CN109551479B - Reconfigurable modular flexible mechanical arm trajectory tracking control method based on parameter optimization - Google Patents

Reconfigurable modular flexible mechanical arm trajectory tracking control method based on parameter optimization Download PDF

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CN109551479B
CN109551479B CN201811450310.8A CN201811450310A CN109551479B CN 109551479 B CN109551479 B CN 109551479B CN 201811450310 A CN201811450310 A CN 201811450310A CN 109551479 B CN109551479 B CN 109551479B
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CN109551479A (en
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张袅娜
韩宗志
李宗昊
孙建伟
秦喜文
杨瀛
张晓芳
呼薇
姜春霞
矫德强
张琪
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Changchun University of Technology
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
    • B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning

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Abstract

A reconfigurable modular flexible mechanical arm trajectory tracking control method based on parameter optimization relates to the field of reconfigurable modular flexible mechanical arm control, establishes a single-joint intelligent flexible mechanical arm system model, and utilizes the idea of redefining output to connect jointsThe linear combination of the motor rotation angle and the flexible modal variable is used as the output of the flexible mechanical arm system, and the single-joint flexible mechanical arm system is decomposed into an input-output subsystem and a zero-dynamic subsystem through input-output linearization. The invention adopts self-adaptive dynamic terminal sliding mode control to lead the input-output subsystem to track the expected reference track within limited time, and designs the parameter lambda by adopting NSGA-II algorithm0i，λ1iPerforming multi-objective optimization design to ensure that a zero dynamic subsystem of the flexible mechanical arm system is gradually stabilized near a balance point, thereby ensuring the tracking requirement of the whole flexible mechanical arm system on an expected track; the invention has better robustness to the nonlinear uncertainty of the system.

Description

Reconfigurable modular flexible mechanical arm trajectory tracking control method based on parameter optimization
Technical Field
The invention relates to the field of reconfigurable modular flexible mechanical arm control, in particular to a trajectory tracking control method of a reconfigurable modular flexible mechanical arm under the condition of uncertain parameters.
Background
According to the concept of module design and the subsystem decentralized control theory, the reconfigurable module mechanical arm is provided with a standard interface and a module, the configuration of the reconfigurable module mechanical arm can be changed according to different task requirements under different external environments and constraints, and a controller does not need to be redesigned. In addition, the joint of the reconfigurable module mechanical arm also comprises units such as communication, driving, control, transmission and the like, so that the reconfigurable mechanical arm has better adaptability to new working environments.
Many scholars have studied the dynamics and control method of the reconfigurable modular manipulator. Document 1 [ Dongbo, Liu Ke Ping, Liyuanchun, etc. ] the optimal control of the distributed reinforcement learning of the reconfigurable modular robot under the dynamic constraint [ J ]. the university of Jilin proceedings (engineering edition), 2014,44(5):1375 + 1384 ] provides an optimal control method of the distributed reinforcement learning of the reconfigurable modular robot under the external dynamic constraint based on the section-critical-identifier (ACI) and the RBF neural network, and solves the problem of the continuous time nonlinear optimal control of the modular robot system with strong coupling uncertainty. In document 2 [ zhao bo, li yuanchun, reconfigurable mechanical arm active dispersion fault-tolerant control based on signal reconstruction [ J ]. automated science and newspaper, 2014,40(9):1942 + 1950 ], a self-adaptive fuzzy dispersion control system is adopted to realize track tracking control of a module joint in a normal working mode aiming at the modularization attribute of the reconfigurable mechanical arm system. Document 3 [ Li Yuan-chun, Dong Bo. decentralized ADRC Control for Reconfigurable managers Based on-VGSTA-ESO of slicing Mode [ J ]. Information-An International inter-disciplineary Journal, 2012,15(6):2453-2465 ] proposes a Reconfigurable modular robot decentralized disturbance rejection Control method Based on VGSTA-ESO, which is used for identifying nonlinear items of subsystem models and subsystem cross-linking items, thereby realizing joint trajectory tracking Control. Document 4 [ zhao jie, wei yanghi, zaa helahu, etc. ] research on a novel distributed control method of a reconfigurable robot [ J ] academic press of harabin industry university, 2008,40(1):39-42 ] proposes a distributed control algorithm of the reconfigurable robot based on variable step pitch, and solves the problem of module motion jitter through differential rounding. Document 5 [ yan relay, guxin, liuyu bin, etc. ] a design and kinematic analysis of a modular robot arm [ J ]. proceedings of harbourne university of industry, 2015,47(1):20-25 ] has no general features for modular robot arm kinematics, and by converting the spatial configuration into a planar geometric relationship, the solution of the modular kinematics of the robot arm and the control of the robot arm in various configurations are realized. Document 6 [ Sun Relay Peng, Mengduan, DumingJun, etc. ] distributed adaptive iterative learning control [ J ] of multiple mechanical arms, Beijing university of aerospace, 2015,41(12): 2384-. In the document 7 [ wucour, duyan, zhangwei, etc. ] the mechanical arm distributed adaptive fuzzy control based on the extended state observer [ J ]. university of southeast university journal (natural science edition), 2012,42(z1): 192-.
