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Design and Optimization of a Rotary Actuator for aTwo Degree-of-Freedom zφ-Module

T. T. Overboom, J. W. Jansen, E. A. Lomonova and F. J. F Tacken

Abstract—The paper concerns the design and optimization ofa rotary actuator of which the rotor is attached to a linearactuator inside a two degree-of-freedom zφ-module, which ispart of a pick-and-place robot. The rotary actuator provides±180° rotation, while the linear actuator offers a z-motion of±5 mm. In the paper, the optimal combinations of magnetpoles and coils are determined for this slotless actuator withconcentrated windings. Based on this analysis, the rotary actuatoris optimized using a multi-physical framework, which contains acoupled electromagnetic, mechanical and thermal model. Becausethe rotation angle is limited, both a moving-coil design witha double mechanical clearance and a moving-magnet designwith a single mechanical clearance have been investigated andcompared. Additionally, the influence of the edge effects of themagnets on the performance of the rotary actuator has beeninvestigated with both 3D FEM simulations and measurements.

I. INTRODUCTION

Pick-and-place (P&P) robots consist of a long-stroke robotarm, which is responsible for moving electrical components(such as Surface-Mounted-Devices or Ball-Grid-Array com-ponents) from a feeder over the Printed-Circuit-Board (PCB).Attached to this arm, a placement module picks up thecomponents, orientates and places them on the PCB. Thishigh-precision, short-stroke, two degree-of-freedom (2 DoF)actuator, therefore, needs to enable rotational and translationalmotion. Moreover, they require a compact and light weighteddesign, due to higher accelerations and operational speeds.Such a module will be referred to as a zφ-module.

Several types for zφ-modules can be distinguished. In thefirst category and classic approach, the linear stroke actuatordrives a trolley which holds a rotary actuator. As an alternative,in the second category the movers of the rotary and linearactuator are attached to the same shaft above each other. Forexample, in [1] a zφ-module from the second category isdiscussed which uses two separate magnet arrays along theaxial length of the rotor; one array for the rotary actuatorand one array for the linear actuator. Instead of stackingtwo magnetization patterns, they can also be integrated in asingle magnet array, as is done in the third category. Using acheckerboard magnet array and by appropriately commutatingthe current inside the coils, the design in [2] offers bothdegrees of freedom.

Based on the second category, a zφ-module is designed,which has to pick-and-place 10.000 BGA-components in one

T. T. Overboom, J. W. Jansen and E. A. Lomonova are with theElectromechanics and Power Electronics group, Eindhoven Universityof Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands.Email: T.T.Overboom@tue.nl

F. J. F Tacken is with Wijdeven B.V., De Scheper 303, 5688 HP, Oirschot,The Netherlands.

Fig. 1. 3D overview of the zφ-module containing a non-commutated short-stroke linear actuator (bottom) and three-phase slotless permanent magnetrotary actuator (top).

hour. Figure 1 shows a 3D overview of this design, where therotary actuator is placed at the top and the linear actuatoris placed at the bottom of the module. The specificationsof the zφ-module are given in Table I. The rotary actuatoris a slotless permanent magnet actuator and provides ±180°degrees rotation. Because the total stroke of the linear actuatoris only 10 mm, a non-commutated short-stroke linear actuatoris selected instead of a three phase linear actuator, like is usedin [1].

This paper concerns the design and optimization of therotary actuator as part of the zφ-module, while the design ofthe non-commutated short-stroke linear actuator is fixed andwill be considered as load. First, the basic design regardingthe rotary actuator is addressed and the test results of a non-optimized pre-prototype are discussed. Prior to the optimiza-tion step, an optimum combination of the number of magnetpoles and coils is selected. A multi-physical framework of therotary actuator is presented and used to optimize the design,which is performed for both a configuration with movingmagnets and a configuration with moving coils. Finally, theoptimized designs are analyzed for the magnet edge-effects.

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Fig. 2. Overview of the rotary actuator for (a) the moving-magnet configuration and (b) the moving-coil configuration.

TABLE ISPECIFICATIONS OF THE zφ-MODULE.

