Finite-time preset performance fault-tolerant control method for two-degree-of-freedom rigid manipulator

Abstract

The application discloses a finite time preset performance fault-tolerant control method for a two-degree-of-freedom rigid mechanical arm, and mainly solves the problem that a conventional trajectory tracking control scheme cannot estimate unknown disturbances in an external environment of a system and cannot reduce the influence of the disturbances on system stability. The method comprises the following steps: constructing a dynamics model of an actuator partial failure and complete failure, designing a finite time preset performance function to constrain a tracking error, constructing a virtual controller and an actual controller based on a backstepping method, introducing an adaptive disturbance observer to compensate for unknown external disturbances, and proving system stability through a Lyapunov function. The application can realize high-precision trajectory tracking of the mechanical arm under actuator failure conditions, ensure that the tracking error converges to a steady-state region within a set time, and reduce the dependence on prior knowledge of external disturbances and system parameters.

Description

Technical Field

[0001] This invention relates to the field of two-degree-of-freedom rigid manipulator control technology, specifically to a finite-time preset performance fault-tolerant control method for a two-degree-of-freedom rigid manipulator. Background Technology

[0002] In recent years, rigid robotic arms have been applied across various industries, enabling them to perform hazardous, repetitive, and precision tasks. Fields such as medical devices, industrial production, welding, and aerospace require robotic arms with high-precision performance requirements, such as small steady-state errors, fast convergence times, and small overshoot. Robotic arms that fail to meet these performance requirements will pose serious economic and safety risks to the tasks they perform. However, two-degree-of-freedom rigid robotic arms are susceptible to unknown external disturbances, actuator malfunctions, and uncertainties in real-world working environments, significantly reducing their performance. Therefore, research on two-degree-of-freedom rigid robotic arms in complex environments is essential to ensure they maintain good dynamic and steady-state performance.

[0003] Currently, existing robotic arm trajectory tracking control methods generally focus on the trajectory tracking of conventional robotic arm systems, rarely considering fault-tolerant control schemes when robotic arm actuators malfunction. Furthermore, most control schemes for robotic arms rely on fully known system information; however, in real-world operating environments, this information is often unavailable, rendering these control schemes ineffective. Conventional trajectory tracking control schemes also fail to anticipate unknown disturbances in the external environment, thus failing to mitigate the impact of disturbances on system stability.

[0004] Therefore, it is urgent to design a control method with fault tolerance to ensure that a two-degree-of-freedom rigid manipulator can still track the reference trajectory according to the preset performance when it encounters actuator failure. Summary of the Invention

[0005] The purpose of this invention is to provide a finite-time preset performance fault-tolerant control method for a two-degree-of-freedom rigid robotic arm, which mainly solves the problem that conventional trajectory tracking control schemes cannot predict unknown disturbances in the external environment of the system and cannot reduce the impact of disturbances on system stability.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] A finite-time preset performance fault-tolerant control method for a two-degree-of-freedom rigid robotic arm includes the following steps:

[0008] S1. Based on the dynamic model of the two-degree-of-freedom rigid manipulator, construct dynamic models with different actuator failures, including a manipulator dynamic model with partial actuator failure and a manipulator dynamic model with complete actuator failure.

[0009] S2, based on the dynamic model of a two-degree-of-freedom rigid manipulator with different actuator failures, uses an asymmetric barrier Lyapunov function and a finite-time preset performance function to constrain the tracking error, so that the actual output trajectory of the manipulator can also track the desired trajectory when the manipulator actuator fails.

[0010] S3, for the first-stage subsystem of a rigid manipulator system with two degrees of freedom and actuator failure, construct an obstacle Lyapunov function V1, and then differentiate the Lyapunov function V1 with respect to time to obtain... To ensure the stability of the first-stage subsystem of a two-DOF robotic arm system with actuator failure within a finite time, design a virtual controller α and substitute it into the system. The virtual controller gain that enables the first-level subsystem to be stable in finite time is obtained by simplifying the inequality.

[0011] S4, for the second-level subsystem of a rigid manipulator system with two degrees of freedom and actuator failure, construct Lyapunov functions V2 and V3, and obtain the time derivatives of Lyapunov functions V2 and V3. and Based on the back-calculation method, a finite-time preset performance tracking virtual controller β, χ and an actual controller τ are designed. a , τ b1 and τ b2 Perturbation Observer Adaptive law and Parameter update law and

[0012] S5, combine the Lyapunov functions V1 and V2 to construct the overall Lyapunov function V of the robotic arm system with partial actuator failure. p =V1+V2, constructing the Lyapunov functions V1 and V3 as the overall Lyapunov function V of the robotic arm system with completely failed actuators. T =V1+V3, find the Lyapunov function V p and V T The first-order time derivative, combined with the designed finite-time preset performance tracking virtual controller β, χ and the actual controller τ a , τ b1 and τ b2 Perturbation Observer Adaptive law and Parameter update law and Prove the stability of the system.

