A flexible robot arm control method based on a predetermined time stable and high order sliding mode algorithm

Abstract

The application discloses a flexible mechanical arm control method based on predetermined time stability, which is used for controlling a flexible mechanical arm with output constraints under the requirements of meeting predetermined time indexes and high-precision control. Firstly, a standard sliding mode integral chain system is extended to a new high-order sliding mode dynamic system with non-matching terms based on the dynamic behavior of the flexible mechanical arm, and a dynamic model of a motor and a joint is established. A Barrier Lyapunov function satisfying an output constraint condition is adopted. In the case that the upper bound of a gain function of the non-matching term is unknown, an original system is converted through a time-varying scaling function, an auxiliary system for designing a predetermined time controller is constructed, a predetermined time controller based on a high-order sliding mode algorithm is proposed, the control method does not need trial-and-error parameter adjustment, the predictability of the system is further enhanced, and the motion precision and stability of the system are improved under the premise of reducing system energy consumption and mechanical loss.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of flexible robot arm control, and particularly relates to a flexible robot arm control method based on predetermined time stability and high-order sliding mode algorithm. BACKGROUND

[0002] Flexible robot arms play an increasingly important role in industrial automation, with high adaptability and adaptability, they can perform tasks in heavy, complex or changing environments. However, due to the nonlinear, time-varying and coupled nature of flexible robot arms, accurate control becomes a challenging problem. In recent years, control methods based on predetermined time and high-order sliding mode have attracted widespread attention in the field of flexible robot arm control, providing a new approach to overcome the limitations of traditional control methods. Because flexible robot arms will produce bending, vibration and other deformations during movement, these deformations will affect control accuracy and even cause system instability. In addition, it also has nonlinear, time-varying and coupled characteristics, and traditional control methods are difficult to effectively handle these problems.

[0003] The predetermined time control is a method that converts the control problem into an optimization problem. In the control of flexible robot arms, the predetermined time control method can achieve accurate trajectory tracking through the pre-set time trajectory, thereby reducing system vibration and instability. In addition, the predetermined time control can also be optimized according to the actual situation of the system to improve the performance and adaptability of the system. High-order sliding mode control is a control method for nonlinear systems, which improves control accuracy by introducing high-order terms. In the control of flexible robot arms, high-order sliding mode control method can effectively suppress system vibration and instability, while having strong robustness, which can cope with external disturbances and parameter uncertainties.

[0004] Combining the predetermined time control method with the high-order sliding mode control method can fully utilize the advantages of both. The predetermined time control can provide accurate time trajectory, while the high-order sliding mode control can ensure the stability and robustness of the system during trajectory tracking. By reasonably designing the parameters of the controller, accurate control of flexible robot arms in complex environments can be achieved. This method not only improves the motion accuracy and stability of the system, but also reduces the energy consumption and mechanical loss of the system, thereby producing significant economic and social benefits in practical applications. SUMMARY

[0005] Inventive purpose: In order to overcome the deformation such as bending and vibration of the flexible manipulator during movement, which affects the control accuracy and even causes system instability and other problems, the application provides a flexible manipulator control method based on predetermined time stability and high-order sliding mode algorithm, which can realize accurate control of the flexible manipulator in a complex environment, under the condition of containing non-matching items, the high-order sliding mode dynamic system is coordinate-transformed by using a time-varying scaling function, the high-order sliding mode control strategy is used to weaken the chattering influence of the flexible manipulator, and the controller is derived based on the backstepping method, and the Barrier Lyapunov function method is used to ensure the realization of the predetermined time stability convergence of the manipulator movement, so that the high-order sliding mode flexible manipulator control method further improves the accurate control of the flexible manipulator in a complex environment and meets the steady-state performance without violating the output constraint condition.

