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<!DOCTYPE html>

<head>

<title>spatialmath.super_pose — Spatial Maths package 0.7.0

documentation</title>

</head><body>

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

# Created by: Aditya Dua, 2017

# Peter Corke, 2020

# 13 June, 2017

import numpy as np

import sympy

from collections import UserList

import copy

from spatialmath.base import argcheck

import spatialmath.base as tr

_eps = np.finfo(np.float64).eps

# colored printing of matrices to the terminal

# colored package has much finer control than colorama, but the latter is available by default with anaconda

try:

from colored import fg, bg, attr

_color = True

print('using colored output')

except:

#print('colored not found')

_color = False

# try:

# import colorama

# colorama.init()

# print('using colored output')

# from colorama import Fore, Back, Style

# except:

# class color:

# def __init__(self):

# self.RED = ''

# self.BLUE = ''

# self.BLACK = ''

# self.DIM = ''

# print(Fore.RED + '1.00 2.00 ' + Fore.BLUE + '3.00')

# print(Fore.BLACK + Style.DIM + '0 0 1')

class SMPose(UserList):

"""

Superclass for SO(N) and SE(N) objects

Subclasses are:

- ``SO2`` representing elements of SO(2) which describe rotations in 2D

- ``SE2`` representing elements of SE(2) which describe rigid-body motion in 2D

- ``SO3`` representing elements of SO(3) which describe rotations in 3D

- ``SE3`` representing elements of SE(3) which describe rigid-body motion in 3D

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.

- ``+`` will add two instances of the same subclass, and the result will be

a matrix, not an instance of the same subclass, since addition is not a group operator.

These classes all inherit from ``UserList`` which enables them to

represent a sequence of values, ie. an ``SE3`` instance can contain

a sequence of SE(3) values. Most of the Python ``list`` operators

are applicable::

>>> x = SE3() # new instance with identity matrix value

>>> len(x) # it is a sequence of one value

1

>>> x.append(x) # append to itself

>>> len(x) # it is a sequence of two values

2

>>> x[1] # the element has a 4x4 matrix value

SE3([

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

[0., 1., 0., 0.],

[0., 0., 1., 0.],

[0., 0., 0., 1.]]) ])

>>> x[1] = SE3.Rx(0.3) # set an elements of the sequence

>>> x.reverse() # reverse the elements in the sequence

>>> del x[1] # delete an element

"""

def __new__(cls, *args, **kwargs):

"""

Create the subclass instance (superclass method)

Create a new instance and call the superclass initializer to enable the

``UserList`` capabilities.

"""

pose = super(SMPose, cls).__new__(cls) # create a new instance

super().__init__(pose) # initialize UserList

return pose

def _arghandler(self, arg, check=True):

"""

Assign value to pose subclasses (superclass method)

:param self: the pose object to be set

:type self: SO2, SE2, SO3, SE3 instance

:param arg: value of pose

:param check: check type of argument, defaults to True

:type check: TYPE, optional

:raises ValueError: bad type passed

The value ``arg`` can be any of:

# a numpy.ndarray of the appropriate shape and value which is valid for the subclass

# a list whose elements all meet the criteria above

# an instance of the subclass

# a list whose elements are all instances of the subclass

Examples::

SE3( np.identity(4))

SE3( [np.identity(4), np.identity(4)])

SE3( SE3() )

SE3( [SE3(), SE3()])

"""

if isinstance(arg, np.ndarray):

# it's a numpy array

assert arg.shape == self.shape, 'array must have valid shape for the class'

assert type(self).isvalid(arg), 'array must have valid value for the class'

self.data.append(arg)

elif isinstance(arg, list):

# construct from a list

if isinstance(arg[0], np.ndarray):

#print('list of numpys')

# possibly a list of numpy arrays

s = self.shape

if check:

checkfunc = type(self).isvalid # lambda function

assert all(map(lambda x: x.shape == s and checkfunc(x), arg)), 'all elements of list must have valid shape and value for the class'

else:

self.data = arg

elif type(arg[0]) == type(self):

