Ude control method for multi-joint robot arm with input saturation and output constraint

Abstract

The application discloses a kind of multi-joint robot arm UDE control method with input saturation and output constraint, specific steps include: (1) establish the multi-joint robot arm dynamics model with input saturation and time-varying output constraint under joint space;(2) using the Jacobian matrix of robot arm, the dynamics model obtained in step (1) is converted, and the multi-joint robot arm dynamics model under task space is obtained;(3) design nonlinear state constraint function to ensure that time-varying output constraint is not violated.(4) design UDE, for approximating the unknown term of robot arm dynamics;(5) define tracking error signal;(6) combined with backstepping method, a stable trajectory tracking controller based on UDE is designed for the robot arm system under task space.The application can realize accurate trajectory tracking of robot arm under the condition of modeling error, input saturation and time-varying output constraint, and ensure the satisfaction of input saturation and time-varying output constraint.

Description

Technical Field

[0001] This invention relates to the field of automation, and in particular to a UDE control method for a multi-joint robotic arm with input saturation and output constraints. Background Technology

[0002] The rapid development of technology has facilitated the further application and promotion of robotic arms. Multi-joint robotic arms are the most widely used automated mechanical equipment, finding broad applications in industrial manufacturing, military, medical, entertainment, and even space exploration. In practical applications, robotic arm control systems face various constraints, such as input saturation, limited task space, and speed limitations. Violation of these constraints can lead to a decline in the performance of the robotic arm system, even causing system damage and threatening the safety of personnel involved. With further technological advancements, the concept of human-computer interaction has emerged, and more and more robots will work alongside humans in the future. Therefore, designing controllers for robotic arm systems with specified constraints has significant theoretical and practical value. However, most current research focuses primarily on the stability and control accuracy of robot systems, neglecting controller design that considers input saturation and time-varying output constraints. Using existing recursive design methods, most research results can only solve robotic arm control problems with constant output constraints, and the upper and lower bounds of the constrained output are usually set relatively loosely. This increases the conservatism of the algorithm while limiting its practicality.

[0003] As a time-varying, coupled, multi-input multi-output complex nonlinear system, the motion control of a multi-joint robotic arm is extremely complex. Furthermore, in practical control design, the parameters of the robotic arm are often unknown, or the parameter measurements contain significant errors. Therefore, controller design tools for unknown parameter models are needed. UDE (Unified Logic Device) approximates the unknown dynamic model and external disturbances of the robotic arm system through filters, thereby achieving precise control even when the system has unknown parameters.

[0004] Based on the UDE (Understanding Variable Impedance) control algorithm, a variable impedance control algorithm has been proposed for multi-joint robotic arm systems under model uncertainty (Y. Dong and B. Ren, “UDE-Based variable impedance control of uncertain robot systems,” IEEE Trans. Syst. Man Cybernet. Syst. 49(12), 2487–2498(2019)). This algorithm helps the robotic arm complete a given interactive task in an unknown environment, improves the overall performance of the robot environment system, and only requires knowledge of the system's bandwidth when using the UDE to approximate the uncertainty of the system. However, although the above algorithm achieves good control results, it does not consider the problems of input saturation and output constraints. Based on neural network theory and considering the input dead zone, an adaptive neural network algorithm for uncertain multi-joint robotic arms has been proposed (Q. Zhou, S. Zhao, H. Li, R. Lu and C. Wu, “Adaptive neural network tracking control for robotic manipulators with deadzone,” IEEE Trans. Neural Netw. Learn. Syst. 30(12), 3611–3620(2019)). This algorithm ensures the stability of the robotic arm system and achieves better control performance by designing an adaptive neural network controller. In the above algorithm, the neural network is used to approximate the uncertainties of the robotic arm, and the approximation effect is good. However, as the number of joints of the robotic arm increases, the computational load of the neural network will be huge, and the hardware requirements will be high. Compared with the neural network, the UDE used in this invention is simpler to design, easier to implement, requires fewer parameters, and requires less computation.

