se(3) twist

class Twist3(arg=None, w=None, check=True)[source]

Bases: BaseTwist

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. \([\omega, \vec{v}]\).

Ad()[source]

Adjoint of 3D twist

Returns:: adjoint matrix
Return type:: 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:

>>> from spatialmath import Twist3
>>> S = Twist3.Rx(0.3)
>>> S.Ad()
array([[ 1.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
       [ 0.    ,  0.9553, -0.2955,  0.    ,  0.    ,  0.    ],
       [ 0.    ,  0.2955,  0.9553,  0.    ,  0.    ,  0.    ],
       [ 0.    ,  0.    ,  0.    ,  1.    ,  0.    ,  0.    ],
       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.9553, -0.2955],
       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.2955,  0.9553]])

Note

This method computes the equivalent SE(3) matrix, then the adjoint of that.

Seealso:: Twist3.ad(), Twist3.SE3(), Twist3.exp()

classmethod Alloc(n=1)

Construct an instance with N default values (BasePoseList superclass method)

Parameters:: n (int, optional) – Number of values, defaults to 1
Return type:: Self
Returns:: pose instance with n default values

X.Alloc(N) creates an instance of the pose class X with N default values, ie. len(X) will be N.

X can be considered a vector of pose objects, and those elements can be referenced X[i] or assigned to X[i] = ....

Note

The default value depends on the pose class and is the result of the empty constructor. For SO2, SE2, SO3, SE3 it is an identity matrix, for a twist class Twist2 or Twist3 it is a zero vector, for a UnitQuaternion or Quaternion it is a zero vector.

Example:

>>> x = X.Alloc(10)
>>> len(x)
10

where X is any of the SMTB classes.

classmethod Empty()

Construct an empty instance (BasePoseList superclass method)

Return type:: Self
Returns:: pose instance with zero values

Example:

>>> x = X.Empty()
>>> len(x)
0

where X is any of the SMTB classes.

classmethod RPY(*pos, **kwargs)[source]

Create a new 3D twist from roll-pitch-yaw angles

Parameters:

𝚪 (array_like or numpy.ndarray with shape=(N,3)) – roll-pitch-yaw angles
unit (str) – angular units: ‘rad’ [default], or ‘deg’
order (str) – rotation order: ‘zyx’ [default], ‘xyz’, or ‘yxz’

Returns:

3D twist vector

Return type:

Twist3 instance

Twist3.RPY(𝚪) is a 3D rotation defined by a 3-vector of roll, pitch, yaw angles \(\Gamma=(r, p, y)\) 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:

>>> from spatialmath import Twist3
>>> Twist3.RPY(0.1, 0.2, 0.3)
Twist3([0, 0, 0, 0.068925, 0.21323, 0.28875])
>>> Twist3.RPY([0.1, 0.2, 0.3])
Twist3([0, 0, 0, 0.068925, 0.21323, 0.28875])
>>> Twist3.RPY(0.1, 0.2, 0.3, order='xyz')
Twist3([0, 0, 0, 0.30875, 0.18343, 0.12892])
>>> Twist3.RPY(10, 20, 30, unit='deg')
Twist3([0, 0, 0, 0.077525, 0.38485, 0.48648])

Seealso:: RPY()
SymPy:: supported

classmethod Rand(*, xrange=(-1, 1), yrange=(-1, 1), zrange=(-1, 1), N=1)[source]

Create a new random 3D twist

Parameters:

xrange (2-element sequence, optional) – x-axis range [min,max], defaults to [-1, 1]
yrange (2-element sequence, optional) – y-axis range [min,max], defaults to [-1, 1]
zrange (2-element sequence, optional) – z-axis range [min,max], defaults to [-1, 1]
N (int) – number of random transforms

Returns:

SE(3) matrix

Return type:

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:

>>> from spatialmath import Twist3
>>> Twist3.Rand(N=2)
Twist3([
  [0.70508, -0.42951, -0.75073, -1.2984, 1.9579, -1.9477],
  [0.95669, 0.20993, 0.56213, 0.14053, -1.0451, -0.21306]
])

Seealso:: Rand()

classmethod Rx(theta, unit='rad')[source]

Create a new 3D twist for pure rotation about the X-axis

Parameters:

θ (float) – rotation angle about X-axis
unit (str) – angular units: ‘rad’ [default], or ‘deg’

Returns:

3D twist vector

Return type:

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:

>>> from spatialmath import Twist3
>>> Twist3.Rx(0.3)
Twist3([0, 0, 0, 0.3, 0, 0])
>>> Twist3.Rx([0.3, 0.4])
Twist3([
  [0, 0, 0, 0.3, 0, 0],
  [0, 0, 0, 0.4, 0, 0]
])

