Flexible joint mechanical arm preset time fuzzy output feedback control method under output constraint

Abstract

The invention discloses a flexible joint mechanical arm preset time fuzzy output feedback control method under output constraint. The method comprises the following steps that a state space model of a flexible joint mechanical arm is established; based on a preset time stability theory, designing a Lyapunov function framework which ensures that the system converges to a bounded set within preset time; a fuzzy logic system is adopted to carry out approximation on an unknown nonlinear dynamic function of the mechanical arm; a barrier Lyapunov function is introduced to process system output constraints, and it is ensured that output does not violate physical constraints; constructing a state observer based on a fuzzy logic system to estimate an unmeasurable state; designing a virtual control function, a controller and a parameter adaptive law; and a preset time filter is designed to reduce the calculation complexity, and finally stable control of the closed-loop system within the preset time is realized. According to the invention, the preset time stabilization control of the closed-loop system is successfully realized.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of mechanical arm control, and particularly relates to a preset time fuzzy output feedback control method for a flexible joint mechanical arm under output constraint. BACKGROUND

[0002] With the development of science and technology, traditional rigid mechanical arms have gradually shown limitations in complex, limited space and human-machine interaction scenarios. Flexible joint mechanical arms emerge as the times require. They are endowed with better flexibility and adaptability by flexible materials or structures, can reduce collision damage, improve operation safety and flexibility, and have great application potential in the fields of medical treatment, aerospace, industrial automation and the like. In order to achieve the high performance targets of improving positioning accuracy, reducing energy consumption, reducing weight, and achieving safer operation through inertia attenuation, the dynamics and control of flexible joint robots have been widely studied in recent years.

[0003] However, the flexible joint will produce parameter drift in the movement process due to factors such as material deformation, friction loss and load change. Not only is it difficult to obtain complete model parameters through sensors, but also most parameters and motion states present a strong nonlinear mapping relationship, which cannot be approximated by a simple linear function. In order to solve the above problems, a large number of tracking control researches have been carried out in the academic field, including sliding mode control, variable structure control, backstepping control and the like. At present, the existing technology still has the following problems:

[0004] Firstly, most control methods do not fully consider the physical limits of the flexible joint mechanical arm in motion, such as the output constraints of joint rotation angle range and end effector speed threshold. When the system approaches the constraint boundary, control quantity saturation or trajectory deviation may easily occur, and even excessive deformation of the flexible part may be caused, reducing the operation accuracy and equipment life.

[0005] Secondly, the convergence speed of the traditional finite time control is affected by the initial error. When the mechanical arm moves in a large range, the larger the initial error is, the slower the convergence will be. Although the fixed time control can guarantee that the convergence time has an upper limit, the control strength is set according to the maximum error, which will cause waste, increase energy consumption and vibration when the error is small, and it is also difficult to adapt to the stiffness change caused by the deformation of the flexible joint, and the performance will decrease under complex conditions. SUMMARY

[0006] The purpose of the present application is to provide a preset time fuzzy output feedback control method for a flexible joint mechanical arm under output constraint, so as to realize the preset time stabilization control of a closed-loop system.

[0007] In order to achieve the above purpose, the present application adopts the following technical scheme:

[0008] A preset time fuzzy output feedback control method for a flexible joint mechanical arm under output constraint, comprising the following steps:

[0009] Step A, establishing a state space model of the flexible joint robot arm;

[0010] Step B, designing a Lyapunov function framework based on the preset time stability theory to ensure that the system converges to a bounded set within a preset time;

[0011] Step C, using a fuzzy logic system to approximate the unknown nonlinear dynamic function of the robot arm;

[0012] Step D, introducing barrier Lyapunov functions to handle system output constraints to ensure that the output does not violate physical limitations;

[0013] Step E, constructing a state observer based on the fuzzy logic system to estimate the unobservable state; designing a virtual control function, a controller, and a parameter adaptive law; and designing a preset time filter to reduce computational complexity, ultimately realizing stable control of the closed-loop system within a preset time.

