A tracking control method for a single-link flexible robotic arm

Abstract

A single-link flexible manipulator tracking control method relates to a manipulator tracking control method. The present invention is a single-link flexible manipulator trajectory tracking control method under asymmetric time-varying full-state constraints. The dynamic model of the single-link flexible manipulator system controlled by a DC motor is used. By selecting appropriate states and making appropriate constraints on unknown bounded disturbances and tracking trajectories, in the design of an actual controller, the backstepping method is applied to the design of an adaptive fuzzy actual controller. In order to prevent the constraint from crossing the bounds, the present invention also introduces a new tan-type asymmetric time-varying hindered Lyapunov function in each step of the backstepping method design, designs corresponding control laws and adaptive laws, and ensures that no matter what the initial conditions are, a final uniformly bounded tracking control effect is achieved, and the asymmetric time-varying constraints can be achieved after a finite time.

Description

Technical Field

[0001] The invention relates to a tracking control method for a mechanical arm, in particular to a tracking control method for a single-link flexible mechanical arm with output asymmetric time-varying constraints. Background Art

[0002] With the rapid development of social productivity, robotic arms are increasingly used in various fields, especially in aerospace, service and other fields, and the research on robotic arms is becoming more and more important. Compared with rigid robotic arms, single-link flexible robotic arms have obvious advantages such as light weight, low energy consumption and high flexibility, so the research on single-link flexible flexible robotic arms has also become a hot topic.

[0003] Robots are increasingly used in modern industrial automation production and have become an important role in the industrial production system. As a type of robot, industrial robotic arms can replace humans in some repetitive high-intensity or high-precision tasks such as assembly, handling and welding. Compared with manual labor, industrial robotic arms have higher work efficiency and can work in some dangerous environments, which can effectively improve the productivity of the factory.

[0004] Faced with increasingly complex production processes, the control requirements for industrial robotic arms are also constantly increasing. How to effectively control industrial robotic arms has always been a hot research direction, and many research results have been achieved in recent years. In some work tasks that require interaction with people or high precision, in order to ensure production safety and control accuracy, it is necessary to limit the motion space, motion speed and tracking error of the robotic arm. Therefore, the study of constraint control of robotic arms has important practical significance.

[0005] However, the advantages of the single-link flexible robot's own structure and materials are also the source of its research difficulties. During the movement, the single-link flexible robot itself will produce elastic deformation and blur, which makes its motion positioning and trajectory tracking difficult. Therefore, it is of great significance to track the motion trajectory of the single-link flexible robot and suppress its blur. In addition, constraint problems are often encountered in motion control, such as the rated torque, rated speed, mechanical limit, motion interference, etc. of the drive motor, and there are special constraint problems in some specific occasions, such as the rope traction mechanism needs to consider the traction force constraint of the rope during movement, the cutting force needs to be constrained by optimizing the tool motion trajectory in mechanical processing, the visual servo system needs to consider the visual visibility constraint, the mobile car needs to consider the curvature constraint during the turning process, and the rigid robot needs to constrain the angular velocity and angle of each joint in the movement.

[0006] The prior art studies the problem of adaptive neural network trajectory tracking control of uncertain underactuated surface ships with output constraints. At the same time, the prior art studies the adaptive neural network tracking control of manipulator systems with output constraints and input dead zones. However, the prior art cannot solve the tracking control problem of a single-link flexible manipulator with output asymmetric time-varying constraints, and these new technologies are yet to be proposed. Summary of the invention

[0007] The purpose of the present invention is to provide a single-link flexible manipulator tracking control method. The present invention is a single-link flexible manipulator trajectory tracking control method under asymmetric time-varying full-state constraints. Under the existing constraints, the method ensures that no matter what the initial conditions are, the final consistent and bounded tracking control effect can be achieved. Through the dynamic model of the single-link flexible manipulator system controlled by a DC motor, the backstepping method is applied to the design of the adaptive fuzzy actual controller in the design of the actual controller to achieve the final consistent and bounded tracking control effect, solving the problem that the finite-dimensional model describes the single-link flexible manipulator system model, resulting in the instability of the single-link flexible manipulator system.

