Adaptive super-twisting multivariable fast terminal sliding mode control method based on TDE

Abstract

The application provides a TDE-based adaptive super-helical multivariable fast terminal sliding mode control method, which guarantees high-precision trajectory tracking control of a cable-driven manipulator under complex unknown uncertainty conditions. First, the error range of a joint is designed, the safety performance of the manipulator is guaranteed, the deviation of the joint position from the expected contour is not too large, and time delay estimation is used to estimate and compensate the residual lumped system dynamics. Second, a safety constraint function is used to design and analyze the angle error expectation, so that the constraints of different sections are processed in a unified system architecture; through strict analysis, the problem of the manipulator trajectory with output constraints and uncertainty is solved. The super-helical adaptive control effectively guarantees the fast, accurate and robust convergence of the algorithm under all constraint conditions in operation, and the stability of the closed-loop control system is analyzed by using the Lyapunov method.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of mechanical arm control, and relates to a TDE-based adaptive super-spiral multivariable fast terminal sliding mode control method for a cable-driven mechanical arm with error safety constraints. BACKGROUND

[0002] Aiming at the control precision problem of a mechanical arm, many scholars have carried out a large amount of research. Han SI designed a robust PID controller by combining robust control and PID control to realize trajectory tracking control of a mechanical arm 【1】. J. Xu proposed a prescribed performance controller based on a linear extended state observer (LESO), which solved the problem of stabilizing a complex transformation system and further improved the control precision 【2】. B. Xu proposed an orthogonal fuzzy PID intelligent control method 【3】. Compared with the fuzzy PID, the orthogonal fuzzy PID can further improve the precision of the system. However, most of the works in the literature considering joint tracking problems do not solve the error constraint problem in the operation process. However, in order to ensure the accurate and safe operation of the cable-driven manipulator, the error constraint requirement cannot be ignored. In the usual control case, when the error of the system can gradually decrease from the initial position to the desired position to a very small value, that is, the error is within the control range, it can be shown that the control purpose is achieved. Due to the limitation of the mechanical control structure, some position range is unsafe, and this range changes with time. If a sudden event occurs, resulting in a large deviation, the error floats in an unpredictable range, and we cannot respond in time, which may cause the mechanical system to have a large movement error, collide with the outside world, and cause safety problems and economic losses.

[0003] In order to solve the error constraint problem and to ensure the prescribed position tracking performance of the robot manipulator, Zhou proposed a new constraint error variable similar to a sliding mode surface 【4】. Li Y et al. solved the output constraint problem caused by physical and environmental limitations by constructing an integral barrier Lyapunov function 【5】. W. He designed a controller by using a traditional logarithmic Lyapunov function to realize trajectory tracking control of a robot arm under full state constraints. 【6】However, most of these methods focus on the safety constraints of input or output, and do not meet the time-varying and asymmetric safety constraints of position error.

[0004] There are many achievements in the control method based on the asymmetric constraint of the robot arm. Z. Kai proposed a neural adaptive tracking control method for uncertain robot manipulator with asymmetric and time-varying full state constraints without involving the feasibility conditions. This scheme can adapt to asymmetric but time-varying motion constraints 【7】. J. Xu proposed a new iterative learning control (ILC) scheme for tracking non-repetitive reference trajectory problems of robot manipulators with different trial lengths on the iteration domain and under the requirement of asymmetric constraints of joint angles 【8】. C. Zhu proposed an adaptive timing estimation algorithm for an uncertain robot, which can avoid measuring acceleration signals during the estimation process 【9】.

[0005] In general, the residual linkage dynamics in the model parameters, motor dynamics and lumped uncertainty are difficult to obtain using traditional methods, and the use of TDE is affected by estimation errors, especially when the system contains fast time-varying dynamics, which can lead to a decline in control performance. Therefore, it is necessary to study a control scheme based on TDE with fast convergence and high control accuracy.

