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Electromagnetic Actuators Featuring Multiple Degrees of Freedom: a Survey P. Bolognesi, O. Bruno, A. Landi, L. Sani, L. Taponecco Department of Electric Systems & Automation - University of Pisa Via Diotisalvi 2, 56122 Pisa, Italy - tel.: +39-050-2217300, fax +39-050-2217333 e-mail:
[email protected] Abstract — Nowadays, advanced applications of motion control often require a coordinated management of several mechanical degrees-of-freedom (DoF). This is conventionally achieved by using suitable mechanical devices and a separate actuator per each DoF. Electric actuators able to directly manage multiple DoF may provide appreciable benefits in terms of performances, volume, weight and cost. This paper aims to provide an overview of several configurations of multi-DoF electromechanical actuators considered in the recent technical literature and presently under study.
I.
INTRODUCTION
The requirements for sophisticated motion control are steadily increasing in several advanced application fields, such as robotics, tooling machines, pick-and-place systems etc. In such ambits, complex motion trajectories are often desired involving several mechanical DoF, which must be properly managed developing suitable torques/forces. Excluding micro-systems, presently in such applications single-DoF actuators are usually employed, each charged to manage 1 of the mechanical DoF of the system. The elementary motions developed by such actuators are then combined by means of suitable mechanical devices. Since the performances, volume, weight, cost and maintenance requirements of similar systems are rather relevant, appreciable benefits might be expected from adopting actuators able to inherently manage multiple DoF. This paper aims to provide a brief overview of the main typologies of electromagnetic actuators featuring a multiDoF structure, recently investigated in the technical literature. In particular, 3 classes of machines will be considered: planar motors; rotary-linear motors; spherical motors. For each of these typologies, the most common or promising machine structures will be synthetically described recalling their main characteristics.
II.
movers, which may be independently operated to simultaneously manage more tasks: this is particularly useful when relatively short strokes are required, such as in semiconductor manufacturing processes.
II.1 - MULTI-UNIT STEP MOTOR The stator of a planar step motor, usually named "platen", consists in a bulk ferromagnetic rectangular plate featuring a regular grid of small identical teeth having square section, separated by air slots having the same width (e.g. fig.1) with a resulting pitch in the order of 1 mm. The mover consists in a payload tray equipped with at least 2 basic units, often named "forcers", attached below it in orthogonal directions. The forcers usually consist in stators of linear hybrid step motors featuring the same tooth pitch as the platen (e.g. fig. 3). When the 2 structures are properly aligned, the complete machine results then able to develop forces in both versa along both directions, allowing to move the payload over the workplane. In most cases, the machine is operated horizontally: a pneumatic closed-loop controlled suspension system balances the weight of the mover, the vertical load force and the magnetic attraction between platen and forcers. This way, the gap between stator and mover, possibly as narrow as 10 µm, is maintained by ejecting compressed air from the mover towards the platen through suitable nozzles. As the machine size is usually fairly small, flexible cables and tubes are used to provide the forcers with the electric and pneumatic supply. By using 4 forcers attached below the payload tray in a loop arrangement (e.g. fig. 2), it becomes also possible to develop of a net torque by separately regulating the force generated by each forcer. This permits to rotate the mover around the orthogonal -vertical- axis,
PLANAR ACTUATORS
Planar motors, also called surface motors, are machines able to translate on a plane, usually kept horizontal, and possibly also to rotate around the orthogonal - usually vertical - direction, managing so 2+1 DoF. They may be usefully employed for precision positioning in various manufacturing systems. In fact, the first multi-DoF machine developed into an industrial product was the Sawyer planar step motor. It was conceived for precision high-speed positioning tasks, even in open-loop mode, under limited passive load, as required in the semiconductors manufacturing industry. Presently, such type of applications still constitutes the most diffused example of actual employ of multi-DoF machines. Moreover, as a general rule planar machines employing passive stators may be equipped with multiple active
Figure 1. Toothed stator plate for planar step motor
Figure 2. Loop arrangement of 4 forcers below payload tray
Figure 3. Cross section of a forcer in planar step motor [1]
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although within a small angular range; anyway, such additional possibility is valuable in precision positioning applications. The specific force density of these machines is fairly good; anyway, it may be improved by adopting composite stator structures including a conventional backiron plate to high-saturation FeCo alloy tooth (e.g. [1]). The step nature of such machines permits to operate them in the classical open-loop mode, achieving also a detent force, due to permanent magnets, possibly useful for position-keeping at null currents. Anyway, the adoption of precise sensors, such as laser interferometers, joined to micro-stepping control techniques, permits to obtain excellent positioning performances down to sub-micron range. On the other hand, the fine tooth pitch makes such machines less suited to perform high-speed travels as often required for initial fast raw positioning.
