Design Method of a Predetermined Time Controller for a Dual-Joint Manipulator

Abstract

The invention discloses a design method for a preset time controller of a dual-joint manipulator. Based on the mathematical model of the dual-joint manipulator, an adjustment function is designed based on the transformed model formula to obtain the first-step error, a virtual control law is designed, and finally the second-step error is obtained to design the preset time controller. The invention can not only ensure the state consistency and parameter consistency of the convergence time of the system; at the same time, in the proposed preset time controller, the convergence time of the system is a parameter in the controller, so the convergence time of the system can be directly set and adjusted through this parameter.

Description

Technical Field

[0001] The present invention relates to the technical field of robotic arms, and specifically to a design method for a preset time controller of a dual-joint robotic arm. Background Art

[0002] Robotic arms are an important part of the manufacturing industry and play a crucial role in fields such as automotive, military, and medical. In current research methods for robotic arms, many control methods have been proposed.

[0003] In solving the problem of rapid position tracking of robotic arms, finite-time control and fixed-time control have been proposed successively. Although the finite-time control method can make the convergence time of the system bounded, its convergence time is related to the initial value of the system and cannot guarantee the consistency of the convergence time with respect to the state; although the fixed-time control method can guarantee the consistency of the convergence time with respect to the state, it is difficult to guarantee the consistency of the convergence time with respect to the parameters. Summary of the Invention

[0004] The purpose of the present invention is to provide a design method for a preset time controller of a dual-joint robotic arm, and to solve the following problems in the prior art: the finite-time control method can make the convergence time of the system bounded, but its convergence time is related to the initial value of the system and cannot guarantee the consistency of the convergence time with respect to the state; although the fixed-time control method can guarantee the consistency of the convergence time with respect to the state, it is difficult to guarantee the consistency of the convergence time with respect to the parameters. The present invention overcomes the above problems. The present invention can not only guarantee the state consistency and parameter consistency of the convergence time of the system; at the same time, in the proposed preset time controller, the convergence time of the system is a parameter in the controller, so the convergence time of the system can be directly set and adjusted through this parameter.

[0005] To achieve the above object, the present invention provides the following technical solutions:

[0006] A design method for a preset time controller of a dual-joint robotic arm, comprising the following steps:

[0007] S1: Establish a mathematical model of the dual-joint robotic arm and perform conversion to obtain a converted model formula;

[0008] S2: Design an adjustment function according to the model formula obtained in S1;

[0009] S3: Define the control objective, and obtain the first-step error according to the model formula obtained in S1 and the adjustment function of S2;

[0010] S4: Design a virtual control law according to the first-step error obtained in S3;

[0011] S5: Obtain the second - step error according to the virtual control law in S4 and the model formula in S1.

[0012] S6: Design a predefined - time controller according to the model formula in S1 and the second - step error obtained in S5.

[0013] The mathematical model of the two - joint robotic arm established in S1 is:

[0014]

[0015] where q is the actual angular position of the robotic arm; is the angular velocity of the robotic arm; is the angular acceleration of the robotic arm, M(q) is the inertia matrix of the robotic arm; is the matrix of the centrifugal force and Coriolis force of the robotic arm; G(q) represents the gravity term of the robotic arm; u represents the predefined - time controller to be designed, and d(t) is the uncertain disturbance of the robotic arm varying with time.

[0016] The transformed model formula in S1 is:

[0017]

[0018]

[0019] where q1 = q, q2 is the angular velocity of the robotic arm, is the angular acceleration of the robotic arm, C is the matrix of the centrifugal force and Coriolis force of the transformed robotic arm, G is the gravity term of the robotic arm, and d is the uncertain disturbance of the robotic arm.

[0020] The adjustment - function design formula in S2 is:

[0021]

[0022] where μ i t, T is the adjustment function, m0 and m1 are initial constant vectors, the subscript i represents the i - th component of the vector, t is the operating time of the robotic arm, and T is the set value of the convergence time.

[0023] The acquisition of the first - step error is based on the backstepping control principle.

[0024] The first - step error in S3 is:

[0025] z1 = q1 - q d -μ(t, T)

[0026] where z1 is the first - step error, q1 is the actual angular position of the robotic arm, q dLet the target tracking trajectory be, \(T\) be the set value of the convergence time, and \(\mu(t, T)\) be the time-varying adjustment function vector.

[0027] The setting method of the virtual control law in S4 is as follows:

[0028]

[0029] Among them, \(\alpha\) is the virtual control law, \(c1\) is the control parameter, \(z1\) is the first-step error, is the first derivative of the target tracking trajectory, is the first derivative of the time-varying adjustment function vector.

