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<!DOCTYPE html>

<head>

<title>spatialmath.pose3d — Spatial Maths package 0.11.0

documentation</title>

<!--[if lt IE 9]>

<![endif]-->

</head>

<a href="../../index.html" class="icon icon-home"> Spatial Maths package

</a>

0.9

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<ul>

<li class="toctree-l1"><a class="reference internal" href="../../intro.html">Introduction</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../spatialmath.html">Class reference</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../functions.html">Function reference</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../indices.html">Indices</a></li>

</ul>

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</nav>

<a href="../../index.html">Spatial Maths package</a>

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<li><a href="../index.html">Module code</a> »</li>

<li>spatialmath.pose3d</li>

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</div>

<h1>Source code for spatialmath.pose3d</h1><div class="highlight"><pre>

# Part of Spatial Math Toolbox for Python

# MIT Licence, see details in top-level file: LICENCE

"""

Classes to abstract 3D pose and orientation using matrices in SE(3) and SO(3)

To use::

from spatialmath.pose3d import *

T = SE3.Rx(0.3)

import spatialmath as sm

T = sm.SE3.Rx(0.3)

.. inheritance-diagram:: spatialmath.pose3d

:top-classes: collections.UserList

:parts: 1

.. image:: ../figs/pose-values.png

"""

# pylint: disable=invalid-name

import numpy as np

from spatialmath import base

from spatialmath.baseposematrix import BasePoseMatrix

# ============================== SO3 =====================================#

<div class="viewcode-block" id="SO3"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3">[docs]</a>class SO3(BasePoseMatrix):

"""

SO(3) matrix class

This subclass represents rotations in 3D space. Internally it is a 3x3

orthogonal matrix belonging to the group SO(3).

.. inheritance-diagram:: spatialmath.pose3d.SO3

:top-classes: collections.UserList

:parts: 1

"""

<div class="viewcode-block" id="SO3.__init__"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.__init__">[docs]</a> def __init__(self, arg=None, *, check=True):

"""

Construct new SO(3) object

:rtype: SO3 instance

There are multiple call signatures:

- ``SO3()`` is an ``SO3`` instance with one value -- a 3x3 identity

matrix which corresponds to a null rotation

- ``SO3(R)`` is an ``SO3`` instance with with the value ``R`` which is a

3x3 numpy array representing an SO(3) rotation matrix. If ``check``

is ``True`` check the matrix belongs to SO(3).

- ``SO3([R1, R2, ... RN])`` is an ``SO3`` instance wwith ``N`` values

given by the elements ``Ri`` each of which is a 3x3 NumPy array

representing an SO(3) matrix. If ``check`` is ``True`` check the

matrix belongs to SO(3).

- ``SO3([X1, X2, ... XN])`` is an ``SO3`` instance with ``N`` values

given by the elements ``Xi`` each of which is an SO3 or SE3 instance.

:SymPy: supported

"""

super().__init__()

if isinstance(arg, SE3):

self.data = [base.t2r(x) for x in arg.data]

elif not super().arghandler(arg, check=check):

raise ValueError('bad argument to constructor')</div>

@staticmethod

def _identity():

return np.eye(3)

# ------------------------------------------------------------------------ #

@property

def shape(self):

"""

Shape of the object's interal matrix representation

:return: (3,3)

:rtype: tuple

Each value within the ``SO3`` instance is a NumPy array of this shape.

"""

return (3, 3)

@property

def R(self):

"""

SO(3) or SE(3) as rotation matrix

:return: rotational component

:rtype: numpy.ndarray, shape=(3,3)

``x.R`` is the rotation matrix component of ``x`` as an array with

shape (3,3). If ``len(x) > 1``, return an array with shape=(N,3,3).

.. warning:: The i'th rotation matrix is ``x[i,:,:]`` or simply

``x[i]``. This is different to the MATLAB version where the i'th

rotation matrix is ``x(:,:,i)``.

