Hydraulic arm position sliding mode control method based on expansion state observer

Abstract

The invention discloses a hydraulic arm position sliding mode control method based on an expansion state observer, which belongs to the technical field of electro-hydraulic position servo control, and comprises the following steps: step 1, establishing a state space model of a hydraulic mechanical arm joint position system; step 2, establishing a mathematical model of the extended state observer; 3, designing an improved sliding-mode active disturbance rejection controller based on an extended state observer, and replacing linear feedback with an improved sliding-mode control law while performing real-time observation and compensation on the comprehensive disturbance quantity of a joint position system of the hydraulic mechanical arm; and 4, carrying out stability proving on the improved sliding-mode active-disturbance-rejection controller by using a Lyapunov stability theory. According to the method, the disturbance quantity of the joint position system of the hydraulic mechanical arm is observed and compensated in real time, meanwhile, the improved sliding mode control law is adopted to replace linear feedback, the response lag phenomenon can be improved, and the control precision and robustness of the joint position system of the hydraulic mechanical arm are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of hydraulic mechanical arm position control, and in particular to a hydraulic arm position sliding mode control method based on an extended state observer. Background Art

[0002] As an advanced industrial automation solution, robotic arms have broad application prospects in various industrial production environments. Based on their drive type, they can be broadly categorized as motor-driven, hydraulic-driven, and pneumatic-driven. Hydraulic robotic arms have a relatively simple structure, and their core components, such as hydraulic cylinders and hydraulic pumps, are ruggedly designed and easy to maintain. Furthermore, the hydraulic system can accommodate long periods of continuous operation, reducing downtime due to equipment failures, ensuring the continuity of mining operations, and improving production efficiency. Based on these advantages, hydraulically driven robotic arms have become the preferred equipment for many harsh and dangerous industrial scenarios, such as mining projects.

[0003] Hydraulic manipulators often experience ample power but struggle to precisely control them in conditions requiring highly precise operations. In practice, the control performance of these devices is often limited by the operator's reaction speed and proficiency. A model that relies heavily on manual operation clearly cannot meet the demands of refined, intelligent mining operations. In this context, achieving fully automated control, autonomous remote control, and intelligent control of hydraulic manipulators requires more sophisticated position control technologies.

[0004] As a high-load and high-inertia mechanical equipment driven by a hydraulic system, the hydraulic robotic arm, coupled with the nonlinear characteristics of the electro-hydraulic system, will experience problems such as vibration, lag, and high energy consumption during operation. Therefore, in order to achieve automated and intelligent operation of the hydraulic arm, further requirements are put forward for its control strategy.

[0005] Therefore, studying the position control strategy of hydraulically driven robotic arms is of great significance for achieving the goal of intelligent operation. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to provide a hydraulic arm position sliding mode control method based on an extended state observer. While real-time observation and compensation of the disturbance of the hydraulic manipulator arm joint position system are carried out, an improved sliding mode control law is used instead of linear feedback, which can improve the response lag phenomenon and improve the control accuracy and robustness of the hydraulic manipulator arm joint position system.

[0007] In order to solve the above technical problems, the technical solution adopted by the present invention is:

[0008] A sliding mode control method for hydraulic arm position based on an extended state observer comprises the following steps:

[0009] Step 1: Establish a state space model of the hydraulic manipulator joint position system;

[0010] Step 2, establish a mathematical model of the extended state observer;

[0011] Step 3: Design an improved sliding mode active disturbance rejection controller based on the extended state observer. While observing and compensating the comprehensive disturbance of the hydraulic manipulator joint position system in real time, an improved sliding mode control law is used instead of linear feedback.

[0012] Step 4: Use Lyapunov stability theory to prove the stability of the improved sliding mode active disturbance rejection controller.

