.. _mobilePose:

Tracking the Robot Pose
===========================

.. topic:: Pose Equations

    For each motor:
        
        .. math:: \Delta tick = tick - old\_tick

        .. math:: D = 2\,\pi\,R\frac{\Delta tick}{N}

    To update the pose:

    .. math:: d_{center} = \frac{d_{left} + d_{right}}{2}

    .. math:: \phi = \frac{d_{right} - d_{left}}{L}

    .. math:: \mathcal{P}_{i+1} = \left[ \begin{array}{c}
            X_i \\ Y_i \\ \theta_i \end{array} \right]
             + \left[ \begin{array}{c}
             d_{center}\,\cos \theta_i\\ 
             d_{center}\,\sin \theta_i\\ 
             \phi
             \end{array} \right]

Two Frames of Reference
---------------------------

One of the *gotchas* of robotics is keeping track of which frame of reference
is being used. The **global** coordinate frame is relative to the robot's
surrounding. We start robotics experiments with the robot at position (0, 0)
and facing in the direction of the :math:`X` axis (:math:`\theta = 0`).  The
**local** coordinate frame is relative to the robot. We use the local
coordinates when measuring the movement or setting the velocity of the wheels.

.. figure:: kinematic1.png
   :align: center

.. _poseDef:

The Robot's Global Position
-------------------------------

.. index:: pose, global coordinate

We call the robot's location and orientation in the global coordinate frame the
position, or **pose**, of the robot. The robot's position is represented as a
vector (:math:`\mathcal{P}`) containing its :math:`x` and :math:`y` Cartesian
coordinate pair and its angle of orientation (:math:`\theta`).

.. math:: \mathcal{P} = \left[ \begin{array}{c}
                                x \\ y \\ \theta
                                \end{array} \right]
                                

.. toctree::
   :maxdepth: 2

   encoders
   odometry