# transform_utils

Utility functions of matrix and vector transformations.

NOTE: convention for quaternions is (x, y, z, w)

## `anorm(x, axis=None, keepdims=False)`

Compute L2 norms alogn specified axes.

Source code in `omnigibson/utils/transform_utils.py`
 ```1100 1101 1102``` ``````def anorm(x, axis=None, keepdims=False): """Compute L2 norms alogn specified axes.""" return np.linalg.norm(x, axis=axis, keepdims=keepdims) ``````

## `axisangle2quat(vec)`

Converts scaled axis-angle to quat.

Parameters:

Name Type Description Default
`vec` `array`

(ax,ay,az) axis-angle exponential coordinates

required

Returns:

Type Description

np.array: (x,y,z,w) vec4 float angles

Source code in `omnigibson/utils/transform_utils.py`
 ```492 493 494 495 496 497 498 499 500 501 502``` ``````def axisangle2quat(vec): """ Converts scaled axis-angle to quat. Args: vec (np.array): (ax,ay,az) axis-angle exponential coordinates Returns: np.array: (x,y,z,w) vec4 float angles """ return R.from_rotvec(vec).as_quat() ``````

## `cartesian_to_polar(x, y)`

Convert cartesian coordinate to polar coordinate

Source code in `omnigibson/utils/transform_utils.py`
 ```1111 1112 1113 1114 1115``` ``````def cartesian_to_polar(x, y): """Convert cartesian coordinate to polar coordinate""" rho = np.sqrt(x ** 2 + y ** 2) phi = np.arctan2(y, x) return rho, phi ``````

## `check_quat_right_angle(quat, atol=0.05)`

Check by making sure the quaternion is some permutation of +/- (1, 0, 0, 0), +/- (0.707, 0.707, 0, 0), or +/- (0.5, 0.5, 0.5, 0.5) Because orientations are all normalized (same L2-norm), every orientation should have a unique L1-norm So we check the L1-norm of the absolute value of the orientation as a proxy for verifying these values

Parameters:

Name Type Description Default
`quat` `4 - array`

(x,y,z,w) quaternion orientation to check

required
`atol` `float`

Absolute tolerance permitted

`0.05`

Returns:

Name Type Description
`bool`

Whether the quaternion is a right angle or not

Source code in `omnigibson/utils/transform_utils.py`
 ```1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140``` ``````def check_quat_right_angle(quat, atol=5e-2): """ Check by making sure the quaternion is some permutation of +/- (1, 0, 0, 0), +/- (0.707, 0.707, 0, 0), or +/- (0.5, 0.5, 0.5, 0.5) Because orientations are all normalized (same L2-norm), every orientation should have a unique L1-norm So we check the L1-norm of the absolute value of the orientation as a proxy for verifying these values Args: quat (4-array): (x,y,z,w) quaternion orientation to check atol (float): Absolute tolerance permitted Returns: bool: Whether the quaternion is a right angle or not """ return np.any(np.isclose(np.abs(quat).sum(), np.array([1.0, 1.414, 2.0]), atol=atol)) ``````

## `clip_rotation(quat, limit)`

Limits a (delta) rotation to a specified limit

Converts rotation to axis-angle, clips, then re-converts back into quaternion

Parameters:

Name Type Description Default
`quat` `array`

(x,y,z,w) rotation being clipped

required
`limit` `float`

Value to limit rotation by -- magnitude (scalar, in radians)

required

Returns:

Type Description

2-tuple:

• (np.array) Clipped rotation quaternion (x, y, z, w)
• (bool) whether the value was clipped or not
Source code in `omnigibson/utils/transform_utils.py`
 ```805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845``` ``````def clip_rotation(quat, limit): """ Limits a (delta) rotation to a specified limit Converts rotation to axis-angle, clips, then re-converts back into quaternion Args: quat (np.array): (x,y,z,w) rotation being clipped limit (float): Value to limit rotation by -- magnitude (scalar, in radians) Returns: 2-tuple: - (np.array) Clipped rotation quaternion (x, y, z, w) - (bool) whether the value was clipped or not """ clipped = False # First, normalize the quaternion quat = quat / np.linalg.norm(quat) den = np.sqrt(max(1 - quat[3] * quat[3], 0)) if den == 0: # This is a zero degree rotation, immediately return return quat, clipped else: # This is all other cases x = quat[0] / den y = quat[1] / den z = quat[2] / den a = 2 * math.acos(quat[3]) # Clip rotation if necessary and return clipped quat if abs(a) > limit: a = limit * np.sign(a) / 2 sa = math.sin(a) ca = math.cos(a) quat = np.array([x * sa, y * sa, z * sa, ca]) clipped = True return quat, clipped ``````

## `clip_translation(dpos, limit)`

Limits a translation (delta position) to a specified limit

Scales down the norm of the dpos to 'limit' if norm(dpos) > limit, else returns immediately

Parameters:

Name Type Description Default
`dpos` `n - array`

n-dim Translation being clipped (e,g.: (x, y, z)) -- numpy array

required
`limit` `float`

Value to limit translation by -- magnitude (scalar, in same units as input)

required

Returns:

Type Description

2-tuple:

• (np.array) Clipped translation (same dimension as inputs)
• (bool) whether the value was clipped or not
Source code in `omnigibson/utils/transform_utils.py`
 ```785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802``` ``````def clip_translation(dpos, limit): """ Limits a translation (delta position) to a specified limit Scales down the norm of the dpos to 'limit' if norm(dpos) > limit, else returns immediately Args: dpos (n-array): n-dim Translation being clipped (e,g.: (x, y, z)) -- numpy array limit (float): Value to limit translation by -- magnitude (scalar, in same units as input) Returns: 2-tuple: - (np.array) Clipped translation (same dimension as inputs) - (bool) whether the value was clipped or not """ input_norm = np.linalg.norm(dpos) return (dpos * limit / input_norm, True) if input_norm > limit else (dpos, False) ``````

## `convert_quat(q, to='xyzw')`

Converts quaternion from one convention to another. The convention to convert TO is specified as an optional argument. If to == 'xyzw', then the input is in 'wxyz' format, and vice-versa.

Parameters:

Name Type Description Default
`q` `array`

a 4-dim array corresponding to a quaternion

required
`to` `str`

either 'xyzw' or 'wxyz', determining which convention to convert to.

