Ky Ngac

Eric Nguyen

N Semail

F Bracikowski

Gillon

F Tan

J Gong

Q Zhao

H Wang

N-K Nguyen

E Semail

N Bracikowski

F Gillon

Optimal Design of a Five

HAL is a

Φ Abstract --This paper proposes the design and development of a five-phase external rotor permanent magnet synchronous machine (PMSM) which is used for direct drive convey application. Firstly, the slot/pole combination of fraction slot concentrated winding (FSCW) is selected according to three criteria; secondly, the analytical model of the proposed machine topology is built and optimized, the initial machine parameters are then obtained; thirdly, the machine is optimized by combing the finite element model and the Kriging model, and the final optimal results are compared to the initial one. Detail design principles and performance characteristics of the proposed machine topology are presented and validated with finite element models. Index Terms-Five-phase machine, external rotor, permanent magnet, convey application.

I. INTRODUCTION

ITH redundant design in machine phases, multiphase machines (>3 phases) are essentially characterized with high reliability which makes them widely used in in critical applications, such as historically marine electric propulsion [START_REF] Zhang | Electromagnetic and Structural Design of a Novel Low-Speed High-Torque Motor With Dual-Stator and PM-Reluctance Rotor[END_REF], aerospace [START_REF] Zhao | Multiphase PMSM With Asymmetric Windings for More Electric Aircraft[END_REF] and automotive traction [START_REF] Yang | Development of an External Rotor V-Shape Permanent Magnet Machine for E-Bike Application[END_REF], with multiphase machines supplied by a voltage source inverter. Compared to classical wye-coupled three-phase machines, which possess the minimum number of independent currents for achieving a rotating field in normal condition, multiphase machines have more degrees of freedom than the minimum necessary, thus allowing a rotating field even with one opened phase.

In the context of high reliability requirement, this paper proposes and develops an external five-phase PMSM. The design adopts fractional-slot concentrated winding which is characterized by short end windings, high power density, high efficiency, high slot-filling factor, small cogging torque, etc. [START_REF] Farshadnia | Design of Optimal Winding Configurations for Symmetrical Multiphase Concentrated-Wound Surface-Mount PMSMs to Achieve Maximum Torque Density Under Current Harmonic Injection[END_REF][START_REF] Bianchi | Design and Tests of a Four-Layer Fractional-Slot Interior Permanent-Magnet Motor[END_REF][START_REF] Zhu | Novel 24-slots14-poles fractionalslot concentrated winding topology with lowspace harmonics for electrical machine[END_REF]. Therefore, the proposed machine can be benefits with high power, high torque and high efficiency capability.

The paper will be organized as follows. After a brief introduction about the multiphase machine, the selection of the proposed machine topology is presented in section II. The analytical model is built, and optimization problem is formulated and optimized in section III. Based on the initial design obtained in section II, a finite element model (FEM) F. Tan, J. Gong are with the School of Electrical Engineering, Shandong University, Jinan, 250061, China (e-mail: gongjinlin@sdu.edu.cn).

H. Wang, Q. is constructed and optimized by combining a Kriging model in section IV. The performance of the proposed machine is analyzed using FEM in section V.

II. MOTOR TOPOLOGY

For the direct drive convey application, the external rotor structure with surface mounted permanent magnet is choosed due to the high torque density. The selection of the slot/pole combination is the first step for the design of machine. The relationship between the number of slots and the number of poles for a 5-phase machine should be respected as follows [START_REF] Demir | Design, analysis and validation of a dual three-phase 72-slot, 12-pole permanent magnet synchronous motor[END_REF]:

, 5 { 2p} s s N k GCD N = ( 1
)
Where Ns is the number of slots, GCD is the greatest common multiple, p is the number of poles, and k is an integer.

A. LCM and GCD

Greatest common divisor (GCD) and least common multiple (LCM) of the slot/pole combination are firstly taken into account. The larger the value of the GCD, the smaller the torque ripple [START_REF] Yang | Development of an External Rotor V-Shape Permanent Magnet Machine for E-Bike Application[END_REF]; and the larger the LCM, the smaller the noise and vibration. TABLE I lists the different combinations of the slots and poles for 5-phase machines. The former number is GCD, the latter number is LCM. It can be noted that, the families have been sorted by background color in TABLE I. Orange family exhibits low LCD, and blue family is with low LCM. The one in black is not feasible, and the red values represent the high LCM and GCD, i.e. potential candidates.

B. Winding factor

The Winding factor of high value is also an important parameter which is related to the torque density. The windings factors of the three candidates selected in part A are calculated and shown in TABLE II. In which, kw1 represents the fundamental winding factor, and kw3 represents the third harmonic winding factor which is also considered due to the specialty of multiphase machines. As can be seen from the Table II, 40s/36p, 40s/44p have higher winding factors which will be taken as candidates for further study.

C. MMF distribution

The mmf spectrum of the last two candidates are shown in Fig. 1. It can be seen that the same spectrum is obtained. However, the working harmonic is the 18 th and the 22 th respectively. It should be noted that the small value of the thickness of the back-iron can be designed with bigger value of poles. First determine the magnetic density of the motor, which is the basis of the motor design parameters and radial dimensions are all obtained from the magnetic density. Fig. 2 is the air gap flux density waveform. ξR is the mechanical angle in the rotor coordinate system, bE1 is the pole arc coefficient, BRE1 is the air gap magnetic density amplitude. Combined with Fig. 2, the expression of the airgap flux density BE1 is as follows:

( )

1 1 1 0 4 1 ( , ) sin cos ( ) 2 RE E E S S i B b B h hp h π ξ θ ξ θ π ∞ =   = -      (2) 2 1, S R h i ξ ξ θ = + = + (3)
Where ξS is the mechanical angle in the stator coordinate system, θ is the rotor position angle.

According to (2) (3), the expression of the magnetic flux of one turn of the coil ( )

θ Φ
can then be obtained:

1 1 3 2 0 sin( ) 8 2 ( ) sin( ) cos( ) 2 p RE E i h k B b R l h hp p h π π θ θ π ∞ = Φ =  ( 4
)
Where kp is the pitch factor, l is the axial length of the motor,R3 is the average radius of the air gap, which is signed in the Fig. Φ can be obtained:

1 1 3 2 0 sin( ) 16 2 ( ) sin( ) cos( ) 2 w s RE E ph i h k N B b R l h hp h π π θ θ π ∞ = Φ =  (5)
According to [START_REF] Bianchi | Design and Tests of a Four-Layer Fractional-Slot Interior Permanent-Magnet Motor[END_REF], the electromotive force ( ) e t can be expressed:

1 1 3 0 sin( ) 16 2 ( ) sin( )sin( ) 2 w s RE E i h k N B b e t R l h h t h π ω π ω π ∞ = =  (6)
Motor design adopts virtual motor design. Each virtual two-phase motor corresponds to a set of harmonics. The torque of a five-phase motor Tem can be expressed as:

1 1 2 em F s E T K AR N π ω = (7) 2 1 1 E I F I K K K K + = + (8) 1 1 3 3
,

E I E I K K E I = = ( 9
)
Where En is the rms value of the n th harmonic of EMF, and In is the rms value of the n th harmonic of the injected current. A is the line load. Substituting ( 6) into [START_REF] Demir | Design, analysis and validation of a dual three-phase 72-slot, 12-pole permanent magnet synchronous motor[END_REF], the torque expression Tem can be rewritten as:

1 1 1 1 3 8 sin( ) 2 2 w E em RE F k b T B K AR R l π = (10)
B. Determination of the working point of PM 4. The working point of the permanent magnet *Blue is the load line. Red is the characteristic line Fig. 4 is the working point of the permanent magnet, which is the intersection of the load line and the characteristic line of the permanent magnet.

1 AE B ' 1 AE B ' 1 AE H 1 AE H Fig.
According to Ampere's loop theorem(C1 is signed in Fig. 3), assuming that all the magnetic flux of the permanent magnet enter the air gap and the stator and there is no magnetic leakage, the thickness of the permanent magnet LAE is obtained as:

' ' ' 2 ( ) 2(
)

AE RME KE AE RME AE ROT ROT STA STA t E RME ROT STA t RME AE B K e L J B B L B L B L J B µ µ µ µ µ = + - + + - ( 11
)
Where JRME is the remanence of the permanent magnet. e is the air gap length. BROT , BSTA and Bt are the magnetic densities of rotor yoke, stator yoke and teeth respectively. LROT , LSTA and LE are the magnetic path length of rotor yoke, stator yoke and teeth respectively. μRME, μROT, μSTA, μt are the relative permeability of permanent magnet, rotor yoke, stator yoke and teeth respectively. KKE is Carter coefficient, which is affected by the shape of the slot.

As long as the magnetic density and torque of the motor are known, the radial size of the motor can be obtained. Therefore, the inverse model of the motor can be derived according to the direct model, so as to prepare for the optimization of the motor.

C. Analytical Model Optimization

Fmincon is a matlab function for solving the minimum value of nonlinear multivariate functions. According to the principle of motor design, taking the maximum torque as the optimization goal, the motor design parameters are optimized.

The optimization conditions are as follows:

{ }

1 min s.t. 1.0, 1.5, 1.732 CSE, CRE, E, AE, tooth L L L L L RE STA t f B B B  -    ≤ ≤ ≤  (12) with [ ] [ ] [ ] [ ]
25, 45 , 60,120 , 8,15 , 20,30

CSE E AE tooth em L L L L f T ∈ ∈ ∈ ∈ = (13 )
According to (12) (13), The optimization problem contains one objective function and three constraint functions, the purpose is to get the maximum torque as possible. At the same time, magnetic density must be limited.

After the optimization of the fmincon function, the initial motor parameter values can be obtained in TABLE III. On the basis of these initial values, the variable range is further narrowed, and then the Kriging proxy model is optimized. Kriging model is an interpolation method for optimal linear unbiased estimation of spatially distributed data. It can estimate the error distribution of the model, has strong local refinement ability, and is suitable for highly nonlinear responses. It is based on finite element model analysis, which can greatly shorten the time period of finite element calculation and improve work efficiency [START_REF] Xiao | Multi-Objective Pareto Optimization of Electromagnetic Devices Exploiting Kriging With Lipschitzian Optimized Expected Improvement[END_REF], The flowchart of kriging model Optimization as shown in Fig. 5.

