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Augmentation of Propulsion Based on Coil Array Commutation for Magnetically Levitated Stage Yu Zhu, Shengguo Zhang, Haihua Mu, Kaiming Yang, and Wensheng Yin Institute of Manufacturing Engineering, Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, China This paper focuses on augmenting the propulsion via commutation of coil array for the long-stroke magnetically levitated stage with moving coils, whose mechatronics structure have been defined. The used commutation of coil array is based on the analytical force/ torque-decomposing model of the stage and it is characterized by bounding the coil currents. Through this current-bounded commutation, the 1-norm of commutated coil current vector is increased so that the propulsion can be augmented, and simultaneously the infinite norm of commutated coil current vector is limited so that the amplitudes of commutated coil currents are not beyond the capacity of selected coil power amplifiers. By the investigation example of a long-stroke magnetically levitated stage with moving coils, it is theoretically verified that the propulsion (acceleration) can be augmented by 125% as well as the commutated coil currents can be kept within the capacity of selected coil power amplifiers, 3 A. The study results indicate that the propulsion of a magnetically levitated stage can be augmented via current-bounded commutation of coil array rather than via reconfiguring the mechatronics structure of stage or reselecting coil power amplifiers of larger capacity. Index Terms—Augmentation of propulsion, capacity of coil power amplifiers, commutation of coil array, current bounding, force/ torque-decomposing model, magnetically levitated stage.
I. INTRODUCTION
M
AGNETICALLY LEVITATED STAGES (MLS) as alternatives to air bearing stages constructed of stacked linear motors have been developed in recent years. This is largely due to the fact that the MLS can operate in vacuum, for example in extreme-ultraviolet lithography (EUVL) equipment, or nanoimprint lithography equipment. Whereas, this alternative still requires that the MLS could obtain good dynamic performances and especially could reach higher propulsive acceleration and velocity. For obtaining higher propulsive acceleration and velocity, more propulsion is needed. Some hardware measures of reconfiguring the mechatronics structure such as permanent magnet array of stronger residual magnetism, coil power amplifiers (CPA) of larger capacity, and the redesign of coil array can be used to accomplish this goal. However, they are quite costly. In contrast, a software measure—the commutation of coil array, which is indispensible to decouple the force components and the torque components corresponding to the multiple degrees of freedom (DOF) motion of the MLS [1], [2], may be used to attain the same goal. In literature [3]–[6], DQ0-decomposition (or Park-transformation) was used to commutate coils for decoupling the forces and the torques while controlling the MLS. By this method, however, it is difficult to take into account the additional torques originating from the distribution force along the turn width and the turn thickness of the coil, which may lead to the torque errors [4], [7], [8]. In addition, the result of this commutation is unique so that it is difficult to augment the propulsion by this method. In literature [2], [9], [10], Lierop et al. put forward a direct Manuscript received June 09, 2011; accepted August 19, 2011. Date of publication October 03, 2011; date of current version December 23, 2011. Corresponding author: S. Zhang (e-mail:
[email protected]). Digital Object Identifier 10.1109/TMAG.2011.2166559
wrench-current decoupling method including smooth switching of coil currents, which enables combined long-stroke propulsion and active magnetic bearing control of an ironless multi-DOF moving magnet actuator. This method can also be used to commutate a MLS with moving coils, whereas each coil of the stage has to be excited with a single phase power amplifier. Dominated by least squares principle, the coil currents are commutated optimally in the sense of minimum global heat-loss of coil array. For a MLS of definite mechatronics structure, the commutated coil currents must be increased synchronously for generating higher propulsion. When commutated currents of some coils exceed the capacity of CPA, these CPA will be saturated or even protectively cut off in some spots of workspace, especially in the spots which are defined by the worst-case acceleration (or force) constraint of the MLS [10]. This may cause the errors of all generated forces/torques, and cause the reduction of motion precision, and even cause the control failure of multi-DOF motion. It seems that the problem could be solved by the reselection of CPA of larger capacity. Yet, it is not as simple as that. The local over-heat of some coils, the redesign of coil array, the modification of cooling system, and even the reconfiguration of whole mechatronics structure may have to be taken into consideration. The solutions of all these considerations will greatly increase the total development cost of the MLS. In this paper, a 6-DOF and long stroke MLS with moving coils is investigated. Our investigations attempt to augment the propulsion through the commutation of coil array, under the condition that neither reselects CPA of larger capacity nor reconfigures the mechatronics structure of MLS. In Section II, the basic principle of propulsion augmentation is analyzed and the feasibility of propulsion augmentation is introduced based on the force/torque-decomposing model of the MLS firstly. Then the current-bounded commutation, which is an improvement of direct force/torque-current decoupling, is described. In Section III, an example of investigated MLS is formulated in detail to theoretically verify the validity of propulsion augmentation. Section IV concludes the paper.