`'xyzw'`
Source code in `omnigibson/utils/transform_utils.py`
 ```117 118 119 120 121 122 123 124 125 126 127 128 129 130 131``` ``````def convert_quat(q, to="xyzw"): """ Converts quaternion from one convention to another. The convention to convert TO is specified as an optional argument. If to == 'xyzw', then the input is in 'wxyz' format, and vice-versa. Args: q (np.array): a 4-dim array corresponding to a quaternion to (str): either 'xyzw' or 'wxyz', determining which convention to convert to. """ if to == "xyzw": return q[[1, 2, 3, 0]] if to == "wxyz": return q[[3, 0, 1, 2]] raise Exception("convert_quat: choose a valid `to` argument (xyzw or wxyz)") ``````

## `euler2mat(euler)`

Converts euler angles into rotation matrix form

Parameters:

Name Type Description Default
`euler` `array`

(r,p,y) angles

required

Returns:

Type Description

np.array: 3x3 rotation matrix

Raises:

Type Description
`AssertionError`

[Invalid input shape]

Source code in `omnigibson/utils/transform_utils.py`
 ```413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430``` ``````def euler2mat(euler): """ Converts euler angles into rotation matrix form Args: euler (np.array): (r,p,y) angles Returns: np.array: 3x3 rotation matrix Raises: AssertionError: [Invalid input shape] """ euler = np.asarray(euler, dtype=np.float64) assert euler.shape[-1] == 3, "Invalid shaped euler {}".format(euler) return R.from_euler("xyz", euler).as_matrix() ``````

## `euler2quat(euler)`

Converts euler angles into quaternion form

Parameters:

Name Type Description Default
`euler` `array`

(r,p,y) angles

required

Returns:

Type Description

np.array: (x,y,z,w) float quaternion angles

Raises:

Type Description
`AssertionError`

[Invalid input shape]

Source code in `omnigibson/utils/transform_utils.py`
 ```505 506 507 508 509 510 511 512 513 514 515 516 517 518``` ``````def euler2quat(euler): """ Converts euler angles into quaternion form Args: euler (np.array): (r,p,y) angles Returns: np.array: (x,y,z,w) float quaternion angles Raises: AssertionError: [Invalid input shape] """ return R.from_euler("xyz", euler).as_quat() ``````

## `ewma_vectorized(data, alpha, offset=None, dtype=None, order='C', out=None)`

Calculates the exponential moving average over a vector. Will fail for large inputs.

Parameters:

Name Type Description Default
`data` `Iterable`

Input data

required
`alpha` `float`

scalar in range (0,1) The alpha parameter for the moving average.

required
`offset` `None or float`

If specified, the offset for the moving average. None defaults to data[0].

`None`
`dtype` `None or type`

Data type used for calculations. If None, defaults to float64 unless data.dtype is float32, then it will use float32.

`None`
`order` `None or str`

Order to use when flattening the data. Valid options are {'C', 'F', 'A'}. None defaults to 'C'.

`'C'`
`out` `None or array`

If specified, the location into which the result is stored. If provided, it must have the same shape as the input. If not provided or `None`, a freshly-allocated array is returned.

`None`

Returns:

Type Description

np.array: Exponential moving average from @data

Source code in `omnigibson/utils/transform_utils.py`
 ``` 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114``` ``````def ewma_vectorized(data, alpha, offset=None, dtype=None, order="C", out=None): """ Calculates the exponential moving average over a vector. Will fail for large inputs. Args: data (Iterable): Input data alpha (float): scalar in range (0,1) The alpha parameter for the moving average. offset (None or float): If specified, the offset for the moving average. None defaults to data[0]. dtype (None or type): Data type used for calculations. If None, defaults to float64 unless data.dtype is float32, then it will use float32. order (None or str): Order to use when flattening the data. Valid options are {'C', 'F', 'A'}. None defaults to 'C'. out (None or np.array): If specified, the location into which the result is stored. If provided, it must have the same shape as the input. If not provided or `None`, a freshly-allocated array is returned. Returns: np.array: Exponential moving average from @data """ data = np.array(data, copy=False) if dtype is None: if data.dtype == np.float32: dtype = np.float32 else: dtype = np.float64 else: dtype = np.dtype(dtype) if data.ndim > 1: # flatten input data = data.reshape(-1, order) if out is None: out = np.empty_like(data, dtype=dtype) else: assert out.shape == data.shape assert out.dtype == dtype if data.size < 1: # empty input, return empty array return out if offset is None: offset = data[0] alpha = np.array(alpha, copy=False).astype(dtype, copy=False) # scaling_factors -> 0 as len(data) gets large # this leads to divide-by-zeros below scaling_factors = np.power(1.0 - alpha, np.arange(data.size + 1, dtype=dtype), dtype=dtype) # create cumulative sum array np.multiply(data, (alpha * scaling_factors[-2]) / scaling_factors[:-1], dtype=dtype, out=out) np.cumsum(out, dtype=dtype, out=out) # cumsums / scaling out /= scaling_factors[-2::-1] if offset != 0: offset = np.array(offset, copy=False).astype(dtype, copy=False) # add offsets out += offset * scaling_factors[1:] return out ``````

## `force_in_A_to_force_in_B(force_A, torque_A, pose_A_in_B)`

Converts linear and rotational force at a point in frame A to the equivalent in frame B.

Parameters:

Name Type Description Default
`force_A` `array`

(fx,fy,fz) linear force in A

required
`torque_A` `array`

(tx,ty,tz) rotational force (moment) in A

required
`pose_A_in_B` `array`

4x4 matrix corresponding to the pose of A in frame B

required

Returns:

Type Description

2-tuple:

• (np.array) (fx,fy,fz) linear forces in frame B
• (np.array) (tx,ty,tz) moments in frame B
Source code in `omnigibson/utils/transform_utils.py`
 ```704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724``` ``````def force_in_A_to_force_in_B(force_A, torque_A, pose_A_in_B): """ Converts linear and rotational force at a point in frame A to the equivalent in frame B. Args: force_A (np.array): (fx,fy,fz) linear force in A torque_A (np.array): (tx,ty,tz) rotational force (moment) in A pose_A_in_B (np.array): 4x4 matrix corresponding to the pose of A in frame B Returns: 2-tuple: - (np.array) (fx,fy,fz) linear forces in frame B - (np.array) (tx,ty,tz) moments in frame B """ pos_A_in_B = pose_A_in_B[:3, 3] rot_A_in_B = pose_A_in_B[:3, :3] skew_symm = _skew_symmetric_translation(pos_A_in_B) force_B = rot_A_in_B.T.dot(force_A) torque_B = -rot_A_in_B.T.dot(skew_symm.dot(force_A)) + rot_A_in_B.T.dot(torque_A) return force_B, torque_B ``````

## `frustum(left, right, bottom, top, znear, zfar)`

Create view frustum matrix.

Source code in `omnigibson/utils/transform_utils.py`
 ```1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071``` ``````def frustum(left, right, bottom, top, znear, zfar): """Create view frustum matrix.""" assert right != left assert bottom != top assert znear != zfar M = np.zeros((4, 4), dtype=np.float32) M[0, 0] = +2.0 * znear / (right - left) M[2, 0] = (right + left) / (right - left) M[1, 1] = +2.0 * znear / (top - bottom) # TODO: Put this back to 3,1 # M[3, 1] = (top + bottom) / (top - bottom) M[2, 1] = (top + bottom) / (top - bottom) M[2, 2] = -(zfar + znear) / (zfar - znear) M[3, 2] = -2.0 * znear * zfar / (zfar - znear) M[2, 3] = -1.0 return M ``````

## `get_orientation_diff_in_radian(orn0, orn1)`

Returns the difference between two quaternion orientations in radian

Parameters:

Name Type Description Default
`orn0` `array`

(x, y, z, w)

required
`orn1` `array`

(x, y, z, w)

required

Returns:

