Online parameter identification of a giant magnetostrictive actuator based on the dynamic Jiles–Atherton model

Abstract

Giant magnetostrictive actuator (GMA) suffers dominant hysteresis nonlinearity while working. In order to better predict its output, a dynamic model of GMA based on the J–A model is established. On the foundation of the model, an online parameter identification method is proposed. Classified into two types, some parameters are determined by physical theories, while the others are identified by a sensitivity-based identification method. Serving as the identification algorithm, the particle swarm optimization (PSO) algorithm is modified to accelerate the identification process. Moreover, to validate the effectiveness of the proposed method, an online parameter identification system is designed as well as identification software. Experimental results prove that the proposed system performs well and is fit for online identification.

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1 Introduction

Giant magnetostrictive material (GMM) is an intelligent material firstly invented by A. E. Clark in the 1970s.¹ Compared with other smart materials, GMM has many excellent features such as large deformation, great output force as well as high energy density.² Due to its good performance, actuators based on giant magnetostrictive materials, usually called GMAs, have been widely applied in many areas stretching from sonar systems³ to precision machining.^4,5 However, just like actuators based on piezoelectric materials and shape memory alloys, GMAs also possess some inherent defects: strong nonlinearity, multi-field coupling and vulnerability to outside factors. These properties are detrimental to improve their control precision.⁶ Therefore, it is of significant importance to establish an accurate model and identify parameters precisely.

As hysteresis nonlinearity exists in GMA between the input current and the output displacement, a thorough depiction of the hysteresis property helps to better predict its final displacement. At present, hysteresis model of GMA can be roughly classified into two types: namely, mathematical model and physical model.⁶ Mathematical model is a kind of black box model, which only cares about the relationship between input and output. Preisach model is one of the most famous mathematical models.⁷ Due to its general applicability and good performance in fitting hysteresis curves, it is widely used to depict the hysteresis loop of GMA. Similar to Presaich model, Prandtl–Ishlinskii model (P–I model) and Bouc–Wen model are other representative mathematical models.^8–10 Since their inverse can be obtained analytically, they have been applied to control of GMA. However, mathematical model usually fails to match the hysteresis loop well in dynamic conditions, even if the eddy-current loss and abnormal loss are counted. Moreover, identification of the parameters in these models often relies on large quantities of experiment data, making it sensitive to experiment error inevitably. Different from mathematical model, physical model is set up based on correlative physical process, which can reveal the mechanism of hysteresis. Therefore, physical model not only describes the hysteresis loop, but also contributes to design, optimizing and control of GMA. Established depending on micromagnetics and Weiss molecular field theory, Jiles–Atherton model (J–A model) has some advantages over the other models such as relatively simple expressions and high precision.¹¹ So it is widely used to predict both static and dynamic output of GMA.

Advantages as J–A model has, it can't be neglected that the parameters of J–A model are strongly coupled, making them difficult to determine. Jiles first established five implicit equations via five special points in the hysteresis loop, and parameters of J–A model can be identified by iterative algorithm through these equations.¹² Calkins divided the identification process into two step: in the first step, sequential quadratic programming was used to fit the magnetization model, and the second step attempted to get reasonable estimation of parameters in the magnetostrictive model.¹³ In recent studies, many efforts are made to identify the J–A model including adjoint method,¹⁴ Taguchi method¹⁵ and several intelligent algorithms.^16–18 These methods get good results in the offline case. Among them, particle swarm optimization (PSO) algorithm possesses its own advantages such as fast convergence speed, high precision and insensitivity to initial values.^19,20

Nevertheless, when GMA works in dynamic status, the problem becomes more complicated. The parameters in J–A model may vary in real time. The reasons can be summarized as follows.

(1) Change of working temperature may cause parameter altering.

Aiming to provide excitation magnetic field, coil is an essential part of GMA. In dynamic condition, a large amount of heat is generated by coil. Enclosed by a housing, GMA performs poorly in heat dissipation. So the temperature of GMM may rise a lot while working. Although self-cooling system is designed,²¹ the temperature still can't remain constant for the space inside the GMA is too small. Literatures published before show that the saturation magnetization, coercivity and remanence of GMM may change with temperature, similar to most of the ferromagnetic materials. This phenomenon leads to four temperature dependent parameters in J–A model, which are M_s, k, α and c.²²

(2) Change of excitation frequency may cause parameter altering.

