Research on the constant output force control system for giant magnetostrictive actuator disturbed by external force

Research on the constant output force control system for giant magnetostrictive actuator disturbed by external force

Mechatronics 22 (2012) 911–922 Contents lists available at SciVerse ScienceDirect Mechatronics journal homepage: www.elsevier.com/locate/mechatronic...

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Mechatronics 22 (2012) 911–922

Contents lists available at SciVerse ScienceDirect

Mechatronics journal homepage: www.elsevier.com/locate/mechatronics

Research on the constant output force control system for giant magnetostrictive actuator disturbed by external force Huifang Liu, Zhenyuan Jia ⇑, Fuji Wang, Fucai Zong Key Laboratory for Precision and Non-traditional Machining Technology of Ministry of Education, Dalian University of Technology, Dalian 116024, PR China

a r t i c l e

i n f o

Article history: Received 22 July 2011 Accepted 28 May 2012 Available online 13 August 2012 Keywords: Control system Giant magnetostrictive actuator Constant output force External force

a b s t r a c t In this paper, a constant output force control system for giant magnetostrictive actuator which is disturbed by external force continuously was developed. Based on the giant magnetostrictive actuator model related to the combined action of magnetic field and external force, a controller is designed. The controller is composed of giant magnetostrictive actuator model control algorithm and PID control algorithm to realize the output force reaching the constant force target quickly and finely adjustment. Control system’s hardware and related software were designed and software package was developed. Experiment results show that the control system has perfect response performance and stability, and favorable controllability. It can effectively realize the giant magnetostrictive actuator which is disturbed by external force continuously keeping outputting a constant force through adjusting the working current. It lays a certain theoretical foundation for giant magnetostrictive actuator being used in the fields of cutting with invariableness cutting force. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction In the field of precision, ultra-precision machining and other manufacturing industry, microdisplacement actuator which has a certain high performances of high position resolution, large output force, fast response and low creep is one of the key elements. With the emergence of giant magnetostrictive materials (GMMs), a kind of actuator with perfect performances has been developed [1,2]. Giant magnetostrictive actuator (GMA) is a transducer which uses the characteristics of GMM producing deformation under the action of external magnetic field to realize the conversion from electromagnetic energy to mechanical energy. Currently, GMA has been applied in many areas, such as high power underwater acoustic transducers, linear motors, rail diesel injection system, vibration absorber of turboprop aircraft, and hybrid magnetostrictive device [3,4]. Recently, GMM and its actuators also attract attentions of the researchers of mechanical processing field. Tong et al. developed a dual stage feed drive system in which a linear motor is used as the coarse actuator and a magnetostrictive actuator is used as the fine actuator [5]. Filipovic and Sutherland presented a new technology for dry deep hole drilling of aluminum. In the drilling process, a magnetostrictive actuator is used to create modulated motions at the tip of the drill to vary the chip ⇑ Corresponding author. Address: School of Mechanical Engineering, Dalian University of Technology, No. 2, Linggong Road, Ganjingzi District, Dalian 116024, Liaoning Province, PR China. Tel.: +86 13940925372; fax: +86 41184707743. E-mail address: [email protected] (Z. Jia). 0957-4158/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechatronics.2012.05.008

size [6]. Wu and Xiang presented a non-circular piston pinhole machining system through embedding GMM into an appropriate position of the tool and using GMM’s characteristics to control the deformation of arbor [7]. According to Jiles–Atherton nonlinear hysteretic model and combining the Kelvin–Voigt damping theory, Sun et al. established a magneto-elastic model of GMA for non-circular cutting [8]. At present in the application of machining system, it mainly uses the microdisplacement outputted by GMA to achieve high resolution micro-feed, small-size non-circular turning, deep-hole and profiled hole machining, and active vibration control of machines. Moreover, at present researches on the structure and related control system of this type of actuator, and its characteristics analysis are relatively mature. Kim and Sadighi designed a low-power linear magnetostrictive actuator which allows the flexibility to operate it in various configurations depending on the type of applications. The linear magnetostrictive motor in the local three-phase operation mode demonstrates its travel range of 45 mm with power consumption of 95 W [9]. Karunanidhi and Singaperumal designed a magnetostrictive actuator with flexure amplifier and integrated it into an existing flapper-nozzle servo valve to replace the torque motor. Compared with conventional valve, the valve has satisfactory static and dynamic characteristics for applications in high-speed actuation systems [10]. Moon et al. designed a linear magnetostrictive actuator and a real-time digital control system with a linear quadratic feedback controller. The actuator displacement can reach up to about 27 lm in the linear range [11]. A general magnetostrictive constitutive model for the