Seealso:: trotx()
SymPy:: supported

classmethod Ry(theta, unit='rad', t=None)[source]

Create a new 3D twist for pure rotation about the Y-axis

Parameters:

θ (float) – rotation angle about X-axis
unit (str) – angular units: ‘rad’ [default], or ‘deg’

Returns:

3D twist vector

Return type:

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:

>>> from spatialmath import Twist3
>>> Twist3.Ry(0.3)
Twist3([0, 0, 0, 0, 0.3, 0])
>>> Twist3.Ry([0.3, 0.4])
Twist3([
  [0, 0, 0, 0, 0.3, 0],
  [0, 0, 0, 0, 0.4, 0]
])

Seealso:: troty()
SymPy:: supported

classmethod Rz(theta, unit='rad', t=None)[source]

Create a new 3D twist for pure rotation about the Z-axis

Parameters:

θ (float) – rotation angle about Z-axis
unit (str) – angular units: ‘rad’ [default], or ‘deg’

Returns:

3D twist vector

Return type:

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:

>>> from spatialmath import Twist3
>>> Twist3.Rz(0.3)
Twist3([0, 0, 0, 0, 0, 0.3])
>>> Twist3.Rz([0.3, 0.4])
Twist3([
  [0, 0, 0, 0, 0, 0.3],
  [0, 0, 0, 0, 0, 0.4]
])

Seealso:: trotz()
SymPy:: supported

SE3(theta=1, unit='rad')[source]

Convert 3D twist to SE(3) matrix

Returns:: an SE(3) representation
Return type:: SE3 instance

S.SE3() is an SE3 object representing the homogeneous transformation equivalent to the Twist3. This is the exponentiation of the twist vector.

Example:

>>> from spatialmath import Twist3
>>> S = Twist3.Rx(0.3)
>>> S.SE3()
SE3(array([[ 1.    ,  0.    ,  0.    ,  0.    ],
           [ 0.    ,  0.9553, -0.2955,  0.    ],
           [ 0.    ,  0.2955,  0.9553,  0.    ],
           [ 0.    ,  0.    ,  0.    ,  1.    ]]))

Seealso:: Twist3.exp()

classmethod Tx(x)[source]

Create a new 3D twist for pure translation along the X-axis

Parameters:: x (float) – translation distance along the X-axis
Returns:: 3D twist vector
Return type:: Twist3 instance

Twist3.Tx(x) is an se(3) translation of x along the x-axis

Example:

>>> from spatialmath import Twist3
>>> Twist3.Tx(2)
Twist3([2, 0, 0, 0, 0, 0])
>>> Twist3.Tx([2,3])
Twist3([
  [2, 0, 0, 0, 0, 0],
  [3, 0, 0, 0, 0, 0]
])

Seealso:: transl()
SymPy:: supported

classmethod Ty(y)[source]

Create a new 3D twist for pure translation along the Y-axis

Parameters:: y (float) – translation distance along the Y-axis
Returns:: 3D twist vector
Return type:: Twist3 instance

Twist3.Ty(y) is an se(3) translation of y along the y-axis

Example:

>>> from spatialmath import Twist3
>>> Twist3.Ty(2)
Twist3([0, 2, 0, 0, 0, 0])
>>> Twist3.Ty([2, 3])
Twist3([
  [0, 2, 0, 0, 0, 0],
  [0, 3, 0, 0, 0, 0]
])

Seealso:: transl()
SymPy:: supported

classmethod Tz(z)[source]

Create a new 3D twist for pure translation along the Z-axis

Parameters:: z (float) – translation distance along the Z-axis
Returns:: 3D twist vector
Return type:: Twist3 instance

Twist3.Tz(z) is an se(3) translation of z along the z-axis

Example:

>>> from spatialmath import Twist3
>>> Twist3.Tz(2)
Twist3([0, 0, 2, 0, 0, 0])
>>> Twist3.Tz([2, 3])
Twist3([
  [0, 0, 2, 0, 0, 0],
  [0, 0, 3, 0, 0, 0]
])

Seealso:: transl()
SymPy:: supported

classmethod UnitPrismatic(a)[source]

Construct a new 3D unit prismatic twist

Parameters:: a (array_like(3)) – Twist axis or line of action
Returns:: a prismatic twist
Return type:: Twist instance

A prismatic twist with a line of action in the z-direction would be:

classmethod UnitRevolute(a, q, pitch=None)[source]

Construct a new 3D rotational unit twist

Parameters:

a (array_like(3)) – Twist axis or line of action
q (array_like(3)) – Point on the line of action
p (float, optional) – pitch, defaults to None

Returns:

a rotational or helical twist

Return type:

Twist instance

A revolute twist with a line of action in the z-direction and passing through (1, 2, 0) would be:

__eq__(right)

Overloaded == operator (superclass method)

Returns:: Equality of two operands
Return type:: bool or list of bool

S1 == S2 is True if S1` is elementwise equal to ``S2.

