CN114200837B - Layering sliding mode control method for interference unknown spherical robot - Google Patents
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Abstract
The invention provides a layering sliding mode control method of an unknown spherical robot system, which can be used for controlling the movement of the spherical robot system by adopting a self-adaptive neural network theory, so that the accurate control of the unknown spherical robot system is realized. The technical scheme of the invention comprises the following steps: and establishing a mathematical model of the spherical robot system containing unknown items aiming at the controlled spherical robot system, wherein the unknown items are unknown interferences. And approximating unknown items in a mathematical model of the spherical robot system based on the neural network, and adaptively estimating the weight parameters of the neural network based on the control error information. A sliding mode controller with interference compensation is designed based on unknown items of the adaptive neural network approximation and a defined sliding mode surface. And controlling the unknown spherical robot by using a sliding mode controller.
Description
Technical Field
The invention relates to the technical field of robot motion control, in particular to a layering sliding mode control method for an unknown spherical robot.
Background
For the motion control of the spherical robot, the existing research is mostly based on the premise of assuming the spherical robot to move along a straight line, and the premise of the spherical robot to move along the straight line is ignored. For the control of keeping the linear motion of the spherical robot, the sliding mode control with simple structure and high response speed can be adopted, and the state feedback is carried out by designing a sliding mode surface formed by control errors and measuring and collecting the posture of the spherical robot in real time. For an unknown interference part in a control object dynamics model, the nonlinear interference observer has complex design form, depends on the model and has more design parameters, so that a neural network approximator can be designed to estimate the unknown interference part.
In particular, in recent years, research on a motion control method of the spherical robot has been greatly advanced, but research results for estimating an interference part by using an adaptive neural network theory to maintain motion control of the spherical robot have not been found.
Disclosure of Invention
In view of the above, the invention provides a layered sliding mode control method for an unknown spherical robot system, which can adopt the adaptive neural network theory to control the movement of the spherical robot system, thereby realizing the accurate control of the unknown spherical robot system.
In order to achieve the above purpose, the technical scheme of the invention comprises the following steps:
step one, establishing a mathematical model of the spherical robot system containing unknown items aiming at the controlled spherical robot system, wherein the unknown items are unknown interferences.
And secondly, approximating unknown items in a mathematical model of the spherical robot system based on the neural network, and adaptively estimating the weight parameters of the neural network based on the control error information.
And thirdly, designing a sliding mode controller with interference compensation based on unknown items of the self-adaptive neural network approximation and a defined sliding mode surface.
And fourthly, controlling the unknown spherical robot by utilizing a sliding mode controller.
Further, in the first step, a mathematical model of the spherical robot system containing unknown interference is built for the controlled spherical robot system, specifically as follows:
wherein M is 11 ,M 12 ,M 21 ,M 22 Elements of an inertial matrix of the spherical robotic system, respectively; v (V) 11 And V 21 Is an element in a gravity moment vector of the spherical robot; phi is the corner of the spherical shell,the angular acceleration of the corner of the spherical shell, ζ is the corner of the inner pendulum of the spherical shell, and ++>Angular acceleration which is the corner of the spherical shell inner pendulum; τ y Inputting torque to a motor of the spherical robot;
setting four state variables x 1 ,x 2 ,x 3 ,x 4 Let x 1 ＝φ，x 3 ＝ζ/>The mathematical model of the spherical robotic system, equation (1), is converted into a system state space expression:
wherein the spherical robot system comprises a spherical shell subsystem and a spherical shell inner pendulum subsystem, f 1 A time-varying function of a system state variable contained in the spherical shell subsystem; beta 1 Time-varying coefficients for control inputs in the spherical shell subsystem; b 1 Time-varying coefficients for unknown items in the spherical shell subsystem; f (f) 2 The spherical shell inner pendulum subsystem comprises a time-varying function of a system state variable; beta 2 The time-varying coefficient is controlled and input in the spherical shell inner swing subsystem; b 2 Time-varying coefficients of unknown items in the spherical shell subsystem; delta y Unknown terms of a mathematical model of a spherical robotic system;
wherein the unknown term in the spherical shell subsystem is d 1 ＝b 1 Δ y Unknown item in 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 Elements of the inertial matrix of the spherical robotic system, in particular:
wherein M is s Is the mass of the spherical shell, m p R is the mass of the inner pendulum of the ball s The radius of the spherical shell is l, the length of the inner pendulum of the spherical shell is l, and g is gravity acceleration.
