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# Part of Spatial Math Toolbox for Python
# Copyright (c) 2000 Peter Corke
# MIT Licence, see details in top-level file: LICENCE
import numpy as np
import spatialmath.base as smb
from spatialmath.baseposelist import BasePoseList
from spatialmath.geom3d import Line3
class BaseTwist(BasePoseList):
"""
Superclass for 3D and 2D twist objects
Subclasses are:
- ``Twist3`` representing rigid-body motion in 3D as a 6-vector
- ``Twist2`` representing rigid-body motion in 2D as a 3-vector
A twist is the unique elements of the logarithm of the corresponding SE(N)
matrix.
Arithmetic operators are overloaded but the operation they perform depend
on the types of the operands. For example:
- ``*`` will compose two instances of the same subclass, and the result will be
an instance of the same subclass, since this is a group operator.
These classes all inherit from ``UserList`` which enables them to
represent a sequence of values, ie. a ``Twist3`` instance can contain
a sequence of twists. Most of the Python ``list`` operators
are applicable:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> x = Twist3() # new instance with zero value
>>> len(x) # it is a sequence of one value
>>> x.append(x) # append to itself
>>> len(x) # it is a sequence of two values
>>> x[1] # the element has a 4x4 matrix value
>>> x[1] = SE3.Rx(0.3).Twist3() # set an elements of the sequence
>>> x.reverse() # reverse the elements in the sequence
>>> del x[1] # delete an element
:References:
- "Mechanics, planning and control"
Park & Lynch, Cambridge, 2016.
This class is subclassed for the 3D and 2D cases
.. inheritance-diagram:: spatialmath.twist.Twist3 spatialmath.twist.Twist2
:top-classes: collections.UserList
:parts: 2
"""
def __init__(self):
super().__init__() # enable UserList superpowers
@property
def S(self):
"""
Twist as a vector (superclass property)
:return: Twist vector
:rtype: ndarray(N)
- ``X.S`` is a 3-vector if X is a ``Twist2`` instance, and a 6-vector if
X is a ``Twist3`` instance.
.. note::
- the vector is the unique elements of the se(N) representation.
- the vector is sometimes referred to as the twist coordinate vector.
- if ``len(X)`` > 1 then return a list of vectors.
"""
# get the underlying numpy array
if len(self.data) == 1:
return self.data[0]
else:
return self.data
@property
def isprismatic(self):
r"""
Test for prismatic twist (superclass property)
:return: Whether twist is purely prismatic
:rtype: bool
A prismatic twist has :math:`\vec{\omega} = 0`.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> x = Twist3.UnitPrismatic([1,2,3])
>>> x.isprismatic
>>> x = Twist3.UnitRevolute([1,2,3], [4,5,6])
>>> x.isprismatic
"""
if len(self) == 1:
return smb.iszerovec(self.w)
else:
return [smb.iszerovec(x.w) for x in self.data]
@property
def isrevolute(self):
r"""
Test for revolute twist (superclass property)
:return: Whether twist is purely revolute
:rtype: bool
A revolute twist has :math:`\vec{v} = 0`.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> x = Twist3.UnitPrismatic([1,2,3])
>>> x.isrevolute
>>> x = Twist3.UnitRevolute([1,2,3], [0,0,0])
>>> x.isrevolute
"""
if len(self) == 1:
return smb.iszerovec(self.v)
else:
return [smb.iszerovec(x.v) for x in self.data]
@property
def isunit(self):
r"""
Test for unit twist (superclass property)
:return: Whether twist is a unit-twist
:rtype: bool
A unit twist is one with a norm of 1, ie. :math:`\| S \| = 1`.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> S = Twist3([1,2,3,4,5,6])
>>> S.isunit()
>>> S = Twist3.UnitRevolute([1,2,3], [4,5,6])
>>> S.isunit()
"""
if len(self) == 1:
return smb.isunitvec(self.S)
else:
return [smb.isunitvec(x) for x in self.data]
@property
def theta(self):
"""
Twist angle (superclass method)
:return: magnitude of rotation (1x1) about the twist axis in radians
:rtype: float
"""
if self.N == 2:
return abs(self.w)
else:
return smb.norm(np.array(self.w))
def inv(self):
"""
Inverse of Twist (superclass method)
:return: inverse
:rtype: Twist instance
Compute the inverse of each of the values within the twist instance.
