJ3(-0.353553128699, 0.000000000000, 0.612372616786, -0.353553102817), // NOLINT
MakeJ3( 0.866025628186, 0.000000000000, 0.499999611325, 0.000000000000), // NOLINT
MakeJ3( 0.353553102817, 0.000000000000, -0.612372571957, -0.353553128699) // NOLINT
};
// clang-format on
EXPECT_THAT(rotation_matrix,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_euler[1], euler_angles);
EulerAnglesToRotation<IntrinsicZXY>(euler_angles, rotation_matrix);
{
// clang-format off
const J3 expected[] = {
MakeJ3( 0.533493553520, -0.808012821828, -0.124999913397, -0.808012821828), // NOLINT
MakeJ3(-0.249999816229, -0.433012359189, 0.433012832394, 0.000000000000), // NOLINT
MakeJ3( 0.808012821828, 0.399519181706, 0.216506188745, 0.533493553520), // NOLINT
MakeJ3( 0.808012821828, 0.533493553520, 0.216506188745, 0.399519181706), // NOLINT
MakeJ3( 0.433012359189, -0.249999816229, -0.750000183771, 0.000000000000), // NOLINT
MakeJ3(-0.399519181706, 0.808012821828, -0.374999697927, 0.808012821828), // NOLINT
MakeJ3(-0.249999816229, 0.000000000000, 0.433012832394, -0.433012359189), // NOLINT
MakeJ3( 0.866025628186, 0.000000000000, 0.499999611325, 0.000000000000), // NOLINT
MakeJ3( 0.433012359189, 0.000000000000, -0.750000183771, -0.249999816229) // NOLINT
};
// clang-format on
EXPECT_THAT(rotation_matrix,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_euler[2], euler_angles);
EulerAnglesToRotation<IntrinsicZXY>(euler_angles, rotation_matrix);
{
// clang-format off
const J3 expected[] = {
MakeJ3( 0.047366781483, -0.659739427619, -0.530330235247, -0.789149143778), // NOLINT
MakeJ3(-0.612372449483, -0.612372404654, 0.353553418477, 0.000000000000), // NOLINT
MakeJ3( 0.789149143778, -0.435596057906, 0.306185986727, 0.047366781483), // NOLINT
MakeJ3( 0.659739427619, 0.047366781483, 0.530330196424, -0.435596057906), // NOLINT
MakeJ3( 0.612372404654, -0.612372449483, -0.353553392595, 0.000000000000), // NOLINT
MakeJ3( 0.435596057906, 0.789149143778, -0.306185964313, 0.659739427619), // NOLINT
MakeJ3(-0.750000183771, 0.000000000000, 0.433012832394, -0.433012359189), // NOLINT
MakeJ3( 0.500000021132, 0.000000000000, 0.866025391584, 0.000000000000), // NOLINT
MakeJ3( 0.433012359189, 0.000000000000, -0.249999816229, -0.750000183771) // NOLINT
};
// clang-format on
EXPECT_THAT(rotation_matrix,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
}
// Test ZXY/312 Extrinsic Euler Angles to rotation matrix conversion using Jets
TEST(EulerAngles, Extrinsic312EulerSequenceToRotationMatrixForJets) {
J3 euler_angles[3];
J3 rotation_matrix[9];
ArrayToArrayOfJets(sample_euler[0], euler_angles);
EulerAnglesToRotation<ExtrinsicZXY>(euler_angles, rotation_matrix);
{
// clang-format off
const J3 expected[] = {
MakeJ3( 0.918558725988, 0.176776842652, 0.176776571821, -0.306186150563), // NOLINT
MakeJ3( 0.176776842652, -0.918558725988, 0.306185986727, 0.883883614902), // NOLINT
MakeJ3( 0.353553128699, 0.000000000000, -0.612372616786, 0.353553102817), // NOLINT
MakeJ3( 0.249999816229, 0.433012359189, -0.433012832394, 0.000000000000), // NOLINT
MakeJ3( 0.433012359189, -0.249999816229, -0.750000183771, 0.000000000000), // NOLINT
MakeJ3(-0.866025628186, 0.000000000000, -0.499999611325, 0.000000000000), // NOLINT
MakeJ3(-0.306186150563, 0.883883614902, 0.176776558880, -0.918558725988), // NOLINT
MakeJ3( 0.883883614902, 0.306186150563, 0.306185964313, -0.176776842652), // NOLINT
MakeJ3( 0.353553102817, 0.000000000000, -0.612372571957, -0.353553128699) // NOLINT
};
// clang-format on
EXPECT_THAT(rotation_matrix,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_euler[1], euler_angles);
EulerAnglesToRotation<ExtrinsicZXY>(euler_angles, rotation_matrix);
{
// clang-format off
const J3 expected[] = {
MakeJ3( 0.966506404215, -0.058012606358, 0.124999913397, -0.058012606358), // NOLINT
MakeJ3(-0.058012606358, -0.966506404215, 0.216506188745, 0.899519223971), // NOLINT
MakeJ3( 0.249999816229, 0.000000000000, -0.433012832394, 0.433012359189), // NOLINT
MakeJ3( 0.249999816229, 0.433012359189, -0.433012832394, 0.000000000000), // NOLINT
