Event triggering preset time control method for flexible joint robot under fault

Abstract

The invention relates to the technical field of flexible joint robots, and discloses an event triggering predetermined time control method for a flexible joint robot under a fault, and the method comprises the following steps: S1, constructing an event triggering mechanism so as to reduce the communication frequency, S2, introducing a hyperbolic tangent function to design a control signal of a nonsingular problem, and carrying out the control of the nonsingular problem. The method comprises the following steps of: S1, carrying out multi-time instruction filtering, and simultaneously adopting fixed time instruction filtering to avoid the problem of complex explosion, S3, realizing output constraint through a conversion function and coordinate conversion to ensure the safety during interaction with people, S4, processing unknown sensor faults and unknown system dynamic states by utilizing an error compensation updating method and a neural weight updating method, and S5, carrying out multi-time instruction filtering on the basis of a predetermined time Lyapunov stability theory. It is proved that the proposed method ensures the predefined time stability of the closed-loop system. According to the event triggering mechanism suitable for the flexible joint robot, communication between the controller and the actuator is reduced, and finally the communication load is reduced.

Description

Technical Field

[0001] The present invention relates to the technical field of flexible joint robots, and in particular to a method for controlling a predetermined time of an event triggering of a flexible joint robot under fault conditions. Background Art

[0002] With the development of science and technology, robots are increasingly used in various fields such as medical treatment, mechanical equipment and services, and human labor is replaced by robots. In these processes, robots need to interact with humans, and flexible joint robots have important compliant behaviors in human-machine interaction to ensure human safety. Therefore, it is very meaningful to obtain system information from flexible joint robot systems and achieve effective control.

[0003] The control design of flexible joint robot manipulators is usually a challenging problem due to their high nonlinearity, strong coupling and time-varying characteristics. Their kinematics and dynamics inevitably have uncertainties, so it is impossible to provide an accurate model for control design. In addition, in actual operation, the problem of the state variables required by the controller cannot be measured may be encountered. In the literature [1], the control system design using neural network observers is studied to solve the problem of unmeasurable state.

[0004] [1] Liu X, Yang C, Chen Z, Wang M, Su CY. Neuro-adaptive observer based control of flexible jointtrobot. Neurocomputing. 2018; 275:73–82.

[0005] Reference [2] studies the adaptive asymptotic tracking control problem of a flexible joint robot system. In the initial stage of system operation, the output tracking error can be kept within the specified range. As time approaches infinity, an asymptotic tracking result can be obtained.

[0006] [2]Wang L, Sun W, Su SF, et al. Adaptive asymptotic tracking control for flexible-joint robots with prescribed performance: Design and experiments [J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2023, 53(6): 3707-3717.

[0007] For existing tracking control schemes of flexible joint robots, accurate sensor measurement is an implicit premise. However, due to manufacturing technology or other external factors, ideal sensors are difficult to achieve. Therefore, reference [3] further studies the tracking control problem of flexible joint robots with unknown measurement sensitivity.

[0008] [3]Diao S, Sun W, Su SF, et al. Novel adaptive control for flexible-joint robots with unknown measurement sensitivity [J]. IEEE Transactions onAutomation Science and Engineering, 2023, 21(2): 1445-1456.

[0009] Reference [4] proposed an adaptive radial basis function (RBF) neural network (NN) backstepping controller for a class of input-constrained flexible joint robots represented in strict feedback form with unknown terms, external random disturbances, and output disturbances. The proposed method is robust to both deterministic and random uncertainties and disturbances, and can cope with control input amplitude saturation.

[0010] [4]Arefi MM, Vafamand N, Homayoun B, et al.Command filteredbackstepping control of constrained flexible joint robotic manipulator[J]. IETControl Theory&Applications, 2023,17(18):2506-2518.

[0011] Reference [5] mainly studies the trajectory tracking control of a multi-link flexible joint robot with time-varying full-state constraints and uncertain dynamics. To avoid violating the time-varying constraints, an obstacle Lyapunov function is constructed. A fuzzy logic system is used to deal with unknown nonlinear problems in the flexible joint system.

