An adaptive boundary anti-disturbance vibration reduction control method for a flexible manipulator system

Abstract

The present invention discloses an adaptive boundary anti-disturbance vibration reduction control method for a flexible manipulator system, which belongs to the field of flexible manipulator vibration suppression. First, the Hamilton principle is used to model the flexible manipulator system as a distributed parameter system; the input saturation and dead zone problems are transformed into new saturation problems, and an auxiliary system is designed to forward compensate the difference between the control input and the actuator output; the composite influence of matching and non-matching interference on the distributed parameter system is compensated by designing an adaptive law; the Lyapunov direct method is used to combine the barrier Lyapunov function to propose an adaptive boundary controller, and the system stability is proved. Under the action of the proposed control scheme, the slope of the angular displacement tracking error of the flexible manipulator and the arm end deflection is limited to a safe range, and the angular displacement tracking error and the arm deflection can converge to a small neighborhood of zero, thereby realizing the angular displacement tracking and anti-disturbance vibration reduction control of the flexible manipulator system under constraints.

Description

Technical Field

[0001] The invention belongs to the technical field of flexible manipulator vibration suppression, and in particular relates to an adaptive boundary anti-disturbance vibration reduction control method for a flexible manipulator system. Background Art

[0002] Compared with traditional rigid robotic arms, flexible robotic arms have unique advantages in practical applications due to their light weight, high toughness, low energy consumption and other characteristics. They have been widely used in many fields such as aerospace, industrial manufacturing, and medicine. However, the vibration problem caused by the flexible characteristics is the most critical problem to be overcome in the research and application of flexible robotic arm systems.

[0003] In recent years, research on flexible manipulators has achieved certain scientific research results. Existing results usually model the flexible structure system as a concentrated parameter system or a distributed parameter system. Although the latter has a complex model, it can avoid the control overflow problem. Interference is one of the main factors causing vibration of flexible manipulators. Most existing research is devoted to solving the matching interference in the same channel as the controller. However, due to the parameter disturbance, unmodeled dynamics, and external wind field, the non-matching interference caused by factors such as external wind field can directly act on the system state. Therefore, the non-matching interference of the flexible manipulator system in practical applications cannot be ignored. Effective control methods are urgently needed to solve the combined effects of matching and non-matching interference acting on the flexible manipulator system represented by the distributed parameter system.

[0004] In addition, due to the limitations of physical factors, the actuator will experience saturation and dead zone phenomena in response to the input control signal, resulting in a deviation between the actual control input of the system and the ideal control input, affecting the accuracy and stability of the control system; for safety reasons, the slope of the deflection of the end of the robot arm and the angular displacement tracking error need to be limited to a safe range, and output limitation is also a hot topic in the current research field of flexible robot arms; therefore, the research on anti-disturbance and vibration reduction of flexible robot arm systems under input and output limitations has important theoretical and application value. Summary of the invention

[0005] In view of the key issues that need to be considered in the study of the above-mentioned flexible robotic arm system, the present invention proposes an adaptive boundary anti-disturbance vibration reduction control method for a flexible robotic arm system under input and output constraints, which effectively suppresses the vibration of the flexible robotic arm and enables it to stably track the set angular displacement signal.

[0006] An adaptive boundary anti-disturbance vibration reduction control method for a flexible manipulator system comprises the following steps:

[0007] Step 1: Using Hamilton's principle, the flexible robotic arm system is modeled as a distributed parameter system;

[0008] Step 2: Convert the input saturation and dead zone problem into a new saturation problem by designing an auxiliary system to forward compensate the difference between the control input and the actuator output;

[0009] Step 3: Design a parameter adaptive law to compensate for the combined effects of matching and non-matching interference on the distributed parameter system;

[0010] Step 4: Using Lyapunov direct method and combining barrier Lyapunov function, an adaptive boundary controller is proposed and the system stability is proved.