Further, 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, specifically:
the unknown term in the mathematical model of the spherical robotic system comprises the unknown term d in the spherical shell subsystem 1 Unknown item d in spherical shell pendulum subsystem 2 ：
Wherein W is 1 ，W 2 For the set two neural network weights, h 1 (X)＝[h i ] T And h 2 (X)＝[h j ] T Respectively radial basis functions, h i ，h j As an element of a radial basis function, X is a state vector, which is composed of four state variables, x= [ X ] 1 ,x 2 ,x 3 ,x 4 ] T ；ε 1 、ε 2 To approximate the error, the unknown item d in the spherical shell subsystem 1 Unknown item d in spherical shell pendulum subsystem 2 The estimated values of (a) are respectivelyAnd->The expression is as follows:
the unknown item estimate in the system state space expression (2) isThe system tracking error is
Wherein e 1 For tracking error, e, of the position of the spherical shell 3 Tracking error, x for spherical shell inner swinging position 1d X is the desired position of the spherical shell 3d Swinging the spherical shell inwards to a desired position;
defining a system sliding die surface including a first sliding die surface S 1 And a second slide surface S 2 ：
Wherein e i (i=1, 3) represents tracking error, c 1 E is 1 Coefficient of c 2 E is 2 Coefficients of (2);
the combined sliding die surface is
S y ＝aS 1 +bS 2 (8)
a and b are respectively a first sliding die surface S 1 And a second slide surface S 2 Weight coefficient of (2);
the self-adaptive law of the weight parameter of the neural network is
Wherein the method comprises the steps ofThe change rate of the weight estimation value of the neural network is calculated; />The change rate of the weight estimation value of the neural network is calculated; gamma ray 1 Is a coefficient; gamma ray 2 Is a 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 in the formula (5) are adoptedThe network approximation form is obtained as an unknown term of approximation
Further, by Is S i To obtain the equivalent controller as
Wherein τ y1 A first equivalent controller; τ y2 A second equivalent controller;is the desired angular velocity of the spherical shell; />A desired angular velocity for the spherical shell inner pendulum;
the final slip-form 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)
Wherein k is 1 Error coefficients preset for the joint sliding mode surface symbol functions; k (k) 2 The error coefficient is preset for the combined sliding mode surface.
The beneficial effects are that:
1. the invention provides a layering sliding mode control method of an interference unknown spherical robot system, which is based on a sliding mode control method of a self-adaptive neural network, and introduces a design method of a controller for keeping a spherical robot to move along a straight line by applying a neural network interference approximator. Firstly, aiming at a spherical robot system model with unknown interference, a neural network is applied to approach an unknown part; establishing a self-adaptive weight updating law based on control errors; a sliding mode controller with interference compensation is designed according to the approximated interference and a system model of the spherical robot, so that the spherical robot can quickly and stably execute the expected motion.
2. The self-adaptive neural network approaches to the interference unknown item of the spherical robot, and can realize accurate modeling on the spherical robot system with unknown physical model. The recognition system can be used to analyze the partial expected signal of the spherical robot and solve the motor control output.
3. The sliding mode control method provided by the invention can ensure that a plurality of states of the spherical robot are stable under the condition of single input, namely, when only a single motor is input, the corner positions of the spherical shell and the spherical inner pendulum can be converged to the expected positions.
Drawings
FIG. 1 is a schematic diagram of a spherical robotic system;
FIG. 2 is a schematic diagram of a controller design;
fig. 3 is a flow chart of a design method of a linear motion sliding mode controller of an interference unknown spherical robot based on a self-adaptive neural network.
Detailed Description
The invention will now be described in detail by way of example with reference to the accompanying drawings.
Example 1:
fig. 1 is a diagram showing a system structure of a spherical robot, and the invention provides a design method of a linear motion sliding mode controller of an unknown spherical robot based on self-adaptive neural network interference, which comprises the following steps:
step one, analyzing a controlled spherical robot system, mainly performing stress analysis on a robot, and establishing a mathematical model of the spherical robot system with unknown interference according to a mechanical structure and a physical law of the robot. The purpose of the model is to better understand the characteristics of the spherical robotic system, and then design the spherical robotic system interference approximator and the sliding mode controller.
