Preset performance control method of robotic arm based on segmented threshold event triggering

Abstract

The present invention relates to robotic arm control, and in particular to a method for controlling a preset performance of a robotic arm based on segmented threshold event triggering. The event triggering mechanism of the present invention can reduce data transmission and improve resource utilization, and the control law structure is simple and easy to implement. It comprises the following steps: step 1. establishing a mathematical model of a single-joint robotic arm; step 2. designing a preset performance function to limit the motion trajectory of the trajectory tracking error generated by the mathematical model of the single-joint robotic arm in combination with nonlinear transformation; step 3. constructing an instruction filter with filtering error compensation to process the time derivative of the virtual control law based on the intermediate variable obtained by the nonlinear transformation; step 4. designing an event-driven control law and an adaptive law based on the output of the instruction filter in combination with a neural network observer, a segmented threshold triggering mechanism and adaptive technology.

Description

Technical Field

[0001] The present invention relates to robotic arm control, and in particular to a robotic arm preset performance control method based on segmented threshold event triggering. Background Art

[0002] Currently, robotic arms are widely used in healthcare, agriculture, and industry due to their ease of manipulation and high efficiency. However, due to cost, gravity, and space constraints, it is often difficult to equip control systems with joint angular velocity sensors. Achieving trajectory tracking control with preset performance without relying on these sensors has always been a key challenge in robotic arm control. Furthermore, to reduce data transmission and improve resource utilization, designing novel event-driven mechanisms to reduce the frequency of control signal updates while maintaining preset trajectory tracking performance remains a research challenge in robotic arm control.

[0003] In practical applications, it is essential to consider both the transient and steady-state performance indicators of the trajectory tracking error. Traditional control methods can force the tracking error to converge within a bounded small region, but the size of this region is regulated by design parameters. Therefore, when the preset performance indicators of a given system are clearly defined, traditional control methods are powerless. Among existing methods, the guaranteed cost control method uses a single exponential decay function to limit the motion trajectory of the tracking error, which can achieve the tracking error converging to a predetermined region at a pre-arranged convergence rate. However, this method cannot guarantee that the tracking error converges within a finite time, and has certain constraints on the initial tracking error, which has limitations. In addition, when designing the control law using the backstepping method, it is necessary to take the time derivative of the virtual control function. As the system order increases, this will lead to a complexity explosion problem, which is not conducive to the practical application of the control method.

[0004] Event-driven strategies are gradually replacing traditional periodic sampling strategies because they can significantly reduce the frequency of control signal updates. Compared to traditional time-driven strategies, event-driven mechanisms only update control instructions when the driving conditions are met. Existing event-driven mechanisms mainly include fixed threshold driving and relative threshold driving. Compared with the fixed threshold mechanism, the relative threshold mechanism further reduces data transmission by using a variable threshold. However, because the relative threshold trigger condition includes the control signal, the complexity of the control law increases and it has a certain degree of conservatism. Summary of the Invention

[0005] The present invention addresses the shortcomings of existing technologies by providing a method for controlling the preset performance of a robotic arm based on segmented threshold event triggering. This method, without relying on an angular velocity sensor, achieves convergence of the tracking error to a preset range within a set time, while relaxing the constraints on convergence time and initial tracking error. Furthermore, this novel event-triggered mechanism reduces data transmission and improves resource utilization, while also providing a simple control law structure that is easy to implement.

[0006] To achieve the above object, the present invention adopts the following technical solutions, including:

[0007] Step 1. (Considering external environmental disturbances), establish a mathematical model of a single-joint robotic arm.

[0008] Step 2. Aiming at the trajectory tracking error e1 generated by the mathematical model of the single-joint robotic arm, a preset performance function is designed to limit its motion trajectory in combination with nonlinear transformation.

[0009] Step 3. Based on the intermediate variable z1 obtained by nonlinear transformation, construct a command filter with filtering error compensation to process the time derivative of the virtual control law.

[0010] Step 4. Based on the output of the command filter, the event-driven control law u and the adaptive law are designed by combining the neural network observer, the segmented threshold trigger mechanism and the adaptive technology.

[0011] Furthermore, the mathematical model of the single-joint robotic arm is:

[0012]

[0013] Where q represents the joint angle position, represents the joint angular velocity, represents the joint angular acceleration, J m >0 is the moment of inertia of the motor, V m is the viscous friction coefficient, M m is the mass of the joint, l m Represents the distance from the joint axis to the center of mass, g is the acceleration due to gravity, G τ is the external uncertainty disturbance, and u is the control torque provided by the motor; (the event-driven control law to be designed).