The inverse is the negative of the twist vector.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> S = Twist3(SE3.Rand())
>>> S
>>> S.inv()
>>> S * S.inv()
"""
return self.__class__([-t for t in self.data])
def prod(self):
r"""
Product of twists (superclass method)
:return: Product of elements
:rtype: Twist2 or Twist3
For a twist instance with N values return the matrix product of those
elements :math:`\prod_i=0^{N-1} S_i`.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> S = Twist3.Rx([0.2, 0.3, 0.4])
>>> len(S)
>>> S.prod()
>>> Twist3.Rx(0.9)
"""
if self.N == 2:
log = smb.trlog2
exp = smb.trexp2
else:
log = smb.trlog
exp = smb.trexp
twprod = exp(self.data[0])
for tw in self.data[1:]:
twprod = twprod @ exp(tw)
return self.__class__(log(twprod))
def __eq__(left, right): # lgtm[py/not-named-self] pylint: disable=no-self-argument
"""
Overloaded ``==`` operator (superclass method)
:return: Equality of two operands
:rtype: bool or list of bool
``S1 == S2`` is True if ``S1` is elementwise equal to ``S2``.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist2
>>> S1 = Twist3([1,2,3,4,5,6])
>>> S2 = Twist3([1,2,3,4,5,6])
>>> S1 == S2
>>> S2 = Twist3([1,2,3,4,5,7])
>>> S1 == S2
:seealso: :func:`__ne__`
"""
if type(left) != type(right):
raise TypeError("operands to == are of different types")
return left.binop(right, lambda x, y: all(x == y), list1=False)
def __ne__(left, right): # lgtm[py/not-named-self] pylint: disable=no-self-argument
"""
Overloaded ``!=`` operator (superclass method)
:rtype: bool
``S1 == S2`` is True if ``S1` is not elementwise equal to ``S2``.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> S1 = Twist3([1,2,3,4,5,6])
>>> S2 = Twist3([1,2,3,4,5,6])
>>> S1 != S2
>>> S2 = Twist3([1,2,3,4,5,7])
>>> S1 != S2
:seealso: :func:`__ne__`
"""
if type(left) != type(right):
raise TypeError("operands to != are of different types")
return left.binop(right, lambda x, y: not all(x == y), list1=False)
def __truediv__(
left, right
): # lgtm[py/not-named-self] pylint: disable=no-self-argument
if smb.isscalar(right):
return left.__class__(left.S / right)
else:
raise ValueError("Twist /, incorrect right operand")
# ======================================================================== #
class Twist3(BaseTwist):
r"""
3D twist class
A Twist class holds the parameters of a twist, a representation of a
3D rigid body transformation which is the unique elements of the Lie
algebra se(3) of the corresponding SE(3) matrix.
:References:
- Robotics, Vision & Control for Python, Section 2.3.2.3, P. Corke, Springer 2023.
- Modern Robotics, Lynch & Park, Cambridge 2017
.. note:: Compared to Lynch & Park this module implements twist vectors
with the translational components first, followed by rotational
components, ie. :math:`[\omega, \vec{v}]`.
"""
def __init__(self, arg=None, w=None, check=True):
"""
Construct a new 3D twist object
- ``Twist3()`` is a Twist3 instance representing null motion -- the
identity twist
- ``Twist3(S)`` is a Twist3 instance from an array-like (6,)
- ``Twist3(v, w)`` is a Twist3 instance from a moment ``v`` (3,) and
direction ``w`` (3,)
- ``Twist3([S1, S2, ... SN])`` where each ``Si`` is a numpy array (6,)
- ``Twist3(X)`` is a Twist3 instance with the same value as ``X``, ie.