II.2 - MULTI-UNIT INDUCTION MOTORS The same basic principle above described, i.e., using elementary stators of linear motors assembled in loop fashion below the mover plate and facing a unique common stator, may be applied straightforwardly also to realize induction machines. In fact, in this case the stator may be constituted by a homogeneous conductive plate surrounding a bulk ferromagnetic plate, whereas the mover may be equipped with 4 separate sub-stators of 3-phase linear machines. Alternatively, since in this case no preferential direction exists on the stator plane (except along the borders), the concept may be indeed generalized to a polygonal or annular arrangements of the active parts of the mover, possibly using a larger number of elementary 3-phase windings (e.g. fig. 4). In fact, when these basic units are suitably independently regulated, such machine becomes able to actually provide a full 3-DoF motion capability along the whole plane. Moreover, the absence of anisotropy and permanent magnets eliminates any cogging effect. Nevertheless, augmenting the number of separate windings increases accordingly the complexity and cost of
Figure 4. Motion regulation in an annular induction motor [2]
Figure 5. Magnetization patterns for planar motors [3]
Figure 6. Layout of 4-units brushless planar motor [3]
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the supply system, as a separate inverter is required per each winding. Moreover, the flexible connections used to feed the mover limit the actual motion capability. This problem may be overcome by swapping the roles of mover and stator, though notably increasing the inertia of the moving part thus worsening the dynamic response. In any case, the specific performances developed by induction machines are rather low, especially when mechanical bearings are used requiring larger air-gap values.
II.3 - MULTI-UNIT BRUSHLESS MOTORS The multi-unit basic principle above described can be also applied to realize brushless machines. In this case, the stator consists in a large layer of permanent magnets arranged according to one of the various schemes proposed (e.g. fig. 5), determining a regular checkerboard-like grid of alternating polarities perpendicular to the plane with uniform pole pitches along the 2 main directions. The magnets are usually supported by a bulk ferromagnetic plate acting as magnetic yoke, although this last function would be ideally not necessary for Hallbach-like arrays. The basic units, arranged in square loop fashion, consist in elementary yet complete stators of 3-phase linear machines featuring the same pole pitch as the magnets array, yet with a smaller overall length. Usually, the structure of the basic units includes simple 3-poles ferromagnetic slotted cores made up of laminations hosting 3 non-overlapping coils whose pitch equals 2/3 of the stator pitch (e.g. fig. 6); the core transversal depth is instead about equal to the stator pole pitch. Therefore, when suitably faced to the magnets plate with aligned main directions, each sub-stator is able to develop a tangential force along its axis, acting as a linear brushless machine. Nevertheless, such force depends on the relative position of the magnets array along the orthogonal direction; moreover, appreciable cogging forces also arise due to the magnets-cores interaction. Therefore, a coordinated control of the 4 basic units is required to achieve a smooth translation capability over the whole stator plane. Anyway, such regulation permits to generate also a net torque, allowing so both to compensate external loads and to achieve a limited rotation capability around the orthogonal axis. These machines exhibit good force densities; nevertheless, the stator dispersed flux, the limited rotation capability, the necessity to feed the mover with 4 controlled 3-phase inverters through flexible wires, and the cogging wrenches due to magnets-cores interaction, represent not negligible drawbacks.
II.4 - MULTI-CORE BRUSHLESS MOTOR Another machine design using as stator a rectangular array of permanent magnets with uniform pole pitch, possibly laid on a back-iron plate as above described, is based on the employ of multiple units constituted by separate C-shaped cores equipped with independent coils. Such cores, featuring a square cross section and a pitch that is an even multiple of stator pole pitch (e.g. fig. 7), when facing the stator are thus able to provide forces along both of the main directions, depending on their position relative to the magnets rows, acting as mono-phase brushless machines. A mover equipped with several of such units located in suitable positions (e.g. fig. 8) below the payload
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Figure 7. Single coil-core in Figure 8. 8-cores brushless motor brushless motor [4] with mechanical bearings [4]
tray may then provide a 2-DoF translation capability along the stator plane, while a limited rotation may be also achieved when permitted by the suspension system.
