Standardize "Ref:" to "Reference" across files (#1210)

Updated all instances of "Ref:" to "Reference" for consistency in both code and documentation. This change improves clarity and aligns with standard terminology practices.
2026-01-11 07:18:00 -05:00 · 2025-05-02 10:01:19 +09:00
parent a2c42c3d68
commit 22cbee4921
45 changed files with 179 additions and 161 deletions
--- a/AerialNavigation/rocket_powered_landing/rocket_powered_landing.py
+++ b/AerialNavigation/rocket_powered_landing/rocket_powered_landing.py
@@ -5,7 +5,7 @@ A rocket powered landing with successive convexification
 author: Sven Niederberger
        Atsushi Sakai

-Ref:
+Reference:
 - Python implementation of 'Successive Convexification for 6-DoF Mars Rocket Powered Landing with Free-Final-Time' paper
 by Michael Szmuk and Behcet Acıkmese.

--- a/ArmNavigation/two_joint_arm_to_point_control/two_joint_arm_to_point_control.py
+++ b/ArmNavigation/two_joint_arm_to_point_control/two_joint_arm_to_point_control.py
@@ -5,7 +5,7 @@ Left-click the plot to set the goal position of the end effector
 Author: Daniel Ingram (daniel-s-ingram)
        Atsushi Sakai (@Atsushi_twi)

-Ref: P. I. Corke, "Robotics, Vision & Control", Springer 2017,
+Reference: P. I. Corke, "Robotics, Vision & Control", Springer 2017,
 ISBN 978-3-319-54413-7 p102
 - [Robotics, Vision and Control]
 (https://link.springer.com/book/10.1007/978-3-642-20144-8)
--- a/Localization/ensemble_kalman_filter/ensemble_kalman_filter.py
+++ b/Localization/ensemble_kalman_filter/ensemble_kalman_filter.py
@@ -4,7 +4,7 @@ Ensemble Kalman Filter(EnKF) localization sample

 author: Ryohei Sasaki(rsasaki0109)

-Ref:
+Reference:
 Ensemble Kalman filtering
 (https://rmets.onlinelibrary.wiley.com/doi/10.1256/qj.05.135)

--- a/Mapping/DistanceMap/distance_map.py
+++ b/Mapping/DistanceMap/distance_map.py
@@ -3,7 +3,7 @@ Distance Map

 author: Wang Zheng (@Aglargil)

-Ref:
+Reference:

 - [Distance Map]
 (https://cs.brown.edu/people/pfelzens/papers/dt-final.pdf)
--- a/Mapping/rectangle_fitting/rectangle_fitting.py
+++ b/Mapping/rectangle_fitting/rectangle_fitting.py
@@ -4,7 +4,7 @@ Object shape recognition with L-shape fitting

 author: Atsushi Sakai (@Atsushi_twi)

-Ref:
+Reference:
 - Efficient L-Shape Fitting for Vehicle Detection Using Laser Scanners -
 The Robotics Institute Carnegie Mellon University
 https://www.ri.cmu.edu/publications/
--- a/MissionPlanning/BehaviorTree/behavior_tree.py
+++ b/MissionPlanning/BehaviorTree/behavior_tree.py
@@ -3,7 +3,7 @@ Behavior Tree

 author: Wang Zheng (@Aglargil)

-Ref:
+Reference:

 - [Behavior Tree](https://en.wikipedia.org/wiki/Behavior_tree_(artificial_intelligence,_robotics_and_control))
 """
--- a/MissionPlanning/StateMachine/state_machine.py
+++ b/MissionPlanning/StateMachine/state_machine.py
@@ -3,7 +3,7 @@ State Machine

 author: Wang Zheng (@Aglargil)

-Ref:
+Reference:

 - [State Machine]
 (https://en.wikipedia.org/wiki/Finite-state_machine)
@@ -23,7 +23,7 @@ def deflate_and_encode(plantuml_text):
    """
    zlib compress the plantuml text and encode it for the plantuml server.

-    Ref: https://plantuml.com/en/text-encoding
+    Reference https://plantuml.com/en/text-encoding
    """
    plantuml_alphabet = (
        string.digits + string.ascii_uppercase + string.ascii_lowercase + "-_"
--- a/PathPlanning/ElasticBands/elastic_bands.py
+++ b/PathPlanning/ElasticBands/elastic_bands.py
@@ -3,7 +3,7 @@ Elastic Bands

 author: Wang Zheng (@Aglargil)

-Ref:
+Reference:

 - [Elastic Bands: Connecting Path Planning and Control]
 (http://www8.cs.umu.se/research/ifor/dl/Control/elastic%20bands.pdf)
--- a/PathPlanning/Eta3SplinePath/eta3_spline_path.py
+++ b/PathPlanning/Eta3SplinePath/eta3_spline_path.py
@@ -5,7 +5,7 @@ eta^3 polynomials planner
 author: Joe Dinius, Ph.D (https://jwdinius.github.io)
        Atsushi Sakai (@Atsushi_twi)

-Ref:
+Reference:
 - [eta^3-Splines for the Smooth Path Generation of Wheeled Mobile Robots]
 (https://ieeexplore.ieee.org/document/4339545/)

