V Ciarla

C Canudas De Wit

J Dumon

F Quaine

V Cahouet

INTRODUCTION

In order to study the general methodology to adapt the classical amplification curves to disabled people, an interview has been done to a sample of manufacturers that prepare the vehicles to the patient. The following points emerged from these investigations:

• The adaptation of the vehicle is decided by the Prefecture, when releases the driving license to the patient. This organ decides what is the adaptation mechanism that the patient needs (for example, the ball on the steering wheel for drivers with one hand).

• In case of disabilities to the upper limbs, the following mechanisms are used:

-Removable ball on the steering wheel, in case of total/partial invalidity to one of the two arms. About the position of the ball on the steering wheel, there is not a clear policy, because some manufacturers (such as Hand-drive) mount the ball at the top/center of the steering wheel. Some others (such as Hand-mobile or Ceremh) mount the ball on the side of the healthy arm. In particular, they consider the steering wheel as a clock and they mount the ball at 2 PM for the right hand and at 10 AM for the left hand.

A radio command can be mounted close to the ball to allow the driver to activate headlights/flashes and show the position. In some cases, a lever is mounted on the opposite side to brake/accelerate the vehicle. This lever is directly linked to the brake/acceleration pedals. This is the case of patients without problems of prehension (gripping ability) or quadriplegic. This last case is not clear, because some manufacturers (Hand-drive) prefer to use a double steering wheel mounted upper (or over) the normal steering wheel, instead of the ball and the lever. -Forks on the steering wheel, in case of a good mobility of the wrist, but bad mobility of the fingers.

-Tripod, in case of partial/complete invalidity of the wrist and the fingers.

All these adaptation mechanisms are compatible with the standard assistance systems (ABS, EPS, etc..), that is possible to find on a commercial vehicle. Starting from these considerations, it is possible to divide the drivers in two large clusters:

1. standard drivers, who have the correct functionality of the upper limb;

2. non-standard drivers, who are obliged to drive with only one hand, with the use of one of the previous mechanism described above.

The adaptation methodology has been developed starting from a control framework, designed for a standard driver, and has been extended to non-standard drivers with the introduction of additional blocks. This document has the goal to introduce the results of the work packages 5.3 and 5. [START_REF] De | Fun-to-drive by feedback[END_REF]. In particular, the first aim is to validate the theoretical results concerning the oscillation annealing control and the observer using a HIL setup and to verify the compatibility of these elements with the classical booster stage (part of the task 5.4). In this section, an additional validation is done, concerning the tire-road friction model. A comparison between the simulated friction torque and the real one is done, in order to show that the model prediction is realistic. The second part of this document concerns the analysis of the methodology to adapt the general architecture for non-standard drivers. An experimental validation is also proposed at the end of this part (task 5.4).

General architecture of the EPAS system

The general architecture, that is proposed for the driver's profiles, is shown in Fig. 3 and can be described as follows:

1. The model of the vehicle, that includes both the model of the steering wheel column and the bicycle model to obtain informations about the lateral dynamics of vehicle.

The exogenous inputs acting on the vehicle are the driver's exerted torque and the tire-road friction torque. Both these sub-systems are modelled in order to obtain a realistic behaviour. The driver is modelled as a position tracking control. The friction model is used to describe the Sticking and the Self-Alignment phenomena. The first one appears at lows speed of the vehicle, in opposition to the movement of the wheels during parking manoeuvre. The second one brings the wheels at their vertical axes during high speed driving.

2. In open loop, the steering system is affected by undesired oscillations that can be compensated with an optimal inner control loop. From the available measures of this system, it is possible to design an observer to estimate the exogenous inputs acting on it.

3.

Booster stage: it provides the necessary amplification torque during the driving. It takes in input the driver's exerted torque and the vehicle speed and provides in output the assistance torque, necessary to improve the driving comfort. The behaviour differs according to the driver: for a standard driver, the classical amplification curves are employed. For non-standard drivers, the standard parameters change to let the booster curve to provide a more adapted assistance. A tuning mechanism is employed to provide additional informations to the booster stage and it is under development. Note that the adaptation can be implemented via software, without modifications to the hardware of the system.

Y u

Vehicle

τ v Driver's Torque + Observer x1) 2)

MODELS FOR SIMULATIONS

This section is dedicated to the description of the mathematical models used to simulate the general architecture for the EPAS system. First, the mechanical model of the steering column and the electrical motor is introduced; then, the description of the tire-road friction torque and of the driver's model is done. The friction model takes in input the measure of the slip angle of the vehicle, so an introduction to the simple bicycle model used to obtain this measure is done.