[0013] The further improvement of the technical solution of the present invention is that: in step 1, specifically including: the hydraulic mechanical arm joint position system is composed of a valve-controlled hydraulic cylinder, the load flow of the valve is about the valve core displacement x v and load voltage drop p L The function of load flow increment is obtained as formula (1):

[0014]

[0015] In formula (1), K q is the flow gain; K c is the flow-pressure coefficient; q L is the average flow rate of the two working ports of the proportional valve; q L0 is the first term of the Taylor series expansion of the hydraulic cylinder at the working point;

[0016] The flow continuity equation of the hydraulic cylinder is as follows:

[0017]

[0018] In formula (2), A p is the effective area of the hydraulic cylinder piston; x p is the displacement of the hydraulic cylinder piston; β e is the effective bulk elastic modulus of the hydraulic oil; C tp is the total leakage coefficient of the hydraulic cylinder; V tp is the total volume of the hydraulic cylinder V tp =V a +V b According to Newton's second law, the force balance equation of the hydraulic cylinder is as follows:

[0019]

[0020] In formula (3), m t is the total mass of the piston, piston rod and external load; B pis the system viscous damping coefficient; k is the stiffness of the load spring; F L is the external force acting on the load;

[0021] Perform Laplace transformation on formula (1), formula (2) and formula (3), and take k = 0, the system viscous damping coefficient B p is zero, and the transfer function of the hydraulic cylinder system is obtained as formula (4):

[0022]

[0023] In formula (4), s is a complex frequency domain variable; other parameters are specifically expressed as shown in formula (5):

[0024]

[0025] According to the geometric structure analysis of the robot joint, the drive space and the joint space are connected so that the joint angle can be selected as the state variable. According to the cosine theorem, formula (6) is obtained:

[0026]

[0027] Angles α and β are fixed parameters of the robot arm, L OA 、L OC is the joint parameter, L AC is the total length of the two-axis cylinder, q a The angle between OA and OC is also the angle measured by the sensor. The state space model of the hydraulic manipulator joint position system is established and the joint angle q is selected. a is the state variable, and the state vector is as shown in formula (7):

[0028]

[0029] In formula (7), the state variables x1, x2, and x3 represent the angular displacement, angular velocity, and angular acceleration of the joint, respectively;

[0030] When considering external disturbances, the state equation of the hydraulic manipulator joint position system is expressed as formula (8):

[0031]

[0032] In formula (8), u represents the control input; the various parameters are shown in formula (9):

[0033]

[0034] The factors that are difficult to accurately model in the hydraulic manipulator joint position system are shown in formula (10):

[0035]

[0036] In formula (10), δ(t) represents the hysteresis and dead zone in the motion; Λ(t) represents the comprehensive disturbance that reflects the uncertainty of the electro-hydraulic system parameters.

[0037] A further improvement of the technical solution of the present invention is that in step 2, it specifically includes: according to the state space model in step 1, expanding a new disturbance state, such as formula (11):

[0038]

[0039] The estimated state variables are The estimated error function is e i , then it is expressed as formula (12):

[0040]

[0041] Design the extended state observer as shown in formula (13):

[0042]

[0043] The error estimation model is as shown in formula (14):

[0044]

[0045] In formula (14), υ1, υ2, υ3 and υ4 are the expansion state observation gains. When the hydraulic manipulator joint position system tends to be stable, each term in the above formula tends to 0.

[0046] The further improvement of the technical solution of the present invention is that: in step 3, specifically including: designing an improved sliding mode active disturbance rejection controller based on an extended state observer, Represents the observation result obtained by the state observer, and defines q c is the expected value, and the cylinder angle error vector is defined as formula (15):

[0047]

[0048] In formula (15), e c1 、e c2 、e c3 is the error value of each dimension of the error vector;

[0049] In traditional active disturbance rejection control strategies, the extended state observer often cannot achieve completely accurate estimation of complex disturbances and modeling errors in hydraulic drives. Sliding mode control has high adaptability to the joint position system of hydraulic manipulators because it does not rely on the precise model of the system and has low parameter sensitivity. The sliding mode control strategy is used to replace the traditional linear state feedback to improve the overall performance of active disturbance rejection control. Formula (16) is obtained from Formula (13) and Formula (15):

[0050]

[0051] The sliding mode switching function is defined as formula (17):

[0052] s=c1·e c1 +c2·e c2 +e c3 (17)

[0053] In formula (17), c1>0, c2>0;

[0054] Taking the derivative of both ends of the above equation and substituting equation (16) into it, we get equation (18):

[0055]

[0056] An improved reaching law is adopted. Based on the power reaching law, the improved reaching law adds a variable speed reaching term and an exponential factor, which strengthens the convergence characteristics of the hydraulic arm joint position system when it is far away from the sliding surface, further accelerates the approach process, and maintains good stability. In order to reduce chattering and get closer to the control goal of smooth and accurate convergence, the hyperbolic tangent function tanh(s) is used instead of sgn(s). The improved reaching law is designed as shown in formula (19):

[0057]

[0058] In formula (19), k1, k2 and ε are the parameters of the sliding mode controller.