[0014] Further, in step A, the state space model of the flexible joint robot arm is described by the following equation:

[0015]

[0016] wherein q, respectively represent link position, velocity and acceleration, q m , respectively represent rotor angular position, velocity and acceleration, M is link mass, g is gravitational acceleration, L is link length, K represents joint stiffness coefficient, J represents joint flexibility, B is natural damping, u represents control input signal, and y represents system output;

[0017] Let x1=q, x3=q m ,

[0018] Equation (1) is rewritten as a state space expression:

[0019]

[0020] wherein w2=K / ML 2 , w4=J -1 , g4=-Bx4 / J-K(x3-x1) / J, R is the set of real numbers.

[0021] Further, in step B, for a nonlinear system:

[0022]

[0023] wherein t is time;

[0024] If N(N0,t) satisfies ||N(N0,t)-N e ||≤ε, then N e is practically preset-time stable; where N is the state vector, g(N,t) is a continuous function, N e is the equilibrium state of the system;

[0025] If a positive definite continuously differentiable function V satisfies:

[0026]

[0027] where T m is the preset time, △>0 is a constant, ψ and m0 are design parameters satisfying 0<ψ<1 and then the nonlinear system (3) is practically preset-time stable, and the state trajectory will lie in a bounded set Ω:

[0028]

[0029] Further, in step C, the fuzzy rule base of the fuzzy logic system consists of a series of fuzzy If-Then inference rules of the following form:

[0030] R l : If l1 is and l2 is is

[0031] then j is J l ,k=1,2,...,M;

[0032] where, and J k are fuzzy sets, and μ Jk (y) are fuzzy functions of and J k respectively, and M is the rule number;

[0033] By singleton function, center average defuzzification and product inference, the fuzzy logic system j(x) is expressed as:

[0034]

[0035] where,

[0036] Define the fuzzy basis function as:

[0037]

[0038] where, And δ(x)=[δ1(l),δ2(l),...,δ N (l)] T ; Then the fuzzy logic system (6) is rewritten as:

[0039]

[0040] Let g(n) be a continuous function defined on a compact set Ω; then for any constant ι>0, such that

[0041]

[0042] Furthermore, in step D, the barrier Lyapunov function is introduced to solve the output limitation problem, which is defined as follows:

[0043]

[0044] Among them, r a1 represents a constraint on ζ1 and Where ζ1 is the conversion error, r a1 is a constant greater than 0;

[0045] For any constant r a1 >0,|ζ1|≤r a1 Satisfies the following inequality:

[0046]

[0047] Furthermore, in step E, the constructed fuzzy state observer is:

[0048]

[0049] Among them, k1, k2, k3, k4 are constants greater than 0;

[0050] The constructed preset time filter is:

[0051]

[0052] in, ρ=(3n+1) η / 2 , η=η1 / η2;η1 and η2 are positive even and positive odd respectively and η1<η2;T m is the preset time;

[0053] The constructed Lyapunov function is:

[0054]

[0055]

[0056] wherein, i = 2, 3, 4; ζ i is the virtual tracking error of the system, is the estimation of x i , and is the filter output signal, and alpha i-1 is the virtual control signal, is the filter error;

[0057] xi, x4 are ideal weight vectors, is the estimation of ideal weight vectors; Theta2, Theta4, H2 are adaptive parameters, is the estimation of adaptive parameters;

[0058] The constructed virtual control function is:

[0059]

[0060] wherein, c1 is a normal number, y r is the desired reference signal, and phi i (·) represents the fuzzy basis function vector;

[0061] The constructed controller u is:

[0062]

[0063] The constructed parameter adaptive law is:

[0064]

[0065] Advantages: compared with the prior art, the present application has the following advantages:

[0066] Firstly, the preset time control problem of the flexible joint robot system is studied in the present application. Unlike the conventional finite time control method, the method proposed in the present application can directly preset the system stable time and ensure that the system is stable within the preset time. Compared with the previous fixed time control scheme, the preset time control proposed in the present application does not need to adjust multiple parameters.

