# Wind Speed Measurement via Visual Recognition of Wind-Induced Waving Light Stick Target
## 4 Experimental Validation
### Model Validation  
This section describes the implementation of the derived formulations through a validation experiment, where a system was set up as shown in Figure 17 below. The camera was employed in capturing the target's displacement in bottom, side, and front views after swinging it in REV1# and/or REV2# direction(s) by a known magnitude with the aid of two protractors. Attached to the right and front sides of the cantilevered arm are the side and front plates on which the 360\({}^{\circ}\) and 180\({}^{\circ}\) protractors are placed, respectively. The front plate can be rotated (REV1#) up to 360\({}^{\circ}\) about the revolute joint, as in Figure 17 (). The side plate is fixed and acts to perform the swing of the front plate in the REV1# direction, whereas the stick on the front protractor can be regulated in the swinging range of \(\pm\)90\({}^{\circ}\) at REV2# by double-faced adhesive tape between the stick and the front protractor plate, as in Figure 17 (). The side and front protractors are designed to change the stick's orientations. However, there is an indirect relationship between REV1#, REV2#, and the yaw angle, the wind direction angle. In the coupled imaging cases, given that the stick OA is first rotated in the front plate by \(\alpha_{\text{Rev2#}}\) up to OA\({}^{\prime}\) according to the front protractor, and the front plate is then  
Figure 16: Stick target segmentation. (**a**) Original image; (**b**) segmented target with coordinates at the two ends. Figure 17: Validation experiment set-up. rotated by \(\alpha_{\rm Rev1\#}\) to a new position OA\" according to the side protractor, the two steps can be illustrated in Figure 18a below, in terms of a fan-shaped cylinder. When mathematically geometrized, the model can be expressed as a triangular prism in Figure 18b where \(\angle\)AOA\({}^{\prime}\) describes the stick swing angle \(\alpha_{\rm Rev2\#}\) in the front plate and \(\angle\)AOD denotes the front plate swing angle \(\alpha_{\rm Rev1\#}\) along the side plate. Accordingly, the stick yaw angle \(\alpha\) is the angle between the initial and ultimate position \(\angle\)AOC. The wind direction is the angle between \(\triangle\)OAC and \(\triangle\)OAD, denoted as \(\alpha_{\rm WD}\), which is found by constructing a perpendicular plane between them. In light of the stick length \(L\), \(\mid\)OA \(\mid\) = \(\mid\)OD \(\mid\) = \(L\). From the equivalent right triangles \(\triangle\)ODC and \(\triangle\)OAB, we can write:  
\[\mid\text{AB}\mid\,=\,\mid\text{CD}\mid\,=L\text{-}\text{tan}\alpha_{\rm Rev2\#} \tag{34}\] \[\mid\text{OC}\mid\,=L/\text{cos}\alpha_{\rm Rev2\#}\]  
In \(\triangle\)OAD, adopting the law of cosines:  
\[\Big{\lvert}\text{AD}\Big{\rvert}_{2}=\Big{\lvert}\text{OA}\Big{\rvert}_{2 }+\Big{\lvert}\text{OD}\Big{\rvert}_{2}-2\cos\alpha_{\rm Rev1\#}\Big{\lvert} \text{OA}\Big{\rvert}\cdot\Big{\lvert}\text{OD}\Big{\rvert}=2L^{2}\cdot(1- \cos\alpha_{\rm Rev1\#}) \tag{35}\]  
In the Rt\(\triangle\)CDA, we have:  
\[\mid\text{AC}\mid\,^{2}\,=\,\mid\text{CD}\mid\,^{2}\,+\,\mid\text{AD}\mid\,^{ 2}\,=L^{2}\cdot(\text{tan}^{2}\alpha_{\rm Rev2\#}-2\text{cos}\alpha_{\rm Rev1 \#}+2) \tag{36}\]  
In the \(\triangle\)OAC, employing the law of cosines again based on (34) and (36) above, we have:  
\[\text{Cos}\alpha=(\,\mid\text{OA}\mid\,^{2}\,+\,\mid\text{OC}\mid\,^{2}\,-\, \mid\text{AC}\mid\,^{2})/(2\,\mid\text{OA}\mid\,\cdot\,\mid\text{OC}\mid\,)= \text{cos}\alpha_{\rm Rev1\#}\cdot\text{cos}\alpha_{\rm Rev2\#} \tag{37}\]  
The yaw angle \(\alpha\) is thus given in terms of REV1# and REV2# in Equation (37) as below. \[\alpha=\cos^{-1}(\cos\alpha_{\rm Rev1\#}\cdot\text{cos}\alpha_{\rm Rev2\#}) \tag{38}\]  
To find a joint perpendicular plane of both \(\triangle\)OAC and \(\triangle\)OAD, DF is created in \(\triangle\)OAD and vertically intersects with OA at F. Since the face OABE rotates about OE to face ODEC, CD\(\,\perp\triangle\)OAD, thus OA\(\,\perp\)CD. Since OA\(\,\perp\)DF, then OA\(\,\perp\triangle\)CDF.