Hybrid FEM-analytical force and torque models of a reaction sphere actuator

Hybrid FEM-Analytical Force and Torque Models

of a Reaction Sphere ActuatorLeopoldo Rossini, Emmanuel Onillon, Olivier Chetelat, and Yves Perriard

Abstract—This paper presents a hybrid FEM-analyticalmodel for the magnetic flux density, the force and torque ofa Reaction Sphere (RS) actuator for satellite attitude control.The RS is a permanent magnet synchronous spherical actuatorwhose rotor is magnetically levitated and can be acceleratedabout any desired axis. The spherical actuator is composed ofan 8-pole permanent magnet spherical rotor and of a 20-coilstator. Due to the highly complex geometry of the sphericalrotor, consisting of 8 bulk permanent magnet poles withtruncated spherical shape adjusted on the back-iron structurewith truncated octahedral shape, a pure analytical approachis not possible. Therefore, in this article we adopt a hybridapproach in which FEM or measured derived values are com-bined with other boundary conditions on a known analyticalstructure to derive expressions for the magnetic flux density,the force, and the torque. The Laplace equation is solved byexploiting powerful properties of spherical harmonic functionsunder rotation to derive closed-form linear expressions forall possible orientations of the rotor. The proposed modelsare experimentally validated using a developed laboratoryprototype and with finite element simulations.

Index Terms—Spherical actuator, attitude control, electro-magnetic modeling, electromagnetic forces, magnetic levitation.

I. INTRODUCTION

A Reaction Sphere (RS) has been proposed as a po-

tential alternative to Attitude and Orbit Control System

(AOCS) actuators based on Reaction Wheels (RWs) or

Control Moment Gyroscopes (CMGs) [1]. The RS can be

categorized into the family of electromagnetic spherical

actuators, mainly proposed in the robotic literature [2]–[4].

The RS consists of an 8-pole permanent magnet spherical

rotor that is magnetically levitated and can be accelerated

about any axis by a 20-pole stator with electromagnets. A

schematic illustration of the RS is depicted in Fig. 1. A

RS laboratory prototype has been manufactured to validate

force and torque analytical models [5]. In this prototype,

the 8 permanent magnet poles of the spherical rotor have

been discretized using a mosaic of 728 cylindrical magnets

to approximate the desired fundamental spherical harmonic

of the magnetic flux density. More recently, a spherical

rotor optimized to improve its manufacturability has been

designed and manufactured [6]. As illustrated in Fig. 2, this

rotor has 8 bulk permanent magnet poles with truncated

spherical shape that are parallel magnetized and adjusted

on the back-iron structure, which has truncated octahedral

This work was supported in part by the European Space Agency(ESA), under the GSTP contract 20227/06/NL/SFe and the NPI contract40000101313/10/NL/PA, and by maxon motor ag.

L. Rossini, E. Onillon, and O. Chetelat are with the SystemsDivision of the Swiss Center for Electronics and Microtechnology(CSEM), Rue Jacquet-Droz 1, CH-2000 Neuchatel, Switzerland (email:[email protected]).

Y. Perriard is Professor at the Integrated Actuators Laboratory (LAI),Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne, Switzerland(email: [email protected]).

shape. The optimization of design parameters was carried out

to minimize the magnetic flux density distortion with respect

to the fundamental spherical harmonic of degree 3 and order

2 (octupole), which has interesting properties relevant to the

actuator control [5].

In this article, the magnetic flux density, the force and the

torque models of the optimized spherical rotor are derived

and proposed in Section II and Section III, respectively.

As firstly proposed in [5], the procedure to derive these

models exploits powerful properties of spherical harmonic

functions under rotation so that force and torque models for

all possible orientations of the rotor are expressed in closed

form as linear combination of mutually orthogonal force and

torque characteristic matrices. However, contrary to [5], a

pure analytical approach is no longer possible due to the

highly complex geometry of the optimized rotor. Therefore,

in this article we adopt a hybrid approach in which Finite

Element Method (FEM) or measured derived values are com-

bined with other boundary conditions on a known analytical

structure to derive expressions for the magnetic flux den-

sity, the force, and the torque. Subsequently, procedures to

compute the spherical harmonic decomposition coefficients

offline and online are proposed in Section IV and Section

V, respectively. As it will be illustrated, these coefficients

fully determine the magnetic flux density, the force, and the

toque in stator coordinates for any possible orientation of

the rotor. Finally, experimental and numerical verifications

of the proposed models using the developed prototype and

finite element simulations are proposed in Section VI.

