# Synthesis of Four-Link Initial Kinematic Chains with Spherical Pairs for Spatial Mechanisms
## 6 Conclusions  
The synthesis of spatial kinematic chains, particularly those involving spherical kinematic pairs, is a complex yet essential problem in the structural-kinematic design of spatial mechanisms. This problem was effectively addressed using an iterative algorithm that minimizes the objective function \(S\), representing the difference between the current and desired configurations of the kinematic chain. The algorithm leverages the Weierstrass theorem to ensure convergence to a local minimum, providing a robust solution for spatial mechanism synthesis. The flexibility of the algorithm enables the creation of different modifications of the original kinematic chains by setting different desired synthesis parameters, making it a valuable tool for structural-kinematic synthesis. The approach also provides an effective means of ensuring the desired functional reproduction of the kinematic chain, with the prescribed function accurately satisfied at the final iteration. The global minimum of the objective function \(S_{min}\) = 0.0042921 signifies that the algorithm has successfully identified the optimal configuration within the required tolerance, confirming the effectiveness and efficiency of the iterative approach. By following the iterative algorithm and using the provided conditions for minimizing the objective function \(S\), the algorithm can synthesize the desired spatial kinematic chains (both open and closed forms). The iterative nature ensures convergence to a local minimum, and the Weierstrass theorem guarantees the existence of the minimum within the specified accuracy. The various modifications of the kinematic chain allow for flexible configurations based on the given parameters. In summary, this methodology represents a significant advancement in solving the problem of synthesizing spatial kinematic chains, offering a reliable and flexible tool for the design of complex spatial mechanisms in various engineering applications.  
**Author Contributions:** Conceptualization, A.Z. and S.K.; methodology, A.Z.; software, A.Z.; validation, A.Z.; formal analysis, A.Z. and K.A.; investigation, A.Z.; resources, A.A. and S.A.; data curation, A.O.; writing--original draft preparation, A.Z., S.A., S.K. and K.A.; writing--review and editing, A.Z., A.A. and A.O.; visualization, A.Z. and A.O.; supervision, A.Z.; project administration, A.Z., S.K. and A.A.; funding acquisition, A.Z., S.A. and K.A. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Almaty University of Power Engineering and Telecommunications, named after [PERSON], grant number AP19677356.  
Not applicable.  
The data presented in this study are available in the article.  
This work was supported financially by the research project (To develop systems for controlling the orientation of nanosatellites with flywheels as executive bodies based on linearization methods) of the Ministry of Education and Science of the Republic of Kazakhstan and was performed at Research Institute of Communications and Aerospace Engineering at Almaty University of Power Engineering and Telecommunications, named after [PERSON], which is gratefully acknowledged by the authors.  
The authors declare no conflicts of interest.