
************
Introduction
************


Spatial maths capability underpins all of robotics and robotic vision. 
It provides the means to describe the relative position and orientation of objects in 2D or 3D space.  
This package provides Python classes and functions to represent, print, plot, manipulate and covert between such representations.
This includes relevant mathematical objects such as rotation matrices :math:`\mat{R} \in \SO{2}, \SO{3}`, 
homogeneous transformation matrices :math:`\mat{T} \in \SE{2}, \SE{3}`, unit quaternions :math:`\q \in \mathrm{S}^3`,
and twists :math:`S \in \se{2}, \se{3}`.

For example, we can create a rigid-body transformation that is a rotation about the x-axis of 30 degrees:

.. runblock:: pycon

      >>> from spatialmath.base import *
      >>> rotx(30, 'deg')


which results in a NumPy :math:`4 \times 4` array that belongs to the group
:math:`\SE{3}`.  We could also create a class instance:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> T = SE3.Rx(30, 'deg')
    >>> type(T)
    >>> print(T)                

which is *internally* represented as a :math:`4 \times 4` NumPy array.

While functions and classes can provide similar functionality the class provide the benefits of:

- type safety, it is not possible to mix a 3D rotation matrix with a 2D rigid-body motion, even though both are represented
  by a :math:`3 \times 3` matrix
- operator overloading allows for convenient and readable expression of algorithms
- representing not a just a single value, but a sequence of values, which are handled by the operators with implicit broadcasting of values


Spatial math classes
====================

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

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

Additional classes include:

- ``Quaternion`` a general quaternion, and parent class to ``UnitQuaternion``
- ``Plucker`` to represent a line in 3D space
- ``Plane`` to represent a plane in 3D space

These classes abstract, and implement appropriate operations, for the following
groups:

================================  ==============================  ======================
Group                             Name                            Class
================================  ==============================  ======================
:math:`\SE{3}`                    rigid-body transformaton in 3D  ``SE3``
:math:`\se{3}`                    twist in 3D                     ``Twist3``
:math:`\SO{3}`                    orientation in 3D               ``SO3``
:math:`\mathrm{S}^3`              unit quaternion                 ``UnitQuaternion``
:math:`\SE{2}`                    rigid-body transformaton in 2D  ``SE2``
:math:`\se{2}`                    twist in 2D                     ``Twist2``
:math:`\SO{2}`                    orientation in 2D               ``SO2``
:math:`\mathbb{H}`                quaternion                      ``Quaternion``
:math:`P^5`                       Plücker lines                   ``Plucker``
:math:`M^6`                       spatial velocity                ``SpatialVelocity``
:math:`M^6`                       spatial acceleration            ``SpatialAcceleration``
:math:`F^6`                       spatial force                   ``SpatialForce``
:math:`F^6`                       spatial momentum                ``SpatialMomentum``
:math:`\mathbb{R}^{6 \times 6}`   spatial inertia                 ``SpatialInertia``
================================  ==============================  ======================


In addition to the merits of classes outlined above, classes ensure that the numerical value is always valid because the 
constraints (eg. orthogonality, unit norm) are enforced when the object is constructed.  For example::

  >>> SE3(np.zeros((4,4)))
  Traceback (most recent call last):
    .
    .
  AssertionError: array must have valid value for the class

Type safety and type validity are particularly important when we deal with a sequence of values.  
In robotics we frequently deal with a multiplicity of objects (poses, cameras), or a trajectory of
objects moving over time.
However a list of these items, for example::

  >>> X = [SE3.Rx(0), SE3.Rx(0.2), SE3.Rx(0.4), SE3.Rx(0.6)]

has the type `list` and the elements are not guaranteed to be homogeneous, ie. a list could contain a mixture of classes.
This requires careful coding, or additional user code to check the validity of all elements in the list.
We could create a NumPy array of these objects, the upside being it could be more than one-dimensional, but again NumPy does not
enforce homogeneity of objects within an array (with ``dtype='O'``).