Parameter Value Descriptionzstroke ±5 mm Linear strokeφstroke ±180° degrees Angular stroke

zerr 5 µm Linear accuracyφerr 3 mrad Angular accuracy

αz 150 m s−2 Linear accelerationαφ 7700 rad s−2 Angular acceleration

vmax 3 m s−1 Maximum linear velocityωmax 135 rad s−1 Maximum angular velocity

dz 0.22 Duty cycle linear actuatordφ 0.39 Duty cycle rotary actuator

Lzφ 105 mm Maximum length of zφ-modulero,max 30 mm Maximum outer radius of zφ-moduleri,min 18 mm Minimum inner radius of zφ-module

II. BASIC DESIGN CONSIDERATIONS AND PRE-PROTOTYPETESTING

The basic design of the three-phase AC brushless permanentmagnet rotary actuator is shown in Fig. 2. It has a slotlessstructure in order to provide a low torque ripple. To increasethe magnetic loading inside the airgap, a two segmented quasi-Halbach magnet array is used. Concentrated windings areselected for their ease of manufacturing. Because the rotaryactuator also needs to accommodate the translational motionof the rotor, the coils are elongated in order to deliver constanttorque for any given axial position. Because the rotationalangle of the rotary actuator is limited to ±180°, both a moving-magnet and a moving-coil configuration will be optimized andcompared. In the moving-magnet configuration the coils areattached to the back-iron leaving only a single mechanicalclearance between the coils and magnets. For moving coils,however, a double mechanical clearance is required; oneadditional clearance between the coils and back-iron. Figure 2shows an overview of the rotary actuator for the moving-magnet and moving-coil configuration. In both cases, the rotoris coupled to the rotor of the linear actuator.

A non-optimized pre-prototype of the rotary actuator hasalready been designed, built and tested. Also a prototype of thenon-commutated short-stroke actuator is built, but as the linearactuator is out of the scope of this work, its specifications are

TABLE IISPECIFICATIONS OF THE NON-COMMUTATED SHORT-STROKE LINEAR

ACTUATOR.

Parameter Value Descriptionkt,lin 14 N A−1 Force constantRlin 8.2 Ω Coil resistanceJlin 78 kg mm2 InertiaMlin 190 g Moving mass

given in Table II. The design of the rotary actuator is basedon a configuration with moving coils and its dimensions aregiven in Table III. Figure 3 shows the pre-prototype. Each coilin the slotless PM actuator contains 92 turns and the magnetsare anisotropic sintered NdFeB, with a remanent flux densityof 1.33 T and a relative recoil permeability of 1.1. The coreand back-iron are made of steel N398, which is a low-carbonsteel (99.8% Fe).

Using the 2D semi-analytical magneto-static model, asdescribed in Section IV, a torque constant of 0.93 Nm A−1

is estimated, however, tests on the pre-prototype showed atorque constant of 0.76 Nm A−1. To eliminate the influenceof friction, electro-motive force (EMF) measurements wereperformed, which showed a reduced amplitude of 11.7 %compared to the 2D semi-analytical model. This reductionis caused by the lower magnetic loading near the edges ofthe magnets, as will be discussed in Section V. The test alsoshowed that the helical wound electrical wires (see Fig. 1)were insufficient to guarantee a long life operation.

III. SELECTION NUMBER OF MAGNET POLES AND COILS

As a first step in the design process of the rotary actuator,the appropriate combinations of the number of magnet poles(2p) and coils (Q) are selected. First of all, only combinationsresulting in a balanced structure will be evaluated during theoptimization step. Next, the selection is further reduced bychoosing magnet pole and coil combinations based on thewinding factor, kw. The winding factor is a measure for theflux linkage of a certain winding layout and ranges from 0(no linkage) to 1 (optimal linkage). The torque produced bya three-phase AC brushless PM actuator is related to the flux

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TABLE IIIDIMENSIONS OF INITIAL DESIGN ROTARY ACTUATOR AND PERFORMANCE

OF THE zφ-MODULE.

