Exponential tracking control method for parallel five-bar manipulator based on event-triggered strategy

Abstract

The application discloses an event-triggered strategy-based exponential tracking control method for a parallel five-link mechanical arm, and comprises the following steps: establishing an Euler-Lagrange differential equation of the parallel five-link mechanical arm; converting the Euler-Lagrange differential equation into an error state differential equation through coordinate transformation; adopting the certain equivalent control principle to complete the design of a certain equivalent controller for the error state differential equation; obtaining the optimal solution of unknown parameters by using the optimal control principle; generating a specific implementation algorithm of the optimal solution by using Fermat's lemma; performing stability analysis on a closed-loop system formed by the error state differential equation and the designed actual controller; and giving an exponential tracking control semi-physical simulation experiment of the parallel five-link mechanical arm. The application can model complex dynamics of the parallel five-link mechanical arm and realize exponential tracking control, and can effectively improve the safety, reliability and steering stability of the operation of the parallel five-link mechanical arm.

Description

Technical Field

[0001] This invention belongs to the field of robotic arm control technology, and relates to an exponential tracking control method for a parallel five-bar robotic arm based on an event-triggered strategy. Background Technology

[0002] Parallel five-bar linkage robotic arms outperform serial robotic arms in terms of dynamic behavior, load capacity, precision, rigidity, and construction cost, and are widely used in various fields, including non-industrial sectors such as space, seabed, nuclear power plants, and medical services, as well as industrial sectors such as loading and unloading, welding, and assembly. Today, remotely operated robotic arms are increasingly prevalent, enabling autonomous task execution in dangerous or hard-to-reach locations.

[0003] Parallel five-bar linkage robotic arms, as an important branch of remotely controlled robotic arms, play a crucial role in autonomous task execution. It is worth noting that parallel five-bar linkage robotic arms typically operate in environments with various uncertainties, including but not limited to payload, length, mass, and inertia. Furthermore, the structural and dynamic characteristics of parallel five-bar linkage robotic arms are more complex than those of serial robotic arms, with key features including a large working range, complex system dynamics, and therefore strong nonlinear characteristics. These factors make the modeling and control research of parallel five-bar linkage robotic arms extremely scientific and challenging.

[0004] Currently, researchers have conducted a series of modeling, adjustment, and control studies on robotic arms with various uncertainties. These include: adaptive parameter control for tracking and prediction errors; adaptive fuzzy output feedback control for single-link robotic arms with non-rigid joint brushed DC motors; optimal control for robotic arms with uncertain loads; and online gravity-compensated adaptive control for n-link robotic arms. It is worth noting that all of these control schemes are based on continuous-time control. In practice, most control algorithms are implemented using digital computers, and the control of robotic arms inevitably involves considering encoder resolution and sampling rate. Due to the limitations and shortcomings of traditional continuous-time control, a significant amount of control resources are wasted during robotic arm control, which has a significant negative impact on both control cost and effectiveness, thus failing to adequately meet control requirements. Furthermore, existing tracking control technologies cannot effectively estimate the coupling uncertainties of robotic arms, resulting in compromised safety and operational stability. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the prior art and provide an exponential tracking control method for a parallel five-bar linkage manipulator based on an event-triggered strategy. This method can model the complex dynamics of the parallel five-bar linkage manipulator and realize exponential tracking control of the parallel five-bar linkage manipulator based on an event-triggered strategy. It ensures that the unknown parameter vector is estimated to an unknown constant vector within a finite time, thereby improving the safety, reliability and operational stability of the parallel five-bar linkage manipulator.

[0006] To solve the above-mentioned technical problems, the present invention adopts the following technical solution.

[0007] An exponential tracking control method for a parallel five-bar linkage robotic arm based on an event-triggered strategy is disclosed. The parallel five-bar linkage robotic arm has two degrees of freedom, exhibiting two independent motion numbers on a three-dimensional coordinate axis. It includes two motors, both located on a base serving as one of the linkages. The weight and reaction force of one motor do not directly affect the other motor. The method includes the following steps:

[0008] Step 1: Based on the model structure of the parallel five-bar linkage manipulator, establish the Euler-Lagrange differential equation of the parallel five-bar linkage manipulator; after coordinate transformation, rewrite the Euler-Lagrange differential equation of the parallel five-bar linkage manipulator into the error dynamic equation of the parallel five-bar linkage manipulator; use matrix operations to establish the error state differential equation of the parallel five-bar linkage manipulator.

[0009] Step 2: For the established error state differential equation of the parallel five-bar linkage robot arm, the back-reasoning method, positive definite Lyapunov function and necessary equivalent control principle are used to design the necessary equivalent controller by designing virtual control signals, thereby forming the corresponding closed-loop system.

[0010] Step 3: Apply the least squares method based on the event-triggered mechanism, combined with Fermat's lemma and the optimal control principle, to obtain the optimal solution of the unknown parameter θ of the parallel five-bar linkage robot arm; use calculus theory and Fermat's lemma to generate a specific implementation algorithm for the optimal solution of the unknown parameter θ of the parallel five-bar linkage robot arm; design an exponential tracking controller for the parallel five-bar linkage robot arm based on the event-triggered strategy.

[0011] Step 4: For the closed-loop system formed by the error state differential equation of the parallel five-bar manipulator and the exponential tracking controller of the parallel five-bar manipulator based on the event-triggered strategy designed in Steps 2 and 3, a stability analysis is performed: According to the threshold conditions (11) and (12) of the event-triggered strategy, with the help of the Lyapunov function and the derivative form of the Lyapunov function, using the Lyapunov stability theory and matrix theory, it is proved that the Zeno phenomenon will not occur; then, using the existence and continuity of the system solution and (10), and with the help of the fact that the Zeno phenomenon will not occur, it is proved that e(t) will not have a finite escape phenomenon on the interval [0, +∞), that is: the angle tracking error of the parallel five-bar manipulator The angular velocity tracking error is defined and globally bounded in the interval [0, +∞). Based on the Lyapunov function, the derivative form of the Lyapunov function, and the error variable formula (14), the norms of the angle tracking error e1(t) and angular velocity tracking error e2(t) of the parallel five-bar linkage are directly obtained, and then it is concluded that the angle tracking error e1(t) and angular velocity tracking error e2(t) of the parallel five-bar linkage converge to the origin in exponential form. With the help of the event triggering strategy threshold conditions (11) and (12), the Lyapunov function, the derivative form of the Lyapunov function, and formulas (24)-(26), it is proved that the unknown parameter vector of the parallel five-bar linkage converges in finite time t. * =max 1＜i≤6 {r i , iT} converges to an unknown constant parameter vector.