MakeJ3( 0.433012359189, -0.249999816229, -0.750000183771, 0.000000000000), // NOLINT
MakeJ3(-0.866025628186, 0.000000000000, -0.499999611325, 0.000000000000), // NOLINT
MakeJ3(-0.058012606358, 0.899519223971, 0.216506188745, -0.966506404215), // NOLINT
MakeJ3( 0.899519223971, 0.058012606358, 0.374999697927, 0.058012606358), // NOLINT
MakeJ3( 0.433012359189, 0.000000000000, -0.750000183771, -0.249999816229) // NOLINT
};
// clang-format on
EXPECT_THAT(rotation_matrix,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_euler[2], euler_angles);
EulerAnglesToRotation<ExtrinsicZXY>(euler_angles, rotation_matrix);
{
// clang-format off
const J3 expected[] = {
MakeJ3( 0.659739424151, -0.047366829780, 0.530330235247, -0.435596000136), // NOLINT
MakeJ3(-0.047366829780, -0.659739424151, 0.530330196424, 0.789149175666), // NOLINT
MakeJ3( 0.750000183771, 0.000000000000, -0.433012832394, 0.433012359189), // NOLINT
MakeJ3( 0.612372449483, 0.612372404654, -0.353553418477, 0.000000000000), // NOLINT
MakeJ3( 0.612372404654, -0.612372449483, -0.353553392595, 0.000000000000), // NOLINT
MakeJ3(-0.500000021132, 0.000000000000, -0.866025391584, 0.000000000000), // NOLINT
MakeJ3(-0.435596000136, 0.789149175666, 0.306185986727, -0.659739424151), // NOLINT
MakeJ3( 0.789149175666, 0.435596000136, 0.306185964313, 0.047366829780), // NOLINT
MakeJ3( 0.433012359189, 0.000000000000, -0.249999816229, -0.750000183771) // NOLINT
};
// clang-format on
EXPECT_THAT(rotation_matrix,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
}
// Test ZXZ/313 Intrinsic Euler Angles to rotation matrix conversion using Jets
// The two ZXZ test cases specifically cover handling of proper Euler Sequences
// i.e. last axis of rotation is same as the first
TEST(EulerAngles, Intrinsic313EulerSequenceToRotationMatrixForJets) {
J3 euler_angles[3];
J3 rotation_matrix[9];
ArrayToArrayOfJets(sample_euler[0], euler_angles);
EulerAnglesToRotation<IntrinsicZXZ>(euler_angles, rotation_matrix);
{
// clang-format off
const J3 expected[] = {
MakeJ3( 0.435595832833, -0.659739379323, 0.306186321334, -0.789149008363), // NOLINT
MakeJ3(-0.789149008363, 0.047367454164, 0.306186298920, -0.435595832833), // NOLINT
MakeJ3( 0.433012832394, 0.750000183771, 0.249999816229, 0.000000000000), // NOLINT
MakeJ3( 0.659739379323, 0.435595832833, -0.530330235247, -0.047367454164), // NOLINT
MakeJ3(-0.047367454164, -0.789149008363, -0.530330196424, -0.659739379323), // NOLINT
MakeJ3(-0.750000183771, 0.433012832394, -0.433012359189, 0.000000000000), // NOLINT
MakeJ3( 0.612372616786, 0.000000000000, 0.353553128699, 0.612372571957), // NOLINT
MakeJ3( 0.612372571957, 0.000000000000, 0.353553102817, -0.612372616786), // NOLINT
MakeJ3( 0.499999611325, 0.000000000000, -0.866025628186, 0.000000000000) // NOLINT
};
// clang-format on
EXPECT_THAT(rotation_matrix,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_euler[1], euler_angles);
EulerAnglesToRotation<IntrinsicZXZ>(euler_angles, rotation_matrix);
{
// clang-format off
const J3 expected[] = {
MakeJ3( 0.625000065470, -0.649518902838, 0.216506425348, -0.649518902838), // NOLINT
MakeJ3(-0.649518902838, -0.124999676795, 0.375000107735, -0.625000065470), // NOLINT
MakeJ3( 0.433012832394, 0.750000183771, 0.249999816229, 0.000000000000), // NOLINT
MakeJ3( 0.649518902838, 0.625000065470, -0.375000107735, 0.124999676795), // NOLINT
MakeJ3( 0.124999676795, -0.649518902838, -0.649519202838, -0.649518902838), // NOLINT
MakeJ3(-0.750000183771, 0.433012832394, -0.433012359189, 0.000000000000), // NOLINT
MakeJ3( 0.433012832394, 0.000000000000, 0.249999816229, 0.750000183771), // NOLINT
MakeJ3( 0.750000183771, 0.000000000000, 0.433012359189, -0.433012832394), // NOLINT
MakeJ3( 0.499999611325, 0.000000000000, -0.866025628186, 0.000000000000) // NOLINT
};
// clang-format on
EXPECT_THAT(rotation_matrix,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_euler[2], euler_angles);
EulerAnglesToRotation<IntrinsicZXZ>(euler_angles, rotation_matrix);
{
// clang-format off
const J3 expected[] = {
MakeJ3(-0.176777132430, -0.883883325124, 0.306186321334, -0.918558558685), // NOLINT