[0006] The relevant references are as follows:

[0007] 【1】Han S I, Lee J M. Decentralized neural network control for guaranteed tracking error constraint of a robot manipulator [J]. International Journal of Control, Automation and Systems, 2015, 13(4): 906-915.

[0008] 【2】Xu J, Qiao L. Robust Adaptive PID Control of Robot Manipulator with Bounded Disturbances [J]. Mathematical Problems in Engineering, 2013, 2013 (pt. 13): 1-13.

[0009] Xu B, Ji S, Zhang C, et al. Linear-extended-state-observer-based prescribed performance control for trajectory tracking of a robotic manipulator [J]. Industrial Robot, 2021, ahead-of-print (ahead-of-print).

[0010] Zhou, Chen, Liu. Design and Analysis of a Drive System for a Series Manipulator Based on Orthogonal-Fuzzy PID Control [J]. Electronics, 2019, 8(9): 1051-

[0011] Li Y, Yang C, Yan W, et al. Admittance-based adaptive cooperative control for multiple manipulators with output constraints [J]. IEEE transactions on neural networks and learning systems, 2019, 30(12): 3621-3632.

[0012] He W, Chen Y, Yin Z. Adaptive neural network control of an uncertain robot with full-state constraints [J]. IEEE transactions on cybernetics, 2015, 46(3): 620-629.

[0013] Kai Z, Song Y. Neuroadaptive Robotic Control Under Time Varying Asymmetric Motion Constraints: A Feasibility-Condition-Free Approach [J]. IEEE Transactions on Cybernetics, 2018, PP: 1-10.

[0014] [8] Xu J. Iterative Learning Control for Robot Manipulators with NonRepetitive Reference Trajectory, Iteration Varying Trial Lengths, and Asymmetric Output Constraints [C] / / 2020 American Control Conference (ACC). 2020.

[0015] [9] Zhu C, Jiang Y, Yang C. Fixed-time Parameter Estimation and Control Design for Unknown Robot Manipulators with Asymmetric Motion Constraints [J]. International Journal of Control, Automation and Systems, 2022, 20(1): 268-282. SUMMARY

[0016] In order to overcome the shortcomings of the prior art, the present application aims at the high-precision trajectory tracking problem of cable-driven manipulators, and designs a super-helix controller based on asymmetric error constraints using time delay estimation (TDE) method. The method uses safety constraints to control the error range, and the super-helix design of the controller ensures that the manipulator meets the safety constraints in terms of control error. At the same time, the joint position error constraints are designed using barrier functions. This method reduces the position error of the manipulator, and the trajectory of the manipulator can largely coincide with the desired trajectory.

[0017] The technical scheme adopted by the present application is a TDE-based adaptive super-helix multivariable fast terminal sliding mode control method, including the following steps:

[0018] Step 1: Establish a dynamic model of the manipulator to describe the state of the manipulator.

[0019] Step 2: Use TDE to estimate the parameters in the dynamic model.

[0020] Step 3: Design a sliding surface to facilitate the design of the control method in the next step, including the following sub-steps:

[0021] Step 3.1, introduce the safety constraint for joint position error, set a safety range for the error between actual joint position and desired joint position, so as to have an upper limit and a lower limit, to ensure the accuracy and normal operation of cable-driven manipulator;

[0022] Step 3.2, use the safety constraint function to design and analyze the angle error expectation combined with the barrier function, so that different sections of constraints are processed in a unified architecture;

[0023] Step 3.3, design the sliding mode surface according to the kinematics principle of the joint position of the manipulator;

[0024] Step 4, use the sliding mode surface to design the terminal sliding mode controller.