# possibly a list of objects of same type

assert all(map(lambda x: type(x) == type(self), arg)), 'all elements of list must have same type'

self.data = [x.A for x in arg]

else:

raise ValueError('bad list argument to constructor')

elif type(self) == type(arg):

# it's an object of same type, do copy

self.data = arg.data.copy()

else:

raise ValueError('bad argument to constructor')

@classmethod

def Empty(cls):

"""

Construct a new pose object with zero items (superclass method)

:param cls: The pose subclass

:type cls: SO2, SE2, SO3, SE3

:return: a pose with zero values

:rtype: SO2, SE2, SO3, SE3 instance

This constructs an empty pose container which can be appended to. For example::

>>> x = SO2.Empty()

>>> len(x)

0

>>> x.append(SO2(20, 'deg'))

>>> len(x)

1

"""

X = cls()

X.data = []

return X

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

@property

def A(self):

"""

Interal array representation (superclass property)

:param self: the pose object

:type self: SO2, SE2, SO3, SE3 instance

:return: The numeric array

:rtype: numpy.ndarray

Each pose subclass SO(N) or SE(N) are stored internally as a numpy array. This property returns

the array, shape depends on the particular subclass.

Examples::

>>> x = SE3()

>>> x.A

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

[0., 1., 0., 0.],

[0., 0., 1., 0.],

[0., 0., 0., 1.]])

:seealso: `shape`, `N`

"""

# get the underlying numpy array

if len(self.data) == 1:

return self.data[0]

else:

return self.data

@property

def shape(self):

"""

Shape of the object's matrix representation (superclass property)

:return: matrix shape

:rtype: 2-tuple of ints

(2,2) for ``SO2``, (3,3) for ``SE2`` and ``SO3``, and (4,4) for ``SE3``.

Example::

>>> x = SE3()

>>> x.shape

(4, 4)

"""

return (2, 2)

elif type(self).__name__ == 'SO3':

return (3, 3)

return (4, 4)

@property

def about(self):

"""

Succinct summary of object type and length (superclass property)

:return: succinct summary

:rtype: str

Displays the type and the number of elements in compact form, for

example::

>>> x = SE3([SE3() for i in range(20)])

>>> len(x)

20

>>> print(x.about)

SE3[20]

"""

return "{:s}[{:d}]".format(type(self).__name__, len(self))

@property

def N(self):

"""

Dimension of the object's group (superclass property)

:return: dimension

:rtype: int

Dimension of the group is 2 for ``SO2`` or ``SE2``, and 3 for ``SO3`` or ``SE3``.

This corresponds to the dimension of the space, 2D or 3D, to which these

rotations or rigid-body motions apply.

Example::

>>> x = SE3()

>>> x.N

3

"""

if type(self).__name__ == 'SO2' or type(self).__name__ == 'SE2':

return 2

else:

return 3

#----------------------- tests

@property

def isSO(self):

"""

Test if object belongs to SO(n) group (superclass property)

:param self: object to test

:type self: SO2, SE2, SO3, SE3 instance

:return: ``True`` if object is instance of SO2 or SO3

:rtype: bool

"""

return type(self).__name__ == 'SO2' or type(self).__name__ == 'SO3'

@property

def isSE(self):

"""

Test if object belongs to SE(n) group (superclass property)

:param self: object to test

:type self: SO2, SE2, SO3, SE3 instance

:return: ``True`` if object is instance of SE2 or SE3

:rtype: bool

"""

return type(self).__name__ == 'SE2' or type(self).__name__ == 'SE3'

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

def __getitem__(self, i):

"""

Access value of a pose object (superclass method)

:param i: index of element to return

:type i: int

:return: the specific element of the pose

:rtype: SO2, SE2, SO3, SE3 instance

:raises IndexError: if the element is out of bounds

Note that only a single index is supported, slices are not.

