CN114200837B - A hierarchical sliding mode control method that interferes with unknown spherical robots - Google Patents

Abstract

The invention provides a hierarchical sliding mode control method for a spherical robot system with unknown interference, which can use adaptive neural network theory to perform motion control on the spherical robot system, thereby achieving precise control of the spherical robot system with unknown interference. The technical solution of the present invention includes the following steps: establishing a mathematical model of the spherical robot system containing unknown items for the controlled spherical robot system, and the unknown items are unknown interferences. The unknown items in the mathematical model of the spherical robot system are approximated based on the neural network, and the weight parameters of the neural network are adaptively estimated based on the control error information. Based on the unknown items approximated by the adaptive neural network and the defined sliding mode surface, a sliding mode controller with interference compensation is designed. A sliding mode controller is used to control the spherical robot with unknown interference.

Description

A hierarchical sliding mode control method that interferes with unknown spherical robots

Technical field

The invention relates to the technical field of robot motion control, and in particular to a hierarchical sliding mode control method that interferes with an unknown spherical robot.

Background technique

Regarding the motion control of spherical robots, most existing research assumes that the spherical robot moves along a straight line, ignoring the premise that the spherical robot moves along a straight line. For the control of maintaining the linear motion of the spherical robot, sliding mode control with simple structure and fast response speed can be used. By designing the sliding mode surface composed of control errors, the attitude of the spherical robot can be measured and collected in real time for state feedback. For the unknown interference part in the dynamic model of the control object, the design form of the nonlinear interference observer is complex, model-dependent, and has many design parameters. Therefore, a neural network approximator can be designed to estimate it.

Especially in recent years, great progress has been made in the research on motion control methods of spherical robots. However, there have been no research results that apply adaptive neural network theory to estimate the interference part to maintain motion control of spherical robots.

Contents of the invention

In view of this, the present invention provides a hierarchical sliding mode control method for a spherical robot system with unknown interference, which can use adaptive neural network theory to control the motion of the spherical robot system, thereby achieving precise control of the spherical robot system with unknown interference.

In order to achieve the above objects, the technical solution of the present invention includes the following steps:

Step 1: Establish a mathematical model of the spherical robot system containing unknown items for the controlled spherical robot system. The unknown items are unknown interference.

Step 2: Approximate the unknown items in the mathematical model of the spherical robot system based on the neural network, and adaptively estimate the weight parameters of the neural network based on the control error information.

Step 3: Design a sliding mode controller with interference compensation based on the unknown items approximated by the adaptive neural network and the defined sliding mode surface.

Step 4: Use the sliding mode controller to control the interference unknown spherical robot.

Further, in step one, a mathematical model of the spherical robot system containing unknown interference is established for the controlled spherical robot system, as follows:

Among them, M ₁₁ , M ₁₂ , M ₂₁ and M ₂₂ are the elements of the inertia matrix of the spherical robot system respectively; V ₁₁ and V ₂₁ are the elements of the gravity moment vector of the spherical robot; Φ is the rotation angle of the spherical shell, is the angular acceleration of the rotation angle of the spherical shell, ζ is the rotation angle of the internal pendulum of the spherical shell,/> is the angular acceleration of the internal pendulum angle of the spherical shell; τ _y is the motor input torque of the spherical robot;

Set four state variables x ₁ , x ₂ , x ₃ , x ₄ , let x ₁ =φ, x ₃ =ζ/> Then the mathematical model of the spherical robot system, that is, formula (1), is transformed into a system state space expression:

The spherical robot system includes a spherical shell subsystem and a spherical shell pendulum subsystem. f ₁ is the time-varying function of the system state variables in the spherical shell subsystem; β ₁ is the time-varying coefficient of the control input in the spherical shell subsystem; b ₁ is the time-varying coefficient of the unknown item in the spherical shell subsystem; f ₂ is the time-varying function of the system state variables included in the spherical shell pendulum subsystem; β ₂ is the time-varying coefficient of the control input in the spherical shell pendulum subsystem; b _2. The time-varying coefficient of the unknown item in the spherical shell subsystem; Δ _y is the unknown item of the mathematical model of the spherical robot system;