[0025] Further, the dynamics model of the manipulator with n degrees of freedom established in step 1 is represented as:

[0026]

[0027]

[0028]

[0029] In the formula, J and d m are the motor inertia and damping matrix, q and θ are the joint and motor position vectors, , respectively, represent the first and second derivatives of q and θ, M(q) is the inertia matrix, is the Coriolis / centrifugal matrix, g(q) is the gravity, is the friction vector, τ m and τ s are the control torque given by the motor and the joint flexibility torque, d s is the damping matrix, k s is the joint stiffness matrix, τ d represents the lumped unknown uncertainty;

[0030] In order to facilitate the use of TDE scheme, formula (2) is substituted into formula (1), and constant parameter is applied to obtain:

[0031]

[0032] Wherein, the expression of f is:

[0033]

[0034] Further, the estimated value of f is obtained by using TDE scheme in step 2 :

[0035]

[0036] where Δt is the delay time, and substituting equation (4) into equation (6) gives

[0037]

[0038] From equation (6) and equation (7), it can be seen that the purpose of the TDE scheme is to estimate the lumped system dynamics using only the time delay values of the control and acceleration signals, and then to give a model-free scheme;

[0039] In engineering applications, u s (t-Δt) can be obtained from the direct time delay of u s , and the numerical differentiation method is used to obtain :

[0040]

[0041] where the time node t0≥2Δt, at the initial stage t≤2Δt, q(t) has an actual measured value, and q(t-2Δt) will be manually set to zero, which may cause strong fluctuations, so equation (8) is used to alleviate the possible strong fluctuations.

[0042] Further, the specific implementation mode of step 3.1 is as follows:

[0043] The pose tracking error of the mechanical arm is defined as:

[0044] e Q =[e q1 ,e q2 ]=Q-Q di (10)

[0045] where Q=[q1,q2] T , e q1 ,e q2 are the errors of joint one and joint two respectively, Q di (t)=[q d1 (t),q d2 (t)] T , according to the start and end positions, velocities, accelerations and ballistic time t, 6 equations can be constructed to solve 6 coefficients of a 5th order polynomial to obtain Q di (t);

[0046] The pose tracking error must satisfy the following conditions:

[0047] -Ω Li (t)<e qi (t)<Ω Hi (t) (11)

[0048] where, for all t≥0, the constraint function Ω Li and Ω Hi are twice differentiable functions satisfying

[0049]

[0050] The safety constraint equations (10)-(12) require the pose tracking error to be within a user-defined range, where the constraint function is different for each degree of freedom. If the constraint equation (10) is violated, it will affect the motion performance of the robot arm, making the dynamic system unstable and leading to system failure.

[0051] In this step, the error e Q of the joint angle Q of the robot arm is defined, which satisfies the equation e Q = Q-Q di , where q d is the artificially given joint position expectation, and the value range of q is -π < q i < π.

[0052] Define ω as the derivative of q, and use the backstepping method to get -π < q i < π, and e ω is the error of ω, whose derivative is:

[0053]

[0054] Further, the barrier function in step 3.2 is designed as follows:

[0055] For the pose tracking error constraint condition of the robot arm, the transformed error variable is introduced:

[0056]

[0057] Ω qL is the lower bound of the asymmetric constraint, Ω qH is the upper bound of the asymmetric constraint, and η qi can achieve asymmetric constraint on the error e qi , and η qi represents the barrier function of the i-th degree of freedom.

[0058] In this step, the safety constraint of the joint position error is considered, and the derivative of the barrier function η qi is as follows:

[0059]

[0060] where,

[0061] Define the asymmetric constraint function:

[0062]

[0063]

[0064] Further, the sliding mode surface in step 3.3 is designed as:

[0065]

[0066] For the parameter, the derivative of both sides of formula (21) is

[0067]

[0068] For the convenience of expression, let:

[0069]

[0070] For external disturbance, wherein,

[0071]

[0072] Further, the design of the controller in step 4 is as follows:

[0073]

[0074] ξ1(S) and ξ2(S) are represented as follows:

[0075]

[0076] k q1 ,k q2 is an adaptive gain and is greater than 0, k q3 is a constant parameter;

[0077]

[0078] Accordingly, formula (25) can be represented as:

[0079]

[0080] Wherein, μ, η, ε are parameters.

