# Analysis of a Customized Clutch Joint Designed for the Safety Management of an Ultrasound Robot
## 3 Results and Discussion  
### Clutch Performance  
Based on the analytical method for each individual spring-ball detent introduced in Section 2.2, we calculated the exact value of the triggering force of each detent and the overall triggering torque of the clutch using the following parameters from the design: \(R=3.9\) mm and \(w=2.5\) mm. The selected spring is a 10-turn music wire spring (made of high-carbon steel alloy, free length of 21.336 mm, outer diameter of 7.137 mm, inner diameter of 4.851 mm, and spring constant \(k=9.811\) N/mm). The sliding force for each spring-ball detent at the triggering point is 24.55 N, which assumes the clutch cover pushes the spring fully into the detent hole (3 mm initial compressed length of the spring) for preloading (preloaded force \(P_{0}=29.433\) N). The according triggering torque of the clutch is 3928 N-mm, with four spring-ball pairs on each side of the link. Analyzing the triggering process of the clutch based on Equations (1)-(5), we can also identify the change of parameters with regard to the relative horizontal movement of the ball (Figure 6), including the vertical displacement of the ball, the reaction force angle, the vertical force, and the side force. It is observed that, in our design, the side force decreases linearly from the triggering point and becomes  
Figure 5: Schematic representation of the selected probe positions for the upper abdominal US scan using the robot based on the phantom geometry, with the pose numbered 1–5 for the patient middle, 6–10 for the patient right, and 11–15 for the patient left. The joint configuration is shown for each pose. zero force when the ball is out of the detent hole. Thus, the maximum force during the triggering process appears when the ball contacts with the notch and is about to move. For the kinematic-based analyses, the first analysis using the exhaustive method identified the relationship between the three angles (the main tilt, yaw, and pitch) and the corresponding triggering force at the US probe end (Figure 7a). This was based on the selection ranges of the main tilt angle from \(-5^{\circ}\) to \(70^{\circ}\), the yaw angle from \(-35^{\circ}\) to \(35^{\circ}\), and the pitch angle from \(-20^{\circ}\) to \(40^{\circ}\) with two-degree intervals. The histogram of the triggering force is shown in Figure 7b. It is observed that the mean triggering force is \(23.01\pm 2.21\) N (mean \(\pm\) standard deviation). The maximum triggering force is 32.06 N and the minimum triggering force is 19.29 N. Analyzing Figure 7b, it can be seen that the majority of the triggering forces are within the range of 19-26 N (91.72%) and Figure 7a shows that those high values only appear at the margin of the plot where the joint angles come to their maximum values. Compared with the force limited values reported in the literature [13; 14; 28; 30], we believe that the mean value of the maximum triggering force set at 23 N is reasonable in terms of safety as well as considering the adequate force for imaging. The variation of the value with the standard deviation of 2.21 N is also acceptable. This indicates that the posture changes of the robot would not significantly  
Figure 6: Parameter changes during the process of the ball moving out from the detent hole after the clutch is triggered based on the theoretical calculation: (**a**) the vertical displacement of the ball, (**b**) the reaction force angle, (**c**) the vertical force, and (**d**) the side force. Figure 7: Kinematic-based clutch performance analysis using the exhaustive method: (**a**) the relationship between different joint angles (\(x\), \(y\), and \(z\) axes) and the resulting triggering force at the US probe end (color coded, in N), (**b**) the histogram of the triggering force based on the result of the exhaustive method, which covers the entire joint spaces of J\({}_{5}\), J\({}_{6}\), and J\({}_{7}\). influence the maximum allowable force at the US probe end with a fixed design torque triggering value of the clutch joint.