The working principle of the macro-motion part of the actuator is the same as that of the voice coil motor. At the working air gap, the permanent magnet provides a constant magnetic field source. The energized macro-motion coil is acted by the ampere force F to promote the mover movement, in which the ampere force F is the driving force of the macro-motion part, and its calculation expression is 1F=BNLIa, where B is the magnetic field intensity generated by the macro-motion permanent magnet at the air gap, unit: Wb m-2; L is the effective length of each coil in a magnetic field, unit: mm; Ia is the working current of coil winding, unit: A; and N is the number of turns of the coil.

Since the uniformity of the magnetic field intensity B at the working air gap determines the stability of the driving force when the macro-motion part works, the magnetic field intensity distribution of the macro–micro composite actuator when the macro-motion coil is electrified is obtained through finite element simulation, as shown in Fig. 4.

It can be seen from the color distribution in Fig. 4 that when the macro-motion coil is energized, the magnetic flux density of the internal and external yokes of the macro-motion part is the highest, but the magnetic flux density of the middle section of the internal and external yokes is similar to that at the working air gap of the macro-motion coil, and the magnetic flux density value is low. The magnetic flux density of the micro-motion yoke is higher, but the magnetic flux density at the position of the GMM rod is the same as that in the air domain. It can be seen from the arrow direction in Fig. 4 that the internal and external yokes constitute the magnetic conduction circuit of the macro-motion part and provide the working magnetic field for the macro-motion part.

In order to further explore the magnetic field direction and uniformity of the magnetic field intensity at the working air gap of the macro-motion part, the moving axis system shown in Fig. 5 is established. When the working current of the macro-motion coil is 2 A, the magnetic field distribution map and isopleth map at the working air gap of the macro-motion part are shown in Figs. 6 and 7, respectively.

It can be seen from Fig. 6 that the direction of the magnetic field intensity at the working air gap of the macro-motion part is radial, so the energized macro-motion coil generates an axial ampere force, thereby pushing the mover to move in a straight line along the axial direction.

From Fig. 7, it can be seen that the contour of radial magnetic flux density in the working air gap of the macro-motion part is gradually inclined. Therefore, the average value of the magnetic flux at the inner circle and outer circle generatrix of the macro-motion coil is taken as the magnetic flux density value at the working air gap. The simulation results show that when the working current of the macro-motion coil is (0–4) A, the magnetic flux density at the working air gap is shown in Fig. 8.

Figure 8 shows that the curve of the macro-motion coil when the working current of 0 A is flat, indicating that the uniformity of the magnetic flux density at the working air gap is the best at this time, but with the increase of the working current, the greater the change of the magnetic flux density at the working air gap, indicating that the macro-motion part is suitable for working in the small current driving state.

In order to obtain a more accurate driving force model, it is necessary to obtain the relationship between the magnetic field intensity at the working air gap and the position of the mover, and the working current of the macro-motion coil. By simulating the mover within the travel range of (0–50) mm, passing (0–4) A current to the macro-motion coil every 10 mm, the three-dimensional surface of the magnetic field intensity on the outer circular bus of the macro-motion coil is obtained as shown in Fig. 9.

It can be seen from Fig. 9 that under different working currents, when the position of the mover changes, the magnetic field intensity at the working air gap of the macro-motion part is almost unchanged. With the increase of the coordinate value of the macro-motion coil bus, the magnetic field intensity at the working air gap first decreases and then increases. When the working current is (0–1) A, the magnetic field intensity changes gently, and when the working current is (3–4) A, the magnetic field intensity changes greatly. It can be seen that the magnetic field intensity at the working air gap is almost independent of the position of the mover and is greatly affected by the working current of the macro-motion coil.

Based on the above analysis, the magnetic field intensity at the working air gap of the macro-motion part is almost independent of the position of the mover. It is greatly affected by the working current of the macro-motion coil. The greater the working current, the greater the change of the magnetic field at the working air gap. Since the residual magnetic flux density of the permanent magnet used in the experiment cannot be accurately measured, and the residual magnetic flux density of the permanent magnet is set to 1.21 Wb m-2 in the simulation. The magnetic field intensity at the working air gap of the macro-motion part obtained by simulation is set as Bc, while the actual magnetic field intensity at the working air gap of the macro-motion part is set as K⋅Bc.

It can be seen from Fig. 9 that Bc is related to the current I and axial size of the macro-motion coil, so it is expressed as Bc(mL, I). Therefore, the calculation formula of ampere force generated by the macro-motion coil is rewritten as follows: 2F=BcmL,IKNLI=BcmL,IKtI, where Kt=KNL, K is a constant, unit: mm, Kt is the parameter to be identified, and mL is the position of the mover.