II. MAGNETIC FLUX DENSITY MODEL

For the derivation of the magnetic flux density model,

we use the rotor and stator schematics in Fig. 3 with

the respective relevant dimensions. We are interested in

computing the magnetic flux density in the airgap (region

1), which is enclosed in the stator (region 2), which in turn

is surrounded by air (region 3).

Fig. 1: Schematic illustration of the 3-D motor of the RS,

which is composed of a 8-pole rotor (cube) and a 20-pole

stator (dodecahedron).

978-1-4673-4974-1/13/$31.00 ©2013 IEEE 694

Fig. 2: Schematic illustration of the optimized spherical rotor

of the RS.

Fig. 3: Rotor and half stator schematic illustration with

relevant dimensions and region numbers.

A. Constitutive Relations

The constitutive relations describing the magnetic effects

inside the three regions are characterized by

Bi = [Bir Biθ Biφ]T=

{

µ0Hi, for i = 1, 3µ0µrHi, for i = 2

(1)

where the subscript i denotes the region numbers; B and

H are the magnetic flux density and the magnetic field

respectively; µ0 is the vacuum magnetic permeability while

µr is the magnetic permeability of the stator material.

B. Governing Equations

Starting from the magnetostatic field equation for current-

free regions ∇×H = 0, the magnetic field H in region i is

calculated as the gradient of a magnetic scalar potential ϕ[7]

Hi = [Hir Hiθ Hiφ]T= −∇ϕi. (2)

Then, substituting the constitutive relations in equation (1)

into ∇ ·B = 0, and employing equation (2), we obtain the

Laplace equations for regions i = 1, 2, 3

∇2ϕi = 0. (3)

Solutions ϕi of equations (3) are the magnetic scalar po-

tentials inside the three regions. Once these equations are

solved, the magnetic flux density within region i can be

calculated by first employing definition (2) and subsequently

applying the respective constitutive relations (1).

C. General Solutions

The general solution of Laplace’s equation in spherical

coordinates within region i = 1, 2, 3 is [2]

ϕi =

∞∑

n=0

n∑

m=−n

(

κm

n,irn + ξmn,ir

−(n+1))

Y m

n (θ, φ) (4)

where κmn,i

and ξmn,i

are coefficients to be defined using

boundary conditions while Y mn (θ, φ) are complex-valued

spherical harmonic functions.

D. Analytical Boundary Conditions

A set of boundary conditions is necessary to determine

coefficients κmn,i

and ξmn,i

, i = 1, 2, 3, that provide particular

solutions to (3). Boundary conditions can be summarized as

B3r|r→∞ = 0, B3θ|r→∞ = 0, B3φ|r→∞= 0, (5)

Bir|r=Ri= Bi+1,r|r=Ri

, and (6)

Hiθ|r=Ri= Hi+1,θ|r=Ri

, Hiφ|r=Ri= Hi+1,φ|r=Ri

. (7)

Expressions (5) are the boundary conditions to be satisfied

at the far field, where the magnetic flux density approaches

zero when r → ∞. Furthermore, boundary conditions (6)

and (7) specify that the radial component of B must be

continuous across the interfaces and that, in absence of a

free surface current density, the tangential component of H

must also be continuous [7].

E. Measured or Simulated Boundary Conditions

In the proposed approach, the analytical expression of the

magnetic flux density within the rotor cannot be calculated

due to the complex geometry of the optimized rotor. There-

fore, no analytical expression of the magnetic flux density

is available as boundary condition at r = R0. Similarly as

[8], in absence of an analytical expression, this boundary

condition can be either simulated (for instance during the

design optimization phase) or measured. Suppose that the

simulated (or measured) radial component of the magnetic

flux density Bmeas1r is available on the entire spherical surface

at r = Rs with Rs ∈ [R0, R1]. Then, we can decompose

Bmeas1r on a spherical harmonic basis up to degree N as

Bmeas1r =

N∑

n=0

n∑

m=−n

cmn,imm (Rs)Ym

n (θ, φ) (8)

where cmn,imm are spherical harmonic decomposition co-

efficients for the immobile rotor. Proposed techniques to

compute these coefficients for offline analysis or online

operation are proposed in Section IV and Section V. Hence,

by comparing the measurement equation (8) to

B1r|r=Rs= −µ0

∂ϕ1

∂r

∣

∣

∣

∣

r=Rs

, (9)

the measured boundary condition for the immobile rotor can

be expressed as

cmn,imm (Rs) = −µ0nκm

n,1R(n−1)s + µ0 (n+ 1) ξmn,1R

−(n+2)s .