[0067] Secondly, based on the preset time stability theory and the barrier Lyapunov function, the present application proposes an adaptive control method for the flexible joint robot with output constraint, which not only ensures the control accuracy and the satisfaction of the constraint condition, but also ensures that the system can remain stable within the preset time. BRIEF DESCRIPTION OF DRAWINGS

[0068] Figure 1 is a schematic diagram of a flexible joint robot arm model.

[0069] Figure 2 is a flexible joint robot model trajectory tracking effect diagram.

[0070] Figure 3 is a flexible joint robot model tracking error effect diagram.

[0071] Figure 4 is a flexible joint robot model system output x1 and its estimated value effect diagram.

[0072] Figure 5 is a flexible joint robot model system state x2 and its estimated value effect diagram.

[0073] Figure 6 is a flexible joint robot model system state x3 and its estimated value effect diagram.

[0074] Figure 7 is a flexible joint robot model system state x4 and its estimated value effect diagram.

[0075] Figure 8 is a flexible joint robot model filtering error effect diagram.

[0076] Figure 9 is a flexible joint robot model control input u effect diagram. DETAILED DESCRIPTION

[0077] The present application will be further explained in conjunction with the accompanying drawings.

[0078] The flexible joint robot model involved in the present application is shown in the schematic diagram Figure 1 .

[0079] A flexible joint robot preset time fuzzy output feedback control method under output constraint, comprising the following steps:

[0080] Step A, establish the state space model of the flexible joint robot;

[0081] The state space model of the flexible joint robot is described by the following equation:

[0082]

[0083] Where, q, respectively represent the link position, velocity and acceleration, q m , ​where x1=x, x2= x, x3= x, x4= x, respectively, M is the link mass, g is the gravitational acceleration, L is the link length, K represents the joint stiffness coefficient, J denotes the joint flexibility, B is the natural damping, u represents the control input signal, y denotes the system output;

[0084] Let x1=q, x3=q m ,

[0085] Equation (1) is rewritten as a state-space expression:

[0086]

[0087] where w2=K / ML 2 , w4=J -1 , g4=-Bx4 / J-K(x3-x1) / J, R is the real number set.

[0088] Step B, based on the preset time stability theory, a Lyapunov function framework is designed to ensure that the system converges to a bounded set within a preset time;

[0089] For a nonlinear system:

[0090]

[0091] where t is time;

[0092] If N(N0,t) satisfies ||N(N0,t)-N e ||≤ε, then N e is actually preset time stable; where N is the state vector, g(N,t) is a continuous function, N e is the equilibrium state of the system;

[0093] If a positive definite continuous differentiable function V satisfies:

[0094]

[0095] where T m is the preset time, △>0 is a constant, ψ and m0 are design parameters satisfying 0<ψ<1 and ; then, the nonlinear system (3) is actually preset time stable, and the state trajectory will be located in a bounded set Ω:

[0096]

[0097] Step C, a fuzzy logic system is used to approximate the unknown nonlinear dynamic function of the mechanical arm;

[0098] The fuzzy rule base of the fuzzy logic system consists of a set of fuzzy If-Then rules of the following form:

[0099] R l : If l1 is and l2 is is

[0100] then j is J l , k = 1, 2,..., M.

[0101] where, and J k are fuzzy sets, and μ Jk (y) are the membership functions of and J k , respectively, and M is the rule number.