Description Parameter ValueDimensions rotary actuatorNumber of pole pairs p 8Number of coils Q 12Inner radius ri [mm] 18.0Outer radius ro [mm] 30.0Radial length core lc [mm] 1.5Radial length magnets lm [mm] 5.7Radial length coils lw [mm] 1.9Radial length back-iron lb [mm] 2.5Radial length mechanical clearance lcl [mm] 0.2Active length actuator Lact [mm] 15.0Magnet arc to pole arc ratio α 0.67Coil opening angle βo [degree] 1.70Volume V [cm3] 104.6Peak airgap flux density Bgap [T] 0.87Coil temperature Tw [°C] 42.9Magnet temperature Tm [°C] 39.7Performance zφ-moduleTotal copper losses Pcu,tot [W] 16.44Rotary copper losses Pcu,rot [W] 10.90Linear copper losses Pcu,lin [W] 5.55Total inertia Jtot [kg.mm2] 142Total moving mass Mtot [g] 282

Fig. 3. Pre-prototype of the rotary actuator with (a) the test set-up and (b)the coil array.

linkage according to

T =32ω

IE =32ω

IkwdΛmax

dt, (1)

where I is the peak phase current, E is the peak phase back-EMF, ω is the angular velocity and Λmax is the maximumpossible flux linkage of a single phase. As the expressionshows, the torque is directly linked to the winding factor, andit would be desirable to select a combination of magnet polesand coils resulting in the highest possible winding factor.

The winding factor can be split into

kw = kp · kd · kskew, (2)

where kp is the pitch factor, kd is the distribution factor andkskew is the skewing factor. Since no skewing is applied, theskewing factor is assumed to be unity. The winding factors canonly be found for slotted structures, though [3], [4], [5], [6].Therefore, for slotless machines with concentrated windings,the fundamental winding factor is calculated for differentcombinations of the number of magnet poles and coils. Below,

Fig. 4. Illustration of a slotless actuator for predicting the pitch factor.

both the pitch and distribution factor will be determined, whenonly the fundamental harmonic of the magnetic flux densitydue to the magnets is considered.

A. Pitch factor

The pitch factor is a measure for the flux linkage of a singlecoil. Ideally, the total flux through one magnet pole is linked byall turns in a coil, which is achieved in slotted machines havingone coil per magnet pole and per phase. Whereas the teeth ina slotted machine provides a low reluctance path for the fluxthrough the coil and all turns in the coil link the same amountof flux, in slotless machines however, this low reluctance pathis not provided and, hence, not all turns show the same fluxlinkage. To determine the pitch factor in a slotless machine asis shown in Fig. 4, the average flux linkage of a single turnis calculated by varying the angular span, 2αt, of a turn atradius r

Ψav =1

αc − βo

∫ αc

βo

Ψt(αt)dαt

=1

αc − βo

∫ αc

βo

∫ αt

−αt

rLactB1(r) cos(pφ)dφdαt

=2rLactB1(r)

αc − βo

1p2

[cos(pβo)− cos(pαc)] , (3)

where 2βo is the opening angle of a coil, 2αc is the angularspan of a coil, Lact is the length in axial direction and B1 isthe amplitude of the fundamental harmonic of the magneticflux density. With the maximum flux linkage of a single turnbeing equal to the flux through one magnet pole, Ψmax =2prLactB1(r), the pitch factor, kp, for a slotless machine withconcentrated windings becomes

kp =Ψav

Ψmax=

1pαc − pβo

[cos(pβo)− cos(pαc)] . (4)

B. Distribution factor

The distribution factor is a measure of the electrical align-ment of all coils in a single phase and can be calculated by

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Fig. 5. EMF phasor representation of a three-phase system with p=7 andQ=12.

writing the back-EMF of a single coil in phasor representation(in per unit) −→

E i,pu = ej(γi),

where γi = 2πpQ · i is the electrical angle offset of coil i.

Figure 5 shows an example of a phasor representation ofthe back-EMF for the individual coils and also the resultingphase back-EMF phasors, which are symmetrically shifted by120° electrical degrees from each other. The minus sign for aphasors means that the windings of the coil are connected inthe opposite direction. The phase back-EMF phasors (in perunit) can be calculated according

−→E ph,pu =

∑

Q/3

−→E i,pu.

