## 5 Optimization Result  
### Problem Data  
In order to analyze the dynamics of the 6-DoF industrial robot with the above algorithm, kinematic parameters such as the coordinate transformation matrices at home position \(M_{i}=\left[ \begin{array}{cc}R_{i}&p_{i}\\ 0&1\end{array} \right]\) and the screw vectors \(S_{i}\) are necessary that appears in Equation (2). Here, \(R_{i}\) represents a rotation matrix to convert a 3D spatial vector represented in the \(i\)-th link coordinates into the \(i-1\)-th link coordinates. \(p_{i}\) is a vector in which the origin of the \(i\)-th link coordinates is represented in the \(i-1\)-th link coordinates. \(S_{i}\) represents a screw vector to express the degree of freedom of the \(i\)-th joint. Table 1 shows the kinematic parameters for the industrial robot used in this study. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Index (\(i\))** & **Rotation (\(R_{i}\))** & **Translation (\(p_{i}\))** & **Screw Vector (\(S_{i}\))** \\ \hline
**1** & \(\left[\begin{array}{cc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right]\) & \(\left[\begin{array}{cc}0\\ 0.108\\ \end{array}\right]\) \\
**2** & \(\left[\begin{array}{cc}0&1&0\\ 0&0&1\\ 1&0&0\end{array}\right]\) & \(\left[\begin{array}{c}0.200\\ 0.177\\ 0.250\end{array}\right]\) \\
**3** & \(\left[\begin{array}{cc}0&1&0\\ 1&0&0\\ 0&0&-1\end{array}\right]\) & \(\left[\begin{array}{c}0\\ 0\\ 0.108\\ \end{array}\right]\) \\
**4** & \(\left[\begin{array}{cc}0&0&-1\\ 1&0&0\\ 0&-1&0\end{array}\right]\) & \(\left[\begin{array}{c}0.178\\ 0.140\\ 0\end{array}\right]\) \\
**5** & \(\left[\begin{array}{cc}0&-1&0\\ 0&0&1\\ -1&0&0\\ -1&0&0\end{array}\right]\) & \(\left[\begin{array}{c}0\\ 0\\ -0.472\\ 0\\ 0\end{array}\right]\) \\
**6** & \(\left[\begin{array}{cc}0&0&1\\ 0&-1&0\\ 1&0&0\\ 0&0\end{array}\right]\) & \(\left[\begin{array}{c}0\\ 0\\ 0\\ 0.016\end{array}\right]\) \\
**7** & \(\left[\begin{array}{cc}0&-1&0\\ 1&0&0\\ 0&0&1\end{array}\right]\) & \(\left[\begin{array}{c}0\\ 0\\ 0.016\end{array}\right]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Coordinate transformation matrices and screw vectors for the 6-DoF robot. In the dynamics analysis with inward recursion, an inertial parameter expressed as  
\[J_{i}=\left[ \begin{array}{cc}I_{i}-m_{i}[r_{i}]^{2}&m_{i}[r_{i}]\\ -m_{i}[r_{i}]&m_{i}\cdot 1\end{array} \right]\]  
is required. Here, \(I_{i}\) is the rotational inertia at the center of gravity of the \(i\)-th link, \(r_{i}\) is the position of the center of gravity of the \(i\)-th link expressed in the \(i\)-th link coordinates, and \(m_{i}\) is the mass of the i -th link. Table 2 is the inertial parameters for the industrial robot used in this study. The inertial matrix \(J_{7}\) for the object fixed to link 6 was converted into a representation of the coordinates of link 6 as  
\[Ad_{T_{6,7}}^{\tau}J_{w}Ad_{T_{6,7}}^{\tau}\]  
and the analysis was performed by summing it up to the rotational inertia of link 6. The limit on displacement, speed and torque of each joint is given in Table 3, where the rated torque \(\tau_{rate}\) for each joint is assumed to be same with the maximum torque \(\tau_{max}\). Table 4 presents information on the start and end postures of the robot joint displacement. The initial value of the B-spline control parameter, which is the optimization variable, was set to the motion profile moving at a constant velocity by applying the value obtained by dividing the start and end posture of the robot joint displacement by equal intervals. The order of the B-spline describing the displacement of each joint was set to 3 to ensure \(C^{2}\) continuity of the motion profile, and the number of spline control points for each joint was set to 10.