## Appendix A
Based on the geometric coordinate of the FLHBR system shown in Figure 1, the coordinate \((x_{ci},y_{ci})\) and velocity \(\big{(}\dot{x}_{ci},\dot{y}_{ci}),i=1,2,3,4,5\) of each link is written as  
\[x_{c1}=D  
\[K_{kinetic\_1}\equiv\tfrac{1}{2}M_{1}(v_{\text{c}1})^{2}+\tfrac{1}{2}I_{1} \Big{(}\dot{\theta}_{1}\Big{)}^{2}=\tfrac{1}{2}M_{1}\big{(}\dot{x}_{\text{c}1} \big{)}^{2}+\tfrac{1}{2}M_{1}\big{(}\dot{y}_{\text{c}1}\big{)}^{2}+\tfrac{1}{2} I_{1}\Big{(}\dot{\theta}_{1}\Big{)}^{2} \tag{117}\] \[=\tfrac{1}{2}M_{1}\Big{(}D_{1}\cos\theta_{1}\Big{(}\dot{\theta}_{ 1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}M_{1}\Big{(}D_{1}\sin\theta_{1}\Big{(}\dot{ \theta}_{1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}I_{1}\Big{(}\dot{\theta}_{1}\Big{)} ^{2}=\tfrac{1}{2}(M_{1}D_{1}^{2}+I_{1})\Big{(}\dot{\theta}_{1}\Big{)}^{2}\] (118) \[\qquad K_{kinetic\_2}\equiv\tfrac{1}{2}M_{2}(v_{\text{c}2})^{2}+ \tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}=\tfrac{1}{2}M_{2}\big{(} \dot{x}_{\text{c}2}\big{)}^{2}+\tfrac{1}{2}M_{2}\big{(}\dot{y}_{\text{c}2} \big{)}^{2}+\tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}\] \[=\tfrac{1}{2}\big{(}M_{2}D_{2}^{2}+I_{2}\Big{)}\Big{(}\dot{\theta }_{2}\Big{)}^{2}+\tfrac{1}{2}(M_{2}L_{1}^{2})\Big{(}\dot{\theta}_{1}\Big{)}^{ 2}+(M_{2}D_{2}L_{1})\cos(\theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)} \Big{(}\dot{\theta}_{2}\Big{)}\] \[K_{kinetic\_3}\equiv\tfrac{1}{2}M_{3}(v_{\text{c}3})^{2}+\tfrac{1 }{2}I_{3}\Big{(}\dot{\theta}_{3}\Big{)}^{2}=\tfrac{1}{2}M_{3}(\dot{x}_{\text{c }3})^{2}+\tfrac{1}{2}M_{3}(\dot{y}_{\text{c}3})^{2}+\tfrac{1}{2}I_{3}\Big{(} \dot{\theta}_{3}\Big{)}^{2}\] (119) \[=\tfrac{1}{2}\big{(}M_{3}D_{3}^{2}+I_{3}\big{)}\Big{(}\dot{ \theta}_{3}\Big{)}^{2}+\tfrac{1}{2}M_{3}\Big{\{}L_{1}^{2}\Big{(}\dot{\theta} _{1}\Big{)}^{2}+L_{2}^{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}+2L_{1}L_{2}\cos( \theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)}\Big{(}\dot{\theta}_{2}\Big{)}\] \[+2L_{1}D_{3}\cos(\theta_{1}-\theta_{3})\Big{(}\dot{\theta}_{1} \Big{)}\Big{(}\dot{\theta}_{3}\Big{)}+2L_{2}D_{3}\cos(\theta_{2}-\theta_{3}) \Big{(}\dot{\theta}_{2}\Big{)}\Big{(}\dot{\theta}_{3}\Big{)}\] \[K_{kinetic\_4}\equiv\tfrac{1}{2}M_{4}(v_{\text{c}4})^{2}+\tfrac{1 }{2}I_{4}\Big{(}\dot{\theta}_{4}\Big{)}^{2}=\tfrac{1}{2}M_{4}\big{(}\dot{x}_{ \text{c}4}\big{)}^{2}+\tfrac{1}{2}M_{4}\big{(}\dot{y}_{\text{c}4}\big{)}^{2}+ \tfrac{1}{2}I_{4}\Big{(}\dot{\theta}_{4}\Big{)}^{2}\] (120) \[=\tfrac{1}{2}\big{(}M_{4}(L_{4}-D_{4})^{2}+I_{4}\Big{)}\Big{(}\dot {\theta}_{4}\Big{)}^{2}+\tfrac{1}{2}M_{4}\Big{\{}L_{1}^{2}\Big{(}\dot{\theta }_{1}\Big{)}^{2}+L_{2}\big{(}\dot{\theta}_{2}\Big{)}^{2}+2L_{1}L_{2}\cos( \theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)}\Big{(}\dot{\theta}_{2}\Big{)}\] \[+2L_{1}(L_{4}-D_{4})\cos(\theta_{1}+\theta_{4})\Big{(}\dot{\theta }_{1}\Big{)}\Big{(}\dot{\theta}_{4}\Big{)}+2L_{2}(L_{4}-D_{4})\cos(\theta_{2}- \theta_{4})\Big{(}\dot{\theta}_{2}\Big{)}\Big{(}\dot{\theta}_{4}\Big{)}\] \[=\tfrac{1}{2}M_{1}\Big{(}D_{1}\cos\theta_{1}\Big{(}\dot{\theta}_{ 1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}M_{1}\Big{(}D_{1}\sin\theta_{1}\Big{(}\dot{ \theta}_{1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}I_{1}\Big{(}\dot{\theta}_{1}\Big{)} ^{2}=\tfrac{1}{2}(M_{1}D_{1}^{2}+I_{1})\Big{(}\dot{\theta}_{1}\Big{)}^{2}\]0 \[=\tfrac{1}{2}M_{1}\Big{(}D_{1}\cos\theta_{1}\Big{(}\dot{\theta}_{ 1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}M_{1}\Big{(}D_{1}\sin\theta_{1}\Big{(}\dot{ \theta}_{1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}I_{1}\Big{(}\dot{\theta}_{1}\Big{)} ^{2}=\tfrac{1}{2}(M_{1}D_{1}^{2}+I_{1})\Big{(}\dot{\theta}_{1}\Big{)}^{2}\]1 (121) \[=\tfrac{1}{2}M_{1}\Big{(}D_{1}\cos\theta_{1}\Big{(}\dot{\theta}_{ 1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}M_{1}\Big{(}D_{1}\sin\theta_{1}\Big{(}\dot{ \theta}_{1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}I_{1}\Big{(}\dot{\theta}_{1}\Big{)} ^{2}=\tfrac{1}{2}(M_{1}D_{1}^{2}+I_{1})\Big{(}\dot{\theta}_{1}\Big{)}^{2}\]2 \[=\tfrac{1}{2}M_{1}\Big{(}D_{1}\cos\theta_{1}\Big{(}\dot{\theta}_{ 1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}M_{1}\Big{(}D_{1}\sin\theta_{1}\Big{(}\dot{ \theta}_{1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}I_{1}\Big{(}\dot{\theta}_{1}\Big{)} ^{2}=\tfrac{1}{2}(M_{1}D_{1}^{2}+I_{1})\Big{(}\dot{\theta}_{1}\Big{)}^{2}\]3 \[=\tfrac{1}{2}M_{1}\Big{(}D_{1}\cos\theta_{1}\Big{(}\dot{\theta}_{ 1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}M_{1}\Big{(}D_{1}\sin\theta_{1}\Big{(}\dot{ \theta}_{1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}I_{1}\Big{(}\dot{\theta}_{1}\Big{)} ^{2}=\tfrac{1}{2}(M_{1}D_{1}^{2}+I_{1})\Big{(}\dot{\theta}_{1}\Big{)}^{2}\]4  
