Decoupling and levitation control of a six-degree ... - Semantic Scholar
Original Article
Decoupling and levitation control of a six-degree-of-freedom magnetically levitated stage with moving coils based on commutation of coil array
Proc IMechE Part I: J Systems and Control Engineering 226(7) 875–886 Ó IMechE 2012 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0959651812445860 pii.sagepub.com
Shengguo Zhang, Yu Zhu, Haihua Mu, Kaiming Yang and Wensheng Yin
Abstract The research in the present paper focuses on decoupling the six-degrees-of-freedom motions of a magnetically levitated stage with moving coils and controlling the levitation motion. The decoupling is based on commutation of coil array, which is obtained reversely from the electromagnetic force/torque model of the stage. The control of levitation motion is carried out by a phase lead-lag controller designed on the criterion of minimum integral of time-weighted absolute error after gravity compensation modeling of the plant dynamics. Comprehensive simulations of the control system in MATLAB/Simulink and real levitation experiments on a digital signal processor-centered test platform are made to verify the six-degrees-of-freedom decoupling effect and the closed-loop control performances of the levitation degree of freedom. The results indicate that the six-degrees-of-freedom motions of the stage can be decoupled through coil array commutation and the levitation motion can be controlled by the phase lead-lag controller based on the gravity compensation model and designed on the criterion of minimum integral of time-weighted absolute error.
Keywords Decoupling, levitation control, commutation of coil array, gravity compensation modeling, phase lead-lag controller, magnetically levitated stage
Date received: 9 September 2011; accepted: 28 March 2012
Introduction Magnetically levitated stages have been developed in recent years as an alternative to air bearing stages constructed of stacked linear motors. This is mainly due to the fact that the magnetically levitated stage can operate in vacuum, for example in extreme-ultraviolet lithography equipment or nanoimprint lithography equipment. However, this alternative still demands that the stage obtains good motion control performance. This is a great challenge for the magnetically levitated stage. In a Euclid space, the magnetically levitated stage has six-degrees-of-freedom (6-DOF) motions because of the active magnetic bearing. It has to be controlled in six degrees of freedom (DOFs) although its mover can move over long stroke only in the xyplane. In order to provide adequate energy for levitation and propulsion, the magnetically levitated stage is usually over-actuated, i.e. the number of coils is greater than the number of DOFs. Each coil current contributes to the three force components and the three torque components corresponding to the 6-DOF motions.
The magnetic flux density distribution of the permanent magnet array varies in a non-strictly sinusoidal waveform in the horizontal direction and decays exponentially in the vertical direction. This implies fundamentally larger coupling among the 6-DOF motions and the position dependency of the coil currents. In order to control the 6-DOF motions, the three force components and the three torque components corresponding to the 6-DOF motions, which are defined by the interaction of the magnetic flux density distribution and the coil currents, have to be decoupled. Kim and colleagues1–3 have used DQ-0 decomposition in combination with the design symmetries of the
Department of Precision Instruments and Mechanology, Tsinghua University, People’s Republic of China Corresponding author: Shengguo Zhang, Department of Precision Instruments and Mechanology, Tsinghua University, Tsinghua Campus, Beijing 100084, People’s Republic of China. Email: [email protected]
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stage to decouple the three force components and three torque components, thus the 6-DOF motions of the stage can be controlled under six single-input and single-output (SISO) controllers. Compter4 adopted the same method to decouple the three force components and the three torque components of an electro-dynamic planar motor with moving coils. However, the distribution of the forces over the mover surface cannot be taken into account by this decoupling method, which may lead to disturbance torques. Van Lierop and colleagues5,6 and De Boeij et al.7 proposed the direct wrench-current decoupling method, which removes the disturbance torques and still includes smooth switching of coil currents. By this method, the three force components and three torque components corresponding to the 6-DOF motions can be decoupled and so the 6DOF motions can be controlled under their own SISO controllers. However, this method applies chiefly to the magnetically levitated stage with moving magnets. The present paper investigates the motion decoupling and control of a 6-DOF magnetically levitated stage with moving coils. The paper is organized as follows. First, the electromagnetic force/torque is modeled based on the Lorenz force law and the commutation of coil array is derived from the reflective generalized inverse of the electromagnetic force/torque model, thus the three force components and three torque components corresponding to the 6-DOF motions can be decoupled. Next, the plant dynamics of the 6-DOF motions, especially the gravity compensation dynamics of levitation motion, are modeled and analyzed, and then the controllers are designed based on the criterion of minimum integral of time-weighted absolute error (ITAE). The whole motion control system is next simulated in MATLAB/Simulink, following which the decoupling effect and levitation control performances are verified by experimenting on a real digital signal processor (DSP)-centered test platform. After comparing and discussing the simulation and experiment results, conclusions are drawn.
Electromagnetic force/torque model and commutation of coil array The investigated stage is a configuration after optimal design of structure. Its stator is a kind of Halbach permanent magnet array4–8 and its mover contains 20 ironless coils arranged in order. Figure 1(a), (b) and (c) shows the concept of the stage, the top view of the stator (partial) and the perspective view of the mover, respectively. The electromagnetic force/torque model of the stage is established based on the Lorenz force law. Figure 2 shows the coordinate system and the graphical representation of the model. For a Halbach permanent magnet array, neglecting all higher order harmonics, its first order harmonic magnetic flux density distribution in the O–XYZ coordinate system can be represented analytically. For a
Figure 1. Investigated magnetically levitated stage with moving coils: (a) concept of the stage; (b) stator—permanent magnet array; (c) mover—coil array.
tightly wound rectangular coil, neglecting the impact of its corner segments and winding gaps, it can be equivalent to four current-carrying surfaces. Thus, the forces and torques acting on the mover, produced by a single coil in the Halbach permanent magnet field, can be integrated using the Lorenz force law. Then, the three force components and three torque components acting on the mover of the stage, defined at the center of mass and around the inertia axes of the mover, can be calculated using the force/torque superposition principle of a rigid body.9 This electromagnetic force/torque model can be represented as
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ð3Þ
In equation (3) the rank of G(q) is 6, but the number of unknown i is n and n . 6, so there are infinitely many solutions. Based on the pseudo-inverse G–(q) of the electromagnetic force/torque model matrix G(q) G (q) = GT (q) ½G(q) GT (q)1
ð4Þ
the coil current vector, i, can be obtained, in the sense of least squares,10 as i = G (q)Wr
ð5Þ
This maps the six desired force/torque components corresponding to the 6-DOF motions to the currents of n coils, so it is called ‘‘the commutation of coil array’’. The coil currents commutated by equations (5), i, not only can generate the desired force/torque vector, Wr, but also can globally guarantee minimum heat loss of the coil array.
Dynamics and controller design
Figure 2. Graphical representation of the electromagnetic force/torque model: (a) top view; (b) cut view.
ð1Þ
Wr (q) = G(q)i
where Wr (q) = ½Fx , Fy , Fz , Tx , Ty , Tz T
denotes the force/torque vector acting on the mover
After the 6-DOF motion decoupling based on the commutation of coil array and the electromagnetic force/torque model, the complete motion control system model can be configured as shown in Figure 3(a). However, due to the position dependency of commutated coil currents, the implementation of coil array commutation needs the real-time feedback of positions and orientations of the mover just as the implementation of controllers does. Assuming the controller output is Wrc Wrc = ½fx , fy , fz , t x , t y , t z T
q = ½x, y, z, c, u, fT
is the position and orientation vector of the mover in the O–XYZ coordinate system
then the actually produced force/torque vector is Wr = G(q)i = G(q)G (q)Wrc = Wrc
ð6Þ
T
i = ½i1 , i2 , . . . , in
is the coil current vector and 2 Fx1 (q) Fx2 (q) . . . 6 Fy1 (q) Fy2 (q) . . . 6 6 Fz1 (q) Fz2 (q) . . . G(q) = 6 6 Tx1 (q) Tx2 (q) . . . 6 4 Ty1 (q) Ty2 (q) . . . Tz1 (q) Tz2 (q) . . .