MakeJ3(-0.918558558685, 0.306186652473, 0.176776571821, 0.176777132430), // NOLINT
MakeJ3( 0.353553418477, 0.353553392595, 0.612372449483, 0.000000000000), // NOLINT
MakeJ3( 0.883883325124, -0.176777132430, -0.306186298920, -0.306186652473), // NOLINT
MakeJ3(-0.306186652473, -0.918558558685, -0.176776558880, -0.883883325124), // NOLINT
MakeJ3(-0.353553392595, 0.353553418477, -0.612372404654, 0.000000000000), // NOLINT
MakeJ3( 0.433012832394, 0.000000000000, 0.750000183771, 0.249999816229), // NOLINT
MakeJ3( 0.249999816229, 0.000000000000, 0.433012359189, -0.433012832394), // NOLINT
MakeJ3( 0.866025391584, 0.000000000000, -0.500000021132, 0.000000000000) // NOLINT
};
// clang-format on
EXPECT_THAT(rotation_matrix,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
}
// Test ZXZ/313 Extrinsic Euler Angles to rotation matrix conversion using Jets
TEST(EulerAngles, Extrinsic313EulerSequenceToRotationMatrixForJets) {
J3 euler_angles[3];
J3 rotation_matrix[9];
ArrayToArrayOfJets(sample_euler[0], euler_angles);
EulerAnglesToRotation<ExtrinsicZXZ>(euler_angles, rotation_matrix);
{
// clang-format off
const J3 expected[] = {
MakeJ3( 0.435595832833, -0.659739379323, 0.306186321334, -0.789149008363), // NOLINT
MakeJ3(-0.659739379323, -0.435595832833, 0.530330235247, 0.047367454164), // NOLINT
MakeJ3( 0.612372616786, 0.000000000000, 0.353553128699, 0.612372571957), // NOLINT
MakeJ3( 0.789149008363, -0.047367454164, -0.306186298920, 0.435595832833), // NOLINT
MakeJ3(-0.047367454164, -0.789149008363, -0.530330196424, -0.659739379323), // NOLINT
MakeJ3(-0.612372571957, 0.000000000000, -0.353553102817, 0.612372616786), // NOLINT
MakeJ3( 0.433012832394, 0.750000183771, 0.249999816229, 0.000000000000), // NOLINT
MakeJ3( 0.750000183771, -0.433012832394, 0.433012359189, 0.000000000000), // NOLINT
MakeJ3( 0.499999611325, 0.000000000000, -0.866025628186, 0.000000000000) // NOLINT
};
// clang-format on
EXPECT_THAT(rotation_matrix,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_euler[1], euler_angles);
EulerAnglesToRotation<ExtrinsicZXZ>(euler_angles, rotation_matrix);
{
// clang-format off
const J3 expected[] = {
MakeJ3( 0.625000065470, -0.649518902838, 0.216506425348, -0.649518902838), // NOLINT
MakeJ3(-0.649518902838, -0.625000065470, 0.375000107735, -0.124999676795), // NOLINT
MakeJ3( 0.433012832394, 0.000000000000, 0.249999816229, 0.750000183771), // NOLINT
MakeJ3( 0.649518902838, 0.124999676795, -0.375000107735, 0.625000065470), // NOLINT
MakeJ3( 0.124999676795, -0.649518902838, -0.649519202838, -0.649518902838), // NOLINT
MakeJ3(-0.750000183771, 0.000000000000, -0.433012359189, 0.433012832394), // NOLINT
MakeJ3( 0.433012832394, 0.750000183771, 0.249999816229, 0.000000000000), // NOLINT
MakeJ3( 0.750000183771, -0.433012832394, 0.433012359189, 0.000000000000), // NOLINT
MakeJ3( 0.499999611325, 0.000000000000, -0.866025628186, 0.000000000000) // NOLINT
};
// clang-format on
EXPECT_THAT(rotation_matrix,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_euler[2], euler_angles);
EulerAnglesToRotation<ExtrinsicZXZ>(euler_angles, rotation_matrix);
{
// clang-format off
const J3 expected[] = {
MakeJ3(-0.176777132430, -0.883883325124, 0.306186321334, -0.918558558685), // NOLINT
MakeJ3(-0.883883325124, 0.176777132430, 0.306186298920, 0.306186652473), // NOLINT
MakeJ3( 0.433012832394, 0.000000000000, 0.750000183771, 0.249999816229), // NOLINT
MakeJ3( 0.918558558685, -0.306186652473, -0.176776571821, -0.176777132430), // NOLINT
MakeJ3(-0.306186652473, -0.918558558685, -0.176776558880, -0.883883325124), // NOLINT
MakeJ3(-0.249999816229, 0.000000000000, -0.433012359189, 0.433012832394), // NOLINT
MakeJ3( 0.353553418477, 0.353553392595, 0.612372449483, 0.000000000000), // NOLINT
MakeJ3( 0.353553392595, -0.353553418477, 0.612372404654, 0.000000000000), // NOLINT
MakeJ3( 0.866025391584, 0.000000000000, -0.500000021132, 0.000000000000) // NOLINT
};
// clang-format on
EXPECT_THAT(rotation_matrix,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
}
using J9 = Jet<double, 9>;
// The following 4 tests Tests the conversion of rotation matrices to Euler
// Angles for Jets.
