On the use of natural coordinates in optimal synthesis of mechanisms

This paper deals with the use of natural coordinates for the synthesis of mechanisms using optimization methods. It will be shown that an approach based on this kind of coordinates has many interesting aspects. The modeling of a mechanism with natural coordinates, like any other multi-body system, is carried out by means of a system of algebraic constraint equations. These are complemented by additional equations describing the requirements of the mechanism. All types of requirements - paths, function generation, body guidance, correlation between members - may be given in this way, so that there is a unified method for treating any kind of synthesis. An interesting method is developed here for kinematic analysis of candidate mechanisms. According to this method, kinematic analysis is carried out in the sense that only constraint equations are satisfied exactly, while requirements are satisfied at best. This corresponds to finding the motion of the candidate mechanism that is `closest' to established requirements. This method is then reduced to the solution of the Initial Value Problem (IVP) of a proper system of Ordinary Differential Equations (ODEs). Lastly, the design space (i.e., the space of the design parameters) also takes advantage of the natural coordinates approach. It is based on the initial values of the natural coordinates themselves rather than on link lengths. This avoids the need to assemble the mechanism in the initial position (and associated branching problems), gives more uniform spanning of the solution space, and guarantees that at least the starting configuration for the ODEs IVP exists and is known. Three examples are given: a four-bar linkage generating a straight path, the same type of linkage generating a square angle (both without correlation), and a six-link Stephenson's mechanism producing a function with a dwell range.