Example:

.. runblock:: pycon

>>> from spatialmath import SO3

>>> x = SO3.Rx(0.3)

>>> x.R

:SymPy: supported

"""

if len(self) == 1:

return self.A[:3, :3]

else:

return np.array([x[:3, :3] for x in self.A])

@property

def n(self):

"""

Normal vector of SO(3) or SE(3)

:return: normal vector

:rtype: numpy.ndarray, shape=(3,)

This is the first column of the rotation submatrix, sometimes called the

*normal vector*. It is parallel to the x-axis of the frame defined by

this pose.

"""

@property

def o(self):

"""

Orientation vector of SO(3) or SE(3)

:return: orientation vector

:rtype: numpy.ndarray, shape=(3,)

This is the second column of the rotation submatrix, sometimes called

the *orientation vector*. It is parallel to the y-axis of the frame

defined by this pose.

"""

@property

def a(self):

"""

Approach vector of SO(3) or SE(3)

:return: approach vector

:rtype: numpy.ndarray, shape=(3,)

This is the third column of the rotation submatrix, sometimes called the

*approach vector*. It is parallel to the z-axis of the frame defined by

this pose.

"""

# ------------------------------------------------------------------------ #

<div class="viewcode-block" id="SO3.inv"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.inv">[docs]</a> def inv(self):

"""

Inverse of SO(3)

:return: inverse

:rtype: SO2 instance

Efficiently compute the inverse of each of the SO(3) values taking into

account the matrix structure. For an SO(3) matrix the inverse is the

transpose.

"""

return SO3(self.A.T, check=False)

else:

return SO3([x.T for x in self.A], check=False)</div>

<div class="viewcode-block" id="SO3.eul"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.eul">[docs]</a> def eul(self, unit='rad', flip=False):

r"""

SO(3) or SE(3) as Euler angles

:param unit: angular units: 'rad' [default], or 'deg'

:type unit: str

:return: 3-vector of Euler angles

:rtype: ndarray(3,), ndarray(n,3)

``x.eul`` is the Euler angle representation of the rotation. Euler angles are

a 3-vector :math:`(\phi, \theta, \psi)` which correspond to consecutive

rotations about the Z, Y, Z axes respectively.

If ``len(x)`` is:

- 1, return an ndarray with shape=(3,)

- N>1, return ndarray with shape=(3,N)

:seealso: :func:`~spatialmath.pose3d.SE3.Eul`, :func:`~spatialmath.base.transforms3d.tr2eul`

:SymPy: not supported

"""

return base.tr2eul(self.A, unit=unit, flip=flip)

else:

return np.array([base.tr2eul(x, unit=unit, flip=flip) for x in self.A])</div>

<div class="viewcode-block" id="SO3.rpy"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.rpy">[docs]</a> def rpy(self, unit='rad', order='zyx'):

"""

SO(3) or SE(3) as roll-pitch-yaw angles

:param order: angle sequence order, default to 'zyx'

:type order: str

:param unit: angular units: 'rad' [default], or 'deg'

:type unit: str

:return: 3-vector of roll-pitch-yaw angles

:rtype: ndarray(3,), ndarray(n,3)

``x.rpy`` is the roll-pitch-yaw angle representation of the rotation. The angles are

a 3-vector :math:`(r, p, y)` which correspond to successive rotations about the axes

specified by ``order``:

- ``'zyx'`` [default], rotate by yaw about the z-axis, then by pitch about the new y-axis,

then by roll about the new x-axis. Convention for a mobile robot with x-axis forward

and y-axis sideways.

- ``'xyz'``, rotate by yaw about the x-axis, then by pitch about the new y-axis,

then by roll about the new z-axis. Convention for a robot gripper with z-axis forward

and y-axis between the gripper fingers.

- ``'yxz'``, rotate by yaw about the y-axis, then by pitch about the new x-axis,

then by roll about the new z-axis. Convention for a camera with z-axis parallel

to the optic axis and x-axis parallel to the pixel rows.

