simulation/modules/IsaacLab/source/isaaclab/test/utils/test_math.py

# Copyright (c) 2022-2026, The Isaac Lab Project Developers (https://github.com/isaac-sim/IsaacLab/blob/main/CONTRIBUTORS.md).
# All rights reserved.
#
# SPDX-License-Identifier: BSD-3-Clause

import math
from math import pi as PI

import numpy as np
import pytest
import scipy.spatial.transform as scipy_tf
import torch
import torch.utils.benchmark as benchmark

import isaaclab.utils.math as math_utils

DECIMAL_PRECISION = 5
"""Precision of the test.

This value is used since float operations are inexact. For reference:
https://github.com/pytorch/pytorch/issues/17678
"""


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
@pytest.mark.parametrize("size", ((5, 4, 3), (10, 2)))
def test_scale_unscale_transform(device, size):
    """Test scale_transform and unscale_transform."""

    inputs = torch.tensor(range(math.prod(size)), device=device, dtype=torch.float32).reshape(size)

    # test with same size
    scale_same = 2.0
    lower_same = -scale_same * torch.ones(size, device=device)
    upper_same = scale_same * torch.ones(size, device=device)
    output_same = math_utils.scale_transform(inputs, lower_same, upper_same)
    expected_output_same = inputs / scale_same
    torch.testing.assert_close(output_same, expected_output_same)
    output_unscale_same = math_utils.unscale_transform(output_same, lower_same, upper_same)
    torch.testing.assert_close(output_unscale_same, inputs)

    # test with broadcasting
    scale_per_batch = 3.0
    lower_per_batch = -scale_per_batch * torch.ones(size[1:], device=device)
    upper_per_batch = scale_per_batch * torch.ones(size[1:], device=device)
    output_per_batch = math_utils.scale_transform(inputs, lower_per_batch, upper_per_batch)
    expected_output_per_batch = inputs / scale_per_batch
    torch.testing.assert_close(output_per_batch, expected_output_per_batch)
    output_unscale_per_batch = math_utils.unscale_transform(output_per_batch, lower_per_batch, upper_per_batch)
    torch.testing.assert_close(output_unscale_per_batch, inputs)

    # test offset between lower and upper
    lower_offset = -3.0 * torch.ones(size[1:], device=device)
    upper_offset = 2.0 * torch.ones(size[1:], device=device)
    output_offset = math_utils.scale_transform(inputs, lower_offset, upper_offset)
    expected_output_offset = (inputs + 0.5) / 2.5
    torch.testing.assert_close(output_offset, expected_output_offset)
    output_unscale_offset = math_utils.unscale_transform(output_offset, lower_offset, upper_offset)
    torch.testing.assert_close(output_unscale_offset, inputs)


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
@pytest.mark.parametrize("size", ((5, 4, 3), (10, 2)))
def test_saturate(device, size):
    "Test saturate of a tensor of differed shapes and device."

    num_elements = math.prod(size)
    input = torch.tensor(range(num_elements), device=device, dtype=torch.float32).reshape(size)

    # testing with same size
    lower_same = -2.0 * torch.ones(size, device=device)
    upper_same = 2.0 * torch.ones(size, device=device)
    output_same = math_utils.saturate(input, lower_same, upper_same)
    assert torch.all(torch.greater_equal(output_same, lower_same)).item()
    assert torch.all(torch.less_equal(output_same, upper_same)).item()
    # testing with broadcasting
    lower_per_batch = -2.0 * torch.ones(size[1:], device=device)
    upper_per_batch = 3.0 * torch.ones(size[1:], device=device)
    output_per_batch = math_utils.saturate(input, lower_per_batch, upper_per_batch)
    assert torch.all(torch.greater_equal(output_per_batch, lower_per_batch)).item()
    assert torch.all(torch.less_equal(output_per_batch, upper_per_batch)).item()


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
@pytest.mark.parametrize("size", ((5, 4, 3), (10, 2)))
def test_normalize(device, size):
    """Test normalize of a tensor along its last dimension and check the norm of that dimension is close to 1.0."""

    num_elements = math.prod(size)
    input = torch.tensor(range(num_elements), device=device, dtype=torch.float32).reshape(size)
    output = math_utils.normalize(input)
    norm = torch.linalg.norm(output, dim=-1)
    torch.testing.assert_close(norm, torch.ones(size[0:-1], device=device))


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
def test_copysign(device):
    """Test copysign by copying a sign from both a negative and positive value and
    verify that the new sign is the same.
    """

    size = (10, 2)

    input_mag_pos = 2.0
    input_mag_neg = -3.0

    input = torch.tensor(range(20), device=device, dtype=torch.float32).reshape(size)
    value_pos = math_utils.copysign(input_mag_pos, input)
    value_neg = math_utils.copysign(input_mag_neg, input)
    torch.testing.assert_close(abs(input_mag_pos) * torch.ones_like(input), value_pos)
    torch.testing.assert_close(abs(input_mag_neg) * torch.ones_like(input), value_neg)

    input_neg_dim1 = input.clone()
    input_neg_dim1[:, 1] = -input_neg_dim1[:, 1]
    value_neg_dim1_pos = math_utils.copysign(input_mag_pos, input_neg_dim1)
    value_neg_dim1_neg = math_utils.copysign(input_mag_neg, input_neg_dim1)
    expected_value_neg_dim1_pos = abs(input_mag_pos) * torch.ones_like(input_neg_dim1)
    expected_value_neg_dim1_pos[:, 1] = -expected_value_neg_dim1_pos[:, 1]
    expected_value_neg_dim1_neg = abs(input_mag_neg) * torch.ones_like(input_neg_dim1)
    expected_value_neg_dim1_neg[:, 1] = -expected_value_neg_dim1_neg[:, 1]

    torch.testing.assert_close(expected_value_neg_dim1_pos, value_neg_dim1_pos)
    torch.testing.assert_close(expected_value_neg_dim1_neg, value_neg_dim1_neg)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_is_identity_pose(device):
    """Test is_identity_pose method."""
    # Single row identity pose (xyzw format)
    identity_pos = torch.zeros(3, device=device)
    identity_rot = torch.tensor((0.0, 0.0, 0.0, 1.0), device=device)
    assert math_utils.is_identity_pose(identity_pos, identity_rot) is True

    # Modified single row pose (not identity)
    identity_pos = torch.tensor([1.0, 0.0, 0.0], device=device)
    identity_rot = torch.tensor((0.0, 1.0, 0.0, 1.0), device=device)
    assert math_utils.is_identity_pose(identity_pos, identity_rot) is False

    # Multi-row identity pose (xyzw format)
    identity_pos = torch.zeros(3, 3, device=device)
    identity_rot = torch.tensor([[0.0, 0.0, 0.0, 1.0], [0.0, 0.0, 0.0, 1.0], [0.0, 0.0, 0.0, 1.0]], device=device)
    assert math_utils.is_identity_pose(identity_pos, identity_rot) is True

    # Modified multi-row pose (not identity)
    identity_pos = torch.tensor([[1.0, 0.0, 0.0], [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]], device=device)
    identity_rot = torch.tensor([[0.0, 1.0, 0.0, 1.0], [0.0, 0.0, 0.0, 1.0], [0.0, 0.0, 0.0, 1.0]], device=device)
    assert math_utils.is_identity_pose(identity_pos, identity_rot) is False


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_axis_angle_from_quat(device):
    """Test axis_angle_from_quat method."""
    # Quaternions of the form (2,4) and (2,2,4) in xyzw format
    quats = [
        torch.Tensor([[0.0, 0.0, 0.0, 1.0], [0.142006, 0.0, 0.5206887, 0.8418536]]).to(device),
        torch.Tensor(
            [
                [[0.0, 0.0, 0.0, 1.0], [0.142006, 0.0, 0.5206887, 0.8418536]],
                [[0.0, 0.0, 0.0, 1.0], [0.0995007, 0.0995007, 0.0995007, 0.9850375]],
            ]
        ).to(device),
    ]

    # Angles of the form (2,3) and (2,2,3)
    angles = [
        torch.Tensor([[0.0, 0.0, 0.0], [0.3, 0.0, 1.1]]).to(device),
        torch.Tensor([[[0.0, 0.0, 0.0], [0.3, 0.0, 1.1]], [[0.0, 0.0, 0.0], [0.2, 0.2, 0.2]]]).to(device),
    ]

    for quat, angle in zip(quats, angles):
        torch.testing.assert_close(math_utils.axis_angle_from_quat(quat), angle)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_axis_angle_from_quat_approximation(device):
    """Test the Taylor approximation from axis_angle_from_quat method.