[0013] Furthermore, in step S1, the dynamic model of the robotic arm with partial actuator failure is as follows:

[0014]

[0015] The dynamic model of the robotic arm with complete actuator failure is as follows:

[0016]

[0017] Where t is time, q, These represent the position vector, velocity vector, and acceleration vector of the robotic arm joint, respectively; M(q) is the inertia matrix. G is the eccentric-Coriolis matrix; G(q) is the gravity vector; K t ρ1, ρ2 are the actuator torque coefficient matrices; τ a (t) represents the command control input vector of the dynamic model of the robotic arm with partial actuator failure; τ1 and τ2 are the actual input vectors of the two actuators on a single robotic arm in the redundant control of the robotic arm; τ b1 (t), τ b2 d(t) is the command control input vector of the redundant control in the dynamic model of the robotic arm with complete actuator failure; d(t) is the unknown external disturbance vector. Let M(q) be an unknown constant vector representing the position where the actuator is stuck; let M(q) = M, G(q) = G, d(t) = d.

[0018] Furthermore, the specific implementation method of step S2 is as follows:

[0019] S21, based on the robotic arm dynamics model and the robotic arm dynamics model with complete actuator failure, the following coordinate transformation is constructed:

[0020]

[0021] Where q = [q1 q2] T The actual output trajectory of the robotic arm is represented by q = [q1 q2]. T Let represent the desired trajectory of the robotic arm, e represent the tracking error between the actual output trajectory and the desired trajectory, and S represent the speed tracking error.

[0022] S22 uses an asymmetric barrier Lyapunov function to constrain the tracking error, i.e. and k i (t) represent the upper and lower boundaries of the constraint tracking error, respectively; where k i(t) is a monotonically decreasing, continuous, and differentiable finite-time preset performance function, as shown below:

[0023]

[0024] Among them, k0 and T represents the initial and steady-state values ​​of the finite-time preset performance function, respectively. h The convergence time is the design time.

[0025] Furthermore, in step S3, the expression for constructing the Lyapunov function V1 in the first-level subsystem is:

[0026]

[0027] In the formula: Differentiating the Lyapunov function V1 with respect to time, we get:

[0028]

[0029] Virtual controller α = [α1α2] T The expression is:

[0030]

[0031] In the formula: k 1i and k 2i For the gain of the virtual controller to be designed, k ai For a suitable constant, and k 1i k 2i and k ai All are positive numbers.

[0032] Furthermore, in step S4, for the second-level subsystem of the rigid manipulator system with two degrees of freedom due to actuator failure, based on the manipulator dynamics model with partial actuator failure, the expression for the Lyapunov function V2 of the second-level subsystem is constructed as follows:

[0033]

[0034] In the formula: and All are estimation errors, and satisfy the following conditions: and For parameter b i W i W di The estimated value of d is defined. and Represent matrices Γ iand Γ di The inverse matrix, Γ i ,Γ di All are diagonal matrices;

[0035] Differentiating the Lyapunov function V² with respect to time, we get:

[0036]

[0037] Where: β i This is the virtual controller introduced next to address actuator failure. To ensure the stability of the second-level subsystem within a finite time, based on finite-time stability theory, the designed virtual controller is β = [β1, β2]. T as follows:

[0038]

[0039] In the formula: K3=diag[k 31 k 32 ], K4 = diag[k 41 k 42 ], k 3i k 4i The gain of the controller to be designed.

[0040] The unknown external disturbance is estimated for the constructed disturbance observer, which has the following structure:

[0041]

[0042] In the formula: k a =diag{k a1 ,k a2}, k d =diag{k d1 ,k d2}, and k a ≥hk d ≥k d M -1 (q)>0, k ai k di Let h be the appropriate parameter to be designed, and h be the adaptive parameter. As an auxiliary variable;

[0043] Actual controller τ a Parameter adaptive law and neural network adaptive law The expressions are as follows:

[0044]

[0045] In the formula: γi , σ i , σ di Let Γ be a suitable positive constant to be designed. i and Γ di It is the gain matrix to be designed.

