# A Method of Estimating an Object's Parameters Based on Simplication with Momentum for a Manipulator
## 4 Parameter Estimation Simulation
### Simulation Result and Analysis  
The simulation results are presented in Tables 3-6. Tables 3-5 depict the error between the actual and estimated values by each motion range and noise level. Specifically, Table 3 presents the estimation error for Object 1, Table 4 for Object 2, and Table 5 for Object 3. The estimation error is quantified using the following equations:  
\[e_{m}=\left|\frac{\hat{m}-m}{m}\right|\times 100(\%) \tag{17}\] \[e_{r_{G/R,b,i}}=\left|\frac{\hat{r}_{G/R,b,i}-r_{G/R,b,i}}{r_{G/R,b,i}}\right|\times 100(\%)\] (18) \[e_{l_{i}}=\left|\frac{\hat{l}_{i}-I_{i}}{I_{i}}\right|\times 100( \%), \tag{19}\]  
where \(\hat{m},\hat{r}_{G/R,b,i}\), and \(\hat{l}_{i}\) represent the estimated values of mass, the elements of \(\mathbf{r}_{G/R,b}\) along each axis, and each element of \(\mathbf{I}\), respectively. Table 6 presents the range of joint angles, defined as the difference between the maximum and minimum angles of each joint while the robot generates motion, along with the maximum magnitude of joint velocities and accelerations during the estimation process with the proposed method. The maximum joint velocity of UR16e is \(\frac{2}{3}\pi\) rad/s (120 \({}^{\circ}\)/s) for the first and second joints and \(\pi\) rad/s (180 \({}^{\circ}\)/s) for all other joints [34]. None of the motion range exceeded these limits. The average time taken to estimate the parameters with the taken measurements was 0.31 s.  
The results reveal that \(m\) and \(\mathbf{r}_{G/R,b}\) were estimated with the maximum estimation error of 9.8% in all motion range and noise levels, indicating a low estimation error. However, the estimation error of \(\mathbf{I}\) increased as the motion range decreased. This is attributed to the decrease in signal-to-noise ratio (SNR) as the motion range decreases, even though noise from the force-torque sensor was minimized by integrating the data over time to neglect noisy acceleration data. Interestingly, reducing the motion range of linear motion to 0.025 m did not appear to affect the estimation error of \(m\) and \(\mathbf{r}_{G/R,b}\), likely due to the different units for linear and angular motion. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{3}{*}{
\begin{tabular}{c} **Estimation** \\ **Errors** \\ **(\%)** \\ \end{tabular} } & \multicolumn{4}{c}{**Motion Range**} \\ \cline{2-10}  & \multicolumn{4}{c}{**0.1**} & \multicolumn{4}{c}{**0.05**} & \multicolumn{4}{c}{**0.025**} \\ \cline{2-10}  & \multicolumn{4}{c}{**Noise Level**} & \multicolumn{4}{c}{**Noise Level**} & \multicolumn{4}{c}{**Noise Level**} \\ \cline{2-10}  & **Low** & **Mode** & **High** & **Low** & **Mode** & **High** & **Low** & **Mode** & **High** \\ \hline \(e_{m}\) & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ \hline \(e_{r_{G/R,b,x}}\) & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 \\ \hline \(e_{r_{G/R,b,y}}\) & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.3 \\ \hline \(e_{r_{G/R,b,x}}\) & 6.2 & 6.2 & 4.8 & 7.0 & 6.6 & 1.7 & 5.7 & 5.1 & 5.3 \\ \hline \(e_{l_{x}}\) & 3.3 & 3.0 & 1.9 & 3.4 & 3.7 & 0.6 & 3.0 & 1.8 & 9.0 \\ \hline \(e_{l_{y}}\) & 1.1 & 1.7 & 9.0 & 4.2 & 0.4 & 16.3 & 6.3 & 6.4 & 42.1 \\ \hline \(e_{l_{z_{z}}}\) & 0.2 & 1.2 & 11.5 & 1.9 & 7.6 & 13.0 & 3.3 & 3.2 & 14.5 \\ \hline \(e_{l_{z_{y}}}\) & 3.6 & 3.6 & 1.3 & 4.8 & 4.3 & 0.3 & 3.5 & 4.7 & 5.2 \\ \hline \(e_{l_{z_{z}}}\) & 1.2 & 0.9 & 4.8 & 1.4 & 1.6 & 9.3 & 1.0 & 3.6 & 14.8 \\ \hline \(e_{l_{z_{z}}}\) & 0.1 & 0.0 & 2.0 & 0.1 & 0.2 & 2.0 & 0.5 & 0.2 & 3.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Estimation error for Object 1.