Parametric design and experiment of maglev actuators for microgravity vibration isolation system

Abstract

The International Space Station (ISS) has been regarding as a laboratory for experiments in numerous microgravity science discipline. However, the microvibration has a serious impact on science experiments on the ISS as well as existing electromagnetic actuators could not satisfy the requirement of good linearity needed for large stroke of low frequency vibration isolation platforms. This paper aims to design a maglev actuator with good linearity, which could be applied to microgravity vibration isolation platforms. Based on the principle of Lorentz force and self-demagnetization effect, the structural form and preliminary design indices of the actuator were presented. In order to minimize the weight and heat consumption of its coil, parametric design was carried out and multi-objective optimization was adopted for it. Moreover, to investigate the dynamic characteristics of the actuator, system identification was performed to obtain a mathematical model of its control channel, which has good fitting degree of the time and frequency domain signals. Therefore, the result provides an important basis for structural optimal design of the actuator for microgravity vibration isolation system.

Keywords

Maglev actuator multi-objective optimization dynamic characteristic system identification mathematical model

1. Introduction

A good microgravity environment is very important for conducting scientific experiment in space station. From the test and assessment conducted on the acceleration environment of the ISS, it can been found that the vibration of aviation equipment such as pumps, fans, motors and compressor are featured with low frequency (0.01 Hz∼100 Hz), small amplitude (acceleration magnitude in the vicinity of 10⁻⁴ g), high randomness and so forth [1]. Therefore, all of that makes the environment difficult to meet a good level of microgravity acceleration. There are two solutions come up to reduce the influence of vibration on experimental payloads: the first one is to optimize the structure and layout of vibration source; the second one is to adopt the vibration isolation technology to mitigate the effects of the flutter transfer [2]. At present, the vibration isolation technology can be divided into passive vibration isolation and active vibration isolation [3–5].

Passive vibration isolation is to place damping structure, such as metal rubber, air spring, etc., on the path of vibration transfer to reduce or isolate vibration [6,7]. Conventional passive vibration isolation device has simple structure but valid only when the external disturbance frequency is $\sqrt{2}$ times the natural frequency [8]. However, in space shuttles, passive vibration isolation cannot significantly suppress low frequency vibration (≤10 Hz). Not only could active vibration isolation adopted certain control law control actuators to isolate external disturbances, but it could also suppress inertial disturbances once any movement of isolation object is measured by the sensors [9].

Currently, the actuators for active vibration isolation could be separated into several types, such as pneumatic/hydrodynamic type, piezoelectric type, magnetostrictive type, electro-rheological fluid type, magneto-rheological fluid type and electromagnetic type [10–12]. Maglev actuators belong to the electromagnetic type with many advantages of wide frequency range, fast response and no mechanical contact, which could satisfy application requirements of the microgravity vibration isolation platform. They can also be divided into two categories according to the basic principle of electromagnetic force. The first type is based on the Maxwell’s electromagnetic force. As the electromagnetic force has a quadratic nonlinear relationship with the current and air gap, it will rapidly decrease with the increase of air gap. Moreover, the current cannot control the direction of electromagnetic force that makes this kind of actuators be often used for high-speed and precise manufacturing facilities [13]. The second type is based on the Lorentz force. Thers is a good linear relationship between the electromagnetic force and current of the actuator in the working range, and the stroke could reach dozens of millimeters. Therefore, it is ideal for active vibration isolation systems. For instance, a Lorentz motor (LM) with small volume, given force coefficient and limited heat dissipation is designed [14]. The performance of the optimized LM is described by simulation and experimental results, which confirm effectiveness of the optimized design for active isolation system.

The vibration isolation platform with magnetic suspension not only removes mechanical contact between the vibration source and payload but also features with simple and compact structured, anti-interference and low power consumption. Furthermore, vibration isolation requirements with ultra-low frequency and large stroke need to be used in many fields, such as the aerospace, microelectronics manufacturing, precision positioning and micro-machining [15,16]. Currently, there are four main active vibration isolation systems for space science experiments, such as STABLE (Suppression of Transient Accelerations by Levitation Evaluation) [17], ARIS (Active Rack Isolation System) [18], MIM (Microgravity-vibration Isolation Mount) [19] and g-LIMIT (Glovebox Integrated Microgravity Isolation Technology) [20]. These active vibration isolation platforms are not only featured with large stroke, high precision and low frequency isolation but their actuators are also self-designed Lorentz force actuators.