    This test checks for unstable conversions where theta is very small.
    """
    # Generate a small rotation quaternion
    # Small angle
    theta = torch.Tensor([0.0000001]).to(device)
    # Arbitrary normalized axis of rotation in rads, (x,y,z)
    axis = [-0.302286, 0.205494, -0.930803]
    # Generate quaternion in xyzw format: [qx, qy, qz, qw]
    qw = torch.cos(theta / 2)
    quat_vect = [d * torch.sin(theta / 2) for d in axis] + [qw]
    quaternion = torch.tensor(quat_vect, dtype=torch.float32, device=device)

    # Convert quaternion to axis-angle
    axis_angle_computed = math_utils.axis_angle_from_quat(quaternion)

    # Expected axis-angle representation
    axis_angle_expected = torch.tensor([theta * d for d in axis], dtype=torch.float32, device=device)

    # Assert that the computed values are close to the expected values
    torch.testing.assert_close(axis_angle_computed, axis_angle_expected)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_quat_error_magnitude(device):
    """Test quat_error_magnitude method."""
    # No rotation (xyzw format)
    q1 = torch.Tensor([0, 0, 0, 1]).to(device)
    q2 = torch.Tensor([0, 0, 0, 1]).to(device)
    expected_diff = torch.Tensor([0.0]).to(device)
    q12_diff = math_utils.quat_error_magnitude(q1, q2)
    assert math.isclose(q12_diff.item(), expected_diff.item(), rel_tol=1e-5)

    # PI/2 rotation around x (xyzw format)
    q1 = torch.Tensor([0, 0, 0, 1.0]).to(device)
    q2 = torch.Tensor([0.7071068, 0, 0, 0.7071068]).to(device)
    expected_diff = torch.Tensor([PI / 2]).to(device)
    q12_diff = math_utils.quat_error_magnitude(q1, q2)
    assert math.isclose(q12_diff.item(), expected_diff.item(), rel_tol=1e-5)

    # PI rotation around y (xyzw format)
    q1 = torch.Tensor([0, 0, 0, 1.0]).to(device)
    q2 = torch.Tensor([0.0, 1.0, 0, 0.0]).to(device)
    expected_diff = torch.Tensor([PI]).to(device)
    q12_diff = math_utils.quat_error_magnitude(q1, q2)
    assert math.isclose(q12_diff.item(), expected_diff.item(), rel_tol=1e-5)

    # Batched inputs (xyzw format)
    q1 = torch.stack(
        [torch.Tensor([0, 0, 0, 1]), torch.Tensor([0, 0, 0, 1.0]), torch.Tensor([0, 0, 0, 1.0])], dim=0
    ).to(device)
    q2 = torch.stack(
        [torch.Tensor([0, 0, 0, 1]), torch.Tensor([0.7071068, 0, 0, 0.7071068]), torch.Tensor([0.0, 1.0, 0, 0.0])],
        dim=0,
    ).to(device)
    expected_diff = (
        torch.stack([torch.Tensor([0.0]), torch.Tensor([PI / 2]), torch.Tensor([PI])], dim=0).flatten().to(device)
    )
    q12_diff = math_utils.quat_error_magnitude(q1, q2)
    torch.testing.assert_close(q12_diff, expected_diff)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_quat_unique(device):
    """Test quat_unique method."""
    # Define test cases
    quats = math_utils.random_orientation(num=1024, device=device)

    # Test positive real quaternion
    pos_real_quats = math_utils.quat_unique(quats)

    # Test that the real part (w, at index 3 in xyzw format) is positive
    assert torch.all(pos_real_quats[:, 3] > 0).item()

    non_pos_indices = quats[:, 3] < 0
    # Check quaternion is negated if real part was negative
    torch.testing.assert_close(pos_real_quats[non_pos_indices], -quats[non_pos_indices])
    torch.testing.assert_close(pos_real_quats[~non_pos_indices], quats[~non_pos_indices])


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_quat_mul_with_quat_unique(device):
    """Test quat_mul method with different quaternions.

    This test checks that the quaternion multiplication is consistent when using positive real quaternions
    and regular quaternions. It makes sure that the result is the same regardless of the input quaternion sign
    (i.e. q and -q are same quaternion in the context of rotations).
    """

    quats_1 = math_utils.random_orientation(num=1024, device=device)
    quats_2 = math_utils.random_orientation(num=1024, device=device)
    # Make quats positive real
    quats_1_pos_real = math_utils.quat_unique(quats_1)
    quats_2_pos_real = math_utils.quat_unique(quats_2)

    # Option 1: Direct computation on quaternions
    quat_result_1 = math_utils.quat_mul(quats_1, math_utils.quat_conjugate(quats_2))
    quat_result_1 = math_utils.quat_unique(quat_result_1)

    # Option 2: Computation on positive real quaternions
    quat_result_2 = math_utils.quat_mul(quats_1_pos_real, math_utils.quat_conjugate(quats_2_pos_real))
    quat_result_2 = math_utils.quat_unique(quat_result_2)

    # Option 3: Mixed computation
    quat_result_3 = math_utils.quat_mul(quats_1, math_utils.quat_conjugate(quats_2_pos_real))
    quat_result_3 = math_utils.quat_unique(quat_result_3)

    # Check that the result is close to the expected value
    torch.testing.assert_close(quat_result_1, quat_result_2)
    torch.testing.assert_close(quat_result_2, quat_result_3)
    torch.testing.assert_close(quat_result_3, quat_result_1)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_quat_error_mag_with_quat_unique(device):
    """Test quat_error_magnitude method with positive real quaternions."""

    quats_1 = math_utils.random_orientation(num=1024, device=device)
    quats_2 = math_utils.random_orientation(num=1024, device=device)
    # Make quats positive real
    quats_1_pos_real = math_utils.quat_unique(quats_1)
    quats_2_pos_real = math_utils.quat_unique(quats_2)

    # Compute the error
    error_1 = math_utils.quat_error_magnitude(quats_1, quats_2)
    error_2 = math_utils.quat_error_magnitude(quats_1_pos_real, quats_2_pos_real)
    error_3 = math_utils.quat_error_magnitude(quats_1, quats_2_pos_real)
    error_4 = math_utils.quat_error_magnitude(quats_1_pos_real, quats_2)

    # Check that the error is close to the expected value
    torch.testing.assert_close(error_1, error_2)
    torch.testing.assert_close(error_2, error_3)
    torch.testing.assert_close(error_3, error_4)
    torch.testing.assert_close(error_4, error_1)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_convention_converter(device):
    """Test convert_camera_frame_orientation_convention to and from ros, opengl, and world conventions."""
    # Quaternions in xyzw format (converted from original wxyz test values)
    quat_ros = torch.tensor([[0.33985114, 0.82047325, -0.42470819, -0.17591989]], device=device)
    quat_opengl = torch.tensor([[0.17591988, 0.42470818, 0.82047324, 0.33985113]], device=device)
    quat_world = torch.tensor([[-0.27984815, -0.1159169, 0.88047623, -0.3647052]], device=device)

    # from ROS
    torch.testing.assert_close(
        math_utils.convert_camera_frame_orientation_convention(quat_ros, "ros", "opengl"), quat_opengl
    )
    torch.testing.assert_close(
        math_utils.convert_camera_frame_orientation_convention(quat_ros, "ros", "world"), quat_world
    )
    torch.testing.assert_close(math_utils.convert_camera_frame_orientation_convention(quat_ros, "ros", "ros"), quat_ros)
    # from OpenGL
    torch.testing.assert_close(
        math_utils.convert_camera_frame_orientation_convention(quat_opengl, "opengl", "ros"), quat_ros
    )
    torch.testing.assert_close(
        math_utils.convert_camera_frame_orientation_convention(quat_opengl, "opengl", "world"), quat_world
    )
    torch.testing.assert_close(
        math_utils.convert_camera_frame_orientation_convention(quat_opengl, "opengl", "opengl"), quat_opengl
    )
    # from World
    torch.testing.assert_close(
        math_utils.convert_camera_frame_orientation_convention(quat_world, "world", "ros"), quat_ros
    )
    torch.testing.assert_close(
        math_utils.convert_camera_frame_orientation_convention(quat_world, "world", "opengl"), quat_opengl
    )
    torch.testing.assert_close(
        math_utils.convert_camera_frame_orientation_convention(quat_world, "world", "world"), quat_world
    )


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
@pytest.mark.parametrize("size", ((10, 4), (5, 3, 4)))
def test_convert_quat(device, size):
    """Test convert_quat from "xyzw" to "wxyz" and back to "xyzw" and verify the correct rolling of the tensor.

