A Novel 2-Freedom-Degrees Brushless Motor

A Novel 2-Freedom-Degrees Brushless Motor Paolo BOLOGNESI, Member, IEEE, Lucio TAPONECCO University of Pisa - Dept. Electric Systems & Automation - Via Diotisalvi 2, 56126 Pisa - Italy tel. +39-050-565309 - fax +39-050-565333 - email: [email protected] Abstract – In apparatuses requiring articulated movements involving multiple freedom degrees, usually each one is managed by means of a conventional actuator charged to control the corresponding elementary motion, and the required comprehensive motion is obtained with the aid of suitable mechanisms. Nevertheless, weight, space and cost savings, besides structural simplifications, could be achieved by employing actuators able to simultaneously manage more freedom degrees. This paper proposes a novel rotary-linear 4-phase permanent magnets brushless motor. The machine structure and its basic principles are first described, obtaining then an approximated analytical model by means of a simplified electromagnetic analysis based on the equivalent magnetic circuit approach. The machine regulation is then examined, demonstrating the ideal possibility to achieve a fully decoupled control of the electromagnetic force and torque, joined to the suppression of the potentially correlated ripple phenomena, by adopting a suitable yet simple supply logic. Several theoretical waveforms are finally reported. I. INTRODUCTION In a growing number of applications; such as modern tooling machines and robotics, apparatuses able to perform articulated movements involving multiple mechanical freedom degrees are increasingly required. In such applications, the control of each freedom degree is usually performed separately by means of a dedicated rotary or linear actuator, often coupled to suitable mechanical devices to achieve the actually required motion. Nevertheless, the employ of compact actuators, able to directly manage several mechanical freedom degrees, could offer significant improvements in terms of weight, volume, response time, losses and overall cost reduction, with precision and reliability improvements. Electromechanical actuators result potentially very interesting for such applications, since they should be potentially able to exhibit the same interesting features that already made electric machines far the most diffused single-freedom-degree actuators, i.e. good versatility, controllability, specific performances, efficiency and reliability with minimal maintenance requirements. Presently, the most diffused typology of multi-freedomdegree electric machine is probably constituted by the Sawyer step motor (e.g. [1]), which is widely used as a precision positioning actuator for planar motion especially in the ambit of semiconductors manufacturing processes. Moreover, spherical motors for small torque applications, such as micro-cameras pointing in robotics ambit, have been proposed mainly basing on permanent magnets structures (e.g. [2]). For what concerns instead rotary-linear applications involving arbitrary rotation angles, most of the solutions till now proposed consist in mechanically coupling a rotary machine to a linear one keeping substantially independent their electromagnetic structures. For example, in [3] a 3-phase induction motor is coupled to a voice-coil linear actuator, while in [4] a unique bulk cylindrical mover is shared between 2 adjacent, yet independent, 3-phase induction stators, whose structures are respectively a standard rotary and a linear-tubular one. Although easily applicable, such solutions usually determine low specific performances, resulting in cumbersome non-optimized actuators requiring a dual output converter is to supply the 2 machines. In [5] it was instead investigated the use of multiple narrow linear induction stators regularly disposed around a cylindrical bulk mover, obtaining the rotational torque by suitably phase shifting the windings' supply terns. Nevertheless, low specific torque performances are achieved with poor dynamic characteristics, while the complexity of both the power and control parts is notable. In [6] it was proposed to obtain an integrated rotary-linear machine from the structure of the “screw-thread” reluctance linearcylindrical motor, by exploiting the asynchronous torque due to the eddy currents induced in the structure of the bulk iron anisotropic mover. In fact, it was reported that this torque increases with stator frequency while the axial force decreases, meaning that in a suitably sized machine these 2 thrusts can be controlled by injecting in the unique 3-phase stator winding 2 sets of 3-phase currents having respectively high and low