[0014] definition Then the state equation expression of the single-joint robotic arm can be obtained:

[0015]

[0016] in,

[0017] Furthermore, in step 2, the trajectory tracking error e1 is:

[0018] e1=x1-q d (3)

[0019] Among them, q d is the desired joint angle position.

[0020] The default performance functions are:

[0021]

[0022] Among them, h p and l p is a positive constant, T f is the set time for error convergence, β f is the preset range for error convergence; the nonlinear transformation is:

[0023]

[0024] Specifically, according to formula (4), we can get and The traditional preset performance control method requires an accurate initial tracking error e1(0) to determine the preset performance function β p (t) to ensure that e1(0)<β p (0). However, it is sometimes difficult to obtain an accurate initial tracking error e1(0), which brings challenges to the preset performance control. The present invention presets the performance function and adjusts the design parameter l p Thus, e1(0)<β p (0), which relaxes the constraints on the initial tracking error.

[0025] Furthermore, in step 3, the instruction filter with filtering error compensation is constructed as follows:

[0026]

[0027] in, is a positive constant, and the expression of the virtual control law α1 is:

[0028]

[0029] Among them, a1, c1 are positive constants, The compensation signal generated by the following filter error compensation system:

[0030]

[0031] Among them, a2, ι1, ι2 are positive constants, sign(·) represents the sign function,

[0032] Furthermore, in step 4, the neural network observer is:

[0033]

[0034] Among them, l o1 ,l o2 is a positive constant, represents the neural network's estimation of the uncertain dynamics F(x1,x2), θ a =[θ a1 ,...,θ aM ] T represents the neural network weight vector, is the Gaussian basis function, M>1 is the node of the neural network, k=1,...,M,ν ak =[ν ak1 ,ν ak2 ] T represents the center vector, γ ak represents the width of the Gaussian basis function, For the state variable X = [x1, x2] T Estimates.

[0035] According to the output of the command filter, the event-driven control law u and the adaptive law are designed by combining the neural network observer, the segmented threshold trigger mechanism and the adaptive technology. for:

[0036]

[0037] Among them, π a ,δ a is a positive constant, k is a positive integer, t k represents the update time of the control law, and the designed segmented threshold trigger mechanism is:

[0038]

[0039] Among them, δ ma ,δ mi , represents a positive constant and satisfies δ ma >δ mi , ξ m (t) = τ(t) - u(t) represents the measurement error between the intermediate control law τ(t) and the event-driven control law u(t). The expression of τ(t) is:

[0040]

[0041] in, is a positive constant and satisfies The expression of α2 is:

[0042]

[0043] Where c2 is a positive constant.

[0044] Furthermore, the proof method of the preset performance control method of the robotic arm based on segmented threshold event triggering is as follows:

[0045] The proof method is:

[0046] According to the nonlinear transformation (5), we get

[0047]

[0048] From equation (14) and considering the state equation (2) of the single-joint manipulator, we can get

[0049]

[0050] Taking the time derivative of χ1 and considering (8) and (15) we get

[0051]

[0052] in, Construct the first Lyapunov function L1 as:

[0053]

[0054] in, is the estimation error of the neural network observer, P is a symmetric positive definite matrix; by appropriately selecting l in the neural network observer o1 ,l o2 , then there exist positive definite symmetric matrices P and Q such that the following relationship holds:

[0055]

[0056] The time derivative of the Lyapunov function L1 is:

[0057]

[0058] in, Considering that the external interference is bounded, according to the universal approximation characteristics of the neural network, there exists a positive constant Make Established; According to Young's inequality,

[0059]

[0060] Substituting equation (20) into equation (19), we get

[0061]

[0062] Substituting the virtual control law α1 into equation (21) we get

[0063]

[0064] Among them, a x =λ min (Q)-1 / (2c1)-1 / 2>0,

[0065] Taking the time derivative of χ2 we get

[0066]

[0067] Construct the second Lyapunov function L2 as:

[0068]

[0069] in, Represents the optimal weight vector of the neural network; the time derivative of the Lyapunov function L2 is:

[0070]

[0071] According to Young's inequality:

[0072]

[0073] Substituting equation (26) into equation (25), we obtain:

[0074]

[0075] According to the segmentation threshold trigger mechanism, there is a time-varying variable |κ e (t)|≤1 makes the following equation true:

[0076] u(t)=τ(t)-κ e (t)δ ma (28)

[0077] Substituting equations (12) and (28) into (27), we obtain:

[0078]

[0079] α2 and Substituting into formula (29) we get

[0080]

[0081] according to And Young's inequality is:

[0082]

[0083] For the hyperbolic tangent function tanh(·), the following properties hold:

[0084]

[0085] because Substituting equations (31) and (32) into (30), we obtain

[0086]

[0087] in,

[0088] In order to verify the stability of the filtering error compensation system, the Lyapunov function is constructed right Taking the time derivative we get:

[0089]

[0090] For the command filter, there is ε γ >0 such that |γ2-α1|<ε γ Therefore, by choosing ι1>ε γ ,get:

[0091]

[0092] Constructing Lyapunov functions To prove the boundedness of all signals in the closed-loop system, take the time derivative of L3 and get:

[0093]

[0094] Where Λ=min{2a x / λ max (P),2a1,2a2,(δ a -π a M / c2)}>0; According to the above formula,

[0095]

[0096] From the above formula, we can get that for k=1,2, χ k , θ a is bounded; since Get z k is bounded; because the expected trajectory q d is bounded, so x1 and is bounded, which means that α1 and γ2 are bounded. Similarly, we can get α2,x2, τ(t),u(t) are bounded, so the closed-loop system is stable.

[0097] Since z1 is bounded, the trajectory tracking error e1 meets the preset performance requirements, that is, |e1|<β p .

[0098] Next, we prove that there exists t H >0, so that {t k+1 -t k}≥t H holds, where k represents a positive integer.

[0099] For t∈[t k ,t k+1 ), according to ξ m (t) = τ(t) - u(t) and we get:

[0100]

[0101] because is a bounded function, so there exists a positive constant γ τ >0 makes holds; in addition, because ξ m (t k )=0 and So the time interval between adjacent triggers is t H Satisfy t H ≥δ mi / γ τ >0.

[0102] It can be seen that under the segmented threshold event triggering mechanism, there is a strict positive lower bound on the time interval between adjacent triggering events, which avoids Zeno behavior.

[0103] Compared with the prior art, the present invention has beneficial effects.

[0104] The present invention constructs a neural network observer to ensure that the trajectory tracking error converges to a preset range within a set time without relying on an angular velocity sensor.

[0105] The present invention designs a preset performance function to limit the motion trajectory of the tracking error, thereby relaxing the restrictions on the tracking error convergence time and the initial tracking error.

[0106] The present invention designs a segmented threshold event trigger mechanism, which effectively reduces data transmission and the number of actuator operations, and does not include a control signal in the event trigger condition, which makes the control law structure simple and easy to implement.

[0107] The present invention constructs an instruction filter with filtering error compensation to process the time derivative of the virtual control law, thereby solving the complexity explosion problem in the traditional backstepping design method. BRIEF DESCRIPTION OF THE DRAWINGS

[0108] The present invention is further described below with reference to the accompanying drawings and specific embodiments. The scope of protection of the present invention is not limited to the following description.

[0109] Figure 1 This is the technical roadmap of the present invention.

[0110] Figure 2 Schematic diagram of a model of a single-joint robotic arm in an embodiment of the present invention.

[0111] Figure 3 This is a diagram showing the trajectory tracking effect of a single-joint robotic arm in an embodiment of the present invention.

[0112] Figure 4 2 is a diagram showing the joint angle position and its estimated effect in an embodiment of the present invention.

[0113] Figure 5 Graph showing the joint angular velocity and its estimation effect in an embodiment of the present invention.

[0114] Figure 6 This is a trajectory tracking error effect diagram in an embodiment of the present invention.

[0115] Figure 7 Graph showing the control torque in an embodiment of the present invention.

[0116] Figure 8 4 is a graph showing the weight norm of a neural network in an embodiment of the present invention.

[0117] Figure 9 Comparison of control performance of different event triggering methods. DETAILED DESCRIPTION

[0118] This paper addresses the problem of event-driven, uncertain robotic arm trajectory tracking. By combining nonlinear transformations, a preset performance function is designed to constrain the trajectory of tracking errors. A command filter with filter error compensation is constructed to process the time derivative of the virtual control law. Based on the command filter output, a neural network observer, a segmented threshold trigger mechanism, and adaptive technology are used to design an event-driven, preset performance control method for the robotic arm.