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giant magnetostrictive film and rod which can describe the nonlinear change of the maximum and saturation magnetostrictive strain with the pre-stress was proposed by Zhou et al. [12]. Based on the Jiles–Atherton model, a constitutive magnetostrictive response model to an average diffused magnetic field is quantified through the dynamic eddy current losses being modeled as a one-dimensional magnetic diffusion problem in cylindrical coordinates [13]. Linnemann et al. used the magnetostrictive delay line method to propose an analytical modeling which can be used for the determination of magnetization function as well as their uniformity distribution along the length of magnetostrictive ribbons and wires [14]. When GMA which works in mechanical clamping state is under the action of magnetic field, GMM will occur magnetostrictive effect and GMA will output magnetostrictive deformation in the form of force. Meanwhile, reaction force of the output force can make GMM occur magnetostrictive inverse effect and it has effect on the magnetization state of GMM. The coexisting of magnetostrictive effect and inverse effect makes magnetic system and mechanical system be coupled. At this time, for the GMA which outputs a force under the action of a current, the output force not only depend the working current, but also is related to the coupling effect between magnetostrictive effect and inverse effect. Moreover, its output force will change when it is disturbed by an external force. Therefore, utilizing the bidirectional reversible magneto-mechanical energy conversion property of GMM and the coupling relationship of magnetostrictive effect and inverse effect, it can adjust output force through changing the working current to realize GMA output a same size force with the force before the action of external force. Based on this performance, GMA can be used in cutting with invariableness cutting force and other fields. For ensuring GMA which is disturbed by external force can keep the output force constant accurately, it is crucial to decouple the coupling action of magnetostriction effect and inverse effect. At present, researches related to the coupling characteristics of these two effects have just started, and most of them are reference to the implementation method of piezoelectric self-sensing actuator. Jones et al. proposed using an AC bridge composed of resistance and inductance to isolate the induced voltage signal generated by magnetostriction inverse effect in the GMA coil [15]. Zhejiang University and Hebei University of Technology also started to study the mechanism and implementation method of giant magnetostrictive self-sensing actuator [16,17]. Moreover, theoretical model and analysis for the output force of GMA are inadequate currently. Reports on using the output force of GMA for cutting with invariableness cutting force are even fewer. Aiming at these problems and expanding the application fields of GMA, our studying team propose that using the method of mutual calling the magnetization results calculated by magnetostrictive effect model and inverse magnetostrictive model respectively as the initial magnetization state of calculation again to rescind the coupling effect between the magnetostrictive effect and inverse magnetostrictive effect in GMM rod repeatedly. Our studying team developed a GMA using a GMM rod as the core element. The change law of internal magnetization state of GMA working in mechanical clamping condition when it is under the combined action of magnetic field and external force is analyzed. Moreover, taking the actuator as research object and taking keeping the output force of GMA constant when it is disturbed by external force exactly and real time as the target, a control method which is a combination of control current solution method based on the magnetostrictive effect model and inverse effect model and PID method is presented. The control method is on the basis of the above decoupling method for magnetostrictive effect and inverse effect. Then, a closed-loop constant output force control system for GMA is developed. It lays a certain theoretical foundation

for GMA being used in the fields of cutting with invariableness cutting force. This paper is organized as follows. In Section 2, GMA structure is introduced and the working principle of keeping output constant force is presented. In Section 3, models for GMA working both under the action of magnetic field and external force are expounded. Section 4 elaborates the detailed design process of the whole control system, including hardware structure, controller and software. In Section 5, the control system performance is tested through a series of experiment. Section 6 concludes. 2. Structure and working principle of GMA with constant output force 2.1. Structure of GMA The structure of GMA using a GMM cylindrical rod as the core element is shown in Fig. 1. Under the action of magnetic field generated by working current in excitation coil, the axial dimension of GMM rod changes. It outputs displacement or force through transferring spindle. GMM rod as well as lower magnetizer and lower guide pad are installed in the stainless steel tube. It ensures that magnetic conductive elements and GMM rod are in the axial direction of distributing magnetic field. Upper magnetizer, upper guide pad, lower magnetizer, lower guide pad, lower magnetic conductive gasket, cylindrical magnetic yoke made of high permeable electrical pure iron DT4 and GMM rod form a closed magnetic circuit. It makes that there is larger and more uniform magnetic flux density in GMM rod under the action of a certain current. The average magnetic flux density in GMM rod is measured by a Hall sensor installed at the bottom of GMM rod. Moreover, a stainless steel ring is added around Hall sensor in accordance with the magnetic circuit properties. The special structure improves the measuring sensitivity of magnetic flux density. Disk spring is a preload device which provides an adjustable pre-tightening force to GMM rod by adjusting the thread fit distance between pre-tightening thread sleeve and head cover. In order to guarantee the stationary of whole structure and realize fine adjustment of pre-tightening force conveniently, two disk springs with involution combination mode are chosen. Outer sleeve, head cover and bottom cover made of antimagnetic stainless steel material form a closed shell. It packages the whole actuator in the internal to prevent the internal magnetic circuit interfering from external environment. 2.2. Implementation principle of GMA outputting constant force For GMM, there are two important physical effects. On the one hand, under the action of magnetic field, a deformation of the ferromagnetic material occurs, which is called magnetostrictive effect. On the other hand, it is the inverse magnetostrictive effect also known as the Villari effect. It specifies that under the action of external force, it generates mechanical stress in ferromagnetic materials and leads to relative permeability and magnetic flux density change. When a GMM rod being under the action of a magnetic field is subjected to an external force, magnetostrictive effect and inverse effect will occur simultaneously. The change process of magnetic state in GMM rod being in mechanical clamping boundary condition is shown in Fig. 2. When external magnetic field H is applied to the GMM rod, magnetic flux density in GMM rod is B0. Magnetostrictive effect occurs in GMM rod, and there will generate stress in GMM rod. The stress is output in the form of magnetostrictive force Fo1. Then, if an external force Fe is imposed on GMM rod, it leads to the occurrence of inverse magnetostrictive effect. The action of inverse magnetostrictive effect cause

