Modeling and controller design of a novel large-gap magnetic suspension and balance system for 3D non-contact manipulation

Abstract

Magnetic suspension is a good choice for high precision manipulation. Several magnetic suspension systems have been proposed and shown to be effective for precision positioning. Most of these systems work with small air gaps and have a small movement range for suspended-load. Increasing the air gap will introduce uncertainty into the modeling and control of the system but is required for specific applications. The Large-Gap Magnetic Suspension and Balance System (LGMSBS) is a conceptual design for experiments which could be used to investigate the technology issues associated with magnetic suspension and accurate suspended-load control at large gaps. This paper investigates regulation and the control methods of magnetic suspension, and proposed a novel design comprised of a cylindrical element containing a permanent magnet, a planar electromagnet array, a position sensing system, a control system. In order to increase the uniformity of the magnetic field and have control over the distribution of the magnetic field, a new structure called planar electromagnet array is established. The magnetic properties are investigated through the mathematical modeling method. Since the magnetic suspension system is highly nonlinear and inherently unstable in operation, this paper deals with the problems of modeling and control strategy for the LGMSBS with 3D-movement control, and numerical simulations and experiments are provided to demonstrate the effectiveness and validity of our conclusions.

Keywords

Large-gap magnetic suspension and balance system planar array of electromagnets 3D-movement control

1. Introduction

A magnetic suspension and balance system (MSBS) is one of the experimental systems among the magnetic levitation technologies [1, 2, 3]. It supports a model by the magnetic force generated by the interaction between a magnet inside the model and coils outside the test section [4]. Thus, support interference associated with mechanical supports can be eliminated. Aerodynamic forces acting on the model can also be evaluated by measuring the electric current of the electromagnets. In addition to this, the MSBS can also be introduced to the area of high precision 3D remote positioning, which recently has a wide range of technical applications. These actuators can be used in special environments without problems caused by friction, abrasion or lubrication. These advantages have provided motivation to use magnetic levitation in wafer transportation, photolithography, teleoperation, magnetic bearings and many other applications [5, 6, 7, 8, 9].

The concept of MSBS was presented in the 1940s, and ONERA – The French Aerospace Lab first succeeded in magnetic levitation for wind-tunnel testing in the 1950s. Since then, the MSBSs have been studied in many laboratories, Sawada et al. [10], Owen et al. and Higuchi et al. have made outstanding contributions in this area respectively [11, 12, 13]. Regardless of the advancements made in regards to MSBSs, further development of an improved MSBS is still needed, requiring greater effort and increasing costs. Here, we can see that most of MSBSs were unable to realize 3D motion control, the workspace available was small, and had conventional limitations on the configuration of the model. The problems mentioned above may be based on the fact that these MSBSs couldn’t generate a moving and uniform field that would always keep a moving permanent magnet stable if the field source was below the permanent magnet.

In this paper, we propose a novel type of LGMSBS having the characteristics of large workspace, 3D-movement control. Through the intelligent control of a planar electromagnet array, we can not only compensate the gravity of the permanent magnet, but also drive the motion of the permanent magnet. Due to the particularity of the proposed system, some problems need to be solved. Firstly, for the large gap suspension, the analysis method is different from the ones with small gap. In a small gap magnetic suspension system, the field at the surface of the actuator may be assumed uniform, and magnetic circuit theory may be utilized [14, 15, 16]. A large gap magnetic suspension system may require a full three-dimensional analysis of the magnetic field at the point of suspension. Secondly, the electromagnet exhibits nonlinear force-current-distance characteristics, and the principal difficulty in the magnetic suspension over a wide range of orientation lies in the feedback control of the suspended-load. This is due to the fact that the magnetic coupling between the suspension electromagnets and the suspended-load is a strong, nonlinear functions of orientation [17, 18, 19]. Thirdly, due to the choice of levitation by repulsive force rather than attractive force more usually employed, strong open-loop unstable modes will be present [20, 21, 22, 23]. Fourthly, the configuration of planar array exhibits none of the usual “decoupling” between 3-dimensions [24, 25, 26].

In this study, a new magnetic modeling of LGMSBS is proposed and a complete controller design procedure is also provided. The simulation results and real world experimental results are guaranteed by the proposed scheme. This paper is structured as follows: In Section 2, a novel LGMSBS is proposed and the system model is built; In Section 3, theoretical models of magnetic force and stable levitation are then derived; Section 4 describes the controller design; Section 5 presents the experiments and results; and Section 6 is conclusions and prospect of this work.