According to the stress condition of the spherical robot, a mathematical model of the spherical robot system containing unknown interference is established according to the structure and the physical law, and the mathematical model is specifically as follows:
wherein M is 11 ,M 12 ,M 21 ,M 22 Elements of inertial matrix of 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 ,V 21 Element M being a gravity moment vector of a spherical robot s ,m p The mass of the spherical shell and the spherical inner pendulum are respectively R s L is the radius of the spherical shell and the length of the spherical shell inner pendulum respectively, phi,the rotational angle and the rotational speed of the spherical shell, the angular acceleration of the spherical shell rotational angle, ζ and +.>The rotation angle and the rotation speed of the inner pendulum of the spherical shell are respectively +.>Angular acceleration, delta, of the spherical shell inner swing angle y The uncertainty term is modeled for a spherical robot. τ y Inputting torque to a motor of the spherical robot; g gravitational acceleration.
Let four state variables x 1 ＝φ，x 3 ＝ζ，/>Then the spherical robot model (1) is transformed into a system state space expression:
f 1 a time-varying function of a system state variable contained in the spherical shell subsystem; beta 1 Time-varying coefficients for control inputs in the spherical shell subsystem; b 1 Time-varying coefficients for unknown items in the spherical shell subsystem; f (f) 2 The spherical shell inner pendulum subsystem comprises a time-varying function of a system state variable; beta 2 The time-varying coefficient is controlled and input in the spherical shell inner swing subsystem; b 2 Time-varying coefficients of unknown items in spherical shell subsystem
Let unknown item in spherical shell subsystem be d 1 ＝b 1 Δ y Unknown item in spherical shell pendulum subsystem is d 2 ＝b 2 Δ y The system can be described as
Thereby simplifying the controller design method.
And secondly, the model interference of the spherical robot system is unknown, approximation is carried out on the unknown items of the system based on the neural network, and self-adaptive estimation is carried out on the weight parameters of the neural network based on the control error information.
The unknown items in the approximation system (3) can be obtained by using the neural network
Wherein W is 1 And W is 2 For the neural network weight, the specific weight can be set according to the actual situation, and in one embodiment of the invention, W can be set 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 As radial basis function, h i ，h j Elements that are radial basis functions; wherein the elements areWherein c j And b j The parameters of the radial basis function are respectively the mean value and the mean square error of the radial basis function, and the values are set according to the actual situation, such as c in one embodiment of the invention j ＝0.5*[-2,-1,0,1,2],b j =3, x is a state vector, consisting of four state variables; epsilon 1 、ε 2 To approximate the error, the value is set according to the actual situation, e.g. epsilon is set in one embodiment of the invention 1 ＝0.01，ε 2 ＝0.001。
The unknown estimate is expressed as follows:
then the unknown item estimate in system (2) isThe system tracking error is
Wherein e 1 For tracking error, e, of the position of the spherical shell 3 Tracking error, x for spherical shell inner swinging position 1d For the desired position of the spherical shell, the value is artificially given, and can be set according to the actual situation, for example, the invention is provided in one embodimentx 1d =0, at which time the spherical robot moves in a straight line; x is x 3d To swing the spherical shell inwards to the expected position, x 3d And (3) withIs related to the size of (a), in particularMeanwhile, for designing a neural network weight parameter self-adaptive law, a system sliding mode surface is defined firstly: first slide surface S 1 Second sliding surface S 2 ，
Wherein e i (i=1, 3) represents tracking error, c 1 E is 1 Coefficient of c 2 E is 2 The value of which is set according to the actual situation, e.g. c is set in one embodiment of the invention 1 ＝0.1c 2 ＝0.1。
The combined sliding die surface is
S y ＝aS 1 +bS 2 (8)
a and b are respectively a first sliding die surface S 1 And a second slide surface S 2 The values of the weight coefficients of (a) and (b) are set according to the actual situation, for example, a=1 and b=2 are set in one embodiment of the present invention.
The adaptive law of the weight parameters of the neural network can be designed as
The change rate of the weight estimation value of the neural network is calculated; />The change rate of the weight estimation value of the neural network is calculated; gamma ray 1 Is the coefficient, gamma 2 Is a coefficient whose value is set according to the actual situation, e.g. gamma is set in one embodiment of the invention 1 ＝1γ 2 ＝0.5。
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 items delta y Can be approximated with an approximate exact approximation.
And thirdly, designing a sliding mode controller with interference compensation based on the self-adaptive neural network approaching unknown item and the defined sliding mode surface.
