CN113341724B - Asynchronous sliding mode control method of multi-mode mechanical arm - Google Patents

Asynchronous sliding mode control method of multi-mode mechanical arm Download PDF

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CN113341724B
CN113341724B CN202110677946.1A CN202110677946A CN113341724B CN 113341724 B CN113341724 B CN 113341724B CN 202110677946 A CN202110677946 A CN 202110677946A CN 113341724 B CN113341724 B CN 113341724B
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sliding mode
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mechanical arm
matrix
switching
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CN113341724A (en
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齐文海
宗广灯
候雅琨
孙海滨
杨东
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Qufu Normal University
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    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
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Abstract

The invention discloses an asynchronous sliding mode control method of a multi-mode mechanical arm, which comprises the following steps: constructing a hidden semi-Markov switching system of the multi-modal mechanical arm based on the influence of hidden modes and uncertain parameters on the multi-modal mechanical arm; designing an integral sliding mode switching surface based on the state and modal information of a hidden half Markov switching system; carrying out stability analysis on sliding mode dynamics in an integral sliding mode switching surface; calculating parameters of a sliding mode controller which meet the target of the finite time stability of the hidden semi-Markov switching system based on a linear matrix inequality obtained from a stability analysis result; designing an asynchronous sliding mode control law of observation mode dependence based on sliding mode controller parameters; and performing accessibility analysis on the sliding mode switching surface based on an asynchronous sliding mode control law depending on a designed observation mode. The method effectively inhibits the influence of parameter uncertainty and nonlinearity on the mechanical arm, solves the asynchronous sliding mode control problem of the multi-mode mechanical arm, and improves the task execution precision of the mechanical arm.

Description

Asynchronous sliding mode control method of multi-mode mechanical arm
Technical Field
The invention relates to the technical field of mechanical arms, in particular to an asynchronous sliding mode control method of a multi-mode mechanical arm.
Background
With the development of modern industrial level, the living standard of human beings is continuously improved, and the demand for productivity and advanced degree of production tools is increased. Robots have gained widespread attention as a representative product of the fourth industrial technological revolution in its advanced form of presentation. The mechanical arm is the most critical execution link of the robot, independently completes task arrangement in actual production by simulating the motion form of human joints, and is in an indispensable important position in modern production. In addition to performing daily activities such as grasping, moving objects, etc., robotic arms are widely used in important fields such as machine manufacturing, medical care, space exploration, disaster relief, etc. In the development process of mechanical arm control, early control methods such as PID are often used in low-speed and low-precision mechanical arm work occasions. However, with the increasing development of industrial production, the requirements for the speed and precision of the mechanical arm joint are continuously increased, and the traditional PID control does not meet the control requirement any more.
As is well known, the sliding mode control has stronger anti-interference performance, and the control precision of the mechanical arm system is improved to a certain degree. For example, aerospace astronauts are required to face a great deal of risk when working in order to cope with severe conditions such as solar wind, extreme cold and extreme heat, and oxygen-free vacuum in an extraterrestrial environment. Under the scene, the high-precision space mechanical arm can replace an astronaut to carry out experimental operation, so that the environmental difficulty is effectively overcome; in the medical field, the high-precision mechanical arm can help a doctor to have more free in complex operations, so that human errors are effectively reduced, the success rate of the precision operations is improved, and the life of a patient is saved. Therefore, the method has important research significance on effectively inhibiting the influence of complex environmental factors on the mechanical arm, accurately describing the dynamic characteristics of the multi-mode mechanical arm and solving the sliding mode control problem of the multi-mode mechanical arm.
