Feedforward drive control method for 3D printer based on pseudo-input Gaussian process

Abstract

The present invention discloses a 3D printer feedforward drive control method based on a pseudo-input Gaussian process. The specific steps are as follows: Step 1: Represent the 3D printer drive system P as a discrete causal system formula and a non-causal system formula; Step 2: Under the action of a feedforward compensation controller, the control system obtains an output trajectory that matches the desired trajectory; Step 3: Inversely obtain the inverse model required for system feedforward compensation; Step 4: Re-express the drive system inverse model as Gaussian feedforward compensation; Step 5: Construct a sparse pseudo-input Gaussian process feedforward control architecture; Step 6: Introduce the Gaussian process algorithm in Step 4 into the system inverse model identification, and optimize the Gaussian feedforward control in combination with Step 5. Then, load the optimized inverse model into the feedforward control architecture to complete the 3D printer feedforward drive control based on the pseudo-input Gaussian process. This method solves the problem of difficulty in achieving high-efficiency and high-precision control of 3D printing weak-rigidity drive systems.

Description

Technical Field

[0001] The present invention belongs to the technical field of mechanical design and manufacturing, and in particular relates to a feedforward drive control method for a 3D printer based on a pseudo-input Gaussian process. Background Art

[0002] 3D printing technology (also known as additive manufacturing) can rapidly complete the process from prototyping to physical realization, eliminating tedious tasks such as CNC programming and fixture design, significantly shortening product design cycles. This provides a significant competitive advantage in the face of complex and efficient production demands, and is applicable across a wide range of sectors, including mechanical manufacturing, engineering and construction, biomedicine, and the food industry. Currently, 3D printing has significantly increased its accuracy, repeatability, and material range, making it a reliable processing technology for industrial production. However, with the rapid adoption of 3D printing technology across various industries, the demands for ever-changing products and customer demands are increasingly stringent for printer build quality and print speed. In recent years, numerous researchers and companies have conducted extensive research on improving the build accuracy of 3D printed products. However, challenges remain, such as weak stiffness characteristics of printer drive systems (e.g., belt drives) and nonlinear friction disturbances, resulting in substandard print quality and low efficiency. Therefore, there is an urgent need to develop drive technologies that can significantly improve print efficiency while maintaining print accuracy, thereby increasing part production efficiency and reducing costs. Summary of the Invention

[0003] The purpose of the present invention is to provide a feedforward drive control method for a 3D printer based on a pseudo-input Gaussian process, which solves the problem that existing methods are difficult to meet the requirements of high-precision motion control of weak-rigidity drive systems, thereby resulting in poor molding quality and low efficiency of printed parts.

[0004] The technical solution adopted by the present invention is a 3D printer feedforward drive control method based on pseudo-input Gaussian process, and the specific steps are as follows:

[0005] Step 1: Express the 3D printer drive system P as a discrete causal system formula and a non-causal system formula:

[0006] Step 2: Under the action of the feedforward compensation controller, the control system obtains an output trajectory that matches the desired trajectory;

[0007] Step 3: Obtain the inverse model required for system feedforward compensation by inversion;

[0008] Step 4: Reformulate the inverse model of the drive system as Gaussian feedforward compensation;

[0009] Step 5: Construct a sparse pseudo-input Gaussian process feedforward control architecture;

[0010] Step 6: Introduce the Gaussian process algorithm in step 4 into the system inverse model identification, and optimize the Gaussian feedforward control in combination with step 5. Then load the optimized inverse model into the feedforward control architecture to complete the 3D printer feedforward drive control based on pseudo-input Gaussian process.

[0011] The present invention is also characterized in that

[0012] In step 1, the 3D printer drive system P is represented as a discrete causal system formula and a non-causal system formula, as follows:

[0013] y t- =f t- {u(t),u(t-1),…,u(tn d )} (1)

[0014] y t+ =f t+ {u(t+n f ),u(t+n f -1),…,u(t)} (2)

[0015] Where t is time, u(·) is the control input signal, n d and n f are the number of delayed samples and preview samples, respectively, t- {·} is the mapping function between the delayed sample as input signal and the 3D printer physical system P, f t+ {·} is the mapping function between the preview sample as input signal and the 3D printer physical system P, y t- and y t+ The output tracks are obtained after the delayed samples and preview samples are input into the system respectively.