[0046] Furthermore, in step S4, for the second-level subsystem of the rigid manipulator system with two degrees of freedom due to actuator failure, based on the manipulator dynamics model with all actuators failing, the expression for the Lyapunov function V3 of the second-level subsystem is constructed as follows:

[0047]

[0048] In the formula: To estimate the error, and satisfy the following conditions: For parameter K ij The estimated value, Represent matrices Γ Di The inverse matrix, Γ Di It is a diagonal matrix;

[0049] Differentiating the Lyapunov function V3 with respect to time, we get:

[0050]

[0051] Where: χ i A virtual controller was introduced to address actuator failure, and a controller β was designed to handle partial actuator failure. i Same structure, β i and χ i Only external disturbances There is a difference: the virtual controller χ = [χ1χ2] T The structure is as follows:

[0052]

[0053] In the formula: K5=diag[k 51 k 52 ], K6 = [k 61 k 62 ], k 5i k 6i The gain of the controller to be designed;

[0054] The disturbance observer structure is constructed as follows when the actuator fails completely:

[0055]

[0056] The dynamic uncertainty term is approximated using radial basis function neural network technology, specifically expressed as follows:

[0057]

[0058] In the formula: F D (Z)=[f D1 (Z D ),f D2 (Z D )] T Applying radial basis function neural networks to approximate f Di (Z D ):

[0059]

[0060] In the formula: W Di Represents the ideal weight vector. For the ideal weight vector W Di The estimated value, N Di (Z D ) indicates that the input is Z D radial basis functions, ε Di (Z D ) represents the approximation error, and satisfies ||N D (Z D )|| 2 ≤n D , n D All are positive numbers;

[0061] Actual controller τ b1 , τ b2 Parameter adaptive law and neural network adaptive law The expressions are as follows:

[0062]

[0063]

[0064] In the formula: γ here i and σ i Parameter γ of the robotic arm system with actuator failure i and σ i Same, σ Di K is a suitable positive constant to be designed. i1 K is a 2×2 matrix. i2 It is a 2×1 vector.

[0065] Compared with the prior art, the present invention has the following beneficial effects:

[0066] (1) The steady-state region of the performance range of the constraint tracking error of the present invention decreases as time increases, and the time to enter the steady-state region cannot be quantitatively expressed. The tracking error only converges to the preset steady-state region as time increases.

[0067] (2) The finite-time preset performance function introduced in this invention can ensure that the tracking error converges to the boundary region of the performance function within any time to be set, and the dynamic performance and steady-state performance of the system can be designed according to the requirements to meet the performance requirements under different conditions.

[0068] (3) The present invention uses a neural network to construct an adaptive perturbation observer, which avoids the disadvantage of needing to obtain the parameters of a two-degree-of-freedom rigid manipulator system to construct a traditional observer.

[0069] (4) This invention uses neural networks to approximate the dynamic uncertainty in the system. The proposed controllers do not require a precise two-degree-of-freedom rigid manipulator system model or prior knowledge of external disturbances. Even when the physical parameters of the manipulator system are completely unknown, the designed controllers still ensure that the position and velocity tracking errors of the first and second manipulators converge to the stable region within a finite time. Attached Figure Description

[0070] Figure 1 This is a schematic diagram of a two-degree-of-freedom rigid robotic arm;

[0071] Figure 2 This is a schematic diagram of the design process for a finite-time preset performance fault-tolerant controller for a two-degree-of-freedom rigid robotic arm.

[0072] Figure 3 This is a structural diagram of a fault-tolerant control method for a two-degree-of-freedom rigid robotic arm;

[0073] Figure 4 This is a graph showing the trajectory tracking performance of a two-degree-of-freedom rigid robotic arm under partial failure conditions.

[0074] Figure 5 This is a tracking error trajectory diagram of the first robotic arm of a two-degree-of-freedom rigid robotic arm under partial failure conditions;

[0075] Figure 6 This is a tracking error trajectory diagram of the second robotic arm of a two-degree-of-freedom rigid robotic arm under partial failure conditions;

[0076] Figure 7 This is a control input curve diagram of the two robotic arms in the case of partial failure of a two-degree-of-freedom rigid robotic arm;

[0077] Figure 8 This is a graph showing the trajectory tracking performance of two rigid manipulators under the condition of complete failure.

[0078] Figure 9 This is a tracking error trajectory diagram of the first robotic arm of a two-degree-of-freedom rigid robotic arm under the condition of complete failure.

[0079] Figure 10 This is a tracking error trajectory diagram of the second robotic arm of a two-degree-of-freedom rigid robotic arm under the condition of complete failure.