Considering that the reconfigurable modular mechanical arm has increased joint and connecting rod flexibility effects in the motion process, the reconfigurable modular flexible mechanical arm deforms, so that the task execution precision is reduced, and therefore the reconfigurable modular flexible mechanical arm is controlled to be one of the hot spots of the research in the robot control field in a high-precision mode. At present, the research on flexible mechanical arms is very mature, but all the flexible mechanical arms are based on a fixed structural form, and when the structure of the mechanical arm is changed, the controller needs to be designed again; at present, research on reconfigurable mechanical arms is also achieved with certain research results, but the influence of the flexibility and the flexible mode of joints on the tracking precision of the system is less considered.
The sliding mode can be designed and is irrelevant to the parameters and the disturbance of an object, so that the sliding mode variable structure control has the advantages of insensitive parameter change and disturbance, no need of system online identification, simple realization and the like. The terminal sliding mode variable structure control improves the convergence characteristic of the system by purposefully introducing a nonlinear term into a sliding mode, so that the system state converges to a given track within a limited time, and the terminal sliding mode variable structure control has the advantages of quick response, high steady-state tracking precision and particular suitability for high-precision control.
Disclosure of Invention
The invention provides a reconfigurable modular flexible mechanical arm track tracking control method based on parameter optimization aiming at the problems that the flexible effect of joints and connecting rods of a reconfigurable modular flexible mechanical arm is increased in the motion process, so that the structure is deformed, the task execution precision is reduced, the reusability is low and the like, firstly, a mechanical arm dynamic model is described as a set of cross-linking subsystems, then, the system is decomposed into an input-output subsystem and a zero-dynamic subsystem by adopting a method for redefining system output, and an adaptive dynamic terminal sliding mode control strategy is provided for the input-output subsystem to adaptively identify the subsystem association items, the system uncertainty items and the interferences; the zero dynamic subsystem is approximately linearized at a balance point, the design parameters of the controller are subjected to multi-objective optimization by adopting an NSGA-II algorithm, so that the zero dynamic subsystem is gradually stabilized near the balance point, the reconfigurable modular flexible mechanical arm subsystem with strong coupling and uncertain parameters is enabled to gradually track an expected track, the method can change the configuration of the reconfigurable modular flexible mechanical arm subsystem according to different task requirements under different external environments and constraints, the controller does not need to be redesigned, and the reconfigurable flexible mechanical arm subsystem has strong robustness to uncertainty in the system. In addition, the optimization of the controller parameters is realized by considering a plurality of targets such as the requirements of the system state, the control and the end point state, and the like, and the redefined system output has more design parameters, so that the design parameter selection margin for ensuring the stability of the zero dynamic subsystem is increased, the system convergence speed is improved, and the tracking error is converged and bounded.
The technical scheme adopted by the invention for solving the technical problem is as follows:
the reconfigurable modular flexible mechanical arm trajectory tracking control method based on parameter optimization comprises the following steps:
establishing a dynamic model of an ith joint intelligent body of a reconfigurable modular flexible mechanical arm, taking a linear combination of a joint motor corner and a flexible modal variable as the output of a single-joint flexible mechanical arm system, and decomposing the system into an input-output subsystem and a zero-dynamic subsystem through input-output linearization;
step two, aiming at the input and output subsystem of the ith joint intelligent body, a system equation of the input and output subsystem is obtained according to the joint angle, the joint angle change rate and the flexible mode of the reconfigurable flexible mechanical arm detected by the signal acquisition and conditioning module and the dynamic model of the ith joint intelligent body obtained in the step one; under the conditions of interference and parameter uncertainty of the system, the difference value of the defined parameter and the nominal quantity represents the parameter uncertainty existing in the system to obtain a flexible mechanical arm system equation, and the tracking of the input and output subsystem state of the n-joint reconfigurable modular flexible mechanical arm on the expected reference track is realized;
step three, redefining the output of the ith joint agent to obtain an input-output subsystem equation; the flexible mechanical arm system is changed from a non-minimum phase system to a minimum phase system which is easy to control near an equilibrium point; when the specific control input enables the output of the input-output subsystem to be zero, the internal subsystem is a zero dynamic subsystem; approximately linearizing a zero dynamic subsystem of the ith joint intelligent agent at a balance point to ensure that the whole flexible mechanical arm system quickly tracks an expected reference track;
designing a self-adaptive dynamic terminal sliding mode control strategy aiming at an input/output subsystem of an ith joint intelligent body of the reconfigurable flexible mechanical arm to ensure that the state of the input/output subsystem tracks an expected reference track within a limited time;
step five, approximately linearizing the zero dynamic subsystem of the ith joint intelligent agent at the balance point by Ai(λ0i,λ1i) The characteristic value of the complex plane is strictly on the left half plane of the complex plane as a constraint condition, the comprehensive optimal target of the dynamic deviation of the system in the control process, the energy consumption and the system steady-state deviation at the end of control is ensured, and the design parameter lambda is subjected to the NSGA-II algorithm0i，λ1iAnd performing parameter optimization design, and ensuring that a zero dynamic subsystem of the ith intelligent joint subsystem of the flexible arm is asymptotically stable at a balance point by using an elite retention strategy and a diversity maintenance mechanism specific to the algorithm, so that the reconstructed flexible mechanical arm system is ensured to track an expected reference track.