2. System modeling

2.1 System design

The large-gap magnetic suspension and balance system (LGMSBS), as originally defined, is shown schematically in Fig. 1. This invention relates to the suspension of a load in space by magnetic levitation and, more particularly to the use of a set of electromagnets array which are symmetrically positioned about a central vertical axis and are selectively energized, for providing both lift and directional stability to the load. The suspended-load, to be referred to herein as a model, consists of permanent magnet material with the magnetization in the direction of the vertical axis ( $z$ -axis).

Figure 1.

Schematic representation of the LGMSBS.

In order to realize the three-dimensional motion control of model, the distinct design feature of the LGMSBS called “planar electromagnets array” is proposed. For the complex structure and weight, the planar electromagnet array are arranged below the model. So the present invention is a repulsive force conversion drive. The rectangular electromagnets array include the lift coils assemblies (just directly below the model) and sets of control coils for providing positional and directional stability. In the three-axis coordinate system, the $z$ axis is the central vertical axis, while the $x$ axis and the $y$ axis define a horizontal plane. The lift coils assembly establishes a magnetic field which is symmetrical about the $z$ axis, and has a potential energy profile along the $z$ axis with a depression at the location of the model to stabilize the $z$ -axis coordinate of the model’s location. In the $x$ and the $y$ directions, the equilibrium provided by the magnetic field of the lift coils assembly is unstable, and must be supplemented by the operation of the control coils assembly to attain stability. The control coils assembly is located around the lift coils assembly. Combination of magnetic fields of these control coils, with adjustment of intensity of their respective fields produces translational energy as may be required to adjust the horizontal position of the model.

The position sensing system does not follow traditional approach of multiple light beams partially interrupted by the model. The beams are arranged ahead of the model. The light sources are miniature infra-red light-emitting diodes, intended for use with fiber-optics. The light receivers are matching infra-red phototransistors. Due to beam dispersion, collimating lenses were added to both transmitter and receiver. The complete sensor system is mounted on a framework which is initially fixed in $z$ -axis orientation relative to the suspension electromagnets, and has the ability to move horizontally.

2.2 System modeling

Consider the LGMSBS shown in Fig. 1, the translational acceleration of the model in model coordinates can be written as

$\dot{\vec{V}}=\left({1\mathord{\left/{\vphantom{1m}}\right.\kern-1.2pt}m}% \right)\vec{F}$ (1)

Where $\vec{V}$ is the velocity of the model, $m$ is the mass of the model, and $\vec{F}$ denotes the total forces on the model. The force $\vec{F}$ can be expanded as

$\vec{F}=\vec{F}_{m}+\vec{F}_{d}+\vec{F}_{g}$ (2)

where $\vec{F}_{m}$ denotes control forces on the model produced by the electromagnets, $\vec{F}_{d}$ denotes external disturbance forces, and $\vec{F}_{g}$ consists of the force acting on the model, due to gravity, transformed into model coordinates.

3. Magnetic torques and force

The first object of this study is to find the precise currents in the electromagnet array that satisfy a required $F_{Z}$ (lift), and control $F_{X}$ , $F_{Y}$ at any position for the model. A schematic representation of the proposed system which shows the coordinate systems and initial alignment is shown in Fig. 1. A set of orthogonal $x$ , $y$ , and $z$ axes, as the inertial space, defines the location of the electromagnet array. A set of orthogonal $x$ ’, $y$ ’, and $z$ ’, model fixed axes, defines the motion with respect to inertial space. The $x$ , $y$ , and $z$ axes are parallel to the $x$ ’, $y$ ’, and $z$ ’ axes respectively. The equations of electromagnetic forces have been presented and discussed in the following section.

3.1 Magnetic forces

The torque on the model can be approximated as

$T\cong\int{(d\vec{M}\times\vec{B})}\mbox{d}V$ (3)

where $\vec{M}$ is the magnetization of the permanent magnet attached on the model, and $\vec{B}$ is the flux density at the centroid of the permanent magnet produced by the electromagnets.