3 Fxn (q) Fyn (q) 7 7 Fzn (q) 7 7 Txn (q) 7 7 Tyn (q) 5 Tzn (q)
So the motion control system model can be equivalent to the model shown in Figure 3(b) and controllers of motions with different DOFs can be designed based on their own dynamics. ð2Þ
is the electromagnetic force/torque model matrix of 6 3 n dimension, whose six elements of column n represent the three force components and three torque components produced by the coil n commutated per unit current (1 A) when the mass center of the translator is at the position and orientation q in the O–XYZ coordinate system. On the basis of equation (1), if the desired force/torque vector is Wr = ½Fx , Fy , Fz , Tx , Ty , Tz T
Figure 3. (a) Basic control configuration including the commutation of coil array and the electromagnetic force/torque model of the investigated magnetically levitated stage motion system; (b) equivalent control configuration.
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Dynamics models and gravity compensation of levitation motion For 6-DOF motions of a magnetically levitated stage, except for the long moving travel in the x- and y-directions, its z-direction moving travel is usually several millimeters, and its rotation travels around the x-, yand z-axes are usually quite short (maximum of a few : : : : hundred microradians). So c ¼ 0, u ¼ 0, f ¼ 0, c_ ¼ 0, : : u_ ¼ 0 and f_ ¼ 0 can be assumed in modeling the linear dynamics. Under this small angle rotation condition, the 6-DOF dynamics can be represented as 8 m x€ = fx > > > > m y€ = fy > > < m z€ + mg = f z ð7Þ € = tx c I > x > > > u = ty > Iy € > : € Iz f = t z where m is the mass of the mover, g is acceleration due to gravity, and Ix, Iy and Iz are the mover’s moments of inertia around the x-, y- and z-axis, respectively. Equations (7) indicate that the dynamics of the x- and y-direction moving DOFs, and the x-, y- and z-axis rotation DOFs, can be modeled as pure mass (moment) plants and that all five of these models in the frequency domain are double integrators. On the other hand, the dynamics of the z-direction moving DOF cannot be modeled directly as pure mass plant due to the constant gravity. According to equations (1) and (6), the dynamics of the z-direction moving DOF can be reconfigured as m€ z + mg = fz = Fz = G(3) (q)i pz i = K3 exp tn
ð8Þ
i = i^+ i0
expanding the exponential term of equation (8) to be a Taylor series on the set-point z0 and truncating all the terms after (including) the second order term and then rearranging equation (8), we can derive m€z^ + K^ z = K0 G(3) (q)i^
Letting z in place of z^ and i in place of ^i, the linearized dynamics model of the z-direction moving DOF, including gravity compensation, can be derived as m€ z + Kz = K0 fz
ð9Þ
where
pz0 p K = mg exp tn tn pz0 K0 = exp tn
Design of controllers Taking account of the plant dynamics and the motion control requirements, the controllers are designed based on the criterion of minimum ITAE11,12 ðts min½ITAE =
tje(t)jdt
ð12Þ
0
where ts is the settling time of the closed-loop system and e(t) is the deviation between the reference input and the controlled output e(t) = r(t) y(t)
where G(3)(q) denotes the third row of G(q) and K3 is a quantity independent of z. On the control set-point z0, assuming that z = z^ + z0 ,
From equations (9) and (10), the dynamic behavior of the z-direction moving DOF (‘‘levitation DOF’’, in short form below) can be observed and analyzed. It is a mass– spring plant first. It is a plant without damping and it is of open-loop unstability (critical stability). In addition, it has uncertainty of model parameters because K and K0 change with the change of set-point z0. Thus all 6-DOF linear dynamics have been derived as equations (11) show, which can be used to design the corresponding controllers 8 m€ x = fx > > > > m€ y = fy > > < m€ z + Kz = K0 fz ð11Þ € I > x c = tx > > > Iy €u = t y > > : € Iz f = t z
This criterion, on the one hand gives consideration to both dynamic performance and static precision; while on the other hand it emphasizes the effect of the timenearest response and reduces the effect of initial error in the dynamic process. The geometric significance of the minimum ITAE criterion is that the error has the minimum area in the dynamic process of the closed-loop system. It is one minimum of the multi-dimension phase space surface and yet it is difficult to obtain the analytical solution for a high order closed-loop system.12 Xue11 gives the performances, characteristic polynomials and their optimal coefficients of the first order to the sixth order standardization closed-loop system based on the criterion of minimum ITAE, which are derived by the experimental method. For the fourth order standardization closed-loop system, assuming its natural frequency is vn and its settle time of step response is ts, if its characteristic polynomial satisfies Ds4 = s4 + 2:1vn s3 + 2:4v2n s2 + 2:7v3n s + v4n
ð10Þ
ð13Þ
then the maximum overshoot of the closed-loop system’s step response, d, will be 1.9% and the standardizing time of the closed-loop system’s step response, vnts, will be 5.4.
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In accordance with equation (13), for the levitation DOF, to match a fourth order standardization closedloop system, a phase lead-lag controller can be designed as shown in equation (14) Gz (s) =
Kg (s + z1 )(s + z2 ) s(s + p1 )
ð14Þ
From equations (11), the plant transfer function of the levitation DOF is K0 Pz (s) = 2 ms + K
ð15Þ
then the characteristic polynomial of the designed closed-loop system is D4 = s4 + p1 s3 + ½(K + K0 Kg )=ms2 + ½Kp1 =m + K0 Kg (z1 + z2 )=ms + K0 Kg z1 z2 =m
Let D4 = Ds4 , thus equations (17) are established 8 p1 = 2:1vn > > < (K + K0 Kg )=m = 2:4v2n ½Kp1 + K0 Kg (z1 + z2 )=m = 2:7v3n > > : K0 Kg z1 z2 =m = v4n
ð16Þ
dynamics. Therefore four controllers need to be designed in all. The controllers of other DOFs can also be designed by the above method. After modifying and rounding slightly, the final phase lead-lag controllers designed for the investigated 6-DOF magnetically levitation stage with moving coils are shown in equations (18). They will be used in the following control system simulations and experiments 8 780, 000(s + 55)(s + 1) > > Gx, y (s) = > > s(s + 1200) > > > > 3, 000, 000(s + 160)(s + 1) > > > < Gz (s) = s(s + 2000) > 9000(s + 90)(s + 1) > > Gc, u (s) = > > s(s + 2000) > > > > > 2500(s + 55)(s + 1) > G (s) = : f s(s + 1200)
ð18Þ
Control system simulation ð17Þ
Then given a desired settle time ts, the natural frequency of the closed-loop system can be defined as vn = 5:4=ts . Solving equations (17), the parameters of the designed phase lead-lag controller shown in equation (14), namely Kg, z1, z2 and p1, can be derived. Because of the symmetries of structure and movement, the x- and y-direction moving DOFs have completely the same dynamics, and the rotation DOFs around the x- and y-axis also have completely the same
6-DOF motion decoupling simulation The simulation model of the complete control system is built based on MATLAB/Simulink. As shown in Figure 4, it consists mainly of four parts: plant dynamics, electromagnetic force/torque model, commutation of coil array and phase lead-lag controllers, plus input and output modules. In order to simulate the disturbance rejection performance, random disturbances, whose amplitudes are 0.001 N, 0.001 N, 0.001 N, 0.001 Nmm, 0.001 Nmm and 0.001 Nmm, respectively, are added to the inputs of dynamics. In addition, band-limited white noises, whose power spectrum densities are 100 nm2 and
Figure 4. Control system simulation model based on MATLAB/Simulink.
Proc IMechE Part I: J Systems and Control Engineering 226(7)
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Figure 5. Closed-loop system responses when the levitation DOF is given 10 mm step signal at set-point z0 = 0.0 mm.
