CN110815225B - Point-to-point iterative learning optimization control method of motor-driven single mechanical arm system - Google Patents

Point-to-point iterative learning optimization control method of motor-driven single mechanical arm system Download PDF

Info

Publication number
CN110815225B
CN110815225B CN201911119240.2A CN201911119240A CN110815225B CN 110815225 B CN110815225 B CN 110815225B CN 201911119240 A CN201911119240 A CN 201911119240A CN 110815225 B CN110815225 B CN 110815225B
Authority
CN
China
Prior art keywords
point
input
tracking
matrix
motor
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Active
Application number
CN201911119240.2A
Other languages
Chinese (zh)
Other versions
CN110815225A (en
Inventor
陶洪峰
李健
黄彦德
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Zhongkexin Micro Intelligent Equipment Shenyang Co ltd
Original Assignee
Jiangnan University
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Jiangnan University filed Critical Jiangnan University
Priority to CN201911119240.2A priority Critical patent/CN110815225B/en
Publication of CN110815225A publication Critical patent/CN110815225A/en
Application granted granted Critical
Publication of CN110815225B publication Critical patent/CN110815225B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • 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
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/163—Programme controls characterised by the control loop learning, adaptive, model based, rule based expert control

Landscapes

  • Engineering & Computer Science (AREA)
  • Robotics (AREA)
  • Mechanical Engineering (AREA)
  • Feedback Control In General (AREA)

Abstract

The invention discloses a point-to-point iterative learning optimization control method of a motor-driven single mechanical arm system, and relates to the field of mechanical arm optimization control. The method comprises the steps of converting a single mechanical arm system which runs repeatedly into an input and output matrix model of a time sequence based on a lifting technology, then designing a point-to-point trajectory tracking robust iterative learning optimization method based on performance indexes, obtaining an optimized iterative learning control law by solving a quadratic optimal solution of a multi-target performance index function, proving the robust convergence of an algorithm when bounded uncertainty exists in the model according to a maximum singular value theory, and proving the convergence of the mechanical arm system under the condition that input is constrained.