[0080] Figure 11 This is a control input curve diagram of the two robotic arms in the case of complete failure of a two-degree-of-freedom rigid robotic arm. Detailed Implementation

[0081] The present invention will be further described below with reference to the accompanying drawings and embodiments. The embodiments of the present invention include, but are not limited to, the following embodiments.

[0082] Example

[0083] Please refer to Figure 1 , Figure 2 and Figure 3 This invention provides a finite-time preset performance fault-tolerant control method for a two-degree-of-freedom rigid robotic arm, comprising the following steps:

[0084] Step 1: Based on the dynamic model of the two-degree-of-freedom rigid manipulator, construct the following dynamic models with different actuator failures:

[0085] Dynamic model of a robotic arm with partial actuator failure:

[0086]

[0087] Dynamics model of a robotic arm with complete actuator failure:

[0088]

[0089] In the formula K t =diag{k t1 ,k t2}, 0 < k ti ≤1; where C 22 =0; G1=(m1l c2 +m2l1)gcosq1+m2l c2 gcos(q1+q2); G2=m2l c2 gcos(q1+q2); ρ1=diag{ρ 11 , ρ 12}, ρ2=diag{ρ21 , ρ 22},0≤ρ ij ≤1, i=1,2, j=1,2; t is time; q,

[0090] q1, q2, and q3 represent the position vector, velocity vector, and acceleration vector of the robotic arm joints, respectively; M(q) is the inertia matrix; q1, q2, q3, q4, q5, q6, q7, q8, q9, q1 ... and These are the position, velocity, and acceleration of the first robotic arm; q2, and These are the position, velocity, and acceleration of the second robotic arm; G is the eccentric-Coriolis matrix; G(q) is the gravity vector; K t ρ1, ρ2 are the actuator torque coefficient matrices; τ a (t) represents the command control input vector of the dynamic model of the robotic arm with partial actuator failure; τ1 and τ2 are the actual input vectors of the two actuators on a single robotic arm in the redundant control of the robotic arm; τ b1 (t), τ b2 (t) is the command control input vector of the redundant control of the dynamic model of the robotic arm with complete actuator failure; d(t) is the unknown external disturbance vector; Let m1 and m2 be the unknown constant vectors representing the actuator's stuck position; m1 and m2 are the masses of the first and second robotic arms, respectively; l1 and l2 are the lengths of the first and second robotic arms, respectively; l c1 and l c2 I1 and I2 are the distances from the center of mass of the first and second robotic arms to their preceding joints, respectively; I1 and I2 are the moments of inertia of the first and second robotic arms, respectively; g is the acceleration due to gravity. diag[·] represents a diagonal matrix.

[0091] In the actuator complete failure dynamics model of this embodiment, τ b1 (t), τ b2 (t) indicates that each joint of the robotic arm is controlled by two actuators, and simultaneously satisfies This indicates that partial failure and complete failure of the actuator will not occur simultaneously on the same actuator.

[0092] Detailed system parameters are shown in Table 1:

[0093] Table 1: Model Parameters of a Two-DOF Rigid Robotic Arm System

[0094]

[0095] Step 2: Based on the dynamic model obtained in Step 1, construct the following coordinate transformation:

[0096]

[0097] In the formula: α=[α1α2] T Design a virtual controller for the finite-time stability of the first-level subsystem, where S represents the speed tracking error.

[0098] The fault-tolerant control objective for a two-degree-of-freedom rigid robotic arm with different actuator failures is to ensure that the system output q = [q1 q2] for both the first and second robotic arms under different types of actuator failures. T Track the reference trajectory q as much as possible d =[q 1d q 2d ] T . q 1d It is the reference trajectory of the first robotic arm, q 2d This is the reference trajectory for the second robotic arm. This results in a tracking error e. i =q i -q id In order to achieve a shorter convergence time, smaller overshoot, and smaller steady-state error in tracking error, i.e. and k i (t) represents the finite-time preset performance function for constraining the upper and lower boundaries of the tracking error. k i (t)=-η i k i (t), where and For the parameters to be designed,

[0099] k i (t) is a monotonically decreasing, continuous, and differentiable finite-time preset performance function, as shown below:

[0100]

[0101] In the formula: k i,0 and T represents the initial and steady-state values ​​of the finite-time preset performance function, respectively. h The convergence time is designed to be small, where the tracking error boundary is an asymmetric performance range, i.e.