Various uncertain factors in the multi-mode mechanical arm system, such as friction damping, unknown interference, signal drift, original errors caused by measurement uncertainty of mechanical arm joint physical parameters and the like, bring great difficulty to the design of a mechanical arm control algorithm. Meanwhile, considering the cost factor in production application and the limitation of the physical structure of the mechanical arm, the modal information of the original system is not directly available in the transmission process, and the hidden half Markov model is introduced to describe the dynamic characteristics of the multi-modal mechanical arm under an asynchronous switching mechanism. In addition, transient performance requirements often exist in a plurality of practical engineering systems, particularly mechanical arm systems in the process of executing tasks, and the asynchronous sliding mode control law is designed by utilizing observation mode information to ensure that the mechanical arm systems meet better transient performance within a limited time interval.
In conclusion, how to establish a dynamic model of a multi-modal mechanical arm under an asynchronous switching mechanism, design a new integral sliding mode switching surface by using available information of a system switching mode, and further design an asynchronous sliding mode control law of the mechanical arm to ensure the transient performance of the mechanical arm is a key problem to be solved urgently, and the method has important theoretical research significance and application value.
Disclosure of Invention
In order to solve the technical problem, the invention provides an asynchronous sliding mode control method of a multi-mode mechanical arm. In order to achieve the purpose, the technical scheme of the invention is as follows:
an asynchronous sliding mode control method of a multi-mode mechanical arm comprises the following steps:
constructing a hidden semi-Markov switching system of the multi-modal mechanical arm based on the influence of hidden modes and uncertain parameters on the multi-modal mechanical arm;
designing an integral sliding mode switching surface based on the state and modal information of a hidden half Markov switching system;
carrying out stability analysis on sliding mode dynamics in an integral sliding mode switching surface;
calculating parameters of a sliding mode controller which meet the target of the finite time stability of the hidden semi-Markov switching system based on a linear matrix inequality obtained from a stability analysis result;
designing an asynchronous sliding mode control law of observation mode dependence based on sliding mode controller parameters;
and carrying out accessibility analysis on the sliding mode switching surface based on an asynchronous sliding mode control law of design observation mode dependence.
Preferably, the integral sliding mode switching surface:
Figure BDA0003121571230000021
wherein F α As parameters of the sliding mode controller,
when the state reaches the switching plane, according to s (t) 0,
Figure BDA0003121571230000022
obtaining an equivalent sliding mode controller u eq (t) is:
Figure BDA0003121571230000023
further, it can be seen that the dynamic trajectory of the sliding phase system is:
Figure BDA0003121571230000024
y(t)＝G ρ z(t),
wherein
Figure BDA0003121571230000025
Preferably, the stability analysis of the sliding mode dynamics in the integral sliding mode switching surface specifically includes the following steps: for each original system modality
Figure BDA0003121571230000026
And observing the modality α ∈ S, in a matrix P ρ ＞0,Z＞0,V ρα ,X α ,Y α Sum positive scalar quantity
Figure BDA0003121571230000027
γ,θ,τ βρ (β ═ 1,2, …,5) as an unknown, the following linear matrix inequality was solved:
Figure BDA0003121571230000031
Figure BDA0003121571230000032
Π 5 ＜0,
Figure BDA0003121571230000033
wherein
Figure BDA0003121571230000034
Figure BDA0003121571230000035
Figure BDA0003121571230000036
Figure BDA0003121571230000037
Figure BDA0003121571230000038
Figure BDA0003121571230000039
Figure BDA00031215712300000310
Figure BDA00031215712300000311
Preferably, the formula of the sliding mode controller parameters is as follows:
Figure BDA00031215712300000312
preferably, the formula of the discrete-time sliding-mode control law is as follows:
Figure BDA00031215712300000313
wherein
Figure BDA00031215712300000314
Figure BDA00031215712300000315
Figure BDA00031215712300000316
Where T is a predetermined finite time.
Preferably, the reachability analysis of the sliding mode switching surface is performed based on the asynchronous sliding mode control law depending on the design observation mode, and specifically includes the following steps:
selecting a Lyapunov function as
Figure BDA00031215712300000317
The following can be obtained:
Figure BDA00031215712300000318
wherein
Figure BDA0003121571230000041
Further, the method can be obtained as follows:
Figure BDA0003121571230000042
wherein T is * Is the instant the system state trajectory is driven to the sliding mode switching surface.
