## 4 Design of Controller
** Introducing the Lyapunov function:  
\[V=\frac{1}{2}\mathbf{r}^{\text{T}}M_{\text{h}}\mathbf{r}+\frac{1}{2}\sum\limits_{k=1}^ {m}\frac{1}{k_{\text{wck}}}\widetilde{\mathbf{w}}_{\text{SR}}^{\text{T}}\hat{\mathbf{ w}}_{\text{SR}}+\frac{1}{2}\sum\limits_{k=1}^{m}\frac{1}{k_{\text{wck}}} \widetilde{\mathbf{w}}_{\text{CK}}^{\text{T}}\widetilde{\mathbf{w}}_{\text{CK}}+\frac{ 1}{2k_{\text{\omega}}}\widetilde{\mathbf{\omega}}^{\text{T}}\widetilde{\mathbf{\omega }}+\frac{1}{2k_{l}}\widetilde{I}^{\text{T}}\widetilde{I} \tag{36}\]  
Then, derivative of the Equation (36)  
\[\dot{V}=\mathbf{r}^{\text{T}}M_{\text{h}}\dot{\mathbf{r}}+\frac{1}{2}\mathbf{r}^{\text{T}} \dot{M}_{\text{h}}\mathbf{r}-\sum\limits_{k=1}^{m}\frac{1}{k_{\text{wck}}} \widetilde{\mathbf{w}}_{\text{SR}}^{\text{T}}\dot{\mathbf{w}}_{\text{SR}}-\sum\limits_{ k=1}^{m}\frac{1}{k_{\text{wck}}}\widetilde{\mathbf{w}}_{\text{CK}}^{\text{T}}\dot{\mathbf{ w}}_{\text{CR}}-\frac{1}{k_{\text{\omega}}}\widetilde{\mathbf{\omega}}^{ \text{T}}\dot{\mathbf{\omega}}-\frac{1}{k_{l}}\widetilde{I}^{\text{T}}\dot{\mathbf{l}} \tag{37}\]  
Combined with Property 1 and Equations (27)-(29), the Equation (37) can be rewrite:  
\[ \begin{array}{ll}\dot{V}=&\mathbf{r}^{\text{T}}(\mathbf{\Xi}_{\text{X}}+\mathbf{W}_{ \text{S}}^{\text{T}}\hat{\mathbf{y}})-\mathbf{r}^{\text{T}}\tau_{\text{H}}-(\mathbf{W}_{ \text{S}}^{\text{T}}\hat{\mathbf{y}})^{\text{T}}\Gamma\hat{\mathbf{W}}_{\text{C}}^{ \text{T}}\hat{\mathbf{y}}+(\hat{\mathbf{W}}_{\text{S}}^{\text{T}}\hat{\mathbf{y}})^{\text{ T}}\Gamma\hat{\mathbf{W}}_{\text{C}}^{\text{T}}\hat{\mathbf{y}}-\\ &\sum\limits_{k=1}^{m}(r_{sk}+r_{sk}\hat{\mathbf{w}}_{\text{CK}}^{\text{T}}\hat{\mathbf{y }})|_{sk}\frac{\widetilde{\mathbf{w}}_{\text{SR}}^{\text{T}}\hat{\mathbf{w}}_{\text{SR}} \hat{\mathbf{y}}}{\|\hat{\mathbf{w}}_{\text{SR}}\|^{2}}-\mathbf{r}^{\text{T}}K_{\text{DL} }\mathbf{r}-\frac{1}{k_{\text{\omega}}}\widetilde{\mathbf{\omega}}^{\text{T}}\hat{\mathbf{ \omega}}-\frac{1}{k_{l}}\widetilde{I}^{\text{T}}\dot{\mathbf{l}}+\\ &\sum\limits_{k=1}^{m}r_{sk}(\hat{\mathbf{w}}_{\text{SR}}^{\text{T}}\hat{\mathbf{y}}) \bigg{(}\widetilde{\mathbf{w}}_{\text{CK}}^{\text{T}}(\hat{\mathbf{y}}-\alpha^{\text{T}} \hat{\mathbf{\sigma}}-\beta^{\text{T}}l)-I_{ck}\frac{\widetilde{\mathbf{w}}_{\text{CK}}^{ \text{T}}\hat{\mathbf{w}}_{\text{CK}}\hat{\mathbf{w}}_{\text{CK}}^{\text{T}}(\hat{\mathbf{y}} -\alpha^{\text{T}}\hat{\mathbf{\sigma}}-\beta^{\text{T}}l)}{\|\hat{\mathbf{w}}_{\text{CK} }\|^{2}}\bigg{)}\end{array}  \tag{38}\]  
where \(K_{\text{DL}\,r}=K_{\text{r}}+D_{\text{LC}}\). Substituting Equations (26), (30) and (31) into Equation (38) can obtain:  
\[\begin{array}{ll}\dot{V}=&-(W_{\text{S}}^{\text{T}}\hat{\mathbf{y}})^{\text{T}} \Gamma\hat{\mathbf{W}}_{\text{C}}^{\text{T}}\hat{\mathbf{y}}-(\hat{\mathbf{W}}_{\text{S}}^{ \text{T}}\hat{\mathbf{y}})^{\text{T}}\Gamma\Big{[}\widetilde{\mathbf{W}}_{\text{C}}^{ \text{T}}(\hat{\mathbf{y}}-\alpha^{\text{T}}\hat{\mathbf{\sigma}}-\beta^{\text{T}}l)+ \hat{\mathbf{W}}_{\text{C}}^{\text{T}}(\mathbf{\alpha}^{\text{T}}\widehat{\mathbf{\sigma}}+ \beta^{\text{T}}l)+\Xi_{\text{x}}\Big{]}-\\ &\mathbf{r}^{\text{T}}K_{\text{DL}\,r}+\mathbf{r}^{\text{T}}(\Xi_{\text{X}}+W_{\text{S}} ^{\text{T}}\hat{\mathbf{y}})-\mathbf{r}^{\text{T}}\tau_{\text{H}}+\sum\limits_{k=1}^{m}(r_ {sk}+r_{sk}\hat{\mathbf{w}}_{\text{CK}}^{\text{T}}\hat{\mathbf{y}}))I_{sk}\frac{ \widetilde{\mathbf{w}}_{\text{CK}}^{\text{T}}\hat{\mathbf{w}}_{\text{SR}}\hat{\mathbf{w}}_{ \text{SR}}\hat{\mathbf{y}}}{\|\hat{\mathbf{w}}_{\text{SR}}\|^{2}}+\\ &\sum\limits_{k=1}^{m}r_{sk}(\hat{\mathbf{w}}_{\text{SR}}^{\text{T}}\hat{\mathbf{y}}) \bigg{(}\widetilde{\mathbf{w}}_{\text{CK}}^{\text{T}}(\hat{\mathbf{y}}-\alpha^{\text{T}} \hat{\mathbf{\sigma}}-\beta^{\text{T}}l)-I_{ck}\frac{\widetilde{\mathbf{w}}_{\text{CK}}^{ \text{T}}\hat{\mathbf{w}}_{\text{CK}}\hat{\mathbf{w}}_{\text{CK}}^{\text{T}}(\hat{\mathbf{y}}- \alpha^{\text{T}}\hat{\mathbf{\sigma}}  
Then  
\[