[0011] Furthermore, the distributed parameter system is established using the Hamilton principle, and the main control equation of the system is:

[0012] ; (1)

[0013] The boundary conditions are:

[0014] ; (2)

[0015] ; (3)

[0016] ; (4)

[0017] ; (5)

[0018] ; (6)

[0019] in, , , , , , , , ; is the arm length; It represents the mass per unit length of the arm; is the mass of the effective load at the end of the boom; represents the bending stiffness of the flexible manipulator; is the moment of inertia of the rigid hub; Indicates the offset of the flexible manipulator; Indicates that the flexible robotic arm is Moment, in The vibration amplitude or deflection at ; represents the angular displacement of the flexible manipulator; It is the distributed interference to the system; The control force provided by the end effector of the flexible robot arm is The control torque provided to the actuator at the rigid hub of the flexible manipulator; represents the non-matching interference of the position loop subsystem, represents the matching disturbance of the position loop subsystem, represents the non-matching disturbance of the attitude loop subsystem, represents the matching disturbance of the attitude loop subsystem; express , express , express , express .

[0020] Furthermore, in step 2, the saturation function Described as:

[0021] ; (7)

[0022] in, is a saturation function, and the known constant and denote the upper and lower saturation levels of the actuator output, respectively, and as well as , , represents the control input of the end effector, represents the control input of the actuator at the rigid hub;

[0023] The dead zone function is expressed as:

[0024] ; (8)

[0025] in, The control force or torque provided to the actuator, represents the dead zone function, with known constants and Respectively represent the upper and lower limits of the dead zone range, and satisfy and ; Normal number and represents a known slope;

[0026] In order to deal with input nonlinearity, the right inverse function of the dead zone function is defined as for:

[0027] ; (9)

[0028] in, The control law for the design;

[0029] Combining equations (7), (8) and (9), we get a new saturation function:

[0030] ; (10)

[0031] Therefore, the saturation and dead zone problem of the actuator is transformed into a new saturation problem; in order to compensate for the difference between the control input and the actuator output under the new form of saturation, the form of the designed auxiliary system is described as:

[0032] ; (11)

[0033] ; (12)

[0034] in, and They are the state quantities of the auxiliary systems designed for the position loop and attitude loop, and is the difference between the control input and the actuator output and The upper bound of , , and The control laws designed for the end effector of the flexible manipulator and the actuator at the rigid hub are respectively; is a positive constant; , , and is an undetermined normal number; a normal number , are the upper bounds of the range in which the slope of the flexible manipulator end deflection and the angular displacement tracking error are limited, namely, , , is the angular displacement tracking error; represents a symbolic function; Auxiliary variables for construction.

[0035] Furthermore, in step 3, the form of the parameter adaptation law is described as:

[0036] ; (13)

[0037] ; (14)

[0038] ; (15)

[0039] in, , is a composite interference defined, and there exists a positive constant , and satisfy , as well as , ; , and They are the unknown constants , and The estimated value of , and ; and Auxiliary variables for construction; is a positive constant, , is a positive constant to be determined, .

[0040] Furthermore, in step 4, the form of the adaptive boundary controller is:

[0041] ; (16)

[0042] ; (17)

[0043] The barrier Lyapunov function has the form:

[0044] ; (18)

[0045] ; (19)

[0046] ; (20)

[0047] ;(twenty one)

[0048] in, is the energy term, is the cross term, and For auxiliary items; is a positive constant to be determined, .

[0049] Furthermore, by adopting the proposed adaptive boundary controller, the flexible manipulator can stably track the set angular displacement signal. During this process, the slope of the arm end deflection and the angular displacement tracking error are constrained within the set range. and angular tracking error Converges to the following range:

[0050] ;(twenty two)

[0051] ;(twenty three)

[0052] in,

[0053] ,

[0054] ,

[0055] ,

[0056] ,

[0057] in, , and is a normal number and satisfies ; is a positive constant, .

[0058] Beneficial technical effects brought by the present invention:

[0059] Based on the designed auxiliary system and adaptive law, the adaptive boundary anti-disturbance vibration reduction control scheme is proposed, which effectively compensates for the error between the actuator output and the control input caused by input saturation and dead zone, as well as the combined influence of matching and non-matching interference on the system; the slope of the flexible robot arm's angle tracking error and the arm end deflection are limited to the set safety range; at the same time, by selecting appropriate parameters, the angular displacement tracking error and the arm deflection converge to a small neighborhood of zero, thereby realizing the angular displacement tracking and anti-disturbance vibration reduction control of the flexible robot arm system under constraints. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 It is a schematic structural diagram of the flexible robotic arm system of the present invention.

[0061] Figure 2 The flowchart is a method for adaptive boundary anti-disturbance vibration reduction control of a flexible manipulator of the present invention.

