Enhance dubins path docs (#679)

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

* Enhance dubins path docs

docs/conf.py

@@ -39,6 +39,7 @@ release = ''
 # extensions coming with Sphinx (named 'sphinx.ext.*') or your custom
 # ones.
 extensions = [
+    'matplotlib.sphinxext.plot_directive',
     'sphinx.ext.autodoc',
     'sphinx.ext.mathjax',
     'sphinx.ext.viewcode',

docs/modules/path_planning/dubins_path/dubins_path_main.rst

@@ -9,40 +9,68 @@ Dubins path
 ~~~~~~~~~~~~
 Dubins path is a analytical path planning algorithm for a simple car model.
 
-It can generates a shortest path between 2D poses (x, y, yaw) with maximum curvture comstraint and tangent(yaw angle) constraint.
+It can generates a shortest path between two 2D poses (x, y, yaw) with maximum curvature constraint and tangent(yaw angle) constraint.
 
-The path consist of 3 segments of maximum curvature curves or a straight line segment.
+Generated paths consist of 3 segments of maximum curvature curves or a straight line segment.
 
 Each segment type can is categorized by 3 type: 'Right turn (R)' , 'Left turn (L)', and 'Straight (S).' 
 
-Possible path will be at least one of these six types: RSR, RSL, LSR, LSL, RLR, LRL. 
+Possible path will be at least one of these six types: RSR, RSL, LSR, LSL, RLR, LRL.
 
-For example, one of RSR Dubins paths is:
+Dubins path planner can output each segment type and distance of each course segment.
+
+For example, a RSR Dubins path is:
 
 .. image:: RSR.jpg
    :width: 400px
 
-one of RLR Dubins paths is:
+Each segment distance can be calculated by:
+
+:math:`\alpha = mod(-\theta)`
+
+:math:`\beta = mod(x_{e, yaw} - \theta)`
+
+:math:`p^2 = 2 + d ^ 2 - 2\cos(\alpha-\beta) + 2d(\sin\alpha - \sin\beta)`
+
+:math:`t = atan2(\cos\beta - \cos\alpha, d + \sin\alpha - \sin\beta)`
+
+:math:`d_1 = mod(-\alpha + t)`
+
+:math:`d_2 = p`
+
+:math:`d_3 = mod(\beta - t)`
+
+where :math:`\theta` is tangent and d is distance from :math:`x_s` to :math:`x_e`
+
+A RLR Dubins path is:
 
 .. image:: RLR.jpg
    :width: 200px
 
-Dubins path planner can output three types and distances of each course segment.
+Each segment distance can be calculated by:
+
+:math:`t = (6.0 - d^2 + 2\cos(\alpha-\beta) + 2d(\sin\alpha - \sin\beta)) / 8.0`
+
+:math:`d_2 = mod(2\pi - acos(t))`
+
+:math:`d_1 = mod(\alpha - atan2(\cos\beta - \cos\alpha, d + \sin\alpha - \sin\beta) + d_2 / 2.0)`
+
+:math:`d_3 = mod(\alpha - \beta - d_1 + d_2)`
 
 You can generate a path from these information and the maximum curvature information.
 
-In the example code, a path which is minimum course length one among 6 course type is selected and then a path is constructed.
+A path type which has minimum course length among 6 types is selected,
+and then a path is constructed based on the selected type and its distances.
 
-Code
+API
 ~~~~~~~~~~~~~~~~~~~~
 
-.. automodule:: PathPlanning.DubinsPath.dubins_path_planner
-    :members:
+.. autofunction:: PathPlanning.DubinsPath.dubins_path_planner.plan_dubins_path
 
 
 Reference
 ~~~~~~~~~~~~~~~~~~~~
-
+-  `On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents <https://www.jstor.org/stable/2372560?origin=crossref>`__
 -  `Dubins path - Wikipedia <https://en.wikipedia.org/wiki/Dubins_path>`__
 -  `15.3.1 Dubins Curves <http://planning.cs.uiuc.edu/node821.html>`__
 -  `A Comprehensive, Step-by-Step Tutorial to Computing Dubin’s Paths <https://gieseanw.wordpress.com/2012/10/21/a-comprehensive-step-by-step-tutorial-to-computing-dubins-paths/>`__