[0102] Step 3: Based on the coordinate transformation constructed in Step 2, construct the Lyapunov function V1 for the first-stage subsystem of a two-degree-of-freedom rigid manipulator with different actuator faults:

[0103]

[0104] In the formula: and k i (t) is the error constraint boundary of the rigid robotic arm.

[0105] Differentiating the Lyapunov function V1 with respect to time, we get:

[0106]

[0107] To ensure that the first-stage subsystem of the rigid robotic arm tends to stabilize in finite time, the following virtual controller α = [α1α2] is designed. T :

[0108]

[0109] In the formula: k 1i and k 2i For the gain of the virtual controller to be designed, k ai For a suitable constant, and k 1i k 2i and k ai All are positive numbers.

[0110] Substituting the designed virtual controller α into formula (6), we get:

[0111]

[0112] In the formula: For simplicity, let e i (t)=e i .

[0113] Step 4: Based on Step 3, for the second-stage subsystem of a two-degree-of-freedom rigid manipulator with different actuator failures, construct different Lyapunov functions, and then perform classification analysis:

[0114] Step 4.1: Based on the robotic arm dynamics model (1) of the actuator failure in Step 1, construct the Lyapunov function V2 for the second-level subsystem:

[0115]

[0116] In the formula:

[0117] and All are estimation errors, and satisfy the following conditions: and For parameter b i W i W di The estimated value of d is defined. and Represent matrices Γ i and Γ di The inverse matrix, Γ i ,Γ di All are diagonal matrices.

[0118] Differentiating the Lyapunov function V² with respect to time, we get:

[0119]

[0120] Where: β i This is the virtual controller introduced next to address actuator failure. To ensure the stability of the second-level subsystem within a finite time, based on finite-time stability theory, the designed virtual controller is β = [β1, β2]. T as follows:

[0121]

[0122] In the formula: K3=diag[k 31 k 32 ], K4 = diag[k 41 k 42 ], k 3i k 4i The gain of the controller to be designed.

[0123] The unknown external disturbance is estimated for the constructed disturbance observer, which has the following structure:

[0124]

[0125] In the formula: k a =diag{k a1 ,k a2}, k d =diag{k d1 ,k d2}, and k a ≥hk d ≥k d M -1 (q)>0, k ai k di Let h be the appropriate parameter to be designed, and h be the adaptive parameter. It is an auxiliary variable.

[0126] To avoid the impact of system parameter uncertainties on the use of the virtual controller, a radial basis function neural network technique is used to approximate the dynamic uncertainties, specifically expressed as follows:

[0127]

[0128] In the formula: F(Z)=[f1(Z),f2(Z)] T F d (Z d )=[f d1 (Z d ),f d2 (Z d )] T Radial basis function neural networks are used to approximate f respectively. i (Z), f di (Z d ):

[0129]

[0130]

[0131] In the formula: W, W d Represents the ideal weight vector. W respectively i W di The estimated value, N i (Z) and N di (Z d ) represent inputs Z and Z respectively d radial basis functions, ε i (Z), ε di (Z d ) represents the approximation error, and satisfies ||N d (Z d )|| 2 ≤n d 。 ||·| 2 Let be the squared norm of the vector ·.

[0132] To ensure that a two-degree-of-freedom rigid robotic arm system with partial actuator failure tends to stabilize in finite time, the virtual controller β introduced earlier is used... i Based on this, the actual controller τ is designed according to the system model (1). a Parameter adaptive law and neural network adaptive law

[0133]

[0134] In the formula: γ i , σ i , σ di Let Γ be a suitable positive constant to be designed. i and Γ di It is the gain matrix to be designed.

[0135] The designed finite-time real controller τ is adopteda Perturbation Observer Parameter adaptive law and neural network adaptive law Substituting into formula (10), we can simplify using inequalities to obtain:

[0136]

[0137] In the formula: I is the identity matrix.

[0138] Step 4.2: Similar to the failure type of a rigid robotic arm with partial actuator failure, based on the dynamic model (2) of the robotic arm with complete actuator failure in Step 1, a Lyapunov function V3 is constructed for the second-level subsystem:

[0139]

[0140] In the formula: To estimate the error, and satisfy the following conditions: For parameter K ij The estimated value, Represent matrices Γ Di The inverse matrix, Γ Di It is a diagonal matrix.

[0141] Differentiating the Lyapunov function V3 with respect to time, we get:

[0142]

[0143] Where: χ i A virtual controller was introduced to address actuator failure, and a controller β was designed to handle partial actuator failure. i Same structure, β i and χ i Only external disturbances There is a difference: the virtual controller χ = [χ1χ2] T The structure is as follows:

[0144]

[0145] In the formula: K5=diag[k 51 k 52 ], K6 = [k 61 k 62 ], k 5i k 6i The gain of the controller to be designed.