The distribution factor is found by dividing the magnitude ofthe resulting phase EMF phasor by the number of coils perphase, as given by

kd =|−→E ph,pu|

Q/3. (5)

The method presented in [5] is used to obtain the windinglayout which results in the highest distribution factor for agiven number of magnets poles and coils.

C. Analysis of the winding factor

The winding factor inside a slotless machine with concen-trated windings is calculated for different combinations ofthe number of magnets poles and coils using (2), (4) and(5). Combinations resulting in the same number of coils permagnet pole and per phases, q, have the same winding factorand, therefore, the winding factor as function of q is shownin Fig. 6. As the winding factor also depends on the openingangle of the coil, it is calculated for the case when the openingangle is zero and half the total coil span (βo=0 and βo=0.5αc,respectively). For both cases it can be noticed that machineshaving q=1/4 give the highest winding factor, kw=0.716 andkw=0.955, respectively. These types of machines have kd=1and, therefore, create the peaks in Fig. 6.

Fig. 6. Winding factors for slotless machines with concentrated windingsversus the number of coils per magnet pole per phase.

In the further analysis of the rotary actuator, only magnetpoles and coils combinations resulting in q=1/4 are consideredbecause they provide a high winding factor and can be ob-tained with every number of magnet poles which is a multipleof four. During optimization the number of magnet poles isvaried and the number of coils is determined according toQ=6qp.

IV. MULTI-PHYSICAL FRAMEWORK

A multi-physical framework, containing a 2D semi-analytical magneto-static, thermal and mechanical model, isderived to obtain an optimized design of the rotary actuator.In the following subsections these models are discussed.

A. 2D semi-analytical magneto-static model

The magnetic field distribution inside the slotless permanentmagnet actuator, as depicted in Fig. 7, is obtained by usingthe method as described in [7]. This method provides anaccurate and fast means of determining the torque capabilities,while accounting for the relative recoil permeability, µr, ofthe magnets. In this subsection the models obtained fromthis technique are further extended to account for the two-segmented Halbach array with straight magnetization, whichis shown in Fig. 8.

The method assumes two-dimensional fields in cylindricalcoordinates, infinite permeable iron and linear demagnetizationof the magnets in the second-quadrant of the B-H curve. Dueto the infinite permeability of the iron parts, only the magneticfields inside the airgap (rm ≤ r ≤ rb, indicated by subscriptI) and the magnets (rc ≤ r ≤ rm, indicated by subscriptII) have to be determined. By introducing the vector potentialaccording to

B = ∇×A, (6)

and assuming no current density inside the airgap, Ampere’slaw for the airgap region can be rewritten into the Laplaceequation

∇2AI = 0. (7)

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Fig. 7. Schematic of the slotless permanent magnet moving-coil actuatorincluding the coils.

(a)

(b)Fig. 8. Straight (parallel) magnetized Halbach array with (a) the magnetarray and (b) the magnetization waveform.

Likewise, the Poisson equation for the magnet region can beobtained

∇2AII = −µ0∇×M0. (8)

For both regions the following constitutive relations are used

BI = µ0HI, (9)BII = µ0µrHII + µ0M0. (10)

where M0 is Fourier series representation of the remanentmagnetization vector and is given by

M0 =∞∑

n=odd

[Mnr cos(npφ)er −Mnφ sin(npφ)eφ] . (11)

Fig. 9. Comparison of the flux density inside the airgap calculated analyti-cally and simulated using 2D FLUX (rc=19.5mm, rm=25.7mm, rb=27.5mm,α=2/3, Brem=1.33T, µr=1.1).