[0005] With the increasing prevalence of human-machine collaboration scenarios, constraint control is crucial for ensuring operator safety. Extensive research has been conducted to address the issues of input saturation and output constraints. For robotic arms with input constraints, a neural network-based model predictive control method has been proposed (E.Kang, H.Qiao, J.Gao and W.Yang, “Neural network-based model predictive tracking control of an uncertainrobotic manipulator with input constraints,” ISA Trans. 109(3), 89–101(2021)), which introduces a non-quadratic cost function to solve for the input constraints. Based on neural network theory, a fixed-time control method for robotic arms has been proposed (D.Zhang, L.Kong, S.Zhang, Q.Li and Q.Fu, “Neural networks-based fixed-time control for a robot with uncertainties and input deadzone,” Neurocomputing. 390(5), 139–147(2020)), which utilizes neural networks to compensate for the input deadzone. For robotic arms with symmetric output constraints, an adaptive neural network tracking controller has been proposed (Z.-L. Tang, SSGe, KPTee and W. He, “Adaptive neural control for an uncertain robotic manipulator with jointspace constraints,” Int. J. Control 89(7), 1428–1446 (2016).), which utilizes a barrier Lyapunov function to solve the output constraint problem. Compared to the general form of the barrier Lyapunov function used in the aforementioned algorithms, the NSDF algorithm used in this invention removes the feasibility conditions required by the barrier Lyapunov function and is simpler in design. Furthermore, NSDF can solve both symmetric and asymmetric output constraints, thus avoiding the trouble of converting the barrier Lyapunov function into a piecewise barrier Lyapunov function when symmetric constraints are converted into asymmetric constraints.

[0006] Currently, most research on trajectory tracking control for robotic arms is conducted in joint space. However, in reality, the given tasks of robotic arms are all performed in task space. For robotic arms with uncertain models, a task space trajectory tracking controller based on delay control was designed (X.Liang, Y.Wan and C.Zhang, “Task space trajectorytracking control of robot manipulators with uncertain kinematics and dynamics,” Math.Probl.Eng.2017(1),1–19(2017).), and the effectiveness of the algorithm was verified through simulation experiments. However, this algorithm does not consider the problems of input saturation and output constraints. Summary of the Invention

[0007] The purpose of this invention is to overcome the shortcomings of existing technologies and provide a UDE control method for multi-joint robotic arms with input saturation and output constraints. This invention enables high-precision tracking control of multi-joint robotic arms with input saturation and time-varying output constraints even under conditions of inaccurate parameter measurements and unknown external disturbances. First, the invention transforms the dynamic model of the joint space into the task space. Then, a nonlinear state constraint function is used to ensure that the time-varying output constraints are not violated. UDE is used to address the problems of inaccurate parameter measurements and unknown external disturbances. An auxiliary variable is introduced to address the input saturation problem, and a UDE-based controller is established using backstepping to achieve trajectory tracking control. In the task space, this invention achieves fast, stable, and precise tracking control of the robotic arm while ensuring that input saturation and time-varying output constraints are not violated.

[0008] The present invention is achieved by at least one of the following technical solutions.

[0009] The UDE control method for multi-joint robotic arms with input saturation and output constraints includes the following steps:

[0010] (1) Establish a dynamic model of a multi-joint robotic arm in joint space with input saturation and time-varying output constraints;

[0011] (2) Using the Jacobian matrix of the robotic arm, the dynamic model obtained in step (1) is transformed to obtain the dynamic model of the multi-joint robotic arm in the task space.

[0012] (3) Establish a nonlinear state constraint function to ensure that the time-varying output constraint is not violated;

[0013] (4) Establish an Uncertainty and Disturbance Estimator (UDE) to approximate the unknowns in the dynamics of the robotic arm;

[0014] (5) Define the tracking error signal;

[0015] (6) Combining the backstepping method, a stable UDE-based trajectory tracking controller is established for the robotic arm system in the task space.