[0012] [5] Y. Zhu, J. Liu, J. Yu and Q.-G. Wang, CommandFiltering-Based AdaptiveFuzzy Control of Flexible-Joint Robots With Time-Varying Full-StateConstraints[J], IEEE Transactions on Circuits and Systems II: Express Briefs, 2024, 71(2): 682-686. Summary of the invention

[0013] The purpose of the present invention is to provide an event-triggered predetermined time control method for a flexible joint robot under fault conditions, aiming at (1) Unlike the existing constraint control, the present invention adopts fixed instruction filtering and filtering error compensation mechanism to avoid the explosion of computational complexity. (2) Unlike the use of dynamic surface technology / instruction filter to perform fuzzy / neural tracking control on flexible joint robots, the present invention takes into account sensor failures and event triggering mechanisms and is more suitable for practical engineering applications. Although the existing methods have used event triggering mechanisms and second-order instruction filtering for flexible joint robots, the method proposed in the present invention simplifies the adjustment of parameters and allows the adjustment time to be set in advance as needed. (3) Different from the existing robot output performance constraints, the present invention introduces a hyperbolic tangent function to design a virtual control signal to eliminate the singularity problem. Unlike the method using a split function, this method avoids tedious parameter setting and analytical research.

[0014] To achieve the above object, the present invention provides the following technical solution: a method for controlling the event-triggered predetermined time of a flexible joint robot under fault, the method comprising the following steps:

[0015] S1: An event trigger mechanism was built to reduce the frequency of communication

[0016] The dynamic process formula of the flexible joint robot is:

[0017]

[0018] in, represent the angular position, velocity and acceleration of the connecting rod respectively; Represent the angular position, velocity and acceleration of the motor respectively; M(q)∈R n×n is a symmetric positive definite inertia matrix; Represents the Coriolis force and centripetal force matrix, G(q)∈R n is the gravity vector, is the uncertain friction term; the positive definite diagonal matrix K∈Rn×n ,J∈R n×n and B∈R n×n They represent joint flexibility, brake inertia and natural damping respectively. The torque input to the brake is u∈R n express;

[0019] S2: Introducing the hyperbolic tangent function to design the control signal for non-singular problems, while using fixed-time instruction filtering to avoid complexity explosion problems

[0020] The formula for the fixed time command filter is:

[0021]

[0022] Among them, λ1, λ2, λ3, λ4 and γ>1 are some positive parameters, and the output and Within a fixed time T, α and Estimate to arbitrary precision;

[0023] S3: Output constraints are implemented through conversion functions and coordinate transformations to ensure safety when interacting with people

[0024] The conversion function formula is:

[0025]

[0026] S4: Using error compensation update and neural weight update to handle unknown sensor failures and unknown system dynamics

[0027] First, the tracking error definition formula is:

[0028]

[0029] Among them, z i =[z i,1 ,z i,2 ,...,z i,n ] T , x i-1,c =[x i-1,c,1 ;x i-1,c,2 ;...;x i-1,c,n ]. i,c is a virtual control signal α i The command filter output.

[0030] Then, the tracking error with the error compensation signal is defined as

[0031] v i =z i -ξ i ,i=1,...,4 (7)

[0032] where v i =[v i,1 ;v i,2 ;...;v i,n ],ξ i =[ξ i,1 ξ i,2 ;...;ξ i,n ] is the filter error compensation signal, designed as

[0033]

[0034] in, η1 is an even positive number, η2 is an odd positive number, satisfying η1<η2, T s It is called the scheduled time and is a positive design parameter; k1 and k2 are positive design parameters.

[0035] The design virtual control signal formula is:

[0036]

[0037] Designing sensor fault compensation update rate for

[0038]

[0039] Designing Neural Network Adaptive Update Rate and for

[0040]

[0041] The actual controller is constructed as

[0042]

[0043] S5: Based on the predefined time Lyapunov stability theory, it is proved that the proposed method guarantees the predefined time stability of the closed-loop system

[0044] Even if the sensor fails, the output y(t) can still track the reference trajectory y d (t); the system output y is subject to asymmetric constraints that vary over time Among them, H1, is a quadratically differentiable constraint function.

[0045] Preferably, the event triggering mechanism defined in step S1

[0046]

[0047] Where γ(t) = u e(t)-u(t) represents the measurement error.