In this paper, the main contents of this study are as follows: a maglev actuator is designed based on the principle of Lorentz force, its magnetic circuit is designed in accordance with the self-demagnetization effect, and parametric design of its coil was carried out and multiobjective optimization is adopted in Section 2; in Section 3, the variation trend of magnetic field and electromagnetic force along each axis of the actuator had been analyzed by using finite element software Ansys Workbench12.1; the system identification method was utilized to establish a mathematical model for the control channel of maglev actuator in Section 4, and experimental study were carried out to verify its dynamic characteristic; finally, the conclusions are summarized in Section 5.

2. Actuator structure

2.1. Determining basic indicators

In order to meet the microgravity acceleration level of space science experiments, the schematic diagram of a vibration isolation platform is shown in Fig. 1. It mainly consists of a floater primarily for placing experimental payloads and a stator including a baseboard and an annular side plate. The self-designed Lorentz force actuator is mainly composed of a coil component and a magnet yoke component. Among them, the actuator adopted dual mutually perpendicular coil components which could generate horizontal and vertical forces, respectively, that makes its structure more simple and compact. The stator is equipped with 3 position sensitive detectors (PSD) and 3 coil components. 3 magnet yoke components and 3 sensor modules are located at the bottom of the floater. Each sensor module primarily includes 1 laser source and 2 mutually perpendicular accelerometers. A large stroke is often required for low-frequency vibration isolation platform that leads to the strong non-linear output characteristics of the actuator. Therefore, the mentioned actuator is only equipped with one coil in this paper.

The maglev actuator is designed based on the Lorentz force principle. The Lorentz force would be created when a current-carrying wire is cutting the magnetic induction lines. Moreover, the input current and the output force of the actuator have a good linear relationship in the working range, and the stroke can reach dozens of millimeters. Its structure as well as the winding method of its coil are shown in Figs 2 and 3, respectively. Two cylindrical permanent magnets are respectively placed on both end faces of the yoke according to their magnetic pole SN-SN. The magnetic flux density would be generated in the air gap between two permanent magnets and it is mainly along the axial of the permanent magnet and horizontally to right. The current direction and magnitude of straight conduct wire in the static magnetic field of the air gap are of the same, and Lorentz force would be created in the vertical direction. Therefore, the controller could be able to control the size and direction of the output force by changing the size and direction of the coil current.

As an executing agency, the actuator is one of most important part in active vibration isolation system. To meet the requirements of space scientific experiments and take Shenzhou spacecraft microgravity disturbance levels as a reference [21], further research has been undertaken on the maglev actuator and preliminary design indices of the actuator are put forward, such as stroke, maximum output force, current density, current changes frequency, et al. Assuming the quality of the payload and floating platform as 60 kg, the acceleration of each axis and the maximum force in a single direction are determined to be 10⁻² N/kg and 6 N, respectively. There are two output forces in each direction and the maximum force of each actuator is 4 N with 20% margin. Since the vibration frequency ranging from 0.1∼10 Hz, the frequency of the current changes must be an order of the magnitude greater than vibration frequency in order to effectively isolate vibration at low frequencies. Table 1 summarizes the relevant calculation of design indices.

2.2. Magnetic circuit design

The high-grade permanent magnet material N50M is utilized, of which the remanence: B _r = 1.412 T; coercivity: H _cb = 1065.048 KA/m; and magnetic energy product: (BH)_max = 394.26 KI/m³. In addition, on the basis of the self-demagnetization effect [22,23], this paper designs a magnetic circuit, of which the magnetic flux density B _d at the maximum magnetic energy product is: $\begin{eqnarray}\displaystyle B_{d}=\displaystyle \frac{B_{r}H_{d}}{H_{cb}}=\sqrt{\displaystyle \frac{B_{r}(BH)_{\max }}{H_{cb}}}. & & \displaystyle\end{eqnarray}$ (1)

The size ratio coefficient (A) of magnetic circuit is: $\begin{eqnarray}\displaystyle A^{(1+0.05A)}=0.41\times 10^{4}\cdot \displaystyle \frac{B_{d}^{2}}{(BH)_{\max }}. & & \displaystyle\end{eqnarray}$ (2)