D. Kriging model Optimization

In the case, firstly, 100 sample points are selected for finite element analysis, and these 100 sample points are selected by the Latin hypercube method. After the simulation of 100 sample points, it is substituted into the model. According to the evaluation value of the finite element model, an initial model is established for each objective function and constraint function, and the pareto optimal solution set is finally obtained, so as to obtain the optimal solution for motor optimization.

The kriging model optimization results are shown in TABLE III.

IV. PERFORMANCE ANALYSIS

A. No-Load Analysis

The final design of kriging model optimization results in the Table III is analyzed using FEM. Fig. 6 shows the 1/2 structure of the design, and the magnetic field line and the magnetic density are obtained by FE method. In the context of the direct drive convey application, there exists a mechanical constraint on the thickness of the rotor back-iron, which has been taken into account during the design step. It explains the lower value of the flux density in the rotor backiron. Cogging torque is the torque generated by the interaction between the permanent magnet and the stator teeth, which can influence the quality of the output torque and thus cause vibration of the drive system. It is desired to as small as possible and is taken into account within design step. Fig. 7 shows the cogging torque waveform. It can be seen from the Fig. 7 that the maximum cogging torque is about 80 N‧m, which is about 0.39% of the average output torque. The no-load trapezoidal back-EMF waveform and its spectrum are shown in Fig. 8 and Fig. 9 respectively. It can be seen from Fig. 9 that the amplitude of the fundamental harmonic is about 930V, and the amplitude of the 3 rd harmonic is about 135V, the other harmonics are negligible. 11 shows the wave forms of the torques at the speed 75 rpm. As expected, the torque with the two harmonic current injection (1 st + 3 rd ) is 11.8% higher than the one obtained with only the 1 st harmonic current. It can be observed that the torque ripples are only 1.01% and 1.33%, respectively. This paper presents the design and development of a fivephase PMSM with external rotor structure for direct convey application. The 5-phase/40-slot/44-pole topology of FSCW is firstly selected according to three criteria: torque ripple, winding factor and mmf distribution. Then, the analytical model of the proposed structure is built and the initial design is obtained using optimization process. The FEM of the initial design is constructed and optimized by integrating a Kriging model. The performances of the final optimized design are shown based on the FE analysis. A prototype is producing and all the analysis will be compared to the experiment results in the final paper.

Fig. 2 .

 2 Fig. 2. Air gap flux density waveform

Fig. 3 .

 3 Fig. 3. Motor radial dimension drawing According to (4), the expression of the magnetic flux of each phase ( ) ph θ

Fig. 5 .

 5 Fig. 5. Kriging Model Optimization Flowchart

Fig. 6 .

 6 Fig. 6. magnetic field distribution under no-load condition

Fig. 7 .

 7 Fig. 7. The cogging torque waveform.

Fig. 8 .

 8 Fig. 8. The no-load back EMF waveform.

Fig. 9 .

 9 Fig. 9. Fourier decomposition of the no-load back EMF of phase A

Fig. 10 .

 10 Fig.10. The two kinds current supply Fig.11shows the wave forms of the torques at the speed 75 rpm. As expected, the torque with the two harmonic current injection (1 st + 3 rd ) is 11.8% higher than the one obtained with only the 1 st harmonic current. It can be observed that the torque ripples are only 1.01% and 1.33%, respectively.

Fig. 11 .

 11 Fig. 11. The wave forms of the torques V. CONCLUSION

  Zhao are with the Excellent Electric Co. Ltd., Shandong Province, Jining City 272000, China.

	ParisTech,	Centrale	Lille,	HEI,	EA	2697-L2EP-Laboratoire
	d'Electrotechnique et d'Electronique de Puissance, F-59000 Lille, France.
	N. Bracikowski is with IREENA University of Nantes, 44600 Saint-
	Nazaire, France.

N-k. Nguyen, E. Semail, F. Gillon are with Univ. Lille, Arts et Métiers


Fabio Briscese

Francesco Calogero

Conjectures analogous to the Collatz conjecture

In this paper we introduce some conjectures analogous to the well-known Collatz conjecture.

Introduction

The Collatz conjecture (also called "3n + 1 conjecture", or with various other names) [1] [2] is, so far, neither proven, nor shown to be unprovable. It is considered "one of the most famous unsolved problems in mathematics" [1]. In this paper we briefly describe the Collatz conjecture, some of its known variants, and some related, presumably new, analogous conjectures.

, and it was him who first brought the attention of FC to this topic, and who made the numerical

]ocnmp[ F Briscese and F Calogero calculations reported in a joint paper never published (the last typed copy of this paper is dated February 4th, 2019, and we recently put it on arXiv [START_REF] Bruschi | Variations on a Theme of Collatz[END_REF]).

Also Decio Levi got his Laurea at the Department of Physics of the University of Rome "La Sapienza" by defending a dissertation prepared under the supervision of FC. Both Mario and Decio were only half a generation younger than FC, so we subsequently became quite good personal friends among ourselves; they both co-authored some papers with FC and many more were co-authored by the 2 of them, and several papers also with other members (of their generation or somewhat younger) belonging to the small group of Italian theoretical physicist working in Rome on analogous topics (say, on "integrable systems"); including in particular the 4 co-editors of this book dedicated to Decio (but it should also be mentioned that the main mentor of Decio, as well as his close friend and co-author of very many papers, was Pavel Winternitz ). It is sad that all 3 of them-Mario, Decio, Pavel (all of them younger than FC)-are no more with us.

The Collatz conjecture

Consider the sequence of positive integers n j (j = 0, 1, 2, ...) generated by the following 2 simple rules (hereafter we assume all these rules to be applied sequentially, so that they generate a uniquely defined sequence of positive integers):

n j+1 = n j /2 if n j is even, n j+1 = 3n j + 1 if n j is odd.
The Collatz conjecture states that, for every positive integer initial value n 0 , this sequence ends up into the cycle {1, 4, 2, 1}. This conjecture is attributed to Lothar Collatz , who perhaps first thought of it in 1937 [START_REF] Lagarias | The 3n+1 problem: an overview[END_REF]; it has been checked (in 2020) for all positive integers numbers n 0 up to 2 68 ≈ 2.95 • 10 20 , and some computer experts continue to labor in order to increase this upper bound [1]; it is obviously true for certain arbitrarily large initial values (for instance for n 0 = 2 N with N any positive integer ); but it has not yet been proven nor disproved, nor shown to be unprovable; recently it has been shown that it is extremely probable that this conjecture is true, in an appropriate mathematical framework where this statement has a precise meaning [START_REF] Tao | Almost all orbits of the Collatz map attain almost bounded values[END_REF]. The reason why it might not be true is, of course, because there might exist an initial positive integer n 0 such that the sequence generated by the simple rules detailed above either diverges or leads to a different cycle than the single one mentioned above.

Many analogous conjectures have been proposed over time (see for instance the book [START_REF] Lagarias | The Ultimate Challenge: The 3x + 1 Problem[END_REF]). In this paper we mention some, possibly new, analogous conjectures.

3 The "5n + 1 Conjecture" Consider the sequence of positive integers n j (j = 0, 1, 2, ...) generated by the following 3 simple rules: n j+1 = n j /2 if n j is even, n j+1 = n j /3 if n j is odd and divisible by 3, n j+1 = 5n j + 1 if n j is odd and not divisible by 3. The "5n + 1 Conjecture" states that, for every positive integer initial value n 0 , this sequence ends up into the cycle {1, 6, 3, 1}. This conjecture is not new : it is in fact sometimes called the Kakutani 5n + 1 conjecture, and it has been checked for all positive integers up to 10 16 , see for instance the Chapter entitled Empirical verification of the 3x + 1 and related conjectures by Tomás Oliveira e Silva (kindly made available to us by its author) in the book [START_REF] Lagarias | The Ultimate Challenge: The 3x + 1 Problem[END_REF].

A modified version of the "5n + 1 Conjecture"

Note that, if we eliminate the second rule namely we do not divide n j by 3 if n j is divisible by 3, we find several cycles: (e.g. {3, 16, 8, 4, 2, 1, 6, 3}; {26, 13, 66, 33, 166, 83, 416, 208, 104, 52, 26}; {13, 66, 33, 166, 83, 416, 208, 104, 52, 26, 13}); while starting from other numbers (e.g. 7, 11,...) the sequence seems to diverge. [START_REF] Bruschi | A generalization of the Collatz problem and conjecture[END_REF] The "7n + 1 Conjecture" Consider the sequence of positive integers n j (j = 0, 1, 2, ...) generated by the following 4 simple rules: n j+1 = n j /2 if n j is even, n j+1 = n j /3 if n j is odd and divisible by 3, n j+1 = n j /5 if n j is odd and divisible by 5, n j+1 = 7n j + 1 if n j is odd and not divisible by 3 and 5. The "7n + 1 Conjecture" states that, for every positive integer initial value n 0 , this sequence ends up into the cycle {1, 8, 4, 2, 1}. But also this conjecture is not new : it is called the Kakutani 7n + 1 conjecture, and it has been checked for all positive integers up to 10 14 , see again the Chapter entitled Empirical verification of the 3x + 1 and related conjectures by Tomás Oliveira e Silva in [START_REF] Lagarias | The Ultimate Challenge: The 3x + 1 Problem[END_REF].

A modified version of the "7n + 1 Conjecture"

Consider the sequence of positive integers n j (j = 0, 1, 2, ...) generated by the following 3 simple rules: n j+1 = n j /2 if n j is even, n j+1 = n j /3 if n j is odd and divisible by 3, n j+1 = 7n j + 1 if n j is odd and not divisible by 3 and 5. Then, for every positive integer initial value n 0 , this sequence ends up into one of the two cycles {1, 8, We checked this statement (only) for all positive integers n 0 up to 10 9 . This is the first case we found, of a conjecture, somewhat analogous to the Collatz conjectures, which seems to yield (only) 2 cycles. [START_REF] Bruschi | Variations on a Theme of Collatz[END_REF] The "11n + 1 Conjecture" Following the same idea that had led us to write the "5n + 1 Conjecture" and the "7n + 1 Conjecture", one might be tempted to propose many quite analogous conjectures (based on primes); even stating that, starting from any positive integer, one always reaches a unique cycle containing 1. However, the following observation disproves this naive idea (although some variations of this idea might merit further exploration; see below).