//
// The dual parts (with dimension 9) of the resultant array of Jets contain the
// derivative of each Euler angle w.r.t. each of the 9 elements of the rotation
// matrix, or a 9-by-1 array formed from flattening the rotation matrix. In
// other words, for each element in angles = RotationMatrixToEulerAngles(R), we
// have angles.v = jacobian(angles, [R11 R12 R13 R21 ... R32 R33]);
//
// Note: the order of elements in v depend on row/column-wise flattening of
// the rotation matrix
//
// The test data (dual parts of the Jets) is generated by analytically
// differentiating the formulas for Rotation Matrix to Euler Angle conversion
// clang-format off
static double sample_matrices[][9] = {
{ 0.433012359189, 0.176776842652, 0.883883614902, 0.249999816229, 0.918558725988, -0.306186150563, -0.866025628186, 0.353553128699, 0.353553102817}, // NOLINT
{ 0.433012359189, -0.058012606358, 0.899519223971, 0.249999816229, 0.966506404215, -0.058012606358, -0.866025628186, 0.249999816229, 0.433012359189}, // NOLINT
{ 0.612372404654, -0.047366829780, 0.789149175666, 0.612372449483, 0.659739424151, -0.435596000136, -0.500000021132, 0.750000183771, 0.433012359189} // NOLINT
};
// clang-format on
// Test rotation matrix to ZXY/312 Intrinsic Euler Angles conversion using Jets
// The two ZXY test cases specifically cover handling of Tait-Bryan angles
// i.e. last axis of rotation is different from the first
TEST(EulerAngles, RotationMatrixToIntrinsic312EulerSequenceForJets) {
J9 euler_angles[3];
J9 rotation_matrix[9];
ArrayToArrayOfJets(sample_matrices[0], rotation_matrix);
RotationMatrixToEulerAngles<IntrinsicZXY>(rotation_matrix, euler_angles);
{
// clang-format off
const J9 expected[] = {
MakeJet<9>(-0.190125743401, 0.000000000000, -1.049781178951, 0.000000000000, 0.000000000000, 0.202030634558, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000), // NOLINT
MakeJet<9>( 0.361366843930, 0.000000000000, -0.066815309609, 0.000000000000, 0.000000000000, -0.347182270882, 0.000000000000, 0.000000000000, 0.935414445680, 0.000000000000), // NOLINT
MakeJet<9>( 1.183200015636, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, -0.404060603418, 0.000000000000, -0.989743365598) // NOLINT
};
// clang-format on
EXPECT_THAT(euler_angles,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_matrices[1], rotation_matrix);
RotationMatrixToEulerAngles<IntrinsicZXY>(rotation_matrix, euler_angles);
{
// clang-format off
const J9 expected[] = {
MakeJet<9>( 0.059951064811, 0.000000000000, -1.030940063452, 0.000000000000, 0.000000000000, -0.061880107384, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000), // NOLINT
MakeJet<9>( 0.252680065344, 0.000000000000, 0.014978778808, 0.000000000000, 0.000000000000, -0.249550684831, 0.000000000000, 0.000000000000, 0.968245884001, 0.000000000000), // NOLINT
MakeJet<9>( 1.107149138016, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, -0.461879804532, 0.000000000000, -0.923760579526) // NOLINT
};
// clang-format on
EXPECT_THAT(euler_angles,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_matrices[2], rotation_matrix);
RotationMatrixToEulerAngles<IntrinsicZXY>(rotation_matrix, euler_angles);
{
// clang-format off
const J9 expected[] = {
MakeJet<9>( 0.071673287221, 0.000000000000, -1.507976776767, 0.000000000000, 0.000000000000, -0.108267107713, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000), // NOLINT
MakeJet<9>( 0.848062356818, 0.000000000000, 0.053708966648, 0.000000000000, 0.000000000000, -0.748074610289, 0.000000000000, 0.000000000000, 0.661437619389, 0.000000000000), // NOLINT
MakeJet<9>( 0.857072360427, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, -0.989743158900, 0.000000000000, -1.142857911244) // NOLINT
};
// clang-format on
EXPECT_THAT(euler_angles,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
}
// Test rotation matrix to ZXY/312 Extrinsic Euler Angles conversion using Jets
TEST(EulerAngles, RotationMatrixToExtrinsic312EulerSequenceForJets) {
J9 euler_angles[3];
J9 rotation_matrix[9];
ArrayToArrayOfJets(sample_matrices[0], rotation_matrix);
RotationMatrixToEulerAngles<ExtrinsicZXY>(rotation_matrix, euler_angles);
{
// clang-format off
const J9 expected[] = {
MakeJet<9>( 0.265728912717, 0.000000000000, 0.000000000000, 0.000000000000, 1.013581996386, -0.275861853641, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000), // NOLINT