Based on the technical scheme, the invention has the beneficial effects that: the invention designs a sliding mode switching surface related to a hidden half Markov model, and further provides a corresponding asynchronous sliding mode control method.
Drawings
The following describes embodiments of the present invention in further detail with reference to the accompanying drawings.
FIG. 1 is a flow chart of an asynchronous sliding mode control method of a multi-modal manipulator in one embodiment;
FIG. 2 is a graph of experimental simulation results for a sliding mode switching surface in one embodiment;
FIG. 3 is a graph of experimental simulation results for a velocity constraint signal in one embodiment.
Detailed Description
The technical solution in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.
Based on the technical background, the multi-modal mechanical arm under the asynchronous switching mechanism can be modeled as a hidden semi-Markov switching system, and has strong mixing performance and uncertainty. Aiming at the characteristic of asynchronous switching, the invention designs a sliding mode switching surface related to a hidden semi-Markov model, and further provides a corresponding asynchronous sliding mode control method. Aiming at strong mixing, the invention designs a modal-dependent sliding mode control method of a multi-mode mechanical arm based on a statistical theory and a stochastic system theory, and the specific implementation mode of the method is described by combining an implementation flow diagram shown in fig. 1:
step (1): according to the difference of the actual load mass and the moment of inertia of the mechanical arm in the actual industrial process, a control system of the mechanical arm is divided into three modes, wherein the mode 1 corresponds to the mode of the mechanical arm when the load mass is 0.5kg and the moment of inertia is 0.15 N.m, the mode 2 corresponds to the mode of the mechanical arm when the load mass is 1.0kg and the moment of inertia is 0.25 N.m, and the mode 3 corresponds to the mode of the mechanical arm when the load mass is 1.5kg and the moment of inertia is 0.5 N.m. The mechanical arm model is as follows:
Figure BDA0003121571230000043
where ψ (t) and u (t) denote the angular position of the arm and the control input, M (δ) t ) To load mass, J (delta) t ) The moment of inertia is G, the gravity acceleration is L, the length of the mechanical arm is L, and the viscous friction coefficient is W.
Definition of
Figure BDA0003121571230000051
z 1 (t)＝ψ(t),
Figure BDA0003121571230000052
The obtained hidden half Markov switching system model of the multi-modal mechanical arm is as follows:
Figure BDA0003121571230000053
y(t)＝G(δ t )z(t),
z(0)＝z 0 ,
wherein z (t), u (t), y (t) represent status, input and output, respectively, z 0 Indicating the initial state of the system. Parameter uncertainty Δ A (δ) t T) satisfies Δ A (δ) t ,t)＝M(δ t )H(δ t ,t)N(δ t ) Where M (δ) t ),N(δ t ) Being a matrix of appropriate dimensions, H T (δ t ,t)H(δ t T) is less than or equal to I; the nonlinear term f (δ) t Z (t), t) satisfies | | | f (δ) t ,z(t),t)||≤ψ(δ t ) H | | z (t) | |, where ψ (δ) t ) Is a known positive scalar quantity. System matrix B (delta) t ) Is column full rank. E (delta) t ),G(δ t )，A(δ t )，B(δ t ) For a matrix of appropriate dimensions, in particular:
G(δ t )＝[0 1],E(δ t )＝[0 0.2] T ,ΔA(δ t ,t)＝M(δ t )H(δ t ,t)N(δ t ),
wherein M (delta) t )＝[0 0.1] T ,N(δ t )＝[0.1 0.1],H(δ t )(t)＝sin(t)，
Figure BDA0003121571230000054
The interference vector iota (t) satisfies
Figure BDA0003121571230000055
When not in lineProperty term f (δ) t )＝0.1sin(x 1 )x 1 The interference vector iota (t) satisfies
Figure BDA0003121571230000056
The parameter is given as G-9.81 m/s 2 ,L＝0.5m,W＝2。
Definition (δ) t ,φ t ) The hidden semi-Markov process is described as follows: stochastic process
Figure BDA0003121571230000057
Is represented in a set
Figure BDA0003121571230000058
A homogeneous semi-Markov chain of continuous time discrete states of the medium value. Definition of
Figure BDA0003121571230000059