By passing throughEquivalent controller
τ y1 A first equivalent controller; τ y2 A second equivalent controller;is the desired angular velocity of the spherical shell; />A desired angular velocity for the spherical shell inner pendulum;
the final slip-form 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 Error coefficients such as k preset for joint sliding mode surface sign functions 1 ＝0.5；k 2 For error coefficients preset for joint slip-form surfaces, e.g. k 2 ＝2；
For spherical robotic systems (2) with unknown items, if neural networksInput vector h 1 ，h 2 By applying the adaptive law (9), a sliding mode controller (11) can be obtained, and the spherical robot system containing unknown items is controlled to ensure that the spherical robot moves towards a set direction, such as setting x 1d At 0, the spherical robot can move along a straight line, thereby achieving the purpose of the invention. Fig. 2 is a controller design constructed according to the above steps.
Example 2:
step one, according to the stress condition of the spherical robot, establishing a mathematical model of the spherical robot system containing unknown interference according to the structure and the physical law, wherein the mathematical model is specifically as follows:
wherein x is 1 ＝φ,Respectively the rotation angle and the rotation speed of the spherical shell, x 3 ＝ζ，/>The rotation angle and the rotation speed delta of the inner pendulum of the spherical shell are respectively y The uncertainty term is modeled for a spherical robot. Let d 1 ＝b 1 Δ x ，d 2 ＝b 2 Δ x The system can be described as
Thereby simplifying the controller design method.
And secondly, assuming that the model interference of the spherical robot system is unknown, approximating the unknown items of the system based on the neural network, and carrying out self-adaptive estimation on the weight parameters of the neural network based on the control error information.
The unknown items in the approximation system (13) can be obtained by using a neural network
Wherein W is 1 ，W 2 For neural network weights, the unknown term estimates are expressed as follows:
then the unknown item estimate in the system (12) isThe system tracking error is
Wherein, when the spherical robot moves along a straight line, x is as follows 1d ＝0，Meanwhile, in order to design a neural network heavy parameter self-adaptive law, a system sliding mode surface is defined firstly:
wherein e i (i=1, 3) represents tracking error, then the joint slide surface is
S y ＝aS 1 +bS 2 (18)
The adaptive law of the weight parameters of the neural network 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 method is unknownTerm delta y Can be approximated with an approximate exact approximation.
And thirdly, designing a sliding mode controller with interference compensation based on the self-adaptive neural network approaching unknown item and the defined sliding mode surface.
By passing throughEquivalent controller
The final slip-form 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 spherical robotic systems (12) with unknown terms, if the neural network inputs a vector h 1 ，h 2 A sliding mode controller (21) can be obtained by applying a self-adaptive law (19), and the spherical robot system containing unknown items is controlled to ensure that the spherical robot moves along a straight line, so that the aim of the invention is fulfilled.
Simulation results
And simulating the processing result. The spherical robot dynamics model is assumed to be:
wherein the method comprises the steps of
In the simulation analysis, it is assumed that only Δ is in the spherical robot model y Unknown. Firstly, approaching unknown interference items by using a neural network, then updating the weight parameters of the neural network in real time by using an adaptive law, and setting the input vector of the neural network as h 1 ＝h 2 ＝[x 1 ,x 2 ,x 3 ,x 4 ] T The initial value of the system state is set as x 1 (0)＝0,x 2 (0)＝0,x 3 (0)＝0,x 4 (0) =0. Other parameters are properly adjusted to estimate approximate unknown delta y The estimated unknown term may converge to its true value.