Similarly, the force on the model can be approximated as

$F_{m}\cong\int{(\vec{M}\cdot\nabla)\vec{B}}\mbox{d}V$ (4)

Since the size of the model is small relative to the size of the electromagnets and the gaps involved, a reasonable approximation of the forces which are produced can be obtained by making the assumption that the field and gradient components are uniform over the volume of the electromagnet. Under this assumption

$\vec{M}\cdot\nabla=\vec{M}^{T}\nabla=\left({M_{x}\frac{\partial}{\partial x}+M% _{y}\frac{\partial}{\partial y}+M_{z}\frac{\partial}{\partial z}}\right)$ (5)

Taking the dot product, $M\cdot\nabla$ results in the scalar. Equation (4) then becomes

$F_{m}=\int_{vol}{\left[{\begin{array}[]{l}M_{x}\frac{\partial B_{x}}{\partial x% }+M_{y}\frac{\partial B_{x}}{\partial y}+M_{z}\frac{\partial B_{x}}{\partial z% }\\ M_{x}\frac{\partial B_{y}}{\partial x}+M_{y}\frac{\partial B_{y}}{\partial y}+% M_{z}\frac{\partial B_{y}}{\partial z}\\ M_{x}\frac{\partial B_{z}}{\partial x}+M_{y}\frac{\partial B_{z}}{\partial y}+% M_{z}\frac{\partial B_{z}}{\partial z}\\ \end{array}}\right]}$ (6)

Where $V o l$ is the volume of the permanent magnet.

The notation can be simplified considerably by letting $B_{ij}$ represent ${\partial B_{i}}\mathord{\left/{\vphantom{{\partial B_{i}}{\partial j}}}\right% .\kern-1.2pt}{\partial j}$ . By factoring $\vec{M}$ out as a vector, Eq. (6) can be written in the form

$\vec{F}_{m}=\int_{\textit{vol}}{[\partial\vec{B}]\vec{M}}$ (7)

where

$\partial\vec{B}=\left[{{\begin{array}[]{*{20}c}{B_{xx}^{\prime}}\hfill&{B_{xy}% ^{\prime}}\hfill&{B_{xz}^{\prime}}\hfill\\ {B_{yx}^{\prime}}\hfill&{B_{yy}^{\prime}}\hfill&{B_{yz}^{\prime}}\hfill\\ {B_{zx}^{\prime}}\hfill&{B_{zy}^{\prime}}\hfill&{B_{zz}^{\prime}}\hfill\\ \end{array}}}\right]$ (8)

in simplified notation. From Maxwell’s equations, in the region of the permanent magnet, $\nabla\times\vec{B}=0$ , which results in

$\displaystyle B_{xy}^{\prime}=B_{yx}^{\prime}$ $\displaystyle B_{xz}^{\prime}=B_{zx}^{\prime}$ (9) $\displaystyle B_{yz}^{\prime}=B_{zy}^{\prime}$

Also $\nabla\cdot B=0$ , which results in

$B_{xx}^{\prime}+B_{yy}^{\prime}+B_{zz}^{\prime}=0$ (10)

In the LGMSBS, $M$ is defined in model coordinates and $B$ is defined in inertial coordinates. Therefore, in order to calculate torque in model coordinates, $B$ has to be transformed into the same coordinate system. This results in

$\vec{T}=\int_{\textit{vol}}{\vec{M}\left[{T_{m}}\right]\vec{B}}$ (11)

Where $T_{m}$ is the vector transformation matrix from inertial to model coordinates.

The assumptions made in developing the model are that the model is held fixed at an initial nominal operating point which only can be move in yaw ( $\theta_{z}$ ), but will be zero in pitch and roll. So the coordinate transformation [ $T_{m}$ ] becomes

$\displaystyle[T_{m}]=\left[\begin{array}[]{ccc}\cos\theta_{z}&\sin\theta_{z}&0% \\ -\sin\theta_{z}&\cos\theta_{z}&0\\ 0&0&1\\ \end{array}\right]$ (12)

Where $\theta_{z}$ is a given yaw angle.