[0081] Further, the design method of the controller is proved by using Lyapunov method.

[0082] Compared with the prior art, the present application has the following beneficial effects:

[0083] (1) A new adaptive super-twisting multivariable fast terminal sliding mode control scheme based on TDE is proposed, which has high control accuracy and robustness.

[0084] (2) The super-twisting algorithm is improved by considering asymmetric error constraints, ensuring fast convergence and high control accuracy.

[0085] (3) A general barrier function is used to solve the safety constraints of time-varying and asymmetric position errors.

[0086] (4) By using time delay estimation and compensating the overall system dynamics, the error between the actual position and the desired position is reduced. DETAILED DESCRIPTION

[0087] The technical solutions of the present application are described below.

[0088] The present application provides a TDE-based adaptive super-twisting multivariable fast terminal sliding mode control method, comprising the following steps:

[0089] Step 1: Establish a system model to describe the state of the mechanical arm;

[0090] The dynamics model of a mechanical arm with n degrees of freedom is represented as:

[0091]

[0092]

[0093]

[0094] In the formula, J and d m are the motor inertia and damping matrix, q and θ are the joint and motor position vectors, denote the first and second derivatives of q and θ, respectively, M(q) is the inertia matrix, is the Coriolis / centrifugal matrix, g(q) is the gravitational force, is the friction vector, τ m and τ s are the control torque given by the motor and the joint flexibility torque, d s is the damping matrix, k s is the joint stiffness matrix, τ d represents the lumped unknown uncertainty.

[0095] In order to facilitate the use of TDE scheme, formula (2) is substituted into formula (1), and constant parameters are applied to obtain:

[0096]

[0097] The expression for f is:

[0098]

[0099] The three main components of f, including residual link dynamics, motor dynamics, and total uncertainty, are difficult to obtain using traditional methods. Therefore, TDE is used to estimate the value of f.

[0100] Step 2: Use TDE delay estimation to obtain the estimated value of f. ;

[0101] f is particularly complex and difficult to calculate. In this part, we will use the TDE scheme to calculate it. value:

[0102]

[0103] In the formula, Δt is the delay time. Substituting the comprehensive system dynamics formula (4) into formula (6), we can obtain:

[0104]

[0105] As can be seen from Equations (6) and (7), the main purpose of the TDE scheme is to estimate the dynamics of the lumped system using only the time delay values ​​of the control and acceleration signals, and then give a model-free scheme.

[0106] In engineering applications, u s (t-Δt) can be derived from u s The direct time delay is derived, and the numerical differentiation method is used to obtain... :

[0107]

[0108] Where the time node t0≥2Δt, in the initial stage t≤2Δt, q(t) has an actual measured value, while q(t-2Δt) will be manually set to zero, which may lead to strong fluctuations. Therefore, formula (8) is used to mitigate the possible strong fluctuations. In addition, attention should be paid to formula (8) and its initial version, namely:

[0109]

[0110] It is widely used in many robust control schemes based on TDE. Without intervention, the numerical differential equation (9) will significantly amplify the noise effect, thereby reducing control performance. However, it has been theoretically proven that this can be mitigated by reducing the gain. Alternatively, an additional low-pass filter can be used to solve this problem.

[0111] From equation (8), the current value of the dynamic model equation (4) is obtained by the TDE scheme using the state estimation of time-delay system, so when a large disturbance occurs, the estimation error of this method will be large, but our proposed method can effectively reduce the estimation error.

[0112] Step 3: Design the sliding mode surface for the system model, which facilitates the design of the next control method, including the following sub-steps:

[0113] Step 3.1: Introduce a safety constraint on the joint position error in the control loop, set a safety range for the error between the actual joint position and the desired joint position, so that it has an upper limit and a lower limit. The upper limit can be given by the actual situation, which can be asymmetric or time-varying. The purpose of our control is to make it meet the safety constraint to ensure the accuracy and normal operation of the cable-driven manipulator.

