# Introduction

Spatial maths capability underpins all of robotics and robotic vision by describing the relative position and orientation of objects in 2D or 3D space. This package:

- provides Python classes and functions to manipulate matrices that represent relevant mathematical objects such as rotation matrices \(R \in SO(2), SO(3)\), homogeneous transformation matrices \(T \in SE(2), SE(3)\) and quaternions \(q \in \mathbb{H}\).

- replicates, as much as possible, the functionality of the Spatial Math Toolbox for MATLAB ® which underpins the Robotics Toolbox for MATLAB. Important considerations included:

  - being as similar as possible to the MATLAB Toolbox function names and semantics

  - but balancing the tension of being as Pythonic as possible

  - use Python keyword arguments to replace the MATLAB Toolbox string options supported using tb_optparse`

  - use `numpy` arrays for rotation and homogeneous transformation matrices, quaternions and vectors

  - all functions that accept a vector can accept a list, tuple, or np.ndarray

  - The classes can hold a sequence of elements, they are polymorphic with lists, which can be used to represent trajectories or time sequences.

Quick example:

    >>> import spatialmath as sm
    >>> R = sm.SO3.Rx(30, 'deg')
    >>> R
       1         0         0
       0         0.866025 -0.5

which constructs a rotation about the x-axis by 30 degrees.

## High-level classes

These classes abstract the low-level numpy arrays into objects of class SO2, SE2, SO3, SE3, UnitQuaternion that obey the rules associated with the mathematical groups SO(2), SE(2), SO(3), SE(3) and H. Using classes has several merits:

- ensures type safety, for example it stops us mixing a 2D homogeneous transformation with a 3D rotation matrix – both of which are 3x3 matrices.

- ensure that an SO(2), SO(3) or unit-quaternion rotation is always valid because the constraints (eg. orthogonality, unit norm) are enforced when the object is constructed.

    >>> from spatialmath import *
    >>> SO2(.1)
    [[ 0.99500417 -0.09983342]
     [ 0.09983342  0.99500417]]

Type safety and type validity are particularly important when we deal with a sequence of such objects. In robotics we frequently deal with trajectories of poses or rotation to describe objects moving in the world. However a list of these items has the type list and the elements are not enforced to be homogeneous, ie. a list could contain a mixture of classes. Another option would be to create a numpy array of these objects, the upside being it could be a multi-dimensional array. The downside is that again the array is not guaranteed to be homogeneous.

The approach adopted here is to give these classes list super powers so that a single SE3 object can contain a list of SE(3) poses. The pose objects are a list subclass so we can index it or slice it as we would a list, but the result each time belongs to the class it was sliced from. Here’s a simple example of SE(3) but applicable to all the classes

    T = transl(1,2,3) # create a 4x4 np.array

    a = SE3(T)
    len(a)
    type(a)
    a.append(a)  # append a copy
    a.append(a)  # append a copy
    type(a)
    len(a)
    a[1]  # extract one element of the list
    for x in a:
      # do a thing

These classes are all derived from two parent classes:

- RTBPose which provides common functionality for all

- UserList which provdides the ability to act like a list

### Operators for pose objects

Standard arithmetic operators can be applied to all these objects.

Operator | dunder method
--- | ---
`*` | __mul__ , __rmul__
`*=` | __imul__
`/` | __truediv__
`/=` | __itruediv__
`**` | __pow__
`**=` | __ipow__
`+` | __add__, __radd__
`+=` | __iadd__
`-` | __sub__, __rsub__
`-=` | __isub__

This online documentation includes just the method shown in bold. The other related methods all invoke that method.

The classes represent mathematical groups, and the rules of group are enforced. If this is a group operation, ie. the operands are of the same type and the operator is the group operator, the result will be of the input type, otherwise the result will be a matrix.

#### SO(n) and SE(n)

For the groups SO(n) and SE(n) the group operator is composition represented by the multiplication operator. The identity element is a unit matrix.

Operands | `*`
--- | ---
left | right | type | result
Pose | Pose | Pose | composition [1]
Pose | scalar | matrix | elementwise product
scalar | Pose | matrix | elementwise product
Pose | N-vector | N-vector | vector transform [2]
Pose | NxM matrix | NxM matrix | vector transform [2] [3]

Notes:

1. Composition is performed by standard matrix multiplication.

2. N=2 (for SO2 and SE2), N=3 (for SO3 and SE3).

3. Matrix columns are taken as the vectors to transform.

Operands | `/`
--- | ---
left | right | type | result
Pose | Pose | Pose | matrix * inverse #1
Pose | scalar | matrix | elementwise product
scalar | Pose | matrix | elementwise product

Notes:

1. The left operand is multiplied by the `.inv` property of the right operand.

Operands | `**`
--- | ---
left | right | type | result
Pose | int >= 0 | Pose | exponentiation [1]
Pose | int <=0 | Pose | exponentiation [1] then inverse

Notes:

1. By repeated multiplication.

Operands | `+`
--- | ---
left | right | type | result
Pose | Pose | matrix | elementwise sum
Pose | scalar | matrix | add scalar to all elements
scalar | Pose | matrix | add scalarto all elements

Operands | `-`
--- | ---
left | right | type | result
Pose | Pose | matrix | elementwise difference
Pose | scalar | matrix | subtract scalar from all elements
scalar | Pose | matrix | subtract all elements from scalar

#### Unit quaternions and quaternions

Quaternions form a ring and support the operations of multiplication, addition and subtraction. Unit quaternions form a group and the group operator is composition represented by the multiplication operator.