Description

Point-to-point iterative learning optimization control method of motor-driven single mechanical arm system
Technical Field
The invention relates to the field of mechanical arm optimization control, in particular to a point-to-point iterative learning optimization control method of a motor-driven single mechanical arm system.
Background
The robot arm is an automatic operation device which can simulate some action functions of a human arm and is used for grabbing, carrying objects or operating tools according to a fixed program. It can replace the heavy labor of people to realize the mechanization and automation of production, and can be operated under harmful environment to protect the personal safety, thus being widely applied to the departments of mechanical manufacturing, metallurgy, electronics, light industry, atomic energy and the like.
A robot arm often does not need to track a complete desired trajectory when performing repetitive process tasks, but only needs to track a given desired value at certain specific points in time, such as "pick" and "put" operations of the robot arm, and only needs to focus on the outputs of the pick and place points, while the outputs at other points in time often do not need much consideration, and this type of control problem for a robot arm is called point-to-point tracking control.
There are two common solutions to the point-to-point tracking problem: firstly, an arbitrary track passing through a specific tracking point is designed, so that the problem is converted into general full-track tracking, but the method has limitations, enough prior knowledge is needed to determine the optimal fixed reference track, and the degree of freedom of a non-tracking time point is not fully utilized to design a controller; the other is a point-to-point control method based on reference track updating, although the convergence is faster than that of a fixed reference track method, the method is essentially a full-track tracking problem, the full-track tracking needs to track the output of the whole track, and some industrial processes such as mechanical arm operation only need to track the expected values of a plurality of points on the track to meet the control requirements.
Iterative Learning Control (ILC) is a high-performance control method widely used for executing repetitive tasks, and a point-to-point ILC method arises from the above point-to-point tracking problem.
Disclosure of Invention
The invention provides a point-to-point iterative learning optimization control method of a motor-driven single mechanical arm system aiming at the problems and the technical requirements, and the tracking error information at the appointed time point is used for updating the input, so that unnecessary non-key point tracking constraint is eliminated, and the degree of freedom at the non-key point increases the degree of freedom for algorithm design and simultaneously increases the promotion space of the overall performance of the system.
The technical scheme of the invention is as follows:
the point-to-point iterative learning optimization control method of the motor-driven single mechanical arm system comprises the following steps:
establishing a model of a motor-driven single mechanical arm control system; constructing a discrete state space equation of the motor-driven single mechanical arm control system; establishing a point-to-point trajectory tracking model; designing a point-to-point iterative learning trajectory tracking optimization algorithm; analyzing the convergence and robustness of the point-to-point iterative learning trajectory tracking optimization algorithm; analyzing the convergence and robustness of the point-to-point iterative learning trajectory tracking optimization algorithm under the input constraint condition; point-to-point trajectory tracking of the motor-driven single mechanical arm control system under the condition of input constraint is realized;
firstly, establishing a dynamic model of a motor-driven single mechanical arm control system:
the actual physical model of the motor-driven single mechanical arm is shown as the formula (1):
Figure GDA0002760047140000021
wherein N isl＝m2gl+m1gl，
Figure GDA0002760047140000022
g is gravitational acceleration, theta is connecting rod angle, i is current, KtIs torque, KbIs the back electromotive force coefficient, BcIs the viscous friction coefficient of the bearing, DcIs the load factor, l is the link length, m1For loading mass, m2For connecting rod mass, RrIs a resistor, u is a motor control voltage, xi is an actuator rotational inertia and is a reactor;
secondly, constructing a discrete state space equation of the motor-driven single mechanical arm system:
the method comprises the following steps of defining state variables by using a connecting rod angle and a current of a motor-driven single mechanical arm control system:
Figure GDA0002760047140000023
defining the motor control voltage u as an input variable, a motor-driven single robot control system shown in equation (1) can be described as:
Figure GDA0002760047140000024
obviously, the formula (2) is a continuous system model, so the formula (2) needs to be discretized, and a sampling period T meeting the Shannon sampling theorem is selectedsAnd taking the connecting rod angle theta as output, further obtaining a discrete state space model of the motor-driven single mechanical arm control system:
Figure GDA0002760047140000025
wherein T and k represent sampling time and batch respectively, and the time T epsilon [0, T of the batch process in a running period T]The range contains N sampling points; u. ofk(t)，yk(t) and xk(t) input, output and state vectors of the motor-driven single mechanical arm control system at the kth batch time t are respectively; a, B and C are discrete system parameter matrixes corresponding to the formula (2), and CB is not equal to 0; and assuming the initial state of system operation to be consistent, i.e. xk(0)＝x0；
Step three, establishing a point-to-point trajectory tracking model:
for a linear discrete system of the form of equation (3), its state space expression can be converted into a time-series input-output matrix model:
yk＝Guk+d (4)
wherein:
Figure GDA0002760047140000031
d＝[CA CA2 CA3 … CAN]Tx0
uk＝[uk(0),uk(1),...,uk(N-1)]T
yk＝[yk(1),yk(2),...,yk(N)]T
g is an input and output transfer matrix on a time sequence, and d is the influence of the initial state of the system on the output;