(10)

This equation shall remain valid for any possible orientation

of the rotor inside the stator parametrized using ZYZ Euler

angles α, β, and γ. The effect of such a rotation on the

spherical harmonic decomposition coefficients cmn,imm of

degree n can be formulated as

cmn (α, β, γ) =∑

l

Dn

ml (α, β, γ) cl

n,imm (11)

where Dn

ml(α, β, γ) are unitary rotation matrices [9]. There-

fore, the measured boundary condition for the rotated rotor

can be expressed as

cmn (Rs) = −µ0nκm

n,1Rn−1s + µ0 (n+ 1) ξmn,1R

−(n+2)s .

(12)

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F. Solution

Because the magnetic flux density is used to compute

forces and torques, only the magnetic scalar potential ϕ1

within the airgap (region 1) will be calculated. Invoking

the orthogonal property of spherical harmonics, using the

definition in equation (2) together with general solution

(4), analytical boundary conditions (5)-(7) and measured

boundary condition (12) result in

κmn,3 = 0

nµrκmn,2R

n−12 − (n+ 1)µrξ

mn,2R

−(n+2)2 =

nκmn,3R

n−12 − (n+ 1) ξmn,3R

−(n+2)2

nκmn,1R

n−11 − (n+ 1) ξmn,1R

−(n+2)1 =

nµrκmn,2R

n−11 − (n+ 1)µrξ

mn,2R

−(n+2)1

κmn,2R

n2 + ξmn,2R

−(n+1)2 = κm

n,3Rn2 + ξmn,3R

−(n+1)2

κmn,1R

n1 + ξmn,1R

−(n+1)1 = κm

n,2Rn1 + ξmn,2R

−(n+1)1

cmn (Rs) = −µ0nκmn,1R

n−1s + µ0 (n+ 1) ξmn,1R

−(n+2)s ,

(13)

which is a system of 6 linear equations for the 6 unknown

coefficients κmn,i

and ξmn,i

, i = 1, 2, 3. Although the values

of κmn,i

and ξmn,i

can be easily calculated using a program of

symbolic calculation to solve the linear system of equations

(13), their expressions are relatively long for a finite value

of stator permeability µr. Therefore, solving (13) for an

infinitely permeable stator (µr → ∞), using general solu-

tion (4), expression (2), and constitutive relations (1), the

magnetic flux density within the airgap (region 1) can be

formulated as

B1 =N∑

n=0

n∑

m=−n

cmn (Rs)Bm

n,1 (14)

where

Bm

n,1 = ∇

{

rn −R2n+11 r−(n+1)

nRn−1s + (n+ 1)R

−(n+2)s R2n+1

1

Y m

n (θ, φ)

}

.

(15)

III. FORCE AND TORQUE MODELS

In the previous section we derived a hybrid FEM-

analytical expression for the magnetic flux density within

the airgap of the RS. In this section, given the magnetic flux

density, analytical expressions for the force and the torque

will be derived. The procedure is based on the Lorentz force

law and is an extension of the one presented in [5], where

spherical harmonics of degree 3 only are taken into account.

By exploiting the superposition principle and applying the

Lorentz force law, the force and torque developed by the RS

can be expressed in matrix form as [5]

F = KF i, and T = KT i (16)

where i ∈ R20×1 is the current vector flowing in the coils

while KF ∈ R3×20 and KT ∈ R

3×20 are respectively force

and torque characteristic matrices defined as

KF =N∑

n=0

n∑

m=−n

cmn (Rs)Km

F,n, and

KT =N∑

n=0

n∑

m=−n

cmn (Rs)Km

T,n. (17)