The difference between the mover of the macro-motion part and the mover of the moving coil voice coil motor is that a micro giant magnetostrictive actuator is embedded in it, which increases the mass of the mover. Since the materials of the micro-motion yoke of the mover part and the internal and external yokes of the stator part are pure iron with high magnetic permeability, it is easy to be magnetized by the permanent magnet, resulting in magnetic force between the internal yoke and the micro-motion yoke, resulting in greater resistance of the mover part during startup. The magnetization curve of the internal yoke is obtained through simulation, as shown in Fig. 10.

It can be seen from Fig. 10 that the magnetization of the yoke in the macro-motion stator decreases with the increase of its axial distance, and the magnetization is directly proportional to the force on the micro-motion yoke, indicating that the force on the micro-motion yoke will decrease with the increase of the positive displacement of the mover. According to the principle of magnetic flux conservation, the simulation results show that when the mover is at the displacement of 0 mm, the axial force on the micro-motion yoke is about 1.89 N, and its force direction is opposite to the direction of macro-motion positive displacement.

By testing the experimental prototype, it is obtained that the static friction of the mover is about 10 N at the displacement of 0 mm and about 7 N at the displacement of 5 mm, indicating that the force received by the micro-motion yoke due to magnetization is related to the macro-motion displacement. By curve fitting the experimental measurement data, the expression of static friction is approximately obtained, 3fs=10cos⁡(0.157mL), where fs is the static friction and mL is the position of the mover.

In order to obtain the self-inductance change model of the macro-motion coil (Manh and Chen, 2020; Luo et al., 2019), the equivalent circuit of the macro-motion coil is established, as shown in Fig. 11.

It is assumed that the control current of the macro-motion part is I, the equivalent resistance of the loop is R, the equivalent inductance is LB, and the working current in the loop is IA. When the macro-motion part works, the macro-motion coil moves to cut the magnetic induction line in the magnetic field to produce the induced current Ie opposite to the driving current, which is expressed as 4Ie=BLBvR=Kev, where v is the speed of mover, unit: m s-1; B is the average magnetic field intensity at the working air gap of the macro-motion part, unit: Wb m-2; LB is the equivalent inductance of macro-coil; and Ke is the back electromotive force coefficient. Since the variation of magnetic field intensity B is small, the equivalent length and resistance of the macro-motion coil are constants, and the relative variation of the Ke value is very small, so it can be defined as a constant.

Then the loop equation of the macro-motion part is 5Ia=I-Ie=I-Kev.

In the series circuit, the inductance LB and the resistance R are equivalent to the first-order inertia link, the expression is as follows, Eq. (6): 6L(s)=1Ts+1, where T is the time constant related to the self-induction of the macro-motion part.

When the current controller drives the macro-motion coil, it will cause the self-inductance of the macro-motion coil, and the self-inductance can affect the electromechanical time constant T of the macro-motion part. By analyzing the experimental data and combining the definition of the electromechanical time constant, it is obtained that the electromechanical time constant T is not a fixed value but related to the macro-motion displacement. Therefore, Eq. (6) is further equivalent to the following Eq. (7): 7L(s)=1λTL(mL)s+1, where λTL(mL)=T,λ is the parameter to be identified, and TL(mL) is a function of electromechanical time and mover position.

According to the working mode of the macro-motion part, the driving force F generated by the macro-motion coil not only overcomes the friction Fs generated during the movement of the mover but also needs to provide the driver with the inertial force FM and sliding friction Fc for acceleration and deceleration. Therefore, the kinetic model of the macro-motion part is 8F=Fc+FM+fs=Kcv+Md2mLdt2+10cos⁡0.157mL, where M is the total mass of the macro-motion mover, unit: kg; mL is the macro-motion displacement, unit: mm; and Kc is the coefficient of viscous dynamic friction.

By quoting the Stribeck friction model, the expression of friction Ff describing the startup and movement of the macro-motion part is obtained as follows: 9Ff=Kcv+fs-Kcv⋅e-v/vs, where Ff is the friction overcome during the operation of the macro-motion part, Kc is the sliding friction force, v is the speed of the mover, and vs is the maximum speed.

Therefore, the calculation formula of driving force F can be rewritten as 10F=Md2mLdt2+Kcv+10cos⁡0.157mL-Kcv⋅e-v/vs,11F=Md2mLdt2+Kcv+10cos⁡0.157mL-Kcv⋅e(-v/vs)Ia=I-IeF=BcmL,IKtIaIe=Kev.

According to the mechanical equation and loop equation of the motor, the basic equations of the dynamic characteristics of the macro-motion part are obtained as follows: the Eq. (11) is transformed by Laplace transform and combined with the control principle of the actuator controller, and the kinetic model block diagram of the macro-motion part is obtained as shown in Fig. 12.