in the design of the traditional iterative learning control algorithm, a reference track is usually fixed, and a track tracking target requires that the output of the system is gradually close to a set expected output value, namely y, at each moment along with the continuous running of a batch processk(t)→yd(t), where t ∈ {0,1,2, …, N }, yk(t) actual output of the system at time t of the kth batch, yd(t) is the system expected output at time t for the kth batch; for the point-to-point tracking problem, only the expected values of some specified key time points, namely y, need to be trackedk(ti)＝yd(ti) I is 1,2, …, M, and has a value of 0. ltoreq.t1<t2<…<tMN is less than or equal to N; assume that the expected values for these M tracking points are:
ydM＝[yd(t1),yd(t2),...,yd(tM)]T (5)
defining a matrix psi as a tracking point selection matrix of M rows and N columns, where psiijFor the ith row in the matrix psij elements of column, when tracking time t at iiWhen selected, all N elements of the ith row of the matrix are divided by j to tiThe time is 1, and the rest are all 0, and M tracking time points are selected, so that psi y can be obtainedkActual output values of M tracking time points selected for N sampling time points of the entire trajectory are expressed in the following specific form:
Figure GDA0002760047140000032
the tracking error at a tracking point can thus be defined as:
ekM＝ydM-ψyk (7)
wherein,
ekM＝[ek(t1),ek(t2),...,ek(tM)]T (8)
fourthly, designing a point-to-point iterative learning trajectory tracking optimization algorithm:
in practical engineering application, the system inputs uk+1Generally reflecting the energy consumed in the control process, the smaller control energy and the control oscillation thereof are crucial to the loss of the controller and the final actuator thereof, so that the optimization condition of the control quantity needs to be considered on the basis of the error in the optimization process, and the following performance index function is considered:
Figure GDA0002760047140000041
wherein Q is a symmetric positive definite weight array, R and S are corresponding symmetric non-negative definite weight arrays, and
Figure GDA0002760047140000042
substituting the formula (4), the formula (7) and the formula (10) into the formula (9), and obtaining an optimized iterative learning control law by solving a quadratic optimal solution of the formula:
uk+1＝Luuk+LeekM (11)
wherein,
Lu＝[(ψG)TQ(ψG)+R+S]-1[(ψG)TQ(ψG)+R]，Luis the gain of the iterative learning control law controller input term,
Le＝[(ψG)TQ(ψG)+R+S]-1(ψG)TQ，Leis the gain of the error term;
fifthly, analyzing the convergence and robustness of the point-to-point iterative learning trajectory tracking optimization algorithm:
if the optimized iterative learning control law (11) meets the condition eta | | | | Lu-Le(ψG)||i2<1，||·||i2Representing the maximum singular value of the matrix, then when the iteration batch k → ∞ the tracking output error of the system at the finite expected tracking point converges;
the formula (11) can be rewritten from the formulae (4) and (7):
Figure GDA0002760047140000043
defining system steady state control inputs
Figure GDA0002760047140000044
According to formula (12):
u∞-uk+1＝(Lu-Le(ψG))(u∞-uk)+ξ (13)
wherein xi ═ I- (L)u-Le(ψG)))u∞-Le(ydMPhi d) is a zero matrix, and the norm of two sides of the formula (13) is taken to obtain phi u∞-uk+1||＝||(Lu-Le(ψG))(u∞-uk)+ξ||≤η||u∞-ukIf the condition eta is satisfied<1, then the control input converges in the norm sense, further yielding:
Figure GDA0002760047140000051
where I is the identity matrix of the corresponding dimension, then:
Figure GDA0002760047140000052
when each weight matrix parameter is determined, the steady-state error norm can be obtained by the equation (15) to be converged to a constant value finally, if S is smaller, the point-to-point tracking steady-state error of the system under the action of the optimization iterative learning control law (11) is smaller, and particularly when S is taken as a zero matrix, the tracking output error norm at the tracking point of the system can be converged to zero;
in the actual process production process, a controlled system usually has certain modeling uncertainty; considering the uncertainty Δ of the original system G plus a variation in the time domainGAnd further defining the expression of the uncertain controlled system as Gθ＝G(I+ΔG) And define ΔGThe bounded condition of the uncertainty factor is | | | Δ | | | luminancei2≤1，
The conditions for the system robust convergence are:
||Lu-Le(ψGθ)||i2<1
if the symmetric non-negative constant weight matrix R is taken as a zero matrix, the following formula is provided:
Figure GDA0002760047140000053
wherein W is an uncertain weight matrix;
it can be obtained if the parameters satisfy the condition:
||[(ψG)TQ(ψG)+S]-1(ψG)TQ(ψG)W||i2<1 (16)
the system robustness converges; if the symmetric non-negative fixed weight array R is equal to rI, R is more than or equal to 0, if the condition (16) is met, robust convergence of the linear discrete system (3) under the action of the optimized iterative learning control law (11) can still be ensured; because (psi G)TQ (psi G) + S is a symmetric positive definite matrix, so that it can be decomposed into singular values by (psi G)TQ(ψG)+S＝U∑UTWhere U is unitary matrix and Σ is element σiThe full-rank diagonal matrix of (1) is expressed as Σ ═ diag { σ }iJ, for convenience of representation, let
Figure GDA0002760047140000054
And conditions of
Figure GDA0002760047140000055
Satisfy the requirement of obtaining | | H | non-conducting phosphori2＝α<1。
Then one can get:
Figure GDA0002760047140000061
if the condition (16) is satisfied, the system robustness is converged, that is, the design of the parameter R ═ rI does not affect the robustness of the system, and on the other hand, the design of the parameter S needs to satisfy the condition (16);
sixthly, analyzing the convergence and robustness of the point-to-point iterative learning trajectory tracking optimization algorithm under the input constraint condition:
in many industrial process control applications, in order to ensure that an industrial process operates safely and smoothly, certain constraints need to be imposed on input variables, the input signal constraints include constraints on input amplitude, constraints on input amplitude with respect to time direction and batch direction, and the constraints are often expressed in inequalities;
saturation constraint of controller input:
ulow≤uk+1≤uhi (18)
wherein u islow，uhiAre respectively controller input uk+1The lower and upper bound values of (1);
input variation constraint between two adjacent sampling times:
ulow≤uk+1≤uhi (19)
wherein u islow、uhiAre respectively an input variable uk+1Lower and upper bound values, u, varying along the time axisk+1(t)＝uk+1(t-1)-uk+1(t-2)；
Input variation constraints between two adjacent batches:
Δulow≤Δuk+1≤Δuhi (20)
wherein Δ ulow、ΔuhiAre respectively input variables Δ uk+1Lower and upper bound values, Δ u, that vary along the axis of iterationk+1＝uk+1-uk；
