CN114740735A - Variable-length feedback-assisted PD type iterative learning control method of single-joint robot - Google Patents

Variable-length feedback-assisted PD type iterative learning control method of single-joint robot Download PDF

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CN114740735A
CN114740735A CN202210493291.7A CN202210493291A CN114740735A CN 114740735 A CN114740735 A CN 114740735A CN 202210493291 A CN202210493291 A CN 202210493291A CN 114740735 A CN114740735 A CN 114740735A
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陶洪峰
官上雷
庄志和
郑月昌
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Abstract

The invention discloses a variable-length feedback-assisted PD type iterative learning control method of a single-joint robot, relating to the field of iterative learning control; the method is used for storing the operation information of the latest iteration batch at each time point of the system by constructing an update error and an input sequence generated by recursion, and then jointly repairing the update error and the input sequence into an input signal of the system by combining the feedback error information of the current iteration batch of the system, thereby constructing a feedback-assisted PD type iterative learning control method; in addition, the convergence of the designed feedback-assisted PD type iterative learning control algorithm on a variable-length system is proved by an inductive analysis method. The method can be suitable for the track tracking control problem of the single-joint robot system under the variable batch length problem, and the accurate tracking of the system output to the expected track is realized.

Description

Variable-length feedback-assisted PD type iterative learning control method of single-joint robot
Technical Field
The invention relates to the field of iterative learning control, in particular to a variable-length feedback-assisted PD type iterative learning control method for a single-joint robot.
Background
Industrial robots are widely used in the fields of industrial manufacturing and the like. To the production process of repeatability, industrial robot can replace manpower to carry out the work of some repetitive nature exquisitely, can also carry out stable work in dangerous abominable environment simultaneously, has greatly promoted industrial process's production efficiency and product quality. Therefore, there is an increasing demand for the control performance of industrial robots. The control of the industrial robot mainly focuses on the accurate tracking control of the track of the industrial robot, and the target has important significance for ensuring the production safety and improving the product quality.
The single joint robot system is a basic and very important robot system, and the complex robot system is usually built on the basis of the single joint robot system, and the motion of a plurality of joints jointly forms the overall action of the complex robot system. A single-joint robot system is taken as a typical time-varying and nonlinear system, and a system control strategy is a key technology of the system. Aiming at the problem of trajectory tracking of a single-joint robot system, the control effect can be doubled with half the effort by selecting a proper control algorithm. In view of its characteristic of repeatedly performing the same task, Iterative Learning Control (ILC) is widely applied to a trajectory tracking task of a robot. The core idea of the ILC is to generate the current control input by using the previous tracking error and the control input information to adjust the tracking estimation of the system, so that the control system has the capability of self-learning and self-improvement. The strategy defines each execution of the system repeated task as an iteration batch, and the actual output track of the robot is gradually converged to the expected track along with the rise of the iteration batch. This requires that the length of each iteration batch during the operation of the repeatedly operated single-joint robot system is the same as the expected length, so as to ensure the learning efficiency and tracking effect of the system in the whole expected time. However, in some practical application scenarios of single-joint robots, this requirement may not always be met. For example, a single-joint robot system is widely used in a transfer robot, a gantry crane, and other cargo robot systems, and since a load is allowed to move only in a specific area of a reference object, when the output of the system deviates out of a constraint range, the operation of the system for the lot is immediately interrupted. Thus. The batch lengths of all the iterative batches are not exactly the same. However, even for incomplete trials, the available tracking sections may still be used to improve the learning performance of the control system. In addition, the single-joint robot can be applied to a lower limb auxiliary rehabilitation robot, the lower limbs of a patient are driven to move by a proper angle by controlling the movement of the arm of force of the robot, and the recovery of the lower limb movement function of the patient with injured lower limbs or a stroke is facilitated. However, in practical applications, due to physical ability of the patient, interference from external factors or safety considerations, the lower limb movements of the patient cannot be kept completely consistent, i.e. the length of the iteration batch of the single-joint robot system varies. The variable batch length problem in the single-joint robot control system is formed, a proper control strategy is designed, effective outputs generated by the system under various different batch lengths are fully utilized for learning, and the control signal of the ILC is corrected to realize accurate tracking of the single-joint robot system on the expected track, so that the method is the key point for solving the problem.
In many conventional ILC designs, a key requirement is that as the number of iterations of the control system increases, the batch length for each iteration is a fixed value, referred to as the desired length. When the actual batch length of the system changes, the system operation fails to reach the expected length, and the information of the part of the actual batch length smaller than the expected batch length is lost. It is common practice to set the error information for this part that cannot be obtained to 0, so that all iteration batches of the system can be analyzed with reference to the desired length under the conventional ILC framework. In addition, some researches are dedicated to constructing a high-order error signal compensation missing part by using the operation information of previous iteration of the system, the robustness of the system to the variable batch length is enhanced to a certain extent, the system can provide full-length error information for each iteration batch, and various control algorithm designs of the ILC are provided based on the idea. However, such control strategies may dilute errors that are large or from the most recent iteration batch, thereby affecting the convergence speed of the system.