Among them, the unknown item in the spherical shell subsystem is d ₁ =b ₁ Δ _y , and the unknown item in the spherical shell pendulum subsystem is d ₂ =b ₂ Δ _y . Then the system state space expression (2) is described as

Preferably, M ₁₁ , M ₁₂ , M ₂₁ , M ₂₂ are elements of the inertia matrix of the spherical robot system, specifically:

Among them, M _s is the mass of the spherical shell, m _p is the mass of the ball's inner pendulum, R _s is the radius of the spherical shell, l is the length of the ball's inner pendulum, and g is the acceleration of gravity.

Furthermore, the unknown terms in the mathematical model of the spherical robot system are approximated based on the neural network, and the weight parameters of the neural network are adaptively estimated based on the control error information, specifically as follows:

The unknown items in the mathematical model of the spherical robot system include the unknown item d ₁ in the spherical shell subsystem and the unknown item d ₂ in the pendulum subsystem of the spherical shell:

Among them, W ₁ and W ₂ are the two neural network weights set, h ₁ (X) = [h _i ] ^T and h ₂ (X) = [h _j ] ^T are the radial basis functions respectively, h _i , h _j is the element of the radial _basis function _{, X} is _the state vector _{, which} is composed of four state variables. The state ^vector The estimated values of the unknown term d ₁ in the subsystem and the unknown term d ₂ in the spherical shell pendulum subsystem are respectively and/> Expressed as follows:

Then the estimated value of the unknown item in the system state space expression (2) is Then the system tracking error is

Among them, e ₁ is the position tracking error of the spherical shell, e ₃ is the position tracking error of the inner swing of the spherical shell, x _1d is the expected position of the spherical shell, and x _3d is the expected position of the inner swing of the spherical shell;

First define the system sliding mode surface, including the first sliding mode surface S ₁ and the second sliding mode surface S ₂ :

Among them, e _i (i=1,3) represents the tracking error, c ₁ is the coefficient of e ₁ , and c ₂ is the coefficient of e ₂ ;

Then the joint sliding mode surface is

S _y =aS ₁ +bS ₂ (8)

a, b are the weight coefficients of the first sliding mode surface S ₁ and the second sliding mode surface S ₂ respectively;

The adaptive law of neural network weight parameters is

in is the rate of change of the neural network weight estimate;/> is the rate of change of the neural network weight estimate; γ ₁ is the coefficient; γ ₂ is the coefficient;

For the system state space expression (2) of the spherical robot system containing unknown terms, if the neural network weight adaptation law (9) in the joint sliding mode surface (8) and the neural network approximation form in formula (5) are used, Then the approximate unknown term is obtained, which is

further, through is the first derivative of S _i , the equivalent controller is

Among them, τ _y1 is the first equivalent controller; τ _{y2 is} the second equivalent controller; is the expected angular velocity of the spherical shell;/> is the expected angular velocity of the internal pendulum of the spherical shell;

Then the final sliding mode controller is designed as

τ _y =(aβ ₁ +bβ ₂ ) ^-1 (aβ ₁ τ _y1 +bβ ₂ τ _y2 -k ₁ sign(S _y )-k ₂ S _y ) (11)

Among them, k ₁ is the error coefficient preset for the joint sliding mode surface sign function; k ₂ is the error coefficient preset for the joint sliding mode surface.

Beneficial effects:

1. The present invention provides a hierarchical sliding mode control method that interferes with an unknown spherical robot system. The sliding mode control method is based on an adaptive neural network. It applies a neural network interference approximator and introduces the controller design to keep the spherical robot moving along a straight line. method. First, for the spherical robot system model with unknown interference, a neural network is applied to approximate the unknown part; an adaptive weight update law based on control error is established; a sliding mode controller with interference compensation is designed based on the approximate interference and the system model of the spherical robot, so that Spherical robots can perform desired movements quickly and stably.

