Variable length feedback assisted PD-type iterative learning control method for single-joint robot

Abstract

The present invention discloses a variable length feedback-assisted PD-type iterative learning control method for a single-joint robot, and relates to the field of iterative learning control; the method constructs a recursively generated update error and input sequence to store the operation information of the most recent iteration batch of the system at each time point, and then combines it with the feedback error information of the current iteration batch of the system to jointly repair the input signal 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 for variable length systems is proved by inductive analysis. The method can be applied to the trajectory tracking control problem of a single-joint robot system under the variable batch length problem, and realizes the accurate tracking of the system output to the desired trajectory.

Description

Technical Field

[0001] 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 Art

[0002] Industrial robots are widely used in industrial manufacturing and other fields. For repetitive production processes, industrial robots can replace human labor to perform some repetitive work, and can also work stably in dangerous and harsh environments, greatly improving the production efficiency and product quality of industrial processes. Therefore, there are increasingly higher requirements for the control performance of industrial robots. The control of industrial robots mainly focuses on the precise tracking control of their trajectories, which is of great significance to ensuring production safety and improving product quality.

[0003] The single-joint robot system is a basic and very important robot system. Complex robot systems are often built on the basis of single-joint robot systems. The movement of multiple joints together constitutes the overall action of the complex robot system. As a typical time-varying and nonlinear system, the system control strategy is its key technology. For the trajectory tracking problem of the single-joint robot system, choosing a suitable control algorithm can make the control effect more effective. In view of its characteristics of repeatedly performing the same task, iterative learning control (ILC) has been widely used in the robot's trajectory tracking task. The core idea of ​​ILC is to generate the current control input by using the previous trajectory tracking error and control input information to adjust the system tracking estimate, so that the control system has the ability of self-learning and self-improvement. This strategy defines each execution of the system's repeated tasks as an iterative batch. As the iterative batch increases, the actual output trajectory of the robot should converge to the expected trajectory asymptotically. This requires that for the single-joint robot system that runs repeatedly, the length of each iterative batch during its operation must be the same as the expected length, so as to ensure the learning efficiency and tracking effect of the system within the entire expected time. However, in some practical application scenarios of single-joint robots, this requirement may not always be met. For example, single-joint robot systems are widely used in freight robot systems such as handling robots and gantry cranes. Since the load is only allowed to move within a specific area of ​​the reference object, when the output of the system deviates from the constraint range, the system operation of the batch will be interrupted immediately. Therefore. The batch lengths of all iterative batches are not exactly the same. However, even for incomplete experiments, the available tracking part can still be used to improve the learning performance of the control system. In addition, single-joint robots can be applied to lower limb auxiliary rehabilitation robots. By controlling the movement of the robot arm, the patient's lower limbs are driven to move to an appropriate angle, which helps to restore the lower limb motor function of patients with lower limb injuries or strokes. However, in actual applications, due to the patient's own physical fitness, interference from external factors or safety considerations, the patient's lower limb movements cannot remain completely consistent, that is, the iterative batch length of the single-joint robot system will change. These constitute the variable batch length problem in the single-joint robot control system. The key to solving this problem is to design a suitable control strategy, make full use of the effective output generated by the system under various batch lengths for learning, and correct the ILC control signal to achieve accurate tracking of the desired trajectory by the single-joint robot system.

[0004] In many conventional ILC designs, a key requirement is that when the number of iterations of the control system increases, the batch length of each iteration is a fixed value, called the expected 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 where the actual batch length is less than the expected batch length will be missing. For this part of the error information that cannot be obtained, a common practice is to set it to 0, so that all iterative batches of the system can be analyzed with reference to the expected length under the traditional ILC framework. In addition, some studies are committed to using the operation information of the previous iterations of the system to construct a high-order error signal to compensate for the missing part, which to a certain extent enhances the robustness of the system to the changing batch length, and also enables each iterative batch of the system to provide full-length error information, and based on this idea, a variety of ILC control algorithm designs are proposed. However, such a control strategy may dilute larger errors or errors from the most recent iterative batches, thereby affecting the convergence speed of the system. Summary of the invention

[0005] In view of the above problems and technical requirements, the inventors proposed a variable length feedback-assisted PD-type iterative learning control method for a single-joint robot. An update error and input sequence generated by instant recursion are constructed, and the latest error and the latest valid input signal generated by the system at each time point are stored respectively, so that the system can better utilize the latest operation information from the most recent iterations. Based on this design, a feedback-assisted PD-type ILC control law for a variable batch length system is proposed, which improves the tracking performance by using the derivation of the update sequence and the tracking error of the current iteration as the correction term of the input signal.

