# Synthesis of Four-Link Initial Kinematic Chains with Spherical Pairs for Spatial Mechanisms
## 3 Algorithm and Methodology
If the desired accuracy is not reached, the algorithm adjusts the reference points and continues to iterate until the solution meets the specified criteria. Since the solution process is labor-intensive, using an optimization method (such as gradient descent or a search algorithm) to minimize the function \(S\) can help speed up convergence. The choice of \(\varepsilon\) (tolerance for convergence) will affect how precise the solution needs to be before stopping the iteration. This method ensures that we can iteratively refine the positions of \(A\), \(B\), and \(C\) and check the accuracy of the prescribed kinematic relationships. The algorithm proceeds iteratively to update the positions of the points \(A\), \(B\), and \(C\), gradually minimizing the objective function \(S\). The steps involved in each iteration are as follows:  
Step 1: Solve the system for \(A^{(1)}\)--using the initial reference point \(B^{(0)}\) and \(C^{(0)}\), solve the system of linear equations to determine the new positions of point A(\(X_{A}^{(1)}\), \(Y_{A}^{(1)}\), \(Z_{A}^{(1)}\)) and the associated distance \(R_{1}^{(1)}\). This is achieved using matrix formulations and the optimization process defined for the system. Step 2: Update point \(A^{(1)}\)--once the positions of point \(A\) have been determined, update the reference point \(A^{(1)}\) for use in the next steps of the iteration. Step 3: Solve the system for \(B^{(1)}\)--with the updated position of point \(A^{(1)}\), solve the system of equations to determine the new positions of point \(B(x_{B}^{(1)},y_{B}^{(1)},z_{B}^{(1)})\) and the associated distance \(R_{2}^{(1)}\). Step 4: Update point \(B^{(1)}\)--once the position of point \(B\) has been updated, proceed to update the reference point \(B^{(1)}\) for use in the next iteration. Step 5: Solve the system for \(C^{(1)}\)--with the updated positions of points \(A^{(1)}\) and \(B^{(1)}\), solve the system of equations to determine the new positions of point \(C(x_{C}^{(1)},y_{C}^{(1)},z_{C}^{(1)})\) and the associated distance \(R_{3}^{(1)}\). Step 6: Check for convergence--after updating all the points, check if the changes in the positions of the points are sufficiently small, indicating convergence of the algorithm: \(\max\left(\left|\mathbf{R}^{(i)}-\mathbf{R}^{(i-1)}\right|\cdot\left|\mathbf{ X}_{A}^{(i)}-\mathbf{X}_{A}^{(i-1)}\right|\right|\), \(\left|\mathbf{Y}_{A}^{(i)}-\mathbf{Y}_{A}^{(i-1)}\right|\), \(\left|Z_{A}^{(i)}-Z_{A}^{(i-1)}\right|\right)\leq\varepsilon\), where \(\varepsilon\) is the specified tolerance (the desired calculation accuracy) \(\varepsilon=10^{-4}\). If convergence is reached, the algorithm terminates. Step 7: Repeat if necessary--if convergence is not achieved, return to Step 1 and replace the reference points \(B^{(0)}\) and \(C^{(0)}\) with the updated positions \(B^{(1)}\) and \(C^{(0)}\), and proceed with the next iteration. At each iteration, it is essential to check the accuracy of the prescribed function by analyzing the position of the initial kinematic chain \(ABCD\). This is done by evaluating the kinematic relationship: \(\overline{r}_{D_{0}}=T_{10}\cdot T_{21}\cdot T_{32}\cdot\overline{r}_{D_{3}}\), where \(T_{10}\), \(T_{21}\), \(T_{32}\) are the transformation matrices that relate the positions of the points in the kinematic chain. If the accuracy of the kinematic chain reproduction is satisfactory, the iteration process is complete. If the accuracy does not meet the specified criteria, the algorithm returns to Step 1, adjusting the reference points and continuing the iterations [23]. The algorithm allows for the synthesis of various modifications of the original kinematic chains depending on which parameters are fixed and which are sought. The possible modifications include the following:  
Modification 1: the coordinates of point \(A(X_{A},Y_{A},Z_{A})\) and Euler angles of \(Q_{1}\) and the coordinates of point \(D(X_{D},Y_{D},Z_{D})\) and Euler angles of \(Q_{2}\). Objective: Synthesize a three-link open chain \(ABCD\).