Model of the steering column

The mathematical model of the steering column that authors take into account is based on works in [START_REF] Canudas De Wit | Stability analysis via passivity of the lateral actuator dynamics of a heavy vehicle[END_REF], [START_REF] Claeys | Modeling and control of steering actuator for heavy duty vehicle[END_REF] and [START_REF] Canudas De Wit | Control design for an electro power steering system: Part i the reference model[END_REF]. The model is shown in Fig. 4 and can be described by the following set of linear differential equations:

J v θv = τ v -k(θ v -θ s ) -B v θv (1)
J T θs = -k(θ s -θ v ) -N 2 2 B m θs + τ a N 1 + N 2 u (2)
with

J T = J c + N 2 2 J m + Jw N 2 1
is the total inertia at the column, motor and rack. The constants of the model are defined in Table 1, while θ v , θ s and θ m correspond to the angular positions of the steering wheel, the steering column and the motor. Let introduce the following state-space variables:

τ v J v B v θ v k θ s = θ m /N 2 J m B m N 2 N 1 u τ a θ m
x T = x 1 , x 2 , x 3 T = θv , θs , θ v -θ s T (3)
The model can be formulated into the state-space form ẋ = Ax + Bu + Gw with the exogenous inputs w = τ v , τ a T and the state-space matrices

A =    -Bv Jv 0 -k Jv 0 - N 2 2 Bm J T k J T 1 -1 0    ; B =   0 N2 J T 0   ; G =   1 Jv 0 0 1 N1J T 0 0   = g 1 g 2 ; (4)
The control input u is actuated from the electrical motor at the basis of the steering column. It has two goals: to anneal the oscillations of the steering column and to amplify the driver's exerted torque, in order to guarantee a better driving feeling.

Linear bicycle model

The model under consideration is well known in literature ( [START_REF] Ackermann | Yaw disturbance attenuation by robust decoupling of car steering[END_REF], [START_REF] Mammar | Vehicle handling improvement by active steering[END_REF], [START_REF] Poussot-Vassal | Commande robuste lpv multivariable de chassis automobile[END_REF]) and is based on the following assumptions:

• Linear lateral tire friction curve, i.e.:

F tyf = C yf β f F tyr = C yr β r (5)
where C yf and C yr are the linear stiffness of the lateral tire characteristics at the front and rear, respectively.

β f = θ v -β - l f ψ v β r = β + lr ψ v (6)
where ψ denotes the vehicle yaw rate, β f and β r denote the slip angles at the front and rear, while θ v is the steering angle of the front wheel.

Under these assumptions, the linearized model is given by (where V is a model parameter):

ψ β = - l 2 f C yf +l 2 r Cyr IzV lrCyr-l f C yf Iz -1 + lrCyr-l f C yf mV 2 - (C yf +Cyr) mV ψ β + l f C yf Iz C yf mv θ v (7)
Note that this model is not defined for speeds of the vehicle that are null. This is not a problem, because the Self-Alignment torque appears at high speeds. The values of the parameters are given in Table 1.

Dynamic tire-road friction model

The friction model used in this paper refers to the dynamic LuGre friction model, developed in [START_REF] Velenis | Dynamic Tire Friction Models for Combined Longitudinal and Lateral Vehicle Motion[END_REF], [START_REF] Canudas De Wit | Dynamic friction models for road/tire longitudinal interaction[END_REF] and [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF]. Most important components captured from this model are:

• Sticking torque M stick , that opposes to the movement of the tires when turning them around the z axis at low speeds of the vehicle.

• Self-Alignment M self -align torque, that appears at highs speeds and tends to rotate the wheels around the y axis, letting them to return to their vertical position.

Both these models derive from models at distributed parameters, where the friction force depends from the deflections z i (t, ζ) of the patch element along x and y directions respectively, located at the point ζ with respect to the patch frame at a certain time t [START_REF] Velenis | Dynamic Tire Friction Models for Combined Longitudinal and Lateral Vehicle Motion[END_REF]. Nevertheless, the distributed models may not be easy to use for analysis and control design. For this reason, the corresponding lumped models are introduced.