[0006] The technical solution of the present invention is as follows:

[0007] A variable length feedback assisted PD type iterative learning control method for a single joint robot, the method comprising:

[0008] The first step is to analyze and model the input and output of the single-joint robot system:

[0009] The typical structure of a single-joint robot system is considered to be composed of a single rod connected at both ends. The cross-section of the single rod is rectangular, and it is a slender rigid structure, with the two ends connected being the axis end and the free end. The axis end is often used as a trajectory reference point, and is composed of a motor and a matching reducer, which together form the joint in the single-joint robot. With the axis end as the center of the circle and the reference point, the single rod can rotate within its 360 degrees. By determining the angle between the single rod and the horizontal plane or other reference plane, the length of the single rod can be mastered, and then the specific position of the other end of the single rod in space can be located. The other end of the single rod is called the free end. The control goal of the single-joint robot is to accurately control the trajectory of the free end to achieve precise and repeatable operations in industrial processes. Depending on the specific application scenario, the free end can be equipped with a manipulator for handling work, or connected to the axis end of another single-joint robot system to form a multi-joint robot system.

[0010] In the single-joint robot system, the angle between the single rod and the horizontal plane is used as the operating state of the system and is measured by an absolute angle encoder. The input signal of the system is the torque acting on the axis end of the single rod, which is generated by the axis end motor driven by an external control signal to drive the single rod to rotate. For the sake of simplicity in calculation, the weight of the motor and gear can be ignored, as well as the inertia of the single arm itself. Taking the axis end as the origin and the translation of the absolute coordinate system as the corresponding coordinate axis, the single rod rotates around the Z axis with the origin as the center. Based on this setting, the dynamic model of the single-joint robot system is expressed as:

[0011]

[0012] Where: t represents time, J m =1.33Ml 2 , is the moment of inertia, θ(t) is the rotation angle of the 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 from the center of mass to the axis end, and τ(t) is the torque acting on the axis end.

[0013] In this way, the dynamic relationship between the single-arm rotation angle and the axis end torque of the single-joint robot system is constructed. By adjusting the rotational torque, the rotation angle of the free end can be accurately controlled, thereby further controlling its running trajectory and realizing the specific application of the single-joint robot.

[0014] Step 2: Construct the discrete state space equation of the single-joint robot system:

[0015] In order to apply iterative learning control to the single-joint robot system and perform its convergence analysis, the dynamic model of the single-joint robot torque-rotation angle 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) = x (1) (t),θ(t+1)=x(2) (t),u(t)=τ(t), then the system is further described as:

[0016]

[0017] Considering the repeated operation of the system, equation (2) is further expressed by a nonlinear function f(x k (t),t) represents. and Represents the state information of the system at the kth iteration. Establish the nonlinear discrete state space equation of the single-joint robot system:

[0018]

[0019] Among them, f(x k (t),t) belongs to Non-linear function. Represent the state, input and output of the system at the kth iteration respectively. is the time variable. k=0,1,2,... represents the number of iterations of the system, B t , C t is a time-varying system matrix of appropriate dimensions, and C t+1 B t is full rank. The system has a unique ideal control input That is, under the action of this signal,

[0020]

[0021] Among them, x d (t) and y d (t) are the expected state and expected output, respectively, A t is the system matrix of appropriate dimensions.

[0022] For this iterative system, there are two assumptions: First, the initial state of each batch of the repeated process is equal to the expected initial state, that is, x k (0) = x d (0). Second, the system satisfies the global Lipschitz condition, that is, there exists a real number k f ≥0, for all t∈[0,N d ], all satisfy:

[0023]

[0024] Step 3: Design update error and input sequence for variable batch length problem:

[0025] Since the batch length is not uniform in actual operation of the system, let the actual batch length of the kth iteration be N k , it is reasonable to assume that N k There is a minimum value N L With a maximum value N H Since we only focus on the tracking performance within the expected time, we assume that N d =N H , where N d is the expected batch length of the system. Thus, we get N k ∈[N L ,N d ], that is, t=0,1,2,...,N L ,...,N k ,...,N d .