Indeed, consider the sequence of positive integer numbers n j (j = 0, 1, 2, ...) generated by the following 5 simple rules: n j+1 = n j /2 if n j is even, n j+1 = n j /3 if n j is odd and divisible by 3, n j+1 = n j /5 if n j is odd and divisible by 5, n j+1 = n j /7 if n j is odd and divisible by 7, n j+1 = 11n j + 1 if n j is odd and not divisible by 3, 5 and 7. The "11n + 1 Conjecture" states that, for every positive integer initial value n 0 , this sequence ends up into one of the (only) 2 cycles {1, 12, 6, 3, 1} and {17, 188, 94, 47, 518, 259, 37, 408, 204, 102, 51, 17}. We checked its validity (only) for all positive integers n 0 up to 10 9 .

]ocnmp[ F Briscese and F Calogero

The "Modified 11n + 1 Conjecture"

Surprisingly, if we modify the "11n + 1 Conjecture" by eliminating the fourth rule, we seem to only get 1 cycle. Indeed, consider the sequence of positive integer numbers n j (j = 0, 1, 2, ...) generated by the following 4 simple rules: n j+1 = n j /2 if n j is even, n j+1 = n j /3 if n j is odd and divisible by 3, n j+1 = n j /5 if n j is odd and divisible by 5, n j+1 = 11n j + 1 if n j is odd and not divisible by 3 and 5. The "Modified 11n + 1 Conjecture" states that, for every positive integer initial value n 0 , this sequence ends up into the cycle {1, 12, 6, 3, 1}. We checked this statement (only) for all positive integers n 0 up to 10 9 .

Final remarks

Note that, for the "Modified 7n + 1 conjecture" and the "11n + 1 conjecture", one seems to have only 2 cycles. So-assuming these conjectures are indeed true-this entails the possibility to divide, in 2 different ways, all positive integers in 2 classes, identified by the property to generate-when used as initial value-a sequence that eventually ends up into one of these 2 cycles.

The results mentioned above clearly suggest the possibility to introduce many-indeed, presumably an infinity-of somewhat analogous conjectures to those described above. A very interesting open issue is whether the hypothesis that one of these conjecture be true (or, alternatively, false) might be shown to imply that another one of these conjectures is as well true (or, alternatively, false); and whether progress in this direction might eventually provide an avenue to finally prove or disprove the validity of some, or even of all, these conjectures.

Let us end this paper by recalling the simple idea [START_REF] Bruschi | Variations on a Theme of Collatz[END_REF], that if the Collatz conjecture is true, then the 3-times iterated Collatz procedure (as described in Section 2) implies that all positive integers can be divided into 3 distinct categories: those such that, when taken as initial values of the sequence generated by that procedure, entail that it converge eventually to 1, or to 2 or to 4; hence the related conjecture -relevant of course only if the Collatz conjecture is true-that these 3 categories of integers are equally populated, a statement meant to be true in the same sense as the assertion that there are as many even positive integers as odd positive integers [START_REF] Bruschi | Variations on a Theme of Collatz[END_REF]. Clearly analogous ideas and conjectures can be associated to all the conjectures formulated above.


Sven Goffin

Silvère Bonnabel

Olivier Brüls

Pierre Sacré

Invariant Kalman Filtering with Noise-Free Pseudo-Measurements

In this paper, we focus on developing an Invariant Extended Kalman Filter (IEKF) for extended pose estimation for a noisy system with state equality constraints. We treat those constraints as noise-free pseudo-measurements. To this aim, we provide a formula for the Kalman gain in the limit of noisefree measurements and rank-deficient covariance matrix. We relate the constraints to group-theoretic properties and study the behavior of the IEKF in the presence of such noise-free measurements. We illustrate this perspective on the estimation of the motion of the load of an overhead crane, when a wireless inertial measurement unit is mounted on the hook.

I. INTRODUCTION

Since its introduction in the 1960s, the Extended Kalman Filter (EKF) is probably the most used filter in the industry [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF]. Despite its use in many real world applications, it relies on strong assumptions that are rarely met in practice. Moreover, its development is solely based on a probabilistic description of a dynamical and a measurement model, disregarding the geometrical structure of the considered problem. The latter concern gave birth to a proper field that focuses on geometric filtering methods. This research direction provided improved solutions for many useful tasks, like Simultaneous Localization And Mapping (SLAM) [START_REF] Mahony | A geometric nonlinear observer for simultaneous localisation and mapping[END_REF]- [START_REF] Mahony | An homogeneous space geometry for simultaneous localisation and mapping[END_REF], inertial navigation and localization [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF]- [START_REF] Barrau | The geometry of navigation problems[END_REF], or attitude and pose estimation [START_REF] Bonnabel | On invariant observers[END_REF]- [START_REF] Cohen | Navigation and control of unconventional vtol uavs in forward-flight with explicit wind velocity estimation[END_REF]. The invariance and equivariance properties of frameworks developed in [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF], [START_REF] Barrau | The geometry of navigation problems[END_REF], [START_REF] Barrau | Non-linear state error based extended kalman filters with applications to navigation[END_REF], [START_REF] Mahony | Equivariant filter design for kinematic systems on lie groups[END_REF] are reminiscent of those of linear systems in many regards.

In various contexts, some additional (deterministic) information further constrain the state to belong to a subspace of the state space. Incorporating equality constraints in probabilistic filtering constitutes a significant challenge that has led to the realm of state constrained extended Kalman filtering, see [START_REF] Simon | Kalman filtering with state equality constraints[END_REF]. In this paper, we consider nonlinear constraints, dictated by kinematic of mechanical relations, which arise for instance when estimating the configuration of a multibody mechanical system whose parts are rigidly linked and equipped with one or several Inertial Measurement Units (IMUs). Especially, we consider the equations associated to an IMU that may move freely in space and assume it is mounted at the end of a rigid "link" whose other end's location is known, or fixed. This may be the case when mounting an IMU on a robotic arm, or on the hook of a Sven Goffin is a FRIA grantee of the Fonds de la Recherche Scientifique -FNRS.

S. Goffin and P. Sacré are with the Department of Electrical Engineering and Computer Science, University of Liège, Belgium (sven.goffin@uliege.be, p.sacre@uliege.be).

O. Brüls is with the Department of Aerospace and Mechanical Engineering, University of Liège, Belgium (o.bruls@uliege.be).

S. Bonnabel is with MINES ParisTech, PSL Research University, France (silvere.bonnabel@mines-paristech.fr). crane (a pendulum), as in [START_REF] Rauscher | Motion control of an overhead crane using a wireless hook mounted IMU[END_REF]. In those cases, the known location of the other end of the link provides an (nonlinear) equality constraint that may help estimating the state of the IMU.

One may then resort to state constrained Kalman filtering, see [START_REF] Simon | Kalman filtering with state equality constraints[END_REF]. In the nonlinear case, such methods based on (re)projections are a means of enforcing the constraint. However, they come with no guarantee of consistency whatsoever with the estimation problem at hand, and a brutal (re)projection onto the constraint contains a degree of arbitrariness. Herein, we advocate that such nonlinear constraints may rather be incorporated in the extended Kalman filter and its variants through pseudo-measurements without noise, as they constitute information known with certainty. This technique is perfectly justified in the linear case, and allows for optimal fusion of noisy sensors and deterministic side information. This poses two challenges, though. First, the matrix that must be inverted in the Kalman filter may become rank-deficient, see e.g., [START_REF] Gurumoorthy | Rank deficiency of kalman error covariance matrices in linear time-varying system with deterministic evolution[END_REF]. Second, in the nonlinear case, there is no guarantee the constraint is respected after the pseudo-measurement has been incorporated, contrary to the linear case.

We derive a formulation of the Kalman gain that accommodates the rank deficiency issues that may stem from the noise-free setting. Furthermore, we cast the problem into the framework of the Invariant Extended Kalman Filter (IEKF) by embedding the state space into a Lie group. Although [START_REF] Barrau | Extended kalman filtering with nonlinear equality constraints: a geometric approach[END_REF] provides us with both an inspiration and a mathematical framework, we consider different problems: [START_REF] Barrau | Extended kalman filtering with nonlinear equality constraints: a geometric approach[END_REF] considers the limit case of a noise-free (deterministic) dynamics with noisy measurements, along with an initial condition lying in a constrained set, so that deterministic information is propagated. Herein, we consider arbitrary noisy dynamics (and possibly noisy measurements) starting from an unrestricted initial configuration, and investigate how to incorporate noise-free measurements corresponding to deterministic side information. This comes with different challenges and allows for addressing different problems. Recently, [START_REF] Chauchat | Robust linearly constrained invariant filtering for a class of mismatched nonlinear systems[END_REF] proposed to linearly constrain the Riccati equation for robust invariant filtering, which is a wholly different problem.

The paper is structured as follows. Section II specifies the considered class of estimation problems, provides two motivating examples of engineering interest, and highlights the properties one shall pursue and the challenges that are raised in both the linear and nonlinear cases. Section III derives a formulation for the Kalman gain in the limit of noise-free measurements in the linear case and shows its corresponding properties. Section IV extends the use of this gain formulation in the framework of invariant Kalman filtering, and proposes an alternative update for the IEKF in the presence of noise-free measurements. Finally, Section V illustrates the performance of the proposed approach for the estimation of the extended pose of the hook of a crane.

II. MOTIVATING PROBLEM

Since the advent of smartphones, the use of cheap IMUs has increased significantly. In this paper, we consider the problem of fusing the IMU measurements with deterministic constraints, treated as noise-free pseudo-measurements.