[0016] Step 2 is implemented as follows:

[0017] Input the sampling trajectory into R={q1,…,q n} is designed as a set of reference trajectories of different proportions or frequencies, where the trajectory of the i-th sampling is q i =[q(0),…,q(n-1)], i∈[1,…,n], where n is the number of samples of the sampling trajectory q(·); then, R is input into the printer drive system, and the output value y of the position sensor on the printer is obtained according to the open-loop system least squares forward identification process. i =[y(0),…,y(n-1)] T And control input signal u i =[u(0),…,u(n-1)] T , and then the data generated by n sampling trajectories are superimposed into data sets (3) and (4):

[0018] Y=[y1,y2,…,y n ] T (3)

[0019] U=[u1,u2,…,u n ] T (4).

[0020] Step 3 is implemented as follows:

[0021] The 3D printing inverse model is obtained based on the open-loop system least squares reverse identification method. At this time, the input u(·) in the above formula (2) will be used as the reverse identification output, and the original output y will be used as the reverse identification input. The inverse model of the drive system is expressed as formula (5):

[0022] u i =f P (x i )+ε (5)

[0023] In formula (5), ε is the vibration and nonlinear friction disturbance noise existing in the weak stiffness belt drive; the reverse identification input of the inverse model is x i =[y(t+n f ), y(t+n f -1),…,y(t)];

[0024] Then, formula (3) is formatted by formula (6) to obtain the inverse model training data set S = {U, X}, where X is the training matrix and U is the training target; the specific expression of the training matrix X is as follows:

[0025]

[0026] Step 4 is implemented as follows:

[0027] The inverse model (5) of the drive system is reformulated as Gaussian feedforward compensation, as shown in Equation (7):

[0028] U={f P (x1), f P (x2),…,f P (x n )} T +ε (7)

[0029] Among them, f P (·) is the training matrix X={x1,x2,…,x n} T The mapping relationship between the training target U and ε is the white noise of the external system to which the output signal is subjected. For input data x iThe local variance of the adjacent region; the Gaussian prior of the inverse model is defined as formula (8):

[0030] p(U|X,θ)=N(M,K) (8)

[0031] Where p(U|X, θ) is the output probability objective function with respect to input X, N(·) is the Gaussian distribution function, M is the mean function, and θ is the kernel function adjustment hyperparameter.

[0032] Step 5 is implemented as follows:

[0033] Step 5.1: Based on the Gaussian process definition in step 4, the sparse pseudo-input Gaussian process regression method uses the sparse pseudo data set Replace the real data set S = {U, X};

[0034] Step 5.2: Given a new printer system reference input signal x*, obtain the predicted distribution u* through the likelihood function (9), where u* is the printer feedforward signal u f ;

[0035]

[0036] in, Λ=diag(λ), K ** =K(x * ,u * ),k * =K(x * , x)Q=K ** +k(Λ+σ 2 I) -1 k * , k=K(x,x * ).

[0037] The beneficial effects of the present invention are:

[0038] The method of the present invention takes into account the characteristics of the printer drive structure such as weak stiffness response lag and nonlinear disturbance. By analyzing the printer drive system, a 3D printer feedforward control method based on pseudo-input Gaussian process is constructed. This data-driven feedforward control can not only obtain an accurate inverse model of the drive system, but also overcome the non-minimum phase zero point problem in the classic inverse model feedforward control, solving the problem of difficulty in achieving high-efficiency and high-precision control of 3D printing weak stiffness drive systems (such as belt drives). BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 This is a principle block diagram of the precision drive control strategy of the present invention;

[0040] Figure 21. It is a schematic diagram of the open-loop system least squares forward identification setting of the present invention;

[0041] Figure 3 It is a schematic diagram of the reverse identification setting of the open-loop system least squares method of the present invention. DETAILED DESCRIPTION