frequencies. Nevertheless, only a partly decoupled control of the force and torque effects is achieved with modest dynamic performances, also due to the not negligible influence of the axial speed. Moreover, high machine manufacturing costs can be expected, due to its complex structure. In this paper a novel typology of 2-freedom-degrees rotary-linear electric motor is described and analyzed, highlighting its most interesting features: a relatively simple and robust structure, including a permanent magnets mover and a salient poles ferromagnetic stator with a 4-phase symmetrical winding; the possibility to easily adapt popular converters for supplying; the ideal possibility to achieve a decoupled linear regulation of force and torque and to eliminate thrusts ripple by means of a suitable yet simple supply strategy; a performance capability ideally independent on the mover status. II. M ACHINE STRUCTURE II.A Basic structure The simplest version of the considered rotary-linear machine structure, which can be conceptually obtained as a generalization of a planar configuration previously proposed (e.g. [7]) for linear machines, consists of the following parts: - a stator ferromagnetic core consisting in an external yoke (for example having cylindrical shape) and 8 identical salient poles featuring cylindrical internal surfaces, that are regularly arranged onto the internal yoke surface in a 4 (circumferential) by 2 (axial) array disposition; these poles result so separated by 4 straight axial slots, evenly spaced in the tangential direction, and by 1 circumferential slot, located at half stator length, whose inlet widths are assumed to be relatively small with respect to the poles dimensions; - a winding consisting in 8 identical coils, each one wound around a different stator pole, which are grouped into 4 independent phases by connecting in series each couple of coils located at diametrically corresponding positions, in such a way to achieve opposite magnetic polarities with respect to the yoke; - a mover composed of a cylindrical ferromagnetic sleeve, having a length about triple of the stator poles one, onto which is fitted, in central position, a permanent magnet toroidal ring featuring an about rectangular cross section and an axial length equal to the poles one: the torus is circumferentially divided into 2 symmetrical sections, which are magnetized in radial direction with equal uniform strength but opposite polarities, and are separated by 2 small transition zones having suitable characteristics; - a mechanical output shaft, which can be constituted either by the mover sleeve prolongation or by a smaller bar, even not ferromagnetic, centrally fitted inside the sleeve itself; the shaft is supported and kept coaxial with the stator core by means of a set of suitable mechanical guides, such as a sleeve bearings, able to permit the mover to freely rotate and axially translate up to the useful stroke limits, which are such to keep anyway the magnets inside the stator core. The cross and axial sections of this basic structure result so as sketched respectively in fig. 1, 2, where different hatching styles are used to highlight the various parts. The mover position with respect to stator poles can also be schematized by the air-gap equivalent planar sketch depicted in fig. 3, where the rounding of corners of poles expansions is enhanced to improve visibility. Moreover, the poles and coils are sequentially numbered, and the versa of field lines separately determined from permanent magnets and winding phases, composed of coils pairs 1-5, 2-6, 3-7, 4-8, are also highlighted for positive currents entering the reference terminals. The basic structure above described can be expected to offer interesting features analogously to its planar singlefreedom-degree predecessor ([7]), but also to suffer of similar performance limitations around the boundary Fig. 1 - Basic machine structure: cross section Fig. 2 - Basic machine structure: axial section Fig. 3 - Basic structure: equivalent planar sketch Fig. 4 - Proposed structure: equivalent planar sketch positions. In fact, from symmetry considerations it can be argued that the mover angular positions determining the alignment of the transition zones with axial slots represent critical points, since their crossing involves a significant variation of the force and torque dependence on all the currents. Such phenomenon could be highlighted by means of an electromagnetic analysis as developed in III, which is not reported here for brevity. Therefore, although some partial remedies could be assumed, the basic machine structure above described presents a significant risk to produce