.. image:: ../figs/pose-values.png
      :width: 600
      :alt: A SpatialMath pose class can hold multiple values


The Spatial Math package give these classes list *super powers* so that, for example, a single `SE3` object can contain a sequence of SE(3) values:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> X = SE3.Rx([0, 0.2, 0.4, 0.6])
    >>> len(X)
    >>> print(X[1])          


The classes form a rich hierarchy

.. inheritance-diagram:: spatialmath.SE3 spatialmath.SO3 spatialmath.SE2 spatialmath.SO2 spatialmath.Twist3 spatialmath.Twist2 spatialmath.UnitQuaternion spatialmath.spatialvector
    :top-classes: collections.UserList
    :parts: 1


Ultimately they all inherit from ``collections.UserList`` and have all the functionality of Python lists, and this is discussed further in
section :ref:`list-powers`
The pose objects are a list subclass so we can index it or slice it as we
would a list, but the result always belongs to the class it was sliced from.  


Operators for pose objects
--------------------------

Group operations
^^^^^^^^^^^^^^^^

The classes represent mathematical groups, and the group arithmetic rules are enforced.
The operator ``*`` denotes composition and the result will be of the same type as the operand:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> T = SE3.Rx(0.3)
    >>> type(T)
    >>> X = T * T
    >>> type(X)

The implementation of composition depends on the class:

* for SO(n) and SE(n) composition is imatrix multiplication of the underlying matrix values,
* for unit-quaternions composition is the Hamilton product of the underlying vector value,
* for twists it is the logarithm of the product of exponentiating the two twists

The ``**`` operator denotes repeated composition, so the exponent must be an integer.  If the negative exponent the repeated multiplication
is performed then the inverse is taken.

The group inverse is given by the ``inv()`` method:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> T = SE3.Rx(0.3)
    >>> T * T.inv()

and ``/`` denotes multiplication by the inverse:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> T = SE3.Rx(0.3)
    >>> T / T

Vector transformation
^^^^^^^^^^^^^^^^^^^^^


The classes ``SE3``, ``SO3``, ``SE2``, ``SO2`` and ``UnitQuaternion`` support vector transformation when 
premultiplying a vector (or a set of vectors columnwise in a NumPy array) using the ``*`` operator.
This is either rotation about the origin (for ``SO3``, ``SO2`` and ``UnitQuaternion``) or rotation and translation (``SE3``, ``SE2``).  
The implementation depends on the class of the object involved:

- for ``UnitQuaternion`` this is performed directly using Hamilton products
  :math:`\q \circ \mathring{v} \circ \q^{-1}`.
- for ``SO3`` and ``SO2`` this is a matrix-vector product
- for ``SE3`` and ``SE2`` this is a matrix-vector product with the vectors
  being first converted to homogeneous form, and the result converted back to
  Euclidean form.

.. runblock:: pycon

    >>> from spatialmath import *
    >>> v = [1, 2, 3]
    >>> SO3.Rx(0.3) * v
    >>> SE3.Rx(0.3) * v
    >>> UnitQuaternion.Rx(0.3) * v

Non-group operations
^^^^^^^^^^^^^^^^^^^^

Addition, subtraction and scalar multiplication are not defined group operations
so the result will be a NumPy array rather than a class. The operations are
performed elementwise, for example:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> T = SE3.Rx(0.3)
    >>> T - T

or, in the case of a scalar, broadcast to each element:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> T - 1
    >>> 2 * T

The exception is the ``Quaternion`` class which supports these since a
quaternion is a ring not a group:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> q = Quaternion([1, 2, 3, 4])
    >>> 2 * q

Compare this to the unit quaternion case:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> q = UnitQuaternion([1, 2, 3, 4])
    >>> q
    >>> 2 * q

Noting that unit quaternions are denoted by double angle bracket delimiters of their vector part,
whereas a general quaternion uses single angle brackets.  The product of a general quaternion and a 
unit quaternion is always a general quaternion.