[0059] The further improvement of the technical solution of the present invention is that: in step 4, it specifically includes: proving the stability of the improved sliding mode active disturbance rejection controller, performing Lyapunov stability analysis on the constructed improved sliding mode active disturbance rejection controller, and constructing a Lyapunov function such as formula (20):

[0060]

[0061] In order to ensure the stability of the improved sliding mode active disturbance rejection controller, it is necessary to improve the sliding mode active disturbance rejection controller to satisfy the negative derivative of the Lyapunov function. The derivative of the Lyapunov function is obtained as formula (21):

[0062]

[0063] The hydraulic manipulator arm joint position system meets the stability conditions, so the hydraulic manipulator arm joint position system is stable;

[0064] According to formula (16) and formula (18), the control input u is designed as:

[0065]

[0066] Due to the adoption of the above technical solution, the technical advancements achieved by the present invention are:

[0067] 1. The improved sliding mode anti-disturbance rejection controller designed in the present invention not only observes and compensates the disturbance of the hydraulic arm joint position system in real time, but also adopts an improved sliding mode control law instead of linear feedback, thereby reducing the controller's dependence on the hydraulic arm joint position system model and improving the control accuracy and robustness of the hydraulic manipulator arm electro-hydraulic position system.

[0068] 2. The present invention aggregates the uncertainty of the hydraulic manipulator joint position system and the load disturbance, which are difficult to accurately model, into a total disturbance, estimates it in real time by designing an extended state observer, and designs an improved sliding mode active disturbance rejection controller in combination with the improved sliding mode control law. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. Those skilled in the art can also derive other drawings based on these drawings without inventive efforts.

[0070] Figure 1 Schematic diagram of the hydraulic arm joint position system in an embodiment of the present invention;

[0071] Figure 2 is a schematic diagram of the two-dimensional geometric structure of the joint in an embodiment of the present invention;

[0072] Figure 3 is a control strategy block diagram in an embodiment of the present invention;

[0073] Figure 4 1 is a simulation diagram comparing the controller designed in the embodiment of the present invention with other controllers;

[0074] Figure 5 This is a simulation error analysis diagram comparing the controller designed in the embodiment of the present invention with other controllers. DETAILED DESCRIPTION

[0075] It should be noted that the terms "include" and "have" and any variations thereof in the specification and claims of the present invention and the above-mentioned drawings are intended to cover non-exclusive inclusions. For example, a process, method, system, product or apparatus that includes a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units that are not explicitly listed or are inherent to these processes, methods, products or apparatuses.

[0076] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments:

[0077] like Figure 3 As shown, a hydraulic arm position sliding mode control method based on an extended state observer includes the following steps:

[0078] Step 1: Establish a state space model of the hydraulic manipulator joint position system;

[0079] Specifically include the following: The hydraulic arm mechanical joint position system is composed of valve-controlled hydraulic cylinders. Figure 1 As shown, the load flow of the valve is related to the valve core displacement x v and load voltage drop p L The function of load flow increment is obtained as formula (1):

[0080]

[0081] In formula (1), K q is the flow gain; K c is the flow-pressure coefficient; q L is the average flow rate of the two working ports of the proportional valve; q L0 is the first term of the Taylor series expansion of the hydraulic cylinder at the working point;

[0082] The flow continuity equation of the hydraulic cylinder is as follows:

[0083]

[0084] In formula (2), A p is the effective area of the hydraulic cylinder piston; x p is the displacement of the hydraulic cylinder piston; β e is the effective bulk elastic modulus of the hydraulic oil; C tp is the total leakage coefficient of the hydraulic cylinder; V tp is the total volume of the hydraulic cylinder V tp =V a +V b According to Newton's second law, the force balance equation of the hydraulic cylinder is as follows:

[0085]

[0086] In formula (3), m t is the total mass of the piston, piston rod and external load; B p is the system viscous damping coefficient; k is the stiffness of the load spring; F L is the external force acting on the load.