Sensitive to the outside factors, GMM can be easily influenced by the mechanical structure of GMA in high frequency. In turn, the mechanical properties of GMM may also exert a tremendous influence on the mechanical structure. The two components of GMA interact with each other, leading to a result that the parameters change along with the altering of excitation frequency. Some scholars discover a phenomenon while conducting experiments: if the parameters are kept constant, the model of GMA can't fit the experiment data well with frequency altering even though eddy current loss and abnormal loss are considered.^23,24

Therefore, constant parameters can't predict the output of GMA precisely. Online identification and adjustment of these parameters is important and necessary. As far as we are concerned, there are still few literatures concerning this issue.

The framework of our work is set as follows: Section 1 introduces the background and necessity of this work; Section 2 describes the structure and working principle of GMA; Section 3 establishes the model of GMA considering the dynamic properties of both the GMM and the mechanical structure; Section 4 analyzes each parameter of the established model and proposes an online identification method; in Section 5, an improved PSO algorithm suitable for online identification is introduced; in Section 6, an online parameter identification system is set up; in Section 7, experimental validation is conducted and the results are discussed; in Section 8, some conclusions are drawn.

2 Structure of GMA

The structure of GMA is detailed in Fig. 1. It mainly contains excitation coil, GMM rod, permanent magnets, housing, disc spring and push-rod. When the GMA works, working current, flowing in excitation coil, generates driving magnetic field along the axial dimension of GMM rod. Under the action of the magnetic field, the GMM rod is deformed, which is called magnetostrictive effect. Followed with the GMM rod, the push-rod outputs displacement and force. Permanent magnets, pasted on both ends of the GMM rod, are used to provide a bias magnetic field so as to avoid frequency doubling of GMM. Disc spring exerts pre-pressure on GMM rod, since larger deformation can be obtained on condition that the GMM rod is preloaded. The housing provides a container for other components and ensures each element has its fixed position in it.

Fig. 1 Structure drawing of GMA (a) 3D model of GMA (b) 2D sketch of GMA.

3 Dynamic model of GMA

3.1 Magnetic field model

Magnetic field model describes the relationship between excitation current I and magnetic field H, which can be expressed aswhere N denotes the quantity of excitation coil turns, I denotes the excitation current, l_r denotes the length of GMM rod and c_r is a number between 0 and 1, which describes the leakage of magnetic flux.

3.2 Magnetization model

The hysteresis between magnetic field H and magnetization M can be described by dynamic J–A model, which is established based on the energy principle. The total energy available to the material can be decomposed into two parts including the energy that the GMM rod absorbs and the energy dissipated, shown as follows.

E_tol = E_abs + E_dis (2)

The total energy can be calculated by

where M_an is the anhysteretic magnetization, H_e is the effective magnetic field internal the GMM rod and μ₀ is the vacuum permeability. The energy GMM rod absorbs to change its magnetization state can be expressed by the following equation.where M denotes the total magnetization of GMM rod. Shown as the following equation, the energy dissipated mainly includes three parts.

The first term on the right hand denotes the hysteresis loss the same as that in static J–A model. The second one denotes the eddy current loss and the third one denotes the abnormal loss. In this equation, k is a parameter quantifying the energy needed to break pinning sites, c is the reversible coefficient, δ is 1 when Ḣ > 0 and is −1 when Ḣ < 0, ρ is the resistivity of excitation coil, d is the diameter of the cross section, β is a geometrical factor, for cylinder it is 16, G is a constant independent of dimension, w is the width of laminations, and H₀ is a parameter related to domain walls.

Combine eqn (5) with other equations in J–A model shown as eqn (6).

The dynamic J–A model can be finally simplified down to a differential equation

To solve the differential equation, 4-order Runge–Kutta method needs to be used, with Newton–Raphson method applied to solve the step length of Runge–Kutta.

3.3 Magnetostriction model

The magnetostriction of GMM rod along the axial dimension can be calculated by the quadratic domain rotation model expressed as followswhere λ_s denotes the saturation magnetostriction and M_s denotes the saturation magnetization of the GMM rod.

