Source code for concepts.utils.rotationlib

#! /usr/bin/env python3
# -*- coding: utf-8 -*-
# File   : rotationlib.py
# Author : Jiayuan Mao
# Email  : maojiayuan@gmail.com
# Date   : 12/04/2019
#
# This file is part of Project Concepts.
# Distributed under terms of the MIT license.

"""
Tools for converting between rotation representations.

Conventions
-----------

- All functions accept batches as well as individual rotations.
- All rotation conventions match respective MuJoCo defaults (e.g., quaternions use wxyz convention).
    Note that this is DIFFERENT from PyBullet (which uses xyzw).
- All angles are in radians.
- Matricies follow LR convention.
- Euler Angles are all relative with 'xyz' axes ordering.
- See specific representation for more information.

Representations
---------------

Euler
    There are many euler angle frames -- here we will strive to use the default
    in MuJoCo, which is eulerseq='xyz'.

    This frame is a relative rotating frame, about x, y, and z axes in order.
    Relative rotating means that after we rotate about x, then we use the
    new (rotated) y, and the same for z.

Quaternions
    These are defined in terms of rotation (angle) about a unit vector (x, y, z)
    We use the following <q0, q1, q2, q3> convention:

    .. code-block:: python

        q0 = cos(angle / 2)
        q1 = sin(angle / 2) * x
        q2 = sin(angle / 2) * y
        q3 = sin(angle / 2) * z

    This is also sometimes called qw, qx, qy, qz.

    Note that quaternions are ambiguous, because we can represent a rotation by
    angle about vector <x, y, z> and -angle about vector <-x, -y, -z>.
    To choose between these, we pick "first nonzero positive", where we
    make the first nonzero element of the quaternion positive.

    This can result in mismatches if you're converting an quaternion that is not
    "first nonzero positive" to a different representation and back.

Axis Angle
    .. warning::

        (Not currently implemented)
        These are very straightforward.  Rotation is angle about a unit vector.

XY Axes
    .. warning::

        (Not currently implemented)
        We are given x axis and y axis, and z axis is cross product of x and y.

Z Axis
    .. warning::

        This is NOT RECOMMENDED.  Defines a unit vector for the Z axis,
        but rotation about this axis is not well defined.

    Instead pick a fixed reference direction for another axis (e.g. X)
    and calculate the other (e.g. Y = Z cross-product X),
    then use XY Axes rotation instead.
SO3
    .. warning::

        (Not currently implemented)
        While not supported by MuJoCo, this representation has a lot of nice features.
        We expect to add support for these in the future.

TODOs/Missings
    - Rotation integration or derivatives (e.g. velocity conversions)
    - More representations (SO3, etc)
    - Random sampling (e.g. sample uniform random rotation)
    - Performance benchmarks/measurements
    - (Maybe) define everything as to/from matricies, for simplicity
"""

# Copyright (c) 2009-2017, Matthew Brett and Christoph Gohlke
#    All rights reserved.
#
#    Redistribution and use in source and binary forms, with or without
#    modification, are permitted provided that the following conditions are
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# Many methods borrow heavily or entirely from transforms3d:
# https://github.com/matthew-brett/transforms3d
# They have mostly been modified to support batched operations.

