## 3 Results and Discussion
### Results of Kinematic Analysis in Selected Kinematic Pairs of the CPM Mechanism  
Mathematical equations describing how to solve the forward kinematics of the CPM mechanism are derived and detailed in [PERSON] et al. [9]. For this reason, the analysis of the kinematics in this thesis will only add to the knowledge of the displacements, velocities, and accelerations in the specific kinematic pairs that have the greatest impact on the rehabilitative movement that initiates the continuous passive movement of the positioner of the CPM device. Figure 3 presents the results of the analysis of displacements, velocities, and accelerations in the kinematic pair \(C^{\prime}\). Figure 3: Results of the kinematics analysis of the \(C^{\prime}\) kinematic pair (position, velocity, and acceleration in a range of motion) and the graphical representation of the path of movement. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **316 STAINLESS STEEL** \\ \hline Density, [kg/m\({}^{3}\)] & 8000 \\ Young’s Modulus, [Pa] & \(2.05\times 10^{8}\) \\ Poisson’s Ratio & 0.28 \\ Yield Stress, [Pa] & \(2.9\times 10^{8}\) \\ Shear Stiffness, [Pa] & \(7.37\times 10^{10}\) \\ Thermal Expansion, [1/K] & \(2.34\times 10^{-5}\) \\ Conductivity, [W/(mK)] & 21.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Material properties used in Generative Design simulation for links 1 and 2. The graph shows the changes in the kinematic parameters excluding the \(Z\)-axis which, due to the flat nature of the mechanism, has a constant value and is irrelevant for the purposes of the analysis, so it was deliberately omitted. The analysis was carried out assuming maximum actuator speed. In rehabilitation practice, such speeds can only be applied in the final stages of rehabilitation, when the ROM in the knee has already been deepened. The actual actuator velocity values at the beginning of the rehabilitation process are four times lower, so the obtained parameter values are also four times lower than those shown. The above comment applies to all kinematic analysis results presented below. When analyzing the result in Figure 3, special attention should be paid to the maximum value of the \(X\)-axis velocity, which is 8.8897 \(\times\) 10\({}^{-2}\) m/s, and the maximum value of the \(Y\)-axis velocity, which is 9.2152 \(\times\) 10\({}^{-2}\) m/s. The maximum acceleration in this case on the \(X\)-axis is 1.2826 \(\times\) 10\({}^{-4}\) m/s\({}^{2}\), and on the \(Y\)-axis is 3.7237 \(\times\) 10\({}^{-2}\) m/s\({}^{2}\). As the movement of the mechanism corresponds to the flexion and extension phase of the knee, the minimum values of the given parameters are the same, but with opposite signs. Figure 3 also shows a graphical representation of the path of the C\({}^{\prime}\) point of the positioner mechanism of the CPM device. Figure 4 shows the results of the kinematic analysis for the kinematic pair \(D^{\prime}\) of the positioner mechanism of the CPM rehabilitation device. Consideration of this kinematic pair is very important because it is through this kinematic pair that the main flexion movement of the knee that we rehabilitate after knee arthroscopy or knee arthroplasty is built. The results show maximum velocity values in the \(X\)-axis of 7.3364 \(\times\) 10\({}^{-2}\) m/s\({}^{2}\) and the \(Y\)-axis of 9.1371 \(\times\) 10\({}^{-2}\) m/s\({}^{2}\), respectively. The maximum acceleration, in this case, in the \(X\)-axis is 2.7614 \(\times\) 10\({}^{-2}\) m/s\({}^{2}\), and in the \(Y\)-axis is 1.4632 \(\times\) 10\({}^{-3}\) m/s\({}^{2}\). The minimum values of the given parameters are the same, but with the opposite sign, in the same way as above. Figure 4: Results of the kinematics analysis of the \(D^{\prime}\) kinematic pair (position, velocity, and acceleration in a range of motion) and the graphical representation of the path of movement. Figure 5 shows the result of the analysis of the positions, velocities, and accelerations of the translational kinematic pair \(E^{\prime}\) of the slider connected to the linear guide.