II.5 - MULTI-COIL BRUSHLESS MOTORS A third possible variant of brushless machine using as stator a rectangular array of permanent magnets, as above described, involves the use of a mover composed of an array of ferromagnetic cylindrical poles equipped with independent coils. The poles are attached to a common back-iron bulk yoke, constituting also the payload tray (e.g. fig. 9). Several variants have been considered about the number of poles, their arrangement and the pitch ratio with respect to the stator array. It is reported that a 3x3 arrangement with pitch ratio of 2/3 provides a good 2-DoF translation capability with a limited rotation range around the vertical axis, while a 4x(2x2) square grouped arrangement provides a full 3-DoF operation capability, yet requiring a more complex control and supply system. When the above configuration is modified increasing the number of pole expansions up to cover the whole workspace required, while the permanent magnets array is reduced to a few elements, a different structure is obtained. Such machine is naturally supposed to be operated using as stator the excited part, while the permanent magnets are attached below the payload tray. The mover is supposed to be levitated by using one of the methods above recalled, or simply to slide above the pole expansions supported by 4 spherical bearings sliding on a rigid smooth non-magnetic surface covering main side of the stator (e.g. fig. 10). This last configuration fully eliminates the problem of flexible connections, as the mover requires no supply, thus permitting to achieve a full 3-DoF motion capability. Moreover, the low inertia of the mover might permit to achieve good dynamic performances. Nevertheless, not negligible cogging forces are present, while the support surface increases the air-gap thickness, lowering the achievable performances. Moreover, maximum flexibility is achieved by controlling separately each coil, requiring so a large number of independent supply converters.
Figure 9. Multi-coil static magnet brushless motor [5]
Figure 10 Multi-coil moving magnet planar motor [6]
Figure 11.
II.6 - SLOTLESS MOVING-MAGNET MOTOR A different machine structure may be obtained by using as stator a bulk ferromagnetic rectangular plate, around which 2 groups of identical coils are wound in orthogonal directions. The coils of each group are disposed side-byside, possibly in multiple layers intercalated with the coils belonging to the other group. The active part of the mover consists instead in a pair of identical permanent magnets attached to a ferromagnetic yoke with opposite axial (vertical) magnetization versa. The mover may be suspended above the main side of the stator by means of linear bearings (e.g. fig.6), or may directly slide over an interposed rigid surface of low-friction plastic material. The mover is suitably shaped and oriented diagonally, in such a way that the 2 magnets face different coils along both of the main directions. By properly supplying these coils, the desired combination of forces can be then achieved, as it may be easily realized by observing that in this case the reaction forces associated to the Lorentz effect directly act on the magnets. Therefore, in case of free sliding mover a 3-DoF motion capability may be achieved. While providing a good linearity in the force-current relationship, such machine requires a complex supply system able to separately regulate each coil current, while the expected specific force and torque is inherently limited by the large equivalent air-gap.
III.
ROTARY-LINEAR ACTUATORS
Rotary-linear machines, able to rotate and translate along their axis (2 DoF), are attracting an increasing interest as such motion capability is required, or would be desirable, for several pick-and-place and tooling tasks commonly found in industrial processes, such as drilling, threading, screwing, mounting, etc., and for the actuation of robotic arms and various types of end-effectors.
III.1 - COUPLED PAIRS OF MOTORS The simplest - yet less effective - solutions proposed for integrated rotary-linear units consist in mechanically coupling linear and rotary motors, suitably designed to minimize their reciprocal influence. Considering that the maximum axial speed achieved is usually limited due to the short stroke length, thus making small the motional voltages due to translation, this may be achieved using induction machines with bulk prolonged movers. For example, a linear and a rotary induction motor may be coupled, using different parts of the same mover without significant electromagnetic interactions (e.g. fig.12).