Name Type Description
`orn_diff` `float`

Source code in `omnigibson/utils/transform_utils.py`
 ```948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963``` ``````def get_orientation_diff_in_radian(orn0, orn1): """ Returns the difference between two quaternion orientations in radian Args: orn0 (np.array): (x, y, z, w) orn1 (np.array): (x, y, z, w) Returns: orn_diff (float): orientation difference in radian """ vec0 = quat2axisangle(orn0) vec0 /= np.linalg.norm(vec0) vec1 = quat2axisangle(orn1) vec1 /= np.linalg.norm(vec1) return np.arccos(np.dot(vec0, vec1)) ``````

## `get_orientation_error(target_orn, current_orn)`

Returns the difference between two quaternion orientations as a 3 DOF numpy array. For use in an impedance controller / task-space PD controller.

Parameters:

Name Type Description Default
`target_orn` `array`

(x, y, z, w) desired quaternion orientation

required
`current_orn` `array`

(x, y, z, w) current quaternion orientation

required

Returns:

Name Type Description
`orn_error` `array`

(ax,ay,az) current orientation error, corresponds to (target_orn - current_orn)

Source code in `omnigibson/utils/transform_utils.py`
 ```924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945``` ``````def get_orientation_error(target_orn, current_orn): """ Returns the difference between two quaternion orientations as a 3 DOF numpy array. For use in an impedance controller / task-space PD controller. Args: target_orn (np.array): (x, y, z, w) desired quaternion orientation current_orn (np.array): (x, y, z, w) current quaternion orientation Returns: orn_error (np.array): (ax,ay,az) current orientation error, corresponds to (target_orn - current_orn) """ current_orn = np.array([current_orn[3], current_orn[0], current_orn[1], current_orn[2]]) target_orn = np.array([target_orn[3], target_orn[0], target_orn[1], target_orn[2]]) pinv = np.zeros((3, 4)) pinv[0, :] = [-current_orn[1], current_orn[0], -current_orn[3], current_orn[2]] pinv[1, :] = [-current_orn[2], current_orn[3], current_orn[0], -current_orn[1]] pinv[2, :] = [-current_orn[3], -current_orn[2], current_orn[1], current_orn[0]] orn_error = 2.0 * pinv.dot(np.array(target_orn)) return orn_error ``````

## `get_pose_error(target_pose, current_pose)`

Computes the error corresponding to target pose - current pose as a 6-dim vector. The first 3 components correspond to translational error while the last 3 components correspond to the rotational error.

Parameters:

Name Type Description Default
`target_pose` `array`

a 4x4 homogenous matrix for the target pose

required
`current_pose` `array`

a 4x4 homogenous matrix for the current pose

required

Returns:

Type Description

np.array: 6-dim pose error.

Source code in `omnigibson/utils/transform_utils.py`
 ```966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997``` ``````def get_pose_error(target_pose, current_pose): """ Computes the error corresponding to target pose - current pose as a 6-dim vector. The first 3 components correspond to translational error while the last 3 components correspond to the rotational error. Args: target_pose (np.array): a 4x4 homogenous matrix for the target pose current_pose (np.array): a 4x4 homogenous matrix for the current pose Returns: np.array: 6-dim pose error. """ error = np.zeros(6) # compute translational error target_pos = target_pose[:3, 3] current_pos = current_pose[:3, 3] pos_err = target_pos - current_pos # compute rotational error r1 = current_pose[:3, 0] r2 = current_pose[:3, 1] r3 = current_pose[:3, 2] r1d = target_pose[:3, 0] r2d = target_pose[:3, 1] r3d = target_pose[:3, 2] rot_err = 0.5 * (np.cross(r1, r1d) + np.cross(r2, r2d) + np.cross(r3, r3d)) error[:3] = pos_err error[3:] = rot_err return error ``````

## `invert_pose_transform(pos, quat)`

Inverts a pose transform

Parameters:

Name Type Description Default
`pos`

(x,y,z) position to transform

required
`quat`

(x,y,z,w) orientation to transform

required

Returns:

Type Description

2-tuple: - (np.array) (x,y,z) position array in cartesian coordinates - (np.array) (x,y,z,w) orientation array in quaternion form

Source code in `omnigibson/utils/transform_utils.py`
 ```610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627``` ``````def invert_pose_transform(pos, quat): """ Inverts a pose transform Args: pos: (x,y,z) position to transform quat: (x,y,z,w) orientation to transform Returns: 2-tuple: - (np.array) (x,y,z) position array in cartesian coordinates - (np.array) (x,y,z,w) orientation array in quaternion form """ # Get pose mat = pose2mat((pos, quat)) # Invert pose and convert back to pos, quat return mat2pose(pose_inv(mat)) ``````

## `l2_distance(v1, v2)`

Returns the L2 distance between vector v1 and v2.

Source code in `omnigibson/utils/transform_utils.py`
 ```1050 1051 1052``` ``````def l2_distance(v1, v2): """Returns the L2 distance between vector v1 and v2.""" return np.linalg.norm(np.array(v1) - np.array(v2)) ``````

## `make_pose(translation, rotation)`

Makes a homogeneous pose matrix from a translation vector and a rotation matrix.

Parameters:

Name Type Description Default
`translation` `array`

(x,y,z) translation value

required
`rotation` `array`

a 3x3 matrix representing rotation

required

Returns:

Name Type Description
`pose` `array`

a 4x4 homogeneous matrix

Source code in `omnigibson/utils/transform_utils.py`
 ```848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863``` ``````def make_pose(translation, rotation): """ Makes a homogeneous pose matrix from a translation vector and a rotation matrix. Args: translation (np.array): (x,y,z) translation value rotation (np.array): a 3x3 matrix representing rotation Returns: pose (np.array): a 4x4 homogeneous matrix """ pose = np.zeros((4, 4)) pose[:3, :3] = rotation pose[:3, 3] = translation pose[3, 3] = 1.0 return pose ``````

## `mat2euler(rmat)`

Converts given rotation matrix to euler angles in radian.

Parameters:

Name Type Description Default
`rmat` `array`

3x3 rotation matrix

required

Returns:

Type Description

np.array: (r,p,y) converted euler angles in radian vec3 float

Source code in `omnigibson/utils/transform_utils.py`
 ```433 434 435 436 437 438 439 440 441 442 443 444``` ``````def mat2euler(rmat): """ Converts given rotation matrix to euler angles in radian. Args: rmat (np.array): 3x3 rotation matrix Returns: np.array: (r,p,y) converted euler angles in radian vec3 float """ M = np.array(rmat, dtype=np.float32, copy=False)[:3, :3] return R.from_matrix(M).as_euler("xyz") ``````

## `mat2pose(hmat)`

Converts a homogeneous 4x4 matrix into pose.

Parameters:

Name Type Description Default
`hmat` `array`

a 4x4 homogeneous matrix

required

Returns:

Type Description

2-tuple: - (np.array) (x,y,z) position array in cartesian coordinates - (np.array) (x,y,z,w) orientation array in quaternion form

Source code in `omnigibson/utils/transform_utils.py`
 ```365 366 367 368 369 370 371 372 373 374 375 376 377 378 379``` ``````def mat2pose(hmat): """ Converts a homogeneous 4x4 matrix into pose. Args: hmat (np.array): a 4x4 homogeneous matrix Returns: 2-tuple: - (np.array) (x,y,z) position array in cartesian coordinates - (np.array) (x,y,z,w) orientation array in quaternion form """ pos = hmat[:3, 3] orn = mat2quat(hmat[:3, :3]) return pos, orn ``````

## `mat2quat(rmat)`

Converts given rotation matrix to quaternion.

