## 6 Deviation State Space Model for the MAP of Three-Dimensional Weakly Rigid Parts
### Deviation State Equation of Weakly Rigid Parts in a MAP  
In order to reflect the dimensional deviations of the product, features that have a significant impact on the quality of the product are usually selected as objects for monitoring and control, such as the dimensions of the parts and sub-assemblies that affect the assembly accuracy and assembly force, which are usually referred to as key characteristics (KC). KC are divided into two types: key product characteristics (KPC) and key control characteristics (KCC). KPC refer to dimensional characteristics that affect the performance and appearance of assembled components, such as the coordinates of measurement points reflecting the coaxiality of mating holes in sub-assembly components, etc. ; KCC refer to the relevant tooling, processes, etc. that have an impact on assembly accuracy, and can be taken as points that are not on the product, such as the key location control characteristics (KLCC) of fixtures acting on the product, interference or clearance support elements (key contact control characteristics (KCCC)) where assembly forces are applied to the product. In Section 3.2, the three main sources of part dimensional deviations were described. By combining the above sources of deviations, the equation of state for the transfer of part deviation flow at station \(k\) is obtained as  
\[ \begin{array}{cccccccc}X(k)=X(k-1)+D^{P}(k)+D^{Q}(k)+R(k)&&\\ X(k){=}\begin{bmatrix}{\Delta X_{A_{1}}}(k)&{\Delta X_{B_{1}}}(k)&\cdots&{ \Delta X_{N_{1}}}(k)&\cdots&{\Delta X_{A_{w}}}(k)&\cdots&{\Delta X_{N_{w}}}(k) \end{bmatrix}^{T}\end{array}  \tag{37}\]  
In Equation (37), \(k=\{1,2,3,\cdots\}\) denotes the number of stations in the overall assembly process, \(X(k)\) is a state variable indicating the dimensional deviation of all parts within \(L_{\text{gs}}\), \(\{1,2,\cdots,w\}\) denotes all part numbers in the overall assembly process and \(\{A_{1},B_{1},\cdots,N_{1},A_{w},\cdots,N_{w}\}\) denotes the KPC points on parts 1-\(w\) with numbered \(A_{1}\sim N_{w}\). When \(k=0\), \(X(0)\) denotes the manufacturing deviation of the part. \(D^{P}(k)\) is the part deviation due to fixture deviation, \(D^{Q}(k)\) is the part deviation due to interference or clearance errors at the contact locations between the parts to be assembled and \(R(k)\) is the random noise error due to uncertainties in the assembly, all three of which lie within \(L_{\text{gs}}\). In \(D^{Q}(k)\), the rigid rotation deviation \({\Delta a}(k)\) and the flexible rotation deviation \({\Delta\beta}(k)\) are noted as zero, since only part displacement information exists. In Equation (37), \(D^{P}(k)\) is the deviation of the part due to fixture deviation, and the deviation of the KCC point on the part in relation to the KPC point can be seen in Section 4.2:  
\[ \begin{array}{cccccccc}&D^{P}(k)=B(k)\cdot U(k)&&\\ U(k){=}\begin{bmatrix}{\Delta U_{P_{1}}}(k)&{\Delta U_{P_{2}}}(k)&{\Delta U_{P _{3}}}(k)&\cdots&{\Delta U_{P_{n}}}(k)\end{bmatrix}^{T}\end{array}  \tag{38}\]\[B(k)= \begin{bmatrix}\Pi_{1}E_{A_{1},P_{1}}F_{A_{1},P_{1}}G_{P_{1},P_{2}}&0^{5 \times 6}&0^{5\times 6}&\cdots&0^{5\times 6}\\ 0^{5\times 6}&\Pi_{2}E_{A_{2},P_{2}}F_{A_{2},P_{3}}G_{P_{3},P_{4}}&0^{5\times 6}& \cdots&0^{5\times 6}\\ 0^{5\times 6}&0^{5\times 6}&\Theta_{3}&0^{5\times 6}\\ \vdots&\vdots&\vdots&\vdots&\\ 0^{5\times 6}&0^{5\times 6}&0^{5\times 6}&\cdots&\Theta&\\ &&&&&\ddots\\ &&&&&&&0^{5\times 6}\\ \end{bmatrix}  \tag{39}\]  
\[\Theta=\Pi_{w}\cdot E_{A_{w},P_{1}}\cdot F_{A_{w},P_{1}}\cdot G_{P_{1},P_{1 +1}}\]  
In Equations (38) and (39), \(\{P_{1},P_{2},P_{3},\cdots,P_{n}\}\) denotes all KCC point numbers in the entire assembly process, \(\left\{\Delta U_{P_{k}}(\kappa),\Delta U_{P_{k+1}}(k)\right\}\subseteq\left\{ \Delta U_{P_{1}}(k),\Delta U_{P_{2}}(k),\Delta U_{P_{3}}(k),\cdots,\Delta U_{ P_{n}}(k)\right\}\) and \(\{P_{l},P_{l+1}\}\subseteq\{P_{1},P_{2},P_{3},\cdots,P_{n}\}\).