The coefficients Mnr and Mnφ are obtained from the Fouriertransformation of the magnetization waveform, which for thetwo-segmented Halbach array with straight magnetization isgiven by

Mr = −Brem

µ0cos(φ + π/p)

Mφ = Brem

µ0sin(φ + π/p)

−π

p <φ <− (2−α)π2p ,

Mr = Brem

µ0sin(φ + π/2p)

Mφ = Brem

µ0cos(φ + π/2p)

− (2−α)π

2p <φ <−απ2p ,

Mr = Brem

µ0cos(φ)

Mφ = −Brem

µ0sin(φ)

−απ

2p <φ <απ2p ,

Mr = −Brem

µ0sin(φ− π/2p)

Mφ = −Brem

µ0cos(φ− π/2p)

απ2p <φ < (2−α)π

2p ,

Mr = −Brem

µ0cos(φ− π/p)

Mφ = Brem

µ0sin(φ− π/p)

(2−α)π

2p <φ <πp .

Here, Brem is the remanent flux density of the magnets andα = φp

φp+φcis the radial magnet arc to the total magnet pole

arc ratio.A description of the magnetic fields inside the the magnet

and airgap region is obtained by finding solutions for the vectorpotential which are governed by the Laplace and Poissonequations and by applying the appropriate boundary conditions[7]. The semi-analytical field solutions show good agreementwith results obtained from finite element modeling (FEM) in2D FLUX (error <1%), as can be seen in Fig. 9 where theflux density inside the air gap is shown.

For a machine with 1/4 coil per magnet pole per phasethe winding arrangement for a machine with moving coils isshown in Fig. 7. The torque is calculated by using the Lorentz’force law

T =∫

V

r× (J×BI)dv =∫

V

JzBrIrdvez, (12)

where V is the volume occupied by the coils, r is the vector tothe point about which the torque is computed, J is the current

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Fig. 10. Radial heat flow inside the actuator with moving coils.

density vector and BI is the airgap flux density vector, which isassumed to be constant along the axial length of the magnets.

B. Thermal modelThe amplitude of the current density inside the coils is

constrained by the temperature distribution inside the actuator.A similar approach like is proposed in [8], is used andmodified to predict the temperature distribution inside therotary actuator. In this approach a thermal equivalent circuit(TEC) of the actuator is created, which offers a fast way topredict the thermal behavior. The actuator geometry is dividedinto lumped components and the thermal behavior of thesecomponents is presented in a network of thermal resistances,capacitances and heat sources.

While the transient thermal behavior of a machine canbe simulated using TEC, in this case only the steady-statetemperature distribution is of concern, and hence, only thethermal resistances and heat sources need to be determined.The thermal model is further simplified by considering onesection of the actuator and assuming only radial heat flow.For the moving coil configuration, Fig. 10 shows the resultingradial heat flow by conduction, convection and radiation. Heatis produced inside the coils and is determined from the ohmiclosses. The linear dependency of the electrical resistivity ofcopper on temperature is accounted for by modeling it asa negative thermal resistance. Iron losses inside the magnetsare neglected because of the low magnetic field created bythe coreless coils. In the moving coil concept, no changingmagnetic field is induced inside the back-iron, because themagnets and back-iron are fixed relative to each other. Inthe moving magnet concept, the iron losses inside the back-iron, which are caused due to the rotating magnets, areapproximately 12 W kg−1 at 1.5 T and 50 Hz [9]. Comparedto the ohmic losses, these iron losses are small and can beignored in the thermal model.

The conductive heat flow across the actuator is modeledby thermal resistances which are determined for the differentcylindrical components. To account for radiation in the TEC,this mode of heat flow is linearized and modeled like convec-tion with a heat transfer coefficient hε ≈ 6ε, where ε is theemissivity. Convection at inner and outer radii of the actuatoris modeled by a heat transfer coefficient, which is estimated tobe 10 W m−2 K−1 [10]. Inside the airgap, forced air coolingis applied, which is also modeled by convection with a heattransfer coefficient, hgap.

Fig. 11. Influence of the airgap heat transfer coefficient on the coil andmagnet temperatures when 0.5 W of heat is produced in the coils at an ambienttemperature of 20 °C.