    Also check the correct exceptions are raised for bad inputs for the quaternion and the 'to'.
    """

    quat = torch.zeros(size, device=device)
    quat[..., 0] = 1.0

    value_default = math_utils.convert_quat(quat)
    expected_default = torch.zeros(size, device=device)
    expected_default[..., -1] = 1.0
    torch.testing.assert_close(expected_default, value_default)

    value_to_xyzw = math_utils.convert_quat(quat, to="xyzw")
    expected_to_xyzw = torch.zeros(size, device=device)
    expected_to_xyzw[..., -1] = 1.0
    torch.testing.assert_close(expected_to_xyzw, value_to_xyzw)

    value_to_wxyz = math_utils.convert_quat(quat, to="wxyz")
    expected_to_wxyz = torch.zeros(size, device=device)
    expected_to_wxyz[..., 1] = 1.0
    torch.testing.assert_close(expected_to_wxyz, value_to_wxyz)

    bad_quat = torch.zeros((10, 5), device=device)

    with pytest.raises(ValueError):
        math_utils.convert_quat(bad_quat)

    with pytest.raises(ValueError):
        math_utils.convert_quat(quat, to="xwyz")


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
def test_quat_conjugate(device):
    """Test quat_conjugate by checking the sign of the imaginary part changes but the magnitudes stay the same."""

    quat = math_utils.random_orientation(1000, device=device)

    value = math_utils.quat_conjugate(quat)
    # In xyzw format: xyz is imaginary (indices 0-2), w is real (index 3)
    expected_imag = -quat[..., :3]
    expected_real = quat[..., 3]
    torch.testing.assert_close(expected_imag, value[..., :3])
    torch.testing.assert_close(expected_real, value[..., 3])


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
@pytest.mark.parametrize("num_envs", (1, 10))
@pytest.mark.parametrize(
    "euler_angles",
    [
        [0.0, 0.0, 0.0],
        [math.pi / 2.0, 0.0, 0.0],
        [0.0, math.pi / 2.0, 0.0],
        [0.0, 0.0, math.pi / 2.0],
        [1.5708, -2.75, 0.1],
        [0.1, math.pi, math.pi / 2],
    ],
)
def test_quat_from_euler_xyz(device, num_envs, euler_angles):
    """Test quat_from_euler_xyz against scipy."""

    angles = torch.tensor(euler_angles, device=device).unsqueeze(0).repeat((num_envs, 1))
    quat_value = math_utils.quat_unique(math_utils.quat_from_euler_xyz(angles[:, 0], angles[:, 1], angles[:, 2]))
    # scipy's as_quat() already returns xyzw format, no conversion needed
    expected_quat = (
        torch.tensor(
            scipy_tf.Rotation.from_euler("xyz", euler_angles, degrees=False).as_quat(),
            device=device,
            dtype=torch.float,
        )
        .unsqueeze(0)
        .repeat((num_envs, 1))
    )
    torch.testing.assert_close(expected_quat, quat_value)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_wrap_to_pi(device):
    """Test wrap_to_pi method."""
    # No wrapping needed
    angle = torch.Tensor([0.0]).to(device)
    expected_angle = torch.Tensor([0.0]).to(device)
    wrapped_angle = math_utils.wrap_to_pi(angle)
    torch.testing.assert_close(wrapped_angle, expected_angle)

    # Positive angle
    angle = torch.Tensor([PI]).to(device)
    expected_angle = torch.Tensor([PI]).to(device)
    wrapped_angle = math_utils.wrap_to_pi(angle)
    torch.testing.assert_close(wrapped_angle, expected_angle)

    # Negative angle
    angle = torch.Tensor([-PI]).to(device)
    expected_angle = torch.Tensor([-PI]).to(device)
    wrapped_angle = math_utils.wrap_to_pi(angle)
    torch.testing.assert_close(wrapped_angle, expected_angle)

    # Multiple angles
    angle = torch.Tensor([3 * PI, -3 * PI, 4 * PI, -4 * PI]).to(device)
    expected_angle = torch.Tensor([PI, -PI, 0.0, 0.0]).to(device)
    wrapped_angle = math_utils.wrap_to_pi(angle)
    torch.testing.assert_close(wrapped_angle, expected_angle)

    # Multiple angles from MATLAB docs
    # fmt: off
    angle = torch.Tensor([-2 * PI, - PI - 0.1, -PI, -2.8, 3.1, PI, PI + 0.001, PI + 1, 2 * PI, 2 * PI + 0.1]).to(device)
    expected_angle = torch.Tensor([0.0, PI - 0.1, -PI, -2.8, 3.1 , PI, -PI + 0.001, -PI + 1 , 0.0, 0.1]).to(device)
    # fmt: on
    wrapped_angle = math_utils.wrap_to_pi(angle)
    torch.testing.assert_close(wrapped_angle, expected_angle)


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
@pytest.mark.parametrize("shape", ((3,), (1024, 3)))
def test_skew_symmetric_matrix(device, shape):
    """Test skew_symmetric_matrix."""

    vec_rand = torch.zeros(shape, device=device)
    vec_rand.uniform_(-1000.0, 1000.0)

    if vec_rand.ndim == 1:
        vec_rand_resized = vec_rand.clone().unsqueeze(0)
    else:
        vec_rand_resized = vec_rand.clone()

    mat_value = math_utils.skew_symmetric_matrix(vec_rand)
    if len(shape) == 1:
        expected_shape = (1, 3, 3)
    else:
        expected_shape = (shape[0], 3, 3)

    torch.testing.assert_close(
        torch.zeros((expected_shape[0], 3), device=device), torch.diagonal(mat_value, dim1=-2, dim2=-1)
    )
    torch.testing.assert_close(-vec_rand_resized[:, 2], mat_value[:, 0, 1])
    torch.testing.assert_close(vec_rand_resized[:, 1], mat_value[:, 0, 2])
    torch.testing.assert_close(-vec_rand_resized[:, 0], mat_value[:, 1, 2])
    torch.testing.assert_close(vec_rand_resized[:, 2], mat_value[:, 1, 0])
    torch.testing.assert_close(-vec_rand_resized[:, 1], mat_value[:, 2, 0])
    torch.testing.assert_close(vec_rand_resized[:, 0], mat_value[:, 2, 1])


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_orthogonalize_perspective_depth(device):
    """Test for converting perspective depth to orthogonal depth."""
    # Create a sample perspective depth image (N, H, W)
    perspective_depth = torch.tensor([[[10.0, 0.0, 100.0], [0.0, 3000.0, 0.0], [100.0, 0.0, 100.0]]], device=device)

    # Create sample intrinsic matrix (3, 3)
    intrinsics = torch.tensor([[500.0, 0.0, 5.0], [0.0, 500.0, 5.0], [0.0, 0.0, 1.0]], device=device)

    # Convert perspective depth to orthogonal depth
    orthogonal_depth = math_utils.orthogonalize_perspective_depth(perspective_depth, intrinsics)

    # Manually compute expected orthogonal depth based on the formula for comparison
    expected_orthogonal_depth = torch.tensor(
        [[[9.9990, 0.0000, 99.9932], [0.0000, 2999.8079, 0.0000], [99.9932, 0.0000, 99.9964]]], device=device
    )

    # Assert that the output is close to the expected result
    torch.testing.assert_close(orthogonal_depth, expected_orthogonal_depth)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_combine_frame_transform(device):
    """Test combine_frame_transforms function."""
    # create random poses
    pose01 = torch.rand(1, 7, device=device)
    pose01[:, 3:7] = torch.nn.functional.normalize(pose01[..., 3:7], dim=-1)

    pose12 = torch.rand(1, 7, device=device)
    pose12[:, 3:7] = torch.nn.functional.normalize(pose12[..., 3:7], dim=-1)

    # apply combination of poses
    pos02, quat02 = math_utils.combine_frame_transforms(
        pose01[..., :3], pose01[..., 3:7], pose12[:, :3], pose12[:, 3:7]
    )
    # apply combination of poses w.r.t. inverse to get original frame
    pos01, quat01 = math_utils.combine_frame_transforms(
        pos02,
        quat02,
        math_utils.quat_rotate(math_utils.quat_inv(pose12[:, 3:7]), -pose12[:, :3]),
        math_utils.quat_inv(pose12[:, 3:7]),
    )

    torch.testing.assert_close(pose01, torch.cat((pos01, quat01), dim=-1))


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
@pytest.mark.parametrize("seed", [0, 1, 2, 3, 4])
def test_interpolate_poses(device, seed):
    """Test interpolate_poses function.