[0102] The fuzzy logic system j(x) is represented by singleton function, center average defuzzification and product inference as:

[0103]

[0104] where,

[0105] The fuzzy basis functions are defined as:

[0106]

[0107] where, and δ(x) = [δ1(l), δ2(l),..., δ N (l)] T ; then the fuzzy logic system (6) is rewritten as:

[0108]

[0109] Let g(n) be a continuous function defined on a compact set Ω; then for any constant ε > 0, there exists a δ > 0 such that

[0110]

[0111] Step D, introduce barrier Lyapunov function to deal with system output constraints, to ensure that the output does not violate the physical limit;

[0112] Introduce barrier Lyapunov function to solve the output limit problem, defined as follows:

[0113]

[0114] where, r a1denotes the constraint on ζ1and where ζ1is the conversion error, r a1 is a constant greater than 0;

[0115] for any constant r a1 > 0, |ζ1|≤r a1 satisfies the following inequality:

[0116]

[0117] Step E, constructing a state observer based on fuzzy logic system to estimate the unmeasurable state; designing a virtual control function, a controller and a parameter adaptive law; and designing a preset time filter to reduce the computational complexity, finally realizing the stable control of the closed-loop system within the preset time;

[0118] The constructed fuzzy state observer is:

[0119]

[0120] where k1, k2, k3, k4 are constants greater than 0;

[0121] The constructed preset time filter is:

[0122]

[0123] where, ρ = (3n + 1) η / 2 , η = η1 / η2; η1 and η2 are positive even and positive odd numbers respectively and η1 < η2; T m is the preset time;

[0124] The constructed Lyapunov function is:

[0125]

[0126] where, i = 2, 3, 4; ζ i is the virtual tracking error of the system, is the estimated value of x i , is the filter output signal, α i-1 is the virtual control signal, is the filter error; ξ2, ξ4 are ideal weight vectors, is the estimated value of the ideal weight vector; Θ2, Θ4, H2 are adaptive parameters, is the estimated value of the adaptive parameter;

[0127] The constructed virtual control function is:

[0128]

[0129] Among them, c1 is a positive constant, y r is the desired reference signal, φ i (·) represents the fuzzy basis function vector;

[0130] The constructed controller u is:

[0131]

[0132] The constructed parameter adaptive law is:

[0133]

[0134] Figure 2 The figure shows the trajectory tracking effect of the flexible joint robotic arm model. Figure 3 Shows the tracking error of the flexible joint robot model 's effect diagram. Figure 4 Shows the output x1 of the flexible joint manipulator model system and its estimated value 's effect diagram. Figure 5 Shows the state x2 of the flexible joint manipulator model system and its estimated value 's effect diagram. Figure 6 Shows the state x3 of the flexible joint manipulator model system and its estimated value 's effect diagram. Figure 7 Shows the state x4 of the flexible joint manipulator model system and its estimated value 's effect diagram. Figure 8 Shows the filtering error of the flexible joint robot model 's effect diagram. Figure 9 The effect diagram of the control input u of the flexible joint robot arm model is shown.

[0135] It can be seen from the above simulation result graph that under this strategy, the system can achieve stability within the actual preset time and the trajectory tracking effect is good.

[0136] The application discloses a preset time fuzzy output feedback control method for a flexible joint robot arm under output constraint, which is used for approximatively modeling unknown nonlinear dynamic functions by means of a fuzzy logic system (FLS), and a state observer based on the FLS is constructed to estimate the unmeasured state. In solving the output constraint problem, a barrier Lyapunov function is introduced, which can effectively ensure that the robot arm movement does not exceed the physical limit. By constructing a preset time filter, the calculation complexity can be significantly reduced. The control strategy successfully realizes the stable control of the closed-loop system within the preset time, and finally, a series of simulation results fully verify that the control strategy has good effectiveness and superiority.

[0137] The above only describes the preferred embodiments of the present application, and it should be noted that those skilled in the art can make several improvements and refinements without departing from the principles of the present application, and these improvements and refinements should also be considered as the protection scope of the present application.

Claims

1. A preset time fuzzy output feedback control method for a flexible joint manipulator under output constraints, characterized in that: The following steps are involved: Step A, establishing a state space model of the flexible joint robotic arm; Step B: Based on the preset time stability theory, a Lyapunov function framework is designed to ensure that the system converges to a bounded set within a preset time; Step C, using a fuzzy logic system to approximate the unknown nonlinear dynamic function of the manipulator; Step D, introduce the barrier Lyapunov function to deal with the system output constraints to ensure that the output does not violate physical limitations; Step E: constructing a state observer based on a fuzzy logic system to estimate the unmeasurable state; designing a virtual control function, a controller, and a parameter adaptation law; A preset time filter is designed to reduce the computational complexity and ultimately achieve stable control of the closed-loop system within the preset time.