import itertools
import numpy as np

# For testing whether a number is close to zero
_FLOAT_EPS = np.finfo(np.float64).eps
_EPS4 = _FLOAT_EPS * 4.0


[docs]def as_rotation(r): """Convert a 3x3 matrix or a quaternion into a standard 3x3 matrix representation.""" if isinstance(r, np.ndarray) and r.shape == (3, 3): return r if isinstance(r, np.ndarray) and r.shape == (4,): return quat2mat(r) raise ValueError('Invalid rotation: {}.'.format(r))
[docs]def euler2mat(euler): """ Convert Euler Angles to Rotation Matrix. See rotation.py for notes """ euler = np.asarray(euler, dtype=np.float64) assert euler.shape[-1] == 3, "Invalid shaped euler {}".format(euler) ai, aj, ak = -euler[..., 2], -euler[..., 1], -euler[..., 0] si, sj, sk = np.sin(ai), np.sin(aj), np.sin(ak) ci, cj, ck = np.cos(ai), np.cos(aj), np.cos(ak) cc, cs = ci * ck, ci * sk sc, ss = si * ck, si * sk mat = np.empty(euler.shape[:-1] + (3, 3), dtype=np.float64) mat[..., 2, 2] = cj * ck mat[..., 2, 1] = sj * sc - cs mat[..., 2, 0] = sj * cc + ss mat[..., 1, 2] = cj * sk mat[..., 1, 1] = sj * ss + cc mat[..., 1, 0] = sj * cs - sc mat[..., 0, 2] = -sj mat[..., 0, 1] = cj * si mat[..., 0, 0] = cj * ci return mat
[docs]def euler2quat(euler): """ Convert Euler Angles to Quaternions. See rotation.py for notes """ euler = np.asarray(euler, dtype=np.float64) assert euler.shape[-1] == 3, "Invalid shape euler {}".format(euler) ai, aj, ak = euler[..., 2] / 2, -euler[..., 1] / 2, euler[..., 0] / 2 si, sj, sk = np.sin(ai), np.sin(aj), np.sin(ak) ci, cj, ck = np.cos(ai), np.cos(aj), np.cos(ak) cc, cs = ci * ck, ci * sk sc, ss = si * ck, si * sk quat = np.empty(euler.shape[:-1] + (4,), dtype=np.float64) quat[..., 0] = cj * cc + sj * ss quat[..., 3] = cj * sc - sj * cs quat[..., 2] = -(cj * ss + sj * cc) quat[..., 1] = cj * cs - sj * sc return quat
[docs]def mat2euler(mat): """ Convert Rotation Matrix to Euler Angles. See rotation.py for notes """ mat = np.asarray(mat, dtype=np.float64) assert mat.shape[-2:] == (3, 3), "Invalid shape matrix {}".format(mat) cy = np.sqrt(mat[..., 2, 2] * mat[..., 2, 2] + mat[..., 1, 2] * mat[..., 1, 2]) condition = cy > _EPS4 euler = np.empty(mat.shape[:-1], dtype=np.float64) euler[..., 2] = np.where( condition, -np.arctan2(mat[..., 0, 1], mat[..., 0, 0]), -np.arctan2(-mat[..., 1, 0], mat[..., 1, 1]) ) euler[..., 1] = np.where(condition, -np.arctan2(-mat[..., 0, 2], cy), -np.arctan2(-mat[..., 0, 2], cy)) euler[..., 0] = np.where(condition, -np.arctan2(mat[..., 1, 2], mat[..., 2, 2]), 0.0) return euler
[docs]def mat2quat(mat): """ Convert Rotation Matrix to Quaternion. See rotation.py for notes """ mat = np.asarray(mat, dtype=np.float64) assert mat.shape[-2:] == (3, 3), "Invalid shape matrix {}".format(mat) Qxx, Qyx, Qzx = mat[..., 0, 0], mat[..., 0, 1], mat[..., 0, 2] Qxy, Qyy, Qzy = mat[..., 1, 0], mat[..., 1, 1], mat[..., 1, 2] Qxz, Qyz, Qzz = mat[..., 2, 0], mat[..., 2, 1], mat[..., 2, 2] # Fill only lower half of symmetric matrix K = np.zeros(mat.shape[:-2] + (4, 4), dtype=np.float64) K[..., 0, 0] = Qxx - Qyy - Qzz K[..., 1, 0] = Qyx + Qxy K[..., 1, 1] = Qyy - Qxx - Qzz K[..., 2, 0] = Qzx + Qxz K[..., 2, 1] = Qzy + Qyz K[..., 2, 2] = Qzz - Qxx - Qyy K[..., 3, 0] = Qyz - Qzy K[..., 3, 1] = Qzx - Qxz K[..., 3, 2] = Qxy - Qyx K[..., 3, 3] = Qxx + Qyy + Qzz K /= 3.0 # TODO: vectorize this -- probably could be made faster q = np.empty(K.shape[:-2] + (4,)) it = np.nditer(q[..., 0], flags=['multi_index']) while not it.finished: # Use Hermitian eigenvectors, values for speed vals, vecs = np.linalg.eigh(K[it.multi_index]) # Select largest eigenvector, reorder to w,x,y,z quaternion q[it.multi_index] = vecs[[3, 0, 1, 2], np.argmax(vals)] # Prefer quaternion with positive w # (q * -1 corresponds to same rotation as q) if q[it.multi_index][0] < 0: q[it.multi_index] *= -1 it.iternext() return q
[docs]def quat2euler(quat): """ Convert Quaternion to Euler Angles. See rotation.py for notes """ return mat2euler(quat2mat(quat))
[docs]def subtract_euler(e1, e2): assert e1.shape == e2.shape assert e1.shape[-1] == 3 q1 = euler2quat(e1) q2 = euler2quat(e2) q_diff = quat_mul(q1, quat_conjugate(q2)) return quat2euler(q_diff)