2. The adaptive neural network used in the present invention approximates the interference unknown items of the spherical robot, and can achieve accurate modeling of the spherical robot system with unknown physical models. The identification system can be used to analyze part of the expected signals of the spherical robot and solve the motor control output.

3. The sliding mode control method proposed by the present invention can ensure the stability of multiple states of the spherical robot under a single input condition, that is, when there is only a single motor input, the angular positions of the spherical shell and the inner pendulum of the ball can converge to the desired position. .

Description of the drawings

Figure 1 is a schematic structural diagram of the spherical robot system;

Figure 2 is a schematic diagram of the controller design structure;

Figure 3 is a flow chart of a design method of a sliding mode controller for maintaining linear motion of a spherical robot with unknown interference based on an adaptive neural network provided by the present invention.

Detailed ways

The present invention will be described in detail below with reference to the accompanying drawings and examples.

Example 1:

Figure 1 shows the structure diagram of a spherical robot system. The present invention provides a design method of a sliding mode controller for linear motion of a spherical robot with unknown interference based on an adaptive neural network, which includes the following steps:

Step 1: Analyze the controlled spherical robot system, mainly analyze the force of the robot, and establish a mathematical model of the spherical robot system with unknown interference based on its mechanical structure and physical laws. The purpose of establishing this model is to better understand the characteristics of the spherical robot system and then design the interference approximator and sliding mode controller of the spherical robot system.

According to the stress situation of the spherical robot, according to its structure and physical laws, a mathematical model of the spherical robot system with unknown interference is established, as follows:

Where M ₁₁ , M ₁₂ , M ₂₁ , M ₂₂ are the elements of the inertia matrix of the spherical robot system

M ₁₂ =M ₂₁ =m _p R _s lcos(ζ)

M ₂₂ = m _p l ²

V ₂₁ =m _p glsin(ζ)

V ₁₁ and V ₂₁ are the elements of the gravity moment vector of the spherical robot. M _s and m _p are the masses of the spherical shell and the inner pendulum of the ball respectively. R _s and l are respectively the radius of the spherical shell and the length of the inner pendulum of the spherical shell, Φ, are the spherical shell rotation angle and rotation speed respectively, and the angular acceleration of the spherical shell rotation angle, ζ,/> are the rotation angle and rotation speed of the internal pendulum of the spherical shell/> is the angular acceleration of the swing angle in the spherical shell, and Δ _y is the uncertainty term in the modeling of the spherical robot. τ _y is the motor input torque of the spherical robot; g is the gravity acceleration.

Let the four state variables x ₁ =φ, x ₃ =ζ,/> Then convert the spherical robot model (1) into a system state space expression:

f ₁ is the time-varying function of the system state variable in the spherical shell subsystem; β ₁ is the time-varying coefficient of the control input in the spherical shell subsystem; b ₁ is the time-varying coefficient of the unknown item in the spherical shell subsystem; f ₂ sphere The pendulum subsystem inside the shell contains the time-varying function of the system state variables; β ₂ is the time-varying coefficient of the control input in the pendulum subsystem inside the spherical shell; b ₂ is the time-varying coefficient of the unknown item in the spherical shell subsystem

Let the unknown item in the spherical shell subsystem be d ₁ = b ₁ Δ _y , and the unknown item in the spherical shell pendulum subsystem be d ₂ = b ₂ Δ _y , then the system can be described as

This simplifies the controller design method.

Step 2: The interference of the spherical robot system model is unknown. The unknown items of the system are approximated based on the neural network, and the weight parameters of the neural network are adaptively estimated based on the control error information.

Using neural network to approximate the unknown items in system (3), we can get

Among them, W ₁ and W ₂ are the neural network weights. The specific weights can be set according to the actual situation. In one embodiment of the present invention, W ₁ =W ₂ =[0.1,0.1,0.1,0.1,0.1] ^T ) ; h ₁ (X) = [h _i ] ^T , h ₂ (X) = [h _j ] ^T is the radial basis function, h _i , h _j are the elements of the radial basis function; where the elements where c _j and b _j are parameters of the radial basis function, representing the mean and mean square error of the radial basis function respectively. Their values are set according to the actual situation. For example, in one embodiment of the present invention, c _j =0.5*[-2 ,-1,0,1,2] _, b _j =3 _, In one embodiment of the present invention, ε ₁ =0.01 and ε ₂ =0.001 are set.

