#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from collections import UserList
import numpy as np
import math
from spatialmath.base import argcheck
import spatialmath.base as tr
from spatialmath import super_pose as sp
import spatialmath.pose3d as p3
# ============================== SO2 =====================================#
[docs]class SO2(sp.SMPose):
# SO2() identity matrix
# SO2(angle, unit)
# SO2( obj ) # deep copy
# SO2( np ) # make numpy object
# SO2( nplist ) # make from list of numpy objects
# constructor needs to take ndarray -> SO2, or list of ndarray -> SO2
[docs] def __init__(self, arg=None, *, unit='rad', check=True):
"""
Construct new SO(2) object
:param unit: angular units 'deg' or 'rad' [default] if applicable
:type unit: str, optional
:param check: check for valid SO(2) elements if applicable, default to True
:type check: bool
:return: SO(2) rotation
:rtype: SO2 instance
- ``SO2()`` is an SO2 instance representing a null rotation -- the identity matrix.
- ``SO2(theta)`` is an SO2 instance representing a rotation by ``theta`` radians. If ``theta`` is array_like
`[theta1, theta2, ... thetaN]` then an SO2 instance containing a sequence of N rotations.
- ``SO2(theta, unit='deg')`` is an SO2 instance representing a rotation by ``theta`` degrees. If ``theta`` is array_like
`[theta1, theta2, ... thetaN]` then an SO2 instance containing a sequence of N rotations.
- ``SO2(R)`` is an SO2 instance with rotation described by the SO(2) matrix R which is a 2x2 numpy array. If ``check``
is ``True`` check the matrix belongs to SO(2).
- ``SO2([R1, R2, ... RN])`` is an SO2 instance containing a sequence of N rotations, each described by an SO(2) matrix
Ri which is a 2x2 numpy array. If ``check`` is ``True`` then check each matrix belongs to SO(2).
- ``SO2([X1, X2, ... XN])`` is an SO2 instance containing a sequence of N rotations, where each Xi is an SO2 instance.
"""
super().__init__() # activate the UserList semantics
if arg is None:
# empty constructor
if type(self) is SO2:
self.data = [np.eye(2)]
elif argcheck.isvector(arg):
# SO2(value)
# SO2(list of values)
self.data = [tr.rot2(x, unit) for x in argcheck.getvector(arg)]
elif isinstance(arg, np.ndarray) and arg.shape == (2,2):
self.data = [arg]
else:
super()._arghandler(arg, check=check)
[docs] @classmethod
def Rand(cls, *, range=[0, 2 * math.pi], unit='rad', N=1):
r"""
Construct new SO(2) with random rotation
:param range: rotation range, defaults to :math:`[0, 2\pi)`.
:type range: 2-element array-like, optional
:param unit: angular units as 'deg or 'rad' [default]
:type unit: str, optional
:param N: number of random rotations, defaults to 1
:type N: int
:return: SO(2) rotation matrix
:rtype: SO2 instance
- ``SO2.Rand()`` is a random SO(2) rotation.
- ``SO2.Rand([-90, 90], unit='deg')`` is a random SO(2) rotation between
-90 and +90 degrees.
- ``SO2.Rand(N)`` is a sequence of N random rotations.
Rotations are uniform over the specified interval.
"""
rand = np.random.uniform(low=range[0], high=range[1], size=N) # random values in the range
return cls([tr.rot2(x) for x in argcheck.getunit(rand, unit)])
[docs] @classmethod
def Exp(cls, S, check=True):
"""
Construct new SO(2) rotation matrix from so(2) Lie algebra
:param S: element of Lie algebra so(2)
:type S: numpy ndarray
:param check: check that passed matrix is valid so(2), default True
:type check: bool
:return: SO(2) rotation matrix
:rtype: SO2 instance
- ``SO2.Exp(S)`` is an SO(2) rotation defined by its Lie algebra
which is a 2x2 so(2) matrix (skew symmetric)
:seealso: :func:`spatialmath.base.transforms2d.trexp`, :func:`spatialmath.base.transformsNd.skew`
"""
if argcheck.ismatrix(S, (-1,2)) and not so2:
return cls([tr.trexp2(s, check=check) for s in S])
else:
return cls(tr.trexp2(S, check=check), check=False)
[docs] @staticmethod
def isvalid(x):
"""
Test if matrix is valid SO(2)
:param x: matrix to test
:type x: numpy.ndarray
:return: True if the matrix is a valid element of SO(2), ie. it is a 2x2
orthonormal matrix with determinant of +1.
:rtype: bool
:seealso: :func:`~spatialmath.base.transform3d.isrot`
"""
return tr.isrot2(x, check=True)
[docs] def inv(self):
"""
Inverse of SO(2)
:return: inverse rotation
:rtype: SO2 instance
- ``x.inv()`` is the inverse of `x`.