Example:

  File "<input>", line 1, in <module>
NameError: name 'Twist3' is not defined
Traceback (most recent call last):
  File "<input>", line 1, in <module>
NameError: name 'S1' is not defined
Traceback (most recent call last):
  File "<input>", line 1, in <module>
NameError: name 'Twist3' is not defined
Traceback (most recent call last):
  File "<input>", line 1, in <module>
NameError: name 'S1' is not defined

Seealso:: __ne__()

__init__(arg=None, w=None, check=True)[source]

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

__mul__(right)[source]

Overloaded * operator

Parameters:

left – left multiplicand
right – right multiplicand

Returns:

product

Raises:

ValueError

Twist composition or scaling:

X * Y compounds the twists X and Y
X * s performs elementwise multiplication of the elements of X by s
s * X performs elementwise multiplication of the elements of X by s

Multiplicands		Product
left	right	type	operation
Twist3	Twist3	Twist3	product of exponentials
Twist3	scalar	Twist3	element-wise product
scalar	Twist3	Twist3	element-wise product
Twist3	SE3	Twist3	exponential x SE3

Note

scalar x Twist is handled by __rmul__

#. scalar multiplication is commutative but the result is not a group operation so the result will be a matrix #. Any other input combinations result in a ValueError.

For pose composition the left and right operands may be a sequence

len(left)	len(right)	len	operation
1	1	1	`prod = left * right`
1	M	M	`prod[i] = left * right[i]`
N	1	M	`prod[i] = left[i] * right`
M	M	M	`prod[i] = left[i] * right[i]`

__ne__(right)

Overloaded != operator (superclass method)

Return type:: bool

S1 == S2 is True if S1` is not elementwise equal to ``S2.

Example:

>>> from spatialmath import Twist3
>>> S1 = Twist3([1,2,3,4,5,6])
>>> S2 = Twist3([1,2,3,4,5,6])
>>> S1 != S2
False
>>> S2 = Twist3([1,2,3,4,5,7])
>>> S1 != S2
True

Seealso:: __ne__()

__truediv__(right)

ad()[source]

Logarithm of adjoint of 3D twist

Returns:: logarithm of adjoint matrix
Return type:: 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:

>>> from spatialmath import Twist3
>>> S = Twist3.Rx(0.3)
>>> S.ad()
array([[ 0. , -0. ,  0. ,  0. , -0. ,  0. ],
       [ 0. ,  0. , -0.3,  0. ,  0. , -0. ],
       [-0. ,  0.3,  0. , -0. ,  0. ,  0. ],
       [ 0. ,  0. ,  0. ,  0. , -0. ,  0. ],
       [ 0. ,  0. ,  0. ,  0. ,  0. , -0.3],
       [ 0. ,  0. ,  0. , -0. ,  0.3,  0. ]])

Note

An alternative approach to computing the adjoint is to exponentiate this 6x6 matrix.

Seealso:: Twist3.Ad()

append(item)

Append a value to an instance (BasePoseList superclass method)

Parameters:: x (Quaternion or UnitQuaternion instance) – the value to append
Raises:: ValueError – incorrect type of appended object
Return type:: None

Appends the argument to the object’s internal list of values.

Example:

>>> x = X.Alloc(10)
>>> len(x)
10
>>> x.append(X())   # append to the list
>>> len(x)
11

where X is any of the SMTB classes.

clear() → None -- remove all items from S

exp(theta=1, unit='rad')[source]

Exponentiate a 3D twist

Parameters:

theta (float, optional) – rotation magnitude, defaults to None
units (str, optional) – rotational units, defaults to ‘rad’

Returns:

SE(3) matrix

Return type:

SE3 instance

X.exp() is the homogeneous transformation equivalent to the twist, \(e^{[S]}\)
X.exp(θ) as above but with a rotation of ``θ about the twist axis, \(e^{ heta[S]}\)

If len(X)==1 and len(θ)==N then the resulting SE3 object has N values equivalent to the twist \(e^{ heta_i[S]}\).

