Spatial Maths for Python

A Python implementation of the Spatial Math Toolbox for MATLAB® GitHub repository Documentation Recent changes Wiki (examples and details) Installation

Spatial mathematics capability underpins all of robotics and robotic vision where we need to describe the position, orientation or pose of objects in 2D or 3D spaces.

What it does

The package provides classes to represent pose and orientation in 3D and 2D space:

Represents	in 3D	in 2D
pose	SE3 Twist3 UnitDualQuaternion	SE2 Twist2
orientation	SO3 UnitQuaternion	SO2

More specifically:

• SE3 matrices belonging to the group $\mathbf{SE}(3)$ for position and orientation (pose) in 3-dimensions
• SO3 matrices belonging to the group $\mathbf{SO}(3)$ for orientation in 3-dimensions
• UnitQuaternion belonging to the group $\mathbf{S}^3$ for orientation in 3-dimensions
• Twist3 vectors belonging to the group $\mathbf{se}(3)$ for pose in 3-dimensions
• UnitDualQuaternion maps to the group $\mathbf{SE}(3)$ for position and orientation (pose) in 3-dimensions
• SE2 matrices belonging to the group $\mathbf{SE}(2)$ for position and orientation (pose) in 2-dimensions
• SO2 matrices belonging to the group $\mathbf{SO}(2)$ for orientation in 2-dimensions
• Twist2 vectors belonging to the group $\mathbf{se}(2)$ for pose in 2-dimensions

These classes provide convenience and type safety, as well as methods and overloaded operators to support:

• composition, using the * operator
• point transformation, using the * operator
• exponent, using the ** operator
• normalization
• inversion
• connection to the Lie algebra via matrix exponential and logarithm operations
• conversion of orientation to/from Euler angles, roll-pitch-yaw angles and angle-axis forms.
• list operations such as append, insert and get

These are layered over a set of base functions that perform many of the same operations but represent data explicitly in terms of numpy arrays.

The class, method and functions names largely mirror those of the MATLAB toolboxes, and the semantics are quite similar.

Citing

Check out our ICRA 2021 paper on IEEE Xplore or get the PDF from Peter's website. This describes the Robotics Toolbox for Python as well Spatial Maths.

If the toolbox helped you in your research, please cite

@inproceedings{rtb,
  title={Not your grandmother’s toolbox--the Robotics Toolbox reinvented for Python},
  author={Corke, Peter and Haviland, Jesse},
  booktitle={2021 IEEE International Conference on Robotics and Automation (ICRA)},
  pages={11357--11363},
  year={2021},
  organization={IEEE}
}

Using the Toolbox in your Open Source Code?

If you are using the Toolbox in your open source code, feel free to add our badge to your readme!

Simply copy the following

[![Powered by the Spatial Math Toolbox](https://github.com/bdaiinstitute/spatialmath-python/raw/master/.github/svg/sm_powered.min.svg)](https://github.com/bdaiinstitute/spatialmath-python)

Installation

Using pip

Install a snapshot from PyPI

pip install spatialmath-python

From GitHub

Install the current code base from GitHub and pip install a link to that cloned copy

git clone https://github.com/bdaiinstitute/spatialmath-python.git
cd spatialmath-python
pip install -e .
# Optional: if you would like to contribute and commit code changes to the repository,
# pre-commit install

Dependencies

numpy, scipy, matplotlib, ffmpeg (if rendering animations as a movie)

Examples

High-level classes

These classes abstract the low-level numpy arrays into objects that obey the rules associated with the mathematical groups SO(2), SE(2), SO(3), SE(3) as well as twists and quaternions.

Using classes ensures type safety, for example it stops us mixing a 2D homogeneous transformation with a 3D rotation matrix -- both of which are 3x3 matrices. It also ensures that the internal matrix representation is always a valid member of the relevant group.

