## 4 Control-Law Design
### Design of Active Disturbance Rejection Controller
#### 4.2.3 Fourth-Order LADRC-SMC Comparison Group with or without SP Loop  
Delays, including control and sensor delays, are inevitable in the design of robotic controllers. When the sensor delays are significant, they can potentially lead to controller instability (as shown in Figures 12 and 14 in Section 5.3). If we define the total system delay as \(D\), control delay as \(D^{c}\), and sensor delay as \(D^{s}\), then \(D=D^{c}+D^{s}\). However, increasing this assumed delay did not yield satisfactory results after addressing the tracking issues of the joint angles and contact forces under disturbances using the previously designed LADRC-SMC. There are various approaches to address this issue, and we choose a simple and classical solution, SP [34, 35], which forms a multi-loop control strategy (see Figure 4). This strategy decomposes the single-leg control system of a robot into multiple independent control loops, each of which is responsible for controlling specific variables or subsystems within the system. Finally, mathematical simulations verified that this strategy could enhance the system's control performance, stability, and robustness. It allows for flexible adaptation to system variations and faults, improving control effectiveness (as shown in Figures 12-16 in Section 5.3). interference and delay, the engagement of the switching mechanism with CD signifies the execution of the LADRC-SMC-SP scheme. In comparison, its connection to CE denotes the implementation of the LADRC-SMC scheme. Regarding Figure 4, the closed-loop transfer function \(G(S)\) of the LADRC-SMC-SP scheme can be ascertained in accordance with the SP principle:  
\[G(S)=\frac{H_{\text{c}}(\text{s})S(\text{s})\text{e}^{-D^{\text{c}}}}{1+H_{ \text{c}}(\text{s})S^{\prime}(\text{s})+H_{\text{c}}(\text{s})\Big{(}S(\text{ s})\text{e}^{-D^{S}}-S^{\prime}(\text{s})\text{e}^{-D^{S}}\Big{)}}, \tag{29}\]  
Here, \(H_{\text{c}}(\text{s})\) represents the transfer function of the residual portion in the LADRC-SMC-SP scheme, which does not include the sensor, whereas \(S(\text{s})\) represents the transfer function of the sensor in the signal feedback loop. \(D^{S}\) is the actual delay of the sensor. Pressure sensors with second-order system dynamics were used for the robot joint-angle and contact-force control. In the presented theoretical and applied research presented, \(S^{\prime}(\text{s})\text{e}^{-D^{S}}\) is the introduced predictive compensation transfer function. The sensor transfer function, denoted as \(S(\text{s})\), represents a second-order system. When there is no model mismatch, we can get \(S^{\prime}(\text{s})=S(\text{s}),D^{\prime S}=D^{S}\). The \(D^{\text{c}}\) can be eliminated by controller design and parameter adjustment. \(G(S)\) in the LADRC-SMC-SP can be simplified as follows:  
\[\left\{ \begin{array}{l}G(S)=H_{\text{c}}(\text{s})S(\text{s})/(1+H_{\text{ c}}(\text{s})S(\text{s}))\\ S(\text{s})=N(\text{s})/D(\text{s})=\epsilon_{1}/\left(c_{2}S^{2}+c_{3}s+c_{4} \right)\end{array} \right., \tag{30}\]  
where the parameters \(\sum\limits_{i=1}^{4}c_{i}=[5,1,3,5]\) (given randomly, with substitutability) are the coefficients of the sensor transfer function. In Equation (30), the closed-loop system does not include a lag term, and the lag component does not affect the characteristic equation of the system. Compared to systems without the SP, the system compensated by the SP eliminates the \(D^{\text{s}}\) during the control process. The control block diagram of LADRC-SMC-SP scheme finally proposed is shown in Figure 5.  
Equation (31) elucidates the transition mechanism on the third set of comparative strategies--namely, the LADRC-SMC and LADRC-SMC-SP schemes--as depicted in Figure 4. Signal \(u_{ALADRC}\) from the LADRC loop is the input signal of the SMC loop in the LADRC-SMC and LADRC-SMC-SP schemes. The output signal \(\theta_{1}\), of the delay loop or predictor loop is the input signal of the LESO in the LADRC loop.