[0026] According to the characteristics of iterative learning control and the above settings, the control signal u is adjusted using the historical information of past batches. k (t) Update and correct so that the control target (4) can be achieved.

[0027] Introduce an indicator function I k (t),t∈[0,N d ], if and only if the system reaches time t in the kth iteration, the function is equal to 1, otherwise it is 0, that is, I k (t)=0,t≤N k It should be noted that under the influence of the unfixed batch length, for batch length N k For the kth iteration, at time N k +1,...,N d The output of cannot be obtained. This means that the error in this time period cannot be calculated. In view of this situation, the error of this part is set to 0, so the error of the kth batch is expressed as:

[0028]

[0029] Among them, e k (t) = y d (t)-y k (t) is the error between the system's expected trajectory and the actual output trajectory. Using the indicator function proposed earlier, the above formula can be simply expressed as

[0030] For the upcoming iteration, the batch length cannot be predicted, so all u k (t), t∈[0,N d -1] is necessary. To do this, for all Construct the following update error sequence:

[0031]

[0032] E in the formula k (t) is generated recursively and updated with the information of the current iteration batch after each run.

[0033] When the iteration length of the system is unevenly distributed, E k The composition of (t) contains error information of multiple batches, and the error sequence E is updated using k (t), the error signal is long enough to calculate the input of the next iteration. However, it should be noted that for each upcoming iteration, there is a long enough error signal to update all u k+1 (t), t∈[0,N d -1], regardless of whether it has actually been put into operation. For this purpose, it is necessary to introduce k (t) Construct an update input sequence with the same idea:

[0034]

[0035] The indicator function is I k The reason for (t+1) is that the input at time t can only obtain the error at time t+1, so the recursively generated updated input sequence stores the input signal that was most recently updated and has the output result at each time point.

[0036] Step 4: Design a feedback-assisted PD-type iterative learning control algorithm for the variable batch length problem:

[0037] Using the updated error sequence and updated input sequence constructed in the third step, the system stores the latest input signal and obtains its corresponding output signal after each batch, providing a new framework for the control of systems with variable batch lengths. In order to improve the tracking performance of the system and speed up the convergence of the error, Hou built the following feedback-assisted PD-type iterative learning control algorithm by using the updated error sequence and updated input sequence and combining the error of the current iterative batch:

[0038] u k+1 (t) = U k (t)+L t E k (t+1)+Γ t E k (t)+K t E k+1 (t),t∈[0,N d ], (9)

[0039] Among them, L t, Γ t and K t is the learning gain of the system. By applying the learning law to the discrete nonlinear time-varying system (3), the following theorem is obtained: For the discrete nonlinear time-varying system (3), the iterative learning control update algorithm (9) is applied. If a suitable learning gain is selected so that:

[0040] 0<||IC t+1 B t L t ||<1,

[0041] Then when the number of iterations k approaches infinity, in the time interval [0,N d ] to achieve the tracking goal described in (2), that is, where I is the identity matrix of appropriate dimensions.

[0042] Step 5: Analyze the convergence of the feedback-assisted PD-type iterative learning control algorithm under variable batch length:

[0043] Before analyzing the convergence of the feedback-assisted PD-type iterative learning control algorithm proposed in the fourth step, it is necessary to divide the batch length of the system iteration into two parts. The first part is the minimum length range that the system can reach in each iteration. 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 changes. Due to the influence of the variable batch length, the actual batch length of each iterative batch may be any value among them. For this reason, a reasonable assumption three is given: for an arbitrary iterative batch k and time t∈{0,1,...,N k}, in the past σ batches, there is at least one batch whose batch length is greater than or equal to t.

[0044] The first part of the proof of convergence of the control algorithm is as follows:

[0045] Prove that the theorem is L ] is true. Because for all iterations, N L The moment will arrive, Then the error on the time interval can be obtained and updated in each iteration, and we can get In addition, according to formula (1), when t∈ψ,ψ=[0,N L -1], using assumption 2, we get:

[0046]

[0047] In the formula is the time-varying system matrix B t The maximum value in the time interval ψ, and according to assumption 1, x k+1 (0)-xk (0)=0, Therefore, according to The definition of is:

[0048]

[0049] Taking the norm on both sides of the above equation, we get:

[0050]

[0051] make is the time-varying system matrix C t+1 The maximum value within the time interval ψ, further obtained:

[0052]

[0053] The parameter α ≥ k f For simplicity, let and Therefore, it can be further concluded from the above formula:

[0054]

[0055] To construct the λ-norm, define a constant λ>0 and multiply both sides of the above equation by α -λ(t+1) And taking the least supremum, we have:

[0056]

[0057] Where ρ = sup t∈ψ ||IC t+1 B t L t ||. According to the definition of λ-norm, we get:

[0058]

[0059] in, Combining (15) and (16) we obtain:

[0060] ||e k+1 (t+1)|| λ ≤ρ0||e k (t+1)|| λ (17)

[0061] In the formula When 0<||IC t+1 B t L t When ||<1, 0<ρ<1, so when λ is large enough, ρ0<1 and 1-k3ρ1>0. Further conclusion

[0062] At the same time, due to So we can conclude It is verified that t∈[0,N L ] when iteratively learning the control law.

[0063] The second part of the proof of convergence of the control algorithm is as follows:

[0064] Through inductive analysis, we make the inductive hypothesis: for any T∈[N L ,N d -1], Therefore, we need to prove that when t=T+1,

[0065] Define a subset It contains all batches of length N k ≥T+1. Rearrange the elements in the order of appearance to get And from Assumption 3, we know that k i With k i+1 The distance is less than or equal to σ, that is, k i+1 -k i ≤σ. Using this subsequence, when t=T+1, we get:

[0066]

[0067] and:

[0068]

[0069] Using the above formula we get:

[0070]

[0071] Because for all I k (T+1)=0, that is, after these batches are completed, U k (T) and E k (T+1) has not been updated, that is:

[0072]

[0073] According to the ILC update law, we define get:

[0074]

[0075] Next, substitute equation (22) into equation (20) and define Further we get:

[0076]

[0077] for By recursion we get:

[0078]

[0079] in From the generation rule of updating the input sequence, we know that when N ki ≥t+1, U ki (t) will be updated. Taking this into account, define a series of subsets Similarly, The elements in the order of appearance are It can be seen that for the input signal at time t∈[0,T] In fact, it will only After the batch in is finished, it is updated, so that for all t∈[0,T], With k i For the same batch, With k i+1 is the same batch. Therefore, when t∈[0,T-1], we get:

[0080]

[0081] At the same time, by and The definition of k i →∞, And k i →∞ as k→∞. According to the previous inductive hypothesis, And, according to Assumption 3, in k i+1 With k i There are at most σ-1 iteration batches between batches, that is, for each time t∈[0,T], The elements in are finite, which ensures As a result, we get

[0082] Next, taking the norms on both sides of equation (23), we get the following inequality:

[0083]

[0084] in,

[0085]

[0086] Combining the above derivations, we can get Applying the lemma to inequality (26): for an iterative system z k+1 =Dz k +d k , where z k and are the state and bounded internal input respectively, is the system matrix; when When If and only if 0<||D||<1.

[0087] It follows that if and only if 0<||IC T+1 B T L T When ||<1, Right now Based on the inductive analysis method, we further conclude that This completes the second part of the proof.

[0088] Combining the first and second parts, assume that the currently selected learning gain matrix satisfies 0<||IC T+1 B T L T When ||<1, It is established and the tracking target is achieved. This completes the convergence proof of the feedback-assisted PD-type iterative learning control algorithm under variable batch length.

[0089] Step 6: Realize trajectory tracking of single joint robot system under variable batch length

[0090] According to the designed feedback-assisted PD-type iterative learning control algorithm, the input signal required by the system at each time point in each iteration process is calculated, and the signal is applied to the single-joint robot system for system control to obtain the response output, so that the motion trajectory of the free end of the system can track the expected output trajectory set in advance by the system, thereby achieving precise trajectory tracking control.

[0091] The beneficial technical effects of the present invention are:

[0092] This application discloses a nonlinear time-varying system with repetitive motion characteristics such as a single-joint robot system. The single-joint robot system is taken as the controlled object. For the variable batch length problem that may occur in the actual application of the controlled object, a recursively generated update error and input sequence is constructed, and the sequence is combined with the current iterative error signal to construct a feedback-assisted PD-type iterative learning control algorithm, so that the system can achieve a faster convergence speed while ensuring the tracking requirements under the variable batch length problem. At the same time, the convergence of the designed feedback-assisted PD-type iterative learning control algorithm is analyzed by inductive analysis, thereby ensuring the convergence of the system tracking error. BRIEF DESCRIPTION OF THE DRAWINGS

[0093] Figure 1 It is a principle block diagram of the single-joint robot control system disclosed in this application.

