## 4 Singular Perturbation Decomposition of the System
\begin{array}{ccc}0&1&\mathbf{0}_{1\times 2}& \mathbf{0}_{1\times 2}\\ -\overline{\mathbf{N}}_{11}k_{\mathrm{bf}}&0&\mathbf{N}_{12}^{\mathrm{T}}K_{ \mathrm{mf}}&\mathbf{0}_{1\times 2}\\ \mathbf{0}_{2\times 1}&\mathbf{0}_{2\times 1}&\mathbf{0}_{2\times 2}&\mathbf{I}_{2 \times 2}\\ \mathbf{0}_{2\times 1}&\mathbf{0}_{2\times 1}&-J_{m}^{-1}(\mathbf{I}+\mathbf{K}_{ \mathrm{c}})K_{\mathrm{mf}}&\mathbf{0}_{2\times 2}\end{array} \right]\)  
By introducing the second equation of Equation (18) into the first equation, the rigid model of the slow subsystem is obtained [28; 29] as follows:  
\[M(q,\delta)\ddot{q}+C(q,\delta,\dot{q},\dot{\delta})\dot{q}+\xi(q,\delta,\dot {q},\dot{\delta})=u \tag{20}\]  
where \(\mathbf{M}=\mathbf{Q}_{11}-\mathbf{Q}_{12}\mathbf{Q}_{22}^{-1}\mathbf{Q}_{21}\), \(C=\mathbf{R}_{11}-\mathbf{Q}_{12}\mathbf{Q}_{22}^{-1}\mathbf{R}_{21}\), \(\xi=\left(\mathbf{R}_{12}-\mathbf{Q}_{12}\mathbf{Q}_{22}^{-1}\mathbf{R}_{22} \right)\dot{\delta}-\mathbf{Q}_{12}\mathbf{Q}_{22}^{-1}\mathbf{K}_{\delta}\), \(\mathbf{u}=\left[\mathbf{\gamma}_{0},\tau_{\mathrm{ns}}^{\mathrm{T}}\right]^ {\mathrm{T}}\), and \(C(\mathbf{x},\mathbf{y})\) meet \(C(\mathbf{x},\mathbf{y})z=C(\mathbf{x},\mathbf{z})y,\forall\mathbf{y},z\in \mathbf{R}^{3\times 1}\). **5. Controller Design**  
_5.1. Controller Design of the Slow Subsystem_  
5.1.1. Repetitive Learning Sliding Mode Control  
For the slow subsystem rigid model Equation (20), define the continuous reference signal with period \(T\) as \(\mathbf{q}_{\mathrm{d}}=\left[\mathbf{\theta}_{\mathrm{d}0},\mathbf{\theta}_{ \mathrm{d}1},\mathbf{\theta}_{\mathrm{d}2}\right]^{\mathrm{T}}\) and substitute it into Equation (20) to get:  
\[\left\{ \begin{array}{l}u_{\mathrm{d}}=\mathbf{M}(\mathbf{q}_{\mathrm{d}}) \ddot{\mathbf{q}}_{\mathrm{d}}+\boldsymbol{C}\big{(}\mathbf{q}_{\mathrm{d}}, \dot{\mathbf{q}}_{\mathrm{d}}\big{)}\dot{\mathbf{q}}_{\mathrm{d}}+\boldsymbol{ \xi}\big{(}\mathbf{q}_{\mathrm{d}},\dot{\mathbf{q}}_{\mathrm{d}}\big{)}\\ u_{\mathrm{d}}(t)=u_{\mathrm{d}}(t+T)\end{array} \right. \tag{21}\]  
Expand the periodic function \(\mathbf{u}_{\mathrm{d}}\) with an infinite-dimensional Fourier series as:  
\[\mathbf{u}_{\mathrm{d}}=\overline{\mathbf{\vartheta}}_{0}+\sum_{k=1}^{\infty} \left[\overline{\mathbf{\vartheta}}_{k}\cos(k\omega t)+\overline{\mathbf{ \beta}}_{k}\sin(k\omega t)\right] \tag{22}\]  
where \(\omega=2\pi/T\) is the fundamental frequency, and \(\overline{\mathbf{\vartheta}}_{0}\), \(\overline{\mathbf{\vartheta}}_{k}\) and \(\overline{\mathbf{\beta}}_{k}\) are the unknown constant vector. After defining the trajectory tracking error \(\boldsymbol{e}=\mathbf{q}_{\mathrm{d}}-\mathbf{q}\), the exponential approach law sliding mode vector \(\boldsymbol{s}\in\mathbf{R}^{3\times 1}\) is:  
\[\boldsymbol{s}=\dot{\boldsymbol{e}}+a\boldsymbol{e}+b\boldsymbol{e}^{(q/p)} \tag{23}\]  
where \(a,b,p\) and \(q\) are positive constants. Equation (23) takes the derivative of time \(t\) to get:  
\[\dot{\boldsymbol{s}}=\ddot{\boldsymbol{e}}+a\dot{\boldsymbol{e}}+b\frac{q}{p} \boldsymbol{e}^{(\frac{q}{p}-1)} \tag{24}\]  
Multiply both sides of Equation (24) to the left by \(\mathbf{M}\) at the same time, and we obtain the error equation of the open loop system as follows:  
\[\mathbf{M}\ddot{\boldsymbol{s}}=-\boldsymbol{C}\boldsymbol{s}+\mathbf{Y}+a \mathbf{M}\dot{\boldsymbol{e}}+b\frac{q}{p}\mathbf{M}\boldsymbol{e}^{(\frac{q}{p} -1)}+\mathbf{u}_{\mathrm{d}}-\mathbf{u} \tag{25}\]  
Taking the differentiation of Equation (35) with respected to time \(t\), and substituting into Equations (27), (32) and (33) and \(\mathbf{\eta}\), we have:  
\[