 Original Article
 Open access
 Published:
Active Disturbance Rejection Control of Hydraulic Quadruped Robots Rotary Joints for Improved Impact Resistance
Chinese Journal of Mechanical Engineering volumeÂ 37, ArticleÂ number:Â 103 (2024)
Abstract
Hydraulic actuated quadruped robots have bright application prospects and significant research values in unmanned area investigation, disaster rescue and other scenarios, due to the advantages of high payload and high power to weight ratio. Among these fields, inevitable collision of robots may occur when contact with unknown objects, step on empty objects, or collapse, all of which have an impact on the working hydraulic system. To overcome the unknown external disturbances, this paper proposes an active disturbance rejection control (ADRC) strategy of double vane hydraulic rotary actuators for the hip joints of the quadruped robots. Considering the order of the valvecontrolled actuator model, a threestage tracking differentiator, a fourstage extended state observer, and a state error feedback controller are designed relatively, and the extended state observer is adopted to observe and compensate the uncertainty of external load torque of the system. The effectiveness of the ADRC method is verified in simulation environment and a single joint experimental platform. Moreover, the impact experiments of the limb leg unit are carried out after introducing the proposed ADRC strategy into hip joint, the limb leg unit of quadruped robots presents better impact resistance ability.
1 Introduction
Recently, legged robots have been a hot topic for domestic and foreign researchers to carry out scientific research work, such as imitating the galloping cheetahs for higher motion speed [1], versatile climbing on the ferromagnetic surfaces [2], multimodal locomotion ability [3]. Compared to the wheeled or tracked robots, which rely on continuous contact with the ground, legged robots have the ability to traverse unknown and unstructured terrains easily via dynamically adjusting their leg movements and gaits. Along with the urgent demand for heavy payload and highspeed locomotion, many hydraulic driven quadruped robot prototypes have been developed and applied in payload transportation and uNÂ·manned surveying, including BigDog [4] from Boston Dynamics, HyQ [5] series from the Italian Institutes of Technology, Baby elephant [6] from Shanghai Jiao Tong University.
As the most critical motion unit of quadruped robot, limb leg units play an important role in the locomotion ability of the robots [7]. When working in the complex and rugged terrains [8, 9], the foot ends of limb leg units inevitably contact with the objects in the enviroNÂ·ment, and the expectant control of joint actuators would be affected by the external force transferred from the foot end. In addition, the collision between the foot end and the ground occurs sometimes [10] due to the discontinuous support characteristic although compliance control strategies are employed. If no measures are taken when collision happens, it may affect the position/force tracking performance of the limb leg units, furtherly, resulting in more serious damage to the quadruped robots due to excessive joint output torque. Therefore, it is necessary to estimate and compensate the interference torque in realtime. Furthermore, the joints directly determine the output capacity and the dynamic performance of the legged robots, so optimizing the control method for joints is a key step in improving the impact resistance of the limb leg unit and even quadruped robots.
The control methods of robot joints mainly include adaptive control [11], sliding mode control [12], fuzzy control [13], neural networkbased control [14] and other control strategies. Refs.Â [15,16,17] designed an adaptive sliding mode controller, which combined the advantages of adaptive backstepping control strategy and sliding mode control strategy to enhance the tracking performance of the whole system. ZhangÂ et al. [18] proposed a multimodal robust adaptive control theory, which involved designing multiple identification models based on the range of parameter uncertainty and adding nonlinear robust terms to the model to suppress the influence of uncertainty and nonlinear factors. NaÂ et al. [19, 20] proposed a control method that need not require backstepping schemes and function approximators, as well as proposing another simple and effective estimation method to handle unknown dynamics and external disturbances in robot system motion control. However, the aforementioned control strategies are often burdened with complicated control laws that proved difficult to finetune. On the other hand, the computational complexity associated with these approaches can hinder their applicability in industrial settings. To address these concerns, Han [21, 22] proposed the ADRC algorithm based on indepth analysis of PID and presented that ADRC was capable of replacing PID strategy with obvious advantage in performance and providing solutions for external disturbances.