[0146] The disturbance observer structure is constructed as follows when the actuator fails completely:

[0147]

[0148] The dynamic uncertainty term is approximated using radial basis function neural network technology, specifically expressed as follows:

[0149]

[0150] In the formula: F D (Z)=[f D1 (Z D ),f D2 (Z D )] T Applying radial basis function neural networks to approximate f Di (Z D ):

[0151]

[0152] In the formula: W Di Represents the ideal weight vector. For the ideal weight vector W Di The estimated value, N Di (Z D ) indicates that the input is Z D radial basis functions, ε Di (Z D ) represents the approximation error, and satisfies ||N D (Z D )|| 2 ≤n D , n D All are normal numbers.

[0153] To ensure that a two-degree-of-freedom rigid robotic arm system with complete actuator failure tends to stabilize in finite time, the virtual controller χ introduced earlier is used... i Based on this, the actual controller τ is designed according to the system model (2). b1 , τ b2 Parameter adaptive law and neural network adaptive law

[0154]

[0155] In the formula: γ here i and σ i Parameter γ of the robotic arm system with actuator failure i and σ i Same, σ Di K is a suitable positive constant to be designed. i1 K is a 2×2 matrix. i2 It is a 2×1 vector.

[0156] Employing the designed finite-time practical controller τ b1 τ b2 Parameter adaptive law and neural network adaptive law Substituting into formula (23), we can simplify using inequalities to obtain:

[0157]

[0158] Step 5: Based on formulas (8), (21), and (34) obtained in steps 3 and 4, construct the global Lyapunov function V for a two-degree-of-freedom rigid manipulator system with actuator failure. p =V1+V2 and V T =V1+V3, V p It is the global Lyapunov function V corresponding to the actuator failure of the robotic arm system. T It is the global Lyapunov function corresponding to the robotic arm system where the actuators have completely failed.

[0159] Step 5.1, V p Taking the derivative with respect to time, we get:

[0160]

[0161] In the formula:

[0162]

[0163] In the formula: λ min (·) and λ max (*) represent the minimum and maximum eigenvalues ​​of matrices · and *, respectively.

[0164] According to formula (35), a virtual controller α is designed. i β i finite-time preset performance actual controller τ a Perturbation Observer Parameter adaptive law and neural network adaptive law This ensures the finite-time boundedness of all signals in the closed-loop system of a two-degree-of-freedom rigid robotic arm under partial actuator failure, with the upper bound of the settling time being:

[0165]

[0166] In the formula: V(0) is the value of the initial time, 0<θ0<1, and according to the above stability analysis, all signals of the two-degree-of-freedom rigid manipulator system with actuator partial failure are finite-time bounded.

[0167] Step 5.2, V T Differentiating with respect to time, we get:

[0168]

[0169] In the formula:

[0170]

[0171] Similarly, using the actual controller τ b1 , τ b2 Parameter adaptive law and neural network adaptive law This ensures the finite-time boundedness of all signals in a closed-loop robotic arm system where the actuators completely fail, with the upper bound of the settling time being:

[0172]

[0173] Based on the stability analysis above, all signals of a two-degree-of-freedom rigid robotic arm system with complete actuator failure are finite-time bounded.

[0174] To illustrate the control effect of the method of the present invention in detail, a simulation experiment will be conducted in MATLAB, with the reference trajectory set as q. 1d =sin(t), q 2d =cos(2t), the initial conditions of the system are set as q1(0)=1, q2(0)=0; The unknown external disturbances are set as d1(t) = -0.2sin(t) and d2(t) = 0.2cos(2t).

[0175] The boundary of the tracking error of the constraint system is Under partial actuator failure, the actuator torque coefficient matrix is: In the case of complete actuator failure, the torque coefficient matrix of the actuator is ρ1=K t , The selected controller gain is shown in Table 2.