It can be seen from Fig. 12 that the parameters λ, Kt, Kc, and Ke in the kinetic model of the macro-motion part are unknown. It is necessary to identify the parameters according to the experimental data, where M is the mass of the mover and M is 4.5 kg by measurement.

In order to obtain the parameter values of the model and verify the validity of the kinetic model, an experimental test platform of a macro–micro composite actuator is built, as shown in Fig. 13. It is mainly composed of a macro–micro composite actuator, positioning workbench, switching power supply, a grating displacement encoder, and current controller. The minimum resolution of the grating displacement encoder is 20 nm and the rise time of the current driver is 1 µs. LabVIEW is used as the upper computer control system and displacement display interface to adjust the output pulse of the current controller, drive the mover of the actuator, and display the real-time data collected by the grating displacement sensor.

During the experiment, the positioning workbench is placed at different starting positions. The variation curve of the output displacement of the positioning workbench with time is measured through the given current value, as shown in Fig. 14.

It can be seen from Fig. 14 that the overall response time of the macro-motion part of the driver is about 50 ms, and when the positioning workbench reaches a certain speed value, its motion displacement changes approximately linearly with time. At the starting position of 9 and 30 mm, the starting time of the actuator is relatively longer, which may be due to the large change of friction and self-inductance of the macro-motion coil, which increases the electromechanical time constant of the macro-motion part.

Cubic spline interpolation is performed on the experimental data. After fitting with the least square method, the first-order derivation is carried out to obtain the maximum speed reached at different starting positions, as shown in Fig. 15.

As can be seen from Fig. 15, the maximum value of maximum speed vs is 234 mm s-1. Combining the data in Fig. 15 with the experimental data, by solving the time corresponding to the maximum speed value and combined with the definition of electromechanical time constant, The function of electromechanical time and mover position is obtained as follows: 12TLmL=-0.0000038067⋅mL3+0.00030573⋅mL2-0.010827⋅mL+0.38807.

As shown in Fig. 16, the parameter identification model block diagram of the macro-motion part is built by using the MATLAB/Simulink module. The relevant parameter values of the macro-motion part are obtained by using the parameter identification toolbox, as shown in Table 1.

The identified parameters are used for simulation and compared with the experimental data, and the comparison diagram between the simulation curve and the experimental curve is obtained, as shown in Fig. 17.

It can be seen from Fig. 17 that the displacement curve obtained by simulation using the parameters obtained by parameter identification is consistent with the changing trend of the displacement curve obtained by experimental measurement. The differences are as follows: in the initial response stage, the displacement of the actuator is almost unchanged. Combined with the simulation analysis of the magnetic field and current magnetic field at the working air gap of the mover in Fig. 8, the main reason may be due to the input current of the macro-motion coil gradually increasing, causing the magnetic field from the working air gap to decrease, resulting in a decrease in the driving force. When the mover is started, the mass of the mover part is larger than that of the general voice coil motor, and its static friction force is larger. Therefore, the displacement of the actuator changes little during the start-up stage of the mover. In addition, the overall displacement curve shows a non-linear trend. The reason may be that the self-inductance of the macro-motion coil prevents the increase of the current, which leads to the change of the driving force, and the nonlinear change of the friction force, which increases the nonlinearity of the macro-motion part. Therefore, when the macro-motion part is started, the current needs to be appropriately increased to increase the thrust value of the macro-motion coil. At the same time, combined with the analysis of the phenomenon that the middle value of the magnetic field intensity of the internal and external magnetic yokes is low in Fig. 4, the changing trends of the two curves are consistent, but the data do not overlap, indicating that the kinematics model of the macro-motion part needs to overcome friction and self-inductance. It is also necessary to consider the nonlinear influence of the eddy current loss generated between the macro-motion coil and the internal and external magnetic yokes due to the strong magnetic field environment and high-frequency pulse driving.

To quantitatively compare the approximation degree between the identification model and the actual kinetic model, the model matching degree is calculated as follows according to the model matching index formula: 13Km=1-∑i=1Nyit-Yit2∑i=1Nyi2t, where Km is the model matching index; yi(t) is the actual output at the i sampling time of the actual structure; and Yi(t) is the calculation output of the identification model at i sampling times.

The closer the model matching value is to 100 %, the closer the identification model is to the actual structure. The matching degree between the identification model and the actual structure is 92.97 %. The results show that the calculated results of the identification model are consistent with the response of the actual structure.

In order to verify the identified transfer function model, the velocity curve is obtained according to the experimental data and compared with the simulation velocity curve, as shown in Fig. 18.

It can be seen from Fig. 18 that the velocity curve obtained by simulation is very close to the velocity curve obtained by experiment, which indicates that the established kinetic model of the macro-motion part can approximately characterize the nonlinear influence law of the macro-motion part, and provides a reference value for further design of the control system.