A. Mathematical formulation

Free from any prior knowledge about the motion, the general dynamics of an IMU write (e.g., [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF])

R k+1 = R k exp((ω k + w ω k ) × dt), (1a)
v k+1 = v k + (R k (a k + w a k ) + g) dt, (1b)
p k+1 = p k + v k dt, (1c)
where we neglected sensors' biases. The matrix R k ∈ SO(3) is the rotation matrix between the IMU and world frames, and the vectors v k , p k ∈ R 3 are respectively the IMU velocity and position vectors expressed in the world frame at time instant k. The angular velocity ω k and linear acceleration a k output by the IMU are used as inputs to the dynamical system. The vector g denotes the gravity vector expressed in the world frame. The map exp(•) denotes the matrix exponential map and the notation (b) × denotes the skew-symmetric matrix associated with cross product with vector b ∈ R 3 . Even if no assumption is made on the dynamics, the structural and mechanical properties of the body to which the IMU is attached may constrain the IMU pose. When dealing with a rigid body, this comes as noise-free equality constraints of the form

R k r k + α k v k + β k p k = y k , (2)
where r k , y k ∈ R 3 and α k , β k ∈ R. Considered problem We aim at devising a meaningful (Kalman) filter to estimate the state of the noisy system (1) from noise-free measurements (2) in real time.

B. Application examples

Estimating the pose of an IMU under constraints like (2) applies to a wide range of applications. We provide two concrete applications for this problem, which is nonlinear, owing to the state variable R k being a rotation matrix.

1) Crane state estimation: As a first example, consider a crane on a construction site (see Figure 1). It is technically very feasible nowadays to mount an IMU on the hook that transmits its sensor readings [START_REF] Rauscher | Motion control of an overhead crane using a wireless hook mounted IMU[END_REF]. Estimating the position and velocity of the load from the IMU measurements may open the door to new automation capabilities since it allows for feedback. However, the underlying dynamical model is unknown, as the hook does not necessarily follow a simple pendulum motion, because of external forces like friction and wind. The IMU dynamics (1), though, consist of kinematic relations which are satisfied at all times, whatever the motion. Moreover, when the cable is hanging, the distance between the hook and the cable hang-up point is the cable length l k , which is very accurately measured in modern cranes for safety reasons, see, e.g., [START_REF] Bonnabel | The industrial control of tower cranes: An operator-in-the-loop approach [applications in control[END_REF]. This provides a constraint of the form of (2) with r k = (0, 0, -l k ), α k = 0, β k = 1, and y k = 0 3×1 . 𝑂 Fig. 1. Crane with a wireless IMU mounted on its hook.

2) Robotic arm state estimation: The field of robotics also uses IMUs extensively. As a second example, robotic arms are usually equipped with IMUs in order to precisely estimate the pose of their end effector. Although the dynamics of the arm cannot be predicted in advance, kinematic constraints enforce that the segments of the arm are attached to each other at the corresponding joints, providing constraints of the form [START_REF] Mahony | A geometric nonlinear observer for simultaneous localisation and mapping[END_REF]. This is what is exploited in [START_REF] Hartley | Contactaided invariant extended kalman filtering for robot state estimation[END_REF] in the form of noisy measurements.

C. Kalman filtering with noise-free pseudo-measurements

Our problem of estimating noisy systems while treating noise-free equality constraints like (2) fits into the framework of probabilistic filtering. In this framework, the belief at time index k, taking into account measurements up to time j, consists of a multivariate Normal distribution parameterized by a mean xk|j ∈ R n (the state estimate) and a covariance matrix P k|j ∈ R n×n (the associated uncertainty). We advocate indeed that the constraints can be incorporated in a probabilistic filter by using y k as a noise-free pseudomeasurement

y k = h(x k ), letting h(x k ) = h(R k , v k , p k ) = R k r k + α k v k + β k p k . (3)
From the recent theory of two-frames [START_REF] Barrau | The geometry of navigation problems[END_REF], it can readily be checked (3) defines an output that fits into the framework of invariant filtering, and so do the IMU equations [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF].

Noise-free pseudo-measurements convey information that is known for certain. Hence there are two desirable properties one would expect from a (Normal) probabilistic filter.

• Property 1: After a noise-free pseudo-measurement y k , the state is known to belong to the subset

{x ∈ R n | h(x) = y k }
owing to noise-free measurement. This should translate into h(x k|k ) = y k . • Property 2: After a noise-free pseudo-measurement y k , the error covariance matrix P k|k should encode some deterministic information: null variance is to be expected in the perfectly observed directions. Mathematically, it should translate into H k P k|k H T k = 0 m×m with H k ∈ R m×n the Jacobian matrix of h at the estimate xk|k .

D. The linear case

In the linear case, the Kalman filter-whose equations are recalled in Appendix I-is optimal. Thus, we can expect both properties above to be satisfied. We have

H k xk|k = H k K k y k + (H k -H k K k H k )x k|k-1 (4)
and

H k P k|k H T k = (H k -H k K k H k )P k|k-1 H T k . (5)
Both desirable properties above are satisfied indeed, as soon as

H k K k = I m , since it ensures that H k -H k K k H k = 0 m×n . Recalling that K k = P k|k-1 H T k (H k P k|k-1 H T k ) -1 whenever measurements are noise-free, the desired relation holds if H k P k|k-1 H T k is invertible.
If it is not, the Kalman gain is not even defined. This may occur when a noisefree measurement is made for the second time: the rank of H k P k|k-1 H T k has dropped and the matrix cannot be inverted anymore. This may also occur when performing a subsequent noise-free measurement along directions that overlap with the span of H k so that the matrix to be inverted is not fullrank.

E. The nonlinear case

In the nonlinear case, that is, when using an Extended Kalman Filter (EKF), problems accumulate. Indeed, even if

H k P k|k-1 H T k is invertible leading to H k K k = I m ,
there is no reason one should have h(x k|k ) = y k , as we see that (4) relies on h being linear, which is not the case in our problem, see (3), as R k is a rotation matrix that cannot be treated linearly. Moreover, the Jacobian matrix H k depends on the linearization point in EKF design, so that the desired relation H k P k|k H T k = 0 m×m is not properly defined: the Jacobian matrices at different linearization points span different directions, making the condition unclear, in the sense that the dispersion encoded by P k|k has no clear relation to the subset defined by h(x) = y k .

In the sequel, we address the issues that arise both in the linear and nonlinear case.

III. HANDLING RANK DEFICIENCY IN THE LINEAR CASE

Let us consider the discrete-time linear system

x k+1 = F k x k + B k u k + w k , (6a) y k = H k x k . ( 6b
)
where u k ∈ R b is the system input, and where

F k ∈ R n×n , B k ∈ R n×b
, and H k ∈ R m×n . When the measurement y k is noise-free, as in (6b), the Kalman gain (35c) is not defined if H k P k|k-1 H T k is singular, since the measurement noise covariance N k is null. This is logical as then the same measurements does not bring additional information. However, in the noisy setting we may need to update the state regularly to combat the dispersion due to noise. Then H k P k|k-1 H T k may be nearly singular, and we may run into numerical issues. We now address this problem.

A. Solving for the Kalman gain in the noise-free limit The following result proves the limit of the Kalman gain when N k → 0 is well defined and provides its expression.

Theorem 1: Assume P k|k-1 to be of rank l ≤ n and write it as P k|k-1 = L k L T k where L k ∈ R n×l has linearly independent columns. The following Kalman gain

K k := L k (H k L k ) † , (7)
is the limit of

P k|k-1 H T k (H k P k|k-1 H T k + N k ) -1
as the measurement noise covariance matrix N k shrinks to 0 m×m , where A † denotes the Moore-Penrose pseudo-inverse of A.

Note that the given expression is valid whatever the invertibility of the innovation covariance

H k P k|k-1 H T k as (H k L k ) † always exists.
Proof: Given any matrix A, its Moore-Penrose pseudoinverse can be expressed as the limit

A † = lim δ→0 A T (AA T + δ I) -1 . ( 8
)
The limit is finite and exists even if AA T is singular [START_REF] Golub | Matrix computations[END_REF].

We pose A k = H k L k and assume for now that N k = δ I m .

lim δ→0 K k = lim δ→0 P k|k-1 H T k (H k P k|k-1 H T k + δ I m ) -1 , (9) = lim δ→0 L k A T k (A k A T k + δ I m ) -1 , (10)
= L k lim δ→0 A T k A k A T k + δ I m -1 , (11) (8)
= L k A † k , (12)
= L k (H k L k ) † . (13)
When N k is not of the form δ I m , we may upper and lower bound its eigenvalues by matrices of the desired form, and use the squeeze theorem (in the sense of positive semidefinite matrix inequalities), using the fact matrices A k A T k and N k are symmetric.

B. Properties of the Riccati update (Property 2)

When a noise-free measurement (6b) has been made, it is logical that the corresponding variance drops to zero, that is, H k P k|k H T k = 0 m×m , along the lines of Property 2. We now show this is the case with the obtained limit gain.

Theorem 2: Assume P k|k-1 to be of rank l ≤ n and write it as P k|k-1 = L k L T k where L k ∈ R n×l has linearly independent columns. Consider the limit gain [START_REF] Hartley | Contactaided invariant extended kalman filtering for robot state estimation[END_REF] and the corresponding Riccati update (35e). Then we have H k P k|k H T k = 0 or equivalently H k P k|k = 0.

Proof: We have

H k (I -K k H k )P k|k-1 (14) = H k (I -L k (H k L k ) † H k )P k|k-1 (15) = (H k L k -H k L k (H k L k ) † H k L k )L T k (16) = 0 m×n , (17)
where we used the property of the pseudo-inverse that that AA † A = A. This proves the result.

IV. INVARIANT KALMAN FILTERING WITH NOISE-FREE PSEUDO-MEASUREMENTS

Consider the discrete-time nonlinear system (1) along with nonlinear output map [START_REF] Barrau | Invariant kalman filtering[END_REF]. The EKF first linearizes the system before applying the machinery of linear Kalman filtering. Gain formulation [START_REF] Hartley | Contactaided invariant extended kalman filtering for robot state estimation[END_REF] may thus be applied in the nonlinear case too. Nevertheless, the dependence of Jacobian matrix H k = ∂h(x) ∂x on the linearization point makes the interpretation obscure: the update ensures Property 2 at the linearization point, which is not the true state. The invariant filtering framework brings clarification in this regard.