In order to perform the operations involved in the force calculations, $\vec{M}$ is transformed into inertial coordinates

$\vec{M}=\left[{T_{m}}\right]^{-1}\vec{M}_{\textit{inertial}}$ (13)

And the resulting force can be transformed back into model coordinates

$\vec{F}_{m}=\int_{\textit{vol}}{\left[{T_{m}}\right]\left[{\partial\vec{B}}% \right]\left[{T_{m}}\right]^{-1}\vec{M}_{\textit{inertial}}}$ (14)

The translational acceleration of the suspension load, in inertial coordinates, can be written as

$\dot{\vec{V}}=\left({1\mathord{\left/{\vphantom{1m}}\right.\kern-1.2pt}m}% \right)\left({\int_{\textit{vol}}{\left[{T_{m}}\right]\left[{\partial\vec{B}}% \right]\left[{T_{m}}\right]^{-1}\vec{M}_{\textit{inertial}}}}\right)$ (15)

Since the permanent magnet is magnetized along the $z$ axis, $\vec{M}$ can be written as

$\vec{M}=[0\ \ \ 0\ \ \ \ M_{z}]$ (16)

3.2 System equations

In this section the magnetic torques and forces combined with the equations of motion are produced in an open-loop model of the LGMSBS. Since the model is assumed to be accurately held to a fixed point in translation with only small displacements being allowed about the operating point, $B$ and $\partial B$ at this point are assumed to be essentially constant. This means that $B$ and $\partial B$ can be represented as functions of electromagnet currents only.

If $B$ and $\partial B$ are calculated at the operating point, in inertial coordinates, for the maximum current in each electromagnet, then $B$ can be written as

$B=\left({1\mathord{\left/{\vphantom{1{I_{\max}}}}\right.\kern-1.2pt}{I_{\max}}% }\right)\left[{K_{B}}\right]u$ (17)

Where $I_{\max}$ is the maximum electromagnet current, $K_{B}$ is a 3*n matrix elements represents the values of $B$ produced by a corresponding electromagnet driven by the current, and

$u^{T}=\left[{{\begin{array}[]{*{20}c}{I_{1}}\hfill&{I_{2}}\hfill&\ldots\hfill&% {I_{n}}\hfill\\ \end{array}}}\right]$ (18)

Figure 2.

Block diagram of analytical model of LGMSBS.

The gradients can be put in the same form by arranging the elements of $\partial B$ as a column vector. This results in

$\partial B=\left({1\mathord{\left/{\vphantom{1{I_{\max}}}}\right.\kern-1.2pt}{% I_{\max}}}\right)\left[{K_{\partial B}}\right]\left\{I\right\}$ (19)

Where $\partial B$ is a nine element column vector containing the gradients of $B$ , and $K_{\partial B}$ is a 9*n matrix whose elements represent the values of $\partial B$ produced by a corresponding coil driven by the maximum current. Each elements of $\partial B$ , can be written in the form like this,

$B_{xx}=\left({1\mathord{\left/{\vphantom{1{I_{\max}}}}\right.\kern-1.2pt}{I_{% \max}}}\right)\left[{K_{xx}}\right]u$ (20)

Where $K_{xx}$ is a 1*n matrix containing values of $B_{xx}$ produced by a corresponding coil driven by the maximum current.

These equations can be used to calculate feedback gains to stabilize and control the model about the selected operating point. Figure 2 is the analytical diagram for the LGMSBS, which represents the aerodynamic couplings between the forces and the corresponding model motion. The “magnet coupling” is an algebraic relation between the gradient electromagnet currents and the forces. According to the experimental specifications, it is usually required $x$ , $y$ and $z$ -axis, positions to follow a desired trajectory. The input-output decoupling technique can satisfy these requirements. The magnetic force $F_{x}$ , $F_{y}$ and $F_{z}$ applied to the model depend on respective currents produced by appropriate power supplies, the relations between them are shown in the figure below.

4. Controller design procedure

Here, a decoupled, multi-input multi-output (MIMO) control concept is developed for wide range operation of s three-dimensional position control of an LGMSBS. The proposed control concept combines a MIMI-decoupled control strategy and LQR $+$ STF controller. In the following, we first construct the control strategy, then develop the controller from intuitively physical observation.

4.1 MIMO-decoupled control design

In the case of LGMSBS, one of its major design features is its three-dimension motion control, the input and output need to be sufficiently decoupled for the precise control of the model. The decoupled motions can be deduced from the sensor signals. For the output-current, the situation will become more complicated. The currents are not decoupled with respect to the degrees of freedom because of the nonlinear distribution of magnetic fields.

All complex trajectories can be dispersed into the combination of a simple movement. If the electromagnet coil is small enough relative to the permanent magnet and the electromagnet coils array are closely arranged, the driving force can be produced on any planar direction. That is to say, any complex motion can be achieved. Here, the space motion of model can be divided into these three categories, namely, one-dimension motion ( $z$ -axis); two-dimension motion ( $x$ - $y$ plane); three-dimension motion.

