This is a representation of a fixture at the level of fixture elements.

    A fixture consists of several sub-assemblies . Each sub-assembly performs one or more fixturing functions (usually one). These kinds of sub-assemblies in a fixture are considered as fixture functional units. In a fixture units, all elements are connected directly with the baseplate and one or more elements in the subset are contacted directly with workpiece serving as locator , clamp or support.

    Let   denote a fixture units in a fixture. From above description we have:

                                                      (2)

    Where   is the number of elements in unit  .

    Therefore a representation of a fixture at the level of fixture units can be written in the following way:

    Where   is the number of units in fixture F.

    Dividing a fixture structure into functional units and giving detailed analyses on the functional units plays a key role in automated modular fixture designs.

    A fixture element consists of several surfaces which can either serve as a locating , clamping or supporting surface in contact directly with workpiece ( which is named a fixturing-functional surface ) or serve as supporting or supported surfaces in contact with other fixture elements (which are called assembly-functional surfaces) . Therefore an element can be represented by:

                                                     (5)

    Where   denotes the functional surface   on fixture element  , and   is the number of functional surfaces the element   contains.

    By combining formulars 4 and 5, a fixture can be represented at the level of fixture surfaces in the following form:

                                      (6)   

In this way, a fixture structure is decomposed into three levels , i.e., units , element, and functional surface levels . A conceptual sketch of the fixture structure decomposition is shown in Fig..5.

Based on the investigation of various application examples of dowel-pin modular fixtures and also for the purpose of automated fixture configuration design, a fixture structure can be classified into seven types of units (sub-structures): Vertical Locating Unit (VLU), Horizontal Locating Unit(HLU)，Vertical-Horizontal Combination Locating Unit(VHCLU), Vertical Claming Unit(VCU), Horizontal Clamping Unit (HCU), Vertical Supporting Unit (VSU), Horizontal Supporting Unit (HSU). Fixture units are composed of modular fixture elements. The functional surfaces of a fixture element perform the task of locating , supporting and clamping. All the above units are mounted on a baseplate . Figure 6 shows the fixture structure decomposition for dowel-pin modular fixture systems.

3.2 Fixture Units 

    In general  , a fixture unit consists of several fixture elements where usually only one element is in contact with the workpiece by its fixturing-functional surface to serve as locator , supporter , or clamp . All fixture elements in a fixture unit are connected together their assembly-functional surfaces. This fixturing-functional surface in a fixture unit is defined as an acting surface of the fixture unit. Each unit must have at least one acting surface which perform the fixturting function. Usually the acting surface is a plane or a cylindrical surface . The acting plane of a fixture unit can be described by a point on the plane and the external normal vector of the plane . The acting cylindrical surface of a fixture unit can be described by a point on the axis of the cylinder and the vector of axis. The center point of the acting surface is defined as an acting point of the unit and the distance between the surface of baseplate and the acting point is defined as an acting height of the fixture unit . The acting direction of a fixture unit can also be defined by the direction of the external normal vector of the acting surface .