[0042] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0043] The present invention provides a 3D printer feedforward drive control method based on pseudo-input Gaussian process. Figure 1 This is the principle block diagram of the precision drive control strategy of the present invention. The control block diagram consists of traditional PID control and Gaussian feedforward control (dashed box), including the 3D printer drive system P, input reference trajectory r, tracking error e, Gaussian process feedforward controller f P , PID feedback controller, feedforward controller signal u f , feedback controller signal u fb and output signal y; the specific implementation steps include:

[0044] Step 1: To avoid the instability problem of the Gaussian inverse model, a Gaussian feedforward controller f is constructed. P (·) fails, the present invention needs to express the Gaussian feedforward controller of the printer drive axis as a non-causal finite impulse response system. Therefore, the 3D printer drive system P is expressed as a discrete form of the causal system formula (1) and the non-causal system formula (2):

[0045] y t- =f t- {u(t),u(t-1),…,u(tn d )} (1)

[0046] y t+ =f t+ {u(t+n f ),u(t+n f -1),…,u(t)} (2)

[0047] Where t is time, u(·) is the control input signal, n d and n f are the number of delayed samples and preview samples, respectively, t- {·} is the mapping function between the delayed sample as input signal and the 3D printer physical system P, f t+ {·} is the mapping function between the preview sample as input signal and the 3D printer physical system P, y t- and y t+ The output tracks are obtained after the delayed samples and preview samples are input into the system respectively.

[0048] Step 2: In order to ensure the accuracy and stability of the inverse model compensation, the control system can obtain an output trajectory that matches the desired trajectory under the action of the feedforward compensation controller. The sampling trajectory can be input into R = {q1,…,q n} is designed as a set of reference trajectories of different proportions or frequencies, where the trajectory of the i-th sampling is q i =[q(0),…,q(n-1)], i∈[1,…,n], n is the number of samples of the sampling trajectory q(·); then, R is input to the printer drive system, according to Figure 2 The open-loop system least squares forward identification process shown in the figure obtains the output value y of the position sensor on the printer. i =[y(0),…,y(n-1)] T And control input signal u i =[u(0),…,u(n-1)] T , and then the data generated by n sampling trajectories are superimposed into data sets (3) and (4):

[0049] Y=[y1,y2,…,y n ] T (3)

[0050] U=[u1,u2,…,u n ] T (4);

[0051] Step 3: To obtain the inverse model required for system feedforward compensation, we need to Figure 3 The open-loop system least squares reverse identification method shown in the figure is used to obtain the 3D printing inverse model; at this time, the input u(·) in the above formula (2) will be used as the reverse identification output, and the original output y will be used as the reverse identification input, then the drive system inverse model is expressed as formula (5):

[0052] u i =f P (x i )+ε (5) In formula (5), ε is the vibration and nonlinear friction disturbance noise existing in the weak stiffness belt drive; the reverse identification input of the inverse model is x i =[y(t+n f ), y(t+n f -1),…,y(t)];

[0053] Then, we format Equation (3) by using Equation (6) to obtain the inverse model training data set S = {U, X}, where X is the training matrix and U is the training target. Based on this, the mapping relationship between the training matrix X and the training target U is the precise inverse model f of the control system. P ;

[0054]

[0055] Step 4: In order to significantly reduce the model training time while ensuring the training accuracy, the drive system inverse model (5) is reformulated as Gaussian feedforward compensation, as shown in Equation (7):

[0056] U={f P (x1)f P (x2),…,f P (x n )} T +ε (7)

[0057] Among them, f P (·) is the training matrix X={x1,x2,…,x n} T The mapping relationship between the training target U and ε is the white noise of the external system to which the output signal is subjected. For input data x i The local variance of the adjacent region; the Gaussian prior of the inverse model is defined as formula (8):

[0058] p(U|X,θ)=N(M,K) (8)

[0059] Where p(U|X,θ) is the output probability objective function with respect to input X, N(·) is the Gaussian distribution function, M is the mean function, θ is the kernel function tuning hyperparameter, and K represents the mean square exponential kernel function (SE). The K(·) kernel function is used to indicate that the training data x i and test data u i The correlation between f By controlling the output size of the kernel function, the characteristic length Λ can control the degree of influence of the characteristic attributes of each dimension of the input variable on the output result. Therefore, the kernel function also determines the potential structure of the inverse model under the Gaussian process prior.