Model of the Sticking torque

The internal state of the model is obtained from the differential equation

żz = φ - σ 0z | φ| g z ( φ) z z (8)
with the angular velocity of the wheel rim is φ = θs /N 1 . The function

g z φ = µ kz + (µ sz -µ kz ) e -φ φs 2 (9)
determines the steady-state behaviour of the model. The parameters µ kz and µ sz are, respectively, the kinetic and static friction coefficients across the z-axis, while φs is the Stribeck velocity. The corresponding torque is given by

M sticking = -LF n (σ 0z z z + σ 1z żz + σ 2z φ)e -|v|/v k (10)
The dependency from the cruise speed of the vehicle v allows to simulate the effect that at high speeds the sticking toque is almost negligible, if compared to the self-alignment torque.

Model of the Self-Alignment torque

Let zi (i = x, y) and ẑy denote the internal states of the system:

żi = v ri -C 0i (v r )z i -κ ss i |ωr|z i (t) for (i = x, y) (11) ży
= G F n L v ry -C 0y (v r )ẑ y -ν ss |ωr|ẑ y + |ωr| L zy (12)
In Eqs. ( 11) and [START_REF] Hoon | Control logic for an electric power steering system using assist motor[END_REF], it is possible to recognize the vector of the relative speeds v ri (i = x, y):

v r = v rx v ry = ωr -vcos(β) -vsin(β) (13)
This vector depends on the angular velocity ω of the wheel, the radius r and the slip angle β.

The scalar function C 0i (v r ) (with i = x, y) is given by:

C 0i (v r ) = ||M 2 k v r ||σ 0i g(v r )µ 2 ki (14) with g (v r ) = M 2 k v r M k v r + M 2 s v r M s v r - M 2 k v r M k v r e -( vr vs ) γ (15)
It depends on the diagonal matrices M k and M s of the kinetic and static friction coefficients

M k = µ kx 0 0 µ ky > 0 (16)
M s = µ sx 0 0 µ sy > 0 (17)
The constants κ ss i and ν ss in Eqs. [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF] and ( 12) can be obtained from the hypothesis that the steadystate solution of the lumped model is the same as the steady-state solution of the distributed one. The complete expression of these parameters is given from:

κ ss i = 1 |ωr| v ri zss i -C 0i (v r ) ( 18
)
ν ss = 1 |ωr| 1 ẑss y Gv ry F n L + |ωr|z ss y L -C 0y (19)
with zss i and ẑss y , that are the steady-states of the system. The explicit expression for these parameters is given in the appendix of [START_REF] Velenis | Dynamic Tire Friction Models for Combined Longitudinal and Lateral Vehicle Motion[END_REF]. The self-alignment torque can be written in terms of the mean states zy and ẑy as follows:

M self -alig = F n L σ 0y 1 2 zy (t) -ẑy + σ 1y 1 2 ży -ży + σ 2y 1 2 v ry - G F n L (20)
The parameters used in this model are reported in Table 2. Hence, the total torque generated by the contact between the tires and the road is:

τ a = M stick + M self -alig (21)
The validation of this model was proposed in [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF], where authors show that the proposed models respect the expected behavior: at low speeds the sticking torque M stick is more important than the self-alignment torque M self -alig ; at higher speeds of the vehicle, the self-alignment torque is higher and turns the wheels to their vertical position.

Model of the driver

A standard driver, that is not affected by any pathology to the upper limbs, represents a closed kinematic chain with the steering wheel. He is symmetric around his upper limb and can choose to use the best placed arm to turn the wheel. In literature, he is considered as a feedback controller that takes into account of the path previewed information [START_REF] Jiang | A new mathematical model for car drivers with spatial preview[END_REF][START_REF] Hess | A control theoretic model of driver steering behavior[END_REF]. The control takes in input the reference trajectory θ ref and the steering wheel position θ v and calculates the tracking error at each time instant t:

e = θ ref -θ v ( 22
)
The output of the controller corresponds to the driver's brain response and it is multiplied by the physical action time delay T . This element can be identified as the muscular torque τ m and depends on the cognitive ability of the driver. As the main goal of this report is not the precise modeling of a standard driver's behavior, a simple Proportional Integrative Derivative (PID) control with a time delay is used to model it. In addition of the muscular torque τ m , the weight of each arm produces a torque on the wheel. This element can be assumed as a disturb input, depending on the steering wheel position θ v and others constant terms, that project the weight force on the plane of the steering wheel. Let define this disturb τ w (θ v ), as follows:

τ g (θ v ) = mgR cos(φ) sin(θ v -θ 0 ) ( 23
)
where φ defines the inclination of the steering wheel with respect to the z axis and θ 0 corresponds to the position of the arm on the top of the steering wheel at 12. Under these considerations, the total driver's exerted torque can be defined as the sum:

τ v = τ m + τ g (θ v ) (24)
Note that the only unknown parameter is the mass of the arm1 . It can be obtained throughout the use of anthropometric tables [START_REF] Winter | Biomechanics and Motor Control of Human Movement[END_REF] or with an off-line estimation procedure.

