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docs/modules/path_planning/clothoid_path/clothoid_path_main.rst

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+.. _clothoid-path-planning:
+
+Clothoid path planning
+--------------------------
+
+.. image:: https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathPlanning/ClothoidPath/animation1.gif
+.. image:: https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathPlanning/ClothoidPath/animation2.gif
+.. image:: https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/PathPlanning/ClothoidPath/animation3.gif
+
+This is a clothoid path planning sample code.
+
+This can interpolate two 2D pose (x, y, yaw) with a clothoid path,
+which its curvature is linearly continuous.
+In other words, this is G1 Hermite interpolation with a single clothoid segment.
+
+This path planning algorithm as follows:
+
+Step1: Solve g function
+~~~~~~~~~~~~~~~~~~~~~~~
+
+Solve the g(A) function with a nonlinear optimization solver.
+
+.. math::
+
+    g(A):=Y(2A, \delta-A, \phi_{s})
+
+Where
+
+* :math:`\delta`: the orientation difference between start and goal pose.
+* :math:`\phi_{s}`: the orientation of the start pose.
+* :math:`Y`: :math:`Y(a, b, c)=\int_{0}^{1} \sin \left(\frac{a}{2} \tau^{2}+b \tau+c\right) d \tau`
+
+
+Step2: Calculate path parameters
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+We can calculate these path parameters using :math:`A`,
+
+:math:`L`: path length
+
+.. math::
+
+        L=\frac{R}{X\left(2 A, \delta-A, \phi_{s}\right)}
+
+where
+
+* :math:`R`: the distance between start and goal pose
+* :math:`X`: :math:`X(a, b, c)=\int_{0}^{1} \cos \left(\frac{a}{2} \tau^{2}+b \tau+c\right) d \tau`
+
+
+- :math:`\kappa`: curvature
+
+.. math::
+
+        \kappa=(\delta-A) / L
+
+
+- :math:`\kappa'`: curvature rate
+
+.. math::
+
+        \kappa^{\prime}=2 A / L^{2}
+
+
+Step3: Construct a path with Fresnel integral
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+The final clothoid path can be calculated with the path parameters and Fresnel integrals.
+
+.. math::
+        \begin{aligned}
+        &x(s)=x_{0}+\int_{0}^{s} \cos \left(\frac{1}{2} \kappa^{\prime} \tau^{2}+\kappa \tau+\vartheta_{0}\right) \mathrm{d} \tau \\
+        &y(s)=y_{0}+\int_{0}^{s} \sin \left(\frac{1}{2} \kappa^{\prime} \tau^{2}+\kappa \tau+\vartheta_{0}\right) \mathrm{d} \tau
+        \end{aligned}
+
+
+References
+~~~~~~~~~~
+
+-  `Fast and accurate G1 fitting of clothoid curves <https://www.researchgate.net/publication/237062806>`__