[0012] Specifically, in step 1, based on the model of the parallel five-bar linkage manipulator, the Euler-Lagrange differential equation of the parallel five-bar linkage manipulator is established; after coordinate transformation, the Euler-Lagrange differential equation of the parallel five-bar linkage manipulator is rewritten as the error dynamic equation of the parallel five-bar linkage manipulator; using matrix operations, the error state differential equation of the parallel five-bar linkage manipulator is established, the process of which includes:

[0013] Step 1-1: Based on the structure of the parallel five-bar linkage manipulator and combined with the knowledge of manipulator dynamics modeling, the Euler-Lagrange differential equation of the parallel five-bar linkage manipulator is established as follows:

[0014]

[0015] in, Represents the angle of the robotic arm. Represents the angular velocity of the robotic arm. Represents the angular acceleration of the robotic arm. Represents driving torque. Represents an unknown parameter; Represents the inertia matrix. Represents the Coriolis force matrix. Represents the gravity matrix, where

[0016]

[0017]

[0018]

[0019] in, θ3 = m3l2d3, θ4 = m4l1d4, θ5 = m1d1 + m3d3 + m4l1, θ6 = m2d2 + m3l2 - m4d4; for i = 1, 2, 3, 4, m i Let l represent the mass of the i-th link. i I represents the length of the i-th link. i Let represent the moment of inertia of the i-th link; D(q, θ) is a symmetric positive definite matrix, and has the following formula:

[0020]

[0021] Step 1-2: Based on the Euler-Lagrange differential equation of the parallel five-bar linkage manipulator established in Step 1-1, after coordinate transformation, the Euler-Lagrange differential equation of the parallel five-bar linkage manipulator is rewritten as the error dynamic equation of the parallel five-bar linkage manipulator; using matrix operations, the error state differential equation of the parallel five-bar linkage manipulator is established:

[0022] For ease of description, the following coordinate transformation is performed:

[0023]

[0024] in, It is a continuously differentiable ideal reference signal; therefore, the Euler-Lagrange differential equation (1) of the parallel five-bar linkage manipulator can be written as:

[0025]

[0026] The error dynamic equation of the parallel five-bar linkage robotic arm is:

[0027]

[0028] Through matrix operations, the error state differential equation of the parallel five-bar linkage robotic arm is established:

[0029]

[0030] in,

[0031]

[0032]

[0033] Specifically, in step 2, the error state differential equation of the established parallel five-bar linkage robotic arm is used to design a necessary equivalent controller by employing the back-reasoning method, positive definite Lyapunov function, and the principle of necessary equivalent control, after designing virtual control signals, thus forming a corresponding closed-loop system. The process includes:

[0034] Step 2-1: Introduce two new error variables: error variable z1 is the output signal, and error variable z2 is the difference between the second state signal and the virtual control signal.

[0035]

[0036] Introducing positive definite Lyapunov functions Combining the backstepping method, a virtual control law α1(e1) = -Ae1 is designed, where A is a two-dimensional constant matrix; according to the error dynamic equation (6) and error variable formula (14) of the parallel five-bar linkage robotic arm, the derivative formula of V1 is:

[0037]

[0038] Step 2-2: Combining the designed virtual control law α1, introduce a positive definite Lyapunov function V, and design an actual controller such that the derivative of the Lyapunov function V... The negative definite function is used, and the necessary equivalence principle is applied to design a necessary equivalent controller, thereby realizing exponential tracking control of the parallel five-bar linkage robotic arm; a positive definite Lyapunov function is chosen as... According to the derivative formula (15) of V1, we can obtain:

[0039]

[0040] The actual controller is designed as follows:

[0041]

[0042] Where k(θ, e) is a smooth function and satisfies k(θ, 0) = 0; substituting equation (17) into equation (16), the derivative of the Lyapunov function V satisfies:

[0043]

[0044] Where, λ i (A) and λ i (B) represent the eigenvalues ​​of A and B respectively; according to the principle of necessary equivalence, the necessary equivalent controller is:

[0045]

[0046] Specifically, in step 3, the application of the least squares method based on the event-triggered strategy, combined with Fermat's lemma and the optimal control principle, yields the optimal solution for the unknown parameter θ of the parallel five-bar linkage robotic arm; a specific implementation algorithm for generating the optimal solution for the unknown parameter θ of the parallel five-bar linkage robotic arm is generated using calculus theory and Fermat's lemma; and an exponential tracking controller for the parallel five-bar linkage robotic arm based on the event-triggered strategy is designed, the process of which includes:

[0047] Step 3-1: Integrate both sides of the error state differential equation of the parallel five-bar linkage robot, define new variables, and use Fermat's lemma and optimal control principles to obtain the optimal solution for the unknown parameter θ of the parallel five-bar linkage robot:

[0048] For any s and t ≥ 0, integrating both sides of the error state differential equation (7) of the parallel five-bar linkage robotic arm yields:

[0049]

[0050] make

[0051]

[0052] Combining (20) and (21), we can obtain:

[0053]

[0054] make It is a closed compact set, and is defined as follows:

[0055]

[0056] We can obtain: when hour, It has a minimum value, and the minimum value is γ. i (θ) = 0; in addition, the following equation holds:

[0057] H(t i+1 )=M(t i+1 )θ (24)

[0058] in, M(t i ) is symmetric and positive definite, therefore, when satisfying Optimization problem under given conditions There is a unique solution:

[0059]

[0060] If det(M(t) i+1 ))≠0, according to (24) we can get θ=(M(t) i+1 )) -1 H(ti+1 However, if det(M(t) i+1 If ))=0, then the specific expression for θ cannot be obtained from (24); to solve this problem, the optimal solution for the unknown parameter θ of the parallel five-bar linkage robot arm is obtained:

[0061]

[0062] Where μ>0 is a constant;

[0063] Step 3-2: For the optimal solution of the unknown parameter θ of the parallel five-bar linkage robot, a specific implementation algorithm for generating the optimal solution of the unknown parameter θ of the parallel five-bar linkage robot is generated using calculus theory and Fermat's lemma; an exponential tracking controller for the parallel five-bar linkage robot based on an event-triggered strategy is designed; the following integrator set is defined:

[0064]

[0065] in, According to (21), we can obtain Furthermore, the following formula holds true:

[0066]

[0067] Where λ(())=0; similarly, we can obtain:

[0068]

[0069] Where ξ(0)=0; using (21) again, we can obtain:

[0070]

[0071]

[0072] Where, v(0)=ρ(0)=0; using It can be obtained

[0073]

[0074] in, Based on the definitions of H(t), M(t), ρ(t) and χ(t) and Fermat's lemma, the specific implementation algorithm for generating the optimal solution (26) of the unknown parameters θ of the parallel five-bar linkage manipulator is as follows:

[0075]

[0076] Design an exponential tracking controller for a parallel five-bar linkage robotic arm based on an event-triggered strategy:

[0077]

[0078]

[0079]

[0080] Among them, t i >0 is the trigger point, T>0 is a positive integer, and r i >t i It is a moment determined by an event trigger, ω(s) is a continuous function and satisfies ω(0)=0, and for all s>0, ω(s>0). ∈>0 is a suitable constant, and μ>0 is a large positive constant. I is a positive definite constant matrix, and I2 and I6 are identity matrices of the corresponding dimensions.

[0081] Specifically, in step 4, a stability analysis is performed on the closed-loop system formed by the error state differential equation of the parallel five-bar linkage robotic arm and the event-triggered exponential tracking controller for the parallel five-bar linkage robotic arm designed in steps 2 and 3. The process includes:

[0082] Step 4-1: Based on the event triggering strategy threshold conditions (11) and (12), and with the help of the Lyapunov function and its derivative, using Lyapunov stability theory and matrix theory, prove that the Zeno phenomenon will not occur.