Notes:
- for elements of SO(2) this is the transpose.
- if `x` contains a sequence, returns an `SO2` with a sequence of inverses
"""
if len(self) == 1:
return SO2(self.A.T)
else:
return SO2([x.T for x in self.A])
@property
def R(self):
"""
SO(2) or SE(2) as rotation matrix
:return: rotational component
:rtype: numpy.ndarray, shape=(2,2)
``x.R`` returns the rotation matrix, when `x` is `SO2` or `SE2`. If `len(x)` is:
- 1, return an ndarray with shape=(2,2)
- N>1, return ndarray with shape=(N,2,2)
"""
return self.A[:2, :2]
[docs] def theta(self, units='rad'):
"""
SO(2) as a rotation angle
:param unit: angular units 'deg' or 'rad' [default]
:type unit: str, optional
:return: rotation angle
:rtype: float or list
``x.theta`` is the rotation angle such that `x` is `SO2(x.theta)`.
"""
if units == 'deg':
conv = 180.0 / math.pi
else:
conv = 1.0
if len(self) == 1:
return conv * math.atan2(self.A[1,0], self.A[0,0])
else:
return [conv * math.atan2(x.A[1,0], x.A[0,0]) for x in self]
[docs] def SE2(self):
"""
Create SE(2) from SO(2)
:return: SE(2) with same rotation but zero translation
:rtype: SE2 instance
"""
return SE2(tr.rt2tr(self.A, [0, 0]))
# ============================== SE2 =====================================#
[docs]class SE2(SO2):
# constructor needs to take ndarray -> SO2, or list of ndarray -> SO2
[docs] def __init__(self, x=None, y=None, theta=None, *, unit='rad', check=True):
"""
Construct new SE(2) object
:param unit: angular units 'deg' or 'rad' [default] if applicable
:type unit: str, optional
:param check: check for valid SE(2) elements if applicable, default to True
:type check: bool
:return: homogeneous rigid-body transformation matrix
:rtype: SE2 instance
- ``SE2()`` is an SE2 instance representing a null motion -- the identity matrix
- ``SE2(x, y)`` is an SE2 instance representing a pure translation of (``x``, ``y``)
- ``SE2(t)`` is an SE2 instance representing a pure translation of (``x``, ``y``) where``t``=[x,y] is a 2-element array_like
- ``SE2(x, y, theta)`` is an SE2 instance representing a translation of (``x``, ``y``) and a rotation of ``theta`` radians
- ``SE2(x, y, theta, unit='deg')`` is an SE2 instance representing a translation of (``x``, ``y``) and a rotation of ``theta`` degrees
- ``SE2(t)`` is an SE2 instance representing a translation of (``x``, ``y``) and a rotation of ``theta`` where ``t``=[x,y,theta] is a 3-element array_like
- ``SE2(T)`` is an SE2 instance with rigid-body motion described by the SE(2) matrix T which is a 3x3 numpy array. If ``check``
is ``True`` check the matrix belongs to SE(2).
- ``SE2([T1, T2, ... TN])`` is an SE2 instance containing a sequence of N rigid-body motions, each described by an SE(2) matrix
Ti which is a 3x3 numpy array. If ``check`` is ``True`` then check each matrix belongs to SE(2).
- ``SE2([X1, X2, ... XN])`` is an SE2 instance containing a sequence of N rigid-body motions, where each Xi is an SE2 instance.
"""
super().__init__() # activate the UserList semantics
if x is None and y is None and theta is None:
# SE2()
# empty constructor
self.data = [np.eye(3)]
elif x is not None:
if y is not None and theta is None:
# SE2(x, y)
self.data = [tr.transl2(x, y)]
elif y is not None and theta is not None:
# SE2(x, y, theta)
self.data = [tr.trot2(theta, t=[x, y], unit=unit)]
elif y is None and theta is None:
if argcheck.isvector(x, 2):
# SE2([x,y])
self.data = [tr.transl2(x)]
elif argcheck.isvector(x, 3):
# SE2([x,y,theta])
self.data = [tr.trot2(x[2], t=x[:2], unit=unit)]
else:
super()._arghandler(x, check=check)
else:
raise ValueError('bad arguments to constructor')
[docs] @classmethod
def Rand(cls, *, xrange=[-1, 1], yrange=[-1, 1], trange=[0, 2 * math.pi], unit='rad', N=1):
r"""
Construct a new random SE(2)
: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 trange: theta range [min,max], defaults to :math:`[0, 2\pi)`
:type yrange: 2-element sequence, optional
:param N: number of random rotations, defaults to 1
:type N: int
:return: homogeneous rigid-body transformation matrix
:rtype: SE2 instance
Return an SE2 instance with random rotation and translation.
- ``SE2.Rand()`` is a random SE(2) rotation.