Parameters:

Name Type Description Default
`rmat` `array`

(..., 3, 3) rotation matrix

required

Returns:

Type Description

np.array: (..., 4) (x,y,z,w) float quaternion angles

Source code in `omnigibson/utils/transform_utils.py`
 ```382 383 384 385 386 387 388 389 390 391 392``` ``````def mat2quat(rmat): """ Converts given rotation matrix to quaternion. Args: rmat (np.array): (..., 3, 3) rotation matrix Returns: np.array: (..., 4) (x,y,z,w) float quaternion angles """ return R.from_matrix(rmat).as_quat() ``````

## `mat4(array)`

Converts an array to 4x4 matrix.

Parameters:

Name Type Description Default
`array` `n - array`

the array in form of vec, list, or tuple

required

Returns:

Type Description

np.array: a 4x4 numpy matrix

Source code in `omnigibson/utils/transform_utils.py`
 ```352 353 354 355 356 357 358 359 360 361 362``` ``````def mat4(array): """ Converts an array to 4x4 matrix. Args: array (n-array): the array in form of vec, list, or tuple Returns: np.array: a 4x4 numpy matrix """ return np.array(array, dtype=np.float32).reshape((4, 4)) ``````

## `matrix_inverse(matrix)`

Helper function to have an efficient matrix inversion function.

Parameters:

Name Type Description Default
`matrix` `array`

2d-array representing a matrix

required

Returns:

Type Description

np.array: 2d-array representing the matrix inverse

Source code in `omnigibson/utils/transform_utils.py`
 ```1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010``` ``````def matrix_inverse(matrix): """ Helper function to have an efficient matrix inversion function. Args: matrix (np.array): 2d-array representing a matrix Returns: np.array: 2d-array representing the matrix inverse """ return np.linalg.inv(matrix) ``````

## `normalize(v, axis=None, eps=1e-10)`

L2 Normalize along specified axes.

Source code in `omnigibson/utils/transform_utils.py`
 ```1105 1106 1107 1108``` ``````def normalize(v, axis=None, eps=1e-10): """L2 Normalize along specified axes.""" norm = anorm(v, axis=axis, keepdims=True) return v / np.where(norm < eps, eps, norm) ``````

## `ortho(left, right, bottom, top, znear, zfar)`

Create orthonormal projection matrix.

Source code in `omnigibson/utils/transform_utils.py`
 ```1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088``` ``````def ortho(left, right, bottom, top, znear, zfar): """Create orthonormal projection matrix.""" assert right != left assert bottom != top assert znear != zfar M = np.zeros((4, 4), dtype=np.float32) M[0, 0] = 2.0 / (right - left) M[1, 1] = 2.0 / (top - bottom) M[2, 2] = -2.0 / (zfar - znear) M[3, 0] = -(right + left) / (right - left) M[3, 1] = -(top + bottom) / (top - bottom) M[3, 2] = -(zfar + znear) / (zfar - znear) M[3, 3] = 1.0 return M ``````

## `perspective(fovy, aspect, znear, zfar)`

Create perspective projection matrix.

Source code in `omnigibson/utils/transform_utils.py`
 ```1091 1092 1093 1094 1095 1096 1097``` ``````def perspective(fovy, aspect, znear, zfar): """Create perspective projection matrix.""" # fovy is in degree assert znear != zfar h = np.tan(fovy / 360.0 * np.pi) * znear w = h * aspect return frustum(-w, w, -h, h, znear, zfar) ``````

## `pose2mat(pose)`

Converts pose to homogeneous matrix.

Parameters:

Name Type Description Default
`pose` `2 - tuple`

a (pos, orn) tuple where pos is vec3 float cartesian, and orn is vec4 float quaternion.

required

Returns:

Type Description

np.array: 4x4 homogeneous matrix

Source code in `omnigibson/utils/transform_utils.py`
 ```447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462``` ``````def pose2mat(pose): """ Converts pose to homogeneous matrix. Args: pose (2-tuple): a (pos, orn) tuple where pos is vec3 float cartesian, and orn is vec4 float quaternion. Returns: np.array: 4x4 homogeneous matrix """ homo_pose_mat = np.zeros((4, 4), dtype=np.float32) homo_pose_mat[:3, :3] = quat2mat(pose[1]) homo_pose_mat[:3, 3] = np.array(pose[0], dtype=np.float32) homo_pose_mat[3, 3] = 1.0 return homo_pose_mat ``````

## `pose_in_A_to_pose_in_B(pose_A, pose_A_in_B)`

Converts a homogenous matrix corresponding to a point C in frame A to a homogenous matrix corresponding to the same point C in frame B.

Parameters:

Name Type Description Default
`pose_A` `array`

4x4 matrix corresponding to the pose of C in frame A

required
`pose_A_in_B` `array`

4x4 matrix corresponding to the pose of A in frame B

required

Returns:

Type Description

np.array: 4x4 matrix corresponding to the pose of C in frame B

Source code in `omnigibson/utils/transform_utils.py`
 ```536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554``` ``````def pose_in_A_to_pose_in_B(pose_A, pose_A_in_B): """ Converts a homogenous matrix corresponding to a point C in frame A to a homogenous matrix corresponding to the same point C in frame B. Args: pose_A (np.array): 4x4 matrix corresponding to the pose of C in frame A pose_A_in_B (np.array): 4x4 matrix corresponding to the pose of A in frame B Returns: np.array: 4x4 matrix corresponding to the pose of C in frame B """ # pose of A in B takes a point in A and transforms it to a point in C. # pose of C in B = pose of A in B * pose of C in A # take a point in C, transform it to A, then to B # T_B^C = T_A^C * T_B^A return pose_A_in_B.dot(pose_A) ``````

## `pose_inv(pose_mat)`

Computes the inverse of a homogeneous matrix corresponding to the pose of some frame B in frame A. The inverse is the pose of frame A in frame B.

