# Synthesis of Four-Link Initial Kinematic Chains with Spherical Pairs for Spatial Mechanisms
## 4 Results  
The conducted analysis involves a systematic examination of kinematic transformations, function approximations [3], and convergence behaviors in the context of spatial mechanism synthesis and optimization. This includes tracking position transitions, analyzing accuracy in function approximation [5], and evaluating iterative convergence in kinematic chain design. The programming language chosen for use was Python 3.13.0. The Weierstrass approximation theorem [8] ensures that if the transformation between \(B^{(0)}\) and \(C^{(0)}\) is continuous, a mathematical function (polynomial or spline) can approximate it. This means the motion path from \(B^{(0)}\) to \(C^{(0)}\) can be modeled smoothly; the linkage structure can be adjusted iteratively to minimize error [23]. If the difference between \(B^{(0)}\) and \(C^{(0)}\) reduces over iterations, the iterative kinematic synthesis algorithm is converging, as shown in Figure 2. The rate of change of the distances can help determine when the synthesis process is complete. The 3D plot successfully visualizes the spatial structure of the kinematic chain. Correspondence between \(B^{(0)}\) and \(C^{(0)}\) is established, confirming the initial synthesis step. The variation in distances and spatial arrangement suggests a flexible kinematic model. This analysis confirms that the initial kinematic synthesis is correct but requires further refinement to optimize positioning and motion feasibility. Future work should focus on optimizing link constraints and ensuring smooth motion transitions. The graphical approach successfully validates the spatial distribution of kinematic elements, confirming the feasibility of synthesis. This 3D visualization illustrates the relationships between the spatial coordinates \(X_{A}^{(1)}\), \(Y_{A}^{(1)}\), \(Z_{A}^{(1)}\) and the function \(R_{1}^{(1)}\), which is color-mapped to indicate time progression (parameter \(t\)), as shown in Figure 3. The trajectory of \(X_{A}^{(1)}\), \(Y_{A}^{(1)}\), \(Z_{A}^{(1)}\) represents the spatial motion of a kinematic linkage. The cyclic behavior of \(R_{1}^{(1)}\) suggests a periodic constraint, possibly a rotating or oscillating mechanism. The gradual increase in \(Z_{A}^{(1)}\) confirms that the motion follows a smooth and bounded trajectory. The bounded oscillation of \(R_{1}^{(1)}\) prevents unstable or unbounded growth, ensuring mechanism feasibility. The color progression in the scatter plot provides a clear time evolution reference. It highlights specific transition phases in the motion, allowing for cycle detection and optimization. Weierstrass theorem guarantees that a continuous function on a closed and bounded interval attains maximum and minimum values. The polynomial approximation property ensures that \(X_{A}^{(1)}\), \(Y_{A}^{(1)}\), \(Z_{A}^{(1)}\) can be accurately modeled by a suitable function [16]. The bounded behavior of \(R_{1}^{(1)}\) aligns with the theorem's existence conditions for extrema. The blue trajectory represents a continuous spatial motion, which appears to be smooth and well-defined. The use of a logarithmic function for \(Z_{A}^{(1)}\) leads to slow growth in the beginning (\(t\approx 0\)) and saturation as \(t\) increases (logarithmic behavior prevents excessive growth). The scatter plot represents \(R_{1}^{(1)}\), which follows an oscillatory pattern. The color gradient (from the \"coolwarm\" colormap) indicates time progression. The function \(R_{1}^{(1)}=0.5 sin(2t)+0.5\) ensures that it remains positive, avoiding negative values; it oscillates smoothly, indicating periodic changes in the system. The logarithmic function for \(Z_{A}^{(1)}\) is a good choice for systems that require growth control. The oscillatory function \(R_{1}^{(1)}\) ensures that the motion does not diverge or collapse. Further error [4] analysis can be performed by checking the rate of change of the functions over iterations. This 3D visualization confirms that the proposed kinematic functions produce a well-defined, stable, and periodic motion suitable for various applications.