Analysis of critical boundaries of closed loop mechanisms via serial chain decomposition

Mechanisms inherently have boundaries that establish impediments with the desired actuation. These boundaries, often called critical boundaries or singular configurations, define the workspace of the mechanism. Input and output critical boundaries of closed-loop mechanisms are found by intersecting the boundaries of decomposed serial chains with the mechanism regular solution set. Output critical boundaries are associated with difficulty in determining an input that yields a desired output. Input critical boundaries are closed-loop mechanism configurations where a change in the input fails to uniquely characterize the output. A system of equations is presented to map input critical boundaries with continuation methods. The simplicity and availability of serial chain mechanism critical boundaries is leveraged for the design and analysis of closed-loop mechanisms.; Output critical boundaries are found by decomposing the closed-loop topology into serial chains connecting the reference body to the body-of-interest, establishing the associated serial chain critical boundaries, and then searching for intersections with the closed-loop mechanism regular solution set. The approach is shown to yield the mechanism configuration and necessary rank deficiency vector for continuation at the intersections with the output critical boundary system of equations. The new method furthers design knowledge by revealing that the intersected serial chain critical boundary is the source of the closed-loop mechanism output critical boundary. Alternate mechanism assemblies, and subsequently workspaces, are identified by the decomposition approach. Multiple serial chain critical boundary intersections with the mechanism regular solution set are revealed to be output critical boundary bifurcation points. Spatial and planar single and multi closed-loop mechanisms are used to illustrate the novel approach.; Input critical boundaries are found by searching the mechanism topology for passively actuated serial chains, establishing the associated serial chain critical boundaries according to a temporary reference, and then finding intersections with the closed-loop mechanism. The resulting intersections are then examined for input critical boundaries. The method provides the designer with knowledge of uncontrollable mechanism configurations. Preceding alternatives to locating input critical boundaries were random encounters, a priori knowledge, or comprehensive examination of all input combinations. Planar examples demonstrate the approach.