TABLE IVMATERIAL PROPERTIES.

m k ρ[kg m−3] [W m−1 K−1] [Ω m]

NdFeB 7350 10 -N398 7850 73 -Copper 8900 1 1.678×10−8

Air 1.067 0.0285 -@T=25°C

The resulting TEC is solved as described in [8] and usingthe dimensions of the initial design of the slotless actuatorand the material properties from Table IV, the temperaturesinside the coils and magnets as function of the heat transferfunction, hgap, for both the moving-coil and moving-magnetconfiguration are shown in Fig. 11. As the figure shows, thetemperature drops significantly when hgap is increased from 5to 30 W m−2 K−1. Although high values for the heat transferfunction can be achieved by forced air cooling [11], hgap isset to a value of 15 W m−2 K−1 during the optimizationprocedure. This value was estimated from measurements onthe pre-prototype with an air flow of 25 liters per minutethrough the airgap.

C. Mechanical model

The performance of the actuator strongly depends on themechanical load on it. When no external load is considered,the required torque and force to achieve a certain accelerationare linked to the inertia and total mass of the moving partsof both the rotary and linear actuator. While the design ofthe linear actuator is fixed, the inertia and moving mass ofthe rotary actuator are determined for the moving-magnet andmoving-coil configuration.

V. DESIGN OPTIMIZATION

The multi-physical framework is implemented in MATLABand used in the optimization procedure, which is performedwith sequential quadratic programming (SQP). In this section

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TABLE VLIST OF ALL CONSTRAINTS FOR THE ROTARY ACTUATOR.

Constraint DescriptionBcore<1.4 T Saturation in coreBback<1.4 T Saturation in back-ironri>18 mm Minimum inner radiusro<30 mm Maximum outer radiusL<37 mm Maximum length actuator

lw ≥0.9 mm Radial length coils (windings)Tw<100 °C Maximum coil temperatureTm<60 °C Maximum magnet temperature

the optimization objective, design constraints and results arediscussed.

A. Optimization objective

To create an efficient design of the zφ-module the rotaryactuator is optimized with the objective to minimize the copperlosses inside the rotary and linear actuator combined. Theselosses are calculated for a third order motion profile. By takingthe losses inside the linear actuator into account, the movingmass of the rotary actuator is limited. The objective functioncan be written as

f(α, βo, ri, lc, lm, lw, lb, Lact) = Pcu,rot + I2linRlin, (13)

where α, βo, ri, lc, lm, lw, lb and Lact are the optimizationvariables, which are described in Table III and indicated inFig. 2 and 7. Ilin is the current and Rlin is the resistance ofthe linear actuator, as is given in Table II. The relatively smalliron losses in the back-iron are again ignored.

The initial design of the rotary actuator already showedthat the specifications given in Table I are easily met withinthe volume constraints, making this design oversized for theapplication. Therefore, the design is also optimized with theobjective to minimize the volume of the rotary actuator. Theobjective function can be expressed as

f(α, βo, ri, lc, lm, lw, lb, Lact) = πr2oL, (14)

where ro is the outer radius of the actuator and L is the totallength of the coils.

B. Constraints

The design of the rotary actuator is subject to severalconstraints. First, the volume of the actuator is constrainedby the available height inside the zφ-module, the radius ofthe shaft and the outside radius. Furthermore, a lower boundis set to the radial length of the coils. To guarantee a linearresponse of the actuator to the injected current, saturationof the steel is avoided. Finally, to prevent damage of thewinding insulation, possible irreversible demagnetization ofthe magnets or substantial loss of performance due to a lowervalue of the intrinsic magnetization, the coil and magnettemperatures are constrained. Table V lists all the constraints.

C. Optimization results

The optimization is performed for the moving-magnet con-figuration and moving-coil configuration and Fig. 12 shows

Fig. 12. Minimized copper losses for different magnet pole pairs.

Fig. 13. Minimized volume for different magnet pole pairs.

the minimized copper losses for different magnet counts. Bothconfigurations have a minimum at p=14 and are constrainedby the volume. In Fig. 13, the results of the optimizationwith the objective to minimize the volume of the rotaryactuator, are shown. The figure shows a minimum for p=18 andp=16 for the moving-magnet and moving-coil configuration,respectively. In this case the minimization is limited by themagnet temperature.