Example::

>>> x = SE3.Rx([0, math.pi/2, math.pi])

>>> len(x)

3

>>> x[1]

1 0 0 0

0 0 -1 0

0 1 0 0

0 0 0 1

"""

if isinstance(i, slice):

return self.__class__([self.data[k] for k in range(i.start or 0, i.stop or len(self), i.step or 1)])

else:

return self.__class__(self.data[i])

def __setitem__(self, i, value):

"""

Assign a value to a pose object (superclass method)

:param i: index of element to assign to

:type i: int

:param value: the value to insert

:type value: SO2, SE2, SO3, SE3 instance

:raises ValueError: incorrect type of assigned value

Assign the argument to an element of the object's internal list of values.

This supports the assignement operator, for example::

>>> x = SE3([SE3() for i in range(10)]) # sequence of ten identity values

>>> len(x)

10

>>> x[3] = SE3.Rx(0.2) # assign to position 3 in the list

"""

if not type(self) == type(value):

raise ValueError("cant append different type of pose object")

if len(value) > 1:

raise ValueError("cant insert a pose sequence - must have len() == 1")

self.data[i] = value.A

def append(self, x):

"""

Append a value to a pose object (superclass method)

:param x: the value to append

:type x: SO2, SE2, SO3, SE3 instance

:raises ValueError: incorrect type of appended object

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

Examples::

>>> x = SE3()

>>> len(x)

1

>>> x.append(SE3.Rx(0.1))

>>> len(x)

2

"""

#print('in append method')

raise ValueError("cant append different type of pose object")

if len(x) > 1:

raise ValueError("cant append a pose sequence - use extend")

super().append(x.A)

def extend(self, x):

"""

Extend sequence of values of a pose object (superclass method)

:param x: the value to extend

:type x: SO2, SE2, SO3, SE3 instance

:raises ValueError: incorrect type of appended object

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

Examples::

>>> x = SE3()

>>> len(x)

1

>>> x.append(SE3.Rx(0.1))

>>> len(x)

2

"""

#print('in extend method')

raise ValueError("cant append different type of pose object")

if len(x) == 0:

raise ValueError("cant extend a singleton pose - use append")

super().extend(x.A)

def insert(self, i, value):

"""

Insert a value to a pose object (superclass method)

:param i: element to insert value before

:type i: int

:param value: the value to insert

:type value: SO2, SE2, SO3, SE3 instance

:raises ValueError: incorrect type of inserted value

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

Examples::

>>> x = SE3()

>>> x.inert(0, SE3.Rx(0.1)) # insert at position 0 in the list

>>> len(x)

2

"""

raise ValueError("cant append different type of pose object")

raise ValueError("cant insert a pose sequence - must have len() == 1")

super().insert(i, value.A)

def pop(self):

"""

Pop value of a pose object (superclass method)

:return: the specific element of the pose

:rtype: SO2, SE2, SO3, SE3 instance

:raises IndexError: if there are no values to pop

Removes the first pose value from the sequence in the pose object.

Example::

>>> x = SE3.Rx([0, math.pi/2, math.pi])

>>> len(x)

3

>>> y = x.pop()

>>> y

SE3(array([[ 1.0000000e+00, 0.0000000e+00, 0.0000000e+00, 0.0000000e+00],

[ 0.0000000e+00, -1.0000000e+00, -1.2246468e-16, 0.0000000e+00],

[ 0.0000000e+00, 1.2246468e-16, -1.0000000e+00, 0.0000000e+00],

[ 0.0000000e+00, 0.0000000e+00, 0.0000000e+00, 1.0000000e+00]]))

>>> len(x)

2

"""

return self.__class__(super().pop())

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

# --------- compatibility methods

def isrot(self):

"""

Test if object belongs to SO(3) group (superclass method)

:return: ``True`` if object is instance of SO3

:rtype: bool

For compatibility with Spatial Math Toolbox for MATLAB.

In Python use ``isinstance(x, SO3)``.

Example::

>>> x = SO3()

>>> x.isrot()

True

>>> x = SE3()

>>> x.isrot()

False

"""

def isrot2(self):

"""

Test if object belongs to SO(2) group (superclass method)

:return: ``True`` if object is instance of SO2

:rtype: bool

For compatibility with Spatial Math Toolbox for MATLAB.