[0176] Table 2: Controller Gain

[0177]

[0178] The following results were obtained through simulation experiments using MATLAB. Figure 4 This is a graph showing the trajectory tracking performance of a two-degree-of-freedom rigid manipulator under partial failure conditions. It can be seen that the system output trajectories q1 and q2 can quickly track the reference trajectory q even when the actuator experiences partial failure. 1d and q 2d ; Figure 5 and Figure 6 It is a tracking error trajectory diagram of a two-degree-of-freedom rigid manipulator under partial failure conditions. The tracking error can converge to the envelope of the finite-time performance function in a finite time. Figure 7 This is a graph showing the control input curves of the first and second robotic arms in a two-degree-of-freedom rigid robotic arm under partial failure conditions, τ. a τ represents the controller's command input, and τ is the actuator's actual input, satisfying τ = K t τ a ; Figure 8 This is a graph showing the trajectory tracking performance of a two-degree-of-freedom rigid manipulator under complete failure conditions. The system output trajectories q1 and q2 can quickly track the reference trajectory q even when the actuators experience complete failure. 1d and q 2d ; Figure 9 and Figure 10 It is the tracking error trajectory diagram of the second robotic arm of the two-degree-of-freedom rigid robotic arm under the condition of complete failure. The tracking error can converge to the preset steady-state error region within a finite time. Figure 11 This is a control input curve diagram of the first and second robotic arms in the case of complete failure of a two-degree-of-freedom rigid robotic arm.

[0179] Simulation results show that the finite-time preset performance tracking controller design method proposed in this invention ensures the finite-time stability of a two-degree-of-freedom rigid manipulator system with actuator failure; the tracking error converges to the preset steady-state range within a given finite time; and the tracking performance requirements under different conditions can be met by adjusting the design parameters of the finite-time performance function; even when the actuator fails, the two-degree-of-freedom manipulator system can still accurately track the reference trajectory.

[0180] The above embodiments are merely one of the preferred embodiments of the present invention and should not be used to limit the scope of protection of the present invention. Any modifications or refinements made to the main design concept and spirit of the present invention that are not of substantial significance, but solve the same technical problem as the present invention, should be included within the scope of protection of the present invention.

Claims

1. A finite-time preset performance fault-tolerant control method for a two-degree-of-freedom rigid robotic arm, characterized in that, Includes the following steps: S1. Based on the dynamic model of the two-degree-of-freedom rigid manipulator, construct dynamic models with different actuator failures, including a manipulator dynamic model with partial actuator failure and a manipulator dynamic model with complete actuator failure. S2, based on the dynamic model of a two-degree-of-freedom rigid manipulator with different actuator failures, uses an asymmetric barrier Lyapunov function and a finite-time preset performance function to constrain the tracking error, so that the actual output trajectory of the manipulator can also track the desired trajectory when the manipulator actuator fails. S3, constructing a barrier Lyapunov function for the first-stage subsystem of a rigid manipulator system with a two-degree-of-freedom actuator failure. Lyapunov function Taking the derivative with respect to time yields To stabilize the first-level subsystem of a two-degree-of-freedom robotic arm system with actuator failure within a finite time, a virtual controller is designed. Substitution The virtual controller gain that enables the first-level subsystem to be stable in finite time is obtained by simplifying the inequality. S4, Constructing a Lyapunov function for the second-level subsystem of a rigid manipulator system with two degrees of freedom and actuator failure. and For Lyapunov functions and The time derivative is obtained and Design of a finite-time preset performance tracking virtual controller based on the back-propagation method , and actual controller , and Perturbation Observer Adaptive law , and Parameter update law and In particular, for the second-stage subsystem of a two-degree-of-freedom rigid manipulator with different actuator failures, different Lyapunov functions are constructed, and a classification analysis is performed: Based on the dynamic model of the robotic arm with partial actuator failure, a Lyapunov function is constructed for the second-level subsystem. In the introduced virtual controller Based on this, an actual controller is designed according to the dynamic model of the robotic arm. Parameter adaptive law and neural network adaptive law , ; Based on the dynamics model of the robotic arm with complete actuator failure, a Lyapunov function is constructed for the second-level subsystem. In the introduced virtual controller Based on this, a practical controller is designed according to the dynamic model of a completely failed robotic arm. , Parameter adaptive law and neural network adaptive law , ; S5, Lyapunov function and The global Lyapunov function is constructed by combining the components of a robotic arm system with partially failed actuators. Lyapunov function and The overall Lyapunov function of the robotic arm system is created when the actuators completely fail. Find the Lyapunov function. and The first-order time derivative, combined with the designed finite-time preset performance tracking virtual controller. , and actual controller , and Perturbation Observer Adaptive law , and Parameter update law and This proves the stability of the system.