2. The preset time fuzzy output feedback control method for a flexible joint manipulator under output constraints according to claim 1 is characterized by: In step A, the state space model of the flexible joint manipulator is described by the following equation: Among them, q, Represent the link position, velocity and acceleration respectively, q m , Represent the rotor angular position, velocity and acceleration respectively, M is the mass of the connecting rod, g is the acceleration due to gravity, L is the link length, K represents the joint stiffness coefficient, J represents the joint flexibility, B is the natural damping, u represents the control input signal, and y represents the system output; Let x1 = q, x3=q m , Formula (1) is rewritten as a state space expression: Where w2 = K / ML 2 , w4=J -1 , g4=-Bx4 / JK(x3-x1) / J, R is the set of real numbers.

3. The preset time fuzzy output feedback control method for a flexible joint manipulator under output constraints according to claim 1 is characterized by: In step B, for nonlinear systems: Where t is time; If N(N0,t) satisfies ||N(N0,t)-N e ||≤ε, then N e is stable in actual preset time; where N is the state vector, g(N,t) is a continuous function, N e is the equilibrium state of the system; If a positive definite continuously differentiable function V satisfies: Among them, T m is the preset time, △>0 is a constant, ψ and m0 satisfy 0<ψ<1 and design parameters; then, the nonlinear system (3) is stable in practical preset time, and the state trajectory will be located in the bounded set Ω:

4. The preset time fuzzy output feedback control method for a flexible joint manipulator under output constraints according to claim 1 is characterized in that: In step C, the fuzzy rule base of the fuzzy logic system consists of a series of fuzzy If-Then reasoning rules of the following form: R l : If l1 is And l2 is yes Then j is J l ,k=1,2,...,M; in, and J k is a fuzzy set, and They are and J k The fuzzy function, M is the rule number; Through singleton function, central average defuzzification and product reasoning, the fuzzy logic system j(x) is expressed as: in, The fuzzy basis function is defined as: in, And δ(x)=[δ1(l),δ2(l),...,δ N (l)] T ; Then the fuzzy logic system (6) is rewritten as: Let g(n) be a continuous function defined on a compact set Ω; then for any constant ι>0, such that 5. The preset time fuzzy output feedback control method for a flexible joint manipulator under output constraints according to claim 1 is characterized in that: In step D, the barrier Lyapunov function is introduced to solve the output limitation problem, which is defined as follows: Among them, r a1 represents a constraint on ζ1 and Where ζ1 is the conversion error, r a1 is a constant greater than 0; For any constant r a1 >0,|ζ1|≤r a1 Satisfies the following inequality:

6. The preset time fuzzy output feedback control method for a flexible joint manipulator under output constraints according to claim 1 is characterized by: In step E, the constructed fuzzy state observer is: Among them, k1, k2, k3, k4 are constants greater than 0; The constructed preset time filter is: in, ρ=(3n+1) η / 2 , η=η1 / η2;η1 and η2 are positive even and positive odd respectively and η1<η2;T m is the preset time; The constructed Lyapunov function is: in, i=2,3,4; ζ i is the virtual tracking error of the system, is x i The estimated value of is the filtered output signal, α i-1 is a virtual control signal, is the filtering error; ξ2, ξ4 are ideal weight vectors, is the estimate of the ideal weight vector; Θ2, Θ4, H2 are adaptive parameters, is the estimated value of the adaptive parameter; The constructed virtual control function is: Among them, c1 is a positive constant, y r is the desired reference signal, φ i (·) represents the fuzzy basis function vector; The constructed controller u is: The constructed parameter adaptive law is:

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