## 5 Case Study  
The gear pump [9,23] in Figure 5 is composed of (a) pump body, (b) leading shaft and gear, and (c) shaft and gear. The gear is made of 45\({}^{\#}\) steel. The number of teeth is \(z=10\) and the number of modules is \(m=1.5\). Rated pressure is 25 Mpa and rated speed is 1450 r/min. The proportional factor of temperature deviation is 1. In the following text, the proposed model is used for pose decoupling analysis of the gear axis. In Figure 5, local coordinate systems are constructed at TF of each part. Since the nominal coordinate system is very close to the tracking local coordinate system and actual local coordinate system, only the nominal coordinate system is expressed [19]. Through analysis, this gear pump can construct two open-looped dimension chains which have a consistent starting point. The starting point of two open-looped dimension chains is in the origin of the reference coordinate system, \(O_{0}\) in Figure 5.  
Dimension chain 1 included three links: tolerance feature C1 (corresponding to position tolerance \(t1\)), tolerance features C1 & C2 (corresponding to clearance fit tolerance \(t2\)) and C3 (corresponding to coaxiality tolerance \(t2\)), while dimension chain 2 only included two links: tolerance features E1 & E2 (corresponding to gap fit tolerance \(t4\)) and E3 (corresponding to coaxiality tolerance \(t5\)). Tolerance and the type of TF of dimension chain 2 could be found in dimension chain 1 which was more complicated with greater representativeness. Therefore, only dimension chain 1 was selected for explanation. Dimension chain 1 is formed by connecting nominal coordinate systems \(O_{0}\), \(O_{1}\), \(O_{2}\) and \(O_{3}\), whereas dimension chain 2 is formed by connecting nominal coordinate systems \(O_{0}\), \(O_{4}\) and \(O_{5}\). Here, the dimension chain 1 was used as the example. Transformation among the nominal local coordinate systems (\(T_{1}\)) is:  
\[T_{1}=P_{O_{1}\to O_{0}}D_{1}P_{O_{2}\to O_{1}}D_{2}P_{O_{3}\to O_{2}}D_{3} \tag{18}\]  
Figure 5: Gear pump (Unit:mm): (**a**) Pump body; (**b**) Leading shaft and gear; (**c**) Shaft and gear. The homogeneous coordinate value of end point, M\({}_{1}\) in the nominal local coordinate system, \(O_{3}\) is \(\left[\mathbf{M_{1}}\right]_{O_{3}}=\left(0,0,-8,1\right)^{T}\). Then, the coordinate value in the reference coordinate system \(O_{0}\) is \(\left[\mathbf{M_{1}}\right]_{O_{0}}\):  
\[\left[\mathbf{M_{1}}\right]_{O_{0}}=\mathbf{T_{1}}\left[\mathbf{M_{1}}\right]_ {O_{3}} \tag{19}\]  
Combined with Equation (16), \(\left[\mathbf{M_{1}}\right]_{O_{0}}\) which is expressed by the area coordinate value can be obtained:  
\[\left[\mathbf{M_{1}}\right]_{O_{0}}=\left[ \begin{array}{c}\frac{\lambda_{2} \cdot 1_{1}^{t}+\lambda_{2}\cdot 2_{2}\cdot 2_{3}\cdot 3_{3}}{2}+\frac{15 \lambda_{3}\cdot 1_{1}+15\lambda_{5}\cdot 2_{2}^{t}}{d_{1}}-\frac{8\lambda_{3} \cdot 3_{3}}{d_{3}}+15\\ \frac{\lambda_{3}\cdot 1_{1}+\lambda_{3}\cdot 2_{2}+\lambda_{3}\cdot 3_{3}}{2}- \frac{15\lambda_{4}\cdot 1_{1}+15\lambda_{4}\cdot 2_{2}}{d_{1}}\\ \frac{8\lambda_{5}\cdot 3_{3}\left(\lambda_{5}\cdot 1_{1}+\lambda_{5}\cdot 2_{2} \right)}{d_{1}d_{1}}+30\end{array} \right] \tag{20}\]  
Equation (20) not only establishes relationships between any pose of point M\({}_{1}\) and the tolerance value and clearance value, but it can also find the limits of target feature variations when \(\lambda_{i}=\pm 1\) (\(i=1\)-\(5\)), that is, the vertex of the convex hull. After calculation, permutation and combination of basic area coordinate values in Equation (17) are brought into Equation (20), then the three-dimensional scatter points in Euclidean space can be acquired. Each scatter point represents one possible limit value. Next, using MATLAB (R2012b, MathWorks, Natick, MA, USA), the smallest convex hull model which covers all scatter points is searched for Vertexes of the convex hull are the pose limits of point M\({}_{1}\).