[0094] Figure 2 This is a mechanical structure diagram of the single-joint robot disclosed in this application.

[0095] Figure 3 It is a model block diagram of the single-joint robot control system disclosed in this application.

[0096] Figure 4 It is an output curve chart of different batches of the single-joint robot system disclosed in this application along the time axis.

[0097] Figure 5 It is an error curve diagram of different batches of the single-joint robot system disclosed in this application along the time axis.

[0098] Figure 6 It is a maximum error norm curve graph of the single-joint robot system along the batch axis disclosed in this application. DETAILED DESCRIPTION

[0099] The specific implementation of the present invention will be further described below in conjunction with the accompanying drawings.

[0100] Please refer to Figure 1, which shows the principle block diagram of the single joint robot control system used in this application. The system designs the control law through a human-machine interaction device, applies the control algorithm to the control system of the single joint robot, generates a corresponding signal to drive the actuator of the axis end, and finally realizes the trajectory tracking of the free end. The current state of the free end and the axis end is detected by an angle measuring device to update the control law for the next iteration. The system uses a controller area network (CAN) bus to communicate with a human-machine interaction device such as a host computer in a two-way manner, and feeds back the current operating state of the system to the user end in real time, so that the user can monitor the state of the system and correct the parameters. The control system can be controlled by an embedded ARM microprocessor of model STM32F103RCT6, which has a highly integrated CAN controller inside the chip and provides a standard CAN protocol interface. It is connected to a CAN transceiver of model SN65HVD230 to realize the function of sending system information to the CAN network. In addition, the chip is used to generate an adjustable PWM signal to drive the axis end drive system, so that the output torque of the axis end actuator can meet the expected value, thereby realizing the accurate tracking of the free end trajectory. The core of the axis end drive system uses a dual-phase DMOS full-bridge driver chip model A3997SBT. By connecting the PWM signal from the control system to the ENABLE pin of the chip, the DC motor can be precisely controlled. The motion information of the free end of the system is calculated from its rotation angle and can be obtained through the AS5048B magnetic rotary encoder. By setting its resolution to 14 bits, that is, 0.022 degrees, the angle value of the free end of the single-joint robot relative to the horizontal plane within 0 to 360° can be obtained and transmitted back to the control system to calculate the running trajectory and error of the free end and display the running status. The measurement frequency of the single-joint robot control system is set to 10Hz, that is, the sampling time is 0.1s, to achieve real-time control of the trajectory of the free end of the robot.

[0101] Figure 2 The mechanical structure diagram of the single-joint robot system used in the present application is shown. As shown in the figure, in the single-joint robot system targeted by the present application, the 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 connection between the robot single arm and the shaft end to obtain the real-time operating status of the system. The motor and the reducer constitute the actuator at the shaft end, which is used to obtain the current signal from the drive system, change the output torque of the shaft end and finally change the angle change trajectory of the free end.

[0102] Figure 3 The single-joint robot control system model block diagram disclosed in this application is shown. The input signal of the single-joint robot axis end at time t in batch k is u k(t), the system axis end drive mechanism applies the input signal to the robot system, and changes the angle between the free end and the horizontal plane. The expected trajectory of the system is pre-stored in the expected trajectory memory, and the difference is calculated in real time with the actual output of the system. Due to the change in batch length, the system error will be calculated and stored in the update error sequence E k (t), and update the input sequence U k (t) Combined with the feedback-assisted iterative learning control algorithm, the controller input signal u for the next batch is calculated k+1 (t) is transmitted to the control signal memory, so that the system runs iteratively and makes full use of the operation information of the past batches to update the input signals of the next batch to achieve accurate tracking of the desired trajectory.

[0103] Combination Figure 4-6 As shown in Figure 1, it shows a numerical simulation of a single-joint robot system. Combining the system hardware requirements and actual application, the expected time of each iteration batch of the single-joint robot system is set to 5.5s, and the sampling interval is set to 0.1s, then N d =5.5. For the actual mechanical physical model of the single-joint robot system shown in formula (1), the specific parameters used in this numerical simulation are: M = 10kg, g = 9.8m / s 2 and l = 2.5 m.