Is the index value of the l-th switching modality;
Figure BDA00031215712300000510
representing the running time from the (l-1) th switching instant to the l-th switching instant. Corresponding modal transition probability
Figure BDA00031215712300000511
Wherein λ represents the residence time;
Figure BDA00031215712300000512
χ ρκ (λ) ≧ 0 represents a transition probability from the switching mode ρ to the switching mode κ (ρ ≠ κ) and satisfies
Figure BDA00031215712300000513
Corresponding mode delta t The transfer rate matrix is
Figure BDA0003121571230000061
Obeying a Weber distribution in view of residence time
Figure BDA0003121571230000062
When rho is 1, pi is 1, iota is 2, g 1 (λ)＝2λexp[-λ 2 ](ii) a When rho is 2, pi is 1, iota is 3, g 2 (λ)＝3λ 2 exp[-λ 3 ](ii) a When ρ is 3, pi is 1, l is 4, g 2 (λ)＝4λ 3 exp[-λ 4 ]. Computing a transition probability matrix
Figure BDA0003121571230000063
Similarly, is calculated to obtain
Figure BDA0003121571230000064
Because the hidden mode exists, the original system mode is not directly available, so the lower layer observation mode phi is introduced t And e, observing the upper original system mode by {1,2 and 3 }. Mode delta from original system t The associated known conditional probability is defined as
Figure BDA0003121571230000065
Wherein
Figure BDA0003121571230000066
Mode delta from original system t The related known conditional probabilities are as follows
Figure BDA0003121571230000067
Step (2): based on available information of a system mode in the step (1), designing a novel integral sliding mode switching surface:
Figure BDA0003121571230000068
wherein F α Are sliding mode controller parameters.
When the state reaches the switching plane, according to s (t) 0,
Figure BDA0003121571230000069
obtaining an equivalent sliding mode controller u eq (t) is:
Figure BDA00031215712300000610
further, the dynamic trajectory of the sliding phase system is as follows:
Figure BDA00031215712300000611
y(t)＝G ρ z(t),
wherein
Figure BDA00031215712300000612
And (3): further, stability analysis is carried out on the sliding mode dynamic state in the step (2). For each original system mode rho E {1,2,3} and observation mode alpha E {1,2,3}, taking matrix P ρ ＞0,Z＞0,V ρα ,X α ,Y α Sum positive scalar quantity
Figure BDA00031215712300000613
γ,θ,τ βρ (β ═ 1,2, …,5) as unknowns, solve the following linear matrix inequality:
Figure BDA0003121571230000071
Figure BDA0003121571230000072
Π 5 ＜0,
Figure BDA0003121571230000073
wherein
Figure BDA0003121571230000074
Figure BDA0003121571230000075
Figure BDA0003121571230000076
Figure BDA0003121571230000077
Figure BDA0003121571230000078
Figure BDA0003121571230000079
Figure BDA00031215712300000710
Figure BDA00031215712300000711
And (4): for the
Figure BDA00031215712300000712
ι＝0.4,T＝15,
Figure BDA00031215712300000713
θ＝0.1,γ＝0.4,
Figure BDA00031215712300000714
Ψ ═ I. Based on the four conditions of the conditional probability matrix in the step (1) and the matrix X obtained in the step (3) α ,Y α Further calculating sliding mode controller parameters F meeting the system finite time stability target α ：
Case 1 (full sync):
Figure BDA00031215712300000715
case 2 (partially asynchronous):
Figure BDA00031215712300000716
case 3 (partially asynchronous):
Figure BDA00031215712300000717
case 4 (fully asynchronous):
Figure BDA00031215712300000718
and (5): for case 4 (fully asynchronous). Sliding mode controller parameter F obtained based on step (4) α Designing an asynchronous sliding mode control law on which an observation mode depends:
Figure BDA0003121571230000081
wherein
Figure BDA0003121571230000082
φ＝0.05,υ＝0.571,ξ＝1。
And (6): performing reachability analysis based on the asynchronous sliding mode control law designed in the step (5):
selecting the Lyapunov function as
Figure BDA0003121571230000083
The following can be obtained:
Figure BDA0003121571230000084
wherein
Figure BDA0003121571230000085
Further, the method can be obtained as follows:
Figure BDA0003121571230000086
wherein T is * Is the instant the system state trajectory is driven to the sliding mode switching surface.