[0083] ①If According to (18), we can obtain in right Integrating both sides from 0 to t, we obtain the following inequality:

[0084]

[0085] Formulas (12) and (34) show that event triggering will not be activated; according to (11), t i+1 =t i +T, similarly, if in At this point, it can be obtained Therefore, Zeno's phenomenon will not occur in this situation;

[0086] ②If According to (26), we can obtain Subtracting formula (24) from this inequality yields in addition, The existence means Using (24) again, we can obtain definition Where t∈[0,τ), For any η∈Δ(M(t)), M(t)η=0 holds; note that Therefore, for all s∈[0, t], This is equivalent to M(t)η=0; according to (21) and the continuity of g(t, e(t), u(t), θ), for t∈[0, τ), we can obtain: g(t, e(t), u(t), θ)η=0 holds if and only if M(t)η=0 holds; therefore, Established; noted Head We can obtain Δ(M(t) i+1 )) in Δ(M(t) i The definition of Δ(M(t1)) indicates that there are at most 6 linearly independent solutions, combined with... We can conclude that Δ(M(t6)) is an empty set, which indicates that That is: parameter estimation A maximum of 6 switches can occur; after a switch is completed, the event triggers the execution of step ①; therefore, the Zeno phenomenon will not occur.

[0087] Step 4-2: Utilizing the existence and continuity of system e(t) and (10), and with the aid of the fact that Zeno's phenomenon will not occur as proven in Step 4-1, prove that e(t) will not experience finite escape on the interval [0, +∞). Combining the fact that neither Zeno's phenomenon nor finite escape occurs, further prove that the angle tracking error and angular velocity tracking error of the parallel five-bar linkage are defined and globally bounded on the interval [0, +∞). Based on the existence and continuity of the solution, if no event is triggered, e(t) can be defined in the largest time interval. Above, among which For finite positive constants or ∞; to prove that e(t) lies in the interval... The boundedness of e(t) on the interval t suffices to prove that e(t) lies on the interval t t suffices. Boundedness on; in fact, Furthermore, based on the boundedness of e(t) on the interval [t1, t2), we can obtain definition At this time, with the help of It can be proven that e(t) is bounded on the interval [t3, t4), where The following proof can be performed by repeatedly using the same method and steps as above: According to (34), we can obtain e(t) in the interval [t] i , t i+1 ), The boundedness of the interval; therefore, e(t) can be defined on the interval. Above; based on the proof in step 4-1 that Zeno's phenomenon will not occur, we can conclude that... This shows that e(t) will not experience finite escape on the interval [0, +∞). Combining the fact that neither Zeno's phenomenon nor finite escape occurs, it further proves that the angle tracking error and angular velocity tracking error of the parallel five-bar linkage robot are defined and globally bounded on the interval [0, +∞).

[0088] Step 4-3: Based on the Lyapunov function, the derivative form of the Lyapunov function, and the error variable formula (14), the norms of the angle tracking error e1(t) and angular velocity tracking error e2(t) of the parallel five-bar linkage are directly obtained, and then it is concluded that the angle tracking error e1(t) and angular velocity tracking error e2(t) of the parallel five-bar linkage converge to the origin in exponential form; Integrating both sides of (18) from 0 to t, we can obtain:

[0089]

[0090] According to (14) and (35), we can obtain:

[0091]

[0092] Therefore, the angle tracking error e1(t) and angular velocity tracking error e2(t) of the parallel five-bar linkage robotic arm converge to the origin in an exponential manner;

[0093] Step 4-4: Using the event-triggered strategy threshold conditions (11) and (12), the Lyapunov function, the derivative form of the Lyapunov function, and formulas (24)-(26), it is proved that the unknown parameter vector of the parallel five-bar linkage robot arm is within a finite time t. * =max 1＜i≤6 {r i , iT} converges to an unknown constant parameter vector; according to step 4-1, if Parameter estimation A maximum of 6 handovers will occur; assuming Where i∈{2, 3, 4, 5, 6}; if i=6, according to (18) we can obtain:

[0094] V(e(t))≤V(e(t i )) <V(e(t i ))+ω(e(t i ))+∈,t∈[t i , t i+1 )

[0095] Where i≥6; therefore, t can be obtained by means of (11) and (12). i+1 =t i +T; According to (24)-(26), we can obtain i≥6; similarly, if i takes the values ​​5, 4, 3, and 2, the same conclusion can be reached, that is: for t≥t i , i∈{2, 3, 4, 5, 6}, Established.

[0096] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0097] 1. The present invention is based on Figure 2 The model of the parallel five-bar linkage manipulator shown is used to establish the Euler-Lagrange differential equations of the manipulator, based on knowledge of manipulator dynamics modeling. Through a clever coordinate transformation, the Euler-Lagrange differential equations of the parallel five-bar linkage manipulator are rewritten as the error dynamic equations of the manipulator. Matrix operations are then used to establish the error state differential equations of the parallel five-bar linkage manipulator. This modeling method can accurately describe the dynamic properties of the parallel five-bar linkage manipulator and also opens up new avenues for controlling the parallel five-bar linkage manipulator using backstepping.

[0098] 2. This invention addresses the error state differential equations of a parallel five-bar linkage manipulator. Utilizing the back-reasoning method, positive definite Lyapunov functions, the principle of necessary equivalent control, and the optimal control principle, a necessary equivalent controller is designed by creating a virtual control signal. This control strategy enables exponential tracking control of the parallel five-bar linkage manipulator while avoiding computational explosion, thus improving the reliability of the manipulator's operation.

[0099] 3. Based on the requirements of operational safety and maneuverability of a parallel five-bar linkage manipulator, this invention, utilizing backstepping, optimal control principles, necessary equivalent control principles, and event triggering mechanisms, designs a control scheme that estimates the unknown parameter vector to an unknown constant vector within a finite time for the established error state differential equation of the parallel five-bar linkage manipulator. This control scheme ensures that the unknown parameter vector of the parallel five-bar linkage manipulator converges to an unknown constant vector within a finite time. This characteristic not only prevents the deterioration of the parallel five-bar linkage manipulator structure but also ensures the operational safety of the parallel five-bar linkage manipulator and saves control resources.

[0100] 4. This invention addresses the exponential tracking control problem of a parallel five-bar linkage robotic arm, employing novel mathematical tools, including a series of coordinate transformations, proof by contradiction, matrix operation theory, Fermat's lemma, optimal control principles, and least squares methods. The use of these mathematical tools simplifies controller design and stability analysis, and enables precise capture of the nonlinear characteristics of the parallel five-bar linkage robotic arm. Attached Figure Description

[0101] Figure 1 This is a flowchart of a method according to an embodiment of the present invention.

[0102] Figure 2 This is a model diagram of a parallel five-bar linkage robotic arm according to an embodiment of the present invention.

[0103] Figure 3-a This is a schematic diagram of a serial structure model in the prior art.

[0104] Figure 3-b This is a schematic diagram of a parallel structure model according to an embodiment of the present invention.

[0105] Figure 4 The parameter estimation response curve is shown in the hardware-in-the-loop simulation of the exponential tracking control of a parallel five-bar linkage robotic arm according to an embodiment of the present invention.

