# 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} \hline \hline  & & & & \multicolumn{3}{c}{**The Newton Algorithm**} \\ \hline
**Output** & **Control** & \multirow{2}{*}{\(\delta\)} & \multicolumn{3}{c}{**Basic**} & \multicolumn{3}{c}{**Null-Space Optimization**} \\
**Function** & **Param.** & & **Length** & **Energy** & **Length** & **Energy** \\ \hline \multirow{8}{*}{(22)} & & 0.05 & 0.41 \(\div\) 3.03 & 0.18 \(\div\) 1.04 & 0.36 \(\div\) 0.62 & 0.35 \(\div\) 0.50 \\  & & 0.1 & 0.76 \(\div\) 3.01 & 0.36 \(\div\) 1.18 & 0.72 \(\div\) 1.28 & 0.50 \(\div\) 0.70 \\  & & 0.2 & 1.65 \(\div\) 4.07 & 0.60 \(\div\) 1.27 & 1.83 \(\div\) 1.99 & 0.86 \(\div\) 0.90 \\  & & 0.5 & 4.35 \(\div\) 6.28 & 1.21 \(\div\) 1.63 & 4.80 \(\div\) 4.80 & 1.39 \(\div\) 1.39 \\  & & 0.7 & 6.45 \(\div\) 9.79 & 1.40 \(\div\) 1.88 & — & — \\  & & 1 & 9.00 \(\div\) 13.46 & 1.72 \(\div\) 2.23 & — & — \\ \hline \multirow{8}{*}{(18)} & & 0.05 & 0.38 \(\div\) 1.91 & 0.18 \(\div\) 0.90 & 0.36 \(\div\) 0.41 & 0.34 \(\div\) 0.41 \\  & & 0.1 & 0.73 \(\div\) 2.07 & 0.33 \(\div\) 0.91 & 0.72 \(\div\) 0.73 & 0.54 \(\div\) 0.54 \\  & & 0.2 & 1.51 \(\div\) 2.89 & 0.61 \(\div\) 0.98 & — & — \\  & & 0.5 & 4.31 \(\div\) 5.12 & 1.13 \(\div\) 1.30 & — & — \\  & & 0.7 & 6.51 \(\div\) 6.78 & 1.44 \(\div\) 1.47 & — & — \\ \hline \multirow{8}{*}{(24)} & & 0.05 & 0.63 \(\div\) 2.18 & 0.22 \(\div\) 1.13 & 0.62 \(\div\) 0.67 & 0.48 \(\div\) 0.60 \\  & & 0.1 & 1.26 \(\div\) 2.88 & 0.45 \(\div\) 1.39 & 1.24 \(\div\) 1.28 & 0.68 \(\div\) 0.71 \\  & & 0.2 & 2.52 \(\div\) 4.33 & 0.65 \(\div\) 1.42 & 2.50 \(\div\) 2.51 & 1.01 \(\div\) 1.01 \\  & & 0.5 & 6.37 \(\div\) 7.91 & 1.26 \(\div\) 1.94 & — & — \\  & & 0.7 & 9.04 \(\div\) 10.42 & 1.56 \(\div\) 2.32 & — & — \\  & & 1 & 12.88 \(\div\) 14.18 & 1.90 \(\div\) 2.65 & — & — \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & \multicolumn{3}{c}{**The Newton Algorithm**} \\ \hline
**Output** & **Control** & \multirow{2}{*}{\(\delta\)} & \multicolumn{3}{c}{**Basic**} & \multicolumn{3}{c}{**Null-Space Optimization**} \\
**Function** & **Param. ** & & **Length** & **Energy** & **Length** & **Energy** \\ \hline \multirow{8}{*}{(22)} & & 0.05 & 6912 \(\div\) 49,083 & 74 \(\div\) 199 & 5166 \(\div\) 5392 & 64 \(\div\) 66 \\  & & 0.1 & 18,281 \(\div\) 18,281 & 121 \(\div\) 121 & 9788\(\div\) 10,735 & 89 \(\div\) 93 \\  & & 0.2 & — & — & 18,057 \(\div\) 19,637 & 120 \(\div\) 126 \\  & & 0.5 & 44,864 \(\div\) 135,750 & 190 \(\div\) 331 & 41,005 \(\div\) 45,333 & 182 \(\div\) 191 \\  & & 0.7 & — & — & 59,707 \(\div\) 75,423 & 219 \(\div\) 247 \\  & & 1 & — & — & 82,748 \(\div\) 87,747 & 258 \(\div\) 266 \\ \hline \multirow{8}{*}{ident.} & & 0.05 & 21,332 \(\div\) 45,418 & 131 \(\div\) 191 & 4049 \(\div\) 24,218 & 57 \(\div\) 140 \\  & & 0.1 & 76,600 \(\div\) 79,107 & 249 \(\div\) 253 & 8310 \(\div\) 22,968 & 82 \(\div\) 136 \\  & & 0.2 & — & — & 18,926 \(\div\) 137,973 & 123 \(\div\) 334 \\  & & 0.5 & — & — & 40,082 \(\div\) 162,380 & 180 \(\div\) 362 \\  & & 0.7 & — & — & 55,316 \(\div\) 113,002 & 211 \(\div\) 302 \\  & & 1 & — & — & 80,542 \(\div\) 197,181 & 255 \(\div\) 399 \\ \hline \multirow{8}{*}{(24)} & & 0.05 & 21,146 \(\div\) 123,282 & 130 \(\div\) 316 & 4020 \(\div\) 86,003 & 57 \(\div\) 264 \\  & & 0.1 & 86,087 \(\div\) 163,986 & 264 \(\div\) 364 & 8229 \(\div\) 105,744 & 81 \(\div\) 292 \\  & & 0.2 & — & — & 17,406 \(\div\) 176,427 & 118 \(\div\) 378 \\  & & 0.5 & — & — & 54,373 \(\div\) 142,871 & 209 \(\div\) 340 \\  & & 0.7 & — & — & 76,363 \(\div\) 211,192 & 248 \(\div\) 413 \\  & & 1 & — & — & 106,666 \(\div\) 219,503 & 294 \(\div\) 421 \\ \hline \multirow{8}{*}{(18)} & & 0.05 & 2.63 \(\div\) 5.96 & 0.94 \(\div\) 1.70 & 5.28 \(\div\) 5.28 & 1.40 \(\div\) 1.40 \\  & & 0.1 & 4.43 \(\div\) 10.33 & 1.20 \(\div\) 1.61 & 9.00 \(\div\) 9.00 & 1.71 \(\div\) 1.71 \\ \cline{1-1}  & & 0.2 & 8.17 \(\div\) 17.8 Based on results of the tests, some observations can be formulated:  
1.