It can be seen from the Fig. 2 that the air gap length between two permanent magnets (L _g = 10 + b) expressed by the empirical formula might be derived: $\begin{eqnarray}\displaystyle L_{g}=\displaystyle \frac{(10+b)\left[1+0.04(10+b)\right]\times \sqrt[3.2]{\left(\displaystyle \frac{{\pi}}{4}D^{2}\right)}}{\displaystyle \frac{{\pi}}{4}D^{2}}. & & \displaystyle\end{eqnarray}$ (3)

By considering the magnetic flux leakage, the utilization coefficient (ϵ) of the permanent magnetic flux is: $\begin{eqnarray}\displaystyle {\varepsilon}=\displaystyle \frac{1}{1+Q\cdot L_{g}}. & & \displaystyle\end{eqnarray}$ (4)

The magnetic circuit type constant (Q) is an empirical constant and it is set as 6 here. On the basis of principle of magnetic flux continuity, this paper integrates the above-mentioned Eqs (1)--(4) so as to calculate the magnetic flux density in the air gap: $\begin{eqnarray}\displaystyle B_{g}=\displaystyle \frac{{\varepsilon}B_{d}\cdot S_{m}}{S_{g}}=\displaystyle \frac{0.72}{1+6\times \displaystyle \frac{(10+b)\left[1+0.04(10+b)\right]\times \sqrt[3.2]{\left(\frac{{\pi}}{4}D^{2}\right)}}{\displaystyle \frac{{\pi}}{4}D^{2}}}. & & \displaystyle\end{eqnarray}$ (5)

Figure 4 gives the relationship between the magnetic flux density (B _g) of the air gap vs. b and D. It can be found that when b declines or D increases, the B _g increases. As the disturbance force or torque may be relatively large when the coil deviates from its center, the value of D should not be too large. The structural parameters are preliminary set as b = 5 mm and D = 15 mm, and B _g is 0.10313 T in the air gap center. The empirical formula of the total length of the permanent magnet is: $\begin{eqnarray}\displaystyle L_{m}=\displaystyle \frac{1}{2}\left(A\cdot \displaystyle \frac{{\pi}D}{4}+2L_{g}\right). & & \displaystyle\end{eqnarray}$ (6)

After substituting various values into the above equation, it can be obtained that L _m = 38.56 mm (rounded to 40 mm).

2.3. Multi-objective optimal design of the coil

Owing to the dual O-shaped winding coil shown in Fig. 3, the magnitude and direction of the coil current in magnetic field are of the same. See the followings for the relevant parameters: electrical resistivity of copper wire: 𝜌 = 2 × 10⁻⁸Ω ⋅ mm²∕m; coil space factor: K _u = 0.65; copper core wire diameter: d ₀ = 0.51 mm, and coating diameter: d = 0.56 mm. The parameters w, l, and b are the width, effective length and thickness of the coil, respectively. The volume of the coil is $\begin{eqnarray}\displaystyle V_{\text{coil}}=2\times b\{{\pi}\cdot [(w+2.5)^{2}-2.5^{2}]+2bl\}. & & \displaystyle\end{eqnarray}$ (7)

The turns of the coil are: $\begin{eqnarray}\displaystyle n=4\cdot \displaystyle \frac{K_{u}\cdot wb}{{\pi}d^{2}}. & & \displaystyle\end{eqnarray}$ (8)

The average length of each turn of the coil is: $\begin{eqnarray}\displaystyle l_{\text{mean}}=2{\pi}\left(2.5+\displaystyle \frac{w}{2}\right)+2l. & & \displaystyle\end{eqnarray}$ (9)

The parameters J, S ₀ and n are the maximum current density, coil cross-sectional area and turns of the coil, respectively. The maximum current of the coil is: $\begin{eqnarray}\displaystyle I_{\max }=J\times \displaystyle \frac{{\pi}}{4}d_{0}^{2}=\displaystyle \frac{F_{\max }}{S_{0}\cdot 2nB_{g}l}\times \displaystyle \frac{{\pi}}{4}d_{0}^{2}=\displaystyle \frac{F_{\max }}{B_{g}K_{u}\cdot wlb}\times \displaystyle \frac{{\pi}}{4}d_{0}^{2}. & & \displaystyle\end{eqnarray}$ (10)