Displaying values
-----------------

Each class has a compact text representation via its ``__repr__`` method and its
``str()`` method. The ``printline()`` methods prints a single-line for tabular
listing to the console, file and returns a string:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> X = SE3.Rand()
    >>> _ = X.printline()

The classes ``SE3``, ``SO3``, ``SE2`` and ``SO2`` can provide colorized text output to the console::

  >>> T = SE3()
  >>> T.print()

.. image:: ../figs/colored_output.png
   :width: 300
with rotational elements in red, translational elements in blue and constants in grey.

The foreground and background colors can be controlled using the following
class variables for the ``SMPose`` subclasses

===============  ===================  ============================================
Variable         Default              Description
===============  ===================  ============================================
_color           True                 Enable all colorization
_rotcolor        'red'                Foreground color of rotation submatrix
_transcolor      'blue'               Foreground color of rotation submatrix
_constcolor      'grey_50'            Foreground color of matrix constant elements
_bgcolor         None                 Background color of matrix
_indexcolor      (None, 'yellow_2')   Foreground, background color of index tag
_format          '{:< 12g}'           Format string for each matrix element
_suppress_small  True                 Suppress *small* values, set to zero
_suppress_tol    100                  Threshold for *small* values in eps units
_ansimatrix      False                Display as a matrix with brackets
===============  ===================  ============================================

For example::

    >>> SE3._rotcolor = 'green'   # rotation part in green

or to supress color, perhaps for inclusion in documentation::

  >>> SE3._color = False


Graphics
--------

Each class has a ``plot`` method that displays the corresponding pose as a coordinate frame, for example::

  >>> X = SE3.Rand()
  >>> X.plot()

.. image:: ../figs/fig1.png


and there are many display options.

The ``animate`` method animates the motion of the coordinate frame from the null-pose, for example:

  >>> X = SE3.Rand()
  >>> X.animate(frame='A', arrow=False)

.. image:: ../figs/animate.gif


Constructors
------------

The constructor for each class can accept:

* no arguments, in which case the identity element is created:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> UnitQuaternion()
    >>> SE3()

* class specific values, eg. ``SE2(x, y, theta)`` or ``SE3(x, y, z)``, for example:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> SE2(1, 2, 0.3)
    >>> UnitQuaternion([1, 0, 0, 0])

* a numeric value for the class as a NumPy array or a 1D list or tuple which will be checked for validity:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> import numpy as np
    >>> SE2(np.identity(3))

* a list of numeric values, each of which will be checked for validity:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> import numpy as np
    >>> X = SE2([np.identity(3), np.identity(3), np.identity(3), np.identity(3)])
    >>> X
    >>> len(X)

Other constructors are implemented as class methods and are common to ``SE3``,
``SO3``, ``Twist3``, ``SE2``, ``SO2`` ``Twist2`` and ``UnitQuaternion`` and
begin with an uppercase letter:

===========   ==================================================
Constructor   Meaning
===========   ==================================================
Rx            Pure rotation about the x-axis
Ry            Pure rotation about the y-axis
Rz            Pure rotation about the z-axis
RPY           specified as roll-pitch-yaw angles
Eul           specified as Euler angles
AngVec        specified as rotational axis and rotation angle
Rand          random rotation
Exp           specified as se(2) or se(3) matrix
empty         no values
Alloc         N identity values
===========   ==================================================

.. _list-powers:

List capability
---------------

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

.. runblock:: pycon

    >>> from spatialmath import *
    >>> import numpy as np
    >>> R = SO3.Rx(0.3)
    >>> len(R)
    >>> R = SO3.Rx(np.arange(0, 2*np.pi, 0.2))
    >>> len(R)
    >>> R[0]  
    >>> R[-1]
    >>> R[2:4]

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 supports iteration which allows looping and comprehensions:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> import numpy as np
    >>> R = SO3.Rx(np.arange(0, 2*np.pi, 0.2))
    >>> len(R)
    >>> eul = [x.eul() for x in R]
    >>> len(eul)
    >>> eul[10]
  
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
-------------

.. image:: ../figs/broadcasting.png

For most methods, if applied to an object that contains N elements, the result
will be the appropriate return object type with N elements.  In MATLAB this is
referred to as *vectorization* and in NumPy as *broadcasting*.