In Python use ``isinstance(x, SO2)``.

Example::

>>> x = SO2()

>>> x.isrot()

True

>>> x = SE2()

>>> x.isrot()

False

"""

def ishom(self):

"""

Test if object belongs to SE(3) group (superclass method)

:return: ``True`` if object is instance of SE3

:rtype: bool

For compatibility with Spatial Math Toolbox for MATLAB.

In Python use ``isinstance(x, SE3)``.

Example::

>>> x = SO3()

>>> x.isrot()

False

>>> x = SE3()

>>> x.isrot()

True

"""

def ishom2(self):

"""

Test if object belongs to SE(2) group (superclass method)

:return: ``True`` if object is instance of SE2

:rtype: bool

For compatibility with Spatial Math Toolbox for MATLAB.

In Python use ``isinstance(x, SE2)``.

Example::

>>> x = SO2()

>>> x.isrot()

False

>>> x = SE2()

>>> x.isrot()

True

"""

#----------------------- functions

def log(self):

"""

Logarithm of pose (superclass method)

:return: logarithm

:rtype: numpy.ndarray

:raises: ValueError

An efficient closed-form solution of the matrix logarithm.

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

Input Output

----- ---------------------------------------

Pose Shape Structure

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

SO2 (2,2) skew-symmetric

SE2 (3,3) augmented skew-symmetric

SO3 (3,3) skew-symmetric

SE3 (4,4) augmented skew-symmetric

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

Example::

>>> x = SE3.Rx(0.3)

>>> y = x.log()

>>> y

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

[ 0. , 0. , -0.3, 0. ],

[-0. , 0.3, 0. , 0. ],

[ 0. , 0. , 0. , 0. ]])

:seealso: :func:`~spatialmath.base.transforms2d.trlog2`, :func:`~spatialmath.base.transforms3d.trlog`

"""

print('in log')

if self.N == 2:

log = [tr.trlog2(x) for x in self.data]

else:

log = [tr.trlog(x) for x in self.data]

if len(log) == 1:

return log[0]

else:

return log

def interp(self, s=None, T0=None):

"""

Interpolate pose (superclass method)

:param T0: initial pose

:type T0: SO2, SE2, SO3, SE3

:param s: interpolation coefficient, range 0 to 1

:type s: float or array_like

:return: interpolated pose

:rtype: SO2, SE2, SO3, SE3 instance

- ``X.interp(s)`` interpolates the pose X between identity when s=0

and X when s=1.

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

len(X) len(s) len(result) Result

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

1 1 1 Y = interp(identity, X, s)

M 1 M Y[i] = interp(T0, X[i], s)

1 M M Y[i] = interp(T0, X, s[i])

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

Example::

>>> x = SE3.Rx(0.3)

>>> print(x.interp(0))

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

[0., 1., 0., 0.],

[0., 0., 1., 0.],

[0., 0., 0., 1.]]))

>>> print(x.interp(1))

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

[ 0. , 0.95533649, -0.29552021, 0. ],

[ 0. , 0.29552021, 0.95533649, 0. ],

[ 0. , 0. , 0. , 1. ]]))

>>> y = x.interp(x, np.linspace(0, 1, 10))

>>> len(y)

10

>>> y[5]

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

[ 0. , 0.98614323, -0.16589613, 0. ],

[ 0. , 0.16589613, 0.98614323, 0. ],

[ 0. , 0. , 0. , 1. ]]))

Notes:

#. For SO3 and SE3 rotation is interpolated using quaternion spherical linear interpolation (slerp).