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10 mm, 10 mm, 10 mm, 10 mrad, 10 mrad and 10 mrad step signals in turn at set-point z0=2.0 mm. Figure 6 shows the controller force/torque outputs Wrc, the model force/torque outputs Wr and their errors. Figure 7 shows the output responses of all six DOFs. As seen from Figure 6, errors between Wrc and Wr are within 10213 m/rad. As seen from Figure 7, the output of each DOF is only the response to its own input. Summarized from the simulations above, relative to the input of 10 mm, the decoupling error of order 1 nm/
φ−torque error (N⋅mm)
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z−force (N)
y−force (N)
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100 nrad2, are added in the feedback channel to simulate the measuring noises. First, a 10 mm step signal is given to the input of the levitation DOF (other inputs are given constant signals of 0) at set-point z0=0.0 mm. As can be seen from Figure 5, except for the response of the levitation DOF, the responses of the other five DOFs are within 6 2 nm or 6 2 nrad. Further, the x-, y- and z-direction moving DOFs, and the x-, y- and z-axis rotation DOFs, are given
−15 x 10 2 0 −2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 −15 x 10 5 0 −5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 −15 x 10 5 0 −5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 −13 x 10 2 0 −2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 −13 x 10 1 0 −1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 −13 x 10 1 0 −1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 t/s
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Figure 6. Closed-loop system controller force/torque outputs, model force/torque outputs and their errors when the x-, y- and zdirection moving DOFs, and the x-, y- and z-axis rotation DOFs, are given 10 mm, 10 mm, 10 mm, 10 mrad, 10 mrad and 10 mrad step signals in turn at set-point z0 = 2.0 mm.
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10 0 −5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 t(s) 10
5
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x (μm)
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0 −5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 t(s)
Figure 7. Closed-loop system responses when x-, y- and z-direction moving DOFs, and x-, y- and z-axis rotation DOFs, are given as 10 mm, 10 mm, 10 mm, 10 mrad, 10 mrad and 10 mrad step signals in turn at set-point z0 = 2.0 mm.
nrad is enough to be neglected. So, the 6-DOF motions are decoupled completely.
Levitation control simulation
1 0.9 0.8 0.7 0.6 0.5 0.4
14 12 10 0
0.5
1
1.5 2 set−point z0 (m)
2.5
3 −3
x 10
z(μm)
7000 6000 5000 4000 3000 2000
(b) K0
(a) K (N/m)
Equations (9) and (10) indicate that the model parameters of the levitation DOF, K and K0, are both functions of the control set-points, z0. Changing trends of K and K0 with the change of z0 are as shown in Figure 8. This means that the model of the levitation DOF is changeable when the mover is levitated at different gaps. In order to verify the stability of the closed-loop control system of the levitation DOF, letting z0=0.0 mm, 1.0 mm, 2.0 mm and 3.0 mm, respectively, four different plant models under different model parameters, K and K0, are obtained. These four models and the controller of levitation motion shown in equations (18) compose four closed-loop systems, respectively. Figure 9 shows
the 10 mm step responses of the levitation DOF in different plant model parameters when the control set-point z0=0.0 mm, 1.0 mm, 2.0 mm and 3.0 mm. It illustrates that the closed-loop control system of the levitation DOF is always stable in different plant model parameters depending upon different control set-points within levitation DOF travel of 0–3 mm, even though there are measuring noises. Further, the trajectory tracking performances of the levitation DOF are simulated by giving a sine input signal. Figure 10 shows the trajectory tracking result and tracking error at the set-point z0=1.0 mm when the levitation DOF is given a sine signal whose frequency is 2 Hz and amplitude is 1.0 mm. It shows that the given sine signal is tracked precisely while there is a time delay of about 0.02 s. The tracking error is within 6 30 mm after the time delay stage of trajectory. Figure 11 indicates that the other DOF motions have been decoupled
8 6 z0=0.0mm z0=1.0mm z0=2.0mm z0=3.0mm
4 2 0
0.5
1
1.5 2 set−point z0 (m)
2.5
3 −3
x 10
Figure 8. Changes of plant model parameters of the levitation DOF when the control set-point, z0, changes: (a) change of K; (b) change of K0.
0 0
0.01
0.02
0.03
t(s)
0.04
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Figure 9. Closed-loop system step responses of the levitation DOF with different plant model parameters when the control set-point z0 = 0.0 mm, 1.0 mm, 2.0 mm and 3.0 mm.
Proc IMechE Part I: J Systems and Control Engineering 226(7)
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Figure 10. (a) Trajectory tracking result and (b) error when the levitation DOF is given a sine trajectory of 1.0 mm amplitude at the set-point z0 = 1.0 mm.
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Figure 11. Motions of the other DOFs when the levitation DOF is given a sine trajectory of 1.0 mm amplitude at the control setpoint z0 = 1.0 mm.
with the levitation DOF motion and there are decoupling errors of about 6 30 nm/nrad.
Experiments and discussion Control system experiments The configured test platform consists of a magnetically levitated stage with moving coils (the main parameters and dimensions of which are shown in Table 1), a DSP board (TMS320C6713, SEED-XDS510Plus Emulator), measuring sensors, an analog-to-digital board, a digital-to-analog board and 20 single phase power amplifiers. The controller algorithm and coil array commutation algorithm are operated on the DSP. After digital-to-analog conversion, the current output of the DSP is given to 20 power amplifiers and their output ports are connected to 20 coils. The three positions and three orientations of the mover are measured by seven eddy sensors and their measured data are
transferred back to the DSP by analog-to-digital conversion. Figure 12 is a photograph of the test platform (except for the electronic and electrical parts). Figure 13 shows the main mechanical setup of the test platform. A mounting rack, which is used to mount and adjust the three z-direction sensors, is fixed to a marble beam. On the four corners of the mounting rack are four through-holes through which four long bolts are bolted to the mover. The diameter of the throughholes is 2 mm larger than the diameter of the long bolt so that the mover can move in the z-direction (from 0.25 mm to 2.25 mm), move in x- and y-directions within short travel (6 1 mm) and rotate around the x-, y- and z-axes in small angle (maximum 6 0.1 rad around x- and y-axes and 6 0.017 rad around z-axis). The placement of seven eddy sensors is shown in Figure 14. From the readings given by these sensors, displacements of the x-, y- and z-directions can be calculated precisely, and angles around the x-, y- and
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Table 1. Main parameters and dimensions of the investigated magnetically levitated stage with moving coils. Quantity
Symbol
Dimension
Permanent magnet array stage Permanent magnet array
– –
Magnet-pole pitch Residual magnetism Translator mass Coil array stage Number of coils Number of coil turns Coil resistance Coil inductance
tn M0 M – – N R L
245 mm 3 245 mm 20 3 20 magnet poles 10.6 mm 1.23 T 0.92 kg 155 mm 3 155 mm 435 526 1 0.66 0.05 O 0.1 mH
Figure 14. Placement of the seven eddy sensors.
Figure 12. Photograph of the test platform (except for the electronic and electrical parts).