0018-9464/$26.00 © 2011 IEEE
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denotes the force/torque vector acting on the translator, and is the position and orientation vector of the translator in fixed coordinate system, and Fig. 1. Basic control configuration including the commutation (algorithm) of coil array and the force/torque-decomposing model of investigated MLS motion system.
II. THE BASIC PRINCIPLE AND METHOD OF PROPULSION AUGMENTATION
is the coil current vector, and
(2)
A. The Basic Principle of Propulsion Augmentation The MLS has 6-DOF movements in Euclid space. It has to be controlled in 6-DOF because of active magnetic bearing, although its translator can only move over relatively long stroke in the -plane [2], [9]. In order to provide adequate energy for levitation and propulsion, the MLS is often over-actuated, i.e., the number of coils is generally larger than the number of . Each coil current contributes to the three force DOF: components and three torque components corresponding to the 6-DOF motion. Thus, the 6-DOF motions of the MLS are coupled fundamentally. Because the quasi-static magnetic field of permanent magnet array (For example, the Halbach permanent magnet array) is unalterable anyway, the current distribution of coil array has to be introduced to decouple the force components and the torque components corresponding to 6-DOF while controlling the MLS [1], [2]. This current distribution is called the commutation of coil array. It is based on the force/torque-decomposing model of the MLS. Therefore, a complete control system model of MLS necessarily includes the commutation of coil array and the force/torque-decomposing model besides the plant dynamics and the controllers. Fig. 1 shows the basic control configuration of MLS motion control system, where CPA and represent the denotes the coil power amplifiers, force/torque-decomposing model (matrix) and the commutation (algorithm) of coil array, respectively. The force/torque-decomposing model of the MLS is based on the Lorenz force law. For a Halbach permanent magnet array, neglecting all the higher order harmonics, its first order harmonic magnetic flux density distribution in fixed coordinate system can be represented analytically. For a tightly wound rectangular coil, neglecting its corner segments, it can be equivalent to four current-carrying surfaces when commutated [11]. Because the coil is ironless, the acting force/torque on the translator, which is produced by each coil in Halbach permanent magnet field, can be integrated using Lorenz force law. Then the three forces and three torques acting on the translator of the stage, which are defined at the center of mass and around the inertia axes of the translator, can be calculated by force/torque Superposition Principle of rigid body. This force/torque-decomposing model can be constructed and represented as follows: (1) where
is the force/torque decomposing matrix of dimension, whose six elements of column represent the three force components and the three torque components produced by the coil commutated per unit current (1 A) when the mass center of translator is at the position and orientation [11]. Based on the conservation law of energy, the power input of system equals the power output of system. For the MLS, its power input includes the magnetic field power of permanent magnet array and the electric power of coil array. Its power output includes the mechanical power, the heat-loss (or ohmic heat) of coil resistance, and the power storage of coil inductance (neglecting the eddy current loss). So the power conservation equation of MLS can be represented as follows:
(3) On the right of (3), the first component is the mechanical power output, the second component is heat-loss (or ohmic heat) of coil resistance, and the third component is the power storage of coil inductance. The magnetic field power input is unalterable so we can only alter the electric power input of coils for increasing the mechanical power output. Observing from (3), the mechanical power output depends on the 1-norm of coil current vector and the heat-loss depends on the 2-norm of coil current vector:
Whereas, the capacity of CPA is associated with the infinite norm of coil current vector:
ZHU et al.: AUGMENTATION OF PROPULSION BASED ON COIL ARRAY COMMUTATION
If we want to augment the propulsion, levitation or torques of MLS:
the 1-norm of commutated coil current vector should be increased:
In accordance with the force/torque-decomposing model (1), should satisfy following the commutated coil current vector equations:
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be divided into two sets: coil set 1 and coil set 2, whose current and : vector is denoted by
And
meet the demand (7) and (8), respectively: (7) (8)
, let its ampliFor the commutated currents of coil set 2, tude of each element equal to the capacity of CPA, , and let its sign of each element unchanged: (9)
(4) Simultaneously, for assuring that all CPA are not saturated, the infinite norm of commutated coil current vector (maximum current amplitude of all coils) must not be larger than the ca: pacity of CPA, (5) Assuming that the total number of commutated coils is , and the number of coils whose commutated current amplitudes , is . If , there exceed the capacity of CPA, , there is unique sois no any solution for (4). If , there always are infinitely many lution for (4). If solutions for (4). In order to provide adequate energy for stage, is usually several times of number of DOF, 6 (it is 20 for is usually a small number because the investigated MLS). number of coils, which are involved in the worst-case acceleration constraint, is usually small (it involves two coils at most for investigated MLS). So it is feasible to find an appropriate solution among infinitely many solutions of (4) that meets (5) at the same time. To obtain this one appropriate solution, we try the current-bounded commutation, by which the commutated coil currents can generate desired forces and torques and any coil current can be bounded within a limit, as well as the global heat-loss of all coils can be minimized as much as possible. B. Current-Bounded Commutation of MLS Based on the reflective generalized inverse (or pseudo-inverse) of force/torque decomposing matrix , we can in the sense of least squares, derive the initial current vector which can globally guarantee the minimum heat-loss of commutated coil array:
(6) Yet it is quite possible that
In this case, we compare each element of the initial current vector with the capacity of CPA, , so that all coils can
For the commutated currents of coil set 1, following equation:
, it satisfies the
(10) and are two blocks of matrix where respond to coil set 1 and coil set 2, respectively:
From (10), well:
and cor-
can be solved, in the sense of least squares as (11)
So the whole commutated current vector
is
(12) These commutated currents can generate the desired forces/ torques as well as their amplitudes are not larger than the ca. pacity of CPA, III. THE EXAMPLE OF PROPULSION AUGMENTATION BASED ON THE CURRENT-BOUNDED COMMUTATION OF MLS The investigated MLS is a configuration after optimal structure design [12]. Its stator is a kind of Halbach permanent magnet array [1], [2], [4], and its translator contains 20 ironless coils arranged in definite order. Fig. 2 (a), (b), and (c) show the concept diagram of the stage, the top view of the (partial) stator, and the bottom view of translator, respectively. The related parameters and dimensions of investigated MLS system are shown in Table I. The force/torque-decomposing model and model-based commutation of MLS have been simulated and verified in some literature [8], [10], [11]. The force/torque-decomposing model and coil array commutation of investigated MLS with moving coils
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Fig. 2. Investigated magnetically levitated stage with moving coils. (a) Concept diagram of the stage. (b) Top view of (partial) stator. (c) Bottom view of translator.
Fig. 3. (a) The desired propulsion (9N) and levitation (9N), and (b) the coil currents commutated by the direct wrench-current decoupling method for obtaining propulsive acceleration of 1 g.