If len(X)==N and len(θ)==1 then the resulting SE3 object has N values equivalent to the twist \(e^{ heta[S_i]}\).

If len(X)==N and len(θ)==N then the resulting SE3 object has N values equivalent to the twist \(e^{ heta_i[S_i]}\).

Example:

>>> from spatialmath import SE3, Twist3
>>> T = SE3(1, 2, 3) * SE3.Rx(0.3)
>>> S = Twist3(T)
>>> S.exp(0)
SE3(array([[1., 0., 0., 0.],
           [0., 1., 0., 0.],
           [0., 0., 1., 0.],
           [0., 0., 0., 1.]]))
>>> S.exp(1)
SE3(array([[ 1.    ,  0.    ,  0.    ,  1.    ],
           [ 0.    ,  0.9553, -0.2955,  2.    ],
           [ 0.    ,  0.2955,  0.9553,  3.    ],
           [ 0.    ,  0.    ,  0.    ,  1.    ]]))

Note

For the second form, the twist must, if rotational, have a unit rotational component.

Seealso:: spatialmath.smb.trexp()

extend(iterable)

Extend sequence of values in an instance (BasePoseList superclass method)

Parameters:: x (instance of same type) – the value to extend
Raises:: ValueError – incorrect type of appended object
Return type:: None

Appends the argument’s values to the object’s internal list of values.

Example:

>>> x = X.Alloc(10)
>>> len(x)
10
>>> x.append(X.Alloc(5))   # extend the list
>>> len(x)
15

where X is any of the SMTB classes.

insert(i, item)

Insert a value to an instance (BasePoseList superclass method)

Parameters:

i (int) – element to insert value before
item (instance of same type) – the value to insert

Raises:

ValueError – incorrect type of inserted value

Return type:

None

Inserts the argument into the object’s internal list of values.

Example:

>>> x = X.Alloc(10)
>>> len(x)
10
>>> x.insert(0, X())   # insert at start of list
>>> len(x)
11
>>> x.insert(10, X())   # append to the list
>>> len(x)
11

where X is any of the SMTB classes.

Note

If i is beyond the end of the list, the item is appended to the list

inv()

Inverse of Twist (superclass method)

Returns:: inverse
Return type:: Twist instance

Compute the inverse of each of the values within the twist instance. The inverse is the negative of the twist vector.

Example:

>>> from spatialmath import Twist3
>>> S = Twist3(SE3.Rand())
>>> S
Twist3([0.56869, -0.21742, 0.74689, 0.60958, 1.7309, 0.14935])
>>> S.inv()
Twist3([-0.56869, 0.21742, -0.74689, -0.60958, -1.7309, -0.14935])
>>> S * S.inv()
Twist3([2.2204e-16, -5.5511e-17, 7.6328e-17, 0, 0, 0])

static isvalid(v, check=True)[source]

Test if matrix is valid twist

Parameters:: x (ndarray) – array to test
Returns:: Whether the value is a 6-vector or a valid 4x4 se(3) element
Return type:: bool

A twist can be represented by a 6-vector or a 4x4 skew symmetric matrix, for example:

>>> from spatialmath import Twist3
>>> from spatialmath.base import skewa
>>> import numpy as np
>>> Twist3.isvalid([1, 2, 3, 4, 5, 6])
True
>>> a = skewa([1, 2, 3, 4, 5, 6])
>>> a
array([[ 0., -6.,  5.,  1.],
       [ 6.,  0., -4.,  2.],
       [-5.,  4.,  0.,  3.],
       [ 0.,  0.,  0.,  0.]])
>>> Twist3.isvalid(a)
True
>>> Twist3.isvalid(np.random.rand(4,4))
False

line()[source]

Line of action of 3D twist as a Plucker line

Returns:: the 3D line of action
Return type:: Line instance

X.line() is a Plucker object representing the line of the twist axis.

Example:

  File "/opt/hostedtoolcache/Python/3.8.18/x64/lib/python3.8/site-packages/spatialmath/twist.py", line 968, in <listcomp>
    return Line3([Line3(-tw.v + tw.pitch * tw.w, tw.w) for tw in self])
  File "/opt/hostedtoolcache/Python/3.8.18/x64/lib/python3.8/site-packages/spatialmath/geom3d.py", line 314, in __init__
    raise ValueError("invalid Plucker coordinates")
ValueError: invalid Plucker coordinates

pop(i=-1)

Pop value from an instance (BasePoseList superclass method)

Parameters:: i (int) – item in the list to pop, default is last
Returns:: the popped value
Return type:: instance of same type
Raises:: IndexError – if there are no values to pop

Removes a value from the value list and returns it. The original instance is modified.