[0030] The second-step error in S5 is:

[0031] \(z2 = q2 - \alpha\)

[0032] Among them, \(\alpha\) is the virtual control law, \(z2\) is the second-step error, and \(q2\) is the angular velocity of the robotic arm.

[0033] The design method of the predefined-time controller in S6 is as follows:

[0034]

[0035] Among them, \(u\) is the predefined-time controller, \(c2\) is the control parameter, \(z2\) is the second-step error, \(C\) is the matrix of the centrifugal force and Coriolis force of the robotic arm, \(q2\) is the angular velocity of the robotic arm, \(G\) is the gravity term of the robotic arm, \(D\) is a known constant, \(\tanh(\cdot)\) is the hyperbolic tangent function, \(\kappa = 0.2785\), \(\kappa1\) is a constant, \(\varepsilon\) is the upper bound control parameter of the final convergence error, \(z1\) is the first-step error, and \(M\) is the moment of inertia matrix of the robotic arm, is the first derivative of the virtual control law.

[0036] After S4 and S6 are completed, the Lyapunov function is used for stability analysis.

[0037] Compared with the prior art, the present invention has the following beneficial effects: The present invention provides a design method for a predefined-time controller of a two-joint robotic arm. Based on the mathematical model of the two-joint robotic arm, an adjustment function is designed based on the transformed model formula, the first-step error is obtained, a virtual control law is designed, and finally the second-step error is obtained to design the predefined-time controller. The present invention can not only ensure the state consistency and parameter consistency of the convergence time of the system; at the same time, in the proposed predefined-time controller, the convergence time of the system is a parameter in the controller, so the convergence time of the system can be directly set and adjusted through this parameter. Description of the Drawings

[0038] Figure 1 is the design flow chart of a design method for a predefined-time controller of a two-joint robotic arm according to the present invention;

[0039] Figure 2 This is the structure diagram of the double-joint robotic arm of the present invention;

[0040] Figure 3 This is the structure diagram of the predetermined time controller designed by the present invention;

[0041] Figure 4 This is the angle tracking diagram of the double-joint robotic arm of the present invention;

[0042] Figure 5 This is the angle tracking error diagram of the double-joint robotic arm of the present invention;

[0043] Figure 6 This is the angular velocity diagram of the double-joint robotic arm of the present invention;

[0044] Figure 7 This is the control torque input diagram of the double-joint robotic arm of the present invention. Detailed implementation manners

[0045] The following further elaborates on the present invention in conjunction with specific embodiments, which is an explanation rather than a limitation of the present invention.

[0046] As Figure 1-7 shown, the present invention provides a design method for a predetermined time controller of a double-joint robotic arm,

[0047] For the convenience of describing the problem, mark to represent the set of real numbers, to represent the set of positive real numbers, to represent the set of non-negative real numbers; to represent an n-dimensional column vector, to represent an m×n matrix; λ min (A) and λ max (A) respectively represent the minimum eigenvalue and the maximum eigenvalue of the bounded positive definite matrix A; for the vector ||x||1 = [|x1|, |x2|,..., |x n |] T represents the 1-norm of the vector, represents the Euclidean norm of the vector, and ||A|| represents the Euclidean norm of the matrix.

[0048] Definition of globally predetermined time boundedness:

[0049] For the nonlinear system

[0050]

[0051] In the formula is the system state, T d is the known system parameter, is a non - linear function, which is piece - wise continuous with respect to t and locally Lipschitz continuous with respect to x. The trajectory of the solution of the above - mentioned system can be expressed as x(t, x0, T d ), where x0 = x(0, x0, T d ) represents the initial value of the system.

[0052] Then the definition of globally pre - determined - time boundedness can be given as:

[0053] For the system (1), if there exists an arbitrary constant such that for the trajectory x(t, x0, T ) starting from any initial value d it always satisfies ||x(t, x0, T d )|| ≤ δ after time T d ), then the solution of the system (1) is said to be pre - determined - time globally bounded.

[0054] The system (1) can be equivalent to a control system:

[0055]

[0056] where u = u(x, t, T d ) is the control vector, and Substituting u(x, t, T d ) into (2), we can get f(x, t, T d ) = g(x, u(x, t, T d )).

[0057] Without loss of generality, the model of a two - joint robotic arm can be described as:

[0058]

[0059] where is the actual angular position of the robotic arm, q = [q1, q2] T ; is the angular velocity of the robotic arm; is the angular acceleration of the robotic arm. is the inertia matrix of the robotic arm; is the matrix of centrifugal and Coriolis forces of the robotic arm; represents the gravity term of the robotic arm; represents the joint control torque of the robotic arm, is the uncertain disturbance of the robotic arm.