MakeJet<9>( 0.311184173598, 0.000000000000, 0.000000000000, -0.284286741927, 0.000000000000, 0.000000000000, -0.951971659874, 0.000000000000, 0.000000000000, -0.113714586405), // NOLINT
MakeJet<9>( 1.190290284357, 0.000000000000, 0.000000000000, 0.390127543992, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, -0.975319806582) // NOLINT
};
// clang-format on
EXPECT_THAT(euler_angles,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_matrices[1], rotation_matrix);
RotationMatrixToEulerAngles<ExtrinsicZXY>(rotation_matrix, euler_angles);
{
// clang-format off
const J9 expected[] = {
MakeJet<9>( 0.253115668605, 0.000000000000, 0.000000000000, 0.000000000000, 0.969770129215, -0.250844022378, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000), // NOLINT
MakeJet<9>( 0.058045195612, 0.000000000000, 0.000000000000, -0.052271487648, 0.000000000000, 0.000000000000, -0.998315850572, 0.000000000000, 0.000000000000, -0.025162553041), // NOLINT
MakeJet<9>( 1.122153748896, 0.000000000000, 0.000000000000, 0.434474567050, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, -0.902556744846) // NOLINT
};
// clang-format on
EXPECT_THAT(euler_angles,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_matrices[2], rotation_matrix);
RotationMatrixToEulerAngles<ExtrinsicZXY>(rotation_matrix, euler_angles);
{
// clang-format off
const J9 expected[] = {
MakeJet<9>( 0.748180444286, 0.000000000000, 0.000000000000, 0.000000000000, 0.814235652244, -0.755776390750, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000), // NOLINT
MakeJet<9>( 0.450700288478, 0.000000000000, 0.000000000000, -0.381884322045, 0.000000000000, 0.000000000000, -0.900142280234, 0.000000000000, 0.000000000000, -0.209542930950), // NOLINT
MakeJet<9>( 1.068945699497, 0.000000000000, 0.000000000000, 0.534414175972, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, -0.973950275281) // NOLINT
};
// clang-format on
EXPECT_THAT(euler_angles,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
}
// Test rotation matrix to ZXZ/313 Intrinsic Euler Angles conversion using Jets
//// The two ZXZ test cases specifically cover handling of proper Euler
/// Sequences
// i.e. last axis of rotation is same as the first
TEST(EulerAngles, RotationMatrixToIntrinsic313EulerSequenceForJets) {
J9 euler_angles[3];
J9 rotation_matrix[9];
ArrayToArrayOfJets(sample_matrices[0], rotation_matrix);
RotationMatrixToEulerAngles<IntrinsicZXZ>(rotation_matrix, euler_angles);
{
// clang-format off
const J9 expected[] = {
MakeJet<9>( 1.237323270947, 0.000000000000, 0.000000000000, 0.349926947837, 0.000000000000, 0.000000000000, 1.010152467826, 0.000000000000, 0.000000000000, 0.000000000000), // NOLINT
MakeJet<9>( 1.209429510533, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, -0.327326615680, 0.133630397662, -0.935414455462), // NOLINT
MakeJet<9>(-1.183199990019, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.404060624546, 0.989743344897, 0.000000000000) // NOLINT
};
// clang-format on
EXPECT_THAT(euler_angles,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_matrices[1], rotation_matrix);
RotationMatrixToEulerAngles<IntrinsicZXZ>(rotation_matrix, euler_angles);
{
// clang-format off
const J9 expected[] = {
MakeJet<9>( 1.506392616830, 0.000000000000, 0.000000000000, 0.071400104821, 0.000000000000, 0.000000000000, 1.107100178948, 0.000000000000, 0.000000000000, 0.000000000000), // NOLINT
MakeJet<9>( 1.122964310061, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, -0.416024849727, 0.120095910090, -0.901387983495), // NOLINT
MakeJet<9>(-1.289761690216, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.307691969119, 1.065877306886, 0.000000000000) // NOLINT
};
// clang-format on
EXPECT_THAT(euler_angles,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_matrices[2], rotation_matrix);
RotationMatrixToEulerAngles<IntrinsicZXZ>(rotation_matrix, euler_angles);
{
// clang-format off
const J9 expected[] = {
MakeJet<9>( 1.066432836578, 0.000000000000, 0.000000000000, 0.536117958181, 0.000000000000, 0.000000000000, 0.971260169116, 0.000000000000, 0.000000000000, 0.000000000000), // NOLINT
MakeJet<9>( 1.122964310061, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, -0.240192006893, 0.360288083393, -0.901387983495), // NOLINT