    This test checks the output from the :meth:`~isaaclab.utils.math_utils.interpolate_poses` function against
    the output from :func:`scipy.spatial.transform.Slerp` and :func:`np.linspace`.
    """
    torch.manual_seed(seed)
    np.random.seed(seed)
    for _ in range(100):
        mat1 = math_utils.generate_random_transformation_matrix()
        mat2 = math_utils.generate_random_transformation_matrix()
        pos_1, rmat1 = math_utils.unmake_pose(mat1)
        pos_2, rmat2 = math_utils.unmake_pose(mat2)

        # Compute expected results using scipy's Slerp
        key_rots = scipy_tf.Rotation.from_matrix(np.array([rmat1, rmat2]))

        # Create a Slerp object and interpolate create the interpolated rotation matrices
        num_steps = np.random.randint(3, 51)
        key_times = [0, 1]
        slerp = scipy_tf.Slerp(key_times, key_rots)
        interp_times = np.linspace(0, 1, num_steps)
        expected_quat = slerp(interp_times).as_matrix()

        # Test interpolation against expected result using np.linspace
        expected_pos = np.linspace(pos_1, pos_2, num_steps)

        # interpolate_poses using interpolate_poses and quat_slerp
        interpolated_poses, _ = math_utils.interpolate_poses(
            math_utils.make_pose(pos_1, rmat1), math_utils.make_pose(pos_2, rmat2), num_steps - 2
        )
        result_pos, result_quat = math_utils.unmake_pose(interpolated_poses)

        # Assert that the result is almost equal to the expected quaternion
        np.testing.assert_array_almost_equal(result_quat, expected_quat, decimal=DECIMAL_PRECISION)
        np.testing.assert_array_almost_equal(result_pos, expected_pos, decimal=DECIMAL_PRECISION)


def test_pose_inv():
    """Test pose_inv function.

    This test checks the output from the :meth:`~isaaclab.utils.math_utils.pose_inv` function against
    the output from :func:`np.linalg.inv`. Two test cases are performed:

    1. Checking the inverse of a random transformation matrix matches Numpy's built-in inverse.
    2. Checking the inverse of a batch of random transformation matrices matches Numpy's built-in inverse.
    """
    # Check against a single matrix
    for _ in range(100):
        test_mat = math_utils.generate_random_transformation_matrix(pos_boundary=10, rot_boundary=(2 * np.pi))
        result = np.array(math_utils.pose_inv(test_mat))
        expected = np.linalg.inv(np.array(test_mat))
        np.testing.assert_array_almost_equal(result, expected, decimal=DECIMAL_PRECISION)

    # Check against a batch of matrices
    test_mats = torch.stack(
        [
            math_utils.generate_random_transformation_matrix(pos_boundary=10, rot_boundary=(2 * math.pi))
            for _ in range(100)
        ]
    )
    result = np.array(math_utils.pose_inv(test_mats))
    expected = np.linalg.inv(np.array(test_mats))
    np.testing.assert_array_almost_equal(result, expected, decimal=DECIMAL_PRECISION)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_quat_to_and_from_angle_axis(device):
    """Test that axis_angle_from_quat against scipy and that quat_from_angle_axis are the inverse of each other."""
    n = 1024
    q_rand = math_utils.quat_unique(math_utils.random_orientation(num=n, device=device))
    rot_vec_value = math_utils.axis_angle_from_quat(q_rand)
    # Our quaternions are already in xyzw format, which scipy expects
    rot_vec_scipy = torch.tensor(
        scipy_tf.Rotation.from_quat(q_rand.to(device="cpu").numpy()).as_rotvec(),
        device=device,
        dtype=torch.float32,
    )
    torch.testing.assert_close(rot_vec_scipy, rot_vec_value)
    axis = math_utils.normalize(rot_vec_value.clone())
    angle = torch.linalg.norm(rot_vec_value.clone(), dim=-1)
    q_value = math_utils.quat_unique(math_utils.quat_from_angle_axis(angle, axis))
    torch.testing.assert_close(q_rand, q_value)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_quat_box_minus(device):
    """Test quat_box_minus method.

    Ensures that quat_box_minus correctly computes the axis-angle difference
    between two quaternions representing rotations around the same axis.
    """
    axis_angles = torch.tensor([0.0, 0.0, 1.0], device=device)
    angle_a = math.pi - 0.1
    angle_b = -math.pi + 0.1
    quat_a = math_utils.quat_from_angle_axis(torch.tensor([angle_a], device=device), axis_angles)
    quat_b = math_utils.quat_from_angle_axis(torch.tensor([angle_b], device=device), axis_angles)

    axis_diff = math_utils.quat_box_minus(quat_a, quat_b).squeeze(0)
    expected_diff = axis_angles * math_utils.wrap_to_pi(torch.tensor(angle_a - angle_b, device=device))
    torch.testing.assert_close(expected_diff, axis_diff, atol=1e-06, rtol=1e-06)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_quat_box_minus_and_quat_box_plus(device):
    """Test consistency of quat_box_plus and quat_box_minus.

    Checks that applying quat_box_plus to accumulate rotations and then using
    quat_box_minus to retrieve differences results in expected values.
    """

    # Perform closed-loop integration using quat_box_plus to accumulate rotations,
    # and then use quat_box_minus to compute the incremental differences between quaternions.
    # NOTE: Accuracy may decrease for very small angle increments due to numerical precision limits.
    for n in (2, 10, 100, 1000):
        # Define small incremental rotations around principal axes
        delta_angle = torch.tensor(
            [
                [0, 0, -math.pi / n],
                [0, -math.pi / n, 0],
                [-math.pi / n, 0, 0],
                [0, 0, math.pi / n],
                [0, math.pi / n, 0],
                [math.pi / n, 0, 0],
            ],
            device=device,
        )

        # Initialize quaternion trajectory starting from identity quaternion (xyzw format)
        quat_trajectory = torch.zeros((len(delta_angle), 2 * n + 1, 4), device=device)
        quat_trajectory[:, 0, :] = torch.tensor([[0.0, 0.0, 0.0, 1.0]], device=device).repeat(len(delta_angle), 1)

        # Integrate incremental rotations forward to form a closed loop trajectory
        for i in range(1, 2 * n + 1):
            quat_trajectory[:, i] = math_utils.quat_box_plus(quat_trajectory[:, i - 1], delta_angle)

        # Validate the loop closure: start and end quaternions should be approximately equal
        torch.testing.assert_close(quat_trajectory[:, 0], quat_trajectory[:, -1], atol=1e-04, rtol=1e-04)

        # Validate that the differences between consecutive quaternions match the original increments
        for i in range(2 * n):
            delta_result = math_utils.quat_box_minus(quat_trajectory[:, i + 1], quat_trajectory[:, i])
            torch.testing.assert_close(delta_result, delta_angle, atol=1e-04, rtol=1e-04)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
@pytest.mark.parametrize("t12_inputs", ["True", "False"])
@pytest.mark.parametrize("q12_inputs", ["True", "False"])
def test_combine_frame_transforms(device, t12_inputs, q12_inputs):
    """Test combine_frame_transforms such that inputs for delta translation and delta rotation
    can be :obj:`None` or specified.
    """
    n = 1024
    t01 = torch.zeros((n, 3), device=device)
    t01.uniform_(-1000.0, 1000.0)
    q01 = math_utils.quat_unique(math_utils.random_orientation(n, device=device))

    mat_01 = torch.eye(4, 4, device=device).unsqueeze(0).repeat(n, 1, 1)
    mat_01[:, 0:3, 3] = t01
    mat_01[:, 0:3, 0:3] = math_utils.matrix_from_quat(q01)

    mat_12 = torch.eye(4, 4, device=device).unsqueeze(0).repeat(n, 1, 1)
    if t12_inputs:
        t12 = torch.zeros((n, 3), device=device)
        t12.uniform_(-1000.0, 1000.0)
        mat_12[:, 0:3, 3] = t12
    else:
        t12 = None

    if q12_inputs:
        q12 = math_utils.quat_unique(math_utils.random_orientation(n, device=device))
        mat_12[:, 0:3, 0:3] = math_utils.matrix_from_quat(q12)
    else:
        q12 = None

    mat_expect = torch.einsum("bij,bjk->bik", mat_01, mat_12)
    expected_translation = mat_expect[:, 0:3, 3]
    expected_quat = math_utils.quat_from_matrix(mat_expect[:, 0:3, 0:3])
    translation_value, quat_value = math_utils.combine_frame_transforms(t01, q01, t12, q12)

    torch.testing.assert_close(expected_translation, translation_value, atol=1e-3, rtol=1e-5)
    torch.testing.assert_close(math_utils.quat_unique(expected_quat), math_utils.quat_unique(quat_value))


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
@pytest.mark.parametrize("t02_inputs", ["True", "False"])
@pytest.mark.parametrize("q02_inputs", ["True", "False"])
def test_subtract_frame_transforms(device, t02_inputs, q02_inputs):
    """Test subtract_frame_transforms with specified and unspecified inputs for t02 and q02.