TABLE I MAIN PARAMETERS AND DIMENSIONS OF THE INVESTIGATED STAGE
as represented respectively by (1) and (4) also have been simulated [13] and specially verified via abundant experiments [14] by a multidimensional force/torque sensor on a DSP-centered test system (TMS320C6713 DSP, A/D, D/A, and CPA). So, in this paper we only present the theoretically commutated currents and the theoretical forces/torques actuated by these commutated currents instead of the practical force/torque output measured by the multidimensional force/torque sensor. For a MLS system, its needed propulsion depends on the demand of motion profile. A most likely motion profile is that the translator moves straight from one point to another in horizontal direction. So we desire the translator is always levitated at the clearance of 2 mm (which is the maximum clearance of invesin -direction of fixed tigated MLS) and moves from 0 to coordinate system at the definite acceleration. Above all, we desire the translator can derive the propulsive acceleration of 1 g. By the direct wrench-current decoupling method, we can obtain the commutated coil currents. Fig. 3(a) and (b) show the desired propulsion (9 N) and levitation (9 N)
(other force and torques are zero), and the commutated coil currents, respectively. We can observe from Fig. 3(b) that the commutated currents of some coils are larger than 3 A in somewhere of the travel from 0 to . When commutated currents of some coils exceed 3 A, the coil currents of these coils will still be 3 A (as Fig. 4(a) shows) because of the saturation of CPA. They result in not only the errors of propulsion and levitation, but also the obvious errors of other torques (as Fig. 4(b) shows). These are totally unacceptable for the control of MLS. In fact, for assuring the amplitude of commutated coil currents not to exceed the capacity of CPA, 3 A (as Fig. 5(a) shows), the desired maximum propulsion can only be 5.6N (as Fig. 5(b) shows) and the obtained maximum acceleration can only be about 0.62 g. In order to augment the propulsion and obtain the acceleration of 1 g without reselecting CPA of larger capacity, the current-bounded commutation is to be tried. Fig. 6(a) shows the currents by this commutation and Fig. 6(b) shows the theoretically generated -direction propulsion and levitation ( -direction propulsion and other torques all are 0). From Fig. 6, we can observe that the coil currents, whose amplitudes are larger than 3 A, are bounded within 3 A as well as the currents of other coils have been redistributed. Thus, the desired propulsion (9N) and levitation (9N) can be generated whereas the maximum current amplitude is not larger than the capacity of CPA. Furthermore, two sets of infinite norms are compared so that the maximum value, which the propulsion can be augmented to, can be defined. One set is the infinite norms of current vectors when the coil array is commutated by the direct wrench-current
ZHU et al.: AUGMENTATION OF PROPULSION BASED ON COIL ARRAY COMMUTATION
Fig. 4. (a) The coil currents because of the saturation of CPA and (b) generated propulsions, levitation, and torques.
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Fig. 6. (a) Coil currents commutated by current-bounded commutation and (b) theoretically generated propulsion (9N), and levitation (9N).
Fig. 7. The infinite norms of commutated current vectors for generating different propulsions in the stage’s valid workspace of ( 2 ) ( 2 ) when the coil array of MLS is commutated by the direct wrench-current decoupling (nobound commutation) and by the current-bounded commutation, respectively.
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Fig. 5. (a) Coil currents commutated by the direct wrench-current decoupling and (b) generated maximum propulsion, levitation when the amplitudes of commutated coils currents are not larger than the capacity of CPA, 3 A.
decoupling (no-bound commutation) and another set is the infinite norms of current vectors when the coil array is commutated by the current-bounded commutation, both for the purpose of generating different propulsions in the stage’s valid workspace . This relation between the propulsion and of
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the maximum amplitude of coil currents in the whole valid workspace is shown in Fig. 7. Observing from Fig. 7, so far as the stage’s valid workspace is concerned, the propulsion can be augof mented from 5.6N up to 12.6N (increase of 125 percent) by the current-bounded commutation. This implies the propulsive acceleration can be increased from 0.62 g to 1.4 g under the con, is 3 A. dition that the capacity of CPA, Of course, the augmentation of propulsion is not unlimited. The propulsion can only be augmented up to 12.6N and the corresponding acceleration can only be increased to 1.4 g even if the current-bounded commutation is used. This is because there is no any solution among infinitely many solutions of (4) that meets (5) at the same time.