All of the above constraint equations can be converted to Δ uk+1First, the saturation constraint (18) of the controller input can be translated into:
ulow-uk≤Δuk+1≤uhi-uk (21)
let u (0) be u0Then u isk+1Can be expressed as:
uk+1＝μuk+1 (22)
wherein:
Figure GDA0002760047140000071
the input variation constraint (19) between two adjacent sampling times can therefore be translated into:
ulow-μuk≤μΔuk+1≤uhi-μuk (24)
the constraints described above can be combined into the following linear inequalities:
ζuΔuk+1≥ζk+1 (25)
wherein:
Figure GDA0002760047140000072
theoretically, the input described by equation (25)Income Delta uk+1The constraint of (1) is a convex set, here denoted by Ω; the error equation for model uncertainty can be expressed as:
e(k+1)M＝ekM-ψGθΔuk+1 (26)
wherein G isθIs an input-output transfer matrix with uncertainty on a time sequence, corresponding to a deterministic system G;
substituting equation (26) into the performance indicator function (9) yields:
Figure GDA0002760047140000073
and (3) convergence and robustness analysis:
under the constraint, assume that condition 1) there is one feasible expected input u∞And corresponding expected error e ∞0, and satisfies ulow≤u∞≤uhi，ulow≤u∞≤uhiAnd Δ ulow<0，Δuhi>0; assuming the condition 2) Q is a symmetrical positive definite weight array, and R and S are corresponding symmetrical non-negative definite weight arrays; the system robustness converges and the system steady state error converges to 0 when S takes the zero matrix, i.e., k → ∞ Δ uk→0，ekM→0；
Defining the optimal performance indicator function of the (k + 1) th batch as:
Figure GDA0002760047140000074
first consider Δ uk+1Case 0, when uk+1＝uk，e(k+1)M＝ekM,uk+1＝ukSo that (e) can be derivedkM,0,uk) E Ω, because the performance indicator function is at point (e)kM,0,uk) The value of (b) is always greater than or equal to the optimum value, so the following relationship can be obtained:
Figure GDA0002760047140000081
by analogy, the following results are obtained:
Figure GDA0002760047140000082
as is clear from the formula (30), Δ u when k → ∞ is satisfiedk→ 0 because Δ u when k is sufficiently largek→ 0, and satisfies the condition Δ ulow<0，Δuhi>0, so the constraint Δ ulow≤Δuk+1≤ΔuhiThe same is satisfied; from the assumption condition 1) that u is knownk+1＝u∞Time e(k+1)MWhen the value is equal to 0, then
Figure GDA0002760047140000083
Is also a feasible point within a restricted area, wherein
Figure GDA0002760047140000084
Is obvious (e)(k+1)M,Δuk+1,uk+1) Is also a feasible point in a constrained region, because Ω is a convex set, it can be deduced that a point between any two feasible points in Ω is a feasible point;
meanwhile, the performance index function J is solvedk+1From (e)(k+1)M,Δuk+1,uk+1) To
Figure GDA0002760047140000085
The directional derivatives of (a):
Figure GDA0002760047140000086
from equation (31):
Figure GDA0002760047140000087
since Δ u when k → ∞ isk→ 0, so when k → ∞ time formula(32) Can be simplified as follows:
Figure GDA0002760047140000088
as can be seen from equation (33) and assumption condition 2), when k → ∞ is reached, the steady-state output error of the system converges to a certain finite value, and particularly when S takes a zero matrix, the steady-state output error reaches a minimum value and becomes 0; the result shows that when the assumed conditions are satisfied, the input constraint model uncertain system can still be converged under the action of the optimization iterative learning control law (11); in particular, when the system is a nominal system, i.e. GθThe same holds true when G;
and seventhly, determining the input vector of each iteration batch of the motor-driven single mechanical arm system according to a robust optimization iterative learning control law, inputting the obtained input vector into the motor-driven single mechanical arm system to perform point-to-point trajectory tracking control, and tracking the expected output at the specified tracking point by the motor-driven single mechanical arm system under the control action of the input vector.
The beneficial technical effects of the invention are as follows:
the application discloses a method for designing a point-to-point trajectory tracking iterative control algorithm aiming at a linear system with repetitive motion characteristics, such as a motor-driven single mechanical arm control system, wherein the motor-driven single mechanical arm control system is used as a controlled object, the control algorithm can update control input by utilizing tracking error information at a limited time point of trajectory tracking, so that unnecessary non-key point tracking constraint is eliminated, and the degree of freedom at the non-key point increases the degree of freedom for ILC algorithm design and simultaneously increases the promotion space of the overall performance of the system. The method is based on a norm optimal iterative learning framework, a multi-point multi-target performance function under the condition of input constraint is introduced, an iterative learning control law is obtained by solving a quadratic optimal solution of a target function, meanwhile, the convergence condition of a robust iterative learning algorithm is analyzed under the condition that a system model has uncertainty, and bounded convergence of a system steady-state tracking output error at a limited expected time point is guaranteed.
Drawings
Fig. 1 is a model block diagram of a motor-driven single robot control system disclosed in the present application.
FIG. 2 is a graph of actual output versus reference point tracking for the motor driven single robot control system of the present application.
FIG. 3 is a graph comparing nominal and uncertainty system root mean square error under no input constraints in the present application.
FIG. 4 is a graph comparing nominal and uncertainty system performance indicators without input constraints in the present application.
Fig. 5 is a comparative plot of nominal system root mean square error with and without input constraints in the present application.
FIG. 6 is a graph comparing nominal system performance indicators with and without input constraints in the present application.
FIG. 7 is a graph comparing nominal and uncertainty system root mean square error under the input constraints of the present application.
FIG. 8 is a graph comparing nominal and uncertainty system performance indicators under the input constraints of the present application.
Detailed Description
The following further describes the embodiments of the present invention with reference to the drawings.