[0006] Technical scheme: The application discloses a flexible manipulator control method based on predetermined time stability and high-order sliding mode algorithm, which is used for controlling the flexible manipulator with output constraint and comprises the following steps:

[0007] Step 1: a dynamic model of a motor and a joint is established based on the dynamic behavior of the flexible manipulator according to a high-order sliding mode dynamics equation with output constraint and non-matching items;

[0008] Step 2: a Barrier Lyapunov function meeting the output constraint condition is used, in the case that the upper bound of the gain function of the non-matching items is unknown, the original system is converted by using a time-varying scaling function, and an auxiliary system for designing a predetermined time controller is constructed;

[0009] Step 3: a virtual control variable and an actual control input are designed by using a backstepping recursive method;

[0010] Step 4: the virtual control variable and the actual control input in step 3 are substituted into the derivative of the Barrier Lyapunov function, and it is verified whether the derivative of the Barrier Lyapunov function can meet Continue step 4, if the closed-loop system variable cannot be ultimately consistent and bounded, return to step 3 to redesign the virtual control variable and the actual control input; wherein, V n is used for describing the system, which is a continuous positive scalar function; is the rate of change of the system with time, which is continuous negative; c is a negative feedback gain, which is a normal number; alpha is a nonlinear response of the system, and is in (0, 1); delta is an external excitation, which represents the influence on the system;

[0011] Step 5: Lyapunov second method and pre-time stability analysis are performed for the closed-loop system, which proves that the output constraint condition is not violated, the pre-time control parameters remain consistent and bounded, and the control performance index meets the pre-time control requirements.

[0012] Further, the model of the nonlinear high-order sliding mode system with output constraints is as follows:

[0013]

[0014] In the formula, s i ∈R,i=1,...,n is the output (sliding variable), u∈R is the control input; h0(t,x) and g0(t,x) are unknown smooth functions; x is the system state, t is the time, f i (s i ),i=1,...,n-1 is the non-matching term; let the sliding variable s i The relative degree of the control input u is n, n is the order of the system, that is,

[0015]

[0016] The system model is rewritten as

[0017]

[0018] Further, the nonlinear high-order sliding mode system with output constraints has the following assumptions and lemmas:

[0019] Assumption 1: There exists a known normal number G m and a known positive definite function such that the following conditions are met:

[0020]

[0021] Assumption 2: The upper bound of the gain function of the non-matching term f i (s i ) does not require to be known, but meets the following requirements:

[0022] f i ≤ρ i (t,s i )|s i |,i=1,...,n-1,ρ i ≤d i (t)ω i (s i )

[0023] where ω i (s i )≥0 is a known continuous function, d i(t) bounded but upper bounded by d max Not required to be known;

[0024] Assumption 3: The output s i Satisfies one constraint condition as

[0025] |s i | < δ, i = 1,..., n

[0026] where δ > 0 represents the output constraint;

[0027] Lemma 1: The following inequality holds:

[0028]

[0029] where λ ∈ R + and a, b ∈ R;

[0030] Lemma 2: If x, y are real variables, then for any constant satisfying m1 ≥ 0 and 0 ≤ m2 ≤ 1, we have

[0031]

[0032] holds;

[0033] Lemma 3: If x, y are real variables, c, d > 0 are constants, then for any given function γ > 0, we have

[0034]

[0035] holds;

[0036] Lemma 4: If are real variables, then for any real number p satisfying 0 < p < 1, we have

[0037] (|x1|+…+|x n |) p ≤|x1| p +…+|x n | p

[0038]

[0039] holds.

[0040] Further, the time-varying scaling function constructed in step 2 has the following form:

[0041]

[0042] where T is a given arbitrary specified time, and obviously, the function μ1(t, T) has the following two states:

[0043]

[0044] The time-varying scaling function is improved, defined as

[0045]

[0046] where m is a positive integer, and obviously

[0047] and when t→0, there are the following properties

[0048]

[0049] where is the derivative of μ with respect to time t; μ is a short form of μ(t, T); in order to design the scheduled HOSM controller, the following time-varying coordinate transformation is considered:

[0050] w i = μ n+1-i s i , i = 1, 2,... n

[0051] It is noted that μ(t, T) is a monotonically increasing function, and when t→T, μ(t, T) will tend to infinity, through the time-varying coordinate inverse transformation:

[0052] s i = v n+1-i w i , i = 1, 2,... n

[0053] where v = μ -1 , v n+1-i = μ -(n+1-i) ; when t→T, v n+1-i will monotonically decrease to 0; therefore, if w i is bounded, s i →0 when t→T;

[0054] Rewrite the system model using the coordinate transformation:

[0055]

[0056] where h0(t, x) and g0(t, x) are unknown smooth functions; u ∈ R is the control input; f1, f2, f i are the short forms of f i (s i ), i = 1,..., n-1, and are the unmatched terms.

[0057] Further, the step 3 of designing the virtual control variable and the actual control input comprises the following steps:

[0058] (31) Select a Lyapunov function that can meet the output constraint, the expression is as follows:

[0059]

[0060] Wherein, And σ i Satisfies w i Is a time-varying coordinate transformation function, the transformed sliding mode variable s i With w i Substitute, Is a virtual control law; Lyapunov function V1(σ1) is defined in the region D1 = {σ1: |σ1| < δ}, δ > 0 represents the output constraint;

[0061] (32) Along the transformed system V1, the second virtual control law is designed by the backstepping method So that Satisfies Form, and repeat the derivation of the virtual control law and the control input u; therefore, equivalent to design a non-continuous predetermined time HOSM controller to stabilize the system;

[0062] (33) Apply the following virtual control (2) (3), the actual control input (4), to ensure that all closed-loop system signals are semi-global uniformly bounded without violating the output constraint in the closed-loop system, and the closed-loop system control performance meets the predetermined time stability:

[0063]

[0064]

[0065]

[0066] Wherein M is a positive integer, λ, c n,1 ,c n,2 ,c n,3 Are the designed control parameters, G m Is a normal number, k0 > 0, is a negative feedback coefficient, Is a positive definite function Abbreviation.

[0067] Further, in step 3, the n-order expression of the Lyapunov function is as follows:

[0068]

[0069] Beneficial effects:

[0070] The present application greatly reduces the uncertainty of the system by retaining useful items or unknown items in the previous derivative by introducing non-matching items; get rid of the disadvantage that the convergence time depends on the initial conditions of the system in the traditional sliding mode control, improve the robustness of the control system; considering the case of output constraint, the actuator is constrained by Barrier Lyapunov function, to adapt to more common and more demanding working conditions. The high-order sliding mode algorithm is adopted to retain the robustness and anti-interference performance of the traditional sliding mode, and to realize the effective vibration attenuation of the flexible manipulator system state, and to solve the problem of relative order limitation. The present application realizes the precise control of the flexible manipulator in complex environment, under the condition of containing non-matching items, combining the time-varying scaling function to perform coordinate transformation on the high-order sliding mode dynamic system, the high-order sliding mode control strategy weakens the chattering influence of the flexible manipulator, and the Barrier Lyapunov function method is used to ensure the realization of the predetermined time stable convergence of the manipulator movement on the basis of the controller derived by the backstepping method, further improve the precise control of the flexible manipulator in complex environment, and meet the steady-state performance of the high-order sliding mode flexible manipulator control method without violating the output constraint condition. BRIEF DESCRIPTION OF DRAWINGS

[0071] Figure 1 It is a simplified model of the single-joint double-mass flexible manipulator system of the present application.

[0072] Figure 2 It is an implementation step flow chart of the present application. DETAILED DESCRIPTION

[0073] The technical solutions of the present application will be further described below in combination with the drawings.

[0074] The present application discloses a flexible manipulator control method based on predetermined time stability and high-order sliding mode algorithm, which is used for the control of flexible manipulator with output constraint, and comprises the following steps:

[0075] Step 1: According to the high-order sliding mode dynamics equation with output constraint and non-matching item, the dynamics model of motor and joint is established based on the dynamics behavior of flexible manipulator.

