## 6 Conclusions  
This paper presented the dynamic balance optimization of a six-bar Watt linkage modeled using _FCC_ (also known as natural coordinates), comparing the performance of single- and multi-objective optimization algorithms. While much of the current state-of-the-art mechanism balancing optimization relies on Cartesian coordinates and single-objective algorithms, the balancing problem involves two quantities: \(ShM\) and **ShF**. Using multi-objective algorithms offers a more efficient approach to balancing optimization, enabling the identification of a wider range of solutions in a shorter time frame.  
This research compared the performance of single-objective (GA, ES, DE, SRES, and ISRES) and multi-objective (NSGA-II, NSGA-III, R-NSGA-II, R-NSGA-III, UNSGA-III, MOEA/D, AGE-MOEA, C-TAEA, SMS-EMOA, and RVEA) algorithms. SMS-EMOA emerged as the top performer, exhibiting the highest hypervolume and an execution time of 10.44 min. The experiments demonstrated that multi-objective algorithms are significantly more efficient at identifying diverse solutions than single-objective algorithms, establishing them as the preferred method for balancing mechanical systems.  
Sensitivity analysis revealed that two counterweights could be eliminated without affecting the optimization performance achieved with all five counterweights. Further analysis identified the \(y\) position of the third counterweight and the thickness (\(t\)) of the second counterweight as critical parameters influencing whether the optimization prioritizes \(ShM\) or **ShF** reduction.  
The analysis presented in this research demonstrates that the SMS-EMOA optimization algorithm can achieve either a 91.19% reduction in **ShF** (with a corresponding 1.33% reduction in \(ShM\)) or a 77.60% reduction in \(ShM\) (with a corresponding 9.54% reduction in **ShF**).  
Conceptualization, M.T.O.-G. and C.N.S.; methodology, M.T.O.-G. and C.N.S.; software--mechanical development, M.T.O.-G.; software--optimization, C.N.S., L.E.R.-J. and S.C.G.-D.; validation, M.T.O.-G. and C.N.S.; formal analysis, M.T.O.-G. and C.N.S.; investigation, M.T.O.-G., C.N.S., L.E.R.-J. and S.C.G.-D.; writing--original draft preparation, M.T.O.-G. and C.N.S.; writing--review and editing, M.T.O.-G. and C.N.S.; supervision, M.T.O.-G. and C.N.S.; project administration, M.T.O.-G. and C.N.S.; funding acquisition, M.T.O.-G., through the Instituto de Ciencia y Tecnologia del Estado de Aguascalientes (INCyTEA) and the Faculty of Engineering at Universidad Panamericana, Aguascalientes campus. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Instituto de Ciencia y Tecnologia del Estado de Aguascalientes (INCyTEA FEIT-2024-MTOG) and the Faculty of Engineering, Universidad Panamericana, Aguascalientes campus.  
Data is contained within the article.  
The authors gratefully acknowledge the financial support provided by the Instituto de Ciencia y Tecnologia del Estado de Aguascalientes (INCyTEA) for equipment procurement through the project titled \"Analisis y optimizacion del balanceo de mecanismos utilizando coordenadas naturales\". [PERSON] extends her sincere appreciation to INCyTEA for their generous support.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
FCC Fully Cartesian Coordinates  
ShF Shaking Force