[0060] It can be seen that if the dataset S has sufficient prior knowledge of the 3D printer physical system P, the trained Gaussian prior model will be the accurate inverse model f of the printer drive control system. p However, in actual use, if a large data set is used to train a model directly based on the SE kernel function, the optimization of hyperparameters often consumes a lot of time. To address this problem, the present invention adopts the sparse pseudo-input Gaussian process method (SPGP) to reduce the original data set, and uses the conjugate gradient method to optimize the sparse pseudo sample set and hyperparameters, so that the pseudo sample set has the characteristics of the complete data set, thereby reducing the model training time while ensuring the accuracy of the model.

[0061] Step 5: Construct a sparse pseudo-input Gaussian process feedforward control architecture; according to the Gaussian process definition in step 4, the sparse pseudo-input Gaussian process regression method uses the sparse pseudo-input data set Instead of the real data set S = {U, X}, the sparse pseudo input data m is much smaller than the number of real data n.

[0062] Based on this, given a new printer system reference input signal x*, the predicted distribution u* is obtained through the likelihood function (9), u* is Figure 1 Printer feedforward signal u f ;

[0063]

[0064] in Λ=diag(λ), K ** =K(x * ,u * ),k * =K(x * , x)Q=K ** +k(Λ+σ 2 I) -1 k * , k=K(x,x * )

[0065] In formula (9), the training time complexity of the SPGP model is mainly composed of the matrix multiplication k(Λ+σ 2 I) -1 k * Determine, where the matrix Λ+σ 2 Since I is a diagonal matrix, the time complexity of the inversion is negligible. This significantly improves computational efficiency while maintaining model accuracy, and also avoids the problem of low prediction accuracy caused by sensitivity to random initial values.

[0066] Step 6: Introduce the Gaussian process algorithm in step 4 into the system inverse model identification, and optimize the Gaussian feedforward control in combination with step 5, thereby greatly reducing the computational complexity, and then use the optimized inverse model f p Load to Figure 1 In the feedforward control architecture, the feedforward drive control of the 3D printer based on the pseudo-input Gaussian process is completed.

[0067] Experimental results

[0068] The trajectory tracking errors of the printer's x-axis and y-axis motion systems measured using different control schemes are shown in Tables 1 and 2:

[0069] Table 1 Experimental results of x-axis

[0070]

[0071] Table 2 Y-axis experimental results

[0072]

[0073] As can be seen from the table, the tracking accuracy of the Gaussian process feedforward control has been significantly improved. At the same time, when using sparse pseudo-input Gaussian process feedforward control, the training time of the feedforward controller is greatly reduced while ensuring the tracking accuracy of the printer motion system. This shows that precise driving of the printer can be achieved by using this method.

[0074] Compared with traditional control methods, the feedforward control method of the present invention can significantly reduce the trajectory tracking errors of the x-axis and y-axis of the 3D printer, solving the problem that existing methods are difficult to meet the requirements of high-precision motion control of weak-rigidity drive systems, thereby resulting in poor molding quality and low efficiency of printed parts; without using a physical model of the printer, precise drive control of a low-cost 3D printer is achieved, achieving the expected control effect, and providing a new idea for the practical application of intelligent feedforward control algorithms in the engineering field.

Claims

1. A 3D printer feedforward drive control method based on pseudo-input Gaussian process, characterized in that: The specific steps are as follows: Step 1: Express the 3D printer drive system P as a discrete causal system formula and a non-causal system formula: Step 2: Under the action of the feedforward compensation controller, the control system obtains an output trajectory that matches the desired trajectory; Step 3: Obtain the inverse model required for system feedforward compensation by inversion; Step 4: Reformulate the inverse model of the drive system as Gaussian feedforward compensation; Step 5: Construct a sparse pseudo-input Gaussian process feedforward control architecture; Step 6: Introduce the Gaussian process algorithm in step 4 into the system inverse model identification, and optimize the Gaussian feedforward control in combination with step 5. Then load the optimized inverse model into the feedforward control architecture to complete the 3D printer feedforward drive control based on pseudo-input Gaussian process.