a copy
- ``Twist3([X1, X2, ... XN])`` where each Xi is a Twist3 instance, is a
Twist3 instance containing N motions
"""
from spatialmath.pose3d import SE3
super().__init__()
if w is None:
# zero or one arguments passed
if super().arghandler(arg, check=check):
return
elif isinstance(arg, SE3):
self.data = [arg.twist().A]
elif w is not None and smb.isvector(w, 3) and smb.isvector(arg, 3):
# Twist(v, w)
self.data = [np.r_[arg, w]]
return
else:
raise ValueError("bad value to Twist constructor")
# ------------------------ SMUserList required ---------------------------#
@staticmethod
def _identity():
return np.zeros((6,))
def _import(self, value, check=True):
if isinstance(value, np.ndarray) and self.isvalid(value, check=check):
if value.shape == (4, 4):
# it's an se(3)
return smb.vexa(value)
elif value.shape == (6,):
# it's a twist vector
return value
elif smb.ishom(value, check=check):
return smb.trlog(value, twist=True, check=False)
raise TypeError("bad type passed")
@staticmethod
def isvalid(v, check=True):
"""
Test if matrix is valid twist
:param x: array to test
:type x: ndarray
:return: Whether the value is a 6-vector or a valid 4x4 se(3) element
:rtype: bool
A twist can be represented by a 6-vector or a 4x4 skew symmetric matrix,
for example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> from spatialmath.base import skewa
>>> import numpy as np
>>> Twist3.isvalid([1, 2, 3, 4, 5, 6])
>>> a = skewa([1, 2, 3, 4, 5, 6])
>>> a
>>> Twist3.isvalid(a)
>>> Twist3.isvalid(np.random.rand(4,4))
"""
if smb.isvector(v, 6):
return True
elif smb.ismatrix(v, (4, 4)):
# maybe be an se(3)
if not smb.iszerovec(v.diagonal()): # check diagonal is zero
return False
if not smb.iszerovec(v[3, :]): # check bottom row is zero
return False
if check and not smb.isskew(v[:3, :3]):
# top left 3x3 is skew symmetric
return False
return True
return False
# ------------------------ properties ---------------------------#
@property
def shape(self):
"""
Shape of the object's internal array representation
:return: (6,)
:rtype: tuple
"""
return (6,)
@property
def N(self):
"""
Dimension of the object's group
:return: dimension
:rtype: int
Dimension of the group is 3 for ``Twist3`` and corresponds to the
dimension of the space (3D in this case) to which these
rigid-body motions apply.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> x = Twist3()
>>> x.N
"""
return 3
@property
def v(self):
"""
Moment vector of twist
:return: Moment vector
:rtype: ndarray(3)
``X.v`` is a 3-vector representing the moment vector of the twist.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> t = Twist3([1, 2, 3, 4, 5, 6])
>>> t.v
"""
return self.data[0][:3]
@property
def w(self):
"""
Direction vector of twist
:return: Direction vector
:rtype: ndarray(3)
``X.w`` is a 3-vector representing the direction vector of the twist.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> t = Twist3([1, 2, 3, 4, 5, 6])
>>> t.w
"""
return self.data[0][3:6]
# -------------------- variant constructors ----------------------------#
@classmethod
def UnitRevolute(cls, a, q, pitch=None):
"""
Construct a new 3D rotational unit twist
:param a: Twist axis or line of action
:type a: array_like(3)
:param q: Point on the line of action
:type q: array_like(3)
:param p: pitch, defaults to None
:type p: float, optional
:return: a rotational or helical twist
:rtype: Twist instance
A revolute twist with a line of action in the z-direction and passing