As is already mentioned in Section II, the reduced amplitudeof the EMF measurements compared to the 2D semi-analyticalmagneto-static model, is caused by the magnet edge-effects.For the design of the pre-prototype, the peak airgap fluxdensity along the axial length of the magnets is simulatedwith 3D FEM, and Fig. 14 shows a comparison to theflux density predicted with the 2D semi-analytical model. Areduction of 10.3 % of the flux linkage by the coils and,hence, a similar reduction of the EMF, is predicted from thisanalysis. Likewise, the magnet edge-effects are predicted forthe optimized designs and the copper losses in the optimizeddesigns are recalculated while the reduced magnetic loadingis accounted for. The sizes and specifications of these designs

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Fig. 14. 3D FEM to 2D semi-analytical magneto-static modeling comparisonof the airgap flux density along the axial length of the permanent magnet.

are listed in Table VI for both minimized copper losses andminimized volume.

With the objective to minimize the copper losses, theoptimized design with moving coils produces the lowestlosses, which is 30 % less compared to the moving-magnetconfiguration. This is due to a lower inertia, a smaller movingmass and a higher magnetic loading. On the other hand, froma practical point of view, the electrical wiring and additionalmechanical clearance with the moving-coil configuration maybe of a concern. As Table VI shows, these two designs occupythe maximum available space inside the zφ-module. In thenew designs with minimized volume, both the moving-coiland moving-magnet configuration occupy 50 % and 45 %,respectively, less volume than the first two designs. Becausethe magnet edge-effects were analyzed after the optimizationstep, the magnet temperature has slightly exceeded its thermalconstraint. In this case the moving-magnet configuration ispreferred because it has lower copper losses. In all fournew designs, it can be noticed that the linear actuator is thedominating mechanical load and can account for 85 % and87 % of the total inertia and moving mass, respectively. In bothmoving magnet designs, the iron losses are less than 2.2 % ofthe copper losses. Therefore, it is justified to ignore these ironlosses in the thermal model and in (13).

VI. CONCLUSIONS

In this paper an optimized design of a rotary actuator, whichis coupled to a linear actuator, has been obtained using amulti-physical framework. Only combinations of the numbermagnet poles and coils resulting in 1/4 coil per magnet poleper phase have been considered during optimization, because,for slotless machines, they have the highest winding factor.Within the available volume, a design with moving coils resultsin the lowest combined copper losses of the rotary and linearactuator. This design has 30 % less copper losses compared toa configuration with moving magnets. Although only a limitedrotational stroke is required, electrical wiring with movingcoils may still form a point of concern. It is also shown that

TABLE VIOPTIMIZED DESIGNS OF THE ROTARY ACTUATOR WITH MINIMIZED

COPPER LOSSES OR MINIMIZED VOLUME (MC = MOVING COIL, MM =MOVING MAGNET).

Objective Min. Copper losses Min. VolumeConfiguration MC MM MC MMp 14 14 18 16Q 21 21 27 24ri [mm] 21.2 24.7 18.0 18.0ro [mm] 30.0 30.0 22.9 22.6lc [mm] 0.5 0.5 0.5 0.5lm [mm] 5.4 2.2 2.5 2.2lw [mm] 0.9 1.2 0.9 0.9lb [mm] 1.6 1.2 0.6 0.8L [mm] 37 37 31.7 36.2Lact [mm] 18.7 18.6 16.7 20.7α 0.5 0.7 0.57 0.59βo [degree] 3.4 1.5 1.27 1.60V [cm3] 104.6 104.6 52.3 57.9Bgap [T] 1.09 0.77 0.72 0.81Tw [°C] 30.2 42.7 72.4 71.1Tm [°C] 28.8 39.4 65.1 63.8Pcu,tot [W] 7.74 11.08 20.83 16.17Pcu,rot [W] 4.28 6.69 17.52 12.15Pcu,lin [W] 3.46 4.39 3.32 4.02Jtot [kg.mm2] 103 120 92 97Mtot [g] 223 251 218 240

the same specifications for the zφ-module can be met withina 50 % smaller design of the rotary actuator.

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