[0062] Figure 3 It is a principle diagram of the adaptive boundary anti-disturbance vibration reduction control method of the present invention. DETAILED DESCRIPTION

[0063] The specific implementation of the present invention is further described below in conjunction with specific embodiments:

[0064] In order to avoid the control overflow problem caused by the centralized parameter model, the present invention uses a distributed parameter model to characterize the flexible manipulator system; for the flexible manipulator system, due to the limitation of physical factors, the actuator often has input saturation and dead zone phenomenon, which makes the deviation between the ideal control input and the actual actuator, affecting the accuracy and stability of the control system. The present invention takes into account the input limitation of the actuator; for safety reasons, the slope of the deflection of the end of the flexible manipulator of the present invention and the angular displacement tracking error will be limited to a safe range. Since the non-matching interference caused by factors such as parameter disturbance, unmodeled dynamics and external wind field can directly act on the system state, the present invention takes into account the combined impact of matching and non-matching interference on the system.

[0065] Based on the above analysis, the present invention proposes an adaptive boundary anti-disturbance vibration reduction control method for a flexible manipulator system under input and output constraints.

[0066] like Figure 1 The flexible robotic arm system shown in the figure consists of a single arm, a rigid hub and an end load; two controllers are located at the rigid hub and at the end of the arm respectively; is the effective load mass at the end of the boom, is the moment of inertia of the rigid hub; is the inertial coordinate system, is the coordinate system of the flexible manipulator arm. Indicates that the flexible robotic arm is at time t. The vibration amplitude or deflection at axis s; Indicates the offset of the flexible manipulator; It is the distributed interference to the system; represents the angular displacement of the flexible manipulator; The control force provided by the end effector of the flexible robot arm is The control torque provided to the actuator at the rigid hub of the flexible manipulator; represents the non-matching interference of the position loop subsystem, represents the matching disturbance of the position loop subsystem, represents the non-matching disturbance of the attitude loop subsystem, Represents the matching interference of the attitude loop subsystem.

[0067] like Figure 2 and Figure 3 As shown, the control method of the present invention specifically includes the following steps:

[0068] Step 1: Using Hamilton's principle, the flexible robotic arm system is modeled as a distributed parameter system;

[0069] Step 1.1, considering the distributed interference and boundary matching interference, the flexible manipulator system is modeled as a distributed parameter system, and the main control equation of the system is:

[0070] ; (1)

[0071] And the boundary conditions are:

[0072] ; (2)

[0073] ; (3)

[0074] ; (4)

[0075] in, , , , , , , , ; is the arm length; represents the mass per unit length of the arm, represents the bending stiffness of the flexible manipulator; , They represent the matching interference of the position loop and attitude loop subsystems respectively.

[0076] Step 1.2: Consider the non-matching interference at the boundary and write the boundary conditions (2) and (3) in the form of state equations:

[0077] ; (5)

[0078] ; (6)

[0079] ; (7)

[0080] ; (8)

[0081] in, Indicates the non-matching interference of the position loop subsystem, Indicates the non-matching interference of the attitude loop subsystem; express , express , express , express .

[0082] Step 2: For input limitation, the input saturation and dead zone problems are transformed into new saturation problems, and the difference between the control input and the actuator output is forward compensated by designing an auxiliary system;

[0083] Step 2.1: First, consider the input saturation and dead zone phenomenon of the actuator, and transform the saturation and dead zone problems into new saturation problems. Saturation function Described as:

[0084] ; (9)

[0085] in, is a saturation function, and the known constant and denote the upper and lower saturation levels of the actuator output, respectively, and as well as , , represents the control input of the end effector represents the control input of the actuator at the rigid hub;

[0086] The dead zone function is expressed as:

[0087] ; (10)

[0088] in, The control force or torque provided to the actuator, represents the dead zone function, with known constants and Respectively represent the upper and lower limits of the dead zone range, and satisfy and ; Normal number and represents a known slope;

[0089] In order to deal with input nonlinearity, the right inverse function of the dead zone function is defined as for:

[0090] ; (11)

[0091] in, The control law for the design;

[0092] Combining equations (7), (8) and (9), we get a new saturation function:

[0093] ; (12)

[0094] Therefore, the actuator saturation and dead zone problem is transformed into a new saturation problem.

[0095] Step 2.2. Design an auxiliary system to forward compensate the difference between the control input and the actuator output. Define the error , , and The control laws designed for the flexible manipulator end effector and the rigid hub actuator are given, and the following assumptions are given:

[0096] Assumption 1: Error , Bounded, that is, there is a normal number , satisfy , , .

