Kinematic Bicycle Model

Abstract

A python library for the Kinematic Bicycle model. The Kinematic Bicycle is a compromise between the non-linear and linear bicycle models for high-speed integration of the library with little configuration. The model can be defined with the following state-space representation,

$$ \frac{d}{dt} \begin{pmatrix} x \\ y \\ \theta \\ v \end{pmatrix} = \begin{pmatrix} v\cos{\theta} \\ v\sin{\theta} \\ \frac{v\tan{\delta}}{L} \\ a - f \end{pmatrix} $$

where $v$ is the vehicle's velocity in the x-axis, $\theta$ is the vehicle's yaw, $\delta$ is the steering angle, $L$ is the vehicle's wheelbase, $a$ is the acceleration/throttle, $f$ is friction in the x-axis.

At initialisation
:param wheelbase:       (float) vehicle's wheelbase [m]
:param max_steer:       (float) vehicle's steering limits [rad]
:param dt:              (float) discrete time period [s]
:param c_r:             (float) vehicle's coefficient of resistance 
:param c_a:             (float) vehicle's aerodynamic coefficient

At every time step  
:param x:               (float) vehicle's x-coordinate [m]
:param y:               (float) vehicle's y-coordinate [m]
:param yaw:             (float) vehicle's heading [rad]
:param velocity:        (float) vehicle's velocity in the x-axis [m/s]
:param throttle:        (float) vehicle's accleration [m/s^2]
:param delta:           (float) vehicle's steering angle [rad]

:return x:              (float) vehicle's x-coordinate [m]
:return y:              (float) vehicle's y-coordinate [m]
:return yaw:            (float) vehicle's heading [rad]
:return velocity:       (float) vehicle's velocity in the x-axis [m/s]
:return delta:          (float) vehicle's steering angle [rad]
:return omega:          (float) vehicle's angular velocity [rad/s]

Advantages

The model allows the vehicle to come to rest without passing the model a negative acceleration; similar to the non-linear bicycle.
This lightweight model is able to accurately represent a vehicle with no slip or tire stiffness.

Limitations

Just like with all other bicycle models, this model is a discrete model and loses its accuracy when the time step is set too large or the vehicle is made to travel at unreasonably high speeds. Usually, the FPS of the simulation should be set to the highest possible value for the greatest accuracy. However, for rendering high-quality GIFs, 50 FPS is found to be most optimal. See the GIF89a specification.

Requirements

The model itself has zero dependencies and can run its fastest on vanilla python. However, to support type-hinting, Python 3.10 or greater is required.

Demo

Recursively git clone the repository

git clone --recursive https://github.com/winstxnhdw/KinematicBicycleModel.git

Install the requirements

pip install -r requirements.txt

Play the animation

python animate.py

Concept

Though our implementation is titled the Kinematic Bicycle Model, it does take into account some forward friction. This was never intended to improve the accuracy of the model. Instead, it provides a more intuitive API by removing the need to constantly control the input throttle. However, that is where the differences end. You can read about the bicycle model in full detail by Theers et al., here.

ayushmankumar7 / KinematicBicycleModel