2. The finite-time preset performance fault-tolerant control method for a two-degree-of-freedom rigid robotic arm according to claim 1, characterized in that, In step S1, the dynamic model of the robotic arm with partial actuator failure is as follows: The dynamic model of the robotic arm with complete actuator failure is as follows: in, For time, , , These represent the position vector, velocity vector, and acceleration vector of the robotic arm joints, respectively. The inertia matrix; It is an eccentric-Coriolis matrix; It is the gravity vector; , , This is the actuator torque coefficient matrix; The command control input vector is used for the dynamic model of the robotic arm with partially failed actuators. and For redundant control of a robotic arm, the actual input vectors of the two actuators on a single robotic arm are given. , The command control input vector is used for redundant control in the dynamic model of a robotic arm where the actuator has completely failed. The external disturbance vector is unknown. , Let be an unknown constant vector representing the position where the actuator is stuck; let , , , .

3. The finite-time preset performance fault-tolerant control method for a two-degree-of-freedom rigid robotic arm according to claim 2, characterized in that, The specific implementation method of step S2 is as follows: S21, based on the robotic arm dynamics model and the robotic arm dynamics model with complete actuator failure, the following coordinate transformation is constructed: in, This represents the actual output trajectory of the robotic arm. Let represent the desired trajectory of the robotic arm, e represent the tracking error between the actual output trajectory and the desired trajectory, and S represent the speed tracking error. S22 uses an asymmetric barrier Lyapunov function to constrain the tracking error, i.e. , and These represent the upper and lower boundaries of the constraint tracking error, respectively; where, It is a monotonically decreasing, continuous, and differentiable finite-time preset performance function, as shown below: in, and These represent the initial and steady-state values ​​of the finite-time preset performance function, respectively. The convergence time is the design time.

4. The finite-time preset performance fault-tolerant control method for a two-degree-of-freedom rigid robotic arm according to claim 3, characterized in that, In step S3, the first-level subsystem constructs the Lyapunov function. The expression is: In the formula: Lyapunov functions Taking the derivative with respect to time, we get: Virtual Controller The expression is: In the formula: , , and For the gain of the virtual controller to be designed, For a suitable constant, and , and All are positive numbers.

5. The finite-time preset performance fault-tolerant control method for a two-degree-of-freedom rigid robotic arm according to claim 4, characterized in that, In step S4, for the second-level subsystem of a rigid manipulator system with a two-degree-of-freedom actuator failure, a Lyapunov function of the second-level subsystem is constructed based on the manipulator dynamics model with partial actuator failure. The expression is: In the formula: , , and All are estimation errors, and satisfy the following conditions: , , , ; , , and For parameters , , and The estimated value, defined , and Represent matrices respectively and The inverse matrix, , All are diagonal matrices; Lyapunov function Taking the derivative with respect to time, we get: In the formula: The virtual controller introduced next is designed to address actuator failures. To ensure the stability of the second-level subsystem within a finite time, the virtual controller is designed based on finite-time stability theory. as follows: In the formula: , , , The gain of the controller to be designed; The unknown external disturbance is estimated for the constructed disturbance observer, which has the following structure: In the formula: , ,and , , For the appropriate parameters to be designed, For adaptive parameters, As an auxiliary variable; Actual controller Parameter adaptive law and neural network adaptive law , The expressions are as follows: In the formula: , , For the appropriate positive constants to be designed, and It is the gain matrix to be designed.

6. The finite-time preset performance fault-tolerant control method for a two-degree-of-freedom rigid robotic arm according to claim 4, characterized in that, In step S4, for the second-level subsystem of a rigid manipulator system with two degrees of freedom due to actuator failure, a Lyapunov function of the second-level subsystem is constructed based on the dynamic model of the manipulator with all actuators failing. The expression is: In the formula: , To estimate the error, and satisfy the following conditions: , ; For parameters The estimated value, Represent matrices respectively The inverse matrix, It is a diagonal matrix; Lyapunov function Taking the derivative with respect to time, we get: In the formula: A virtual controller introduced to address actuator failure, and a controller designed to handle partial actuator failure. Same structure and Only external disturbances There are differences, virtual controller The structure is as follows: In the formula: , , , The gain of the controller to be designed; The disturbance observer structure is constructed as follows when the actuator fails completely: The dynamic uncertainty term is approximated using radial basis function neural network technology, specifically expressed as follows: In the formula: Approximation using radial basis function neural networks : In the formula: Represents the ideal weight vector. Ideal weight vector The estimated value, Indicates that the input is radial basis functions, Let represent the approximation error, and satisfy . , , , All are positive numbers; Actual controller , Parameter adaptive law and neural network adaptive law , The expressions are as follows: In the formula: and Parameters of a robotic arm system with partial actuator failure and same, For the appropriate positive constants to be designed, for The matrix, for The vector.

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