Disclosure of Invention
The invention provides a variable-length feedback-assisted PD type iterative learning control method of a single-joint robot aiming at the problems and the technical requirements. An updating error and an input sequence generated by real-time recursion are constructed, and the latest generated error and the latest effective input signal generated by the system at each time point are respectively stored, so that the system can better utilize the latest running information from a plurality of recent iterations. Based on the design, a feedback-assisted PD type ILC control law for a variable batch length system is provided, and the tracking performance is improved by using an update sequence and the derivation of the tracking error of the current iteration as a correction term of an input signal.
The technical scheme of the invention is as follows:
a variable-length feedback-assisted PD type iterative learning control method of a single-joint robot comprises the following steps:
firstly, carrying out analysis modeling on input and output of a single-joint robot system:
a typical structure of a single-joint robot system is considered to be constituted by connecting both ends by a single rod. The cross section of the single rod is rectangular and is a slender rigid structure, and two connected ends of the single rod are an axle center end and a free end respectively. The axis end is used as a track reference point, and is composed of a motor and a speed reducer matched with the motor, so that joints in the single-joint robot are formed together. The axis end is used as a circle center and a reference point, the single rod can rotate within 360 degrees, the length of the single rod is mastered by determining the angle between the single rod and a horizontal plane or other reference planes, and then the specific position of the other end of the single rod in the space is positioned. The other end of the single rod is called as a free end, and the control target of the single-joint robot is to accurately control the track of the free end so as to realize accurate and repetitive operation of the industrial process. According to different application scenes, the free end can be provided with a mechanical hand for carrying, or is connected with the axis end of another single-joint robot system to form a multi-joint robot system.
In a single-joint robot system, the included angle between a single rod and a horizontal plane is used as the running state of the system, and is measured by an absolute value angle encoder. The input signal of the system is the torque acting on the axle center end of the single rod, and the external control signal drives the axle center end motor to generate the torque to drive the single rod to rotate. For simplicity of calculation, the weight of the motor and gears can be neglected, as well as the inertia of the single arm itself. And taking the axis end as an original point, taking the translation of the absolute coordinate system as a corresponding coordinate axis, and rotating the single rod around the Z axis by taking the original point as a circle center. Based on this setting, the dynamic model of the single-joint robot system is expressed as:
Figure BDA0003632549100000031
wherein: t represents time, Jm＝1.33Ml2The moment is the moment of inertia, theta (t) is the rotation angle of a single rod of the single-joint robot, M is the overall mass of the single rod, g is the gravitational acceleration, l is the rotation distance of the center of mass from the axis end, and tau (t) is the moment acting on the axis end.
Therefore, a dynamic relation between the single-arm rotation angle of the single-joint robot system and the moment of the axis end is established, and the rotation angle of the free end can be accurately controlled by adjusting the rotation moment, so that the running track of the single-joint robot is further controlled, and the specific application of the single-joint robot is realized.
Secondly, constructing a discrete state space equation of the single-joint robot system:
in order to apply iterative learning control to the single-joint robot system and carry out convergence analysis of the single-joint robot system, the dynamic model of the moment-rotation angle of the single-joint robot constructed in the first step is rewritten into a nonlinear discrete state space equation. Let the sampling period of the system be 0.1, and θ (t) be x(1)(t),θ(t+1)＝x(2)(t), u (t) ═ τ (t), then the system is further described as:
Figure BDA0003632549100000032
equation (2) further uses a non-linear function f (x) to account for repeated operation of the systemk(t), t) is shown. Therein
Figure BDA0003632549100000041
Figure BDA0003632549100000042
And
Figure BDA0003632549100000043
representing state information for the kth iteration of the system. Establishing a nonlinear discrete state space equation of the single-joint robot system:
Figure BDA0003632549100000044
wherein, f (x)k(t), t) is
Figure BDA0003632549100000045
Is a non-linear function of (a).
Figure BDA0003632549100000046
Respectively representing the state, input and output of the kth iteration of the system,
Figure BDA0003632549100000047
is a time variable. k is 0,1,2t，CtIs a system matrix of appropriate dimensions, and Ct+1BtIs of full rank. The system has only ideal control input
Figure BDA0003632549100000048
I.e. under the influence of this signal, such that:
Figure BDA0003632549100000049
wherein x isd(t) and yd(t) desired state and desired output, AtIs a system matrix with appropriate dimensions.
For this iterative system, there are two assumptions: once the initial state of each batch of the iterative process is equal to the desired initial state, xk(0)＝xd(0). Secondly, the system meets the global Lipschitz condition, namely, a real number k existsfIs more than or equal to 0, and belongs to [0, N ] for all td]All the requirements are that:
Figure BDA00036325491000000410
thirdly, designing an updating error and input sequence aiming at the variable batch length problem:
the actual run length of the k-th iteration is N because the batch lengths are not uniform in the actual operation of the systemkReasonably consider NkThere is a minimum value NLAnd a maximum value NH. Since only the tracking performance in the desired time is concerned, N is consideredd＝NHWherein N isdIs the desired run length of the system. From this, N is obtainedk∈[NL,Nd]I.e., t is 0,1,2L,...,Nk,...,Nd。
According to the characteristics of iterative learning control and the setting, historical information of past batches is used for controlling the signal uk(t) update correction is performed so that the control target (4) can be achieved.