[docs]def quat2mat(quat): """ Convert Quaternion to Euler Angles. See rotation.py for notes """ quat = np.asarray(quat, dtype=np.float64) assert quat.shape[-1] == 4, "Invalid shape quat {}".format(quat) w, x, y, z = quat[..., 0], quat[..., 1], quat[..., 2], quat[..., 3] Nq = np.sum(quat * quat, axis=-1) s = 2.0 / Nq X, Y, Z = x * s, y * s, z * s wX, wY, wZ = w * X, w * Y, w * Z xX, xY, xZ = x * X, x * Y, x * Z yY, yZ, zZ = y * Y, y * Z, z * Z mat = np.empty(quat.shape[:-1] + (3, 3), dtype=np.float64) mat[..., 0, 0] = 1.0 - (yY + zZ) mat[..., 0, 1] = xY - wZ mat[..., 0, 2] = xZ + wY mat[..., 1, 0] = xY + wZ mat[..., 1, 1] = 1.0 - (xX + zZ) mat[..., 1, 2] = yZ - wX mat[..., 2, 0] = xZ - wY mat[..., 2, 1] = yZ + wX mat[..., 2, 2] = 1.0 - (xX + yY) return np.where((Nq > _FLOAT_EPS)[..., np.newaxis, np.newaxis], mat, np.eye(3))
[docs]def quat_conjugate(q): inv_q = -q inv_q[..., 0] *= -1 return inv_q
[docs]def quat_mul(q0, q1): assert q0.shape == q1.shape assert q0.shape[-1] == 4 assert q1.shape[-1] == 4 w0 = q0[..., 0] x0 = q0[..., 1] y0 = q0[..., 2] z0 = q0[..., 3] w1 = q1[..., 0] x1 = q1[..., 1] y1 = q1[..., 2] z1 = q1[..., 3] w = w0 * w1 - x0 * x1 - y0 * y1 - z0 * z1 x = w0 * x1 + x0 * w1 + y0 * z1 - z0 * y1 y = w0 * y1 + y0 * w1 + z0 * x1 - x0 * z1 z = w0 * z1 + z0 * w1 + x0 * y1 - y0 * x1 q = np.array([w, x, y, z]) if q.ndim == 2: q = q.swapaxes(0, 1) assert q.shape == q0.shape return q
[docs]def quat_rot_vec(q, v0): q_v0 = np.array([0, v0[0], v0[1], v0[2]]) q_v = quat_mul(q, quat_mul(q_v0, quat_conjugate(q))) v = q_v[1:] return v
[docs]def quat_identity(): return np.array([1, 0, 0, 0])
[docs]def axisangle2quat(axis, angle): quat = np.zeros(4, dtype='float64') quat[0] = np.cos(angle / 2) quat[1:] = np.sin(angle / 2) * axis return quat
[docs]def quat2axisangle(quat): theta = 0 axis = np.array([0, 0, 1]) sin_theta = np.linalg.norm(quat[1:]) if sin_theta > 0.0001: theta = 2 * np.arcsin(sin_theta) theta *= 1 if quat[0] >= 0 else -1 axis = quat[1:] / sin_theta return axis, theta
[docs]def euler2point_euler(euler): _euler = euler.copy() if len(_euler.shape) < 2: _euler = np.expand_dims(_euler, 0) assert _euler.shape[1] == 3 _euler_sin = np.sin(_euler) _euler_cos = np.cos(_euler) return np.concatenate([_euler_sin, _euler_cos], axis=-1)
[docs]def point_euler2euler(euler): _euler = euler.copy() if len(_euler.shape) < 2: _euler = np.expand_dims(_euler, 0) assert _euler.shape[1] == 6 angle = np.arctan(_euler[..., :3] / _euler[..., 3:]) angle[_euler[..., 3:] < 0] += np.pi return angle
[docs]def quat2point_quat(quat): # Should be in qw, qx, qy, qz _quat = quat.copy() if len(_quat.shape) < 2: _quat = np.expand_dims(_quat, 0) assert _quat.shape[1] == 4 angle = np.arccos(_quat[:, [0]]) * 2 xyz = _quat[:, 1:] xyz[np.squeeze(np.abs(np.sin(angle / 2))) >= 1e-5] = (xyz / np.sin(angle / 2))[ np.squeeze(np.abs(np.sin(angle / 2))) >= 1e-5 ] return np.concatenate([np.sin(angle), np.cos(angle), xyz], axis=-1)
[docs]def point_quat2quat(quat): _quat = quat.copy() if len(_quat.shape) < 2: _quat = np.expand_dims(_quat, 0) assert _quat.shape[1] == 5 angle = np.arctan(_quat[:, [0]] / _quat[:, [1]]) qw = np.cos(angle / 2) qxyz = _quat[:, 2:] qxyz[np.squeeze(np.abs(np.sin(angle / 2))) >= 1e-5] = (qxyz * np.sin(angle / 2))[ np.squeeze(np.abs(np.sin(angle / 2))) >= 1e-5 ] return np.concatenate([qw, qxyz], axis=-1)
[docs]def normalize_angles(angles): """Puts angles in [-pi, pi] range.""" angles = angles.copy() if angles.size > 0: angles = (angles + np.pi) % (2 * np.pi) - np.pi assert -np.pi - 1e-6 <= angles.min() and angles.max() <= np.pi + 1e-6 return angles
[docs]def round_to_straight_angles(angles): """Returns closest angle modulo 90 degrees """ angles = np.round(angles / (np.pi / 2)) * (np.pi / 2) return normalize_angles(angles)
[docs]def get_parallel_rotations(): """Returns a list of all possible rotations that are parallel to the canonical axes.""" mult90 = [0, np.pi / 2, -np.pi / 2, np.pi] parallel_rotations = [] for euler in itertools.product(mult90, repeat=3): canonical = mat2euler(euler2mat(euler)) canonical = np.round(canonical / (np.pi / 2)) if canonical[0] == -2: canonical[0] = 2 if canonical[2] == -2: canonical[2] = 2 canonical *= np.pi / 2 if all([(canonical != rot).any() for rot in parallel_rotations]): parallel_rotations += [canonical] assert len(parallel_rotations) == 24 return parallel_rotations