Example:

>>> x = X.Alloc(10)
>>> len(x)
10
>>> y = x.pop()  # pop the last value x[9]
>>> len(x)
9
>>> y = x.pop(0)  # pop the first value x[0]
>>> len(x)
8

where X is any of the SMTB classes.

printline(**kwargs)[source]

prod()

Product of twists (superclass method)

Returns:: Product of elements
Return type:: Twist2 or Twist3

For a twist instance with N values return the matrix product of those elements \(\prod_i=0^{N-1} S_i\).

Example:

>>> from spatialmath import Twist3
>>> S = Twist3.Rx([0.2, 0.3, 0.4])
>>> len(S)
3
>>> S.prod()
Twist3([0, 0, 0, 0.9, 0, 0])
>>> Twist3.Rx(0.9)
Twist3([0, 0, 0, 0.9, 0, 0])

reverse(): S.reverse() – reverse IN PLACE

skewa()[source]

Convert 3D twist to se(3)

Returns:: An se(3) matrix
Return type:: 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:

unit()[source]

Unit twist

S.unit() is a Twist2 objec3 representing a unit twist aligned with the Twist S.

Example:

NameError: name 'S' is not defined

property A: List[ndarray[Any, dtype[ScalarType]]] | ndarray[Any, dtype[ScalarType]]

Array value of an instance (BasePoseList superclass method)

Returns:: NumPy array value of this instance
Return type:: ndarray

X.A is a NumPy array that represents the value of this instance, and has a shape given by X.shape.

Note

This assumes that len(X) == 1, ie. it is a single-valued instance.

property N

Dimension of the object’s group

Returns:: dimension
Return type:: 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:

>>> from spatialmath import Twist3
>>> x = Twist3()
>>> x.N
3

property S

Twist as a vector (superclass property)

Returns:: Twist vector
Return type:: 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.

property isprismatic

Test for prismatic twist (superclass property)

Returns:: Whether twist is purely prismatic
Return type:: bool

A prismatic twist has \(\vec{\omega} = 0\).

Example:

>>> from spatialmath import Twist3
>>> x = Twist3.UnitPrismatic([1,2,3])
>>> x.isprismatic
True
>>> x = Twist3.UnitRevolute([1,2,3], [4,5,6])
>>> x.isprismatic
False

property isrevolute

Test for revolute twist (superclass property)

Returns:: Whether twist is purely revolute
Return type:: bool

A revolute twist has \(\vec{v} = 0\).

Example:

>>> from spatialmath import Twist3
>>> x = Twist3.UnitPrismatic([1,2,3])
>>> x.isrevolute
False
>>> x = Twist3.UnitRevolute([1,2,3], [0,0,0])
>>> x.isrevolute
True

property isunit

Test for unit twist (superclass property)

Returns:: Whether twist is a unit-twist
Return type:: bool

A unit twist is one with a norm of 1, ie. \(\| S \| = 1\).

Example:

  File "<input>", line 1, in <module>
TypeError: 'bool' object is not callable

property pitch

Pitch of a 3D twist

Returns:: the pitch of the twist
Return type:: 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:

>>> from spatialmath import SE3, Twist3
>>> T = SE3(1, 2, 3) * SE3.Rx(0.3)
>>> S = Twist3(T)
>>> S.pitch
0.3

property pole

Pole of a 3D twist

Returns:: the pole of the twist
Return type:: ndarray(3)

X.pole() is a point on the twist axis. For a pure translation this point is at infinity.

Example:

>>> from spatialmath import SE3, Twist3
>>> T = SE3(1, 2, 3) * SE3.Rx(0.3)
>>> S = Twist3(T)
>>> S.pole
array([ 0.    , -2.6775,  2.435 ])

property shape

Shape of the object’s internal array representation

Returns:: (6,)
Return type:: tuple

property theta

Twist angle (superclass method)

Returns:: magnitude of rotation (1x1) about the twist axis in radians
Return type:: float

property v

Moment vector of twist

Returns:: Moment vector
Return type:: ndarray(3)

X.v is a 3-vector representing the moment vector of the twist.

Example:

>>> from spatialmath import Twist3
>>> t = Twist3([1, 2, 3, 4, 5, 6])
>>> t.v
array([1, 2, 3])

property w

Direction vector of twist

Returns:: Direction vector
Return type:: ndarray(3)

X.w is a 3-vector representing the direction vector of the twist.

Example:

>>> from spatialmath import Twist3
>>> t = Twist3([1, 2, 3, 4, 5, 6])
>>> t.w
array([4, 5, 6])