clean up clothoidal_paths.py (#638)

* clean up clothoidal_paths.py

* add unit tests for clothoid_paths

* add unit tests for clothoid_paths

* add unit tests for clothoid_paths

* code clean up

* code clean up

* code clean up

* code clean up

* code clean up

* code clean up

* code clean up

* code clean up

docs/modules/control/inverted_pendulum_control/inverted_pendulum_control.rst

@@ -34,7 +34,7 @@ So
     & \ddot{\theta} = \frac{g(M + m)sin{\theta} - \dot{\theta}^2lmsin{\theta}cos{\theta} + ucos{\theta}}{l(M + m - mcos^2{\theta})}
 
 
-Linearlied model when :math:`\theta` small, :math:`cos{\theta} \approx 1`, :math:`sin{\theta} \approx \theta`, :math:`\dot{\theta}^2 \approx 0`.
+Linearized model when :math:`\theta` small, :math:`cos{\theta} \approx 1`, :math:`sin{\theta} \approx \theta`, :math:`\dot{\theta}^2 \approx 0`.
 
 .. math::
     & \ddot{x} =  \frac{gm}{M}\theta + \frac{1}{M}u\\
@@ -68,16 +68,20 @@ If control x and \theta
 LQR control
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
-The LQR cotroller minimize this cost function defined as:
+The LQR controller minimize this cost function defined as:
 
 .. math::  J = x^T Q x + u^T R u
+
 the feedback control law that minimizes the value of the cost is:
 
 .. math::  u = -K x
+
 where:
 
 .. math::  K = (B^T P B + R)^{-1} B^T P A
-and :math:`P` is the unique positive definite solution to the discrete time `algebraic Riccati equation <https://en.wikipedia.org/wiki/Inverted_pendulum#From_Lagrange's_equations>`__  (DARE):
+
+and :math:`P` is the unique positive definite solution to the discrete time
+`algebraic Riccati equation <https://en.wikipedia.org/wiki/Inverted_pendulum#From_Lagrange's_equations>`__  (DARE):
 
 .. math::  P = A^T P A - A^T P B ( R + B^T P B )^{-1} B^T P A + Q
 
@@ -85,13 +89,13 @@ and :math:`P` is the unique positive definite solution to the discrete time `alg
 
 MPC control
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~
-The MPC cotroller minimize this cost function defined as:
+The MPC controller minimize this cost function defined as:
 
 .. math:: J = x^T Q x + u^T R u
 
 subject to:
 
-- Linearlied Inverted Pendulum model
+- Linearized Inverted Pendulum model
 - Initial state
 
 .. image:: https://github.com/AtsushiSakai/PythonRoboticsGifs/raw/master/Control/InvertedPendulumCart/animation.gif