V V Zakharov

S L Ivanovski

Jean-François Crifo

V Della Corte

A Rotundi

M Fulle

J.-F Crifo

J.-F Crifo

V Della Corte

Asymptotics for spherical

Keywords: comets, general -dust dynamics -methods, numerical simulations

published or not. The documents may come

The study of dust-gas outflow from cometary nuclei has a long history and has now reached a remarkable degree of sophistication. The pioneering dust coma studies (Whipple (1951), [START_REF] Dobrovolskiy | Nonstationary process in comets and solar activity[END_REF], [START_REF] Dobrovolskiy | Komety[END_REF], Probstein (1969 ), Shulman (1972), Wallis (1982), [START_REF] Gombosi | Dust and Neutral Gas Modeling of the Inner Atmospheres of Comets[END_REF]) tried to derive analytical expressions keeping as much as possible a meaningful physical description, but often using exceedingly crude simplifications (e.g. omitting of gravity and/or solar pressure, assuming constancy of gas expansion velocity etc.).

More recently, numerical models of much higher complexity for the description of dust dynamics in the cometary coma appeared (e.g. [START_REF] Crifo | Multidimensional physicochemical models of the near-nucleus coma: Present achievements and requested future developments[END_REF], Rodionov et al. (2002)). These models either solve a system of differential equations describing the dust motion, or, perform a numerical simulation of the process. To predict the spatial and temporal distribution of physical variables (density, velocity etc.) in the coma, such models introduce a large number of governing parameters characterizing processes. Therefore the relative role of these processes is not easy to ascribe and relevant intercomparison of model results becomes difficult.

Our present work may look surprising. Following older models, we study the motion of a single spherical particle in a spherically expanding flow of an ideal perfect gas. However, contrary to previous models, we always keep in mind that, in all possible cases, the motion of the particle is governed by at least three forces: the aerodynamic drag, the nucleus gravity and the solar radiation pressure force. The aim of this study is to reveal the general properties of dust motion in an expanding flow and to reveal the relative influence of the three principal forces -drag, gravity and radiation pressure.

The model

The nucleus gravity and solar pressure force acting on a grain have simple classical expressions (see below) -as long as we assume the nucleus to be a homogeneous sphere.

The assessment of the aerodynamic drag force on the grain is more complex. It requires, first, computing the whole gas flow field. In general, the gas expansion from a spherical source into vacuum has three characteristic regions (see Lukianov and Khanlarov (2000)). The non-equilibrium nearsurface layer (Knudsen layer), where initially non-equilibrium flow (due to the boundary conditions)

relaxes to an equilibrium state, then a region of equilibrium flow, and finally once more a region of non-equilibrium flow resulting from the lack of intermolecular collisions. This scheme is shown in Fig. 1. The sizes of these three regions depend on the gas production of the source and may vary considerably. In the present study, we simplify the problem by assuming that the Knudsen layer is thin and the rest of the flow is in equilibrium. The gas flow is postulated as an adiabatic expansion of ideal perfect gas from a sonic spherical source, which means that on the nucleus surface we assume that the gas velocity is equal to the local sound velocity ( γT g k b /m g , where T g is the gas temperature, m g is the mass of the molecule, k B is the Boltzmann constant and γ is the specific heat ratio). This assumption about sonic conditions on the initial surface is justified by the fact that the expanding flow into vacuum is always supersonic and therefore irrespective of the conditions of gas production, the sonic surface surrounds the nucleus.

This approximation has the merit of having a fully analytical solution (Landau and Lifshitz (1966)), which we use here.

In conclusion, the motion of a particle with invariable size and mass under the influence of aerodynamic drag, nucleus gravity, and solar pressure forces is described in our model by:

m d dv d dt = (v g -v d ) 2 2 ρ g πa 2 C D (s, T d T g ) - GM N m d r 2 -πa 2 Q ef c cr 2 h cos(z ) (1)
dr dt = v d (2)
where a, m d , v d , T d are the particle radius, mass, velocity and temperature, ρ g , v g are the gas density and velocity, M N is the mass of the nucleus, G is the gravitational constant, Q ef is the radiation pressure efficiency, c is the velocity of light, c is the solar energy flux, z is the solar zenith angle, r h is the heliocentric distance (in AU), C D is the drag coefficient, and r and t are radial distance and time, respectively. To use once more an existing analytical expression, we assume that the particle size a is everywhere much smaller than the mean free path of the gas molecules. The drag coefficient C D is then given by the "free-molecular" expression for a spherical particle (see [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF]):

A C C E P T E D M A N U S C R I P T s C D (s,τ)
C D s, T d T g = 2s 2 + 1 s 3 √ π exp(-s 2 ) + 4s 4 + 4s 2 -1 2s 4 erf(s) + 2 √ π 3s T d T g (3)
where

s = | v g -v d |/ 2T g k B /
m g is a speed ratio. The variation of C D in a wide range of s for three ratios of T d /T g is shown in Fig. 2.

In order to let chacteristic dimensionless parameters appear, let us rewrite the governing dust equations in dimensionless form, introducing the following dimensionless variables:

θ = T d /T * , ṽg = v g /v max g , ṽd = v d /v max g , r = r/R N , t = t/∆t, τ = T d /T g , ρg = ρ g /ρ * . Here v max g = γ γ+1 γ-1 k b mg T * is
the theoretical maximal velocity of gas expansion, ρ * and T * are the gas density and temperature on the sonic surface (i.e. on the surface of the nucleus) and ∆t = R N /v max g .

Then equations ( 1) and ( 2) can be rewritten as:

dṽ d d t = ρg (ṽ g -ṽd ) 2 C D (s, τ ) • Iv -Fu • 1 r2 -Ro • cos(z ) (4) A C C E P T E D M A N U S C R I P T r/R N M v g /v g max , T g /T * ρ g /
dr d t = ṽd (5)
Where ṽg , r, τ and s in adiabatic spherical expansion (Landau and Lifshitz (1966)) are:

ṽg = (γ -1)M 2 2 + (γ -1)M 2 1/2 (6) r2 = 1 M 2 + (γ -1)M 2 γ + 1 γ+1 2(γ-1) (7) s = |ṽ g -ṽd | γ γ -1 + γ 2 M 2 (8) τ = θ 2 + (γ -1)M 2 γ + 1 (9) ρg = 2 + (γ -1)M 2 γ + 1 -1 γ-1 (10)
Here M = v g / γT g k b /m g is the Mach number. An example of the spatial distribution of gas parameters for γ=1.33 is given in Fig. 3.

