updating README

Author: lucasrm25

README.md

@@ -1,21 +1,48 @@
-# Gaussian-Process-based-Model-Predictive-Control
+# Gaussian-Process based Model Predictive Control
 Project for the course "Statistical Learning and Stochastic Control" at University of Stuttgart
 
 
+For detailed information about the project, please refer to the [Report]().
+
 Supported Matlab Version **>= R2019a**
 
 
+## Control of a Race Vehicle with unkown complex dynamics
 
-## Control of an Inverted Pendulum with deffect motor
+To run the Race Car example execute:
+> main_singletrack.m
 
-To run the Inverted Pendulum please execute
-> main_invertedPendulum.m
+<br>
 
+A Gaussian process is used to learn unmodeled dynamics
+```math
+    x_{k+1} = f_d(x_k,u_k) + B_d * ( GP(z_k) + w )
 
-## Control of a Race Vehicle with unkown true dynamics
+    , where z_k = [Bz_x*xk ; Bz_u*uk] is the vector of selected features
+            f_d is the dicrete nominal model
+            w ~ N(0,\sigma_n) is the process WG noise
+            GP is the Gaussian Process model reponsible for learning the unmodeled dynamics
+
+```
+The Gaussian Process model GP is then fed with data (X,Y+w) collected online, such that:
+```math
+    X = [x_k,u_k]
+    Y + w = pinv(B_d) * ( x_{k+1} - f_d(x_k,u_k) )
+```
+and it is trained (hyperparameter optimization) by maximizing the log Likelihood p(Y|X,theta), where theta is the vector of hyperparameters.
 
-To run the Race Car example execute:
-> main_singletrack.m
 
-![](./simresults/trackAnimVideo-13-Jan-2020.gif)
 
+### Results
+
+
+| MPC controller with Gaussian Process **deactivated**        | MPC controller with **TRAINED** and **ACTIVATED** Gaussian Process         |
+| ------------- |-------------|
+| <img src="./simresults/trackAnimVideo-16-Jan-2020-without-GP.gif" alt="drawing" width="400"/>      | <img src="./simresults/trackAnimVideo-16-Jan-2020-with-GP-optimized.gif" alt="drawing" width="400"/>    |
+
+<br>
+
+## Control of an Inverted Pendulum with deffect motor
+
+To run the Inverted Pendulum please execute
+> main_invertedPendulum.m