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Figure 12. Induction motors without magnetic coupling [8]
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Figure 13. 3-phase stators with common mover [9]
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Figure 15. Helical machine with axially laminated stator [11]
III.2 - MULTI-UNIT INDUCTION MOTORS When induction machines of the same type, sharing a unique bulk mover, are located in close proximity, the eddy currents induced in the mover determine a coupling that may be exploited to achieve extra motion capability. For example, 4 identical stators of conventional 3-phase rotary induction machines may be axially aligned in close proximity, sharing a unique prolonged bulk conductive ferromagnetic mover (e.g. fig. 13). In this case, the main electromechanical effect is constituted by the torque generated almost independently by the 4 units. Anyway, it is reported that under sinusoidal symmetric steady-state conditions with equal supply amplitude and frequency for the 4 units, a subsidiary force can be generated by time displacing the 4 supply terns, achieving the maximum effect with quadrature displacements. Analogously, 4 identical stators of 3-phase linear induction machines, adapted to feature a round-shaped active surface, may be evenly disposed around a bulk cylindrical conductive mover with ferromagnetic yoke (e.g. fig. 14). In this case, the main wrench developed consists in the axial forces generated about independently by the 4 sub-stators, but under symmetric steady state conditions a subsidiary torque may be generated again by using suitable phase displacements among the 4 supply terns. Nevertheless, in both structures a decoupled regulation of force and torque is quite problematic and is notably affected by the rotary speed, while the required supply system is not very simple.
III.3 - HELICAL INDUCTION MOTOR A different configuration of rotary-linear induction motor can be obtained by markedly skewing the stator slots of a conventional rotary machine, while retaining the conventional 3-phase structure of the windings. In this case, the mover is supposed to be fully isotropic, i.e., composed as a cylindrical ferromagnetic yoke surrounded by a bulk uniform conductive layer. The stator core may be also assembled putting side-by-side around the mover several packets of axially oriented and evenly slotted rectangular laminations, suitably cut to achieve the desired skewing and packed in such a way to determine a
Figure 14. 3-phase linear stators with common mover [10]
Figure 16. 3-phase helical winding in assembled stator [11]
cylindrical surface on the main side (e.g. fig. 15,16). As a result, when the 3-phase stator winding operates in symmetrical sinusoidal steady-state, the resulting magnetic field distribution, in terms of equivalent planar layout, exhibits an equivalent wave motion oriented orthogonal to the slots, becoming an helical motion inside the actual airgap. Consequently, the eddy currents induced in the mover determine an electromagnetic interaction tending to make the mover follow the field wave, developing so both resultant force and torque depending on the supply parameters and on the mover speed. A complete machine, providing enough regulation possibilities, may be then obtained by mechanically coupling 2 identical basic units as above describes helical slotted in opposite versa (righthand and left-hand); this may be simply achieved by using the same prolonged mover for both units. In fact, by separately adjusting the supply parameters of both windings in terms of sequence, amplitude and frequency, a decoupled regulation of the resulting force and torque, able to compensate for motional effects, may be achieved, although rather difficultly and with variable capability.
III.4 - HYBRID RELUCTANCE-INDUCTION MOTOR The helical motion of the resulting air-gap m.m.f. distribution obtained using a helical slotted stator hosting a conventional 3-phase winding, as above recalled, may give rise also to reluctance actions when an anisotropic mover is employed. In particular, under symmetric steady-state conditions a ferromagnetic mover, featuring annular slots having the same pole pitch as the axial pitch of stator slots (e.g. fig. 17,18), may give rise to a stable axial force when it moves synchronously with the axial speed of the air-gap field [12]. On the opposite, if the field axial speed is much higher than the mover speed, the resulting interaction turns into an oscillating force whose effect on the motion profile is more and more filtered by the mechanical inertia as the supply frequency raises. Anyway, if the mover is also conductive, as it is the case when it is made-up of bulk iron, then also a torque may be generated if any slip exists with respect to the angular speed of the field wave, due to the eddy currents induced in the mover itself. Assuming
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that the mover angular speed is lower than the field speed, within a suitable range the effect results so increasing with frequency. Therefore, a possibility to regulate in grossly decoupled way the developed force and torque exists if the 3-phase machine winding is fed with 2 superimposed symmetrical terns having markedly different frequencies. In fact, in this case the low frequency component will mainly influence the force thus ruling the axial synchronous- motion, while the high frequency tern will mainly determine the asynchronous torque, ruling the rotational motion and inducing only a vibration disturbance on the axial motion. Anyway, achieving a truly decoupled force/torque control in any condition is rather difficult, also due to the high bandwidth required to the supply converters to achieve the required frequency separation.