[0104] For this single-joint robot system, an expected tracking trajectory is set as y(t)=20+1.5tt 2 , the output signal of the system is y(t)=(0.4+0.2sint)x (2) (t), that is, B t =[0 0.012] T ,C t =[0 0.4+0.2sint]. At the same time, the initial state of each iteration batch is set to x k (0) = [0 50] T , the total number of iterations is set to 80. Due to external factors and safety considerations in actual applications, the actual iteration batch length of the system may be inconsistent with the expected length. The actual running time of each batch of the system varies between 4.5s and 5.5s, that is, N L =4.5, for the sake of simplicity, the actual batch length N k It is set to satisfy the discrete uniform distribution, that is, the value is uniformly random between 4.5 and 5.5s, and the control law of the system is set to L t =100,Γ t =10,K t =20,

[0105] Figure 4 It shows the trajectory tracking effect diagram of different iteration batches of the single-joint robot control system using the feedback-assisted PD-type iterative learning control algorithm proposed in this application. It can be seen that the actual iteration batch length of the system is not always equal to the expected batch length. The batch length of the 10th iteration of the system is 4.5s, and its input is close to the expected tracking trajectory, but there is still a certain error. The batch length of the 80th iteration of the system is 5.2s, and it can be seen that the output trajectory of this batch almost coincides with the expected trajectory. Figure 5 The actual output error of different batches of the system along the time axis is shown. The error of the system decreases significantly with the increase of iteration batches. In detail, the error of the system at the 40th iteration is significantly lower than that of the 10th iteration, and the error of the 80th iteration is very close to 0 during the entire operation period. Figure 6 The norm image of the maximum tracking error of each batch of the feedback-assisted PD-type iterative learning control algorithm along the batch axis is shown. Due to the application of the feedback-assisted mechanism, the tracking error of the system can be reduced rapidly, and the system can track the desired reference trajectory more accurately after a certain number of batches of iterations.

[0106] This application proposes a feedback-assisted PD-type iterative learning control tracking trajectory algorithm for the variable batch length problem. The algorithm constructs an update error and input sequence by utilizing the system's past operating information, and updates the system input together with the feedback signal of the system's current batch. For the variable batch length problem existing in the actual system, the application of the constructed sequence can fully guarantee the learning efficiency of the part of the batch length that changes, so that the system can track the desired trajectory within the expected length range, thereby improving the tracking performance of the system.

[0107] The above is only a preferred embodiment of the present application, and the present invention is not limited to the above embodiments. It is understood that other improvements and changes directly derived or associated by those skilled in the art without departing from the spirit and concept of the present invention should be considered to be included in the protection scope of the present invention.