Referring to fig. 1, referring to fig. 8, a model block diagram of a motor-driven single robot control system disclosed in the present application is shown. Controller input for the k-th batch is ukActing on the robot to obtain the actual output y of the k-th lot of the systemkThe actual output at the specified tracking point is obtained via the tracking point selector, and the error from the set desired value stored in the desired trajectory memory is transmitted to ekMComparing the error with a set precision value, and if the error does not reach the set precision, determining the error ekMWith current controller input ukController input u passed to the optimized iterative learning controller to generate the next batchk+1And the operation is circularly performed until the error between the actual output and the expected value of the system reaches the precision requirement, and the operation is stopped, wherein the input of the controller at the moment is the optimal control input.
For the actual physical model of the motor-driven single mechanical arm shown in the formula (1), variable parameters are respectively set as:
Kt＝1N·m,Kb＝0.085V·s/rad,
Rr＝0.075Ω,Bc＝0.015kg·m2/s,Dc＝0.05,
l＝0.6m,m1＝0.05kg,m2＝0.01kg,
Ξ＝0.05kg·m2,＝0.0008Ω,g＝9.8m/s2。
the system simulation time is set to be T-2 s, and the sampling time is set to be Ts0.1s, the parameter matrices of the discrete state space expression of the system are respectively:
Figure GDA0002760047140000101
C＝[1 0 0]
without loss of generality, 5 points (M is 5) in a sampling interval are taken to carry out point-to-point tracking control on the system, and the selected sampling point is tiE { 26101418 }, i.e., tracking times of 0.2s, 0.6s, 1s, 1.4s, 1.8s, respectively, the expected values at the tracking points are set to:
ydM＝[1.5 0.3 0 -0.3 1.5]T
in units of rad, while setting the initial state to xk(0)＝[-0.01 0.02 -0.03]T. The symmetric positive definite matrix Q and the symmetric nonnegative definite matrix R and S are respectively selected as Q being 10I, R being 0.5I and S being 0.001I. Optimizing L in an iterative learning control law (11) when a symmetric positive definite matrix Q, a symmetric non-negative definite matrix R, S and a sampling point are determinedu，LeAs determined accordingly. The optimized iterative learning controller is realized on a STM32F103RCT6 chip, and the input of the chip is motor control voltage u which is acquired by a voltage sensor. The input signal enters an STM32F103RCT6 chip through a conditioning circuit to be stored and calculated, an iterative learning updating law is constructed, and the signal obtained after CPU calculation is an optimal control input signal uk+1The input signal acts on the motor through the RS232 communication moduleAnd driving a single mechanical arm control system, and continuously correcting the tracking track of the mechanical arm until the expected track at the specified point on the tracking. When the dynamic model (1) of the motor-driven single-robot control system operates, please refer to fig. 2, which shows that when the motor-driven single-robot system is accurately modeled, i.e., is a nominal system, the system applies a point-to-point trajectory tracking effect diagram of an optimized iterative learning control law (11), and after a certain batch k, the output value of the system at a specified tracking point can be accurately tracked to a desired value, and fig. 3 and 4 show that the system can be converged after a certain iterative batch. When each physical parameter of the motor-driven single mechanical arm system model has a certain deviation, a parameter matrix after the system is dispersed has a certain uncertainty, and then the uncertainty delta exists in a system matrix G, wherein delta is a bounded lower triangular random number matrix. When the physical parameters of the system are changed, the uncertainty delta is between-0.8 and 0.2]At this time, fig. 3 and fig. 4 respectively show comparison graphs of the root mean square error and the performance index of the nominal system and the uncertain system respectively without input constraint, which shows that when parameters of the motor-driven single mechanical arm system are uncertain, the optimized iterative learning control law can still ensure system convergence, the rate of convergence of the root mean square error and the performance index of the nominal system is faster than that of the uncertain system, and the rationality and the effectiveness of the algorithm are verified. The input of the motor-driven single mechanical arm system can be restrained in the actual control process, and the input amplitude restraint is considered to be applied to the system, namely the range of the motor control voltage is [ -0.9V]Fig. 5 and fig. 6 are comparison graphs of the root mean square error and the performance index of the nominal system with or without input constraint, and fig. 7 and fig. 8 are comparison graphs of the root mean square error and the performance index of the nominal system with or without input constraint, respectively, which indicate that the input signal of the system can still complete the point-to-point tracking task after being subjected to certain constraint conditions, but the convergence speed of the root mean square error curve of the system is slower than that of the system without constraint, but is finally convergent, thereby further verifying the rationality and the effectiveness of the algorithm.
According to the method, the point-to-point tracking problem of the repeated process of the motor-driven mechanical arm system is solved by directly obtaining the optimal control input signal through the optimization algorithm without introducing a reference track, and the iterative learning algorithm and the optimization algorithm are combined to construct the optimization controller. When uncertainty of certain conditions exists in system modeling, an optimization algorithm based on performance indexes has robustness on the uncertainty of the model, and in addition, when a motor drives a mechanical arm system to execute repeated process tasks, input signals of a controller can be restrained. The point-to-point robust iterative learning optimization method can ensure that the high-precision tracking of the expected track can be realized no matter whether the input of the motor-driven single mechanical arm system is constrained or not and under the condition that certain uncertainty exists in modeling.
What has been described above is only a preferred embodiment of the present application, and the present invention is not limited to the above embodiment. It is to be understood that other modifications and variations directly derivable or suggested by those skilled in the art without departing from the spirit and concept of the present invention are to be considered as included within the scope of the present invention.

