## 3 Active Disturbance Rejection Control Design
### Trajectory Planner Design  
It is worth noting that for the conventional backstepping, the desired reference trajectory should be carefully designed to be sufficiently smooth and bounded, and its derivatives are also continuous and bounded. Therefore, it is extremely challenging to maintain a high-accuracy tracking performance of the closed-loop system in the case of step reference trajectories. Inherited from [19], a modified nonlinear filter-based trajectory planner for online trajectory planning is utilized to transform the original trajectory into an appropriate one by considering the system constraints, whose structure is depicted in Figure 3.  
Therefore, the trajectory planner guarantees its output, i.e., newly reference trajectory, is smooth, tracks the original reference trajectory in minimum time without overshoot, and satisfies the prescribed kinematic constraints, i.e., lower bound and upper bound of velocity and acceleration, depending on the system specifications. The mathematical representation of this planner is as follows:The integrator chains are mathematically represented as  
\[\left[ \begin{array}{c}x_{1d}(k+1)\\ \dot{x}_{1d}(k+1)\end{array} \right]=\left[ \begin{array}{cc}1&T\\ 0&1\end{array} \right]\left[ \begin{array}{c}x_{1d}(k)\\ \dot{x}_{1d}(k)\end{array} \right]+\left[ \begin{array}{c}T^{2}/2\\ T\end{array} \right]u(k) \tag{31}\]  
where the pair \((x_{1d}(k),\dot{x}_{1d}(k))\) represents the state vector, \(T\) is the system sampling time, \(u(k)=\dot{x}_{1d}(k)\) is the control action of the integrator chain, and \(k\) denotes the sample number, i.e., \(t(k)=kT\). Compared to [19], the control law \(u(k)\) is redesigned as  
\[u(k)=\left\{ \begin{array}{c}\ddot{x}_{1d}^{-}\text{tanh}(C_{tp}\sigma(k)), \text{ if }\sigma(k)\geq 0\\ -\dot{x}_{1d}^{+}\text{tanh}(C_{tp}\sigma(k)),\text{ otherwise}\end{array} \right. \tag{32}\]  
where \(C_{tp}\) is a positive constant, \(\sigma(k)=\dot{z}(k)-\dot{z}(k)\) with \(\dot{z}(k)\) and \(\dot{z}(k)\) are determined via following equations  
\[\dot{z}^{+} =-\frac{\dot{x}_{1d}^{+}-\dot{x}_{r}(k)}{Tx_{1d}^{-}};z^{+}=\left. \left[z^{+}\right]\left[z^{+}-\frac{\left[\dot{z}^{+}\right]-1}{2}\right]\] \[\dot{z}^{-} =-\frac{\dot{x}_{1d}^{-}-\dot{x}_{r}(k)}{Tx_{1d}^{+}};z^{+}=\left. \left[-z^{-}\right]\left[-\dot{z}^{-}-\frac{\left[-\dot{z}^{-}\right]-1}{2}\right]\] \[\left[\alpha,\beta\right] =\left\{ \begin{array}{c}\left[\dot{x}_{1d}^{+}\dot{x}_{1d}^{-} \right],\text{ if }\frac{y(k)}{T}+\frac{y(k)}{2}>0\\ \left[\dot{x}_{1d}^{-}\dot{x}_{1d}^{+}\right],\text{ otherwise}\end{array} \right.\] \[z(k) =\frac{1}{Ta}\left|\frac{y(k)}{T}+\frac{\dot{y}(k)}{2}\right|\] \[\gamma(k) =\left\{ \begin{array}{c}z^{+},\text{ if }z(k)<z^{+}\\ z(k),\text{ if }z^{+}\leq z(k)\leq z^{-}\\ z^{-},\text{ if }z(k)>z^{-}\end{array} \right. \tag{33}\] \[m(k) =\left[\frac{1+\sqrt{1+8|\gamma(k)|}}{2}\right]\] \[\dot{z}(k) =-\frac{\gamma(k)}{m(k)}-\frac{m(k)-1}{2}\text{sign}(\gamma(k))\] \[z(k) =\left\{ \begin{array}{c}\frac{\dot{y}(k)}{T|\alpha|},\text{ if }\left[\left(z(k)\geq 0\delta\frac{\dot{y}(k)}{T|\alpha|}\leq\dot{z}(k) \right)\text{ or }\left(z_{k}<0\delta\frac{\dot{y}(k)}{T|\alpha|}\right)\right]\\ \frac{\dot{y}(k)}{T|\beta|}+\left(\frac{m(k)-1}{2}+\frac{|\gamma(k)|}{m(k)} \right)\frac{\alpha+\beta}{|\beta|},\text{otherwise}\end{array} \right.\]  
where \(\ddot{x}_{1d}^{-},\dot{x}_{1d}^{-}\), \(\ddot{x}_{1d}^{+}\), and \(\dot{x}_{1d}^{+}\) are the lower bounds and upper bounds of the acceleration and velocity so-called kinematic constraints of the system, i.e., \(\dot{x}_{1d}^{-}\leq\dot{x}_{1d}\leq\dot{x}_{1d}^{+}\) and \(\dot{x}_{1d}^{-}\leq\dot{x}_{1d}^{-}\). Figure 3: The structure of the nonlinear filter-based trajectory planner.