--- a/PathPlanning/FrenetOptimalTrajectory/cartesian_frenet_converter.py
+++ b/PathPlanning/FrenetOptimalTrajectory/cartesian_frenet_converter.py
@@ -4,7 +4,7 @@ A converter between Cartesian and Frenet coordinate systems

 author: Wang Zheng (@Aglargil)

-Ref:
+Reference:

 - [Optimal Trajectory Generation for Dynamic Street Scenarios in a Frenet Frame]
 (https://www.researchgate.net/profile/Moritz_Werling/publication/224156269_Optimal_Trajectory_Generation_for_Dynamic_Street_Scenarios_in_a_Frenet_Frame/links/54f749df0cf210398e9277af.pdf)
--- a/PathPlanning/FrenetOptimalTrajectory/frenet_optimal_trajectory.py
+++ b/PathPlanning/FrenetOptimalTrajectory/frenet_optimal_trajectory.py
@@ -4,7 +4,7 @@ Frenet optimal trajectory generator

 author: Atsushi Sakai (@Atsushi_twi)

-Ref:
+Reference:

 - [Optimal Trajectory Generation for Dynamic Street Scenarios in a Frenet Frame]
 (https://www.researchgate.net/profile/Moritz_Werling/publication/224156269_Optimal_Trajectory_Generation_for_Dynamic_Street_Scenarios_in_a_Frenet_Frame/links/54f749df0cf210398e9277af.pdf)
--- a/PathPlanning/PotentialFieldPlanning/potential_field_planning.py
+++ b/PathPlanning/PotentialFieldPlanning/potential_field_planning.py
@@ -4,7 +4,7 @@ Potential Field based path planner

 author: Atsushi Sakai (@Atsushi_twi)

-Ref:
+Reference:
 https://www.cs.cmu.edu/~motionplanning/lecture/Chap4-Potential-Field_howie.pdf

 """
--- a/PathPlanning/QuinticPolynomialsPlanner/quintic_polynomials_planner.py
+++ b/PathPlanning/QuinticPolynomialsPlanner/quintic_polynomials_planner.py
@@ -4,7 +4,7 @@ Quintic Polynomials Planner

 author: Atsushi Sakai (@Atsushi_twi)

-Ref:
+Reference:

 - [Local Path planning And Motion Control For Agv In Positioning](https://ieeexplore.ieee.org/document/637936/)

--- a/PathPlanning/StateLatticePlanner/state_lattice_planner.py
+++ b/PathPlanning/StateLatticePlanner/state_lattice_planner.py
@@ -8,7 +8,7 @@ author: Atsushi Sakai (@Atsushi_twi)
 https://github.com/AtsushiSakai/PythonRobotics/blob/master/PathPlanning
 /ModelPredictiveTrajectoryGenerator/lookup_table_generator.py

-Ref:
+Reference:

 - State Space Sampling of Feasible Motions for High-Performance Mobile Robot
 Navigation in Complex Environments
--- a/PathTracking/cgmres_nmpc/cgmres_nmpc.py
+++ b/PathTracking/cgmres_nmpc/cgmres_nmpc.py
@@ -4,7 +4,7 @@ Nonlinear MPC simulation with CGMRES

 author Atsushi Sakai (@Atsushi_twi)

-Ref:
+Reference:
 Shunichi09/nonlinear_control: Implementing the nonlinear model predictive
 control, sliding mode control https://github.com/Shunichi09/PythonLinearNonlinearControl

--- a/PathTracking/move_to_pose/move_to_pose.py
+++ b/PathTracking/move_to_pose/move_to_pose.py
@@ -76,7 +76,7 @@ class PathFinderController:
        # [-pi, pi] to prevent unstable behavior e.g. difference going
        # from 0 rad to 2*pi rad with slight turn

-        # Ref: The velocity v always has a constant sign which depends on the initial value of α.
+        # The velocity v always has a constant sign which depends on the initial value of α.
        rho = np.hypot(x_diff, y_diff)
        v = self.Kp_rho * rho

--- a/PathTracking/stanley_controller/stanley_controller.py
+++ b/PathTracking/stanley_controller/stanley_controller.py
@@ -4,7 +4,7 @@ Path tracking simulation with Stanley steering control and PID speed control.

 author: Atsushi Sakai (@Atsushi_twi)

-Ref:
+Reference:
    - [Stanley: The robot that won the DARPA grand challenge](http://isl.ecst.csuchico.edu/DOCS/darpa2005/DARPA%202005%20Stanley.pdf)
    - [Autonomous Automobile Path Tracking](https://www.ri.cmu.edu/pub_files/2009/2/Automatic_Steering_Methods_for_Autonomous_Automobile_Path_Tracking.pdf)

--- a/README.md
+++ b/README.md
@@ -168,7 +168,7 @@ All animation gifs are stored here: [AtsushiSakai/PythonRoboticsGifs: Animation

 <img src="https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/Localization/extended_kalman_filter/animation.gif" width="640" alt="EKF pic">

-Ref:
+Reference

 - [documentation](https://atsushisakai.github.io/PythonRobotics/modules/localization/extended_kalman_filter_localization_files/extended_kalman_filter_localization.html)

@@ -186,7 +186,7 @@ It is assumed that the robot can measure a distance from landmarks (RFID).