The dimensionless parameters are:

Iv = 1 2 ρ * σ d R N m d = 3Q g m g 32R N aρ d π T * γk B /m g (11) A C C E P T E D M A N U S C R I P T and Fu = GM N R N 1 (v max g ) 2 (12)
and

Ro = 1 m d (v max g ) 2 R N σ d Q ef c cr 2 h (13)
Here Q g is the total gas production rate, σ d and ρ d are the particle cross-section and specific density respectively.

We note that the theoretical maximum velocity of gas expansion could be expressed also in terms of a stagnation temperature T 0 and a heat capacity C p as v max g = 2C p T 0 . In this way the gas velocity on the sonic surface is v * = 2C p T 0 γ-1 γ+1 and the gas production in [kg m -2 s -1 ] is

q = ρ * v * .
Therefore we can rewrite Iv as:

Iv = γ + 1 2(γ -1) 3qR N 8 C p T 0 aρ d (14)
The second term in this equation is the reciprocal of a dimensionless similarity parameter characterizing the ability of a dust particle to adjust to the local gas velocity introduced in Probstein (1969 ).

As can be seen the three parameters have the following meaning:

1. Iv represents the ratio of the gas mass present in a flow tube with the cross section of the particle and a characteristic length, R N , to the particle mass. This parameter characterizes the efficiency of entrainment of the particle within the gas flow (i.e. the ability of a dust particle to adjust to the gas velocity);

2. Fu represents the ratio of the comet surface gravitational potential to the flow thermodynamic potential (enthalpy, C p T 0 ). This parameter characterizes the efficiency of gravitational attraction;

3. Ro represents the ratio of the specific work done by the solar pressure force on the characteristic length R N to the flow thermodynamic potential. This parameter characterizes the contribution of solar radiation pressure.

In order to define Iv, Fu, Ro it is necessary to know: m g , γ, Q g (or ρ * ), T * , R N , M N , a and ρ d (or

σ d and m d ), Q ef and r h . A C C E P T E D M A N U S C R I P T

Results

The present study covers the range of Iv = 10 5 -10 -10 , Fu = 0.1-10 -10 , Ro/Fu = 1.0-0.0 and γ = 1.33 and θ = 1. For simplicity, we study only the case when cos(z )=1 (i.e. sunward direction).

The gas velocity is permanently increasing -initially rapidly and then quite slowly. For γ=1.33 it reaches 0.9 and 0.99 of v max g at r=6.4 and 185 correspondingly (see fig. 3). Therefore we limited the radial size of the computational domain to r ≤ 10 3 . For numerical integration of the system defined by equations ( 4) and ( 5) we used the Bulirsch-Stoer method (Press (1986)).

The condition that the grain could be lifted from the surface is:

γ -1 γ + 1 C D ( γ/2, θ) • Iv > Fu + Ro (15)
For θ=1, γ=1.33 and Ro = 0 this condition is: 0.794 Iv > Fu.

If we neglect the solar pressure force (i.e. Ro = 0), the maximum liftable size a max is:

a max = 3Q g m g C D ( γ/2, θ) T * γk B /m g 32ρ d πM N G (16)
and a max = 1.075 • 10 -2 Q g T * m g /(ρ d M N ) for θ=1, γ=1.33.

Case when Ro = 0

Let us first study the case when the solar pressure force could be neglected (i.e. Ro = 0). The accelerating aerodynamic force decreases with the same rate as the decelerating gravity attraction and therefore if a particle is lifted from the surface then it is continuously and monotonously accelerating.

Fig. 4 shows the particle velocity ṽd and transit time t at the distances r=10, 10 2 and 10 3 (the particle velocities at r = 10 2 and 10 3 are practically indistinguishable and therefore shown only for r = 10 3 ). For the maximum values of Iv the grain velocity at r = 10 3 and the gas maximal velocity are very close. For Iv > 1 the velocity of a particle at r = 10 3 is practically the same for any Fu < 1. For Iv < 1 the velocity of the particle is very sensitive to Fu and is considerably smaller then the gas velocity. This is intuitively correct from inspection of equation ( 14), which shows that Iv is proportional to q/a.

Though the gas and dust velocity permanently increase at large distance their acceleration is very small and therefore as a terminal velocity of the particle (v max d

) we take v d at r = 10 3 . Fig. 5 A shows the distance and transit time when ṽd reaches 0.9 and 0.99 of the velocity at r = 10 3 . For the maximum values of Iv the particle reaches 0.9 and 0.99 of the terminal velocity at r = 6 and 116.5, correspondingly. For Iv > 1 this distance does not depend on Fu (for Fu < 1) but for Iv < 1 it is very sensitive to Fu. It is interesting to note that the distance where the particle acceleration practically finishes (i.e. v d ≈ const) has three characteristic ranges of Iv. For Iv > 10 4 , it assymptotically approaches r ≈ 6 for 0.9 of v max d (or r ≈ 116.5 for 0.99

C C E P T E D M A N U S C R I P T log 10 (Iv) v d /v g max (r/R N =10) v d /v
• v max d
). For Iv < 10 -2 it assymptotically approaches r ≈ 5.6 (or r ≈ 51.7 for 0.99

• v max d
). In the range 10 -2 < Iv < 10 4 the variation of this distance is non-monotonic with minimum r ≈ 3.5 (or r ≈ 30 for 0.99

• v max d ) at Iv ≈ 3.162.
In the above discussed cases the initial velocity of the particle was ṽ0 d =0. Now we study the cases when the particle has initial velocities from 0 up to 0.5. For a sonic source the gas velocity on the surface is: ṽ * = γ-1 γ+1 = 0.376, for γ=1.33. Note that we included also the case when the initial particle velocity is greater than the gas velocity at the surface (case ṽ0 d = 0.5). Fig. 6 shows the particle velocity ṽd at the distances r=10 and 10 3 and it allows us to estimate the influence of the initial velocity at various Iv and Fu. For Iv > 1 a non-zero initial velocity is undetectable already at r = 10. But for Iv < 1, depending on Fu, the non-zero initial velocity may affect the particle velocity up to the external boundary of the region. The lower is Fu, the more noticeable is the influence of an initial velocity; this is particularly evident for the low values of the initial velocity. d for γ = 1.33 and θ=1.0. The color of the lines and labels corresponds to log 10 (Fu), the pattern of lines and values of labels correspond to ṽ0 d .