Parameters:

Name Type Description Default
`pose_mat` `array`

4x4 matrix for the pose to inverse

required

Returns:

Type Description

np.array: 4x4 matrix for the inverse pose

Source code in `omnigibson/utils/transform_utils.py`
 ```557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583``` ``````def pose_inv(pose_mat): """ Computes the inverse of a homogeneous matrix corresponding to the pose of some frame B in frame A. The inverse is the pose of frame A in frame B. Args: pose_mat (np.array): 4x4 matrix for the pose to inverse Returns: np.array: 4x4 matrix for the inverse pose """ # Note, the inverse of a pose matrix is the following # [R t; 0 1]^-1 = [R.T -R.T*t; 0 1] # Intuitively, this makes sense. # The original pose matrix translates by t, then rotates by R. # We just invert the rotation by applying R-1 = R.T, and also translate back. # Since we apply translation first before rotation, we need to translate by # -t in the original frame, which is -R-1*t in the new frame, and then rotate back by # R-1 to align the axis again. pose_inv = np.zeros((4, 4)) pose_inv[:3, :3] = pose_mat[:3, :3].T pose_inv[:3, 3] = -pose_inv[:3, :3].dot(pose_mat[:3, 3]) pose_inv[3, 3] = 1.0 return pose_inv ``````

## `pose_transform(pos1, quat1, pos0, quat0)`

Conducts forward transform from pose (pos0, quat0) to pose (pos1, quat1):

pose1 @ pose0, NOT pose0 @ pose1

Parameters:

Name Type Description Default
`pos1`

(x,y,z) position to transform

required
`quat1`

(x,y,z,w) orientation to transform

required
`pos0`

(x,y,z) initial position

required
`quat0`

(x,y,z,w) initial orientation

required

Returns:

Type Description

2-tuple: - (np.array) (x,y,z) position array in cartesian coordinates - (np.array) (x,y,z,w) orientation array in quaternion form

Source code in `omnigibson/utils/transform_utils.py`
 ```586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608``` ``````def pose_transform(pos1, quat1, pos0, quat0): """ Conducts forward transform from pose (pos0, quat0) to pose (pos1, quat1): pose1 @ pose0, NOT pose0 @ pose1 Args: pos1: (x,y,z) position to transform quat1: (x,y,z,w) orientation to transform pos0: (x,y,z) initial position quat0: (x,y,z,w) initial orientation Returns: 2-tuple: - (np.array) (x,y,z) position array in cartesian coordinates - (np.array) (x,y,z,w) orientation array in quaternion form """ # Get poses mat0 = pose2mat((pos0, quat0)) mat1 = pose2mat((pos1, quat1)) # Multiply and convert back to pos, quat return mat2pose(mat1 @ mat0) ``````

## `quat2axisangle(quat)`

Converts quaternion to axis-angle format. Returns a unit vector direction scaled by its angle in radians.

Parameters:

Name Type Description Default
`quat` `array`

(x,y,z,w) vec4 float angles

required

Returns:

Type Description

np.array: (ax,ay,az) axis-angle exponential coordinates

Source code in `omnigibson/utils/transform_utils.py`
 ```478 479 480 481 482 483 484 485 486 487 488 489``` ``````def quat2axisangle(quat): """ Converts quaternion to axis-angle format. Returns a unit vector direction scaled by its angle in radians. Args: quat (np.array): (x,y,z,w) vec4 float angles Returns: np.array: (ax,ay,az) axis-angle exponential coordinates """ return R.from_quat(quat).as_rotvec() ``````

## `quat2euler(quat)`

Converts euler angles into quaternion form

Parameters:

Name Type Description Default
`quat` `array`

(x,y,z,w) float quaternion angles

required

Returns:

Type Description

np.array: (r,p,y) angles

Raises:

Type Description
`AssertionError`

[Invalid input shape]

Source code in `omnigibson/utils/transform_utils.py`
 ```521 522 523 524 525 526 527 528 529 530 531 532 533 534``` ``````def quat2euler(quat): """ Converts euler angles into quaternion form Args: quat (np.array): (x,y,z,w) float quaternion angles Returns: np.array: (r,p,y) angles Raises: AssertionError: [Invalid input shape] """ return R.from_quat(quat).as_euler("xyz") ``````

## `quat2mat(quaternion)`

Converts given quaternion to matrix.

Parameters:

Name Type Description Default
`quaternion` `array`

(..., 4) (x,y,z,w) float quaternion angles

required

Returns:

Type Description

np.array: (..., 3, 3) rotation matrix

Source code in `omnigibson/utils/transform_utils.py`
 ```465 466 467 468 469 470 471 472 473 474 475``` ``````def quat2mat(quaternion): """ Converts given quaternion to matrix. Args: quaternion (np.array): (..., 4) (x,y,z,w) float quaternion angles Returns: np.array: (..., 3, 3) rotation matrix """ return R.from_quat(quaternion).as_matrix() ``````

## `quat_conjugate(quaternion)`

Return conjugate of quaternion.

E.g.:

q0 = random_quaternion() q1 = quat_conjugate(q0) q1[3] == q0[3] and all(q1[:3] == -q0[:3]) True

Parameters:

Name Type Description Default
`quaternion` `array`

(x,y,z,w) quaternion

required

Returns:

Type Description

np.array: (x,y,z,w) quaternion conjugate

Source code in `omnigibson/utils/transform_utils.py`
 ```163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182``` ``````def quat_conjugate(quaternion): """ Return conjugate of quaternion. E.g.: >>> q0 = random_quaternion() >>> q1 = quat_conjugate(q0) >>> q1[3] == q0[3] and all(q1[:3] == -q0[:3]) True Args: quaternion (np.array): (x,y,z,w) quaternion Returns: np.array: (x,y,z,w) quaternion conjugate """ return np.array( (-quaternion[0], -quaternion[1], -quaternion[2], quaternion[3]), dtype=np.float32, ) ``````

## `quat_distance(quaternion1, quaternion0)`

Returns distance between two quaternions, such that distance * quaternion0 = quaternion1

Parameters:

Name Type Description Default
`quaternion1` `array`

(x,y,z,w) quaternion

required
`quaternion0` `array`

(x,y,z,w) quaternion

required

Returns:

Type Description

np.array: (x,y,z,w) quaternion distance

Source code in `omnigibson/utils/transform_utils.py`
 ```204 205 206 207 208 209 210 211 212 213 214 215``` ``````def quat_distance(quaternion1, quaternion0): """ Returns distance between two quaternions, such that distance * quaternion0 = quaternion1 Args: quaternion1 (np.array): (x,y,z,w) quaternion quaternion0 (np.array): (x,y,z,w) quaternion Returns: np.array: (x,y,z,w) quaternion distance """ return quat_multiply(quaternion1, quat_inverse(quaternion0)) ``````

## `quat_inverse(quaternion)`

Return inverse of quaternion.

E.g.:

q0 = random_quaternion() q1 = quat_inverse(q0) np.allclose(quat_multiply(q0, q1), [0, 0, 0, 1]) True

Parameters:

Name Type Description Default
`quaternion` `array`

(x,y,z,w) quaternion

required

Returns:

Type Description

np.array: (x,y,z,w) quaternion inverse

Source code in `omnigibson/utils/transform_utils.py`
 ```185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201``` ``````def quat_inverse(quaternion): """ Return inverse of quaternion. E.g.: >>> q0 = random_quaternion() >>> q1 = quat_inverse(q0) >>> np.allclose(quat_multiply(q0, q1), [0, 0, 0, 1]) True Args: quaternion (np.array): (x,y,z,w) quaternion Returns: np.array: (x,y,z,w) quaternion inverse """ return quat_conjugate(quaternion) / np.dot(quaternion, quaternion) ``````

## `quat_multiply(quaternion1, quaternion0)`

Return multiplication of two quaternions (q1 * q0).