[0106] Figure 5 The image shows the tracking response curve of the joint angle q1 in a hardware-in-the-loop simulation of the exponential tracking control of a parallel five-bar linkage robotic arm according to an embodiment of the present invention.

[0107] Figure 6 The image shows the tracking response curve of the joint angle q2 in a hardware-in-the-loop simulation of the exponential tracking control of a parallel five-bar linkage robotic arm according to an embodiment of the present invention.

[0108] Figure 7 A hardware-in-the-loop simulation of joint angular velocity for exponential tracking control of a parallel five-bar linkage robotic arm according to an embodiment of the present invention. The tracking response curve.

[0109] Figure 8 A hardware-in-the-loop simulation of joint angular velocity for exponential tracking control of a parallel five-bar linkage robotic arm according to an embodiment of the present invention. The tracking response curve.

[0110] Figure 9 The response curve of the Lyapunov function V(e) in the hardware-in-the-loop simulation of the exponential tracking control of a parallel five-bar linkage robotic arm according to an embodiment of the present invention. Detailed Implementation

[0111] The parallel five-bar linkage robotic arm described in this invention has two degrees of freedom. The base serves as one of the links, and both motors are located on the base. Therefore, the weight and reaction force of one motor have no direct impact on the other motor. Figure 3-a As shown, in the serial drive mechanism, the lower link is driven by motor 1 located on the base, and the upper link is driven by motor 2 located between the lower and upper links. The weight of motor 2 also becomes the load on motor 1, and the reaction force of motor 2 also affects motor 1. Compared with a serial drive structure with the same motor power and the same working space, as... Figure 2 , Figure 3-b As shown, the parallel drive structure has lower power consumption, especially for Figure 2The parallel five-bar linkage shown has invariant and decoupled inertia tensors in its links. These characteristics simplify the dynamics of the parallel five-bar linkage and result in greater structural stiffness. Compared with serial structures, parallel five-bar linkages offer advantages such as lower power consumption, simpler dynamics, higher structural stiffness, and no direct influence between the two generators. Based on the principles of optimal control and necessary equivalent control, this invention designs an exponential tracking control method for the parallel five-bar linkage based on an event-triggered strategy. This method achieves exponential tracking of the angle and angular velocity of the parallel five-bar linkage with respect to the desired signal, significantly reducing the costly computation. Furthermore, because it can effectively estimate unknown parameters within a finite time, it exhibits robustness to coupling uncertainties.

[0112] The present invention will now be described in further detail with reference to the accompanying drawings.

[0113] Figure 1 This is a flowchart illustrating a method according to an embodiment of the present invention. Figure 1 As shown, the present invention provides an exponential tracking control method for a parallel five-bar linkage robotic arm based on an event-triggered strategy, comprising the following steps:

[0114] Step 1, based on Figure 2 The model structure of the parallel five-bar linkage manipulator shown is used to establish the Euler-Lagrange differential equations of the parallel five-bar linkage manipulator, based on knowledge of manipulator dynamics modeling. Through a clever coordinate transformation, the Euler-Lagrange differential equations of the parallel five-bar linkage manipulator are rewritten as the error dynamic equations of the parallel five-bar linkage manipulator. Using matrix operations, the error state differential equations of the parallel five-bar linkage manipulator are established.

[0115] The specific process of step 1 is as follows:

[0116] Step 1-1, based on Figure 2 The structure of the parallel five-bar linkage manipulator shown is used to establish the Euler-Lagrange differential equation of the parallel five-bar linkage manipulator by combining the knowledge of manipulator dynamics modeling.

[0117] The Euler-Lagrange differential equation for a parallel five-bar linkage robotic arm is:

[0118]

[0119] in, Represents the angle of the robotic arm. Represents the angular velocity of the robotic arm. Represents the angular acceleration of the robotic arm. Represents driving torque. Represents an unknown parameter. Represents the inertia matrix. Represents the Coriolis force matrix. Represents the gravity matrix, where

[0120]

[0121]

[0122]

[0123] in, θ3 = m3l2d3, θ4 = m4l1d4, θ5 = m1d1 + m3d3 + m4l1, θ6 = m2d2 + m3l2 - m4d4. For i = 1, 2, 3, 4, m i Let l represent the mass of the i-th link. i I represents the length of the i-th link. i Let q represent the moment of inertia of the i-th link. D(q, θ) is a symmetric positive definite matrix and has the following formula:

[0124]

[0125] Step 1-2: Based on the Euler-Lagrange differential equation of the parallel five-bar linkage manipulator established in Step 1-1, through a clever coordinate transformation, the Euler-Lagrange differential equation of the parallel five-bar linkage manipulator is rewritten as the error dynamic equation of the parallel five-bar linkage manipulator. Using matrix operations, the error state differential equation of the parallel five-bar linkage manipulator is established. First, for ease of description, the following coordinate transformation is performed:

[0126]

[0127] in, It is a continuously differentiable ideal reference signal. Therefore, the Euler-Lagrange differential equation (1) for the parallel five-bar linkage manipulator can be written as:

[0128]

[0129] The error dynamic equation of the parallel five-bar linkage robotic arm is:

[0130]

[0131] Through matrix operations, the error state differential equation of the parallel five-bar linkage robotic arm is established:

[0132]

[0133] in,

[0134]

[0135]

[0136] Step 2: For the established error state differential equation of the parallel five-bar linkage robot arm, the design of the necessary equivalent controller is completed by using the back-reasoning method, positive definite Lyapunov function and necessary equivalent control principle after designing virtual control signal, thereby forming the corresponding closed-loop system.

[0137] Step 2 includes the following process:

[0138] Step 2-1: Introduce two new error variables: error variable z1 is the output signal, and error variable z2 is the difference between the second state signal and the virtual control signal. This prepares for the application of the backstepping method. The formulas for the introduced error variables are as follows:

[0139]

[0140] Introducing positive definite Lyapunov functions Combining the backstepping method, a virtual control law α1(e1) = -Ae1 is designed, where A is a two-dimensional constant matrix. Based on the error dynamic equation (6) and error variable formula (14) of the parallel five-bar linkage robotic arm, the derivative formula of V1 is:

[0141]

[0142] Step 2-2: Combining the designed virtual control law α1, introduce a positive definite Lyapunov function V, and design an actual controller such that the derivative of the Lyapunov function V... The negative definite Lyapunov function is used to design a necessary equivalent controller based on the principle of necessary equivalence, thereby achieving exponential tracking control of the parallel five-bar linkage robotic arm. A positive definite Lyapunov function is chosen as... According to the derivative formula (15) of V1, we can obtain:

[0143]

[0144] Next, the actual controller is designed as follows:

[0145]

[0146] Where k(θ, e) is a smooth function and satisfies k(θ, 0) = 0. Substituting equation (17) into equation (16), the derivative of the Lyapunov function V satisfies:

[0147]

[0148] Where, λ i (A) and λ i (B) represent the eigenvalues ​​of A and B, respectively. According to the principle of necessary equivalence, the necessarily equivalent controller is:

[0149]

[0150] At this point, the design of the necessary equivalent controller for the exponential tracking control of the parallel five-bar linkage robotic arm based on the event-triggered strategy has been completed.

