# An Orthogonal Wheel Odometer for Positioning in a Relative Coordinate System on a Floating Ground
## 3 Experiments
### Simulated Floating Deck Experiment  
The orthogonal wheels positioning system is designed at the beginning to deal with the common positioning methods, such as a gyroscope on the ship deck and other floating carriers, which do not have a good positioning effect. The environment of the above experiments is a fixed world coordinate system. In the world coordinate system, through the above experiments, we can verify the positioning accuracy and robustness of the orthogonal wheel positioning system in the absolute coordinate system. Next, the relative coordinate system experiment is carried out on the simulated floating deck experimental platform. Because the experimental scenes, such as the hull deck, are not convenient to obtain, the platform as shown in Figure 31 is adopted as the floating platform for this experiment. The floating hull is simulated by people lifting the plank and walking around at random. The floating platform is fitted with universal wheels to simulate the movement of the deck at sea. On the floating test platform, the desired motion curve should be the actual curve of the positioning system relative to the floating test platform, not the actual curve in the world coordinate system. The tracker reflects the \(x,y,z\) coordinates and yaw, pitch, roll angles of the current moment relative to the world coordinate system. So we put a tracker on the moving chassis and a tracker on the corner of the platform. Through the tracker on the chassis, we can obtain the coordinates \((x,y)\) of the chassis in the world coordinate system at every moment. Through the tracker fixed in the corner of the platform, we can obtain the coordinates \((x_{T},y_{T})\) of the origin of the floating platform coordinate system in the world coordinate system and the rotation angle \(\theta\) of the floating platform coordinate system in the world coordinate system at every moment. The position of a point \((x,y)\) in world coordinates in relative coordinates \((x^{\prime},y^{\prime})\) is:  
\[\left[ \begin{array}{c}x^{\prime}\\ y^{\prime}\end{array} \right]=\left[ \begin{array}{cc}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{array} \right]\left[ \begin{array}{c}x\\ y\end{array} \right]+\left[ \begin{array}{c}x_{T}\\ y_{T}\end{array} \right]\]  
The actual curve of the moving chassis relative to the floating platform can be obtained by the above method. Then the positioning curve is compared with that of the orthogonal wheel positioning system developed by ourselves. Figure 32 shows the actual process of this floating platform experiment. Two students control the floating platform to move irregularly, during which they shake the floating platform up and down to produce a floating effect. Another student remotely controls the chassis, moving around the floating platform in an irregular circle. Figure 30: (**a**) Comparison diagram of the gyroscope position curve, orthogonal wheels position curve and HTC VIVE position curve. (**b**) The error curve of orthogonal wheel relative to HTC VIVE. Figure 33a shows the motion track of the orthogonal wheel positioning system and HTC VIVE positioning system on the mobile floating platform. Figure 33b reflects the trajectory of the gyroscope positioning in this case. As can be seen from the figure, the positioning curve of the orthogonal wheel positioning system tends to be consistent with the real curve of HTC VIVE. In the process of floating motion, the angle read by the gyroscope is not that of the orthogonal wheels relative to the floating platform, but that of the world coordinate system. As a result, the curve of the gyroscope positioning is unpredictable and inconsistent with the actual motion curve. Figure 34 shows that the floating degree of the mobile floating platform and the error curve of the orthogonal wheel positioning system relative to HTC VIVE in the process of movement. On the mobile floating platform used in this experiment, the maximum error is 2.43 (\(\pm\)0.075) cm.