# Vehicle Articulated Linear

Linear articulated bicycle model with 4 degrees of freedom.

The code of this class can be found in VehicleArticulatedLinear. It inherits methods from abstract class VehicleArticulated. The complete list of class codes is in API.

## Theory

The development of the equations of motion of this model can be found in TheoryVehicleArticulated.

## Sintax

dx = VehicleModel.Model(t,states,tspan)

dx = VehicleModel.MassMatrix(t,states,tspan)

## Arguments

The following table describes the input arguments:

 t Time states Model state variables: [XT YT PSI PHI VT ALPHAT dPSI dPHI] tspan Time span

## Description

Bicycle model

Free body diagram

The center of gravity of the tractor and semitrailer are located at the point $$T$$ and $$S$$, respectively. The front and rear axles are located at the points $$F$$ and $$R$$, respectively. $$A$$ is the articulation point and $$M$$ is the axle of the semitrailer. The constant $$a$$ measures the distance of point $$F$$ to $$T$$ and $$b$$ the distance of point $$T$$ to $$R$$. The distance of the articulation from the rear axle of the tractor is given by $$c$$. $$d$$ and $$e$$ are the distances from the semitrailer. The angles $$\alpha_F$$ e $$\alpha_R$$ are the front and rear slip angles, respectively. $$\alpha_T$$ is the vehicle side slip angle and $$\psi$$ is the vehicle yaw angle. $$\delta$$ is the steering angle.

Este modelo escrito na forma:

${\bf M}({\bf x}) \dot{{\bf x}} = {\bf f}({\bf x})$

Where $${\bf x}$$ is the state vector, $${\bf M}({\bf x})$$ the mass matrix and $${\bf f}({\bf x})$$ is the vector function. Therefore, a function that allows the integration of the system with an explicit mass matrix is necessary. In this package the ode45 function is used. Details: ode45 (Mass matrix).