E.g.:

q = quat_multiply([1, -2, 3, 4], [-5, 6, 7, 8]) np.allclose(q, [-44, -14, 48, 28]) True

Parameters:

Name Type Description Default
`quaternion1` `array`

(x,y,z,w) quaternion

required
`quaternion0` `array`

(x,y,z,w) quaternion

required

Returns:

Type Description

np.array: (x,y,z,w) multiplied quaternion

Source code in `omnigibson/utils/transform_utils.py`
 ```134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160``` ``````def quat_multiply(quaternion1, quaternion0): """ Return multiplication of two quaternions (q1 * q0). E.g.: >>> q = quat_multiply([1, -2, 3, 4], [-5, 6, 7, 8]) >>> np.allclose(q, [-44, -14, 48, 28]) True Args: quaternion1 (np.array): (x,y,z,w) quaternion quaternion0 (np.array): (x,y,z,w) quaternion Returns: np.array: (x,y,z,w) multiplied quaternion """ x0, y0, z0, w0 = quaternion0 x1, y1, z1, w1 = quaternion1 return np.array( ( x1 * w0 + y1 * z0 - z1 * y0 + w1 * x0, -x1 * z0 + y1 * w0 + z1 * x0 + w1 * y0, x1 * y0 - y1 * x0 + z1 * w0 + w1 * z0, -x1 * x0 - y1 * y0 - z1 * z0 + w1 * w0, ), dtype=np.float32, ) ``````

## `quat_slerp(quat0, quat1, fraction, shortestpath=True)`

Return spherical linear interpolation between two quaternions.

E.g.:

q0 = random_quat() q1 = random_quat() q = quat_slerp(q0, q1, 0.0) np.allclose(q, q0) True

q = quat_slerp(q0, q1, 1.0) np.allclose(q, q1) True

q = quat_slerp(q0, q1, 0.5) angle = math.acos(np.dot(q0, q)) np.allclose(2.0, math.acos(np.dot(q0, q1)) / angle) or np.allclose(2.0, math.acos(-np.dot(q0, q1)) / angle) True

Parameters:

Name Type Description Default
`quat0` `array`

(x,y,z,w) quaternion startpoint

required
`quat1` `array`

(x,y,z,w) quaternion endpoint

required
`fraction` `float`

fraction of interpolation to calculate

required
`shortestpath` `bool`

If True, will calculate the shortest path

`True`

Returns:

Type Description

np.array: (x,y,z,w) quaternion distance

Source code in `omnigibson/utils/transform_utils.py`
 ```218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268``` ``````def quat_slerp(quat0, quat1, fraction, shortestpath=True): """ Return spherical linear interpolation between two quaternions. E.g.: >>> q0 = random_quat() >>> q1 = random_quat() >>> q = quat_slerp(q0, q1, 0.0) >>> np.allclose(q, q0) True >>> q = quat_slerp(q0, q1, 1.0) >>> np.allclose(q, q1) True >>> q = quat_slerp(q0, q1, 0.5) >>> angle = math.acos(np.dot(q0, q)) >>> np.allclose(2.0, math.acos(np.dot(q0, q1)) / angle) or \ np.allclose(2.0, math.acos(-np.dot(q0, q1)) / angle) True Args: quat0 (np.array): (x,y,z,w) quaternion startpoint quat1 (np.array): (x,y,z,w) quaternion endpoint fraction (float): fraction of interpolation to calculate shortestpath (bool): If True, will calculate the shortest path Returns: np.array: (x,y,z,w) quaternion distance """ q0 = unit_vector(quat0[:4]) q1 = unit_vector(quat1[:4]) if fraction == 0.0: return q0 elif fraction == 1.0: return q1 d = np.dot(q0, q1) if abs(abs(d) - 1.0) < EPS: return q0 if shortestpath and d < 0.0: # invert rotation d = -d q1 *= -1.0 angle = math.acos(np.clip(d, -1, 1)) if abs(angle) < EPS: return q0 isin = 1.0 / math.sin(angle) q0 *= math.sin((1.0 - fraction) * angle) * isin q1 *= math.sin(fraction * angle) * isin q0 += q1 return q0 ``````

## `random_axis_angle(angle_limit=None, random_state=None)`

Samples an axis-angle rotation by first sampling a random axis and then sampling an angle. If @angle_limit is provided, the size of the rotation angle is constrained.

If @random_state is provided (instance of np.random.RandomState), it will be used to generate random numbers.

Parameters:

Name Type Description Default
`angle_limit` `None or float`

If set, determines magnitude limit of angles to generate

`None`
`random_state` `None or RandomState`

RNG to use if specified

`None`

Raises:

Type Description
`AssertionError`

[Invalid RNG]

Source code in `omnigibson/utils/transform_utils.py`
 ```305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336``` ``````def random_axis_angle(angle_limit=None, random_state=None): """ Samples an axis-angle rotation by first sampling a random axis and then sampling an angle. If @angle_limit is provided, the size of the rotation angle is constrained. If @random_state is provided (instance of np.random.RandomState), it will be used to generate random numbers. Args: angle_limit (None or float): If set, determines magnitude limit of angles to generate random_state (None or RandomState): RNG to use if specified Raises: AssertionError: [Invalid RNG] """ if angle_limit is None: angle_limit = 2.0 * np.pi if random_state is not None: assert isinstance(random_state, np.random.RandomState) npr = random_state else: npr = np.random # sample random axis using a normalized sample from spherical Gaussian. # see (http://extremelearning.com.au/how-to-generate-uniformly-random-points-on-n-spheres-and-n-balls/) # for why it works. random_axis = npr.randn(3) random_axis /= np.linalg.norm(random_axis) random_angle = npr.uniform(low=0.0, high=angle_limit) return random_axis, random_angle ``````

## `random_quat(rand=None)`

Return uniform random unit quaternion.

E.g.:

q = random_quat() np.allclose(1.0, vector_norm(q)) True q = random_quat(np.random.random(3)) q.shape (4,)

Parameters:

Name Type Description Default
`rand` `3 - array or None`

If specified, must be three independent random variables that are uniformly distributed between 0 and 1.

`None`

Returns:

Type Description

np.array: (x,y,z,w) random quaternion

Source code in `omnigibson/utils/transform_utils.py`
 ```271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302``` ``````def random_quat(rand=None): """ Return uniform random unit quaternion. E.g.: >>> q = random_quat() >>> np.allclose(1.0, vector_norm(q)) True >>> q = random_quat(np.random.random(3)) >>> q.shape (4,) Args: rand (3-array or None): If specified, must be three independent random variables that are uniformly distributed between 0 and 1. Returns: np.array: (x,y,z,w) random quaternion """ if rand is None: rand = np.random.rand(3) else: assert len(rand) == 3 r1 = np.sqrt(1.0 - rand[0]) r2 = np.sqrt(rand[0]) pi2 = math.pi * 2.0 t1 = pi2 * rand[1] t2 = pi2 * rand[2] return np.array( (np.sin(t1) * r1, np.cos(t1) * r1, np.sin(t2) * r2, np.cos(t2) * r2), dtype=np.float32, ) ``````

## `relative_pose_transform(pos1, quat1, pos0, quat0)`

Computes relative forward transform from pose (pos0, quat0) to pose (pos1, quat1), i.e.: solves:

pose1 = pose0 @ transform

Parameters:

Name Type Description Default
`pos1`

(x,y,z) position to transform

required
`quat1`

(x,y,z,w) orientation to transform

required
`pos0`

(x,y,z) initial position

required
`quat0`

(x,y,z,w) initial orientation

required

Returns:

Type Description

2-tuple: - (np.array) (x,y,z) position array in cartesian coordinates - (np.array) (x,y,z,w) orientation array in quaternion form

Source code in `omnigibson/utils/transform_utils.py`
 ```630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652``` ``````def relative_pose_transform(pos1, quat1, pos0, quat0): """ Computes relative forward transform from pose (pos0, quat0) to pose (pos1, quat1), i.e.: solves: pose1 = pose0 @ transform Args: pos1: (x,y,z) position to transform quat1: (x,y,z,w) orientation to transform pos0: (x,y,z) initial position quat0: (x,y,z,w) initial orientation Returns: 2-tuple: - (np.array) (x,y,z) position array in cartesian coordinates - (np.array) (x,y,z,w) orientation array in quaternion form """ # Get poses mat0 = pose2mat((pos0, quat0)) mat1 = pose2mat((pos1, quat1)) # Invert pose0 and calculate transform return mat2pose(pose_inv(mat0) @ mat1) ``````

## `rotation_matrix(angle, direction, point=None)`

Returns matrix to rotate about axis defined by point and direction.

E.g.: >>> angle = (random.random() - 0.5) * (2math.pi) >>> direc = numpy.random.random(3) - 0.5 >>> point = numpy.random.random(3) - 0.5 >>> R0 = rotation_matrix(angle, direc, point) >>> R1 = rotation_matrix(angle-2math.pi, direc, point) >>> is_same_transform(R0, R1) True

``````>>> R0 = rotation_matrix(angle, direc, point)
>>> R1 = rotation_matrix(-angle, -direc, point)
>>> is_same_transform(R0, R1)
True

>>> I = numpy.identity(4, numpy.float32)
>>> numpy.allclose(I, rotation_matrix(math.pi*2, direc))
True

>>> numpy.allclose(2., numpy.trace(rotation_matrix(math.pi/2,
...                                                direc, point)))
True
``````

Parameters:

Name Type Description Default
`angle` `float`

Magnitude of rotation

required
`direction` `array`

(ax,ay,az) axis about which to rotate

required
`point` `None or array`

If specified, is the (x,y,z) point about which the rotation will occur

`None`

Returns:

Type Description

np.array: 4x4 homogeneous matrix that includes the desired rotation

Source code in `omnigibson/utils/transform_utils.py`
 ```727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782``` ``````def rotation_matrix(angle, direction, point=None): """ Returns matrix to rotate about axis defined by point and direction. E.g.: >>> angle = (random.random() - 0.5) * (2*math.pi) >>> direc = numpy.random.random(3) - 0.5 >>> point = numpy.random.random(3) - 0.5 >>> R0 = rotation_matrix(angle, direc, point) >>> R1 = rotation_matrix(angle-2*math.pi, direc, point) >>> is_same_transform(R0, R1) True >>> R0 = rotation_matrix(angle, direc, point) >>> R1 = rotation_matrix(-angle, -direc, point) >>> is_same_transform(R0, R1) True >>> I = numpy.identity(4, numpy.float32) >>> numpy.allclose(I, rotation_matrix(math.pi*2, direc)) True >>> numpy.allclose(2., numpy.trace(rotation_matrix(math.pi/2, ... direc, point))) True Args: angle (float): Magnitude of rotation direction (np.array): (ax,ay,az) axis about which to rotate point (None or np.array): If specified, is the (x,y,z) point about which the rotation will occur Returns: np.array: 4x4 homogeneous matrix that includes the desired rotation """ sina = math.sin(angle) cosa = math.cos(angle) direction = unit_vector(direction[:3]) # rotation matrix around unit vector R = np.array(((cosa, 0.0, 0.0), (0.0, cosa, 0.0), (0.0, 0.0, cosa)), dtype=np.float32) R += np.outer(direction, direction) * (1.0 - cosa) direction *= sina R += np.array( ( (0.0, -direction[2], direction[1]), (direction[2], 0.0, -direction[0]), (-direction[1], direction[0], 0.0), ), dtype=np.float32, ) M = np.identity(4) M[:3, :3] = R if point is not None: # rotation not around origin point = np.array(point[:3], dtype=np.float32, copy=False) M[:3, 3] = point - np.dot(R, point) return M ``````

## `unit_vector(data, axis=None, out=None)`

Returns ndarray normalized by length, i.e. eucledian norm, along axis.

E.g.: >>> v0 = numpy.random.random(3) >>> v1 = unit_vector(v0) >>> numpy.allclose(v1, v0 / numpy.linalg.norm(v0)) True

``````>>> v0 = numpy.random.rand(5, 4, 3)
>>> v1 = unit_vector(v0, axis=-1)
>>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=2)), 2)
>>> numpy.allclose(v1, v2)
True

>>> v1 = unit_vector(v0, axis=1)
>>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=1)), 1)
>>> numpy.allclose(v1, v2)
True

>>> v1 = numpy.empty((5, 4, 3), dtype=numpy.float32)
>>> unit_vector(v0, axis=1, out=v1)
>>> numpy.allclose(v1, v2)
True

>>> list(unit_vector([]))
[]

>>> list(unit_vector([1.0]))
[1.0]
``````

Parameters:

Name Type Description Default
`data` `array`

data to normalize

required
`axis` `None or int`

If specified, determines specific axis along data to normalize

`None`
`out` `None or array`

If specified, will store computation in this variable

`None`

Returns:

Type Description

None or np.array: If @out is not specified, will return normalized vector. Otherwise, stores the output in @out

Source code in `omnigibson/utils/transform_utils.py`
 ```866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921``` ``````def unit_vector(data, axis=None, out=None): """ Returns ndarray normalized by length, i.e. eucledian norm, along axis. E.g.: >>> v0 = numpy.random.random(3) >>> v1 = unit_vector(v0) >>> numpy.allclose(v1, v0 / numpy.linalg.norm(v0)) True >>> v0 = numpy.random.rand(5, 4, 3) >>> v1 = unit_vector(v0, axis=-1) >>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=2)), 2) >>> numpy.allclose(v1, v2) True >>> v1 = unit_vector(v0, axis=1) >>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=1)), 1) >>> numpy.allclose(v1, v2) True >>> v1 = numpy.empty((5, 4, 3), dtype=numpy.float32) >>> unit_vector(v0, axis=1, out=v1) >>> numpy.allclose(v1, v2) True >>> list(unit_vector([])) [] >>> list(unit_vector([1.0])) [1.0] Args: data (np.array): data to normalize axis (None or int): If specified, determines specific axis along data to normalize out (None or np.array): If specified, will store computation in this variable Returns: None or np.array: If @out is not specified, will return normalized vector. Otherwise, stores the output in @out """ if out is None: data = np.array(data, dtype=np.float32, copy=True) if data.ndim == 1: data /= math.sqrt(np.dot(data, data)) return data else: if out is not data: out[:] = np.array(data, copy=False) data = out length = np.atleast_1d(np.sum(data * data, axis)) np.sqrt(length, length) if axis is not None: length = np.expand_dims(length, axis) data /= length if out is None: return data ``````

## `vec(values)`

Converts value tuple into a numpy vector.

Parameters:

Name Type Description Default
`values` `n - array`

a tuple of numbers

required

Returns:

Type Description

np.array: vector of given values

Source code in `omnigibson/utils/transform_utils.py`
 ```339 340 341 342 343 344 345 346 347 348 349``` ``````def vec(values): """ Converts value tuple into a numpy vector. Args: values (n-array): a tuple of numbers Returns: np.array: vector of given values """ return np.array(values, dtype=np.float32) ``````