:seealso: :func:`~spatialmath.base.transforms3d.trinterp`, :func:`spatialmath.base.quaternions.slerp`, :func:`~spatialmath.base.transforms2d.trinterp2`

"""

s = argcheck.getvector(s)

if T0 is not None:

assert len(T0) == 1, 'len(X0) must == 1'

T0 = T0.A

if self.N == 2:

if len(s) > 1:

assert len(self) == 1, 'if len(s) > 1, len(X) must == 1'

return self.__class__([tr.trinterp2(self.A, T0, _s) for _s in s])

else:

assert len(s) == 1, 'if len(X) > 1, len(s) must == 1'

return self.__class__([tr.trinterp2(x, T0, s) for x in self.data])

elif self.N == 3:

return self.__class__([tr.trinterp(self.A, T1=T0, s=_s) for _s in s])

else:

return self.__class__([tr.trinterp(x, T1=T0, s=s) for x in self.data])

def norm(self):

"""

Normalize pose (superclass method)

:return: pose

:rtype: SO2, SE2, SO3, SE3 instance

- ``X.norm()`` is an equivalent pose object but the rotational matrix

part of all values has been adjusted to ensure it is a proper orthogonal

matrix rotation.

Example::

>>> x = SE3()

>>> y = x.norm()

>>> y

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

[0., 1., 0., 0.],

[0., 0., 1., 0.],

[0., 0., 0., 1.]]))

Notes:

#. Only the direction of A vector (the z-axis) is unchanged.

#. Used to prevent finite word length arithmetic causing transforms to

become 'unnormalized'.

:seealso: :func:`~spatialmath.base.transforms3d.trnorm`, :func:`~spatialmath.base.transforms2d.trnorm2`

"""

if self.N == 2:

return self.__class__([tr.trnorm2(x) for x in self.data])

else:

return self.__class__([tr.trnorm(x) for x in self.data])

# ----------------------- i/o stuff

def printline(self, **kwargs):

"""

Print pose as a single line (superclass method)

:param label: text label to put at start of line

:type label: str

:param file: file to write formatted string to. [default, stdout]

:type file: str

:param fmt: conversion format for each number as used by ``format()``

:type fmt: str

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

:type unit: str

:return: optional formatted string

:rtype: str

For SO(3) or SE(3) also:

:param orient: 3-angle convention to use

:type orient: str

- ``X.printline()`` print ``X`` in single-line format to ``stdout``, followed

by a newline

- ``X.printline(file=None)`` return a string containing ``X`` in

single-line format

Example::

>>> x=SE3.Rx(0.3)

>>> x.printline()

t = 0, 0, 0; rpy/zyx = 17, 0, 0 deg

"""

if self.N == 2:

tr.trprint2(self.A, **kwargs)

else:

tr.trprint(self.A, **kwargs)

def __repr__(self):

"""

Readable representation of pose (superclass method)

:return: readable representation of the pose as a list of arrays

:rtype: str

Example::

>>> x = SE3.Rx(0.3)

>>> x

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

[ 0. , 0.95533649, -0.29552021, 0. ],

[ 0. , 0.29552021, 0.95533649, 0. ],

[ 0. , 0. , 0. , 1. ]]))

"""

name = type(self).__name__

if len(self) == 0:

return name + '([])'

elif len(self) == 1:

# need to indent subsequent lines of the native repr string by 4 spaces

return name + '(' + self.A.__repr__().replace('\n', '\n ') + ')'

else:

# format this as a list of ndarrays

return name + '([\n' + ',\n'.join([v.__repr__() for v in self.data]) + ' ])'

def __str__(self):

"""

Pretty string representation of pose (superclass method)

:return: readable representation of the pose

:rtype: str

Convert the pose's matrix value to a simple grid of numbers.

Example::

>>> x = SE3.Rx(0.3)

>>> print(x)

1 0 0 0

0 0.955336 -0.29552 0

0 0.29552 0.955336 0

0 0 0 1

Notes:

- By default, the output is colorised for an ANSI terminal console:

* red: rotational elements

* blue: translational elements

* white: constant elements

"""

return self._string(color=True)

def _string(self, color=False, tol=10):

"""

Pretty print the matrix value

:param color: colorise the output, defaults to False

:type color: bool, optional

:param tol: zero values smaller than tol*eps, defaults to 10

:type tol: float, optional

:return: multiline matrix representation

:rtype: str

Convert a matrix to a simple grid of numbers with optional

colorization for an ANSI terminal console:

* red: rotational elements

* blue: translational elements

* white: constant elements

Example::

>>> x = SE3.Rx(0.3)

>>> print(str(x))

1 0 0 0

0 0.955336 -0.29552 0

0 0.29552 0.955336 0

0 0 0 1

"""

#print('in __str__')

FG = lambda c: fg(c) if _color else ''

BG = lambda c: bg(c) if _color else ''

ATTR = lambda c: attr(c) if _color else ''

def mformat(self, X):

# X is an ndarray value to be display

# self provides set type for formatting

out = ''

n = self.N # dimension of rotation submatrix

for rownum, row in enumerate(X):

rowstr = ' '

# format the columns

for colnum, element in enumerate(row):

if isinstance(element, sympy.Expr):

s = '{:<12s}'.format(str(element))

else:

if tol > 0 and abs(element) < tol * _eps:

element = 0

s = '{:< 12g}'.format(element)

if rownum < n:

if colnum < n:

# rotation part

s = FG('red') + BG('grey_93') + s + ATTR(0)

else:

# translation part

s = FG('blue') + BG('grey_93') + s + ATTR(0)

else:

# bottom row

s = FG('grey_50') + BG('grey_93') + s + ATTR(0)

rowstr += s

out += rowstr + BG('grey_93') + ' ' + ATTR(0) + '\n'

return out

output_str = ''

output_str = '[]'

elif len(self.data) == 1:

# single matrix case

output_str = mformat(self, self.A)

else:

# sequence case

for count, X in enumerate(self.data):

# add separator lines and the index

output_str += fg('green') + '[{:d}] =\n'.format(count) + attr(0) + mformat(self, X)

return output_str

# ----------------------- graphics

def plot(self, *args, **kwargs):

"""

Plot pose object as a coordinate frame (superclass method)

:param `**kwargs`: plotting options

- ``X.plot()`` displays the pose ``X`` as a coordinate frame in either

2D or 3D axes. There are many options, see the links below.

Example::

>>> X = SE3.Rx(0.3)

>>> X.plot(frame='A', color='green')

:seealso: :func:`~spatialmath.base.transforms3d.trplot`, :func:`~spatialmath.base.transforms2d.trplot2`

"""

if self.N == 2:

tr.trplot2(self.A, *args, **kwargs)

else:

tr.trplot(self.A, *args, **kwargs)

def animate(self, *args, T0=None, **kwargs):

"""

Plot pose object as an animated coordinate frame (superclass method)

:param `**kwargs`: plotting options

- ``X.plot()`` displays the pose ``X`` as a coordinate frame moving

from the origin, or ``T0``, in either 2D or 3D axes. There are

many options, see the links below.

Example::

>>> X = SE3.Rx(0.3)

>>> X.animate(frame='A', color='green')

:seealso: :func:`~spatialmath.base.transforms3d.tranimate`, :func:`~spatialmath.base.transforms2d.tranimate2`

"""

if T0 is not None:

T0 = T0.A

if self.N == 2:

tr.tranimate2(self.A, T0=T0, *args, **kwargs)

else:

tr.tranimate(self.A, T0=T0, *args, **kwargs)

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

#----------------------- arithmetic

def __mul__(left, right):

"""

Overloaded ``*`` operator (superclass method)

:arg left: left multiplicand

:arg right: right multiplicand

:return: product

:raises: ValueError

Pose composition, scaling or vector transformation:

- ``X * Y`` compounds the poses ``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``

- ``X * v`` linear transform of the vector ``v``

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

Multiplicands Product

------------------------------- -----------------------------------

left right type operation

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

Pose Pose Pose matrix product

Pose scalar NxN matrix element-wise product

scalar Pose NxN matrix element-wise product

Pose N-vector N-vector vector transform

Pose NxM matrix NxM matrix transform each column

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

Notes:

#. Pose is ``SO2``, ``SE2``, ``SO3`` or ``SE3`` instance

#. N is 2 for ``SO2``, ``SE2``; 3 for ``SO3`` or ``SE3``

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