appreciable force and torque ripple phenomena around the alignment points. While this problem results not relevant for a linear motor, since the stroke is bounded in nature, it becomes significant for a rotational one, which would be expected to perform properly for any angular position. II.B Proposed structure The transition problem above highlighted can be overcome by suitably modifying the machine structure. In fact, the presence of 2 layers of stator poles along the axial direction can be exploited displacing angularly the 2 half-stators by a suitable quantity, whose value can be conveniently chosen as π/4 to take advantage of the resulting structural symmetry. This spatial displacement determine a corresponding time separation of the boundary transitions relative to the 2 groups of poles, permitting so to better manage these events as described in section IV. Consequently, in the proposed machine structure the number of axial slots becomes 8, while their length is reduced to about half the stator core. The air-gap equivalent planar sketch results then modified as depicted in fig. 4 . III. T HEORETICAL ANALYSIS The fundamental characteristics of the proposed machine, and their basic dependence on the materials properties and main structural dimensions, can be highlighted by performing an approximated theoretical electromagnetic analysis, assuming suitable simplifying hypotheses able to lead to handy expressions. III.A Assumed hypotheses For the analysis developed in the following, the hypotheses below listed will be assumed: - negligible m.m.f. drops inside the stator core and the mover sleeve, excluding so saturation phenomena; - negligible width of both slots inlets and transition zones with respect to the poles dimensions; - small thickness of the air-gap, allowing to assume a prevalent radial orientation of the magnetic field lines inside it with about uniform field strength along them; - permanent magnets featuring a practically linear B-H relationship with stable values of residual induction Br and coercive field Hc and relative differential permeability differing negligibly from 1 ; - negligible losses, due to parasitic currents and hysteresis, inside stator core, mover sleeve and magnets; - negligible variations of the windings resistances directly due to electric phenomena, such as skin effect; - mover axial excursion bounded in such a way that the overlapping zone of magnets with both half-stators is never shorter than several times the circumferential slot opening, to permit to employ a unique analysis approach. III.B Electromagnetic analysis The position of the mover with respect to the stator can be uniquely singled out by a coordinates vector χ featuring 2 components, a translational and a rotational one, which can be conveniently chosen as indicated in fig. 4: the axial ( displacement x ∈ − a , a 2 2 ) of the mover from the central symmetrical position towards the odd poles, where a is the axial length of the magnets, and the overlapping angle θ ∈ [ 0, 2π ) of the north magnet (as seen from the stator) with pole expansion 1, in the direction towards pole 2. As previously stated, the actual axial stroke limits will have to be intended as suitably inner to the ideal ones above indicated. The significant range of the angular coordinate can be conveniently divided into the 8 sectors S k = [ θ k , θ k +1 ] where θ k = k2−1 ⋅ π , k = 1,..8 , (1) inside which can be defined the local angular coordinates [ γ k = θ − θk − π ∈ − π , π 8 8 8 ] k = 1,..8 . (2) In fact, the bounds of these sectors correspond to the alignment points between transition zones and axial slots, meaning that inside each sector no qualitative variation takes place in the reciprocal stator-mover configuration. Keeping into account the above hypotheses and considerations, the proposed machine structure can be conveniently analyzed by applying the equivalent magnetic circuit method and focusing on the main flux tubes. Firstly, it must be kept into account the main field lines that, starting from the yoke, pass through the stator poles, cross the main geometrical air gap, cross then either the magnets or the corresponding air space, and finally turn back inside the mover sleeve, returning to the stator going through a similar path in reversed order. In approximated terms, these field lines determine so 3 distinct flux tubes per each stator pole, depending on the fact that they pass through either the north magnet (as seen from the stator), or the south magnet, or the adjacent air space. The portions of these tubes delimited by the 2 equipotential surfaces corresponding to pole expansion and sleeve can be so represented