## Appendix A
\begin{array}{ll}C_{11}=&2d_{1}\dot{\theta}_{1}[m_{2}\lambda_{1}+(m_{3}+m_{4}+ m_{5}+m_{6})l_{1}+(m_{3}l_{3}+m_{4}\lambda_{2}+m_{5}l_{2}+m_{6}l_{2})\cos(\theta_{2}) \\ &+(m_{5}l_{5}+m_{6}l_{3})\cos(\theta_{2}+\theta_{3})]+2d_{2}(\dot{\theta}_{1}+ \dot{\theta}_{2})[m_{4}\lambda_{2}+(m_{5}+m_{6})l_{2}+(m_{4}+m_{5}\\ &+m_{6})l_{1}\cos(\theta_{2})+(m_{5}l_{5}+m_{6}l_{3})\cos(\theta_{3})]+2d_{3}( \dot{\theta}_{1}+\dot{\theta}_{2}+\dot{\theta}_{3})[m_{6}l_{3}+m_{6}l_{2}\cos( \theta_{3})\\ &+m_{6}l_{1}\cos(\theta_{2}+\theta_{3})]-\dot{\theta}_{2}(2\dot{\theta}_{1}+ \dot{\theta}_{2})[(m_{3}l_{3}+m_{4}\lambda_{2}+m_{5}l_{2}+m_{6}l_{2})l_{1}\sin (\theta_{2})+(m_{5}l_{5}\\ &+m_{6}l_{3})l_{1}\sin(\theta_{2}+\theta_{3})]-\dot{\theta}_{3}(2\dot{\theta}_ {1}+2\dot{\theta}_{2}+\dot{\theta}_{3})[(m_{5}l_{5}+m_{6}l_{3})l_{2}\sin( \theta_{3})+(m_{5}l_{5}\\ &+m_{6}l_{3})l_{1}\sin(\theta_{2}+\theta_{3})];\\ \end{array}  \tag{100}\]  
\[ \begin{array}{ll}C_{21}=&-(\dot{\theta}_{1})^{2}[m_{2}\lambda_{1}+(m_{3}+m_{ 4}+m_{5}+m_{6})l_{1}]-(\dot{\theta}_{1}+\dot{\theta}_{2})^{2}(m_{3}l_{3}+m_{4} \lambda_{2}+m_{5}l_{2}+m_{6}l_{2})\\ &\cos(\theta_{2})-(\dot{\theta}_{1}+\dot{\theta}_{2}+\dot{\theta}_{3})^{2}(m_ {5}l_{5}+m_{6}l_{3})\cos(\theta_{2}+\theta_{3})-2\dot{d}_{2}(\dot{\theta}_{1}+ \dot{\theta}_{2})(m_{4}+m_{5}+m_{6})\\ &\sin(\theta_{2})-2\dot{d}_{3}(\dot{\theta}_{1}+\dot{\theta}_{2}+\dot{\theta }_{3})m_{6}\sin(\theta_{2}+\theta_{3});\\ \end{array}  \tag{101}\]  
\[ \begin{array}{ll}C_{31}=&2d_{1}\dot{\theta}_{1}[(m_{3}l_{3}+m_{4}\lambda_{2} +m_{5}l_{2})\cos(\theta_{2})+(m_{5}l_{5}+m_{6}l_{3})\cos(\theta_{2}+\theta_{3 })]+2d_{2}(\dot{\theta}_{1}+\dot{\theta}_{2})\\ &[m_{3}\lambda_{2}+(m_{5}+m_{6})l_{2}+(m_{5}l_{5}+m_{6}l_{3})\cos(\theta_{3}) ]+2d_{3}(\dot{\theta}_{1}+\dot{\theta}_{2}+\dot{\theta}_{3})[m_{6}l_{3}\\ &+m_{6}l_{2}cos(\theta_{3})]+(\dot{\theta}_{1})^{2}[(m_{3}l_{5}+m_{4}\lambda_{2 }+m_{5}l_{2}+m_{6}l_{2})l_{1}\sin(\theta_{2})+(m_{5}l_{5}\\ &+m_{6}l_{3})\sin(\theta_{2}+\theta_{3})]-\dot{\theta}_{3}(2\dot{\theta}_{1}+ 2\dot{\theta}_{2}+\dot{\theta}_{3})[(m_{5}l_{5}+m_{6}l_{3})l_{2}\sin(\theta_{3 })];\\ \end{array}  \tag{102}\]  
\[ \begin{array}{ll}C_{41}=&-(\dot{\theta}_{1})^{2}(m_{4}+m_{5}+m_{6})l_{1} \cos(\theta_{2})-(\dot{\theta}_{1}+\dot{\theta}_{2})^{2}(m_{4}\lambda_{2}+m_{5 }l_{2}+m_{6}l_{2})-(\dot{\theta}_{1}+\dot{\theta}_{2}\\ &+\dot{\theta}_{3})^{2}(m_{5}l_{5}+m_{6}l_{3})\cos(\theta_{3})+2d_{1}\dot{ \theta}_{1}(m_{4}+m_{5}+m_{6})\sin(\theta_{2})-2d_{3}(\dot{\theta}_{1}+\dot{ \theta}_{2}+\dot{\theta}_{3})\\ &m_{6}\sin(\theta_{3});\\ \end{array}  \tag{103}\]  
\[ \begin{array}{ll}C_{51}=&2\dot{d}_{1}\dot{\theta}_{1}(m_{5}l_{5}+m_{6}l_{3} )\cos(\theta_{2}+\theta_{3})+2\dot{d}_{2}(\dot{\theta}_{1}+\dot{\theta}_{2})(m_ {5}l_{5}+m_{6}l_{3})\cos(\theta_{3})+2\dot{d}_{3}(\dot{\theta}_{1}\\ &+\dot{\theta}_{3})m_{6}l_{3}+(\dot{\theta}_{1})^{2}(m_{5}l_{5}+m_{6}l_{3})l_{1} \sin(\theta_{2}+\theta_{3})+(\dot{\theta}_{1}+\dot{\theta}_{2})^{2}(m_{5}l_{5} +m_{6}l_{3})l_{2}\sin(\theta_{3});\\ \end{array}  \tag{104}\]  
\[ \begin{array}{ll}C_{61}=&-(\dot{\theta}_{1})^{2}m_{6}l_{1}\cos(\theta_{2}+ \theta_{3})-(\dot{\theta}_{1}+\dot{\theta}_{2})^{2}m_{6}l_{2}\cos(\theta_{3})-( \dot{\theta}_{1}+\dot{\theta}_{2}+\dot{\theta}_{3})^{2}m_{6}l_{3}\\ &+2d_{1}\dot{\theta}_{1}m_{6}\sin(\theta_{2}+\theta_{3})+2d_{2}(\dot{\theta}_{1}+ \dot{\theta}_{2})m_{6}\sin(\theta_{3})\\ \end{array}  \tag{105}\]  
with: \(\lambda_{1}=L_{1}+d_{1}+l_{2}\); \(\quad l_{1}=L_{1}+d_{1}+L_{2}\); \(\quad\lambda_{2}=L_{1}+d_{2}+l_{4}\); \(\quad l_{2}=L_{1}+d_{2}+L_{2}\); \(\quad l_{3}=L_{1}+d_{3}+l_{6}\), where \(m_{i}\): mass of the link \(i\) (kg), \(i\)=1,2,3,4,5,6;* [PERSON] et al. (2019) [PERSON]; [PERSON]; [PERSON] Optimum manipulator path generation based on improved differential evolution constrained optimization algorithm. _Int. J. Adv. Robot. Syst._**2019**, _16_, 1729881419872060.