[0119] like Figure 1 As shown, the present invention provides a method for controlling the preset performance of a robotic arm based on segmented threshold event triggering, comprising the following steps:

[0120] Step 1. Considering external environmental disturbances, establish a mathematical model of a single-joint robotic arm.

[0121] The mathematical model of the single-joint robotic arm with external environmental interference and parameter uncertainty is:

[0122]

[0123] Where q represents the joint angle position, represents the joint angular velocity, represents the joint angular acceleration, J m >0 is the moment of inertia of the motor, V m is the viscous friction coefficient, M m is the mass of the joint, l m Represents the distance from the joint axis to the center of mass, g is the acceleration due to gravity, G τ is the external uncertainty disturbance, u is the control torque provided by the motor, and the event-driven control law to be designed is defined as x1=q, Then the state equation expression of the single-joint robotic arm can be obtained:

[0124]

[0125] in,

[0126] Step 2. Based on the tracking error e1 generated by the mathematical model of the single-joint robotic arm, a preset performance function is designed to limit its motion trajectory in combination with nonlinear transformation.

[0127] The tracking error is defined as:

[0128] e1=x1-q d (41)

[0129] Among them, q d is the desired joint angle position.

[0130] The default performance functions are:

[0131]

[0132] Among them, h p and l p is a positive constant, T f is the set time for error convergence, β f is the preset range for error convergence. The nonlinear transformation is:

[0133]

[0134] According to formula (42), we can get and The traditional preset performance control method requires an accurate initial tracking error e1(0) to determine the preset performance function β p (t) to ensure that e1(0)<βp (0).

[0135] However, it is sometimes difficult to obtain an accurate initial tracking error e1(0), which brings challenges to the preset performance control. The present invention designs a novel preset performance function, by adjusting the design parameter l p Thus, e1(0)<β p (0), which relaxes the constraints on the initial tracking error.

[0136] Step 3. Based on the intermediate variable z1 obtained by nonlinear transformation, construct a command filter with filtering error compensation to process the time derivative of the virtual control law.

[0137] Construct the instruction filter as:

[0138]

[0139] in, is a positive constant, and the expression of the virtual control law α1 is:

[0140]

[0141] Among them, a1, c1 are positive constants, The compensation signal generated by the following filter error compensation system:

[0142]

[0143] Among them, a2, ι1, ι2 are positive constants, sign(·) represents the sign function,

[0144] Step 4. Based on the output of the command filter, the event-driven control law u and the adaptive law are designed by combining the neural network observer, the segmented threshold trigger mechanism and the adaptive technology.

[0145] Construct the neural network observer as:

[0146]

[0147] Among them, l o1 ,l o2 is a positive constant, represents the neural network's estimation of the uncertain dynamics F(x1,x2), θ a =[θ a1 ,...,θ aM ] T represents the neural network weight vector, is the Gaussian basis function vector, M>1 is the node of the neural network, k=1,...,M,νak =[ν ak1 ,ν ak2 ] T represents the center vector, γ ak represents the width of the Gaussian basis function, For the state variable X = [x1, x2] T Estimates.

[0148] According to the output of the command filter, the event-driven control law u and the adaptive law are designed by combining the neural network observer, the segmented threshold trigger mechanism and the adaptive technology.

[0149]

[0150] Among them, π a ,δ a is a positive constant, k is a positive integer, t k Represents the update time of the control law, and the novel segmented threshold trigger mechanism is designed as follows:

[0151]

[0152] Among them, δ ma ,δ mi , represents a positive constant and satisfies δ ma >δ mi , ξ m (t) = τ(t) - u(t) represents the measurement error between the intermediate control law τ(t) and the event-driven control law u(t). The expression of τ(t) is:

[0153]

[0154] in, is a positive constant and satisfies The expression of α2 is:

[0155]

[0156] Where c2 is a positive constant.

