# An Orthogonal Wheel Odometer for Positioning in a Relative Coordinate System on a Floating Ground
## 2 Materials and Methods
### The Motion Model of a Single Orthogonal Wheel  
Counting in the X and Y directions of the orthogonal wheel depends on a magnetic encoder to complete. The dividing value of magnetic encoder is 16,384, that is, one turn  
Figure 4: Main view and left view of a single orthogonal omnidirectional wheel.  
Figure 5: (**a,b**) The status of the sliding mechanism of the positioning system before and after complete pressure, respectively, which can withstand a movement of approximately 6 cm up and down. (**c**) The positioning system can still maintain contact with the ground under the plane simulating the inclined ground. From the positioning system structure, the maximum allowable slope is approximately 15\({}^{\circ}\).  
corresponds to 16,384. The radius of the orthogonal wheel is 25.4 mm. The displacement of the robot in the X and Y directions can be obtained by integrating the short distance. To determine the real-time accurate pose of the robot in the constructed map, the scheme adopts the relative positioning method. The positioning sensor system is installed on the bottom of the car, and the orthogonal wheels sense the prior position of the robot. For the algorithm of orthogonal wheel odometer, Figure 6 shows the odometer motion model in the X and Y directions of a single orthogonal wheel.  
The displacement of the car in the world coordinates and the speed \(v_{xxw}\) and \(v_{yw}\) of the car's own coordinate system are calculated with the counting of the orthogonal code plate. The coordinate axis \(X_{\text{ws}}\)-O-\(Y_{\text{ws}}\) shown in the diagram is the coordinate system of the orthogonal wheel group and records the velocity and displacement of the orthogonal wheel group. The angle between the two frames is \(\theta\). The distance from the center of the trolley to the installation of the orthogonal wheels is \(l\), and the velocity \(v_{xw}\) and \(v_{yw}\) of the orthogonal wheel set in the X-O-Y coordinate system is:  
\[v_{xw}=v_{xw}\cos(\theta)-v_{yw}\sin(\theta) \tag{1}\]  
\[v_{yw}=v_{yw}\cos(\theta)-v_{xw}\sin(\theta) \tag{2}\]  
The displacement \(s_{xw}\) and \(s_{yw}\) of the orthogonal wheels in the X-O-Y coordinate system can be calculated by (1) and (2):  
\[s_{xw}=\int v_{xw}\text{d}t \tag{3}\]  
\[s_{yw}=\int v_{yw}\text{d}t \tag{4}\]  
Through (3) and (4), the displacement \(s_{xr}\) and \(s_{yr}\) of the trolley in the X-O-Y coordinate system can be calculated:  
\[s_{xr}=s_{xw}+l\cos(\theta) \tag{5}\]  
\[s_{yr}=s_{yw}+l\sin(\theta) \tag{6}\]  
By differentiating both sides of (5) and (6), the velocity \(v_{xr}\) and \(v_{yr}\) of the car in the X-O-Y coordinate system can be calculated, where \(\omega\) is the angular velocity of the car:  
\[v_{xr}=v_{xw}-\omega l\sin(\theta) \tag{7}\]  
Figure 6: The motion model of single omnidirectional wheel in world coordinate system and self coordinate system.  
\[v_{yr}=v_{yw}+\omega l\cos(\theta) \tag{8}\]  
Through (7) and (8), the velocity \(v_{xrs}\) and \(v_{yrs}\) of the car in the coordinate system of the car itself can be obtained:  
\[v_{xrs}=v_{xr}\cos(\theta)+v_{yr}\sin(\theta) \tag{9}\]  
\[v_{yrs}=v_{yr}\cos(\theta)-v_{yr}\sin(\theta) \tag{10}\]  
From the above, the data received through the magnetic encoders on the orthogonal wheels can be converted into the position and velocity information in the world coordinate system. In other words, the velocity and the displacement of the moving chassis in the X-O-Y coordinate system are obtained.  
In a simpler sense, the use of two vertically positioned omni-wheels and encoders results in accurate coordinate information relative to the world's coordinate system, rather than four. In this case, the angle information of the robot can be obtained by using a gyroscope. However, the disadvantages of such positioning systems can be shown when gyroscopes are exposed to floating ground. In subsequent chapters, we prove the defect through experiment.