[0016] Furthermore, in step (1), the multi-joint manipulator dynamic model under the constraints of input saturation and time-varying output is a manipulator dynamic model with strong nonlinear coupling, expressed as:

[0017]

[0018] in, Let M(q) represent angular displacement, angular velocity, and angular acceleration, respectively; M(q) = M0(q) + ΔM0(q) ∈ R n ×n Represents the inertia matrix. Let G(q) = G0(q) + ΔG0(q) ∈ R represent the centripetal force matrix. n The gravitational vector is represented by τ = [τ1,...,τ]. n ] T τ represents the input torque vector. i ,i=1,...,n represents the i-th item of τ, U(τ)=[U(τ1),...,U(τ n )] T U(τ) represents the input torque saturation function vector. i ), i = 1, ..., n represent the i-th term of U(τ), τ d Let represent the disturbance terms from people and external factors; n represents the number of joints in a rigid robotic arm with time-varying output constraints; m represents the dimension of the robotic arm's task space; ΔM0(q) represents the disturbance term from people and external factors. ΔG0(q) represents the unknown parts of the inertia matrix, centripetal force matrix, and gravitational force, respectively, and M0(q), G0(q) represents the known parts of the inertia matrix, centripetal force matrix, and gravitational force, respectively.

[0019] Furthermore, establish the input torque saturation function vector U(τ)=[U(τ1),...,U(τ)] n )] T any one of them, U(τ) i The following is represented:

[0020]

[0021] Where sign(.) is the standard sign function, and the known positive number U mi is U(τ i ) boundary.

[0022] Furthermore, in step (2), the Jacobian matrix J∈R is used. m×n , combined The dynamic model obtained in step (1) is transformed to obtain the multi-joint robotic arm dynamic model in the task space, which is expressed as:

[0023]

[0024] Where x = [x1,...,x] m ] T , These represent the position, velocity, and acceleration of the robotic arm's end effector, respectively. M represents angular velocity; x (x)=J +T M0(q)J + , G x =J +T G0, Unknown Item J represents the derivative matrix of the Jacobian matrix. + This represents the pseudo-inverse of the Jacobian matrix.

[0025] Furthermore, the time-varying output constraint of the robotic arm is expressed as:

[0026] -F i1 (t)<x i <F i2 (t), i = 1, ..., m

[0027] Among them, -F i1 (t) and F i2 (t) represents the output x of the robotic arm. i The lower and upper bounds.

[0028] Furthermore, in step (3), the nonlinear state constraint function vector ζ = [ζ1,...,ζ] m ] T and ζ d =[ζ d1 ,...,ζ dm ] T Represented as:

[0029]

[0030]

[0031] Where, ζ i With ζ di Representing ζ and ζ respectively d The i-th term, K represents the positive constant to be set, x di (t) represents xd The i-th term, x d =[x d1 ,...,x dm ] T x represents the desired motion trajectory of the robotic arm's end effector. i (t) represents the i-th term of x, where x = [x1,...,x] m ] T This indicates the position of the end effector of the robotic arm.

[0032] Furthermore, the nonlinear state constraint function satisfies the property that for any initial value x satisfying the time-varying output constraint... i (0), if and only if x i (t) Approximates the upper bound F of the time-varying output constraint i2 (t) or lower bound -F i1 When (t), ζ i Only then will it approach infinity, that is, as long as ζ i Boundedness ensures that time-varying output constraints are not violated.

[0033] Furthermore, in step (4), the uncertainty term and the Laplace function matrix G of the disturbance estimator UDE are... f (s) is represented as:

[0034]

[0035] Among them, G fi (s), i = 1, ..., m represents G f (s) is the i-th element on the diagonal, where s represents the complex frequency, and T i ,i=1,...,m represent the time constants that need to be set.

[0036] Furthermore, the uncertainty term and the interference estimator UDE are filters. For an input signal with frequency w, s = jw, where j is the imaginary unit, and the amplitude gain is:

[0037]

[0038] In theory, the smaller the time constant, the closer the amplitude gain is to 1, meaning the higher the accuracy of the approximation.

[0039] Furthermore, the time constant T i The selection methods are as follows:

[0040] In time constant T i Within the range of values, first set the time constant T. i The value is chosen to be close to the upper limit of the range, and the control accuracy of the system is verified to meet the requirements. If the requirements are met, the selection stops; if not, the value of T is further decreased.i Adjust the value accordingly.

[0041] Compared with the prior art, the present invention has the following beneficial effects:

[0042] 1. The control method provided by this invention does not require accurate robotic arm system parameters, and can perform high-performance tracking control of the robotic arm under the condition that the system has unknown external disturbances, while ensuring that input saturation and time-varying output constraints are not violated.

[0043] 2. This invention uses a UDE, which is essentially a filter, to estimate unknown terms in the system model and unknown external disturbances. The implementation method is simple and reduces the complexity of the control scheme.