[0097] In order to compensate for the difference between the control input and the actuator output under the new form of saturation, the auxiliary system proposed by the present invention is described as follows:

[0098] ; (13)

[0099] ; (14)

[0100] in, and They are the state quantities of the auxiliary systems designed for the position loop and attitude loop, and is the difference between the control input and the actuator output and The upper bound of is a positive constant; , , and is an undetermined normal number; a normal number , are the upper bounds of the range in which the slope of the flexible manipulator end deflection and the angular displacement tracking error are limited, namely, , , is the angular displacement tracking error; Represents a symbolic function.

[0101] Step 3: Design a parameter adaptive law to compensate for the combined effects of various interferences including non-matching interference on the distributed parameter system.

[0102] First, the following assumptions are made:

[0103] Assumption 2: Unknown boundary interference Bounded, that is , there is an unknown positive constant satisfy , .

[0104] Defining composite interference , , from Assumption 2, we know that there is an unknown positive constant and satisfy as well as ; In order to better analyze the stability of the flexible robotic arm system, two auxiliary variables are constructed: , ,in is the angular displacement tracking error, , To set the angular displacement signal.

[0105] The parameter adaptation law is described as:

[0106] ; (15)

[0107] ; (16)

[0108] ; (17)

[0109] in, , and They are the unknown constants , and The estimated value of , , , ; Define the estimation error , and ; and Auxiliary variables for construction; is a positive constant, , is a positive constant to be determined, ;

[0110] Step 4: For output limitation, Lyapunov direct method is used, combined with barrier Lyapunov function, to propose an adaptive boundary controller, and the system stability is proved. The following assumptions and lemmas are used in the process:

[0111] Assumption 3: Unknown distributed interference is bounded, that is, there exists a normal number Make , .

[0112] Lemma 1: For any function , there exists a normal number So that the following inequality holds:

[0113] ; (18)

[0114] Lemma 2: For , and any , , the following inequality holds:

[0115] ; (19)

[0116] in, is a positive real number. If and , then the following inequality holds:

[0117] ; (20)

[0118] ;(twenty one)

[0119] Step 4.1, the form of the adaptive boundary controller is:

[0120] ;(twenty two)

[0121] ;(twenty three)

[0122] in, is a positive constant to be determined, .

[0123] The form of the selected barrier Lyapunov function is:

[0124] ;(twenty four)

[0125] in,

[0126] ; (25)

[0127] ; (26)

[0128] ; (27)

[0129] ; (28)

[0130] in, is the energy term, is the cross term, and As auxiliary items, , and is a normal number and satisfies .

[0131] Step 4.2, barrier Lyapunov function proves its positive definiteness.

[0132] function The positive definiteness condition needs to be satisfied to prove the uniform boundedness of the flexible robotic arm system. Positive definiteness cannot be determined, and Lemma 2 shows that and Relationship: ; (29)

[0133] in, ,

[0134] Combining (26), we can get:

[0135] ; (30)

[0136] Here the parameters , , The selection makes , so the barrier Lyapunov function is selected It is positive.

[0137] Step 4.3, derive the barrier Lyapunov function to prepare for further proof.

[0138] right Taking the derivative, substituting it into equation (1) and combining it with the method of integration by parts, we can obtain:

[0139] ; (31)

[0140] Depend on Knowable , and substituting the boundary condition (4), equation (31) is further written as:

[0141] ; (32)

[0142] Depend on , square both sides of the equation and move the terms to get:

[0143] ; (33)

[0144] Substituting formula (33) into formula (32) and combining it with Lemma 2, we can obtain:

[0145] ; (34)

[0146] in, and is a normal number.

[0147] right The derivative is:

[0148] ; (35)

[0149] make:

[0150] ; (36)

[0151] ; (37)

[0152] ; (38)

[0153] ; (39)

[0154] Substituting equation (1) into equation (36), combined with assumption 3, Lemma 2 and integration by parts, and substituting boundary condition (4), we can obtain:

[0155] ; (40)

[0156] in, is a positive constant. Consider , combining the method of integration by parts, Lemma 2 and the boundary condition (4), we can obtain:

[0157] ; (41)

[0158] in, , is a positive constant. Substituting into equation (1), combining the method of integration by parts, Lemma 2 and Assumption 3, and bringing in boundary conditions (4), we obtain:

[0159] ; (42)

[0160] in, is a positive constant. Consider , combining the method of integration by parts, Lemma 2 and the boundary condition (4), we can obtain: ; (43)

[0161] in, , is a positive constant. Combining (40), (41), (42) and (43), we can obtain:

[0162] ; (44)

[0163] right Regarding the time derivative, substitute into equations (5), (6), (13), (15), (18) and , combined with Lemma 1 and Lemma 2, we can get:

[0164] ; (45)

[0165] in, is a normal number.