Then the estimated value of the unknown item is expressed as follows:

Then the estimated value of the unknown item in system (2) is Then the system tracking error is

Among them, e ₁ is the position tracking error of the spherical shell, e ₃ is the inward swing position tracking error of the spherical shell, and x _1d is the desired position of the spherical shell. Their values are artificially given and can be set according to the actual situation. For example, the present invention In one embodiment, x _1d = 0 is set. At this time, the spherical robot moves _{along a straight line; x 3d is} the desired position of the spherical shell's internal swing. related to the size, specifically At the same time, in order to design the adaptive law of the weight parameters of the neural network, the system sliding mode surface is first defined: the first sliding mode surface S ₁ , the second sliding mode surface S ₂ ,

Among them, e _i (i=1,3) represents the tracking error, c ₁ is the coefficient of e ₁ , c ₂ is the coefficient of e ₂ , and its value is set according to the actual situation. For example, in one embodiment of the present invention, c is set ₁ =0.1c ₂ =0.1.

Then the joint sliding mode surface is

S _y =aS ₁ +bS ₂ (8)

a and b are the weight coefficients of the first sliding mode surface S ₁ and the second sliding mode surface S ₂ respectively, and their values are set according to the actual situation. For example, in one embodiment of the present invention, a=1 and b=2 are set.

The adaptive law of neural network weight parameters can be designed as

is the rate of change of the neural network weight estimate;/> is the rate of change of the neural network weight estimate; γ ₁ is a coefficient, γ ₂ is a coefficient, and their values are set according to the actual situation. For example, in one embodiment of the present invention, γ ₁ =1 γ ₂ =0.5 is set.

For the spherical robot system (2) containing unknown items, if the neural network weight adaptive law in (8) and the neural network approximation form in (5) are adopted, the unknown item Δ _y can be approximately accurately approximated.

Step 3: Design a sliding mode controller with interference compensation based on the adaptive neural network to approximate the unknown items and the defined sliding mode surface.

pass Get equivalent controller

τ _y1 first equivalent controller; τ _y2 second equivalent controller; is the expected angular velocity of the spherical shell;/> is the expected angular velocity of the internal pendulum of the spherical shell;

Then the final sliding mode controller is designed as

τ _y =(aβ ₁ +bβ ₂ ) ^-1 (aβ ₁ τ _y1 +bβ ₂ τ _y2 -k ₁ sign(S _y )-k ₂ S _y ) (11)

k ₁ is the error coefficient preset for the sign function of the joint sliding mode surface, such as k ₁ =0.5; k ₂ is the error coefficient preset for the joint sliding mode surface, such as k ₂ =2;

For the spherical robot system (2) containing unknown items, if the neural network input vectors h ₁ and h ₂ apply the adaptive law (9), the sliding mode controller (11) can be obtained in a short time. For the spherical robot system containing unknown items The system controls to ensure that the spherical robot moves in the set direction. For example, when x _1d is set to 0, the spherical robot can move along a straight line, thus achieving the purpose of the present invention. Figure 2 shows the controller design structure constructed based on the above steps.

Example 2:

Step 1. According to the force situation of the spherical robot, according to its structure and physical laws, establish a mathematical model of the spherical robot system with unknown interference, as follows:

Among them, x ₁ =φ, are the spherical shell rotation angle and rotation speed respectively, x ₃ =ζ,/> are the rotation angle and rotation speed of the internal pendulum of the spherical shell respectively, and Δ _y is the uncertainty term of the spherical robot modeling. Let d ₁ =b ₁ Δ _x , d ₂ =b ₂ Δ _x , then the system can be described as

This simplifies the controller design method.