[0048] Preferably, in step S1, in the dynamic process of the flexible joint robot, let q = x1 = [x 1,1 ,x 1,2 ,...,x 1,n ] T , q m =x3=[x 3,1 ,x 3,2 ,...,x 3,n ] T , When , the dynamic process formula of the n-link flexible joint robot can be rewritten as

[0049]

[0050] Preferably,

[0051] In formula (2), f2(x1,x2,x3)=M -1 (x1)(Kx3-C(x1,x2)x2-G(x1)-F(x2)-Kx1)-x3,f4(x1,...,x4,u)=J -1 (u-Bx4-K(x3-x1))-u.

[0052] Preferably, the output y of the system is under an unknown sensor failure, which is described as

[0053] y=P(t)x1+Q(t) (3)

[0054] Where P(t) and Q(t) represent the sensor fault parameters and are given as follows:

[0055] ①P(t)=1,Q(t)=0, indicating the sensor is normal;

[0056] ② is a constant, P(t) = 1, which means that the sensor has a bias fault;

[0057] ③|Q(t)|=Jt,0<J<<1,P(t)=1, which means the sensor has drift fault;

[0058] ④ Q→0, and P(t)=1, indicating that the sensor has a misalignment fault;

[0059] ⑤ and Q(t) = 0, which indicates that the sensor has a failure fault;

[0060] Definition F y=(P(t)-1)x1+Q(t), further, y=x1+F y ,So,

[0061] Tracking error e1 = yy d , satisfying the constraint condition -H1<e1

[0062] Compared with the prior art, the present invention has the following beneficial effects:

[0063] (1) An event trigger mechanism suitable for flexible joint robots is proposed, which reduces the communication between the controller and the actuator and ultimately reduces the communication load;

[0064] (2) An output sensor fault-tolerant control method was proposed. By leveraging the learning characteristics of neural networks, an adaptive update rate for fault compensation and an update rate for neural network weights were invented, achieving accurate tracking control under unknown output measurement faults and unknown system dynamics.

[0065] (3) A command filtering and filtering compensation mechanism for an n-link flexible joint robot is proposed to reduce the computational complexity explosion problem when implementing the controller;

[0066] (4) A preset time controller strategy was invented, which cleverly used the hyperbolic tangent function to solve the singularity problem of the controller, ensuring that all signal errors of the closed-loop system can converge to zero within the set time content;

[0067] (5) In order to take safety considerations and physical limitations into account during human-computer interaction, the present invention proposes a time-varying output limitation solution to ensure the safety of human-computer interaction. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 It is a design block diagram of the present invention;

[0069] Figure 2 is the x of the link 1 of the boundary constraint of the present invention 1,1 and d1 Time trajectory diagram of

[0070] Figure 3 is the x of the link 2 of the boundary constraint of the present invention 1,2 and d2 Time trajectory diagram of

[0071] Figure 4 is the tracking error e of the connecting rod 1 of the present invention 1,1 Time trajectory diagram of

[0072] Figure 5 is the tracking error e of the connecting rod 2 of the present invention 1,2 Time trajectory diagram of ​

[0073] Figure 6 is the system state x of the present invention 2,1 ,x 3,1 ,x 4,1 picture;

[0074] Figure 7 is the system state x of the present invention 2,2 ,x 3,2 ,x 4,2 picture;

[0075] Figure 8 is the sensor error compensation update rate of the connecting rod 1 of the present invention picture;

[0076] Fig. 9 is the sensor error compensation update rate of the connecting rod 2 of the present invention picture;

[0077] Fig.10 is the adaptive update parameter of the connecting rod 1 neural network of the present invention and picture;

[0078] Fig.11 is the adaptive update parameter of the connecting rod 2 neural network of the present invention and picture;

[0079] Fig.12 is the controller input u of the connecting rod 1 of the present invention 1,e and actuator input u1 graph;

[0080] Fig.13 is the controller input u of the connecting rod 2 of the present invention 2,e and actuator input u2 graph;

[0081] Fig.14 is the event triggering interval t of the connecting rod 1 of the present invention k -t k-1 picture;

[0082] Fig.15 is the event triggering interval t of the connecting rod 2 of the present invention k -t k-1 picture. DETAILED DESCRIPTION

[0083] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0084] A method for controlling a predetermined time of event triggering of a flexible joint robot under fault conditions, the method comprising the following steps:

[0085] S1: An event trigger mechanism was built to reduce the frequency of communication

[0086] The dynamic process formula of the flexible joint robot is:

[0087]

[0088] in, represent the angular position, velocity and acceleration of the connecting rod respectively; Represent the angular position, velocity and acceleration of the motor respectively; M(q)∈R n×n is a symmetric positive definite inertia matrix; Represents the Coriolis force and centripetal force matrix, G(q)∈R n is the gravity vector, is the uncertain friction term; the positive definite diagonal matrix K∈R n×n ,J∈R n×n and B∈R n×n They represent joint flexibility, brake inertia and natural damping respectively. The torque input to the brake is u∈R n express;

[0089] S2: Introducing the hyperbolic tangent function to design the control signal for non-singular problems, while using fixed-time instruction filtering to avoid complexity explosion problems

[0090] The formula for the fixed time command filter is:

[0091]

[0092] Among them, λ1, λ2, λ3, λ4 and γ>1 are some positive parameters, and the output and Within a fixed time T, α and Estimate to arbitrary precision;

[0093] S3: Output constraints are implemented through conversion functions and coordinate transformations to ensure safety when interacting with people

[0094] The conversion function formula is:

[0095]

[0096] S4: Using error compensation update and neural weight update to handle unknown sensor failures and unknown system dynamics

[0097] First, the tracking error definition formula is:

[0098]

[0099] Among them, z i =[z i,1 ,z i,2 ,...,z i,n ] T , x i-1,c =[x i-1,c,1 ;x i-1,c,2 ;...;x i-1,c,n ]. i,c is a virtual control signal α i The command filter output.

[0100] Then, the tracking error with the error compensation signal is defined as

[0101] v i =z i -ξ i ,i=1,...,4 (7)

[0102] where v i =[v i,1 ;v i,2 ;...;v i,n ],ξ i =[ξ i,1 ξ i,2 ;...;ξ i,n ] is the filter error compensation signal, designed as

[0103]

[0104] in, η1 is an even positive number, η2 is an odd positive number, satisfying η1<η2, T s It is called the scheduled time and is a positive design parameter; k1 and k2 are positive design parameters.

[0105] The design virtual control signal formula is:

[0106]

[0107]

[0108] Designing sensor fault compensation update rate for

[0109]

[0110] Designing Neural Network Adaptive Update Rate and for

[0111]

[0112] The actual controller is constructed as

[0113]

[0114] S5: Based on the predefined time Lyapunov stability theory, it is proved that the proposed method guarantees the predefined time stability of the closed-loop system

[0115] Even if the sensor fails, the output y(t) can still track the reference trajectory y d (t); the system output y is subject to asymmetric constraints that vary over time in H 1, is a quadratically differentiable constraint function.

[0116] In this embodiment: Step S1 defines an event trigger mechanism

[0117]

[0118] Where γ(t) = u e (t)-u(t) represents the measurement error.

[0119] In this embodiment: In step S1, in the dynamic process of the flexible joint robot, let q = x1 = [x 1,1 ,x 1,2 ,...,x 1,n ] T , q m =x3=[x 3,1 ,x 3,2 ,...,x 3,n ] T , When , the dynamic process formula of the n-link flexible joint robot can be rewritten as

[0120] In this embodiment:

[0121] In formula (2), f2(x1,x2,x3)=M -1 (x1)(Kx3-C(x1,x2)x2-G(x1)-F(x2)-Kx1)-x3,f4(x1,...,x4,u)=J -1 (u-Bx4-K(x3-x1))-u.

[0122] In this embodiment, the output y of the system is under an unknown sensor failure, which is described as

[0123] y=P(t)x1+Q(t) (3)

[0124] Where P(t) and Q(t) represent the sensor fault parameters and are given as follows:

[0125] ①P(t)=1,Q(t)=0, indicating the sensor is normal;

[0126] ② is a constant, P(t) = 1, which means that the sensor has a bias fault;

[0127] ③|Q(t)|=Jt,0<J<<1,P(t)=1, which means the sensor has drift fault;

[0128] ④ Q→0, and P(t)=1, indicating that the sensor has a misalignment fault;

[0129] ⑤ and Q(t) = 0, which indicates that the sensor has a failure fault;

[0130] Definition F y =(P(t)-1)x1+Q(t), further, y=x1+F y ,So,

[0131] Tracking error e1 = yy d , satisfying the constraints

[0132] In order to verify the effectiveness of the proposed method, a robot system consisting of two flexible joints is used here.