For example, at Iv < 10 -4 and Fu < 10 -4 the initial velocity ṽ0 d ≥ 10 -2 (and at Iv < 10 -2 and Fu < 10 -2 the initial velocity ṽ0 d ≥ 0.1) remains practically constant until exiting from the region.

In the case Iv < 10 -2 and Fu = 10 -1 the particle with an initial very high velocity ṽ0 d = 0.5 even decelerates.

Case when Ro > 0

Here we study only the case when cos(z ) = 1 (i.e. sunward direction) and Fu ≥ Ro. To simplify results representation, we express the solar pressure force as a fraction of the gravity force on the surface, which is equivalent to Ro/Fu. Since gas and dust velocity for r > 6 vary not much and the first term in eq.4 decrease as r-2 , the ratio Ro/Fu allows one to estimate the distance at which the solar pressure force starts to be sizeable.

The presence of non-zero solar pressure force, which does not decrease with distance, leads to a final deceleration of the particle motion. Fig. 7 shows the particle velocity ṽd at distances r=10 and 10 3 , the transit time t at distances r=10, 10 2 and 10 3 , the particle maximal velocity, and the distance and time when it is reached. For this figure, if the particle did not start deceleration before r = 10 3 , the particle velocity at the exit of the region is taken as the maximal velocity ṽmax that since we use same ratios Ro/Fu in all cases, with increasing of Fu grows Ro as well. Fig. 7 shows the distance where ṽmax

Conclusion

The dust flow preserves typical general features regardless the particular coma model and the characteristics of the real cometary coma. We have presented an elementary model of particle motion in a spherically expanding flow parametrized by three dimensionless parameters (Iv, Fu, Ro). This model could be used as a reference model for the comparison of more sophisticated cometary dust coma models. It can also be used for rough estimations of the asymptotic behaviour of particle motion (terminal velocity, and distance and time when it is reached).

We intentionally skip the processes in the layer close to the surface since they are strongly

•A

  Asymptotics for spherical particle motion in a spherically expanding flow Preprint submitted to Icarus April 26, 2018

Figure 1 :

 1 Figure 1: Flow structure in the source (∆ * , ∆ Kn are the thickness of subsonic layer, non-equilibrium or Knudsen layer) .

Figure 2 :

 2 Figure 2: Variation of the drag coefficient C D for 10 -2 ≤ s ≤ 10 2 and τ = T d /Tg=0.1, 1 and 100.

Figure 3 :

 3 Figure 3: Radial distribution of M , vg/v max g , Tg/T * , ρg/ρ * for γ=1.33.

Figure 4 :Figure 5 :

 45 Figure 4: Particle velocity ṽd at r=10 (top left), particle velocity ṽd at r = 10 3 (bottom left) and transit time t (top right) for γ = 1.33 and θ=1.0 (color numbers indicate the corresponding log 10 (Fu)).

Figure 6 :

 6 Figure 6: Velocity at radius r = 10 (left) and r = 10 3 (right) for different initial velocity ṽ0 d for γ = 1.33 and θ=1.0. The color of the lines and labels corresponds to log 10 (Fu), the pattern of lines and values of labels correspond to ṽ0 d .

  r=10 there is no remarkable variation of velocity for 0 ≤ Ro/Fu ≤ 0.1. But at the distance r=10 3 the distinction in velocity increases with increase of Fu. It should be noted

Figure 7 :Figure 8 :

 78 Figure 7: Particle velocity ṽd at r=10 (top left), particle velocity ṽd at r = 10 3 (top right), transit time t (middle left), radius r(ṽ d = ṽmax d ) (middle left), transit time t(ṽ d = ṽmax d ) (bottom left) and maximum particle velocity

d

  is acquired (for Ro/Fu=0 this is r = 10 3 ) and the transit time required to achieve ṽmax d . The transit time depends weakly on Iv < 1 for the given Fu and Ro. Note also that the transit times and the distances where ṽmax d is acquired are similar for the same Ro independently on Fu.

Fig. 8

 8 Fig.8shows the distance and time when the particle decelerates down to zero velocity. With growth of Iv the distance and transit time to the apex of the trajectory (where ṽd = 0) become similar for the same Ro irrespective of Fu (since the value of Fu relative to Iv decreases).

A C C E P T E D M

A N U S C R I P T dependent on the local properties (irregularity and inhomogeneity of the nucleus). For the same reasons, in the case when dust grains have an initial velocity, we have omitted any discussion of the reasons how such an initial velocity might occur.

For evaluation of the aerodynamic force we used a spherically expanding flow of an ideal perfect gas. This idealized model of the flow practically never happens in real cometary atmosphere (nonequilibrium and rarefied) but it physically preserves integral properties of the real flow on a large scale correctly.

We would like to emphasize that the model, being oversimplified, cannot be used as a cometary coma model. But, based on the results presented, it is possible to estimate the relative influence of the main factors (aerodynamics, gravity, solar pressure, initial velocity) affecting every dust flow and reveal processes similarity.

OSIRIS data constrain the dust acceleration limited within six nuclear radii for a broad range of particle sizes [START_REF] Gerig | On deviations from free-radial outflow in the inner coma of comet 67P/Churyumov-Gerasimenko[END_REF], in agreement with our model.


Abdelfetteh Lachtar
email: lachtar.abdelfetteh@gmail.com

Abdennaceur Kachouri

Thierry Val
email: thierry.val@irit.fr

Real-time monitoring of elderly using their connected walking stick

Keywords: m-Health, Thresholding, Accelerometer, Magnetometer

Lots of aging people rely on a caneasan assistive device to ovoid problems such as leg weakness and balance loss, which are presumed to haveinducedmanyfallincidents. In this paper, we present a new design and implementation of Canne3D system, a local monitoring system for elderly using their connected walking stick. Actually, the 3Dcane is modeled through consideration of real-time movements for a deep understanding of the person's state, including steps counting and calculating the distance traveled. We have applied a simple multi-stage thresholdingmethod based on the acceleration value. Experiments show that the algorithms appear to be robust in term of detection rate and false positive rate.