by means of Pnk , Psk , Pak , 3 permeances, respectively named whose values depend on the mover positions; moreover, for the tubes crossing the magnets a magnetic tension generator F n , F s corresponding to the coercive m.m.f. F n = −F s = H c ⋅ σ m (3) where σm is the magnets thickness, has to be considered series connected to the corresponding permeance. Secondly, it has to be considered the lateral field lines that embrace couples of coils by passing through the adjacent stator poles, the yoke and the enclosed slot. It can be so approximately considered 4 equivalent main flux tubes per each stator pole, magnetically linked with its whole coil: 2 tubes crossing the adjacent axial slots and 2 tubes crossing the circumferential slot towards the two facing poles. Due to the machine symmetry, the air portion of these flux tubes can be represented by means of 2 permeances, named Pt , P l respectively for the tubes linking tangentially and longitudinally adjacent poles, which can be considered not influenced from the mover position. Thirdly, it has to be considered the dispersed field lines linking only one coil, that are mainly concentrated at the machine extremities. They can be approximately taken into account by means of an equivalent dispersed flux tube per pole, which can be represented by means of a constant permeance having the same value Pd for each coil. Finally, the winding effect is represented by means of an equivalent magnetic tension generator per each coil, positioned along the path corresponding to the associated flux tubes and having amplitude equal to the corresponding m.m.f. F k = −F k + 4 = N ⋅ ik k ∈ {1,.. 4 } (4) directly expressed in terms of phase current, where N is the number of turns per coil. Therefore, the simplified equivalent magnetic circuit of the proposed machine presents a modular structure as outlined in fig. 5, where the coils numbering has clearly to be intended as cyclic. Assuming for example that the mover angular position θ ∈ S 2 is within sector 2 (e.g. fig. 4), the values of the magnetic circuit parameters result as follows: invariant with respect to the mover position, by solving the magnetic circuit one obtains the general expression of the total fluxes linked to the winding phases ψ (i , χ ) = ψo ( χ ) + L ⋅ i (17) with versa coherent to the positive entering currents convention. In the assumed hypotheses, the inductance matrix results ideally constant 0 Lm   L p − Lm −L L p − Lm 0  m L= (18)  0 − Lm L p − Lm    − Lm 0 L p   Lm i.e. independent on both currents and mover position, since for the phase and mutual inductances one obtains Pn1 = Ps5 = µ o ⋅ r ⋅ 3 π + γ 2 ⋅ a + x σ 8 2 (5) L p = 2 ⋅ N 2 ⋅ P p + 2 ⋅ P l + 2 ⋅ Pt + Pd (6) Lm = 2 ⋅ N ⋅ P l (7) The electric equation of the machine winding results so expressed in compact vector form as d d v (t ) = R ⋅ i (t ) + L ⋅ i (t ) + M ( χ ) ⋅ χ (t ) (21) dt dt where R is the total resistance per phase, while  1 ⋅ ( 1 + 4 ⋅ γ 2 ) 4 ⋅ ( 1 + x ) π 2 a  a 2 π 1 ⋅ ( 1 − 4 ⋅ γ ) 4 ⋅ ( 1 − x )  ∂ ψ (i , χ) 2 π 2 a = ψ m ⋅  a 2 1π M ( χ) =  (22) 0 − ∂ χT   a   1 0 a   is the motional e.m.f. coefficients matrix. In the assumed hypothesis, both the thrusts developed by the machine at zero currents result ideally null: therefore, since M is not dependent on the currents, the general expressions of force and torque results  F (i , χ ) T (23)   = M (χ ) ⋅ i = ( , ) T i χ    1 ⋅ 4 ⋅ γ 2 ⋅ ( i1 − i2 ) + 1 ⋅ ( i1 + i2 ) − i3 + i4  2  . = ψm ⋅  a π 4 x ⋅ ⋅ ( i1 − i2 ) + 1 ⋅ ( i1 + i2 )   a π 2   ( )( ) Ps1 = Pn5 = µ o ⋅ r ⋅ ( π − γ 2 )⋅ ( a + x ) σ 8 2 π r Pn2 = Ps6 = µ o ⋅ ⋅ ( + γ 2 )⋅ ( a − x ) σ 8 2 3 r Ps2 = Pn6 = µ o ⋅ ⋅ ( π − γ 2 )⋅ ( a − x ) σ 8 2 (8) Pn3 = Ps7 = Pn4 = Ps8 = 0 ( ( (9) ) −x) Ps3 = Pn7 = µ o ⋅ r ⋅ π ⋅ a + x σ 2 2 Ps4 = Pn8 = µ o ⋅ r ⋅ π ⋅ a σ 2 2 (10) ( ) Pa2 = Pa4 = Pa6 = Pa8 = µ o ⋅ r ⋅ π ⋅ ( a + x ) σ 2 2 Pa1 = Pa3 = Pa5 = Pa7 = µ o ⋅ r ⋅ π ⋅ a − x σ 2 2 (11) (12) (13) where σ = σ a + σ m (14) is the sum of the thickness of the main air gap σa and of the magnets, while r is the mean radius of the equivalent air gap between sleeve and pole expansion. Introducing the maximum total flux linked to any phase ψ m = 2 ⋅ N ⋅ F m ⋅ P p = π ⋅ N ⋅ Br ⋅ σ m ⋅ r ⋅ a σ due to the sole magnets, where P p = Pnk + Psk + Pak = µ o ⋅ r ⋅ π ⋅ a σ 2 ∀ k ∈ {1,.. 