 These measurements are used for PF localization.

-Ref:
+Reference

 - [PROBABILISTIC ROBOTICS](http://www.probabilistic-robotics.org/)

@@ -207,7 +207,7 @@ The filter integrates speed input and range observations from RFID for localizat

 Initial position is not needed.

-Ref:
+Reference

 - [PROBABILISTIC ROBOTICS](http://www.probabilistic-robotics.org/)

@@ -256,7 +256,7 @@ It can calculate a rotation matrix, and a translation vector between points and

 ![3](https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/SLAM/iterative_closest_point/animation.gif)

-Ref:
+Reference

 - [Introduction to Mobile Robotics: Iterative Closest Point Algorithm](https://cs.gmu.edu/~kosecka/cs685/cs685-icp.pdf)

@@ -275,7 +275,7 @@ Black points are landmarks, blue crosses are estimated landmark positions by Fas
 ![3](https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/SLAM/FastSLAM1/animation.gif)


-Ref:
+Reference

 - [PROBABILISTIC ROBOTICS](http://www.probabilistic-robotics.org/)

@@ -321,7 +321,7 @@ This is a 2D grid based the shortest path planning with D star algorithm.

 The animation shows a robot finding its path avoiding an obstacle using the D* search algorithm.

-Ref:
+Reference

 - [D* Algorithm Wikipedia](https://en.wikipedia.org/wiki/D*)

@@ -346,7 +346,7 @@ This is a 2D grid based path planning with Potential Field algorithm.

 In the animation, the blue heat map shows potential value on each grid.

-Ref:
+Reference

 - [Robotic Motion Planning:Potential Functions](https://www.cs.cmu.edu/~motionplanning/lecture/Chap4-Potential-Field_howie.pdf)

@@ -362,7 +362,7 @@ This script is a path planning code with state lattice planning.

 This code uses the model predictive trajectory generator to solve boundary problem.

-Ref: 
+Reference 

 - [Optimal rough terrain trajectory generation for wheeled mobile robots](https://journals.sagepub.com/doi/pdf/10.1177/0278364906075328)

@@ -390,7 +390,7 @@ Cyan crosses means searched points with Dijkstra method,

 The red line is the final path of PRM.

-Ref:
+Reference

 - [Probabilistic roadmap \- Wikipedia](https://en.wikipedia.org/wiki/Probabilistic_roadmap)

@@ -406,7 +406,7 @@ This is a path planning code with RRT\*

 Black circles are obstacles, green line is a searched tree, red crosses are start and goal positions.

-Ref:
+Reference

 - [Incremental Sampling-based Algorithms for Optimal Motion Planning](https://arxiv.org/abs/1005.0416)

@@ -426,7 +426,7 @@ A double integrator motion model is used for LQR local planner.

 ![LQR_RRT](https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathPlanning/LQRRRTStar/animation.gif)

-Ref:
+Reference

 - [LQR\-RRT\*: Optimal Sampling\-Based Motion Planning with Automatically Derived Extension Heuristics](https://lis.csail.mit.edu/pubs/perez-icra12.pdf)

@@ -441,7 +441,7 @@ Motion planning with quintic polynomials.

 It can calculate a 2D path, velocity, and acceleration profile based on quintic polynomials.

-Ref:
+Reference

 - [Local Path Planning And Motion Control For Agv In Positioning](https://ieeexplore.ieee.org/document/637936/)

@@ -451,7 +451,7 @@ A sample code with Reeds Shepp path planning.

 ![RSPlanning](https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathPlanning/ReedsSheppPath/animation.gif?raw=true)

-Ref:
+Reference

 - [15.3.2 Reeds\-Shepp Curves](http://planning.cs.uiuc.edu/node822.html) 

@@ -477,7 +477,7 @@ The cyan line is the target course and black crosses are obstacles.

 The red line is the predicted path.

-Ref:
+Reference

 - [Optimal Trajectory Generation for Dynamic Street Scenarios in a Frenet Frame](https://www.researchgate.net/profile/Moritz_Werling/publication/224156269_Optimal_Trajectory_Generation_for_Dynamic_Street_Scenarios_in_a_Frenet_Frame/links/54f749df0cf210398e9277af.pdf)

@@ -492,7 +492,7 @@ This is a simulation of moving to a pose control

 ![2](https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathTracking/move_to_pose/animation.gif)

-Ref:
+Reference

 - [P. I. Corke, "Robotics, Vision and Control" \| SpringerLink p102](https://link.springer.com/book/10.1007/978-3-642-20144-8)

@@ -503,7 +503,7 @@ Path tracking simulation with Stanley steering control and PID speed control.

 ![2](https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathTracking/stanley_controller/animation.gif)

-Ref:
+Reference

 - [Stanley: The robot that won the DARPA grand challenge](http://robots.stanford.edu/papers/thrun.stanley05.pdf)

@@ -517,7 +517,7 @@ Path tracking simulation with rear wheel feedback steering control and PID speed

 ![PythonRobotics/figure_1.png at master · AtsushiSakai/PythonRobotics](https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathTracking/rear_wheel_feedback/animation.gif)

-Ref:
+Reference

 - [A Survey of Motion Planning and Control Techniques for Self-driving Urban Vehicles](https://arxiv.org/abs/1604.07446)

@@ -528,7 +528,7 @@ Path tracking simulation with LQR speed and steering control.