Hence, force and torque characteristic matrices KF and KT

are expressed as a linear combination of force and torque

characteristic matrices Km

F,n∈ R

3×20 and Km

T,n∈ R

3×20

given by each spherical harmonic of degree n and order mtaken into account. Notice that these expressions are valid

for any orientation of the rotor, which is directly taken

into account by the decomposition coefficients cmn (Rs).Moreover, Km

F,nand Km

T,nare constant and can be computed

offline. Force and torque characteristic matrices Km

F,nand

Km

T,ncan be expressed as

Km

F,n =[

R1Fm

1,n, R2Fm

2,n, . . . , R20Fm

20,n

]

, and

Km

T,n =[

R1Tm

1,n, R2Tm

2,n, . . . , R20Tm

20,n

]

(18)

where Rk, k = 1, . . . , 20, are rotation matrices defining the

position of each coil while Fm

k,nand T

m

k,nare analytical

expressions of the force and torque given by each coil and

obtained by integrating the spherical harmonic of degree nand order m. Details on the calculation of F

m

k,nand T

m

k,n

can be found in [5].

IV. OFFLINE DETERMINATION OF cmn AND SPHERICAL

HARMONIC ANALYSIS

The magnetic flux density within the airgap in (14), force

and torque models in (17), all depend on the spherical

harmonic decomposition coefficients of the rotated rotor

cmn (Rs). For offline analysis, these values can be determined

starting from the spherical harmonic decomposition coef-

ficients of the immobile rotor cmn,imm through the rotation

transformation in (11). In this section, we propose to mea-

sure the spherical harmonic decomposition coefficients of the

immobile rotor cmn,imm using FEM simulations. Supposing

that the radial component of the simulated magnetic flux

density Bmeas1r in (8) is available on the entire spherical

surface at r = Rs, the decomposition coefficients can be

computed using integration as

cmn,imm (Rs) =

2π∫

0

π∫

0

Bmeas1r (Rs, θ, φ)Y m

n (θ, φ) sin θ dθdφ.

(19)

where Y mn is the complex conjugate of Y m

n . Notice, however,

that if Bmeas1r is only available at discrete points, for instance

on the equiangular latitude-longitude grid, the computation

of cmn,imm (Rs) can also be performed using the sampling

theorem [10].

To compute Bmeas1r with FEM simulations, we use nu-

merical values of the geometry parameters summarized in

Table I. Permanent magnets are simulated by applying their

linear constitutive relations B = µmµ0H + Br, where the

relative magnetic permeability µm is fixed to 1 and the

module of the remanent magnetic flux density Br is equal

to 1.36 T. The non-linear magnetic properties of the back-

iron material (X46Cr13) are modeled through the nonlinear

BH curve and its relative permeability, taking into account

saturation. Notice that, to simplify the following experi-

mental verification, the spherical harmonic decomposition

coefficients cmn,imm are computed using a non-ferromagnetic

stator. Hence, the analytical expression of the magnetic flux

density without the magnetic stator can be easily obtained

from equation (15) by calculating the limit for the inner

stator radius R1 → ∞. Electromagnetic 3-D finite element

simulations are performed using the AC/DC module of

COMSOL Multiphysics v4.3. The module of the normalized

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TABLE I: Parameters for finite elements simulations.

c Eccentricity 23.7 mmRM Magnet Radius 63.8 mmh Octahedron structure height 111.1 mmHR Back-iron thickness 6.0 mmHC Cut depth 3.5 mmRs Radius 98.0 mm

spherical harmonic decomposition coefficients with N = 20are illustrated in Fig. 4. As it can be observed, the funda-

mental component of the magnetic flux density corresponds,

as desired, to the spherical harmonic of degree 3 and order

2 (octupole). However, higher order harmonics with degree

n = 7, 9, 11, 13, 15, 17, and 19 also appear in the spectrum.

In light of the spherical harmonic content, which is virtually

zero for degrees n /∈ I = {3, 7, 9, 11, 13, 15, 17, 19}, the

magnetic flux density, the force, and torque models can be

reformulated and approximated by considering the maximum

spherical harmonic degree Nh taken into account in the

model. Therefore, for a given Nh, the magnetic flux density

in (14) is formulated as

BNh

1 =[

BNh

1r BNh

1θ BNh

1φ

]T

=∑

n∈In≤Nh

n∑

m=−n

cmn (Rs)Bm

n,1

(20)

while force and torque characteristic martrices in (17) are

calculated as

KNh

F=

∑

n∈In≤Nh

n∑

m=−n

cmn (Rs)Km

F,n, and

KNh

T=

∑

n∈In≤Nh

n∑

m=−n

cmn (Rs)Km

T,n. (21)

The influence of the number of harmonics Nh will be

investigated in Section VI.