Operands | `*`
--- | ---
left | right | type | result
Quaternion | Quaternion | Quaternion | Hamilton product
Quaternion | UnitQuaternion | Quaternion | Hamilton product
Quaternion | scalar | Quaternion | scalar product #2
UnitQuaternion | Quaternion | Quaternion | Hamilton product
UnitQuaternion | UnitQuaternion | UnitQuaternion | Hamilton product #1
UnitQuaternion | scalar | Quaternion | scalar product #2
UnitQuaternion | 3-vector | 3-vector | vector rotation #3
UnitQuaternion | 3xN matrix | 3xN matrix | vector transform #2#3

Notes:

1. Composition.

2. N=2 (for SO2 and SE2), N=3 (for SO3 and SE3).

3. Matrix columns are taken as the vectors to transform.

Operands | `/`
--- | ---
left | right | type | result
UnitQuaternion | UnitQuaternion | UnitQuaternion | Hamilton product with inverse #1

Notes:

1. The left operand is multiplied by the `.inv` property of the right operand.

Operands | `**`
--- | ---
left | right | type | result
Quaternion | int >= 0 | Quaternion | exponentiation [1]
UnitQuaternion | int >= 0 | UnitQuaternion | exponentiation [1]
UnitQuaternion | int <=0 | UnitQuaternion | exponentiation [1] then inverse

Notes:

1. By repeated multiplication.

Operands | `+`
--- | ---
left | right | type | result
Quaternion | Quaternion | Quaternion | elementwise sum
Quaternion | UnitQuaternion | Quaternion | elementwise sum
Quaternion | scalar | Quaternion | add to each element
UnitQuaternion | Quaternion | Quaternion | elementwise sum
UnitQuaternion | UnitQuaternion | Quaternion | elementwise sum
UnitQuaternion | scalar | Quaternion | add to each element

Operands | `-`
--- | ---
left | right | type | result
Quaternion | Quaternion | Quaternion | elementwise difference
Quaternion | UnitQuaternion | Quaternion | elementwise difference
Quaternion | scalar | Quaternion | subtract scalar from each element
UnitQuaternion | Quaternion | Quaternion | elementwise difference
UnitQuaternion | UnitQuaternion | Quaternion | elementwise difference
UnitQuaternion | scalar | Quaternion | subtract scalar from each element

Any other operands will raise a `ValueError` exception.

### List capability

Each of these object classes has `UserList` as a base class which means it inherits all the functionality of a Python list

    >>> R = SO3.Rx(0.3)
    >>> len(R)
       1

    >>> R = SO3.Rx(np.arange(0, 2*np.pi, 0.2)))
    >>> len(R)
      32
    >> R[0]
       1         0         0
       0         1         0
       0         0         1
    >> R[-1]
       1         0         0
       0         0.996542  0.0830894
       0        -0.0830894 0.996542

where each item is an object of the same class as that it was extracted from. Slice notation is also available, eg. `R[0:-1:3]` is a new SO3 instance containing every third element of `R`.

In particular it includes an iterator allowing comprehensions

    >>> [x.eul for x in R]
    [array([ 90.        ,   4.76616702, -90.        ]),
     array([ 90.        ,  16.22532292, -90.        ]),
     array([ 90.        ,  27.68447882, -90.        ]),
       .
       .
     array([-90.       ,  11.4591559,  90.       ]),
     array([0., 0., 0.])]

Useful functions that be used on such objects include

Method | Operation
--- | ---
`clear` | Clear all elements, object now has zero length
`append` | Append a single element
`del` |
`enumerate` | Iterate over the elments
`extend` | Append a list of same type pose objects
`insert` | Insert an element
`len` | Return the number of elements
`map` | Map a function of each element
`pop` | Remove first element and return it
`slice` | Index from a slice object
`zip` | Iterate over the elments

### Vectorization

For most methods, if applied to an object that contains N elements, the result will be the appropriate return object type with N elements.

Most binary operations (*, *=, **, +, +=, -, -=, ==, !=) are vectorized. For the case:

    Z = X op Y

the lengths of the operands and the results are given by

operands | results
--- | ---
len(X) | len(Y) | len(Z) | results
1 | 1 | 1 | Z = X op Y
1 | M | M | Z[i] = X op Y[i]
M | 1 | M | Z[i] = X[i] op Y
M | M | M | Z[i] = X[i] op Y[i]

Any other combination of lengths is not allowed and will raise a `ValueError` exception.