[0060] Property 1: The inertia matrix M(q) is a positive - definite matrix, symmetric and bounded, that is

[0061] 0 < λ min(M) ≤ ||M(q)|| ≤ λ max (M) (4)

[0062] Property 2: The matrices of the Coriolis force and the centrifugal force satisfy the following relationship:

[0063]

[0064]

[0065] where C min and C max are unknown positive constants.

[0066] Property 3: is an anti-symmetric matrix, i.e.,

[0067]

[0068] Lemma 1: For the following inequality always holds

[0069]

[0070] where κ = 0.2785.

[0071] Assumption 1: The uncertain disturbance d(t) of the robotic arm is bounded. That is, there exists a known constant D1 such that ||d(t)|| ≤ D1 always holds.

[0072] Assumption 2: The target tracking trajectory q d (t) of the system is continuous and twice differentiable, and q d (t) and its derivative are both known functions.

[0073] Let q1 = q, the system (3) can be transformed into the strict feedback form

[0074]

[0075] Error definition:

[0076]

[0077] where: is the virtual control law, which will be given later; T ∈ [0, +∞) is a parameter of the system, representing the set value of the convergence time of the system; is the time-varying adjustment function vector, which satisfies the following three conditions:

[0078] Condition 1: μ(t, T) is twice continuously differentiable with respect to time t.

[0079] Condition 2: μ(t, T) = 0 always holds when t ∈ [T, +∞).

[0080] Condition 3: μ(0, T) and satisfy

[0081]

[0082] wherein and are two constant vectors, which are related to the initial values of the system.

[0083] Note 1, there are multiple alternative functions that satisfy Conditions 1 to 3. In the present invention, a quartic polynomial will be used as an example. According to Conditions 1 - 3, the selected adjustment function μ i (t, T) (i = 1, 2) needs to satisfy:[[]]END]]

[0084] Initial value condition: μ i (0, T) = m 0i ,

[0085] Final value condition: μ i (T, T) = 0,

[0086] Through the above 5 equations, the following unique quartic polynomial can be obtained:

[0087]

[0088] Step 1:

[0089] Considering the error definition (10), the derivative of the first - step error z1 is obtained as

[0090]

[0091] Virtual control law design:

[0092]

[0093] where the constant is a control parameter.

[0094] First - step stability analysis:

[0095] Select the Lyapunov function:

[0096]

[0097] Taking the derivative of the Lyapunov function (14) and substituting the virtual control law (13) can obtain

[0098]

[0099] The second step:

[0100] Derive the error of the second step and substitute it into (8) to get

[0101]

[0102] Design the control rate:

[0103]

[0104] Where the constant is the control parameter, and the constant is the control parameter representing the upper bound of the final convergence error, and the constant κ1 = min{2c1, 2c2}.

[0105] Stability analysis of the second step:

[0106] Give the Lyapunov function:

[0107]

[0108] Derive the Lyapunov function (18) and substitute it into (15), (16) and (17) to obtain

[0109]

[0110] Where tanh(κDz2 / κ1ε 2 ) = [tanh(κDz 21 / κ1ε 2 ), tanh(κDz 22 / κ1ε 2 )] T , and tanh(·) is the hyperbolic tangent function.

[0111] According to Lemma 1, we can obtain

[0112]

[0113] From Assumption 1 and (21), we know that

[0114]

[0115] According to Property 3, we know that

[0116]

[0117] Therefore, substituting (22) and (23) into (19), we get

[0118]

[0119] Solving the differential equation (24) gives

[0120]

[0121] Substituting (10) in Condition 3 and the virtual control law (13) into the error definition (9), we can obtain

[0122]

[0123] Combined with the Lyapunov function (18), we have V(0) = 0. Therefore, from (25), it can be further obtained that when t ∈ [0, +∞), V(t) ≤ ε 2 / 2. Then, according to Condition 2: μ(t, T) = 0 holds constantly for t ∈ [T, +∞], we can finally obtain that the tracking error of the robotic arm is less than the preset error bound within the predetermined time, that is

[0124]

[0125] To verify the effectiveness of the proposed method, the following simulations are set up:

[0126] System parameters: l1 = 1m, l2 = 0.85m, m1 = 2.04kg, m2 = 1kg, g = 9.8m / s 2 ;

[0127] p1 = (m1 + m2)l1 2 and p2 = m2l2 2 and p3 = m2l1l2, p4 = (m1 + m2)l1, p5 = m2l2,

[0128]

[0129]

[0130]

[0131]

[0132] Given reference signal: q d = [0.5sin(t), 0.5sin(t)] T .