[0151] Step 3: Instead of traditional adaptive compensation techniques, the least squares method based on an event-triggered strategy is applied, combined with Fermat's lemma and optimal control principles, to obtain the optimal solution for the unknown parameters θ of the parallel five-bar linkage manipulator. Next, a specific implementation algorithm for generating the optimal solution for the unknown parameters θ of the parallel five-bar linkage manipulator is developed using calculus theory and Fermat's lemma. Finally, to save communication and computational resources, an exponential tracking controller for the parallel five-bar linkage manipulator based on an event-triggered strategy is designed.

[0152] Step 3 includes the following process:

[0153] Step 3-1: Integrate both sides of the error state differential equation of the parallel five-bar linkage robot, define new variables, and use Fermat's lemma and optimal control principles to obtain the optimal solution for the unknown parameter θ of the parallel five-bar linkage robot:

[0154] For any s and t ≥ 0, integrating both sides of the error state differential equation (7) of the parallel five-bar linkage robotic arm yields:

[0155]

[0156] For ease of expression, let

[0157]

[0158] Combining (20) and (21), we can obtain:

[0159]

[0160] make It is a closed compact set, and is defined as follows:

[0161]

[0162] It can be obtained when hour, It has a minimum value, and the minimum value is γ. i (θ) = 0. Furthermore, the following equation holds:

[0163] H(t i+1 )=M(t i+1 )θ (24)

[0164] in, M(t i ) is symmetric and positive definite, therefore, when satisfying Optimization problem under given conditions There is a unique solution:

[0165]

[0166] If det(M(t) i+1 ))≠0, according to (24) we can get θ=(M(t) i+1 )) -1 H(t i+1 However, if det(M(t) i+1 If ))=0, then the specific expression for θ cannot be obtained from (24). To solve this problem, the optimal solution for the unknown parameter θ of the parallel five-bar linkage robot arm is obtained:

[0167]

[0168] Where μ>0 is a constant.

[0169] Step 3-2: For the optimal solution of the unknown parameter θ of the parallel five-bar linkage manipulator, a specific implementation algorithm is generated using calculus theory and Fermat's lemma. To save communication and computing resources, an exponential tracking controller for the parallel five-bar linkage manipulator based on an event-triggered strategy is designed. The following integrator set is defined:

[0170]

[0171] in, According to (21), we can obtain Furthermore, the following equation holds true:

[0172]

[0173] Where λ(0) = 0. Similarly, we can obtain:

[0174]

[0175] Where ξ(0)=0. Using (21) again, we can obtain:

[0176]

[0177]

[0178] Where, v(0) = ρ(0) = 0. Using... It can be obtained

[0179]

[0180] in, Based on the definitions of H(t), M(t), ρ(t) and χ(t) and Fermat's lemma, the specific implementation algorithm for generating the optimal solution (26) of the unknown parameters θ of the parallel five-bar linkage manipulator is as follows:

[0181]

[0182] To save communication and computing resources, an exponential tracking controller for a parallel five-bar linkage robotic arm based on an event-triggered strategy is designed:

[0183]

[0184]

[0185]

[0186] Among them, t i >0 is the trigger point, T>0 is a positive integer, and r i >t i It is a moment determined by an event trigger, ω(s) is a continuous function and satisfies ω(0)=0, and for all s>0, ω(s>0). ∈>0 is a suitable constant, and μ>0 is a large positive constant. I is a positive definite constant matrix, and I2 and I6 are identity matrices of the corresponding dimensions.

[0187] Step 4: Stability analysis is performed on the closed-loop system formed by the error state differential equation (7) of the parallel five-bar manipulator and the exponential tracking controller of the parallel five-bar manipulator based on the event-triggered strategy designed in Steps 2 and 3. According to the threshold conditions (11) and (12) of the event-triggered strategy, the Zeno phenomenon is proven to be non-existent by means of the Lyapunov function and the derivative form of the Lyapunov function, using Lyapunov stability theory and matrix theory. Then, by means of the existence and continuity of the system solution and (10), and by means of the non-existence of the Zeno phenomenon, it is proven that e(t) will not experience finite escape on the interval [0, +∞). In other words, the angle tracking error and angular velocity tracking error of the parallel five-bar manipulator are defined and globally bounded on the interval [0, +∞). Next, based on the Lyapunov function, the derivative form of the Lyapunov function, and the error variable formula (14), the norms of the angle tracking error e1(t) and angular velocity tracking error e2(t) of the parallel five-bar linkage are directly obtained, thus proving that the angle tracking error e1(t) and angular velocity tracking error e2(t) of the parallel five-bar linkage converge to the origin in exponential form. Finally, with the help of the event triggering strategy threshold conditions (11) and (12), the Lyapunov function, the derivative form of the Lyapunov function, and formulas (24)-(26), it is proved that the unknown parameter vector of the parallel five-bar linkage converges to the origin in a finite time t. * =max 1＜i≤6 {r i The vector converges to an unknown constant parameter vector within the range of iT. Stability analysis in this step proves that the designed event-triggered parallel five-bar linkage exponential tracking controller can achieve exponential tracking control of the parallel five-bar linkage.

[0188] Step 4 includes the following process:

[0189] Step 4-1: Based on the threshold conditions (11) and (12) of the event triggering strategy, and with the help of the Lyapunov function and the derivative form of the Lyapunov function, and using the Lyapunov stability theory and matrix theory, prove that the Zeno phenomenon will not occur.

[0190] ①If According to (18), we can obtain in right Integrating both sides from 0 to t, we obtain the following inequality:

[0191]

[0192] (12) and (34) indicate that event triggering will not be activated. According to (11), t i+1 =t i +T. Similarly, if in At this point, it can be obtained Therefore, Zeno's phenomenon will not occur in this situation.

[0193] ②If According to (26), we can obtain Subtracting equation (24) from this inequality yields... besides, The existence means Using (24) again, we can obtain definition Where t∈[0τ), For any η∈Δ(M(t)), M(t)η=0 holds. Note that Therefore, for all s∈[0, t], This is equivalent to M(t)η=0. Based on (21) and the continuity of g(t, e(t), u(t), θ), for t∈[0, τ), we can obtain that g(t, e(t), u(t), θ)η=0 holds if and only if M(t)η=0 holds. Therefore, Established. Note. Head We can obtain Δ(M(t) i+1 )) in Δ(M(t) i The definition of Δ(M(t1)) indicates that there are at most 6 linearly independent solutions, combined with... We can conclude that Δ(M(t6)) is an empty set, which indicates that In other words, parameter estimation A maximum of 6 switches can occur. After a switch is complete, the event triggers step ①. Therefore, Zeno's phenomenon will not occur.