[0076] Step 2: The Barrier Lyapunov function satisfying the output constraint condition is used, and in the case that the gain function upper bound of non-matching item is unknown, the original system is converted by time-varying scaling function, and an auxiliary system for designing predetermined time controller is constructed.

[0077] Step 3: The virtual control variable and actual control input are designed by using the backstepping recursive method.

[0078] Step 4: The virtual control variable and the actual control input in step 3 are substituted into the derivative of the Barrier Lyapunov function to verify whether the control law can make the derivative form of the Barrier Lyapunov function satisfy Continuing step 4, if the closed-loop system variable cannot be ultimately consistent and bounded, return to step 3 to redesign the virtual control variable and the actual control input; wherein, V n is used to describe the system, which is a continuous positive scalar function; represents the rate of change of the system over time, which is continuous negative; c is a negative feedback gain, which is a normal number; alpha is a nonlinear response of the system, and alpha is in (0, 1); and delta is an external excitation, which represents the influence on the system.

[0079] Step 5: Lyapunov's second method and predetermined time stability analysis are performed on the closed-loop system to prove that the output constraint condition is not violated, the predetermined time control parameter remains bounded at all times, and the control performance index meets the predetermined time control requirements.

[0080] In this embodiment, the dynamics model of the motor and joint of the flexible robot arm is taken as an example for controller design, and the dynamic model of the system is as follows:

[0081]

[0082] Wherein, theta represents the joint rotation angle, q represents the motor inclination angle, tau m represents the input motor driving torque, K represents the stiffness coefficient of the motor torsion, J l represents the inertia of the joint rotation, J m represents the inertia of the motor rotation, tau f represents the viscous friction torque.

[0083] A reasonable sliding variable is selected and derived, and combined with the high-order sliding mode dynamics form containing non-matching terms, the following dynamics equation is obtained:

[0084]

[0085] Combined with the system model, the following function can be selected:

[0086]

[0087] From the above formula, it can be seen that the flexible robot arm model can be regarded as a structure of the high-order sliding mode system researched in the present application when n=4. Therefore, the model of the nonlinear high-order sliding mode system with output constraint is as follows:

[0088]

[0089] In the formula, s i∈ R, i = 1,..., n are output (sliding variable), u ∈ R is control input; h0(t, x) and g0(t, x) are unknown smooth functions; x is system state, t is time, f i (s i ), i = 1,..., n-1 are unmatched terms. Let the relative degree of sliding variable s with respect to control input u be n, n is the order of system, i.e.,

[0090]

[0091] The system model is rewritten as

[0092]

[0093] The control objective of the present application is to design a control scheme based on pre-determined time stability and output constraint for a nonlinear high-order sliding mode system with unmatched disturbance and output constraint, so that the control system satisfies the following control objectives:

[0094] Objective 1: All sliding variables of the system do not violate the output constraint condition, i.e., guarantee |s i |<δ, i = 1, 2,..., n, δ > 0, which means output constraint.

[0095] Objective 2: By a given arbitrary specified time T, it is guaranteed that the system state after the transformation of the time-varying scaling function can converge to 0 within the predetermined time.

[0096] Objective 3: According to Lyapunov stability theorem, it is proved that all closed-loop system variables are ultimately uniformly bounded, so that

[0097] In order to achieve the above control objectives, the following assumption conditions are imposed on the system, and the nonlinear high-order sliding mode system with output constraint has the following assumptions and lemmas:

[0098] Assumption 1: There exists a known normal number G m and a known positive definite function such that the following conditions are met:

[0099]

[0100] Assumption 2: The upper bound of the gain function of the unmatched term f i (s i ) is not required to be known, but meets the following requirements:

[0101] f i ≤ ρ i (t, s i )|s i |, i = 1,..., n-1, ρ i ≤ di (t)ω i (s i )

[0102] where ω i (s i )≥0 is a known continuous function, d i (t) is bounded but the upper bound d max is not required to be known.

[0103] Assumption 3: In addition, the output s i satisfies a constraint condition

[0104] |s i |<δ,i = 1,...,n.