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(a) Two-dimensional design diagram 1- transferring spindle, 2- upper guide pad, 3- upper magnetizer, 4- cylindrical magnetic yoke, 5excitation coil, 6- lower magnetizer, 7- stainless steel tube, 8- stainless steel ring, 9- lower guide pad, 10- Hall sensor, 11- lower magnetic conductive gasket, 12- GMM rod, 13- outer sleeve, 14head cover, 15-bottom cover, 16- pre-tightening thread sleeve, 17-disk spring

(b) Solid diagram of GMA Fig. 1. Structure diagram of GMA.

Fig. 2. Magnetization state of GMM rod under magnetic field and external force.

magnetization state in GMM rod change and generate a new additional magnetic flux density BF. Consequently, compared with initial magnetic flux density B0, the total magnetic flux density in GMM rod has changed. Then, since magnetostrictive effect of GMM rod, the changing magnetic flux density causes the changing of stress in GMM rod, and magnetostrictive force changes from the initial Fo1 to Fo2. According to the Newton’s third law, when GMM rod is subjected to an external force Fe, it generates a same size reaction output force F 0e . The magnetostrictive force Fo2 couples with the reaction force F 0e , and the total output force Fo of GMA is no longer equal to initial force Fo1.

According to the above analysis, for GMA using GMM rod with bilateral energy conversion as the core element and being under the action of a magnetic field, when it is subjected to an external force, the total output force of GMA changes compared with the initial output magnetostrictive force. Furthermore, the total output force is not a simple superposition of initial output magnetostrictive force and the reaction force of external force, but there is an additional output force which is caused by the coupling effect between magnetostrictive effect and inverse effect. In this situation, if it needs to keep the total output force of GMA being equal to the initial output magnetostrictive force, it can change working current to adjust the magnetostrictive force Fo2. Then, the difference value of output force can be offset and GMA keeping output constant force is realized. 3. Model of GMA under the action of magnetic field and external force According to the implementation principle of keeping GMA’s output force being constant, it is known that the keys are rescinding the coupling relationship between magnetostrictive effect and inverse effect, accurately quantifying output force and magnetization of GMA being under the combined actions of current and external force. In this paper, considering the actual working process and magnetic state changing process, it proposes that

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the coupling effect of magnetostrictive effect and inverse effect is decoupled by the method of repeatedly calculating magnetic state through magnetostrictive effect model and inverse effect model until magnetization nearly reaches a balance state. The concrete method is as follows: Magnetization state of GMM rod caused by magnetic field is determined through Jiles–Atherton magnetization model. The influences of external force on magnetization state and magnetic flux density are calculated on the basis of magnetomechanical effect approach. Then, based on the magnetization state results independently calculated by magnetostrictive effect model and inverse effect model, magnetostrictive effect and inverse effect are decoupled by the method of repeatedly mutual calling the magnetization results calculated above as the initial magnetization state for calculation again until magnetization nearly reaches a balance state. Then, GMA’s magnetostrictive output force is quantified by the combination of the gained magnetization state and change law of elastic modulus. Finally, it adopts the inverse model of magnetostrictive effect model and change law of elastic modulus to determine the new working current which is needed to make the actuator maintain outputting constant force.

3.1. Magnetostrictive effect model Jiles–Atherton model is a hysteretic model describing the relationship between external magnetic field intensity and magnetization, and it is established on the basis of ferromagnetic material domain wall theory [18]. Based on the magnetization model, a relationship model among magnetization, GMM rod strain, actuator’s output force and working current is established. In this model, the total magnetization is composed of reversible magnetization and irreversible magnetization. Combining with anhysteretic magnetization, a relationship between magnetization and working current is obtained. In this model, material’s effective magnetic field Heh, anhysteretic magnetization Man, irreversible magnetization Mirr, reversible magnetization Mrev, total magnetization M, and magnetic flux density B are as follows:

4

8 Heh ¼ H þ aM > > > > > H ¼ NI=Lc > > > > > > < Man ¼ Ms ðcothðHeh =aÞ  a=Heh Þ @Mirr @H

irr ¼ @M @H

@Heh @H

irr ¼ dk1 MaanðMM an M

eh > > > > Mrev ¼ c1 ðMan  M irr Þ > > > > > M ¼ Mirr þ M rev > > : B ¼ l0 ðM þ Heh Þ

