For a long time, the technique of the lower deformation energy has been successfully employed for the synthesis of mechanisms. This technique has been seen to be versatile, yet powerful a method for assisting in the design of mechanisms. So far, most of the implementations of this method have utilized the dimensions of the mechanism as the synthesis variables, as it brings along some advantages as well as some drawbacks. For instance, the assembly configuration is not usually considered in the optimization process, a flaw that ensures that the same initial configuration is used when computing the deformed positions in each synthesis point. Consequently, it translates into a reduction of the total search space. A possible solution to this problem is the use of a set of initial coordinates as variables for the synthesis, which has been successfully applied to other methods. This also has some additional advantages, such as: the fact that any generated mechanism can be assembled. Unfortunately, the change from dimensions to initial coordinates coerces a reformulation of the optimization problem when using derivatives if one wants them to be analytically derived.
Recently, Dr. Vanessa García-Marina at University of the Basque Country in collaboration with Dr. Igor Fernández de Bustos, Mr. Gorka Urkullu and Dr. Mikel Abasolo at University of the Basque Country explored the use of initial coordinates for the synthesis of mechanisms using SQP and the deformation energy error function. They presented relevant mathematical developments and validate them using examples. Their work is currently published in the research journal, Meccanica.
In brief, the research method employed commenced with a thorough review of the deformed energy method. Next, the researchers reasoned out the choice of using initial coordinates as synthesis variables along with the deformed energy method. They then developed the energetic error function using initial coordinates and later presented the analytic expressions. Lastly, some remarks on the optimization method were made.
The authors observed that the inclusion of the initial coordinates as optimization variables enabled the assembly configuration to be encompassed in the optimization process, which was seen to be of utmost significance in the definition of the mechanism. They also noted that the coordinates of the fixed points were also variables of the optimization and thus, one did not need to include workarounds to optimize them. They also realized that all of the possible solution vectors defined a mechanism which always could be assembled, which not always remained acceptable when using dimensions.
In summary, the study by Vanessa García-Marina and her colleagues presented a novel approach to the dimensional synthesis of mechanism which, although based in the same concepts as previous developments, introduced fundamental changes in its conception. From the observations presented, it can be concluded that the new algorithm inherits not only the advantages of the former approach, but also some of its drawbacks, specially the problem of the low stiffness mechanisms. All in all, reformulation along with a proper comparison of the use of both alternatives using sequential quadratic programming methods have been seen as worthwhile solutions.
V. Garcia-Marina, I. Fernandez de Bustos, G. Urkullu, M. Abasolo. Using nodal coordinates as variables for the dimensional synthesis of mechanisms. Meccanica (2018) volume 53, page 1981–1996.Go To Meccanica (2018)