III.5 - BRUSHLESS MOTOR An unconventional structure of brushless rotary-linear machine is better analysed in [13]. In synthesis, referring to its simplest configuration, the stator of such machine includes 2 identical ferromagnetic half-cores, coaxially aligned side-by-side with a 45º angular displacement. Each half-core features 4 salient poles equipped with coils: a 4phase winding is obtained by connecting in series the coils located in diametrically opposed positions. The mover includes 2 sector-annular permanent magnets featuring radial magnetization with opposite versa, half-turn angular span and axial length equal to the stator axial pitch (e.g. fig. 19,20). The magnets are surface mounted centrally onto a ferromagnetic sleeve acting as yoke, having a length
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about 3 times the stator pitch. Such machine is ideally able to generate electromagnetic force and torque linearly depending on currents, with negligible cogging wrenches. By using a suitable variables transformation singling out 2 current components separately proportional to force and torque, such machine may be then ideally regulated in fully decoupled linear way in any operating condition.
IV.
SPHERICAL ACTUATORS
The last class of multi-DoF actuators here considered is constituted by spherical motors, able to adjust the pointing direction of the main axis of the mover and possibly also to rotate it around the same axis (2+1 DoF). Presently, such actuators are mainly proposed for pointing of microcameras and laser beams, in robotic, artificial vision, alignment and sensing applications. In larger sizes, they may be also used as active wrist joints for robotic arms.
IV.1 - CORE-LESS BRUSHLESS ACTUATORS In the simplest configurations of spherical motors, the stator is constituted by 3 windings, symmetrically disposed in space to achieve orthogonal magnetic axes, usually kept in place by plastic frames or resin painting. The mover is equipped with axially magnetized permanent magnet/s whose active sides face the coils either from inside or outside the region they delimit (e.g. fig.21). A suitable suspension system permits the mover to rotate and to transmit to the external mechanical load the torques applied to the magnet/s. Such machines, providing only a 2-DoF pointing capability, are able to exhibit fast dynamic responses thanks to the low weight and low inductance values. Nevertheless, only a limited angular range can be actually covered, with torques significantly depending on mover position thus requiring a sophisticated control. Moreover, the achievable torque density is low, making these machines suitable only for low-power applications.
IV.2 - ISOTROPIC BRUSHLESS MOTORS Figure 17. Basic arrangement of helical induction motor [12]
Figure 18. Equivalent airgap layout of helical induction motor [12]
Figure 20. Front view of brushless rotary-linear motor
Figure 19. Section of brushless rot.-lin. motor
Figure 21. Core-less spherical motor [14]
A more technically sound structure of spherical motor involves the use of an isotropic ferromagnetic stator core, usually shaped as a hollow spherical shell enclosing the mover. In this case, the coils are located either inside slots facing the mover, or directly inside the air-gap. The mover may consist either in a unique axially magnetized spherical block, or in a ferromagnetic spherical core equipped with surface mounted magnets suitably shaped. In both cases, a shaft is usually employed to link the mover to the external load passing through suitable openings in the stator core. Several solutions have been proposed for the suspension of the mover, such as self-lubricating plastic membranes filling the air-gap or air-cushion pneumatic systems. In the simplest configuration (e.g. fig. 23), using 3 coils and a mono-axial magnetized mover, the machine provides only the 2-DoF pointing capability. Anyway, by using suitable multi-pole configurations with multiple coil sets (e.g. fig. 23), the further axis-rotation capability can be achieved. Anyway, the angular range in limited by the size of the shaft bores in the the actually usable pointing workspace is limited to the size of the bores in the stator. Moreover, multi-pole arrangements require a more complex control/supply system due to the larger number of coils.
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Figure 22. Spherical isotropic brushless motor [15]
Figure 23. 3-DoF spherical brushless motor mover [15]
IV.3 - ANISOTROPIC BRUSHLESS MOTORS A different configuration for core-equipped brushless spherical machines includes a rather large number of salient poles hosting separated coils and fitted inside a hollow-spherical shell acting as ferromagnetic stator yoke. Usually, the poles are disposed at quasi-constant 2-D pitch onto the spherical surface of the stator, though this is not a trivial task. When high torque densities are required, the stator is designed as rather closed and a shaft is used to move the payload. In this case, the mover is constituted by a large number (even over 100) of permanent magnets attached onto a spherical ferromagnetic yoke connected to the shaft. The magnets may be shaped as quadrilateral curved blocks aligned in a parallel-meridian like lattice (e.g. fig. 24) obtaining so a variable pitch machine. Alternatively, when small torques are required with a fast dynamic response and larger curvature radii, the stator may be also shaped as an open part of a sphere. In this case, the mover may include simple disc-shaped magnets disposed, in quasi-regular way, onto a matching spherical partial surface, possibly non-ferromagnetic and also constituting the payload tray (e.g. fig. 25). Depending on the actual machine configuration, its behavior may then resemble either poly-phase synchronous motors or hybrid planar step motors. Anyway, the usually large number of poles used permits a fairly good position regulation even in open-loop operation, though a fine control requires micro-stepping techniques. Nevertheless, the required supply and control systems result clearly complex and expensive.