The parameter R is the resistance of the coil and the maximum heat consumption of the coil is: $\begin{eqnarray}\displaystyle Q_{\max }=2I^{2}R=2I^{2}\times {\rho}_{\text{coil}}\displaystyle \frac{n\cdot l_{\text{mean}}}{S_{0}}. & & \displaystyle\end{eqnarray}$ (11)

The parameter S is the wire core area of the coil and the coil magnetic flux is: $\begin{eqnarray}\displaystyle {\varphi}=n\cdot {\mu}_{0}\displaystyle \frac{n}{b}I_{\max }\cdot S. & & \displaystyle\end{eqnarray}$ (12)

The parameter L is the inductance of the coil and the current frequency of the coil is: $\begin{eqnarray}\displaystyle f_{3dB}=\displaystyle \frac{1}{2{\pi}{\tau}}=\displaystyle \frac{1}{2{\pi}\cdot \displaystyle \frac{L}{R}}=\displaystyle \frac{1}{2{\pi}\cdot \displaystyle \frac{{\varphi}}{I_{\max }\cdot R}}. & & \displaystyle\end{eqnarray}$ (13)

Because of the vibration isolation frequency being among 0.1 ∼ 10 Hz, the current frequency is considered to be selected as f _3dB ≥ 100 Hz to ensure the vibration isolation performance. As the designed actuator needs to have the characteristics of light weight and low power consumption, the structural parameters of the coil were optimized in this part. From the Eqs (7) to (13), it is known that to minimize the quality and power consumption of the coil under the constraint of the maximum coil current (≤2 A) and current frequency (≥100 Hz) is a multi-objective goal attainment problem [24,25]. Equations (7)--(13) are integrated and the mathematical model may be expressed as to: $\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\min :m_{\text{coil}}={\rho}\cdot V_{\text{coil}}\\[6.0pt] \min :Q_{\max }=2I_{\max }^{2}\cdot R\\[6.0pt] s.t.\left\{\begin{array}{@{}l@{}}I_{\max }\leq 2A\\ f_{3dB}\geq 100∼\text{Hz}\end{array}\right..\end{array} & & \displaystyle\end{eqnarray}$ (14)

Therefore, the Matlab fgoalattain function is utilized to optimize three structural parameters (w, l, and b) of the coil, whose syntax is: $\begin{eqnarray}\displaystyle [\text{x},\text{fval}]=\text{fgoalattain}(\text{fun},\text{x0},\text{goal},\text{weight},\text{A},\text{b},\text{Aeq},\text{beq},\text{lb},\text{ub},\text{nonlcon}). & & \displaystyle \nonumber\end{eqnarray}$

Among them, fun is an objective function defined by a M file; x0 are the initial points for x; goal is a vector of values that the objects attempt to attain; weight represents a weighting vector to control the relative underattainment or overattainment of the objectives in fgoalattain; A and b define inequality constraints (A ⋅ x ≤ b); Aeq and beq define equality constraints (Aeq ⋅ x = Beq); lb and ub denote the vectors of lower bounds and upper bounds of x, respectively; nonlcon denotes the nonlinear inequality constraints. The return values x and fval indicate the minimum values of the objective functions computed in fun at the solution x. In this part, the vector of values (goal) is set to [190, 20] in turn, and the same weight percentage is set for each objective function. The optimization parameters of the coil structure are successively w, l and b; the vectors of lower bounds and upper bounds of the structural parameters are set to [15, 20, 3] and [18, 30, 5], respectively. The optimal result of the structural parameters calculated by the Matlab fgoalattain function are rounded up to w = 17 mm, l = 30 mm, b = 5 mm; coil mass (m _coil)_min = 196 g, heat consumption (Q _max) _min = 25. 5 w.

3. Simulation analysis

The electromagnetic force is generated by the energized coil placed in the static magnetic field. The size and direction of the electromagnetic force would change with the size and direction of the current in the coil. According to the Lorentz Force Equation (F = B ⋅ I ⋅ L), and it implies that the actuator output force is primarily related to the spatial magnetic flux density and the current size of the coils. Therefore, the distribution of the magnetic field and force in each direction around the coil should be considered and analyzed thoroughly.