Figure 13. Main mechanical setup of the test platform.
z-axes can also be calculated under the condition of small angle rotation, as shown in equation (19) x = (x1 + x2 )=2 y = (y1 + y2 )=2 z = (z1 + z2 + z3 )=3 c = ½(z1 + z2 )=2 z3 =L1 u = (z2 z1 )=L2
ð19Þ
f = ½(x2 x1 )=L1 + (y2 y1 )=L2 =2
The standard measuring objects of the three z-direction sensors, which are ferromagnetic, are pasted on the
appropriate positions of the mover top. After calibrating, these three sensors can reach 0.25–2.25 mm linear measuring range. The four x- and y-direction sensors are mounted on the appropriate positions outside the stator. Different from the three z-direction sensors, they measure the side plane of the mover directly rather than the ferromagnetic standard measuring objects, which would be too near to the magnetic field. After calibrating, these four sensors can reach only 0.25–1.25 mm linear measuring range. Within linear measuring range, all seven sensors have non-linearity of less than 0.85%. These seven sensors have standard sensitivity of 5 V/ mm and the sensitivity deviation is 1.21%. The coil array commutation algorithm and the discretized controller algorithm are programmed in an interrupt routine in C language. The closed-loop sampling frequency is at 1 kHz currently. On the configured test system, the positioning and the trajectory tracking of the levitation DOF are experimented primarily, so that the effects of decoupling and controllers can be observed and analyzed. Figure 15(a) and (b) shows respectively the measured positioning result and the positioning error of the levitation DOF when 500 mm stair signals are given intermittently. It indicates that the mover can position accurately whereas there is a positioning ripple of about 6 50 mm. Figure16 shows the coupled motions of the other DOFs. It indicates the 6-DOF motions are decoupled but there are still decoupling errors of about 6 100 mm and 6 2 mrad. Figure 17 shows a planned (acceleration, velocity, displacement) trajectory of the levitation DOF, in which the mover first ascends from 0 mm to 2 mm and then descends from 2 mm to 0 mm. Figure 18(a) and (b) shows respectively the trajectory tracking result and tracking error. It indicates that the mover can track the planned trajectory whereas there is a tracking error of about 6 0.25 mm. Figure 19 shows the coupled motions of the other DOFs. It indicates that the 6-DOF motions are decoupled, whereas there are still decoupling errors of about 6 50 mm and 6 1 mrad.
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Discussion Decoupling of 6-DOF motions is the primary goal of our investigations. As can be observed from the simulation results, the influences of step input signals on the other DOFs are below nano-meter/radian order and
the influences of the reference trajectory of the levitation DOF on the other DOFs are below micro-meter/ radian order. From the experiment results it is observed that there are decoupling errors of order about dozens of micro-meters/radians, relative to 2 mm positioning
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or tracking travel of the levitation DOF. Simulation and experiment results can illustrate the 6-DOF motions are decoupled whereas there is deviation between simulation results and experiment results. This deviation may originate from the mismatch between the mathematical model and the physical structure of the stage. Manufactured first time and assembled by hand, the assembly dimensions of the stage and the dimensions of stage parts, such as the winding and assembling of 20 coils and the assembling of a large number of permanent magnets, are really coarse. Thus the electromagnetic force/torque model and the commutation of coil array match the design requirements rather than the real mechatronic structure. Control of levitation motion is another goal of our investigations. No matter whether observing from simulation results or experiment results, the phase lead-lag controllers based on the minimum ITAE criterion have achieved good results; whereas there exist non-negligible positioning or tracking ripples of micro-meter/radian order in simulations and of dozens
of micro-meters/radians order in experiments. The origin of this result can be traced to two aspects. On the one hand, the eddy sensors adopted currently have high amplitude measuring noises and any signals in the control loop have not yet been filtered; on the other hand, the present closed-loop sampling frequency is at 1 kHz, which is a bit low to reject the high frequency disturbances and noises.
Conclusions The 6-DOF motion decoupling and levitation DOF control of a magnetically levitated stage with moving coils have been investigated in this paper. Commutation of coil array is used to decouple the 6DOF motions. This commutation is derived inversely from the electromagnetic force/torque model based on the Lorenz force law. In essence, this commutation is to map the six force/torque components corresponding to the 6-DOF motions to the 20 currents of the coil array. The 6-DOF plant dynamics is modeled to design the controllers. The dynamics of the levitation DOF is
886 modeled especially to compensate the gravity, which is of high importance to the magnetically levitated stage. Based on these dynamic models, the phase lead-lag controllers are designed by using the criterion of minimum ITAE. This criterion is very suitable for, and beneficial to, position tracking control of the magnetically levitated stage. The coil array commutation in combination with the designed controllers is used to decouple the 6-DOF motions and to control the levitation DOF motion of the stage. Both the simulations of the complete control system in MATLAB/Simulink and the experiments on a real DSP-centered test platform demonstrate that the 6-DOF motions are decoupled and the levitation DOF motion is controlled. However, there are still decoupling errors and control errors. Perhaps better matching between the theoretical model (coil array commutation algorithm and electromagnetic force/torque model) and the practical mechatronic structure, the increase of the closed-loop sampling frequency and the introduction of a filter may improve the decoupling effects and the control performances effectively. Moreover, the horizontal-direction motions still need to be controlled. These will be studied further in our next work. Funding This work was supported in part by the National Basic Research Program (973 Program) of China [grant number 2009CB724205] and in part by the National High Technology Research and Development Program (863 Program) of China [grant number 2009AA04Z148]. References 1. Kim WJ, Trumper DL and Jeffrey HL. Modeling and vector control of planar magnetic levitator. IEEE Trans Ind Applic 1998; 34(6): 1254–1262. 2. Kim WJ. Nanoscale dynamics, stochastic modeling, and multivariable control of a planar magnetic levitator. Int J Contr Autom Syst 2003; 1(1): 1–10. 3. Yu H and Kim WJ. Controller design and implementation of six-degree-of-freedom magnetically levitated positioning system with high precision. Proc IMechE, Part I: J Systems and Control Engineering 2008; 222(8): 745–756. 4. Compter JC. Electro-dynamic planar motor. Precis Eng 2004; 28(2): 171–180. 5. Van Lierop CMM, Jansen JW, Damen AAH, et al. Control of multi-degree-of-freedom planar actuators. In: 2006 IEEE international conference on control applications (CCA 2006), Munich, Germany, 4–6 October 2006, pp.2516–2521. New York: IEEE. 6. Van Lierop CMM, Jansen JW, Damen AAH, et al. Model-based commutation of a long-stroke magnetically levitated linear actuator. In: Conference record of the IEEE industry applications conference 41st annual meeting, Tampa, USA, 8–12 October 2006, pp.393–399. New York: IEEE.
Proc IMechE Part I: J Systems and Control Engineering 226(7) 7. De Boeij J, Lomonova EA and Vandenput AJA. Modeling ironless permanent-magnet planar actuator structures. IEEE Trans Magn 2006; 42(8): 2009–2016. 8. Min W, Zhang M, Zhu Y, et al. Analysis and optimization of a new 2-D magnet array for planar motor. IEEE Trans Magn 2010; 46(5): 1167–1171. 9. Jansen JW, Van Lierop CMM, Lomonova EA, et al. Modeling of magnetically levitated planar actuators with moving magnets. IEEE Trans Magn 2007; 43(1): 15–25. 10. Zhang S, Zhu Y, Yin W, et al. Coil array real-time commutation law for a magnetically levitated stage with moving-coils. J Mech Eng 2011; 47(6): 180–185. 11. Xue D. Simulation and computer aided design of control system. 2nd ed. Beijing: Machine Press, 2009. 12. Hu S. Automatic control principle. 5th ed. Beijing: Science Press, 2007.