If `len(x)` is:

- 1, return an ndarray with shape=(3,)

- N>1, return ndarray with shape=(3,N)

:seealso: :func:`~spatialmath.pose3d.SE3.RPY`, :func:`~spatialmath.base.transforms3d.tr2rpy`

:SymPy: not supported

"""

return base.tr2rpy(self.A, unit=unit, order=order)

else:

return np.array([base.tr2rpy(x, unit=unit, order=order) for x in self.A])</div>

<div class="viewcode-block" id="SO3.angvec"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.angvec">[docs]</a> def angvec(self, unit='rad'):

r"""

SO(3) or SE(3) as angle and rotation vector

:param unit: angular units: 'rad' [default], or 'deg'

:type unit: str

:param check: check that rotation matrix is valid

:type check: bool

:return: :math:`(\theta, {\bf v})`

:rtype: float, numpy.ndarray, shape=(3,)

``q.angvec()`` is a tuple :math:`(\theta, v)` containing the rotation

angle and a rotation axis which is equivalent to the rotation of

the unit quaternion ``q``.

By default the angle is in radians but can be changed setting `unit='deg'`.

.. notes::

- If the input is SE(3) the translation component is ignored.

Example:

.. runblock:: pycon

>>> from spatialmath import UnitQuaternion

>>> UnitQuaternion.Rz(0.3).angvec()

:seealso: :func:`~spatialmath.quaternion.AngVec`, :func:`~angvec2r`

"""

return base.tr2angvec(self.R, unit=unit)</div>

# ------------------------------------------------------------------------ #

<div class="viewcode-block" id="SO3.isvalid"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.isvalid">[docs]</a> @staticmethod

def isvalid(x, check=True):

"""

Test if matrix is valid SO(3)

:param x: matrix to test

:type x: numpy.ndarray

:return: ``True`` if the matrix is a valid element of SO(3), ie. it is a 3x3

orthonormal matrix with determinant of +1.

:rtype: bool

:seealso: :func:`~spatialmath.base.transform3d.isrot`

"""

return base.isrot(x, check=True)</div>

# ---------------- variant constructors ---------------------------------- #

<div class="viewcode-block" id="SO3.Rx"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.Rx">[docs]</a> @classmethod

def Rx(cls, theta, unit='rad'):

"""

Construct a new SO(3) from X-axis rotation

:param θ: rotation angle about the X-axis

:type θ: float or array_like

:param unit: angular units: 'rad' [default], or 'deg'

:type unit: str

:return: SO(3) rotation

:rtype: SO3 instance

- ``SE3.Rx(θ)`` is an SO(3) rotation of ``θ`` radians about the x-axis

- ``SE3.Rx(θ, "deg")`` as above but ``θ`` is in degrees

If ``theta`` is an array then the result is a sequence of rotations defined by consecutive

elements.

Example:

.. runblock:: pycon

>>> from spatialmath import SO3

>>> x = SO3.Rx(np.linspace(0, math.pi, 20))

>>> len(x)

>>> x[7]

"""

return cls([base.rotx(x, unit=unit) for x in base.getvector(theta)], check=False)</div>

<div class="viewcode-block" id="SO3.Ry"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.Ry">[docs]</a> @classmethod

def Ry(cls, theta, unit='rad'):

"""

Construct a new SO(3) from Y-axis rotation

:param θ: rotation angle about Y-axis

:type θ: float or array_like

:param unit: angular units: 'rad' [default], or 'deg'

:type unit: str

:return: SO(3) rotation

:rtype: SO3 instance

- ``SO3.Ry(θ)`` is an SO(3) rotation of ``θ`` radians about the y-axis

- ``SO3.Ry(θ, "deg")`` as above but ``θ`` is in degrees

If ``θ`` is an array then the result is a sequence of rotations defined by consecutive

elements.

Example:

.. runblock:: pycon

>>> from spatialmath import UnitQuaternion

>>> x = SO3.Ry(np.linspace(0, math.pi, 20))

>>> len(x)

>>> x[7]

"""

return cls([base.roty(x, unit=unit) for x in base.getvector(theta)], check=False)</div>

<div class="viewcode-block" id="SO3.Rz"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.Rz">[docs]</a> @classmethod

def Rz(cls, theta, unit='rad'):

"""

Construct a new SO(3) from Z-axis rotation

:param θ: rotation angle about Z-axis

:type θ: float or array_like

:param unit: angular units: 'rad' [default], or 'deg'

:type unit: str

:return: SO(3) rotation

:rtype: SO3 instance

- ``SO3.Rz(θ)`` is an SO(3) rotation of ``θ`` radians about the z-axis

- ``SO3.Rz(θ, "deg")`` as above but ``θ`` is in degrees

If ``θ`` is an array then the result is a sequence of rotations defined by consecutive

elements.