I. INTRODUCTION

The Internet of Things (IoT) is the future Internet; it represents a revolution in computing and communications. It enables a world of networked devices, where everything is interconnected and has a virtual representation [START_REF] Pascual | Modeling architecture for collaborative virtual objects based on services[END_REF]. Everyday objects are transformed into smart objects able to sense, interpret and react to the environment thanks to the combination of Internet and emerging technologies such as Radio-frequency Identification (RFID), real-time localization and embedded sensors [START_REF] Ovidiu Vermesan | Internet of Things: Converging Technologies for Smart Environments and Integrated Ecosystems[END_REF].

We strongly believe that the Internet of Things can give the assistance and support to people with disabilities that they need to achieve a good quality of life and allows them to participate in the social and economic life. Assistive IoT technologies are powerful tools to increase independence and improve participation. Real-time monitoring systems will be increasingly useful and necessary in the coming years. The population is aging and the society must solve this new problem and find the best way to take care of elderly [START_REF] Ben | Behavioral wireless sensor network for smart home monitoring[END_REF]. Therefore, mobility devices and personal assistance are strongly desired to ensure a certain independence of the elderly. Possible assistance devices include canes that have a large number of older users because of their simplicity and walking aid they bring. This paper proposes a contribution for the 3DCane system [START_REF] Lachtar | 3DCane: a monitoring system for the elderly using a connected walking stick[END_REF], which is a system that tracks the movements of elderly more visually using their walking stick. This system allows the display of a 3D model of the cane and its graphic animation based on actual movements obtained through a wireless transmission of data from sensors on the cane (accelerometer and magnetometer), it also counts the number of steps and calculates distance in meters.

The following article is organized as follows: Section 2 provides an overview of 3DCane system, including the hardware and software architectures. In Section 3, we offer to present the design and details of 3D sensing and modeling algorithms that we have developed, which are next evaluated experimentally in terms detection rate and of false positive rate in Section 4. Finally, in section 5, we conclude the whole paper and we give an idea of the possible future improvements and extensions of 3DCane system.

II. SYSTEM ARCHITECTURE

Faced with an aging population and the massive use of the walking cane by the elderly, researchers from Toulouse-Blagnac have created the CANet project [START_REF] Val | CANet project : a monitoring system for people with disabilities using their walking stick[END_REF] which is a system for monitoring people with disabilities through their walking cane. Our monitoring system is based on the CANet project, Figure 1 shows the architecture of 3DCane system that we have developed. The system consists of: (i) a transmitter node positioned on the cane, (ii) a receiver node and (iii) a terminal for data analysis and 3DCane application.

A. The transmitter

The transmitter node is used (i) to collect data from the 3D accelerometer and 3D Magnetometer, (ii) calculate the linear Fig. 1. 3DCane System architecture acceleration and acceleration due to gravity, (iii) calculate the angle of rotation on x-axis (roll), the rotation angle on the yaxis (pitch), the rotation angle on the z-axis (yaw), (iv) fall detection, calculate the number of steps and the distance traveled, and (v) ensure the transmission of data.

For this sake, we used an lsm303dlhc which is a system-inpackage featuring a 3D digital linear acceleration sensor and a 3D digital magnetic sensor, a Teensy 3.2 as a processingboard and a Radio RFM22 used for data transmission, the node architecture is explained in figure 2.

The LSM303DLHC has linear acceleration full-scales of ±2g / ±4g / ±8g / ±16g and a magnetic field full-scale of ±1.3 / ±1.9 / ±2.5 / ±4.0 / ±4.7 / ±5.6 / ±8.1 gauss. All full-scales available are fully selectable by the user [START_REF]Ultra-compact high-performance eCompass module: 3D accelerometer and 3D magnetometer[END_REF].The Teensy 3.2 hasa MK20DX256 Processor, 32 bit ARM, Cortex-M4, 72 MHz. Flash Memory: 262144. RAM Memory: 65536. EEPROM: 2048.The RFM22 has a Frequency Range between 240 MHz and 930 MHz, a sensitivity of -118 dBm, a data Rate from 1 to 128 kbps, a +17 dBm Max Output Power and a power supply of 1.8 V to 3.6 V.

B. Comminication protocol

The architecture we proposed is composed of multiple transmitters (cane) that send data to one receiver for local monitoring, so we used aloha as a medium access protocol. The RFM22 radio module uses the 802.15.4 norm for packet format.The MAC frame format is composed of a MHR, a MAC payload, and a MFR as represented in figure 3.

For transmission we used a frequency that belongs to an ISM (Industrial, Scientific and Medical) 411 Mhz .ISM bands [START_REF] Loy | ISM-Band and Short Range Device Regulatory Compliance Overview[END_REF] are frequency bands that can be used in a small space for industrial, scientific, medical, domestic or similar.

The slot time used is 100 ms to enable better monitoring in real time. Frame payload is represented in table 1. Themessage sizeis 337bitsthereforemessagetransmission time is0.81ms.

III. METHOD

In this section, we will detail the algorithms of functioning of our Canne3d system such as the determination of absolute orientation, the algorithm of steps counting and calculate the distance traveled.

X, Y and Z axes of the acceleration are represented on the cane in the following manner as shown in Figure 4.

A. Absolute orientation

To determine the orientation of cane, it is necessary to calculate the angle of rotation on the x-axis (roll), the rotation angle on the y-axis (pitch) and the rotation angle on z axis (yaw) as shown in Figure 5. The accelerometer is intended to provide a measurement based on the linear acceleration and acceleration due to gravity [START_REF] Badura | Acceleration trajectory analysis in remote gait monitoring[END_REF]. Still, a major problem related to the accelerometers consists in the fact that they help measure both of the linear acceleration and that related to gravity. As it cannot distinguish between these two accelerations, it turns out to be critically necessary to separate both of the acceleration types (gravity and motion) through the use of a special filter.

On the X axis, for instance, the gravity relating acceleration can be calculated as follows:

Initially Xg = 0.