[0008] The objective of the present invention is achieved through the following technical solutions:

[0009] A single-link flexible manipulator tracking control method is provided. The method establishes a dynamic model of a single-link flexible manipulator system controlled by a DC motor. The tracking control method for a single-link flexible manipulator with an output asymmetric time-varying constraint according to actual requirements comprises the following steps:

[0010] Step 1:

[0011] Establish the dynamic model of the single-link flexible manipulator system, including the control equation of the single-link flexible manipulator system and the motion equation of the angular displacement of the end of the single-link flexible manipulator;

[0012]

[0013] in,

[0014]

[0015]

[0016] J is the motor rotor inertia, m is the connecting rod mass, m0 is the load mass, L0 is the connecting rod length, R0 is the load radius, G is the acceleration of gravity, and B is the viscous friction coefficient; p(t) represents the motor angular displacement (also the robot arm joint angle), I(t) represents the motor armature current, and K τ is the torque coefficient, L is the armature inductance, R is the armature resistance, K Bis the back electromotive force coefficient, V0 represents the input control voltage;

[0017] Step 2:

[0018] The established dynamic model is transformed into a state equation; by setting x1 = p, x3=I,u=V0 and appropriate electromechanical parameters, the coefficients are equivalent to the following form:

[0019]

[0020] Where F1(x1)=0, and is an unknown function, it is known that G1=1, External disturbance signal d i (i=1,2,3) satisfies and is a constant;

[0021] Step 3: Apply the actual controller to the control of the single-link flexible manipulator, design an asymmetric time-varying constraint interval, and constrain the rotation angle, rotation speed and torque of the single-link flexible manipulator;

[0022] Step 4: According to the dynamic model, determine the actual controller with output constraints, suppress the fuzziness of the single-link flexible manipulator with output constraints, and control the single-link flexible manipulator to rotate to the desired angular position. Then, perform stability analysis on the actual controller by Lyapunov direct method.

[0023] Step 5: After performing stability analysis on the actual controller using the Lyapunov direct method, the feasibility and effectiveness of the actual controller are verified through simulation.

[0024] In the tracking control method for a single-link flexible mechanical arm, the control target of the constraint control in step 3 is:

[0025] (1) All signals in a closed-loop system are bounded;

[0026] (2) System spatial state x i , i = 1, 2, ..., n does not violate the preset constraints;

[0027] (3) The system tracking error can be kept within the specified range e i (t)∈Z i ={-k ai <e i <k bi}, i = 1, 2, ..., n, Inside;

[0028] (4) When only the connecting rod displacement xi is measurable, make y track the reference signal y r , the maximum allowable error is e1, the connecting rod tracks the reference signal y r .

[0029] The tracking control method of a single-link flexible manipulator, the tracking control method for a single-link flexible manipulator with an output asymmetric time-varying constraint, the actual controller is expressed as:

[0030]

[0031] The virtual controllers are:

[0032]

[0033] Adaptive rate:

[0034]

[0035] The described single-link flexible robotic arm tracking control method, the described single-link flexible robotic arm tracking control method with output asymmetric time-varying constraints, the method performs stability analysis on the actual controller through the Lyapunov direct method, including constructing a Lyapunov candidate function for the closed-loop system of the single-link flexible robotic arm. If the Lyapunov candidate function is positive definite and the derivative of the Lyapunov candidate function with respect to time is semi-negative definite, then it is determined that the actual controller can make the closed-loop system of the single-link flexible robotic arm asymptotically stable.

