linearN

linearN is a MATLAB implementation of a general linear single-track model for articulated vehicles with an arbitrary number of vehicle units and axles.

The main function, articABCD, builds a continuous-time state-space model for an articulated vehicle at a constant longitudinal speed. The model is intended for lateral dynamics analysis, controller design, eigenvalue analysis, and step-response simulation of multi-unit road vehicles such as tractor-semitrailers, longer combination vehicles, and other articulated vehicle configurations.

Features

• Builds a linear state-space model for an articulated vehicle with n units and N axles.
• Supports different numbers of axles on different vehicle units.
• Uses vehicle mass, yaw inertia, axle locations, cornering stiffness, joint locations, and vehicle speed as inputs.
• Returns continuous-time state-space matrices:

[Am, Bm, Cm, Dm] = articABCD(M, J, L, Ca, Xf, Xr, U);

• Includes an example parameter file for a three-unit Nordic combination vehicle.

Model description

The model represents the lateral dynamics of an articulated vehicle using a linear single-track formulation.

For an articulated vehicle with n vehicle units and N axles, the reduced independent state vector has 2n states:

x = [beta_1, r_1, r_2, ..., r_n, theta_1, theta_2, ..., theta_(n-1)]'

where:

• beta_1 is the body slip angle of the first unit.
• r_i is the yaw rate of vehicle unit i.
• theta_i is the articulation angle between unit i and unit i+1.

The inputs are the steering angles at each axle:

u = [delta_1, delta_2, ..., delta_N]'

The function returns:

• Am: state matrix
• Bm: input matrix
• Cm: output matrix
• Dm: feedthrough matrix

The output is currently defined as all independent states.

Requirements

This project requires MATLAB.

For basic matrix generation with articABCD.m, no special toolbox is required. For state-space simulation using commands such as ss and step, MATLAB Control System Toolbox is required.

Quick start

Clone the repository:

git clone https://github.com/QingweiLau/linearN.git
cd linearN

Open MATLAB and add the repository folder to the MATLAB path:

addpath(pwd)

Load the example Nordic vehicle parameters:

Nordic01_par

Set the vehicle speed:

kph = 1/3.6;
U = 80 * kph;

Build the state-space model:

[Am, Bm, Cm, Dm] = articABCD(M, J, L, Ca, Xf, Xr, U);

Create a state-space system using front-axle steering as the input:

sys = ss(Am, Bm(:,1), Cm, Dm(:,1));

Simulate and plot the step response:

[y, t] = step(sys, 5);

figure
plot(t, y(:,2:4))
xlabel('Time (s)')
ylabel('Yaw rate (rad/s)')
legend('Unit 1', 'Unit 2', 'Unit 3')
grid on

Input format

M

Masses of the vehicle units.

M = [M1, M2, ..., Mn];

Unit: kg

Example:

M = [23, 3, 34] * 1000;

J

Yaw moments of inertia of the vehicle units.

J = [J1, J2, ..., Jn];

Unit: kg m²

Example:

J = [15, 3, 46] * 1000;

L

Cell array of axle longitudinal positions relative to the center of gravity of each vehicle unit.

L = {L1, L2, ..., Ln};

Each Li is a vector containing the axle positions of unit i.

Unit: m

Example:

L1 = [3.88, -1.02, -2.39];
L2 = [0.65, -0.65];
L3 = [-1.33, -2.64, -3.95];
L = {L1, L2, L3};

Ca

Cell array of axle cornering stiffness values.

Ca = {Ca1, Ca2, ..., Can};

The structure of Ca should match the structure of L.

Unit: N/rad

Example:

Ca1 = [356, 480, 480] * 1000;
Ca2 = [402, 402] * 1000;
Ca3 = [432, 432, 432] * 1000;
Ca = {Ca1, Ca2, Ca3};

Xf

Front joint positions relative to the center of gravity of each vehicle unit.

Xf = [Xf1, Xf2, ..., Xfn];

Unit: m

For the first unit, the front joint value is not used by the model.

Xr

Rear joint positions relative to the center of gravity of each vehicle unit.

Xr = [Xr1, Xr2, ..., Xrn];

Unit: m

U

Constant longitudinal vehicle speed.

Unit: m/s

Example:

U = 80 / 3.6;

Example vehicle

The file Nordic01_par.m provides parameters for a three-unit Nordic combination vehicle. The data are based on the example referenced in:

Kharrazi, S., Gordon, T., & Lidberg, M. “Improving lateral performance of longer combination vehicles—An approach based on eigenstructure assignment.” IEEE Intelligent Vehicles Symposium, 2012.

Notes

• The model assumes constant longitudinal speed.
• Tire lateral forces are represented using linear cornering stiffness.
• The function eliminates dependent constraints and returns a reduced state-space model using independent states.
• Steering inputs are defined at axle level, so the number of inputs equals the total number of axles.

Citation

If you use this code in academic work, please cite the repository and the relevant vehicle dynamics references used in your study.

Suggested repository citation:

Liu, Q. linearN: MATLAB linear single-track model for articulated vehicles. GitHub repository: https://github.com/QingweiLau/linearN