[0044] 3. The controller in this invention can ensure that the robotic arm meets the input saturation and time-varying output constraints under any given known input saturation and time-varying output constraints, thus solving the out-of-bounds problem that may exist in existing conventional trajectory tracking controllers when facing input saturation and time-varying output constraints. Attached Figure Description

[0045] Figure 1 This is a schematic diagram of a linkage planar robotic arm in an embodiment of the present invention;

[0046] Figure 2 This invention provides a UDE-based control method for a multi-joint robotic arm with input saturation and time-varying output constraints.

[0047] Figure 3 A simulation diagram showing the tracking of the robotic arm's end effector at coordinate x1;

[0048] Figure 4 This is a simulation diagram of the tracking situation of the robotic arm's end effector at coordinate x2;

[0049] Figure 5 Given the coordinates x1, x2 of the robotic arm's end effector and the desired tracking trajectory x... d1 ,x d2 Error graph between;

[0050] Figure 6 Simulation diagram of the control input τ1 for the robotic arm;

[0051] Figure 7 Simulation diagram of the control input τ2 for the robotic arm. Detailed Implementation

[0052] The present invention will be further described in detail below with reference to the embodiments and accompanying drawings, but the embodiments of the present invention are not limited thereto.

[0053] like Figure 2The diagram shows a detailed flowchart of a multi-joint robotic arm control method based on UDE with input saturation and time-varying output constraints. The specific steps include:

[0054] (1) Establish a dynamic model of a multi-joint manipulator with input saturation and time-varying output constraints in joint space; in this embodiment, the manipulator is a two-link planar manipulator, the specific structure of which is as follows: Figure 1 As shown.

[0055] The two-bar linkage robotic arm consists of two links. Angular displacement and velocity sensors are installed at each joint point of the links to measure the joint angular position and angular velocity. The dynamic model of the two-bar linkage planar robotic arm is represented as follows:

[0056]

[0057] in, Let M(q) represent angular displacement, angular velocity, and angular acceleration, respectively. M(q) = M0(q) + ΔM0(q) ∈ R 2×2 Represents the inertia matrix. Let G(q) = G0(q) + ΔG0(q) ∈ R represent the centripetal force matrix. 2 Let τ represent the gravitational vector, τ = [τ1, τ2]. T τ represents the input torque vector. d This represents disturbances originating from people and external factors. All unknown. U(τ)=[U(τ1),U(τ2)] T This represents the input torque saturation function vector, where any term U(τ) i The following is represented:

[0058]

[0059] Where sign(.) is the standard sign function, and the known positive number U mi is U(τ i ) boundary.

[0060] In this embodiment, the input saturation is set to: U m1 =15, U m2 =5.

[0061] The representation is as follows:

[0062]

[0063]

[0064]

[0065]

[0066] Where q1 and q2 represent the angular displacements of joint 1 and joint 2, respectively; m1 and m2 represent the masses of link 1 and link 2, respectively; l1 and l2 represent the lengths of link 1 and link 2, respectively; I1 and I2 represent the inertia of link 1 and link 2, respectively; and g represents the acceleration due to gravity.

[0067] In this embodiment, the relevant parameters of the system are as follows:

[0068] l1=0.35m, l2=0.31m, m1=2.0Kg, m2=0.85Kg, g=9.8m / s 2

[0069] I1 = 61.25 × 10 -3 kgm 2 I² = 20.42 × 10 -3 kgm 2

[0070] The uncertainty term in the system model is represented as follows:

[0071] ΔM0(q)=-0.1M0(q)

[0072] ΔC0(q)=-0.2C0(q)

[0073] ΔG0(q)=-0.1G0(q)

[0074] The disturbance term from the environment is represented as:

[0075] τ d =[sin(t),sin(t)] T

[0076] (2) Establish a dynamic model of a multi-joint robotic arm with input saturation and time-varying output constraints in the task space;

[0077] Using the Jacobian matrix J and the formula The dynamic model obtained in step (1) is transformed to obtain the multi-joint robotic arm dynamic model in the task space, which is expressed as:

[0078]

[0079] Where x = [x1, x2] T , These represent the position, velocity, and acceleration of the robotic arm's end effector, respectively. M x (x)=J +T M0(q)J + , G x =J +TG0, Unknown Item The time-varying output constraint of the robotic arm is expressed as:

[0080] -F i1 (t)<x i <F i2 (t), i = 1, 2

[0081] Among them, -F i1 (t) and F i2 (t) represents the output x of the robotic arm. i The lower and upper bounds.