[0114] Define the pose tracking error of the manipulator as:

[0115] e Q =[e q1 ,e q2 ]=Q-Q di (10)

[0116] Where Q = [q1,q2] T , e q1 , e q2 are the errors of joint one and joint two, respectively, Q di (t) = [q d1 (t), q d2 (t)] T , according to the start and end positions, velocities, accelerations and ballistic time t, 6 equations can be constructed to solve 6 coefficients of 5-order polynomials to obtain Q di (t);

[0117] The pose tracking error must satisfy the following conditions:

[0118] -Ω Li (t) < e qi (t) < Ω Hi (t) (11) Where for all t ≥ 0, the constraint functions Ω Li and Ω Hi are second-order derivable functions that satisfy

[0119]

[0120] Safety constraint formulas (10)-(12) require that the attitude tracking error does not exceed the user-defined range. The constraint functions on each degree of freedom profile can be different, and the constraint functions mentioned above can also be time-varying and asymmetric. If constraint formula (10) is violated, it will affect the motion performance of the robotic arm, making its dynamic system unstable and leading to system failure.

[0121] In this step, we define the error e of the robotic arm joint angle Q. Q It satisfies e Q =QQ di The equation, q d The expected joint position is given by the user, and the value of q is in the range of -π < q. i <π;

[0122] Define ω as the derivative of q, and use the backstepping method to obtain -π < q. i <π, e ω For the error ω, its derivative is:

[0123]

[0124] Step 3.2: Use safety constraint functions to perform design analysis on the expected angle error, so that the constraints of different sections are handled in a unified system architecture;

[0125] To meet safety constraints, this paper introduces a barrier function, which is a continuous function that can replace inequality constraints with a more manageable term in the objective function of constraint optimization, making it more conducive to asymmetric constraints.

[0126] First, the error variables are introduced as follows:

[0127]

[0128] Ω Lij Ω is the lower bound of the asymmetric constraint. Hij η is the upper bound of the asymmetric constraint. ij It can achieve the measurement of error e ij The asymmetric constraint. It can be clearly seen that if and only if e ij When η = 0, ij =0. Furthermore, when e ij →Ω Hij η ij →+∞, or, when e ij →Ω Lij , there is η ij →-∞;

[0129] For the universal barrier function V ij If the constraint function is symmetric, i.e., Ω Hij =ΩLij = Ω ij When the obstacle function η ij is:

[0130]

[0131] When there is no constraint on e ij , it can be equivalent to Ω Hij = Ω Lij = Ω ij → +∞, at this time:

[0132]

[0133] The above reasoning shows that we can regard the system without output constraint requirements as a special case of the general case of asymmetric constraint requirements.

[0134] For the pose tracking error constraint condition of the manipulator, the transformed error variable is introduced:

[0135]

[0136] Ω qL is the lower bound of the asymmetric constraint, Ω qH is the upper bound of the asymmetric constraint, η qi can achieve asymmetric constraints on the error e qi . η qi represents the obstacle function of the i-th degree of freedom;

[0137] In this step, we consider the safety constraint of the joint position error, and the derivative of the obstacle function η qi is as follows:

[0138]

[0139] where,

[0140] Define the asymmetric constraint function:

[0141]

[0142]

[0143] Step 3.3, design the sliding mode surface according to the joint position kinematics of the manipulator;

[0144] In this step, we consider the joint position kinematics of the cable-driven manipulator, and design the sliding mode surface as:

[0145]

[0146] For the parameter, the derivative of both sides of equation (21) is taken

[0147]

[0148] The symbol ° represents Hadamard product, i.e. the product of elements between two matrices with the same dimension, let:

[0149]

[0150] is the external disturbance, where,

[0151]

[0152] Step 4: Design the controller using the sliding surface

[0153] In this step, the overall controller will be designed, and the overall control scheme is as follows:

[0154]

[0155] and represent as follows:

[0156]

[0157] k q1 ,k q2 is an adaptive gain and is greater than 0, k q3 is a constant parameter;

[0158]

[0159] Therefore, equation (25) can be expressed as:

[0160]

[0161] Theorem 1: In the system with gain (27), by selecting appropriate parameters μ, η,ε, The sliding surface S of the system can converge to 0 in finite time.