3.4 Dynamics model

To describe the relationship between magnetostriction λ and output displacement y, GMA is often regarded as a second-order system,²⁵ and the transfer equation can help to express its final output.where M, C, and K respectively denote the equivalent mass, damping and stiffness of the system, E denotes Young's modulus of the GMM rod and A denotes the cross-section of the GMM rod.

In conclusion, there are eleven parameters need identifying in total. To list them out, they are α, a, k, c, λ_s, M_s, H₀, E, M, C, and K.

4 Reduction of parameters to be identified

According to the previous researches, if the driving frequency or the working temperature of GMA alters, the parameters of the model may change. And the output of the GMA may vary a lot as well.²⁶ As a result, while working in dynamic condition, it is necessary to identify and adjust the parameters online to better predict the output of the GMA. However, identification methods available usually rely on a large number of iterations to obtain a satisfactory solution, especially intelligent algorithms, which have relatively high precision. Thus, if there is one more parameter to be identified, a large quantity of excess calculation is needed. Therefore, to realize online identification, these parameters have to be reduced in number. For the sake of this target, two methods are proposed in this section, one of which is to determine some parameters by calculation, and the other one is to select out parameters that have little influence on the output.

4.1 Reducing parameters by calculation

As is mentioned above, ascribed to the modeling method based on physical process, each parameter in dynamic model of GMA has its own physical meaning. Hence, some parameters are not difficult to be calculated by basic physical theories such as equivalent mass M, Young's modulus E and equivalent stiffness K.

4.1.1 Calculation of M. When working in dynamic condition, the GMA can be regarded as a second-order system shown as Fig. 2. The equivalent mass of the system can be simply represented as the following equation.²⁷where M_r, M_p and M_l respectively denote the mass of GMM rod, push-rod and load.

Fig. 2 Equivalent mechanical model of GMA.

4.1.2 Calculation of K. Shown as Fig. 2, the equivalent stiffness K can be expressed as follows.where K_r is the equivalent stiffness of the entire GMM rod together with permanent magnets; K_s is that of the disc spring; K_g and K_m are the stiffness of the GMM rod and the permanent magnet respectively.

As for K_g, with the assumption that magnetic field inside the excitation coil is uniform, K_g can be represented by

where E, A and l_r have the same meaning as those in dynamic model of GMA. Combine the equations above, the equivalent stiffness of the system can be calculated by

4.1.3 Calculation of E. As a constitutive parameter of GMM, Young's modulus E is not only determined by the material itself, but also by the magnetic field and preload exerted on it.²⁸ Shown as Fig. 3, the fitting polynomial matches well when magnetic field is less than 40 000 A m⁻¹ with preload of 10 MPa.

Fig. 3 E(H) curve and its fitting equation at 10 MPa.

4.2 Selection of non-sensitive parameters

Even though some parameters can be calculated by methods discussed above, there are still too many parameters need identifying, resulting in large computing amount and impossibility of online identification and adjustment of these parameters.

To reduce the parameters further, every parameter left is analyzed to estimate its influence on the final displacement of the GMA, so that the parameters, which have little influence on the output, will be identified only once and fixed during the process of online identification. Based on the idea, three steps are employed as follows:

Step 1: build a reference model based on the parameters displayed in the Table 1, which is obtained by us through offline identification.

Table 1 Parameters in reference model

Parameter	Value	Parameter	Value
Ms	800 kA m^−1	M	0.06 kg
λs	1000 ppm	C	3305 Ns m^−1
a	7800 A m^−1	G	0.1356
c	0.18	d	8 mm
α	−0.001	w	3.6
k	3200 A m^−1	β	16
H0	225.1 A m^−1	ρ	0.0172 Ω mm^−2 m^−1
f	100 Hz

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Step 2: alter the value of each parameter successively with the range of ±15%, plot comparative graph via Matlab and calculate the output displacement deviation for a certain parameter by the following equation

where θ is defined as deviation factor which can depict the effect of each parameter, n is the number of sampling points, y_ri and y_i are the displacement of the i^th sampling point before and after parameter change. The deviation factor of every parameter can be shown in Table 2.

