Synthesis of Mechanisms Based on Optimal Solution Density

The traditional process for kinematic synthesis of planar mechanisms involves setting a few prescribed positions, then solving a set of equations to identify a vector chain that exactly reproduces those positions. In evaluating these equations, designers often must sift through multiple “infinities” of solutions corresponding to some number of free-choice variables that each have an infinite number of possible values. In this vast solution space, some combination of those variables will produce the most optimal solution, but finding that optimal solution is not trivial. There are two extremes for addressing the impossibility of sifting through infinite possible values. First, one could use analytical techniques to make educated estimates of the optimal values. Or, alternatively, a designer could completely remove their perspective from the process, passing the problem into a computer and programming it to sift through millions (or orders of magnitude more) possible solutions. The present work proposes a novel intermediate step in the analytical synthesis process that functions as a middle ground between these extremes. Optimizing solution density involves a designer manually manipulating the problem definition to increase the percentage of solutions that have pivots in acceptable locations. This is accomplished by changing the values of δ_j and α_j (prescribed translation and rotation of the moving plane, respectively) to manipulate the position of the poles. A physical example, designing a 7-bar parallel-motion generator, shows that applying this method yields more passing solutions when comparing over the same search depth. Specifically, 0.008% of solutions pass the design criteria without applying the method, and 3.154% pass after optimizing. This approach can reduce the computational load placed on a computer running a search script, as designers can use larger increments on the free choices without skipping over a family of solutions.