V. ONLINE DETERMINATION OF cmn - MAGNETIC STATE

ESTIMATION

The spherical harmonic decomposition coefficients cmnfully determine the magnetic state of the actuator within the

Fig. 4: Normalized spherical harmonic decomposition coef-

ficients cmn,imm (Rs) of the immobile rotor.

airgap, the force, and the torque for any possible orientation

of the rotor.

To predict forces and torques during the actuator design

optimization procedure, for analysis purposes, or for offline

operation, these spherical harmonics decomposition coeffi-

cients cmn can be determined, as discussed in the previous

section, either by integration using (19) or, alternatively,

from flux density values taken on the equiangular latitude-

longitude grid [10].

For real-time and control operation, however, computing

these coefficients by integration is not possible because

of the large amount of data required. The approach using

measurements on the equiangular latitude-longitude grid is

attractive but forces the sensors to be placed on the latitude-

longitude grid, which is not feasible due to the presence

of the stator coils. Moreover, according to the sampling

theorem [10], and taking into account symmetries of the

magnetic flux density, B2 measurements are theoretically

necessary to completely reconstruct a band-limited function

with bandwidth B. For instance, in our situation, if we take

into account a maximum spherical harmonic degree Nh, the

total number of necessary measurements will be N2h .

For the fundamental harmonics of degree n = 3, sampling

the radial component of the magnetic flux density at N ≥ 7mutually non-collinear locations equidistant from the rotor

surface to subsequently solve the decomposition problem

using least-squares techniques was proposed in [5]. Extend-

ing this technique for higher spherical harmonic degrees is

possible but currently not practical. As a matter of fact,

the number of necessary measurements to reconstruct up

to a given degree Nh is equal to∑

n∈In≤Nh

(2n+ 1). Hence,

for example, 22 measurements are necessary with Nh = 7.

Therefore, reconstructing the magnetic flux density withing

the airgap to determine the spherical harmonics coefficients

cmn for n > 3 requires a considerable number of measure-

ments and cannot, at present, be taken into consideration for

our real-time framework. For this reason, for the real-time

control model of the force and torque, only the fundamental

harmonic of degree n = 3 will be reconstructed and only

coefficients cm3 will be computed.

The procedure presented in [5] to compute cm3 (Rs) and to

update KF and KT will briefly be summarized here. Starting

from expression (20), for Nh = 3, the radial component

of the magnetic flux density evaluated at r = Rs can be

expressed as

B31r (Rs, θ, φ) =

n∑

m=−3

cm3 (Rs)Ym

3 (θ, φ) . (22)

Then, defining the k-th magnetic flux density measurement

as B⊥1,k = B3

1r (Rs, θk, φk) and decomposing cm3 into real

and imaginary part as

cm3 = am3 + ibm3 , |m| ≤ 3, (23)

we can write

B⊥1,k =

a032R0

3 (θk, φk) +

3∑

m=1

am3 Rm

3 (θk, φk)

+

3∑

m=1

bm3 Im3 (θk, φk) (24)

where Rm3 (θk, φk) = 2Re {Y m

3 (θk, φk)} and

Im3 (θk, φk) = −2Im {Y m3 (θk, φk)}. Then, defining

697

a vector of Nm magnetic flux measurements as

B⊥1 =

[

B⊥1,1, B

⊥1,2, . . . , B

⊥1,Nm

]T, the desired coefficients

x =[

a03, a13, a

23, a

33, b

13, b

23, b

33

]Tcan be computed

uniquely solving the measurement equation

A (Γ)x = B⊥1 . (25)

The matrix A (Γ) has dimension Nm × 7 and contains the

spherical harmonic basis of degree 3 as

A (Γ)T=

12R

03 (γ1)

12R

03 (γ2) . . . 1

2R03 (γNm

)R1

3 (γ1) R13 (γ2) . . . R1

3 (γNm)

R23 (γ1) R2

3 (γ2) . . . R23 (γNm

)R3

3 (γ1) R33 (γ2) . . . R3

3 (γNm)

I13 (γ1) I13 (γ2) . . . I13 (γNm)

I23 (γ1) I23 (γ2) . . . I23 (γNm)

