# Research of Calibration Method for Industrial Robot Based on Error Model of Position
## 3 DCP Position Error Model
### DCP Coordinate System Parameter Errors  
The parameter errors of the DCP coordinate system represent the parameter errors of the transformation model from the FCS to the DCP coordinate system. Assuming that the pose transformation matrix of the robot's DCP coordinate system is \( \begin{array}{c}{{}_{6}^{D}T}\end{array} \) relative to that of the robot's end FCS, the transformation model from the robot's end flange to the DCP coordinate system can also be obtained by imitating the error derivation process of the RBCS:  
\[ \begin{array}{c}{{}_{6}^{D}T^{R}=T(x_{D}+\Delta x_{D},y_{D}+\Delta y_{D},z_{ D}+\Delta z_{D})R(x_{D},\theta x_{D}+\Delta\theta x_{D})}\\ \qquad\qquad\qquad R(y_{D},\theta y_{D}+\Delta\theta y_{D})R(z_{D},\theta z_{ D}+\Delta\theta z_{D}).\end{array}  \tag{17}\]  
After the above three pose models are obtained, they can be integrated into the actual DCP pose model in the MICS. Set \(T\) as the transformation matrix of the MICS to the robot DCP coordinate system:  
\[T=\left( \begin{array}{cc}\mathbf{R}&\mathbf{P}\\ 0&1\end{array} \right). \tag{18}\]  
In the equation, \(\mathbf{R}\) denotes the pose of the DCP coordinate system in the MICS, \(\mathbf{P}\) denotes the position of the DCP coordinate system in the MICS. The actual pose of DCP can be represented as  
\[ \begin{array}{c}{{}_{C}^{R}=\begin{array}{cc}0&T^{R},\theta T^{R}, \end{array}}\begin{array}{c}{{}_{D}^{T}}\end{array}\end{array}  \tag{19}\]  
Parameter errors affecting DCP pose accuracy:  
\[ \begin{array}{c}{{}_{q=[\Delta x_{C},\Delta y_{C},\Delta z_{C},\Delta\theta x _{C},\Delta\theta y_{C},\Delta\theta z_{C},\Delta\theta_{1}\Delta\theta_{6}, \Delta d_{1}\Delta d_{6},}}\\ \Delta a_{1}\Delta a_{6},\Delta a_{1}\Delta a_{6},\Delta\beta_{1}\Delta\theta_{6 },\Delta x_{D},\Delta y_{D},\Delta z_{D},\Delta\theta x_{D},\Delta\theta y_{D },\\ \Delta\theta z_{D}].\end{array} \]Position error model is comparatively more general, so the pose error of DCP is neglected. Assuming the set of parameter errors is \(\Delta q\), based on Equation (19), the actual position model of DCP can be expressed as follows:  
\[\mathbf{P}^{R}=\left( \begin{array}{c}\mathrm{P}_{x}^{R}\\ \mathrm{P}_{y}^{R}\\ \mathrm{P}_{z}^{R}\end{array} \right)=\left( \begin{array}{c}f_{\mathrm{P}_{x }}(q+\Delta q)\\ f_{\mathrm{P}_{y}}(q+\Delta q)\\ f_{\mathrm{P}_{z}}(q+\Delta q)\end{array} \right). \tag{20}\]  
Therefore, the mathematical relationship between the DCP position error and the parameters errors of the model can be represented as  
\[\Delta\mathbf{P}^{R}=\left( \begin{array}{c}\mathrm{\Delta}\mathrm{P}_{x}^{R}\\ \mathrm{\Delta}\mathrm{P}_{y}^{R}\\ \mathrm{\Delta}\mathrm{P}_{z}^{R}\end{array} \right)=\left( \begin{array}{c}f _{\mathrm{P}_{x}}(\mathbf{q}+\Delta q)-f_{\mathrm{P}_{z}}(\mathbf{q})\\ f_{\mathrm{P}_{y}}(\mathbf{q}+\Delta q)-f_{\mathrm{P}_{y}}(\mathbf{q})\\ f_{\mathrm{P}_{z}}(\mathbf{q}+\Delta q)-f_{\mathrm{P}_{z}}(\mathbf{q})\end{array} \right). \tag{21}\]  
In Equation (21), the relationship between the DCP position error and the parameter error is nonlinear, so it is difficult to solve them directly. Therefore, it can be analyzed from the perspectives of partial differential. For the model with \(n\) parameter errors, Equation (21) can be rewritten as follows:  
\[\Delta\mathbf{P}^{R}=\left( \begin{array}{c}\mathrm{\Delta}\mathrm{P}_{x}^{R}\\ \mathrm{\Delta}\mathrm{P}_{y}^{R}\\ \mathrm{\Delta}\mathrm{P}_{z}^{R}\end{array} \right)=\left( \begin{array}{ ccc}\frac{\partial f_{\mathrm{P}_{x}}}{\partial\Delta q_{1}}&\frac{\partial f_{ \mathrm{P}_{x}}}{\partial\Delta q_{2}}&\dots&\frac{\partial f_{\mathrm{P}_{x}}} {\partial\Delta q_{n}}\\ \frac{\partial f_{\mathrm{P}_{y}}}{\partial\Delta q_{1}}&\frac{\partial f_{ \mathrm{P}_{y}}}{\partial\Delta q_{2}}&\dots&\frac{\partial f_{\mathrm{P}_{x}}} {\partial\Delta q_{n}}\\ \frac{\partial f_{\mathrm{P}_{z}}}{\partial\Delta q_{1}}&\frac{\partial f_{ \mathrm{P}_{z}}}{\partial\Delta q_{2}}&\dots&\frac{\partial f_{\mathrm{P}_{z}} }{\partial\Delta q_{n}}\end{array} \right)\left(