4.1.1 Control strategy

For $z$ -axis motion, we only control the input current of the lift coils assembly to adjust the position. The lift coils assembly establishes a magnetic field which is symmetrical about the $z$ axis, and has a potential energy profile along the $z$ axis. The equilibrium provided by the magnetic field of the lift coils assembly is unstable, and must be supplemented by the operation of the control coils assembly to attain stability.

For $x$ - $y$ plane motion, the trajectory planning needs to be completed first. The next work is the stable suspension of the model at the desired initial positon. Through the mechanics model given above, we can calculate the output current for the lift coils assembly and the control coils assembly, it is not a hard work.

The input-currents of the lift coils assembly keep unchanged, and the coils are selectively energized according to the position of the model, the only rule is to ensure the lift coils assembly always directly below the model. Meanwhile, the control coils assembly selectively energized also keep the input unchanged except the ones on the direction of motion, these coils provide the driving force to produce translational movement. The specific way is to increase the input-current of the coils (on the negative direction of the movement), reduce the input-current of the coils (on the direction of the movement). Both of the increased and decreased quantities must be consistent, which ensures the $z$ -axis intensity of magnetic fields unchanged. The specific value depends on the requirement of translational speed.

For the three-dimension movement, it is the combination of the two cases above, which can be achieved by the intelligent control of the electromagnetic coils assembly with different functions.

4.1.2 Control process

During the experiments, there are driving force, compensate force and disturbance. In order to achieve precise control, it is necessary to specify the control rules. This paper takes into account the disturbance of kinematic statement primarily. It adopts the online real-time resistance force measurement and open-loop compensation scheme, which is illustrated in Fig. 3. By measuring (or estimating) the real-time velocity during the motion, together with the position of the tested-model and the configuration of the planar electromagnetic coils array, this scheme distributes the thrust force and compensate force, and then realizes the move control of the model. This scheme uncouples force compensation control quantity and the motion control quantity, which are generated by different controllers. The control instruction fed into the actuating mechanism is the simple superposition of these two control quantities. Hence, ideally, without affecting the model’s own motion control, the force compensation control not only benefits from analyzing the equivalence between experiment control instruments and real spatial experiment instruments, but also directly verifies the validity and feasibility of the spatial control strategy in the ground experimental environment.

Figure 3.

Open-loop control of the tested-model.

4.2 LQR

+

STF controller design

Figure 4 shows the configuration of the proposed LQR $+$ STF controller. The LQR controller and the STF controller are parallel structures. The LQR controller can stabilize the control system, and the STF control is used to compensate the nonlinearity of magnetic suspension system for a wide range operation.

Figure 4.

Configuration of the proposed STF $+$ LQR controller.

4.2.1 Linear quadratic regulator controller design

To facilitate operation of LMGSBS, a LQR controller was constructed, following traditional practices for wind-tunnel model or similar magnetic suspension systems. Position sensor outputs are summed and differenced, where appropriate, to derive motion signals in suspended-load axes. Here, the controller configuration which was investigated is one which uses integral feedback. Integral feedback is often used in systems subjected to constant disturbances in order to minimize steady-state errors. A block diagram of the controller is shown in Fig. 5. The feedback gains are calculated using a linear model, obtained from the nonlinear system model described above, by linearizing about a nominal operating point. And the specific calculation process is shown in the following equations.

Figure 5.

Integral feedback regulator.

With a constant disturbance, $V_{0}$ , the system under consideration becomes

$\dot{d}=Ad+Bu+V_{0}$ (21)

Where $A$ is the system matrix, $B$ is the input matrix, $d$ is the performance parameters of the model, $u$ is input vector.

For the LGMSBS, $d$ is defined as

$d^{T}=\left[{{\begin{array}[]{*{20}c}{{\begin{array}[]{*{20}c}{{\begin{array}[% ]{*{20}c}{V_{x}}\hfill&{V_{y}}\hfill\\ \end{array}}}\hfill&{V_{z}}\hfill&x\hfill\\ \end{array}}}\hfill&y\hfill&z\hfill\\ \end{array}}}\right]$ (22)

The controlled variables are assumed to be

$\lambda=Dd$ (23)

Where $D$ is the matrix of relation between $d$ and $\lambda$ .