[0162] Step 5: Effective proof of the above controller

[0163] For system equation (10), Lyapunov method is used for analysis, and the Lyapunov candidate function is selected as follows:

[0164]

[0165] is a symmetric positive definite matrix, where λ > 0, ε > 0;

[0166] Differentiate both sides of equation (29) with respect to k:

[0167]

[0168] Let can be transformed into:

[0169]

[0170] where Q is a symmetric matrix, where:

[0171] c1=k q1 λ 2 +4k q1 ε-λk2=a1k q1 -λk2

[0172]

[0173] c3=λ

[0174] If is negative definite, then the matrix Q is positive definite, and its determinant is greater than zero. To ensure det(Q)>0, the discriminant λk2(k q1 -λ)>0. Since k2>0, we have k q1 >λ.

[0175] where the range of k2is:

[0176]

[0177] Let k q1 =λ+τ, where τ>0, then the solution of det[Q]=0 is:

[0178]

[0179] If the range of ρ1is then det[Q]>0. When , the existence of roots can be guaranteed. Let k 2 =κ 2 , we have: Thus we have Therefore

[0180] Let λ max {P} and λ min {P} be the maximum and minimum eigenvalues of matrix P, respectively. Then equation (33) can be transformed into:

[0181]

[0182] where ξ'1is a characteristic value of ξ1, Thus

[0183]

[0184] Equation (35) can be transformed into:

[0185]

[0186] where β η = min{η1, β1, β2},

[0187] To ensure that k q1 and k q2 increase with the slope of k q1 · |S| and ηk q1 · |S| respectively, |S| must satisfy |S| > ε when the following conditions are met:

[0188]

[0189] ζ1> 0 and ζ2> 0 can be obtained;

[0190] Thus Therefore, |S| can converge to the interval |S| < ε2within a finite time; if |S| < ε2, then ζ1< 0 and ζ2< 0, The positive and negative of ζ1and ζ2are unknown, and the rate of change of the gain will become -k q1 · |S| and -η · k q1 · |S|, when the gain decreases to the interval |S| < ε2, the gain will increase with the slope of k q1 · |S| and η · k q1 · |S|.

Claims

1. A method of adaptive super-twisting multivariable fast terminal sliding mode control based on TDE, characterized in that, Comprising the following steps: Step 1, a dynamic model of the robot arm is established to describe the state of the robot arm; Step 2, the parameters in the dynamic model are estimated using TDE time delay estimation; Step 3, the design of the sliding surface is carried out to facilitate the design of the control method in the next step, including the following sub-steps: Step 3.1, a safety constraint on the joint position error is introduced, which sets a safety range for the error between the actual joint position and the desired joint position, so that it has an upper limit and a lower limit, to ensure the accuracy and normal operation of the cable-driven robot arm; Step 3.2, the safety constraint function is used to design and analyze the angle error expectation in combination with the barrier function, so that the constraints of different sections are processed in a unified architecture; The design of the barrier function in step 3.2 is as follows: For the attitude tracking error constraint condition of the robot arm, the transformed error variable is introduced: Ω qL is the lower bound of the asymmetric constraint, qH is the upper bound of the asymmetric constraint, qi asymmetric constraints on the error e qi can be implemented, η qi represents the barrier function for the i-th degree of freedom; In this step, the safety constraint of joint position error is considered, and the barrier function η qi The derivative of the barrier function η qi is as follows: wherein Define the asymmetric constraint function: Step 3.3, the sliding surface is designed according to the joint position kinematics principle of the robot arm; The sliding surface in step 3.3 is designed as: For the parameter, take the derivative of both sides of equation (21) For convenience of expression, let: for outside interference, wherein, Step 4, the terminal sliding mode controller is designed using the sliding surface; The design of the controller in step 4 is as follows: ξ1(S) and ξ2(S) are represented as follows: k q1 ,k q2 are adaptive gains and are all greater than 0, k q3 is a constant parameter; Based on the above, formula (25) can be represented as: where μ, η, ε are parameters.