[0194] Step 4-2: Utilizing the existence and continuity of system e(t) and (10), and with the aid of the fact that Zeno's phenomenon will not occur as proven in Step 4-1, prove that e(t) will not experience finite escape on the interval [0, +∞). Combining the fact that neither Zeno's phenomenon nor finite escape occurs, further prove that the angle tracking error and angular velocity tracking error of the parallel five-bar linkage are defined and globally bounded on the interval [0, +∞). Based on the existence and continuity of the solution, if no event is triggered, e(t) can be defined in the largest time interval. Above, among which The integer is a finite positive constant or ∞. To prove that e(t) lies on the interval... The boundedness of e(t) on the interval t suffices to prove that e(t) lies on the interval t t suffices. Boundedness on. In fact, Furthermore, based on the boundedness of e(t) on the interval [t1, t2), we can obtain definition At this time, with the help of It can be proven that e(t) is bounded on the interval [t3, t4), where The following proof can be performed by repeatedly using the same method and steps as above. According to (34), we can obtain e(t) in the interval [t] i , t i+1 ), The boundedness of e(t) on the interval [0, 1]. Therefore, e(t) can be defined on the interval [0, 1]. Above. Furthermore, based on the proof in step 4-1 that Zeno's phenomenon will not occur, we can conclude that... This shows that e(t) will not experience finite escape on the interval [0, +∞). Combining the fact that neither Zeno's phenomenon nor finite escape occurs, it is further proven that the angle tracking error and angular velocity tracking error of the parallel five-bar linkage robotic arm are defined and globally bounded on the interval [0, +∞).

[0195] Step 4-3: Based on the Lyapunov function, the derivative form of the Lyapunov function, and the error variable formula (14), the norms of the angle tracking error e1(t) and angular velocity tracking error e2(t) of the parallel five-bar linkage are directly obtained, and then it is concluded that the angle tracking error e1(t) and angular velocity tracking error e2(t) of the parallel five-bar linkage converge to the origin in exponential form. Integrating both sides of (18) from 0 to t, we can obtain:

[0196]

[0197] According to (14) and (35), we can obtain:

[0198]

[0199] Therefore, the angle tracking error e1(t) and angular velocity tracking error e2(t) of the parallel five-bar linkage robotic arm converge to the origin in an exponential manner.

[0200] Step 4-4: Using the event-triggered strategy threshold conditions (11) and (12), the Lyapunov function, the derivative form of the Lyapunov function, and formulas (24)-(26), it is proved that the unknown parameter vector of the parallel five-bar linkage robot arm is within a finite time t. * =max 1<i≤6 {r i , iT} converges to an unknown constant parameter vector. According to step 4-1, if Parameter estimation A maximum of 6 switches can occur. Now, assume... Where i∈{2, 3, 4, 5, 6}. If i=6, according to (18):

[0201] V(e(t))≤V(e(t i )) <V(e(t i ))+ω(e(t i ))+∈,t∈[t i , t i+1 )

[0202] Where i ≥ 6. Therefore, by means of (11) and (12), we can obtain t. i+1 =t i +T. According to (24)-(26), we can obtain i ≥ 6. Similarly, if i takes the values ​​5, 4, 3, and 2, the same conclusion can be reached. In other words, for t ≥ t... i , i∈{2, 3, 4, 5, 6}, Established. At this point, the unknown parameter vector of the parallel five-bar linkage robotic arm has been successfully implemented within a finite time t. * =max 1<i≤6 {r i Proof that , iT} converges to an unknown constant parameter vector.

[0203] Simulation verification of the present invention:

[0204] Step F1: The parameters of the parallel five-bar linkage robotic arm semi-physical simulation system are as follows: ① Parallel five-bar linkage robotic arm: rated load is 5kg, maximum reach is 646mm, maximum angular velocity is 300° / s, weight is 12kg, m1=2.5kg, l1=0.6m, d1=0.4m, I1=0.15kg·m 2 , m2=2.5kg, l2=0.58m, d2=0.23m, I2=0.25mg·m 2 , m3=2.5kg, l3=0.6m, d3=0.4m, I3=0.25kg·m 2 , m4=2.2kg, l4=0.53m, d4=0.22m, I4=0.28kg·m 2 The maximum permissible moment of inertia is 0.3 kg·m. 2 ② Parallel five-bar linkage robotic arm control cabinet: input voltage is 220V, input frequency is 50 / 60Hz, network port data transmission rate is 100MBit / s, supports RS485 communication, and Ethernet bus is used for control between the controller and the driver. The following are the adjustment parameters for the hardware-in-the-loop simulation of the exponential tracking control of the parallel five-bar linkage robotic arm based on an event-triggered strategy: μ = 10 9 ,∈=0.01, T=3s, V(e)=0.5(||e1|| 2 +1.8||e2|| 2 ), ω(e)=0.8(||e1|| 2+||e2|| 2 In the hardware-in-the-loop simulation of exponential tracking control of a parallel five-bar linkage robotic arm based on an event-triggered strategy, the ideal tracking reference signal is set as...

[0205] Step F2: For the hardware-in-the-loop simulation model of the exponential tracking control of a parallel five-bar linkage robotic arm based on an event-triggered strategy, a virtual control signal α1 and an event-triggered mechanism were designed. Then, the initial value was selected: q0 = [70, 5]T. e 10 = [67, -35] T e 20 = [30 - 6π, 5] T , The parameter estimation response curve of the hardware-in-the-loop simulation of the exponential tracking control of a parallel five-bar linkage robotic arm based on an event-triggered strategy is shown below. Figure 4 As shown, the tracking response curve of joint angle q1 in the hardware-in-the-loop simulation of the exponential tracking control of the parallel five-bar linkage robotic arm based on the event-triggered strategy is as follows: Figure 5 As shown, the tracking response curve of the joint angle q2 in the hardware-in-the-loop simulation of the exponential tracking control of the parallel five-bar linkage robotic arm based on the event-triggered strategy is as follows. Figure 6 As shown, a hardware-in-the-loop simulation of the joint angular velocity of a parallel five-bar linkage robotic arm based on an event-triggered strategy for exponential tracking control is presented. The tracking response curve is as follows Figure 7 As shown, a hardware-in-the-loop simulation of the joint angular velocity of a parallel five-bar linkage robotic arm based on an event-triggered strategy for exponential tracking control is presented. The tracking response curve is as follows Figure 8 As shown, the response curve of the Lyapunov function V(e) of the hardware-in-the-loop exponential tracking control of the parallel five-bar linkage robotic arm based on the event-triggered strategy is as follows: Figure 9 As shown. By Figure 4 It can be seen that the unknown parameter vector of the hardware-in-the-loop simulation of the exponential tracking control of the parallel five-bar linkage robot based on the event-triggered strategy converges to the unknown constant parameter vector in a finite time; from Figure 5 It can be seen that in the hardware-in-the-loop simulation of the exponential tracking control of the parallel five-bar linkage robotic arm based on the event-triggered strategy, the joint angle q1 tracks the ideal reference signal q in an exponential manner. e1 ;Depend on Figure 6 It can be seen that in the hardware-in-the-loop simulation of the exponential tracking control of the parallel five-bar linkage robotic arm based on the event-triggered strategy, the joint angle q2 tracks the ideal reference signal q in an exponential manner. e2 ;Depend on Figure 7 It can be seen that the exponential tracking control of the parallel five-bar linkage robotic arm based on the event-triggered strategy is used in the hardware-in-the-loop simulation of joint angular velocity. Tracking the ideal reference signal in exponential form Depend on Figure 8It can be seen that the exponential tracking control of the parallel five-bar linkage robotic arm based on the event-triggered strategy is used in the hardware-in-the-loop simulation of joint angular velocity. Tracking the ideal reference signal in exponential form Depend on Figure 9 It can be seen that the Lyapunov function V(e) of the exponential tracking control of the parallel five-bar linkage robot arm based on the event-triggered strategy smoothly converges to the origin.