A. The Invariant EKF for the considered problem

Let us modify the representation to cast the problem within the invariant framework of [START_REF] Barrau | Invariant kalman filtering[END_REF]. We embed the state into the matrix Lie group SE 2 (3), namely the group of extended poses [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF], [START_REF] Brossard | Associating uncertainty to extended poses for on lie group imu preintegration with rotating earth[END_REF]. The state turns into

χ k =   R k v k p k 0 1×3 1 0 0 1×3 0 1   . (18)
Its dynamics satisfies Equation ( 11) of [START_REF] Barrau | Invariant kalman filtering[END_REF] and is said to be group affine, insuring it fits the framework of the Invariant Extended Kalman Filter (IEKF). In this embedding, the measurement function may write as

h(χ k ) = χ k d k =   R k r k + α k v k + β k p k α k β k   , (19)
where d k = (r k , α k , β k ), which fits into the theory of [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF] and corresponds to left-invariant outputs (observations in the fixed frame). Along the lines of [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF] we define the innovation as

z k = χ-1 k|k-1 y k -d k .
The invariant framework introduces the nonlinear and linearized errors η k|j ∈ SE 2 (3) and ξ k|j ∈ R 9 defined by

η k|j = χ-1 k|j χ k = exp(ξ k|j ), (20)
where exp(•) now denotes the Lie exponential map of group SE 2 (3). To derive the Jacobian matrices, the fastest way is to readily retrieve them from the formulas of the recent two-frames theory [START_REF] Barrau | The geometry of navigation problems[END_REF]. In this context the left-invariant error is R-

1 k|j R k , R-1 k|j (v k -vk|j ), R-1 k|j (p k -p) , the innovation is z k = ( R-1 k|k-1 R k -I)r k + α k R-1 k|k-1 (v k -vk|k-1 ) + β k R-1 k|k-1 (p k -pk|k-1
) and the theory yields, see Proposition 13 of [START_REF] Barrau | The geometry of navigation problems[END_REF],

H k = -(r k ) × α k I 3 β k I 3 , (21)
and denoting Ω k := exp(dt(ω k ) × ), we have as in [START_REF] Barrau | The geometry of navigation problems[END_REF]]

F k =   Ω -1 k 0 0 -dt Ω -1 k (a k ) × Ω -1 k 0 0 dt Ω -1 k Ω -1 k   . (22)
We see we recover the important property of IEKF that the Jacobian matrices are state-independent. The IEKF updates its estimate according to

χk|k = χk|k-1 exp(K k z k ). (23)
The following sections presents how the invariant framework naturally ensures Property 2 and what can be done to satisfy at best Property 1. Since the problems tackled here arise from the noise-free nature of the considered pseudomeasurements, our work constitutes already a contribution in its own right compared to [START_REF] Barrau | Invariant kalman filtering[END_REF], as they only consider noisy measurements.

B. What about desirable Property 2?

In the formalism of invariant filtering, the constraint (3) writes χ k d k = y k . This noise-free pseudo-measurement informs us that the state belongs to the subset

H = {χ ∈ SE 2 (3) | χd k = y k }. (24)
We first recall how the IEKF encodes uncertainty, see [START_REF] Barrau | Invariant kalman filtering[END_REF]. Definition 1: The IEKF belief at time instant k, accounting for measurements up to time j, for the true state χ k is a concentrated Gaussian distribution on the Lie group [START_REF] Bourmaud | Continuous-discrete extended kalman filter on matrix lie groups using concentrated gaussian distributions[END_REF] χ k = χk|j exp(ξ k|j ), ξ k|j ∼ N (0 n×1 , P k|j ).

(

) 25
with χ the "noise-free" mean estimate, and P the covariance matrix encoding the statistical dispersion around the mean. As a result, for the belief to be consistent with the set H, we would ideally like the entire updated distribution to lie within H. It turns out to be the case indeed, as long as the updated mean is in the right subspace of course. This proves the structure of the invariant filtering framework is consistent with the physical problem indeed. Theorem 3: Consider the IEKF's covariance matrix P k|k after updating its state via observation [START_REF] Barrau | Invariant kalman filtering[END_REF]. Then if the updated estimate χk|k satisfies χk|k ∈ H, all the probabilistic dispersion encoded by the updated covariance matrix according to our belief model [START_REF] Bourmaud | From intrinsic optimization to iterated extended kalman filtering on lie groups[END_REF] lies within H, that is, for all random ξ ∈ N (0, P k|k ), we have χk|k exp(ξ) ∈ H.

Proof: The result is a direct application of the following property from [START_REF] Barrau | Extended kalman filtering with nonlinear equality constraints: a geometric approach[END_REF], [START_REF] Chauchat | Kalman filtering with a class of geometric state equality constraints[END_REF], which can be proved through a series expansion of the exponential, see the proof of (iv) of Theorem 1 in [START_REF] Chauchat | Kalman filtering with a class of geometric state equality constraints[END_REF].

Proposition 1: Consider an element ζ of the Lie algebra of SE 2 (3) identified with R 9 . Let H k denote the Jacobian from invariant filtering associated to map h(χ) := χd k , i.e. given by [START_REF] Brossard | Associating uncertainty to extended poses for on lie group imu preintegration with rotating earth[END_REF] in our problem. Then we have (χ ∈ H and

H k ζ = 0) ⇒ χ exp ζ ∈ H.
Now, the Riccati update with the limit gain [START_REF] Hartley | Contactaided invariant extended kalman filtering for robot state estimation[END_REF] ensures that H k P k|k = 0 m×n , see Theorem 2. Thus any element ξ in the span of P k|k satisfies H k ξ = 0

We thus see that Property 2 is satisfied in a way that is meaningful and leads to a consistent belief: the covariance matrix P k|k being aligned with the actual dispersion, regardless of the linearization point. This is a clear improvement over the conventional EKF, that possesses none of the desired properties. In practice, this explains why the IEKF outperforms the EKF for the considered problem, as can be seen below. But before, let us turn to Property 1.

C. What about Property 1?

In the previous theorem we had to assume Property 1 to be satisfied, that is, χk|k ∈ H for Property 2 to make complete sense. Unfortunately, due to the nonlinear nature of the problem, the IEKF does not satisfy Property 1. All we know is that the residual innovation (in other terms the prediction error) is null up to second order terms after the update, that is,

z k|k = χ-1 k|k y k -d k = O ∥ξ k|k ∥ 2 , (26)
and as a result χk|k may not belong to the expected set H.

To address this issue, we now propose an alternative IEKF update procedure to reduce the norm of the residual innovation. The rationale is the following. Pseudo-measurements come for free, in the sense that they are not actual sensor measurements, but side information that one has for certain. In principle, once a noise-free measurement is made, the drop in uncertainty is fully incorporated in the belief, so that making the same measurement immediately does not change the belief. In practice, we have seen this is not the case, owing to undesirable effects of nonlinearity. However, there is no reason one could not reuse this same information several times in a row.

Our idea is thus to update the estimate with the same noise-free pseudo-measurement y k and the same Kalman gain K k as long as it makes the prediction error decrease, in other words cycle on the noise-free measurement until ∥z k|k ∥ stabilizes. This procedure makes sense if we keep the gain K k constant when performing the same noise-free measurement several times, since as soon as K k is updated it does not correct the state along the measurement direction, owing to Property 2 (as K k z k lies in the span of P k|k ). The corresponding procedure is detailed in Algorithm 1.

Remark 1: Algorithm 1 reduces the residual innovation but is not always able to completely eliminate it. Indeed, the innovation can be decomposed as

z k = z k,∥ + z k,⊥ , (27)
where z k,∥ ∈ R(S k ) and z k,⊥ ∈ R(S k ) ⊥ , with z k,⊥ = O ∥ξ k|k-1 ∥ 2 being entirely due to linearization errors and S k = H k P k|k-1 H T k . The component z k,⊥ lies in the kernel of K k , i.e., K k z k,⊥ = 0 n×1 . This means this part of the innovation accounts for a component ξ k,⊥ of ξ k|k-1 that lives within R(P k|k-1 ) ⊥ and cannot be corrected using Kalman filtering techniques. As a consequence, the updated state χk|k becomes slightly inconsistent with P k|k . In most applications, this inconsistency issue is immediately fixed at the propagation stage as process noise reintroduces Algorithm 1 Update procedure for the IEKF with noise-free pseudo-measurements.

1: Compute K k using (7)

2: z 0 k ← χ-1 k|k-1 y k -d k 3: χ ← χk|k-1 exp(K k z 0 k ) 4: z 1 k ← χ -1 y k -d k 5: i ← 1 6: while ∥z i k -z i-1 k ∥ > tol do 7: χ ← χ exp(K k z i k ) 8: z i+1 k ← χ -1 y k -d k 9:
i ← i + 1 10: end while 11: χk|k ← χ 12:

P k|k ← (I -K k H k )P k|k-1 (I -K k H k ) T
uncertainty in the directions for which the last noise-free update negated the variance. In the absence of process noise, this is problematic as it makes the filter become quickly overconfident. This problem is left unsolved here and will be the subject of future work.

V. ESTIMATION OF THE PENDULUM ANGLE FOR A CRANE

We consider the problem of estimating the extended pose (orientation, position, velocity) of an IMU fixed on the hook of a crane as presented in section II and illustrated in Figure 1, and compare the three following filters:

• EKF: a conventional extended Kalman filter.

• IEKF: a conventional invariant extended Kalman filter.

• Noise-free IEKF: the invariant extended Kalman filter that implements gain [START_REF] Hartley | Contactaided invariant extended kalman filtering for robot state estimation[END_REF] and uses Algorithm 1 to further refine its estimate at each pseudo-measurement. For the conventional EKF and IEKF, the gain was computed by setting the noise covariance N k to the small value of 10 -4 I 2 and letting

K k = P k|k-1 H T k (H k P k|k-1 H T k + N k ) -1 .
Estimating the IMU pose in the 3D space with cable length as the unique constraint raises observability issues that might obscure the big picture of the present paper. For the sake of simplicity, the IMU is assumed to stay in the plane ϕ k = 0 so that we consider the associated 2D problem, letting χ k ∈ SE 2 (2) and ξ k|j ∈ R 5 .