 ![3](https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathTracking/lqr_speed_steer_control/animation.gif)

-Ref:
+Reference

 - [Towards fully autonomous driving: Systems and algorithms \- IEEE Conference Publication](https://ieeexplore.ieee.org/document/5940562/)

@@ -539,7 +539,7 @@ Path tracking simulation with iterative linear model predictive speed and steeri

 <img src="https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathTracking/model_predictive_speed_and_steer_control/animation.gif" width="640" alt="MPC pic">

-Ref:
+Reference

 - [documentation](https://atsushisakai.github.io/PythonRobotics/modules/path_tracking/model_predictive_speed_and_steering_control/model_predictive_speed_and_steering_control.html)

@@ -551,7 +551,7 @@ A motion planning and path tracking simulation with NMPC of C-GMRES

 ![3](https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathTracking/cgmres_nmpc/animation.gif)

-Ref:
+Reference

 - [documentation](https://atsushisakai.github.io/PythonRobotics/modules/path_tracking/cgmres_nmpc/cgmres_nmpc.html)

@@ -591,7 +591,7 @@ This is a 3d trajectory generation simulation for a rocket powered landing.

 ![3](https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/AerialNavigation/rocket_powered_landing/animation.gif)

-Ref:
+Reference

 - [documentation](https://atsushisakai.github.io/PythonRobotics/modules/aerial_navigation/rocket_powered_landing/rocket_powered_landing.html)

--- a/docs/modules/12_appendix/Kalmanfilter_basics_2_main.rst
+++ b/docs/modules/12_appendix/Kalmanfilter_basics_2_main.rst
@@ -331,7 +331,7 @@ are vectors and matrices, but the concepts are exactly the same:
 -  Use the process model to predict the next state (the prior)
 -  Form an estimate part way between the measurement and the prior

-References:
+Reference
 ~~~~~~~~~~~

 1. Roger Labbe’s
--- a/docs/modules/12_appendix/Kalmanfilter_basics_main.rst
+++ b/docs/modules/12_appendix/Kalmanfilter_basics_main.rst
@@ -552,7 +552,7 @@ a given (X,Y) value.
 .. image:: Kalmanfilter_basics_files/Kalmanfilter_basics_28_1.png


-References:
+Reference
 ~~~~~~~~~~~

 1. Roger Labbe’s
--- a/docs/modules/12_appendix/steering_motion_model_main.rst
+++ b/docs/modules/12_appendix/steering_motion_model_main.rst
@@ -91,7 +91,7 @@ the target minimum velocity :math:`v_{min}` can be calculated as follows:
 :math:`v_{min} = \frac{d_{t+1}+d_{t}}{\Delta t} = \frac{d_{t+1}+d_{t}}{(\kappa_{t+1}-\kappa_{t})}\frac{tan\dot{\delta}_{max}}{WB}`


-References:
+Reference
 ~~~~~~~~~~~

 - `Vehicle Dynamics and Control <https://link.springer.com/book/10.1007/978-1-4614-1433-9>`_
--- a/docs/modules/2_localization/ensamble_kalman_filter_localization_files/ensamble_kalman_filter_localization_main.rst
+++ b/docs/modules/2_localization/ensamble_kalman_filter_localization_files/ensamble_kalman_filter_localization_main.rst
@@ -5,3 +5,8 @@ Ensamble Kalman Filter Localization

 This is a sensor fusion localization with Ensamble Kalman Filter(EnKF).

+Code Link
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+.. autofunction:: Localization.ensemble_kalman_filter.ensemble_kalman_filter.enkf_localization
+
--- a/docs/modules/2_localization/extended_kalman_filter_localization_files/extended_kalman_filter_localization_main.rst
+++ b/docs/modules/2_localization/extended_kalman_filter_localization_files/extended_kalman_filter_localization_main.rst
@@ -23,28 +23,40 @@ is estimated trajectory with EKF.

 The red ellipse is estimated covariance ellipse with EKF.

-Code: `PythonRobotics/extended_kalman_filter.py at master ·
-AtsushiSakai/PythonRobotics <https://github.com/AtsushiSakai/PythonRobotics/blob/master/Localization/extended_kalman_filter/extended_kalman_filter.py>`__
-
-Kalman Filter with Speed Scale Factor Correction
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-
-.. image:: ekf_with_velocity_correction_1_0.png
-   :width: 600px
-
-This is a Extended kalman filter (EKF) localization with velocity correction.
-
-This is for correcting the vehicle speed measured with scale factor errors due to factors such as wheel wear.
-
-Code: `PythonRobotics/extended_kalman_ekf_with_velocity_correctionfilter.py
-AtsushiSakai/PythonRobotics <https://github.com/AtsushiSakai/PythonRobotics/blob/master/Localization/extended_kalman_filter/extended_kalman_ekf_with_velocity_correctionfilter.py>`__
-
-Filter design
+Code Link
 ~~~~~~~~~~~~~