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.

In addition:

- ``Plucker`` objects support the ``^`` and ``|`` operators to test intersection
  and parallelity respectively.

- ``SpatialVector`` subclass objects support the ``^`` operator to indicate the
  spatial vector cross product.

Symbolic operations
-------------------

The Toolbox supports SymPy which provides powerful symbolic support for Python
and it works well in conjunction with NumPy, ie. a NumPy array can contain
symbolic elements.  Many the Toolbox methods and functions contain extra logic
to ensure that symbolic operations work as expected. While this also adds to the
overhead it means that for the user, working with symbols is as easy as working
with numbers.  For example:

.. runblock:: pycon

    >>> from spatialmath import *
    >>> import spatialmath.base.symbolic as sym
    >>> theta = sym.symbol('theta')
    >>> SE3.Rx(theta)

SymPy allows any expression to be converted to runnable code in a variety of
languages including C, Python and Octave/MATLAB.

Implementation
--------------

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

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

Low-level spatial math
======================

The classes above abstract the ``base`` package which represent the spatial-math
types as 1D and 2D  arrays implemented by NumPy n-dimensional arrays - the
``ndarray`` class.

=================    ================   ===================
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,)
=================    ================   ===================

.. note:: ``SpatialVector`` and ``Plucker`` objects have no equivalent in the 
  ``base`` package.

Inputs to functions in this package are either floats, lists, tuples or
numpy.ndarray objects describing vectors or arrays.  

NumPy arrays have a shape described by a shape tuple which is a list of the
dimensions.  Typically all ``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 an ``(M,)`` vector.  

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)`` 
where a vector argument is required.  

.. warning:: For a user transitioning from MATLAB the most significant
  differences are:
    - the use of 1D arrays -- all MATLAB arrays have two dimensions,
      even if one of them is equal to one.
    - Iterating over a 1D NumPy array (N,) returns consecutive elements
    - Iterating over a 2D NumPy array is done by row, not columns as in MATLAB.  
    - Iterating over a row vector ``(1,N)`` returns the entire row
    - Iterating a column vector ``(N,1)`` returns consecutive elements (rows).

.. note::
  - Functions that require vector can be passed a list, tuple or numpy.ndarray
    for a vector -- described in the documentation as being of type
    *array_like*. 
  - This toolbox documentation refers to NumPy arrays succinctly as:

    - ``ndarray(N)`` for a 1D array of length ``N``
    - ``ndarray(N,M)`` for a 2D array of dimension :math:`N \times M`.

The 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 container described in the
`high-level spatial math section` <#high-level-classes>`_.

Let's show a simple example:

.. runblock:: pycon
    :linenos:

    >>> from spatialmath.base import *
    >>> rotx(0.3)
    >>> rotx(30, unit='deg')
    >>> R = rotx(0.3) @ roty(0.2)
    >>> R

At line 1 we import all the base functions into the current namespace. In line
10 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

.. runblock:: pycon

    >>> from spatialmath.base import *
    >>> transl2(1, 2)

* as a list or a tuple

.. runblock:: pycon

    >>> from spatialmath.base import *
    >>> transl2( [1,2] )
    >>> transl2( (1,2) )

* or as a NumPy array

.. runblock:: pycon

    >>> from spatialmath.base import *
    >>> transl2( np.array([1,2]) )

There is a single module that deals with quaternions, regular quaternions and
unit quaternions. In both cases, the representation is a NumPy array of four elements.
As above, functions can accept a NumPy array, a list, dict or NumPy row or
column vectors.