Table 2 Deviation factor of every parameter

Parameter	θmax	Parameter	θmax
Ms	2.0311 × 10^−14	α	6.1039 × 10^−15
λs	7.2940 × 10^−12	k	3.2521 × 10^−13
a	7.9395 × 10^−12	C	6.8836 × 10^−14
c	2.9813 × 10^−15	H0	8.964 × 10^−15

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The comparative graphs are shown as Fig. 4.

Fig. 4 Influence of each parameter on output of GMA (a) parameter λ_s, a and k (b) parameter M_s, c, α, C and H₀.

In Fig. 4, it should be noted that there are three curves in every group. The red one represents the input–output curve determined by the parameters displayed in Table 1. With a certain parameter minus or plus 15%, the black and blue curves are obtained respectively. The curves colored red, black and blue may not be distinguished in Fig. 4(b) for they coincide with each other.

Judging from the figures and calculation above, the parameters can be divided into two types. The first type contains λ_s, a, and k, which have a significant impact on the final output of GMA, while the rest, classified as the other type, have little influence.

Step 3: for the suspected non-sensitive parameters selected out, alter the value of them further so as to confirm their effects on output of GMA. The variation range of the suspected non-sensitive parameters and their deviation to reference model is shown in Table 3.

Table 3 Variation range of the suspected non-sensitive parameters and their maximum deviation factor

Parameter	Value	θmax
Ms	[600, 1000] kA m^−1	6.1202 × 10^−14
c	[0.05, 0.40]	9.1527 × 10^−14
α	[−0.01, 0.01]	3.0059 × 10^−11
C	[2000, 4500] Ns m^−1	1.2278 × 10^−13
H0	[100, 350] A m^−1	7.952 × 10^−14

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According to Table 3, M_s, c and H₀ have much less influence on output of GMA than λ_s, a, α, C and k. To summary up, λ_s, a, α, C and k need to be identified and adjusted online, while the others can be fixed to reduce the computation amount of online identification.

5 Identification method based on modified PSO

Particle swarm optimization (PSO) algorithm is an intelligent algorithm widely applied in parameter identification and structure optimization.²⁹ Imitating the feeding behavior of birds, PSO searches optimal solution by moving numerous of searching points, called particles, across the searching space to minimize the fitness function which can be represented as follows.where y_i denotes the experiment data acquired, ŷ_i denotes the displacement predicted by identified model, and N denotes the population size of particles.

Every particle in PSO, which is a potential solution, is attached to two vectors, called velocity vector v_i = [v_i¹, v_i²…v_i^D] and position vector X_i = [x_i¹, x_i², … x_i^D], where D is the dimension of solution space. As for identification problems, it is the number of parameters to be identified. Particles are initialized by a group of random solutions and search solution space for the best solution adhering to the following equation.

where v_i^d and X_i^d denote the v and X vector when number of iteration is d; w denotes the inertia factor of the particle; p_i denotes the best position of a particle called p_best; p_g denotes the best position of the entire swarm called g_best; r₁, r₂ ∈ [0, 1] are random numbers; c₁ and c₂ denote the acceleration coefficients.

The new position of particle is substituted into fitness function, and if the fitness value J(X_i) is less than that of p_best, p_best is updated by X_i. The update of g_best follows the same rule mentioned above.

Despite its faster convergence speed than other intelligent algorithms, traditional PSO still can't meet the requirements of online identification. Therefore, some improvements are needed to accelerate it.

(1) A trigonometric function operator is introduced to searching process of the particles. Then, the algorithm can be modified as follows.³⁰

where γ is an angular value; β is a constant greater than 1; Θ is a trigonometric function operator which can be specified as sin, tan, or cos and Θ[(γ)] is less than 1. The convergence speed of PSO with this improvement can be calculated as follows.where λ_1,2 are determined by trigonometric function operator Θ[(γ)]; V_IPSO and V_PSO denote the convergence speed of PSO with this modification and PSO without modifications respectively.

(2) An error-varying inertia factor is introduced. It is reported that a large inertia factor has a good performance of global searching, while a small one converges to optimal solution more easily. Therefore, a varying inertia factor is necessary in different phase of the algorithm. The error-varying inertia factor can be represented as follows.

where BestJ(1) and BestJ(i) respectively denote the fitness value of the best particle when the number of iteration is 1 and i; w_min and w_max denote the minimum and maximum value of w defined respectively.