-Position Estimation Kalman Filter
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+.. autofunction:: Localization.extended_kalman_filter.extended_kalman_filter.ekf_estimation
+
+Extended Kalman Filter algorithm
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+Localization process using Extended Kalman Filter:EKF is
+
+=== Predict ===
+
+:math:`x_{Pred} = Fx_t+Bu_t`
+
+:math:`P_{Pred} = J_f P_t J_f^T + Q`
+
+=== Update ===
+
+:math:`z_{Pred} = Hx_{Pred}`
+
+:math:`y = z - z_{Pred}`
+
+:math:`S = J_g P_{Pred}.J_g^T + R`
+
+:math:`K = P_{Pred}.J_g^T S^{-1}`
+
+:math:`x_{t+1} = x_{Pred} + Ky`
+
+:math:`P_{t+1} = ( I - K J_g) P_{Pred}`
+
+
+
+Filter design
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

 In this simulation, the robot has a state vector includes 4 states at
 time :math:`t`.
@@ -82,27 +94,9 @@ In the code, “observation” function generates the input and observation
 vector with noise
 `code <https://github.com/AtsushiSakai/PythonRobotics/blob/916b4382de090de29f54538b356cef1c811aacce/Localization/extended_kalman_filter/extended_kalman_filter.py#L34-L50>`__

-Kalman Filter with Speed Scale Factor Correction
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-In this simulation, the robot has a state vector includes 5 states at
-time :math:`t`.
-
-.. math:: \textbf{x}_t=[x_t, y_t, \phi_t, v_t, s_t]
-
-x, y are a 2D x-y position, :math:`\phi` is orientation, v is
-velocity, and s is a scale factor of velocity.
-
-In the code, “xEst” means the state vector.
-`code <https://github.com/AtsushiSakai/PythonRobotics/blob/916b4382de090de29f54538b356cef1c811aacce/Localization/extended_kalman_filter/extended_kalman_ekf_with_velocity_correctionfilter.py#L163>`__
-
-The rest is the same as the Position Estimation Kalman Filter.

 Motion Model
-~~~~~~~~~~~~
-
-Position Estimation Kalman Filter
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+~~~~~~~~~~~~~~~~~

 The robot model is

@@ -137,9 +131,61 @@ Its Jacobian matrix is

 :math:`\begin{equation*} 　= \begin{bmatrix} 1& 0 & -v \sin(\phi) \Delta t & \cos(\phi) \Delta t\\ 0 & 1 & v \cos(\phi) \Delta t & \sin(\phi) \Delta t\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{equation*}`

+Observation Model
+~~~~~~~~~~~~~~~~~
+
+The robot can get x-y position information from GPS.
+
+So GPS Observation model is
+
+.. math:: \textbf{z}_{t} = g(\textbf{x}_t) = H \textbf{x}_t
+
+where
+
+:math:`\begin{equation*} H = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{bmatrix} \end{equation*}`
+
+The observation function states that
+
+:math:`\begin{equation*} \begin{bmatrix} x' \\ y' \end{bmatrix} = g(\textbf{x}) = \begin{bmatrix} x \\ y \end{bmatrix} \end{equation*}`
+
+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'}{ \partial v}\\ \end{bmatrix} \end{equation*}`
+
+:math:`\begin{equation*} 　= \begin{bmatrix} 1& 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ \end{bmatrix} \end{equation*}`
+

 Kalman Filter with Speed Scale Factor Correction
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+This is a Extended kalman filter (EKF) localization with velocity correction.
+
+This is for correcting the vehicle speed measured with scale factor errors due to factors such as wheel wear.
+
+
+In this simulation, the robot has a state vector includes 5 states at
+time :math:`t`.
+
+.. math:: \textbf{x}_t=[x_t, y_t, \phi_t, v_t, s_t]
+
+x, y are a 2D x-y position, :math:`\phi` is orientation, v is
+velocity, and s is a scale factor of velocity.
+
+In the code, “xEst” means the state vector.
+`code <https://github.com/AtsushiSakai/PythonRobotics/blob/916b4382de090de29f54538b356cef1c811aacce/Localization/extended_kalman_filter/extended_kalman_ekf_with_velocity_correctionfilter.py#L163>`__
+
+The rest is the same as the Position Estimation Kalman Filter.
+
+.. image:: ekf_with_velocity_correction_1_0.png
+   :width: 600px
+
+Code Link
+~~~~~~~~~~~~~
+
+.. autofunction:: Localization.extended_kalman_filter.ekf_with_velocity_correction.ekf_estimation
+
+
+Motion Model
+~~~~~~~~~~~~