.. runblock:: pycon

    >>> from spatialmath.base import *
    >>> q = quat.qqmul([1,2,3,4], [5,6,7,8])
    >>> q
    >>> qprint(q)
    >>> qnorm(q)

Functions exist to convert to and from SO(3) rotation matrices and a minimal 3-vector
quaternion 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:

.. code-block:: python
   :linenos:

    >>> 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)

.. figure:: ../figs/transforms2d.png 
   :align: center

   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:

.. code-block:: python
   :linenos:

    >>> 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])

.. figure:: ../figs/transforms3d.png
   :align: center

   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

.. code-block:: python

    >>> plt.show()


Symbolic support
----------------

Some functions have support for symbolic variables, for example:

.. runblock:: pycon

    >>> from spatialmath.base import *
    >>> import spatialmath.base.symbolic as sym
    >>> theta = sym.symbol('theta')
    >>> print(rotx(theta))

The resulting NumPy array is an array of symbolic objects not numbers --
the constants are also symbolic objects.  You can slice out the elements of the
matrix

.. runblock:: pycon

    >>> from spatialmath.base import *
    >>> import spatialmath.base.symbolic as sym
    >>> theta = sym.symbol('theta')
    >>> T = rotx(theta)
    >>> a = T[0,0]
    >>> a
    >>> type(a)
    >>> a = T[1,1]
    >>> a 
    >>> type(a)

We see that the symbolic constants have been converted back to Python numeric
types.

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.

.. runblock:: pycon

    >>> from spatialmath.base import *
    >>> import spatialmath.base.symbolic as sym
    >>> theta = sym.symbol('theta')
    >>> T = trotx(theta)
    >>> T[0,3] = 22
    >>> print(T)

.. warning:: You can't write a symbolic value directly into a floating point matrix (ie.
  one whose ``dtype`` is ``np.float64`` or similar). The array must be first converted
  to *object type* using ``T = T.astype('O')``.

.. note:: Not all functions support symbolic operations.  For those that do,
  this is noted in the last line of the docstring: **SymPy: support**

Relationship to MATLAB tools
----------------------------

This package replicates, as much as possible, the functionality of the `Spatial
Math Toolbox  <https://github.com/petercorke/spatial-math>`__ for MATLAB®
which underpins the `Robotics Toolbox
<https://github.com/petercorke/robotics-toolbox-matlab>`__ for MATLAB®. It
comprises:

* the *classic* functions (which date back to the origin of the Robotics Toolbox
  for MATLAB) such as ``rotx``, ``trotz``, ``eul2tr`` etc. which can be imported
  from the ``base`` package::

    >>> from spatialmath.base import rotx, trotx

  and works with NumPy arrays.  This package also includes a set of functions,
  not present in the MATLAB version, to handle quaternions, unit-quaternions 
  which are represented as 4-element NumPy arrays, and twists.
* the classes (which appeared in Robotics Toolbox for MATLAB release 10 in 2017) such as ``SE3``, ``UnitQuaternion`` etc.  The only significant difference
  is that the MATLAB ``Twist`` class is now called ``Twist3``.

The design considerations included:

  - being as similar as possible to the MATLAB Toolbox function names and semantics
  - while balancing the tension of being as Pythonic as possible
  - using Python keyword arguments to replace the MATLAB Toolbox string options supported using ``tb_optparse()``
  - using NumPy arrays internally to represent rotation and homogeneous transformation matrices, quaternions, twists and vectors
  - allowing all functions that accept a vector can accept a list, tuple, or NumPy array
  - allowing a class instance can hold a sequence of elements, they are polymorphic with lists, which can be used to represent trajectories or time  sequences
  - having classes that are *generally* polymorphic, ie. they share many common constructor options and methods

.. note::  None of the functions in the ``base`` package are *vectorized*, whereas many of the MATLAB
           equivalents are.  Vectorization is done by the classes.

Creating a MATLAB-like environment in Python
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

We can create a MATLAB-like environment by

.. code-block:: python

    >>> from spatialmath  import *
    >>> from spatialmath.base  import *

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

.. code-block:: python

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

    T = SE3(1, 2, 3)