irr Þ

ð1Þ

r ¼ Ek

ð3Þ

F ¼ rAr ¼ Ar Ek

ð4Þ

where Ar and E are the cross sectional area and elastic modulus of GMM rod. However, there is a coupling between magnetic and elastic properties in GMM, therefore, there are significant differences in elastic modulus with different magnetic field and force. Consequently, the elastic modulus cannot be seen as a constant in the process of calculating actuator’s output force. In this paper, the concrete value of elastic modulus is determined by the elastic modulus model which is established based on the internal energy and experimental characteristics of GMM [19]. Eq. (1) is the magnetization model of magnetostrictive effect, and it is used to describe the magnetization state in GMM rod when GMA is driven by current. Eqs. (1) and (2) constitute the nonlinear hysteresis displacement model, and based on this, the strain generated in GMM rod can be calculated. According to the strain, and combining Eqs. (3) and (4), magnetostrictive output force of GMA under the action of excitation current can be quantified. 3.2. Inverse magnetostrictive effect model While GMM is under the action of a force, magnetization and magnetic flux density will change. Based on the assumption that hysteresis originates primarily from domain wall pinning and the freeing of domain walls from their pinning sites cause the magnetization to change in such a way as to approach the anhysteretic state, Jiles proposed a magnetomechanical effect approach which can describe changes in magnetization oM/or that a magnetostrictive material undergoes when subjected to a uniaxial stress effectively. In this model, the effective magnetic field Her related to stress, anhysteretic magnetization Mar related to stress, the elastic energy W of magnetic domain movement caused by stress, total magnetization M, and magnetic flux density B are as follows:

Her ¼ H þ aM þ

3 2

r @k l0 @M

ð5Þ

Substituting Eq. (2) into Eq. (5) yields:

Her ¼ H þ ða þ rð3c1 þ 6c2 M2 Þ

.

l0 ÞM ¼ H þ ar M

ð6Þ

.

where a is the dimension coefficient related to magnetization process, Ms is the saturation magnetization, a represents the effective domain density, k1 is irreversible loss coefficient, c1 quantifies the amount by which domain walls bulge before breaking away from pinning sites, H is the applied magnetic field, N is total turns of the coil, I is current in coil, Lc is the effective height of coil, l0 is the permeability of vacuum. Parameter d is +1 when dH/dt > 0 and 1 when dH/dt < 0. While isotropic material is under the action of a magnetic field parallel to the axis, its dimension changing in length direction is caused by domain wall rotation and motion. The magnetostriction generated along the magnetic field direction on the basis of second order and fourth order magnetostrictive coefficient is:

k ¼ c1 M 2 þ c2 M4

According to the Hooke’s law, under the action of a magnetic field, stress r generated in GMM rod and the output force F are respectively:

ð2Þ

where k is the strain (or magnetostrictive coefficient) of GMM rod, c1 and c2 are second order and fourth order magnetostrictive coefficient respectively.

where ar ¼ a þ rð3c1 þ 6c2 M2 Þ l0 . Anhysteretic magnetization related to stress is quantified using Langevin function [20].

    her a Mar ¼ M s LðHer =aÞ ¼ M s coth  H er a

ð7Þ

The total magnetization is instead of anhysteretic magnetization in Eq. (6), and Eqs. (6) and (7) become:

Her ¼ H þ aa M ar M ar ¼ M s



   H þ aa Mar a coth  H þ ar M ar a

ð8Þ ð9Þ

The elastic energy of per unit volume supplied to the GMM rod by the changing applied stress depends on the square of stress [21,22]:



r2 ð2EÞ

ð10Þ

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According to magnetomechanical effect, irreversible magnetization and reversible magnetization caused by elastic energy are defined as respectively [21,22]:

@M irr 1 ¼ ðM ar  M irr Þ n @W

ð11Þ

M rev ¼ c2 ðMar  M irr Þ

ð12Þ

where n is coefficient with dimensions of energy per unit volume, c2 describes the flexibility of the magnetic domain walls. Combining Eqs. (10) and (11) with the derivative of elastic energy with respect to stress gained from Eq. (10), the derivative of the irreversible magnetization with respect to stress is obtained:

@M irr @M irr dW r r ¼ ¼ ðMar  M irr Þ ¼ 2 ðM ar  Mirr Þ @r @W dr En e

ð13Þ

where e2 = En. The total magnetization M is then dictated by the superposition of the irreversible and reversible magnetization, and the magnetic flux density B can be determined correspondingly:

M ¼ Mirr þ M rev

ð14Þ

B ¼ l0 ðM þ HÞ

ð15Þ

Therefore, the magnetization and magnetic flux density of GMM rod which is under the action of an external force can be determined through the solution of Eqs. (9), (12)–(15).

Fig. 3. Flow chart of parameter identification.

3.3. Parameters identification There are nine unknown magnetic and magnetostrictive parameters involved in the GMA model, that is a, Ms, a, c1, k1, c1, c2, n and c2. These parameters are related to the components and production processes of GMM rod; therefore, it needs to identify these parameters for the GMM rod which is practical application in the actuator. The genetic algorithm is a non-gradient parallel optimization algorithm which is based on Darwin’s survival of the fittest evolutionism. The algorithm has an advantage of rapid convergence. However, it will take a long time to achieve the real optimal solution. Moreover, it is difficult to find the exact minimum value sometimes. The simulated annealing algorithm is originated from the simulation of solid annealing process in physical statistics. The algorithm can escape from local optimum ‘‘trap’’. But, the calculating time is longer and efficiency is lower. Therefore, in view of the defects of genetic algorithm and simulated annealing algorithm, these above parameters are identified by an approach which is a combination of genetic algorithm and simulated annealing algorithm in this paper. The concrete process of parameters identification is shown in Fig. 3. Firstly, the initial parameter populations are crossed and mutated using the fast search ability of genetic algorithm and a group of optimum parameters are obtained. Then the parameter populations are optimized and adjusted using the sudden jump ability of simulated annealing algorithm. In the process of parameters identification, optimum reserved strategy and dynamic step search method are adopted. It not only makes the algorithm converge faster, but also improves identification precision and quality of the optimal solution. According to the experimental measuring values of GMA’s working current and GMM rod’s strain, external compressive force acted on GMA and magnetic flux density in GMM rod, these parameters are identified and the results are shown in Table 1. 3.4. Model verification Based on the magnetostrictive effect model established in Section 3.1 and above parameters identification results, elongation