 The robot model is

@@ -178,33 +224,7 @@ Its Jacobian matrix is
 Observation Model
 ~~~~~~~~~~~~~~~~~

-Position Estimation Kalman Filter
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-The robot can get x-y position infomation from GPS.
-
-So GPS Observation model is
-
-.. math:: \textbf{z}_{t} = g(\textbf{x}_t) = H \textbf{x}_t
-
-where
-
-:math:`\begin{equation*} H = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{bmatrix} \end{equation*}`
-
-The observation function states that
-
-:math:`\begin{equation*} \begin{bmatrix} x' \\ y' \end{bmatrix} = g(\textbf{x}) = \begin{bmatrix} x \\ y \end{bmatrix} \end{equation*}`
-
-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'}{ \partial v}\\ \end{bmatrix} \end{equation*}`
-
-:math:`\begin{equation*} 　= \begin{bmatrix} 1& 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ \end{bmatrix} \end{equation*}`
-
-Kalman Filter with Speed Scale Factor Correction
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-The robot can get x-y position infomation from GPS.
+The robot can get x-y position information from GPS.

 So GPS Observation model is

@@ -225,32 +245,8 @@ Its Jacobian matrix is
 :math:`\begin{equation*} 　= \begin{bmatrix} 1& 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ \end{bmatrix} \end{equation*}`


-Extended Kalman Filter
-~~~~~~~~~~~~~~~~~~~~~~

-Localization process using Extended Kalman Filter:EKF is
-
-=== Predict ===
-
-:math:`x_{Pred} = Fx_t+Bu_t`
-
-:math:`P_{Pred} = J_f P_t J_f^T + Q`
-
-=== Update ===
-
-:math:`z_{Pred} = Hx_{Pred}`
-
-:math:`y = z - z_{Pred}`
-
-:math:`S = J_g P_{Pred}.J_g^T + R`
-
-:math:`K = P_{Pred}.J_g^T S^{-1}`
-
-:math:`x_{t+1} = x_{Pred} + Ky`
-
-:math:`P_{t+1} = ( I - K J_g) P_{Pred}`
-
-Ref:
-~~~~
+Reference
+^^^^^^^^^^^

 -  `PROBABILISTIC-ROBOTICS.ORG <http://www.probabilistic-robotics.org/>`__
--- a/docs/modules/2_localization/histogram_filter_localization/histogram_filter_localization_main.rst
+++ b/docs/modules/2_localization/histogram_filter_localization/histogram_filter_localization_main.rst
@@ -16,6 +16,11 @@ The filter uses speed input and range observations from RFID for localization.

 Initial position information is not needed.

+Code Link
+~~~~~~~~~~~~~
+
+.. autofunction:: Localization.histogram_filter.histogram_filter.histogram_filter_localization
+
 Filtering algorithm
 ~~~~~~~~~~~~~~~~~~~~

@@ -107,7 +112,7 @@ There are two ways to calculate the final positions:



-References:
+Reference
 ~~~~~~~~~~~

 - `_PROBABILISTIC ROBOTICS: <http://www.probabilistic-robotics.org>`_
--- a/docs/modules/2_localization/particle_filter_localization/particle_filter_localization_main.rst
+++ b/docs/modules/2_localization/particle_filter_localization/particle_filter_localization_main.rst
@@ -15,6 +15,12 @@ It is assumed that the robot can measure a distance from landmarks

 This measurements are used for PF localization.

+Code Link
+~~~~~~~~~~~~~
+
+.. autofunction:: Localization.particle_filter.particle_filter.pf_localization
+
+
 How to calculate covariance matrix from particles
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

@@ -30,7 +36,7 @@ The covariance matrix :math:`\Xi` from particle information is calculated by the

 - :math:`\mu_j` is the :math:`j` th mean state of particles.

-References:
+Reference
 ~~~~~~~~~~~

 - `_PROBABILISTIC ROBOTICS: <http://www.probabilistic-robotics.org>`_
--- a/docs/modules/2_localization/unscented_kalman_filter_localization/unscented_kalman_filter_localization_main.rst
+++ b/docs/modules/2_localization/unscented_kalman_filter_localization/unscented_kalman_filter_localization_main.rst
@@ -7,7 +7,13 @@ This is a sensor fusion localization with Unscented Kalman Filter(UKF).

 The lines and points are same meaning of the EKF simulation.

-References:
+Code Link
+~~~~~~~~~~~~~
+
+.. autofunction:: Localization.unscented_kalman_filter.unscented_kalman_filter.ukf_estimation
+
+
+Reference
 ~~~~~~~~~~~

 - `Discriminatively Trained Unscented Kalman Filter for Mobile Robot Localization <https://www.researchgate.net/publication/267963417_Discriminatively_Trained_Unscented_Kalman_Filter_for_Mobile_Robot_Localization>`_
--- a/docs/modules/4_slam/ekf_slam/ekf_slam_main.rst
+++ b/docs/modules/4_slam/ekf_slam/ekf_slam_main.rst
@@ -578,7 +578,7 @@ reckoning and control functions are passed along here as well.

 .. image:: ekf_slam_1_0.png

-References:
+Reference
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

 - `PROBABILISTIC ROBOTICS <http://www.probabilistic-robotics.org/>`_
--- a/docs/modules/4_slam/graph_slam/graph_slam_main.rst
+++ b/docs/modules/4_slam/graph_slam/graph_slam_main.rst
@@ -17,7 +17,7 @@ The black stars are landmarks for graph edge generation.
 .. include:: graphSLAM_formulation.rst
 .. include:: graphSLAM_SE2_example.rst

-References:
+Reference
 ~~~~~~~~~~~