For the sake of simplicity, we choose to restrict the domain of the study to patients who can drive with only one hand, because of total or partial invalidity to the other one. The European directive 2007/46/CE imposes that the adaptation of the steering wheel for these patients consists into the use of a ball, mounted at the upper centrer of the steering wheel or at ±60 • on the side of the healthy arm. If we make the comparison with the model of a standard driver, it is possible to make the hypothesis that the cognitive ability is the same, meaning that both behave as feedback controller. The difference is the ability to act halved. Moreover, the arm having an aymmetric capacity to act, the standard driver can choose at every moment arm best suited to apply a given torque to the steering wheel. In the case of conducting with one arm, some tasks are more difficult to accomplish than others.

Equivalence of the gravity torque

From a trigonometric point of view, driving with 2 hands at 10h10 (θ 0 = ±60 • ) induces a gravity torque, that is equivalent to the gravity torque induced with 1 arm at 12h (θ 0 = ±0 • ). The gravity torque imposed with the two hands at θ 0 = 0 • is:

τ g0 = mgR cos(φ) sin θ v (25)
while, the gravity torque imposed with the two hands at θ 0 = ±60 • is:

τ g60 = mgR cos(φ) sin(θ v -60) + mgR cos(φ) sin(θ v + 60) (26)
We want to proof that:

τ g0 = τ g60 (27)
Let apply the formulas of addition and subtraction for the sinus function in Eq. ( 26):

mgR cos(φ)[sin θ v cos 60 + cos θ v sin 60 + sin θ v cos 60 -cos θ v sin 60] (28) = mgR cos(φ)[2 sin θ v cos 60] (29) = mgR cos(φ) sin θ v (30) = τ g0 (31)
This implies that the proposed methodology can be used in presence of patients driving with the 2 arms, so as for patients driving with only one arm.

CONTROL AND OBSERVER DESIGN

This section is dedicated to recall the design of a state-feedback control and of an exogenous input observer for the EPS system, shown in [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF]. For sake of simplicity, in this paper the details concerning the control/observer synthesis are not reported. Nevertheless, these elements are part of the general architecture proposed for a standard driver profile and can be used also for a non-standard driver. In particular, the state control contribute to anneal the low frequency oscillations of the steering column. The estimation of the driver's exerted torque is used as input for the driver's assistance blocks, described in Section 4.

State feedback control and torque observer

In [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF], authors show that EPS system becomes unstable when there is a variation of the driver's exerted torque τ v . The instability is physically due to the torsion of the steering column and it is possible to verify it looking at the frequency response of the system. To compensate these oscillations, an optimal LQR controller is designed. The resonance peak that might cause undesirable oscillations has been eliminated thanks to the proposed optimal linear control, as shown in Fig. 5. Note also that the low-frequency gain has kept unchanged. Details concerning the synthesis of this controller are shown in [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF].

In [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF], authors designed a full state observer for the exogenous inputs estimation, for instance the driver's exerted torque and the tire-road load friction torque. The observer was designed under the hypothesis Open loop peak at 14 Hz of a slow-time variation of the exogenous inputs τv ≈ τv = 0, that have been added as state variables to the original EPS system in Eq. ( 4). The measures of the column-shaft rate θs and the column torsion

F t = k(θ v -θ s
) are taken into account to guarantee the observability of the system. In this paper, the estimation of the driver's torque τv is used as input for the assistance system shown in Section 4, instead of the torsion measure, that is commonly used in automotive. The advantage is that the estimated measure is not affected by mechanical noise and its precision is not limited to the inertia and the viscosity of the steering column, as it happens when the column torsion is considered.

DRIVER'S ASSISTANCE SYSTEM

Main goal of the adaptation methodology is to generalize the standard assistance to the case of patients with a partial or total disability to the upper limb.