Introducing an indication function
Figure BDA00036325491000000411
If and only if the system has reached time t in the operation of the kth iteration, the function is equal to 1 and, conversely, 0, i.e. it is a function
Figure BDA00036325491000000412
It is noted that under the influence of a batch length that is not fixed, for a batch length of NkFor the kth iteration of (1), time Nk+1,...,NdIs not available. This means that the error in this time period cannot be calculated. In view of this, the error of this portion is set to 0, so that the error of the kth lot is expressed as:
Figure BDA00036325491000000413
wherein e isk(t)＝yd(t)-ykAnd (t) is the error of the expected track and the actual output track of the system. Using the indicator function previously proposed, the above equation is simply expressed as
Figure BDA00036325491000000414
For the upcoming iteration, its run length cannot be predicted, so all u's are calculatedk(t)，t∈[0,Nd-1]Is necessary. For this reason, all
Figure BDA0003632549100000051
The following update error sequence was constructed:
Figure BDA0003632549100000052
in the formula Ek(t) is generated recursively and updated with the run information of the current iteration after the run ends at each time.
When the iteration length distribution of the system is not uniform, Ek(t) composition comprising error information for a plurality of batches, using the updated error sequence Ek(t) calculating the input for the next iteration by a sufficiently long error signal. However, it should be noted that for each upcoming iteration, there is a sufficiently long error signal used to update all uk+1(t)，t∈[0,Nd-1]Whether or not it is actually put into operation, for which purpose E is introducedk(t) constructing an updated input sequence with the same concept:
Figure BDA0003632549100000053
wherein the indicator function is
Figure BDA0003632549100000054
The reason for this is that the input at time t needs to have an error at time t +1, so that the recursively generated updated input sequence stores the input signal that was updated last at each time point and has the output result.
Fourthly, designing a feedback-assisted PD type iterative learning control algorithm aiming at the variable batch length problem:
by utilizing the update error sequence and the update input sequence constructed in the third step, the system stores the latest input signal and acquires the corresponding output signal after running each time, thereby providing a new framework for the control of the system with variable batch length. In order to improve the tracking performance of the system and accelerate the convergence speed of errors, the following feedback-assisted PD type iterative learning control algorithm is built by utilizing an updated error sequence and an updated input sequence and combining the errors of the current batch operation:
uk+1(t)＝Uk(t)+LtEk(t+1)+ΓtEk(t)+KtEk+1(t),t∈[0,Nd], (9)
wherein Lt，ΓtAnd KtIs the learning gain of the system. By applying the learning law to a discrete nonlinear time varying system (3), the following theorem is obtained: for discrete nonlinear time-varying systems (3), an iterative learning control update algorithm (9) is applied, if a suitable learning gain is selected such that:
0<||I-Ct+1BtLt||<1,
then as the number of iterations k approaches infinity, at time interval t e 0, Nd]The tracking target described in (2) above, i.e.
Figure BDA0003632549100000055
Where I is a dimensionally appropriate identity matrix.
Fifthly, analyzing the convergence of the feedback-assisted PD type iterative learning control algorithm under the variable batch length:
before analyzing the convergence of the feedback-assisted PD type iterative learning control algorithm proposed in the fourth step, the method needs to be implementedThe run length of the system iteration is divided into two parts. The first part is the minimum length range that the system can reach for each iteration, and the learning of this part can be equivalent to the traditional ILC control problem. The second part is the part where the system batch length varies, and the actual run length of each iteration batch may be at any value due to the influence of the varying batch length. A reasonable assumption three is given for this: for an arbitrary iteration batch k and time t e {0,1kAnd f, in the past sigma batches, the batch length of at least one batch is greater than or equal to t.
The first part of the control algorithm convergence verification is as follows:
proving the theorem at t ∈ [0, NL]This is true. Since for all iterations NLAll the time comes, i.e.
Figure BDA0003632549100000061
Then the error over the time interval is obtained and updated every iteration to yield
Figure BDA0003632549100000062
Further, according to the formula (1), when t ∈ ψ, ψ [0, N ]L-1]Then, using hypothesis two, we obtain:
Figure BDA0003632549100000063
in the formula
Figure BDA0003632549100000064
As a time-varying system parameter BtMaximum value in time ψ, and x is found from hypothesis onek+1(0)-xk(0)＝0，
Figure BDA0003632549100000065
Thereby, according to
Figure BDA0003632549100000066
The definition of (c) yields:
Figure BDA0003632549100000067
taking norm of two sides of the above formula to obtain:
Figure BDA0003632549100000068
order to
Figure BDA0003632549100000069
As a time-varying system parameter Ct+1Maximum value within time ψ, further yields:
Figure BDA00036325491000000610
wherein the parameter alpha is more than or equal to kfFor the sake of brevity, let
Figure BDA00036325491000000611
And
Figure BDA0003632549100000071
it follows from the above equation:
Figure BDA0003632549100000072
to construct the lambda norm, a constant lambda is defined>0, the two ends of the above formula are multiplied by alpha-λ(t+1)And taking the minimum supremum, including:
Figure BDA0003632549100000073
wherein ρ ═ supt∈ψ||I-Ct+1BtLtL. According to the definition of the lambda norm, the following results:
Figure BDA0003632549100000074
wherein the content of the first and second substances,
Figure BDA0003632549100000075
combining (15) and (16) to obtain:
||ek+1(t+1)||λ≤ρ0||ek(t+1)||λ (17)
in the formula
Figure BDA0003632549100000076
When 0 is present<||I-Ct+1BtLt||<1 time, 0<ρ<1, so when λ is large enough, let ρ be0<1 and 1-k3ρ1>0. Thus, the device
Figure BDA0003632549100000077
Further obtain
Figure BDA0003632549100000078
At the same time, because
Figure BDA0003632549100000079
Thus obtaining
Figure BDA00036325491000000710
Then it verifies t e [0, N ∈L]And (5) performing time iteration to learn the convergence of the control law.