3.1. Simulation analysis of magnetic field distribution

High-grade magnet materials N50M and pure iron DT4 were utilized to increase the magnetic flux density as well as ensure greater output force. Due to the large air gap between the two permanent magnets, the static magnetic field is not uniform at the air gap. Therefore, the Ansys Workbench 12.1 was used in the simulating process. The air model of the actuator is shown in Fig. 5 with the 15 × 35 × 35 mm³ between the gaps of two permanent magnets. The coordinate origin is at the geometric center of the air model. Different cross sections paralleled to the YZ plane were selected in the air model corresponding to each different x offset in step of 1 mm within the range of ±7 mm on the x-axis. After that, the distribution of magnetic flux density in the air model was analyzed. The distribution of each magnetic flux density components (B_xB_y and B_z) of the air model is shown in Fig. 6.

Figure 6 shows that, due to the non-uniformity of the static magnetic field, B_x is maximum at the end face of the magnet and gradually reduced while away from the permanent magnet; the minimum value is in the middle of the air gap. The B_y has a trend of smaller decrement and the magnetic flux density changes in the positive direction along the y-axis firstly, and then changed in the negative direction. The overall trend showed central symmetry distribution. The value of the B_z is minimum and the maximum value is only −2.3119 × 10⁻³ mT. It is indicated that there is a slight non-linearity in the magnetic flux density component B_x of the air model mainly along the x direction and the component values in y, z direction are very small, not exceeding 2 mT.

3.2. Simulation analysis of electromagnetic forces

The relationship between the axial force components (F_x, F_y and F_z) of the coil and displacement of the coil that move along each axis was determined after the simulation model was established in Ansys Workbench12.1. The simulation model and result of the simulated force are shown in the Figs 7 and 8, respectively; where the red arrows in Fig. 7 shows the direction of current flow. The current in the coil is constant and set to 2 A. The N-pole of two permanent magnets are towards the positive x-axis.

From Fig. 8, it could be found that the value of the force F_y is always greater than 4 N when the coil moves along each axis. In the static magnetic field, both the electromagnetic force direction and current direction are perpendicular to each other. The simulation results show that the force F_z is the smallest as same as the theoretical analysis. The maximum value along the x-axis direction does not exceed 0.2 N, which is the 4% of the F_y under the same conditions. The change in the value of F_x and F_z is negligible to all position of the coil. They can be seen as an interference to the major driving force of F_y. When the coil moves along the x-axis to both ends of the permanent magnet, the value of F_y increases as the value of the magnetic flux density increased significantly; when it moved along the z-axis, there is no significant change in the value of F_y, however, when it moves along the y-axis, the magnetic flux density in the center of the coil gradually decreases thus the force F_y becomes smaller.

4. Modeling of control channel of the actuator

For investigating the dynamic output characteristics of the maglev actuator and different control methods could be used to the actuator in the simulation control and actual applications, the mathematical model of control channel of the maglev actuator was established by using the system identification method. A single degree-of-freedom (DOF) dynamic characteristic experimental system was developed, whose real-time data acquisition system flow chart based on dSPACE is shown in Fig. 9, which is made up of a PC, a dSPACE and a single DOF dynamic characteristic experimental platform. During the test of system identification, a 1.5 V swept signal (0.1 ∼ 200 Hz) was inputted to the power amplifier, and then the dSPACE was adopted to collect the data of the experimental subject in the time and frequency domain. In order to investigate the dynamic characteristics of the controlled object in the frequency domain and set up a control channel mathematical model for the vibration isolation system, the system identification method is adopted by using the Matlab System Identification Toolkit. The system experimental devices and single DOF dynamic characteristic experimental platform are shown in Figs 10 and 11, respectively. The platform consists of the above-mentioned actuator, a linear rolling unit, a quartz flexible accelerometer, a PSD, a laser source, a vibration motor, a power amplifier, a constant current voltage source and a dSPACE real-time simulation system. The coil component and a laser source are fixed to the substrate; and the holder installed on the linear rolling unit could travel horizontally and is equipped with the magnet yoke component, the accelerometer, the PSD and the vibration motor.

