backstepping controller design for a planar maglev positioning system

Proceedings of the 2005 IEEE/ASME International Conference on Advanced Intelligent Mechatronics Monterey, California, USA, 24-28 July, 2005

WB4-05

BACKSTEPPING CONTROLLER DESIGN FOR A PLANAR MAGLEV POSITIONING SYSTEM Mei-Yung Chen1, Shao-Kang Hung2, Li-Chen Fu2 1.Department of Electrical Engineering, China Institute of Technology, Taipei 115, Taiwan, E-mail: [email protected] 2.Department of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan, E-mail:[email protected]

and high-precision 2-dimensional positioning. To attain the former goal, we first build a prototype of one-axis electro-magnetic actuator to test and verify the feasibility. From the experimental results, the EMA positioning system was successfully designed and implemented [1].

ABSTRACT In the previous research, an electro-magnetic actuator (EMA) system has to be designed and implemented successfully [1]. The system can demonstrate good performances including stiffness and resolution. Based on those experiences, a prototype of a novel planar Maglev positioning system is designed in this research. In the new structure, the carrier motion (both levitation and propulsion) results from a sum of repelling forces each exerted on some magnet from its corresponding coil. Likewise, the associated full-DOF (degree-of-freedom) model is initially derived and analyzed, and then a backstepping controller for this Maglev positioning system is developed. Finally, from the experimental results, satisfactory performances including regulation, tracking accuracy and control stiffness have all been demonstrated.

Based on those experiences, a prototype of a novel planar Maglev positioning system is designed and implemented in this research. In the new structure, the carrier motion (both levitation and propulsion) results from a sum of repelling forces each exerted on some magnet from its corresponding coil. But in this structure, we provide enough gap space between the magnet and the coil to let the carrier possess enough moving room. However, the larger gap will generally cause the higher magnetic loss. Therefore, it is intuitively clear that the repelling force between the magnet and the coil will vary when the magnet is situated in different locations inside the coil. Such special phenomenon causes the controller design task much harder than the EMA system. Moreover, in the novel Maglev positioning system, many complex factors need to be taken into consideration. For example, the feedback control should be involved extensively in that system. That means, when the mechanical part is designed, it always needs to take control methods into consideration. In the past decade, the backstepping control strategy has been widely used in dealing with nonlinear system [7][8]. In fact, to control of a Maglev system can also apply such backstepping control scheme to overcome those problems.

Keywords: Planar maglev, Precision positioning, Backstepping controller.

I. INTRODUCTION In the previous research, a double-deck mechanism is adopted in the design of a dual-axis Maglev system [2~4], where the carrier is free to move along the X- or the Y-direction. For the reasons of simplifying the mechanism of the Maglev system while trying to make the power consumption more economical, so a planar, single-layer Maglev positioning system needs to be developed. Although Trumper and Kim [5,6] designed and implemental a world’s first high-precision six DOF magnetic levitator with two-dimensional motion capability for photolithography in semiconductor manufacturing, that system mainly works for small moving range (micro meter). However, as we see, the challenge of designing a planar Maglev system is how to appropriately arrange the sensors and actuators and design an advanced controller so as to achieve the objectives of large moving range (centimeter)

0-7803-9046-6/05/$20.00 ©2005 IEEE.

In this paper, first of all, a full-DOF (degree-of-freedom) model will be derived and then that model is thoroughly analyzed. For the control purpose, a backstepping controller is developed in this Maglev positioning system. Finally, from the experimental results, satisfactory performance indices, including regulation, tracking accuracy, and control stiffness, have all been demonstrated.

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attitude DOFs needs to be analyzed thoroughly. Before derive the model, several reasonable assumptions will be made as follows which will facilitate subsequent derivations.

Carrier

Permanent Magnet

A. Both the carrier and the platform are rigid bodies. B. Every PM (permanent magnet) can be approximated as one single dipole carrying the same magnetic dipole moment and is located at the center of each PM.

Active Coil

C. All the PMs are widely spread so that flux couplings among them can be negligible. Moreover, coils are arranged such that the mutual inductance among them can also be negligible.

Optical Sensor

D. The magnetic forces and torques between PMs and their neighboring upright solenoids coils can be linearized around some appropriately defined operating points. But the PMs and their neighboring (horizontal) rectangular coils will induce linear forces as in the previous research.

Figure 1: Prototype of the proposed Maglev system.