[0133] The dual-joint robot system is as follows

[0134]

[0135] Let x1=q, x3=q m ,and

[0136]

[0137] where x1 = [x 11 ,x 12 ] T ,x2=[x 21 ,x 22 ] T ,x3=[x 31 ,x 32 ] T ,x4=[x41 ,x 42 ] T , and u=[u1,u2] T .

[0138]

[0139] Where c1 = cos(x 11 ),s1=sin(x 11 ),c2=cos(x 12 ),s2=sin(x 12 The corresponding system parameters of the double-link flexible joint robot system are shown in Table 1.

[0140] Table 1 Parameters of the dual-link flexible joint robot system

[0141]

[0142] The expected trajectory is [y d1 ,y d2 ] T =[0.8sin(2t),0.4cos(0.8t)+0.8sin(1.4t)] T , the constraint condition is H 1,1 -y d1 =1.1-0.1sin(t), H 1,2 -y d2 =1.5-0.2sin(0.5t), The initial value is x1(0)=[0.35,0.6] T ,x2(0)=[1,1] T ,x3(0)=[0,0] T ,x4(0)=[1,1] T .[ξ i,1 (0),ξ i,2 (0)] T =[0.01,0.01] T , The design parameters of the command filter are λ1=λ3=4,λ2=λ4=6, The uncertainty term is F(x2)=[0.01cos(x 2,1 ),0.02sin(x 2,1 x 2,2 )].

[0143] In order to simulate the efficiency failure of the sensor, the corresponding model is designed as follows

[0144]

[0145] In the simulation, the design parameters are set as: T s =6, ε1=ε2=ε3=ε4=0.001, m=0.5, Based on the controller design, we constructed the RBF neural network The network has 8 2 nodes, the node centers are evenly distributed on [-1.2, 1.2] × [-1.6, 1.6], and the width is $0.36$; Contains 6 4 There are nodes with centers evenly distributed on [-1.2, 1.2] × [-1.57, 1.57] × [-1.6, 1.6] × [-1.8, 1.8], and a width of 0.45.

[0146] The simulation results are as follows Figure 2-Figure 15 As shown. Figure 2 and Figure 3 As shown in the figure, the real-time position y(t) of the dual-link flexible joint robot tracks the ideal trajectory y d (t), and the output constraint is not violated. Tracking error yy d The time trajectory of Figure 4 and Figure 5 shown. Figure 6 and Figure 7 Describes the system state x 2,i 、x 3,i 、x 4,i Adaptive fault compensation update parameters and like Figure 8 and Fig. 9 As shown. Neural network weight update parameters and like Fig.10 and Fig.11 shown. Fig.12 and Fig.13 The control input u of the two-link flexible joint robot is shown i,e and actuator input u i The event triggering time interval is as follows. Fig.14 and Fig.15 As shown, it can be inferred that the proposed strategy for the dual-link flexible joint robot significantly reduces the communication frequency. From the simulation results, it can be seen that the proposed controller realizes event-triggered output constraint tracking control and has good tracking effect.

[0147] The processing results show that the output tracking control method developed by the present invention for a multi-flexible joint robot system has unknown system dynamics and unknown output strategy failures, which is more in line with the actual situation. The proposed controller reduces the communication burden between the controller and the actuator, realizes fault-tolerant control, solves the problem of time-varying constraints of the output, and finally enables the system tracking signal to be completed within the set time, and the set time is set separately according to actual needs, and does not depend on the setting of design parameters. The present invention is in the framework of flexible joint robot tracking control, integrating event triggering, fault-tolerant control, neural network learning, time-varying constraints, command filtering, and preset time convergence methods. Based on the above control framework, the controller performance, stability and human-computer interaction safety are improved. Therefore, no other related solutions can replace it.

[0148] It should be noted that, in this article, relational terms such as first and second, etc. are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Moreover, the terms "include", "comprise" or any other variants thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device.