The ground-truth simulation is the trajectory followed by the hook when it starts with an initial angle θ 0 = 20 • and no angular velocity. No other external force than gravity is simulated. The crane cable is assumed to stay straight during the entire simulation and its length varies according to the profile displayed in Figure 2. The simulation is stopped after 2 s. The IMU and the three filters operate at the same frequency of 100 Hz. The initial error covariance matrix is set to

P 0|0 = 0.05 2 0 1×4 0 4×1 0.5 2 I 4 . (28)
The gyroscope and accelerometer of the IMU are affected by Normal noise of zero mean and covariance where E(•) denotes the expectation operator. The error

E (w ω k ) 2 = (0.005) 2 , ( 29
) E w a k (w a k ) T = (0.005) 2 I 2 , (30)
ξ k = log χ-1 k|k χ k (31)
is used to compare the performances of the three filters, where log(•) : SE 2 (2) → R 5 is the logarithmic map of SE 2 (2). The comparison is carried over 30 simulations in which the initial error ξ 0 is drawn randomly from the distribution N (0 5×1 , P 0|0 ). The average and standard deviations of the norm of the error ξ k are computed for the three filters.

Results are plotted in Figure 3. Two general comments can be made:

1) The conventional extended Kalman filter is outperformed by its invariant filtering counterparts: its convergence is way slower, with a large variability across simulations.

2) The noise-free IEKF exhibits the best convergence rate, with only 6 time steps needed on average for ∥ξ k ∥ to go bellow 1% of the initial error ∥ξ 0 ∥, against 59 and 191 for the classical IEKF and EKF. It is also the method with the lowest variability from one simulation to another. Note that with a tolerance tol = 10 -7 , Algorithm 1 performs 2.03 cycles on average before ∥z k|k ∥ stabilizes.

This simulation shows that treating equality constraints like (2) as proper noise-free pseudo-measurements clearly benefits the estimation and results in extremely fast convergence.

Remark 2: We indicate the failure to ensure Property 1 can sometimes make the noise-free IEKF diverge, when the residual innovation is too large. This leaves room for improvement, which is the subject of ongoing work.

VI. CONCLUSION

This work presented a practical and efficient solution for incorporating equality constraints of the form (2) in the (extended) Kalman filtering framework. We advocated such constraints should be expressed as noise-free pseudo-measurements. We also derived a formulation of the Kalman gain that solves singular matrix inversion issues that may be encountered when dealing with noise-free pseudomeasurements. Using a proper matrix Lie group embedding and the theory of invariant filtering [START_REF] Barrau | Invariant kalman filtering[END_REF] leads to Normal filters that correctly encode the actual physical uncertainty in the problem: their belief is consistent with the constraint in that the covariance matrix after update corresponds to an uncertainty that is wholly contained within the constrained subset. Finally, an alternative update for the IEKF was proposed in order to mitigate the impact of linearization errors in the noise-free update process. The performance of this method was evaluated on the task of estimating the extended pose of the hook of a crane, and was proved to outperform both the EKF and IEKF in this context.

As a perspective, we would like to thoroughly treat the case of the 3D crane, which poses some potential observability issues. We also intend to derive a general theory for noise-free measurements in the context of invariant filtering, namely for two-frame systems [START_REF] Barrau | The geometry of navigation problems[END_REF]. In this context, we would like to explore how to fully enforce the constraint y k = h( χk|k ) at update in a natural way, that is, while retaining all the information that has been acquired before update. This might be achieved via the iterated Kalman filter [START_REF] Bell | The iterated kalman filter update as a gauss-newton method[END_REF], [START_REF] Bourmaud | From intrinsic optimization to iterated extended kalman filtering on lie groups[END_REF]. To do iterated EKF in the present context, we anticipate the results developed at Theorem 2 may prove useful.

APPENDIX I KALMAN FILTER

Let x k ∈ R n be the state to estimate. The discrete-time Kalman Filter (KF) is an optimal probabilistic filter that fuses information coming from two noisy sources: the state dynamics model

x k+1 = F k x k + B k u k + w k , (32)
with F k ∈ R n×n , B k ∈ R n×b , input u k ∈ R b , and process noise w k ∈ R n with covariance Q k , and measurements

y k = H k x k + n k (33)
with H k ∈ R m×n and measurement noise n k ∈ R m with covariance N k . The KF makes the assumption that ξ k|j = x k -xk|j ∼ N (0 n×1 , P k|j ), where x, ξ and P respectively denote the filter estimate, the estimation error and the associated covariance matrix. One iteration of the KF consists of two stages: the propagation and the update. a) Propagation: The KF propagates its belief through the state dynamics model as follows:

xk+1|k = F k xk|k + B k u k , (34a)
P k+1|k = F k P k|k F T k + Q k . (34b)
b) Update: Receiving measurement y k , the KF updates its belief as follows:

z k = y k -H k xk|k-1 , (35a)
S k = H k P k|k-1 H T k + N k , (35b)
K k = P k|k-1 H T k S -1 k , (35c) xk|k = xk|k-1 + K k z k , (35d)
P k|k = (I n -K k H k )P k|k-1 (35e)
where z k is called the innovation, S k is the innovation covariance, and K k is the Kalman gain.

Fig. 2 .

 2 Fig. 2. Evolution of the length of the crane cable l k as a function of the time index.

Fig. 3 .

 3 Fig. 3. Mean norm of the error function ξ k as a function of the time index, computed over 30 simulations. The standard deviation of the norm is displayed in light colors.

Authorized licensed use limited to: ECOLE DES MINES PARIS. Downloaded on January 22,2024 at 10:55:12 UTC from IEEE Xplore. Restrictions apply.


J Gong

F Tan

B Zhao

E Semail
email: eric.semail@ensam.eu

N K Nguyen

F Gillon

N Bracikowski
email: nicolas.bracikowski@univ-nantes.fr

Proposal of a Novel Topology of Five-Phase Fractional Slot Concentrated Non-Overlapping Winding with Selective MMF Harmonic Elimination

Keywords: 5-phase PM machine, FSCW, FEA, Selective Harmonic Elimination. I

In this paper, a novel topology of five-phase fractional-slot concentrated winding (FSCW) with selective magnetomotive force (MMF) harmonic elimination is proposed. Compared to traditional FSCW with stator shift technique, not only the parasitic sub-or sup-order harmonics of the MMF can be selectively reduced or eliminated, the non-overlapping characteristic is also kept. Firstly, based on 20-slot/22-pole singlelayer (SL) winding topology, a dual five-phase 40-slot/22-pole one of SL winding with 1 st harmonic elimination is introduced. Secondly, a new 40-slot/22-pole of double-layer with selective harmonic reduction is presented. Finally, the high performances of the machine with the proposed winding topology are evaluated using finite element analysis (FEA), including air gap flux density, output torque, machines losses, etc.

INTRODUCTION

Compared to traditional integer slot winding, the FSCW are more and more concerned for electrical vehicle application, due to the short end-winding, field-weakening capability, the facilities of making, recycling and repairing. In addition, the machine stator with FSCW can be modulated designed, which can improve the fault-tolerant capability and the efficiency of mass market production.

The common disadvantage of FSCW is the rich harmonic content of MMF [START_REF] Reddy | Generalized Approach of Stator Shifting in Interior Permanent-Magnet Machines Equipped With Fractional-Slot Concentrated Windings[END_REF] which, in certain case, can cause high eddy-current magnet rotor losses and furthermore restrict the performance of the machines, especially for high-speed applications.

Reference [START_REF] Zhao | A Novel Five-Phase Fractional Slot Concentrated Winding with Low Space Harmonic Contents[END_REF] proposed a novel 5-phase/20-slot/22-pole winding topology by a stator shift technique combined with a star/pentagon winding connection. Compared to the conventional configuration, the first harmonic of MMF in the new winding is eliminated. Moreover, the non-overlapping characteristics of FSCW are kept. However, the method cannot selectively eliminate harmonics. This paper proposed a new topology of five-phase FSCW with selective harmonic elimination, which can completely eliminated the 1 st harmonic and significantly reduce the 9 th harmonic. Moreover, the output torque is almost the same value. This method can also be extended to higher number of phase winding design.

II. BASIC WINDING LAYOUT

The SL winding is often preferred with high fault-tolerant machine design due to the isolation features, and the DL solution is preferred due to its advantages of rotor losses for the high speed applications. Fig. 1 presents the winding layouts of 20-slot/22-pole SL machine and 40-slot/22-pole SL machine. For the two both configurations, the 11 th harmonic of MMF is the working harmonic. And the 1 st MMF harmonics of 40-slot/22-pole SL machine is greatly reduced. However, the pitch factor of the 40slot/22-pole SL winding is slightly reduced. Moreover, there are still many non-working harmonics, e.g. the 9 th , 29 th and 31 st etc. In order to further improve the MMF harmonic content, the selective harmonic elimination technique will be proposed and presented in the following section.

III. NEW 40-SLOT/22-POLE DOUBLE-LAYER WINDING

A. Elimination of the 1st harmonic

The MMF expression of the conventional five-phase 40slot/22-pole SL winding shown in Fig. 1(b) can be expressed as Fourier series as (1) (2).

( )

1, 9,11... , sin 40 k k k F t v k t      =-   = --      (1)
20 sin sin 40 40

k NI k k v k       =         ( 2
)
Where vk is the amplitude of the k th MMF space harmonic; N is the number of turns of each coil; I is the RMS value of the injected fundamental current; θ is the space angle; ω is the angular velocity.

Fig. 1(b) shows that the phase difference in space of the two adjacent coils A1 and A2 is 18° in mechanical, and the corresponding electrical angle is also 18° (18×11-180). If the injected currents for the two groups of winding are also shifted by 18° respectively in time, the winding of machine can be considered as dual five-phase one. The resultant MMF of the dual five-phases winding can be expressed as (3) (4).