(1)

With alpha = 0.5 (as presented in [START_REF] Vargha | Motion Processing Technology Driving New Innovations in Consumer Products[END_REF]) and X is the value of acceleration on the x-axis provided by the accelerometer.

Using the accelerometer output, the rotation around the X axis (roll) and the Y axis (pitch) can actually be calculated. On considering the Accel_X, Accel_Y and Accel_Z as the accelerometer measurements (gravity acceleration) in the X, Y and Z, then equations ( 2) and (3) blow, would demonstrate how to calculate the angles Roll and Pitch [START_REF] Abyarjoo | Implementing a Sensor Fusion Algorithm for 3D Orientation Detection with Inertial/Magnetic Sensors[END_REF]:

² (2) ² (3)
If mx, my and mz are calibrated and normalized magnetometer outputs, and , and respectively stand for roll, pitch and yaw, the heading appears to be computable through equation [START_REF]Ultra-compact high-performance eCompass module: 3D accelerometer and 3D magnetometer[END_REF]. Equations 4 and 5 are applied to transform the magnetometer reading to the horizontal plane. Once the magnetometer data appear on the flat plane, equation 6 would then get a reliable calculation.

B. Steps counting

We can use the linear acceleration to determine if the person is walking using his walking stick by observing the change of the linear acceleration in the X, Y and Z as shown in Figure 6.

When an elderly person is walking with his cane, the acceleration value due to gravity on the y-axis remains above 9 m / s.

After analyzing the changes in linear acceleration and acceleration due to gravity we implemented an automatic step detection algorithm:

Steps = 0. Count = 0.
Store the 3 oldvaluesoflinear accelerationonthe z axisin the variablesLZ1, LZ2andLZ3. If(ly>0.5)and(Iz>0.5 orLZ1>0.5orLZ2>0.5orLZ3>0.5)and(gy>9) and

(count == 0) then steps++ Count =5 Endif. If(count > 0) then Count = Count -1. EndIf.
Ly is the linear acceleration on the y-axis and gy is the acceleration related to the gravity of the y axis. The minimum period between steps of a person is equal to 0.5 seconds. For this, we added a counter variable to not confuse one step to another activity.

C. Disstance traveled

The distance is relative to the movement of the Cane on the z axis. In fact, when the person walks with his cane, the acceleration on the z-axis increases and then decreases and then returns to 0. This is the same reasoning for calculating the number of steps.

In general, to calculate a distance from the acceleration, we have to use the following rule:

If we consider a = constant acceleration during a time t then: Speed v (t) = a * t + v0; with v0 = initial speed; Distance s (t) = ½ * a * t² + v0 * t;

In our case t = 100 ms, and we calculated the speed and distance traveled each time a step is detected.

IV. RESULTS AND PERFORMANCE ANALYSES

In a first test, we evaluated the GUI of the application in many situations then we performed the test of the fall detection algorithm and the calculation of the number of steps and distance traveled.

A. The GUI

The 3DCane GUI resulting application allows a visualizing of the real time Cane status using a 3D display representing its absolute orientation. It also helps recognize the number of steps and the traveled distance. It helps also recognizing the direction of the cane (North, East, West, and South) like in figure 7.

B. Steps counting

For the test of the steps counting algorithm; we have chosen four different people. Each one of those persons have made 100 meters of walking and then we have measured the distance given by the application. Table 2 shows that with a slow walks or fast walk, the number of detected steps is almost the same as the number of steps made by the four persons.

C. Distance traveled

We have also chosen four different people in order to test the measurement of the distance traveled, results are shown in the next table .   Table 4 shows that the algorithm of calculation the distance is efficient, in fact the average error doesn't exceed 6% so the measured distance can be used to monitor the elderly. The cane 3D modeling, together with its movements, appears to be helpful in identifying its condition as well as the user's status. Once a fall is detected, the application triggers a voice alert as shown in figure 8. In addition, such an interface enables us to recognize whether the user is walking or in a standing still position. V. CONCLUSION

In this work, we have developed a novel method for realtime monitoring of disabled people, using their walking cane equipped with an accelerometer and a magnetometer. we implemented an algorithm for calculating the steps number and the distance traveled, and then we implemented the 3DCaneapplication, to visualize the state of the walking stick in real-time and to control the elderly movement. Experimentations show that Canne3d application is efficient and useful for elderly monitoring.

As the current application can be used for local monitoring, we plan to extend our work and enable the monitoring via the internet.

Fig. 2 .

 2 Fig. 2. Sender node composition

Fig. 5 .

 5 Fig. 5. Rotation on x, y and z axes.

Fig. 6 .

 6 Fig. 6. Detection of a step through the linear acceleration.

Fig. 8 .

 8 Fig. 8. GUI Cane3d application when falling with the cane

TABLE I .

 I DATA COMPOSITION

	Name	Description	Size
	CaneId	unique identifier of the cane	16
	Roll	Rotation angle on the x axis	32
	Pitch	Rotation angle on the y axis	32
	Lx	Linear acceleration on the x-axis	32
	Ly	Linear acceleration on the y-axis	32
	Lz	Linear acceleration on the z-axis	32
	Gx	Acceleration due to gravity on the x axis	32
	Gy	Acceleration due to gravity on the y axis	32
	Gz	Acceleration due to gravity on the z axis	32
	Pas	Number of steps	16
	Yaw	Rotation angle on the z axis	32
	Distance	Distance traveled	16
	Fall	Boolean indicating if a fall is detected	1

TABLE II .

 II AVERAGE STEPS MESURED FOR 100 STEPS TAKEN BY FOUR

		PERSONS
	Walk types	Result
	Slow walk	96
	Fast walk	94

TABLE III .

 III FALSE POSITIVE RATE FOR VARIOUS ACTIVITIES

	Activity	Percentage
	Fall	0%
	Swing	5%
	Sit and stand	5%
	Lay on lap	0%

Table 3reveals

 3reveals well that steps detection algorithm is robust since the false positive rate is almost equal to 0 for various activities.

TABLE IV .