II. MAGLEV SYSTEM DESCRIPTION

From the above, assumption C clearly suggests that the mutual effects among all the sub-systems are negligible. In assumption D, an important implication is that the performance of the carrier’s translation, which does not have to be small, is the main focus, whereas its rotation is simply subject to regulation. That means the angular displacement of the carrier around each axis, except Z-axis, should be very small. After these assumptions are validated, the complex analysis of the overall Maglev system will become simpler. The procedure steps are:

In this section, the key ideas about design of this system will be introduced. Figure 1 shows a prototype of a magnetically levitation system which has developed. From there, the moving carrier is a center stage of size 10cmu10cmu0.3cm and three supporting beams rigidly affixed to the stage. Moreover, there are totally 6 permanent magnets which are attached to the three beams, two on each beam. An ultimate goal for the design is to allow a planer motion over a range 50mmu50mm with the positioning accuracy up to 10Pm. Besides the carrier, the Maglev platform has another major part which consists of three cylindrical solenoids and three rectangular coils as shown in Fig.2, of which the lateral three rectangular coils {A, B, C} will generate as push-pull (lateral) force whereas the underside three cylindrical solenoids {D, E, F}can generate suspension (vertical) force. From the viewpoint of operation, the three magnets suspended from the three beams mainly provide levitation force and the three side magnets attached to the end of three beams; respectively, mainly supply the horizontal positioning force. After appropriate control, the system not only can levitate the carrier to attain the correct attitude but also can move it to the designated position with up to sufficient accuracy at the desired speed.

1. finding the dynamics of the carrier; 2. finding the magnetic forces and torques between the levitating PMs and the cylindrical solenoids, as well as between the propelling PMs and the rectangular coils; 3. substituting the force and torque equations into the dynamics of the rigid body and simplifying them to obtain the find set of equations of motion. The forces and torques exerted on the levitation PMs and the positioning PMs are due to their interaction with the cylindrical solenoids and rectangular coils, respectively. the equations of force and torque will be derived.

After the structure of the Maglev platform is described, the material of the PM is also an important factor to the success of the proposed system. From the previous discussion, NdFeB is deemed as the best choice for magnetic materials at present. On the contrary, drawbacks remain for the other materials. For example, AlNiCos has low coercivity, Ferrites has low remanence, and Samarium magnets are quite expensive.

III.

ANALYTICAL MODEL MAGLEV SYSTEM

OF

{O}

PLANAR

For the sake of achieving the goal of high-precision positioning, the novel system must control not only the translation but also the attitude of the carrier. Therefore, a complex model including three lateral DOFs and three

Figure 2: The locations of PMs relative to their associated electromagnets

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we obtain the carrier’s translational displacement [X ,Y ,Z]T and rotational displacement [M , I , T ]T, then the six pair of relative displacements have to be calculated; each between a PM and its associated electromagnet.

Force and Torque between the PM and the Coil The expression of the Lorentz force [9] can be simplified as

G F G T

G

mG ˜ ’ B ,

(1)

In the following context, let PMi’|o for i{A,B…,F} denote the new coordinate of PMi|o value after the carrier undergoes translation [X ,Y ,Z]T and rotation [M , I , T ]T. New, the relative displacement for each pair of PM and electromagnet relative to the associated local coordinate system have to be obtained. Combining the previous deriK K vations f g ( x, y , z )u , then every magnetic force relative to its associated local coordinate system can be obtained successfully. Take the pair of PMA and the coordinate system {A}, for example, the force can be expressed as:

G G mu B ,

(2) K G where m is the dipole moment and B is the magnetic flux. As has been mentioned earlier [9], to obtain a compact force model for a PM with an arbitrary shape coil is rather difficult. Therefore, a simpler and probably more accurate approach through direct measurement is adopted here in this paper. However, this approach requests, a force detector to measure the force exerted on the PM within the coil’s magnetic field. The windings of two kinds of coils used in this system are made of copper with diameter equal to 0.65mm. In the previous research [1], we have shown that the magnitude of the magnetic force applied to a PM at a fixed location by an electromagnet is proportional to the current flowing into the electromagnet. Besides, the location of the PM relative to the electromagnet will also affect the magnetic force. To sum up, if this K force f [ f x , f y , f z ]T can be expressed as:

K f

K g ( x , y , z )u ,

f PM A

(3)

grec, y ( y)urec (a6 y6  a5 y5  a4 y4  a3 y3  a2 y2  a1 y  a0 )urec

System Dynamics By Newton’s Law, the system dynamics can be obtained as:

f cyl , z

(b4 z  b3 z  b2 z  b1 z  b0 )ucyl , 3

2

where b4=-4.87 ͪ 106, b3=-6.522 ͪ 105, b1=-736.02, and b0= -4.166.