From Eq. (21) above, a constant input $u_{0}$ must be found to hold the states at $d_{0}$ , so that

$0=Ad_{0}+Bu_{0}+V_{0}$ (24)

Next, define the shifted input, shifted state, and shifted controlled variable, respectively, as

$\displaystyle u^{\prime}=u-u_{0}$ $\displaystyle\lambda^{\prime}=\lambda-\lambda_{0}$ (25) $\displaystyle d^{\prime}=d-d_{0}$

To develop the integral feedback controller equations, assume the system with the controlled variables. To the system add the integral states, defined by

$\dot{q}=\lambda^{\prime}$ (26)

A performance function can now be defined as

$J=\int\limits_{0}^{\infty}{\left({d^{T}\bar{Q}d+q^{T}Qq+u^{T}Ru}\right)}dt$ (27)

Where the weighting matrices, $Q$ and $R$ are symmetric and positive-semidefinite and positive-definite, respectively.

This functions use integral feedback to minimize steady-state position error. The required states are suspended-load positions and orientations together with the corresponding rates. Rates are derived from the sensor position signals by using differentiators. The feedback gains are calculated using a linear model, by linearizing about a nominal operating point.

Assume that the time-invariant control law

$u=-F_{1}d-F_{2}q$ (28)

can be found which stabilizes the augmented system described by Eqs (21), (23), and (26), and which minimizes Eq. (27). Since the presence of the constant disturbance vector has no effect on the asymptotic stability of the system.

$\mathop{\lim}\limits_{t\to\infty}\dot{q}=0$ (29)

And form Eq. (29)

$\mathop{\lim}\limits_{t\to\infty}\dot{d}=0$ (30)

This means that the control system with the control law Eq. (28) has the property that the error in the controlled variables, due to the constant input disturbance, eventually goes to zero.

4.2.2 Self-tuning fuzzy controller design

The controller designed based on the linearized equation can only be operated near the operation point. If the operating range is large, especially tracking a time-varying signal for different operating point, the conventional controllers can no longer work well, because the system is inherently an unstable and nonlinear system.

From the above discussion, if the $f(u_{0},d_{0})$ can be changed with respect to the controlled position in real time, then the linearized equation may still simulate the dynamics of the magnetic suspension system very well. Constructing a function as the force model was considered, however the force model has some relationship with input-current, position, environment temperature and so on. And it is difficult to measure $f(u_{0},d_{0})$ term exactly, because the magnetic suspension system is originally an unstable system. This is the main idea to construct a self-tuning fuzzy controller to compensate the nonlinear term $f(u,d)$ . The procedures are following:

Define fuzzy labels corresponding to some operating points and their associated membership functions.

Construct database, corresponding to these fuzzy labels, which record the information of the nonlinear term $f(u_{0},d_{0})$ .

Determine the compensate term $f(u_{0},d_{0})$ from database.

Tune the database in real time by the reference input and state information-self-tuning process.

Assume the input is fuzzy singleton, and the sup-min operator is used for fuzzy reasoning. Outputs are also fuzzy singletons, then the modified weighted combination method is used for defuzzification process in order to speed up the computation time.

Figure 6.

Specifications of LGMSBS.

The tuning process is as the following: If input is $P_{i}$ then change of $V_{pi}$ is $G_{vi}(u,d)$ . This means that the modified term of $V_{pi}$ should be a function of the reference input $u$ , and the position of the suspended-load $d$ .

For simplicity, $V_{vi}(u,d)$ is assume as:

$\displaystyle G_{vi}(u,d)=\Delta V_{pi}=\alpha f_{pi}(u)(u-d)$ (31)

Where $\alpha$ is a tuning factor, and $f_{pi}\left(u\right)$ is a membership function. Then, the nonlinear effect of the wide-range operation can be compensated by the self-tuning fuzzy controller.

5. Simulation and experiment results

A series of central pole suspension experiments (both of simulation and experiment test) were investigated in order to find out the validity and performance of the proposed method. In this work, the maximum operating gap has been restricted to 150 mm due to operational constraints of existing position sensor, actuator, and power amplifier.