Appendix Notation fx, fy, fz Fx, Fy, Fz g i i^ i0 Ix, Iy, Iz K K0 Kg m p1 q t ts Tx, Ty, Tz Wr(q) Wr Wrc x, y, z z^ z0 z1 z2
control force output (N) model force (N) acceleration due to gravity (m/s2) current vector (A) current variable vector (A) gravity compensation current vector (A) moment of inertia (kgm2) stiffness coefficient (N/m) coefficient (–) gain (–) mass (kg) pole point of transfer function (–) position and orientation vector (m, rad) time (s) settle time (s) model torque (Nm) force/torque vector (N, Nm) model force/torque vector (N, Nm) control output vector (N, Nm) position (m) control variable (m) control set-point (m) zero point of transfer function (–) zero point of transfer function (–)
d D4
overshoot (%) characteristic polynomial of the fourth order closed-loop system (–) characteristic polynomial of the fourth order standardization closed-loop system (–) force/torque model matrix (N/A, Nm/A) pseudo-inverse of force/torque model matrix (A/N, A/Nm) magnet-pole pitch (m) control torque output (Nm) orientation angle (rad) natural frequency of the closed-loop system (rad/s)
Ds4
G(q) G–(q) tn tx, ty, tz c, u, f vn
Decoupling and levitation control of a six-degree-of-freedom magnetically levitated stage with moving coils based on commutation of coil array
Proc IMechE Part I: J Systems and Control Engineering 226(7) 875–886 Ó IMechE 2012 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0959651812445860 pii.sagepub.com
Shengguo Zhang, Yu Zhu, Haihua Mu, Kaiming Yang and Wensheng Yin
Abstract The research in the present paper focuses on decoupling the six-degrees-of-freedom motions of a magnetically levitated stage with moving coils and controlling the levitation motion. The decoupling is based on commutation of coil array, which is obtained reversely from the electromagnetic force/torque model of the stage. The control of levitation motion is carried out by a phase lead-lag controller designed on the criterion of minimum integral of time-weighted absolute error after gravity compensation modeling of the plant dynamics. Comprehensive simulations of the control system in MATLAB/Simulink and real levitation experiments on a digital signal processor-centered test platform are made to verify the six-degrees-of-freedom decoupling effect and the closed-loop control performances of the levitation degree of freedom. The results indicate that the six-degrees-of-freedom motions of the stage can be decoupled through coil array commutation and the levitation motion can be controlled by the phase lead-lag controller based on the gravity compensation model and designed on the criterion of minimum integral of time-weighted absolute error.
Keywords Decoupling, levitation control, commutation of coil array, gravity compensation modeling, phase lead-lag controller, magnetically levitated stage
Date received: 9 September 2011; accepted: 28 March 2012
Introduction Magnetically levitated stages have been developed in recent years as an alternative to air bearing stages constructed of stacked linear motors. This is mainly due to the fact that the magnetically levitated stage can operate in vacuum, for example in extreme-ultraviolet lithography equipment or nanoimprint lithography equipment. However, this alternative still demands that the stage obtains good motion control performance. This is a great challenge for the magnetically levitated stage. In a Euclid space, the magnetically levitated stage has six-degrees-of-freedom (6-DOF) motions because of the active magnetic bearing. It has to be controlled in six degrees of freedom (DOFs) although its mover can move over long stroke only in the xyplane. In order to provide adequate energy for levitation and propulsion, the magnetically levitated stage is usually over-actuated, i.e. the number of coils is greater than the number of DOFs. Each coil current contributes to the three force components and the three torque components corresponding to the 6-DOF motions.
The magnetic flux density distribution of the permanent magnet array varies in a non-strictly sinusoidal waveform in the horizontal direction and decays exponentially in the vertical direction. This implies fundamentally larger coupling among the 6-DOF motions and the position dependency of the coil currents. In order to control the 6-DOF motions, the three force components and the three torque components corresponding to the 6-DOF motions, which are defined by the interaction of the magnetic flux density distribution and the coil currents, have to be decoupled. Kim and colleagues1–3 have used DQ-0 decomposition in combination with the design symmetries of the
Department of Precision Instruments and Mechanology, Tsinghua University, People’s Republic of China Corresponding author: Shengguo Zhang, Department of Precision Instruments and Mechanology, Tsinghua University, Tsinghua Campus, Beijing 100084, People’s Republic of China. Email: [email protected]
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stage to decouple the three force components and three torque components, thus the 6-DOF motions of the stage can be controlled under six single-input and single-output (SISO) controllers. Compter4 adopted the same method to decouple the three force components and the three torque components of an electro-dynamic planar motor with moving coils. However, the distribution of the forces over the mover surface cannot be taken into account by this decoupling method, which may lead to disturbance torques. Van Lierop and colleagues5,6 and De Boeij et al.7 proposed the direct wrench-current decoupling method, which removes the disturbance torques and still includes smooth switching of coil currents. By this method, the three force components and three torque components corresponding to the 6-DOF motions can be decoupled and so the 6DOF motions can be controlled under their own SISO controllers. However, this method applies chiefly to the magnetically levitated stage with moving magnets. The present paper investigates the motion decoupling and control of a 6-DOF magnetically levitated stage with moving coils. The paper is organized as follows. First, the electromagnetic force/torque is modeled based on the Lorenz force law and the commutation of coil array is derived from the reflective generalized inverse of the electromagnetic force/torque model, thus the three force components and three torque components corresponding to the 6-DOF motions can be decoupled. Next, the plant dynamics of the 6-DOF motions, especially the gravity compensation dynamics of levitation motion, are modeled and analyzed, and then the controllers are designed based on the criterion of minimum integral of time-weighted absolute error (ITAE). The whole motion control system is next simulated in MATLAB/Simulink, following which the decoupling effect and levitation control performances are verified by experimenting on a real digital signal processor (DSP)-centered test platform. After comparing and discussing the simulation and experiment results, conclusions are drawn.
Electromagnetic force/torque model and commutation of coil array The investigated stage is a configuration after optimal design of structure. Its stator is a kind of Halbach permanent magnet array4–8 and its mover contains 20 ironless coils arranged in order. Figure 1(a), (b) and (c) shows the concept of the stage, the top view of the stator (partial) and the perspective view of the mover, respectively. The electromagnetic force/torque model of the stage is established based on the Lorenz force law. Figure 2 shows the coordinate system and the graphical representation of the model. For a Halbach permanent magnet array, neglecting all higher order harmonics, its first order harmonic magnetic flux density distribution in the O–XYZ coordinate system can be represented analytically. For a
Figure 1. Investigated magnetically levitated stage with moving coils: (a) concept of the stage; (b) stator—permanent magnet array; (c) mover—coil array.
tightly wound rectangular coil, neglecting the impact of its corner segments and winding gaps, it can be equivalent to four current-carrying surfaces. Thus, the forces and torques acting on the mover, produced by a single coil in the Halbach permanent magnet field, can be integrated using the Lorenz force law. Then, the three force components and three torque components acting on the mover of the stage, defined at the center of mass and around the inertia axes of the mover, can be calculated using the force/torque superposition principle of a rigid body.9 This electromagnetic force/torque model can be represented as
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877 then the coil current vector should satisfy the equation G(q)i = Wr
ð3Þ
In equation (3) the rank of G(q) is 6, but the number of unknown i is n and n . 6, so there are infinitely many solutions. Based on the pseudo-inverse G–(q) of the electromagnetic force/torque model matrix G(q) G (q) = GT (q) ½G(q) GT (q)1
ð4Þ
the coil current vector, i, can be obtained, in the sense of least squares,10 as i = G (q)Wr
ð5Þ
This maps the six desired force/torque components corresponding to the 6-DOF motions to the currents of n coils, so it is called ‘‘the commutation of coil array’’. The coil currents commutated by equations (5), i, not only can generate the desired force/torque vector, Wr, but also can globally guarantee minimum heat loss of the coil array.
Dynamics and controller design
Figure 2. Graphical representation of the electromagnetic force/torque model: (a) top view; (b) cut view.
ð1Þ
Wr (q) = G(q)i
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After the 6-DOF motion decoupling based on the commutation of coil array and the electromagnetic force/torque model, the complete motion control system model can be configured as shown in Figure 3(a). However, due to the position dependency of commutated coil currents, the implementation of coil array commutation needs the real-time feedback of positions and orientations of the mover just as the implementation of controllers does. Assuming the controller output is Wrc Wrc = ½fx , fy , fz , t x , t y , t z T
q = ½x, y, z, c, u, fT
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then the actually produced force/torque vector is Wr = G(q)i = G(q)G (q)Wrc = Wrc
ð6Þ
T
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is the coil current vector and 2 Fx1 (q) Fx2 (q) . . . 6 Fy1 (q) Fy2 (q) . . . 6 6 Fz1 (q) Fz2 (q) . . . G(q) = 6 6 Tx1 (q) Tx2 (q) . . . 6 4 Ty1 (q) Ty2 (q) . . . Tz1 (q) Tz2 (q) . . .