The common way to provide the steering assistance is throughout the use of static booster curves, that correspond to working points of the electrical motor ( [START_REF] Zhang | Modeling and characteristic curves of electric power steering system[END_REF], [START_REF] Chabaan | Optimal control and gain scheduling of electrical power steering systems[END_REF], [START_REF] Jiang | Low-pass filter based automotive eps controller and comparative full-vehicle tests[END_REF], [START_REF] Tsung Hsien | Design of control logic and compensation strategy for electric power steering systems[END_REF], [START_REF] Burton | Innovation drivers for electric power-assisted steering[END_REF], [START_REF] Hoon | Control logic for an electric power steering system using assist motor[END_REF])). For a given value of speed of the vehicle V and of the driver's exerted torque τ v , the electrical motor provide in output the corresponding assistance torque u ass . Indeed, these curves do not provide the same steering feel as the widely diffused hydraulic technology. The reason is that the hydraulic cycle is characterized by an hysteresis curve, that is not reproduced by the static booster curves. Starting from this consideration, some authors developed a model of assistance based on the hydraulic behavior ([4], [START_REF] Canudas De Wit | Stability analysis via passivity of the lateral actuator dynamics of a heavy vehicle[END_REF], [START_REF] Claeys | Modeling and control of steering actuator for heavy duty vehicle[END_REF]), To adapt the standard assistance to patients with disabilities, it is possible to refer to the theory of the position control with gravity compensation, that is widely diffused in robotics for regulation tasks of robot manipulators with elastic joints [START_REF] Luo | Gravity compensation and compliance based force control for auxiliarily easiness in manipulating robot arm[END_REF][START_REF] De Luca | Pd control with on-line gravity compensation for robots with elastic joints: Theory and experiments[END_REF][START_REF] Ulrich | Passive mechanical gravity compensation for robot manipulators[END_REF]. The same principle can be applied to provide the assistance in case of disabled drivers, under the hypothesis that it can help the driver on his driving accuracy and, in some cases, to reduce his driving effort. The compensation represents an additional torque, that modifies the standard assistance according to the cruise speed of the vehicle. At high speeds, the main goal is the tracking position: for instance, the weight compensation helps to perform a predefined path during an emergency manoeuvre or to maintain the equilibrium position on the steering wheel during cruise control. At low speeds, the compensation improve the driver's comfort. The muscular torque is an active load, that acts according to the direction of the movement. The torque due to the weight is a passive load, acting always in the same direction. According to the direction of the movement, it accelerates or brakes the muscular torque. For this reason, it needs to be compensated when he behaves as a brake, but it is useful to improve the driver's comfort in the opposite case. To join these requirements with the diffused

ξ = -aξ -b |ξ| θv + c( |ξ| + ε)τ m , if |ξ| ≤ ξ max 0, else (32)
with the assistance torque τ ass

τ g = (h 1 (V ) × ξ) Assistance -τ g ( θv ) × (1 -h 1 (V )) + h 1 (V ) 2 (1 -sign(τ m × τ g ( θv )) Compensation Ug (33)
where θv and τm correspond, respectively, to the estimated steering rate and the estimated muscular torque, obtained from the observer developed in Section 3.

The function h 1 (V ) determines the transitions between the assistance provided at high speeds and the assistance at lows speeds of the vehicle V . This function is defined as follows:

h 1 (V ) = 1, for V = 0 km/h; 0, for V = 30 km/h. ( 34
)
The disturb τ g (θ v ) to be compensated is defined in Eq. ( 23). The output of the previous differential equation saturates, as soon as ξ reaches the maximum ξ max , admitted from the electrical motor. The values to tune the assistance model allows to the workpackage Gipsa-Tuning.

The assistance provided by this model in steady-state ( ξ = θv = 0) for two range of speed of the vehicle (V = 0 and 30 km/h) is shown in Figs. 6 and7, for a driver's torque between ±5N m all over the steering plane. The weight disturb is positive for all θ v > 0, while it is negative when θ v < 0. For V = 0 km/h, when the muscular torque τ m > 0 and we are in the negative plane of the disturb, then the compensation acts, otherwise it is not active (see Fig. 6). At V = 30 km/h, the compensation is always active, modifying everywhere the classical assistance (see Fig. 7).