[0036] The advantages and effects of the present invention are:

[0037] The present invention is a single-link flexible manipulator trajectory tracking control method under asymmetric time-varying full-state constraints. Under the existing constraints, the method ensures that no matter what the initial conditions are, the final consistent and bounded tracking control effect can be achieved. The dynamic model of the single-link flexible manipulator system controlled by a DC motor is applied to the design of an adaptive fuzzy actual controller by selecting a suitable state and making suitable constraints on unknown bounded disturbances and tracking trajectories. In order to prevent the constraint from crossing the bounds, the present invention also introduces a new tan-type asymmetric time-varying obstruction Lyapunov function in each step of the backstepping method design, designs corresponding control laws and adaptive laws, ensures that no matter what the initial conditions are, the final consistent and bounded tracking control effect is achieved, and the asymmetric time-varying constraints can be realized after a finite time. The present invention is widely applicable to the field of automatic control.

[0038] The adaptive tracking controller of the present invention can make the single-link flexible manipulator have good tracking performance, and the system output can be kept within the asymmetric time-varying constraint interval to meet the constraint conditions. From the simulation results, it can be seen that the asymmetric time-varying constraint control method of the present invention can effectively control the single-link flexible manipulator, and perform asymmetric time-varying constraints on the rotation angle, rotation speed and torque of the manipulator, while reducing the time for the single-link flexible manipulator to stabilize. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 A three-dimensional diagram of a single-link flexible mechanical arm of the present invention;

[0040] Figure 2 This is a schematic diagram of a single-link flexible mechanical arm controlled by a DC motor of the present invention;

[0041] Figure 3 It is a flow chart of the design principle of the present invention;

[0042] Figure 4 The output y of the system of the present invention and the reference signal y r and its trajectory diagram of constraint interval;

[0043] Figure 5 is the trajectory diagram of the system state x2 and its constraint interval of the present invention;

[0044] Figure 6 is the system state x3 and its constraint interval trajectory diagram of the present invention;

[0045] Figure 7 It is the trajectory diagram of the adaptive parameters θ1, θ2 and θ3 of the present invention;

[0046] Figure 8 A trajectory diagram of input u to the system of the present invention;

[0047] Fig. 9 This is the trajectory diagram of the tracking error e1 and its constraint interval of the present invention. DETAILED DESCRIPTION

[0048] The present invention will be described in detail below with reference to the embodiments shown in the accompanying drawings.

[0049] Step 1: A trajectory tracking control method for a single-link flexible manipulator under asymmetric time-varying full-state constraints, comprising:

[0050] Establish the dynamic model of the single-link flexible manipulator system controlled by a DC motor;

[0051] According to the actual requirements, the tracking control method for a single-link flexible manipulator with output asymmetric time-varying constraints is described.

[0052] The dynamic model of the single-link flexible robotic arm system includes the control equations of the single-link flexible robotic arm system and the motion equations of the angular displacement of the end of the single-link flexible robotic arm.

[0053]

[0054] in,

[0055]

[0056] J is the motor rotor inertia, m is the mass of the connecting rod, m0 is the load mass, L0 is the length of the connecting rod, R0 is the load radius, G is the acceleration of gravity, and B is the viscous friction coefficient. p(t) represents the motor angular displacement (also the load displacement), I(t) represents the motor armature current, and K τ is the torque coefficient, L is the armature inductance, R is the armature resistance, K B is the back electromotive force coefficient, and V0 represents the input control voltage.

[0057] Step 2: Convert the established kinetic model into a state equation;

[0058] By setting x1=p, x3=I,u=V0 and appropriate electromechanical parameters can make the coefficient equivalent to the following form:

[0059]

[0060] Where F1(x1)=0, and is an unknown function, it is known that G1=1, External disturbance signal d i (i=1,2,3) satisfies and is a constant.

[0061] Step three: Apply the actual controller designed by the present invention to the control of the single-link flexible robotic arm, design an asymmetric time-varying asymmetric constraint interval, and constrain the rotation angle, rotation speed and torque of the single-link flexible robotic arm.