3 Fxn (q) Fyn (q) 7 7 Fzn (q) 7 7 Txn (q) 7 7 Tyn (q) 5 Tzn (q)
So the motion control system model can be equivalent to the model shown in Figure 3(b) and controllers of motions with different DOFs can be designed based on their own dynamics. ð2Þ
is the electromagnetic force/torque model matrix of 6 3 n dimension, whose six elements of column n represent the three force components and three torque components produced by the coil n commutated per unit current (1 A) when the mass center of the translator is at the position and orientation q in the O–XYZ coordinate system. On the basis of equation (1), if the desired force/torque vector is Wr = ½Fx , Fy , Fz , Tx , Ty , Tz T
Figure 3. (a) Basic control configuration including the commutation of coil array and the electromagnetic force/torque model of the investigated magnetically levitated stage motion system; (b) equivalent control configuration.
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Dynamics models and gravity compensation of levitation motion For 6-DOF motions of a magnetically levitated stage, except for the long moving travel in the x- and y-directions, its z-direction moving travel is usually several millimeters, and its rotation travels around the x-, yand z-axes are usually quite short (maximum of a few : : : : hundred microradians). So c ¼ 0, u ¼ 0, f ¼ 0, c_ ¼ 0, : : u_ ¼ 0 and f_ ¼ 0 can be assumed in modeling the linear dynamics. Under this small angle rotation condition, the 6-DOF dynamics can be represented as 8 m x€ = fx > > > > m y€ = fy > > < m z€ + mg = f z ð7Þ € = tx c I > x > > > u = ty > Iy € > : € Iz f = t z where m is the mass of the mover, g is acceleration due to gravity, and Ix, Iy and Iz are the mover’s moments of inertia around the x-, y- and z-axis, respectively. Equations (7) indicate that the dynamics of the x- and y-direction moving DOFs, and the x-, y- and z-axis rotation DOFs, can be modeled as pure mass (moment) plants and that all five of these models in the frequency domain are double integrators. On the other hand, the dynamics of the z-direction moving DOF cannot be modeled directly as pure mass plant due to the constant gravity. According to equations (1) and (6), the dynamics of the z-direction moving DOF can be reconfigured as m€ z + mg = fz = Fz = G(3) (q)i pz i = K3 exp tn
ð8Þ
i = i^+ i0
expanding the exponential term of equation (8) to be a Taylor series on the set-point z0 and truncating all the terms after (including) the second order term and then rearranging equation (8), we can derive m€z^ + K^ z = K0 G(3) (q)i^
Letting z in place of z^ and i in place of ^i, the linearized dynamics model of the z-direction moving DOF, including gravity compensation, can be derived as m€ z + Kz = K0 fz
ð9Þ
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Design of controllers Taking account of the plant dynamics and the motion control requirements, the controllers are designed based on the criterion of minimum ITAE11,12 ðts min½ITAE =
tje(t)jdt
ð12Þ
0
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where G(3)(q) denotes the third row of G(q) and K3 is a quantity independent of z. On the control set-point z0, assuming that z = z^ + z0 ,
From equations (9) and (10), the dynamic behavior of the z-direction moving DOF (‘‘levitation DOF’’, in short form below) can be observed and analyzed. It is a mass– spring plant first. It is a plant without damping and it is of open-loop unstability (critical stability). In addition, it has uncertainty of model parameters because K and K0 change with the change of set-point z0. Thus all 6-DOF linear dynamics have been derived as equations (11) show, which can be used to design the corresponding controllers 8 m€ x = fx > > > > m€ y = fy > > < m€ z + Kz = K0 fz ð11Þ € I > x c = tx > > > Iy €u = t y > > : € Iz f = t z
This criterion, on the one hand gives consideration to both dynamic performance and static precision; while on the other hand it emphasizes the effect of the timenearest response and reduces the effect of initial error in the dynamic process. The geometric significance of the minimum ITAE criterion is that the error has the minimum area in the dynamic process of the closed-loop system. It is one minimum of the multi-dimension phase space surface and yet it is difficult to obtain the analytical solution for a high order closed-loop system.12 Xue11 gives the performances, characteristic polynomials and their optimal coefficients of the first order to the sixth order standardization closed-loop system based on the criterion of minimum ITAE, which are derived by the experimental method. For the fourth order standardization closed-loop system, assuming its natural frequency is vn and its settle time of step response is ts, if its characteristic polynomial satisfies Ds4 = s4 + 2:1vn s3 + 2:4v2n s2 + 2:7v3n s + v4n
ð10Þ
ð13Þ
then the maximum overshoot of the closed-loop system’s step response, d, will be 1.9% and the standardizing time of the closed-loop system’s step response, vnts, will be 5.4.
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In accordance with equation (13), for the levitation DOF, to match a fourth order standardization closedloop system, a phase lead-lag controller can be designed as shown in equation (14) Gz (s) =
Kg (s + z1 )(s + z2 ) s(s + p1 )
ð14Þ
From equations (11), the plant transfer function of the levitation DOF is K0 Pz (s) = 2 ms + K
ð15Þ
then the characteristic polynomial of the designed closed-loop system is D4 = s4 + p1 s3 + ½(K + K0 Kg )=ms2 + ½Kp1 =m + K0 Kg (z1 + z2 )=ms + K0 Kg z1 z2 =m
Let D4 = Ds4 , thus equations (17) are established 8 p1 = 2:1vn > > < (K + K0 Kg )=m = 2:4v2n ½Kp1 + K0 Kg (z1 + z2 )=m = 2:7v3n > > : K0 Kg z1 z2 =m = v4n
ð16Þ
dynamics. Therefore four controllers need to be designed in all. The controllers of other DOFs can also be designed by the above method. After modifying and rounding slightly, the final phase lead-lag controllers designed for the investigated 6-DOF magnetically levitation stage with moving coils are shown in equations (18). They will be used in the following control system simulations and experiments 8 780, 000(s + 55)(s + 1) > > Gx, y (s) = > > s(s + 1200) > > > > 3, 000, 000(s + 160)(s + 1) > > > < Gz (s) = s(s + 2000) > 9000(s + 90)(s + 1) > > Gc, u (s) = > > s(s + 2000) > > > > > 2500(s + 55)(s + 1) > G (s) = : f s(s + 1200)
ð18Þ
Control system simulation ð17Þ
Then given a desired settle time ts, the natural frequency of the closed-loop system can be defined as vn = 5:4=ts . Solving equations (17), the parameters of the designed phase lead-lag controller shown in equation (14), namely Kg, z1, z2 and p1, can be derived. Because of the symmetries of structure and movement, the x- and y-direction moving DOFs have completely the same dynamics, and the rotation DOFs around the x- and y-axis also have completely the same
6-DOF motion decoupling simulation The simulation model of the complete control system is built based on MATLAB/Simulink. As shown in Figure 4, it consists mainly of four parts: plant dynamics, electromagnetic force/torque model, commutation of coil array and phase lead-lag controllers, plus input and output modules. In order to simulate the disturbance rejection performance, random disturbances, whose amplitudes are 0.001 N, 0.001 N, 0.001 N, 0.001 Nmm, 0.001 Nmm and 0.001 Nmm, respectively, are added to the inputs of dynamics. In addition, band-limited white noises, whose power spectrum densities are 100 nm2 and
Figure 4. Control system simulation model based on MATLAB/Simulink.
Proc IMechE Part I: J Systems and Control Engineering 226(7)
−3 x 10 2 1 0 −1 −2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 t(s) −3 x 10 2 1 0 −1 −2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 t(s)
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Figure 5. Closed-loop system responses when the levitation DOF is given 10 mm step signal at set-point z0 = 0.0 mm.