MakeJet<9>(-0.588002509965, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.923076812076, 0.615384416607, 0.000000000000) // NOLINT
};
// clang-format on
EXPECT_THAT(euler_angles,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
}
// Test rotation matrix to ZXZ/313 Extrinsic Euler Angles conversion using Jets
TEST(EulerAngles, RotationMatrixToExtrinsic313EulerSequenceForJets) {
J9 euler_angles[3];
J9 rotation_matrix[9];
ArrayToArrayOfJets(sample_matrices[0], rotation_matrix);
RotationMatrixToEulerAngles<ExtrinsicZXZ>(rotation_matrix, euler_angles);
{
// clang-format off
const J9 expected[] = {
MakeJet<9>(-1.183199990019, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.404060624546, 0.989743344897, 0.000000000000), // NOLINT
MakeJet<9>( 1.209429510533, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, -0.327326615680, 0.133630397662, -0.935414455462), // NOLINT
MakeJet<9>( 1.237323270947, 0.000000000000, 0.000000000000, 0.349926947837, 0.000000000000, 0.000000000000, 1.010152467826, 0.000000000000, 0.000000000000, 0.000000000000) // NOLINT
};
// clang-format on
EXPECT_THAT(euler_angles,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_matrices[1], rotation_matrix);
RotationMatrixToEulerAngles<ExtrinsicZXZ>(rotation_matrix, euler_angles);
{
// clang-format off
const J9 expected[] = {
MakeJet<9>(-1.289761690216, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.307691969119, 1.065877306886, 0.000000000000), // NOLINT
MakeJet<9>( 1.122964310061, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, -0.416024849727, 0.120095910090, -0.901387983495), // NOLINT
MakeJet<9>( 1.506392616830, 0.000000000000, 0.000000000000, 0.071400104821, 0.000000000000, 0.000000000000, 1.107100178948, 0.000000000000, 0.000000000000, 0.000000000000) // NOLINT
};
// clang-format on
EXPECT_THAT(euler_angles,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
ArrayToArrayOfJets(sample_matrices[2], rotation_matrix);
RotationMatrixToEulerAngles<ExtrinsicZXZ>(rotation_matrix, euler_angles);
{
// clang-format off
const J9 expected[] = {
MakeJet<9>(-0.588002509965, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.923076812076, 0.615384416607, 0.000000000000), // NOLINT
MakeJet<9>( 1.122964310061, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, 0.000000000000, -0.240192006893, 0.360288083393, -0.901387983495), // NOLINT
MakeJet<9>( 1.066432836578, 0.000000000000, 0.000000000000, 0.536117958181, 0.000000000000, 0.000000000000, 0.971260169116, 0.000000000000, 0.000000000000, 0.000000000000) // NOLINT
};
// clang-format on
EXPECT_THAT(euler_angles,
testing::Pointwise(JetClose(kLooseTolerance), expected));
}
}
TEST(Quaternion, RotatePointGivesSameAnswerAsRotationByMatrixCanned) {
// Canned data generated in octave.
double const q[4] = {
+0.1956830471754074,
-0.0150618562474847,
+0.7634572982788086,
-0.3019454777240753,
};
double const Q[3][3] = {
// Scaled rotation matrix.
{-0.6355194033477252, +0.0951730541682254, +0.3078870197911186},
{-0.1411693904792992, +0.5297609702153905, -0.4551502574482019},
{-0.2896955822708862, -0.4669396571547050, -0.4536309793389248},
};
double const R[3][3] = {
// With unit rows and columns.
{-0.8918859164053080, +0.1335655625725649, +0.4320876677394745},
{-0.1981166751680096, +0.7434648665444399, -0.6387564287225856},
{-0.4065578619806013, -0.6553016349046693, -0.6366242786393164},
};
// Compute R from q and compare to known answer.
double Rq[3][3];
QuaternionToScaledRotation<double>(q, Rq[0]);
ExpectArraysClose(9, Q[0], Rq[0], kTolerance);
// Now do the same but compute R with normalization.
QuaternionToRotation<double>(q, Rq[0]);
ExpectArraysClose(9, R[0], Rq[0], kTolerance);
}
TEST(Quaternion, RotatePointGivesSameAnswerAsRotationByMatrix) {
// Rotation defined by a unit quaternion.
double const q[4] = {
+0.2318160216097109,
-0.0178430356832060,
+0.9044300776717159,
-0.3576998641394597,
};
double const p[3] = {
+0.11,
-13.15,
1.17,
};
double R[3 * 3];
QuaternionToRotation(q, R);
double result1[3];
UnitQuaternionRotatePoint(q, p, result1);
double result2[3];
VectorRef(result2, 3) = ConstMatrixRef(R, 3, 3) * ConstVectorRef(p, 3);
ExpectArraysClose(3, result1, result2, kTolerance);
}
// Verify that (a * b) * c == a * (b * c).