through (1, 2, 0) would be:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> Twist3.Revolute([0, 0, 1], [1, 2, 0])
"""
w = smb.unitvec(smb.getvector(a, 3))
v = -np.cross(w, smb.getvector(q, 3))
if pitch is not None:
v = v + pitch * w
return cls(v, w)
@classmethod
def UnitPrismatic(cls, a):
"""
Construct a new 3D unit prismatic twist
:param a: Twist axis or line of action
:type a: array_like(3)
:return: a prismatic twist
:rtype: Twist instance
A prismatic twist with a line of action in the z-direction would be:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> Twist3.Prismatic([0, 0, 1])
"""
w = np.r_[0, 0, 0]
v = smb.unitvec(smb.getvector(a, 3))
return cls(v, w)
@classmethod
def Rx(cls, theta, unit="rad"):
"""
Create a new 3D twist for pure rotation about the X-axis
:param θ: rotation angle about X-axis
:type θ: float
:param unit: angular units: 'rad' [default], or 'deg'
:type unit: str
:return: 3D twist vector
:rtype: Twist3 instance
- ``Twist3.Rx(θ)`` is an SE(3) rotation of θ radians about the x-axis
- ``Twist3.Rx(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 Twist3
>>> Twist3.Rx(0.3)
>>> Twist3.Rx([0.3, 0.4])
:seealso: :func:`~spatialmath.smb.transforms3d.trotx`
:SymPy: supported
"""
return cls([np.r_[0, 0, 0, x, 0, 0] for x in smb.getunit(theta, unit=unit)])
@classmethod
def Ry(cls, theta, unit="rad", t=None):
"""
Create a new 3D twist for pure rotation about the Y-axis
:param θ: rotation angle about X-axis
:type θ: float
:param unit: angular units: 'rad' [default], or 'deg'
:type unit: str
:return: 3D twist vector
:rtype: Twist3 instance
- ``Twist3.Ry(θ)`` is an SO(3) rotation of θ radians about the y-axis
- ``Twist3.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 Twist3
>>> Twist3.Ry(0.3)
>>> Twist3.Ry([0.3, 0.4])
:seealso: :func:`~spatialmath.smb.transforms3d.troty`
:SymPy: supported
"""
return cls([np.r_[0, 0, 0, 0, x, 0] for x in smb.getunit(theta, unit=unit)])
@classmethod
def Rz(cls, theta, unit="rad", t=None):
"""
Create a new 3D twist for pure rotation about the Z-axis
:param θ: rotation angle about Z-axis
:type θ: float
:param unit: angular units: 'rad' [default], or 'deg'
:type unit: str
:return: 3D twist vector
:rtype: Twist3 instance
- ``Twist3.Rz(θ)`` is an SO(3) rotation of θ radians about the z-axis
- ``Twist3.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 Twist3
>>> Twist3.Rz(0.3)
>>> Twist3.Rz([0.3, 0.4])
:seealso: :func:`~spatialmath.smb.transforms3d.trotz`
:SymPy: supported
"""
return cls([np.r_[0, 0, 0, 0, 0, x] for x in smb.getunit(theta, unit=unit)])
@classmethod
def RPY(cls, *pos, **kwargs):
r"""
Create a new 3D twist from roll-pitch-yaw angles
:param 𝚪: roll-pitch-yaw angles
:type 𝚪: array_like or numpy.ndarray with shape=(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: 3D twist vector
:rtype: Twist3 instance
- ``Twist3.RPY(𝚪)`` is a 3D rotation defined by a 3-vector of roll,
pitch, yaw angles :math:`\Gamma=(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. This is the **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. This is the **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. This is the **convention** for a camera with z-axis parallel
to the optical axis and x-axis parallel to the pixel rows.
If ``𝚪`` is an Nx3 matrix then the result is a sequence of rotations each defined by RPY angles
corresponding to the rows of ``𝚪``.
- ``Twist3.RPY(⍺, β, 𝛾)`` as above but the angles are provided as three
scalars.
Foo bar!