The dynamic output characteristics of the actuator is studied and a mathematical model of the control channel of the actuator is established. Different orders of mathematical models could be obtained by setting different numbers of zeros and poles. Moreover, the fitting degree is utilized to represent the similarity of the mathematical model with the frequency and the time domain signal, respectively. The fitting degree of frequency domain and time domain for various-order identification model as shown in the Figs 12 and 13, respectively. It can be seen that the higher the order of mathematical model is, the higher the fitting degree of mathematical model is with the frequency and time domain signals (namely the error of the model to frequency and time domain signals would be smaller). At the same time, it is easy to find that the fitting degree of the mathematical model with the frequency and time domain signals is not very well when the order of the identification model does not exceed 4. However, when the order of the mathematical model is greater than 5, the fitting degree of the model with the frequency domain signal is increasing gradually, but the fitting degree of the model with the time domain signal does not rise significantly. Meanwhile, the higher the model order is, the higher the complexity of mathematical model is. By taking the conciseness of the identification model into account and for ensuring the relatively good fitting degree of the time/frequency domain and the correspondingly measured data as well as reflecting the dynamic characteristics of the experimental subject, thus, the following 5-order identification model is utilized and the fitting degrees of time and frequency domains are 84.26% and 90.63%, respectively, whose zero-pole equation is expressed by: $\begin{eqnarray}\displaystyle G=\displaystyle \frac{1702.8(s^{2}-1065s+5.759\times 10^{5})(s^{2}-389s+1.55\times 10^{6})}{(s+419)(s^{2}+701.7s+5.736\times 10^{5})(s^{2}+276.5s+1.447\times 10^{6})}. & & \displaystyle\end{eqnarray}$ (15)

Moreover, the concept of predominant poles has been utilized to approximate identification model by using the 1st or 2nd order system and the zero-pole distribution of the identification model shown in Fig. 14. The figure shows that poles are located mostly at the left of the imaginary axis but zeros are distributed at the right of the imaginary axis; especially, the size of the real part of the non-predominant pole (−419, 0) marked in green in the real axis is about 3 times of that of the real part of the predominant poles. If the transfer function without the green pole was utilized, the Bode diagram and step response of the identification model are shown in Fig. 15. If the transfer function without the green pole and 2 red complex-conjugate zeroes was utilized, the Bode diagram and step response of the identification model are shown in Fig. 16. It could be found that significant differences in the frequency and time domains occur while lowering the order of the identification model. Thus, the order of the identification model could not be lowered based on the zero-pole distribution diagram.

It can be found from Fig. 17 that the 5-order mathematical model obtained by system identification has a good fitting degree with the experimental data in the amplitude and phase of the Bode graph within the range of 0 ∼ 200 Hz. The model could be considered to reflect the dynamic characteristics of the control channel of the actuator and different control methods could be used to the actuator in the simulation control and actual applications.

5. Conclusions

This paper proposes a structure of a maglev actuator based on the Lorentz force and the magnetic circuit was designed in accordance with the self-demagnetization effect as well as multiobjective optimization is adopted to optimize structure parameters of its coil. The minimum weight and heat consumption of the coil are 196 g and 25.5 W, respectively. The distribution of magnetic flux density and the tendency of each force component distributed of the actuator at the air gap are analyzed by Ansys Workbench 12.1. In addition, a 5-order mathematical model of control channel of the maglev actuator was obtained based on system identification and the mathematical model has a good fitting degree (the fitting degrees of time and frequency domains are 84.26% and 90.63%, respectively) with the experimental data in the range of 0 ∼ 200 Hz. Therefore, the model can be considered to reflect the dynamic characteristics of the control channel of the actuator and different control methods could be used to the actuator in the simulation control and actual applications.

In the future, more work on the realization of a single DOF active control for the maglev actuator would be carried out. The feedback controllers, such as the cascade PID controller and the robust controller, could be researched for single/six DOF vibration isolation system to make sure that the active vibration isolation platform could achieve excellent isolation performance in microgravity environment.

Footnotes

Acknowledgements

The authors would like to thank anonymous reviewers and handling editors for their useful comments and constructive suggestions, which do help improving the quality of this paper.

Funding

This work was supported by the National Natural Science Foundation of China (No. 51205296 and 51275368).

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