[0087] Perform Laplace transformation on formula (1), formula (2) and formula (3), and take k = 0, the system viscous damping coefficient Bp is zero, and the transfer function of the hydraulic cylinder system is obtained as formula (4):

[0088]

[0089] Its parameters are specifically expressed as formula (5):

[0090]

[0091] according to Figure 2 The geometric structure analysis of the robot joint is used to connect the drive space and the joint space so that the joint angle can be selected as the state variable. According to the cosine theorem, formula (6) is obtained:

[0092]

[0093] Angles α and β are fixed parameters of the robot arm, L OA 、L OC for Figure 2 Middle joint parameter, L AC is the total length of the two-axis cylinder, q a The angle between OA and OC is also the angle measured by the sensor. The state space model of the joint position system is established and the joint angle q is selected. a is the state variable, and the state vector is as shown in formula (7):

[0094]

[0095] The state variables x1, x2, and x3 represent the angular displacement, angular velocity, and angular acceleration of the joint, respectively.

[0096] When considering external disturbances, the state equation of the hydraulic manipulator joint position system is expressed as formula (8):

[0097]

[0098] In formula (6), the various parameters are shown in formula (9):

[0099]

[0100] Factors that are difficult to accurately model in the hydraulic manipulator joint position system, such as hysteresis and dead zone in motion, are represented by δ(t), and Λ(t) represents the comprehensive disturbance that reflects the uncertainty of the electro-hydraulic system parameters, as shown in formula (10):

[0101]

[0102] Step 2, establish a mathematical model of the extended state observer;

[0103] Specifically, according to the state space model in step 1, a new disturbance state is expanded, such as formula (11):

[0104]

[0105] The estimated state variables are The estimated error function is e i , then it is expressed as formula (12):

[0106]

[0107] Design the extended observer as shown in formula (13):

[0108]

[0109] The error estimation model is as shown in formula (14):

[0110]

[0111] Among them, υ1, υ2, υ3 and υ4 are the expansion state observation gains. When the hydraulic manipulator joint position system tends to be stable, each term in the above formula tends to 0.

[0112] Step 3: Design an improved sliding mode active disturbance rejection controller based on the extended state observer. While observing and compensating the comprehensive disturbance of the hydraulic manipulator joint position system in real time, an improved sliding mode control law is used instead of linear feedback.

[0113] Specifically include: designing an improved sliding mode active disturbance rejection controller based on the extended state observer, Represents the observation result obtained by the state observer, and defines q c is the expected value, and the cylinder angle error vector is defined as formula (15):

[0114]

[0115] In formula (15), e c1 、e c2 、e c3 is the error value of each dimension of the error vector.

[0116] In traditional ADRC strategies, the extended state observer often fails to accurately estimate complex disturbances and modeling errors in hydraulic drives. Sliding mode control, due to its independence from the precise model of the system and low parameter sensitivity, is highly compatible with hydraulic manipulator joint position systems. This paper uses a sliding mode control strategy to replace traditional linear state feedback to improve the overall performance of ADRC. Formula (16) is obtained from Formula (13) and Formula (15):

[0117]

[0118] The sliding mode switching function is defined as formula (17):

[0119] s=c1·e c1 +c2·e c2 +e c3 (17)

[0120] In the formula, c1>0, c2>0.

[0121] Taking the derivative of both ends of the above equation and substituting equation (16) into it, we get equation (18):

[0122]

[0123] A new improved reaching law is adopted. This improved reaching law adds a variable speed reaching term and an exponential factor to the power reaching law. This strengthens the convergence characteristics of the hydraulic manipulator joint position system when it is far away from the sliding surface, further accelerates the approach process, and maintains good stability. To reduce chattering and get closer to the controller control target, the hyperbolic tangent function tanh(s) is used instead of sgn(s). The reaching law design is as shown in formula (19):

[0124]

[0125] In formula (19), k1, k2 and ε are the parameters of the sliding mode controller.

[0126] Step 4: Use Lyapunov stability theory to prove the stability of the improved sliding mode active disturbance rejection controller.

