CN113341724A - 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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CN113341724A
CN113341724A CN202110677946.1A CN202110677946A CN113341724A CN 113341724 A CN113341724 A CN 113341724A CN 202110677946 A CN202110677946 A CN 202110677946A CN 113341724 A CN113341724 A CN 113341724A
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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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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 carrying out accessibility analysis on the sliding mode switching surface based on an asynchronous sliding mode control law of design observation mode dependence. 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 known, the sliding mode control has stronger anti-interference performance, and the control precision of the mechanical arm system is improved to a certain extent. 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 margin in complex operations, so that human errors are effectively reduced, the success rate of the precision operations is improved, and the lives of patients are 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 FαAs 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 ueq(t) is:
Figure BDA0003121571230000023
further, it can be seen that the dynamic trajectory of the sliding phase system is:
Figure BDA0003121571230000024
y(t)＝Gρz(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 Pρ＞0,Z＞0,Vρα,Xα,YαSum positive scalar quantity
Figure BDA0003121571230000027
γ,θ,τβρ(β ═ 1,2, …,5) as an unknown, the following linear matrix inequality was solved:
Figure BDA0003121571230000031
Figure BDA0003121571230000032
Π5＜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 the 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.
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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 according to an 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 clutter 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 acceleration of gravity is L, the length of the mechanical arm is L, and the viscous friction coefficient is W.
Definition of
Figure BDA0003121571230000051
z1(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(δt)z(t),
z(0)＝z0,
wherein z (t), u (t), y (t) represent status, input and output, respectively, z0Indicating the initial state of the system. Parameter uncertainty Δ A (δ)tT) satisfies Δ A (δ)t,t)＝M(δt)H(δt,t)N(δt) Where M (δ)t),N(δt) Being a matrix of appropriate dimensions, HT(δt,t)H(δtT) is less than or equal to I; the nonlinear term f (δ)tZ (t), t) satisfies | | | f (δ)t,z(t),t)||≤ψ(δt) - | z (t) | l, where ψ (δ)t) Is a known positive scalar quantity. System matrix B (delta)t) Is column full rank. E (delta)t),G(δt)，A(δt)，B(δt) For a matrix of appropriate dimensions, in particular:
G(δt)＝[0 1],E(δt)＝[0 0.2]T,ΔA(δt,t)＝M(δt)H(δt,t)N(δt),
wherein M (delta)t)＝[0 0.1]T,N(δt)＝[0.1 0.1],H(δt)(t)＝sin(t)，
Figure BDA0003121571230000054
The interference vector iota (t) satisfies
Figure BDA0003121571230000055
When the non-linear term f (δ)t)＝0.1sin(x1)x1The interference vector iota (t) satisfies
Figure BDA0003121571230000056
The parameter is given as G-9.81 m/s2,L＝0.5m,W＝2。
Definition (δ)t,φt) 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
χρκ(λ) ≧ 0 represents a transition probability from the switching mode ρ to the switching mode κ (ρ ≠ κ) and satisfies
Figure BDA00031215712300000513
Corresponding mode deltatThe 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, g1(λ)＝2λexp[-λ2](ii) a When rho is 2, pi is 1, iota is 3, g2(λ)＝3λ2exp[-λ3](ii) a When ρ is 3, pi is 1, l is 4, g2(λ)＝4λ3exp[-λ4]. Computing a transition probability matrix
Figure BDA0003121571230000063
Similarly, is calculated to obtain
Figure BDA0003121571230000064
Because of the existence of hidden mode, the original system mode is not directly available, so the lower layer observation mode phi is introducedte.S ═ {1,2,3} corresponding to the upper layer original system modeAnd (5) observing. Mode delta from original systemtThe associated known conditional probability is defined as
Figure BDA0003121571230000065
Wherein
Figure BDA0003121571230000066
Mode delta from original systemtThe related known conditional probabilities are as follows
Figure BDA0003121571230000067
Step (2): based on available information of system mode in the step (1), designing a novel integral sliding mode switching surface:
Figure BDA0003121571230000068
wherein FαAre sliding mode controller parameters.