## `vec2quat(vec, up=(0, 0, 1.0))`

Converts given 3d-direction vector @vec to quaternion orientation with respect to another direction vector @up

Parameters:

Name Type Description Default
`vec` `3 - array`

(x,y,z) direction vector (possible non-normalized)

required
`up` `3 - array`

(x,y,z) direction vector representing the canonical up direction (possible non-normalized)

`(0, 0, 1.0)`
Source code in `omnigibson/utils/transform_utils.py`
 ```395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410``` ``````def vec2quat(vec, up=(0, 0, 1.0)): """ Converts given 3d-direction vector @vec to quaternion orientation with respect to another direction vector @up Args: vec (3-array): (x,y,z) direction vector (possible non-normalized) up (3-array): (x,y,z) direction vector representing the canonical up direction (possible non-normalized) """ # See https://stackoverflow.com/questions/15873996/converting-a-direction-vector-to-a-quaternion-rotation # Take cross product of @up and @vec to get @s_n, and then cross @vec and @s_n to get @u_n # Then compose 3x3 rotation matrix and convert into quaternion vec_n = vec / np.linalg.norm(vec) # x up_n = up / np.linalg.norm(up) s_n = np.cross(up_n, vec_n) # y u_n = np.cross(vec_n, s_n) # z return mat2quat(np.array([vec_n, s_n, u_n]).T) ``````

## `vecs2axisangle(vec0, vec1)`

Converts the angle from unnormalized 3D vectors @vec0 to @vec1 into an axis-angle representation of the angle

Parameters:

Name Type Description Default
`vec0` `array`

(..., 3) (x,y,z) 3D vector, possibly unnormalized

required
`vec1` `array`

(..., 3) (x,y,z) 3D vector, possibly unnormalized

required
Source code in `omnigibson/utils/transform_utils.py`
 ```1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026``` ``````def vecs2axisangle(vec0, vec1): """ Converts the angle from unnormalized 3D vectors @vec0 to @vec1 into an axis-angle representation of the angle Args: vec0 (np.array): (..., 3) (x,y,z) 3D vector, possibly unnormalized vec1 (np.array): (..., 3) (x,y,z) 3D vector, possibly unnormalized """ # Normalize vectors vec0 = normalize(vec0, axis=-1) vec1 = normalize(vec1, axis=-1) # Get cross product for direction of angle, and multiply by arcos of the dot product which is the angle return np.cross(vec0, vec1) * np.arccos((vec0 * vec1).sum(-1, keepdims=True)) ``````

## `vecs2quat(vec0, vec1, normalized=False)`

Converts the angle from unnormalized 3D vectors @vec0 to @vec1 into a quaternion representation of the angle

Parameters:

Name Type Description Default
`vec0` `array`

(..., 3) (x,y,z) 3D vector, possibly unnormalized

required
`vec1` `array`

(..., 3) (x,y,z) 3D vector, possibly unnormalized

required
`normalized` `bool`

If True, @vec0 and @vec1 are assumed to already be normalized and we will skip the normalization step (more efficient)

`False`
Source code in `omnigibson/utils/transform_utils.py`
 ```1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047``` ``````def vecs2quat(vec0, vec1, normalized=False): """ Converts the angle from unnormalized 3D vectors @vec0 to @vec1 into a quaternion representation of the angle Args: vec0 (np.array): (..., 3) (x,y,z) 3D vector, possibly unnormalized vec1 (np.array): (..., 3) (x,y,z) 3D vector, possibly unnormalized normalized (bool): If True, @vec0 and @vec1 are assumed to already be normalized and we will skip the normalization step (more efficient) """ # Normalize vectors if requested if not normalized: vec0 = normalize(vec0, axis=-1) vec1 = normalize(vec1, axis=-1) # Half-way Quaternion Solution -- see https://stackoverflow.com/a/11741520 cos_theta = np.sum(vec0 * vec1, axis=-1, keepdims=True) quat_unnormalized = np.where(cos_theta == -1, np.array([1.0, 0, 0, 0]), np.concatenate([np.cross(vec0, vec1), 1 + cos_theta], axis=-1)) return quat_unnormalized / np.linalg.norm(quat_unnormalized, axis=-1, keepdims=True) ``````

## `vel_in_A_to_vel_in_B(vel_A, ang_vel_A, pose_A_in_B)`

Converts linear and angular velocity of a point in frame A to the equivalent in frame B.

Parameters:

Name Type Description Default
`vel_A` `array`

(vx,vy,vz) linear velocity in A

required
`ang_vel_A` `array`

(wx,wy,wz) angular velocity in A

required
`pose_A_in_B` `array`

4x4 matrix corresponding to the pose of A in frame B

required

Returns:

Type Description

2-tuple:

• (np.array) (vx,vy,vz) linear velocities in frame B
• (np.array) (wx,wy,wz) angular velocities in frame B
Source code in `omnigibson/utils/transform_utils.py`
 ```681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701``` ``````def vel_in_A_to_vel_in_B(vel_A, ang_vel_A, pose_A_in_B): """ Converts linear and angular velocity of a point in frame A to the equivalent in frame B. Args: vel_A (np.array): (vx,vy,vz) linear velocity in A ang_vel_A (np.array): (wx,wy,wz) angular velocity in A pose_A_in_B (np.array): 4x4 matrix corresponding to the pose of A in frame B Returns: 2-tuple: - (np.array) (vx,vy,vz) linear velocities in frame B - (np.array) (wx,wy,wz) angular velocities in frame B """ pos_A_in_B = pose_A_in_B[:3, 3] rot_A_in_B = pose_A_in_B[:3, :3] skew_symm = _skew_symmetric_translation(pos_A_in_B) vel_B = rot_A_in_B.dot(vel_A) + skew_symm.dot(rot_A_in_B.dot(ang_vel_A)) ang_vel_B = rot_A_in_B.dot(ang_vel_A) return vel_B, ang_vel_B ``````

## `z_angle_from_quat(quat)`

Get the angle around the Z axis produced by the quaternion.

Source code in `omnigibson/utils/transform_utils.py`
 ```1143 1144 1145 1146``` ``````def z_angle_from_quat(quat): """Get the angle around the Z axis produced by the quaternion.""" rotated_X_axis = R.from_quat(quat).apply([1, 0, 0]) return np.arctan2(rotated_X_axis[1], rotated_X_axis[0]) ``````

## `z_rotation_from_quat(quat)`

Get the quaternion for the rotation around the Z axis produced by the quaternion.

Source code in `omnigibson/utils/transform_utils.py`
 ```1149 1150 1151``` ``````def z_rotation_from_quat(quat): """Get the quaternion for the rotation around the Z axis produced by the quaternion.""" return R.from_euler("z", z_angle_from_quat(quat)).as_quat() ``````