The second part of the control algorithm convergence verification is as follows:
by inductive analysis, inductive assumptions were made: for arbitrary T e [ N [ ]L,Nd-1]，
Figure BDA00036325491000000711
It is thus to be demonstrated that when T is T +1,
Figure BDA00036325491000000712
defining a subset
Figure BDA00036325491000000713
It contains all running times NkBatch number of not less than T +1. Rearranging the elements in the solution according to the appearance order to obtain
Figure BDA00036325491000000714
And k is given by the hypothesis threeiAnd k isi+1Is less than or equal to σ, i.e., ki+1-kiSigma is less than or equal to. Using this subsequence, when T ═ T +1, we obtain:
Figure BDA00036325491000000715
and:
Figure BDA00036325491000000716
using the formula to obtain:
Figure BDA0003632549100000081
since for all
Figure BDA0003632549100000082
That is, after the batch runs are completed, Uk(T) and Ek(T +1) is not updated, i.e.:
Figure BDA0003632549100000083
update laws according to ILC while defining
Figure BDA0003632549100000084
Obtaining:
Figure BDA0003632549100000085
next, formula (22) is brought into formula (20) while defining
Figure BDA0003632549100000086
Further obtaining:
Figure BDA0003632549100000087
for the
Figure BDA0003632549100000088
By recursion we obtain:
Figure BDA0003632549100000089
wherein
Figure BDA00036325491000000810
By updating the generation rule of the input sequence, when
Figure BDA00036325491000000811
When the temperature of the water is higher than the set temperature,
Figure BDA00036325491000000812
is updated. In view of this, a series of subsets are defined
Figure BDA00036325491000000813
In a similar manner to that described above,
Figure BDA00036325491000000814
are arranged in the order of appearance
Figure BDA00036325491000000815
It can be seen that for time T e 0, T]Input signal of
Figure BDA00036325491000000816
That is, it will only be at
Figure BDA00036325491000000817
After the end of the batch in (c), update is performed to obtain the value of [0, T ] for all T e],
Figure BDA00036325491000000818
And k isiIs the same as the raw material of the batch,
Figure BDA00036325491000000819
and k isi+1Are in the same batch. Whereby when T ∈ [0, T-1 ]]Obtaining the following results:
Figure BDA00036325491000000820
at the same time, from
Figure BDA00036325491000000821
And
Figure BDA00036325491000000822
further get kiThe time → ∞ of the time,
Figure BDA00036325491000000823
and k isi→ ∞ with k → ∞. According to the generalised assumptions above, there are
Figure BDA00036325491000000824
And, from the hypothesis three, at ki+1And k isiAt most, there are σ -1 iterative batches between batches, i.e., for each time T ∈ [0, T]，
Figure BDA0003632549100000091
Is limited, thereby ensuring
Figure BDA0003632549100000092
As a result obtain
Figure BDA0003632549100000093
Next, taking norm at both ends of equation (23) at the same time, the following inequality is obtained:
Figure BDA0003632549100000094
wherein the content of the first and second substances,
Figure BDA0003632549100000095
combined with the above derivation
Figure BDA0003632549100000096
Thereby applying the theorem to the inequality (26) for an iterative system zk+1＝Dzk+dk,
Figure BDA0003632549100000097
Wherein z iskAnd
Figure BDA0003632549100000098
respectively a state and a bounded internal input,
Figure BDA0003632549100000099
is a system matrix; when in use
Figure BDA00036325491000000910
When it is obtained
Figure BDA00036325491000000911
If and only if 0<||D||<1。
It follows that if and only if 0<||I-CT+1BTLT||<When the pressure of the mixture is 1, the pressure is lower,
Figure BDA00036325491000000912
namely, it is
Figure BDA00036325491000000913
Based on the inductive analysis method, further summarize
Figure BDA00036325491000000914
This is so far the second part is certified.
Combining the first part and the second part, assuming that the currently selected learning gain matrix satisfies 0<||I-CT+1BTLT||<When the pressure of the mixture is 1, the pressure is lower,
Figure BDA00036325491000000915
it is true that the tracking objective is achieved. Therefore, the convergence certification of the feedback-assisted PD type iterative learning control algorithm under the variable batch length is completed.
Sixthly, realizing the track tracking of the single-joint robot system under the variable batch length
And calculating an input signal required by each time point in each iteration process of the system according to the designed feedback-assisted PD type iterative learning control algorithm, and applying the signal to the single-joint robot system to perform system control so as to obtain response output, so that the motion trail of the free end of the system can track the preset expected output trail of the system, and accurate trail tracking control is realized.
The beneficial technical effects of the invention are as follows:
the application discloses a nonlinear time-varying system with repetitive motion characteristics, such as a single-joint robot system, the single-joint robot system serves as a controlled object, a recursively generated updating error and input sequence are constructed for the variable batch length problem of the controlled object possibly occurring in practical application, and a feedback-assisted PD type iterative learning control algorithm is constructed by utilizing the sequence and combining a current iterative error signal, so that the system can obtain higher convergence speed while the tracking requirement is ensured under the variable batch length problem. Meanwhile, convergence analysis is carried out on the designed feedback-assisted PD type iterative learning control algorithm by utilizing an inductive analysis method, so that the convergence of the tracking error of the system is ensured.