I33 (γ1) I33 (γ2) . . . I33 (γNm)

,

(26)

where Γ is the parameter vector of sensor spherical angular

coordinates

Γ = [γ1, γ2, . . . , γNm] , (27)

with γk = (θk, φk), k = 1, 2, . . . , Nm. Finally, using (23),

the force characteristic matrix KF in (21) can be readily

calculated as

K3F =

x1

2·K0

F,R + x2 ·K1F,R + x3 ·K

2F,R + x4 ·K

3F,R

+ x5 ·K1F,I + x6 ·K

2F,I + x7 ·K

3F,I (28)

where xk is the kth entry of the solution vector x, Km

F,R =

2Re{

Km

F,3

}

, and Km

F,I = −2Im{

Km

F,3

}

, with m = 1, 2, 3.

Although not reported, the torque characteristic matrix can

be derived as in (28). Finally, notice that the matrices Km

F,R,

Km

F,I, Km

T,R, and Km

T,I are constant and are computed offline.

VI. VERIFICATION WITH FEM SIMULATIONS AND

LABORATORY PROTOTYPE

A. Magnetic Flux Density Model

In this section, the magnetic flux density model is verified

using FEM simulations and experimental measurements for

various values of the maximum number of harmonics Nh

taken into account. The setup for the rotor magnetic flux

density measurement is reported in Fig. 5. The magnetic flux

density is measured with a Gaussmeter (MAGNETPHYSIK

FH55) at J = 8 inclination angles θ = [15, 25, ..., 85]degrees. The rotor is supported by a rigid guiding axis and

connected to an AC electric motor so that, for each angle θ,

a total of K = 600 points equally distributed on the interval

φ ∈ [0, 360] degrees are recorded. The magnetic flux density

is measured at I = 4 radial distances r = [92, 94, 96, 98]mm. Given values of the radial component of the measured

magnetic flux density B1r,meas (r, θ, φ) and values of the

developed model BNh

1r (r, θ, φ) computed as in (20), we

define the Mean Normalized Relative Error (MNRE) as

MNRE(%) =

I∑

i=1

J∑

j=1

K∑

k=1

ǫ (ri, θj , φk)

IJK· 100 %, (29)

where

ǫ (ri, θj , φk) =

∣

∣

∣BNh

1r (ri, θj , φk)−B1r,meas (ri, θj , φk)∣

∣

∣

maxj,k

|B1r,meas (ri, θj , φk)|.

(30)

Fig. 5: Setup for rotor magnetic flux density measurements.

The MNRE can be computed similarly for the remaining tan-

gential components of the flux density, which are evaluated

using FEM simulations.

In Fig. 6, we report the measured magnetic flux density

compared to FEM simulations and values derived using the

developed model at inclination angles θ equal to 55, 65, and

75 degrees. Values computed using the developed model are

reported for maximum spherical harmonics degrees Nh = 3and Nh = 19. As can be observed, measurements are in good

agreement with the expected simulated values. Furthermore,

the magnetic flux density profiles are sinusoidal although

higher order harmonics are observed for θ equal to 65 and 75

degrees. These fluctuations are due to the gaps at the vertexes

of the truncated octahedron structure (see Fig. 2). Finally,

notice that the profiles computed using the hybrid FEM-

analytical model with Nh = 3 and Nh = 19 are in good

agreement with measured values for θ equal to 55 and 65

degrees. For θ equal to 75 degrees, where fluctuations due to

the vertexes occur more singificantly, with Nh = 3 the model

can only capture the fundamental component of the profile,

which is completely approximated using Nh = 19. In Fig.

(7) we report the MNRE values computed using (29) as a

function of the maximum spherical harmonic reconstruction

degree Nh. As expected, the higher the number of harmonics

taken into account in the model is, the smaller the MNRE

becomes.

B. Force and Torque Models

In this section, force and torque direct models are verified

with FEM simulations for 5 randomly-generated orientations

of the rotor. Specifically, for each randomly-generated rotor

orientation, parametrized using ZYZ Euler angles α, β, and

γ, 3 different current vectors i are computed using the

inverse model of (16) as i =[

K3F

]+F +

[

K3T

]+T [5].