For example, to create an object representing a rotation of 0.3 radians about the x-axis is simply

>>> from spatialmath import SO3, SE3
>>> R1 = SO3.Rx(0.3)
>>> R1
   1         0         0          
   0         0.955336 -0.29552    
   0         0.29552   0.955336         

while a rotation of 30 deg about the z-axis is

>>> R2 = SO3.Rz(30, 'deg')
>>> R2
   0.866025 -0.5       0          
   0.5       0.866025  0          
   0         0         1    

and the composition of these two rotations is

>>> R = R1 * R2
   0.866025 -0.5       0          
   0.433013  0.75     -0.5        
   0.25      0.433013  0.866025 

We can find the corresponding Euler angles (in radians)

>> R.eul()
array([-1.57079633,  0.52359878,  2.0943951 ])

Frequently in robotics we want a sequence, a trajectory, of rotation matrices or poses. These pose classes inherit capability from the list class

>>> R = SO3()   # the null rotation or identity matrix
>>> R.append(R1)
>>> R.append(R2)
>>> len(R)
 3
>>> R[1]
   1         0         0          
   0         0.955336 -0.29552    
   0         0.29552   0.955336             

and this can be used in for loops and list comprehensions.

An alternative way of constructing this would be (R1, R2 defined above)

>>> R = SO3( [ SO3(), R1, R2 ] )       
>>> len(R)
 3

Many of the constructors such as .Rx, .Ry and .Rz support vectorization

>>> R = SO3.Rx( np.arange(0, 2*np.pi, 0.2))
>>> len(R)
 32

which has created, in a single line, a list of rotation matrices.

Vectorization also applies to the operators, for instance

>>> A = R * SO3.Ry(0.5)
>>> len(R)
 32

will produce a result where each element is the product of each element of the left-hand side with the right-hand side, ie. R[i] * SO3.Ry(0.5).

Similarly

>>> A = SO3.Ry(0.5) * R 
>>> len(R)
 32

will produce a result where each element is the product of the left-hand side with each element of the right-hand side , ie. SO3.Ry(0.5) * R[i] .

Finally

>>> A = R * R 
>>> len(R)
 32

will produce a result where each element is the product of each element of the left-hand side with each element of the right-hand side , ie. R[i] * R[i] .

The underlying representation of these classes is a numpy matrix, but the class ensures that the structure of that matrix is valid for the particular group represented: SO(2), SE(2), SO(3), SE(3). Any operation that is not valid for the group will return a matrix rather than a pose class, for example

>>> SO3.Rx(0.3) * 2
array([[ 2.        ,  0.        ,  0.        ],
       [ 0.        ,  1.91067298, -0.59104041],
       [ 0.        ,  0.59104041,  1.91067298]])

>>> SO3.Rx(0.3) - 1
array([[ 0.        , -1.        , -1.        ],
       [-1.        , -0.04466351, -1.29552021],
       [-1.        , -0.70447979, -0.04466351]])

We can print and plot these objects as well

>>> T = SE3(1,2,3) * SE3.Rx(30, 'deg')
>>> T.print()
   1         0         0         1          
   0         0.866025 -0.5       2          
   0         0.5       0.866025  3          
   0         0         0         1          

>>> T.printline()
t =        1,        2,        3; rpy/zyx =       30,        0,        0 deg

>>> T.plot()

printline is a compact single line format for tabular listing, whereas print shows the underlying matrix and for consoles that support it, it is colorised, with rotational elements in red and translational elements in blue.

For more detail checkout the shipped Python notebooks:

• gentle introduction
• deeper introduction

You can browse it statically through the links above, or clone the toolbox and run them interactively using Jupyter or JupyterLab.

