diff --git a/.ipynb_checkpoints/README-checkpoint.md b/.ipynb_checkpoints/README-checkpoint.md
index 217e39a..bc6c2b7 100644
--- a/.ipynb_checkpoints/README-checkpoint.md
+++ b/.ipynb_checkpoints/README-checkpoint.md
@@ -1,36 +1,53 @@
 # mpc_python
 
-Python implementation of mpc controller for path tracking.
+Python implementation of a mpc controller for path tracking using **[CVXPY](https://www.cvxpy.org/)**.
 
 ## About
 
-The MPC is a model predictive path following controller which does follow a predefined reference path Xref and Yref by solving an optimization problem. The resulting optimization problem is shown in the following equation:
+The MPC is a model predictive path following controller which does follow a predefined reference by solving an optimization problem. The resulting optimization problem is shown in the following equation:
 
-![](img/quicklatex1.png)
+![](img/quicklatex_equation.png)
 
-The vehicle dynamics are described by the differential drive model:
+The terns of the cost function are the sum of the **reference tracking error**, **heading effort** and **actuaction rate of change**.
+
+Where R,P,Q are the cost matrices used to tune the response.
+
+The vehicle model is described by the bicycle kinematics model using the state space matrices A and B:
 
 ![](img/quicklatex2.png)
 
-The state variables of the model are:
+The state variables **(x)** of the model are:
 
 * **x** coordinate of the robot
 * **y** coordinate of the robot
+* **v** velocuty of the robot
 * **theta** heading of the robot
 
-The inputs of the model are:
+The inputs **(u)** of the model are:
 
-* **v** linear velocity of the robot
-* **w** angular velocity of the robot
+* **a** linear acceleration of the robot
+* **delta** steering angle of the robot
 
 ## Demo
 
+The MPC implementation is tested using **[bullet](https://pybullet.org/wordpress/)** physics simulator. Racing car model is from: *https://github.com/erwincoumans/pybullet_robots*.
+
+![](img/f10.png)
+
+Results:
+
 ![](img/demo.gif)
 
-To run the demo:
+To run the pybullet demo:
 
 ```bash
-python3 mpc_demo/main.py
+python3 mpc_demo/mpc_demo_pybullet.py
+```
+
+To run the simulation-less demo:
+
+```bash
+python3 mpc_demo/mpc_demo_pybullet.py
 ```
 
 ## Requirements
@@ -38,3 +55,8 @@ python3 mpc_demo/main.py
 ```bash
 pip3 install --user --requirement requirements.txt
 ```
+
+## References
+* [mpc paper](https://borrelli.me.berkeley.edu/pdfpub/IV_KinematicMPC_jason.pdf)
+* [pythonrobotics](https://github.com/AtsushiSakai/PythonRobotics/)
+* [pybullet](https://pybullet.org/wordpress/)
\ No newline at end of file
diff --git a/README.md b/README.md
index 36418b6..bc6c2b7 100644
--- a/README.md
+++ b/README.md
@@ -6,11 +6,11 @@ Python implementation of a mpc controller for path tracking using **[CVXPY](http
 
 The MPC is a model predictive path following controller which does follow a predefined reference by solving an optimization problem. The resulting optimization problem is shown in the following equation:
 
-![](img/quicklatex1.png)
+![](img/quicklatex_equation.png)
 
-The terns of the cost function are the sum of the **cross-track error**, **heading error**, **velocity error** and **actuaction effort**.
+The terns of the cost function are the sum of the **reference tracking error**, **heading effort** and **actuaction rate of change**.
 
-Where R,P,K,Q are the cost matrices used to tune the response.
+Where R,P,Q are the cost matrices used to tune the response.
 
 The vehicle model is described by the bicycle kinematics model using the state space matrices A and B:
 
@@ -55,3 +55,8 @@ python3 mpc_demo/mpc_demo_pybullet.py
 ```bash
 pip3 install --user --requirement requirements.txt
 ```
+
+## References
+* [mpc paper](https://borrelli.me.berkeley.edu/pdfpub/IV_KinematicMPC_jason.pdf)
+* [pythonrobotics](https://github.com/AtsushiSakai/PythonRobotics/)
+* [pybullet](https://pybullet.org/wordpress/)
\ No newline at end of file
diff --git a/img/quicklatex_equation.png b/img/quicklatex_equation.png
new file mode 100644
index 0000000..74eb6f9
Binary files /dev/null and b/img/quicklatex_equation.png differ
diff --git a/img/quicklatex_equation_white.png b/img/quicklatex_equation_white.png
new file mode 100644
index 0000000..e1f6b8e
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diff --git a/notebooks/MPC_racecar_tracking.ipynb b/notebooks/MPC_racecar_tracking.ipynb
index 5b94dce..c5cb2a0 100644
--- a/notebooks/MPC_racecar_tracking.ipynb
+++ b/notebooks/MPC_racecar_tracking.ipynb
@@ -634,7 +634,7 @@
     "$\n",
     "\\begin{equation}\n",
     "\\begin{aligned}\n",
-    "\\min_{} \\quad &  \\sum^{t+T-1}_{j=t} (x_{j} - x_{j,ref})^TQ(x_{j} - x_{j,ref}) + \\sum^{t+T-1}_{j=t+1} u^T_{j}Ru_{j} + (u_{j} - u_{j-1})^T(u_{j} - u_{j-1}) \\\\\n",
+    "\\min_{} \\quad &  \\sum^{t+T-1}_{j=t} (x_{j} - x_{j,ref})^TQ(x_{j} - x_{j,ref}) + \\sum^{t+T-1}_{j=t+1} u^T_{j}Ru_{j} + (u_{j} - u_{j-1})^TP(u_{j} - u_{j-1}) \\\\\n",
     "\\textrm{s.t.} \\quad &  x(0) = x0\\\\\n",
     "  & x_{j+1} = Ax_{j}+Bu_{j} \\quad  \\textrm{for} \\quad t<j<t+T-1  \\\\\n",
     "  & u_{MIN} < u_{j} < u_{MAX} \\quad  \\textrm{for} \\quad t<j<t+T-1 \\\\\n",