# Measurement of Wheel Radius in an Automated Guided Vehicle
## 2 Determination of the Vehicle Position and the Role of the Rolling Radius  
### Movement of a Wheeled Vehicle  
The basic method of determining the position in motion of an autonomous vehicle is odometry. Its rules are described in detail in the works [1; 2; 3; 4; 12]. There are many ways to drive and steer the vehicle.  
Most autonomous transport vehicles use two modes of driving and steering. In the first one, independently powered rear wheels are used for driving and steering. In the second case, the front wheel is driven and steered. In both cases, the data necessary to control the vehicle, such as the rotational speed of the wheels or the steering angle of the front wheel, come from encoders installed at the respective wheels. Figure 1 shows the vehicle in the X\({}_{0}\)O\({}_{0}\)Y\({}_{0}\) reference system and the individual quantities characterizing the vehicle movement are marked.  
The position of a selected point O in the base reference system \(X_{0}O_{0}Y_{0}\) (Figure 1) is defined by the state vector \(\mathbf{X}_{j}\). The next position \(j+1\) of the vehicle point O is determined by the state vector \(\mathbf{X}_{j+1}\).  
State vector \(\mathbf{X}_{j+1}\) in the \(j+1\) iteration is expressed by:  
\[\mathbf{X}_{j+1}=\mathbf{X}_{j}+\mathbf{V}_{j+1}dt \tag{1}\]  
where:  
\[\mathbf{X}_{j+1}=\left[\begin{array}{c}x_{j+1}\\ y_{j+1}\\ \theta_{j+1}\end{array}\right]\mathbf{X}_{j}=\left[\begin{array}{c}x_{j}\\ y_{j}\\ \theta_{j}\end{array}\right]\mathbf{V}_{j+1}=\left[\begin{array}{c}v_{\text{ O},j+1}\cos\bigl{(}\theta_{j}+\omega_{j+1}dt\bigr{)}\\ v_{\text{O},j+1}\sin\bigl{(}\theta_{j}+\omega_{j+1}dt\bigr{)}\\ \omega_{j+1}\end{array}\right]. \tag{2}\]  
Velocities \(v_{\text{O},j+1}\) and \(\omega_{j+1}\) can be determined from the following relations, for a vehicle using measured data from the rear wheels:  
\[v_{\text{O},j+1}=\bigl{(}v_{\text{R},j+1}+v_{\text{L},j+1}\bigr{)}/2, \tag{3}\]  
\[\omega_{j+1}=\bigl{(}v_{\text{R},j+1}-v_{\text{L},j+1}\bigr{)}/\text{b}, \tag{4}\]  
where:  
\(v_{\text{R},j+1}\)--speed of the right wheel in \(j+1\) step;  
\(v_{\text{L},j+1}\)--speed of the left wheel in \(j+1\) step;  
\(\omega_{j+1}\)--angular velocity of the vehicle relative to the reference system \(X_{0}O_{0}Y_{0}\);  
b--wheelbase of the driven wheels.  
For a vehicle using measurement data from the front wheel:  
\[v_{\text{O},j+1}=v_{\text{K},j+1}\cos\bigl{(}\alpha_{j+1}\bigr{)}, \tag{5}\]  
\[\omega_{j+1}=v_{\text{K},j+1}\sin\bigl{(}\alpha_{j+1}\bigr{)}/l. \tag{6}\]  
where:  
\(v_{\text{K},j+1}\)--speed of the front wheel in \(j+1\) step;  
\(\alpha_{\text{K},j+1}\)--turning angle of the front wheel in \(j+1\) step.  
To determine the speed of individual wheels: \(v_{\text{R},j+1}\), \(v_{\text{L},j+1}\)\(v_{\text{K},j+1}\) the data from the encoders and the assumed values of the radius \(r\) of the vehicle wheel are used.  
In the above consideration, it was assumed that the wheels are rigid and they roll without slipping, the contact between the wheel and the floor is a point contact and the radii \(r\) of the rear wheels are the same.  
Figure 1: Vehicle movement in the reference system \(X_{0}O_{0}Y_{0}\).