    This test verifies that :meth:`~isaaclab.utils.math_utils.subtract_frame_transforms` is the inverse operation
    to :meth:`~isaaclab.utils.math_utils.combine_frame_transforms`.
    ."""
    n = 1024
    t01 = torch.zeros((n, 3), device=device)
    t01.uniform_(-1000.0, 1000.0)
    q01 = math_utils.quat_unique(math_utils.random_orientation(n, device=device))

    mat_01 = torch.eye(4, 4, device=device).unsqueeze(0).repeat(n, 1, 1)
    mat_01[:, 0:3, 3] = t01
    mat_01[:, 0:3, 0:3] = math_utils.matrix_from_quat(q01)

    if t02_inputs:
        t02 = torch.zeros((n, 3), device=device)
        t02.uniform_(-1000.0, 1000.0)
        t02_expected = t02.clone()
    else:
        t02 = None
        t02_expected = torch.zeros((n, 3), device=device)

    if q02_inputs:
        q02 = math_utils.quat_unique(math_utils.random_orientation(n, device=device))
        q02_expected = q02.clone()
    else:
        q02 = None
        q02_expected = math_utils.default_orientation(n, device=device)

    t12_value, q12_value = math_utils.subtract_frame_transforms(t01, q01, t02, q02)
    t02_compare, q02_compare = math_utils.combine_frame_transforms(t01, q01, t12_value, q12_value)

    torch.testing.assert_close(t02_expected, t02_compare, atol=1e-3, rtol=1e-4)
    torch.testing.assert_close(math_utils.quat_unique(q02_expected), math_utils.quat_unique(q02_compare))


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
@pytest.mark.parametrize("rot_error_type", ("quat", "axis_angle"))
def test_compute_pose_error(device, rot_error_type):
    """Test compute_pose_error for different rot_error_type."""
    n = 1000
    t01 = torch.zeros((n, 3), device=device)
    t01.uniform_(-1000.0, 1000.0)
    t02 = torch.zeros((n, 3), device=device)
    t02.uniform_(-1000.0, 1000.0)
    q01 = math_utils.quat_unique(math_utils.random_orientation(n, device=device))
    q02 = math_utils.quat_unique(math_utils.random_orientation(n, device=device))

    diff_pos, diff_rot = math_utils.compute_pose_error(t01, q01, t02, q02, rot_error_type=rot_error_type)

    torch.testing.assert_close(t02 - t01, diff_pos)
    if rot_error_type == "axis_angle":
        torch.testing.assert_close(math_utils.quat_box_minus(q02, q01), diff_rot)
    else:
        axis_angle = math_utils.quat_box_minus(q02, q01)
        axis = math_utils.normalize(axis_angle)
        angle = torch.linalg.norm(axis_angle, dim=-1)

        torch.testing.assert_close(
            math_utils.quat_unique(math_utils.quat_from_angle_axis(angle, axis)),
            math_utils.quat_unique(diff_rot),
        )


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_rigid_body_twist_transform(device):
    """Test rigid_body_twist_transform method.

    Verifies correct transformation of twists (linear and angular velocity) between coordinate frames.
    """
    num_bodies = 100
    # Frame A to B
    t_AB = torch.randn((num_bodies, 3), device=device)
    q_AB = math_utils.random_orientation(num=num_bodies, device=device)

    # Twists in A in frame A
    v_AA = torch.randn((num_bodies, 3), device=device)
    w_AA = torch.randn((num_bodies, 3), device=device)

    # Get twists in B in frame B
    v_BB, w_BB = math_utils.rigid_body_twist_transform(v_AA, w_AA, t_AB, q_AB)

    # Get back twists in A in frame A
    t_BA = -math_utils.quat_rotate_inverse(q_AB, t_AB)
    q_BA = math_utils.quat_conjugate(q_AB)
    v_AA_, w_AA_ = math_utils.rigid_body_twist_transform(v_BB, w_BB, t_BA, q_BA)

    # Check
    torch.testing.assert_close(v_AA_, v_AA)
    torch.testing.assert_close(w_AA_, w_AA)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_yaw_quat(device):
    """
    Test for yaw_quat methods.
    """
    # 90-degree (pi/2 radians) rotations about the Y-axis (xyzw format)
    quat_input = torch.tensor([0, 0.7071, 0, 0.7071], device=device)
    cloned_quat_input = quat_input.clone()

    # Calculated output that the function should return (identity in xyzw)
    expected_output = torch.tensor([0.0, 0.0, 0.0, 1.0], device=device)

    # Compute the result using the existing implementation
    result = math_utils.yaw_quat(quat_input)

    # Verify original quat is not being modified
    torch.testing.assert_close(quat_input, cloned_quat_input)

    # check that the output is equivalent to the expected output
    torch.testing.assert_close(result, expected_output)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_quat_slerp(device):
    """Test quat_slerp function.

    This test checks the output from the :meth:`~isaaclab.utils.math_utils.quat_slerp` function against
    the output from :func:`scipy.spatial.transform.Slerp`.
    """
    # Generate 100 random rotation matrices
    random_rotation_matrices_1 = [math_utils.generate_random_rotation() for _ in range(100)]
    random_rotation_matrices_2 = [math_utils.generate_random_rotation() for _ in range(100)]

    tau_values = np.random.rand(10)  # Random values in the range [0, 1]

    for rmat1, rmat2 in zip(random_rotation_matrices_1, random_rotation_matrices_2):
        # Convert the rotation matrices to quaternions
        q1 = scipy_tf.Rotation.from_matrix(rmat1).as_quat()  # (x, y, z, w)
        q2 = scipy_tf.Rotation.from_matrix(rmat2).as_quat()  # (x, y, z, w)

        # Compute expected results using scipy's Slerp
        key_rots = scipy_tf.Rotation.from_quat(np.array([q1, q2]))
        key_times = [0, 1]
        slerp = scipy_tf.Slerp(key_times, key_rots)

        for tau in tau_values:
            expected = slerp(tau).as_quat()  # (x, y, z, w)
            result = math_utils.quat_slerp(torch.tensor(q1, device=device), torch.tensor(q2, device=device), tau)
            # Assert that the result is almost equal to the expected quaternion
            np.testing.assert_array_almost_equal(result.cpu(), expected, decimal=DECIMAL_PRECISION)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_matrix_from_quat(device):
    """test matrix_from_quat against scipy."""
    # prepare random quaternions and vectors
    n = 1024
    # prepare random quaternions and vectors
    q_rand = math_utils.quat_unique(math_utils.random_orientation(num=n, device=device))
    rot_mat = math_utils.matrix_from_quat(quaternions=q_rand)
    # Our quaternions are already in xyzw format, which scipy expects
    rot_mat_scipy = torch.tensor(
        scipy_tf.Rotation.from_quat(q_rand.to(device="cpu").numpy()).as_matrix(),
        device=device,
        dtype=torch.float32,
    )
    torch.testing.assert_close(rot_mat_scipy.to(device=device), rot_mat)
    q_value = math_utils.quat_unique(math_utils.quat_from_matrix(rot_mat))
    torch.testing.assert_close(q_rand, q_value)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
@pytest.mark.parametrize(
    "euler_angles",
    [
        [0.0, 0.0, 0.0],
        [math.pi / 2.0, 0.0, 0.0],
        [0.0, math.pi / 2.0, 0.0],
        [0.0, 0.0, math.pi / 2.0],
        [1.5708, -2.75, 0.1],
        [0.1, math.pi, math.pi / 2],
    ],
)
@pytest.mark.parametrize(
    "convention", ("XYZ", "XZY", "YXZ", "YZX", "ZXY", "ZYX", "ZYZ", "YZY", "XYX", "XZX", "ZXZ", "YXY")
)
def test_matrix_from_euler(device, euler_angles, convention):
    """Test matrix_from_euler against scipy for different permutations of the X,Y,Z euler angle conventions."""

    num_envs = 1024
    angles = torch.tensor(euler_angles, device=device).unsqueeze(0).repeat((num_envs, 1))
    mat_value = math_utils.matrix_from_euler(angles, convention=convention)
    expected_mag = (
        torch.tensor(
            scipy_tf.Rotation.from_euler(convention, euler_angles, degrees=False).as_matrix(),
            device=device,
            dtype=torch.float,
        )
        .unsqueeze(0)
        .repeat((num_envs, 1, 1))
    )
    torch.testing.assert_close(expected_mag, mat_value)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_quat_apply(device):
    """Test for quat_apply against scipy."""
    # prepare random quaternions and vectors
    n = 1024
    q_rand = math_utils.random_orientation(num=n, device=device)
    # Our quaternions are already in xyzw format, which scipy expects
    Rotation = scipy_tf.Rotation.from_quat(q_rand.to(device="cpu").numpy())

    v_rand = math_utils.sample_uniform(-1000, 1000, (n, 3), device=device)

    # compute the result using the new implementation
    scipy_result = torch.tensor(Rotation.apply(v_rand.to(device="cpu").numpy()), device=device, dtype=torch.float)
    apply_result = math_utils.quat_apply(q_rand, v_rand)
    torch.testing.assert_close(scipy_result.to(device=device), apply_result, atol=2e-4, rtol=2e-4)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_quat_apply_inverse(device):
    """Test for quat_apply against scipy."""