Example:

.. runblock:: pycon

>>> from spatialmath import SE3

>>> x = SE3.Rz(np.linspace(0, math.pi, 20))

>>> len(x)

>>> x[7]

"""

return cls([base.rotz(x, unit=unit) for x in base.getvector(theta)], check=False)</div>

<div class="viewcode-block" id="SO3.Rand"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.Rand">[docs]</a> @classmethod

def Rand(cls, N=1):

"""

Construct a new SO(3) from random rotation

:param N: number of random rotations

:type N: int

:return: SO(3) rotation matrix

:rtype: SO3 instance

- ``SO3.Rand()`` is a random SO(3) rotation.

- ``SO3.Rand(N)`` is a sequence of N random rotations.

Example:

.. runblock:: pycon

>>> from spatialmath import SO3

>>> x = SO3.Rand()

>>> x

:seealso: :func:`spatialmath.quaternion.UnitQuaternion.Rand`

"""

return cls([base.q2r(base.rand()) for _ in range(0, N)], check=False)</div>

<div class="viewcode-block" id="SO3.Eul"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.Eul">[docs]</a> @classmethod

def Eul(cls, *angles, unit='rad'):

r"""

Construct a new SO(3) from Euler angles

:param 𝚪: Euler angles

:type 𝚪: array_like or numpy.ndarray with shape=(N,3)

:param unit: angular units: 'rad' [default], or 'deg'

:type unit: str

:return: SO(3) rotation

:rtype: SO3 instance

``SO3.Eul(𝚪)`` is an SO(3) rotation defined by a 3-vector of Euler

angles :math:`\Gamma = (\phi, \theta, \psi)` which correspond to

consecutive rotations about the Z, Y, Z axes respectively. If ``𝚪``

is an Nx3 matrix then the result is a sequence of rotations each

defined by Euler angles corresponding to the rows of ``angles``.

``SO3.Eul(φ, θ, ψ)`` as above but the angles are provided as three

scalars.

Example:

.. runblock:: pycon

>>> from spatialmath import SO3

>>> SO3.Eul(0.1, 0.2, 0.3)

>>> SO3.Eul([0.1, 0.2, 0.3])

>>> SO3.Eul(10, 20, 30, 'deg')

:seealso: :func:`~spatialmath.pose3d.SE3.eul`, :func:`~spatialmath.pose3d.SE3.Eul`, :func:`~spatialmath.base.transforms3d.eul2r`

"""

if len(angles) == 1:

angles = angles[0]

if base.isvector(angles, 3):

return cls(base.eul2r(angles, unit=unit), check=False)

else:

return cls([base.eul2r(a, unit=unit) for a in angles], check=False)</div>

<div class="viewcode-block" id="SO3.RPY"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.RPY">[docs]</a> @classmethod

def RPY(cls, *angles, unit='rad', order='zyx', ):

r"""

Construct a new SO(3) from roll-pitch-yaw angles

:param angles: roll-pitch-yaw angles

:type angles: array_like(3), array_like(n,3)

:param unit: angular units: 'rad' [default], or 'deg'

:type unit: str

:param order: rotation order: 'zyx' [default], 'xyz', or 'yxz'

:type order: str

:return: SO(3) rotation

:rtype: SO3 instance

- ``SO3.RPY(angles)`` is an SO(3) rotation defined by a 3-vector of

roll, pitch, yaw angles :math:`(\alpha, \beta, \gamma)`. If ``angles``

is an Nx3 matrix then the result is a sequence of rotations each

defined by RPY angles corresponding to the rows of angles. The angles

correspond to successive rotations about the axes specified by

``order``:

- ``'zyx'`` [default], rotate by yaw about the z-axis, then by pitch about the new y-axis,

then by roll about the new x-axis. Convention for a mobile robot with x-axis forward

and y-axis sideways.

