# Synthesis of Four-Link Initial Kinematic Chains with Spherical Pairs for Spatial Mechanisms
## 4 Results
Future refinements can focus on optimizing link constraints and implementing real-time motion tracking in robotic or mechanical systems. This represents a sinusoidal-cosine function that models a possible kinematic constraint and a cosine-based function that represents a radially symmetric structure, as shown in Figure 4. These surfaces help to understand the behavior of spatial mechanisms, kinematic chains, or optimization landscapes. The function \(A^{(1)}(X,Y)=sin(X)cos(Y)\) generates a wave-like motion. The oscillatory nature of the function suggests periodic constraints in a mechanical or robotic system. It could model angular displacements in a multi-link spatial mechanism. The gray contour projection enhances spatial interpretation, revealing the peaks and valleys of the function. The function \(C^{(0)}(X,Y)=cos\left(\sqrt{X^{2}+Y^{2}}\right)\) exhibits a radially symmetric wave pattern. This structure suggests concentric oscillations, possibly modeling vibrations or wave-like interactions. This could represent a spatial field or a constraint region in a mechanical design. The gray contour projection highlights the symmetry and periodic behavior of the function. Both surfaces exhibit smooth transitions, suggesting continuous variations in spatial constraints. The periodic behavior in \(A^{(1)}(X,Y)\) and radial structure in \(C^{(0)}(X,Y)\) indicate different types of kinematic constraints. The color gradients provide additional insight into function values, making it easier to identify critical regions of interest. Since both functions are continuous and differentiable, Weierstrass theorem guarantees they can be approximated using polynomials [17]. Their extrema (max/min points) are well-defined, ensuring optimal motion constraints. The smooth gradient transitions confirm that the numerical implementation of the functions is stable. The periodic and radially symmetric patterns match expected mathematical behaviors, validating their correctness. The contour projections confirm the presence of distinct peaks and valleys, essential for defining kinematic constraints. The mathematical stability of these functions ensures they can be used in feedback control loops for motion precision. This analysis validates the use of \(A^{(1)}(X,Y)\) and \(C^{(0)}(X,Y)\) in kinematic synthesis, control system optimization, and mechanical design. This 3D visualization represents the kinematic evolution of a spatial system by plotting \(x_{B}^{(1)}\), \(y_{B}^{(1)}\), \(z_{B}^{(1)}\) (blue line curve) the motion trajectory of a kinematic point. It represents a sinusoidal-based motion in 3D space. \(R_{2}^{(1)}\) (color-graded scatter points) encodes an additional constraint or periodic property of the system. It uses a cosine-based function, indicative of oscillatory motion, as shown in Figure 5. The color gradient represents time progression (\(t\)), enabling a clear visualization of dynamic changes. The function \(R_{2}^{(1)}=1+0.5 cos(t)\) describes a periodic fluctuation. It encodes constraint variations in \(R_{2}^{(1)}\). It shows a cyclic dependence over time (\(t\)). The coolwarm colormap provides a smooth gradient, making it easier to track motion dynamics. The trajectory follows \(x_{B}^{(1)}=sin(t)\), which represents a smooth oscillation along the \(X\)-axis, \(y_{B}^{(1)}=cos(t)\) moves out of phase with \(x_{B}^{(1)}\), forming a circular projection in the X-Y plane. \(z_{B}^{(1)}=sin(2t)\) moves at double frequency, introducing vertical variations. The combination of these functions results in a helical motion path. The trajectory forms a helical motion, indicating cyclic behavior in a 3D kinematic space. The oscillatory nature of \(R_{2}^{(1)}\) 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. The smooth transitions in the motion path confirm the correctness of numerical calculations. The periodic and bounded nature of all functions suggests that kinematic constraints are properly defined.