Drawings
Fig. 1 is a schematic block diagram of a single joint robot control system disclosed in the present application.
Fig. 2 is a mechanical structure diagram of the single-joint robot disclosed in the present application.
Fig. 3 is a model block diagram of the single-joint robot control system disclosed in the present application.
Fig. 4 is a graph of the output of different batches of the single-joint robot system disclosed in the present application along a time axis.
Fig. 5 is a graph of error curves along a time axis for different batches of the single-joint robotic system disclosed in the present application.
Fig. 6 is a graph of a maximum error norm along a batch axis for a single joint robotic system as disclosed herein.
Detailed Description
The following further describes the embodiments of the present invention with reference to the drawings.
Referring to fig. 1, a schematic block diagram of a single joint robot control system used in the present application is shown. The system carries out control law design through a human-computer interaction device, applies a control algorithm to a control system of the single-joint robot to generate corresponding signals to drive an actuating mechanism at an axis end, and finally realizes the track tracking of a free end. And detecting the current states of the free end and the axis end through an angle measuring device so as to update the next iteration control law. The system adopts a Controller Area Network (CAN) bus to carry out bidirectional communication with a human-computer interaction device such as an upper computer, the current running state of the system is fed back to a user end in real time so as to be used by the user for carrying out state monitoring and parameter correction of the system, the control system CAN be controlled by an embedded ARM microprocessor with the model of STM32F103RCT6, the CAN controller is highly integrated in the chip, a standard CAN protocol interface is provided at the same time, and the CAN transceiver with the model of SN65HVD230 is connected with the chip, so that the function of sending system information to a CAN network CAN be realized. In addition, the chip is used for generating an adjustable PWM signal to drive the axle center end driving system, so that the output torque of the axle center end actuating mechanism can accord with an expected value, and the accurate tracking of the free end track is realized. The core of the axis end driving system adopts a two-phase DMOS full-bridge driving chip with the model of A3997SBT, and the PWM signal from the control system is accessed into an ENABLE pin of the chip, so that the direct current motor can be accurately controlled. The motion information of the free end of the system, calculated from its rotation angle, can be obtained by the AS5048B magnetic rotary encoder. The resolution ratio is set to be 14 bits, namely 0.022 degrees, an angle value of the free end of the single-joint robot within 0-360 degrees relative to the horizontal plane can be obtained and transmitted back to the control system, and the calculation of the running track and the error of the free end and the display of the running state are carried out. The measuring frequency of the single-joint robot control system is set to be 10Hz, namely the sampling time is 0.1s, so that the real-time control of the free end track of the robot is realized.
Fig. 2 shows a mechanical structure diagram for a single joint robot system of the present application. As shown in the figure, in the single-joint robot system aimed at by the application, the included angle between the free end of the robot and the horizontal plane is the control output of the system, and the angle measuring device is installed at the joint of the single arm of the robot and the axis end so as to obtain the real-time running state of the system. The motor and the speed reducer form an actuating mechanism at the shaft center end, and the actuating mechanism is used for acquiring a current signal from a driving system, changing the output torque at the shaft center end and finally changing the change track of the included angle at the free end.
Fig. 3 shows a model block diagram of a single joint robot control system disclosed in the present application. The shaft center end input signal of the single-joint robot at the kth batch time t is ukAnd (t) applying the input signal to the robot system by the system axis end driving mechanism, and changing an included angle between the free end and the horizontal plane. The expected trajectory of the system is pre-stored in an expected trajectory memory and the difference is calculated in real time from the actual output of the system. Due to the change of the batch length, the error of the system is calculated and stored in the updating error sequence Ek(t) and updating the input sequence Uk(t) combining, and calculating the controller input signal u of the next batch through a feedback-assisted iterative learning control algorithmk+1And (t) transmitting to a control signal storage, so that the system is operated iteratively, and the input signal of the next batch is updated by fully utilizing the operation information of the past batch so as to realize accurate tracking of the expected track.
With reference to fig. 4-6, a numerical simulation of a single joint robot system is shown. KnotSetting the expected time of each iteration batch of the single-joint robot system to be 5.5s according to the system hardware requirements and the practical application condition, setting the sampling interval to be 0.1s, and then setting Nd5.5. For the actual mechanical physical model of the single-joint robot system shown in the formula (1), the numerical simulation adopts the following specific parameters: m10 kg, g 9.8M/s2And l is 2.5 m.