Claims

1. A variable length feedback assisted PD type iterative learning control method for a single joint robot, the method comprising: The first step is to analyze and model the input and output of the single-joint robot system: The typical structure of the single-joint robot system is considered to be composed of a single rod connected to two ends, which are respectively an axial end and a free end; the axial end is used as a trajectory reference point, and is composed of a motor and a reducer matched therewith, which together constitute the joint in the single-joint robot; with the axial end as the center of the circle and the reference point, the single rod rotates within 360 degrees, and the length of the single rod is determined by determining the angle between the single rod and the horizontal plane or other reference planes, thereby locating the specific position of the other end of the single rod in space; the other end of the single rod is called the free end, and the control target of the single-joint robot is to accurately control the trajectory of the free end; In the single-joint robot system, the angle between the single rod and the horizontal plane is taken as the operating state of the system and measured by an absolute angle encoder; the input signal of the system is the torque acting on the shaft end of the single rod, which is generated by the shaft end motor driven by an external control signal to drive the single rod to rotate; ignoring the weight of the motor and the gear, and ignoring the inertia of the single arm itself, taking the shaft end as the origin and the translation of the absolute coordinate system as the corresponding coordinate axis, the single rod rotates around the Z axis with the origin as the center. Based on this setting, the dynamic model of the single-joint robot system is expressed as: Where: t represents time; J m =1.33Ml 2 , is the moment of inertia; θ(t) is the rotation angle of the 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 from the center of mass to the axis end; τ(t) is the torque acting on the axis end; Thus, the dynamic relationship between the single-arm rotation angle and the shaft end torque of the single-joint robot system is constructed, and the rotation angle of the free end is controlled by adjusting the rotation torque, thereby further controlling its running trajectory and realizing the specific application of the single-joint robot; The second step is to construct the discrete state space equation of the single-joint robot system: In order to apply iterative learning control to the single-joint robot system and perform convergence analysis, the dynamic model of the single-joint robot torque-rotation angle constructed in the first step is rewritten into a nonlinear discrete state space equation; the sampling period of the system is set to 0.1, and θ(t) = x (1) (t),θ(t+1)=x (2) (t),u(t)=τ(t), then the system can be further described as: Considering the repeated operation of the system, equation (2) is further expressed by a nonlinear function f(x k (t),t) indicates; and Represents the state information of the system at the kth iteration; establish the nonlinear discrete state space equation of the single-joint robot system: Among them, f(x k (t),t) belongs to Nonlinear function of Represent the state, input and output of the system at the kth iteration respectively. is a time variable composed of discrete moments; k = 0, 1, 2, ... represents the number of iterations of the system; B t , C t is a time-varying system matrix of appropriate dimensions, and C t+1 B t is full rank; the system has a unique ideal control input That is, under the action of this signal, Among them, x d (t) and y d (t) are the expected state and expected output, respectively, A t is a system matrix with appropriate dimensions; For this system, there are two assumptions: First, the initial state of each batch of the repeated process is equal to the expected initial state, that is, x k (0) = x d (0); second, the system satisfies the global Lipschitz condition, that is, there exists a real number k f ≥0, for all t∈[0,N d ], all satisfy: Step 3: Design update error and input sequence for variable batch length problem: Since the batch length is not uniform in the actual operation of the system, let the actual batch length of the kth iteration be N k , that is, the actual moment when each iterative batch ends, it is reasonable to assume that the N k There is a minimum value N L With a maximum value N H , think N d =N H , where N d is the expected batch length of the system; thus, we get N k ∈[N L ,N d ], that is, t=0,1,2,...,N L ,...,N k ,...,,N d ; According to the characteristics of iterative learning control and the above settings, the control signal u is adjusted using the historical information of past batches. k (t) updating and correcting the control target so as to achieve the control target (4); Introduce an indicator function I k (t),t∈[0,N d ], if and only if the system reaches time t in the kth iteration, the function is equal to 1, otherwise it is 0, that is, I k (t)=0,t≤N k ; Under the influence of the unfixed batch length, for batch length N k For the kth iteration, at time N k +1,....,N d The output of cannot be obtained, and the error in this period cannot be calculated; in view of this situation, the error of this part is set to 0, so the error of the kth iteration is expressed as: Among them, e k (t) = y d (t)-y k (t) is the error between the expected trajectory and the actual output trajectory of the system; the above formula can be simply expressed as For the upcoming iteration, the batch length cannot be predicted, so all u k (t), t∈[0,N d -1] is necessary; for this, for all Construct the following update error sequence: E in the formula k (t) is generated recursively and updated with the information of the current iteration batch after each moment of operation. When the iteration lengths of the system are unevenly distributed, E k The composition of (t) includes multiple batches of error signals, and the error sequence E is updated using k (t), a sufficiently long error signal is used to calculate the input of the next iteration; for each upcoming iteration, a sufficiently long error signal is used to update all u k+1 (t), t∈[0,N d -1], regardless of whether it has actually been put into operation, for this purpose, it is necessary to introduce the E k (t) Construct an update input sequence with the same idea: The indicator function is I k The reason for (t+1) is that the input at time t can only obtain the error at time