The core concept of ADRC strategy revolves around the assimilation of both internal and external uncertainties into a unified disturbance, which is then actively eliminated. This approach stands as a modelfree strategy, sparing it from the need for precise knowledge of an accurate parameters and dynamics. To assess the total disturbance, ADRC employs an extended state observer (ESO), the gain selection of which would largely determine the performance of the ADRC [23,24,25]. ADRC has been effectively applied to hydraulic quadruped robots. FanÂ et al. [26] utilized ADRC to improve the control performance and stability of the hydraulic quadruped robot and presented that ADRC controller can keep the robustness, security, and invariability under the indeterminacy dynamic enviroNÂ·ment, which can greatly meet the requirement of the hydraulic control system. GuoÂ et al. [27] used ADRC in valvecontrolled cylinder servo system and demonstrated that ADRC can effectively suppressed the internal parameter changes and external parameter changes of the hydraulic system, and had potential in electrohydraulic systems for highperformance control. Notably, most cases concentrate on position control under unknown disturbances, and the force/torque control of actuators with ADRC has not been reported, especially the hydraulic rotary actuator. Besides, the applications of ADRC in hydraulic quadruped robots require more experimental verification, not just the simulations.
In this paper, a threestage active disturbance rejection controller is designed to the torque control of a valvecontrolled hydraulic rotary actuator, which is to drive the hip joint of the quadruped robots. The input signal is transited via the differential tracker, and a fourstage ESO is expanded to observe the changes of disturbances. Finally, the state error feedback control law is adopted to compensate the error. Simulations are carried out to prove the proposed ADRC strategy is stable and correct. Experiments with the hydraulic actuator are conducted and their results show that the ADRC method has superior performance than that of general PID strategy in overcoming the complicate disturbances. Further experiments are performed in the limb leg unit to show that ADRC can significantly reduce the load torque when the collision occurs.
The main contribution of this paper is to develop a threestage active disturbance rejection control strategy for a valvecontrolled hydraulic rotary actuator and apply it to the hip joint of a hydraulic quadruped robot. This paper is constructed as follows. Section 2 gives the modeling process of the valvecontrolled double vane hydraulic rotary actuator. Section 3 describes the design implementations of ADRC controller consisting of threestage tracking differentiator, fourstage extended state observer, and state error feedback control law. Simulations in the stationary and moving states for hydraulic rotary actuator are shown in Section 4. Then, the experimental results in the level of both hip joint and limb leg unit are expressed in Section 5, followed by the conclusion in Section 6.
2 Mathematical Model of Valvecontrolled Hydraulic Rotary Actuator
2.1 System Overview
Spurlos II, a hydraulic actuated quadruped robot shown in Figure 1, is designed to help humans perform highly challenging tasks such as disaster relief, battlefield reconnaissance, material transportation, and environmental detection on rugged terrain. As the most critical motion unit of quadruped robot, the twodegreeoffreedom limb leg unit depicted in Figure 1, consists of hip and knee joints. The hip joint is driven directly by the double vane rotary actuator, the weight of which is 4.3Â kg. The knee joint is actuated by the embedded linear actuator, and the rotational motion is transferred through a centric slidercrank mechanism. The joints are driven by electrohydraulic servo system, consisting of two servo valves, four pressure sensors, two angle encoders, and so on. The limb leg unit weights about 7.8Â kg while the shank segment weights about 2.3Â kg.
2.2 Mathematical Model
The double vane rotary actuator in hip joint is a key component for the antiinterference control of limb leg unit, whose performance determines the buffering ability of the limb leg unit to various terrains. This paper concentrates on the torque control of the hip joint to realize better impact resistance ability of Spurlos II.