( )

1, 9,11... 1 , sin 20
d dk k k F t v k t      =- -   = --      (3) 20 1 sin sin 40 20 dk NI k k v k    -     =         ( 4
)
Where vdk is the amplitude of the k th MMF space harmonic; It can be seen from Erreur ! Source du renvoi introuvable. that the amplitude of the 1 st -order harmonic vd1 is equal to 0 with k=1, i.e. the 1 st -order harmonic is completely suppressed in dual five-phase winding. The MMF space harmonic spectrum is shown in Fig. 2. It can also be seen that the working harmonic (the 11 th ) is increased by 1.2%, while the 19 th , 21 st , 39 th , and 41 st are both eliminated.

B. New five-phase 40-slot/22-pole DL winding

Based on the concept of stator shifting technique and the novel 40-slot/22-pole SL winding, another dual five-phase winding is added, with the same winding distribution as the first dual five-phase one, but with a specific mechanical shifting angle α between the two windings, and thus the novel 40slot/22-pole DL winding topology is constructed. The final amplitude of MMF of the novel windings topology is expressed in Erreur ! Source du renvoi introuvable. (6). (6) Where vdk is the amplitude of the k th MMF harmonic of a single double five-phase winding; α is the mechanical shift angle which is a discrete number and Q is the number of stator slots; cos(kα/2) is defined as the decay factor. For each harmonic, the decay factor varies in a sinusoidal form with respect to the mechanical shifting angle α as shown in Fig. 3.

In order to reduce or eliminate the 9 th harmonic of MMF, it is necessary to choose an appropriate angle α leading to vα9 close to zero, while keeping vα11 big value. As can be seen from Fig. 3, the angle α=99° is a good choice, which corresponds to 11 slots of a 40-slot/22-pole machine. The related decay factors of vα9 and vα11 are -0.0785 and -0.9969, respectively. Fig. 4 shows the novel winding layout and flux density of the 40-slot/22-pole machine. It is a DL winding without overlapping of end-windings. Each phase consists of eight coils which can be divided into two groups: A1&A2. The four coils of A1 or A2 are connected in series. The current supply of the five-phase machine is achieved using the method by combining the star and pentagram coils connection. Fig. 5 shows the comparison of MMF between the 20slot/22-pole SL winding and the 40-slot/22-pole DL winding. It can be seen that the 1 st harmonic is completely eliminated, and the 9 th harmonic is significantly reduced to 8%. The total harmonic distortion (THD) of the two windings are 220.7% and 38.9%, respectively.

  Fig. 2. MMF harmonic of 40slots/22poles SL machine

Fig. 3 .

 3 Fig. 3. Effect of the shift mechanical angle on winding factors of 1st, 9th, and 11th harmonics


Olivier Hueber
email: olivier.hueber@unice.fr

Online communities of payments and consumer behaviours: The Second Life case

This paper asserts that the online communities of electronic money (e-money) users affect the traditional mechanisms of price determination by introducing anonymity in money payments. By studying the Second Life case it is possible to show the main characteristics of such a communities and raise new questions linked to the online behaviours of the consumers. In the aim of shedding light on the online consumer behaviour we turn to Thorstein Veblen works and to network externalities concepts.

Introduction

The internet hosts an increasing number of large synthetic worlds, like the Second Life World, which users can visit by piloting a computer-generated body, known as an avatar. These virtual worlds exhibit most of the characters we associate with the real world like interpersonal relationships, economic transactions, production of goods and services, political institutions…Inside these synthetic worlds, new economic behaviours can be studied. The price determinations mechanisms and the economic behaviours have their proper rules. Firstly, this paper shows the main benefits for buyers of using electronic monies (e-monies) and entering in a community of payments. Secondly, the economic behaviour of emoney users and the network externalities effects are studied.

The online monetary communities of emoney users

The Internet network makes possible the emergence of new communities of payment using electronic money (e-money). Electronic money is broadly defined as an electronic store of monetary value on a technical device that may be widely used for making payments to undertaking other than the issuer without necessarily involving bank accounts in the transaction, but acting as a prepaid bearer instrument. Communities of emoney have several characteristics. The first one is the anonymity. It can be possible to transfer funds anonymously with the paypal system. Nowadays, over 150 millions of accounts are open in the paypal website (paypal.com). It is possible to receive money while remaining anonymous. A simple e-mail address is sufficient for selling and buying. PayPal lets shop without sharing financial information. Obviously, it is easier to hide money from the government by using the paypal system. However, if you have a non-verified status on your paypal bank account you are limited in your payments. It is impossible to pay more than 1500, 00 €. Such a limit of payment can be crossed by creating, still anonymously numerous paypal accounts. Anonymity in transaction is something that buyers and sellers often want. Forms of electronic money can provide anonymity but still rely on trust between the parties.

In the United Kingdom, it is possible, to buy a prepaid Mastercard called Cashplus without informing your bank. A simple driving licence or a passport is sufficient to buy the Cashplus prepaid card. With such a card, the buyer is not anonymous but, like any paper banknote, the money can circulate apart from the banking system.

In the Europe, the regulation 2002-13 of 21 November 2002 prevented anyone to issue e-money outside the European interbank system. However, the development of the e-commerce is so rapid than the temptation to use anonymous ways of payment is, for many reasons, very strong for many e-buyers and esellers.

The second characteristic of the new communities of e-money users relies on the capacity of creating temporarily private money. A truly private, e-money originates outside the government and outside the banking system.

On that subject, the Second Life network users, is enlightening (see secondlife.com). Second Life has its own currency called Lindens. On any given day people are using it to conduct transactions. People can cash Lindens out of the game on the Lindex. Several online resources allow residents of Second Life to convert Linden Dollars into US Dollars and vice-versa . Rates fluctuate based on supply and demand, but over the last few years they have remained fairly stable at approximately 250 Linden Dollars (L$) to the US Dollar. It is also possible to make money in Second Life but such a money is not generated by private players of the game. All the money circulating in Second Life comes originally from the "real" Central Banks. Nevertheless, players earning Linden Dollars during the game avoid temporarily the official taxation on profits.

Living in an economy, where different monies coexist, forces people to adapt every aspect of their economic activities to this new environment. The most routine daily transactions have to be organized differently. As explained by [START_REF] Heymann | High inflation[END_REF], "Money is not a refrigerator. The picture of money as a service-producing asset is incomplete without a look at transactions practices". Admittedly, with the e-money the transaction costs decrease but new practices of transactions also appear. The monetary regime tends to be parcelled out by different communities of payment. By monetary regime, we adopt the Heymann-Leijonhufvud definition that is on the one hand, a system of expectations on the part of the public that governs their decision and, on the other, that pattern of behaviour on the part of the policy-making authorities that sustains these expectations. The monetary regimes forms a crucial part of the environment in which both the public and the authorities have to make their decisions.

The economy operates with different currencies, which each one used in different types of transactions. Wages are paid in domestic money and households convert them partly in different e-monies or quasi emonies. Different communities of payment threaten social institutions and arrangements existing in a united monetary regime.

The emergence of different communities of payment menaces the public character of the money. Money can be considerate as a public good. On this subject, Wieser clearly regarded money as one of the founding institutions of social economy [START_REF] Weiser | Social economics, reedited in Reprints of Economic Classics[END_REF]. Money is an institution which has social responsibilities. Money ownership must follow the same rules for anyone.

With the Digital bearer money, networks of money users are developing apart from banks and others networks. Digital bearer money is like cash and can be transferred person-to-person without going through a bank or PayPal account. The Digital bearer money creates injustice toward those for practical, legal or financial reasons cannot access it. Moreover, a major change in the nature of money would have significant unintended consequences. For instance, the advantages for criminals and tax evaders could be considerable.

Progressively, different communities of e-payment are setting up. For the moment, the footbridges between the various communities do not exist. Nevertheless, it is enlightening to make a comparison between Smiles and Maximiles.

Smiles or Maximiles are like points, much as the airlines' frequent flyer miles. They are now used as rewards for visiting hundreds web sites. But as more sites are added, these points tend to become world's virtual currency on the internet and can be assimilated as Digital bearer money.

Regarding to the Maximiles provider, it is possible to obtain Maximiles without using euros. The more you are visiting or buying good or services on the Maximiles online network, the more you obtain Maximiles. You can also buy Maximiles online for buying goods or services only available in the Maximiles' shop network. At the present time the EUR/Maximiles exchange rate is 0,02, i.e. with 1 euro, you can buy online 50 maximiles.

The communities of both Paypal banking services and Linden Dollars users are growing very fast. One important consequence results in the decrease in demand for central banks and consequently the decrease of central bank seigniorage revenue [START_REF] Rahn | The Impact of Digital Money on Central Banks[END_REF]. The central banks make large income from issuing paper banknotes, which are non-interest bearing central banks liabilities. The decrease of seigniorage revenues may cause that central banks can not cover their operation costs. As seigniorage revenues are one of the most significant income sources of national treasury at the same time, it can be also stated that there will be a decrease in the income of treasury.

The decrease in money demand due to the spreading of the e-money can be studied by using the Baumol-Tobin framework. Such a framework direct attention to the transaction costs of moving in and out money substitutes. The "shoe-leather" metaphor -the real costs of too many trips to the bank or broker-stems directly from it. The Internet e-money permit a reduction in transaction costs (the "shoe leather cost"). Consequently, on one side, the interest elasticity of money demand increase and on the other side there is a permanent decrease in money demand. Fullenkamp and NSouli demonstrate the analytical consequences of the e-money on the money market [START_REF] Nsouli | The Regulatory Framework for E-Banking[END_REF]. By using the standard IS-LM model, they shows that the e-money shift downward and flatter the LM curve and therefore decrease the interest rate. A flat LM curves represent Keynes's liquidity trap in which the monetary policy is not effective. Consequently, with the development of the e-banking, the fiscal policy becomes more and more inefficient.