    # prepare random quaternions and vectors
    n = 1024
    q_rand = math_utils.random_orientation(num=n, device=device)
    # Our quaternions are already in xyzw format, which scipy expects
    Rotation = scipy_tf.Rotation.from_quat(q_rand.to(device="cpu").numpy())

    v_rand = math_utils.sample_uniform(-1000, 1000, (n, 3), device=device)

    # compute the result using the new implementation
    scipy_result = torch.tensor(
        Rotation.apply(v_rand.to(device="cpu").numpy(), inverse=True), device=device, dtype=torch.float
    )
    apply_result = math_utils.quat_apply_inverse(q_rand, v_rand)
    torch.testing.assert_close(scipy_result.to(device=device), apply_result, atol=2e-4, rtol=2e-4)


@pytest.mark.parametrize("device", ["cpu", "cuda:0"])
def test_quat_inv(device):
    """Test for quat_inv method.

    For random unit and non-unit quaternions q, the Hamilton products
    q ⊗ q⁻¹ and q⁻¹ ⊗ q must both equal the identity quaternion (0,0,0,1)
    within numerical precision.
    """
    num = 2048

    # -------- non-unit sample (average ‖q‖ ≈ 10) --------
    q_nonunit = torch.randn(num, 4, device=device) * 5.0

    # -------- unit sample (‖q‖ = 1) --------
    q_unit = torch.randn(num, 4, device=device)
    q_unit = q_unit / q_unit.norm(dim=-1, keepdim=True)

    identity = torch.tensor([0.0, 0.0, 0.0, 1.0], device=device)  # xyzw format

    for q in (q_nonunit, q_unit):
        q_inv = math_utils.quat_inv(q)

        id_batch = identity.expand_as(q)

        # left and right products must both be identity
        torch.testing.assert_close(math_utils.quat_mul(q, q_inv), id_batch, atol=1e-4, rtol=1e-4)
        torch.testing.assert_close(math_utils.quat_mul(q_inv, q), id_batch, atol=1e-4, rtol=1e-4)


def test_quat_apply_benchmarks():
    """Test for quat_apply and quat_apply_inverse methods compared to old methods using torch.bmm and torch.einsum.
    The new implementation uses :meth:`torch.einsum` instead of `torch.bmm` which allows
    for more flexibility in the input dimensions and is faster than `torch.bmm`.
    """

    # define old implementation for quat_rotate and quat_rotate_inverse
    # Updated for xyzw format (xyz at indices 0-2, w at index 3)

    @torch.jit.script
    def bmm_quat_rotate(q: torch.Tensor, v: torch.Tensor) -> torch.Tensor:
        shape = q.shape
        q_vec = q[:, :3]  # xyz
        q_w = q[:, 3]  # w
        a = v * (2.0 * q_w**2 - 1.0).unsqueeze(-1)
        b = torch.cross(q_vec, v, dim=-1) * q_w.unsqueeze(-1) * 2.0
        c = q_vec * torch.bmm(q_vec.view(shape[0], 1, 3), v.view(shape[0], 3, 1)).squeeze(-1) * 2.0
        return a + b + c

    @torch.jit.script
    def bmm_quat_rotate_inverse(q: torch.Tensor, v: torch.Tensor) -> torch.Tensor:
        shape = q.shape
        q_vec = q[:, :3]  # xyz
        q_w = q[:, 3]  # w
        a = v * (2.0 * q_w**2 - 1.0).unsqueeze(-1)
        b = torch.cross(q_vec, v, dim=-1) * q_w.unsqueeze(-1) * 2.0
        c = q_vec * torch.bmm(q_vec.view(shape[0], 1, 3), v.view(shape[0], 3, 1)).squeeze(-1) * 2.0
        return a - b + c

    @torch.jit.script
    def einsum_quat_rotate(q: torch.Tensor, v: torch.Tensor) -> torch.Tensor:
        q_vec = q[..., :3]  # xyz
        q_w = q[..., 3]  # w
        a = v * (2.0 * q_w**2 - 1.0).unsqueeze(-1)
        b = torch.cross(q_vec, v, dim=-1) * q_w.unsqueeze(-1) * 2.0
        c = q_vec * torch.einsum("...i,...i->...", q_vec, v).unsqueeze(-1) * 2.0
        return a + b + c

    @torch.jit.script
    def einsum_quat_rotate_inverse(q: torch.Tensor, v: torch.Tensor) -> torch.Tensor:
        q_vec = q[..., :3]  # xyz
        q_w = q[..., 3]  # w
        a = v * (2.0 * q_w**2 - 1.0).unsqueeze(-1)
        b = torch.cross(q_vec, v, dim=-1) * q_w.unsqueeze(-1) * 2.0
        c = q_vec * torch.einsum("...i,...i->...", q_vec, v).unsqueeze(-1) * 2.0
        return a - b + c

    # check that implementation produces the same result as the new implementation
    for device in ["cpu", "cuda:0"]:
        # prepare random quaternions and vectors
        q_rand = math_utils.random_orientation(num=1024, device=device)
        v_rand = math_utils.sample_uniform(-1000, 1000, (1024, 3), device=device)

        # compute the result using the old implementation
        bmm_result = bmm_quat_rotate(q_rand, v_rand)
        bmm_result_inv = bmm_quat_rotate_inverse(q_rand, v_rand)

        # compute the result using the old implementation
        einsum_result = einsum_quat_rotate(q_rand, v_rand)
        einsum_result_inv = einsum_quat_rotate_inverse(q_rand, v_rand)

        # compute the result using the new implementation
        new_result = math_utils.quat_apply(q_rand, v_rand)
        new_result_inv = math_utils.quat_apply_inverse(q_rand, v_rand)

        # check that the result is close to the expected value
        torch.testing.assert_close(bmm_result, new_result, atol=1e-3, rtol=1e-3)
        torch.testing.assert_close(bmm_result_inv, new_result_inv, atol=1e-3, rtol=1e-3)
        torch.testing.assert_close(einsum_result, new_result, atol=1e-3, rtol=1e-3)
        torch.testing.assert_close(einsum_result_inv, new_result_inv, atol=1e-3, rtol=1e-3)

    # check the performance of the new implementation
    for device in ["cpu", "cuda:0"]:
        # prepare random quaternions and vectors
        # new implementation supports batched inputs
        q_shape = (1024, 2, 5, 4)
        v_shape = (1024, 2, 5, 3)
        # sample random quaternions and vectors
        num_quats = math.prod(q_shape[:-1])
        q_rand = math_utils.random_orientation(num=num_quats, device=device).reshape(q_shape)
        v_rand = math_utils.sample_uniform(-1000, 1000, v_shape, device=device)

        # create functions to test
        def iter_quat_apply(q: torch.Tensor, v: torch.Tensor) -> torch.Tensor:
            """Iterative implementation of new quat_apply."""
            out = torch.empty_like(v)
            for i in range(q.shape[1]):
                for j in range(q.shape[2]):
                    out[:, i, j] = math_utils.quat_apply(q_rand[:, i, j], v_rand[:, i, j])
            return out

        def iter_quat_apply_inverse(q: torch.Tensor, v: torch.Tensor) -> torch.Tensor:
            """Iterative implementation of new quat_apply_inverse."""
            out = torch.empty_like(v)
            for i in range(q.shape[1]):
                for j in range(q.shape[2]):
                    out[:, i, j] = math_utils.quat_apply_inverse(q_rand[:, i, j], v_rand[:, i, j])
            return out

        def iter_bmm_quat_rotate(q: torch.Tensor, v: torch.Tensor) -> torch.Tensor:
            """Iterative implementation of old quat_rotate using torch.bmm."""
            out = torch.empty_like(v)
            for i in range(q.shape[1]):
                for j in range(q.shape[2]):
                    out[:, i, j] = bmm_quat_rotate(q_rand[:, i, j], v_rand[:, i, j])
            return out

        def iter_bmm_quat_rotate_inverse(q: torch.Tensor, v: torch.Tensor) -> torch.Tensor:
            """Iterative implementation of old quat_rotate_inverse using torch.bmm."""
            out = torch.empty_like(v)
            for i in range(q.shape[1]):
                for j in range(q.shape[2]):
                    out[:, i, j] = bmm_quat_rotate_inverse(q_rand[:, i, j], v_rand[:, i, j])
            return out

        def iter_einsum_quat_rotate(q: torch.Tensor, v: torch.Tensor) -> torch.Tensor:
            """Iterative implementation of old quat_rotate using torch.einsum."""
            out = torch.empty_like(v)
            for i in range(q.shape[1]):
                for j in range(q.shape[2]):
                    out[:, i, j] = einsum_quat_rotate(q_rand[:, i, j], v_rand[:, i, j])
            return out

        def iter_einsum_quat_rotate_inverse(q: torch.Tensor, v: torch.Tensor) -> torch.Tensor:
            """Iterative implementation of old quat_rotate_inverse using torch.einsum."""
            out = torch.empty_like(v)
            for i in range(q.shape[1]):
                for j in range(q.shape[2]):
                    out[:, i, j] = einsum_quat_rotate_inverse(q_rand[:, i, j], v_rand[:, i, j])
            return out

