# Analysis of a Customized Clutch Joint Designed for the Safety Management of an Ultrasound Robot
## 2 Materials and Methods
### Customized Spring-Loaded Ball Clutch  
To simplify the analysis and modelling of the proposed customized spring-loaded ball clutch, a small angle approximation was utilized and each spring-ball detent in the design was treated as a translational detent. This is because the amount of movement of the ball when coming out from the hole as the joint rotates in the clutch triggering stage is small, which results in the arc length subtended by the small rotational angle of \(\mathrm{J_{5}}\) being only slightly longer than the straight line. We started analyzing the individual spring-ball detent using the exact geometry of the design (Figure 3a). When the clutch transmits torque from input to output, the detent structure is in its engaged position with the ball held by a preloaded spring compressed by the clutch cover (Figure 3b). The preload is denoted as \(P_{0}\) and the radius of the ball is denoted as \(R\). The detent case contains the vertical compression spring with a spring constant \(k\). When excessive force occurs, the detent slider slides horizontally over the detent case until the ball completely comes out from the detent hole (Figure 3c). The amount of horizontal movement in the relative movement direction during this process is denoted as \(s\). These two parts develop a horizontal shear force \(F_{S}\) between them. The inside of the slider is the conical notch with its radius denoted as \(w\), which generates the reaction force \(F_{R}\) in the angle of \(\theta\). The detent ball lies between the spring and the conical notch during the disengagement process. The maximum vertical displacement of the ball is as follows:  
\[\delta_{\mathrm{max}}=R-R\sin\theta_{0}, \tag{1}\]  
where \(\theta_{0}\) is the default angle in the initial position. The variable \(\delta\)(s) gives the vertical displacement of the ball defined in the direction and from the starting position as shown in Figure 2a:  
\[\delta(s)=R-\sqrt{R^{2}-\left(w-s\right)^{2}}. \tag{2}\]  
As the ball comes out from the notch, the angle of the reaction force \(\theta\) changes, which is as follows:  
\[\theta(s)=arccos\left(\frac{w-s}{R}\right). \tag{3}\]  
The vertical force \(F_{V}\) generated by the force resulting from the compression of the spring due to the movement of the ball and the preloaded force generated from the clutch is as follows:  
\[F_{V}=k(\delta_{\mathrm{max}}-\delta(s))+P_{0}. \tag{4}\]  
Figure 2: Schematic representation of the engagement and disengagement mechanisms of the clutch joint \(\mathrm{J_{5}}\): **(a)** Links 4 and 5 are connected through the engaged spring-ball pairs and the movements of the driven gear actuate Link 5 rotating in the positive direction (red arrow); **(b)** Links 4 and 5 are disconnected with the ball moving out from the detent hole when excessive force is applied in the reverse direction (blue arrow).  
The horizontal shear force developed during the process can then be calculated as follows:  
\[F_{S}=\frac{F_{V}}{tan\theta}. \tag{5}\]  
Based on the mathematical analysis, we can calculate the side force developed during the process of triggering from the time that the ball starts moving to the time that it completely moves out of the detent hole. The maximum force during this process is treated as the triggering force of each detent hole, with the frictional force omitted in the analysis. The triggering torque of the clutch can be simply calculated based on the triggering force of each detent hole, the radial distance from the J\({}_{5}\) rotational center to the detent hole, and the number of spring-ball pairs.