To compare the convergence speed of the modified PSO with traditional one, numeric simulations are conducted in Matlab. It can be detected from Fig. 5 that PSO with improvements converges within 50 iterations, faster than traditional one, and the precision also performs better.

Fig. 5 Comparative graph of convergence speed.

(3) The previous optimized parameters are used as the initial values of the next identification process. In spite that parameters change during the operating procedure of GMA, the range may not be great. Therefore, if the previous optimized parameters are employed as the initial value, the improved PSO may tend to converge in fewer iterations. To validate it, numeric simulations are carried out by Matlab considering the following six conditions as Table 4.

Table 4 Altering range of excitation frequency conducted in numeric simulations

Condition	Altering range of excitation frequency
1	25 Hz → 75 Hz
2	125 Hz → 175 Hz
3	225 Hz → 275 Hz
4	25 Hz → 125 Hz
5	125 Hz → 225 Hz
6	25 Hz → 225 Hz

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In Table 4, the first three conditions represent where the excitation frequency alters no more than 50 Hz; the conditions numbered 4 and 5 represent where the excitation frequency alters between 50 and 100 Hz; and the last one represents where the excitation frequency alters between 100 and 200 Hz. The simulation results are shown in Fig. 6. The convergence curve can be divided into six parts from left to right. They correspond to the six conditions respectively.

Fig. 6 Converge speed of identification algorithm with different altering range of excitation frequency.

Since the previous optimized parameters are near to the current ones, the number of iterations reduces a lot by using them as initial values. According to the results in Fig. 5, when the excitation frequency changes, the iteration number of PSO can be determined as Table 5.

Table 5 Number of iterations needed in PSO with different altering range of frequency

Range (Hz)	Number of iterations
[0, 50]	10
[50, 100]	20
[100, 200]	30
[200, ∞]	50

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6 Design of online parameter identification system

To validate the proposed method, online parameter identification system is established. Compared with the offline case, online identification is much more difficult to realize. Many more issues need to be considered. We establish our system centering on data acquisition card (DAQ), which possesses advantages of low cost, high precision and good transplantability.

6.1 Design of hardware

As is shown in Fig. 7, the hardware of the identification system mainly consists of four parts: that is, power supply module, GMA, sampling module and master computer.

Fig. 7 Hardware of the online parameter identification system.

The power supply module is designed aiming to provide excitation current for GMA. It is mainly made up by signal generator and power amplifier. While working, it can drive GMA to generate magnetostriction under the required signal.

The sampling module concerns data acquisition of the input excitation current and the output displacement. It mainly contains a current clamp and an eddy current displacement sensor, both of which have high measurement precision and sensitivity.

The master computer, as the core of the system, is employed to get data from the sampling module and run the software developed to complete the identification process. Inside the computer, DAQ, acting as a bridge between the sensors and the master computer, is embedded, which has a precision of 16 bit.

6.2 Design of software

Whether online identification can be conducted effectively relies on not only hardware but also software. And the software, regarded as a bridge between users and hardware, is developed based on Labwindows/CVI and Matlab, for the reason that CVI is convenient for data acquisition and Matlab has powerful features of numeric computing. With modular design thinking, the software is made up of three parts: namely, data acquisition part, data processing part and user interface part. Main flow chart of the software is shown in Fig. 8.

Fig. 8 Main flow chart of the identification software.

6.2.1 Design of data acquisition part. Data acquisition part, as its name suggests, is designed to acquire data from DAQ. Both developed by NI, Labwindows/CVI is well compatible with the DAQ. Using a special data format named TDMS to read and store data simultaneously, the developed software performs well in real-time and precision.

6.2.2 Design of data processing part. Data processing part, namely identification algorithm, is implemented in Matlab. As is analysed above, the algorithm contains calculation part and identification part. The calculation part acquires data from sensors and then calculates parameters by equations represented in Section 4.1, while the identification part, also obtaining data from sensors, identifies parameters based on the modified PSO.