- ``'xyz'``, rotate by yaw about the x-axis, then by pitch about the new y-axis,

then by roll about the new z-axis. Convention for a robot gripper with z-axis forward

and y-axis between the gripper fingers.

- ``'yxz'``, rotate by yaw about the y-axis, then by pitch about the new x-axis,

then by roll about the new z-axis. Convention for a camera with z-axis parallel

to the optic axis and x-axis parallel to the pixel rows.

- ``SO3.RPY(⍺, β, 𝛾)`` as above but the angles are provided as three

scalars.

Example:

.. runblock:: pycon

>>> from spatialmath import SO3

>>> SO3.RPY(0.1, 0.2, 0.3)

>>> SO3.RPY([0.1, 0.2, 0.3])

>>> SO3.RPY(0.1, 0.2, 0.3, order='xyz')

>>> SO3.RPY(10, 20, 30, 'deg')

:seealso: :func:`~spatialmath.pose3d.SE3.rpy`, :func:`~spatialmath.pose3d.SE3.RPY`, :func:`spatialmath.base.transforms3d.rpy2r`

"""

angles = angles[0]

# angles = base.getmatrix(angles, (None, 3))

# return cls(base.rpy2r(angles, order=order, unit=unit), check=False)

return cls(base.rpy2r(angles, unit=unit, order=order), check=False)

else:

return cls([base.rpy2r(a, unit=unit, order=order) for a in angles], check=False)</div>

<div class="viewcode-block" id="SO3.OA"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.OA">[docs]</a> @classmethod

def OA(cls, o, a):

"""

Construct a new SO(3) from two vectors

:param o: 3-vector parallel to Y- axis

:type o: array_like

:param a: 3-vector parallel to the Z-axis

:type o: array_like

:return: SO(3) rotation

:rtype: SO3 instance

``SO3.OA(O, A)`` is an SO(3) rotation defined in terms of

vectors parallel to the Y- and Z-axes of its reference frame. In robotics these axes are

respectively called the *orientation* and *approach* vectors defined such that

R = [N, O, A] and N = O x A.

.. notes::

- Only the ``A`` vector is guaranteed to have the same direction in the resulting

rotation matrix

- ``O`` and ``A`` do not have to be unit-length, they are normalized

- ``O`` and ``A` do not have to be orthogonal, so long as they are not parallel

:seealso: :func:`spatialmath.base.transforms3d.oa2r`

"""

return cls(base.oa2r(o, a), check=False)</div>

<div class="viewcode-block" id="SO3.TwoVectors"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.TwoVectors">[docs]</a> @classmethod

def TwoVectors(cls, x=None, y=None, z=None):

"""

Construct a new SO(3) from any two vectors

:param x: new x-axis, defaults to None

:type x: str, array_like(3), optional

:param y: new y-axis, defaults to None

:type y: str, array_like(3), optional

:param z: new z-axis, defaults to None

:type z: str, array_like(3), optional

Create a rotation by defining the direction of two of the new

axes in terms of the old axes. Axes are denoted by strings ``"x"``,

``"y"``, ``"z"``, ``"-x"``, ``"-y"``, ``"-z"``.

The directions can also be specified by 3-element vectors, but these

must be orthogonal.

To create a rotation where the new frame has its x-axis in -z-direction

of the previous frame, and its z-axis in the x-direction of the previous

frame is::

>>> SO3.TwoVectors(x='-z', z='x')

"""

def vval(v):

if isinstance(v, str):

sign = 1

if v[0] == '-':

sign = -1

v = v[1:] # skip sign char

elif v[0] == '+':

v = v[1:] # skip sign char

v = [sign, 0, 0]

v = [0, sign, 0]

v = [0, 0, sign]

return np.r_[v]

else:

return base.unitvec(base.getvector(v, 3))

if x is not None and y is not None and z is None:

# z = x x y

x = vval(x)

y = vval(y)

z = np.cross(x, y)

elif x is None and y is not None and z is not None:

# x = y x z

y = vval(y)

z = vval(z)

x = np.cross(y, z)

elif x is not None and y is None and z is not None:

# y = z x x

z = vval(z)

x = vval(x)

y = np.cross(z, x)

return cls(np.c_[x, y, z], check=False)</div>

<div class="viewcode-block" id="SO3.AngleAxis"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.AngleAxis">[docs]</a> @classmethod

def AngleAxis(cls, theta, v, *, unit='rad'):

r"""

Construct a new SO(3) rotation matrix from rotation angle and axis

:param theta: rotation

:type theta: float

:param unit: angular units: 'rad' [default], or 'deg'

:type unit: str

:param v: rotation axis, 3-vector

:type v: array_like

:return: SO(3) rotation

:rtype: SO3 instance

``SO3.AngleAxis(theta, V)`` is an SO(3) rotation defined by

a rotation of ``THETA`` about the vector ``V``.

.. note:: :math:`\theta \eq 0` the result in an identity matrix, otherwise

``V`` must have a finite length, ie. :math:`|V| > 0`.

:seealso: :func:`~spatialmath.pose3d.SE3.angvec`, :func:`spatialmath.base.transforms3d.angvec2r`

"""

return cls(base.angvec2r(theta, v, unit=unit), check=False)</div>

<div class="viewcode-block" id="SO3.AngVec"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.AngVec">[docs]</a> @classmethod

def AngVec(cls, theta, v, *, unit='rad'):

r"""

Construct a new SO(3) rotation matrix from rotation angle and axis

:param theta: rotation

:type theta: float

:param unit: angular units: 'rad' [default], or 'deg'

:type unit: str

:param v: rotation axis, 3-vector

:type v: array_like

:return: SO(3) rotation

:rtype: SO3 instance

``SO3.AngVec(theta, V)`` is an SO(3) rotation defined by

a rotation of ``THETA`` about the vector ``V``.

.. deprecated:: 0.9.8

Use :meth:`AngleAxis` instead.

:seealso: :func:`~spatialmath.pose3d.SE3.angvec`, :func:`spatialmath.base.transforms3d.angvec2r`

"""

<div class="viewcode-block" id="SO3.EulerVec"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.EulerVec">[docs]</a> @classmethod

def EulerVec(cls, w):

r"""

Construct a new SO(3) rotation matrix from an Euler rotation vector

:param ω: rotation axis

:type ω: 3-element array_like

:return: SO(3) rotation

:rtype: SO3 instance

``SO3.EulerVec(ω)`` is a unit quaternion that describes the 3D rotation

defined by a rotation of :math:`\theta = \lVert \omega \rVert` about the

unit 3-vector :math:`\omega / \lVert \omega \rVert`.

Example:

.. runblock:: pycon

>>> from spatialmath import SO3

>>> SO3.EulerVec([0.5,0,0])

.. note:: :math:`\theta \eq 0` the result in an identity matrix, otherwise

``V`` must have a finite length, ie. :math:`|V| > 0`.

:seealso: :func:`~spatialmath.pose3d.SE3.angvec`, :func:`~spatialmath.base.transforms3d.angvec2r`

"""

assert base.isvector(w, 3), 'w must be a 3-vector'

w = base.getvector(w)

theta = base.norm(w)

return cls(base.angvec2r(theta, w), check=False)</div>

<div class="viewcode-block" id="SO3.Exp"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.Exp">[docs]</a> @classmethod

def Exp(cls, S, check=True, so3=True):

r"""

Create an SO(3) rotation matrix from so(3)

:param S: Lie algebra so(3)

:type S: numpy ndarray

:param check: check that passed matrix is valid so(3), default True

:type check: bool

:return: SO(3) rotation

:rtype: SO3 instance

- ``SO3.Exp(S)`` is an SO(3) rotation defined by its Lie algebra

which is a 3x3 so(3) matrix (skew symmetric)

- ``SO3.Exp(t)`` is an SO(3) rotation defined by a 3-element twist

vector (the unique elements of the so(3) skew-symmetric matrix)

- ``SO3.Exp(T)`` is a sequence of SO(3) rotations defined by an Nx3 matrix

of twist vectors, one per row.

