## 4 Comparative Evaluation of User Interaction in Wearable AR
### Experimental Setup  
Epson's Moverio BT-200 was used as the AR smart glasses [6; 10]. The transparent screen is located at the front of the user's view (eyewear-central), and a separate touchpad connected to the smart glasses is provided to enable multi-touch interaction. The wearable AR system was implemented using Unity3D [39]. We also used the Vuforia software development kit (SDK) [40] to track natural feature-based AR markers, which was used to overlay virtual objects onto the physical environment.  
We used a Leap Motion sensor [41] as a depth sensor to track the user's actual hands to support the hand gesture interaction. As shown in Figure 8, the Leap Motion sensor was attached to the front of the safety helmet of the worker or the smart glasses, while the AR camera was embedded in the AR smart glasses. The absolute orientation algorithm [42; 43] was used for the calibration of the two coordinate systems.  
Figure 8: Difference between the wearable AR coordinate and the depth sensor coordinate.  
Figure 7: Multi-touch interaction using a touchpad in AR smart glasses.  
In particular, as the origin, transformation, and scale are different between the coordinate of the camera of the smart glasses and that of the depth sensor, a calibration operation for matching the two coordinates must be performed, as shown in Figure 9.  
Let \(X_{Lap}=\{x_{1},\,x_{2},\,\cdots,\,x_{n}\}\), a set of points in the Leap Motion coordinate, and \(Y_{AR}=\{y_{1},\ y_{2},\,\cdots,\,y_{n}\}\), a set of points in the wearable AR coordinate, be corresponding points in 3D space. The absolute orientation algorithm finds the similarity transformation parameters (rotation \(R\), translation \(t\), and scaling \(c\)) that give the least mean squared error between these two point sets. The minimum value \(e^{2}\) of the mean squared error is defined as follows.  
\[e^{2}(R,t,c)=\ \frac{1}{n}\sum_{i=1}^{n}\|y_{i}-(cRx_{i}+t)\|^{2} \tag{1}\]  
In summary, to minimize \(e^{2}\), we have to find \(R\), \(t\), \(c\), and such that  
\[R=USV^{T} \tag{2}\]  
\[c=\frac{1}{\sigma_{Lap}^{2}}trace(DS) \tag{3}\]  
\[t=\mu_{AR}-cR\mu_{Lap} \tag{4}\]  
\[\sum xy=\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\mu_{AR})\Big{(}x_{i}-\mu_{Lap}\Big{)} ^{T} \tag{5}\]  
where \(trace(DS)\) is the sum of the elements on the main diagonal of \(DS\), \(\sum xy\) is a covariance matrix of \(X_{Lap}\) and \(Y_{AR}\), \(\mu_{Lap}\) and \(\mu_{AR}\) are mean vectors of \(X_{Lap}\) and \(Y_{AR}\), \(\sigma_{Lap}^{2}\) and \(\sigma_{AR}^{2}\) are variances of \(X_{Lap}\) and \(Y_{AR}\), respectively. Let a singular value decomposition of \(\sum xy\) be \(UDV^{T}\), \(D=\text{diag}(d_{i})\), \(d_{1}\geq d_{2}\geq\cdots\geq 0\)), and  
\[\text{S}=\left\{\begin{array}{cc}I&if\text{det}(\text{U})\text{det}(\text{ V})=1\\ diag(1,\,1,\,\cdots,\,1,\,-1)&if\text{det}(\text{U})\text{det}(\text{V})=-1 \end{array}\right. \tag{6}\]  
The detailed explanation and the proof are described in Reference [43].  
Figure 9: Calibration between the two coordinates.  
Through the calibration between the AR camera and the Leap Motion sensor, the real hand can be matched with a virtual hand in the wearable AR space, as shown in Figure 10.