TEST(Quaternion, MultiplicationIsAssociative) {
std::mt19937 prng;
std::uniform_real_distribution<double> uniform_distribution{-1.0, 1.0};
double a[4];
double b[4];
double c[4];
for (int i = 0; i < 4; ++i) {
a[i] = uniform_distribution(prng);
b[i] = uniform_distribution(prng);
c[i] = uniform_distribution(prng);
}
double ab[4];
double ab_c[4];
QuaternionProduct(a, b, ab);
QuaternionProduct(ab, c, ab_c);
double bc[4];
double a_bc[4];
QuaternionProduct(b, c, bc);
QuaternionProduct(a, bc, a_bc);
ASSERT_NEAR(ab_c[0], a_bc[0], kTolerance);
ASSERT_NEAR(ab_c[1], a_bc[1], kTolerance);
ASSERT_NEAR(ab_c[2], a_bc[2], kTolerance);
ASSERT_NEAR(ab_c[3], a_bc[3], kTolerance);
}
TEST(AngleAxis, RotatePointGivesSameAnswerAsRotationMatrix) {
std::mt19937 prng;
std::uniform_real_distribution<double> uniform_distribution{-1.0, 1.0};
double angle_axis[3];
double R[9];
double p[3];
double angle_axis_rotated_p[3];
double rotation_matrix_rotated_p[3];
for (int i = 0; i < 10000; ++i) {
double theta = (2.0 * i * 0.0011 - 1.0) * kPi;
for (int j = 0; j < 50; ++j) {
for (int k = 0; k < 3; ++k) {
angle_axis[k] = uniform_distribution(prng);
p[k] = uniform_distribution(prng);
}
const double inv_norm =
theta / std::hypot(angle_axis[0], angle_axis[1], angle_axis[2]);
for (double& angle_axi : angle_axis) {
angle_axi *= inv_norm;
}
AngleAxisToRotationMatrix(angle_axis, R);
rotation_matrix_rotated_p[0] = R[0] * p[0] + R[3] * p[1] + R[6] * p[2];
rotation_matrix_rotated_p[1] = R[1] * p[0] + R[4] * p[1] + R[7] * p[2];
rotation_matrix_rotated_p[2] = R[2] * p[0] + R[5] * p[1] + R[8] * p[2];
AngleAxisRotatePoint(angle_axis, p, angle_axis_rotated_p);
for (int k = 0; k < 3; ++k) {
// clang-format off
EXPECT_NEAR(rotation_matrix_rotated_p[k],
angle_axis_rotated_p[k],
kTolerance) << "p: " << p[0]
<< " " << p[1]
<< " " << p[2]
<< " angle_axis: " << angle_axis[0]
<< " " << angle_axis[1]
<< " " << angle_axis[2];
// clang-format on
}
}
}
}
TEST(Quaternion, UnitQuaternion) {
using Jet = ceres::Jet<double, 4>;
std::array<Jet, 4> quaternion = {
Jet(1.0, 0), Jet(0.0, 1), Jet(0.0, 2), Jet(0.0, 3)};
std::array<Jet, 3> point = {Jet(0.0), Jet(0.0), Jet(0.0)};
std::array<Jet, 3> rotated_point;
QuaternionRotatePoint(quaternion.data(), point.data(), rotated_point.data());
for (int i = 0; i < 3; ++i) {
EXPECT_EQ(rotated_point[i], point[i]);
EXPECT_FALSE(rotated_point[i].v.array().isNaN().any());
}
}
TEST(AngleAxis, NearZeroRotatePointGivesSameAnswerAsRotationMatrix) {
std::mt19937 prng;
std::uniform_real_distribution<double> uniform_distribution{-1.0, 1.0};
double angle_axis[3];
double R[9];
double p[3];
double angle_axis_rotated_p[3];
double rotation_matrix_rotated_p[3];
for (int i = 0; i < 10000; ++i) {
double norm2 = 0.0;
for (int k = 0; k < 3; ++k) {
angle_axis[k] = uniform_distribution(prng);
p[k] = uniform_distribution(prng);
norm2 = angle_axis[k] * angle_axis[k];
}
double theta = (2.0 * i * 0.0001 - 1.0) * 1e-16;
const double inv_norm = theta / sqrt(norm2);
for (double& angle_axi : angle_axis) {
angle_axi *= inv_norm;
}
AngleAxisToRotationMatrix(angle_axis, R);
rotation_matrix_rotated_p[0] = R[0] * p[0] + R[3] * p[1] + R[6] * p[2];
rotation_matrix_rotated_p[1] = R[1] * p[0] + R[4] * p[1] + R[7] * p[2];
rotation_matrix_rotated_p[2] = R[2] * p[0] + R[5] * p[1] + R[8] * p[2];
AngleAxisRotatePoint(angle_axis, p, angle_axis_rotated_p);
for (int k = 0; k < 3; ++k) {
// clang-format off
EXPECT_NEAR(rotation_matrix_rotated_p[k],
angle_axis_rotated_p[k],
kTolerance) << "p: " << p[0]
<< " " << p[1]
<< " " << p[2]
<< " angle_axis: " << angle_axis[0]
<< " " << angle_axis[1]
<< " " << angle_axis[2];
// clang-format on
}
}
}
TEST(MatrixAdapter, RowMajor3x3ReturnTypeAndAccessIsCorrect) {
double array[9] = {1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0};
const float const_array[9] = {
1.0f, 2.0f, 3.0f, 4.0f, 5.0f, 6.0f, 7.0f, 8.0f, 9.0f};
MatrixAdapter<double, 3, 1> A = RowMajorAdapter3x3(array);
MatrixAdapter<const float, 3, 1> B = RowMajorAdapter3x3(const_array);
for (int i = 0; i < 3; ++i) {
for (int j = 0; j < 3; ++j) {
// The values are integers from 1 to 9, so equality tests are appropriate
// even for float and double values.