 IV DISTANCE MESURED FOR 100 METERS OF WALKING

	Person	Result (m)
	1	105
	2	106
	3	103.7
	4	104.2


Hayfa Ayadi
email: ayadihayfa.ing@gmail.com

Ahmed Zouinkhi
email: ahmed.zouinkhi@enig.rnu.tn

Boumedyen Boussaid
email: boumedyen.boussaid@gmail.com

Naceur Abdelkrim

Thierry Val
email: thierry.val@irit.frsabri.khssibi@irit.fr

Sabri Khssibi

An Effective Approach for Energy Aware in Wireless Sensor Network

Keywords: The Standard of IEEE 802.15.4, WSN, The Beacon Enabled Mode, BO, SO

In wireless communication, energy present very serious problem which could be easily affect the lifetime of all the network. The increasing of the wireless connected object lead to the appearance of a new notion which is the Internet of Thing. The growth of network size produce very important quantity of data as a result which request more quantity of energy to consume. The technology of the IEEE 802.15.4 present very powerfull solution for all application dedicated to low energy consumed which justify our choice for it to study. This paper investigate an approach used to postpone the fault energy of node in the network. So we get the energy consumed by every node then we detect the node with lack of power. After that we propose a specific manner to get both the Superframe Order (SO) and the Beacon Order (BO) for this node in order to more maintain the little power lasted in its battery. We use in our work the topology of tree with the mode of beacon enabled for Wireless Sensor Network (WSN).

I. INTRODUCTION

Wireless Sensor Network (WSN) succeed in gaining a great regard from both researchers and engineers thanks to its increased demand in different areas. So many studies were developed to solve the main trouble of it which is the energy efficient. Divers protocols were invented to reduce energy consumption in WSN such as in [START_REF] Mekki | Usee: A uniform data dissemination and energy efficient protocol for communicating materials[END_REF]. In addition to that fault detection present very serious problem for WSN because of the limited resources of the nodes. Multiples methods were assumed to solve that [START_REF] Ayadi | A machine learning methods: Outlier detection in wsn[END_REF], [START_REF] Saihi | Decentralized fault detection in wireless sensor network based on function error[END_REF].

The standard of the IEEE 802.15 Personal Area Network (PAN) is formed by divers protocols destined for little power communication. The IEEE 802.15.4 is considered as sub branch of PAN standard [START_REF] Wang | An improvement of ieee 802.15. 4 mac protocol in high-density wireless sensor networks[END_REF]. This technology is included in the most popular technology which is destined for law data rate. It is increasingly used in many fields blessing to its different benefits. LRWPANs have many important characteristics which are the shortrange operations, the facility of installation, the honest data transfer, cheap cost, objective battery life and the maintain of uncomplicated protocol. The technology of IEEE 802. [START_REF] Mirzoev | Low rate wireless personal area networks (lr-wpan 802.15. 4 standard[END_REF].4 is manipulated in many kinds of networks such as ZigBee, 6LoWPAN and ISA100.11a [START_REF] Reziouk | Practical security overview of ieee 802.15. 4[END_REF].

The LRWPANs is used in many applications and many fields such as agriculture, environment monitoring, security purposes, military, sports ( [START_REF] Lloret | A group-based wireless body sensors network using energy harvesting for soccer team monitoring[END_REF], [START_REF] Garcia | A wireless sensor network for soccer team monitoring[END_REF]) as well as entertainments [START_REF] Moravejosharieh | A survey of ieee 802.15. 4 effective system parameters for wireless body sensor networks[END_REF]. In healthcare fields [START_REF] Ullah | A review of wireless body area networks for medical applications[END_REF], the technology of IEEE 802.15.4 is so used in order to supervise the humans parameters [START_REF] Pantelopoulos | A survey on wearable biosensor systems for health monitoring[END_REF] like the cardiac patients monitoring [START_REF] Kumar | Ubiquitous computing for remote cardiac patient monitoring: a survey[END_REF]. In ( [START_REF] Caldeira | Toward ubiquitous mobility solutions for body sensor networks on healthcare[END_REF], [START_REF] Caldeira | Intra-mobility support solutions for healthcare wireless sensor networks-handover issues[END_REF]) the authors study the case of wearing a WBAN by patient. So they offers a continuous monitoring for him. However, sensor nodes send periodically certain parameters to the PAN coordinator to prove the best operation of the node in which it is implemented.

WPAN is used also in military field. The dress uniform wearing by the battle could consist of a BAN which is a bearable network. This BAN could include so many electronic devices such as support sensors in order to protect the connection of the soldiers wearable computer [START_REF] Qadri | Applications, challenges, security of wireless body area networks (wbans) and functionality of ieee 802.15. 4/zigbee[END_REF]. In addition to that, in sport field the body of players is implemented by sensors in order to detect the players movement. It is destined to explore a new manner to conserve energy and increase the performance of the player to the top of altitude over a long era of time [START_REF] Qadri | Applications, challenges, security of wireless body area networks (wbans) and functionality of ieee 802.15. 4/zigbee[END_REF]. The WPAN is used also in other different domains such as in entertainment. Although the multiple advantages of the technology of IEEE 802.15.4 that person could not ignored, this standard presents many drawbacks such as its restricted communication authenticity, the absence of protection on front of interferences and Powered relay nodes as well as Unbounded Delay. For all theses raisons the IEEE 802.15.4 is not the right choice in terms of accuracy, latency or performing in inaccessible environments [START_REF] Mirzoev | Low rate wireless personal area networks (lr-wpan 802.15. 4 standard[END_REF]. For the reason to exceed such restriction, a Research Group named 802.15 Task Group 4e was lanced in 2008. Consequently, the standard of the 802.15.4e was developed in 2012 [START_REF] Mirzoev | Low rate wireless personal area networks (lr-wpan 802.15. 4 standard[END_REF]. Many other functionality was added in the first standard. This new standard gives us more opportunity such as the Time Slotted Channel Hopping (TSCH). Also an other extension of Multi-channel was added with its Deterministic and Synchronous (DSME) characteristics [START_REF] Mirzoev | Low rate wireless personal area networks (lr-wpan 802.15. 4 standard[END_REF]. Moreover the technology of 802.15.4e is identified by its Low Latency Deterministic Network (LLDN) [START_REF] Mirzoev | Low rate wireless personal area networks (lr-wpan 802.15. 4 standard[END_REF]. So it presents five new MAC protocols destined for distinct operation field and some general application enhancements.