The composition of the valve controlled doublevane rotary cylinder system is shown in Figure 2. The servo valve is the control component, and the swing cylinder is the execution component. According to the oil compressibility and flow continuity, and ignoring internal external leakage [28], the linearization equation of the servo valve flow rate can be written as:
where \(K_{q}\) is the flow gain of the servo valve, \(K_{c}\) is the amplification coefficient of the flow and pressure of the servo valve, \(X_{V}\) is the displacement of the servo valve spool, and \(P_{L}\) is the load pressure.
Due to the fact that the frequency bandwidth of the servo valve spool is much greater than that of the double vane hydraulic rotary actuator, the servo valve is simplified as a proportional link and expressed by the transfer function:
where \(I_{sv}\) is the input current of the servo valve, \(K_{sv}\) is the noload flow gain of the servo valve, and \(K_{v}\) is the controller gain.
The torque sensor and servo amplifier are represented by the following formula:
where \(K_{f}\) is the conversion coefficient of the torque sensor, \(T\) is the torque value feedback, \(I\) is the output current of the torque sensor, and \(K_{a}\) is the amplifier gain.
The continuity equation for the flow rate of the doublevane rotary cylinder is as follows:
where \(D_{m}\) is the theoretical displacement of the doublevane rotary cylinder, \(\theta\) is the output angle, \(C_{tm}\) is the leakage coefficient, \(V_{m}\) is the total volume including the connecting oil pipe, and \(\beta_{e}\) is the elastic modulus of the effective volume.
The load torque balance equation is as follows:
where \(J_{l}\) is the moment of inertia of the load torque, \(B_{m}\) is the viscous damping coefficient, \(G_{m}\) is the torsional stiffness of the connecting shaft, and \(T_{l}\) is unknown external interference.
The output torque expression of the valvecontrolled double vane hydraulic rotary actuator system can be obtained from Eqs. (1â€“6) and is shown as:
where \(K_{ce} = K_{c} + C_{tm}\).
Consider \(x_{1} (t)\) as the output torque of the doublevane rotary cylinder, \(x_{2} (t)\) as the rate of change of \(x_{1} (t)\), and \(x_{3} (t)\) as the derivative of \(x_{2} (t)\). According to Eq. (7), a threestage dynamic system can be obtained as shown in Eq. (8), and the control block diagram of the valvecontrolled hydraulic rotary actuator is shown in Figure 3.
where \(a_{0} ,a_{1} ,a_{2} ,b\) is the system parameter, which determined via parameters tuning analysis. \(y(t)\) is the system output, and \(d(t)\) is the interference term.
3 Active Disturbance Rejection Control Strategy Design
For the ADRC strategy, the significance of the controller order cannot be overstated. An improper order selection of controller can directly influence the handling of internal disturbances within the overarching control framework, resulting in the burden on the state observer. Based on the mathematical model of the valvecontrolled rotary actuator established in Section 2, traditional twostage ADRC method needs to be further upgraded, and a threestage form of ADRC needs to be designed.
As shown in Figure 4, the threestage ADRC comprises three main sections. The tracking differentiator (Part A) processes the reference input signal, providing smoother and more accurate derivatives of the input. The Extended State Observer (ESO) (Part B) is designed to estimate the systemâ€™s state variables and the total disturbance, which includes both internal dynamics and external disturbances acting on the system. Finally, the linear state error feedback (Part C) generates the control input that drives the system towards the desired state, based on the estimated states and disturbances from the ESO.
3.1 Design of ThreeStage Tracking Differentiator
The function of tracking differentiator is to arrange the transition process, reduce the jump of input signal, give a reasonable control signal, and solve the contradiction between response speed and overshoot. Considering the order of the controlled object is three, and accordingly, a third order tracking differentiator needs to be designed by connecting two tracking differentiators in series shown in Figure 5.
The tracking differentiator unit is designed as:
where h is the simulation step size, which is related to the sampling frequency of the system. \(v_{0} (t)\) is the system input, \(v_{1} (t)\) and \(v_{2} (t)\) are the system output derived from tracking differentiator. h_{0} is the filtering factor, which can be equal to or slightly greater than h and reduce the random noise interference. r is the speed factor, which determines the speed of tracking the expected torque signal.