ª X º « » M « Y » , « Z  g » ¬ ¼

K ¦F

ªI K « XX ¦ T # « IYX «¬ I ZX

, (4)

I XY IYY I ZY

(7)

I XZ º ªMº IYZ »» ««I»» , I ZZ »¼ «¬T»¼

(8)

where we will regard IXY, IXZ, IYX, IYZ, IZX, IZY as zero subsequently.

where a6=-1.954 ͪ 109, a5=4.39 ͪ 108, a4=-3.385 ͪ 107, a3=1.16ͪ105, a2=-19338, a1=214.8, and a0=-0.385. Similarly, the same method can be applied to the cylindrical solenoid. The magnetic force component along its z-axis, namely, fcyl,z throughout the control task can be well approximated by the product of coil’s current ucyl and a well defined function gcyl,z(z). Likewise, after measurement a lest-fit 4th order polynomial curve can be obtained, where the specific polynomial function is given below: 4

(6)

fPMA,y|A=grec,y(PMA’|o)uA, where fPMA,x|A=grec,x(PMA’|o)uA, fPMA,z|A=grec,z(PMA’|o)uA, and grec,x(.), grec,y(.) ,grec,z(.) are the force functions for a PM and its associated rectangular coil expressed with respect to the local coordination system {A} which is defined as Eq.(3), and uA is the current input flowing into that coil. Of course, such force expression can also be obtained for other PMs.

where u denote the current flowing into the associated coil, then gK ( x, y, z ) [ gK x ( x, y, z ), gK y ( x, y, z ), gK z ( x, y, z )]T is a function of PM’s location. Therefore, prior to the controller development, the vector gK ( x, y, z ) need to extensively be measured. For practical reason and for simplification, the force function grec,y(x, y, z) in Eq.(3) within the desired traveling range can be approximated by grec,y(y). With this in mind, lest-fit polynomial function over those measured data can then be computed as: frec, y

T

ª f PM , x , f PM , y , f PM , z º , A A A A A A¼ ¬

A

K

First, to obtain the term ¦ F , we have to transfer the forces expressed in local coordinate systems to those in global coordinate system. The new coordinate of each PM in the global coordinate system {O} to it associated local coordinate system have to be transferred, namely, PM A

(5) b2=-33486,

' A

§ ª  D1 º · ¨ ¸ ' R0A ¨ PM A  « 0 » ¸ , « » 0 ¨ ¸ ¬« 0 ¼» ¹ ©

(9)

This task is quite easy as shown below. K ¦F

Note that all the coordinates mentioned above are relative to the global reference coordinate system XYZ, i.e.,{O}. However, in the previous derivations and measurements, the forces exerted on the PMs all depend on their locations relative to their associated local coordinate systems affixed to the electromagnets or coils. Thus, once

K RA0 f PM A  RB0 A K  RD0 f PM D  RE0 D

K

¦T

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K K f PM B  RC0 f PMC B C K K f PM E  RF0 f PM F , E

K K PM A' u RA0 f PM A  PM B ' u RB0 f PM B A

(10)

F

B

K  PM C ' u RC0 f PM C

C

K K K  PMD' u RD0 fPMD  PME' u RE0 fPME  PMF ' u RF0 fPMF . (11) D

E

F

For convenience, the dynamic equations can be rewritten as: K K (12) ¦ F ¦ RI0 f PM I , I

I [ A, B ,..., F ]

K

¦T

¦

I [ A, B ,..., F ]

K PM I ' u RI0 f PM I

.

(13)

I

Fig. 3:The magnetic force between PM and electromagnet

Development of an Overall Model

along the solution trajectories to obtain:

To develop an overall model of the entire system, we turn to specifications referring to the global coordinate system. Its origin is just located at the center of the carrier when it is in its initial nominal configuration. This definition together with the force relations is shown in Fig. 3.

V1

z1 ( z2  D1 )

F

Thus, a Lyapunov function candidate V2 need to be selected and design appropriate control input u to render its time derivative nonpositive. Let V2 be defined as:

V2

ª gA º  2 gC 0 0 0 0 « » M 2 M « » « » gB 2 gC 0 0 0 0 « » ªu º ª 0 º M 2 M « »« A» « » g g g « 0 » « uB » « 0 » D E F 0 0 « » «u » « g » M M M « » C   g E L3 2 g F L2 » «uD » « 0 » « 0 « » « » 0 0 0 « I XX 2 I XX » « uE » « 0 » « »« » « » «  2 g F L2 » ¬« uF ¼» ¬« 0 ¼» g D L3 N N 0 0 0 « 0 G IYY 2 IYY » U « » « » gC L2 0 0 0 0 « 0 » I ZZ » ¬«   

¼

V2

U

V2

x1 x2  D

BU  G  (  c1 z1 )

(15)

.