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commutated coil current vector has been bounded. The propulsion (or propulsive acceleration) is augmented due to the increase of 1-norm of commutated coil current vector. Simultaneously, the amplitude of commutated coil currents is not beyond the capacity of selected coil power amplifiers because the infinite norm of commutated coil current vector is bounded. After the propulsion is augmented, the horizontal motion profile, which is associated with the propulsion, can be planned within the maximum acceleration what the augmented propulsion can produce. Yet the motion profile is still associated with the implementation of controllers. So our next work will be the implementing of control in combination with the currentbounded commutation of coil array and the further verifying of the planned motion profile. ACKNOWLEDGMENT This research was supported in part by the National Basic Research Program (973 Program) of China (2009CB724205), in part by the National High Technology Research and Development Program (“863”Program) of China (2009AA04Z148), and in part by the independent research program of the State Key Laboratory of Tribology of China (SKLT08B04). Fig. 8. (a) Coil currents commutated by current-bounded commutation and (b) generated maximum propulsion (12.6 N), levitation (9 N), and torques (all are zero).
Fig. 8(a) shows the coil currents commutated by the currentbounded commutation for generating the maximum propulsion in one travel from to (this travel has included the spots what the worst-case acceleration defines). Fig. 8(b) shows the theoretically generated propulsions ( -direction 12.6 N, -direction 0 N), levitation (9 N), and torques (all 0 N.mm). As can be seen from Fig. 8, the desired maximum propulsion, 12.6 N, is generated, and simultaneously the amplitudes of coil currents are not yet larger than the capacity of CPA, 3 A. Due to the augmentation of propulsion, higher propulsive accelerations, which are within the maximum acceleration what the augmented propulsion can produce, can be selected in planning the motion profile. This will facilitate the motion profile planning and promote the working efficiency of MLS greatly. IV. CONCLUSION It is important to obtain higher propulsive acceleration and velocity for the MLS. The propulsion (or propulsive acceleration) can be augmented through the coil array commutation rather than through reselection of CPA of larger capacity or reconfiguration of mechatronics structure of the MLS. The coil array commutation used to augment propulsion in this paper could be called current-bounded commutation. In essence, it assigns the currents of coils, whose current amplitudes are larger than the capacity of CPA, to be the capacity bound of CPA, while redistributes the currents of other coils in the least squares sense. By the current-bounded commutation, the 1-norm of commutated coil current vector can be increased but the infinite norm of
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Haihua Mu received the B.S. and Ph.D. degrees in mechanical engineering from Huazhong University of Science & Technology in 2004 and in 2008, respectively. He is currently an Assistant Researcher at the Institute of Manufacturing Engineering of Department of Precision Instruments and Mechanology, Tsinghua University, Beijing, China. His research interests include ultra-precision motion control technology of micro-electronics equipments, CNC technology, and CAD/CAM.
Yu Zhu received the B.S. degree in radio electronics from Beijing Normal University, Beijing, China, in 1983, and the M.S. degree in computer applications and the Ph.D. degree in mechanical design and theory from China University of Mining & Technology in 1993 and 2001, respectively. He is currently a Professor at the Institute of Manufacturing Engineering of Department of Precision Instruments and Mechanology, Tsinghua University, Beijing, China. His research interest covers parallel mechanism and theory, two photon micro-fabrication, ultra-precision motion system and motion control.
Kaiming Yang received the Ph.D. degree in mechanical engineering from Tsinghua University, Beijing, China, in 2005. He is currently an Assistant Researcher at the Institute of Manufacturing Engineering of Department of Precision Instruments and Mechanology, Tsinghua University, Beijing, China. His research interests include precision motion control technology, CNC technology.
Shengguo Zhang received the B.S. degree in mechanical design and manufacturing from Northwest A&F University, Xi’an, China, in 1990 and the M.S. degree in mechanical engineering from Tsinghua University, Beijing, China, in 2007. He is currently working toward the Ph.D. degree in mechanical engineering at Tsinghua University, Beijing, China. His research interests are mechatronics system model and precision motion control.
Wensheng Yin received the B.S. degree in mechanical engineering from Tsinghua University, Beijing, China, in 1993. He is currently an Associate Professor at the Institute of Manufacturing Engineering of Department of Precision Instruments and Mechanology, Tsinghua University, Beijing, China. His research interests include precision motion control technology and CNC technology.