\[=\tfrac{1}{2}M_{1}\Big{(}D_{1}\cos\theta_{1}\Big{(}\dot{\theta}_{ 1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}M_{1}\Big{(}D_{1}\sin\theta_{1}\Big{(}\dot{ \theta}_{1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}I_{1}\Big{(}\dot{\theta}_{1}\Big{)} ^{2}=\tfrac{1}{2}(M_{1}D_{1}^{2}+I_{1})\Big{(}\dot{\theta}_{1}\Big{)}^{2}\]5 \[=\tfrac{1}{2}M_{1}\Big{(}D_{1}\cos\theta_{1}\Big{(}\dot{\theta}_{ 1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}M_{1}\Big{(}D_{1}\sin\theta_{1}\Big{(}\dot{ \theta}_{1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}I_{1}\Big{(}\dot{\theta}_{1}\Big{)} ^{2}=\tfrac{1}{2}(M_{1}D_{1}^{2}+I_{1})\Big{(}\dot{\theta}_{1}\Big{)}^{2}\]6 \[=\tfrac{1}{2}M_{1}\Big{(}D_{1}\cos\theta_{1}\Big{(}\dot{\theta}_{ 1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}M_{1}\Big{(}D_{1}\sin\theta_{1}\Big{(}\dot{ \theta}_{1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}I_{1}\Big{(}\dot{\theta}_{1}\Big{)} ^{2}=\tfrac{1}{2}(M_{1}D_{1}^{2}+I_{1})\Big{(}\dot{\theta}_{1}\Big{)}^{2}\]7 \[=\tfrac{1}{2}M_{1}\Big{(}D_{1}\cos\theta_{1}\Big{(}\dot{\theta}_{ 1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}M_{1}\Big{(}D_{1}\sin\theta_{1}\Big{(}\dot{ \theta}_{1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}I_{1}\Big{(}\dot{\theta}_{1}\Big{)} ^{2}=\tfrac{1}{2}(M_{1}D_{1}^{2}+I_{1})\Big{(}\dot{\theta}_{1}\Big{)}^{2}\]8 \[=\tfrac{1}{2}M_{1}\Big{(}D_{1}\cos\theta_{1}\Big{(}\dot{\theta}_{ 1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}M_{1}\Big{(}D_{1}\sin\theta_{1}\Big{(}\dot{ \theta}_{1}\Big{)}\Big{)}^{2}+\tfrac{1}{2}I_{1}\Big{(}\dot{\theta}_{1}\Big{)} ^{2}=\tfrac{1}{2}(M_{1}D_{1}^{2}+I_{1})\Big{(}\dot{\theta}_{1}\Big{)}^{2}\]9 \[\qquad K_{kinetic\_2}\equiv\tfrac{1}{2}M_{2}(v_{\text{c}2})^{2}+ \tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}=\tfrac{1}{2}M_{2}\big{(} \dot{x}_{\text{c}2}\big{)}^{2}+\tfrac{1}{2}M_{2}\big{(}\dot{y}_{\text{c}2} \big{)}^{2}+\tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}\]0 \[\qquad K_{kinetic\_2}\equiv\tfrac{1}{2}M_{2}(v_{\text{c}2})^{2}+ \tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}=\tfrac{1}{2}M_{2}\big{(} \dot{x}_{\text{c}2}\big{)}^{2}+\tfrac{1}{2}M_{2}\big{(}\dot{y}_{\text{c}2} \big{)}^{2}+\tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}\]1 \[\qquad K_{kinetic\_2}\equiv\tfrac{1}{2}M_{2}(v_{\text{c}2})^{2}+ \tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}=\tfrac{1}{2}M_{2}\big{(} \dot{x}_{\text{c}2}\big{)}^{2}+\tfrac{1}{2}M_{2}\big{(}\dot{y}_{\text{c}2} \big{)}^{2}+\tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}\]2 \[\qquad K_{kinetic\_2}\equiv\tfrac{1}{2}M_{2}(v_{\text{c}2})^{2}+ \tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}=\tfrac{1}{2}M_{2}\big{(} \dot{x}_{\text{c}2}\big{)}^{2}+\tfrac{1}{2}M_{2}\big{(}\dot{y}_{\text{c}2} \big{)}^{2}+\tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}\]3  