Table 1 Parameter identification results. Parameters a Results

7.806  103

Parameters c1 Results

Ms

a

c1

7.905  105

0.007

0.002 1.892  103

c2

n

c2

k1

2.196  1015 1.223  1029 8.792  103 0.883

value of GMM rod is calculated while GMA is under the action of 0–0.5 A, 0–1.0 A, 0–1.5 A, 0–2 A and 0–4 A current. Moreover, magnetic flux density of GMM rod is calculated using the inverse magnetostrictive effect model established in Section 3.2 while GMA is under the action of 0–1000 N compressive force. The calculated and experimental results of elongation value and magnetic flux density are shown in Fig. 4a and b respectively. The results show that, the agreement between calculated quantities and experimental data is good except under the action of smaller current and force. The average relative error of elongation value and magnetic flux density are 4.65% and 3.58% respectively. Therefore, the established models and parameter identification values are able to effectively describe the magnetostrictive effect characteristics and inverse magnetostrictive effect characteristics. 4. Design of control system 4.1. Principle and hardware structure of control system The control system for GMA with a constant output force is mainly composed of master control computer, GMA, data input and output board, high speed bipolar programmable power supply, weighing sensor and auxiliary circuit. Whole structure of the keeping output constant force control system for GMA is shown in Fig. 5. Two couples of input channels of data input and output board are selected to read force signal of weighing sensor and current signal in coil respectively. A couple of output channel is used

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(a) Elongation value of GMM rod

(b) Magnetic flux density of GMM rod

Fig. 4. Comparison of calculated results and experimental data.

Fig. 5. Principle diagram of control system for GMA with constant output force.

Fig. 6. Schematic diagram of the control system controller.

to output control voltage signal. Working principle of the control system is as follows: GMA’s output force signal measured by weighing sensor is converted into 0–5 V voltage signal through the secondary amplifier circuit. Then, the signal is converted into digital signal after being processed by data input and output board and its internal A/D conversion circuit. In the meanwhile, the actual working current in coil after being processed through bleeder circuit is also converted into digital signal by data input and output board and its internal A/D conversion circuit. Computer control system completes digital signal processing and analysis through a complex control algorithm. Moreover, after digital signal which is obtained from the computer control system being processed by D/A converter circuit and data input and output board, 5 V to +5 V voltage signal is obtained. Then, the voltage signal is input to high speed bipolar programmable power supply for power

amplifier and 2–2 A current is gained. The current signal is the new working current of GMA and it is used to generate driving magnetic field. In addition, a 50 KX resistance is connected in series between high speed bipolar programmable power supply and the data input and output board. It makes the output accuracy of control current be able to reach 0.001 A. The design of keeping output constant force control system for GMA mainly includes controller design and the implementation of software program. They are expounded in detail respectively in the following sections. 4.2. Controller design Controller structure of the control system is shown in Fig. 6. According to the constant output force target of GMA system, in this paper, the method of GMA model algorithm combining with

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PID control algorithm is adopted to calculate the control current which is needed to supply to coil. When it is the first time of control system to adjust the current, it uses GMA model discussed in Section 3 to calculate the control current in accordance with the actual current value in coil, the difference value eF between constant output force target Fc and the total output force Fo of GMA being under both action of magnetic field generated by working current and external force Fe. This calculating process is from the view point of the occurrence essence of magnetostrictive effect and inverse effect, and it will make GMA’s output force be close to the constant output force target rapidly. Starting from the second time of adjusting control current, aiming to the calculating error of GMA model, the control current is calculated by PID control algorithm in accordance with the force deviation eF. After the conversion of D/A circuit, control current is input to the high speed bipolar programmable power supply through data input and output board for power amplifier. The current is provided to coil and it is the working current of GMA. Under the action of magnetic field generated by the new working current, GMA’s output force changes. The output force and actual working current measured by measuring devices are feedback to computer control unit. Output force value compares with the constant output force target value and constitutes a closed-loop circuit. The key parts of control system controller is the calculating method and process for control current, that is GMA model algorithm and PID control algorithm. For the concrete calculating process of GMA model algorithm, it is expounded in Section 4.3.2. The design process and parameters tuning of PID control algorithm are discussed as follows. GMA system which can maintain output constant force is mainly used in cutting with invariableness cutting force and other mechanical engineering field, therefore, stability and steady-state error are the system’s main quality. Moreover, the output performance of GMA is influenced by the magnetic history. Consequently, a typical incremental PID control algorithm is adopted in the control system and it is expressed as:

uðkÞ ¼ K p ½eðkÞ  eðk  1Þ þ K I eðkÞ þ K D ½eðkÞ  2eðk  1Þ þ eðk  2Þ

ð16Þ

In which, k is the sampling sequence number. u(k) is the output variable namely the control current at the k th sampling time. e(k), e(k  1) and e(k  2) are the difference values between constant output force target and force feedback value at the k th, k  1th and k  2th sampling time respectively. Kp, KI and KD are proportional coefficient, integral coefficient and differential coefficient. In the controller design process, determining of proportional coefficient Kp, integral coefficient KI and differential coefficient KD of PID control algorithm is the most important. In this paper, the above three parameters are determined through Ziegler-Nichols tuning method [23] combining with trial and error method. These parameters are calculated firstly by PID controller parameter tuning empirical formula proposed by Ziegler and Nichols. Then, trial and error method is adopted to adjust the above three parameters finely on the basis of the actual response curve, and the relative optimal PID controller parameters are obtained ultimately. The concrete process is as follows. For PID controller, the Ziegler-Nichols tuning method formulas are:

8 > < K p ¼ 1:2T=L K I ¼ 1:2T=L2 > : K D ¼ 0:6T

ð17Þ

In which, T is the time constant and L is called delay time. Both of them are determined through observing the system’s actual step response curve. Cursory value of the above three parameters are calculated by Eq. (17), and the calculated results are used to control

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GMA system. Then, system operation situation is observed, and the three parameters are modified simultaneously in accordance with the influence law of controller parameters on system performance until a satisfactory system response performance is obtained. The parameters’ determining step is first proportional coefficient, after the integral coefficient, then the differential coefficient. Firstly, the proportional coefficient Kp is adjusted. Integral coefficient KI and differential coefficient KD are kept invariant. Proportional coefficient Kp is adjusted gradually on the premise of nonoccurrence of vibration in control system, until the system’s response speed and stability are satisfactory and it has a certain range overshoot. If static error of the control system cannot meet design requirement, the integral coefficient KI is adjusted subsequently. Proportional coefficient Kp and differential coefficient KD are kept invariant. Integral coefficient KI is adjusted gradually on the premise of nonoccurrence of vibration in control system, until the system’s control precision and speed of eliminating static error are satisfactory. If there is vibration in the operation process of whole system, differential coefficient KD is adjusted gradually and the overshoot and stability of control system are observed. In the meanwhile, proportional coefficient Kp and integral coefficient KI are adjusted finely until a satisfactory system comprehensive performance is obtained. Finally, the three parameters are determined as follows: Kp = 0.0045, KI = 0.3, KD = 8  108. 4.3. The software design Whether GMA control system can work normally is not only depending on reasonable hardware but also depending on perfect functional software. The software of GMA control system is designed with modular thinking. The mixed programming method of virtual instrument software Labview and mathematical calculation software Matlab is adopted in software design process. Overall design of the software is realized by the high efficient virtual instrument technology of Labview. Internal data processing and computing is completed by the powerful numerical operation ability of Matlab. In the meanwhile, combining with the interface technology for Labview and Matlab, data and parameters transferring is realized and the whole control system software design is completed. The control system software program mainly includes data acquisition and processing program, data analysis and calculating program, power supply output control program, interface technology for Labview and Matlab, data display and storage and other auxiliary programs. Main program flow chart of the control system is shown in Fig. 7. After device and parameters being initialized reasonably, system automatic control button is started and the system is in the control state of keeping output constant force. Control system uses data input and output board to monitor GMA’s output force constantly through calling the data acquisition procedure. If the difference value between output force and constant output force target value is more than difference threshold, the main program automatically calls control algorithm procedure. Firstly, according to the current in coil and GMA’s output force, GMA model control algorithm program is used to calculate the new current supplied to coil. Then, the power supply output control procedure is called by main program and high speed bipolar programmable power supply outputs current to GMA. During this process, GMA’s output force is close to the constant output force target quickly. After that, aiming to the deviation of GMA model control algorithm, the main program calls PID control algorithm procedure to adjust the current value and controls GMA’s output force reaching the constant output force target accurately. When GMA’s output force is equal to the constant output force target, control algorithm program stops and the control system restarts to monitoring GMA’s actual output force.

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Fig. 8. Flow chart of data acquisition.

Fig. 7. Main program flow chart of control system.

process, GMA’s output force can be close to the constant output force target quickly. After the second time of adjusting GMA’s output force, PID control algorithm is used to determine the control current. It realizes the output force being equal to constant output force target accurately.

