Optimum synthesis of mechanisms with uncertainties quantification throughout the maximum likelihood estimators and bootstrap confidence intervals

Abstract

Optimum dimensional synthesis is an interesting and well-known subject in the area of mechanisms. This synthesis process is usually conducted using the least square method (LSQ). Nevertheless, the quantification of the uncertainties for the synthesized parameters is rarely reported. Estimating these uncertainties using a deterministic approach with the standard error is extremely difficult since the derivatives of the mechanism’s parameters with respect to the experimental data (the desired output) are required. When assuming the data as values from a normally distributed random variable, it has been proven that the results from the LSQ method and the maximum likelihood estimators (MLE) coincide. Thus, by using the maximum likelihood method, we are able to not only synthesize a mechanism but we also have a large amount of tools for conducting the statistics. This would allow the quantification of the aforementioned uncertainties. Taking the planar and spherical four bar mechanisms as examples, we present a study of the synthesis of mechanisms following the maximum likelihood method. The path generation task has been chosen to exemplify the synthesis process, obtaining uncertainties for the synthesized parameters with 95% bootstrap confidence intervals.

Keywords:

• Optimum dimensional synthesis
• maximum likelihood estimators
• uncertainties estimation
• 4R mechanisms