 - `A Tutorial on Graph-Based SLAM <http://www2.informatik.uni-freiburg.de/~stachnis/pdf/grisetti10titsmag.pdf>`_
--- a/docs/modules/5_path_planning/bezier_path/bezier_path_main.rst
+++ b/docs/modules/5_path_planning/bezier_path/bezier_path_main.rst
@@ -13,7 +13,7 @@ You can get different Beizer course:

 .. image:: Figure_2.png

-Ref:
+Reference

 -  `Continuous Curvature Path Generation Based on Bezier Curves for
   Autonomous
--- a/docs/modules/5_path_planning/eta3_spline/eta3_spline_main.rst
+++ b/docs/modules/5_path_planning/eta3_spline/eta3_spline_main.rst
@@ -7,7 +7,7 @@ Eta^3 Spline path planning

 This is a path planning with Eta^3 spline.

-Ref:
+Reference

 -  `\\eta^3-Splines for the Smooth Path Generation of Wheeled Mobile
   Robots <https://ieeexplore.ieee.org/document/4339545/>`__
--- a/docs/modules/5_path_planning/frenet_frame_path/frenet_frame_path_main.rst
+++ b/docs/modules/5_path_planning/frenet_frame_path/frenet_frame_path_main.rst
@@ -35,7 +35,7 @@ Low Speed and Merging and Stopping Scenario

 This scenario illustrates the trajectory planning at low speeds with merging and stopping behaviors.

-Ref:
+Reference

 -  `Optimal Trajectory Generation for Dynamic Street Scenarios in a
   Frenet
--- a/docs/modules/5_path_planning/grid_base_search/grid_base_search_main.rst
+++ b/docs/modules/5_path_planning/grid_base_search/grid_base_search_main.rst
@@ -63,7 +63,7 @@ This is a 2D grid based shortest path planning with D star algorithm.

 The animation shows a robot finding its path avoiding an obstacle using the D* search algorithm.

-Ref:
+Reference

 -  `D* search Wikipedia <https://en.wikipedia.org/wiki/D*>`__

@@ -74,7 +74,7 @@ This is a 2D grid based path planning and replanning with D star lite algorithm.

 .. image:: https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathPlanning/DStarLite/animation.gif

-Ref:
+Reference

 - `Improved Fast Replanning for Robot Navigation in Unknown Terrain <http://www.cs.cmu.edu/~maxim/files/dlite_icra02.pdf>`_

@@ -88,7 +88,7 @@ This is a 2D grid based path planning with Potential Field algorithm.

 In the animation, the blue heat map shows potential value on each grid.

-Ref:
+Reference

 -  `Robotic Motion Planning:Potential
   Functions <https://www.cs.cmu.edu/~motionplanning/lecture/Chap4-Potential-Field_howie.pdf>`__
--- a/docs/modules/5_path_planning/model_predictive_trajectory_generator/model_predictive_trajectory_generator_main.rst
+++ b/docs/modules/5_path_planning/model_predictive_trajectory_generator/model_predictive_trajectory_generator_main.rst
@@ -16,7 +16,7 @@ Lookup table generation sample

 .. image:: lookup_table.png

-Ref:
+Reference

 -  `Optimal rough terrain trajectory generation for wheeled mobile
   robots <https://journals.sagepub.com/doi/pdf/10.1177/0278364906075328>`__
--- a/docs/modules/5_path_planning/prm_planner/prm_planner_main.rst
+++ b/docs/modules/5_path_planning/prm_planner/prm_planner_main.rst
@@ -13,7 +13,7 @@ Cyan crosses means searched points with Dijkstra method,

 The red line is the final path of PRM.

-Ref:
+Reference

 -  `Probabilistic roadmap -
   Wikipedia <https://en.wikipedia.org/wiki/Probabilistic_roadmap>`__
--- a/docs/modules/5_path_planning/quintic_polynomials_planner/quintic_polynomials_planner_main.rst
+++ b/docs/modules/5_path_planning/quintic_polynomials_planner/quintic_polynomials_planner_main.rst
@@ -97,7 +97,7 @@ Each velocity and acceleration boundary condition can be calculated with each or

