clean up localization docs (#570)

* clean up localization docs

* clean up localization docs

* clean up localization docs

docs/modules/appendix/Kalmanfilter_basics.rst

@@ -222,9 +222,8 @@ described with two parameters, the mean (:math:`\mu`) and the variance
     
    f(x, \mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi}} \exp\big [{-\frac{(x-\mu)^2}{2\sigma^2} }\big ]
 
- Range is
 
-.. math:: [-\inf,\inf] 
+Range is :math:`[-\inf,\inf]`
 
 This is just a function of mean(\ :math:`\mu`) and standard deviation
 (:math:`\sigma`) and what gives the normal distribution the
@@ -279,7 +278,8 @@ New mean is
 
 .. math:: \mu_\mathtt{new} = \frac{\sigma_z^2\bar\mu + \bar\sigma^2z}{\bar\sigma^2+\sigma_z^2}
 
- New variance is
+
+New variance is
 
 .. math::
 
@@ -336,7 +336,7 @@ of the two.
 .. math::
 
    \begin{gathered}\mu_x = \mu_p + \mu_z \\
-   \sigma_x^2 = \sigma_z^2+\sigma_p^2\, \square\end{gathered}
+   \sigma_x^2 = \sigma_z^2+\sigma_p^2\, \end{gathered}
 
 .. code-block:: ipython3
 

docs/modules/localization/extended_kalman_filter_localization.rst

@@ -2,14 +2,6 @@
 Extended Kalman Filter Localization
 -----------------------------------
 
-.. code-block:: ipython3
-
-    from IPython.display import Image
-    Image(filename="ekf.png",width=600)
-
-
-
-
 .. image:: extended_kalman_filter_localization_files/extended_kalman_filter_localization_1_0.png
    :width: 600px
 
@@ -18,8 +10,6 @@ Extended Kalman Filter Localization
 .. figure:: https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/Localization/extended_kalman_filter/animation.gif
    :alt: EKF
 
-   EKF
-
 This is a sensor fusion localization with Extended Kalman Filter(EKF).
 
 The blue line is true trajectory, the black line is dead reckoning
@@ -127,7 +117,7 @@ The observation function states that
 
 Its Jacobian matrix is
 
-:math:`\begin{equation*} J_g = \begin{bmatrix} \frac{\partial x'}{\partial x} & \frac{\partial x'}{\partial y} & \frac{\partial x'}{\partial \phi} & \frac{\partial x'}{\partial v}\\ \frac{\partial y'}{\partial x}& \frac{\partial y'}{\partial y} & \frac{\partial y'}{\partial \phi} & \frac{\partial y'}{ \partialv}\\ \end{bmatrix} \end{equation*}`
+:math:`\begin{equation*} J_g = \begin{bmatrix} \frac{\partial x'}{\partial x} & \frac{\partial x'}{\partial y} & \frac{\partial x'}{\partial \phi} & \frac{\partial x'}{\partial v}\\ \frac{\partial y'}{\partial x}& \frac{\partial y'}{\partial y} & \frac{\partial y'}{\partial \phi} & \frac{\partial y'}{ \partial v}\\ \end{bmatrix} \end{equation*}`
 
 :math:`\begin{equation*} 　= \begin{bmatrix} 1& 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ \end{bmatrix} \end{equation*}`
 

docs/modules/localization/localization.rst

@@ -15,7 +15,8 @@ This is a sensor fusion localization with Unscented Kalman Filter(UKF).
 
 The lines and points are same meaning of the EKF simulation.
 
-Ref:
+References:
+~~~~~~~~~~~
 
 -  `Discriminatively Trained Unscented Kalman Filter for Mobile Robot
    Localization`_
@@ -37,9 +38,26 @@ It is assumed that the robot can measure a distance from landmarks
 
 This measurements are used for PF localization.
 
-Ref:
+How to calculate covariance matrix from particles
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+The covariance matrix :math:`\Xi` from particle information is calculated by the following equation:
+
+.. math:: \Xi_{j,k}=\frac{1}{1-\sum^N_{i=1}(w^i)^2}\sum^N_{i=1}w^i(x^i_j-\mu_j)(x^i_k-\mu_k)
+
+- :math:`\Xi_{j,k}` is covariance matrix element at row :math:`i` and column :math:`k`.
+
+- :math:`w^i` is the weight of the :math:`i` th particle.
+
+- :math:`x^i_j` is the :math:`j` th state of the :math:`i` th particle.
+
+- :math:`\mu_j` is the :math:`j` th mean state of particles.
+
+References:
+~~~~~~~~~~~
 
 -  `PROBABILISTIC ROBOTICS`_
+-  `Improving the particle filter in high dimensions using conjugate artificial process noise`_
 
 Histogram filter localization
 -----------------------------
@@ -59,12 +77,14 @@ localization.
 
 Initial position is not needed.
 
-Ref:
+References:
+~~~~~~~~~~~
 
 -  `PROBABILISTIC ROBOTICS`_
 
 .. _PROBABILISTIC ROBOTICS: http://www.probabilistic-robotics.org/
 .. _Discriminatively Trained Unscented Kalman Filter for Mobile Robot Localization: https://www.researchgate.net/publication/267963417_Discriminatively_Trained_Unscented_Kalman_Filter_for_Mobile_Robot_Localization
+.. _Improving the particle filter in high dimensions using conjugate artificial process noise: https://arxiv.org/pdf/1801.07000.pdf
 
 .. |2| image:: https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/Localization/unscented_kalman_filter/animation.gif
 .. |3| image:: https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/Localization/particle_filter/animation.gif