[0157] Based on Lyapunov theory analysis, it is proved that the closed-loop system is stable. The method of proving stability is as follows:

[0158] According to the nonlinear transformation (43), we get:

[0159]

[0160] From equation (52) and considering the state equation (40) of the single-joint manipulator, we can obtain:

[0161]

[0162] Taking the time derivative of χ1 and considering (46) and (53) we get

[0163]

[0164] in, Construct the first Lyapunov function L1 as:

[0165]

[0166] in, is the estimated error of the neural network observer, and P is a symmetric positive definite matrix. By properly selecting l in the neural network observer o1 ,l o2 , then there exist positive definite symmetric matrices P and Q such that the following relationship holds:

[0167]

[0168] The time derivative of the Lyapunov function L1 is

[0169]

[0170] in, Considering that the external interference is bounded, according to the universal approximation characteristics of the neural network, there exists a positive constant Make Established. According to Young's inequality,

[0171]

[0172] Substituting equation (58) into (57), we get

[0173]

[0174] Substituting the virtual control law α1 into equation (59) yields

[0175]

[0176] Among them, a x =λ min (Q)-1 / (2c1)-1 / 2>0,

[0177] Taking the time derivative of χ2 yields:

[0178]

[0179] Construct the second Lyapunov function L2 as:

[0180]

[0181] in, Represents the optimal weight vector of the neural network. The time derivative of the Lyapunov function L2 is:

[0182]

[0183] According to Young's inequality:

[0184]

[0185] Substituting equation (64) into (63) yields:

[0186]

[0187] According to the segmentation threshold trigger mechanism (49), there is a time-varying variable |κ e (t)|≤1 makes the following equation hold

[0188] u(t)=τ(t)-κ e (t)δ ma (66)

[0189] Substituting equations (50) and (66) into (65), we obtain

[0190]

[0191] α2 and Substituting into equation (67) we get:

[0192]

[0193] according to And Young's inequality has

[0194]

[0195] For the hyperbolic tangent function tanh(·), the following properties hold:

[0196]

[0197] because Substituting equations (69) and (70) into (68), we obtain

[0198]

[0199] in,

[0200] In order to verify the stability of the filtering error compensation system, the Lyapunov function is constructed right Taking the time derivative we get:

[0201]

[0202] For the command filter (44), there exists ε γ >0 such that |γ2-α1|<ε γ Therefore, by choosing ι1>ε γ ,get:

[0203]

[0204] Constructing Lyapunov functions Prove the boundedness of all signals in the closed-loop system, take the time derivative of L3 and get

[0205]

[0206] Where Λ=min{2a x / λ max (P),2a1,2a2,(δ a -π a M / c2)}>0. According to formula (74), we get:

[0207]

[0208] From (75), we can get that for k = 1, 2, χ k , θ a is bounded. Get z k is bounded. Because the expected trajectory q d is bounded, so x1 and is bounded, which means that α1 and γ2 are bounded. Similarly, we can get α2,x2, τ(t),u(t) are bounded, so the closed-loop system is stable.

[0209] Since z1 is bounded, the trajectory tracking error e1 meets the preset performance requirements, that is, |e1|<β p .

[0210] Next, we prove that there exists t H >0, so that {t k+1 -t k}≥t H holds, where k represents a positive integer.

[0211] For t∈[t k ,t k+1 ), according to ξm (t) = τ(t) - u(t) and we get:

[0212]

[0213] because is a bounded function, so there exists a positive constant γ τ >0 makes In addition, because ξ m (t k )=0 and So the time interval between adjacent triggers is t H Satisfy t H ≥δ mi / γ τ > 0. Therefore, it can be concluded that under the segmented threshold event triggering mechanism proposed in the present invention, there is a strict positive lower bound on the time intervals between adjacent triggering events, that is, Zeno behavior is avoided.

[0214] A simulation experiment was carried out in a virtual environment on the designed robotic arm preset performance control method based on segmented threshold event triggering to verify the feasibility of the proposed method.

[0215] Single joint robotic arm model Figure 2 As shown in the simulation experiment, the model parameters of the robot arm are: J m =1kg·m 2 , V m =1Nm·s / rad, M m gl m =10N·m, G τ =1.3sin(0.5πt)N·m, and the time is set to 10 seconds.

[0216] The initial state of the robot arm is q(0) = -2, The desired trajectory is set as q d =cos(t).

[0217] The parameters of the neural network observer are designed as follows: l o1 =20,l o2 =550, M=60, the center vector is evenly distributed in the range of [-2.5, 1.5] × [-5, 20], the Gaussian basis width is 27.5, and the initial weight vector is the zero vector.