[0206] Step F3, the hardware-in-the-loop simulation tracking response curves of the exponential tracking control of the parallel five-bar linkage robotic arm based on the event-triggered strategy, and the response curve of the Lyapunov function V(e) show that the method proposed in this invention is superior to the adaptive tracking control method and the nonlinear asymptotic tracking control method.

Claims

1. An event-triggered strategy based exponential tracking control method for parallel five-bar manipulator, characterized in that, The parallel five-bar linkage mechanical arm has two degrees of freedom, has two independent movement numbers on three-dimensional space coordinate axes, includes two motors, and the weight and reaction force of one motor do not directly affect the other motor; the method includes the following steps: Step 1, based on the model structure of the parallel five-bar linkage mechanical arm, the Euler-Lagrange differential equation of the parallel five-bar linkage mechanical arm is established; through coordinate transformation, the Euler-Lagrange differential equation of the parallel five-bar linkage mechanical arm is rewritten as the error dynamic equation of the parallel five-bar linkage mechanical arm; the error state differential equation of the parallel five-bar linkage mechanical arm is established by using matrix operation; Step 2, for the error state differential equation of the parallel five-bar linkage mechanical arm established, the inverse method, positive definite Lyapunov function and certain equivalent control principle are used, a virtual control signal is designed, a certain equivalent controller is designed, and a corresponding closed loop system is formed; Step 3, the least square method based on the event trigger mechanism is applied, the Fermat lemma and the optimal control principle are combined, the optimal solution of the unknown parameter θ of the parallel five-bar linkage mechanical arm is obtained; the optimal solution of the unknown parameter θ of the parallel five-bar linkage mechanical arm is generated by using the calculus theory and the Fermat lemma; the parallel five-bar linkage mechanical arm exponential tracking controller based on the event trigger strategy is designed; Step 4, Stability analysis is performed for the closed-loop system formed by the error state differential equations of the parallel five-bar manipulator and the event-triggered strategy based exponential tracking controller designed in Steps 2 and 3: According to the event-triggered strategy threshold conditions (11) and (12), by means of Lyapunov function and its derivative form, using Lyapunov stability theory and matrix theory, it is proved that Zeno phenomenon will not occur; Then, using the existence of system solution and its continuity and formula (10), and by means of the fact that Zeno phenomenon will not occur, it is proved that e(t) will not have finite escape phenomenon in the interval [0, +∞), i.e., the angle tracking error and the angular velocity tracking error of the parallel five-bar manipulator are defined and globally bounded in the interval [0, +∞); According to the Lyapunov function, the derivative form of the Lyapunov function and the error variable formula (14), the norms of the angle tracking error e1(t) and the angular velocity tracking error e2(t) of the parallel five-bar manipulator are directly obtained, and it is further concluded that the angle tracking error e1(t) and the angular velocity tracking error e2(t) of the parallel five-bar manipulator converge to the origin in an exponential form; By means of the event-triggered strategy threshold conditions (11) and (12), the Lyapunov function, the derivative form of the Lyapunov function and formulas (24)-(26), it is proved that the unknown parameter vector of the parallel five-bar manipulator converges to an unknown constant parameter vector within a finite time t * = max 1<i≤6 {r i ,iT} ; In step 1, the model of the parallel five-bar linkage mechanical arm is established, the Euler-Lagrange differential equation of the parallel five-bar linkage mechanical arm is established; through coordinate transformation, the Euler-Lagrange differential equation of the parallel five-bar linkage mechanical arm is rewritten as the error dynamic equation of the parallel five-bar linkage mechanical arm; the error state differential equation of the parallel five-bar linkage mechanical arm is established by using matrix operation, and the specific process includes: Step 1-1, based on the structure of the parallel five-bar linkage mechanical arm, the knowledge of mechanical arm dynamics modeling is combined, and the Euler-Lagrange differential equation of the parallel five-bar linkage mechanical arm is established as: wherein, represents the angle of the robot arm, represents the angular velocity of the robot arm, represents the angular acceleration of the robot arm, represents the driving torque, represents the unknown parameters; represents the inertia matrix, represents the Coriolis force matrix, represents the gravity matrix, wherein wherein, θ3 = m3l2d3, θ4 = m4l1d4, θ5 = ml dl + m3d3 + m4l1, θ6 = m2d2 + m3l2 - m4d4; for i = 1, 2, 3, 4, m i represents the mass of the ith link, l i represents the length of the ith link, I i represents the moment of inertia of the ith link; D(q, θ) is a symmetric positive definite matrix and has the following equation: θ x = (θ3 - θ4) cos(q2 - q1) Step 1-2, according to the Euler-Lagrange differential equation of the parallel five-bar linkage mechanical arm established in step 1-1, the Euler-Lagrange differential equation of the parallel five-bar linkage mechanical arm is rewritten as the error dynamic equation of the parallel five-bar linkage mechanical arm through coordinate transformation; the error state differential equation of the parallel five-bar linkage mechanical arm is established by using matrix operation: In order to describe the convenience, the following coordinate transformation is made: wherein is a continuously differentiable ideal reference signal; then the Euler-Lagrange differential equation (1) for the parallel five-bar robot is written as: The error dynamic equation of the parallel five-bar linkage mechanical arm is: The error state differential equation of the parallel five-bar linkage mechanical arm is established by matrix operation: Wherein, In step 2, for the error state differential equation of the parallel five-bar linkage mechanical arm established, the inverse method, positive definite Lyapunov function and certain equivalent control principle are used, a virtual control signal is designed, a certain equivalent controller is designed, and a corresponding closed loop system is formed, and the specific process includes: Step 2-1, two new error variables are introduced: error variable z1 is the output signal, error variable z2 is the difference between the second state signal and the virtual control signal: Introducing positive definite Lyapunov function Combining with backstepping method, the virtual control law is designed as α1(e1) = -Ae1, where A is a two-dimensional constant matrix; According to the error dynamic equation (6) and the error variable formula (14) of parallel five-bar manipulator, the derivative formula of V1 is: Step 2-2, combined with the designed virtual control law a1, introduce a positive definite Lyapunov function V, design the actual controller, so that the derivative of Lyapunov function V is negative, and complete the design of the necessary equivalent controller by using the necessary equivalent principle, so as to realize the exponential tracking control of parallel five-bar linkage manipulator; the positive definite Lyapunov function is selected as According to the derivative formula (15) of V1, we can get: The actual controller is designed as: where k(θ, e) is a smooth function and satisfies k(θ, 0) = 0; the derivative of Lyapunov function V satisfies: where λ i (A) and λ i (B) represent the eigenvalues of A and B, respectively; and according to the certainty equivalence principle, the certainty equivalence controller is: In step 3, the application obtains the optimal solution of the unknown