[0149] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for controlling the event-triggered predetermined time of a flexible joint robot under fault conditions, characterized in that: The method comprises the following steps: S1: An event trigger mechanism was built to reduce the frequency of communication The dynamic process formula of the flexible joint robot is: Among them, q, Represent the angular position, velocity and acceleration of the connecting rod respectively; q m , Represent the angular position, velocity and acceleration of the motor respectively; M(q)∈R n×n is a symmetric positive definite inertia matrix; Represents the Coriolis force and centripetal force matrix, G(q)∈R n is the gravity vector, is the uncertain friction term; the positive definite diagonal matrix K∈R n×n ,J∈R n×n and B∈R n×n They represent joint flexibility, brake inertia and natural damping respectively. The torque input to the brake is u∈R n express; S2: Introducing the hyperbolic tangent function to design the control signal for non-singular problems, while using fixed-time instruction filtering to avoid complexity explosion problems The formula for the fixed time command filter is: Among them, λ1, λ2, λ3, λ4 and γ>1 are some positive parameters, and the output and Within a fixed time T, α and Estimate to arbitrary precision; S3: Output constraints are implemented through conversion functions and coordinate transformations to ensure safety when interacting with people The conversion function formula is: S4: Using error compensation update and neural weight update to handle unknown sensor failures and unknown system dynamics First, the tracking error definition formula is: Among them, z i =[z i,1 ,z i,2 ,...,z i,n ] T , x i-1,c =[x i-1,c,1 ;x i-1,c,2 ;...;x i-1,c,n ]. i,c is a virtual control signal α i The command filter output. Then, the tracking error with the error compensation signal is defined as v i =z i -ξ i ,i=1,...,4 (7) where v i =[v i,1 ;v i,2 ;...;v i,n ],ξ i =[ξ i,1 ξ i,2 ;...;ξ i,n ] is the filter error compensation signal, designed as in, η1 is an even positive number, η2 is an odd positive number, satisfying η1<η2, T s It is called the scheduled time and is a positive design parameter; k1 and k2 are positive design parameters. The design virtual control signal formula is: Designing sensor fault compensation update rate for Designing Neural Network Adaptive Update Rate and for The actual controller is constructed as S5: Based on the predefined time Lyapunov stability theory, it is proved that the proposed method guarantees the predefined time stability of the closed-loop system Even if the sensor fails, the output y(t) can still track the reference trajectory y d (t); the system output y is subject to asymmetric constraints that vary over time in H 1, is a quadratically differentiable constraint function.

2. The method for controlling the event-triggered predetermined time of a flexible joint robot under fault conditions according to claim 1, characterized in that: Step S1 defines the event trigger mechanism Where γ(t) = u e (t)-u(t) represents the measurement error.

3. The method for controlling the event-triggered predetermined time of a flexible joint robot under fault conditions according to claim 1, characterized in that: Step S1 In the dynamic process of the flexible joint robot, let q = x1 = [x 1,1 ,x 1,2 ,...,x 1,n ] T , q m =x3=[x 3,1 ,x 3,2 ,...,x 3,n ] T , When , the dynamic process formula of the n-link flexible joint robot can be rewritten as 4. The method for controlling the event-triggered predetermined time of a flexible joint robot under fault conditions according to claim 3, characterized in that: Such as (2)f2(x1,x2,x3)=M -1 (x1)(Kx3-C(x1,x2)x2-G(x1)-F(x2)-Kx1)-x3,f4(x1,...,x4,u)=J -1 (Bx4−K(x3−x1))−u。 5. The method for controlling the event-triggered predetermined time of a flexible joint robot under fault conditions according to claim 4, characterized in that: The output y of the system is subject to an unknown sensor failure, which is described as y=P(t)x1+Q(t) (3) Where P(t) and Q(t) represent the sensor fault parameters and are given as follows: ①P(t)=1,Q(t)=0, indicating the sensor is normal; ②Q(t)(Q . (t) = 0) is a constant, P(t) = 1, which means that the sensor has a bias fault; ③|Q(t)|=Jt,0<J<<1,P(t)=1, which means the sensor has drift fault; ④ Q→0, and P(t)=1, indicating that the sensor has a misalignment fault; ⑤ and Q(t) = 0, which indicates that the sensor has a failure fault; Definition F y =(P(t)-1)x1+Q(t), further, y=x1+F y ,So, F yd =F' y ; Tracking error e1 = yy d , satisfying the constraints

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