On the treatment of equality constraints in mechanical systems design subject to uncertainty

In the early phase of product development, uncertainty may be treated by a set-based design method relying on so-called solution spaces. In this approach, all relevant system requirements are used to generate interval-Type component requirements that are sufficient for reaching the overall design goal while allowing for design flexibility in a distributed component development process. The product of these interval-Type requirements on component level is defined as solution box. The aim is not to find an optimal design, but to maximize the solution box that guarantees compliance with the system requirements. Unfortunately, with this approach only inequality-Type requirements on the system level can be treated. However, in mechanical systems design it is often necessary to consider equality-Type requirements, for example when mechanisms need to satisfy certain kinematic specifications. This paper proposes several approaches to incorporate equality-based requirements into the framework that calculates solution spaces. First, a relaxed problem statement is presented that can be solved by existing algorithms to maximize solution boxes. Furthermore, design variables are split into two groups: early-decision variables that are subject to large uncertainty and late-decision variables that are controllable and adjustable in a later stage of the development process. An existing projection technique and a novel approach that both project solution spaces to increase permissible intervals for early-decision variables are introduced. The relaxed problem statement and an inversion-based projection technique are applied to the design of a tailgate system of a passenger car, where the chassis and tailgate are designed in a distributed development process. It is shown that the resulting permissible intervals on the component level are significantly wider when projecting the solution space onto a subset of design variables than by solving the relaxed problem statement without applying a projection operator.