        # benchmarks for iterative calls
        timer_iter_quat_apply = benchmark.Timer(
            stmt="iter_quat_apply(q_rand, v_rand)",
            globals={"iter_quat_apply": iter_quat_apply, "q_rand": q_rand, "v_rand": v_rand},
        )
        timer_iter_quat_apply_inverse = benchmark.Timer(
            stmt="iter_quat_apply_inverse(q_rand, v_rand)",
            globals={"iter_quat_apply_inverse": iter_quat_apply_inverse, "q_rand": q_rand, "v_rand": v_rand},
        )

        timer_iter_bmm_quat_rotate = benchmark.Timer(
            stmt="iter_bmm_quat_rotate(q_rand, v_rand)",
            globals={"iter_bmm_quat_rotate": iter_bmm_quat_rotate, "q_rand": q_rand, "v_rand": v_rand},
        )
        timer_iter_bmm_quat_rotate_inverse = benchmark.Timer(
            stmt="iter_bmm_quat_rotate_inverse(q_rand, v_rand)",
            globals={
                "iter_bmm_quat_rotate_inverse": iter_bmm_quat_rotate_inverse,
                "q_rand": q_rand,
                "v_rand": v_rand,
            },
        )

        timer_iter_einsum_quat_rotate = benchmark.Timer(
            stmt="iter_einsum_quat_rotate(q_rand, v_rand)",
            globals={"iter_einsum_quat_rotate": iter_einsum_quat_rotate, "q_rand": q_rand, "v_rand": v_rand},
        )
        timer_iter_einsum_quat_rotate_inverse = benchmark.Timer(
            stmt="iter_einsum_quat_rotate_inverse(q_rand, v_rand)",
            globals={
                "iter_einsum_quat_rotate_inverse": iter_einsum_quat_rotate_inverse,
                "q_rand": q_rand,
                "v_rand": v_rand,
            },
        )

        # create benchmaks for size independent calls
        timer_quat_apply = benchmark.Timer(
            stmt="math_utils.quat_apply(q_rand, v_rand)",
            globals={"math_utils": math_utils, "q_rand": q_rand, "v_rand": v_rand},
        )
        timer_quat_apply_inverse = benchmark.Timer(
            stmt="math_utils.quat_apply_inverse(q_rand, v_rand)",
            globals={"math_utils": math_utils, "q_rand": q_rand, "v_rand": v_rand},
        )
        timer_einsum_quat_rotate = benchmark.Timer(
            stmt="einsum_quat_rotate(q_rand, v_rand)",
            globals={"einsum_quat_rotate": einsum_quat_rotate, "q_rand": q_rand, "v_rand": v_rand},
        )
        timer_einsum_quat_rotate_inverse = benchmark.Timer(
            stmt="einsum_quat_rotate_inverse(q_rand, v_rand)",
            globals={"einsum_quat_rotate_inverse": einsum_quat_rotate_inverse, "q_rand": q_rand, "v_rand": v_rand},
        )

        # run the benchmark
        print("--------------------------------")
        print(f"Device: {device}")
        print("Time for quat_apply:", timer_quat_apply.timeit(number=1000))
        print("Time for einsum_quat_rotate:", timer_einsum_quat_rotate.timeit(number=1000))
        print("Time for iter_quat_apply:", timer_iter_quat_apply.timeit(number=1000))
        print("Time for iter_bmm_quat_rotate:", timer_iter_bmm_quat_rotate.timeit(number=1000))
        print("Time for iter_einsum_quat_rotate:", timer_iter_einsum_quat_rotate.timeit(number=1000))
        print("--------------------------------")
        print("Time for quat_apply_inverse:", timer_quat_apply_inverse.timeit(number=1000))
        print("Time for einsum_quat_rotate_inverse:", timer_einsum_quat_rotate_inverse.timeit(number=1000))
        print("Time for iter_quat_apply_inverse:", timer_iter_quat_apply_inverse.timeit(number=1000))
        print("Time for iter_bmm_quat_rotate_inverse:", timer_iter_bmm_quat_rotate_inverse.timeit(number=1000))
        print("Time for iter_einsum_quat_rotate_inverse:", timer_iter_einsum_quat_rotate_inverse.timeit(number=1000))
        print("--------------------------------")

        # check output values are the same
        torch.testing.assert_close(math_utils.quat_apply(q_rand, v_rand), iter_quat_apply(q_rand, v_rand))
        torch.testing.assert_close(
            math_utils.quat_apply(q_rand, v_rand), iter_bmm_quat_rotate(q_rand, v_rand), atol=1e-3, rtol=1e-3
        )
        torch.testing.assert_close(
            math_utils.quat_apply_inverse(q_rand, v_rand), iter_quat_apply_inverse(q_rand, v_rand)
        )
        torch.testing.assert_close(
            math_utils.quat_apply_inverse(q_rand, v_rand),
            iter_bmm_quat_rotate_inverse(q_rand, v_rand),
            atol=1e-3,
            rtol=1e-3,
        )


def test_interpolate_rotations():
    """Test interpolate_rotations function.

    This test checks the output from the :meth:`~isaaclab.utils.math_utils.interpolate_rotations` function against
    the output from :func:`scipy.spatial.transform.Slerp`.
    """
    # Generate NUM_ITERS random rotation matrices
    random_rotation_matrices_1 = [math_utils.generate_random_rotation() for _ in range(100)]
    random_rotation_matrices_2 = [math_utils.generate_random_rotation() for _ in range(100)]

    for rmat1, rmat2 in zip(random_rotation_matrices_1, random_rotation_matrices_2):
        # Compute expected results using scipy's Slerp
        key_rots = scipy_tf.Rotation.from_matrix(np.array([rmat1, rmat2]))

        # Create a Slerp object and interpolate create the interpolated matrices
        # Minimum 2 required because Interpolate_rotations returns one extra rotation matrix
        num_steps = np.random.randint(2, 51)
        key_times = [0, 1]
        slerp = scipy_tf.Slerp(key_times, key_rots)
        interp_times = np.linspace(0, 1, num_steps)
        expected = slerp(interp_times).as_matrix()

        # Test 1:
        # Interpolate_rotations using interpolate_rotations and quat_slerp
        # interpolate_rotations returns one extra rotation matrix hence num_steps-1
        result_quat = math_utils.interpolate_rotations(rmat1, rmat2, num_steps - 1)

        # Assert that the result is almost equal to the expected quaternion
        np.testing.assert_array_almost_equal(result_quat.cpu(), expected, decimal=DECIMAL_PRECISION)

        # Test 2:
        # Interpolate_rotations using axis_angle and ensure the result is still the same
        # interpolate_rotations returns one extra rotation matrix hence num_steps-1
        result_axis_angle = math_utils.interpolate_rotations(rmat1, rmat2, num_steps - 1, axis_angle=True)

        # Assert that the result is almost equal to the expected quaternion
        np.testing.assert_array_almost_equal(result_axis_angle.cpu(), expected, decimal=DECIMAL_PRECISION)


def test_euler_xyz_from_quat():
    """Test euler_xyz_from_quat function.

