Kinematic Bicycle Model
This page describes the kinematic bicycle model. The reference point for the vehicle dynamics is the center of gravity (CoG).
See also the vehicle model interface documentation for additional information.
State Vector
The state vector is defined as:
\[
\mathbf{x} =
\begin{bmatrix}
x \\
y \\
v \\
\psi \\
\delta
\end{bmatrix}
\]
where:
| Symbol | Description |
|---|---|
| \(x, y\) | Global position of the vehicle CoG |
| \(v\) | (Total) Velocity |
| \(\psi\) | Heading angle |
| \(\delta\) | Steering angle |
Control Inputs
The control input vector is given by:
\[
\mathbf{u} =
\begin{bmatrix}
a \\
\dot{\delta}
\end{bmatrix}
\]
where:
| Symbol | Description |
|---|---|
| \(a\) | Longitudinal acceleration |
| \(\dot{\delta}\) | Steering angle velocity |
Input bounds are derived from the vehicle parameters a_long_max, steering_angle_velocity_max.
Vehicle Parameters
The model uses the following parameters:
| Symbol | Description |
|---|---|
| \(l_f\) | Distance from CoG to front axle |
| \(l_r\) | Distance from CoG to rear axle |
| \(l\_{\mathrm{wb}}\) | Wheelbase (\(l_f + l_r\)) |
Kinematic Relations
The continuous-time dynamics of the kinematic bicycle are:
Slip Angle
\[
\beta = \arctan\left(\tan(\delta) \frac{l_r}{l\_\mathrm{wb}}\right)
\]
Position Dynamics
\[
\dot{x} = v \cos(\psi + \beta)
\]
\[
\dot{y} = v \sin(\psi + \beta)
\]
Longitudinal and Heading Dynamics
\[
\dot{v} = u_1
\]
\[
\dot{\psi} = \frac{v \sin(\beta)}{l_r}
\]
Steering Dynamics
\[
\dot{\delta} = u_2
\]
Normalized Accelerations
The model provides the normalized longitudinal acceleration
\[
a\_{\mathrm{long,norm}} = \frac{u_1}{a\_{\mathrm{long,max}}}
\]
and the normalized lateral acceleration: $$ a_{\mathrm{lat}} = v \cdot \dot{\psi}$$ $$ a_{\mathrm{lat,norm}} = \frac{a_{\mathrm{lat}}}{a_{\mathrm{lat,max}}} $$