# Analysis of a Customized Clutch Joint Designed for the Safety Management of an Ultrasound Robot
## 2 Materials and Methods
### Kinematic-Based Performance Analysis  
With the clutch mechanism modelled and the triggering torque of the clutch calculated, we further analyzed the performance of the clutch in terms of the resulting triggering force at the US probe end when the robot was positioned at different postures kinematically with different joint parameters, denoted as \(q\). At the triggering pose and with zero joint velocity and acceleration, the torque of the clutch joint \(Q\) comprises the torque required to hold the next links' weight against gravity (\(W_{L}(q)\)), the torque required to hold the probe's weight against gravity (\(W_{P}(q)\)), and the joint torque due to the force applied to the US probe (\(G(q)\)):  
\[Q=W_{L}(q)+W_{P}(q)+G(q). \tag{6}\]  
The torque against gravity due to the links' weight can be estimated based on the mass center of the next links, the weight of the next links, and the angulation changes of J\({}_{5}\). As the angulation changes of the last three joints only slightly influence the mass center of the next links due to the limited range of J\({}_{6}\) and J\({}_{7}\), the mass center was treated as a fixed point on the link and was found using CAD software (SolidWorks 2016, Dassault Systemes, Velizy-Villacoublay, France) with the realistic materials and weights of the components modelled. The weights of the components were separately measured using an electronic scale (Toolour, Shenzhen, Guangdong, China). Figure 3: Schematic representation of an individual spring-ball detent with the geometrical parameters and the force definitions shown: (**a**) the detent assembled with the spring-ball pair, (**b**) the detent with the spring preloaded by the clutch cover, and (**c**) the state when the ball moves out of the detent hole. At the triggering point with different postures of the robot positioning the probe to different locations in the workspace, kinematically the torques resulted from the applied forces to the US probe and the torque required to hold the probe's weight against gravity are different. The sum of these two terms can be calculated based on the joint configuration, and we can transfer the wrench in the probe coordinates \(g(\mathbf{q})\) and the force resulting from the probe's weight in the probe coordinates \(\mathbf{w}(\mathbf{q})\) to the torque experienced in J5 coordinates based on the Jacobian transpose as follows:  
\[\mathbf{W_{P}}(\mathbf{q})+\mathbf{G}(\mathbf{q})=\mathbf{J}(\mathbf{q})^{T}(\mathbf{g}(\mathbf{q})+\mathbf{w}( \mathbf{q})). \tag{7}\]  
It should be noted that the impact of the probe's weight must be transferred to the probe coordinates based on the angulation change of J6 and J7 only. The angulation change of the axial rotation of the probe, controlled by J8, does not have any influence. Based on Equations (6) and (7), the triggering force experienced at the US probe end can be derived and calculated using the fixed value of \(\mathbf{Q}\) and the changing joint parameters \(\mathbf{q}\). To study the impact of the joint parameters' variation on the triggering force at the US probe end, we performed two kinematic analyses. The first analysis employed the forward kinematics of the robot and conducted an exhaustive approach covering the entire joint spaces. According to the configuration of the robot, only the angulation changes of J5, J6, and J7 could have impacts that influence the results calculated in Equations (6) and (7). We denote these three angles as the main tilt angle, yaw angle, and pitch angle. Different combinations of these three angles within their limits were tested and the resulting calculated forces were recorded. The second analysis used the customized closed-form inverse kinematics of the robot and positioned the US probe to clinically realistic locations that are possibly required during a real abdominal US scan using this robot.