The invention has the following beneficial effects:
1) aiming at the problem that the reconfigurable modular flexible mechanical arm with the characteristics of high flexibility, short design period, high reliability, low cost and easy maintenance has reduced task execution precision due to the fact that the structure is deformed due to the fact that the joint and connecting rod flexibility effect is increased in the track tracking process, the dynamic model of the reconfigurable modular flexible mechanical arm is described as a set of cross-linked joint intelligent body subsystems, and therefore modeling of a single-joint intelligent body flexible mechanical arm system is achieved. The method is characterized in that linear combination of the joint motor rotation angle and the flexible modal variable is used as the output of the flexible mechanical arm system by utilizing the idea of redefining the output, and the single-joint flexible mechanical arm system is decomposed into an input-output subsystem and a zero-dynamic subsystem through input-output linearization. The invention provides a method for enabling an input-output subsystem to track an expected reference track within a limited time by adopting self-adaptive dynamic terminal sliding mode control on the input-output subsystem. The invention approximately linearizes the zero dynamic subsystem at the balance point, and designs the parameter lambda by adopting the NSGA-II algorithm0i，λ1iAnd performing multi-objective optimization design to ensure that the zero dynamic subsystem of the flexible mechanical arm system is gradually stabilized near a balance point, thereby ensuring the tracking requirement of the whole flexible mechanical arm system on the expected track.
2) Compared with the prior art, the method has better robustness to the nonlinear uncertainty of the system, realizes the gradual tracking of the expected track of the reconfigurable modular flexible mechanical arm subsystem, and has convergent and bounded tracking error; aiming at different application fields, the flexible mechanical arm can be reconstructed at will without redesigning a controller, and the multiplexing rate is effectively improved; a new idea is provided for solving the problem of trajectory tracking of the modularized flexible mechanical arm. The method is simple and easy to realize, and is suitable for wide popularization and application.
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FIG. 1 is a schematic diagram of a trajectory tracking control method of a reconfigurable modular flexible mechanical arm based on parameter optimization.
FIG. 2 is a structural schematic diagram of the n-joint reconfigurable modular flexible mechanical arm.
FIG. 3 is a schematic diagram of the multi-objective parameter optimization process based on the NSGA-II algorithm.
Detailed Description
The invention is described in further detail below with reference to the figures and specific examples.
As shown in fig. 1, the reconfigurable modular flexible manipulator trajectory tracking control method based on parameter optimization is implemented by system output newly defined by an i (i is a natural number, i is 1,2, …, n) th joint intelligent body model, an i th joint intelligent body input/output subsystem, an i th joint intelligent body zero dynamic subsystem, an i th joint intelligent body controller parameter selection, signal acquisition and conditioning, and an i th joint intelligent body expected trajectory and an i th joint intelligent body weight.
The reconfigurable modular flexible mechanical arm considers each joint as an intelligent subsystem, and factors such as parameter uncertainty, external interference and the like existing in the system are considered to establish an ith joint intelligent model. And the ith joint intelligent agent is decomposed into an ith joint intelligent agent input-output subsystem and an ith joint intelligent agent zero dynamic subsystem by redefining the output of the ith joint intelligent agent into the linear combination of a joint corner and a flexible mode.
As shown in fig. 2, the input/output subsystem of the ith joint intelligent controller adopts adaptive dynamic terminal sliding mode control, and the input is the output theta of the signal acquisition and conditioning modulei,qi,
Figure GDA0003153730940000051
Ith joint intelligent controller parameter selection lambda0i,λ1iAnd selecting according to the conditions meeting the progressive stability of the zero dynamic subsystem and the quick convergence of the input and output subsystem.
The parameter selection module of the ith joint intelligent controller firstly linearizes the zero dynamic subsystem at the balance point, establishes a state equation, adopts the NSGA-II algorithm and ensures the matrix Ai(λ0i,λ1i) Under the precondition that all the characteristic values of (A) are negative values, selecting proper lambda0iAnd λ1iAnd the value ensures that the dynamic deviation of the system in the control process, the energy consumption and the system steady-state deviation at the end of control are comprehensively optimal.
As shown in the flow chart of the NSGA-II algorithm-based controller parameter optimization process of FIG. 3, the design parameter λ is0i，λ1iAnd performing optimization design, and ensuring that a zero dynamic subsystem of the ith intelligent joint subsystem of the flexible arm is asymptotically stable at a balance point by using an elite retention strategy and a diversity maintenance mechanism specific to the algorithm, so that the reconstructed flexible mechanical arm system is ensured to track an expected reference track.