For the single-joint robot system, a desired tracking track y (t) 20+1.5t-t is set2The output signal of the system is y (t) ═ 0.4+0.2sin t) x(2)(t), i.e. Bt＝[0 0.012]T,Ct＝[0 0.4+0.2sin t]. At the same time, the initial state of each iteration batch is set to xk(0)＝[0 50]TThe total number of iterations was set to 80. Due to external factors and safety considerations in practical applications, the actual iteration batch length of the system may not coincide with the expected length. The actual run length of each batch of the system varies between 4.5s and 5.5s, i.e. NLFor simplicity of example, the actual batch length N will be givenkSetting the values to satisfy discrete uniform distribution, namely uniformly and randomly taking values between 4.5s and 5.5s, and setting the control law of the system to be Lt＝100,Γt＝10,Kt＝20,
Figure BDA0003632549100000111
Fig. 4 is a diagram showing the tracking effect of different iteration batches of the single-joint robot control system applying the feedback-assisted PD-type iterative learning control algorithm proposed in the present application, and it can be seen that the actual iteration batch length of the system is not always equal to the expected batch length. The run length of iteration 10 of the system is 4.5s, the input is already close to the desired tracking trajectory, but there is still some error, and the run length of iteration 80 of the system is 5.2s, and it can be seen that the output trajectory of the batch almost coincides with the desired trajectory. Fig. 5 shows the actual output error of different batches of the system along the time axis. The error of the system decreases significantly as the iteration batch increases. In detail, the error of the system at iteration 40 is significantly reduced compared to that at iteration 10, while the error at iteration 80 is very close to 0 during the whole running period. Fig. 6 shows a norm image of the maximum tracking error of each batch along the batch axis direction by the feedback-assisted PD-type iterative learning control algorithm, and due to the application of the feedback-assisted mechanism, the tracking error of the system can be rapidly reduced, and the system can more accurately track the desired reference trajectory after iterating a certain batch.
The application provides a feedback-assisted PD type iterative learning control tracking trajectory algorithm for the variable batch length problem, the algorithm constructs an update error and an input sequence by utilizing past operation information of a system, updates system input by combining feedback signals of the current batch of the system together, and can fully ensure the learning efficiency of a batch length change part by applying the constructed sequence aiming at the variable batch length problem existing in an actual system, so that the system can track an expected trajectory within the range of expected length, and the tracking performance of the system is improved.
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. A variable-length feedback-assisted PD type iterative learning control method for a single-joint robot comprises the following steps:
firstly, carrying out analysis modeling on input and output of a single-joint robot system:
the typical structure of the single-joint robot system is regarded as being formed by connecting two ends by a single rod, wherein the two ends are an axis end and a free end respectively; the axis end is used as a track reference point and consists of a motor and a speed reducer matched with the motor, and joints in the single-joint robot are formed together; taking the axis end as a circle center and a reference point, carrying out rotary motion on the single rod within 360 degrees, mastering the length of the single rod by determining the angle between the single rod and a horizontal plane or other reference planes, and further positioning the specific position of the other end of the single rod in space; the other end of the single rod is called a free end, and the control target of the single-joint robot is to accurately control the track of the free end;
in the single-joint robot system, the included angle between the single rod and the horizontal plane is used as the running state of the system, and is measured by an absolute value angle encoder; the input signal of the system is the torque acting on the axis end of the single rod, and the external control signal drives the motor at the axis end to generate a torque to drive the single rod to rotate; neglecting the weight of the motor and the gear, neglecting the inertia of the single arm, taking the axis end as an origin, taking the translation of the absolute coordinate system as a corresponding coordinate axis, then the single rod rotates around the Z axis with the origin as a circle center, and based on the setting, the dynamic model of the single-joint robot system is expressed as:
Figure FDA0003632549090000011
wherein: t represents time; j. the design is a squarem＝1.33Ml2Is the moment of inertia; theta (t) is the rotation angle of a single rod of the single-joint robot; m is the integral mass of the single rod, g is the gravity acceleration, and l is the rotating distance of the center of mass from the axis end; τ (t) is the moment acting on the axial end;
therefore, a dynamic relation between the rotation angle of the single arm of the single-joint robot system and the moment of the axis end is constructed, and the rotation angle of the free end is controlled by adjusting the rotation moment, so that the running track of the single-joint robot is further controlled, and the specific application of the single-joint robot is realized;
secondly, constructing a discrete state space equation of the single-joint robot system:
in order to apply iterative learning control and convergence analysis to the single-joint robot system, the dynamic model of the moment-rotation angle of the single-joint robot constructed in the first step is rewritten into a nonlinear discrete state space equation; let the sampling period of the system be 0.1, and θ (t) be x(1)(t),θ(t+1)＝x(2)(t), u (t) τ (t), then the system will be further described as:
Figure FDA0003632549090000021
equation (2) further uses a non-linear function f (x) to account for repeated operation of the systemk(t), t) represents; therein
Figure FDA0003632549090000022
Figure FDA0003632549090000023
And
Figure FDA0003632549090000024
state information representing a kth iteration of the system; establishing a nonlinear discrete state space equation of the single-joint robot system:
Figure FDA0003632549090000025
wherein, f (x)k(t), t) is
Figure FDA0003632549090000026
A non-linear function of (d);
Figure FDA0003632549090000027
respectively representing the state, input and output of the kth iteration of the system,
Figure FDA0003632549090000028
is a time variable; k is 0,1, 2.. denotes the number of iterations of the system; b ist，CtIs a system matrix of appropriate dimensions, and Ct+1BtIs of full rank; the system has only ideal control input
Figure FDA0003632549090000029
I.e. under the influence of this signal, such that:
Figure FDA00036325490900000210
wherein x isd(t) and yd(t) desired state and desired output, AtIs a system matrix with appropriate dimensions;
for this system, there are two assumptions: one is that the initial state of each batch of the iterative process is equal to the desired initial state, i.e., xk(0)＝xd(0) (ii) a Secondly, the system meets the global Lipschitz condition, namely, a real number k existsfIs more than or equal to 0, and belongs to [0, N ] for all td]All the requirements are that:
Figure FDA00036325490900000211
thirdly, designing an updating error and input sequence aiming at the variable batch length problem:
the actual operation length of the k iteration is N because the batch lengths of the system are not uniform in actual operationkReasonably consider said NkThere is a minimum value NLAnd a maximum value NHIs considered to be Nd＝NHWherein N isdIs the desired run length of the system; from this, N is obtainedk∈[NL,Nd]I.e., t is 0,1,2L,...,Nk,...,Nd；
According to the characteristics of iterative learning control and the setting, historical information of past batches is used for controlling the signal uk(t) performing update correction so that the control target (4) can be achieved;
introducing an indication function
Figure FDA00036325490900000212
The function is equal if and only if the system reaches time t in the k-th iteration runAt 1 and vice versa 0, i.e.