5.1 System simulation

In this section, the simulation model of proposed nonlinear controller has been established under the MATLAB/Simulink environment for LGMSBS. The parameters are given in Fig. 6. The transient responses of the LGMSBS are then discussed. Figure 7 shows the performance of the position control system when the model was levitated. These figures show that the system response in the vertical direction is faster than the response in the horizontal plane. This is due to the larger magnetic field gradient in the vertical direction. Consequently, the large air-gap is a crucial nonlinear factor impacting the transient and steady performance of the LGMSBS system. This result is consistent with the theoretical analysis.

Figure 7.

Step responses of the model in different direction.

Figure 8.

Experimental verification platform.

5.2 Experiments and results

In this section, the real-world experiments are presented. The setup of the LGMSBS experiment is shown in Fig. 8, where a permanent magnet is levitated, and has multidimensional motion at different operating zones. In order to show the performance of the system, we design these three different experiments. In the experiments, optical range finder will be used to record the positions of model. The demonstration results are shown in Figs 9–11.

The specific design of experiments is respectively as follows:

Continuous vertical motion along $z$ -axis;

Continuous plane motion on the $x$ - $y$ plane;

Continuous three-dimensional motion.

From the experiment results, the following conclusions can be got.

Figure 9.

Vertical trajectory tracking along the $z$ -axis.

Figure 10.

Circle trajectory tracking on the $x$ - $y$ plane.

The Fig. 9 shows that the model can quickly reach the state and stable suspension at setting height (along the axis of electromagnet coils array). When the position requirements change, the model can complete a continuous movement and keep suspension at the desired height. In Fig. 10, the model carrys out a plane-circle from a given point, and it completely moves along the designed route. In Fig. 11, the model moves under continuous different height along the diagonal direction (from the second quadrant to the fourth quadrant). Here, the vector decomposition of three-dimensional trajectory is carried out, three curves in Fig. 12 respectively the trajectory in $x$ , $y$ , $z$ axis. These results indicate that not only the three-dimensional motion control of the model is realized, but also the control system has the fast response time and high control precision.

Figure 11.

Three-dimensional trajectory.

Figure 12.

The trajectory tracking in different axis.

All these show that the proposed methods of modeling as well as controller design for the LGMSBS is practicable and successful. Moreover, in actual practice, the system parameters may vary from the theoretical values due to linear approximation of the electromagnetic force made in the theoretical design, temperature variation, unmodeled dynamics of the electromagnet, and other parametric uncertainties.

6. Conclusions

A brief survey of recent applications for magnetic and balance systems was presented. Most of these MSBs have very low working ranges and performed levitation over small air gaps. In this paper, a novel large-gap magnetic suspension and balance system having the characteristics of large workspace, 3D-movement control, has been successfully developed. Meanwhile, a mathematical approach for modeling and a control scheme for the 3D positional control of LGMSBS are also accomplished. Then the levitation and control of a model containing a permanent magnet, in 3D direction, over a large range of orientation, using a planar array of electromagnets, has been achieved. Experiments on a prototype based on the research is presented in this paper, which show that it is feasible to use proposed LGMSBS for the 3D noncontact manipulation. This technology has potential applications in a wide range of areas including microgravity and vibration isolation systems, magnetically suspended pointing mounts, large-gap magnetic suspension systems for advanced actuators, wind tunnel magnetic suspension systems, and remote manipulation/control/positioning of objects in space.

Footnotes

Acknowledgments

This study was supported by the National High Technology Research and Development Program of China (Grant No. 2015AA7041003) and the Fundamental Research Funds for the Central Universities (Grant No. JB150410). We also thank to the Chinese National Key Laboratory of Aerospace Flight Dynamics for technical support.

References

Lee

D.K.

Lee

J.S.

Han

J.H.

Kawamura

and Chung

S.J.

, Development of a Simulator of a Magnetic Suspension and Balance System, Int’l J. of Aeronautical & Space Sci. 11(3) (2010), 175–183.

Ewald

B.F.R.

, Multi-component force balances for conventional and cryogenic wind tunnel, Measurement Science and Technology 11(6) (2000), 81–94.

Boyden

, A Review of Magnetic Suspension and Balance Systems, in: 15th Aerodynamic Testing Conference, AIAA, Reston, VA, 1988, pp. 94–105.

Sawada

, An aerodynamic force estimation method for winged models at the JAXA 60 cm magnetic suspension and balance system, Journal of the Japan Society for Aeronautical and Space Science, 57(668) (2009), 363–371.

Park

K.H.

Ahn

K.Y.