NeCS-CAR TELEOPERATION SETUP

The proposed methodology can be validated experimentally on the dedicated platform NeCS-CAR funded by the NeCS Team at the Control System department of GIPSA-lab, shown in Fig. 8. A remote operator can drive the car via a hybrid (Ethernet + WLAN) networked communication by observing the video and experiencing the torque feedback. A PC-unit is installed on the control station. The embedded network is switched Ethernet to communicate between the controller, the image processor and IP video camera. These devices are connected with a 100 Mbps switch that is further connected to a WLAN router. Figure 7: Behavior of the assistance system at 30 km/h

Control station

The operator can drive on a steering wheel with its motor-shaft column. At the basis of the steering column, there are a torque sensor and a force sensor. For the current study, the only torque sensor is considered. It sends the measured driver's torque to the PC-unit (signal y 2 = τ v2 in Fig. 8). A brushless motor takes in input the reference torque from the PC-unit (signal u 2 = τ mot1 in Fig. 8) and gives into output the measure of the steering wheel position θ v .

PC-unit

The operating system on the PC-unit is Windows XP R . For real time communication of the controller, Ardence RTX interface is used. Controller is designed in Matlab 2007 R and it is converted to Real Time Execution (RTX), that is a rapid prototyping product by Quanser. Real time TCP/IP, included with RTX, provides tools to embed high performance real-time networking protocols into systems and applications. This is possible with the A/D and D/A converters on the Data Acquisition and Control Buffer (DACB) communicating with WinCon using the RTX Workshop installed in Simulink R . The PC-unit sends the torque feedback to the control station. The torque feedback is the sum of simulated and measured elements. Among the simulated ones, it is possible to find the control input u to compensate the oscillations of the steering wheel. The torsional torque k(θ vθ s ) is also simulated to obtain the value of motor-shaft angle θ s . The driver's assistance systems are also simulated. Concerning the tire-road friction torque, it can be obtained by simulation or throughout the use of the NeCS-CAR. In simulation, the LuGre friction model, shown in Section 2.3 is implemented in Simulink.

NeCS-CAR

When the NeCS-CAR is linked to the pc unit, it receives the input control from the steering wheel via teleoperation; this input allows it to move their wheels according to the position θ s at the desired cruise speed. A torque sensor mounted on the direction column of the NeCS-CAR allows to measure the real friction torque, that is used to replace the LuGre model in the HIL setup.

VALIDATION OF THE GENERAL ARCHITECTURE

This section describes the experimental validation of the proposed general architecture. The first part introduces the steps to adapt the state-space model to the hardware configuration. The second part consists into the validation of the oscillation annealing control and of the torque observer. This LuGre tire-road friction model is also validated experimentally with a comparison with the measured friction torque obtained from the NeCS-CAR. Finally, the adaptation methodology is also tested in simulation and on the hardware setup for some scenarios of interest.

Adaptation of the state-space model to the HIL setup

In Eq. ( 1), authors consider an initial value for the steering wheel inertia J v and viscosity B v , that may not completely correspond to the real value of the system. For this reason, let make the following assumption:

Assumption 1: let define J and B as the sum of the values adopted in simulation (respectively, J v and B v ) and values of the setup system (respectively, J 0 and B 0 ), that have to be identified :

J = J v + J 0 (35)
B = B v + B 0 (36)
Let substitute J v and B v , as resulting from Eqs. ( 35) and (36), in Eq. ( 1), as follows:

(J 0 -J) θv + (B 0 -B) θv = τ v -k(θ v -θ s ) (37)
and let apply the Laplace transform to obtain the value of θ v , as function of the driver's torque τ v and the total torque C m , as follows

(J 0 -J)s 2 θ v + (B 0 -B)sθ v = τ v -k(θ v -θ s ) (38)
Let divide the real inertia and viscosity from the residual ones:

(J 0 s 2 + B 0 s)θ v = τ v -k(θ v -θ s ) + (Js 2 + Bs)θ v (39) it follows that C m = -k(θ v -θ s ) + (Js 2 + Bs)θ v (40)
is the simulated torque given to the motor and

θ v = 1 (J 0 s 2 + B 0 s) (τ v + C m ) (41)
is the real steering wheel angle, obtained from the steering wheel and the column. Let apply the same procedure in Eq. ( 2) to obtain the relation between the steering wheel position θ v and the column-shaft angle θ s . The Laplace transform produces:

J T s 2 θ s + N 2 2 B m sθ s = k(θ v -θ s ) + τ a N 1 + N 2 u (42)
By collecting all the terms in θ s , it is possible to obtain:

θ s = 1 J T s 2 + N 2 2 B m s k(θ v -θ s ) + τ a N 1 + N 2 u (43)
Eqs. ( 41) and ( 43) are sufficient to adapt the state-space representation to the HIL model. Preliminary simulations showed that the two models are equivalent.

