## 2 Kinematic Model of a Multijoint Industrial Robot  
### Forward Kinematics Based on POE Theory  
Figure 1 shows the structure and structural parameters of a multijoint industrial robot, and Figure 2 shows the kinematic chain. BCS is the base coordinate system, TCS is the tool coordinate system, and WCS is the workpiece coordinate system. The structural parameters \(\mathrm{V_{A}}\), \(\mathrm{V_{B}}\), \(\mathrm{V_{C}}\), \(\mathrm{V_{D}}\), and \(\mathrm{V_{W}}\) represent the positions of the A-joint, B-joint, C-joint, D-joint, and the base relative to the end-effector, respectively. \(\mathrm{V_{U}}\) represents the position of the workpiece relative to the base. They were represented as  
\[\left\{ \begin{array}{l}\mathrm{V_{A}}=\left[\mathrm{A_{x}}, \mathrm{A_{y}},0\right]^{\mathrm{T}}=\left[-(\mathrm{a_{1}}+\mathrm{a_{3}}+ \mathrm{a_{4}}),0,0\right]^{\mathrm{T}}\\ \mathrm{V_{B}}=\left[\mathrm{B_{x}},0,\mathrm{B_{z}}\right]^{\mathrm{T}}=\left[ -(\mathrm{a_{3}}+\mathrm{a_{4}}),0,\mathrm{a_{5}}-\mathrm{a_{2}}\right]^{ \mathrm{T}}\\ \mathrm{V_{C}}=\left[\mathrm{C_{x}},0,\mathrm{C_{z}}\right]^{\mathrm{T}}=\left[ -(\mathrm{a_{3}}+\mathrm{a_{4}}),0,\mathrm{a_{5}}\right]^{\mathrm{T}}\\ \mathrm{V_{D}}=\left[\mathrm{D_{x}},0,\mathrm{D_{z}}\right]^{\mathrm{T}}=\left[ -\mathrm{a_{4}},0,\mathrm{a_{5}}\right]^{\mathrm{T}}\\ \mathrm{V_{W}}=\left[\mathrm{W_{x}},\mathrm{W_{y}},\mathrm{W_{z}}\right]^{ \mathrm{T}}=\left[-(\mathrm{a_{1}}+\mathrm{a_{3}}+\mathrm{a_{4}}),0,\mathrm{a_ {5}}-\mathrm{a_{2}}-\mathrm{d_{1}}\right]^{\mathrm{T}}\\ \mathrm{V_{U}}=\left[\mathrm{U_{x}},\mathrm{U_{y}},\mathrm{U_{z}}\right]^{ \mathrm{T}}\end{array} \right. \tag{1}\]The clear geometric meaning of POE theory makes it an effective tool for kinematic modeling. One twist can represent the motion of one rigid body. The twist coordinates can be written as  
\[\xi=\left[\omega^{\mathrm{T}},\gamma^{\mathrm{T}}\right]^{\mathrm{T}} \tag{2}\]  
where \(\omega\) is the unit vector of the rotational axis; \(\gamma\) is the linear velocity of the rotational motion; vector \(\
u\) can be obtained as \(\
u=\mathrm{p}\times\omega\); \(\mathrm{p}\) is the position vector in the base for a point fixed on the axis; and the exponential matrix of the twist is represented as  
\[\mathrm{g}=\mathrm{e}^{\varepsilon,\Theta}=\left\{\left[ \begin{array}{cc} \mathrm{e}^{\omega\Theta}&\left(\mathrm{I}-\mathrm{e}^{\omega\Theta}\right) \left(\omega\times\
u\right)+\omega\omega^{\mathrm{T}}\
u\theta\\ 0&1\end{array} \right]\right.\]  
if \(\omega\
eq 0\)  
\[\left.\left( \begin{array}{cc}\mathrm{I}&\
u\theta\\ 0&1\end{array} \right]\right.\]  
if \(\omega=0\)  
where \(\mathrm{e}^{\omega\Theta}\) can be calculated by following the formula  
\[\mathrm{e}^{\omega\Theta}=\mathrm{I}+\omega\sin\theta+\omega^{2}(1-\cos\theta) \tag{4}\]  
Using the POE formula, the forward kinematics of a multijoint industrial robot with n joints can be expressed as  
\[\mathrm{g}_{\mathrm{st}}(\theta)=\mathrm{e}^{\varepsilon_{1}\theta_{1}} \mathrm{e}^{\varepsilon_{2}\theta_{2}}\cdots\mathrm{e}^{\varepsilon_{\alpha} \theta_{n}} \tag{5}\]  
Figure 1: The structural parameters of one multijoint industrial robot. Figure 2: The kinematic chain of the multijoint industrial robot. In order to obtain the position of the print nozzle (the end-effector) relative to the workpiece, and solve the inverse kinematics conveniently, the base coordinate system was set as the global coordinate system. The obtained kinematic POE formula was translated to the workpiece coordinate system in order to obtain the position of the print nozzle relative to the workpiece. A multijoint industrial robot can be seen as one open chain from the base to the printing nozzle. As shown in Figure 2, the order of the open chain is base\(\rightarrow\)A-joint\(\rightarrow\)B-joint\(\rightarrow\)C-joint\(\rightarrow\)D-joint\(\rightarrow\)printing nozzle. Point R\({}_{\text{A}}\) is the intersection point of the rotation axis of the A-joint.