We interest in our paper to this standard (IEEE 802.15.4). Our approach has for purpose to postpone the fault energy of node in the network. We start by calculating the energy consumed by every node. Then we detect the node wich suffer from the lack of power. However, we propose a specific manner to get the Superframe Order (SO) and the Beacon Order (BO) for this node to maximize the lifetime of the network.

Our paper is organised by the way that section 2 presents some related work about the adjustment of the Beacon Interval (BI) and the Superframe Duration (SD). Then we detail an overview about the standard IEEE 802.15.4. Section 4 explains the different methods used to calculate the divers forms of energy consumption. However section 5 provides the way used to adjust the standard parameters. Moreover we introduce our results of simulation in the next section and in the last part we conclude this paper.

II. RELATED WORK

Since both Beacon interval and super-frame duration have a direct link with the period of activity of the node. So it contributes directly in the lifetime of the node as well as the network. Many works were developed in this field. Some research groups were interested in only the Beacon Interval adjustment other were interested to adjust the Super-frame Duration. Although other tried to access to both the BI and the (SO).

The change of Beacon interval is considered as the most attractive case for research. In [START_REF] Neugebauer | A new beacon order adaptation algorithm for ieee 802.15. 4 networks[END_REF], the writers use a defined amount of (SO) and modify (BO) value. This modification affects directly the duty cycle of the network. This adjustment depends essential from the traffic store of the network. If the data traffic increase then (BO) value become lower however if sleeping period decrease (BO) increase.

In [START_REF] Neugebauer | A new beacon order adaptation algorithm for ieee 802.15. 4 networks[END_REF], the authors proposed a new algorithm called the Beacon Order Adaptive Algorithm (BOAA) which is destined for networks with star topology. The technique intervene only in the(BO) parameter of the superframe structure. It's about a technique which recalculate just (BO) parameter.

The second case is to change just the Super-frame Order SO value and to conserve the Beacon frame Order (BO) fixe. In [17], the searchers develop an original approach named the Dynamic Super-frame Adjustment Algorithm (DSAA). They have a very clear objectives which is the reduction of the energy usage and to boost the canal exploitation. So the coordinator begin by calculating the superframe activity and collision percentage. The first one characterise the period in which the PAN is engaged in collecting or sending information. Despite, the occupation rate is get according to the received data at the PAN coordinator who compares after that theses values to other considered as a thresholds then it selects the Super-frame Order.

There are many other parameters taken into account to choose the most perfect value of (SO) in the network such as throughput during the Contention Access Period (CAP), the queue state as well as the idle period in the technique of Adaptive Duty Cycle Algorithm (ADCA) [18]. The PAN node make a decision based on the duration of the idle period. If it exceeds the half of the CAP, the value of (SO) will be decreased. In the case that it is is fewer than 50 percent of the CAP part, then the PAN go over to examine the number of packets collected and the number of suspended packets over the CAP. In the case that suspended packets are higher than collected packets, the (SO) will be developed for the next Beacon Interval.

An other very different methods used for the adjustment of the (SO) value which is presented in [START_REF] Khanafer | Adaptive sleeping periods in ieee 802.15. 4 for efficient energy savings: markov-based theoretical analysis[END_REF], [START_REF] De Paz Alberola | Duty cycle learning algorithm (dcla) for ieee 802.15. 4 beacon-enabled wireless sensor networks[END_REF]. It is about Markov-base theoretical analysis. The main idea of this technique is based on a mathematical models used. The value of (SO) is manipulated based on network conditions.

In the third case both (BO) and (SO) are adapted simultaneously to gain more fluently. In [START_REF] Oliveira | A duty cycle selfadaptation algorithm for the 802.15. 4 wireless sensor networks[END_REF], the approach of Duty cycle Self-Adaptation Algorithm (DBSAA) was developed based on the duty cycle value. The adjustment depend from four parameters: the Collision Rate (CR), the source nodes number, superframe Occupation Ratio (OR) as well as message collected by the coordinator.

III. THE IEEE 802.15.4 OVERVIEW

The technology of the IEEE 802.15.4 is an unwired network standard which is widely used for the low power environment. It defines the two first layers of the Protocol Stack which are the Physical layer and Medium Access Control layer. This technology gives as the choice of frequency bands from the available one which are 868 MHz, 915 MHz and 2.4 GHz with multiple canals of transmission in every band but the most used band is the 2.4 GHz band. In which a big problem could be lived which is the interference trouble. Two kinds of interferences are presented in this band: internal interference and external interference. In the first one it is simply the mutual interference of WBANs over each other although in the second one always he intervention of an external technology is the responsible for example WiFi or Bluetooth over WSNs [START_REF] Petrova | Performance study of ieee 802.15. 4 using measurements and simulations[END_REF], [START_REF] Vaithiyanathan | Performance evaluation of ieee 802.15. 4 using association process and channel measurement[END_REF]. Also this standard give as the opportunity to choose between three data rates 250 kbps, 40 kbps and 20 kbps. In the Personal Area Network, (PAN) coordinator sets the main practical parameters which could be the duty cycle or the band of frequency. This standard presents two functional status which are the non-Beacon-enabled mode and Beaconenabled one. In first mode, the MAC method used is the well known protocol Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA). So the Clear Channel Assessment CCA is employed before sending data on the medium. Then a decision is made according to its state. If it is not clear, it stays waiting for a period of time before the next attempt. The energy consumed in this mode is very important, showed by figure 1. In the Beacon-enabled mode, the coordinator transmit beacon frame every period for all members of the network in order to synchronise it. Two successive frames (Beacons) represent the Beacon Interval (BI) which its expression is given by equation [START_REF] Mekki | Usee: A uniform data dissemination and energy efficient protocol for communicating materials[END_REF]. This period of time is composed by an operating duration and an optional inactive duration. The Superframe Duration (SD) present the first part. It is formed by two parts: (CAP) and the Contention Free Period (CFP), showed by figures 1. The slotted version of CSMA/CA mechanism is employed for channel access in first part whereas in CFP an scheduled TDMA mechanism manage the network. While in the inactive duration, nodes have the chance to enter in sleep station in order to save the battery power. = 60 *

The Superframe Duration is defined by equation ( 6).