The track tracking control method based on the parameter optimization reconfigurable modular flexible mechanical arm comprises the following specific implementation steps:
1) intelligent body model for ith joint of reconfigurable modular flexible mechanical arm
Considering each joint agent of the reconfigurable mechanical arm as a subsystem, the dynamical model of the ith joint agent subsystem can be described as:
Figure GDA0003153730940000061
in the formula, thetai(t) is a motor rotation angle vector of the ith joint agent; q. q.si(t) Flexible Modal vector of the ith Joint agent, qi＝[qi1,…,qir]T；ui(t) is the control torque vector of the ith joint agent; f. offi(θi,qi) And fri(θi,qi) Items of which the ith joint agent is influenced by gravity, Copenforces and centrifugal forces respectively;
Figure GDA0003153730940000062
and
Figure GDA0003153730940000063
respectively positive definite damping matrix for ith joint intelligent agent, Ki(qi) A stiffness matrix is positively determined for the ith joint agent. r is of flexible modeThe number of the first and second groups is,
Figure GDA0003153730940000064
respectively represent thetaiAnd q isiThe second derivative and the first derivative. Mi＝[MriMrfi；Mfri Mfi]Is the positive definite inertia matrix of the ith joint agent. C1i、C2iThe association of the ith joint agent with other agents.
Figure GDA0003153730940000065
Figure GDA0003153730940000066
In the formula, Mrij、Mrfij、Mfrij、MfijAre respectively MrMrf；Mfr MfThe ijth component of (a). n is the number of joints included in the reconfigurable modular flexible mechanical arm, and j is 1,2, …, n.
When there is uncertainty in the system, assume parameter Mi、fri、E1i、ffi、E2i、Ki、C1i、C2iThe nominal amounts of (A) are respectively: mni、frni、E1ni、ffni、E2ni、Kni、C1ni、C2niDefining: Δ Mi＝Mi-Mni，Δfri＝fri-frni，ΔE1i＝E1i-E1ni，Δffi＝ffi-ffni，ΔE2i＝E2i-E2ni，ΔKi＝Ki-Kni，ΔC1i＝C1i-C1ni，ΔC2i＝C2i-C2niRepresenting the parameter uncertainty present in the system. The ith joint agent of the reconfigurable flexible mechanical arm system can be rewritten into the following form:
Figure GDA0003153730940000071
in the formula (I), the compound is shown in the specification,
Mni＝[MrniMrfni；Mfrni Mfni]
Figure GDA0003153730940000072
in the formula (2), the model of each joint agent of the reconfigurable modular flexible manipulator can be obtained by setting i to 1,2, …, n.
2) I-th joint intelligent agent input and output subsystem
Redefining the output z (t) of the reconfigurable modular flexible robotic arm system as follows
zi＝λ0iθi+λ1iqi (3)
In the formula, λ0iAnd λ1iTo design the parameter, λ1iIs a matrix of dimension 1 × r.
Defining:
Figure GDA0003153730940000073
order to
di(λ0i,λ1i,xi,ui)＝λ0i(Ni11d1i+Ni12d2i)+λ1i(Ni21d1i+Ni22d2i)
di(λ0i,λ1i,xi,ui)＝λ0i(Ni11d1i+Ni12d2i)+λ1i(Ni21d1i+Ni22d2i)
ci(λ0i,λ1i,xi,ui)＝(λ0iNi11+λ1iNi21)C1i+(λ0iNi12+λ1iNi22)C2i
Figure GDA0003153730940000074
βi(λ0i,λ1i,xi)＝λ0iNi11(θ,q)+λ1iNi21(θ,q)
The input-output subsystem of the system formula (1) is obtained as follows
Figure GDA0003153730940000075
In the formula, betai(λ0i,λ1i,xi) It is reversible.
3) Zero-dynamic subsystem of ith joint intelligent agent
When a particular control input ui(t) making the input-output subsystem output zero:
Figure GDA0003153730940000076
the substitution of formula (2) to zero dynamic subsystems is as follows
Figure GDA0003153730940000081
Therefore, the ith joint intelligent system is decomposed into an input-output subsystem and a zero-dynamic subsystem through input-output linearization.
4) Ith joint intelligent controller
For the ith joint agent input-output subsystem, order
Figure GDA0003153730940000082
Order to
Figure GDA0003153730940000083
Is ciIs determined by the estimated value of (c),
Figure GDA0003153730940000084
respectively subsystem association C1i、C2iAn estimate of (d). Let ei1＝zi1-zid，
Figure GDA0003153730940000085
ei＝ei1+ei2，
Figure GDA0003153730940000086
Figure GDA0003153730940000087
zidIs the desired output trajectory for the subsystem. Satisfy the requirement of
Figure GDA0003153730940000088
Is bounded, an
Figure GDA0003153730940000089
L>0。uiAnd (t) is a control torque vector of the ith joint agent.
The following sliding modes are selected:
Figure GDA00031537309400000810
in the formula, τ1Design parameter matrix for sliding mode, piAnd q isiAre all positive odd numbers, satisfy pi>qi。
Design of
Figure GDA00031537309400000811
The adaptive change rate of (2) is:
Figure GDA00031537309400000812
let vi＝βiui，
Figure GDA00031537309400000813
Sliding mode control strategy viThe design is as follows:
Figure GDA00031537309400000814
in the formula (I), the compound is shown in the specification,
Figure GDA00031537309400000815
the adaptive change rate is designed as follows:
Figure GDA00031537309400000816
buffeting items of the sliding mode control strategy of the self-adaptive dynamic terminal are added to the control vi＝βiuiThe derivative of the sliding mode switching value is obtained, and therefore buffeting caused by the sliding mode switching process can be effectively reduced.