Figure FDA00036325490900000213
For a batch length of N under the influence of an unfixed batch lengthkFor the kth iteration of (1), time Nk+1,...,NdThe output of (2) is not available, and the error in this time period cannot be calculated; in view of this, the error of this part is set to 0, so that the error of the k-th iteration is represented as:
Figure FDA0003632549090000031
wherein e isk(t)＝yd(t)-yk(t) is the error between the expected track and the actual output track of the system; the above formula is simply expressed as
Figure FDA0003632549090000032
For the upcoming iteration, its run length cannot be predicted, so all u's are calculatedk(t)，t∈[0,Nd-1]Is necessary; for this reason, all
Figure FDA0003632549090000033
The following update error sequence was constructed:
Figure FDA0003632549090000034
in the formula Ek(t) recursion generation is carried out, and after the operation is finished at each moment, the current iteration operation information is used for updating;
when the iteration length of the system is not evenly distributed, Ek(t) comprises a plurality of batches of error signals, using the updated error sequence Ek(t) calculating the input for the next iteration by a sufficiently long error signal; for eachAt an upcoming iteration, an error signal of sufficient length is used to update all uk+1(t)，t∈[0,Nd-1]Whether or not it is actually put into operation, for which purpose E is introducedk(t) constructing an updated input sequence with the same concept:
Figure FDA0003632549090000035
wherein the indicator function is
Figure FDA0003632549090000036
The reason for this is that the input at the time t can obtain an error only at the time t +1, so that the update input sequence generated recursively stores the input signal that has the latest update at each time point and has the output result;
fourthly, designing a feedback-assisted PD type iterative learning control algorithm aiming at the variable batch length problem:
by utilizing the updating error sequence and the updating input sequence constructed in the third step, the system stores the latest input signal and obtains the corresponding output signal after running each time, thereby providing a new framework for the control of the system with variable batch length; in order to improve the tracking performance of the system and accelerate the convergence rate of errors, the following feedback-assisted PD type iterative learning control algorithm is constructed by utilizing the update error sequence and the update input sequence and combining the errors of the current batch operation:
uk+1(t)＝Uk(t)+LtEk(t+1)+ΓtEk(t)+KtEk+1(t),t∈[0,Nd], (9)
wherein Lt，ΓtAnd KtA learning gain for the system; applying the learning law to a discrete nonlinear time varying system (3) yields the following theorem: for the discrete non-linear time-varying system (3), an iterative learning control update algorithm (9) is applied, if a suitable learning gain is selected such that:
0<||I-Ct+1BtLt||<1,
then as the number of iterations k approaches infinity, at time interval t e 0, Nd]The tracking target described in the above implementation formula (2), i.e.
Figure FDA0003632549090000041
Wherein I is a unit matrix with proper dimension;
fifthly, analyzing the convergence of the feedback-assisted PD type iterative learning control algorithm under the variable batch length:
before analyzing the convergence of the feedback-assisted PD type iterative learning control algorithm proposed in the fourth step, the operation length of system iteration needs to be divided into two parts; the first part is the minimum length range which can be reached by each iteration of the system; the second part is a part with variable system batch length, and the actual running length of each iteration batch can be any value due to the influence of the variable batch length; for this reason, assume three: for an arbitrary iteration batch k and time t e {0,1kIn the past sigma batches, the batch length of at least one batch is greater than or equal to t;
the first part of the control algorithm convergence verification is as follows:
proving the theorem at t ∈ [0, NL]When the condition is satisfied; since for all iterations NLAll the time comes, i.e.