Claims (1)

1. The point-to-point iterative learning optimization control method of the motor-driven single mechanical arm system is characterized by comprising the following steps of: establishing a physical model of the motor-driven single mechanical arm system; constructing a discrete state space model of the motor-driven single mechanical arm system; establishing a point-to-point trajectory tracking model; designing a point-to-point iterative learning trajectory tracking optimization algorithm; analyzing the convergence and robustness of the point-to-point iterative learning trajectory tracking optimization algorithm; analyzing the convergence and robustness of the point-to-point iterative learning trajectory tracking optimization algorithm under the input constraint condition; point-to-point trajectory tracking of the motor-driven single mechanical arm system under the condition of input constraint is realized;
firstly, establishing a physical model of the motor-driven single mechanical arm system:
the physical model of the motor-driven single mechanical arm system is shown as the formula (1):
Figure FDA0002772643880000011
wherein N isl＝m2gl+m1gl，
Figure FDA0002772643880000012
g is gravitational acceleration, theta is connecting rod angle, i is current, KtIs torque, KbIs the back electromotive force coefficient, BcIs the viscous friction coefficient of the bearing, DcIs the load factor, l is the link length, m1For loading mass, m2For connecting rod mass, RrIs a resistor, u is a motor control voltage, xi is an actuator rotational inertia and is a reactor;
secondly, constructing a discrete state space model of the motor-driven single mechanical arm system:
defining a state variable by using a link angle and a current of the motor-driven single-robot arm system:
Figure FDA0002772643880000013
defining the motor control voltage u as an input variable, a motor-driven single robot system represented by equation (1) can be described as:
Figure FDA0002772643880000014
obviously, the formula (2) is a continuous system model, so the formula (2) needs to be discretized, and a sampling period T meeting the Shannon sampling theorem is selectedsAnd taking the connecting rod angle theta as output, further obtaining a discrete state space model of the motor-driven single mechanical arm system:
Figure FDA0002772643880000021
in the formula, T and k respectively represent sampling time and batches, and the batch process contains N sampling points within a time range in one operation period T, namely T belongs to {0,1,2, …, N }; u. ofk(t)，yk(t) and xk(t) the kth lot of the motor-driven single robot systemInputting, outputting and state vectors at the time t; a, B and C are discrete system parameter matrixes corresponding to the formula (2), and CB is not equal to 0; and assuming the initial state of system operation to be consistent, i.e. xk(0)＝x0；
Thirdly, establishing the point-to-point track tracking model:
for a discrete state space model of the form of equation (3), its state space expression can be converted into a time-series input-output matrix model:
yk＝Guk+d (4)
wherein:
Figure FDA0002772643880000022
d＝[CA CA2 CA3…CAN]Tx0
uk＝[uk(0),uk(1),...,uk(N-1)]T
yk＝[yk(1),yk(2),...,yk(N)]T
g is an input and output transfer matrix on a time sequence, and d is the influence of the initial state of the system on the output;
in the design of the traditional iterative learning control algorithm, a reference track is usually fixed, and a track tracking target requires that the output of the system is gradually close to a set expected output value, namely y, at each moment along with the continuous running of a batch processk(t)→yd(t) wherein yk(t) actual output of the system at time t of the kth batch, yd(t) is the system expected output at time t for the kth batch; for the point-to-point tracking problem, only the expected values of some specified key time points, namely y, need to be trackedk(ti)＝yd(ti) I is 1,2, …, M, and has a value of 0. ltoreq.t1<t2<…<tMN is less than or equal to N; assume that the expected values for these M tracking points are:
ydM＝[yd(t1),yd(t2),...,yd(tM)]T (5)
defining a matrix psi as a tracking point selection matrix of M rows and N columns, where psiijFor the ith row and jth column element in the matrix psi, when tracking the time point t at the ithiWhen selected, all N elements of the ith row of the matrix are divided by j to tiThe time is 1, and the rest are all 0, and M tracking time points are selected, so that psi y can be obtainedkActual output values of M tracking time points selected for N sampling time points of the entire trajectory are expressed in the following specific form:
Figure FDA0002772643880000031
the tracking error at a tracking point can thus be defined as:
ekM＝ydM-ψyk (7)
wherein,
ekM＝[ek(t1),ek(t2),...,ek(tM)]T (8)
fourthly, designing the point-to-point iterative learning trajectory tracking optimization algorithm:
in practical applications, the tracking error e(k+1)MAnd system input uk+1The convergence performance and energy consumption of each running batch of the system are usually reflected, and the smaller tracking error, control energy and control oscillation thereof are important to the overall performance of the system and the loss of an actual actuator, so that the error and the control quantity need to be considered and optimized in the iterative learning control optimization process; the following performance indicator functions are defined:
Figure FDA0002772643880000032
wherein Q is a symmetric positive definite weight array, R and S are corresponding symmetric non-negative definite weight arrays, and
Figure FDA0002772643880000033
substituting the formula (4), the formula (7) and the formula (10) into the formula (9), and obtaining an optimized iterative learning control law by solving a quadratic optimal solution of the formula:
uk+1＝Luuk+LeekM (11)
wherein,
Lu＝[(ψG)TQ(ψG)+R+S]-1[(ψG)TQ(ψG)+R]，Luis the gain of the iterative learning control law controller input term,
Le＝[(ψG)TQ(ψG)+R+S]-1(ψG)TQ，Leis the gain of the error term;
fifthly, analyzing the convergence and robustness of the point-to-point iterative learning trajectory tracking optimization algorithm:
if the optimized iterative learning control law (11) meets the condition eta | | | | Lu-Le(ψG)||i2<1，||·||i2Representing the maximum singular value of the matrix, then when the iteration batch k → ∞ the tracking output error of the system at the finite expected tracking point converges;
the formula (11) can be rewritten from the formulae (4) and (7):
Figure FDA0002772643880000041
defining system steady state control inputs
Figure FDA0002772643880000042
According to formula (12):
u∞-uk+1＝(Lu-Le(ψG))(u∞-uk)+ξ (13)
wherein xi ═ I- (L)u-Le(ψG)))u∞-Le(ydMψ d) is a zero matrix, and two sides of the formula (13)Taking norm can obtain | | | u∞-uk+1||＝||(Lu-Le(ψG))(u∞-uk)+ξ||≤η||u∞-ukIf the condition eta is satisfied<1, then the control input converges in the norm sense, further yielding:
Figure FDA0002772643880000043
where I is the identity matrix of the corresponding dimension, then:
Figure FDA0002772643880000044
when each weight matrix parameter is determined, the steady state error norm can be obtained by the equation (15) to be converged to a constant value finally, if S is smaller, the point-to-point tracking steady state error of the system under the action of the optimization iterative learning control law (11) is smaller, and when S is taken as a zero matrix, the tracking output error norm at the tracking point of the system can be converged to zero;