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10 mm, 10 mm, 10 mm, 10 mrad, 10 mrad and 10 mrad step signals in turn at set-point z0=2.0 mm. Figure 6 shows the controller force/torque outputs Wrc, the model force/torque outputs Wr and their errors. Figure 7 shows the output responses of all six DOFs. As seen from Figure 6, errors between Wrc and Wr are within 10213 m/rad. As seen from Figure 7, the output of each DOF is only the response to its own input. Summarized from the simulations above, relative to the input of 10 mm, the decoupling error of order 1 nm/
φ−torque error (N⋅mm)
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100 nrad2, are added in the feedback channel to simulate the measuring noises. First, a 10 mm step signal is given to the input of the levitation DOF (other inputs are given constant signals of 0) at set-point z0=0.0 mm. As can be seen from Figure 5, except for the response of the levitation DOF, the responses of the other five DOFs are within 6 2 nm or 6 2 nrad. Further, the x-, y- and z-direction moving DOFs, and the x-, y- and z-axis rotation DOFs, are given
−15 x 10 2 0 −2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 −15 x 10 5 0 −5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 −15 x 10 5 0 −5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 −13 x 10 2 0 −2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 −13 x 10 1 0 −1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 −13 x 10 1 0 −1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 t/s
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Figure 6. Closed-loop system controller force/torque outputs, model force/torque outputs and their errors when the x-, y- and zdirection moving DOFs, and the x-, y- and z-axis rotation DOFs, are given 10 mm, 10 mm, 10 mm, 10 mrad, 10 mrad and 10 mrad step signals in turn at set-point z0 = 2.0 mm.
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10 0 −5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 t(s) 10
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Figure 7. Closed-loop system responses when x-, y- and z-direction moving DOFs, and x-, y- and z-axis rotation DOFs, are given as 10 mm, 10 mm, 10 mm, 10 mrad, 10 mrad and 10 mrad step signals in turn at set-point z0 = 2.0 mm.
nrad is enough to be neglected. So, the 6-DOF motions are decoupled completely.
Levitation control simulation
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Equations (9) and (10) indicate that the model parameters of the levitation DOF, K and K0, are both functions of the control set-points, z0. Changing trends of K and K0 with the change of z0 are as shown in Figure 8. This means that the model of the levitation DOF is changeable when the mover is levitated at different gaps. In order to verify the stability of the closed-loop control system of the levitation DOF, letting z0=0.0 mm, 1.0 mm, 2.0 mm and 3.0 mm, respectively, four different plant models under different model parameters, K and K0, are obtained. These four models and the controller of levitation motion shown in equations (18) compose four closed-loop systems, respectively. Figure 9 shows
the 10 mm step responses of the levitation DOF in different plant model parameters when the control set-point z0=0.0 mm, 1.0 mm, 2.0 mm and 3.0 mm. It illustrates that the closed-loop control system of the levitation DOF is always stable in different plant model parameters depending upon different control set-points within levitation DOF travel of 0–3 mm, even though there are measuring noises. Further, the trajectory tracking performances of the levitation DOF are simulated by giving a sine input signal. Figure 10 shows the trajectory tracking result and tracking error at the set-point z0=1.0 mm when the levitation DOF is given a sine signal whose frequency is 2 Hz and amplitude is 1.0 mm. It shows that the given sine signal is tracked precisely while there is a time delay of about 0.02 s. The tracking error is within 6 30 mm after the time delay stage of trajectory. Figure 11 indicates that the other DOF motions have been decoupled
8 6 z0=0.0mm z0=1.0mm z0=2.0mm z0=3.0mm
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Figure 8. Changes of plant model parameters of the levitation DOF when the control set-point, z0, changes: (a) change of K; (b) change of K0.
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Figure 9. Closed-loop system step responses of the levitation DOF with different plant model parameters when the control set-point z0 = 0.0 mm, 1.0 mm, 2.0 mm and 3.0 mm.
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Figure 10. (a) Trajectory tracking result and (b) error when the levitation DOF is given a sine trajectory of 1.0 mm amplitude at the set-point z0 = 1.0 mm.
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Figure 11. Motions of the other DOFs when the levitation DOF is given a sine trajectory of 1.0 mm amplitude at the control setpoint z0 = 1.0 mm.
with the levitation DOF motion and there are decoupling errors of about 6 30 nm/nrad.
Experiments and discussion Control system experiments The configured test platform consists of a magnetically levitated stage with moving coils (the main parameters and dimensions of which are shown in Table 1), a DSP board (TMS320C6713, SEED-XDS510Plus Emulator), measuring sensors, an analog-to-digital board, a digital-to-analog board and 20 single phase power amplifiers. The controller algorithm and coil array commutation algorithm are operated on the DSP. After digital-to-analog conversion, the current output of the DSP is given to 20 power amplifiers and their output ports are connected to 20 coils. The three positions and three orientations of the mover are measured by seven eddy sensors and their measured data are
transferred back to the DSP by analog-to-digital conversion. Figure 12 is a photograph of the test platform (except for the electronic and electrical parts). Figure 13 shows the main mechanical setup of the test platform. A mounting rack, which is used to mount and adjust the three z-direction sensors, is fixed to a marble beam. On the four corners of the mounting rack are four through-holes through which four long bolts are bolted to the mover. The diameter of the throughholes is 2 mm larger than the diameter of the long bolt so that the mover can move in the z-direction (from 0.25 mm to 2.25 mm), move in x- and y-directions within short travel (6 1 mm) and rotate around the x-, y- and z-axes in small angle (maximum 6 0.1 rad around x- and y-axes and 6 0.017 rad around z-axis). The placement of seven eddy sensors is shown in Figure 14. From the readings given by these sensors, displacements of the x-, y- and z-directions can be calculated precisely, and angles around the x-, y- and
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Table 1. Main parameters and dimensions of the investigated magnetically levitated stage with moving coils. Quantity
Symbol
Dimension
Permanent magnet array stage Permanent magnet array
– –
Magnet-pole pitch Residual magnetism Translator mass Coil array stage Number of coils Number of coil turns Coil resistance Coil inductance
tn M0 M – – N R L
245 mm 3 245 mm 20 3 20 magnet poles 10.6 mm 1.23 T 0.92 kg 155 mm 3 155 mm 435 526 1 0.66 0.05 O 0.1 mH
Figure 14. Placement of the seven eddy sensors.
Figure 12. Photograph of the test platform (except for the electronic and electrical parts).
Figure 13. Main mechanical setup of the test platform.
z-axes can also be calculated under the condition of small angle rotation, as shown in equation (19) x = (x1 + x2 )=2 y = (y1 + y2 )=2 z = (z1 + z2 + z3 )=3 c = ½(z1 + z2 )=2 z3 =L1 u = (z2 z1 )=L2
ð19Þ
f = ½(x2 x1 )=L1 + (y2 y1 )=L2 =2
The standard measuring objects of the three z-direction sensors, which are ferromagnetic, are pasted on the
appropriate positions of the mover top. After calibrating, these three sensors can reach 0.25–2.25 mm linear measuring range. The four x- and y-direction sensors are mounted on the appropriate positions outside the stator. Different from the three z-direction sensors, they measure the side plane of the mover directly rather than the ferromagnetic standard measuring objects, which would be too near to the magnetic field. After calibrating, these four sensors can reach only 0.25–1.25 mm linear measuring range. Within linear measuring range, all seven sensors have non-linearity of less than 0.85%. These seven sensors have standard sensitivity of 5 V/ mm and the sensitivity deviation is 1.21%. The coil array commutation algorithm and the discretized controller algorithm are programmed in an interrupt routine in C language. The closed-loop sampling frequency is at 1 kHz currently. On the configured test system, the positioning and the trajectory tracking of the levitation DOF are experimented primarily, so that the effects of decoupling and controllers can be observed and analyzed. Figure 15(a) and (b) shows respectively the measured positioning result and the positioning error of the levitation DOF when 500 mm stair signals are given intermittently. It indicates that the mover can position accurately whereas there is a positioning ripple of about 6 50 mm. Figure16 shows the coupled motions of the other DOFs. It indicates the 6-DOF motions are decoupled but there are still decoupling errors of about 6 100 mm and 6 2 mrad. Figure 17 shows a planned (acceleration, velocity, displacement) trajectory of the levitation DOF, in which the mover first ascends from 0 mm to 2 mm and then descends from 2 mm to 0 mm. Figure 18(a) and (b) shows respectively the trajectory tracking result and tracking error. It indicates that the mover can track the planned trajectory whereas there is a tracking error of about 6 0.25 mm. Figure 19 shows the coupled motions of the other DOFs. It indicates that the 6-DOF motions are decoupled, whereas there are still decoupling errors of about 6 50 mm and 6 1 mrad.