[0218] The control law and adaptive law parameters are set to a1 = 0.1, a2 = 15, c1 = 0.1, c2 = 1, ι1=0.1,ι2=0.1,δ ma =20,δ mi =2, T f =2,βf =0.1,h p =1,l p =1 / 6,π a =0.01,δ a =8.

[0219] Figure 3 This is the trajectory tracking diagram of a single-joint robotic arm. From this diagram, we can see that the designed event-driven control method can ensure that the robotic arm tracks the desired trajectory.

[0220] Figure 4 and Figure 5 The diagram below shows the joint angle position and angular velocity and their estimation. It can be seen from the figure that the designed neural network observer can achieve effective estimation of the joint angle position and angular velocity.

[0221] Figure 6 This is the trajectory tracking error diagram of the single-joint robotic arm. From this diagram, we can see that the trajectory tracking error converges to the preset range within the set time.

[0222] Figure 7 This is a control torque curve diagram generated by the control method of the present invention. It can be seen from the diagram that the control signal sent to the actuator is a piecewise constant, which greatly reduces data transmission and improves resource utilization.

[0223] Figure 8 This is the trajectory diagram of the neural network weight norm. From this diagram, we can see that the adaptive parameters are bounded.

[0224] To further illustrate the effectiveness of the control method designed by the present invention, a comparative experiment was conducted with the traditional fixed threshold event-driven method and the relative threshold event-driven method. The expression of the traditional fixed threshold event-driven method is:

[0225]

[0226] Among them, the design parameters are

[0227] The expression of the traditional relative threshold event-driven method is:

[0228]

[0229] Among them, the design parameters are ρ re =0.3,δ re =0.3,

[0230] In order to quantitatively compare the control performance of different methods, the present invention uses the integral time absolute error and the integral absolute error to evaluate the dynamic performance and tracking accuracy of the control system respectively. The expression of the integral time absolute error is: The expression of the integral absolute error is Table 1 lists the control performance indicators corresponding to different methods. It can be seen from Table 1 that compared with the traditional fixed threshold and relative threshold methods, the method of the present invention requires fewer event triggers to achieve faster trajectory tracking and higher tracking accuracy, which greatly reduces data transmission and improves resource utilization.

[0231] The above simulation experimental results show that the control method of the present invention can achieve the convergence of the trajectory tracking error to a preset range within a set time without relying on the angular velocity sensor, thereby improving the transient and steady-state performance of the control system. In addition, the novel event triggering mechanism can reduce data transmission and save system resources.

[0232] It can be understood that the above specific description of the present invention is only used to illustrate the present invention and is not limited to the technical solutions described in the embodiments of the present invention. Those skilled in the art should understand that the present invention can still be modified or replaced by equivalents to achieve the same technical effects; as long as the use requirements are met, they are within the scope of protection of the present invention.

Claims

1. A method for controlling the preset performance of a robotic arm based on segmented threshold events, characterized by: include: Step 1. Establish a mathematical model of a single-joint robotic arm; Step 2. Based on the trajectory tracking error e1 generated by the mathematical model of the single-joint robotic arm, a preset performance function is designed to limit its motion trajectory in combination with nonlinear transformation; Step 3. Based on the intermediate variable z1 obtained by nonlinear transformation, construct a command filter with filter error compensation to process the time derivative of the virtual control law; Step 4. Based on the output of the command filter, the event-driven control law u and the adaptive law are designed by combining the neural network observer, the segmented threshold trigger mechanism and the adaptive technology. The mathematical model of the single-joint robotic arm is: Where q represents the joint angle position, represents the joint angular velocity, represents the joint angular acceleration, J m >0 is the moment of inertia of the motor, V m is the viscous friction coefficient, M m is the mass of the joint, l m Represents the distance from the joint axis to the center of mass, g is the acceleration due to gravity, G τ is the external uncertainty disturbance, u is the control torque provided by the motor; Define x1=q, Then the state equation expression of the single-joint robotic arm can be obtained: in, d m =-G τ / J m ; In step 2, the trajectory tracking error e1 is: e1=x1-q d (3) Among them, q d is the desired joint angle position; The default performance functions are: Among them, h p and l p is a positive constant, T f is the set time for error convergence, β f is the preset range for error convergence; the nonlinear transformation is:

2. The method according to claim 1, wherein: In step 3, the instruction filter with filtering error compensation is constructed as: in, is a positive constant, and the expression of the virtual control law α1 is: Among them, a1, c1 are positive constants, The compensation signal generated by the following filter error compensation system: Among them, a2, ι1, ι2 are positive constants, sign(·) represents the sign function, 3. The method according to claim 1, wherein: In step 4, the neural network observer is: Among them, l o1 ,l o2 is a positive constant, represents the neural network's estimation of the uncertain dynamics F(x1,x2), θ a =[θ a1 ,...,θ aM ] T represents the neural network weight vector, is the Gaussian basis function, M>1 is the node of the neural network, k=1,...,M,ν ak =[ν ak1 ,ν ak2 ] T represents the center vector, γ ak represents the width of the Gaussian basis function, For the state variable X = [x1, x2] T estimates; According to the output of the command filter, the event-driven control law u and the adaptive law are designed by combining the neural network observer, the segmented threshold trigger mechanism and the adaptive technology. for: Among them, π a ,δ a is a positive constant, k is a positive integer, t k represents the update time of the control law, and the designed segmented threshold trigger mechanism is: Among them, δ ma ,δ mi , represents a positive constant and satisfies δ ma >δ mi , ξ m (t) = τ(t) - u(t) represents the measurement error between the intermediate control law τ(t) and the event-driven control law u(t). The expression of τ(t) is: in, is a positive constant and satisfies The expression of α2 is: Where c2 is a positive constant.

4. The method according to claim 1, wherein: The proof method of the preset performance control method of the robotic arm based on segmented threshold event triggering is as follows: The proof method is: According to the nonlinear transformation (5), we get From equation (14) and considering the state equation of the single-joint manipulator, we can get Taking the time derivative of χ1 and considering (8) and (15) we get in, Construct the first Lyapunov function L1 as: in, is the estimation error of the neural network observer, P is a symmetric positive definite matrix; by appropriately selecting l in the neural network observer o1 ,l o2 , then there exist positive definite symmetric matrices P and Q such that the following relationship holds: The time derivative of the Lyapunov function L1 is: in, Considering that the external interference is bounded, according to the universal approximation characteristics of the neural network, there exists a positive constant Make Established; According to Young's inequality, Substituting equation (20) into equation (19), we get Substituting the virtual control law α1 into equation (21) we get Among them, a x =λ min (Q)-1 / (2c1)-1 / 2>0, Taking the time derivative of χ2 we get Construct the second Lyapunov function L2 as: in, Represents the optimal weight vector of the neural network; the time derivative of the Lyapunov function L2 is: According to Young's inequality: Substituting equation (26) into equation (25), we obtain: According to the segmentation threshold trigger mechanism, there is a time-varying variable |κ e (t)|≤1 makes the following equation true: u(t)=τ(t)-κ e (t)d ma (28) Substituting equations (12) and (28) into (27), we obtain: α2 and Substituting into formula (29) we get according to And Young's inequality is: For the hyperbolic tangent function tanh(·), the following properties hold: because Substituting equations (31) and (32) into (30), we obtain in, In order to verify the stability of the filtering error compensation system, the Lyapunov function is constructed right Taking the time derivative we get: For the command filter, there is ε γ >0 such that |γ2-α1|<ε γ holds; therefore, by choosing ι1>ε γ ,get: Constructing Lyapunov functions To prove the boundedness of all signals in the closed-loop system, take the time derivative of L3 and get: Where Λ=min{2a x / λ max (P),2a1,2a2,(δ a -π a M / c2)}>0; According to the above formula, From the above formula, we can get that for k=1,2, χ k , θ a is bounded; since Get z k is bounded; because the expected trajectory q d is bounded, so x1 and is bounded, which means that α1 and γ2 are bounded; similarly, we can get α2,x2, τ(t),u(t) are bounded, so the closed-loop system is stable; Since z1 is bounded, the trajectory tracking error e1 meets the preset performance requirements, that is, |e1|<β p ; Next, we prove that there exists t Η >0, so that {t k+1 -t k }≥t Η holds true, where k represents a positive integer; For t∈[t k ,t k+1 ), according to ξ m (t) = τ(t) - u(t) and we get: because is a bounded function, so there exists a positive constant γ τ >0 makes holds; in addition, because ξ m (t k )=0 and So the time interval between adjacent triggers is t Η Satisfy t Η ≥δ mi / γ τ >0; It can be seen that under the segmented threshold event triggering mechanism, there is a strict positive lower bound on the time interval between adjacent triggering events, which avoids Zeno behavior.

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