Model Predictive Control (MPC) for Unmanned Ground Vehicles (UGV): (Course Project)

Problem Statement:

Model Predictive Control(MPC) for trajectory tracking on Unmanned Ground Vehicle (UGV) with waypoint generation in an unknown environment using perception.

Possible Extensions: (Maybe someday in future :P)

1. Dynamic obstacle avoidance
2. Implementation and demonstration on hardware

Setup:

This is the setup for Ubuntu (22.04). Not sure how it works on Windows/Mac.

(Recommended) Make a separate conda environment and install the package in that environment:
conda create -n FOR_Project python=3.8
conda activate FOR_Project

First clone the repository:
git clone https://github.com/prakrutk/FOR_Project.git

Checkout to the branch named 'Prakrut':
git checkout Prakrut

Then go into the directory and install the package using pip:
cd FOR_Project
pip install --upgrade pip
pip install -e .
pip install -r requirements.txt

To run MPC code:
python3 dynamics/MPC.py (disclaimer: Something is working now we have to figure out what exactly is working)

To run Waypoint generation code:
python3 Waypoint_generation/Waypoint_new.py

Slides for more details: [link]

Dynamics Model: (To be updated)

Notations:

State variable: $ X = (x,y,\psi , \dot x, \dot y, \dot \psi )$

Input/control variable: $U = (\delta, \omega )$

Where,
$x,y$ are coordinates of the COM of the car in world frame.
$\psi$ is the heading angle of the car.
$\delta$ is the steering angle of the car.
$\omega$ is the rotational speed of both the wheels.

$f_{f_x},f_{f_y},f_{r_x},f_{r_y}$ are the force acting in the body frame of the front and rear wheels of the car respectively.

$l_f$ is the distance of the front wheel from the COM of the car.
$l_r$ is the distance of the rear wheel from the COM of the car.
$\beta$ is the sideslip angle of the car. (Not used)

Dynamics equation of the car:

$$ m\ddot x = f_{f_x}cos\delta - f_{f_y}sin\delta + f_{r_x} + m\dot y\dot \psi$$

$$m\ddot y = f_{f_y}cos\delta + f_{f_x}sin\delta + f_{r_y} - m\dot x\dot \psi$$

$$I_z\ddot \psi = l_f(f_{f_x}sin\delta + f_{f_y}cos\delta) - f_{r_y}l_r$$

$C_l$ is the cornering stiffness of the tire.
$s_f$ is the slip ratio of the front wheel.
$s_r$ is the slip ratio of the rear wheel.
$c_l$ is the longitudinal stiffness of the tire.
$\alpha_f$ is the slip angle of the front wheel.
$\alpha_r$ is the slip angle of the rear wheel.

Assuming a small slip angle and small slip ratio, the forces acting on the car can be written as:

$$ f_{f_x} = C_ls_f$$

$$ f_{f_y} = C_c\alpha_f$$

With the assumptions and substituting the above equations in the dynamics equation of the car, we get:

$$ m\ddot x = C_ls_f - C_c\alpha_f\delta + c_ls_r + m\dot y\dot \psi$$

$$m\ddot y = C_c\alpha_f + C_ls_f\delta + c_c\alpha_r - m\dot x\dot \psi $$

$$I_z\ddot \psi = l_f(C_ls_f\delta + C_c\alpha_f) - c_c\alpha_rl_r $$

Also,

$$ \dot x = V $$

$$ \dot y = \dot x(l_f + \delta) - l_r\dot \psi $$

$$ \dot \psi = \frac{\dot y}{l_r} $$

MPC Formulation:

$$ \min_{\Delta U, \epsilon } \begin{bmatrix} \Delta U , \epsilon \end{bmatrix}^T H\begin{bmatrix} \Delta U , \epsilon \end{bmatrix} \begin{bmatrix} \Delta U , \epsilon \end{bmatrix} + f\begin{bmatrix} \Delta U , \epsilon \end{bmatrix}$$

$$ \Delta U_{min} \leq \Delta U \leq \Delta U_{max}$$

$$ U_{min} \leq u(t-1) + \sum_{i=t}^{t+N_c-1} \Delta U(i) \leq U_{max}$$

$$ Y_{min} - \epsilon \leq \Phi_{X(t|t)} + \Theta \Delta U(t) \leq Y_{max} + \epsilon$$

Where:

$$H = \begin{bmatrix} \Theta ^T Q \Theta + R & 0 \\ 0 & \rho \end{bmatrix}$$

$$f = \begin{bmatrix} 2E^TQ\Theta & 0\end{bmatrix}$$

$$\epsilon = \text{Slack variable} $$

$$Y = \Phi_{X(t|t)} + \Theta \Delta U(t)$$

Stacking the states and inputs:

$$ \zeta = \begin{bmatrix} x,y,\phi,\dot x, \dot y, \dot \phi \end{bmatrix} $$ $$ U = \begin{bmatrix} V,\delta \end{bmatrix} $$

$$ X = \begin{bmatrix} \zeta \ U \end{bmatrix} $$

$$\eta = C\zeta$$

Now,

$$ X_1 = \bar A X_0 + \bar B \Delta u_0 $$ $$ X_2 = \bar A^2 X_0 + \bar A\bar B \Delta u_0 + \bar B \Delta u_1 $$ $$ X_3 = \bar A^3 X_0 + \bar A^2\bar B \Delta u_0 + \bar A\bar B \Delta u_1 + \bar B \Delta u_3 $$

Where,

$$\bar A = \begin{bmatrix} A & B \\ 0 & I \end{bmatrix}, \bar B = \begin{bmatrix} B \\ I \end{bmatrix} $$

Hence we can write,

$$\begin{bmatrix} X_1 \\ X_2 \\ X_3 \\ \vdots \end{bmatrix} = \begin{bmatrix} \bar A \\ \bar A^2 \\ \bar A^3 \\ \vdots \end{bmatrix}X_0 + \begin{bmatrix}\bar B & 0 & 0 & \cdots \\ \bar A \bar B & \bar B & 0 & \cdots \\ \bar A^2 \bar B & \bar A \bar B & \bar B & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{bmatrix} \begin{bmatrix}\Delta u_0 \\ \Delta u_1 \\ \Delta u_2 \\ \vdots \end{bmatrix}$$

Now writing in a condensed form,

$$Y = \Phi X_0 + \Theta \Delta U$$

Where,

$$Y = \begin{bmatrix} \eta_1 \\ \eta_2 \\ \eta_3 \\ \vdots \end{bmatrix}, \Delta U = \begin{bmatrix} \Delta u_0 \\ \Delta u_1 \\ \Delta u_2 \\ \vdots \end{bmatrix}$$

$$\Phi = \begin{bmatrix} \bar C \bar B & 0 & 0 & \cdots \\ \bar C \bar A \bar B & \bar C \bar B & 0 & \cdots \\ \bar C \bar A^2 \bar B & \bar C \bar A \bar B & \bar C \bar B & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{bmatrix}$$

$$\Theta = \begin{bmatrix} \bar C \bar A \\ \bar C \bar A^2 \\ \bar C \bar A^3 \\ \vdots \end{bmatrix}$$

$$\eta = C\zeta$$

Authors list:

1. Prakrut Kotecha
2. Aastha Mishra
3. Ishita Ganjoo
4. Mehul Nakra
5. Sayli Sawant
6. Tirth D Shiyala