Step 2: Assume that the interference of the spherical robot system model is unknown, approximate the unknown items of the system based on the neural network, and adaptively estimate the weight parameters of the neural network based on the control error information.

Using neural network to approximate the unknown items in system (13), we can get

Among them, W ₁ and W ₂ are the neural network weights, and the estimated value of the unknown item is expressed as follows:

Then the estimated value of the unknown item in system (12) is Then the system tracking error is

Among them, when ensuring that the spherical robot moves along a straight line, x _1d = 0, At the same time, in order to design the adaptive law of neural network heavy parameters, the system sliding mode surface is first defined:

Among them, e _i (i=1,3) represents the tracking error, and the joint sliding mode surface is

S _y =aS ₁ +bS ₂ (18)

The adaptive law of neural network weight parameters can be designed as

For the spherical robot system (2) containing unknown items, if the neural network weight adaptive law in (8) and the neural network approximation form in (5) are adopted, the unknown item Δ _y can be approximately accurately approximated.

Step 3: Design a sliding mode controller with interference compensation based on the adaptive neural network to approximate the unknown items and the defined sliding mode surface.

pass Get equivalent controller

Then the final sliding mode controller is designed as

τ _y =(aβ ₁ +bβ ₂ ) ^-1 (aβ ₁ τ _y1 +bβ ₂ τ _y2 -k ₁ sign(S _y )-k ₂ S _y ) (21)

For the spherical robot system (12) containing unknown items, if the neural network input vectors h ₁ and h ₂ apply the adaptive law (19), the sliding mode controller (21) can be obtained in a short time. For the spherical robot system containing unknown items The system controls to ensure that the spherical robot moves along a straight line, thereby achieving the purpose of the present invention.

Simulation results

Simulate the above processing results. Assume that the spherical robot dynamics model is:

in

In the simulation analysis, it is assumed that only _Δy is unknown in the spherical robot model. First, the neural network is applied to approximate unknown interference terms, and then the adaptive law is used to update the neural network weight parameters in real time. The neural network input vector is set to h ₁ =h ₂ =[x ₁ ,x ₂ ,x ₃ ,x ₄ ] ^T , The initial value of the system state is set to x ₁ (0) = 0, x ₂ (0) = 0, x ₃ (0) = 0, x ₄ (0) = 0. By adjusting other parameters appropriately, the approximate unknown Δ _y can be estimated and the estimated unknown term can converge to its true value.

To sum up, the above are only preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention shall be included in the protection scope of the present invention.

Claims (2)