Note:

- if :math:`\theta \eq 0` the result in an identity matrix

- an input 3x3 matrix is ambiguous, it could be the first or third case above. In this

case the parameter `so3` is the decider.

:seealso: :func:`spatialmath.base.transforms3d.trexp`, :func:`spatialmath.base.transformsNd.skew`

"""

if base.ismatrix(S, (-1, 3)) and not so3:

return cls([base.trexp(s, check=check) for s in S], check=False)

else:

return cls(base.trexp(S, check=check), check=False)</div>

<div class="viewcode-block" id="SO3.angdist"><a class="viewcode-back" href="../../3d_orient_SO3.html#spatialmath.pose3d.SO3.angdist">[docs]</a> def angdist(self, other, metric=6):

r"""

Angular distance metric between rotations

:param other: second rotation

:type other: SO3 instance

:param metric: metric, default is 6

:type metric: int

:raises TypeError: if other is not an SO3

:return: angle in radians

:rtype: float or ndarray

``R1.angdist(R2)`` is the geodesic norm, or geodesic distance between two

rotations.

Several metrics are supported, the first 5 are computed after conversion

to unit quaternions.

====== ===============================================================

Metric Details

====== ===============================================================

0 :math:`1 - | \q_1 \bullet \q_2 | \in [0, 1]`

1 :math:`\cos^{-1} | \q_1 \bullet \q_2 | \in [0, \pi/2]`

2 :math:`\cos^{-1} | \q_1 \bullet \q_2 | \in [0, \pi/2]`

3 :math:`2 \tan^{-1} \| \q_1 - \q_2\| / \|\q_1 + \q_2\| \in [0, \pi/2]`

4 :math:`\cos^{-1} \left( 2 (\q_1 \bullet \q_2)^2 - 1\right) \in [0, 1]`

5 :math:`\|I - \mat{R}_1 \mat{R}_2^T\| \in [0, 2]`

6 :math:`\|\log \mat{R}_1 \mat{R}_2^T\| \in [0, \pi]`

====== ===============================================================

Example:

.. runblock:: pycon

>>> from spatialmath import UnitQuaternion

>>> R1 = SO3.Rx(0.3)

>>> R2 = SO3.Ry(0.3)

>>> print(R1.angdist(R1))

>>> print(R1.angdist(R2))

.. note::

- metrics 1, 2, 4 can throw ValueError "math domain error" due to

numeric errors which push the argument of ``acos()`` marginally

outside its domain [0, 1].

- metrics 2 and 3 are equivalent, but 3 is more robust

:seealso: :func:`UnitQuaternion.angdist`

"""

if metric < 5:

from spatialmath.quaternion import UnitQuaternion

return UnitQuaternion(self).angdist(UnitQuaternion(other), metric=metric)

elif metric == 5:

op = lambda R1, R2: np.linalg.norm(np.eye(3) - R1 @ R2.T)

elif metric == 6:

op = lambda R1, R2: base.norm(base.trlog(R1 @ R2.T, twist=True))

else:

raise ValueError('unknown metric')

ad = self._op2(other, op)

if isinstance(ad, list):

return np.array(ad)

else:

return ad</div></div>

# ============================== SE3 =====================================#

<div class="viewcode-block" id="SE3"><a class="viewcode-back" href="../../3d_pose_SE3.html#spatialmath.pose3d.SE3">[docs]</a>class SE3(SO3):

"""

SE(3) matrix class

This subclass represents rigid-body motion in 3D space. Internally it is a

4x4 homogeneous transformation matrix belonging to the group SE(3).

.. inheritance-diagram:: spatialmath.pose3d.SE3

:top-classes: collections.UserList

:parts: 1

"""

<div class="viewcode-block" id="SE3.__init__"><a class="viewcode-back" href="../../3d_pose_SE3.html#spatialmath.pose3d.SE3.__init__">[docs]</a> def __init__(self, x=None, y=None, z=None, *, check=True):

"""

Construct new SE(3) object

:rtype: SE3 instance

There are multiple call signatures that return an ``SE3`` instance

with one or more values.