\[\qquad K_{kinetic\_2}\equiv\tfrac{1}{2}M_{2}(v_{\text{c}2})^{2}+ \tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}=\tfrac{1}{2}M_{2}\big{(} \dot{x}_{\text{c}2}\big{)}^{2}+\tfrac{1}{2}M_{2}\big{(}\dot{y}_{\text{c}2} \big{)}^{2}+\tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}\]4  
\[\qquad K_{kinetic\_2}\equiv\tfrac{1}{2}M_{2}(v_{\text{c}2})^{2}+ \tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}=\tfrac{1}{2}M_{2}\big{(} \dot{x}_{\text{c}2}\big{)}^{2}+\tfrac{1}{2}M_{2}\big{(}\dot{y}_{\text{c}2} \big{)}^{2}+\tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}\]5  
\[\qquad K_{kinetic\_2}\equiv\tfrac{1}{2}M_{2}(v_{\text{c}2})^{2}+ \tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}=\tfrac{1}{2}M_{2}\big{(} \dot{x}_{\text{c}2}\big{)}^{2}+\tfrac{1}{2}M_{2}\big{(}\dot{y}_{\text{c}2} \big{)}^{2}+\tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}\]6  
\[\qquad K_{kinetic\_2}\equiv\tfrac{1}{2}M_{2}(v_{\text{c}2})^{2}+ \tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}=\tfrac{1}{2}M_{2}\big{(} \dot{x}_{\text{c}2}\big{)}^{2}+\tfrac{1}{2}M_{2}\big{(}\dot{y}_{\text{c}2} \big{)}^{2}+\tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}\]7  
\[\qquad K_{kinetic\_2}\equiv\tfrac{1}{2}M_{2}(v_{\text{c}2})^{2}+ \tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}=\tfrac{1}{2}M_{2}\big{(} \dot{x}_{\text{c}2}\big{)}^{2}+\tfrac{1}{2}M_{2}\big{(}\dot{y}_{\text{c}2} \big{)}^{2}+\tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}\]8  
\[\qquad K_{kinetic\_2}\equiv\tfrac{1}{2}M_{2}(v_{\text{c}2})^{2}+ \tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}=\tfrac{1}{2}M_{2}\big{(} \dot{x}_{\text{c}2}\big{)}^{2}+\tfrac{1}{2}M_{2}\big{(}\dot{y}_{\text{c}2} \big{)}^{2}+\tfrac{1}{2}I_{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}\]9  
\[=\tfrac{1}{2}\big{(}M_{2}D_{2}^{2}+I_{2}\Big{)}\Big{(}\dot{\theta }_{2}\Big{)}^{2}+\tfrac{1}{2}(M_{2}L_{1}^{2})\Big{(}\dot{\theta}_{1}\Big{)}^{ 2}+(M_{2}D_{2}L_{1})\cos(\theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)} \Big{(}\dot{\theta}_{2}\Big{)}\]0  
\[=\tfrac{1}{2}\big{(}M_{2}D_{2}^{2}+I_{2}\Big{)}\Big{(}\dot{\theta }_{2}\Big{)}^{2}+\tfrac{1}{2}(M_{2}L_{1}^{2})\Big{(}\dot{\theta}_{1}\Big{)}^{ 2}+(M_{2}D_{2}L_{1})\cos(\theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)} \Big{(}\dot{\theta}_{2}\Big{)}\]1  
