# Bisector-Based Tracking of In Plane Subpixel Translations and Rotations
## 4 Results
### Experimental Results  
#### 4.2.1 Experimental Setups  
* Rotation stage  
We implemented an experiment consisting of an object attached to a motorized rotation stage (CR1-Z7), which was controlled by a motor controller (TDC001 DC) from Thorlabs. The stage provides continuous 360\({}^{\circ}\) travel, with a wobble and repeatability of less than 2 arcsec and 1 arcmin, respectively. We set a trajectory consisting of a rotation of 5 degrees, with a maximum angular acceleration \(\alpha_{\rm m}=0.5\) deg/s\({}^{2}\) and a maximum angular velocity \(\omega_{\rm m}=1\) deg/s. The motion employed is described by a trapezoidal velocity profile: the stage is gradually ramped at an acceleration to a maximum velocity, and, as the destination is approached, the stage is decelerated so that the final position is slowly reached.  
The object was a sheet with a calibrated grid and the printed logo of our University. The grid was used to facilitate the calculation of the pixel to millimetre ratio. The video sequence was recorded using a Nikon 5100 camera with a Nikkor 18-55 objective, a resolution of 25 fps and 1920 \(\times\) 1080 px at a distance of 1 m from the object, which provided a spatial resolution equal to 4.4 px/mm. The whole scene was illuminated with a halogen projector.  
* Mechanical torsion and bending in a beam  
We have implemented another experiment aimed at measuring the torsion and bending of a cantilever steel beam with a 50 \(\times\) 25 \(\times\) 4 mm U cross-section and a length of 775 mm. In Figure 6,  
Figure 5: Quotient between the computed and the theoretical parameters of the location vector (\(x_{c}\) (crosses), \(y_{c}\) (squares), \(\beta\) (circles)) for the frames of the above simulated sequence.  
we show a scheme of the experimental setup. On the free end, a perpendicular plate is welded. In that plate, one can hang different weights from a hole located at 296.5 mm from point O. In our case, we used a weight \(w=5\) N. A displacement measurer was placed over the plate at \(x_{\text{D}}=200\) mm from the point O in the X direction. It is used to measure the vertical displacement resulting from the combination of torsion and bending forces in the system. We used two Basler AC800-510 uc video cameras with objectives Pentax TV lens of 25 mm recording at 300 frames per second (fps). One captured the distance measurer and the other one recorded the profile of the U-shape beam, specifically point A (0, \(-25\)) mm. The resolution obtained with this last camera was equal to 9.5 px/mm.  
The bending and torsion of the experimental structure caused by a load of known weight \(w\) can be theoretically characterized using the moment-area theorems [17], by which we obtain the vertical displacements due to bending both in the U cross-section beam, \(\Delta y_{BU}\), and in the perpendicular plate, \(\Delta y_{BP}\), as well as the torsional angle \(\vartheta\) in the cross-section beam for the point D, following:  
\[\Delta y_{BU}=-0.01808w;\ \Delta y_{BP}=-3.922\cdot 10^{-3}w;\ \vartheta=-0.00456\ w, \tag{8}\]  
where all distances are in millimeters, \(w\) in Newtons, and \(\vartheta\) in radians. In our case, we got \(\Delta y_{BU}=-0.0904\) mm; \(\Delta y_{BP}=-1.9611\cdot 10^{-2}\) mm; and \(\vartheta=-0.0228\ rad\). Due to the eccentric load, the section rotates around its shear center \(S\) that is located at \(x_{S}=6.56\) mm. Due to the rotation of the U section, all points have an additional displacement, given by  
\[\Delta y_{T}(x,y)=-y+r\ \sin(\gamma+\vartheta), \tag{9}\] \[\Delta x_{T}(x,y)=x_{S}-x+r\ \cos(\gamma+\vartheta),\]  
where \(r=r(x,y)=\sqrt{\left(x-x_{S}\right)^{2}+\left(y-y_{S}\right)^{2}}\) and \(\gamma=\gamma(x,y)=\text{atan}(y/(x-x_{S}))\) are the distance of a point \((x,y)\) to the shearing center, and \(\gamma\) indicates the direction from S to the point.