Kim

S.H.

and Kwak

Y.K.

, Wafer distribution system for a clean room using a novel magnetic suspension technique, IEEE/ASME Trans Mech 3(1) (1998), 73–78.

Holmes

Hocken

and Trumper

D.L.

, Long-range scanning stage: a novel platform for scanned-probe microscopy, J Precision Eng 24(3) (2000), 191–209.

Lin

C.E.

and Yang

C.K.

, Force and moment calibration for NCKU 10 cm ⋅ 10 cm magnetic suspension wind tunnel, in: Proc IEEE Instrument Measur Technol Conf, vol. 2, May 2004; pp. 1060–1065.

Mukhopadhyay

S.C.

Donaldson

Sengupta

Yamada

Chakraborty

and Kacprzak

, Fabrication of a repulsive-type magnetic bearing using a novel arrangement of permanent magnets for verticalrotor suspension, IEEE Trans Magn 39(5) (2003). 3220–3222.

Yasuda

Ohnakaa

Ninomiyaa

Ishiia

Fujitaa

and Kishiob

, Levitation of metallic melt by using the simultaneous imposition of the alternating and the static magnetic fields, J Cryst Growth 260(3–4) (2004), 475–485.

10.

Sawada

Kunimasu

and Suda

, Sphere drag measurements with the NAL 60 cm MSBS, Journal of Wind Engineering (98) (2004), 129–136.

11.

Higuchi

van Langen

Sawada

and Tinney

C.E.

, Axial fow over a blunt circular cylinder with and without shear layer reattachment, Journal of Fluids and Structures 22 (2006), 949–959.

12.

Owen

A.K.

and Owen

F.K.

, Hypersonic free fight measurement techniques, in: 22nd International Congress on Instrumentation in Aerospace Simulation Facilities, Pacifc Grove, CA, 2007, pp. 1–11.

13.

Higuchi

Sawada

and Kato

, Sting-free measurements on a magnetically supported right circular cylinder aligned with the free stream, Journal of Fluid Mechanics 596 (2008), 49–72.

14.

Sinha

P.K.

, Electromagnetic Suspension, Dynamics and Control, Peter Peregrinus Ltd., London, 1987.

15.

Shirazee

N.A.

and Basak

, Electropermanent suspension system for acquiring large air-gaps to suspend loads, IEEE Trans. Magn. 31(6) (1995), 4193–4195.

16.

Banerjee

Prasad

and Pal

, Large gap control in electromagnetic levitation, ISA Transactions 45(2) (2006), 215–224.

17.

Trumper

Olson

S.M.

and Subrahmanyan

P.K.

, Linearizing control of magnetic suspension systems. IEEE Trans. Control Syst. Technol. 5(4) (1997), 427–438.

18.

Kim

Y.C.

, Gain scheduled control of magnetic suspension system, in: Proc. of the American Cont. Conf., 1994, pp. 3127–3131.

19.

Olaru

and Gherca

, Generator with levitated magnet for vibration energy harvesting, International Journal of Applied Electromagnetics and Mechanics 42(3) (2013), 421–435.

20.

Golob

and Tovornik

, Modeling and control of the magnetic suspension system, ISA Trans. 42(1) (2003), 89–100.

21.

Krishnan

, Electrical Motor Drives-Modeling, Analysis and Control. Prentice Hall of India, New Delhi, 2003.

22.

Ogata

, Modern Control Engineering, 3rd ed. Prentice Hall of India, New Delhi, 2000.

23.

Liu

and Li

Y.Y.

, Application of a compound controller based on fuzzy control and support vector machine to ship’s boiler-turbine coordinated control system, J. Marine. Sci. Appl. 8(1) (2009), 33–39.

24.

Noh

M.D.

Cho

S.R.

Kyung

J.H.

S.K.

and Park

J.K.

, Design and implementation of a fault-tolerant magnetic bearing system for turbomolecular vacuum pump, IEEE/ASME Trans. Mechatronics 10(6) (2005), 626–631.

25.

Yang

Z.J.

Miyazaki

Kanae

and Wada

, Robust position control of a magnetic levitation system via dynamic surface control technique, IEEE Trans Ind Electron 51(1) (2004), 26–34.

26.

El Hajjaji

and Ouladsine

, Modeling and nonlinear control of magnetic levitation systems, IEEE Transactions on Industrial Electronics 48(4) (2001), 831–838.