# Synthesis of Four-Link Initial Kinematic Chains with Spherical Pairs for Spatial Mechanisms
## 4 Results
This 3D analysis successfully validates the iterative convergence of spatial points \(X_{A}\), \(Y_{A}\), \(Z_{A}\), ensuring motion stability within an acceptable tolerance. The smooth convergence pattern confirms that the optimization process effectively refines kinematic placements. The 3D visualization represents the transition of spatial points from their initial positions \(B^{(0)}\) and \(C^{(0)}\) to their updated positions \(B^{(1)}\) and \(C^{(1)}\). Each pair of points is shown in Figure 11, connected by a dashed line, highlighting the spatial displacement. The majority of transitions exhibit small displacements, meaning that the transformation follows a controlled movement. The dashed lines connecting each initial and final position indicate the direction and magnitude of transition. Some transitions show slightly longer movement vectors, suggesting non-uniform displacement behavior. Stability in the spatial displacement of the short transition distances indicates a stable motion. The presence of larger displacements for certain points suggests areas where the system undergoes greater changes. Since the motion is continuous and smooth, this ensures that the system follows predictable mathematical behavior. The function transitions can be modeled using polynomial approximations [3]. With structural-kinematic synthesis, the transitions can be used to study deformation mechanics in flexible or compliant mechanisms. Control system optimization analysis helps in understanding how precise and controlled the spatial transitions are, which is critical in precision engineering applications. This 3D analysis effectively visualizes the transition from \(B^{(0)}\!\rightarrow\!B^{(1)}\)) and \(C^{(0)}\!\rightarrow\!C^{(1)}\), validating spatial consistency in motion planning. This graph illustrates the convergence behavior of the objective function \(S\) over a series of iterations, as shown in Figure 3.0 The objective function is expected to decrease over time, approaching a local minimum. The blue curve with markers represents the progressive reduction in \(S\) as iterations proceed. The function follows a monotonic  
decreases rapidly in the beginning, then stabilizes. Minor oscillations are due to numerical noise, and small fluctuations exist, but the overall trend remains downward. The minimum \(S\) achieved is represented by the dashed red line, which highlights the lowest obtained value, indicating convergence. Convergence and stability of the exponential decay suggests that the algorithm effectively minimizes \(S\) over time. The small oscillations near the minimum indicate stability without excessive divergence. Weierstrass theorem guarantees that a continuous function can be approximated with polynomials [3]. Since the function smoothly converges, it ensures the numerical method is stable and reliable. The graph confirms the effective convergence of the objective function \(S\), ensuring optimal system performance. The function \(S\) evaluates how iterative refinements impact system optimization. Convergence of the iterative algorithm is guaranteed by the Weierstrass theorem. Objective function \(S^{(i)}\) decreases until the stopping condition is satisfied. Function approximation with the Weierstrass theorem shows how a polynomial closely approximates a given function [5]. Algorithmically verified kinematic chain synthesis ensures proper parameter tuning. This is shown in Figures 13 and 14 (function approximation using Weierstrass theorem), with the dashed black line indicating the original function \(f(x)\), green curve indicating the polynomial approximation \(P(x)\), and yellow region indicating the approximation error \(|P(x)-f(x)|\). Using the Weierstrass theorem, we can approximate \(f(x)\) with any desired accuracy using a polynomial. Figure 3.1 A graph showing the convergence of objective function \(S\). Figure 3.2 A graph showing the accuracy of an approximation function. This graph visualizes the accuracy of an approximation function compared to a true function [8].