[0062] The control objectives of the present invention are:

[0063] (1) All signals in a closed-loop system are bounded;

[0064] (2) System spatial state x i , i = 1, 2, ..., n does not violate the preset constraints;

[0065] (3) The system tracking error can be kept within the specified range e i (t)∈Z i ={-k ai<e i <k bi}, i = 1, 2, ..., n, Inside;

[0066] (4) When only the connecting rod displacement xi is measurable, make y track the reference signal y r , the maximum allowable error is e1, the connecting rod tracks the reference signal y r ;

[0067] Step 4: To design the controller, you need to define the following tracking error

[0068]

[0069] The first-order derivative of the tracking error is given by

[0070]

[0071] Define q(·)

[0072]

[0073] In order to impose asymmetric time-varying constraints on all states of system (2), the asymmetric time-varying barrier Lyapunov function is selected as follows:

[0074] Among them, ζ is a positive design parameter, θ represents the * The estimate, is the estimated error. a (t) = y r (t)- k c (t), and k c (t) are the time-varying upper and lower bounds of the constraint interval, respectively.

[0075] It turns out that there are some constants K a , K b , so that Calculating the derivative of (5) yields

[0076]

[0077] The actual controller is represented as

[0078]

[0079] in

[0080]

[0081] The virtual controllers are:

[0082]

[0083] Adaptive rate:

[0084]

[0085]

[0086] Step 5: Verify the stability analysis of the single-link flexible robotic arm system

[0087] For the single-link flexible manipulator system with full-state asymmetric time-varying constraints shown in system (2), the actual controller (8), virtual controller (9) and adaptive rate (10) of the system are designed by selecting appropriate design parameters. When the initial condition of the system satisfies e i (0)∈Z i ={-k ai (t)<e i <k bi (t)}, i = 1, 2, ..., n, the following control objectives can be achieved:

[0088] (1) All signals in the single-link flexible manipulator system are bounded;

[0089] (2) System status x i , i = 1, 2, ..., n does not violate the preset constraints;

[0090] (3) The system tracking error can be kept within the specified range e i (t)∈Z i ={-k ai <e i <k bi}, i = 1, 2, ..., n, Inside;

[0091] (4) When only the connecting rod displacement xi is measurable, make y track the reference signal y r , the maximum allowable error is e1, the connecting rod tracks the reference signal y r .

[0092] Proof: For ease of representation, define the following parameters

[0093]

[0094] All variables x in the system i ,θ i , χ i , ei and u are bounded. And we can further get

[0095]

[0096] When q = 1, we can get

[0097]

[0098] Similarly, when q = 0, we can get

[0099]

[0100] Because x1(t)=e1(t)+y r (t), z i (t)∈Z i ={-k ai (t)<z i <k bi (t)}, i=1,2,…,n, we can get

[0101]

[0102] It has been proved in the derivation process that α i , i = 1, 2, ..., n-1 is also bounded, so by the same token, all states in system (2) satisfy

[0103]

[0104] The proof is complete.

[0105] Step 6: Simulation results analysis

[0106] The present invention uses Matlab software to design two simulation programs to verify the effectiveness of the adaptive tracking controller designed by the present invention. First, numerical simulation is used to prove that the designed controller can effectively perform asymmetric time-varying constraint control on the single-link flexible robotic arm system, and then the designed controller is applied to the control of the single-link flexible robotic arm. In order to compare the impact of the dynamic surface control method on the system tracking speed, the following two control methods are applied in each simulation example:

[0107] 1: Apply the method-based full-state asymmetric time-varying constraint control scheme for a single-link flexible robotic arm system proposed in this invention.

[0108] 2: Apply the traditional asymmetric time-varying constraint control scheme to the control of a single-link flexible robotic arm system.