4.3.1. Data acquisition and processing Aiming at the situation that Labview do not support the data input and output board produced by non NI company reading and writing directly, the method of calling dynamic link library file is adopted to drive the data input and output board in the software design process. Moreover, the DMA direct inquiry mode utilizing memory storage technology is used to carry out data acquisition. Flow chart of data acquisition and processing is shown in Fig. 8. After starting A/D component, the program begins to carry out data sampling and acquire DMA status continuously. When the current buffer segment has not been completed by DMA, the corresponding thread enters into sleep wait state automatically. Otherwise program awake the thread immediately and read the data in buffer segment. Then, program is in the waiting state again and reads the data repeatedly to carry out continuous sampling. When acquisition times reaches set times, the system releases and suspends equipment automatically. According to the selected measurement range and relationship between weighing force sensor’s output voltage and force, combining with the conversion rule from LSB data to voltage, LSB original code of collected data is converted into force value. Moreover, the collected data is filtered by Butterworth filter selected in Labview and the real stable signal data is obtained. 4.3.2. Data analysis and calculating Data analysis and calculating process, namely control algorithm program, is implemented in Matlab software. According to the above designed controller of control system, control algorithm program consists two parts correspondingly, GMA model control algorithm and PID control algorithm. When it is the first time of adjusting GMA’s output force, GMA model control algorithm program is called to calculate the control current. In this control

Fig. 9. Control algorithm of GMA model.

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4.3.2.1. GMA model control algorithm. In this paper, considering the actual working process and magnetic state changing process, it proposes the coupling effect of magnetostrictive effect and inverse effect are decoupled by the method of repeatedly calculating magnetic state through magnetostrictive effect model and inverse effect model until magnetization nearly reaches a balance state. Based on this, the paper realizes the quantifying of magnetization state in GMM rod. The concrete process is shown in Fig. 9. The magnetic flux density B0 and magnetization M0 calculated by magnetostrictive effect model are called and used as the initial magnetization state for determining the action of inverse magnetostrictive effect caused by external force Fe, and then the magnetic flux density BF and magnetization MF are obtained by the inverse effect model. Then, magnetic flux density B0 and magnetization M0 are used as the initial magnetization state to determine the action of secondary magnetostrictive effect caused by BF and MF, and the output force Fo2 caused by the coupling effect between magnetostrictive effect and inverse magnetostrictive effect is determined by the combination of magnetostrictive effect model and elastic modulus. Then, using the coupling magnetization state between B0 and BF, M0 and MF as the initial magnetization state, the secondary inverse magnetostrictive effect caused by Fo2 is determined and the final magnetization state B2 and M2 in GMM rod is obtained by the inverse magnetostrictive effect model. Then, using the final magnetization state B2 and M2 as the initial magnetization state, the new control current will be determined. According to the difference value eF, the new control current which is needed to adjust GMA’s output force being equal to the constant output force target is calculated by the inverse model of magnetostrictive effect model through the repeated iteration operation.

4.3.2.2. PID control algorithm. When PID control algorithm is started, coefficients of Kp, KI and KD, and GMA’s actual output force Fo(k), Fo(k  1), Fo(k  2) are transferred from front panel interface of Labview to the Matlab Script of PID control algorithm program. Difference values of e(k), e(k  1) and e(k  2) are calculated in

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accordance with GMA’s actual output force Fo(k), Fo(k  1), Fo(k  2) and constant output force target Fc. Then, control variable u(k) is calculated by Eq. (16) and input to the high speed bipolar programmable power supply. 4.3.3. Power supply output control program The current control variable calculated by control algorithm program is input to GMA coil through power supply output control program. According to the actual output current range and accuracy requirements, output voltage range 5 V to +5 V is selected in the design process of power output control program. Firstly, current control variable calculated by control algorithm program is transferred to Labview program through Matlab Script node. According to the impedance of GMA coil, current control variable is converted into voltage value. Then, combining with the conversion rule from voltage value to LSB original code data, current control variable is converted into LSB original code data eventually. After the device creating and initialization, D/A of data input and output board outputs analog signal in accordance with the LSB original code. The analog signal is written into the selected output channel of data input and output board. The output electric signal of board is input to the high speed bipolar programmable power for signal amplification and used as the working current of GMA coil. Finally, the program suspends and releases all the devices. 4.3.4. Interface technology for Labview and Matlab and other auxiliary functions In the design process of the control system for GMA with constant output force, the mixed programming method is adopted. Front panel interface, data acquisition and other subroutines are designed by virtual instrument software Labview, and the numerical calculating process of control algorithm program is completed by Matlab software. But, the two procedures are not independent. It requires exchanging parameters and calculated results data that between these programs constantly. Therefore, the combining method of Matlab Script nodes and ActiveX function are

Fig. 10. Software interface of the control system.

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used to realize the mixed programming of Labview and Matlab in this paper. It realizes the exchanging dada between Labview and Matlab and completed the mixed programming. In order to operate the control system conveniently, a control system software package is developed in the paper. The software is not only having parameters setting, data acquisition, controlling the output of power supply and other major functions, but also having data and response curve display, data and image saving and other auxiliary functions. In the process of software design, communication between Labview and Excel is established by ActiveX technology. It realizes the data being saved to Excel table in order to carry out a secondary processing of these data. Main software interface of the keeping output constant force control system for GMA is shown in Fig. 10. Fig. 11. Experimental system of GMA with constant output force.