\({\text{fh}}\) is a control synthesis function and is implemented as:
3.2 Design of FourStage Extended State Observer
The ESO is adopted to observe the expanded state variables, estimate the unknown disturbance and the unmodeled part of the control object, realizing the feedback linearization of the dynamic system.
The extended state observer of the system is established by tracking and estimating the system state and interference through the input and output of the system:
where \(a_{0}\), \(a_{1}\), \(a_{2}\), \(b\) are the system parameters, \(y(t)\) are the system output, \(d(t)\) is the torque interference term.
Eq. (11) can be rewritten as
The realtime effect of \(f(x_{1} (t),x_{2} (t),x_{3} (t),d(t))\) on the system is extended to a new state \(x_{4}\), which is expressed as \(x_{4} (t) = f(x_{1} (t),x_{2} (t),x_{3} (t),d(t))\) and \(\dot{x}_{4} (t) = w(t)\), and the thirdstage nonlinear system is further arranged as
Finally, the linear state observer of the extended system is established as follows and shown in Figure 6:
where e is the torque output error, z_{1} is the estimated value of y, z_{2} is the estimated value of the derivative of y, z_{3} is the estimated value of the second derivative of y, z_{4} is the estimated value of the nonlinear term and external disturbance of the model \(\beta_{1} ,\beta_{2} ,\beta_{3} ,\beta_{4}\) and b_{0} are gains.
3.3 Design of State Error Feedback Control Law
The error feedback controller provides the control strategy for the controlled object and compensates for system disturbances through state error feedback. In this system, the design of the state error feedback controller is as follows:
where x_{i} (i=1, 2, 3) is the output value of the tracking differentiator, z_{i} (i=1, 2, 3) is the output value of the extended state observer, \(\alpha_{1i} (i = 1,2,3)\) is the gain coefficient of the controller, and \(u_{0}\) is the control quantity of the error feedback. Considering the external interference of the system, the final output obtained by the linear ADRC controller through compensation is:
4 Simulation Studies
To verify the effectiveness of the ADRC strategy, a simulation model is built through the SIMULINK module of MATLAB, and the parameter settings are shown in Table 1. Three working conditions are defined based on the steady states and impact torque value, and the maximum error of output torque after interference, the average error of output torque after interference, and the time to recover stability after interference are adopted as the quantitative performance indicators. Another three working conditions in dynamic state are carried out, and the maximum error and average error of output torque serve as the quantitative performance indicators. The antiinterference effects of ADRC controller and PID controller under different working conditions are compared, further demonstrating the antiinterference ability and adaptability of ADRC controller.
The simulation results under steady state were depicted in Figure 7. In Condition 1, a 65.0 NÂ·m impact torque was applied to the valvecontrolled doublevane rotary actuator system. After been disturbed, the maximum error of the actuator output torque with ADRC controller was 18.1 NÂ·m, much less than 60.8 NÂ·m with PID controller. The average errors of the PID controller during the disturbance period was 17.8 NÂ·m, almost three times than that of the ADRC controller. The maximum adjustment time of the ADRC controller was 0.14 s, while the PID controller was 0.21 s. Compared to PID controller, the maximum error of ADRC controller was reduced by 70.2%, the average error was reduced by 69.1%, and the maximum adjustment time was reduced by 33.3%.
When the interference torque increased to 160.0 NÂ·m in Condition 2, the maximum errors of the system output torque under the control of PID and ADRC controllers were 156.4 NÂ·m and 51.8 NÂ·m, with an average error of 25.1 NÂ·m and 12.0 NÂ·m. In addition, the maximum adjustment times were 0.23 s and 0.18 s, respectively. Compared with PID controller, the maximum error, the average error, and the maximum adjustment time of ADRC controller reduced by 66.9%, 52.2%, and 21.7%, respectively.