When the reconstruction mechanical arm is assembled by the n joint agents, the track error e of the ith joint agent can be input into and output from the subsystemiThe modification is as follows:
Figure GDA00031537309400000817
in the formula, ai(i-1)、ai(i+1)The correlation coefficients of the ith joint agent, the (i-1) th joint agent and the (i +1) th joint agent are respectively.
And (3) the formula (11) is carried into the formulas (8) to (10), so that the output of the input and output subsystem of the n-joint reconfigurable modular flexible mechanical arm can realize track tracking control, and a reconfigurable modular flexible mechanical arm track tracking controller does not need to be redesigned.
5) Ith joint intelligent controller parameter selection
At zero dynamic sonBalance point x of the systemiThe zero dynamics subsystem is linearized at 0. Define Ω1Is xiNeighborhood of 0, in Ω1The matrix N is formed on the domainiAt xiExpanding the position of 0 according to Taylor series to obtain a constant value matrix Ni0And xiHigher order term f ofhi(x) Form of sum
Figure GDA0003153730940000091
Figure GDA0003153730940000092
Reanalysis ffi(θi,qi) It can be found to be the state variable x onlyiIs a higher order term of (i.e. has
Figure GDA0003153730940000093
Order to
Ai(λ0i,λ1i)＝[0,I；–Pi0ki,–Pi0E2i] (13)
In the formula, Pi0＝Ni220-Ni210(λ0iNi110+λ1iNi210)-1(λ0iNi120+λ1iNi220)
The zero dynamics subsystem can be written as follows:
Figure GDA0003153730940000094
in the formula, GΔi＝–Pi0(fhi+C2i+d2i)。
Suppose there is | | f near zerohi||≤μ3，||C2i||<μ4，||d2i||<μ5Then, then
||GΔι||＝||fhi+C2i+d2i||≤(μ3+μ4+μ5)||-Pi0||
Let constant ε ═ μ3+μ4+μ5)||-Pi0| order
Figure GDA0003153730940000095
Gi＝(0,GΔi)T，GiSatisfy | | Gi||＝||GΔiIf | | < epsilon, then
Figure GDA0003153730940000096
It can be seen that λ is properly selected0iAnd λ1iTo ensure Ai(λ0i,λ1i) The eigenvalues of (a) are strictly in the left half plane of the complex plane, so that the zero-dynamics subsystem converges progressively.
Known by an input-output subsystem self-adaptive dynamic terminal sliding mode controller, a design parameter lambda0i，λ1iAt the same time for design parameters in the input-output subsystem controller, hence lambda0iAnd λ1iThe convergence speed of the zero dynamic subsystem state and the stability of the whole system are related. The invention adopts NSGA-II algorithm to carry out controller parameter lambda0i，λ1iThe multi-objective optimization design is characterized in that the NSGA-II algorithm evolution process is shown as the attached figure 3, and the process is as follows:
(1) initializing a population: parameter lambda0i，λ1iAdopting real number coding to randomly generate an initial parent population P (k) with the population scale of N, wherein k is a genetic algebra;
(2) fitness function: considering the requirements of the flexible mechanical arm on state, control and end point state in the control process, the method selects the parameter lambda0i，λ1iThe optimization target of (2) is as follows, and the dynamic deviation of the system in the control process, the energy consumption and the system steady-state deviation at the end of the control are ensured to be comprehensively optimal.
Figure GDA0003153730940000101
And the constraint conditions are satisfied:
Figure GDA0003153730940000102
|ui(t)|≤mi，miis uiUpper bound of (2)
In the formula (I), the compound is shown in the specification,
Figure GDA0003153730940000103
Qiis a semi-positive definite symmetric array, RiIs a positive constant.
(3) Non-dominated ranking hierarchy and crowding distance calculation: the population is first stratified according to the individual's non-inferior solution level. And then comparing the information crowding degrees of the individuals by adopting a two-dimensional information cyclic crowding mechanism containing an order-magnitude threshold, and deleting the individuals with the minimum crowding degrees until the number of the individuals in the population meets the requirement.
The calculation formula of the information congestion degree is as follows:
Is＝αs,1Ls,1+αs,2Ls,2
in the formula (I), the compound is shown in the specification,
Ls,1＝(|ps-1|+|ps|)/2,s＝2,3,...,h-1；Ls,2＝(|ps|+|ps+1|)/2,s＝2,3,...,h-1；
Figure GDA0003153730940000104
wherein h is the number of population individuals, and the position of the s-th individual is Ps＝(ps1,ps2,…,psm) M is the number of objective functions, m is 3, k is 1,2, 3.