2. The 3D printer feedforward drive control method based on pseudo-input Gaussian process according to claim 1, characterized in that: In step 1, the 3D printer drive system P is represented as a discrete causal system formula and a non-causal system formula, as follows: y t- =f t- {u(t),u(t-1),…,u(t-n d )} (1) y t+ =f t+ {u(t+n f ),u(t+n f -1),…,u(t)} (2) Where t is time, u(·) is the control input signal, n d and n f are the number of delayed samples and preview samples respectively, f t- {·} is the mapping function between the delayed sample as input signal and the 3D printer physical system P, f t+ {·} is the mapping function between the preview sample as input signal and the 3D printer physical system P, y t- and y t+ The output tracks are obtained after the delayed samples and preview samples are input into the system respectively.

3. The feedforward drive control method for a 3D printer based on a pseudo-input Gaussian process according to claim 1, characterized in that: Step 2 is implemented as follows: Input the sampling trajectory into R={q1,...,q n } is designed as a set of reference trajectories of different proportions or frequencies, where the trajectory of the i-th sampling is q i =[q(0),…,q(n-1)], i∈[1,…,n], n is the number of samples of the sampling trajectory q(·); then, R is input to the printer drive system, and the output value y of the position sensor on the printer is obtained according to the open-loop system least squares forward identification process. i =[y(0),...,y(n-1)] T And control input signal u i =[u(0),...,u(n-1)] T , and then the data generated by n sampling trajectories are superimposed into data sets (3) and (4): Y=[y1,y2,...,y n ] T (3) U=[u1,u2,...,u n ] T (4)。 4. The 3D printer feedforward drive control method based on pseudo-input Gaussian process according to claim 3, characterized in that: Step 3 is implemented as follows: The 3D printing inverse model is obtained based on the open-loop system least squares reverse identification method. At this time, the input u(·) in the above formula (2) will be used as the reverse identification output, and the original output y will be used as the reverse identification input. The inverse model of the drive system is expressed as formula (5): you i =f P (x i )+e (5) In formula (5), ε is the vibration and nonlinear friction disturbance noise existing in the weak stiffness belt drive; the reverse identification input of the inverse model is x i =[y(t+n f ), y(t+n f -1),…,y(t)]; Then, we format Equation (3) using Equation (6) to obtain the inverse model training data set S = {U, X}, where X is the training matrix and U is the training target. The specific expression of the training matrix X is as follows: Step 4 is implemented as follows: The inverse model (5) of the drive system is reformulated as Gaussian feedforward compensation, as shown in Equation (7): U={f P (x1),f P (x2),…,f P (x n )} T +ε (7) Among them, f P (·) is the training matrix X={x1,x2,…,x n } T The mapping relationship between the training target U and ε is the white noise of the external system to which the output signal is subjected. For input data x i The local variance of the adjacent region; the Gaussian prior of the inverse model is defined as formula (8): p(U|X,θ)=N(M,K) (8) where p(U|X, θ) is the target function for the probability of output U with respect to input X, N(·) is the Gaussian distribution function, M is the mean function, θ is the kernel function tuning hyperparameter, and K represents the mean square exponential kernel function.

5. The 3D printer feedforward drive control method based on pseudo-input Gaussian process according to claim 4, characterized in that: Step 5 is implemented as follows: Step 5.1: Based on the Gaussian process definition in step 4, the sparse pseudo-input Gaussian process regression method uses the sparse pseudo data set Replace the real data set S = {U, X}; Step 5.2: Given a new printer system reference input signal x*, obtain the predicted distribution u* through the likelihood function (9), where u* is the printer feedforward signal u f ; Among which, Λ = diag(λ), K ** = K(x * , u * ), k * = K(x * , x)Q = K ** + k(Λ + σ 2 I) -1 k * , k = K(x, x * ).

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