## Appendix B  
In this section, we show forward kinematics and inverse kinematics of the six DoF revolute robot manipulators. The results are consistent with the model ABB IRB 4600.  
Forward kinematics refers to the use of kinematic equations of a robot to compute the position of the end-effector from specified values for the joint angles and parameters. Equations (A1)-(A5) are summarized below:  
\[\begin{array}{l}n_{x}=c_{1}s_{23}(s_{4}s_{6}-c_{4}c_{5}c_{6})-s_{1}(s_{4}c_{5} c_{6}+c_{4}s_{6})-c_{1}c_{23}s_{5}c_{6}\\ n_{y}=s_{1}s_{23}(s_{4}s_{6}-c_{4}c_{5}c_{6})+c_{1}(s_{4}c_{5}c_{6}+c_{4}s_{6})-s _{1}c_{23}s_{5}c_{6}\\ n_{z}=c_{23}(s_{4}s_{6}-c_{4}c_{5}c_{6})+s_{23}s_{5}c_{6}\\ \end{array} \tag{1}\]  
\[\begin{array}{l}s_{x}=c_{1}s_{23}(s_{4}c_{6}+c_{4}c_{5}c_{6})+s_{1}(s_{4}c_{5 }s_{6}-c_{4}c_{6})+c_{1}c_{23}s_{5}s_{6}\\ s_{y}=s_{1}s_{23}(s_{4}c_{6}+c_{4}c_{5}c_{6})-c_{1}(s_{4}c_{5}s_{6}-c_{4}c_{6})+ s_{1}c_{23}s_{5}s_{6}\\ s_{z}=c_{23}(s_{4}c_{6}+c_{4}c_{5}c_{6})-s_{23}s_{5}s_{6}\\ \end{array} \tag{2}\]  
\[\begin{array}{l}a_{x}=-c_{1}s_{23}c_{4}s_{5}-s_{1}s_{4}s_{5}+c_{1}c_{23}c_{5 }\\ a_{y}=-s_{1}s_{23}c_{4}s_{5}+c_{1}s_{4}s_{5}+s_{1}c_{23}c_{5}\\ a_{z}=c_{23}c_{4}s_{5}-s_{23}c_{5}\\ \end{array} \tag{3}\]  
\[\begin{array}{l}d_{x}=L_{t}(-c_{1}s_{23}c_{4}s_{5}-s_{1}s_{4}s_{5}+c_{1}c_{ 23}c_{5})+c_{1}(L_{3}s_{23}+L_{2}s_{2}+a_{1})\\ d_{y}=L_{t}(-s_{1}s_{23}c_{4}s_{5}+c_{1}s_{4}s_{5}+s_{1}c_{23}c_{5})+s_{1}(L_{3 }s_{23}+L_{2}s_{2}+a_{1})\\ d_{z}=L_{t}(c_{23}c_{4}s_{5}-s_{23}c_{5})+L_{3}c_{23}+L_{2}c_{2}+L_{1}\\ \end{array} \tag{4}\]  
\[\begin{array}{l}\text{Note : }c_{i}\equiv\cos(q_{i}),\,s_{i}\equiv\sin(q_{i}) \\ c_{i,j}\equiv\cos\big{(}q_{i}+q_{j}\big{)},\,s_{i,j}\equiv\sin(q_{i}+q_{j})\\ i,j\,\in\,\{1,2,3,4,5,6\}\end{array} \tag{5}\]  
where \(\big{[}n_{x},n_{y},n_{z}\big{]}^{T},\)\(\big{[}s_{x},s_{y},s_{z}\big{]}^{T},\) and \(\big{[}a_{x},a_{y},a_{z}\big{]}^{T}\) are the end-effector's directional vectors of yaw, pitch, and rin the base frame \(O_{0}X_{0}Y_{0}Z_{0}\) (Figure A1). And \(\big{[}d_{x},d_{y},d_{z}\big{]}^{T}\) is the vector of absolute position of the center of the end-effector in base frame \(O_{0}X_{0}Y_{0}Z_{0}\). For the specific model ABB IRB 4600-45/2.05 (ABB, Zurich, Switzerland) (handling capacity: 45 kg/reach 2.05 m), the dimensions and mass are summarized in Table A1.  
Inverse kinematics refers to the mathematical process of calculating the variable joint angles needed to place the end-effector in a given position and orientation relative to the inertial base frame. Equations (A6)-(A13) are summarized below:  
\[\begin{array}{l}p_{x}=d_{x}-L_{t}a_{x}\\ p_{y}=d_{y}-L_{t}a_{y}\\ p_{z}=d_{z}-L_{t}a_{z}\end{array} \tag{6}\]\[q_{1}=arctan(\frac{p_{y}}{p_{x}}) \tag{10}\]  
\[q_{2}=\frac{pi}{2}-\arccos(\frac{{L_{2}}^{2}+{L_{3}}^{2}+{L_{4}}^{2}-\left(\sqrt{{ p_{x}}^{2}+{p_{y}}^{2}}-{a_{1}}\right)^{2}+\left({p_{z}}-{L_{1}}\right)^{2}-{L_{3}}^{2 }-{L_{4}}^{2}}{2{L_{2}}\sqrt{{L_{3}}^{2}+{L_{4}}^{2}}}) \tag{11}\]  
\[q_{5}=\arccos(c_{1}c_{23}a_{x}+s_{1}c_{23}a_{y}-s_{23}a_{z}) \tag{12}\]  
\[q_{4}=\arctan(\frac{s_{1}a_{x}-c_{1}a_{y}}{c_{1}s_{23}a_{x}+s_{1}s_{23}a_{y}+c_ {23}a_{z}}) \tag{13}\]  
\[q_{6}=-\arctan(\frac{c_{1}c_{23}s_{x}+s_{1}c_{23}s_{y}-s_{23}s_{z}}{c_{1}c_{23 }{l_{x}}+s_{1}c_{23}{n_{y}}-s_{23}{n_{z}}}) \tag{14}\]  
\[\text{Note : }c_{i}\equiv\cos(q_{i}),\;s_{i}\equiv\sin(q_{i}) \tag{15}\]  
\[c_{i,j}\equiv\cos\bigl{(}q_{i}+q_{j}\bigr{)},\;s_{i,j}\equiv\sin\bigl{(}q_{i}+ q_{j}\bigr{)} \tag{16}\]  
\[i,j\;\in\{1,2,3,4,5,6\}\]  
where \(\bigl{[}n_{x},n_{y},n_{z}\bigr{]}^{T}\), \(\bigl{[}s_{x},s_{y},s_{z}\bigr{]}^{T}\), \(\bigl{[}a_{x},a_{y},a_{z}\bigr{]}^{T}\), and \(\bigl{[}d_{x},d_{y},d_{z}\bigr{]}^{T}\) have been defined above in the forward kinematic discussion.