1. A hierarchical sliding mode control method that interferes with unknown spherical robots, characterized in that the method includes the following steps: Step 1: Establish a mathematical model of the spherical robot system containing unknown items for the controlled spherical robot system, where the unknown items are unknown interference; The mathematical model for establishing a spherical robot system with unknown interference for the controlled spherical robot system is as follows: Among them, M ₁₁ , M ₁₂ , M ₂₁ and M ₂₂ are the elements of the inertia matrix of the spherical robot system respectively; V ₁₁ and V ₂₁ are the elements of the gravity moment vector of the spherical robot; Φ is the rotation angle of the spherical shell, is the angular acceleration of the rotation angle of the spherical shell, ζ is the rotation angle of the internal pendulum of the spherical shell,/> is the angular acceleration of the internal pendulum angle of the spherical shell; τ _y is the motor input torque of the spherical robot; Set four state variables x ₁ , x ₂ , x ₃ , x ₄ , let x ₁ =φ, x ₃ =ζ,/> Then the mathematical model of the spherical robot system, that is, formula (1), is transformed into a system state space expression: The spherical robot system includes a spherical shell subsystem and a spherical shell pendulum subsystem. f ₁ is the time-varying function of the system state variables in the spherical shell subsystem; β ₁ is the time-varying coefficient of the control input in the spherical shell subsystem; b ₁ is the time-varying coefficient of the unknown item in the spherical shell subsystem; f ₂ is the time-varying function of the system state variables included in the spherical shell pendulum subsystem; β ₂ is the time-varying coefficient of the control input in the spherical shell pendulum subsystem; b _2. The time-varying coefficient of the unknown item in the spherical shell subsystem; Δ _y is the unknown item of the mathematical model of the spherical robot system; Among them, the unknown item in the spherical shell subsystem is d ₁ =b ₁ Δ _y , and the unknown item in the spherical shell pendulum subsystem is d ₂ =b ₂ Δ _y . Then the system state space expression (2) is described as Step 2: Approximate the unknown items in the mathematical model of the spherical robot system based on the neural network, and adaptively estimate the weight parameters of the neural network based on the control error information; The second step is specifically: The unknown items in the mathematical model of the spherical robot system include the unknown item d ₁ in the spherical shell subsystem and the unknown item d ₂ in the pendulum subsystem of the spherical shell: d ₁ =W ₁ ^T h ₁ (X)+ε ₁ (4) d ₂ =W ₂ ^T h ₂ (X)+ε ₂ Among them, W ₁ and W ₂ are the two neural network weights set, h ₁ (X) = [h _i ] ^T and h ₂ (X) = [h _j ] ^T are the radial basis functions respectively, h _i , h _j is the element of the radial _basis function _{, X} is _the state vector _{, which} is composed of four state variables. The state ^vector The unknown item in the subsystem is d ₁ , and the estimated values of the unknown item d ₂ in the spherical shell pendulum subsystem are respectively and/> Expressed as follows: Then the estimated value of the unknown item in the system state space expression (2) is Then the system tracking error is Among them, e ₁ is the position tracking error of the spherical shell, e ₃ is the position tracking error of the inner swing of the spherical shell, x _1d is the expected position of the spherical shell, and x _3d is the expected position of the inner swing of the spherical shell; First define the system sliding mode surface, including the first sliding mode surface S ₁ and the second sliding mode surface S ₂ : Among them, e _i (i=1,3) represents the tracking error, c ₁ is the coefficient of e ₁ , and c ₂ is the coefficient of e ₂ ; Then the joint sliding mode surface is S _y =aS ₁ +bS ₂ (8) a, b are the weight coefficients of the first sliding mode surface S ₁ and the second sliding mode surface S ₂ respectively; The adaptive law of neural network weight parameters is in is the rate of change of the neural network weight estimate;/> is the rate of change of the neural network weight estimate; γ ₁ is the coefficient; γ ₂ is the coefficient; For the system state space expression (2) of the spherical robot system containing unknown terms, if the neural network weight adaptive law (9) in the joint sliding mode surface (8) and the neural network approximation form in formula (5) are adopted, Then the approximate unknown term is obtained, which is Step 3: Design a sliding mode controller with interference compensation based on the unknown items approximated by the adaptive neural network and the defined sliding mode surface; pass is the first derivative of S _i , the equivalent controller is Among them, τ _y1 is the first equivalent controller; τ _{y2 is} the second equivalent controller; is the expected angular velocity of the spherical shell;/> is the expected angular velocity of the internal pendulum of the spherical shell; Then the final sliding mode controller is designed as τ _y =(aβ ₁ +bβ ₂ ) ^-1 (aβ ₁ τ _y1 +bβ ₂ τ _y2 -k ₁ sign(S _y )-k ₂ S _y ) (11) where k ₁ is the error coefficient preset for the sign function of the joint sliding mode surface; k ₂ is the error coefficient preset for the joint sliding mode surface; Step 4: Use the sliding mode controller to control the interference unknown spherical robot. 2. A hierarchical sliding mode control method that interferes with unknown spherical robots as claimed in claim 1, wherein M ₁₁ , M ₁₂ , M ₂₁ , and M ₂₂ are respectively elements of the inertia matrix of the spherical robot system. ,specifically: Among them, M _s is the mass of the spherical shell, m _p is the mass of the ball's inner pendulum, R _s is the radius of the spherical shell, l is the length of the ball's inner pendulum, and g is the acceleration of gravity.