[0082] In this embodiment, the robotic arm output reference trajectory is set as follows:

[0083] x d (t)=[0.2sin(πt),0.2cos(πt)] T

[0084] Furthermore, the time-varying output constraints are set as follows:

[0085] -F 11 (t) = -0.21 + 0.03sin(πt)

[0086] F 12 (t) = 0.21 + 0.03cos(πt)

[0087]

[0088]

[0089] In this embodiment, it is necessary to simultaneously achieve trajectory tracking control of the two-bar planar robotic arm while ensuring that the input saturation and time-varying output constraints are not violated.

[0090] (3) Design nonlinear state constraint functions;

[0091] Design a nonlinear state constraint function vector ζ = [ζ1, ζ2] T and ζ d =[ζ d1 ,ζ d2 ] T , is represented as:

[0092]

[0093]

[0094] Where K represents a positive constant that needs to be set, x d =[x d1 ,x d2 ] TThis represents the desired motion trajectory of the robotic arm's end effector.

[0095] In a preferred embodiment, K is set to 0.01.

[0096] (4) Design UDE;

[0097] Design an uncertainty and disturbance estimator UDE, with its Laplace function matrix G. f (s) is represented as:

[0098]

[0099] Among them, T i i = 1, 2 represents the time constant that needs to be set.

[0100] In a preferred embodiment, both T1 and T2 are set to 0.014.

[0101] (5) Define the tracking error signal;

[0102] The tracking error of the robotic arm system is expressed as:

[0103] v1=ζ-ζ d -η1

[0104] v2=X2-α1-η2

[0105] Where v1 and v2 represent error variables used in the backstepping design process, η1 and η2 are auxiliary variables, and α1 is the virtual control law, expressed as:

[0106]

[0107]

[0108]

[0109] Where c1 and c2 are the designed positive feedback gain parameters, Δu=U(τ)-τ, and * denotes convolution operation. μ1, μ2, μ d1 and μ d2 They are represented as follows:

[0110]

[0111]

[0112]

[0113]

[0114] (6) Design a UDE-based multi-joint robotic arm trajectory tracking controller.

[0115] Combining the backstepping method, the designed UDE-based trajectory tracking controller is represented as follows:

[0116]

[0117]

[0118] Where r1 represents a positive constant to be set, I represents the 2nd order identity matrix, and J represents the Jacobian matrix. Here, c2 represents the Laplace transform operator, and c2 represents the design parameters. This indicates the speed of the end effector of the robotic arm. G represents the derivative of the virtual control law. f (s) represents the UDE of the design, v2 represents the error variable of the design, and e1 = ζ - ζ d , ζ and ζ d M represents the vector of nonlinear state constraint functions. x (x)=J +T M0(q)J + , e2 = X2 - α1,

[0119] In this embodiment, the initial system conditions are:

[0120] X(0) = [0.235, 0.205; 0, 0]

[0121] Controller parameters: c1 = 40, c2 = 100, r1 = 1.

[0122] The control method of this invention comprehensively considers the model uncertainty, input saturation and output constraint problems of the robotic arm, and proposes a multi-joint robotic arm task space trajectory tracking controller based on UDE. The effectiveness of the proposed method is verified by simulation.

[0123] Figure 3 This is a simulation diagram of the tracking situation of the end effector of the robotic arm at coordinate x1. Figure 4 This is a simulation diagram of the tracking situation of the end effector of the robotic arm at coordinate x2. Figure 5 Given the coordinates x1, x2 of the robotic arm's end effector and the desired tracking trajectory x... d1 ,x d2 Error graph between them. Figure 6 The simulation diagram shows the control input u1 for the robotic arm. Figure 7 This is a simulation diagram of the robotic arm control input u2. (Through...) Figure 3 , Figure 4 , Figure 6 as well as Figure 7 It can be seen that under the designed controller, the robotic arm can achieve a good output trajectory tracking effect, and its output can meet the given input saturation and time-varying output constraints.