Matrices K3F

and K3T

are computed as in (21) using Nh = 3while F and T are randomly generated forces and torques

with norm equal to 98.1 N and 0.2 Nm, respectively. Finally,

+ is the matrix pseudo-inverse operator. Subsequently, these

15 current vectors i are applied to the finite element model,

to compute FFEM and TFEM, and to the developed hybrid

FEM-analytical force and torque model for various values of

Nh, to compute FNh and T

Nh . Notice that FNh and TNh

are computed with (16) using KF and KT defined as in

(21) with Nh ∈ I. To compare simulated forces FFEM and

698

0 45 90 135 180−500

−250

0

250

500

φ (deg)

Br(m

T)

θ = 55 deg

Model, Nh = 3Model, Nh = 19MeasurementsFEM

0 45 90 135 180−500

−250

0

250

500

φ (deg)

Br(m

T)

θ = 65 deg

Model, Nh = 3Model, Nh = 19MeasurementsFEM

0 45 90 135 180−500

−250

0

250

500

φ (deg)

Br(m

T)

θ = 75 deg

Model, Nh = 3Model, Nh = 19MeasurementsFEM

Fig. 6: Measured magnetic flux density compared to FEM

simulations and values derived using the developed model

at θ = 55 (top), 65 (middle), and 75 (bottom) degrees.

torques TFEM to those computed using the analytical model

FNh and T

Nh , we use the norm relative error defined, for

the force, as

∣

∣

∣

∣

‖FFEM‖−‖FNh‖‖FFEM‖

∣

∣

∣

∣

·100%. The expression for the

torque norm relative error is computed similarly. Moreover,

the angle between FFEM and FNh and between TFEM and

TNh is also used for comparison.

The mean and standard deviation of the norm relative

errors as a function of the maximum spherical harmonic

degree Nh taken into account in force and torque models are

reported in Fig. 8 (top). The mean and standard deviation of

the angle errors are reported in the same figure (bottom). As

it can be noticed, the norm relative errors and angle errors,

3 5 7 9 11 13 15 17 190

1

2

3

4

5

6

7

MNRE(%

)

Maximum spherical harmonic degree Nh

MNRE Br

MNRE Bθ

MNRE Bφ

Fig. 7: MNRE values as a function of the maximum spherical

harmonic reconstruction degree Nh

together with their variances, can be made smaller by in-

creasing the number of harmonics Nh taken in consideration

in the models.

3 5 7 9 11 13 15 17 190

2

4

6

8

10

12

14

16

Maximum spherical harmonic degree Nh

Meannorm

error(%

)

ForceTorqueσ forceσ torque

3 5 7 9 11 13 15 17 190

1

2

3

4

5

6

7

8

9

Maximum spherical harmonic degree Nh

Meanan

gleerror(deg)

ForceTorqueσ forceσ torque

Fig. 8: Force and torque mean norm (top) and mean angle

(bottom) errors as a function of the maximum spherical

harmonic degree Nh. Standard deviations are also depicted

taking into consideration the 15 configurations.

VII. CONCLUSIONS

In this article we presented the development of magnetic

flux density, force, and torque hybrid FEM-analytical models

for the optimized RS rotor. The development of the magnetic

699

flux density model uses the general analytical solution to

the Laplace equation within the RS airgap with analytical

and simulated/measured boundary conditions to determine

the particular solution. Then, based on the derived magnetic

flux density model, analytical expressions for the force and

torque were derived. The calculation of force and torque

models relies on the Lorentz force law and exploits the

superposition principle.

The spherical harmonics analysis performed on the sim-

ulated magnetic flux density in Section IV showed that,

as desired, the magnetic flux density is mainly based on

its principal spherical harmonic of degree 3 and order 2.

However, higher order components of degree 7, 9, 11, 13,

15, 17, and 19 also appear in the spectrum and are taken

into consideration in the developed hybrid FEM-analytical

models.

Experimental verifications on the magnetic flux density

have been performed and showed a good correspondence

between measured data and values computed using the

developed magnetic flux density model. It was illustrated

that the error between measured and analytical values can

be reduced by increasing the maximum spherical harmonic

degree Nh taken into account in the model.

Finally, FEM simulations have been performed to validate

the developed force and torque models. As expected, the av-

erage error between the analytical and simulated forces and

torques decreases by increasing the number of harmonics

taken into consideration in the model.

Experimental measurements to validate the developed

force and torque models together with the online technique

to estimate the spherical harmonics coefficients (as described

in Section V) are currently ongoing.

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