In summary, the above embodiments are only preferred embodiments of the present invention, and are not intended to limit the scope of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (2)
1. The method for controlling the layering sliding mode of the unknown spherical robot is characterized by comprising the following steps of:
step one, establishing a mathematical model of a spherical robot system containing an unknown item aiming at a controlled spherical robot system, wherein the unknown item is unknown interference;
the method comprises the following steps of establishing a mathematical model of the spherical robot system containing unknown interference aiming at the controlled spherical robot system, wherein the mathematical model comprises the following specific steps:
wherein M is 11 ,M 12 ,M 21 ,M 22 Elements of an inertial matrix of the spherical robotic system, respectively; v (V) 11 And V 21 Is an element in a gravity moment vector of the spherical robot; phi is the corner of the spherical shell,the angular acceleration of the corner of the spherical shell, ζ is the corner of the inner pendulum of the spherical shell, and ++>Angular acceleration which is the corner of the spherical shell inner pendulum; τ y Inputting torque to a motor of the spherical robot;
setting four state variables x 1 ,x 2 ,x 3 ,x 4 Let x 1 ＝φ，x 3 ＝ζ，/>The mathematical model of the spherical robotic system, equation (1), is converted into a system state space expression:
wherein the spherical robot system comprises a spherical shell subsystem and a spherical shell inner pendulum subsystem, f 1 A time-varying function of a system state variable contained in the spherical shell subsystem; beta 1 Time-varying coefficients for control inputs in the spherical shell subsystem; b 1 Time-varying coefficients for unknown items in the spherical shell subsystem;f 2 the spherical shell inner pendulum subsystem comprises a time-varying function of a system state variable; beta 2 The time-varying coefficient is controlled and input in the spherical shell inner swing subsystem; b 2 Time-varying coefficients of unknown items in the spherical shell subsystem; delta y Unknown terms of a mathematical model of a spherical robotic system;
wherein the unknown term in the spherical shell subsystem is d 1 ＝b 1 Δ y Unknown item in spherical shell pendulum subsystem is d 2 ＝b 2 Δ y Then the system state space expression (2) is described as
Approximating unknown items in a mathematical model of the spherical robot system based on the neural network, and adaptively estimating the weight parameters of the neural network based on control error information;
the second step is specifically as follows:
the unknown term in the mathematical model of the spherical robot system comprises d in the spherical shell subsystem 1 Unknown item d in spherical shell pendulum subsystem 2 ：
d 1 ＝W 1 T h 1 (X)+ε 1 (4)
d 2 ＝W 2 T h 2 (X)+ε 2
Wherein W is 1 ，W 2 For the set two neural network weights, h 1 (X)＝[h i ] T And h 2 (X)＝[h j ] T Respectively radial basis functions, h i ，h j As an element of a radial basis function, X is a state vector, which is composed of four state variables, x= [ X ] 1 ,x 2 ,x 3 ,x 4 ] T ；ε 1 、ε 2 For the approximation error, the unknown term in the spherical shell subsystem is d 1 Unknown item d in spherical shell pendulum subsystem 2 The estimated values of (a) are respectivelyAnd->The expression is as follows:
the unknown item estimate in the system state space expression (2) isThe system tracking error is
Wherein e 1 For tracking error, e, of the position of the spherical shell 3 Tracking error, x for spherical shell inner swinging position 1d X is the desired position of the spherical shell 3d Swinging the spherical shell inwards to a desired position;
defining a system sliding die surface including a first sliding die surface S 1 And a second slide surface S 2 ：
Wherein e i (i=1, 3) represents tracking error, c 1 E is 1 Coefficient of c 2 E is 2 Coefficients of (2);
the combined sliding die surface is
S y ＝aS 1 +bS 2 (8)
a and b are respectively a first sliding die surface S 1 And a second slide surface S 2 Weight coefficient of (2);
the self-adaptive law of the weight parameter of the neural network is
Wherein the method comprises the steps ofThe change rate of the weight estimation value of the neural network is calculated; />The change rate of the weight estimation value of the neural network is calculated; gamma ray 1 Is a coefficient; gamma ray 2 Is a 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 the formula (5) are adopted, the approximated unknown terms, namely
Step three, designing a sliding mode controller with interference compensation based on unknown items of the self-adaptive neural network approximation and a defined sliding mode surface;
by passing through Is S i To obtain the equivalent controller as
Wherein τ y1 First equivalent ofA controller; τ y2 A second equivalent controller;is the desired angular velocity of the spherical shell; />A desired angular velocity for the spherical shell inner pendulum;
the final slip-form 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)
Wherein k is 1 Error coefficients preset for the joint sliding mode surface symbol functions; k (k) 2 The error coefficient is preset for the combined sliding mode surface;
and fourthly, controlling the unknown spherical robot by using the sliding mode controller.
2. The method for controlling the hierarchical sliding mode of the unknown spherical robot with interference as set forth in claim 1, wherein the M is 11 ,M 12 ,M 21 ,M 22 Elements of the inertial matrix of the spherical robotic system, in particular:
wherein M is s Is the mass of the spherical shell, m p R is the mass of the inner pendulum of the ball s The radius of the spherical shell is l, the length of the inner pendulum of the spherical shell is l, and g is gravity acceleration.
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| 一种摆式球形机器人水中俯仰运动的稳定控制方法;李艳生;《振动与冲击》;20180730;第37卷(第13期);第149-156页 * |
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