## 2 Kinematic Model of a Multijoint Industrial Robot
### Forward Kinematics Based on POE Theory
The coordinates of R\({}_{\text{A}}\) in BCS are R\({}_{\text{A}}=-\text{V}_{\text{W}}+\text{V}_{\text{A}}\). Point R\({}_{\text{B}}\) is the intersection point of the rotation axis of the B-joint. The coordinates relative to BCS are R\({}_{\text{B}}=-\text{V}_{\text{W}}+\text{V}_{\text{B}}\). Point R\({}_{\text{C}}\) is the intersection point of the rotation axis of the C-joint. The coordinates relative to BCS are R\({}_{\text{C}}=-\text{V}_{\text{W}}+\text{V}_{\text{C}}\). Point R\({}_{\text{D}}\) is the intersection point of the rotation axis of the D-joint. The coordinates relative to BCS are R\({}_{\text{D}}=-\text{V}_{\text{W}}+\text{V}_{\text{D}}\). First, motion twists and exponential matrices of all axes were established according to the kinematic chain and POE theory. Unit twists of the A-joint, B-joint, C-joint, and D-joint were expressed as  
\[\left\{ \begin{array}{l}\xi_{1}=\left[\omega_{1}^{\text{T}},\left(\text{P}_{ 1}\times\omega_{1}\right)^{\text{T}}\right]^{\text{T}}=\left[0,0,1,-\text{W}_{ y},-\left(\text{A}_{\text{x}}-\text{W}_{\text{x}}\right),0\right]^{\text{T}} \\ \xi_{2}=\left[\omega_{2}^{\text{T}},\left(\text{P}_{2}\times\omega_{2}\right) ^{\text{T}}\right]^{\text{T}}=\left[0,1,0,-\left(\text{B}_{\text{z}}-\text{W} _{\text{z}}\right),0,\text{B}_{\text{x}}-\text{W}_{\text{x}}\right]^{\text{T}} \\ \xi_{3}=\left[\omega_{3}^{\text{T}},\left(\text{P}_{3}\times\omega_{3}\right)^{ \text{T}}\right]^{\text{T}}=\left[0,1,0,-\left(\text{C}_{\text{z}}-\text{W}_{ \text{z}}\right),0,\text{C}_{\text{x}}-\text{W}_{\text{x}}\right]^{\text{T}} \\ \xi_{4}=\left[\omega_{4}^{\text{T}},\left(\text{P}_{4}\times\omega_{4}\right)^{ \text{T}}\right]^{\text{T}}=\left[0,1,0,-\left(\text{C}_{\text{z}}-\text{W}_{ \text{z}}\right),0,\text{D}_{\text{x}}-\text{W}_{\text{x}}\right]^{\text{T}} \end{array} \right. \tag{6}\]  
According to the order of the open chain, the transformation matrix of a multijoint industrial robot can be obtained with the exponential matrix of each joint. The movements of all joints relative to their local zero positions are positive relative to the global coordinate system (BCS). The POE formula of the tool relative to the base of the multijoint industrial robot was then expressed as  
\[\text{t}^{\text{b}}\text{T}=\text{e}^{\xi_{\text{b}}}\cdot\text{e}^{\xi_{ \text{t}}\alpha}\cdot\text{e}^{\xi_{\text{t}}\beta}\cdot\text{e}^{\xi_{\text{t }}\beta}\cdot\text{e}^{\xi_{\text{t}}\delta}\cdot\text{e}^{\xi_{\text{t}}}= \left[ \begin{array}{cccc}\text{n}_{\text{x}}&\text{o}_{\text{x}}&\text{a}_{ \text{x}}&\text{P}_{\text{x}}^{\prime}\\ \text{n}_{\text{y}}&\text{o}_{\text{y}}&\text{a}_{\text{y}}&\text{P}_{\text{y}}^ {\prime}\\ \text{n}_{\text{z}}&\text{o}_{\text{z}}&\text{a}_{\text{z}}&\text{P}_{\text{z}} ^{\prime}\\ 0&0&0&1\end{array} \right], \tag{7}\]  
where \(\xi_{\text{b}}=\xi_{\text{t}}=\left[0,0,0,0,0,0\right]\) represents the twist of the base and the printing nozzle, respectively. The pose matrix is R = [n,o,a] and the position matrix is P = [P\({}_{\text{x}}\),P\({}_{\text{y}}\),P\({}_{\text{z}}\)]T. However, it can be seen from Figure 1 that the fourth joint of the multijoint industrial robot modeled in this paper is a false joint. Due to this limitation of the mechanical structure, the end-effector is always parallel to the printing platform, meaning that \(\delta=-\beta-\gamma\) can be obtained from the geometry, namely  
\[\left[ \begin{array}{c}\text{P}_{\text{x}}{}^{\prime}\\ \text{P}_{\text{y}}{}^{\prime}\\ \text{P}_{\text{z}}{}^{\prime}\end{array} \right]=\left[ \begin{array}{c} \cos\alpha[\text{a}_{3}\cos(\beta+\gamma)+\text{a}_{2}\cos\beta+\text{a}_{1}+ \text{a}_{4}]\\ \sin\alpha[\text{a}_{3}\cos(\beta+\gamma)+\text{a}_{2}\cos\beta+\text{a}_{1}+ \text{a}_{4}]\\ \text{a}_{3}\sin(\beta+\gamma)+\text{a}_{2}\sin\beta+\text{d}_{1}-\text{a}_{5} \end{array} \right] \tag{8}\]  
According to Equation (3), the position and pose of the multijoint industrial robot can be obtained.