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Figure 16. Decoupling errors of the other DOFs due to the z-direction stair positioning.
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Figure 17. Planned trajectory of the levitation DOF.
Discussion Decoupling of 6-DOF motions is the primary goal of our investigations. As can be observed from the simulation results, the influences of step input signals on the other DOFs are below nano-meter/radian order and
the influences of the reference trajectory of the levitation DOF on the other DOFs are below micro-meter/ radian order. From the experiment results it is observed that there are decoupling errors of order about dozens of micro-meters/radians, relative to 2 mm positioning
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Figure 18. (a) Trajectory tracking result and (b) trajectory tracking error of the levitation DOF.
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Figure 19. Decoupling errors of the other DOFs due to trajectory tracking of the levitation DOF.
or tracking travel of the levitation DOF. Simulation and experiment results can illustrate the 6-DOF motions are decoupled whereas there is deviation between simulation results and experiment results. This deviation may originate from the mismatch between the mathematical model and the physical structure of the stage. Manufactured first time and assembled by hand, the assembly dimensions of the stage and the dimensions of stage parts, such as the winding and assembling of 20 coils and the assembling of a large number of permanent magnets, are really coarse. Thus the electromagnetic force/torque model and the commutation of coil array match the design requirements rather than the real mechatronic structure. Control of levitation motion is another goal of our investigations. No matter whether observing from simulation results or experiment results, the phase lead-lag controllers based on the minimum ITAE criterion have achieved good results; whereas there exist non-negligible positioning or tracking ripples of micro-meter/radian order in simulations and of dozens
of micro-meters/radians order in experiments. The origin of this result can be traced to two aspects. On the one hand, the eddy sensors adopted currently have high amplitude measuring noises and any signals in the control loop have not yet been filtered; on the other hand, the present closed-loop sampling frequency is at 1 kHz, which is a bit low to reject the high frequency disturbances and noises.
Conclusions The 6-DOF motion decoupling and levitation DOF control of a magnetically levitated stage with moving coils have been investigated in this paper. Commutation of coil array is used to decouple the 6DOF motions. This commutation is derived inversely from the electromagnetic force/torque model based on the Lorenz force law. In essence, this commutation is to map the six force/torque components corresponding to the 6-DOF motions to the 20 currents of the coil array. The 6-DOF plant dynamics is modeled to design the controllers. The dynamics of the levitation DOF is
886 modeled especially to compensate the gravity, which is of high importance to the magnetically levitated stage. Based on these dynamic models, the phase lead-lag controllers are designed by using the criterion of minimum ITAE. This criterion is very suitable for, and beneficial to, position tracking control of the magnetically levitated stage. The coil array commutation in combination with the designed controllers is used to decouple the 6-DOF motions and to control the levitation DOF motion of the stage. Both the simulations of the complete control system in MATLAB/Simulink and the experiments on a real DSP-centered test platform demonstrate that the 6-DOF motions are decoupled and the levitation DOF motion is controlled. However, there are still decoupling errors and control errors. Perhaps better matching between the theoretical model (coil array commutation algorithm and electromagnetic force/torque model) and the practical mechatronic structure, the increase of the closed-loop sampling frequency and the introduction of a filter may improve the decoupling effects and the control performances effectively. Moreover, the horizontal-direction motions still need to be controlled. These will be studied further in our next work. Funding This work was supported in part by the National Basic Research Program (973 Program) of China [grant number 2009CB724205] and in part by the National High Technology Research and Development Program (863 Program) of China [grant number 2009AA04Z148]. References 1. Kim WJ, Trumper DL and Jeffrey HL. Modeling and vector control of planar magnetic levitator. IEEE Trans Ind Applic 1998; 34(6): 1254–1262. 2. Kim WJ. Nanoscale dynamics, stochastic modeling, and multivariable control of a planar magnetic levitator. Int J Contr Autom Syst 2003; 1(1): 1–10. 3. Yu H and Kim WJ. Controller design and implementation of six-degree-of-freedom magnetically levitated positioning system with high precision. Proc IMechE, Part I: J Systems and Control Engineering 2008; 222(8): 745–756. 4. Compter JC. Electro-dynamic planar motor. Precis Eng 2004; 28(2): 171–180. 5. Van Lierop CMM, Jansen JW, Damen AAH, et al. Control of multi-degree-of-freedom planar actuators. In: 2006 IEEE international conference on control applications (CCA 2006), Munich, Germany, 4–6 October 2006, pp.2516–2521. New York: IEEE. 6. Van Lierop CMM, Jansen JW, Damen AAH, et al. Model-based commutation of a long-stroke magnetically levitated linear actuator. In: Conference record of the IEEE industry applications conference 41st annual meeting, Tampa, USA, 8–12 October 2006, pp.393–399. New York: IEEE.
Proc IMechE Part I: J Systems and Control Engineering 226(7) 7. De Boeij J, Lomonova EA and Vandenput AJA. Modeling ironless permanent-magnet planar actuator structures. IEEE Trans Magn 2006; 42(8): 2009–2016. 8. Min W, Zhang M, Zhu Y, et al. Analysis and optimization of a new 2-D magnet array for planar motor. IEEE Trans Magn 2010; 46(5): 1167–1171. 9. Jansen JW, Van Lierop CMM, Lomonova EA, et al. Modeling of magnetically levitated planar actuators with moving magnets. IEEE Trans Magn 2007; 43(1): 15–25. 10. Zhang S, Zhu Y, Yin W, et al. Coil array real-time commutation law for a magnetically levitated stage with moving-coils. J Mech Eng 2011; 47(6): 180–185. 11. Xue D. Simulation and computer aided design of control system. 2nd ed. Beijing: Machine Press, 2009. 12. Hu S. Automatic control principle. 5th ed. Beijing: Science Press, 2007.
Appendix Notation fx, fy, fz Fx, Fy, Fz g i i^ i0 Ix, Iy, Iz K K0 Kg m p1 q t ts Tx, Ty, Tz Wr(q) Wr Wrc x, y, z z^ z0 z1 z2
control force output (N) model force (N) acceleration due to gravity (m/s2) current vector (A) current variable vector (A) gravity compensation current vector (A) moment of inertia (kgm2) stiffness coefficient (N/m) coefficient (–) gain (–) mass (kg) pole point of transfer function (–) position and orientation vector (m, rad) time (s) settle time (s) model torque (Nm) force/torque vector (N, Nm) model force/torque vector (N, Nm) control output vector (N, Nm) position (m) control variable (m) control set-point (m) zero point of transfer function (–) zero point of transfer function (–)
d D4
overshoot (%) characteristic polynomial of the fourth order closed-loop system (–) characteristic polynomial of the fourth order standardization closed-loop system (–) force/torque model matrix (N/A, Nm/A) pseudo-inverse of force/torque model matrix (A/N, A/Nm) magnet-pole pitch (m) control torque output (Nm) orientation angle (rad) natural frequency of the closed-loop system (rad/s)
Ds4
G(q) G–(q) tn tx, ty, tz c, u, f vn
