# Optimization Method of Assembly Tolerance Types Based on Degree of Freedom
## 3 Basic Concepts
### Definition and Representation of DOF of Tolerance Zones  
There are 10 types of typical tolerance zone shapes [9], as shown in Figure 1. The difference tolerances have four elements: shape, size (tolerance value), orientation and position [19; 26; 27]. According to the model of DOF [20], it is known that in a three-dimensional coordinate system, when a feature element changes along the direction of the X, Y, and Z axes, the change in the shape, size, and position of the element is called DOF, and vice versa, is called a degree of invariance (DOI) [20]. Similarly, define the DOF or DOI of the tolerance zone in the X, Y, and Z axes directions.  
The variation of geometric features of a part within the tolerance domain can be expressed as a combination of three translational and three rotational DOFs in a Cartesian coordinate system [20,36]. Then the DOF vector of the tolerance band is denoted as V, as shown in Equation (1).  
\[V=\left(T_{Z}\quad T_{Y}\quad T_{Z}\quad R_{X}\quad R_{Y}\quad R_{Z}\right) \tag{1}\]  
where \(T_{X}\), \(T_{Y}\), and \(T_{Z}\), respectively, represent the translational DOF of the X-axis, Y-axis, and Z-axis, \(R_{X}\), \(R_{Y}\), and \(R_{Z}\), respectively, representing the rotational DOF around the X-axis, Y-axis, and Z-axis.  
Then, Boolean representations are given, where \"0\" represents DOI, meaning it does not affect the design structure of the tolerance zone in that direction, and \"1\" represents a DOF. The tolerance zones and DOFs relationship matrix is shown in Table 1 (The two tolerance zones for free curves and free-form surfaces are not investigated in this paper because free-form surfaces are seldom used as assembly locating connections in typical assemblies). From the definition of the DOF of the tolerance zone, it can be seen that the DOF of the tolerance zone affects the direction of changes in assembly feature surfaces, and subsequently affects the changes in target components in the assembly.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Tolerance** & **TS1** & **TS2** & **TS3** & **TS4** & **TS5** & **TS6** & **TS7** & **TS8** \\ \hline \multirow{2}{*}{TZ form} & \multirow{2}{*}{TZ} & \multirow{2}{*}{TZ} & \multirow{2}{*}{TZ} & \multirow{2}{*}{TZ} & \multirow{2}{*}{TZ} & \multirow{2}{*}{TZ} & \multirow{2}{*}{TZ} & \multirow{2}{*}{TZ} \\  & RX, RY, RZ & & & & & & & & & \\ \hline \multirow{2}{*}{DOF} & TX, TY & \multirow{2}{*}{TZ} & \multirow{2}{*}{TZ} & \multirow{2}{*}{TX, TY,} & \multirow{2}{*}{TX, TY,} & \multirow{2}{*}{TX, TY,} & \multirow{2}{*}{TX, TY,} & \multirow{2}{*}{TX, TY,} & \multirow{2}{*}{TX, TY,} \\  & & RY & & & & & & RX, RY, & RX, RY & & RX, RY, RZ \\ \hline \multirow{2}{*}{Representation} & Ti (1,1,0) & Ti (0,0,1) & Ti (0,0,1) & Ti (1,1,0) & Ti (1,1,0) & Ti (1,1,0) & Ti (1,1,1) & Ti (1,1,0) \\  & Ri (0,0,0) & Ri (0,1,0) & Ri (1,1,0) & Ri (0,0,0) & Ri (1,1,0) & Ri (1,1,0) & Ri (0,0,0) & Ri (1,1,1) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Tolerance Zone and DOF Relationship Matrix.  
Figure 1: Ten tolerance zone diagrams. (**a**) Planar circle, (**b**) planar torus, (**c**) sphere, (**d**) toric cylinder, (**e**) free surface, (**f**) square, (**g**) two parallel planes, (**h**) two parallel lines, (**i**) cylinder, (**j**) free curve.