in the actual process production process, a controlled system usually has certain modeling uncertainty; considering the uncertainty Δ of the input/output transfer matrix G in time series plus a variation in the time domainGAnd further defining the expression of the input-output transfer matrix with uncertainty in the time series as Gθ＝G(I+ΔG) And define ΔGThe bounded condition of the uncertainty factor is | | | Δ | | | luminancei2≤1，
The conditions for the system robust convergence are:
||Lu-Le(ψGθ)||i2<1
if the symmetric non-negative constant weight matrix R is taken as a zero matrix, the following formula is provided:
Figure FDA0002772643880000045
wherein W is an uncertain weight matrix;
it can be obtained if the parameters satisfy the condition:
||[(ψG)TQ(ψG)+S]-1(ψG)TQ(ψG)W||i2<1 (16)
the system robustness converges; if the symmetric non-negative fixed weight array R is equal to rI, R is more than or equal to 0, if the condition (16) is met, robust convergence of the discrete state space model (3) under the action of the optimized iterative learning control law (11) can still be ensured; because (psi G)TQ (psi G) + S is a symmetric positive definite matrix, so that it can be decomposed into singular values by (psi G)TQ(ψG)+S＝U∑UTWhere U is unitary matrix and Σ is element σiThe full-rank diagonal matrix of (1) is expressed as Σ ═ diag { σ }iJ, for convenience of representation, let
Figure FDA0002772643880000051
And conditions of
Figure FDA0002772643880000052
Satisfy the requirement of obtaining | | H | non-conducting phosphori2＝α<1，
Then one can get:
Figure FDA0002772643880000053
if the condition (16) is satisfied, the system robustness is converged, that is, the design of the parameter R ═ rI does not affect the robustness of the system, and on the other hand, the design of the parameter S needs to satisfy the condition (16);
sixthly, analyzing the convergence and the robustness of the point-to-point iterative learning trajectory tracking optimization algorithm under the input constraint condition:
in many industrial process control applications, in order to ensure that an industrial process operates safely and smoothly, certain constraints need to be imposed on input variables, the input signal constraints include constraints on input amplitude, constraints on input amplitude with respect to time direction and batch direction, and the constraints are often expressed in inequalities;
saturation constraint of controller input:
ulow≤uk+1≤uhi (18)
wherein u islow，uhiAre respectively controller input uk+1The lower and upper bound values of (1);
input variation constraint between two adjacent sampling times:
ulow≤uk+1≤uhi (19)
wherein u islow、uhiAre respectively an input variable uk+1Lower and upper bound values, u, varying along the time axisk+1(t)＝uk+1(t-1)-uk+1(t-2)；
Input variation constraints between two adjacent batches:
Δulow≤Δuk+1≤Δuhi (20)
wherein Δ ulow、ΔuhiAre respectively input variables Δ uk+1Lower and upper bound values, Δ u, that vary along the axis of iterationk+1＝uk+1-uk；
All of the above constraint equations can be converted to Δ uk+1First, the saturation constraint (18) of the controller input may translate into:
ulow-uk≤Δuk+1≤uhi-uk (21)
let u (0) be u0Then u isk+1Can be expressed as:
uk+1＝μuk+1 (22)
wherein:
Figure FDA0002772643880000061
the input variation constraint (19) between two adjacent sampling times can therefore be translated into:
ulow-μuk≤μΔuk+1≤uhi-μuk (24)
the constraints described above can be combined into the following linear inequalities:
ζuΔuk+1≥ζk+1 (25)
wherein:
Figure FDA0002772643880000062
theoretically, the input Δ u described by equation (25)k+1The constraint of (1) is a convex set, here denoted by Ω; the error equation for model uncertainty can be expressed as:
e(k+1)M＝ekM-ψGθΔuk+1 (26)
wherein G isθThe input and output transfer matrix on the time sequence with uncertainty corresponds to the input and output transfer matrix G on the time sequence;
substituting equation (26) into the performance indicator function (9) yields:
Figure FDA0002772643880000063
and (3) convergence and robustness analysis:
under the constraint, assume that condition 1) there is one feasible expected input u∞And corresponding expected error e∞0, and satisfies ulow≤u∞≤uhi，ulow≤u∞≤uhiAnd Δ ulow<0，Δuhi>0; assuming the condition 2) Q is a symmetrical positive definite weight array, and R and S are corresponding symmetrical non-negative definite weight arrays; the system robustness converges and the system steady state error converges to 0 when S takes the zero matrix, i.e., k → ∞ Δ uk→0，ekM→0；
Defining the optimal performance indicator function of the (k + 1) th batch as:
Figure FDA0002772643880000071
first consider Δ uk+1Case 0, when uk+1＝uk，e(k+1)M＝ekM,uk+1＝ukSo that (e) can be derivedkM,0,uk) E Ω, because the performance indicator function is at point (e)kM,0,uk) The value of (b) is always greater than or equal to the optimum value, so the following relationship can be obtained:
Figure FDA0002772643880000072
by analogy, the following results are obtained:
Figure FDA0002772643880000073
as is clear from the formula (30), Δ u when k → ∞ is satisfiedk→ 0 because Δ u when k is sufficiently largek→ 0, and satisfies the condition Δ ulow<0，Δuhi>0, so the constraint Δ ulow≤Δuk+1≤ΔuhiThe same is satisfied; from the assumption condition 1) that u is knownk+1＝u∞Time e(k+1)MWhen the value is equal to 0, then
Figure FDA0002772643880000074
Is also a feasible point within a restricted area, wherein
Figure FDA0002772643880000075
Is obvious (e)(k+1)M,Δuk+1,uk+1) Is also a feasible point in a constrained region, because Ω is a convex set, it can be deduced that a point between any two feasible points in Ω is a feasible point;
meanwhile, the performance index function J is solvedk+1From (e)(k+1)M,Δuk+1,uk+1) To
Figure FDA0002772643880000076
The directional derivatives of (a):
Figure FDA0002772643880000077
from equation (31):
Figure FDA0002772643880000078
since Δ u when k → ∞ isk→ 0, so when k → ∞ time equation (32) can be simplified to:
Figure FDA0002772643880000079
as can be seen from equation (33) and assumption condition 2), when k → ∞ is reached, the steady-state output error of the system converges to a certain finite value, and when S takes a zero matrix, the steady-state output error reaches a minimum value of 0; the result shows that when the assumed conditions are satisfied, the input constraint model uncertain system can still be converged under the action of the optimization iterative learning control law (11); when the system is a nominal system, i.e. GθThe same holds true when G;
and seventhly, determining an input vector of each iteration batch of the motor-driven single mechanical arm system according to a robust optimization iterative learning control law, inputting the obtained input vector into the motor-driven single mechanical arm system to perform point-to-point trajectory tracking control, and tracking the expected output at the specified tracking point by the motor-driven single mechanical arm system under the control action of the input vector.
CN201911119240.2A 2019-11-15 2019-11-15 Point-to-point iterative learning optimization control method of motor-driven single mechanical arm system Active CN110815225B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN201911119240.2A CN110815225B (en) 2019-11-15 2019-11-15 Point-to-point iterative learning optimization control method of motor-driven single mechanical arm system