[0166] right Regarding the time derivative, substitute into equations (7), (8), (14), (16), (17), (19) and , combined with Lemma 1 and Lemma 2, we can get:

[0167] ; (46)

[0168] in, , , is a normal number.

[0169] consider , as well as , Lemma 2 and Lagrange's mean value theorem, we can get: as well as .in, is a normal number. According to Lemma 2, we can get , is a positive constant. Combining equations (24), (34), (44), (45) and (46), we get Regarding the form of time derivative:

[0170] ; (47)

[0171] in,

[0172] ,

[0173] ,

[0174] ,

[0175] And the parameter selection should satisfy: , , , , , , , , , .

[0176] Step 4.4. Prove that under the action of the proposed control scheme, the deflection and angular displacement tracking errors of the flexible manipulator converge to a small neighborhood of zero.

[0177] Multiply both sides of the inequality (47) by have to:

[0178] ; (48)

[0179] The above formula can be further written as:

[0180] ; (49)

[0181] Taking definite integrals on both sides of the inequality sign in the above equation, we can get:

[0182] ; (50)

[0183] in, yes Combining Lemma 2 and formula (30), we can get:

[0184] ; (51)

[0185] From formula (28), we get:

[0186] ; (52)

[0187] Combining equations (50), (51) and (52), we can obtain:

[0188] ; (53)

[0189] ; (54)

[0190] Further we get:

[0191] ; (55)

[0192] ; (56)

[0193] By selecting appropriate parameters, the deflection of the flexible manipulator arm and angular tracking error It eventually converges to a small neighborhood of zero, and the proof is complete.

[0194] Through the present invention, the following theorem can be obtained:

[0195] Theorem 1: Considering the flexible manipulator system with limited actuator input and output described by equations (1), (4), and (5)-(8), under the proposed adaptive boundary disturbance rejection vibration reduction control schemes (13)-(17), and (22)-(23), assuming that W(0) is bounded, when When the flexible manipulator arm deflection and angular displacement tracking errors converge to a compact set and ,Right now:

[0196] ; (57)

[0197] ; (58)

[0198] The present invention studies the anti-disturbance vibration reduction control problem of a flexible manipulator system considering input-output constraints and the combined influence of multiple interferences, and proposes an adaptive anti-disturbance vibration reduction control scheme: first, the Hamilton principle is used to model the flexible manipulator system as a distributed parameter system; the input saturation and dead zone problems are transformed into new saturation problems, and an auxiliary system is designed to compensate for the error between the control input and the actuator output; then an adaptive law is designed to compensate for the combined influence of matching and non-matching interferences on the system; finally, based on the barrier Lyapunov function, an adaptive boundary controller is proposed to limit the slope of the arm end deflection and the angular displacement tracking error within a safe range, thereby achieving the control goals of arm vibration suppression and tracking the set angular displacement signal; under the proposed adaptive boundary anti-disturbance vibration reduction control scheme, the closed-loop system is proved to be uniformly bounded.

[0199] Of course, the above description is not a limitation of the present invention, and the present invention is not limited to the above examples. Changes, modifications, additions or substitutions made by technicians in this technical field within the essential scope of the present invention should also fall within the protection scope of the present invention.