Figure FDA0003632549090000042
Then the error over the time interval is obtained and updated every iteration to yield
Figure FDA0003632549090000043
Further, according to the formula (1), when t ∈ ψ, ψ [0, N ]L-1]Then, using hypothesis two we obtain:
Figure FDA0003632549090000044
in the formula
Figure FDA0003632549090000045
As a time-varying system parameter BtMaximum value within time ψ, and x is obtained from hypothesis onek+1(0)-xk(0)＝0，
Figure FDA0003632549090000046
Thereby, according to
Figure FDA0003632549090000047
The definition of (a) yields:
Figure FDA0003632549090000048
taking norm of two sides of the above formula to obtain:
Figure FDA0003632549090000049
order to
Figure FDA00036325490900000410
As a time-varying system parameter Ct+1Maximum value over time ψ, further yields:
Figure FDA0003632549090000051
wherein the parameter alpha is more than or equal to kfFor the sake of brevity, order
Figure FDA0003632549090000052
And
Figure FDA0003632549090000053
it follows from the above equation:
Figure FDA0003632549090000054
to construct the lambda norm, a constant lambda is defined>0, the two ends of the above formula are multiplied by alpha-λ(t+1)And taking the minimum supremum, including:
Figure FDA0003632549090000055
wherein ρ ═ supt∈ψ||I-Ct+1BtLtA | l; according to the definition of the lambda norm, the following results:
Figure FDA0003632549090000056
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003632549090000057
combining (15) and (16) to obtain:
||ek+1(t+1)||λ≤ρ0||ek(t+1)||λ (17)
in the formula
Figure FDA0003632549090000058
When 0 is present<||I-Ct+1BtLt||<At 1 time, 0<ρ<1, so that when λ is large enough, ρ is0<1 and 1-k3ρ1>0; thus, it is possible to provide
Figure FDA0003632549090000059
Further obtain
Figure FDA00036325490900000510
At the same time, because
Figure FDA00036325490900000511
Thus obtaining
Figure FDA00036325490900000512
The second part of the control algorithm convergence verification is as follows:
by inductive analysis, inductive assumptions were made: for arbitrary T e [ N [ ]L,Nd-1]，
Figure FDA00036325490900000513
It is thus to be demonstrated that when T is T +1,
Figure FDA00036325490900000514
defining a subsequence
Figure FDA00036325490900000515
It contains all running times NkThe batch number of more than or equal to T + 1; rearranging the elements in the solution according to the appearance order to obtain
Figure FDA00036325490900000516
And k is known from the hypothesisiAnd k isi+1Is less than or equal to σ, i.e. ki+1-kiSigma is less than or equal to; using this subsequence, when T ═ T +1, we obtain:
Figure FDA0003632549090000061
and:
Figure FDA0003632549090000062
using the formula to obtain:
Figure FDA0003632549090000063
for all
Figure FDA0003632549090000064
After the batch runs are finished, Uk(T) and Ek(T +1) is not updated, i.e.:
Figure FDA0003632549090000065
update laws according to ILC while defining
Figure FDA0003632549090000066
Obtaining:
Figure FDA0003632549090000067
next, formula (22) is brought into formula (20) while defining
Figure FDA0003632549090000068
Further obtaining:
Figure FDA0003632549090000069
for the
Figure FDA00036325490900000610
By recursion we obtain:
Figure FDA00036325490900000611
wherein
Figure FDA00036325490900000612
According to the generation rule of the updated input sequence, when
Figure FDA00036325490900000613
When the temperature of the water is higher than the set temperature,
Figure FDA00036325490900000614
will be updated; in view of this, a series of subsets is defined
Figure FDA00036325490900000615
Figure FDA00036325490900000616
Are arranged in the order of appearance
Figure FDA00036325490900000617
For time T e [0, T]Input signal of
Figure FDA00036325490900000618
To say, it will only be at
Figure FDA00036325490900000619
After the end of the batch in (c), update is performed to obtain the value of [0, T ] for all T e],
Figure FDA00036325490900000620
And k isiIs the same as the raw material of the batch,
Figure FDA00036325490900000621
and k isi+1Is the same batch; whereby when T ∈ [0, T-1 ]]Then obtaining the following results:
Figure FDA0003632549090000071
at the same time, composed of
Figure FDA0003632549090000072
And
Figure FDA0003632549090000073
further get kiThe time → ∞ of the time,
Figure FDA0003632549090000074
and k isi→ → ∞ with k → ∞; according to the generalised hypothesis, there are
Figure FDA0003632549090000075
Then, from the three assumptions, at ki+1And k isiAt most, there are σ -1 iterative batches between batches, i.e., for each time T ∈ [0, T]，
Figure FDA0003632549090000076
Is limited, thereby ensuring
Figure FDA0003632549090000077
As a result obtain
Figure FDA0003632549090000078
Taking norm at both ends of equation (23) simultaneously, we get the following inequality:
Figure FDA0003632549090000079
wherein the content of the first and second substances,
Figure FDA00036325490900000710
combined with the above derivation
Figure FDA00036325490900000711
Thereby applying the theorem to the inequality (26) for an iterative system
Figure FDA00036325490900000712
Wherein
Figure FDA00036325490900000713
And
Figure FDA00036325490900000714
respectively a state and a bounded internal input,
Figure FDA00036325490900000715
is a system matrix; when the temperature is higher than the set temperature
Figure FDA00036325490900000716
When it is obtained
Figure FDA00036325490900000717
If and only if 0<||D||<1；
It follows that if and only if 0<||I-CT+1BTLT||<When the pressure of the mixture is 1, the pressure is lower,
Figure FDA00036325490900000718
namely that
Figure FDA00036325490900000719
Based on the induction analysis method, further summarize
Figure FDA00036325490900000720
Combining the first part and the second part, the learning gain matrix selected at present satisfies 0<||I-CT+1BTLT||<When the pressure of the mixture is 1, the pressure is lower,
Figure FDA00036325490900000721
the method is true, namely, the tracking target is realized, so that the convergence proof of the feedback auxiliary PD type iterative learning control algorithm under the variable batch length is completed;
and sixthly, realizing the track tracking of the single-joint robot system under the variable batch length:
and calculating an input signal required by each time point in each iteration process of the system according to the designed feedback-assisted PD type iterative learning control algorithm, and applying the signal to the single-joint robot system to perform system control so as to obtain response output, so that the motion track of the free end of the system can track the preset expected output track of the system, and accurate track tracking control is realized.
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