# Integrated Tolerance and Fixture Layout Design for Compliant Sheet Metal Assemblies
### Assembly Fixture Layout Optimization  
A fixture may be used for different applications in production. Machining fixtures are primarily utilized to withstand against displacement or plastic deformations, due to the machining forces [20]. However, in the design of assembly fixtures, the principle design objective is to minimize the sensitivity of the assembly to variation of the rigid parts [21]. The part variations can be magnified or lessened in the assemblies, depending on the assembly fixture layout.  
The common types of locators are four-way-pins, two-way-pins, and clamps. The four-way-pins are commonly positioned in a hole in the part and the two-way-pins in a slot. Accordingly, four-ways-pins and two-ways-pins can be referred to as a hole and slot, respectively. A hole locks two in-plane translations of the part. A slot locks one in-plane translation, and a clamping unit locks the translation of the part normal to the clamping surface. If a hole or a slot is clamped, their translation will also be locked in the normal direction. Figure 1 depicts an example of a sheet metal part that is locked through a hole, a slot, and three clamping units.  
The assembly is considered to be non-rigid or compliant if the parts deform during the assembly procedure or after releasing the clamps (i.e., spring-back occurs). The studies regarding assembly fixture layout optimization can be divided into two categories of fixtures for rigid assemblies and fixtures for compliant assemblies. A difference between these two categories is the method utilized to determine the relationship between the part variation and assembly variation.  
In rigid assemblies, the effects of part variations in the locating areas are identical to the effects of locator variations in these areas. Hence, the sensitivity of a rigid assembly to a fixture layout can be determined by disturbing its locating positions and evaluating the consequences on the assembly [23]. [PERSON] et al. [24] have developed a state-space model that is based on this technique in order to determine the relationship between the variation of KPCs and variation in the locating points. This model and its other variants are utilized in the fixture layout optimization of rigid assemblies by most of the studies [25; 26; 27; 28; 29; 30]. Most of these studies have utilized state-space modeling to optimize the fixture layouts of all assembly stations simultaneously in multi-station assemblies.  
In compliant assemblies, the parts may be over-constrained by a fixture. A typical locating layout of a compliant part in an assembly fixture is an N-2-1 (N > 3) layout. In this layout, the number of holes and slots are the same as the numbers for rigid parts (i.e., one hole and one slot in each part), but the number of clamping units differs. Consequently, the number of additional clamps is a design parameter in compliant fixture layout optimization that does not exist in rigid fixture layout optimization.  
The relation between the assembly variation and variation of parts is more complicated in compliant assemblies than rigid assemblies. This relation is commonly determined through several finite element simulations that are elaborated in Section 1.3. Contrary to the rigid assemblies, the fixture layout of compliant assemblies is typically conducted for one station.  
The design parameters in most of the studies regarding fixture layout optimization of compliant assemblies are the coordinates of one or several clamping units in a part [31; 32; 33; 34; 35]. [PERSON] et al. [37] utilized the location of the clamping units as the node number of the part and not their coordinates. [PERSON] et al. [38] considered the location of holes, slots, clamping units, in addition to their types and the number of additional clamps as the design parameters for the optimization.