In Condition 3, the impact torque increased to 310.0 NÂ·m, and better torque performance ability was observed in the ADRC controller. Compared with PID controller, the ADRC controller brought about significant performance improvement: 66.1% reduction in the maximum error of the system output, 45.4% reduction in the average error, 44.4% reduction in the maximum adjustment time.
The simulation results under dynamic state were depicted in Figure 8. In working Condition 4, the actuator was in motion, and a 45.0 NÂ·m interference torque was imposed. The maximum errors of the system output torque under the control of ADRC controller and PID controller were 18.9 NÂ·m and 43.5 NÂ·m, respectively, representing a maximum error reduction of 56.6%. The average errors during the disturbance period were 4.2 NÂ·m and 8.5 NÂ·m, with a reduction of 50.6%. On the basis of Condition 4, Condition 5 and Condition 6 increased the impact torque to 75.0 NÂ·m and 95.0 NÂ·m. In condition 5, Compared with PID controller, the ADRC controller reduced the maximum error by 56.2% and the average error by 54.1%. The maximum error in Condition 6 decreased by 56.3%, and the average error decreased by 57.7%.
In summary, whatever states the valvecontrolled double vane hydraulic rotary actuator system were in, the proposed ADRC controller all showed significant improvements in interference resistance.
5 Experimental Studies
To verify the improvement of the antiinterference ability of the controlled object after applying the ADRC controller, experiments were conducted on both the single joint experimental platform and the limb leg unit experimental platform.
5.1 Test Results in the Hip Joint
The framework of the single joint experimental platform system is shown in Figure 9, which consists of a double vane hydraulic rotary cylinder, a servo valve, the CytroPac hydraulic station, a torque sensor, etc. A dynamic torque sensor is used to detect the output torque of the actuator and provide feedback to the ADRC controller to control the opening of the servo valve. The impact experiments were conducted on the single joint experimental platform through sudden loading, and the working conditions were divided by changing the weight of the load and the height of free fall 1.
5.1.1 Steady State Performance
The interference experiments of the double vane rotary cylinder in a stationary state can be divided into three working conditions: Condition 1 (1.0 kg, 0.5 m), Condition 2 (2.5 kg, 0.5 m), Condition 3 (5.0 kg, 0.5 m), shown in Figure 10. In Condition 1, 1.0 kg of load was released at a height of 0.5 m. According to the experimental curve in Figure 11, the maximum output torque error of the system controlled by the PID controller after interference was 57.4 NÂ·m, and the average error during the disturbance period was 15.9 NÂ·m with a maximum adjustment time of 0.25 s. The maximum output torque error of the system under the control of ADRC controller after interference was 18.2 NÂ·m, representing a reduction of 68.3%. The average error during the disturbance period was 5.2 NÂ·m, with a maximum adjustment time of 0.19 s.
Table 2 showed the performance comparison in three different conditions. We can see that the ADRC method had a good antiinterference effect, and brought about the maximum torque error reduction of 68.9% in Condition 2 and 69.4% in Condition 3. In addition, the ADRC method contributed to smaller average error, 9.7 NÂ·m in Condition 2 and 23.7 NÂ·m in Condition 3, representing a reduction of 56.1% and 51.65%. Compared with PID method, the proposed ADRC can shorten the adjustment time by 38.1% in Condition 2 and 31.6% in Condition 3.
5.1.2 Dynamic Performance
When the double vane rotary actuator was in dynamic state that the torque changed from 0â€“20 N. The interference torque was generated by free fall of an object, and three working conditions were given: Condition 4 (1.0 kg, 0.2 m), Condition 5 (1.0 kg, 0.5 m), Condition 6 (2.0 kg, 0.2 m). The experiment results were depicted in Figure 12. In Condition 4, the maximum output torque in interference direction of the system with ADRC method was âˆ’8.5 NÂ·m, contrast toÂ â€“21.7 NÂ·m with PID controller. The maximum output torque error of the PID and ADRC method were 33.4 NÂ·m, 23.9 NÂ·m, respectively. The average error during the disturbance period reduced by 16.9% from 8.9 NÂ·m to 7.4 NÂ·m by changing the PID with ADRC, shown in Table 3.
