# Joints Trajectory Planning of Robot Based on Slime Mould Whale Optimization Algorithm
## 4 Trajectory Optimization of Joints Based on SMWOA
### Path Interpolation of Joints
Therefore, this paper conducts interpolation based on cubic uniform B-spline curves [35], and its mathematical expression can be written as follows:  
\[\theta_{i}(u)=\sum_{j=0}^{3}B_{j,3}(u)V_{i+j-1} \tag{16}\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Functions**} & \multicolumn{1}{c}{**Evaluation**} & \multirow{2}{*}{**SMWOA**} & \multirow{2}{*}{**WOA**} & \multirow{2}{*}{**PSO**} & \multirow{2}{*}{**GSA**} & \multirow{2}{*}{**DE**} & \multirow{2}{*}{**SMA**} \\  & & & & & & & & \\ \hline \multirow{2}{*}{F\({}_{1}\)} & Avg & \(\mathbf{0.00\times 10^{0}}\) & \(7.91\times 10^{-74}\) & \(3.72\times 10^{0}\) & \(1.23\times 10^{-18}\) & \(1.78\times 10^{-92}\) & \(2.74\times 10^{-108}\) \\  & Std & \(\mathbf{0.00\times 10^{0}}\) & \(4.32\times 10^{-74}\) & \(2.65\times 10^{-15}\) & \(5.62\times 10^{-22}\) & \(0.69\times 10^{-91}\) & \(3.11\times 10^{-112}\) \\  & Avg & \(\mathbf{3.57\times 10^{-67}}\) & \(1.86\times 10^{-49}\) & \(2.05\times 10^{-1}\) & \(6.46\times 10^{-10}\) & \(3.09\times 10^{-1}\) & \(3.75\times 10^{-24}\) \\  & Std & \(\mathbf{2.51\times 10^{-64}}\) & \(2.39\times 10^{-48}\) & \(1.52\times 10^{-1}\) & \(0.00\times 10^{0}\) & \(2.78\times 10^{-1}\) & \(2.89\times 10^{-26}\) \\  & Avg & \(\mathbf{3.72\times 10^{-82}}\) & \(4.31\times 10^{-6}\) & \(3.89\times 10^{-3}\) & \(1.13\times 10^{-3}\) & \(3.74\times 10^{-5}\) & \(2.25\times 10^{-45}\) \\  & Std & \(\mathbf{6.59\times 10^{-79}}\) & \(2.93\times 10^{-5}\) & \(2.67\times 10^{-3}\) & \(2.56\times 10^{-25}\) & \(2.78\times 10^{-4}\) & \(5.68\times 10^{-49}\) \\  & Avg & \(2.58\times 10^{-16}\) & \(7.25\times 10^{-12}\) & \(4.56\times 10^{-8}\) & \(7.86\times 10^{-10}\) & \(3.72\times 10^{-14}\) & \(\mathbf{2.74\times 10^{-23}}\) \\  & Std & \(3.67\times 10^{-15}\) & \(3.97\times 10^{-12}\) & \(3.28\times 10^{-9}\) & \(3.57\times 10^{-9}\) & \(2.88\times 10^{-13}\) & \(\mathbf{3.11\times 10^{-26}}\) \\  & Avg & \(\mathbf{2.75\times 10^{-33}}\) & \(2.79\times 10^{1}\) & \(4.52\times 10^{2}\) & \(2.36\times 10^{1}\) & \(3.74\times 10^{0}\) & \(3.75\times 10^{-24}\) \\  & Std & \(\mathbf{3.69\times 10^{-31}}\) & \(7.63\times 10^{-1}\) & \(1.64\times 10^{2}\) & \(1.04\times 10^{-1}\) & \(2.52\times 10^{0}\) & \(2.89\times 10^{-26}\) \\  & Avg & \(5.04\times 10^{-29}\) & \(1.06\times 10^{-21}\) & \(1.29\times 10^{2}\) & \(2.32\times 10^{1}\) & \(5.27\times 10^{-3}\) & \(\mathbf{0.00\times 10^{0}}\) \\  & Std & \(6.52\times 10^{-30}\) & \(2.39\times 10^{-21}\) & \(3.64\times 10^{1}\) & \(7.85\times 10^{-18}\) & \(6.43\times 10^{-6}\) & \(\mathbf{0.00\times 10^{0}}\) \\  & Avg & \(\mathbf{6.54\times 10^{-28}}\) & \(5.39\times 10^{-15}\) & \(3.77\times 10^{-3}\) & \(6.32\times 10^{-8}\) & \(4.47\times 10^{-4}\) & \(2.74\times 10^{-26}\) \\  & Std & \(\mathbf{6.17\times 10^{-27}}\) & \(2.93\times 10^{-6}\) & \(2.58\times 10^{-4}\) & \(2.24\times 10^{-7}\) & \(3.23\times 10^{-4}\) & \(3.11\times 10^{-24}\) \\  & Avg & \(\mathbf{6.24\times 10^{-39}}\) & \(1.26\times 10^{-2}\) & \(5.79\times 10^{-1}\) & \(3.58\times 10^{0}\) & \(3.78\times 10^{-2}\) & \(3.75\times 10^{-24}\) \\  & Std & \(\mathbf{2.63\times 10^{-39}}\) & \(3.97\times 10^{-1}\) & \(2.35\times 10^{-3}\) & \(2.79\times 10^{-1}\) & \(9.01\times 10^{-1}\) & \(2.89\times 10^{-26}\) \\  & Avg & \(4.37\times 10^{-15}\) & \(3.05\times 10^{-3}\) & \(5.69\times 10^{0}\) & \(1.32\times 10^{0}\) & \(7.63\times 10^{-1}\) & \(\mathbf{2.25\times 10^{-15}}\) \\  & Std & \(\mathbf{3.62\times 10^{-15}}\) & \(7.60\times 10^{-2}\) & \(2.29\times 10^{0}\) & \(1.59\times 10^{-1}\) & \(5.22\times 10^{-2}\) & \(\mathbf{5.61\times 10^{-17}}\) \\  & Avg & \(\mathbf{2.47\times 10^{-10}}\) & \(8.97\times 10^{0}\) & \(3.74\times 10^{0}\) & \(5.22\times 10^{-1}\) & \(3.24\times 10^{-1}\) & \(2.74\times 10^{-8}\) \\  & Std & \(\mathbf{3.64\times 10^{-11}}\) & \(6.69\times 10^{0}\) & \(2.28\times 10^{0}\) & \(5.32\times 10^{-1}\) & \(2.56\times 10^{0}\) & \(3.11\times 10^{-9}\) \\  & Avg & \(1.80\times 10^{0}\) & \(3.76\times 10^{0}\) & \(1.89\times 10^{0}\) & \(4.59\times 10^{0}\) & \(2.58\times 10^{0}\) & \(\mathbf{1.75\times 10^{0}}\) \\  & Std & \(1.08\times 10^{0}\) & \(2.59\times 10^{0}\) & \(1.37\times 10^{0}\) & \(3.26\times 10^{0}\) & \(3.97\times 10^{0}\) & \(\mathbf{2.89\times 10^{-1}}\) \\  & Avg & \(0.42034\) & \(0.42718\) & \(0.41581\) & \(0.40278\) & \(\mathbf{0.39997}\) & \(0.  
where \(\theta_{i}(u)\) is the number of vectors at the \(i\)th point corresponding to the parameter \(u\) of spline curve, \(u\in[0,\,1]\).