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN201911119240.2A CN110815225B (en) 2019-11-15 2019-11-15 Point-to-point iterative learning optimization control method of motor-driven single mechanical arm system

Publications (2)

Publication Number Publication Date
CN110815225A CN110815225A (en) 2020-02-21
CN110815225B true CN110815225B (en) 2020-12-25

Family

ID=69555681

Family Applications (1)

Application Number Title Priority Date Filing Date
CN201911119240.2A Active CN110815225B (en) 2019-11-15 2019-11-15 Point-to-point iterative learning optimization control method of motor-driven single mechanical arm system

Country Status (1)

Country Link
CN (1) CN110815225B (en)

Families Citing this family (10)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111510020B (en) * 2020-04-27 2021-04-23 广州大学 Open-close loop iterative learning control method for high-relative-degree piezoelectric motor
CN111722628B (en) * 2020-06-19 2021-07-09 江南大学 Point-to-point tracking control method for multi-agent track updating iterative learning
CN112318505B (en) * 2020-10-28 2021-11-16 江南大学 Mobile robot batch length-variable iterative learning optimization control method
CN113110063B (en) * 2021-05-08 2022-05-31 江南大学 Robust monotonous convergence point-to-point iterative learning control method of single-axis feeding system
CN113625563B (en) * 2021-08-06 2024-03-15 江南大学 DC motor quantization iteration learning fault-tolerant control method
CN113671916B (en) * 2021-08-19 2023-02-28 华东理工大学 Process industrial process optimization control method and device considering input constraints
CN113900377B (en) * 2021-10-19 2023-10-20 国网江苏省电力有限公司盐城供电分公司 Point-to-point iterative learning minimum energy control method for double-rotor pneumatic system
CN115047763B (en) * 2022-06-08 2023-10-13 国网安徽省电力有限公司天长市供电公司 Minimum energy control method for multi-unmanned aerial vehicle system
CN116048085B (en) * 2023-02-03 2023-11-07 江南大学 Fault estimation and fault-tolerant iterative learning control method for mobile robot
CN116810803B (en) * 2023-08-30 2024-04-12 合肥工业大学 Robust control method for joint module of cooperative robot

Family Cites Families (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US3909600A (en) * 1972-06-26 1975-09-30 Cincinnati Milacron Inc Method and apparatus for controlling an automation along a predetermined path
CN100561122C (en) * 2007-11-05 2009-11-18 中山大学 A kind of trailing type measurement mechanical arm calibration method based on distance restraint
CN108638071A (en) * 2018-05-22 2018-10-12 四川超影科技有限公司 A kind of crusing robot optimal path dynamic programming method
CN110307838B (en) * 2019-08-26 2019-12-10 深圳市优必选科技股份有限公司 Robot repositioning method and device, computer-readable storage medium and robot
CN111098309A (en) * 2020-01-06 2020-05-05 江南大学 Hybrid control method, device and system for mechanical arm

Also Published As

Publication number Publication date
CN110815225A (en) 2020-02-21

Similar Documents

Publication Publication Date Title
CN110815225B (en) Point-to-point iterative learning optimization control method of motor-driven single mechanical arm system
Owens et al. Iterative learning control—An optimization paradigm
Xu et al. Linear and nonlinear iterative learning control
Qi et al. Stable indirect adaptive control based on discrete-time T–S fuzzy model
Padhi et al. Model-following neuro-adaptive control design for non-square, non-affine nonlinear systems
Zhang et al. An exponential-type anti-noise varying-gain network for solving disturbed time-varying inversion systems
CN112571420B (en) Dual-function model prediction control method under unknown parameters
Chen et al. Convex polytopic modeling for flexible joints industrial robot using TP-model transformation
Wang et al. An improved koopman-MPC framework for data-driven modeling and control of soft actuators
CN113110063B (en) Robust monotonous convergence point-to-point iterative learning control method of single-axis feeding system
Van Tran et al. Artificial chemical reaction optimization algorithm and neural network based adaptive control for robot manipulator
Umamaheswari et al. ANFIS PD plus I control on simscape model of nonlinear physical system
Lim et al. Pareto iterative learning control: Optimized control for multiple performance objectives
Zebin et al. Dynamic modeling and fuzzy logic control of a two-link flexible manipulator using genetic optimization techniques
Lin et al. Hybrid adaptive fuzzy controllers with application to robotic systems
CN115047763B (en) Minimum energy control method for multi-unmanned aerial vehicle system
Knani Dynamic modelling of flexible robotic mechanisms and adaptive robust control of trajectory computer simulation––Part I
Hou et al. Dynamic neural network predictive compensation-based point-to-point iterative learning control with nonuniform batch length
CN113900377B (en) Point-to-point iterative learning minimum energy control method for double-rotor pneumatic system
Jafari et al. Adaptive PID control of a nonlinear servomechanism using recurrent neural networks
Dobriborsci et al. Discrete robust controller for ball and plate system
Saidi et al. Sliding Mode Control of a 2DOF Robot Manipulator: A Simulation Study Using Artificial Neural Networks with Minimum Parameter Learning
Lazarević et al. Advanced open-closed-loop PIDD 2/PID type ILC control of a robot arm
Tutunji et al. A three-stage PSO-based methodology for tuning an optimal PD-controller for robotic arm manipulators
Ahifar et al. Terminal synergetic design of a nonlinear robot manipulator in the presence of disturbances

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant
TR01 Transfer of patent right

Effective date of registration: 20230626

Address after: No. 77-38-B, No.13 Road, Shenyang Economic and Technological Development Zone, Liaoning Province, 110027

Patentee after: Zhongkexin Micro Intelligent Equipment (Shenyang) Co.,Ltd.

Address before: No. 1800 road 214122 Jiangsu Lihu Binhu District City of Wuxi Province

Patentee before: Jiangnan University

TR01 Transfer of patent right