## 4 Control-Law Design
### Design of Active Disturbance Rejection Controller
#### 4.2.3 Fourth-Order LADRC-SMC Comparison Group with or without SP Loop
\(\hat{\theta}_{i}\) is the estimated value of the LESO in the LADRC loop, and \(\theta_{1d}\) is the reference input of the two schemes. The new switchable scheme (as shown in Figure 4) based on the LADRC-SMC scheme (as shown in Figure 3) is formed by combining Equations (18), (20), (27) and (31). This controller used the signal (\(u_{4 LADRC}\)) from the LADRC loop as the reference input (\(\hat{\theta}_{1d}\)) for the SMC loop. The signal (\(u_{4 LADRC}\)) from the LADRC loop can be expressed using Equation (31):  
Figure 5: The control block diagram of the LADRC-SMC-SP scheme. \[\hat{\theta}_{1d}^{*}=u_{4 LADRC}=\left\{ \begin{array}{l}-\frac{1}{b_{0}}.\left( \beta_{\omega_{c}}(1)\big{(}\hat{\theta}_{1\_[PERSON]}-\theta_{1d}\big{)}+\sum \limits_{i=2}^{5}\beta_{\omega_{c}}(i)\hat{\theta}_{i\_[PERSON]}\right),\text{ Switch to }CD\\ -\frac{1}{b_{0}}.\left(\beta_{\omega_{c}}(1)\big{(}\hat{\theta}_{1\_delay}- \theta_{1d}\big{)}+\sum\limits_{i=2}^{5}\beta_{\omega_{c}}(i)\hat{\theta}_{i \_delay}\right),\text{ Switch to }CE\end{array} \right., \tag{31}\]  
According to Equation (23), \(\hat{\theta}_{1\_[PERSON]}\) is the observed value of \(\theta_{i\_[PERSON]}\) of the LADRC-SMC-SP scheme, and \(\hat{\theta}_{i\_[PERSON]}\) is the \((i-1)\) order derivative of the observed value of \(\hat{\theta}_{1\_[PERSON]}\) of the LADRC-SMC-SP scheme. \(\hat{\theta}_{1\_delay}\) is the observed value of \(\theta_{i\_delay}\) in the LADRC-SMC scheme, and \(\hat{\theta}_{i\_delay}\) is the \((i-1)\) order derivative of the observed value of \(\hat{\theta}_{1\_delay}\) in the LADRC-SMC scheme. According to Equations (28) and (30), the system order of the LADRC-SMC-SP scheme is equivalent to the Fourth order. This multi-loop control strategy optimizes each control loop, thereby enhancing the system's performance and stability. It is worth noting that the LADRC-SMC-SP scheme expands the solution to the sensor delay problem of the second-order LADRC-SMC single leg. The LADRC-SMC-SP scheme also realizes the challenges of weak single-foot contact-force tracking, oscillation problems in joint-angle tracking, and delay elimination (as shown in Figures 12-16 in Section 5.3). According to Equations (4), (18) and (19), the SMC loop law shown in Figure 4 (joint torque) can be reformulated, as shown in Equation (32):  
\[\tau=\left\{ \begin{array}{l}\tau_{i\_[PERSON]}=&M_{1}(\theta_{1\_[PERSON]})\bar{ \theta}_{i\_[PERSON]}+C_{1}\Big{(}\theta_{1\_[PERSON]},\dot{\theta}_{1\_[PERSON]}\Big{)} \hat{\theta}_{i\_[PERSON]}+G_{1}(\theta_{1\_[PERSON]})\\ +K_{P}r_{11}+J_{\phi}^{T}(\theta_{1\_[PERSON]})f_{\intercal}\\ \tau_{i\_delay}=&M_{1}\Big{(}\theta_{1\_delay}\Big{)}\bar{\theta}_{i\_delay }+C_{1}\Big{(}\theta_{1\_delay},\dot{\theta}_{1\_delay}\Big{)}\dot{\theta}_{ i\_delay}+G_{1}\Big{(}\theta_{1\_delay}\Big{)}\end{array} \right.,i=1,2\,, \tag{32}\]  
where \(\tau_{1\_delay}\) is the torque of joint 1 in the LADRC-SMC scheme. \(\tau_{2\_Smith}\) is the torque of joint 2 in the LADRC-SMC-SP scheme (as shown in Figure 14 in Section 5.3). Therefore, this new scheme, which combines SMC, ADRC, and SP, significantly improves the controllability of the research subject in terms of the disturbance rejection, stability, and robustness. The LADRC-SMC-SP scheme addresses the joint-angle and contact-force tracking deficiencies in the MPC and SMC schemes discussed in this study. In addition, it successfully resolved the sensor delay encountered in the LADRC-SMC scheme. An SMC was designed to compare the tracking performance of the joint angles, and a predictive model control was reconstructed based on the existing literature (refer to the control-law design in Sections 4.1.1 and 4.1.2 and the simulation analysis in Figures 6-8 in Section 5.1). A LADRC-SMC scheme was devised to address the issue of contact-force tracking accuracy (refer to the control-law design in Sections 4.1.2, 4.2.1 and 4.2.2 and the simulation analysis in Figures 9-11 in Section 5.2).