As for identification part alone, if it is the first time to identify, it calls for first-time identification program. All parameters except those can be determined by calculation should be identified and the iteration number of the improved PSO is set as 50. After that, online identification program is called for. With non-sensitive parameters fixed, the others should be identified and adjusted online. The main flow chart of online identification program is shown as Fig. 9. According to the altering range of frequency, the number of iterations is determined following the results in Section 5.

Fig. 9 Main flow chart of the online identification method.

6.2.3 Design of user interface part. User interface part, acting as a tool realizing human-computer interaction, is designed by Labwindows/CVI, which provides many controls specialized for virtual instruments. Shown in Fig. 10, the user interface (UI) can be roughly segmented into four areas. Data exhibition area, occupying most of the UI, helps to display data acquired by displacement sensor, predicted by model as well as the deviation between them. Via parameters update area, the parameters identified can be observed. The other areas of UI, including two areas, are used for initial parameter setting and command.

Fig. 10 Main interface of the identification software.

7 Experiment and discussion

With the help of the proposed system, online parameter identification experiments are conducted. According to the operating conditions of GMA, the experiments are classified into two types. In one case, the GMA works in constant frequency and in the other case, the driving frequency changes during the operating procedure of GMA. Besides, it should be noted that both types of experiments are conducted based on the following conditions: (1) the working current is between −3 A and 3 A; (2) the master computer is a workstation with an eight core processor inside. The results are discussed in detail as follows.

7.1 Discussion of identification time

The consequences of parameter identification in constant frequency are shown as Fig. 11. It can be observed from Fig. 11(a) that the predicted output is not displayed until about 12 seconds later, for the parameters are not determined until the first identification process is complicated. Moreover, when the deviation is out of the preset threshold due to raised temperature or other causes, the identification algorithm starts again shown as Fig. 11(b). Since the software can only exhibit data within 0.1 second, Fig. 11(b) just displays part of the identification process. The second-time identification process, owing to good initial values of PSO, costs less than 1.5 second to identify, much less than the first-time identification process.

Fig. 11 Experiment results in constant driving frequency (a) experiment results of first-time identification (b) experiment results of second-time identification.

Fig. 12 shows the experiment results when the driving frequency of GMA changes. The four figures respectively represent the experiment results when the altering range of frequency is 50 Hz, 100 Hz, 200 Hz and more than 200 Hz. To compare the results intuitively, frequency changes are all conducted at the same time, 50 seconds after the beginning of experiments. Observed from the figures, the identification time varies a lot due to different change of frequency. When the frequency alters from 20 Hz to 300 Hz, the identification time is about 6.5 seconds, which is the longest among the experiments.

Fig. 12 Experiment results when driving frequency of GMA changes (a) experiment results from 175 Hz to 225 Hz (b) experiment results from 175 Hz to 275 Hz (c) experiment results from 75 Hz to 275 Hz (d) experiment results from 25 Hz to 300 Hz.

7.2 Discussion of identification precision

The identification precision, acting as another important assessment index, will be discussed in this section. Since the developed software can store the experiment data, it is convenient to draw the hysteresis loop using the data acquired during the experiment. The results are shown as Fig. 13.

Fig. 13 Identification precision in different frequencies.

Learned from Fig. 13, the displacement output predicted by the identified model of GMA matches the test curves well along the frequencies from 50 Hz to 300 Hz despite some deviations in certain points. And the calculation time, which has been discussed before, is much shorter than previous method. Therefore, it is proved that the identification method proposed in this work is suitable for online system.

8 Conclusion

In this paper, a dynamic model of GMA based on J–A model is established. Considering the dynamic properties of GMM and the mechanical structure, the model can better predict the output displacement of GMA in high excitation frequency.

Based on the established model, an online parameter identification method is proposed. Some parameters are determined by calculation through physical theory, and the others are classified into two types according to their sensitivity on final output of GMA. With different identification methods conducted to different type of parameters, the identification time is reduced.

An improved PSO is introduced to complete the process of identification. With some modifications specially for online case, the improved PSO has faster convergence speed than previous ones.

An online parameter identification system is set up to realized online parameter identification for GMA. Both of the hardware and software are developed. Learned from the results, the proposed system performs well and proves its advantage in online identification.

Acknowledgements

The authors acknowledge the financial support from the National Science Foundation of China (Project no. 51275525).

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