[0105] where δ > 0 is the output constraint.

[0106] The time-varying scaling function is constructed by combining the time-varying scaling function with the coordinate transformation of the original system, and the form of the time-varying scaling function is as follows:

[0107]

[0108] where T is a given arbitrary specified time, and obviously, the function μ1(t,T) has the following two states:

[0109]

[0110] The time-varying scaling function is improved, and is defined as

[0111]

[0112] where m is a positive integer, and in addition to the properties of μ(0,T) = 1 and μ(T,T) = +∞, obviously and when t→0, there are the following properties

[0113]

[0114] where denotes the derivative of μ with respect to time t; μ is a short form of μ(t,T).

[0115] In order to design the scheduled time HOSM controller, the following time-varying coordinate transformation is considered:

[0116] w i = μ n+1-i s i ,i = 1,2,...n

[0117] It is noted that μ(t,T) is a monotonically increasing function, and when t→T, μ(t,T) will tend to infinity, and through the time-varying coordinate inverse transformation:

[0118] s i = v n+1-i w i , i = 1, 2,... n

[0119] where v = μ -1 , v n+1-i = μ -(n+1-i) ; v n+1-i monotonically decreases to 0 as t→T; therefore, if w i is bounded,

[0120] s i → 0 as t→T. Rewrite the system model using coordinate transformation:

[0121]

[0122] Lemma 1: The following inequality holds:

[0123]

[0124] where λ ∈ R + and a, b ∈ R.

[0125] Lemma 2: If x, y are real variables, then for any constant satisfying m1≥0 and 0≤m2≤1 have

[0126]

[0127] holds.

[0128] Lemma 3: If x, y are real variables, c, d > 0 are constants, then for any given function γ > 0, have

[0129]

[0130] holds.

[0131] Lemma 4: If are real variables, then for any real number p satisfying 0 < p < 1, have

[0132] (|x1| +... + |x n |) p ≤ |x1| p +... + |x n | p

[0133]

[0134] holds.

[0135] Conclusion 1: Under the assumption conditions 1-3, for the nonlinear high-order sliding mode time-varying system (1) with output constraint, the following virtual control (2), (3), actual control input (4) are applied to ensure that all closed-loop system signals are semi-globally uniformly bounded without violating the output constraint in the closed-loop system, and the closed-loop system control performance meets the predetermined time stability:

[0136]

[0137]

[0138]

[0139] where m is a positive integer, λ, c n,1 ,c n,2 ,c n,3 are designed control parameters, G m is a positive constant, k0>0 is a negative feedback coefficient, is a positive definite function abbreviation.

[0140] The following is the specific proof process of proving that the closed-loop system tracking control performance meets the predetermined time stability for the uncertain nonlinear high-order sliding mode time-varying system model with output constraint combined with the time-varying scaling function μ(t, T).

[0141] First step: Choose Lyapunov function

[0142]

[0143] Taking the derivative of V1(σ1) along the system (1) can get

[0144]

[0145] where According to assumption 2 and lemma 1, the non-matching term μ n f1estimation

[0146]

[0147] Therefore, equation (6) can be written as

[0148]

[0149] To obtain the predetermined time stability expression of , the backstepping method is used to eliminate the residual term, and the first designed virtual control law is obtained

[0150]

[0151] To verify the feasibility of the virtual control law, substitute equation (9) into equation (8) to obtain...

[0152]

[0153] Step 2: Select the Lyapunov function

[0154]

[0155] Taking the time derivative of V2(σ2), we get

[0156]

[0157] Among them, for Item, have

[0158]

[0159] Substituting (13) into (12), we get

[0160]

[0161] for The item has the following estimates.