The same significant performance improvements were observed in Condition 5 and 6. By adopting the ADRC method, the maximum torque in interference direction, the maximum output torque error, and the average error were reduced by 47.7%, 31.3%, and 21.7% (Condition 5); 55.7%, 30.4%, and 21.6% (Condition 6). Consequently, the feasibility of the proposed ADRC strategy was verified from the aspects of maximum output torque in the interference direction, maximum tracking error, and average tracking error.
5.2 Test Results in the Hydraulic Quadruped Robot
After verifying the effectiveness of the ADRC controller at the joint level, it is necessary to verify its effectiveness at the limb leg unit level. An impact experiment based on the limb leg unit is designed in this paper, as shown in Figure 13. The limb leg unit is lifted by the vertical degree of freedom of the limb leg unit experimental platform, and then lifted to a certain height before releasing it at a certain time. The pressure changes in the high and low oil chambers of the joints are measured by installed pressure sensors, and the torque change of the hip joint after the foot touches the ground is recorded. The hydraulic system pressure was set at 5 MPa, the expected torque is 0 NÂ·m, and the foot lifting heights are 0.05 m, 0.10 m, 0.15 m, and 0.20 m, respectively.
As shown inÂ Table 4 and Figure 14,Â when the foot fell 0.05 m off the ground and impacted the hip joint, the difference in the maximum output torque of the system under the control of the two controllers was small, representing 41.5 NÂ·m and 36.2 NÂ·m, respectively. However, the torque fluctuation of the ADRC was relatively minor, and the system restored to the expected torque faster, taking only 0.25 s. Similar results were observed when the foot is 0.10 m off the ground. At a drop height of 0.15 m, the maximum output torque of the hip joint reached 90.0 NÂ·m with PID control, whereas it was 55.3 NÂ·m with the ADRC controller, reducing by 38.6%. When the dropping height reached to 0.20 m, the ADRCcontrolled system exhibited a smaller maximum output torque than the PID control group, demonstrating a more significant effect in resisting external disturbances. The joint maximum output torque reduced from 113.3 NÂ·m to 84.2 NÂ·m, highlighting the capability of the ADRC controller in observing and compensating for interference. Notably, as the drop height increased, the impact force approached the hip actuatorâ€™s maximum output torque, and the adjustable times for both controller were almost the same.
It can be concluded that the maximum torque output of the ADRC control group after being impacted was smaller, which can provide better protection for the hip joint. Moreover, the average torque output before stabilization was diminished, accompanied by a quicker reestablishment of stability postfluctuation in output torque. These observations collectively substantiate the validity of the proposed ADRC controller in strengthening the hip joint against interferences.
6 Conclusions
This paper focused on the torque control of a valvecontrolled double vane hydraulic rotary actuator system, and a three stage ADRC strategy was proposed. The effectiveness of the ADRC method were then verified by simulations, single actuator and limb leg unit experiments. The main conclusions were summarized as follows.

1)
A mathematical model of the doublevane rotary actuator was established, which determines the order of the ADRC controller. According to the separation principle for each part, the model of hydraulic rotary actuator and the ADRC controller were built.

2)
Significant antiinterference ability was observed in both the simulation and the actuator interference test. Moreover, the effectiveness was verified in the steady and dynamic state. Compared with the PID method, the proposed ADRC method presented smaller torque tracking error and shorter recovery time.

3)
The proposed ADRC controller showed a significant improvement in the antiinterference ability of the system at the joint level and limb leg unit level compared to the PID controller. Furtherly, it can improve the locomotion ability of quadruped robots.
Future work will focus on enhancing the performance of the ADRC method, such as adding leakage compensation for instance, introducing nonlinear control law. Furthermore, realizing the application of ADRC to the quadruped robot Spurlos II is worthy of further research.