(4) Genetic operator: the method comprises a selection operator, a cross operator, a mutation operator and a chaotic insertion operator. The selection operator adopts a race-round selection operator, namely 2 individuals are randomly selected, and if the non-dominated sorting sequence numbers are different, the individuals with small sequence numbers (high grade) are selected; if the serial numbers are the same, then the individual with less congestion around is selected. Normal distribution crossover operators are adopted in the crossover operation, and polynomial mutation operators are adopted in the mutation operation; as the chaotic motion can traverse all the states of the space, the small population generated by the chaotic motion has diversity, can supplement the missing effective genes in the evolution process, reserve 10 percent of the genes with higher fitness after genetic operation, and adopt chaotic insertion operators to update the rest 90 percent of the genes in the population, thereby avoiding the evolution stagnation caused by precocity.
(5) An elite strategy. The elite strategy is to keep the good individuals in the parent directly into the offspring, which is a necessary condition for the genetic algorithm to converge with probability 1. The method comprises the following steps: synthesizing all the parents P (k) and the child Q (k) into a uniform population R (k) ═ P (k) · U (q) (k), wherein the number of the individuals of R (k) is 2N; secondly, the population R (k) is sorted quickly and non-dominantly, the local crowding distance of each individual is calculated, the individuals are selected one by one according to the level until the number of the individuals reaches N, a new parent population P (k +1) is formed, and a new round of selection, crossing and variation is started on the basis to form a new child population Q (k + 1).

Claims (4)

1. The reconfigurable modular flexible mechanical arm trajectory tracking control method based on parameter optimization is characterized by comprising the following steps:
establishing a dynamic model of an ith joint intelligent body of a reconfigurable modular flexible mechanical arm, taking a linear combination of a joint motor corner and a flexible modal variable as the output of a single-joint flexible mechanical arm system, and decomposing the system into an input-output subsystem and a zero-dynamic subsystem through input-output linearization;
step two, aiming at the input and output subsystem of the ith joint intelligent body, a system equation of the input and output subsystem is obtained according to the joint angle, the joint angle change rate and the flexible mode of the reconfigurable flexible mechanical arm detected by the signal acquisition and conditioning module and the dynamic model of the ith joint intelligent body obtained in the step one; under the conditions that interference exists in the system and the parameters are uncertain, the difference value between the defined parameters and the nominal quantity represents the parameter uncertainty existing in the system, and a flexible mechanical arm system equation is obtained; the tracking of the input and output subsystem state of the n-joint reconfigurable modular flexible mechanical arm on the expected reference track is realized;
step three, redefining the output of the ith joint agent to obtain an input-output subsystem equation; the flexible mechanical arm system is changed from a non-minimum phase system to a minimum phase system which is easy to control near an equilibrium point; when the specific control input enables the output of the input-output subsystem to be zero, the internal subsystem is a zero dynamic subsystem; approximately linearizing a zero dynamic subsystem of the ith joint intelligent agent at a balance point to ensure that the whole flexible mechanical arm system quickly tracks an expected reference track;
designing a self-adaptive dynamic terminal sliding mode control strategy aiming at an input/output subsystem of an ith joint intelligent body of the reconfigurable flexible mechanical arm to ensure that the state of the input/output subsystem tracks an expected reference track within a limited time;
step five, approximately linearizing the zero dynamic subsystem of the ith joint intelligent agent at the balance point by Ai(λ0i,λ1i) The characteristic value of the complex plane is strictly on the left half plane of the complex plane as a constraint condition, the comprehensive optimal target of the dynamic deviation of the system in the control process, the energy consumption and the system steady-state deviation at the end of control is ensured, and the design parameter lambda is subjected to the NSGA-II algorithm0i，λ1iAnd performing parameter optimization design, and ensuring that a zero dynamic subsystem of the ith intelligent joint subsystem of the flexible arm is asymptotically stable at a balance point by using an elite retention strategy and a diversity maintenance mechanism specific to the algorithm, so that the reconstructed flexible mechanical arm system is ensured to track an expected reference track.
