# A Wireless Hand Grip Device for Motion and Force Analysis
## 3 Results and Validation
### Orientation Degree Error Measurement  
The ITG-3200 datasheets specify that the maximum rotation speed is \(\pm 2000^{\circ}\)/s. This limitation does not pose a problem for the rehabilitation applications, for which the controller has been developed. The error made in the measurements due to noise must be considered. This error is not indicated in the specifications of the remote control, so an experiment to bound it has been carried out.  
In order to measure noise, we have computed the probability of getting a measurement with an error higher than one degree.  
The control has been immobilized for different orientations and the noise of the orientation measurement has been measured for 1000 records. A total of 35 recordings have been made in arbitrary orientations. The measurement results for the highest noise are shown below. They corresponded to the orientation yaw = \(-176.8162\), pitch = \(-17.0508\) and roll = \(109.87212\).  
To carry out the statistical analysis, the orientations with the higher error (high mean error) have been selected. This set of measurements is called \(M\).  
Figure 9 presents the graphical evolution of the orientation signal. Point (0, 0) corresponds to the average orientation (yaw = \(-176.8162\), pitch = \(-17.0508\) and roll = \(109.87212\)). The error measurement is defined as the vector from the unitary orientation vector to the mean center orientation line.  
Figure 8: Virtual object on the screen.  
Figure 7: Force and motion of wireless hand device.  
For each orientation, we have \(n=1000\) stationary measurements, called \(m_{i}\in M\). For each measurement, its error vector \(me_{i}\) is computed as the 2D vector from the orientation to the mean orientation. The greatest scalar error is \(me=\underset{1\leq i\leq n}{max}|me_{i}|\). In the worst scenario, \(me=1.135321\times 10^{-5}\), which corresponds to a difference between measurement and mean measurement orientations of 0.1930557 degrees (asin(\(me\))). Furthermore, this is the maximum error obtained in the 35 arbitrary sets of measurements.  
An analysis of error vectors set \(\{me_{i}|\ 1\leq i\leq n\}\), concludes that there is Gaussian noise along both axes. Therefore, this noise will be modelled as a bidimensional Gaussian noise. This noise has a correlation of \(-0.62904\) according to the Pearson test. The noise error is split into two independent random variables, R\({}_{1}\), obtained from the direction of greatest dispersion, and R\({}_{2}\), the noise in the normal direction.  
The Shapiro test has yielded \(p\)-values of 0.2827 and 0.5851 for R\({}_{1}\) and R\({}_{2}\), respectively. The parameters obtained are  
\[\text{R}_{1}=\text{N}(\mu_{1},\,\sigma_{1})=\text{N}(-3.141921\times 10^{-17}, 0.000958003)\]  
\[\text{R}_{2}=\text{N}(\mu_{2},\,\sigma_{2})=\text{N}(1.134786\times 10^{-16}, 0.0003242265)\]  
Figure 10 shows the sample histograms for R\({}_{1}\) and R\({}_{2}\). The non-zero parameterization of the mean is due to lack of precision, but it can be assumed to be zero.  
Figure 9: Stability measurement for orientation yaw = \(-176.8162\), pitch = \(-17.0508\) and roll = 109.87212. The difference between orientations has been measured by the L2 norm of the projection of the orientation on the unit sphere. The maximum error obtained is \(1.135321\times 10^{-5}\), which is equivalent to a maximum error in degrees of 0.1930557 in degrees.  
From this noise parameterization, which corresponds to the orientations with the higher error, the probability of obtaining noise greater than one degree is calculated as  
\[\mathrm{P}\bigg{(}\sqrt{\mathrm{R}_{1}^{2}+\mathrm{R}_{2}^{2}}\;>\;1^{\circ} \bigg{)} \tag{1}\]  
The probability obtained from (1) is lower than \(1\times 10^{-35}\). From this study, we conclude that noise is less than one degree, which indicates that the data obtained can be used in a rehabilitation environment where orientation measurement errors less than one degree are acceptable [36].