EXPECT_EQ(A(i, j), array[3 * i + j]);
EXPECT_EQ(B(i, j), const_array[3 * i + j]);
}
}
}
TEST(MatrixAdapter, ColumnMajor3x3ReturnTypeAndAccessIsCorrect) {
double array[9] = {1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0};
const float const_array[9] = {
1.0f, 2.0f, 3.0f, 4.0f, 5.0f, 6.0f, 7.0f, 8.0f, 9.0f};
MatrixAdapter<double, 1, 3> A = ColumnMajorAdapter3x3(array);
MatrixAdapter<const float, 1, 3> B = ColumnMajorAdapter3x3(const_array);
for (int i = 0; i < 3; ++i) {
for (int j = 0; j < 3; ++j) {
// The values are integers from 1 to 9, so equality tests are
// appropriate even for float and double values.
EXPECT_EQ(A(i, j), array[3 * j + i]);
EXPECT_EQ(B(i, j), const_array[3 * j + i]);
}
}
}
TEST(MatrixAdapter, RowMajor2x4IsCorrect) {
const int expected[8] = {1, 2, 3, 4, 5, 6, 7, 8};
int array[8];
MatrixAdapter<int, 4, 1> M(array);
// clang-format off
M(0, 0) = 1; M(0, 1) = 2; M(0, 2) = 3; M(0, 3) = 4;
M(1, 0) = 5; M(1, 1) = 6; M(1, 2) = 7; M(1, 3) = 8;
// clang-format on
for (int k = 0; k < 8; ++k) {
EXPECT_EQ(array[k], expected[k]);
}
}
TEST(MatrixAdapter, ColumnMajor2x4IsCorrect) {
const int expected[8] = {1, 5, 2, 6, 3, 7, 4, 8};
int array[8];
MatrixAdapter<int, 1, 2> M(array);
// clang-format off
M(0, 0) = 1; M(0, 1) = 2; M(0, 2) = 3; M(0, 3) = 4;
M(1, 0) = 5; M(1, 1) = 6; M(1, 2) = 7; M(1, 3) = 8;
// clang-format on
for (int k = 0; k < 8; ++k) {
EXPECT_EQ(array[k], expected[k]);
}
}
TEST(RotationMatrixToAngleAxis, NearPiExampleOneFromTobiasStrauss) {
// Example from Tobias Strauss
// clang-format off
const double rotation_matrix[] = {
-0.999807135425239, -0.0128154391194470, -0.0148814136745799,
-0.0128154391194470, -0.148441438622958, 0.988838158557669,
-0.0148814136745799, 0.988838158557669, 0.148248574048196
};
// clang-format on
double angle_axis[3];
RotationMatrixToAngleAxis(RowMajorAdapter3x3(rotation_matrix), angle_axis);
double round_trip[9];
AngleAxisToRotationMatrix(angle_axis, RowMajorAdapter3x3(round_trip));
EXPECT_THAT(rotation_matrix, IsNear3x3Matrix(round_trip));
}
static void CheckRotationMatrixToAngleAxisRoundTrip(const double theta,
const double phi,
const double angle) {
double angle_axis[3];
angle_axis[0] = angle * sin(phi) * cos(theta);
angle_axis[1] = angle * sin(phi) * sin(theta);
angle_axis[2] = angle * cos(phi);
double rotation_matrix[9];
AngleAxisToRotationMatrix(angle_axis, rotation_matrix);
double angle_axis_round_trip[3];
RotationMatrixToAngleAxis(rotation_matrix, angle_axis_round_trip);
EXPECT_THAT(angle_axis_round_trip, IsNearAngleAxis(angle_axis));
}
TEST(RotationMatrixToAngleAxis, ExhaustiveRoundTrip) {
constexpr double kMaxSmallAngle = 1e-8;
std::mt19937 prng;
std::uniform_real_distribution<double> uniform_distribution1{
kPi - kMaxSmallAngle, kPi};
std::uniform_real_distribution<double> uniform_distribution2{
-1.0, 2.0 * kMaxSmallAngle - 1.0};
const int kNumSteps = 1000;
for (int i = 0; i < kNumSteps; ++i) {
const double theta = static_cast<double>(i) / kNumSteps * 2.0 * kPi;
for (int j = 0; j < kNumSteps; ++j) {
const double phi = static_cast<double>(j) / kNumSteps * kPi;
// Rotations of angle Pi.
CheckRotationMatrixToAngleAxisRoundTrip(theta, phi, kPi);
// Rotation of angle approximately Pi.
CheckRotationMatrixToAngleAxisRoundTrip(
theta, phi, uniform_distribution1(prng));
// Rotations of angle approximately zero.
CheckRotationMatrixToAngleAxisRoundTrip(
theta, phi, uniform_distribution2(prng));
}
}
}
} // namespace internal
} // namespace ceres