# Synthesis of Four-Link Initial Kinematic Chains with Spherical Pairs for Spatial Mechanisms
## 4 Results
The time-gradient scatter plot allows for easy tracking of state evolution. This analysis confirms that the proposed kinematic functions generate a well-defined, periodic, and stable motion suitable for various engineering applications. Since \(x_{B}^{(1)}\), \(y_{B}^{(1)}\), \(z_{B}^{(1)}\) and \(R_{2}^{(1)}\) are continuous and differentiable, they can be approximated using polynomials or Fourier series [17]. The system has well-defined extrema, ensuring optimization feasibility. Predictable kinematic behavior makes the system well-suited for control applications. The bounded nature of \(R_{2}^{(1)}\) prevents divergence, ensuring stability. The 3D visualization presents the spatial variation of two functions: \(A^{(1)}(X,Y)\) (left graph--viridis colormap) represents a sinusoidal-cosine function, modeling a kinematic parameter in space. \(B^{(1)}(X,Y)\) (right graph--plasma colormap) represents a cosine-sine function, which could define another kinematic constraint. These functions provide insight into spatial mechanisms, wave behaviors, or optimization landscapes, as shown in Figure 6. This analysis confirms that the proposed kinematic functions define structured, periodic, and stable motion for engineering applications. The 3D visualization shows the kinematic evolution of the spatial system by depicting the motion trajectory of the kinematic point \(x_{C}^{(1)}\), \(y_{C}^{(1)}\), \(z_{C}^{(1)}\) (blue line curve), which is shown in Figure 7. It represents sinusoidal-based motion in 3D space. \(R_{3}^{(1)}\) (color-graded scatter points) encodes an additional constraint or periodic property of the system. It uses a cosine-based function, indicative of oscillatory motion. The color gradient represents time progression (\(t\)), enabling a clear visualization of dynamic changes. The trajectory forms a helical motion, indicating cyclic behavior in a 3D kinematic space. The oscillatory nature of \(R_{3}^{(1)}\) suggests a dynamic variation in link constraints. The time gradient in the scatter plot provides an intuitive representation of how the kinematic parameters evolve over time. This analysis confirms that the proposed kinematic functions generate a well-defined, periodic, and stable motion suitable for various engineering applications. Figure 6: A grid of 3D plots showing the behavior of \(A^{(1)}\in Q\) and \(B^{(1)}\in Q_{1}\). The 3D visualization examines the convergence behavior of the spatial points \(X_{A}\), \(Y_{A}\), \(Z_{A}\), ensuring that \(\left|X_{A}^{i+1}-X_{A}^{i}\right|\leq\varepsilon\), \(\left|Y_{A}^{i+1}-Y_{A}^{i}\right|\leq\varepsilon\) and \(\left|Z_{A}^{i+1}-Z_{A}^{i}\right|\leq\varepsilon\), where \(\varepsilon=0.0004\) is the threshold for convergence. Additionally, the Start and End points are highlighted to understand the trajectory evolution, as shown in Figures 8-10. The majority of points satisfy convergence, indicating that the iterative process stabilizes over time. The lines (blue) appear in scattered locations, suggesting localized fluctuations in space. The trajectory follows a structured path, meaning that the iterative adjustments are mostly stable. Figure 8: The 3D plot above visualizes the iterative convergence of the coordinates \(X_{A},Y_{A},Z_{A}\) under the constraint \(\left|X_{A}^{i+1}-X_{A}^{i}\right|\leq\varepsilon\), \(\left|Y_{A}^{i+1}-Y_{A}^{i}\right|\leq\varepsilon\) and \(\left|Z_{A}^{i+1}-Z_{A}^{i}\right|\leq\varepsilon\). Figure 9: Cont. Figure 10: Cont. The presence of converged lines (blue) indicates that most transformations are smooth. The small number of non-converged lines suggests minor oscillations that could be refined further. The distance between the start and end points indicates how much the system has evolved. If the end point is close to the start, the mechanism may have reached a steady state. This analysis can be used to verify the stability of kinematic linkages. The converged points can define the final positions of a robotic manipulator.