## 3 Topology of Proposed Method
### Linear Extended State Observer Design  
In this section, LESO is developed for a second-order system to reduce lumped disturbances. From Equation (10), the decoupled system is given by  
\[\ddot{q}(t)=f_{0}(\dot{q}_{s}(t),\dot{q}_{e}(t),q_{s}(t),q_{e}(t)+f_{d}(d(t)))+ bu(t) \tag{12}\]  
where \(d(t)\) is lumped disturbance, \(u(t)\) is input, \(q(t)\) is output, and \(b\) is the system parameter. \(f_{d}(d(t))\) is uncertainty resulting from both external and internal modeling and is referred to as a whole disturbance. Assume that \(f_{d}(d(t))\), which represents the ULRRE's nonlinear time-varying dynamics, is differentiable. Equations (13) and (14) are used to rewrite and express the system. \[\left\{ \begin{array}{rl}\dot{q}_{1}=q_{2},\\ \dot{q}_{2}=\ -(f_{10}+f_{1d})P_{f1}+\ \frac{1}{(P_{f1}\text{:}j_{22})}U_{1},\\ \dot{q}_{3}=d_{1}(q,d_{1},\dot{d}_{1}),\\ y_{1}=\ q_{1}\end{array} \right. \tag{13}\]  
\[\left\{ \begin{array}{rl}\dot{q}_{4}=q_{5},\\ \dot{q}_{5}=\ (f_{20}+f_{2d})P_{f1}+\ \frac{1}{(P_{f1}\text{:}j_{11})}U_{2},\\ \dot{q}_{6}=d_{2}(q,d_{2},\dot{d}_{2}),\\ y_{2}=\ q_{4}\end{array} \right. \tag{14}\]  
The LESO is designed for both of the joints individually. Two LESO are  
\[\left\{ \begin{array}{rl}\dot{\hat{q}}_{1}=\widehat{q}_{2}+K_{1}(q_{1}- \widehat{q}_{1}),\\ \dot{\widehat{q}}_{2}=\widehat{q}_{3}+K_{2}(q_{1}-\widehat{q}_{1})+\frac{1}{ (P_{f1}\text{:}j_{22})}U_{1}-(f_{10})P_{f1},\\ \dot{\widehat{q}}_{3}=K_{3}(q_{1}-\widehat{q}_{1})\\ \dot{\widehat{q}}_{4}=\widehat{q}_{5}+K_{4}(q_{4}-\widehat{q}_{4}),\\ \dot{\widehat{q}}_{5}=\widehat{q}_{6}+K_{5}(q_{4}-\widehat{q}_{4})+\frac{1}{ (P_{f1}\text{:}j_{11})}U_{2}+(f_{20})P_{f1},\\ \dot{\widehat{q}}_{6}=K_{6}(q_{4}-\widehat{q}_{4})\end{array} \right. \tag{15}\]  
The observer gain matrices are \(\boldsymbol{L}_{1}=[K_{1},\ K_{2},\ K_{3}]^{T}\) and \(\ \boldsymbol{L}_{2}=[K_{4},\ K_{5},\ K_{6}]^{T}\). All observer poles are set to \(-w_{o}\) for tuning. To calculate observer gain, the characteristic equation given below is used. \[ \begin{array}{rl}s^{3}+K_{1}s^{2}+K_{2}s+K_{3}=(s+w_{o})^{3}\\ s^{3}+K_{4}s^{2}+K_{5}s+K_{6}=(s+w_{o})^{3}\end{array}  \tag{16}\]  
where \(w_{o}\) is the LESO's bandwidth. \(L_{1}\ and\ L_{2}\), gain vectors, are expressed as \(K_{1}=K_{4}=3w_{o}\), \(K_{2}=K_{5}=3w_{o}^{2}\), and \(K_{3}=K_{6}=w_{o}^{3}\). A balance between tracking performance and noise tolerance is achieved by selecting an observer bandwidth \(w_{o}\). ESO-based control law in general is given by \(u=\frac{u_{o}-\hat{f}}{\hat{h}}\), where \(h\) is the system parameter. \(u_{0}=K_{p}(q_{1}-\hat{\hat{q}}_{1})+K_{d}(q_{2}-\hat{\hat{q}}_{2})+\hat{q}_{d}\). Since \(K_{p}=w_{c}^{2}\) and \(\ K_{d}=2w_{c}\)[54], with well-designed ESO, the last term in left-hand side \(\hat{q}_{d}\) is relatively small in a well-designed ESO. The rest of equation is a PD controller. The feedback control law of ADRC is given by \(\ \boldsymbol{U_{0}}=\ [U_{1},U_{2}]^{T}=\boldsymbol{K_{p}}\ \boldsymbol{e}+ \boldsymbol{K_{d}}\ \dot{\boldsymbol{e}}\ \boldsymbol{e}\) and \(\dot{\boldsymbol{e}}\) are the state estimate errors for the position and velocity for both the joints, where \(\boldsymbol{q_{d}}=[q_{s,d}\ q_{e,d}]\) as reference trajectory \(\boldsymbol{e}=[\epsilon_{1}\ \epsilon_{2}]^{T}=[(q_{s,d}-\widehat{q}_{1})\ (q_{e,d}-\widehat{q}_{4})]^{T}\), \(\dot{\boldsymbol{e}}=[\epsilon_{1}\ \dot{\epsilon}_{2}]^{T}=[\widehat{q}_{2}\ \ \widehat{q}_{5}]^{T}\). The ESO-based and FTSTD-based ADRC law for standard second-order integrator \(y\ =\ U_{0}\) can be expressed as \(\boldsymbol{U}=\ [\tau_{1},\tau_{2}]^{T}=\ \boldsymbol{N_{inv}}(\boldsymbol{K_{p}} \ \boldsymbol{e}-\boldsymbol{K_{d}}\ \dot{\boldsymbol{e}}-\boldsymbol{\hat{f}})\), where \(\boldsymbol{\hat{f}}=[\widehat{f}_{1},\ \widehat{f}_{2}]^{T}\) are the estimate of disturbance i.e., \(\boldsymbol{\hat{f}}=[\ \ \hat{\hat{f}}_{1},\ \hat{\hat{f}}_{2}]^{T}=[\hat{\hat{f}}_{3},\ \hat{\hat{f}}_{6}]^{T}\). Controller bandwidth is determined as \(\ w_{c}=\frac{1}{3}w_{o}\). \[\boldsymbol{K_{p}}=