# A Kinematic Model of the Shoulder Complex Obtained from a Wearable Detection System
## 3 Results
### Motion Coupling Relationship in Sagitttal and Coronal Planes  
According to the second experiment, Figures 8 and 9 show the motion curves (blue lines) for the twelve subjects performing humerus elevation movements in the sagittal and coronal planes, and the elevation angle ranges from 0\({}^{\circ}\) to 120\({}^{\circ}\). The movements of the GH joint center displacement variable quantity in the X-, Y- and Z-directions during humeral elevation in the sagittal and coronal planes were observed, which are presented in Figures 8a-c and 9a-c, respectively. Different patterns appear in the two elevation planes relative to the humeral elevation. However, the results are repeated with the same motion trend in the same plane. To summarize the motion coupling relationship, the average data points for the twelve subjects performing humerus elevation movements in the same plane are obtained. To clarify the functional relationship between the GH joint center displacement variable quantity relative to the thorax coordinate system and humeral elevation angle using the average data points, the polynomials of \(x\), \(y\), and \(z\) are acquired in different planes according to the guidelines for curve fitting given in Section 2.4. describing the motion coupling relationship between the X-, Y- and Z-magnification ratios. Figure 6: Diagrams of the glenohumeral (GH) joint motion information relative to the thorax coordinate system during humeral raising and lowering movements of five cycles in the sagittal plane: (**a**) humeral raising and lowering angle, (**b**) X-direction of the GH joint displacement variable quantity, (**c**) Y-direction of the GH joint displacement variable quantity, and (**d**) Z-direction of the GH joint center displacement variable quantity. Figure 8: Diagrams of the magnification ratio of the GH joint center displacement variable quantity relative to the thorax coordinate system during human elevation in the sagittal plane: (**a**) X-magnification ratio and elevation angle, (**b**) Y-magnification ratio and elevation angle, and (**c**) Z-magnification ratio and elevation angle. Figure 7: Diagrams of the GH joint motion information relative to the thorax coordinate system during human raising and lowering movements five cycles in the coronal plane: (**a**) humeral raising and lowering angle, (**b**) X-direction of the GH joint displacement variable quantity, (**c**) Y-direction of the GH joint displacement variable quantity, and (**d**) Z-direction of the GH joint center displacement variable quantity. Humeral elevation in the sagittal plane: Observing the mean data, for the polynomial of \(x\), the degree of the polynomial is four. Although the R-Square and Adjusted R-Square are almost equal to one and the SSE and RMSE are almost equal to zero, the fitting curve first rises and then falls in the initial stage, which does not confirm the daily movement rules and may exhibit overfitting [31]. When the degree of the polynomial is three, the values of the SSE and RMSE are 0.0017 and 0.0038, less than 0.005, and the R-Square and Adjusted R-Square are 0.9967 and 0.9966, slightly greater than 0.995, which is consistent with the evaluation indexes of the criteria, and the fitting curve is consistent with the average data. In this way, the degree of the fitting polynomial is chosen to be three. For the polynomial of \(y\), the degree of the polynomial is four. The indexes of the fitting curve are less than the evaluation indexes of the criteria. However, the effect of the fitting curve is not close to the average data. When the degree of the polynomial is five, the effect of the fitting curve is consistent with the average. Thus, the degree of the polynomial selected is five. The polynomials of \(z\) show a better fit when the degree of polynomials is four.