t+1, so the recursively generated updated input sequence stores the input signal with the most recent update and output result at each time point; Step 4: Design a feedback-assisted PD-type iterative learning control algorithm for the variable batch length problem: Using the updated error sequence and updated input sequence constructed in the third step, the system stores the latest input signal and obtains its corresponding output signal after each batch, providing a new framework for the control of systems with variable batch lengths. In order to improve the tracking performance of the system and accelerate the convergence speed of the error, the updated error sequence and updated input sequence are used, combined with the error of the current iterative batch, to construct the following feedback-assisted PD-type iterative learning control algorithm: u k+1 (t)=U k (t)+L t E k (t+1)+Γ t E k (t)+K t E k+1 (t),t∈[0,N d ], (9) Among them, L t , Γ t and K t is the learning gain of the system; applying the learning law to the discrete nonlinear time-varying system (3), the following theorem is obtained: For the discrete nonlinear time-varying system (3), applying the iterative learning control update algorithm (9), if a suitable learning gain is selected so that: 0<||I-C t+1 B t L t ||<1, Then when the number of iterations k approaches infinity, in the time interval [0,N d ] to achieve the tracking goal described in formula (2), that is, where I is the identity matrix of appropriate dimensions; Step 5: Analyze the convergence of the feedback-assisted PD-type iterative learning control algorithm under variable batch length: Before analyzing the convergence of the feedback-assisted PD-type iterative learning control algorithm proposed in the fourth step, it is necessary to divide the batch length of the system iteration into two parts. The first part is the minimum length range that the system can reach in each iteration. The second part is the part where the system batch length changes. Due to the influence of the variable batch length, the actual batch length of each iterative batch may be any of these values. Therefore, assumption three is given: for an arbitrary iterative batch k and time t∈{0,1,...,N k }, in the past σ batches, there is at least one batch whose batch length is greater than or equal to t; The first part of the proof of convergence of the control algorithm is as follows: Prove that the theorem is L ] holds true; because for all iterations, N L The moment will arrive, Then the error on the time interval can be obtained and updated in each iteration, and we can get In addition, according to formula (1), when t∈ψ,ψ=[0,N L -1], using assumption 2, we get: In the formula is the time-varying system matrix B t The maximum value in the time interval ψ, and according to assumption 1, we get Therefore, according to The definition of is: Taking the norm on both sides of the above equation, we get: make is the time-varying system matrix C t+1 The maximum value within the time interval ψ, further obtained: The parameter α ≥ k f For simplicity, let and Therefore, it can be further concluded from the above formula: To construct the λ-norm, define a constant λ>0 and multiply both sides of the above equation by α -λ(t+1) And taking the least supremum, we have: Where ρ = sup t∈ψ ||IC t+1 B t L t ||; According to the definition of λ-norm, we get: in, Combining (15) and (16) we obtain: ||e k+1 (t+1)|| λ ≤ρ0||e k (t+1)|| λ (17) In the formula When 0<||IC t+1 B t L t When ||<1, 0<ρ<1, so when λ is large enough, ρ0<1 and 1-k3ρ1>0; therefore Further conclusion At the same time, due to So we can conclude The second part of the proof of convergence of the control algorithm is as follows: Through inductive analysis, we make the inductive hypothesis: for any T∈[N L ,N d -1], Therefore, we need to prove that when t=T+1, Define a subsequence It contains all batches of length N k ≥T+1; rearrange the elements in the order of appearance to obtain And from the assumption three, k i With k i+1 The distance is less than or equal to σ, that is, k i+1 -k i ≤σ; Using this subsequence, when t=T+1, we get: and: Using the above formula we get: For all I k (T+1)=0, after these batches are finished, U k (T) and E k (T+1) has not been updated, that is: According to the ILC update law, we define get: Next, substitute equation (22) into equation (20) and define Further we get: for By recursion we get: in It can be seen from the generation rule of the update input sequence that when hour, Will be updated; considering this, define a series of subsets The elements in the order of appearance are For the input signal at time t∈[0,T] In fact, it will only After the batch in is finished, it is updated, so that for all t∈[0,T], With k i For the same batch, With k i+1 is the same batch; thus when t∈[0,T-1], we get: At the same time, by and The definition of k i →∞, And k i →∞As k→∞; According to the inductive hypothesis, Furthermore, according to the third assumption, in k i+1 With k i There are at most σ-1 iteration batches between batches, that is, for each time t∈[0,T], The elements in are finite, thus ensuring As a result, we get Taking the norm on both ends of equation (23) at the same time, we get the following inequality: in, Combining the above derivations, we can get Applying the lemma to inequality (26), we can conclude that for an iterative system in and are the state and bounded internal input respectively, is the system matrix; when When If and only if 0<||D||<1; It follows that if and only if 0<||IC T+1 B T L T When ||<1, Right now Based on the inductive analysis method, we further conclude Combining the first and second parts, the currently selected learning gain matrix satisfies 0<||IC T+1 B T L T When ||<1, It is established, that is, the tracking target is achieved, thus completing the convergence proof of the feedback-assisted PD-type iterative learning control algorithm under variable batch length; Step 6: Realize the trajectory tracking of the joint robot system under variable batch length: According to the designed feedback-assisted PD-type iterative learning control algorithm, the input signal required by the system at each time point in each iteration process is calculated, and the signal is applied to the single-joint robot system for system control to obtain a response output, so that the free end motion trajectory of the system can track the expected output trajectory pre-set by the system, thereby achieving precise trajectory tracking control.

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