# Model predictive speed and steering control — PythonRobotics documentation

# Model predictive speed and steering control

Model predictive speed and steering control

code:

PythonRobotics/model_predictive_speed_and_steer_control.py at master · AtsushiSakai/PythonRobotics

This is a path tracking simulation using model predictive control (MPC).

The MPC controller controls vehicle speed and steering base on linearized model.

This code uses cvxpy as an optimization modeling tool.

- Welcome to CVXPY 1.0 — CVXPY 1.0.6 documentation

## MPC modeling

State vector is:

\[z = [x, y, v,\phi]\]

x: x-position, y:y-position, v:velocity, φ: yaw angle

Input vector is:

\[u = [a, \delta]\]

a: accellation, δ: steering angle

The MPC cotroller minimize this cost function for path tracking:

\[min\ Q_f(z_{T,ref}-z_{T})^2+Q\Sigma({z_{t,ref}-z_{t}})^2+R\Sigma{u_t}^2+R_d\Sigma({u_{t+1}-u_{t}})^2\]

z_ref come from target path and speed.

subject to:

- Linearlied vehicle model

\[z_{t+1}=Az_t+Bu+C\]

- Maximum steering speed

\[|u_{t+1}-u_{t}|<du_{max}\]

- Maximum steering angle

\[|u_{t}|<u_{max}\]

- Initial state

\[z_0 = z_{0,ob}\]

- Maximum and minimum speed

\[v_{min} < v_t < v_{max}\]

- Maximum and minimum input

\[u_{min} < u_t < u_{max}\]

This is implemented at

PythonRobotics/model_predictive_speed_and_steer_control.py at f51a73f47cb922a12659f8ce2d544c347a2a8156 · AtsushiSakai/PythonRobotics

## Vehicle model linearization

Vehicle model is

\[\dot{x} = vcos(\phi)\]
\[\dot{y} = vsin((\phi)\]
\[\dot{v} = a\]
\[\dot{\phi} = \frac{vtan(\delta)}{L}\]

ODE is

\[\dot{z} =\frac{\partial }{\partial z} z = f(z, u) = A'z+B'u\]

where

\(\begin{equation*} A' = \begin{bmatrix} \frac{\partial }{\partial x}vcos(\phi) & \frac{\partial }{\partial y}vcos(\phi) & \frac{\partial }{\partial v}vcos(\phi) & \frac{\partial }{\partial \phi}vcos(\phi)\\ \frac{\partial }{\partial x}vsin(\phi) & \frac{\partial }{\partial y}vsin(\phi) & \frac{\partial }{\partial v}vsin(\phi) & \frac{\partial }{\partial \phi}vsin(\phi)\\ \frac{\partial }{\partial x}a& \frac{\partial }{\partial y}a& \frac{\partial }{\partial v}a& \frac{\partial }{\partial \phi}a\\ \frac{\partial }{\partial x}\frac{vtan(\delta)}{L}& \frac{\partial }{\partial y}\frac{vtan(\delta)}{L}& \frac{\partial }{\partial v}\frac{vtan(\delta)}{L}& \frac{\partial }{\partial \phi}\frac{vtan(\delta)}{L}\\ \end{bmatrix} \\ 　= \begin{bmatrix} 0 & 0 & cos(\bar{\phi}) & -\bar{v}sin(\bar{\phi})\\ 0 & 0 & sin(\bar{\phi}) & \bar{v}cos(\bar{\phi}) \\ 0 & 0 & 0 & 0 \\ 0 & 0 &\frac{tan(\bar{\delta})}{L} & 0 \\ \end{bmatrix} \end{equation*}\)

\(\begin{equation*} B' = \begin{bmatrix} \frac{\partial }{\partial a}vcos(\phi) & \frac{\partial }{\partial \delta}vcos(\phi)\\ \frac{\partial }{\partial a}vsin(\phi) & \frac{\partial }{\partial \delta}vsin(\phi)\\ \frac{\partial }{\partial a}a & \frac{\partial }{\partial \delta}a\\ \frac{\partial }{\partial a}\frac{vtan(\delta)}{L} & \frac{\partial }{\partial \delta}\frac{vtan(\delta)}{L}\\ \end{bmatrix} \\ 　= \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 1 & 0 \\ 0 & \frac{\bar{v}}{Lcos^2(\bar{\delta})} \\ \end{bmatrix} \end{equation*}\)

You can get a discrete-time mode with Forward Euler Discretization with sampling time dt.

\[z_{k+1}=z_k+f(z_k,u_k)dt\]

Using first degree Tayer expantion around zbar and ubar

\[z_{k+1}=z_k+(f(\bar{z},\bar{u})+A'z_k+B'u_k-A'\bar{z}-B'\bar{u})dt\]
\[z_{k+1}=(I + dtA')z_k+(dtB')u_k + (f(\bar{z},\bar{u})-A'\bar{z}-B'\bar{u})dt\]

So,

\[z_{k+1}=Az_k+Bu_k +C\]

where,

\(\begin{equation*} A = (I + dtA')\\ = \begin{bmatrix} 1 & 0 & cos(\bar{\phi})dt & -\bar{v}sin(\bar{\phi})dt\\ 0 & 1 & sin(\bar{\phi})dt & \bar{v}cos(\bar{\phi})dt \\ 0 & 0 & 1 & 0 \\ 0 & 0 &\frac{tan(\bar{\delta})}{L}dt & 1 \\ \end{bmatrix} \end{equation*}\)

\(\begin{equation*} B = dtB'\\ = \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ dt & 0 \\ 0 & \frac{\bar{v}}{Lcos^2(\bar{\delta})}dt \\ \end{bmatrix} \end{equation*}\)

\(\begin{equation*} C = (f(\bar{z},\bar{u})-A'\bar{z}-B'\bar{u})dt\\ = dt( \begin{bmatrix} \bar{v}cos(\bar{\phi})\\ \bar{v}sin(\bar{\phi}) \\ \bar{a}\\ \frac{\bar{v}tan(\bar{\delta})}{L}\\ \end{bmatrix} - \begin{bmatrix} \bar{v}cos(\bar{\phi})-\bar{v}sin(\bar{\phi})\bar{\phi}\\ \bar{v}sin(\bar{\phi})+\bar{v}cos(\bar{\phi})\bar{\phi}\\ 0\\ \frac{\bar{v}tan(\bar{\delta})}{L}\\ \end{bmatrix} - \begin{bmatrix} 0\\ 0 \\ \bar{a}\\ \frac{\bar{v}\bar{\delta}}{Lcos^2(\bar{\delta})}\\ \end{bmatrix} )\\ = \begin{bmatrix} \bar{v}sin(\bar{\phi})\bar{\phi}dt\\ -\bar{v}cos(\bar{\phi})\bar{\phi}dt\\ 0\\ -\frac{\bar{v}\bar{\delta}}{Lcos^2(\bar{\delta})}dt\\ \end{bmatrix} \end{equation*}\)

This equation is implemented at

PythonRobotics/model_predictive_speed_and_steer_control.py at eb6d1cbe6fc90c7be9210bf153b3a04f177cc138 · AtsushiSakai/PythonRobotics

## Reference

- Vehicle Dynamics and Control | Rajesh Rajamani | Springer

- MPC Course Material - MPC Lab @ UC-Berkeley