Low-level spatial math

Import the low-level transform functions

>>> from spatialmath.base import *

We can create a 3D rotation matrix

>>> rotx(0.3)
array([[ 1.        ,  0.        ,  0.        ],
       [ 0.        ,  0.95533649, -0.29552021],
       [ 0.        ,  0.29552021,  0.95533649]])

>>> rotx(30, unit='deg')
array([[ 1.       ,  0.       ,  0.       ],
       [ 0.       ,  0.8660254, -0.5      ],
       [ 0.       ,  0.5      ,  0.8660254]])

The results are numpy arrays so to perform matrix multiplication you need to use the @ operator, for example

rotx(0.3) @ roty(0.2)

We also support multiple ways of passing vector information to functions that require it:

• as separate positional arguments

transl2(1, 2)
array([[1., 0., 1.],
       [0., 1., 2.],
       [0., 0., 1.]])

• as a list or a tuple

transl2( [1,2] )
array([[1., 0., 1.],
       [0., 1., 2.],
       [0., 0., 1.]])

transl2( (1,2) )
Out[444]: 
array([[1., 0., 1.],
       [0., 1., 2.],
       [0., 0., 1.]])

• or as a numpy array

transl2( np.array([1,2]) )
Out[445]: 
array([[1., 0., 1.],
       [0., 1., 2.],
       [0., 0., 1.]])

There is a single module that deals with quaternions, unit or not, and the representation is a numpy array of four elements. As above, functions can accept the numpy array, a list, dict or numpy row or column vectors.

>>> from spatialmath.base.quaternion import *
>>> q = qqmul([1,2,3,4], [5,6,7,8])
>>> q
array([-60,  12,  30,  24])
>>> qprint(q)
-60.000000 < 12.000000, 30.000000, 24.000000 >
>>> qnorm(q)
72.24956747275377

Graphics

The functions support various plotting styles

trplot( transl(1,2,3), frame='A', rviz=True, width=1, dims=[0, 10, 0, 10, 0, 10])
trplot( transl(3,1, 2), color='red', width=3, frame='B')
trplot( transl(4, 3, 1)@trotx(math.pi/3), color='green', frame='c', dims=[0,4,0,4,0,4])

Animation is straightforward

tranimate(transl(4, 3, 4)@trotx(2)@troty(-2), frame='A', arrow=False, dims=[0, 5], nframes=200)

and it can be saved to a file by

tranimate(transl(4, 3, 4)@trotx(2)@troty(-2), frame='A', arrow=False, dims=[0, 5], nframes=200, movie='out.mp4')

At the moment we can only save as an MP4, but the following incantation will covert that to an animated GIF for embedding in web pages

ffmpeg -i out -r 20 -vf "fps=10,scale=640:-1:flags=lanczos,split[s0][s1];[s0]palettegen[p];[s1][p]paletteuse" out.gif

For use in a Jupyter notebook, or on Colab, you can display an animation by

from IPython.core.display import HTML
HTML(tranimate(transl(4, 3, 4)@trotx(2)@troty(-2), frame='A', arrow=False, dims=[0, 5], nframes=200, movie=True))

The movie=True option causes tranimate to output an HTML5 fragment which is displayed inline by the HTML function.

Symbolic support

Some functions have support for symbolic variables, for example

import sympy

theta = sym.symbols('theta')
print(rotx(theta))
[[1 0 0]
 [0 cos(theta) -sin(theta)]
 [0 sin(theta) cos(theta)]]

The resulting numpy array is an array of symbolic objects not numbers – the constants are also symbolic objects. You can read the elements of the matrix

a = T[0,0]

a
Out[258]: 1

type(a)
Out[259]: int

a = T[1,1]
a
Out[256]: 
cos(theta)
type(a)
Out[255]: cos

We see that the symbolic constants are converted back to Python numeric types on read.

Similarly when we assign an element or slice of the symbolic matrix to a numeric value, they are converted to symbolic constants on the way in.

History & Contributors

This package was originally created by Peter Corke and Jesse Haviland and was inspired by the Spatial Math Toolbox for MATLAB. It supports the textbook Robotics, Vision & Control in Python 3e.

The package is now a collaboration with Boston Dynamics AI Institute.