parameter θ of the parallel five-bar linkage robot based on the least square method of the event-triggered strategy, combines Fermat's principle and the optimal control principle, generates the specific implementation algorithm of the optimal solution of the unknown parameter θ of the parallel five-bar linkage robot by using the calculus theory and Fermat's principle, and designs the exponential tracking controller of the parallel five-bar linkage robot based on the event-triggered strategy, and the specific process includes: Step 3-1, the optimal solution of the unknown parameter θ of the parallel five-bar linkage robot is obtained by integrating the error state differential equation of the parallel five-bar linkage robot, defining a new variable, and combining Fermat's principle and the optimal control principle; For any s and t ≥ 0, the error state differential equation (7) of the parallel five-bar linkage robot is integrated to obtain: Let Combining (20) and (21) can obtain: Let be a compact set and define We can obtain: when hour, It has a minimum value, and the minimum value is γ. i (θ) = 0; in addition, the following equation holds: H(t i+1 ) = M(t i+1 ) θ (24) where M(t i ) is symmetric and positive definite, so the optimal problem under the condition has a unique solution: If det(M(t i+1 ))≠0, according to (24), θ=(M(t i+1 )) -1 H(t i+1 ); however, if det(M(t i+1 ))=0, the specific expression of θ cannot be obtained according to (24); to solve this problem, the optimal solution of the unknown parameter θ of the parallel five-bar linkage robot is obtained: where μ > 0 is a constant; Step 3-2, for the optimal solution of the unknown parameter θ of the parallel five-bar linkage robot, the specific implementation algorithm of the optimal solution of the unknown parameter θ of the parallel five-bar linkage robot is generated by using the calculus theory and Fermat's principle, the exponential tracking controller of the parallel five-bar linkage robot based on the event-triggered strategy is designed, and the following integral group is defined: wherein According to (21) we have and the following equation holds: where λ(0) = 0; similarly, the following can be obtained: where ξ(0) = 0; again by (21), the following can be obtained: where v(0) = p(0) = 0; using This gives wherein, According to the definition of H(t), M(t), p(t) and x(t) and Fermat's principle, the specific implementation algorithm of the optimal solution (26) of the unknown parameters of the parallel five-bar linkage robot is as follows: The exponential tracking controller of the parallel five-bar linkage robot based on the event-triggered strategy is designed: where t i > 0 is the triggering point, T > 0 is a positive constant, r i > t i is a time instant determined by the event triggering, ω(s) is a continuous function and satisfies ω(0) = 0, for all s > 0, ω(s) > 0, > 0 is a suitable constant, μ > 0 is a large positive constant, is a positive constant matrix, I2 and I6 are identity matrices of the corresponding dimensions, 2. The exponential tracking control method for a parallel five-bar manipulator based on an event-triggered strategy according to claim 1, characterized in that, In step 4, the stability of the closed-loop system formed by the error state differential equation of the parallel five-bar linkage robot and the exponential tracking controller of the parallel five-bar linkage robot based on the event-triggered strategy designed in steps 2 and 3 is analyzed, and the specific process includes: Step 4-1, according to the event-triggered strategy threshold conditions (11) and (12), by means of the Lyapunov function and the derivative form of the Lyapunov function, the Lyapunov stability theory and the matrix theory, it is proved that the Zeno phenomenon will not occur: ①If According to (18), we have where For Integrating both sides from 0 to t, we obtain the following inequality: Equations (12) and (34) show that the event trigger is not activated; according to (11) there is t i+1 = t i + T, and, similarly, if where At this point we have Thus, in this case, the Zeno phenomenon does not occur; If According to (26), we have Subtracting (24) from this inequality, we have In addition, The existence of Again, using (24), we have Define where t∈[0,τ), For any η∈Δ(M(t)), M(t)η=0 holds; note that Therefore, for all s∈[0,t], is equivalent to M(t)η=0; according to (21) and the continuity of g(t,e(t),u(t),θ), for t∈[0,τ], we have: g(t,e(t),u(t),θ)η=0 holds if and only if M(t)η=0 holds; therefore, holds; note that and we have Δ(M(t i+1 )) is in the interior of Δ(M(t i )); the definition of Δ(M(t1)) shows that there are at most 6 linearly independent solutions, combined with we have Δ(M(t6)) is an empty set, which shows that that is: the parameter estimation switches at most 6 times; after the switch is completed, the event triggers the execution of step ①; therefore, the Zeno phenomenon does not occur; Step 4-2, with the existence of system e(t) and the continuity and (10), and by virtue of the proof of the Zeno phenomenon not occurring in Step 4-1, it is proved that e(t) does not occur finite escape phenomenon on the interval [0, +∞); combining the Zeno phenomenon and the finite escape phenomenon do not occur, further prove that the angle tracking error and angular velocity tracking error of the parallel five-bar manipulator are defined and globally bounded on the interval [0, +∞); according to the existence and continuity of the solution, if there is no event trigger, e(t) can be defined on the largest time interval where is a finite positive constant or ∞; to prove the boundedness of e(t) on the interval , it is only necessary to prove the boundedness of e(t) on the interval ; in fact, and according to the boundedness of e(t) on the interval [t1, t2), it can be obtained that Definition At this time, with the help of it can be proved that e(t) is bounded on the interval [t3, t4), where By repeatedly using the same way as described above and the steps, the following proof can be performed: according to (34), it can be obtained that e(t) is bounded on the interval i , t i+1 ), ; therefore, e(t) can be defined on the interval ; according to the proof of the Zeno phenomenon not occurring in Step 4-1, it can be obtained that This shows that e(t) does not occur finite escape phenomenon on the interval [0, +∞); combining the Zeno phenomenon and the finite escape phenomenon do not occur, further prove that the angle tracking error and angular velocity tracking error of the parallel five-bar manipulator are defined and globally bounded on the interval [0, +∞); Step 4-3, according to the Lyapunov function, the derivative form of the Lyapunov function and the error variable formula (14), the norms of the angle tracking error e1(t) and the angular velocity tracking error e2(t) of the parallel five-bar linkage robot are directly obtained, and then it is concluded that the angle tracking error e1(t) and the angular velocity tracking error e2(t) of the parallel five-bar linkage robot converge to the origin in an exponential form; the integral of (18) from 0 to t is obtained as follows: According to (14) and (35), the following can be obtained: Therefore, the angle tracking error e1(t) and the angular velocity tracking error e2(t) of the parallel five-bar linkage robot converge to the origin in an exponential form; Step 4-4. By means of event-triggered strategy threshold conditions (11) and (12), Lyapunov function, derivative form of Lyapunov function, and equations (24)-(26), it is proved that the unknown parameter vector of the parallel five-bar manipulator converges to a constant unknown parameter vector in finite time t * = max 1＜i≤6 {r i , iT} according to Step 4-1, if Parameter estimation will switch at most 6 times; suppose where i e {2, 3, 4, 5, 6}; if i = 6, according to (18), we have: V(e(t))≤V(e(t i ))<V(e(t i ))+ω(e(t i ))+∈,t∈[t i ,t i+1 ) where i > 6; thus, by means of (11) and (12) it can be obtained that t i+1 = t i + T; according to (24) - (26), it can be obtained that i > 6; likewise, if i takes the values 5, 4, 3 and 2, the same conclusion can be obtained, i.e. for t > t i , i e {2, 3, 4, 5, 6}, holds.

Images