# A Method of Estimating an Object's Parameters Based on Simplication with Momentum for a Manipulator
## 4 Parameter Estimation Simulation
### Comparison with Previous Studies
Table 8 provides the range of joint angles and the corresponding maximum joint velocities and accelerations. To calculate the end-effector motions from previous studies, the kinematic parameters from [36; 37; 38] were used. Additionally, joint angles originally reported in degrees in [13; 18] were converted to radians for consistency. The object dynamics used for the estimation in these studies are detailed in Appendix A. Tables 7 and 8 indicate that the proposed method achieved accurate parameter estimation with smaller estimation motions compared to previous studies. Although certain directional accelerations in the prior studies shown in Table 7 are lower than those in the proposed method, the accelerations in other directions are considerably higher. Furthermore, the proposed method separates linear and rotational motions, with rotational motions applied independently to each axis, whereas previous methods applied multi-directional linear and angular motions simultaneously. This approach allows the proposed method to achieve precise estimation with significantly smaller motions compared to previous techniques, thereby reducing the inertial forces applied to the object and minimizing the risk of collision during the estimation process. The maximum acceleration difference between the proposed method and previous methods is 1172.3 cm/s\({}^{2}\) in the linear x-direction and 23,823 mrad/s\({}^{2}\) in the angular y-direction when comparing the proposed method and the method in [26]. The reduction rates are 95% and 98%, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**Items**} & \multicolumn{1}{c}{**Directions**} & \multicolumn{1}{c}{**Method in**} & \multicolumn{1}{c}{**Method in**} & \multicolumn{1}{c}{**Method in**} & \multicolumn{1}{c}{**Proposed**} \\ \cline{3-6} \multicolumn{1}{c}{} & & \(x\) & 107 & 121 & 99.5 & 10 \\ \cline{3-6} \multirow{4}{*}{Displacement} & \multirow{2}{*}{Linear (cm)} & \(y\) & 59.3 & 42.3 & 89.1 & 10 \\ \cline{3-6}  & & \(z\) & 12.2 & 22.8 & 54.7 & 10 \\ \cline{3-6}  & & \(x\) & 2610 & 124 & 4530 & 100 \\ \cline{3-6}  & & \(y\) & 1660 & 388 & 3100 & 100 \\ \cline{3-6}  & & \(z\) & 914 & 1200 & 3710 & 100 \\ \hline \multirow{4}{*}{Maximum speed} & \multirow{2}{*}{Linear (cm/s)} & \(x\) & 126 & 139 & 239 & 18.8 \\ \cline{3-6}  & & \(y\) & 89.6 & 71.5 & 186 & 18.8 \\ \cline{3-6}  & & \(z\) & 15.0 & 23.9 & 131 & 18.8 \\ \cline{3-6}  & & \(x\) & 3670 & 199 & 3640 & 188 \\ \cline{3-6}  & & \(y\) & 2350 & 424 & 5630 & 188 \\ \cline{3-6}  & & \(z\) & 1500 & 1210 & 4570 & 188 \\ \hline \multirow{4}{*}{Maximum acceleration} & \multirow{2}{*}{Linear (cm/s\({}^{2}\))} & \(x\) & 226 & 242 & 1230 & 57.7 \\ \cline{3-6}  & & \(y\) & 253 & 225 & 905 & 57.7 \\ \cline{3-6}  & & \(z\) & 31.5 & 25.6 & 934 & 57.7 \\ \cline{3-6}  & & \(x\) & 9520 & 548 & 20,900 & 577 \\ \cline{3-6}  & & \(y\) & 5640 & 831 & 24,400 & 577 \\ \cline{3-6}  & & \(z\) & 4770 & 2490 & 21,800 & 577 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Range of displacement, maximum velocity, and acceleration of the end-effector during the estimation in previous studies and this study with motion range of 0.1. The lowest value in each direction is highlighted. A comparison of joint motions in Tables 6 and 8 demonstrates that the proposed method is both safer and more adaptable than previous approaches. The smaller range of joint motion in the proposed method reduces the risk of collisions between the manipulator's links during estimation. Additionally, the lower maximum joint velocity and acceleration indicate that the proposed method can be applied to manipulators with less power. If estimation is required in environments with high noise, increasing the motion range to 0.2 radians or more may enhance accuracy. However, the proposed method requires more estimation time compared to previous methods. While earlier approaches typically estimated parameters within 1-2 or 5 s, the proposed method required 10 s, reflecting a trade-off between safety and estimation time.