2. The method for controlling trajectory tracking of the reconfigurable modular flexible mechanical arm based on parameter optimization according to claim 1, wherein the process of establishing a dynamic model of the ith joint intelligent body of the reconfigurable modular flexible mechanical arm and decomposing the system into an input-output subsystem and a zero-dynamic subsystem through input-output linearization in the step one is as follows:
1) establishing a dynamic model of the ith joint intelligent body of the reconfigurable modular flexible mechanical arm
Taking each joint intelligent agent of the reconfigurable mechanical arm as a subsystem, the dynamic model of the ith joint intelligent agent subsystem can be described as follows:
Figure FDA0003153730930000021
in the formula, thetai(t) is a motor rotation angle vector of the ith joint agent; q. q.si(t) Flexible Modal vector of the ith Joint agent, qi＝[qi1,…,qir]T；ui(t) is the control torque vector of the ith joint agent; f. offi(θi,qi) And fri(θi,qi) Items of which the ith joint agent is influenced by gravity, Copenforces and centrifugal forces respectively;
Figure FDA0003153730930000022
and
Figure FDA0003153730930000023
respectively positive definite damping matrix for ith joint intelligent agent, Ki(qi) Positively determining a stiffness matrix for the ith joint agent; r is the number of the flexible modes,
Figure FDA0003153730930000024
respectively represent thetaiAnd q isiThe second and first derivatives of (d); mi＝[MriMrfi；MfriMfi]A positive definite inertia matrix of the ith joint agent; c1i、C2iThe association of the ith joint agent and other agents;
Figure FDA0003153730930000025
Figure FDA0003153730930000026
in the formula, Mrij、Mrfij、Mfrij、MfijAre respectively MrMrf；MfrMfThe ijth component of (a); n is the number of joints contained in the reconfigurable modular flexible mechanical arm, and j is 1,2, …, n;
when there is uncertainty in the system, assume parameter Mi、fri、E1i、ffi、E2i、Ki、C1i、C2iThe nominal amounts of (A) are respectively: mni、frni、E1ni、ffni、E2ni、Kni、C1ni、C2niDefining: Δ Mi＝Mi-Mni，Δfri＝fri-frni，ΔE1i＝E1i-E1ni，Δffi＝ffi-ffni，ΔE2i＝E2i-E2ni，ΔKi＝Ki-Kni，ΔC1i＝C1i-C1ni，ΔC2i＝C2i-C2niRepresenting the parameter uncertainty existing in the system, the ith joint intelligent body of the reconfigurable flexible mechanical arm system can be rewritten into the following form:
Figure FDA0003153730930000031
in the formula (I), the compound is shown in the specification,
Mni＝[MrniMrfni；MfrniMfni]
Figure FDA0003153730930000032
in the formula (2), i is 1,2, …, n, namely, a model of each joint intelligent body of the reconfigurable modular flexible mechanical arm can be obtained;
2) obtaining the i-th joint intelligent agent input/output subsystem
Redefining the output z (t) of the reconfigurable modular flexible robotic arm system as follows
zi＝λ0iθi+λ1iqi (3)
In the formula, λ0iAnd λ1iTo design the parameter, λ1iIs a matrix of dimension 1 x r;
defining:
Figure FDA0003153730930000036
xi＝[θi，qi]T；
order to
di(λ0i,λ1i,xi,ui)＝λ0i(Ni11d1i+Ni12d2i)+λ1i(Ni21d1i+Ni22d2i)
ci(λ0i,λ1i,xi,ui)＝(λ0iNi11+λ1iNi21)C1i+(λ0iNi12+λ1iNi22)C2i
Figure FDA0003153730930000035
βi(λ0i,λ1i,xi)＝λ0iNi11(θ,q)+λ1iNi21(θ,q)
The input-output subsystem of the system is obtained as follows
Figure FDA0003153730930000033
In the formula, betai(λ0i,λ1i,xi) Reversible;
3) obtaining the zero dynamic subsystem of the ith joint intelligent agent
When a particular control input ui(t) making the input-output subsystem output zero:
Figure FDA0003153730930000034
the substitution of formula (2) to zero dynamic subsystems is as follows
Figure FDA0003153730930000041
And decomposing the ith joint intelligent system into an input-output subsystem and a zero dynamic subsystem through input-output linearization.
3. The method for controlling trajectory tracking of the reconfigurable modular flexible mechanical arm based on parameter optimization according to claim 2, wherein the process of designing the adaptive dynamic terminal sliding mode control strategy to ensure that the state of the input-output subsystem tracks the expected reference trajectory within a limited time is as follows:
for the ith joint agent input-output subsystem, order
Figure FDA0003153730930000042
Order to
Figure FDA0003153730930000043
Is ciIs determined by the estimated value of (c),
Figure FDA0003153730930000044
respectively subsystem association C1i、C2iAn estimated value of (d); let ei1＝zi1-zid，
Figure FDA0003153730930000045
ei＝ei1+ei2，
Figure FDA0003153730930000046
Figure FDA0003153730930000047
zidA desired output trajectory for the subsystem; satisfy the requirement of
Figure FDA00031537309300000415
Is bounded, an
Figure FDA00031537309300000416
L>0；ui(t) is the control torque vector of the ith joint agent;
the following sliding modes are selected:
Figure FDA0003153730930000048
in the formula, τ1Design parameter matrix for sliding mode, piAnd q isiAre all positive odd numbers, satisfy pi>qi；
Design of
Figure FDA0003153730930000049
The adaptive change rate of (2) is:
Figure FDA00031537309300000410
let vi＝βiui，
Figure FDA00031537309300000411
Sliding mode control strategy viThe design is as follows:
Figure FDA00031537309300000412
in the formula (I), the compound is shown in the specification,
Figure FDA00031537309300000413
the adaptive change rate is designed as follows:
Figure FDA00031537309300000414
when the reconstruction mechanical arm is assembled by the n joint agents, the track error e of the ith joint agent can be input into and output from the subsystemiThe modification is as follows:
Figure FDA0003153730930000051
in the formula, ai(i-1)、ai(i+1)Respectively relating coefficients of the ith joint agent, the (i-1) th joint agent and the (i +1) th joint agent;
and (3) the formula (11) is carried into the formulas (8) to (10), so that the output of the input and output subsystem of the n-joint reconfigurable modular flexible mechanical arm can realize track tracking control, and a reconfigurable modular flexible mechanical arm track tracking controller does not need to be redesigned.