5. Experiment and discussion The experimental system of controlling GMA to keeping output constant force is shown in Fig. 11. The GMA is installed between the clamping bolt and clamping device platform, and GMA works in the mechanical clamping state with zero axial strain. A microweighing sensor (LSM-3000) installed between the output end of GMA and clamping bolt is used to measure GMA’s output force real-timely. A data input and output integrated circuit board PCI 8602 with 16 channels is adopted. It is used to read the output force and actual working current, and input current signal to power supply for power amplifier. The power supply selected in the experimental system is a high speed bipolar programmable power (BP4610) which output accuracy is ±0.001 A. The current signal amplified by high speed bipolar programmable power is GMA’s excitation current which is used to generate driving magnetic field. External force is imposed to GMA by rotating the clamping bolt constantly. The main control computer installed the control system software completes the data processing and calculating process of complex control algorithm. Operation performance of the keeping output constant force control system for GMA is tested. Its response performance is evaluated through a comparison between the presented control method in the paper and a PID control algorithm. A set of experiments are performed. GMA’s output force is 102 N only under the action of a 1.3 A current. And the constant output force target is set as 102 N. On this basis, an external force is imposed on GMA using the method of changing the constraint displacement of GMA

(a) GMA disturbed by external forces with samedirection

Fig. 12. Comparison of response result between PID controller and the combined controller.

in longitudinal direction through rotating the clamping bolt. The control system detects GMA’s output force and it is about 141 N. Then, the control system adjusts working current constantly to control GMA’s output force and the control results are shown in

(b) GMA disturbed by external forceswith different direction

Fig. 13. Control result of GMA with constant output force.

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Fig. 12. From the result, it can be seen that PID control result appears a larger overshoot, and the average overshoot reaches 9.8%. The adjusting time of reaching stable state is about 0.5 s. However, from the response curve of the control method designed in the paper, it can be seen that while the working current is controlled through GMA model algorithm only once, GMA’s output force is close to the constant output force target from the initial value rapidly and the output force is about 105 N. Then, working current is controlled by PID algorithm and finally the output force reaches the target value accurately. The overshoot and adjusting time are about 6.1% and 0.3 s respectively. Compared with the control result of PID alone, the overshoot and adjusting time are reduced about 37.8% and 40% respectively. Therefore, the system’s overshoot and response speed are improved obviously. During the experiment, in order to observe the changing process of output force, the control process is conducted a ‘‘slow down’’ treatment. That is some time delays are added in the data acquisition program and other control processes. The constant output force target is set as 102 N. The control result is shown in Fig. 13. Fig. 13a is the output force control result while the actuator is subjected to positive disturbance (that is the output force is increasing). Fig. 13b is the output force control result while the actuator is subjected to positive and negative disturbance (that is the output force is increasing or decreasing randomly). GMA’s output force is monitored by control system in real time. When the actuator is subjected to a positive disturbance, the output force on the moment of a, b and c are detected to be deviating from the constant output force target. Then, the control system adjusts the working current of the actuator and control the output force. Firstly, actuator’s output force is quickly returned to nearby the constant output force target (a0 , b0 , c0 points) through control current solution method using the GMA model algorithm. After that, the working current is adjusted finely through PID algorithm and the output force accurately reaches the target finally. When actuator is subjected to positive or negative disturbance randomly, the output force on the moment of d, e, f, g, h, I, j and k are detected to be deviating from the constant output force target. Then, the control system adjusts the working current and control the output force automatically. Firstly, actuator’s output force is quickly returned to nearby the constant output force target (d0 , e0 , f0 , g0 , h0 , I0 , j0 and k0 points) through control current solution method using the GMA model algorithm. Then, the working current is adjusted through PID algorithm and the output force accurately reaches the target finally. The results show that, when GMA’s output force is changed suddenly by external disturbance, the control system

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can control working current automatically until output force reaches to the constant output force target. It realizes GMA’s output force being kept outputting a constant force through the control system. In order to verify the stability of the constant output force control system, another set of experiment is conducted. The constant output force target is set as 210 N. when the constraint displacement in longitudinal direction is changed through rotating the clamping bolt continuously, GMA’s output force is controlled by the control system and the control result is shown in Fig. 14. The result shows that output force can be kept nearby the constant target with the changing of constraint displacement. The average relative error between the actual output force and the target value is about 0.69%. Therefore, the control method of GMA model algorithm combining with PID control algorithm proposed in the paper can improve the control speed effectively and the control system has a better stability. 6. Conclusions In this paper, change law of the internal magnetization state in GMA when it is under the combined action of magnetic field and external force in the mechanical clamping condition is analyzed. On this basis, a control method of keeping output constant force for GMA which is under the action of magnetic field generated by current constant when it is disturbed by external force is proposed. Based on the GMA model related to the combined action of magnetic field and external force, a controller is designed. The controller is composed of GMA model control algorithm and PID control algorithm to realize the output force reaching the constant force target quickly and finely adjustment respectively. A keeping output constant force control system for GMA is developed in which the sensor, data input and output board, main control computer and other auxiliary circuit are used as the main components. In the meanwhile, the related control system software is designed. According to experiment results, the control system has perfect response performance and stability, and favorable controllability. It can effectively realize the GMA which is disturbed by external force continuously keeping outputting a constant force through adjusting the working current. These research results lay a certain foundation for GMA being used in the fields of cutting with invariableness cutting force. Acknowledgement The work was supported by the National Natural Science Foundation of China under Grant No. 50775021. References

Fig. 14. Control result of GMA with constant output force.

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