## Low-level spatial math

All the classes just described abstract the `base` package which represent the spatial-math object as a numpy.ndarray.

The inputs to functions in this package are either floats, lists, tuples or numpy.ndarray objects describing vectors or arrays. Functions that require a vector can be passed a list, tuple or numpy.ndarray for a vector – described in the documentation as being of type array_like.

Numpy vectors are somewhat different to MATLAB, and is a gnarly aspect of numpy. Numpy arrays have a shape described by a shape tuple which is a list of the dimensions. Typically all `np.ndarray` vectors have the shape (N,), that is, they have only one dimension. The `@` product of an (M,N) array and a (N,) vector is a (M,) array. A numpy column vector has shape (N,1) and a row vector has shape (1,N) but functions also accept row (1,N) and column (N,1) vectors. Iterating over a numpy.ndarray is done by row, not columns as in MATLAB. Iterating over a 1D array (N,) returns consecutive elements, iterating a row vector (1,N) returns the entire row, iterating a column vector (N,1) returns consecutive elements (rows).

For example an SE(2) pose is represented by a 3x3 numpy array, an ndarray with shape=(3,3). A unit quaternion is represented by a 4-element numpy array, an ndarray with shape=(4,).

Spatial object | equivalent class | numpy.ndarray shape
--- | --- | ---
2D rotation SO(2) | SO2 | (2,2)
2D pose SE(2) | SE2 | (3,3)
3D rotation SO(3) | SO3 | (3,3)
3D poseSE3 SE(3) | SE3 | (3,3)
3D rotation | UnitQuaternion | (4,)
n/a | Quaternion | (4,)

Tjhe classes `SO2`, ``SE2`, ``SO3`, `SE3`, `UnitQuaternion` can operate conveniently on lists but the `base` functions do not support this. If you wish to work with these functions and create lists of pose objects you could keep the numpy arrays in high-order numpy arrays (ie. add an extra dimensions), or keep them in a list, tuple or any other python contai described in the [high-level spatial math section](#high-level-classes).

Let’s show a simple example:

     1
     2
     3
     4
     5
     6
     7
     8
     9
    10
    11
    12
     >>> import spatialmath.base.transforms as base
     >>> base.rotx(0.3)
     array([[ 1.        ,  0.        ,  0.        ],
            [ 0.        ,  0.95533649, -0.29552021],
            [ 0.        ,  0.29552021,  0.95533649]])

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

     >>> R = base.rotx(0.3) @ base.roty(0.2)

At line 1 we import all the base functions into the namespae `base`. In line 12 when we multiply the matrices we need to use the @ operator to perform matrix multiplication. The * operator performs element-wise multiplication, which is equivalent to the MATLAB `.*` operator.

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

- or as a numpy array

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

There is a single module that deals with quaternions, regular quaternions and unit quaternions, 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.

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

Functions exist to convert to and from SO(3) rotation matrices and a 3-vector representation. The latter is often used for SLAM and bundle adjustment applications, being a minimal representation of orientation.

### Graphics

If `matplotlib` is installed then we can add 2D coordinate frames to a figure in a variety of styles:

    1
    2
    3
    4
     trplot2( transl2(1,2), frame='A', rviz=True, width=1)
     trplot2( transl2(3,1), color='red', arrow=True, width=3, frame='B')
     trplot2( transl2(4, 3)@trot2(math.pi/3), color='green', frame='c')
     plt.grid(True)

Output of `trplot2`

If a figure does not yet exist one is added. If a figure exists but there is no 2D axes then one is added. To add to an existing axes you can pass this in using the `axes` argument. By default the frames are drawn with lines or arrows of unit length. Autoscaling is enabled.

Similarly, we can plot 3D coordinate frames in a variety of styles:

    1
    2
    3
     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])

Output of `trplot`

The `dims` option in lines 1 and 3 sets the workspace dimensions. Note that the last set value is what is displayed.

Depending on the backend you are using you may need to include

    plt.show()

### 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 &ndash; the constants are also symbolic objects. You can read the elements of the matrix

    >>> a = T[0,0]
    >>> a
      1
    >>> type(a)
     int

    >>> a = T[1,1]
    >>> a
    cos(theta)
    >>> type(a)
     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.

    >>> T[0,3] = 22
    >>> print(T)
    [[1 0 0 22]
     [0 cos(theta) -sin(theta) 0]
     [0 sin(theta) cos(theta) 0]
     [0 0 0 1]]

but you can’t write a symbolic value into a floating point matrix

    >>> T = trotx(0.2)

    >>> T[0,3]=theta
    Traceback (most recent call last):
      .
      .
    TypeError: can't convert expression to float

### MATLAB compatability

We can create a MATLAB like environment by

    from spatialmath  import *
    from spatialmath.base  import *

which has familiar functions like `rotx` and `rpy2r` available, as well as classes like `SE3`

    R = rotx(0.3)
    R2 = rpy2r(0.1, 0.2, 0.3)

    T = SE3(1, 2, 3)