Figure 24. Core-equipped brushless spherical motor [16]
Figure 25. Brushless spherical motor with lightweight mover [17]
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CONCLUSIONS
A growing interest in electric actuators able to directly manage more mechanical DoF is determined by the continuous development of motion control applications. In fact, significant benefits in terms of performances, weight, volume, cost and maintenance may be expected with respect to common solutions using complex mechanisms moved by 1 actuator per each DoF. This paper presented a brief overview of the typologies of multi-DoF macroscopic actuators mainly analyzed in the literature in the recent years, i.e., planar, rotary-linear and spherical machines.
REFERENCES [1] J. Ish-Shalom: "Composite Magnetic Structure for Planar Motors", IEEE Trans. on Magnetics, vol. 31 nº 6 - 1995 [2] N. Fuji, M. Fujitake, K. Hara: "Two-Dimensional Drive Characteristics by Circular-Shaped Motor", IEEE Trans. on Industry Applications, vol. 35 nº 4 - 1999 [3] C. Han-Sam, J. Hyun-Kyo: "Analysis and Design of Synchronous Permanent Magnet Planar Motors", IEEE Trans. on Energy Conversion, vol. 17 nº 4 - 2002 [4] D. Ebihara, T. Watanobe, M. Wasada: "Characteristic Analysis of Surface Motor", IEEE Trans. on Magnetics, vol. 28 nº 5 - 1992 [5] T. Busch, G. Henneberger: "New Stator Pole Arrangement for a Planar Multi-Coordinate Drive", International Conference on Electrical Machines, Helsinki 2000 [6] J. Tsuchija, G. Kimura: "Mover Structure and Thrust Characteristic of Moving-Magnet-Type Surface Motor", IEEE-IES Conference, Denver 2001 [7] A. Flores Filho, A.A. Susin, M.A. Da Silveira, R.P. Homric: "Assessment of a XY Actuator Based on Orthogonal Coils", International Conference on Electric Machines, Bruges 2002 [8] P. de Wit, J. van Dijk, T. Blomer, P. Rutgers: "Mechatronic Design of a z-φ Induction Actuator", IEE Electric Machines and Drives Conference, Cambridge 1997 [9] T. Onuki, W.J. Jeon, M. Tanabiki: "Induction motor with helical motion by phase control", IEEE Trans. on Magnetics, vol. 33 nº 5 - 1997 [10] W.J. Jeon, M. Tanabiki, T. Onuki: "Rotary-Linear Induction Motor Composed of Four Primaries with Independently Energized Ring-Windings", IEEE-IAS Annual Meeting, New Orleans 1997 [11] M. Rabiee, J.J. Cathey: "Verification of a Field-Theory Analysis Applied to a Helical Motion Induction Motor", IEEE Trans. on Magnetics, vol 24 nº 4 - 1988 [12] L. Gobel, W. Hoffman: "Control of a Rotation-Thrust Drive with Helical Motor", IEEE International Conference on Industrial Electronics, Control and Instrumentation, New Orleans 1997 [13] P. Bolognesi: "Analysis and Preliminary Design of a RotaryLinear Brushless Machine", International Conference on Electrical Machines, Krakow 2004 [14] Bederson, R. Wallace, E. Schwartz: "A Miniature Pan-Tilt Actuator: the Spherical Pointing Motor", IEEE Trans. on Robotics and Automation, vol. 10 nº 3 - 1994 [15] J. Wang, K. Mitchell, G.W. Jewell, D. Howe: "Multi-Degree of Freedom Spherical Permanent Magnet Motors", IEEE International Conference on Robotics and Automation, Seoul 2001 [16] K. Kahlen, I. Voss, C. Priebe, R.W. De Doncker: "Torque Control of a Spherical Machine with Variable Pole Pitch", IEEE Power Electronics Society Conference, Cairns 2002 [17] D. Ebihara, N. Katsuyama, Kajioka: "An Approach to Basic Design of the PM-type Spherical Motor", IEEE International Conference on Robotics and Automation, Seoul 2001
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