Data availability
The data are available from the corresponding author on reasonable request.
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Acknowledgements
The authors sincerely thank the Doctoral candidate Kun Zhang, Master candidate Lizhou Fang, for the assistance in processing and experimentation.
Funding
Supported by National Natural Science Foundation of China (Grant No. U21A20124), China Postdoctoral Science Foundation (Grant Nos. 2023T160570, 2022M722737) and the Key Research and Development Program of Zhejiang Province of ChinaÂ (Grant No. 2022C01039).
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Contributions
Conceptualization: HZZ and JHZ; methodology: HZZ, ZXY, and XY; resources: HZZ, and ZYX; validation: HZZ and ZXY; writingâ€”original draft: HZZ and ZXY; writingâ€”review & editing: HZZ, XY, JKA, and FW; supervision: JHZ, FW, BX, and QS; funding acquisition: QXZ, FW, and BX. All authors have read and agreed to the published version of the manuscript.
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Data available on request from the authors.
Appendices
Appendix
The detailed proof of fourstage ESO is given. Firstly, according to the stability proof theorem, if Theorem 1 is satisfied, then the ESO is proven to be stable.
Theorem 1:

(1)
For each normal number a, \(\mathop {\lim }\limits_{\varepsilon \to 0} \left {x_{i} \left( t \right)  z_{i} \left( t \right)} \right\, = \,0\) is always valid, when \(t \in [a,\infty )\);

(2)
\(\mathop {\lim }\limits_{t \to \infty } \left {x_{i} (t)  z_{i} (t)} \right \le O(\varepsilon^{n + 2  i} )\)
where \(x_{i}\), \(z_{i}\) are the solutions of Eq. (13) and the ESO, respectively, and \(i = 1,2,...,n + 1\), \(x_{n + 1} = f + w\) represents the expanded state of system (13).
Proof of Theorem 1:
Based on the designed ESO and the defined error, it is known that
Simultaneously, introducing the scaled estimation error,
Given that the system is thirdorder with parameters, and \(\varvec{\eta} = [\eta_{1} ,\eta_{2} ,\eta_{3} ,\eta_{4} ]^{{\text{T}}}\), Then the differential equation for the system is
To prove the convergence of the ESO, the following two assumptions are further made:
Assumption 1: The solutions of system (13) and the ESO are continuously differentiable with respect to their variables, and
where \(B\,{ > }\,{0}\) and \(t \ge 0\).
Assumption 2: There exist constants \(\lambda_{i} (i = 1,2,3,4)\), \(\alpha\), \(\beta\), a positive definite continuously differentiable function V, and W: \({\mathbb{R}}^{n + 1} \to {\mathbb{R}}\), the following equations also hold true:
where \(y = (y_{1} ,y_{2} ,...,y_{n + 1} )\), \(\ \cdot \\) denotes the Euclidean norm.
From the above assumptions, it can be inferred that once the designed function is selected and determined, it is necessary to prove the existence of a suitable Lyapunov function V whose derivative along the system satisfies the convergence properties. Since the specific form is not given in this paper, the general form of the Lyapunov function cannot be provided. According to Assumption 2, it can be derived that
Based on the relationship between \(\sqrt {V(\eta (t))}\) and \(V(\eta (t))\), it further follows that
Again, according to Assumption 2, it holds that
Finally, based on the relationship between \(\eta (t)\) and \(e(t)\), it is obtained that
Given that \(\varepsilon = 1\), it is evident that \(t \in \left[ {a,\infty } \right)\), \(e_{i} (t) \to 0\). The two conclusions of Theorem 1 can be derived from the above equations. That completes the proof of Theorem 1.
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Zong, H., Yang, Z., Yu, X. et al. Active Disturbance Rejection Control of Hydraulic Quadruped Robots Rotary Joints for Improved Impact Resistance. Chin. J. Mech. Eng. 37, 103 (2024). https://doi.org/10.1186/s10033024010919
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DOI: https://doi.org/10.1186/s10033024010919