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> Twist3.RPY(0.1, 0.2, 0.3)
>>> Twist3.RPY([0.1, 0.2, 0.3])
>>> Twist3.RPY(0.1, 0.2, 0.3, order='xyz')
>>> Twist3.RPY(10, 20, 30, unit='deg')
:seealso: :meth:`~spatialmath.SE3.RPY`
:SymPy: supported
"""
from spatialmath.pose3d import SE3
T = SE3.RPY(*pos, **kwargs)
return cls(T)
@classmethod
def Tx(cls, x):
"""
Create a new 3D twist for pure translation along the X-axis
:param x: translation distance along the X-axis
:type x: float
:return: 3D twist vector
:rtype: Twist3 instance
``Twist3.Tx(x)`` is an se(3) translation of ``x`` along the x-axis
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> Twist3.Tx(2)
>>> Twist3.Tx([2,3])
:seealso: :func:`~spatialmath.smb.transforms3d.transl`
:SymPy: supported
"""
return cls([np.r_[_x, 0, 0, 0, 0, 0] for _x in smb.getvector(x)], check=False)
@classmethod
def Ty(cls, y):
"""
Create a new 3D twist for pure translation along the Y-axis
:param y: translation distance along the Y-axis
:type y: float
:return: 3D twist vector
:rtype: Twist3 instance
``Twist3.Ty(y)`` is an se(3) translation of ``y`` along the y-axis
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> Twist3.Ty(2)
>>> Twist3.Ty([2, 3])
:seealso: :func:`~spatialmath.smb.transforms3d.transl`
:SymPy: supported
"""
return cls([np.r_[0, _y, 0, 0, 0, 0] for _y in smb.getvector(y)], check=False)
@classmethod
def Tz(cls, z):
"""
Create a new 3D twist for pure translation along the Z-axis
:param z: translation distance along the Z-axis
:type z: float
:return: 3D twist vector
:rtype: Twist3 instance
``Twist3.Tz(z)`` is an se(3) translation of ``z`` along the z-axis
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> Twist3.Tz(2)
>>> Twist3.Tz([2, 3])
:seealso: :func:`~spatialmath.smb.transforms3d.transl`
:SymPy: supported
"""
return cls([np.r_[0, 0, _z, 0, 0, 0] for _z in smb.getvector(z)], check=False)
@classmethod
def Rand(
cls, *, xrange=(-1, 1), yrange=(-1, 1), zrange=(-1, 1), N=1
): # pylint: disable=arguments-differ
"""
Create a new random 3D twist
:param xrange: x-axis range [min,max], defaults to [-1, 1]
:type xrange: 2-element sequence, optional
:param yrange: y-axis range [min,max], defaults to [-1, 1]
:type yrange: 2-element sequence, optional
:param zrange: z-axis range [min,max], defaults to [-1, 1]
:type zrange: 2-element sequence, optional
:param N: number of random transforms
:type N: int
:return: SE(3) matrix
:rtype: SE3 instance
Return an SE3 instance with random rotation and translation.
- ``SE3.Rand()`` is a random SE(3) translation.
- ``SE3.Rand(N=N)`` is an SE3 object containing a sequence of N random
poses.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> Twist3.Rand(N=2)
:seealso: :func:`~spatialmath.quaternions.UnitQuaternion.Rand`
"""
from spatialmath.pose3d import SO3
X = np.random.uniform(
low=xrange[0], high=xrange[1], size=N
) # random values in the range
Y = np.random.uniform(
low=yrange[0], high=yrange[1], size=N
) # random values in the range
Z = np.random.uniform(
low=yrange[0], high=zrange[1], size=N
) # random values in the range
R = SO3.Rand(N=N)
def _twist(x, y, z, r):
T = smb.transl(x, y, z) @ smb.r2t(r.A)
return smb.trlog(T, twist=True)
return cls(
[_twist(x, y, z, r) for (x, y, z, r) in zip(X, Y, Z, R)], check=False
)
# ------------------------- methods -------------------------------#
def printline(self, **kwargs):
return self.SE3().printline(**kwargs)
def unit(self):
"""
Unit twist
- ``S.unit()`` is a Twist2 objec3 representing a unit twist aligned with the
Twist ``S``.
