Car-like Vehicle Models
A car-like vehicle resembles completely an automobile. It consists of four wheels for locomotion and is capable of being steered from one place to another. Car-like vehicles model can be classified as rear-wheel, front-wheel and four-wheel driving ground vehicles.
For a rear wheel drive vehicle, the rear tires handle the engine dynamics while the front only needs to handle the steering forces. Figure 2, depicts the vehicle model schematic for a rear drive vehicle. The states of the model are x = [x y 〖 ]〗^T , where (x; y) are the centre point coordinates of the rear axle, is the heading angle of the car body with respect to the x-axis. In figure 2, the angle is the steering angle of the front wheels, and can be referred as a control input. The distance between the front and the rear axles is represented by l. The following mathematical model describes the kinematic relationship of the rear-wheel drive ground vehicle: 
x ̇= v cos
y ̇ = v sin (1)
̇ = v (tan φ )/l
Or, in compact representation, x ̇ = f(x,u); (2)
The steering angle and line velocity v are used as a control input, i.e. u = [ v〖 ]〗^T.
A bicycle model can be used to represent a four wheel vehicle; any vehicle model can be described as a bicycle model . In a bicycle model, the two front wheels are lumped into one wheel and the two back wheels are also lumped into one. For the bicycle model, the complete form of the dynamic model of the vehicle is given by ;
β ̇=(2C_f)/(mv_x ) [δ_f-β-(l_f ( ) ̇)/v_x ]+(2C_r)/(mv_x ) [-β+(l_r ( ) ̇)/v_x ]-( ) ̇
( ) ̇=( ) ̇
( ) ̈=(2l_f C_f)/I_z [δ_f-β-(l_f ( ) ̇)/v_x ]-(2C_r l_r)/I_z [-β+(l_r ( ) ̇)/v_x ]
X ̇=v_x cos()-v_x tan(β)sin()
Y ̇=v_x sin()-v_x tan(β)cos() (3)
the state vector is represented as ϕ=[β ( ) ̇X Y],
β = Vehicle side slip angle,
= Vehicle yaw angle,
( ) ̇= Vehicle yaw rate,
(X ) ̇= x-coordinate of the vehicle's centre of gravity in inertial frame,
Y ̇ = y-coordinate of the vehicle's centre of gravity in inertial frame. While the control vector is represented by u = [δ_f], where δ_f is the front tyre steering angle.
Applying Euler's approximation to equations 1, the discrete-time model for the AGV motion is given below:
x(k + 1) = x(k) + v(k) cos(k)T
y(k + 1) = y(k) + v(k) sin(k)T
(k + 1) = (k) + w(k)T (4)
Where, w = tan/l, T is the sampling period and k a sampling instant. Represented in a compact form as, x(k + 1) = f(x(k),u(k)) (5)
Nonlinear Model Predictive Control (NMPC)
NMPC is a class of MPC theories that are based on the use of nonlinear system models:
x ̇ = f(x(t),u(t)) in the prediction. The cost function can be non-quadratic, and the optimization problem consists of nonlinear constraints on states and controls. Therefore, the optimization problem to be solved at each sampling instant is nonlinear leading to nonlinear model predictive control. Generally, linear MPC could handle problems with multivariable and...