 :math:`v_{xe}=v_ecos(\theta_e), v_{ye}=v_esin(\theta_e)`

-References:
+Reference
 ~~~~~~~~~~~

 -  `Local Path Planning And Motion Control For Agv In
--- a/docs/modules/5_path_planning/reeds_shepp_path/reeds_shepp_path_main.rst
+++ b/docs/modules/5_path_planning/reeds_shepp_path/reeds_shepp_path_main.rst
@@ -380,7 +380,7 @@ Hence, we have:
 - :math:`v = (t - φ)`


-Ref:
+Reference

 -  `15.3.2 Reeds-Shepp
   Curves <https://lavalle.pl/planning/node822.html>`__
--- a/docs/modules/5_path_planning/rrt/rrt_main.rst
+++ b/docs/modules/5_path_planning/rrt/rrt_main.rst
@@ -53,7 +53,7 @@ This is a path planning code with Informed RRT*.

 The cyan ellipse is the heuristic sampling domain of Informed RRT*.

-Ref:
+Reference

 -  `Informed RRT\*: Optimal Sampling-based Path Planning Focused via
   Direct Sampling of an Admissible Ellipsoidal
@@ -68,7 +68,7 @@ Batch Informed RRT\*

 This is a path planning code with Batch Informed RRT*.

-Ref:
+Reference

 -  `Batch Informed Trees (BIT*): Sampling-based Optimal Planning via the
   Heuristically Guided Search of Implicit Random Geometric
@@ -87,7 +87,7 @@ In this code, pure-pursuit algorithm is used for steering control,

 PID is used for speed control.

-Ref:
+Reference

 -  `Motion Planning in Complex Environments using Closed-loop
   Prediction <https://acl.mit.edu/papers/KuwataGNC08.pdf>`__
@@ -109,7 +109,7 @@ A double integrator motion model is used for LQR local planner.

 .. image:: https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathPlanning/LQRRRTStar/animation.gif

-Ref:
+Reference

 -  `LQR-RRT\*: Optimal Sampling-Based Motion Planning with Automatically
   Derived Extension
--- a/docs/modules/5_path_planning/state_lattice_planner/state_lattice_planner_main.rst
+++ b/docs/modules/5_path_planning/state_lattice_planner/state_lattice_planner_main.rst
@@ -22,7 +22,7 @@ Lane sampling

 .. image:: https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathPlanning/StateLatticePlanner/LaneSampling.gif

-Ref:
+Reference

 -  `Optimal rough terrain trajectory generation for wheeled mobile
   robots <https://journals.sagepub.com/doi/pdf/10.1177/0278364906075328>`__
--- a/docs/modules/5_path_planning/vrm_planner/vrm_planner_main.rst
+++ b/docs/modules/5_path_planning/vrm_planner/vrm_planner_main.rst
@@ -11,7 +11,7 @@ Cyan crosses mean searched points with Dijkstra method,

 The red line is the final path of Vornoi Road-Map.

-Ref:
+Reference

 -  `Robotic Motion Planning <https://www.cs.cmu.edu/~motionplanning/lecture/Chap5-RoadMap-Methods_howie.pdf>`__

--- a/docs/modules/6_path_tracking/lqr_speed_and_steering_control/lqr_speed_and_steering_control_main.rst
+++ b/docs/modules/6_path_tracking/lqr_speed_and_steering_control/lqr_speed_and_steering_control_main.rst
@@ -134,7 +134,7 @@ Simulation results



-References:
+Reference
 ~~~~~~~~~~~

 -  `Towards fully autonomous driving: Systems and algorithms <https://ieeexplore.ieee.org/document/5940562/>`__
--- a/docs/modules/6_path_tracking/lqr_steering_control/lqr_steering_control_main.rst
+++ b/docs/modules/6_path_tracking/lqr_steering_control/lqr_steering_control_main.rst
@@ -113,7 +113,7 @@ The optimal control input `u` can be calculated as:

 where `K` is the feedback gain matrix obtained by solving the Riccati equation.

-References:
+Reference
 ~~~~~~~~~~~
 -  `ApolloAuto/apollo: An open autonomous driving platform <https://github.com/ApolloAuto/apollo>`_

--- a/docs/modules/6_path_tracking/pure_pursuit_tracking/pure_pursuit_tracking_main.rst
+++ b/docs/modules/6_path_tracking/pure_pursuit_tracking/pure_pursuit_tracking_main.rst
@@ -9,7 +9,7 @@ speed control.
 The red line is a target course, the green cross means the target point
 for pure pursuit control, the blue line is the tracking.

-References:
+Reference
 ~~~~~~~~~~~

 -  `A Survey of Motion Planning and Control Techniques for Self-driving
--- a/docs/modules/6_path_tracking/rear_wheel_feedback_control/rear_wheel_feedback_control_main.rst
+++ b/docs/modules/6_path_tracking/rear_wheel_feedback_control/rear_wheel_feedback_control_main.rst
@@ -6,7 +6,7 @@ PID speed control.

 .. image:: https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathTracking/rear_wheel_feedback/animation.gif

-References:
+Reference
 ~~~~~~~~~~~
 -  `A Survey of Motion Planning and Control Techniques for Self-driving
   Urban Vehicles <https://arxiv.org/abs/1604.07446>`__
--- a/docs/modules/6_path_tracking/stanley_control/stanley_control_main.rst
+++ b/docs/modules/6_path_tracking/stanley_control/stanley_control_main.rst
@@ -6,7 +6,7 @@ control.

 .. image:: https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathTracking/stanley_controller/animation.gif

-References:
+Reference
 ~~~~~~~~~~~

 -  `Stanley: The robot that won the DARPA grand
--- a/users_comments.md
+++ b/users_comments.md
@@ -10,7 +10,7 @@ This is an electric wheelchair control demo by [Katsushun89](https://github.com/

 ![1](https://github.com/AtsushiSakai/PythonRoboticsGifs/blob/master/Users/WHILL_Model_CR_with_move_to_a_pose/WHLL_Model_CR_with_move_to_a_pose.gif)

-Ref:
+Reference:

 - [toioと同じように動くWHILL Model CR (in Japanese)](https://qiita.com/KatsuShun89/items/68fde7544ecfe36096d2)