= * 2 (4) 0 ≤ SO ≤ BO ≤ 14 (5)
IV. ENERGY EFFICIENT Energy stay again very attractive field for research that is why many works were developed in order to calculate it with more efficiency.In my case i will use some special manner to calculate some kinds of energy which are characterised by theirs link with the parametric of the protocol used to enhance to the canal of transmission which is the CSMA/CA protocol. The emission energy is giving by equation [START_REF] Lloret | A group-based wireless body sensors network using energy harvesting for soccer team monitoring[END_REF].

= ( * * ) + 2 * * * * (6)
present the size of frame. is binary energy, represent the number of data frame present in SD, CCA is the Clear Channel Assessment, U is the Voltage value as well as I expressed current value. Although is presented by equation [START_REF] Caldeira | Toward ubiquitous mobility solutions for body sensor networks on healthcare[END_REF].

= (2 -1) * 20 (7)
With characterise the backoff period.

Concerning energy of collision

, it is get by expression (13).

= * * * (8)
is among of time to access to the transmission canal. is the number of try to transmit information without receiving any acknowledge from destined node. The energy consumed in overhearing and overmitting phase, is mentioned by equation ( 14).

= * * * (9)
present the size of frame (bit). is show the binary energy. PER is the error rate with is obtained as the average of packet transmitted without being well received. Received energy is giving by formula (15)

= * (10)
With show the number of bit received in SD period. The energy in sleep period is giving by formula ( 16).

= * ( - ) (11)
The objective of our work is to make an extension in the survival period of the network via extending the career of the node by more exploit the quantity of energy remaining lasted in the battery of the node. The remaining energy is showed by expression [START_REF] Caldeira | Toward ubiquitous mobility solutions for body sensor networks on healthcare[END_REF] .

= - (12)
is always the som of all types of energy consumed. It is giving by equation [START_REF] Caldeira | Intra-mobility support solutions for healthcare wireless sensor networks-handover issues[END_REF].

= + + + + (13)
V. IEEE 802.15.4 PARAMETERS ADJUSTMENT Our work is based on the adjustment of both the Beacon frame Interval valor and the Superframe Duration. So the PAN coordinator receive the energy consumed by every node than try to select if there is a node in which the battery level is less then the energy threshold . In this case, a new value of and will be calculated and be send to this node in order to decrease its duty cycle and maximize its lifetime as mentioned in figure 2.

We suppose that the energy wasted in the Superframe duration is giving by relation [START_REF] Qadri | Applications, challenges, security of wireless body area networks (wbans) and functionality of ieee 802.15. 4/zigbee[END_REF].The ( ) present the Energy Remaining, ( ) is the size of Beacon Interval period, ( ) is the frame length, ( ) is the Energy of frame as mentioned above.

E {BI} = BI T {tt} * E T (14)
We suppose that energy consumed in BI is always less then the energy remaining in the battery ( 15) The relation ( 16) is expressed used both relations ( 14) and ( 15), * ≤

≤ (15)
As BI is calculated by relation (17). We calculate 2 relation [START_REF] Khanafer | Adaptive sleeping periods in ieee 802.15. 4 for efficient energy savings: markov-based theoretical analysis[END_REF].

2 ≤ * 15, 36 * 10 (-3) * (19)
That is why is mentioned by the formula [START_REF] Vaithiyanathan | Performance evaluation of ieee 802.15. 4 using association process and channel measurement[END_REF].

= ( * 15,36 * 10 (-3) * ) (2) (20)
We propose that we use just 10% of the energy remaining in the node 1 in order to well exploit it, giving by relation [START_REF] Oliveira | A duty cycle selfadaptation algorithm for the 802.15. 4 wireless sensor networks[END_REF]. INETMANET is a Wireless Sensor Network simulator based on the OMNet++ platform [START_REF] Varga | Omnet++[END_REF]. OMNeT++ simulator is adaptable and its components are based on C++ library. Despite, there are many simulators available for Wireless sensor network such as Castalia [START_REF] Khare | On recovery of lost targets in a cluster-based wireless sensor network[END_REF], NS2 [START_REF] Gaur | Comparing the impact of black hole and gray hole attack on leach in wsn[END_REF], NS3 [START_REF] Chanak | Energy-aware distributed routing algorithm to tolerate network failure in wireless sensor networks[END_REF] but INETMANET/OMNET++ present the most realistic wireless environment and mote behaviors. The figures 4, 5, 6, 7 as well as 8 show as the simulation results for all kinds of energy that we calculated which are energy of reception, energy of transmission, energy of overhearing-overmitting and energy consumed in sleep state. All the figures decrease by the same way as a result of the decrease of the Duty Cycle (DC). After intervention the decrease from 10 to 7 and the decrease also from 8 to 4. Therefore the duty cycle DC is reduced from 25% to 12,5%. The (DC) is calculated by formula [START_REF] Varga | Omnet++[END_REF]. Our objective in this work is to postpone the fault of energy in the network. So we are trying to calculate every kinds of energy that could be consumed by the node. Then we detect the node with fault energy. After that we propose a specific manner to get the Beacon Order and the Superframe Order for this node in order to more maintain the little power lasted in the battery. As perspective to this work, it will be very interested if we derive this work to very extended network with mesh topology.

Fig. 1 .( 2 )

 12 Fig. 1. The frame of IEEE 802.15.4

Fig. 2 .

 2 Fig. 2. Method description

= 15 ,

 15 36 * 10 (-3) * 2 (17) Equation (16) become by this way (18).

  tools as shown by figure3.

Fig. 3 .

 3 Fig. 3. Network node

2 *

 2 100 = 25 (24) 2 ( -) * 100 = 1 2 (7-4) * 100 = 1 2 3 * 100 = 12, 5 (25) For the case of the Superframe Duration we just use the 70% from the BI period. So it is expressed by relation (23) Therefore become (22):

Fig. 4 Fig. 6 .

 46 Fig. 4. SLEEP Energy

Fig. 7 .

 7 Fig. 7. Reception Energy

TABLE I

 I

	.	SIMULATION FEATURES
	Parameters		Values
	Simulation time	18000 s
	Network size		(400,200)
	E initial		18720J
	Nodes number	9
	Channel frequency	2.4 GHz
	Radio Type		IEEE 802.15.4 radio
	dispersion		rondom