Parameter identification

The values J 0 and B 0 are obtained with the following procedure:

1) we apply on the open loop system a Pseudo-Random Binary Sequence (PRBS) input sequence of C m , output of the pc-unit.

2) We collect the measured values of the motor angular speed θ m and we make the hypothesis that the model behaves as a first order transfer function of the form:

G 0 (s) = θv C m = k 1 + T p s = 1 B 0 + J 0 s (44)
Under this hypothesis, the parameters to identify correspond to:

J 0 = T p k (45)
B 0 = 1 k (46)
The solution of this problem is obtained by applying a non-linear least squares algorithm, implemented with the software Matlab 2011 R . The resulting values are Table 1.

6.3 Experimental validation of the oscillation annealing control (Gipsa-Comp)

Steering column in open loop

Previous studies [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF] showed that the system becomes unstable, when there is a variation of the driver's exerted torque. The instability is physically due to the torsion of the steering column. From a mathematical point of view, it can be checked with the two imaginary poles of the state matrix of the system (p 1,2 = -1.0023 ± 56.0151i), while the last pole is stable (p 3 = -4.9386).

To test the effect of this instability, in [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF] authors applied a constant torque for a certain time and then they left the steering wheel. The simulations of the system showed important oscillation on the steering wheel rate . To verify this behaviour on the HIL setup, authors applied the input trajectory shown in Fig. 9(a): the operator turns the steering wheel in clockwise of 90 • , then leaves it for 4 s and repeats the manoeuvre in anti-clockwise. Fig. 9(a) shows the response of the real system in terms of angular speed of the steering wheel: when the driver leaves the system, the steering system starts to oscillate (it is evident at times between 4 and 6 s). It stops when all the energy of the system is dissipated. This experimental results is compatible with the simulations, shown in [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF]. Nevertheless, it is possible to make a further check by applying the same input trajectory also on the simulated system. The PID controller acts as the human operator, taking in input the real reference trajectory and applying the corresponding exerted torque. In Fig. 9(a) it is possible to verify the presence of the oscillations on the steering column: note that their amplitude corresponds to the real amplitude that we measure on the experimental setup. As stated in [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF], these oscillations are uncomfortable for the driver and moreover they affects the measure of the torsion, that is commonly used to measure the driver's torque to give in input to the booster stage.

For these reason, it is necessary to compensate them with a proper controller.

Steering column in closed loop

The oscillations shown in previous section can be annealed by introducing an optimal LQR control, under the constraint to not modify the impedance between the driver's and the load torque at low frequency. The theory concerning the synthesis of this controller is shown in [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF]. The exerted driver's trajectory is shown in Fig. 10(a) and is similar to the open loop case: it corresponds to turn the steering wheel from the initial position to 90 • in anticlockwise, then releasing the wheel for 6 s and finally making another turn in clockwise. Fig. 10(b) shows the angular speed of the steering wheel: despite the open loop case, the oscillations are completely annealed by the effect of the inner control. This result validates the simulations, shown in [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF], where the input to the system is the driver's exerted torque. 6.4 Experimental validation of the observer performance (Sect:GipsaObs)

Validation of the driver's torque estimation

In [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF], the observer was designed by using the measures of the steering column rate θs and of the torsion k(θ vθ s ). To validate the performances of the observer, let consider the case of a trajectory exerted by a driver, who can manoeuvre with two hands, with the oscillation annealing control active to prevent oscillations of the steering column, but without the support of the assistance. The applied trajectory on the steering wheel is shown in Fig. 11(a), while the comparison between the estimated torque τv and the measured one C mmeas is shown in Fig. 11(b). The numerical noise (affecting θs ) and the measure noise (affecting the torsion) do not modify the observer performance. They increase when the angular speed and acceleration of the steering column augment. This is due to the influence of the mechanical parameters on the model. For this reason, in Fig. 11(b) it is also shown the profile of the angular speed of the electrical motor. The comparison between the estimated torque τa and the output of the LuGre sticking torque allows to validate the observer performance in this case, as shown in Fig. 12. The estimation error is negligible and the observer performance are satisfying. It is also possible to note that if we divide the value assumed by the friction torque at low frequency (about 30 N m) by the gear ratio N 1 = 13.67, we obtain the value of the driver's torque during the same instant (about 2.2 N m), meaning that the LQR control preserve the impedance relation between the two torques. This property cannot be verified at high frequency, because of the influence of the angular speed and acceleration of the steering wheel.