Example:
.. runblock:: pycon
>>> from spatialmath import SE3, Twist3
>>> T = SE3(1, 2, 0.3)
>>> S = Twist3(T)
>>> S.unit()
"""
if smb.iszerovec(self.w):
# rotational twist
return Twist3(self.S / smb.norm(S.w))
else:
# prismatic twist
return Twist3(smb.unitvec(self.v), [0, 0, 0])
def ad(self):
"""
Logarithm of adjoint of 3D twist
:return: logarithm of adjoint matrix
:rtype: ndarray(6,6)
``S.ad()`` is the 6x6 logarithm of the adjoint matrix of the
corresponding homogeneous transformation.
For a twist representing motion from frame {B} to {A}, the adjoint will
transform a twist relative to frame {A} to one relative to frame {B}.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> S = Twist3.Rx(0.3)
>>> S.ad()
.. note:: An alternative approach to computing the adjoint is to exponentiate this 6x6
matrix.
:seealso: :func:`Twist3.Ad`
"""
return np.block(
[
[smb.skew(self.w), smb.skew(self.v)],
[np.zeros((3, 3)), smb.skew(self.w)],
]
)
def Ad(self):
"""
Adjoint of 3D twist
:return: adjoint matrix
:rtype: ndarray(6,6)
``S.Ad()`` is the 6x6 adjoint matrix of the corresponding
homogeneous transformation.
For a twist representing motion from frame {B} to {A}, the adjoint will
transform a twist relative to frame {A} to one relative to frame {B}.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> S = Twist3.Rx(0.3)
>>> S.Ad()
.. note:: Equivalent to, but faster than, ``self.SE3().Ad()`` -- computes
the adjoint directly from the twist's exponential without
constructing an intermediate ``SE3`` instance.
:seealso: :func:`Twist3.ad`, :func:`Twist3.SE3`, :func:`Twist3.exp`
"""
return smb.tr2adjoint(smb.trexp(self.S, check=False))
def skewa(self):
"""
Convert 3D twist to se(3)
:return: An se(3) matrix
:rtype: ndarray(4,4)
``X.skewa()`` is the twist as a 4x4 augmented skew-symmetric matrix
belonging to the group se(3). This is the Lie algebra of the
corresponding SE(3) element.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3, base
>>> S = Twist3.Rx(0.3)
>>> se = S.skewa()
>>> se
>>> smb.trexp(se)
"""
if len(self) == 1:
return smb.skewa(self.S)
else:
return [smb.skewa(x.S) for x in self]
@property
def pitch(self):
"""
Pitch of a 3D twist
:return: the pitch of the twist
:rtype: float
``X.pitch()`` is the pitch of the twist as a scalar in units of distance
per radian.
If we consider the twist as a screw, this is the distance of
translation along the screw axis for a one radian rotation about the
screw axis.
Example:
.. runblock:: pycon
>>> from spatialmath import SE3, Twist3
>>> T = SE3(1, 2, 3) * SE3.Rx(0.3)
>>> S = Twist3(T)
>>> S.pitch
"""
return np.dot(self.w, self.v)
def line(self):
"""
Line of action of 3D twist as a Plucker line
:return: the 3D line of action
:rtype: Line instance
``X.line()`` is a Plucker object representing the line of the twist axis.
Example:
.. runblock:: pycon
>>> from spatialmath import SE3, Twist3
>>> T = SE3(1, 2, 3) * SE3.Rx(0.3)
>>> S = Twist3(T)
>>> S.line()
"""
return Line3([Line3(-tw.v + tw.pitch * tw.w, tw.w) for tw in self])
@property
def pole(self):
"""
Pole of a 3D twist
:return: the pole of the twist
:rtype: ndarray(3)
``X.pole()`` is a point on the twist axis. For a pure translation
this point is at infinity.
Example:
.. runblock:: pycon
>>> from spatialmath import SE3, Twist3
>>> T = SE3(1, 2, 3) * SE3.Rx(0.3)
>>> S = Twist3(T)
>>> S.pole
"""
return np.cross(self.w, self.v) / self.theta
def SE3(self, theta=1, unit="rad"):
"""
Convert 3D twist to SE(3) matrix
:return: an SE(3) representation
:rtype: SE3 instance
``S.SE3()`` is an SE3 object representing the homogeneous transformation