docs: improve pure pursuit documentation

docs/modules/6_path_tracking/pure_pursuit_tracking/pure_pursuit_tracking_main.rst

@@ -7,16 +7,130 @@ speed control.
 .. image:: https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathTracking/pure_pursuit/animation.gif
 
 The red line is a target course, the green cross means the target point
-for pure pursuit control, the blue line is the tracking.
+for pure pursuit control, and the blue line is the tracked trajectory.
+
+
+
+Algorithm Overview
+~~~~~~~~~~~~~~~~~~
+
+
+Pure Pursuit is a geometric path-tracking algorithm widely used in
+autonomous vehicles and mobile robotics. Instead of steering toward the
+nearest point on the reference path, the vehicle continuously selects a
+target point located ahead on the path (called the *look-ahead point*)
+and steers toward it.
+
+At every control step, the algorithm performs three main operations:
+
+1. Select a target point using a look-ahead distance.
+2. Compute the angular error between vehicle heading and target point.
+3. Compute a steering angle that guides the vehicle toward the target.
+
+The look-ahead distance is dynamically adjusted according to vehicle speed:
+
+.. math::
+
+   L_f = kv + L_{fc}
+
+where :math:`v` is the vehicle speed, :math:`k` is the look-forward gain,
+and :math:`L_{fc}` is the minimum look-ahead distance.
+
+
+
+Geometric Intuition
+~~~~~~~~~~~~~~~~~~~
+
+Pure Pursuit uses the bicycle model, where the rear axle center is treated
+as the reference control point. At each control step, a look-ahead point
+is selected on the reference path at distance :math:`L_f`.
+
+Let the target point be :math:`(x_t, y_t)` and the rear axle position be
+:math:`(x_r, y_r)`. The angle between the vehicle heading and the line
+joining the rear axle to the target point is denoted by :math:`\alpha`.
+
+.. math::
+
+\alpha = \tan^{-1}\left(\frac{y_t - y_r}{x_t - x_r}\right) - \theta
+
+where :math:`\theta` is the current vehicle heading angle.
+
+A larger :math:`\alpha` indicates that the target point lies farther away
+from the current heading direction, requiring a sharper steering command.
+
+
+
+Steering Law Derivation
+~~~~~~~~~~~~~~~~~~~~~~~
+
+
+Under the bicycle model, a vehicle moving with constant steering angle
+follows a circular trajectory of radius :math:`R`.
+
+The relation between steering angle :math:`\delta` and turning radius is:
+
+.. math::
+
+   \tan(\delta) = \frac{L}{R}
+
+where :math:`L` is the wheelbase.
+
+Pure Pursuit assumes the vehicle follows a circular arc passing through
+the look-ahead point. Using circle geometry, the chord length satisfies:
+
+.. math::
+
+   L_f = 2R \sin(\alpha)
+
+Rearranging gives:
+
+.. math::
+
+   R = \frac{L_f}{2\sin(\alpha)}
+
+Substituting into the bicycle model equation gives the Pure Pursuit
+steering law:
+
+.. math::
+
+   \delta = \tan^{-1}\left(\frac{2L\sin(\alpha)}{L_f}\right)
+
+This shows that larger :math:`\alpha` produces sharper turns, while larger
+look-ahead distance :math:`L_f` produces smoother steering.
+
+
+
+Implementation Details
+~~~~~~~~~~~~~~~~~~~~~~
+
+The controller is mainly divided into two stages.
+
+1. Target Point Selection
+
+The function ``search_target_index()`` determines the look-ahead target
+point. It first finds the path point nearest to the rear axle position,
+then moves forward along the path until the distance exceeds
+:math:`L_f`.
+
+2. Steering Computation
+
+The function ``pure_pursuit_steer_control()`` computes the heading error
+:math:`\alpha` and uses the steering law above to calculate the steering
+command.
+
+The control loop repeatedly performs target selection, steering
+computation, and vehicle state updates to smoothly track the reference
+path.
 
 Code Link
-~~~~~~~~~~~~~~~
+~~~~~~~~~
 
 .. autofunction:: PathTracking.pure_pursuit.pure_pursuit.pure_pursuit_steer_control
 
-
 Reference
-~~~~~~~~~~~
+~~~~~~~~~
+
+- `A Survey of Motion Planning and Control Techniques for Self-driving
+  Urban Vehicles <https://arxiv.org/abs/1604.07446>`_
+
 
--  `A Survey of Motion Planning and Control Techniques for Self-driving
-   Urban Vehicles <https://arxiv.org/abs/1604.07446>`_