- ``SE3()`` null motion, value is the identity matrix.

- ``SE3(x, y, z)`` is a pure translation of (x,y,z)

- ``SE3(T)`` where ``T`` is a 4x4 Numpy array representing an SE(3)

matrix. If ``check`` is ``True`` check the matrix belongs to SE(3).

- ``SE3([T1, T2, ... TN])`` has ``N`` values

given by the elements ``Ti`` each of which is a 4x4 NumPy array

representing an SE(3) matrix. If ``check`` is ``True`` check the

matrix belongs to SE(3).

- ``SE3(X)`` where ``X`` is:

- ``SE3`` is a copy of ``X``

- ``SO3`` is the rotation of ``X`` with zero translation

- ``SE2`` is the z-axis rotation and x- and y-axis translation of

``X``

- ``SE3([X1, X2, ... XN])`` has ``N`` values

given by the elements ``Xi`` each of which is an SE3 instance.

:SymPy: supported

"""

if y is None and z is None:

# just one argument passed

if super().arghandler(x, check=check):

return

elif isinstance(x, SO3):

self.data = [base.r2t(_x) for _x in x.data]

elif type(x).__name__ == 'SE2':

def convert(x):

# convert SE(2) to SE(3)

out = np.identity(4, dtype=x.dtype)

out[:2,:2] = x[:2,:2]

out[:2,3] = x[:2,2]

return out

self.data = [convert(_x) for _x in x.data]

elif base.isvector(x, 3):

# SE3( [x, y, z] )

elif isinstance(x, np.ndarray) and x.shape[1] == 3:

# SE3( Nx3 )

self.data = [base.transl(T) for T in x]

else:

raise ValueError('bad argument to constructor')

elif y is not None and z is not None:

# SE3(x, y, z)

self.data = [base.transl(x, y, z)]</div>

@staticmethod

def _identity():

return np.eye(4)

# ------------------------------------------------------------------------ #

@property

def shape(self):

"""

Shape of the object's internal matrix representation

:return: (4,4)

:rtype: tuple

Each value within the ``SE3`` instance is a NumPy array of this shape.

"""

return (4, 4)

@property

def t(self):

"""

Translational component of SE(3)

:return: translational component of SE(3)

:rtype: numpy.ndarray

``x.t`` is the translational component of ``x`` as an array with

shape (3,). If ``len(x) > 1``, return an array with shape=(N,3).

.. runblock:: pycon

>>> from spatialmath import UnitQuaternion

>>> x = SE3(1,2,3)

>>> x.t

array([1., 2., 3.])

>>> x = SE3([ SE3(1,2,3), SE3(4,5,6)])

>>> x.t

array([[1., 2., 3.],

[4., 5., 6.]])

:SymPy: supported

"""

else:

return np.array([x[:3, 3] for x in self.A])

@t.setter

def t(self, v):

if len(self) > 1:

raise ValueError("can only assign translation to length 1 object")

v = base.getvector(v, 3)

self.A[:3, 3] = v

# ------------------------------------------------------------------------ #

<div class="viewcode-block" id="SE3.inv"><a class="viewcode-back" href="../../3d_pose_SE3.html#spatialmath.pose3d.SE3.inv">[docs]</a> def inv(self):

r"""

Inverse of SE(3)

:return: inverse

:rtype: SE3 instance

Efficiently compute the inverse of each of the SE(3) values taking into

account the matrix structure.

.. math::

T = \left[ \begin{array}{cc} \mat{R} & \vec{t} \\ 0 & 1 \end{array} \right],

\mat{T}^{-1} = \left[ \begin{array}{cc} \mat{R}^T & -\mat{R}^T \vec{t} \\ 0 & 1 \end{array} \right]`

Example::

>>> x = SE3(1,2,3)

>>> x.inv()

SE3(array([[ 1., 0., 0., -1.],

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