Spatial Maths for Python
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
|Represents||in 3D||in 2D|
SE3matrices belonging to the group SE(3) for position and orientation (pose) in 3-dimensions
SO3matrices belonging to the group SO(3) for orientation in 3-dimensions
UnitQuaternionbelonging to the group S3 for orientation in 3-dimensions
Twist3vectors belonging to the group se(3) for pose in 3-dimensions
UnitDualQuaternionmaps to the group SE(3) for position and orientation (pose) in 3-dimensions
SE2matrices belonging to the group SE(2) for position and orientation (pose) in 2-dimensions
SO2matrices belonging to the group SO(2) for orientation in 2-dimensions
Twist2vectors belonging to the group 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
- point transformation, using the
- exponent, using the
- 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
The class, method and functions names largely mirror those of the MATLAB toolboxes, and the semantics are quite similar.
Install a snapshot from PyPI
pip install spatialmath-python
Install the current code base from GitHub and pip install a link to that cloned copy
git clone https://github.com/petercorke/spatialmath-python.git cd spatialmath-python pip install -e .
ffmpeg (if rendering animations as a movie)
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
>>> 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
>>> R = SO3() # the identity >>> R.append(R1) >>> R.append(R2) >>> len(R) 3 >>> R 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 (
R2 defined above)
>>> R = SO3( [ SO3(), R1, R2 ] ) >>> len(R) 3
Many of the constructors such as
.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).
>>> 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].
>>> 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
For more detail checkout the shipped Python notebooks:
Low-level spatial math
Import the low-level transform functions
>>> import spatialmath.base as tr
We can create a 3D rotation matrix
>>> tr.rotx(0.3) array([[ 1. , 0. , 0. ], [ 0. , 0.95533649, -0.29552021], [ 0. , 0.29552021, 0.95533649]]) >>> tr.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: array([[1., 0., 1.], [0., 1., 2.], [0., 0., 1.]])
- or as a
transl2( np.array([1,2]) ) Out: 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
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=' 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=' 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
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)]]
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: 1 type(a) Out: int a = T[1,1] a Out: cos(theta) type(a) Out: 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.