When the state reaches the switching plane, according to s (t) 0,
Figure BDA0003121571230000069
obtaining an equivalent sliding mode controller ueq(t) is:
Figure BDA00031215712300000610
further, the dynamic trajectory of the sliding phase system is as follows:
Figure BDA00031215712300000611
y(t)＝Gρz(t),
wherein
Figure BDA00031215712300000612
And (3): further, stability analysis is carried out on the sliding mode dynamic state in the step (2). For theEach original system mode rho E {1,2,3} and observation mode alpha E {1,2,3} are expressed by a matrix Pρ＞0,Z＞0,Vρα,Xα,YαSum positive scalar quantity
Figure BDA00031215712300000613
γ,θ,τβρ(β ═ 1,2, …,5) as an unknown, the following linear matrix inequality was solved:
Figure BDA0003121571230000071
Figure BDA0003121571230000072
Π5＜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,YαFurther calculating sliding mode controller parameters F meeting the system finite time stability targetα：
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)αDesigning 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.
Analysis shows that the integral sliding mode switching surface designed in the step (2) is reachable within a limited time and has a T moment of arrival*. 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 is0Is selected from [ 0.1-0.5 ]]T. In fig. 2 and 3, the horizontal axis represents time. FIG. 2 shows the sliding mode switching surface s (t) at the vertical axis, and the parameters are notUnder the influence of deterministic, nonlinear and asynchronous switching mechanisms, the sliding mode switching surface s (t) can still arrive within a specified limited time. FIG. 3 shows the output speed constraint signal y on the vertical axisT(t) y (t) outputting a speed constraint signal y under the influence of parameter uncertainty, nonlinearity and asynchronous switching mechanismT(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 above description is only a preferred embodiment of the asynchronous sliding mode control method for the multi-mode 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 (6)

1. An asynchronous sliding mode control method of a multi-mode mechanical arm is characterized by comprising 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.
2. The asynchronous sliding mode control method of the multi-mode mechanical arm according to claim 1, wherein the integral sliding mode switching surface:
Figure FDA0003121571220000011
wherein FαAs parameters of the sliding mode controller,
when the state reaches the switching plane, according to s (t) 0,
Figure FDA0003121571220000012
obtaining an equivalent sliding mode controller ueq(t) is:
Figure FDA0003121571220000013
further, it can be seen that the dynamic trajectory of the sliding phase system is:
Figure FDA0003121571220000014
y(t)＝Gρz(t),
wherein
Figure FDA0003121571220000015
3. The asynchronous sliding mode control method of the multi-mode mechanical arm according to claim 1, wherein the stability analysis of the sliding mode dynamics in the integral sliding mode switching surface specifically comprises the following steps: for each original system modality
Figure FDA0003121571220000016
And observing the modality α ∈ S, in a matrix Pρ＞0,Z＞0,Vρα,Xα,YαSum positive scalar quantity
Figure FDA0003121571220000017
As unknowns, the following linear matrix inequality is solved:
Figure FDA0003121571220000018
Figure FDA0003121571220000019
∏5＜0,
Figure FDA00031215712200000110
wherein
Figure FDA0003121571220000021
Figure FDA0003121571220000022
Figure FDA0003121571220000023
Figure FDA0003121571220000024
Figure FDA0003121571220000025
Figure FDA0003121571220000026
Figure FDA0003121571220000027
Figure FDA0003121571220000028
4. The asynchronous sliding-mode control method of the multi-mode mechanical arm according to claim 1, wherein the formula of the parameters of the sliding-mode controller is as follows:
Figure FDA0003121571220000029
5. the asynchronous sliding-mode control method of the multi-mode mechanical arm according to claim 1, wherein the formula of the discrete-time sliding-mode control law is as follows:
Figure FDA00031215712200000210
wherein
Figure FDA00031215712200000211
Figure FDA00031215712200000212
Figure FDA00031215712200000213
Where T is a predetermined finite time.
6. The asynchronous sliding-mode control method of the multi-mode mechanical arm according to claim 1, wherein the sliding-mode switching surface accessibility analysis is performed based on an asynchronous sliding-mode control law of design observation mode dependence, and specifically comprises the following steps:
selecting the Lyapunov function as
Figure FDA00031215712200000214
The following can be obtained:
Figure FDA00031215712200000215
wherein
Figure FDA00031215712200000216
Further, the method can be obtained as follows:
Figure FDA0003121571220000031
wherein T is*Is the instant the system state trajectory is driven to the sliding mode switching surface.
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CN114055463B (en) * 2021-09-26 2023-04-18 曲阜师范大学 Fuzzy sliding mode control method of networked mechanical arm system

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