[0162]

[0163] According to the conditions in Assumption 2, we have so

[0164]

[0165] in, Among them The term adopts Lemma 1 (where...) get

[0166]

[0167] in,

[0168] Similarly, there are:

[0169]

[0170] in, Therefore for The following estimates are available:

[0171]

[0172] in,

[0173] Continuing to estimate the next term μσ1σ2, using Lemma 1 (where a = |σ1|, b = |σ2|), we get

[0174]

[0175] where

[0176] Continuing to estimate the last term using Lemma 1 (where ), we get

[0177]

[0178] where

[0179] Combining (19), (20), (21), we have

[0180]

[0181] w3 * is the virtual control law to be designed, using the backstepping-like method, we get the expression of the virtual control law

[0182] w3 * = -β2σ2 (23)

[0183] where

[0184] Substituting (23) into (22), we have

[0185]

[0186] Third step: Choose Lyapunov function

[0187]

[0188] Taking the derivative of V3 along the system, we get

[0189]

[0190] From which we design w4 * . Repeating the above estimation process, we get

[0191]

[0192] where β3= (n-2)(c 3,1 +c 3,2 +c 3,3 )+(n-2)k0+σ3 2 λω3 2 .

[0193] Substitute (27) into (28) to get

[0194]

[0195] Step k: Choose Lyapunov function

[0196]

[0197] Similarly, differentiate along the system to get

[0198]

[0199] Similarly, estimate the three terms, which are not shown here, and the expression of the kth virtual control law is

[0200] w k+1 * = -β k σ k (31)

[0201] where

[0202] After substitution, we get

[0203]

[0204] Step n: Choose Lyapunov function

[0205]

[0206] Differentiate along the system to get

[0207]

[0208] Estimate the first term

[0209]

[0210] The second term μσ k σ n :

[0211]

[0212] The third term

[0213]

[0214] Substitute (35), (36), and (37) into the design of control input u

[0215]

[0216] Substitute Verification

[0217]

[0218] where D max is the upper bound.

[0219] By

[0220]

[0221] According to the definition So there is

[0222]

[0223] Get

[0224]

[0225] Solve the above differential inequality:

[0226]

[0227] For the integral term in the inequality, the solution is:

[0228]

[0229] For the last term in equation (42), the estimate is:

[0230]

[0231] Substitute (43) and (44) into (42) to get

[0232]

[0233] where ζ(t) is

[0234]

[0235] From (46), it is easy to get ζ(0) = 1 and ζ(T) = 0. Therefore, it is easy to get V n (t) is bounded, so and w1,...,w n are all bounded. At the same time, by the time-varying coordinate inverse transformation, if w1,...,w n are bounded, then

[0236] lim t→T s1 = s2 =... = s n = 0 (47)

[0237] The predetermined time stability performance index is verified. Thus, the predetermined time stability performance index is verified.

[0238] The flexible manipulator control method based on predetermined time and high-order sliding mode provides a new idea for overcoming the limitations of traditional control methods in the face of nonlinear, time-varying and coupled systems. Through reasonable control strategy design, the predictability and stability of the system are improved, the precise control of the flexible manipulator in complex environments is realized, and it has broad application prospects and research value.

