Optimal synthesis of mechanisms using time-varying dimensions and natural coordinates

1. Abstract This paper presents a simple approach to optimize the dimensions and the positions of 2D mechanisms for path or function-generator synthesis. The proposed method is particularly adapted to assembled mechanisms since time-varying dimensions always satisfy the assembly conditions which may represent a real difficulty when dealing with closed-loop mechanisms. The objective is to minimize the strain energy of the bars considered as flexible of the mechanism when this one follows perfectly the desired path. Two optimization strategies are developed and criticized. The first one is based on separated optimizations of design parameters and point coordinates. The second one is more global and is performed in two stages : multiple local synthesis are needed first to find the initial point coordinates, and then a global synthesis stage is undertaken to find both the best dimensions and coordinates. The use of natural coordinates is also particularly interesting since the only non linear functions to optimize are distance functions, and the objective function is rather well-conditioned for a gradient-based optimizer. The question of finding the global optimum is addressed and discussed. Since a standard genetic algorithm may fail to find it, a different approach is proposed: exploring the design space to find several local optima among which the designer will choose the most relevant one taking other design constraints into account. A simple technique is applied which consists in running multiple optimization processes starting from uniformly-distributed initial dimensions (full-factorial design of experiments) across the parameter space. Three applications are presented: a simple four-bar path synthesis to illustrate the optimization strategies, a four-bar steering linkage synthesis for function generation – Ackermann relation – to highlight the limits of both strategies, and eventually a six-bar steering mechanism to explore the design space and find different local optima. 2.