8 } (15) (16) is the total permeance of the stator-mover portions of the primary flux tubes leaving each pole, that results ( 2 ( ) (19) . (20) ( ) ) III.C Results generalization The analytic results above deduced for sector 2 can be easily extended to the other angular sectors. In fact, the only quantity actually varying is constituted by the motional e.m.f. matrix M , since both the inductances and resistances are independent on the mover position. Moreover, the symmetry properties of the machine structure permit to obtain a simple recursive relationship for its expression inside a generic angular sector k , indicated as Mk : in fact, it results M k ( x, γ k ) = M x k ( γ k ) M γ k ( x ) = (24) [ [ ] ] = − Q ⋅ Mxk −1 ( γ k ) Q ⋅ Mγ k −1 (− x) = [ k −2 k −2 ] = (− Q) ⋅ Mx 2 ( γ k ) Q ⋅ Mγ 2 ( − x ) where Q is the permutation and sign change matrix Fig. 5 - Outline of the machine equivalent magnetic circuit 0 0 0 −1 1 0 0 0  Q= . (25) 0 1 0 0   0 0 1 0  Consequently, the developed electromagnetic force and torque can be expressed in general form as F (i , γ k ) = Mxk ( γ k ) T ⋅ i = (26) = Mx 2 ( γ k ) T ⋅ ( − Q T ) k − 2 ⋅ i = ψm a [ ⋅ i F + 4 ⋅ γ k ⋅ id T (i , x) = Mγ k ( x) T ⋅ i = = Mγ 2 ( ( −1) k −2 ⋅x ) T ⋅ QT k −2 ⋅i = 4 ⋅ψ ⋅ m π π [i T ] + x ⋅ id a ] (27) where 3 equivalent currents iF , iT , id have been introduced, whose values are linearly correlated to the phase currents as reported in tab. 1 per each sector S . III.D Secondary phenomena The simplified model analytically deduced for the proposed machine should represent fairly correctly its main functional aspects, as far as its actual design and the considered operation conditions are not in significant contrast with the assumed hypotheses. Nevertheless, the previous approach lacks in describing the behavior around the boundary points of the angular sectors (1). In fact, here the precise geometry of magnets surfaces and slots openings, and the local behavior of the active materials near the facing borders, play a less negligible role. Unfortunately, a reliable analysis of these aspects cannot be performed by means of simple equivalent magnetic circuits, requiring detailed simulations based, for example, on the finite elements method. Anyway, under the given hypotheses it can be stated that the actual transition between the different expressions of the M function, and thus of the force and torque ones, should take place in continuous way inside intervals [ STk = θ k - ε , θ k + 2 ε 2 ] k = 1, ..8 (28) symmetrically spanning around the boundary points and having a relatively small width ε . Moreover, other secondary effects of modest entity should appear, such as a dependence of the inductance matrix on the axial position, a line-up reluctance torque arising within the transition intervals, and an analogous alignment reluctance force at the extremities of the axial stroke. Tab. 1 - Currents for generalized force/torque expressions S 1 2 3 4 5 6 7 8 id i 1 + i4 i1 - i2 i3 - i2 i3 - i4 - i1 - i4 - i1 + i 2 - i3 + i 2 - i3 + i 4 iT ½ ·[ i1 - i4 ] ½ ·[ i1 + i2 ] ½ ·[ i3 + i2 ] ½ ·[ i3 + i4 ] ½ ·[ - i1 + i4 ] ½ ·[ - i1 - i2 ] ½ ·[ - i3 - i2 ] ½ ·[ - i3 - i4 ] iF - iT - i 3 + i 2 i T - i3 + i4 - iT + i 1 + i4 iT + i 1 - i 2 - i T + i 3 - i2 iT + i 3 - i 4 - iT - i1 - i4 i T - i1 + i 2 IV. M ACHINE REGULATION IV.A Decoupled force-torque regulation The general expressions obtained for of the developed electromagnetic force (26) and torque (27) outside the transition intervals present a cross dependence from the mover angular and linear coordinates. Anyway, in both expressions the position dependent terms result proportional to the sole "disturbance" current id . Therefore, if the machine is supplied in such a way that id is kept null id = 0 (29) then the electromagnetic force and torque result linearly proportional respectively to the "force" iF and "torque" iT equivalent currents, permitting so to achieve a fully decoupled regulation of the 2 thrusts. In particular, it can be noted from tab. 1 that applying the decoupling condition (29) forces the 2 phase currents determining iT to assume either its value iT (T ) = 4 π⋅ ψ m ⋅ T = AT ⋅ T (30) or the opposite one, while the other 2 phase currents result constrained only by the required iF current value iF (F ) = a 2⋅ψ m ⋅ F = AF ⋅ F (31) Nevertheless, it must be kept in mind that the relationships between equivalent and phase currents vary inside each transition interval between angular sectors, meaning that suitable supply profiles have to be adopted to minimize possibly correlated torque and force ripple effects. In particular, in tab. 1 one can observe that each phase current appears contemporaneously in the expressions of id and iT for 2 consecutive sectors without variations, while the second phase current involved changes during the intermediate transition. Moreover, for any pair of adjacent sectors the fourth phase current, momentarily not correlated to id and iT , appears in the expression of iF without variations due to the intermediate