Traditionally, Central banks try to influence the households and firms pricing behaviours through interest rate, exchange rate, asset price, credit, expectations and monetarist channels. The success of Central banks monetary policies requires positive demand on government money because central banks can influence real economy mainly through interest rates. As we have seen, the government money demand decrease as a consequence of the spread of electronic money. Therefore, the real effect of central banks monetary policy weakens.

The influence of the e-money on the consumer behaviours Main title

The various e-money networks communities, like the Linden Dollars case raise new questions. Traditionally, games worlds are completely distinct from the real world. In Second Life, a virtual world, many virtual objects -like ideas, software, surveysbought or sold during the game by virtual avatars are directly marketable in the real world. For instance, it is possible to make and enquiry, an opinion pool or any marketing test in the Second Life World and selling in central money the results of these surveys in the real world. Nowadays, most of the biggest international firms are active in Second Life. The borders between the Second Life virtual world and the real world are increasingly fuzzy. On an economic point of view, this opacity complicates the rational calculus of the price vector. In the general neoclassical theory inherited of Walras, Jevons and Menger, the fixation on a price rely on the private utility. In Second Life, the players buy or sell virtual objects at a price based on ostentatious criteria. For instance, you do not buy a virtual coat with linden dollars to avoid a cold in winter but for showing yourself and showing your richness in Linden Dollars. Therefore, goods have not a private utility for agents but a social utility.

The works of Thorstein Veblen (1857Veblen ( -1929) ) are helpful to understand the price determination mechanisms in Second life world. Veblen (1899) argues that economic life is driven not by notions of utility [START_REF] Veblen | The Theory of the Leisure Class[END_REF]. He defined conspicuous consumption as the waste of money by people to display a higher status than others. The conspicuous consumption term has a very popular catchphrase which is "Keeping up with the Joneses". Such a expression refers to the desire to be seen as being as good as one's contemporaries using the comparative benchmarks of social caste or the accumulation of material goods. By Social class, Veblen refers to the hierarchical distinctions between individuals or groups in societies or cultures. Failing to "keep up with the Joneses" is perceived as demonstrating to others socio-economic or cultural inferiority.

The conspicuous consumption term takes root in

The Theory of Moral Sentiments of Adam Smith [START_REF] Smith | The Theory of Moral Sentiments[END_REF]. Adam Smith acknowledged that economic agents value wealth not only for the goods and services that it procure to them, but also for the impression that it makes on others. Veblen analyses displays of wealth as status signals, but Smith proposes a different explanation. According to Smith, the consumer is naturally more disposed to show his richness than his poverty. "It is because mankind are disposed to sympathize more entirely with our joy than with our sorrow, that we make parade of our riches, and conceal our poverty. Nothing is so mortifying as to be obliged to expose our distress to the view of the public, and to feel, that though our situation is open to the eyes of all mankind, no mortal conceives for us the half of what we suffer." [START_REF] Smith | The Theory of Moral Sentiments[END_REF] Following Smith, John Stuart Mill [START_REF] Mill | The Principles of Political Economy[END_REF] observes that many goods are bought for the reputation they procure in regard to the opinion. "a great portion of the expenses of the higher and middle classes in most countries, and the greatest in this, is not incurred for the sake of the pleasure afforded by the things on which the money is spent, but from regard to opinion, and an idea that certain expenses are expected from them, as an appendage of station; and I cannot but think that expenditure of this sort is a most desirable subject of taxation." [START_REF] Mill | The Principles of Political Economy[END_REF] Like Smith and Mill, Veblen recognizes the importance of consumption not for its own sake but what it signals about the consumer. Originally, the conspicuous consumption term was used by Veblen to depict the behavioural characteristic of the nouveau riche, a new class that emerged in the 19th century. Afterwards, the signification of this term was extended to economic agents whose consumption patterns are driven by status seeking rather than their substantial needs. From Veblen works on conspicuous consumption, Paul Nystrom theorised a philosophy of futility which encompassed the emergence addictive or narcissistic behaviours provoked by consumerism [START_REF] Nystrom | Economics of Fashion[END_REF].

For Veblen, the conspicuous leisure is a waste of time by people to give themselves higher status. He asserts that humans are not rational, utility-seeking people who try to maximize their pleasure but are completely irrational creatures that run after social status. Reading Veblen works by taking into consideration what occurs in Second life is very enlightening. In the virtual world of Second Life, the different communities of users behave with an emulation attitude. The term emulation is proposed by Veblen to illustrate the permanent run after the social status. The avatars attempt to mimic the more respected members of their group in order to gain more status for themselves. Such a mimetic process can be studied by using the institutional isomorphic change described by DiMaggio and Powel [START_REF] Dimaggio | The iron cage revisited: Institutional isomorphism and collective rationality in organizational fields[END_REF]. Moreover, the emulation attitude modifies the mechanism of price determination. In Second Life, many goods are Veblen goods. Commodities are Veblen goods, by definition, if people's preference for buying them increases as a direct function of their price. Some types of high-status goods in Second Life , such as expensive virtual clothes, are Veblen goods, in that decreasing their prices decreases people's preference for buying them because they are no longer perceived as exclusive or high status products. Similarly, a price increase may increase that high status and perception of exclusivity, thereby making the good further preferable.

For instance, decreasing the price of the land decreases people's preference for buying it because it is no longer perceived as exclusive or high status products.

The Veblen goods introduce anomalies in the general theory of supply and demand in microeconomics. Among these anomalies, we can observe in Second Life the snob effect which increase the demand for goods because they have a high economic value, but low practical value. In Second Life, the Veblen goods introduce another anomaly called bandwagon effect [START_REF] Leibenstein | Bandwagon, Snob, and Veblen Effects in the Theory of Consumers' Demand[END_REF]. There is a bandwagon effect when the preference for goods increases as the number of people buying them increases. The bandwagon effect disturbs the normal results of the theory of supply and demand , which assumes that consumers make buying decisions solely based on price and their own personal preference. Such an effect introduces network externalities by increasing returns to adoption. Networks externalities can be found in any network because they provoke a positive feedback. The more the network increases, the more the network is attractive for new users.

Network effects become significant after a certain subscription percentage has been achieved, called critical mass.

By studying different electronic purses schemes, Clark found that the most successful networks tend to have the following characteristics: a captive audience that drives critical mass, such as those found in the transportation industry or government sector; an affordable cost structure relative to other payment instruments; compelling incentives to consumers and merchants; and a technology that is well tested and that addresses standard issues before the rollout [START_REF] Clark | Shopping without Cash: The Emergence of the E-Purse[END_REF]. Admittedly, the Second Life community, as any network, is characterized by this positive feedback. Moreover, positive feedbacks can be found inside the network because most a goods in Second life are Veblen goods.

For Veblen, social change can be analysed as a process of institutional coordination and transformation. Any human community, like the Second life community, may be viewed as an economic mechanism. Such an economic mechanism is closely linked to the material environment in which the human community lives.

According to Veblen, the economic and social behaviours are impeded from cultural rigidities. Observing the behaviour and mind sets of the avatars in the Second Life Community confirm the assertion of Veblen.

The Second Life world allow an agent to choose what kind of avatar she or he will live in, allowing a person with any kind of real body to inhabit a completely different body in the virtual world. As studied by Edward Castronova [START_REF] Castronova | Theory of the Avatar[END_REF], the emergence of avatar-mediated living raises both positive and normative questions. Logically, the main aim of creating an avatar in Second Life is to live a "second life" that is, a life which can be completely different of the life in the real world. Surprisingly, most of the avatars are a clone of its owners. For instance, if a man with brown hairs creates his own avatar on Second Life, he will choose a masculine brown hairs avatar.

The cultural rigidities on the real life pass through the virtual second life.

Conclusion

The communities of electronic money users are communities of practice which adopts specific decisions, routines and behaviours. These communities can temporarily move away from the official monetary regime regulated by central banks. Moreover, the classical analysis of price setting behaviour based on the private utility maximisation rule is somewhat unsuited. With e-monies, publics or privates, the monetary policy becomes less effective. Quasi emonies like the Linden Dollars provide socially useful services. The central banks should not prevent them to fulfil their social missions. However, central banks should integrate them in the official interbank system both for fiscal reasons and to protect the lawful aspect of the money. Money must not generate exclusion. All the economic agents must have access to all type of goods and all the virtual shopping centres.

Bandwagon effects, Veblen goods, and networks externalities affect the online computer e-money user behaviour. In the aim of understanding such a specific behaviours, it is useful to move away from the traditional theory of utility maximisation based on private goods because many goods bought online with e-money are social goods.


Claudia Teran-Escobar

Sarah Duché

Sonia Chardonnel

Kamila Tabaka

Aïna Chalabaev

Monika Christofori-Khadka

Nicole Rähle

Claudia Teran
email: claudia.teran.e@gmail.com

Geographic, sociodemographic and psychological factors related to Active and Sustainable mobility

Can we teach an old dog new tricks? Teaching behaviour change to health practitioners in Paraguay

It seems that YES, we can teach an old dog new tricks !

• Health practitioners increased their knowledge, self-efficacy and intention to use theory-and evidence-based behaviour approaches to change behaviour BUT…

• More robust methodologies (e.g., a control group) should be used and data about actual health promotion practices should be collected

Background•••

  The use of theory-and evidence-based approaches for health promotion in the real world is highly recommended 1 Caregivers and practitioners have an essential role in promoting behaviour change, BUT some studies indicate that they lack confidence, motivation, skills and training to deliver behaviour change interventions 2 • Past trainings in behaviour change techniques have increased intention and self-efficacy 3 • To carry a training on how to elaborate, implement and evaluate behaviour change interventions based on theory-and evidencebased approaches • To assess the effects of the training on knowledge, self-efficacy and intention towards using TEAs for behaviour change interventions. Collaboration between research and NGOs (Swiss Red Cross and Tesai Reka Paraguay) • Tesai Reka Paraguay works in promoting community-based family health in rural areas in Paraguay. Module, T= Measurement time. indicates an online survey about Knowledge about behaviour change interventions (multiple-choice test), Self-efficacy and intention towards elaborating and evaluating behaviour change interventions Results β = 3.81*** [2.24 ; 5.39] β = 0.36** [0.12; 0.60] Knowledge about behaviour change Intention towards using behaviour change