Dynamic Bicycle Model

This page describes the dynamic bicycle model with a linear tyre model and longitudinal load transfer. The reference point for the vehicle dynamics is the center of gravity (CoG).

See also the vehicle model interface documentation for additional information.

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State Vector

The state vector is defined as:

\[ \mathbf{x} = \begin{bmatrix} x \\ y \\ v_x \\ v_y \\ \psi \\ \dot{\psi} \\ \delta \end{bmatrix} \]

where:

Symbol	Description
\(x, y\)	Global position of the vehicle CoG
\(v_x\)	Longitudinal velocity in body frame
\(v_y\)	Lateral velocity in body frame
\(\psi\)	Heading angle
\(\dot{\psi}\)	Yaw rate
\(\delta\)	Steering angle

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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.

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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 (\(l_f + l_r\))
\(m\)	mass of the vehicle
\(h\_{\mathrm{cog}}\)	Height of the CoG
\(C_f\)/\(C_r\)	front/ rear cornering stiffness coefficient

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Tyre Model

A linear tyre model is used for both front and rear axles.

Slip Angles

The slip angles are defined as:

\[ \alpha_f = \arctan\left(\frac{v_y + l_f \dot{\psi}}{v_x}\right) - \delta \]

\[ \alpha_r = \arctan\left(\frac{v_y - l_r \dot{\psi}}{v_x}\right) \]

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Normal Forces with Load Transfer

We account for longitudinal load transfer due to the (commanded) longitudinal acceleration:

\[ F\_{z,f} = \frac{m g l_r - m a h\_{\mathrm{cog}}}{l\_{\mathrm{wb}}} \]

\[ F\_{z,r} = \frac{m g l_f + m a h\_{\mathrm{cog}}}{l\_{\mathrm{wb}}} \]

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Lateral Tyre Forces

The lateral tyre forces follow as:

\[ F\_{c,f} = - C_f \alpha_f F\_{z,f} \]

\[ F\_{c,r} = - C_r \alpha_r F\_{z,r} \]

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Nominal Vehicle Dynamics

The nominal (continuous-time) dynamics are governed by the following set of differential equations:

Kinematic Relations

\[ \dot{x} = v_x \cos(\psi) - v_y \sin(\psi) \]

\[ \dot{y} = v_x \sin(\psi) + v_y \cos(\psi) \]

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Longitudinal and Lateral Dynamics

\[ \dot{v}_x = \dot{\psi} v_y + u_1 - \frac{F_{c,f} \sin(\delta)}{m} \]

\[ \dot{v}_y = -\dot{\psi} v_x + \frac{F_{c,f} \cos(\delta) + F\_{c,r}}{m} \]

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Yaw and Steering Dynamics

\[ \ddot{\psi} = \frac{l_f F\_{c,f} \cos(\delta) - l_r F\_{c,r}}{I\_{zz}} \]

\[ \dot{\delta} = u_2 \]

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Normalized Accelerations

The model provides the normalized longitudinal acceleration

\[ a\_{\mathrm{long}} = u_1 - \frac{F\_{c,f} \sin(\delta)}{m} $$ $$ a\_{\mathrm{long,norm}} = \frac{a\_{\mathrm{long}}}{a\_{\mathrm{long,max}}} $$ and the normalized lateral acceleration: $$ a\_{\mathrm{lat}} = \frac{F\_{c,f} \cos(\delta) + F\_{c,r}}{m} $$ $$ a\_{\mathrm{lat,norm}} = \frac{a\_{\mathrm{lat}}}{a\_{\mathrm{lat,max}}} \]

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