[0127] Specifically, it includes: proving the stability of the improved sliding mode active disturbance rejection controller, performing Lyapunov stability analysis on the constructed improved sliding mode active disturbance rejection controller, and constructing the Lyapunov function as shown in formula (20):

[0128]

[0129] In order to ensure the stability of the improved sliding mode active disturbance rejection controller, it is necessary to improve the sliding mode active disturbance rejection controller to satisfy the negative derivative of the Lyapunov function. The derivative of the Lyapunov function is obtained as formula (21):

[0130]

[0131] The hydraulic manipulator arm joint position system meets the stability conditions, so the hydraulic manipulator arm joint position system is stable.

[0132] Control block diagram Figure 3 As shown, according to formula (16) and formula (18), the control input u is designed as:

[0133]

[0134] The present invention is described in detail below with reference to the embodiments and accompanying drawings.

[0135] Example:

[0136] The model parameters of the hydraulic manipulator joint position system (two axes) are as follows:

[0137] K c =7.8×10 -2 , A p =1.45×10 -3 m 2 , V tp =1.21×10 -3 m 3 , β e =8×10 7 , m1=9.6kg,

[0138] K sv =3×10 -3 ,ω sv =314,K q =5.6×10 -2 .

[0139] To verify the control performance of the improved sliding mode active disturbance rejection controller, the controller proposed in this paper is named SM-ADRC controller. The PID controller and LADRC controller are selected for comparison. The parameter values of the controller proposed in this paper and the comparison controller are as follows:

[0140] SM-ADRC controller parameters υ1=46,υ2=70,υ3=35,υ4=110,k s =0.21, k eso =0.02, c1=2.5×10 4 , c2=780,k1=2×10 5 , k1=200.

[0141] PID controller parameter k p =30,k i =0.098, k d =0.002.

[0142] LADRC controller parameters ω0=230,ω c =100, b0=8800.

[0143] The position command signal is y = 12.5(1+tanh(1.2*(t-2)-0.15)+12.5(1+tanh(1.2*(t-5)-0.15))

[0144] Figure 4 This is a simulation diagram comparing the controller designed by the present invention with other controllers. Figure 5 This figure shows a simulation error analysis comparing the controller designed in this paper with other controllers. The simulation results show that, under given conditions of simulated time-varying disturbances, the SM-ADRC compensation strategy, based on the extended state observer and sliding mode control strategy, effectively mitigates the hydraulic system's parameter uncertainty and nonlinear disturbances, demonstrating strong robustness and disturbance compensation performance, and effectively enabling the system to track joint angles.

[0145] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A hydraulic arm position sliding mode control method based on an extended state observer, characterized in that: The following steps are involved: Step 1: Establish a state space model of the hydraulic manipulator joint position system; Step 2, establish a mathematical model of the extended state observer; Step 3: Design an improved sliding mode active disturbance rejection controller based on the extended state observer. While observing and compensating the comprehensive disturbance of the hydraulic manipulator joint position system in real time, an improved sliding mode control law is used instead of linear feedback. Step 4: Use Lyapunov stability theory to prove the stability of the improved sliding mode active disturbance rejection controller.