[0133] Initial values of the system: q = [0.6, -0.4] T ,

[0134] Simulation parameters: T = 10s, c1 = 20, c2 = 30, ε = 0.001.

[0135] The simulation results are as Figure 3-6 shown. Figure 3It is the system tracking curve, from which it can be seen that the controller designed by the present invention has good tracking performance. Figure 4 It is the tracking error curve, where the tracking error and e = q - q d , and it can be seen from the figure that at the moment when the set time T = 10 s, the error converges to near zero. Figure 5 It is the system angular acceleration curve, from which it can be seen that the convergence property of the angular acceleration is good. Figure 6 It is the controller input curve. From Figure 3-6 it can be seen whether the controller designed by the present invention can ensure the convergence of the system within a given time.

[0136] Compared with the prior art, the present patent has the following technical effects:

[0137] It can not only ensure the state consistency and parameter consistency of the system convergence time; at the same time, in the proposed preset time controller, the system convergence time is a parameter in the controller, so the system convergence time can be directly set and adjusted through this parameter. The specific comparison is shown in the following table:

[0138] Table 1 Advantage comparison between the preset time controller and the existing finite time controller and fixed time controller

[0139]

[0140] Although the embodiments of the present invention have been described above in conjunction with the accompanying drawings, the present invention is not limited to the above specific embodiments and application fields. The above specific embodiments are merely illustrative and guiding, rather than restrictive. Under the inspiration of the specification, those of ordinary skill in the art can also make many forms without departing from the scope protected by the claims of the present invention, and these all belong to the scope of protection of the present invention.

Claims

1. A design method for a predetermined time controller of a dual-joint robotic arm, characterized in that, It includes the following steps: S1: Establish a mathematical model of a dual-joint robotic arm, and perform transformation to obtain the transformed model formula; S2: Design an adjustment function according to the model formula obtained in S1; S3: Define the control objective, and obtain the first-step error according to the model formula obtained in S1 and the adjustment function in S2; S4: Design a virtual control law according to the first-step error obtained in S3; S5: Obtain the second-step error according to the virtual control law in S4 and the model formula in S1; S6: Design a predefined-time controller according to the model formula in S1 and the second-step error obtained in S5; The design formula of the adjustment function in S2 is: wherein is a mediation function, and is the initial constant vector, and the subscript i represents the i-th component of the vector, is the operating time of the robotic arm, is the set value of the convergence time; The first-step error in S3 is: Among them, is the first-step error, is the actual angular position of the robotic arm, is the target tracking trajectory, is the set value of the convergence time, is the time-varying adjustment function vector; The setting method of the virtual control law in S4 is: Among them, is the virtual control law, is the control parameter, is the first-step error, is the first derivative of the target tracking trajectory, is the first derivative of the time-varying adjustment function vector; The second-step error in S5 is: wherein, is the virtual control law, is the second-step error, is the angular velocity of the robotic arm; The design method of the predefined-time controller in S6 is: Among them, is a predetermined time controller, is a control parameter, is the second-step error, is the matrix of the centrifugal force and Coriolis force of the robotic arm, is the angular velocity of the robotic arm, is the gravity term of the robotic arm, is a known constant, is the hyperbolic tangent function, , is a constant, is the upper bound control parameter of the final convergence error, is the first-step error, is the moment of inertia matrix of the robotic arm, is the first derivative of the virtual control law.

2. The design method of a predetermined time controller for a double-joint robotic arm according to claim 1, characterized in that The mathematical model of the dual-joint robotic arm established in S1 is: Among them, is the actual angular position of the robotic arm; is the angular velocity of the robotic arm; is the angular acceleration of the robotic arm, is the moment of inertia matrix of the robotic arm; is the matrix of the centrifugal force and Coriolis force of the robotic arm; represents the gravity term of the robotic arm; represents the predetermined time controller to be designed, is the uncertain disturbance of the robotic arm varying with time.

3. The design method of a predetermined time controller for a double-joint robotic arm according to claim 2, characterized in that, The transformed model formula in S1 is: Among them , , is the angular velocity of the robotic arm, is the angular acceleration of the robotic arm, is the matrix of the centrifugal force and Coriolis force of the robotic arm after transformation, is the gravity term of the robotic arm, is the uncertain disturbance of the robotic arm.

4. The design method of a predetermined time controller for a double-joint robotic arm according to claim 1, characterized in that, The acquisition of the first-step error is based on the backstepping control principle.

5. The design method of a predetermined time controller for a double-joint robotic arm according to claim 1, characterized in that, After S4 and S6 are completed, the Lyapunov function is used for stability analysis.

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