- ``SE2.Rand(N)`` is an SE2 object containing a sequence of N random
poses.
Example, create random ten vehicles in the xy-plane::
>>> x = SE3.Rand(N=10, xrange=[-2,2], yrange=[-2,2])
>>> len(x)
10
"""
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
theta = np.random.uniform(low=trange[0], high=trange[1], size=N) # random values in the range
return cls([tr.trot2(t, t=[x, y]) for (t, x, y) in zip(x, y, argcheck.getunit(theta, unit))])
[docs] @classmethod
def Exp(cls, S, check=True, se2=True):
"""
Construct a new SE(2) from se(2) Lie algebra
:param S: element of Lie algebra se(2)
:type S: numpy ndarray
:param check: check that passed matrix is valid se(2), default True
:type check: bool
:param se2: input is an se(2) matrix (default True)
:type se2: bool
:return: homogeneous transform matrix
:rtype: SE2 instance
- ``SE2.Exp(S)`` is an SE(2) rotation defined by its Lie algebra
which is a 3x3 se(2) matrix (skew symmetric)
- ``SE2.Exp(t)`` is an SE(2) rotation defined by a 3-element twist
vector array_like (the unique elements of the se(2) skew-symmetric matrix)
- ``SE2.Exp(T)`` is a sequence of SE(2) rigid-body motions defined by an Nx3 matrix of twist vectors, one per row.
Note:
- an input 3x3 matrix is ambiguous, it could be the first or third case above. In this case the argument ``se2`` is the decider.
:seealso: :func:`spatialmath.base.transforms2d.trexp`, :func:`spatialmath.base.transformsNd.skew`
"""
if isinstance(S, np.ndarray) and S.shape[1] == 3 and not se2:
return cls([tr.trexp2(s) for s in S])
else:
return cls(tr.trexp2(S), check=False)
[docs] @staticmethod
def isvalid(x):
"""
Test if matrix is valid SE(2)
:param x: matrix to test
:type x: numpy.ndarray
:return: true if the matrix is a valid element of SE(2), ie. it is a
3x3 homogeneous rigid-body transformation matrix.
:rtype: bool
:seealso: :func:`~spatialmath.base.transform2d.ishom`
"""
return tr.ishom2(x, check=True)
@property
def t(self):
"""
Translational component of SE(2)
:param self: SE(2)
:type self: SE2 instance
:return: translational component
:rtype: numpy.ndarray
``x.t`` is the translational vector component. If ``len(x)`` is:
- 1, return an ndarray with shape=(2,)
- N>1, return an ndarray with shape=(N,2)
"""
if len(self) == 1:
return self.A[:2, 2]
else:
return np.array([x[:2, 2] for x in self.A])
[docs] def xyt(self):
r"""
SE(2) as a configuration vector
:return: An array :math:`[x, y, \theta]`
:rtype: numpy.ndarray
``x.xyt`` is the rigidbody motion in minimal form as a translation and rotation expressed
in vector form as :math:`[x, y, \theta]`. If ``len(x)`` is:
- 1, return an ndarray with shape=(3,)
- N>1, return an ndarray with shape=(N,3)
"""
if len(self) == 1:
return np.r_[self.t, self.theta()]
else:
return [np.r_[x.t, x.theta()] for x in self]
[docs] def inv(self):
r"""
Inverse of SE(2)
:param self: pose
:type self: SE2 instance
:return: inverse
:rtype: SE2
Notes:
- for elements of SE(2) this takes into account the matrix structure :math:`T^{-1} = \left[ \begin{array}{cc} R & t \\ 0 & 1 \end{array} \right], T^{-1} = \left[ \begin{array}{cc} R^T & -R^T t \\ 0 & 1 \end{array} \right]`
- if `x` contains a sequence, returns an `SE2` with a sequence of inverses
"""
if len(self) == 1:
return SE2(tr.rt2tr(self.R.T, -self.R.T @ self.t))
else:
return SE2([tr.rt2tr(x.R.T, -x.R.T @ x.t) for x in self])
[docs] def SE3(self, z=0):
"""
Create SE(3) from SE(2)
:param z: default z coordinate, defaults to 0
:type z: float
:return: SE(2) with same rotation but zero translation
:rtype: SE2 instance
"Lifts" 2D rigid-body motion to 3D, rotation in the xy-plane (about the z-axis) and
z-coordinate is settable.
"""
def lift3(x):
y = np.eye(4)
y[:2,:2] = x.A[:2,:2]
y[:2,3] = x.A[:2,2]
y[2,3] = z
return y
return p3.SE3([lift3(x) for x in self])
if __name__ == '__main__': # pragma: no cover
import pathlib
import os.path
exec(open(os.path.join(pathlib.Path(__file__).parent.absolute(), "test_pose2d.py")).read())