\[=\tfrac{1}{2}\big{(}M_{2}D_{2}^{2}+I_{2}\Big{)}\Big{(}\dot{\theta }_{2}\Big{)}^{2}+\tfrac{1}{2}(M_{2}L_{1}^{2})\Big{(}\dot{\theta}_{1}\Big{)}^{ 2}+(M_{2}D_{2}L_{1})\cos(\theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)} \Big{(}\dot{\theta}_{2}\Big{)}\]2  
\[=\tfrac{1}{2}\big{(}M_{2}D_{2}^{2}+I_{2}\Big{)}\Big{(}\dot{\theta }_{2}\Big{)}^{2}+\tfrac{1}{2}(M_{2}L_{1}^{2})\Big{(}\dot{\theta}_{1}\Big{)}^{ 2}+(M_{2}D_{2}L_{1})\cos(\theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)} \Big{(}\dot{\theta}_{2}\Big{)}\]3  
\[=\tfrac{1}{2}\big{(}M_{2}D_{2}^{2}+I_{2}\Big{)}\Big{(}\dot{\theta }_{2}\Big{)}^{2}+\tfrac{1}{2}(M_{2}L_{1}^{2})\Big{(}\dot{\theta}_{1}\Big{)}^{ 2}+(M_{2}D_{2}L_{1})\cos(\theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)} \Big{(}\dot{\theta}_{2}\Big{)}\]4  
\[=\tfrac{1}{2}\big{(}M_{2}D_{2}^{2}+I_{2}\Big{)}\Big{(}\dot{\theta }_{2}\Big{)}^{2}+\tfrac{1}{2}(M_{2}L_{1}^{2})\Big{(}\dot{\theta}_{1}\Big{)}^{ 2}+(M_{2}D_{2}L_{1})\cos(\theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)} \Big{(}\dot{\theta}_{2}\Big{)}\]5  
\[=\tfrac{1}{2}\big{(}M_{2}D_{2}^{2}+I_{2}\Big{)}\Big{(}\dot{\theta }_{2}\Big{)}^{2}+\tfrac{1}{2}(M_{2}L_{1}^{2})\Big{(}\dot{\theta}_{1}\Big{)}^{ 2}+(M_{2}D_{2}L_{1})\cos(\theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)} \Big{(}\dot{\theta}_{2}\Big{)}\]6  
\[=\tfrac{1}{2}\big{(}M_{2}D_{2}^{2}+I_{2}\Big{)}\Big{(}\dot{\theta }_{2}\Big{)}^{2}+\tfrac{1}{2}(M_{2}L_{1}^{2})\Big{(}\dot{\theta}_{1}\Big{)}^{ 2}+(M_{2}D_{2}L_{1})\cos(\theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)} \Big{(}\dot{\theta}_{2}\Big{)}\]7 \[=\tfrac{1}{2}\big{(}M_{2}D_{2}^{2}+I_{2}\Big{)}\Big{(}\dot{\theta }_{2}\Big{)}^{2}+\tfrac{1}{2}(M_{2}L_{1}^{2})\Big{(}\dot{\theta}_{1}\Big{)}^{ 2}+(M_{2}D_{2}L_{1})\cos(\theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)} \Big{(}\dot{\theta}_{2}\Big{)}\]8  
\[=\tfrac{1}{2}\big{(}M_{2}D_{2}^{2}+I_{2}\Big{)}\Big{(}\dot{\theta }_{2}\Big{)}^{2}+\tfrac{1}{2}(M_{2}L_{1}^{2})\Big{(}\dot{\theta}_{1}\Big{)}^{ 2}+(M_{2}D_{2}L_{1})\cos(\theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)} \Big{(}\dot{\theta}_{2}\Big{)}\]9 \[K_{kinetic\_3}\equiv\tfrac{1}{2}M_{3}(v_{\text{c}3})^{2}+\tfrac{1 }{2}I_{3}\Big{(}\dot{\theta}_{3}\Big{)}^{2}=\tfrac{1}{2}M_{3}(\dot{x}_{\text{c }3})^{2}+\tfrac{1}{2}M_{3}(\dot{y}_{\text{c}3})^{2}+\tfrac{1}{2}I_{3}\Big{(} \dot{\theta}_{3}\Big{)}^{2}\]0 \[K_{kinetic\_3}\equiv\tfrac{1}{2}M_{3}(v_{\text{c}3})^{2}+\tfrac{1 }{2}I_{3}\Big{(}\dot{\theta}_{3}\Big{)}^{2}=\tfrac{1}{2}M_{3}(\dot{x}_{\text{c }3})^{2}+\tfrac{1}{2}M_{3}(\dot{y}_{\text{c}3})^{2}+\tfrac{1}{2}I_{3}\Big{(} \dot{\theta}_{3}\Big{)}^{2}\]1\[K_{kinetic\_3}\equiv\tfrac{1}{2}M_{3}(v_{\text{c}3})^{2}+\tfrac{1 }{2}I_{3}\Big{(}\dot{\theta}_{3}\Big{)}^{2}=\tfrac{1}{2}M_{3}(\dot{x}_{\text{c }3})^{2}+\tfrac{1}{2}M_{3}(\dot{y}_{\text{c}3})^{2}+\tfrac{1}{2}I_{3}\Big{(} \dot{\theta}_{3}\Big{)}^{2}\]2  
\[K_{kinetic\_3}\equiv\tfrac{1}{2}M_{3}(v_{\text{c}3})^{2}+\tfrac{1 }{2}I_{3}\Big{(}\dot{\theta}_{3}\Big{)}^{2}=\tfrac{1}{2}M_{3}(\dot{x}_{\text{c }3})^{2}+\tfrac{1}{2}M_{3}(\dot{y}_{\text{c}3})^{2}+\tfrac{1}{2}I_{3}\Big{(} \dot{\theta}_{3}\Big{)}^{2}\]3  
\[K_{kinetic\_3}\equiv\tfrac{1}{2}M_{3}(v_{\text{c}3})^{2}+\tfrac{1 }{2}I_{3}\Big{(}\dot{\theta}_{3}\Big{)}^{2}=\tfrac{1}{2}M_{3}(\dot{x}_{\text{c }3})^{2}+\tfrac{1}{2}M_{3}(\dot{y}_{\text{c}3})^{2}+\tfrac{1}{2}I_{3}\Big{(} \dot{\theta}_{3}\Big{)}^{2}\]4  
\[K_{kinetic\_3}\equiv\tfrac{1}{2}M_{3}(v_{\text{c}3})^{2}+\tfrac{1 }{2}I_{3}\Big{(}\dot{\theta}_{3}\Big{)}^{2}=\tfrac{1}{2}M_{3}(\dot{x}_{\text{c }3})^{2}+\tfrac{1}{2}M_{3}(\dot{y}_{\text{c}3})^{2}+\tfrac{1}{2}I_{3}\Big{(} \dot{\theta}_{3}\Big{)}^{2}\]5  
\[K_{kinetic\_3}\equiv\tfrac{1}{2}M_{3}(v_{\text{c}3})^{2}+\tfrac{1 }{2}I_{3}\Big{(}\dot{\theta}_{3}\Big{)}^{2}=\tfrac{1}{2}M_{3}(\dot{x}_{\text{c }3})^{2}+\tfrac{1}{2}M_{3}(\dot{y}_{\text{c}3})^{2}+\tfrac{1}{2}I_{3}\Big{(} \dot{\theta}_{3}\Big{)}^{2}\]6  
\[K_{kinetic\_3}\equiv\tfrac{1}{2}M_{3}(v_{\text{c}3})^{2}+\tfrac{1 }{2}I_{3}\Big{(}\dot{\theta}_{3}\Big{)}^{2}=\tfrac{1}{2}M_{3}(\dot{x}_{\text{c }3})^{2}+\tfrac{1}{2}M_{3}(\dot{y}_{\text{c}3})^{2}+\tfrac{1}{2}I_{3}\Big{(} \dot{\theta}_{3}\Big{)}^{2}\]7  
\[K_{kinetic\_3}\equiv\tfrac{1}{2}M_{3}(v_{\text{c}3})^{2}+\tfrac{1 }{2}I_{3}\Big{(}\dot{\theta}_{3}\Big{)}^{2}=\tfrac{1}{2}M_{3}(\dot{x}_{\text{c }3})^{2}+\tfrac{1}{2}M_{3}(\dot{y}_{\text{c}3})^{2}+\tfrac{1}{2}I_{3}\Big{(} \dot{\theta}_{3}\Big{)}^{2}\]8  
\[K_{kinetic\_3}\equiv\tfrac{1}{2}M_{3}(v_{\text{c}3})^{2}+\tfrac{1 }{2}I_{3}\Big{(}\dot{\theta}_{3}\Big{)}^{2}=\tfrac{1}{2}M_{3}(\dot{x}_{\text{c }3})^{2}+\tfrac{1}{2}M_{3}(\dot{y}_{\text{c}3})^{2}+\tfrac{1}{2}I_{3}\Big{(} \dot{\theta}_{3}\Big{)}^{2}\]9  