Claims

1. A control method for a flexible robotic arm based on predetermined time stability and a high-order sliding mode algorithm, characterized in that, Control of a flexible robotic arm with output constraints includes the following steps: Step 1: Based on the high-order sliding mode dynamics equations with output constraints and non-matching terms, establish a dynamic model of the motor and joints based on the dynamic behavior of the flexible manipulator; Step 2: Using the Barrier Lyapunov function that satisfies the output constraints, and with the upper bound of the gain function of the unmatched term unknown, the original system is transformed by a time-varying scaling function to construct an auxiliary system for designing a predetermined time controller. The time-varying scaling function is constructed in the form shown below: ； Where T is a given arbitrary time, obviously, the function It has the following two states: ； The time-varying scaling function has been improved and defined as follows: ； Where m is a positive integer, except and In addition to its nature, obviously , and when At that time, it has the following properties ； in, refer to The derivative with respect to time t; yes This is an abbreviation; to design a predetermined time HOSM controller, consider the following time-varying coordinate transformation: ； Note It is a monotonically increasing function, and when hour It will tend towards infinity, through inverse time-varying coordinate transformation: ； in, , ;when hour, It will monotonically decrease to 0; therefore, if There are boundaries, when From time to time ; Rewrite the system model using coordinate transformation: ； in, and The smoothing function is unknown; It is a control input; , , for This is an abbreviation for "non-matching item". Step 3: Design virtual control variables and actual control inputs using the backstepping recursive method; Step 4: Substitute the virtual control variables and actual control inputs from Step 3 into the derivative of the Barrier Lyapunov function to verify whether the control law can satisfy the derivative form of the Barrier Lyapunov function. Continue to step 4. If the closed-loop system variables cannot be eventually made uniformly bounded, return to step 3 to redesign the virtual control variables and actual control inputs; among which, Used to describe the system, it is a continuous positive definite scalar function; This represents the rate of change of the system over time, and is a continuous negative constant. For negative feedback gain, a positive number is used. This refers to the nonlinear response of the system. External stimulus, represented as its impact on the system; Step 5: Perform Lyapunov's second method and predetermined time stability analysis on the closed-loop system to prove that the output constraints are not violated, the predetermined time control parameters remain bounded, and the control performance indicators meet the predetermined time control requirements.

2. The flexible robotic arm control method based on predetermined time stability and a high-order sliding mode algorithm according to claim 1, characterized in that, The model of a nonlinear high-order sliding mode system with output constraints is as follows: ； In the formula, It is the output, which is the sliding mode variable. It is a control input; and Let x be an unknown smooth function; x be the system state; and t be time. For non-matching terms; let the sliding variable be... The degree of relative to the control input u is n, where n is the system order, i.e., ； The system model was rewritten as 。 3. The flexible robotic arm control method based on predetermined time stability and a high-order sliding mode algorithm according to claim 2, characterized in that, Nonlinear high-order sliding mode systems with output constraints have the following assumptions and lemmas: Assumption 1: There exist known positive constants. and known positive definite functions This makes the following conditions true: ； Assumption 2: Non-matching items The upper bound of the gain function does not need to be known, but it must satisfy the following requirements: ， ； in, It is a known continuous function. Bounded but with an upper boundary No prior knowledge is required; Assumption 3: Output Satisfy a constraint condition is ； in, This is represented as an output constraint; Lemma 1: The following inequalities hold: ； in and ; Lemma 2: If x and y are real variables, then for any constant, the following holds true: and have ； Established; Lemma 3: If x and y are real variables, If it is a positive constant, then for any given function... ,have ； Established; Lemma 4: If If the variable is a real number, then for any real number... satisfy ,have ； Established.

4. The flexible robotic arm control method based on predetermined time stability and a high-order sliding mode algorithm according to claim 1, characterized in that, Step 3, designing virtual control variables and actual control inputs, includes the following steps: (31) Select a Lyapunov function that satisfies the output constraints. Its expression is as follows: （1）； in, and , satisfy The time-varying coordinate transformation function represents the transformed sliding mode variable. use Replacement For virtual control laws; Lyapunov functions Defined in the region , This is represented as an output constraint; (32) Along the transformed system Differentiate and design a second virtual control law using a backstepping-like method. Make satisfy The virtual control law and control input u are derived repeatedly in this manner; therefore, it is equivalent to designing a non-continuous, predetermined-time HOSM controller to stabilize the system. (33) By applying the following virtual control formulas (2), (3), and actual control input formula (4), it is ensured that all closed-loop system signals are semi-globally consistent and bounded without violating the output constraints of the closed-loop system, and the control performance of the closed-loop system meets the predetermined time stability requirement: （2）； （3）； （4）； in , , m is a positive integer. , For the control parameters of the design, For positive integers, , is the negative feedback coefficient. It is a positive definite function The abbreviation for .

5. The flexible robotic arm control method based on predetermined time stability and a high-order sliding mode algorithm according to claim 4, characterized in that, In step 3, the nth-order expression of the Lyapunov function is as follows: （5）。

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