transition. Therefore, if during the whole transition between any at pair of adjacent sectors the 3 involved phase currents comply the decoupling constraint (29), then within this interval iF results determined exclusively from the fourth phase current. This supply logic permits then to overcome the transitions problem when it can be assumed that no significant force or torque ripple arises under similar conditions. Such conclusion can indeed be analytically deduced assuming that the variation of force and torque expressions take place inside each transition interval in linear symmetric form with respect to the angular coordinate θ , i.e. by adopting a first order interpolation of the actual variation laws which should anyway result acceptable for motion control purposes. The supply logic above described has to be completed with the rules to be applied outside the transition intervals ST , which must be such to provide the required values of iT and iF while complying the decoupling constraint. This can be obtained by exploiting the properties above highlighted, particularly the freedom degree due to the dependence of iF on 2 phase currents not influencing iT . In fact, these Tab. 2 - Supply logic ideally providing the decoupled control of electromagnetic force and torque avoiding transition ripple S 1 2 3 4 5 6 7 8 i1 iT iT iT ↔ iF iF ↔ -iT -iT -iT -iT ↔ -iF -iF ↔ iT i2 iF ↔ iT iT iT iT ↔ -iF -iF ↔ -iT -iT -iT -iT ↔ iF i3 -iT ↔ -iF -iF ↔ iT iT iT i T ↔ iF iF ↔ -iT -iT -iT i4 -iT -iT ↔ iF iF ↔ iT iT iT iT ↔ -iF -iF ↔ -iT -iT currents can be simultaneously varied in such a way to let them assume the correct final values for the following transition interval, swapping their roles without affecting the developed force. The complete scheme of the resulting supply logic is reported in tab. 2, where the complementary variations of the currents pairs determining the developed force are highlighted, per each sector S , by means of double arrow symbols. It can be noted that each phase current is kept steadily at the value ± iT inside 4 sectors, while it varies appropriately inside the other ones and assumes the values ± iF only during 2 transitions. Since the sensing or estimation of the mover position is necessary for a good employ of this machine, a simple and practical solution to manage the complementary variations of the currents consists in linearly correlating them to the local coordinate γk inside each sector Sk , assuming its central positions as the middle point of the corresponding variation interval [ SVk = θ k + π 8 - δ , θk + 2 π 8 + δ 2 ] k = 1,..8 (32) and choosing the width δ narrow enough to avoid any overlapping with the transition intervals, i.e. δ+ε≤ π 4 . (33) In fig. 6 are reported as examples the qualitative ideal waveforms of the rotational e.m.f.s (solid line: phase 1, dashed: phase 2, dot-dashed: phase 3, dotted: phase 4) for a complete revolution of the mover performed at constant positive angular speed with fixed axial position π . x = a6 , assuming a transition intervals width ε = 10 Fig. 5 - E.m.f.s for constant angular speed, fixed axial position Fig. 6 - Ideal phase currents for decoupled force-torque control It can be noted the double amplitude of the voltages induced in the odd phases with respect to the even ones. Moreover, in fig. 7 are reported, for a complete turn of the mover, the qualitative ideal waveforms of the phase currents (same line styles as above) deriving from the application of the decoupling supply logic when the force current is double than the torque one: iF = 2· iT , assuming a width of the variation intervals δ = π . 8 V. CONCLUSIONS A novel 2-freedom-degrees rotary-linear brushless motor has been proposed, featuring a permanent magnets mover and a salient poles ferromagnetic stator core hosting a concentrated 4-phase winding. An analytical approximated model of the machine has been derived from a simplified electromagnetic analysis of its structure. It has been then deduced a suitable yet simple supply logic, which ideally permits to achieve a fully decoupled regulation of the developed force and torque independently on the mover status while also avoiding thrust ripple phenomena. Keeping into account the attractive ideal features and the relative structural simplicity, the proposed machine appears so a potentially interesting alternative to existing rotary-linear actuators. VI. REFERENCES [1] J. Ish-Shalom: “Modeling of Sawyer Planar Sensor and Motor Dependence on Planar Yaw Angle Rotation”, proc. of IEEE ICRA ’97 Conference, Albuquerque 1997, pp.3499÷3504 [2] J. Wang, K. Mitchell, G.W. Jewell, D. 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