[0109] Numerical simulation

[0110] In order to verify the effectiveness of the above system, a verification simulation experiment is carried out for the following single-link flexible robotic arm system with external disturbance and full-state asymmetric time-varying constraints:

[0111]

[0112] Where x1=q is the angular position of the single-link flexible manipulator, is the angular velocity, x3 is the torque, u and y are the input and output of the system respectively, d1=0, d2=0.1cos(x2x3), d3=0 is the external disturbance, G1=0, G2=1, G3=10, F1(x1)=0, F2(x1,x2)=-10sin(x1)-x2, F3(x1,x2,x3)=-2x2-10x3, reference signal y r =0.5cos(t).

[0113] Design the virtual control functions α1, α2 of the system (18), the actual controller u and the adaptive rates θ1, θ2, θ3, and select the relevant parameters as ω1=3, ω2=2, ω3=2, σ1=17, σ2=10, σ3=9, η1=6, η2=5, η3=3, ζ1=0.5, ζ2=0.6, ζ3=0.6, γ1=5, γ2=3, γ2=5, τ1=0.2, τ2=0.02,

[0114] The upper and lower bounds of the system's asymmetric time-varying constraint interval are set as:

[0115]

[0116] The initial state of the system is x1(0)=0.5, x2(0)=0.3, x3(0)=0, θ1(0)=0.01, θ2(0)=0.01, θ3(0)=0.01.

Claims

1. A single-link flexible robotic arm tracking control method, characterized in that: The method establishes a dynamic model of a single-link flexible manipulator system controlled by a DC motor; a tracking control method for a single-link flexible manipulator with an output asymmetric time-varying constraint is provided according to actual requirements, comprising the following steps: Step 1: Establish the dynamic model of the single-link flexible manipulator system, including the control equation of the single-link flexible manipulator system and the motion equation of the angular displacement of the end of the single-link flexible manipulator; in, J is the motor rotor inertia, m is the connecting rod mass, m0 is the load mass, L0 is the connecting rod length, R0 is the load radius, G is the acceleration of gravity, B is the viscous friction coefficient; p(t) represents the motor angular displacement, I(t) represents the motor armature current, K τ is the torque coefficient, L is the armature inductance, R is the armature resistance, K B is the back electromotive force coefficient, V0 represents the input control voltage; Step 2: The established dynamic model is transformed into a state equation; by setting x1 = p, x3=I,u=V0 and appropriate electromechanical parameters, the coefficients are equivalent to the following form: Where F1(x1)=0, and is an unknown function, it is known that G1=1, External disturbance signal d i , i=1,2,3 satisfies and is a constant; Step 3: Apply the actual controller to the control of the single-link flexible manipulator, design an asymmetric time-varying constraint interval, and constrain the rotation angle, rotation speed and torque of the single-link flexible manipulator; Step 4: According to the dynamic model, determine the actual controller with output constraints, suppress the fuzziness of the single-link flexible manipulator with output constraints, and control the single-link flexible manipulator to rotate to the desired angular position. Then, perform stability analysis on the actual controller by Lyapunov direct method. Step 5: After performing stability analysis on the actual controller by Lyapunov direct method, the feasibility and effectiveness of the actual controller are verified by simulation; The control target of the constraint control in step 3 is: (1) All signals in a closed-loop system are bounded; (2) System spatial state x i , i = 1, 2, ..., n does not violate the preset constraints; (3) The system tracking error can be kept within the specified range e i (t)∈Z i ={-k ai <e i <k bi }, i = 1, 2, ..., n, Inside; (4) When only the connecting rod displacement x i If it is measurable, make y track the reference signal y r , the maximum allowable error is e1, the connecting rod tracks the reference signal y r .

2. A single-link flexible robotic arm tracking control method according to claim 1, characterized in that: The tracking control method of the single-link flexible manipulator with an output asymmetric time-varying constraint performs stability analysis on an actual controller through the Lyapunov direct method, including constructing a Lyapunov candidate function for a closed-loop system of the single-link flexible manipulator. If the Lyapunov candidate function is positive definite and the derivative of the Lyapunov candidate function with respect to time is semi-negative definite, then it is determined that the actual controller can make the closed-loop system of the single-link flexible manipulator asymptotically stable.

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