The analysis shows that the integral sliding mode switching surface designed in the step (2) is reachable within a limited time and has a T reaching moment * . For clear demonstration of the limited time accessibility in step (6) and the sliding mode control method in step (5), partial data simulation results are plotted in fig. 2,3, where the initial state z is 0 Is selected from [ 0.1-0.5 ]] T . In fig. 2 and 3, the horizontal axis represents time. The vertical axis of fig. 2 represents a sliding mode switching surface s (t), which can still arrive within a specified limited time under the influence of parameter uncertainty, nonlinearity and asynchronous switching mechanism. FIG. 3 shows the output speed constraint signal y on the vertical axis T (t) y (t) outputting a speed constraint signal y under the influence of parameter uncertainty, nonlinearity and asynchronous switching mechanism T (t) y (t) is still within the specified constraints. According to the method disclosed by the invention, the influence of parameter uncertainty and nonlinearity on the mechanical arm can be effectively inhibited, the asynchronous sliding mode control problem of the multi-mode mechanical arm is solved, and the task execution precision of the mechanical arm is improved.
The foregoing is only a preferred embodiment of the asynchronous sliding mode control method for a multi-modal manipulator disclosed in the present invention, and is not intended to limit the scope of the embodiments of the present disclosure. Any modification, equivalent replacement, improvement and the like made within the spirit and principle of the embodiments of the present disclosure should be included in the protection scope of the embodiments of the present disclosure.
It should also be noted that the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising an … …" does not exclude the presence of other like elements in a process, method, article, or apparatus that comprises the element.
The embodiments in the present specification are all described in a progressive manner, and the same and similar parts among the embodiments can be referred to each other, and each embodiment focuses on the differences from the other embodiments. In particular, for the system embodiment, since it is substantially similar to the method embodiment, the description is simple, and for the relevant points, reference may be made to the partial description of the method embodiment.