V1

2 2  c1 z1  c2 z2 ,

(22)

V. EXPERIMENTAL SETUP AND RESULTS In this section, the experimental result will be provided to demonstrate the validity and performance of the developed planar Maglev positioning system. Here, the output states are X, Y, Z, M, I and T. Moreover, the control currents are treated as inputs, respectively denoted as uA, uB, uC, uD, uE, and uF.

(16)

Evaluate the time derivative of a Lyapunov-like function

1 2 z1 2

(21)

Due to zero convergence of z, it can be readily verified that x converges to zero asymptotically. In other words, the state variable x and its time derivative x all converge to zero eventually, which obviously achieves the goal of precise positioning and regulation.

Now, a backstepping controller will be developed for this system. Redefine the new error variables z1=x1 and z2=x2Į1, where Į1 is a different stabilizing function defined as Į1 = c1z1 and c1 >0. The time derivative of z1 and z2 are now expressed as:

z1 z2

B 1 (G  z1  c1 z1  c2 z2 ) ,

which apparently is nonpositive because c1 and c2 are all positive values. Using arguments of Lyapunov theory [10], it conclude that z1(t), z2(t)Ш0 asymptotically go to 0 as tШɬ.

Based on the dynamical model Eq.(14), which is compactly re-expressed as: .

(20)

where c2>0, then Eq.(20) becomes:

x

BU  G

V  z2 z2

The control law u should be able to cancel the indefinite term in Eq.(20). After substitute the properly designed input U as:

IV. BACKSTEPPING CONTROLLER DESIGN

x  x

(19)

c1 z12  z2 ( z1  BU  G  c1 z1 ).

diag[1 1 1 1 1 1][ X Y Z M I T ] ., (14)   

 

x1 x2

1 2 z2 , 2

c1 z12  z2 ( z1  z2 )

T

C

V1 

whose time derivative can be calculated as:

B

y

(18)

c1 z12  z1 z2 .

By now, the overall system model can be arrived at by collecting Eq.(4) to Eq.(5) and Eq.(12) to Eq.(13). As a result, a state space model of this system can be finally obtained as follows:

ª X º «  » «Y » « Z » « » « M » « I » «  » T ¼» ¬«N

z1 z1

Hardware setup

(17)

The experimental hardware, including the main body, sensor system, driver system and controller hardware,

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will be described here. Figure 1 shows the photographs of the physical set-up.

Experimental results A number of experimental results, including the transient and the steady-state responses in different situations, will also be provided in this sub-section to demonstrate the performance of the developed prototype system with the controller presented in Section IV.

Electromagnetic specifications In the experiment, three rectangular electromagnets and three circular electromagnets (solenoids) was utilized in this system. The material of electromagnet s is copper. The specifications of the rectangular electromagnet and circular eletromagnet used in this experiment are listed in the tables below:

Figure 4 shows the regulation response with initial position on X,Y =-3mm. From the state trajectories, one can see that the error signals converge to their steady states within about 0.5s. In Fig. 4, the final precision is made up to 10Pm in translation, which is suspected to reach a limit of the sensor device. Meanwhile, the response of 100g payload effect is shown in Fig. 5. One might find that there is still a slight perturbation on Z-axis followed by the instant movement of the carrier on X-axis. Due to the successful design of controller, this positioning error converges to zero very soon.

Table 1: The specifications of the rectangular electromagnet Turns 2000

Outer Length 17cm

Inner Length 12cm

Outer Width 10cm

Inner Thickness Width 5cm 6cm

Table 2: The specifications of the circular eletcromagnet. Turns Outer Diameter 890 9cm

Inner Diameter 5cm

Thickness 3cm

Additionally, Fig. 6 shows the experimental results when the desired position was Xd =2.5sin(4.5t)mm and Yd =2.5sin(4.5t)mm. The initial state conditions are randomly assigned. When the guiding system is stable, its stability was slightly affected only at the instant when the coils started to drive the stage in the X- and/or Y-direction; however, the guiding precision is not affected.