\[=\tfrac{1}{2}\big{(}M_{3}D_{3}^{2}+I_{3}\big{)}\Big{(}\dot{ \theta}_{3}\Big{)}^{2}+\tfrac{1}{2}M_{3}\Big{\{}L_{1}^{2}\Big{(}\dot{\theta} _{1}\Big{)}^{2}+L_{2}^{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}+2L_{1}L_{2}\cos( \theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)}\Big{(}\dot{\theta}_{2}\Big{)}\]0  
\[=\tfrac{1}{2}\big{(}M_{3}D_{3}^{2}+I_{3}\big{)}\Big{(}\dot{ \theta}_{3}\Big{)}^{2}+\tfrac{1}{2}M_{3}\Big{\{}L_{1}^{2}\Big{(}\dot{\theta} _{1}\Big{)}^{2}+L_{2}^{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}+2L_{1}L_{2}\cos( \theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)}\Big{(}\dot{\theta}_{2}\Big{)}\]1  
\[=\tfrac{1}{2}\big{(}M_{3}D_{3}^{2}+I_{3}\big{)}\Big{(}\dot{ \theta}_{3}\Big{)}^{2}+\tfrac{1}{2}M_{3}\Big{\{}L_{1}^{2}\Big{(}\dot{\theta} _{1}\Big{)}^{2}+L_{2}^{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}+2L_{1}L_{2}\cos( \theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)}\Big{(}\dot{\theta}_{2}\Big{)}\]2  
\[=\tfrac{1}{2}\big{(}M_{3}D_{3}^{2}+I_{3}\big{)}\Big{(}\dot{ \theta}_{3}\Big{)}^{2}+\tfrac{1}{2}M_{3}\Big{\{}L_{1}^{2}\Big{(}\dot{\theta} _{1}\Big{)}^{2}+L_{2}^{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}+2L_{1}L_{2}\cos( \theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)}\Big{(}\dot{\theta}_{2}\Big{)}\]3  
\[=\tfrac{1}{2}\big{(}M_{3}D_{3}^{2}+I_{3}\big{)}\Big{(}\dot{ \theta}_{3}\Big{)}^{2}+\tfrac{1}{2}M_{3}\Big{\{}L_{1}^{2}\Big{(}\dot{\theta} _{1}\Big{)}^{2}+L_{2}^{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}+2L_{1}L_{2}\cos( \theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)}\Big{(}\dot{\theta}_{2}\Big{)}\]4  
\[=\tfrac{1}{2}\big{(}M_{3}D_{3}^{2}+I_{3}\big{)}\Big{(}\dot{ \theta}_{3}\Big{)}^{2}+\tfrac{1}{2}M_{3}\Big{\{}L_{1}^{2}\Big{(}\dot{\theta} _{1}\Big{)}^{2}+L_{2}^{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}+2L_{1}L_{2}\cos( \theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)}\Big{(}\dot{\theta}_{2}\Big{)}\]5  
\[=\tfrac{1}{2}\big{(}M_{3}D_{3}^{2}+I_{3}\big{)}\Big{(}\dot{ \theta}_{3}\Big{)}^{2}+\tfrac{1}{2}M_{3}\Big{\{}L_{1}^{2}\Big{(}\dot{\theta} _{1}\Big{)}^{2}+L_{2}^{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}+2L_{1}L_{2}\cos( \theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)}\Big{(}\dot{\theta}_{2}\Big{)}\]6  
\[=\tfrac{1}{2}\big{(}M_{3}D_{3}^{2}+I_{3}\big{)}\Big{(}\dot{ \theta}_{3}\Big{)}^{2}+\tfrac{1}{2}M_{3}\Big{\{}L_{1}^{2}\Big{(}\dot{\theta} _{1}\Big{)}^{2}+L_{2}^{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}+2L_{1}L_{2}\cos( \theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)}\Big{(}\dot{\theta}_{2}\Big{)}\]7  