4. The method for controlling trajectory tracking of reconfigurable modular flexible mechanical arm based on parameter optimization according to claim 2, wherein in step five, the NSGA-II algorithm is adopted to carry out design parameter lambda0i，λ1iThe process of performing parameter optimization design is as follows:
balance point x at zero dynamic subsystemiLinearizing the zero dynamics subsystem at 0; define Ω1Is xiAdjacent to 0Domain at Ω1The matrix N is formed on the domainiAt xiExpanding the position of 0 according to Taylor series to obtain a constant value matrix Ni0And xiHigher order term f ofhi(x) Form of sum
Figure FDA0003153730930000052
Figure FDA0003153730930000053
Reanalysis ffi(θi,qi) It can be found to be the state variable x onlyiIs a higher order term of (i.e. has
Figure FDA0003153730930000054
Order to
Ai(λ0i,λ1i)＝[0,I；–Pi0ki,–Pi0E2i] (13)
In the formula, Pi0＝Ni220-Ni210(λ0iNi110+λ1iNi210)-1(λ0iNi120+λ1iNi220)
The zero dynamics subsystem can be written as follows:
Figure FDA0003153730930000055
in the formula, GΔi＝–Pi0(fhi+C2i+d2i)；
Suppose there is | | f near zerohi||≤μ3，||C2i||<μ4，||d2i||<μ5Then, then
||GΔι||＝||fhi+C2i+d2i||≤(μ3+μ4+μ5)||-Pi0||
Let constant ε ═ μ3+μ4+μ5)||-Pi0| order
Figure FDA0003153730930000061
Gi＝(0,GΔi)T，GiSatisfy | | Gi||＝||GΔiIf | | < epsilon, then
Figure FDA0003153730930000062
Suitably selecting λ0iAnd λ1iTo ensure Ai(λ0i,λ1i) The characteristic value of the zero dynamic subsystem is strictly on the left half plane of the complex plane, so that the zero dynamic subsystem gradually converges;
known by an input-output subsystem self-adaptive dynamic terminal sliding mode controller, a design parameter lambda0i，λ1iSimultaneously, the design parameters in the input and output subsystem controller are input and output; controller parameter lambda is carried out by adopting NSGA-II algorithm0i，λ1iThe multi-objective optimization design is carried out, and the NSGA-II algorithm evolution process is as follows;
(1) initializing a population: parameter lambda0i，λ1iAdopting real number coding to randomly generate an initial parent population P (k) with the population scale of N, wherein k is a genetic algebra;
(2) fitness function: selecting a parameter lambda in consideration of the requirements of the flexible mechanical arm on the state, the control and the end point state in the control process0i，λ1iThe optimization target of (2) is as follows, the dynamic deviation of the system in the control process, the energy consumption and the system steady-state deviation at the end of the control are ensured to be comprehensively optimal;
Figure FDA0003153730930000063
and the constraint conditions are satisfied:
Figure FDA0003153730930000064
|ui(t)|≤mi，miis uiUpper bound of (2)
In the formula (I), the compound is shown in the specification,
Figure FDA0003153730930000065
Figure FDA0003153730930000066
Qiis a semi-positive definite symmetric array, RiIs a positive constant;
(3) non-dominated ranking hierarchy and crowding distance calculation: firstly, layering a population according to the non-inferior solution level of an individual; then, comparing the information crowding degrees of the individuals by adopting a two-dimensional information cyclic crowding mechanism containing an order threshold, and deleting the individual with the minimum crowding degree until the number of the individuals in the population meets the requirement;
the calculation formula of the information congestion degree is as follows:
Is＝αs,1Ls,1+αs,2Ls,2
in the formula (I), the compound is shown in the specification,
Ls,1＝(|ps-1|+|ps|)/2,s＝2,3,...,h-1；Ls,2＝(|ps|+|ps+1|)/2,s＝2,3,...,h-1；
Figure FDA0003153730930000071
wherein h is the number of population individuals, and the position of the s-th individual is Ps＝(ps1,ps2,…,psm) M is the number of the objective functions, m is 3, k is 1,2, 3;
(4) genetic operator: the method comprises the following steps of (1) selecting an operator, a crossover operator, a mutation operator and a chaotic interpolation operator; selecting an operator by adopting a race-round selection operator, namely randomly selecting 2 individuals, and selecting the individuals with small sequence numbers and high grade if the non-dominated sorting sequence numbers are different; if the serial numbers are the same, selecting individuals which are less crowded around; normal distribution crossover operators are adopted in the crossover operation, and polynomial mutation operators are adopted in the mutation operation; retaining 10% of the genes with higher fitness generated after genetic operation, and adopting a chaotic insert operator to update the rest 90% of the genes in the population;
(5) elite strategy: keeping good individuals in the parent to directly enter the offspring, which is a necessary condition for the convergence of the genetic algorithm with the probability 1; the method comprises the following steps: synthesizing all the parents P (k) and the child Q (k) into a uniform population R (k) ═ P (k) · U (q) (k), wherein the number of the individuals of R (k) is 2N; secondly, the population R (k) is sorted quickly and non-dominantly, the local crowding distance of each individual is calculated, the individuals are selected one by one according to the level until the number of the individuals reaches N, a new parent population P (k +1) is formed, and a new round of selection, crossing and variation is started on the basis to form a new child population Q (k + 1).
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