# Development of Task-Space Nonholonomic Motion Planning Algorithm Based on Lie-Algebraic Method
## 3 Simulations
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & \multicolumn{8}{c}{**The Newton Algorithm**} \\ \cline{3-10}  & & & \multicolumn{8}{c}{**Basic**} & \multicolumn{8}{c}{**Null-Space Optimization**} \\ \hline
**Output** & **Control** & **Control** & **Param.** & \(\delta\) & **b.a.** & **a.** & **f.** & **Ctime** & **b.a.** & **a.** & **f.** & **Ctime** \\
**Function** & **Param.** & & **[\%]** & **[\%]** & **[\%]** & **[\%]** & **[\%]** & **[\%]** & **[\%]** & **[\%]** \\ \hline  & & 0.05 & 0.001 & 8 & 14 & 31.8 & 0.001 & 6 & 8 & 32.4 \\ \cline{3-10}  & & 0.1 & 0.021 & 1 & 18 & 30.8 & 0.004 & 9 & 8 & 31.5 \\ \cline{3-10}  & & 0.2 & 0.077 & 0 & 18 & 30.5 & 0.022 & 3 & 9 & 31.3 \\ \cline{3-10}  & & 0.5 & 0.017 & 3 & 14 & 28.3 & 0.005 & 7 & 4 & 29.4 \\ \cline{3-10}  & & 0.7 & 0.292 & 0 & 19 & 30.6 & 0.104 & 4 & 3 & 31.5 \\ \cline{3-10}  & & 1 & 0.446 & 0 & 25 & 30.6 & 0.030 & 7 & 4 & 31.7 \\ \cline{3-10}  & & 0.05 & 0.001 & 7 & 0 & 10.0 & 0.001 & 12 & 3 & 10.4 \\ \cline{3-10} ident. & & 0.1 & 0.001 & 4 & 8 & 9.2 & 0.001 & 12 & 31 & 9.5 \\ \cline{3-10}  & & 0.2 & 0.244 & 0 & 10 & 9.5 & 0.000 & 11 & 27 & 9.9 \\ \cline{3-10}  & & 0.5 & 0.507 & 0 & 22 & 8.4 & 0.003 & 7 & 40 & 8.8 \\ \cline{3-10}  & & 0.7 & 0.679 & 0 & 23 & 10.0 & 0.011 & 7 & 36 & 10.3 \\ \cline{3-10}  & & 1 & 0.940 & 0 & 34 & 10.0 & 0.015 & 7 & 35 & 10.5 \\ \hline  & & 0.05 & 0.001 & 47 & 53 & 12.0 & 0.001 & 48 & 52 & 12.8 \\ \cline{3-10}  & & 0.1 & 0.001 & 52 & 48 & 11.1 & 0.001 & 53 & 47 & 12.3 \\ \cline{3-10}  & & 0.2 & — & 0 & 100 & — & 0.000 & 46 & 54 & 14.7 \\ \cline{3-10}  & & 0.5 & — & 0 & 100 & — & 0.002 & 40 & 60 & 15.0 \\ \cline{3-10}  & & 0.7 & — & 0 & 100 & — & 0.003 & 36 & 64 & 14.8 \\ \cline{3-10}  & & 1 & — & 0 & 100 & — & 0.008 & 42 & 58 & 14.1 \\ \hline  & & 0.05 & 0.001 & 6 & 5 & 42.3 & 0.006 & 1 & 0 & 42.4 \\ \cline{3-10}  & & 0.1 & 0.004 & 7 & 3 & 43.1 & 0.018 & 1 & 0 & 43.2 \\ \cline{3-10}  & & 0.2 & 0.007 & 4 & 0 & 38.5 & 0.051 & 1 & 1 & 38.5 \\ \cline{3-10}  & & 0.5 & 0.136 & 1 & 0 & 42.7 & 0.172 & 0 & 0 & 42.8 \\ \cline{3-10}  & & 0.7 & 0.155 & 2 & 2 & 40.3 & 0.257 & 0 & 0 & 40.3 \\ \cline{3-10}  & & 1 & 0.328 & 0 & 3 & 39.2 & 0.381 & 0 & 0 & 39.2 \\ \cline{3-10}  & & 0.05 & 0.005 & 10 & 15 & 13.7 & 0.042 & 0 & 0 & 13.8 \\ \cline{3-10}  & & 0.1 & 0.016 & 2 & 20 & 13.0 & 0.083 & 0 & 1 & 13.2 \\ \cline{3-10}  & & 0.2 & 0.095 & 0 & 21 & 13.0 & 0.164 & 0 & 0 & 13.4 \\ \cline{3-10}  & & 0.5 & 0.388 & 0 & 5 & 12.6 & 0.406 & 0 & 1 & 12.8 \\ \cline{3-10}  & & 0.7 & 0.546 & 0 & 3 & 12.4 & 0.567 & 0 & 2 & 12.5 \\ \cline{3-10}  & & 1 & 0.783 & 0 & 2 & 12.6 & 0.809 & 0 & 1 & 12.7 \\ \hline  & & 0.05 & 0.009 & 7 & 11 & 15.6 & 0.005 & 68 & 2 & 16.6 \\ \cline{3-10}  & & 0.1 & 0.033 & 0 & 13 & 15.0 & 0.017 & 62 & 4 & 15.2 \\ \cline{3-10}  & & 0.2 & 0.069 & 0 & 3 & 15.0 & 0.037 & 43 & 7 & 15.1 \\ \cline{3-10}  & & 0.5 & 0.103 & 1 & 5 & 15.3 & 0.207 & 0 & 8 & 15.4 \\ \cline{3-10}  & & 0.7 & 0.296 & 0 & 2 & 15.0 & 0.329 & 0 & 17 & 15.1 \\ \cline{3-10}  & & 1 & 0.553 & 0 & 4 & 14.7 & 0.539 & 0 & 11 & 14.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Accuracy, numeric failures, and computing time for the kinematic car tasks.