2. The TDE-based adaptive super-twisting multivariable fast terminal sliding mode control method of claim 1, wherein: The dynamic model of the robot arm with n degrees of freedom established in step 1 is represented as: where J and d m are the motor inertia and damping matrices, q and θ are the joint and motor position vectors, denote the first and second derivatives of q and θ, respectively, M(q) is the inertia matrix, is the Coriolis / centrifugal matrix, g(q) is the gravity, is the friction vector, τ m and τ s are the control torque given by the motor and the joint compliance torque, d s is the damping matrix, k s is the joint stiffness matrix, τ d denotes lumped unknown uncertainties; To facilitate the use of TDE scheme, formula (2) is substituted into formula (1) to apply constant parameters resulting in: Wherein, the expression of f is:

3. The TDE-based adaptive super-twisting multivariable fast terminal sliding mode control method of claim 2, wherein: The estimate of f is found in step 2 using the TDE scheme In the formula, Δt is the delay time, and formula (4) is brought into formula (6) to obtain: From formula (6) and formula (7), it can be seen that the purpose of the TDE scheme is to estimate the lumped system dynamics only using the time delay values of the control and acceleration signals, and then to give a model-free scheme; In engineering applications, the numerical differentiation method is used to obtain Wherein, the time node t0≥2Δt, at the initial stage t≤2Δt, q(t) has an actual measurement value, and q(t-2Δt) will be manually set to zero, which may cause strong fluctuations, so formula (8) is used to alleviate the possible strong fluctuations.

4. The TDE-based adaptive super-twisting multivariable fast terminal sliding mode control method of claim 3, wherein: The specific implementation mode of step 3.1 is as follows: Define the attitude tracking error of the robot arm as: e Q = [e q1 , e q2 ] = Q - Q di (10) where Q = [q1, q2] T , e q1 , e q2 , respectively, are the errors of joint one and joint two, Q di (t) = [q d1 (t), q d2 (t)] T From the start and end positions, velocities, accelerations, and ballistic time t, six equations can be constructed to solve for the six coefficients of the fifth order polynomial to obtain Q di (t). The attitude tracking error must satisfy the following conditions: - Ω Li (t) < e qi (t) < Ω Hi (t) (11) where the constraint function Ω Li and Ω Hi is a twice differentiable function satisfying Ω(0) = 0, Ω'(0) = 0, Ω"(0) = 1, and Ω'(t) > 0 for all t > 0. The safety constraint formulas (10)-(12) require that the attitude tracking error does not exceed the range defined by the user, and the constraint functions on each degree of freedom profile are different, if the constraint formula (10) is violated, the motion performance of the robot arm will be affected, the dynamics system will be unstable, and the system will fail; In this step, the error e of the joint angle Q of the robot arm is defined Q , which satisfies the equation e Q = Q - Q di , where q d is the artificially given joint position expectation, and the value range of q is -π < q i < π. Define ω as the derivative of q, and use backstepping to get -π < q i < π, e ω is the error for ω, whose derivative is:

5. The TDE-based adaptive super-twisting multivariable fast terminal sliding mode control method of claim 1, wherein: It also includes the use of Lyapunov method to prove the design method of the controller.

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