Claims

1. An adaptive boundary disturbance rejection vibration reduction control method for a flexible manipulator system, characterized in that: The following steps are involved: Step 1: Using Hamilton's principle, the flexible robotic arm system is modeled as a distributed parameter system; Step 2: Convert the input saturation and dead zone problem into a new saturation problem by designing an auxiliary system to forward compensate the difference between the control input and the actuator output; Step 3: Design a parameter adaptive law to compensate for the combined effects of matching and non-matching interference on the distributed parameter system; Step 4: Using the Lyapunov direct method and combining it with the barrier Lyapunov function, an adaptive boundary controller is proposed, and the system stability is proved; In step 1, a distributed parameter system is established using the Hamilton principle, and the main control equation of the system is: φH tt (s,t)=-EIh ssss (s,t)+g(s,t) (1) The boundary conditions are: h(0,t)=h s (0,t)=h ss (l,t)=0 (6) in, l is the arm length; φ is the arm mass per unit length; m p is the effective load mass at the end of the arm; EI represents the bending stiffness of the flexible manipulator; I r is the moment of inertia of the rigid hub; H(s,t)=h(s,t)+sΩ(t) represents the offset of the flexible manipulator; h(s,t) represents the vibration amplitude of the flexible manipulator at time t at point s, i.e., the deflection. Ω(t) represents the angular displacement of the flexible manipulator; g(s,t) is the distributed disturbance to the system; U m (t) is the control force provided by the end effector of the flexible manipulator, U r (t) is the control torque provided by the actuator at the rigid hub of the flexible manipulator; d1(t) represents the non-matching interference of the position loop subsystem, d2(t) represents the matching interference of the position loop subsystem, d3(t) represents the non-matching interference of the attitude loop subsystem, and d4(t) represents the matching interference of the attitude loop subsystem; x1(t) represents H(s,t), and x2(t) represents H t (s, t), x3(t) represents Ω(t), x4(t) represents In step 2, the saturation function δ c (t) is described as: Among them, Sat(·) is the saturation function, and the known constant and denote the upper and lower saturation levels of the actuator output, respectively, and as well as ,c=m,r,χ m (t) represents the control input of the end effector, χ r (t) represents the control input of the actuator at the rigid hub; The dead zone function is expressed as: Among them, U c (t) is the control force or control torque provided by the actuator, D(·) represents the dead zone function, and the known constant and Respectively represent the upper and lower limits of the dead zone range, and satisfy and <0; normal number ι 1c and 2c represents a known slope; In order to deal with the input nonlinear problem, the right inverse function D of the dead zone function is defined + (·)for: Among them, u c (t) is the designed control law; Combining equations (7), (8) and (9), we get a new saturation function: Therefore, the saturation and dead zone problem of the actuator is transformed into a new saturation problem; in order to compensate for the difference between the control input and the actuator output under the new form of saturation, the form of the designed auxiliary system is described as: Among them, τ1(t) and τ2(t) are the state quantities of the auxiliary systems designed for the position loop and attitude loop, respectively. and is the difference Δu between the control input and the actuator output m (t) and Δu r (t) where Δu m (t) = U m (t)-u m (t), Δu r (t) = U r (t)-u r (t),u m (t) and u r (t) are the control laws designed for the end effector of the flexible manipulator and the actuator at the rigid hub, respectively; ∈3 is a positive constant; λ1, λ4 and λ6 are constants to be determined; the constants γ, are the slope of the deflection of the flexible manipulator end and the upper bound of the range in which the angular displacement tracking error is limited, that is, |h s (l,t)|<γ, e(t) is the angular displacement tracking error; sign(·) represents the sign function; τ4(t) is the constructed auxiliary variable; In step 3, the form of the parameter adaptation law is described as: Among them, D1(t) and D2(t) are defined composite interferences, and there is a positive constant and satisfy as well as and They are the unknown constants and The estimated value of and τ3(t)=τ1(t)+x1(t)+x2(t)-h sss (l, t) and τ4(t) = τ2(t) + e(t) + x4(t) are constructed auxiliary variables; ∈ j is a normal number, μ j is a positive constant to be determined, j=1,2,3.

2. The adaptive boundary disturbance rejection vibration reduction control method of a flexible manipulator system according to claim 1 is characterized in that: In step 4, the form of the adaptive boundary controller is: The barrier Lyapunov function has the form: Among them, W1(t) is the energy term, W2(t) is the cross term, W3(t) and W4(t) are auxiliary terms; λ k is a positive constant to be determined, k=1,2,3,4,5,6.

3. The adaptive boundary disturbance rejection vibration reduction control method of a flexible manipulator system according to claim 2 is characterized in that: By adopting the proposed adaptive boundary controller, the flexible manipulator can stably track the set angular displacement signal. During the process, the slope of the arm end deflection and the angular displacement tracking error are constrained within the set range, and the arm deflection h(s, t) and the angular displacement tracking error e(t) converge to the following range: in, in, and is a normal number and satisfies ν n is a normal number, n=1,3,4,…,14.

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