[0124] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A UDE control method for a multi-joint robotic arm with input saturation and output constraints, characterized in that, The specific steps include: (1) Establish a dynamic model of a multi-joint robotic arm in joint space with input saturation and time-varying output constraints; (2) Using the Jacobian matrix of the robotic arm, the dynamic model obtained in step (1) is transformed to obtain the dynamic model of the multi-joint robotic arm in the task space; (3) Establish nonlinear state constraint functions to ensure that time-varying output constraints are not violated; Nonlinear state constraint function vector and Represented as: in, and They represent and The i-th term, This indicates the positive constant that needs to be set. express The i-th term, This represents the desired motion trajectory of the robotic arm's end effector. express The i-th term, Indicates the position of the end effector of the robotic arm; (4) Establish an Uncertainty and Disturbance Estimator (UDE) to approximate the unknown terms in the robot arm's dynamics; The Laplace function matrix of the uncertainty term and the disturbance estimator UDE Represented as: in, express The i-th element on the diagonal, Represents complex frequency. This indicates the time constant that needs to be set; (5) Define the tracking error signal; (6) Combine the backstepping method to establish a stable UDE-based trajectory tracking controller for the robotic arm system in the task space.

2. The UDE control method for a multi-joint robotic arm with input saturation and output constraints according to claim 1, characterized in that, In step (1), the multi-joint manipulator dynamic model under the constraints of input saturation and time-varying output is a manipulator dynamic model with strong nonlinear coupling, expressed as: in, These represent angular displacement, angular velocity, and angular acceleration, respectively. Represents the inertia matrix. Represents the centripetal force matrix. Represents the gravitational vector. Indicates the input torque vector. express The i-th term, This represents the input torque saturation function vector. express The i item, This represents disturbances originating from both people and external factors; This represents the number of joints in a rigid robotic arm with time-varying output constraints. This represents the dimension of the robotic arm's task space. These represent the unknown parts of the inertia matrix, centripetal force matrix, and gravitational force, respectively. These are the known parts of the inertia matrix, centripetal force matrix, and gravitational force, respectively.

3. The UDE control method for a multi-joint robotic arm with input saturation and output constraints according to claim 2, characterized in that, Establish input torque saturation function vector any one of them It is expressed as follows: in, It is a standard sign function, and the known positive numbers are... yes The boundary.

4. The UDE control method for a multi-joint robotic arm with input saturation and output constraints according to claim 2, characterized in that, In step (2), the Jacobian matrix is ​​used. , combined The dynamic model obtained in step (1) is transformed to obtain the multi-joint robotic arm dynamic model in the task space, which is expressed as: in, These represent the position, velocity, and acceleration of the robotic arm's end effector, respectively. Indicates angular velocity; , , Unknown items ; The derivative matrix of the Jacobian matrix. This represents the pseudo-inverse of the Jacobian matrix.

5. The UDE control method for a multi-joint robotic arm with input saturation and output constraints according to claim 1, characterized in that, The time-varying output constraint of the robotic arm is expressed as: in, and Output of the robotic arm The lower and upper bounds.

6. The UDE control method for a multi-joint robotic arm with input saturation and output constraints according to claim 1, characterized in that, The nonlinear state constraint function satisfies the property that for any initial value satisfying the time-varying output constraint... If and only if Approximating the upper bound of time-varying output constraints Or the lower bound hour, Only then will it approach infinity, that is, as long as Boundedness ensures that time-varying output constraints are not violated.

7. The UDE control method for a multi-joint robotic arm with input saturation and output constraints according to claim 1, characterized in that, The uncertainty term and the interference estimator UDE are filters for frequencies of... The input signal, , The imaginary unit is used, and the amplitude gain is: In theory, the smaller the time constant, the closer the amplitude gain is to 1, meaning the higher the accuracy of the approximation.

8. The UDE control method for a multi-joint robotic arm with input saturation and output constraints according to claim 1, characterized in that, time constant The selection methods are as follows: In time constant Within the range of values, first set the time constant. The selection is made to approach the upper limit of the range, and the control accuracy of the system is verified to meet the requirements. If the requirements are met, the selection stops; if not, the value is further reduced. Adjust the value accordingly.

Images