Sensor system This sensor system, can be further classified into lateral type and vertical type, which respectively provide two translational and three rotational displacement data for the purpose of subsequent control. The lateral sensor devices, measuring X, Y, and Ӱ, are chosen to be ILD 1400-50, which are manufactured by the Micro-Optronic Technology in Germany. Its rise time can be as low as 200 psec whereas the active range is up to 50mm range of measurement (ROM) with a resolution of 0.1% of ROM, which means that sensor has fast enough response and can cover all the traveling range in our application. The scheme of the vertical sensor system is composed of three gap sensors which are needed to measure Z, ӿ, and ӽ comprising the attitude of the carrier. The model number of the vertical gap sensor is Z4W-V25R, and it is manufactured by the OMRON Corporation, Japan. The measurement range is r4mm and the gap sensor possesses 10Ӵm resolution.

VI. CONCLUSIN For this purpose, a novel structure for such system is designed in this paper. We mainly designed a precision planar, single-layer Maglev system for large range two -axis motion. Its dynamics been thoroughly analyzed and then a complete model has also been derived. The system is treated as a MIMO system, and a backstepping controller has been designed here and implemented using a microcomputer. At the same time, the good performance can ensure that the three objectives are satisfactorily obtained: large traveling range, precision positioning, and fast response are achieved.

ACKNOWLEDGEMENT

Driver System and Controller Hardware

This research is sponsored by National Science Council, R.O.C., under the grant NSC-93-2218-E-157 -001.

The function of drivers in the system is to provide sufficient amount of currents to the pulling and the levitating coils. Because the current instead of voltage is used as control input, the current drivers must have enough high bandwidth. The drivers type is C0-502-001Q Torque Amplifiers manufactured by CMC Inc. in USA. They are linear drivers designed to be servo drivers for DC motors. The power is 250 Watts: 5 Amps for ̈́50Volts or 10 Amps for ̈́24Volts. The current microcomputer is an IBM PC with Pentium microprocessor inside. The clock rate is 733MHz which allows the present experimental set-up to accomplish a real-time control implementation. Based on the experiential results, a sampling time between 0.1ms and 0.05ms leads to better behavior of this system. The type of ADC is a 16-bit high resolution data acquisition adapter, and the type of DAC is equipped with six converting channels with 12-bit resolution each.

REFERENCE [1]. M.-Y. Chen, T.-B. Lin, S.-G. Huang and L.-C. Fu, “Design, Analysis and Control of an Electro-Magnetic Actuator,” Proceedings of the American Control Conference, pp.1233-1238, 2003. [2]. M. –Y. Chen, M.-J. Wang and L.-C. Fu, "A Novel Dual-Axis Repulsive Maglev Guiding System with Permanent Magnet : Modeling and Controller Design," IEEE Transactions on Mechtrinacs, Vol. 8, No. 1, pp. 77-86, Mar. 2003. [3]. I.-Y. Wang, A., Busch-Vishniac, I.,”A new repulsive magnetic levitation approach using permanent magnets

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rad

[4]. M. –Y. Chen, K.-N. Wang and L.-C. Fu, "Design, Implementation and Self Tuning Adaptive Control of a Maglv Guiding System," MECHATRONICS, Vol. 10, pp215-237, 2000. [5]. David L. Trumper, Sean M. Olson, and Pradeep K. Subrahmanyan, “Modeling and Vector Control of a Planar Magnetic Levitator” IEEE tran. on Industry Application. vol. 34, no. 6, Nov/Dec., 1998.

Y

I

rad

mm

[6]. W. -J. Kim, Shobhit Verma, and Jie Gu, “MagLev 6-DOF stage for nanopositioing,” Proceedings of IMECE’03, 2003 ASME, Nov. 15-21, 2003.

M

[7]. Kanellakopoulos, I. and P. K. Krein, "Integral-action Nonlinear Control of Induction Motors”, Proc. 12th IFAC World Congress, Sydney, pp. 251-254, 1993. Z

rad

[9]. J. G. David, Introduction to Electrodynamics. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1981.

T

mm

[8]. Kristic, M., I. Kanekkakopoulos and P. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc., 1995.

[10]. Slotine, Li, Applied nonlinear control. Prentice-Hall Inc. 1990. Figure 5: System responses with a 100g payload

mm

rad

I

Z

T

mrad

mm

Z

Y

mm

mm

mrad

I

M

rad

Y

X

mm

mm

mrad

M

T

rad

X

Figure 6: Harmonic motion along the X- and Y- direction with centimeter level

Figure 4: System regulation responses of the carrier

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