2. The hydraulic arm position sliding mode control method based on the extended state observer according to claim 1 is characterized in that: In step 1, the hydraulic manipulator joint position system is composed of a valve-controlled hydraulic cylinder, and the load flow of the valve is about the valve core displacement x v and load voltage drop p L The function of load flow increment is obtained as formula (1): In formula (1), K q is the flow gain; K c is the flow-pressure coefficient; q L is the average flow rate of the two working ports of the proportional valve; q L0 is the first term of the Taylor series expansion of the hydraulic cylinder at the working point; The flow continuity equation of the hydraulic cylinder is as follows: In formula (2), A p is the effective area of the hydraulic cylinder piston; x p is the displacement of the hydraulic cylinder piston; β e is the effective bulk elastic modulus of the hydraulic oil; C tp is the total leakage coefficient of the hydraulic cylinder; V tp is the total volume of the hydraulic cylinder V tp =V a +V b According to Newton's second law, the force balance equation of the hydraulic cylinder is as follows: In formula (3), m t is the total mass of the piston, piston rod and external load; B p is the system viscous damping coefficient; k is the stiffness of the load spring; F L is the external force acting on the load; Perform Laplace transformation on formula (1), formula (2) and formula (3), and take k = 0, the system viscous damping coefficient B p is zero, and the transfer function of the hydraulic cylinder system is obtained as formula (4): In formula (4), s is a complex frequency domain variable; other parameters are specifically expressed as shown in formula (5): According to the geometric structure analysis of the robot joint, the drive space and the joint space are connected so that the joint angle can be selected as the state variable. According to the cosine theorem, formula (6) is obtained: Angles α and β are fixed parameters of the robot arm, L OA 、L OC is the joint parameter, L AC is the total length of the two-axis cylinder, q a The angle between OA and OC is also the angle measured by the sensor. The state space model of the joint position system is established and the joint angle q is selected. a is the state variable, and the state vector is as shown in formula (7): In formula (7), the state variables x1, x2, and x3 represent the angular displacement, angular velocity, and angular acceleration of the joint, respectively; When considering external disturbances, the state equation of the hydraulic manipulator joint position system is expressed as formula (8): In formula (8), u represents the control input; the various parameters are shown in formula (9): The factors that are difficult to accurately model in the hydraulic manipulator joint position system are shown in formula (10): In formula (10), δ(t) represents the hysteresis and dead zone in the motion; Λ(t) represents the comprehensive disturbance that reflects the uncertainty of the electro-hydraulic system parameters.

3. The hydraulic arm position sliding mode control method based on extended state observer according to claim 1, characterized in that: In step 2, specifically, it includes: expanding a new disturbance state according to the state space model in step 1, such as formula (11): The estimated state variables are The estimated error function is e i , then it is expressed as formula (12): Design the extended state observer as shown in formula (13): The error estimation model is as shown in formula (14): In formula (14), υ1, υ2, υ3 and υ4 are the expansion state observation gains. When the hydraulic manipulator joint position system tends to be stable, each term in the above formula tends to 0.

4. The hydraulic arm position sliding mode control method based on extended state observer according to claim 1, characterized in that: In step 3, it specifically includes: designing an improved sliding mode active disturbance rejection controller based on the extended state observer to Represents the observation result obtained by the state observer, and defines q c is the expected value, and the cylinder angle error vector is defined as formula (15): In formula (15), e c1 、e c2 、e c3 is the error value of each dimension of the error vector; In traditional active disturbance rejection control strategies, the extended state observer often cannot achieve completely accurate estimation of complex disturbances and modeling errors in hydraulic drives. Sliding mode control has high adaptability to the joint position system of hydraulic manipulators because it does not rely on the precise model of the system and has low parameter sensitivity. The sliding mode control strategy is used to replace the traditional linear state feedback to improve the overall performance of active disturbance rejection control. Formula (16) is obtained from Formula (13) and Formula (15): The sliding mode switching function is defined as formula (17): s=c1·e c1 +c2·e c2 +e c3 (17) In formula (17), c1>0, c2>0; Taking the derivative of both ends of the above equation and substituting equation (16) into it, we get equation (18): An improved reaching law is adopted. Based on the power reaching law, the improved reaching law adds a variable speed reaching term and an exponential factor, which strengthens the convergence characteristics of the hydraulic manipulator joint position system when it is far away from the sliding surface, further accelerates the approach process, and maintains good stability. In order to reduce chattering and get closer to the control goal of smooth and accurate convergence, the hyperbolic tangent function tanh(s) is used instead of sgn(s). The improved reaching law is designed as shown in formula (19): In formula (19), k1, k2 and ε are the parameters of the sliding mode controller.

5. The hydraulic arm position sliding mode control method based on extended state observer according to claim 1, characterized in that: In step 4, the following steps are specifically performed: proving the stability of the improved sliding mode active disturbance rejection controller, performing Lyapunov stability analysis on the constructed improved sliding mode active disturbance rejection controller, and constructing the Lyapunov function as shown in formula (20): In order to ensure the stability of the improved sliding mode active disturbance rejection controller, it is necessary to improve the sliding mode active disturbance rejection controller to satisfy the negative derivative of the Lyapunov function. The derivative of the Lyapunov function is obtained as formula (21): The hydraulic manipulator arm joint position system meets the stability conditions, so the hydraulic manipulator arm joint position system is stable; According to formula (16) and formula (18), the control input u is designed as:

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