Claims (2)

1. An asynchronous sliding mode control method of a multi-mode mechanical arm is characterized by comprising the following steps:
step S1, constructing a nonlinear hidden semi-Markov switching system of the multi-modal mechanical arm based on the influence of hidden modes and uncertain parameters on the multi-modal mechanical arm;
step S2, designing an integral sliding mode switching surface based on the state and mode information of the hidden semi-markov switching system of step S1:
Figure FDA0003754947050000011
wherein F α Design parameters for sliding mode surfaces, z (t) System State, A ρ Is a system matrix, B ρ Is a control matrix;
when the state reaches the switching plane, according to s (t) 0,
Figure FDA0003754947050000012
to obtainEquivalent sliding mode controller u eq (k) Comprises the following steps:
Figure FDA0003754947050000013
further, the dynamic trajectory of the sliding phase system is as follows:
Figure FDA0003754947050000014
y(t)＝G ρ z(t),
wherein
Figure FDA0003754947050000015
G ρ To be the output matrix, the output matrix is,
step S3, performing stability analysis on the sliding mode dynamic state in the step S2, and performing stability analysis on each original system mode
Figure FDA0003754947050000016
And observation modality
Figure FDA0003754947050000017
In a matrix P ρ ＞0,Z＞0,V ρα ,X α ,Y α Sum positive scalar quantity
Figure FDA0003754947050000018
As unknowns, the following linear matrix inequality is solved:
Figure FDA0003754947050000019
Figure FDA00037549470500000110
Π 5 ＜0,
Figure FDA00037549470500000111
wherein
Figure FDA00037549470500000112
Step S4, based on the matrix parameter X obtained from the stability analysis result α ,Y α Calculating the parameters of sliding mode controller satisfying the target of the finite time stability of the hidden semi-Markov switching system
Figure FDA0003754947050000021
Step S5, designing an asynchronous sliding mode control law of observation mode dependence based on the sliding mode controller parameters obtained in the step S4:
Figure FDA0003754947050000022
wherein
Figure FDA0003754947050000023
Figure FDA0003754947050000024
Figure FDA0003754947050000025
Step S6, based on the asynchronous sliding mode control law designed in step S5, performing accessibility analysis on the sliding mode switching surface:
selecting the Lyapunov function as
Figure FDA0003754947050000026
The following can be obtained:
Figure FDA0003754947050000027
wherein
Figure FDA0003754947050000028
Further, the method can be obtained as follows:
Figure FDA0003754947050000029
wherein T is * Is the instant the system state trajectory is driven to the slip form switching surface,
it can be analyzed that the integral sliding mode surface designed in step S2 is reachable in a limited time and the moment of arrival is T * 。
2. The asynchronous sliding-mode control method of the multi-mode mechanical arm according to claim 1, wherein the step S1 specifically includes the following steps: nonlinear hidden half Markov switching System:
Figure FDA00037549470500000210
y(t)＝G(δ t )z(t),
wherein when is delta t When rho, A (δ) t ) Is a system matrix, B (δ) t ) To control the matrix, E (delta) t ) As an input matrix, G (delta) t ) For the output matrix, the parameter uncertainty term Δ A (δ) t T) satisfies Δ A (δ) t ,t)＝M(δ t )H(δ t ,t)N(δ t ) In which H T (δ t ,t)H(δ t ,t)≤I；M(δ t ) For a known parameter matrix, N (δ) t ) Is a known parameter matrix; the nonlinear term f (δ) t Z (t), t) satisfies | | | f (δ) t ,z(t),t)||≤ψ(δ t ) - | z (t) | l, where ψ (δ) t ) Known positive scalars, z (t), u (t), y (t) respectively represent state, input and output, and the interference vector iota (t) of external input meets the requirement
Figure FDA00037549470500000211
Control matrix
Figure FDA00037549470500000212
Is of a column full rank and,
definition (δ) t ,φ t ) The hidden semi-Markov process is described as follows: stochastic process
Figure FDA0003754947050000031
Is represented in a set
Figure FDA0003754947050000032
Homogeneous semi-Markov chain of discrete state in continuous time with medium value, definition
Figure FDA0003754947050000033
Is the index value of the l-th switching modality;
Figure FDA0003754947050000034
representing the running time from the (l-1) th switching instant to the l-th switching instant, the corresponding modal transition probability is:
Figure FDA0003754947050000035
wherein λ represents the residence time;
Figure FDA0003754947050000036
χ ρκ (λ) ≧ 0 represents a transition probability from the switching mode ρ to the switching mode κ (ρ ≠ κ) and satisfies
Figure FDA0003754947050000037
Because of the existence of hidden mode, the original system mode is not directly available, so the lower layer observation mode phi is introduced t E, observing an upper-layer original system mode by using the E as S ═ {1,2 t The associated known conditional probability is defined as
Figure FDA0003754947050000038
Wherein
Figure FDA0003754947050000039
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