    This test checks the output from the :meth:`~isaaclab.utils.math_utils.euler_xyz_from_quat` function
    against the expected output for various quaternions.
    The test includes quaternions representing different rotations around the x, y, and z axes.
    The test is performed for both the default output range (-π, π] and the wrapped output range [0, 2π).
    Quaternions are in xyzw format.
    """
    quats = [
        torch.Tensor([[0.0, 0.0, 0.0, 1.0]]),  # 0° around x, y, z (identity)
        torch.Tensor(
            [
                [0.3826834, 0.0, 0.0, 0.9238795],  # 45° around x
                [0.0, -0.3826834, 0.0, 0.9238795],  # -45° around y
                [0.0, 0.0, -0.3826834, 0.9238795],  # -45° around z
            ]
        ),
        torch.Tensor(
            [
                [-0.7071068, 0.0, 0.0, 0.7071068],  # -90° around x
                [0.0, 0.0, -0.7071068, 0.7071068],  # -90° around z
            ]
        ),
        torch.Tensor(
            [
                [-0.9238795, 0.0, 0.0, 0.3826834],  # -135° around x
                [0.0, 0.0, -0.9238795, 0.3826834],  # -135° around z
            ]
        ),
    ]

    expected_euler_angles = [
        torch.Tensor([[0.0, 0.0, 0.0]]),  # identity
        torch.Tensor(
            [
                [torch.pi / 4, 0.0, 0.0],  # 45° about x
                [0.0, -torch.pi / 4, 0.0],  # -45° about y
                [0.0, 0.0, -torch.pi / 4],  # -45° about z
            ]
        ),
        torch.Tensor(
            [
                [-torch.pi / 2, 0.0, 0.0],  # -90° about x
                [0.0, 0.0, -torch.pi / 2],  # -90° about z
            ]
        ),
        torch.Tensor(
            [
                [-3 * torch.pi / 4, 0.0, 0.0],  # -135° about x
                [0.0, 0.0, -3 * torch.pi / 4],  # -135° about y
            ]
        ),
    ]

    # Test 1: default no-wrap range from (-π, π]
    for quat, expected in zip(quats, expected_euler_angles):
        output = torch.stack(math_utils.euler_xyz_from_quat(quat), dim=-1)
        torch.testing.assert_close(output, expected)

    # Test 2: wrap to [0, 2π)
    for quat, expected in zip(quats, expected_euler_angles):
        wrapped = expected % (2 * torch.pi)
        output = torch.stack(math_utils.euler_xyz_from_quat(quat, wrap_to_2pi=True), dim=-1)
        torch.testing.assert_close(output, wrapped)


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
def test_create_rotation_matrix_from_view_lookat_along_up_axis_z(device):
    """Camera above target on +Z axis with Z-up should return a valid orthonormal frame."""
    eyes = torch.tensor([[0.0, 0.0, 5.0]], device=device)
    targets = torch.tensor([[0.0, 0.0, 0.0]], device=device)
    R = math_utils.create_rotation_matrix_from_view(eyes, targets, up_axis="Z", device=device)
    assert R is not None
    identity = torch.eye(3, device=device).expand(1, 3, 3)
    torch.testing.assert_close(R @ R.transpose(-1, -2), identity, atol=1e-5, rtol=1e-5)
    torch.testing.assert_close(torch.linalg.det(R), torch.ones(1, device=device), atol=1e-5, rtol=1e-5)


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
def test_create_rotation_matrix_from_view_lookat_along_up_axis_y(device):
    """Camera at +Y looking at origin with Y-up should return a valid orthonormal frame."""
    eyes = torch.tensor([[0.0, 5.0, 0.0]], device=device)
    targets = torch.tensor([[0.0, 0.0, 0.0]], device=device)
    R = math_utils.create_rotation_matrix_from_view(eyes, targets, up_axis="Y", device=device)
    assert R is not None
    identity = torch.eye(3, device=device).expand(1, 3, 3)
    torch.testing.assert_close(R @ R.transpose(-1, -2), identity, atol=1e-5, rtol=1e-5)
    torch.testing.assert_close(torch.linalg.det(R), torch.ones(1, device=device), atol=1e-5, rtol=1e-5)


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
def test_create_rotation_matrix_from_view_lookat_along_negative_up_axis(device):
    """Camera below target looking up (-Z alignment with Z-up) should return a valid orthonormal frame."""
    eyes = torch.tensor([[0.0, 0.0, -5.0]], device=device)
    targets = torch.tensor([[0.0, 0.0, 0.0]], device=device)
    R = math_utils.create_rotation_matrix_from_view(eyes, targets, up_axis="Z", device=device)
    assert R is not None
    identity = torch.eye(3, device=device).expand(1, 3, 3)
    torch.testing.assert_close(R @ R.transpose(-1, -2), identity, atol=1e-5, rtol=1e-5)
    torch.testing.assert_close(torch.linalg.det(R), torch.ones(1, device=device), atol=1e-5, rtol=1e-5)


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
def test_create_rotation_matrix_from_view_zero_forward_returns_nan(device):
    """When eyes == targets the forward direction is undefined; all entries of the row are NaN."""
    eyes = torch.tensor([[1.0, 2.0, 3.0]], device=device)
    targets = eyes.clone()
    R = math_utils.create_rotation_matrix_from_view(eyes, targets, up_axis="Z", device=device)
    assert torch.isnan(R).all()


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
def test_create_rotation_matrix_from_view_batched_partial_failure(device):
    """Mixed batch with one degenerate row should produce NaN in that row and a valid rotation in the other."""
    eyes = torch.tensor([[1.0, 2.0, 3.0], [0.0, 0.0, 5.0]], device=device)
    targets = torch.tensor([[1.0, 2.0, 3.0], [0.0, 0.0, 0.0]], device=device)
    R = math_utils.create_rotation_matrix_from_view(eyes, targets, up_axis="Z", device=device)
    assert torch.isnan(R[0]).any()
    torch.testing.assert_close(R[1] @ R[1].T, torch.eye(3, device=device), atol=1e-5, rtol=1e-5)
    torch.testing.assert_close(torch.linalg.det(R[1]), torch.tensor(1.0, device=device), atol=1e-5, rtol=1e-5)


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
def test_quat_from_matrix_unit_norm_on_valid_input(device):
    """quat_from_matrix should produce unit quaternions for any valid rotation matrix."""
    n = 100
    q_rand = math_utils.random_orientation(num=n, device=device)
    rot_mat = math_utils.matrix_from_quat(q_rand)
    q_value = math_utils.quat_from_matrix(rot_mat)
    norms = torch.linalg.norm(q_value, dim=-1)
    torch.testing.assert_close(norms, torch.ones(n, device=device), atol=1e-5, rtol=1e-5)


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
def test_quat_from_matrix_singular_matrix_returns_nan(device):
    """quat_from_matrix on a singular (non-rotation) matrix should signal NaN, not garbage."""
    singular = torch.tensor([[[0.0, 0.0, 0.0], [0.0, 0.0, 0.0], [0.0, 0.0, 1.0]]], device=device)
    q = math_utils.quat_from_matrix(singular)
    assert torch.isnan(q).all()


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
def test_create_rotation_matrix_from_view_standard(device):
    """Sanity: off-axis eye produces an orthonormal frame whose z-axis points from target back to eye."""
    eyes = torch.tensor([[3.0, 0.0, 4.0]], device=device)
    targets = torch.tensor([[0.0, 0.0, 0.0]], device=device)
    R = math_utils.create_rotation_matrix_from_view(eyes, targets, up_axis="Z", device=device)
    identity = torch.eye(3, device=device).expand(1, 3, 3)
    torch.testing.assert_close(R @ R.transpose(-1, -2), identity, atol=1e-5, rtol=1e-5)
    torch.testing.assert_close(torch.linalg.det(R), torch.ones(1, device=device), atol=1e-5, rtol=1e-5)
    # z_axis is back-of-camera in OpenGL convention: points from target to eye
    expected_z = torch.tensor([[0.6, 0.0, 0.8]], device=device)
    torch.testing.assert_close(R[:, :, 2], expected_z, atol=1e-5, rtol=1e-5)


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
def test_create_rotation_matrix_from_view_non_finite_returns_nan(device):
    """Non-finite input (NaN or Inf in eyes/targets) should produce NaN rows."""
    eyes = torch.tensor([[float("nan"), 0.0, 0.0]], device=device)
    targets = torch.tensor([[0.0, 0.0, 0.0]], device=device)
    R = math_utils.create_rotation_matrix_from_view(eyes, targets, up_axis="Z", device=device)
    assert torch.isnan(R).all()


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
def test_quat_from_matrix_reflection_returns_nan(device):
    """A reflection matrix (det = -1) is not a proper rotation; the safeguard should signal NaN."""
    reflection = torch.diag(torch.tensor([1.0, 1.0, -1.0], device=device)).unsqueeze(0)
    q = math_utils.quat_from_matrix(reflection)
    assert torch.isnan(q).all()


@pytest.mark.parametrize("device", ("cpu", "cuda:0"))
def test_quat_from_matrix_non_orthonormal_returns_nan(device):
    """A non-orthonormal matrix (1% scale error on one axis) is not a valid rotation; expect NaN."""
    R = torch.diag(torch.tensor([1.01, 1.0, 1.0], device=device)).unsqueeze(0)
    q = math_utils.quat_from_matrix(R)
    assert torch.isnan(q).all()