\[=\tfrac{1}{2}\big{(}M_{3}D_{3}^{2}+I_{3}\big{)}\Big{(}\dot{ \theta}_{3}\Big{)}^{2}+\tfrac{1}{2}M_{3}\Big{\{}L_{1}^{2}\Big{(}\dot{\theta} _{1}\Big{)}^{2}+L_{2}^{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}+2L_{1}L_{2}\cos( \theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)}\Big{(}\dot{\theta}_{2}\Big{)}\]8  
\[=\tfrac{1}{2}\big{(}M_{3}D_{3}^{2}+I_{3}\big{)}\Big{(}\dot{ \theta}_{3}\Big{)}^{2}+\tfrac{1}{2}M_{3}\Big{\{}L_{1}^{2}\Big{(}\dot{\theta} _{1}\Big{)}^{2}+L_{2}^{2}\Big{(}\dot{\theta}_{2}\Big{)}^{2}+2L_{1}L_{2}\cos( \theta_{1}-\theta_{2})\Big{(}\dot{\theta}_{1}\Big{)}\Big{(}\dot{\theta}_{2}\Big{)}\]9  
\[+2L_{1}D_{3}\cos(\theta_{1}-\theta_{3})\Big{(}\dot{\theta}_{1} \Big{)}\Big{(}\dot{\theta}_{3}\Big{)}+2L_{2}D_{3}\cos(\theta_{2}-\theta_{3}) \Big{(}\dot{\theta}_{2}\Big{)}\Big{(}\dot{\theta}_{3}\Big{)}\]0  
\[+2L_{1}D_{3}\cos(\theta_{1}-\theta_{3})\Big{(}\dot{\theta}_{1} \Big{)}\Big{(}\dot{\theta}_{3}\Big{)}+2L_{2}D_{3}\cos(\theta_{2}-\theta_{3}) \Big{(}\dot{\theta}_{2}\Big{)}\Big{(}\dot{\theta}_{3}\Big{)}\]1  
\[+2L_{1}D_{3}\cos(\theta_{1}-\theta_{3})\Big{(}\dot{\theta}_{1} \Big{)}\Big{(}\dot{\theta}_{3}\Big{)}+2L_{2}D_{3}\cos(\theta_{2}-\theta_{3}) \Big{(}\dot{\theta}_{2}\Big{)}\Big{(}\dot{\theta}_{3}\Big{)}\]2  
\[+2L_{1}D_{3}\cos(\theta_{1}-\theta_{3})\Big{(}\dot{\theta}_{1} \Big{)}\Big{(}\dot{\theta}_{3}\Big{)}+2L_{2}D_{3}\cos(\theta_{2}-\theta_{3}) \Big{(}\dot{\theta}_{2}\Big{)}\Big{(}\dot{\theta}_{3}\Big{)}\]3  
\[+2L_{1}D_{3}\cos(\theta_{1}-\theta_{3})\Big{(}\dot{\theta}_{1} \Big{)}\Big{(}\dot{\theta}_{3}\Big{)}+2L_{2}D_{3}\cos(\theta_{2}-\theta_{3}) \Big{(}\dot{\theta}_{2}\Big{)}\Big{(}\dot{\theta}_{3}\Big{)}\]4  
where \(M_{1}=M_{5}=4.55\) kg, \(M_{2}=M_{4}=7.63\) kg, \(M_{3}=49.00\) kg are the masses of link1-link5, \(M_{2},M_{4}\) denote the masses of exoskeleton thighs, \(M_{1},M_{5}\) denote the masses of legs, \(M_{3}\) denotes the mass of torso, \(L_{1}=L_{5}=0.502\) m, \(L_{2}=L_{4}=0.431\) m are the lengths of link1, 2, 4, 5, \(D_{1}=D_{5}=0.247\) m, \(D_{2}=D_{4}=0.247\) m, \(D_{3}=0.280\) m are the distances between the mass centers of link1, 2, 3, 4, 5 and those lower joints, \(I_{1}=I_{5}=0.105\) kg \(\cdot\) m\({}^{2}\), \(I_{2}=I_{4}=0.089\) kg \(\cdot\) m\({}^{2}\), \(I_{3}=2.350\) kg \(\cdot\) m\({}^{2}\) are the moments of rotational inertias for link1, 2, 3, 4, 5 and \(G=9.8\) m/s\({}^{2}\) is the acceleration of gravity.