Comparison with the real friction torque at stand-still

A further validation of this tire-road friction model can be done, using the torque measures obtained with the NeCS-CAR. Moreover, the performance of the observer can be tested to verify that it is able to retrieve the real friction torque profile. In Fig. 13(a), the comparison between the simulated friction torque and the real measures obtained from the NeCS-CAR is proposed. It is possible to observe that the simulated torque is realistic, if compared to the measured one. Nevertheless, to improve the model a parameter tuning or identification can be done. In Fig. 13(b), the estimated torque is compared with the measured one. In spite of the measurement error, the observer performances are globally satisfying. It is possible to diminish the transient error at the cost of a high gain of the observer.

Validation of the tire-road contact friction in movement (Self-Alignment torque)

The following additional test has been done to validate the model used to describe the Self-Alignment torque. In [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF], authors showed that this torque appears as the vehicle speed grows up and tends to rotate the wheels around its vertical axis in order to make them return to the straight position. The Self-Alignment model adopted in [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF] needs as input the measure of the slip angle, that is a measure at the origin of the lateral forces acting on the vehicle during turning manoeuvres. To obtain this value, in [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF] authors used an approximated relation between the vehicle speed v and the radius of curvature ρ, given by the curve traced by the vehicle in a given moment, i.e. β(t) = f (v, ρ). To improve this results, in this case, authors use the common form of the linear bicycle model.

Dependency of the simulated models from the speed of the vehicle

In [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF], authors validated the friction torque by verifying that the simulated results follow the expected logic, according to the variations of the speed of the vehicle. At this aim, several simulations were carried out for velocities going from 0 to 30 km h -1 and over a range time of several seconds, in order to see the effects at steady-state. The choice of this speed range is due to the fact that an EPSs operate in this range. The same tests have been carried out also on the HIL setup: in this case, we look at the input steering wheel angle, corresponding at 30 • . The steering ratio N 1 induces an angle of curvature of the wheels of about 2 • . This results is consistent with the actual technologies, where the steering ratio is 15 for power assisted vehicles and 18 for non-assisted ones. The steering angles and the corresponding angles at the wheels of the vehicle are shown in Figs. 14(a [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF]. Note that, even if the two friction models are simulated, they differs in terms of input signals, that are simulated in [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF], while they correspond to real measure in the HIL setup. They differ also because of the slip angle. The effect to have the same behaviour helps to validate the numerical approximation, used in [START_REF] Illán | Oscillation annealing and driver/tire load torque estimation in electric power steering systems[END_REF]. Next step in this sense is to verify this behaviour with the NecsCar, in order to obtain a real friction torque and validate the performance of the observer in this case. 6.5 Scenario for the validation of the driver's assistance system (Gipsa-Booster)

To test the proposed methodology, three scenarios are considered. For each scenario, the simulations results are compared with the corresponding experimental tests and two performance metrics are introduced: the driving precision and the driving strain/comfort. The driving precision is defined as the error e between the reference trajectory and the current value of the steering wheel angle:

D p = T 0 (θ ref -θ v ) 2 dt (47)
and the objective of the assistance is to minimize this error.

A preliminary metrics to validate the assistance into the improvement of the driver's strain is the driver's energy E d exerted during the driving task. It is considered into absolute value, because driver's muscles are not reversible. The definition of this metrics is the following:

E d = T 0 |τ m (t) × θv |dt (48)
Moreover, the effort to maintain the same position over a long time can be shown only with the use of electromyography (EMG), that may reveal the contraction and co-contraction factor of the muscles involved during the driving. An alternative criteria, that can be employed is the driver's strength, defined as follows:

S d = T 0 |τ m (t)| 2 dt (49)
For each study case, the corresponding improvement criteria is shown in Table 3. Criterion at low speed is more successful, when it is tested with a steering angle of 90 • . Moreover, it is common to turn the steering wheel until 90 • . Nevertheless, it is not possible to do this at high speed, because the Self-Algnment drops-out. Each scenario is simulated (and tested) with 3 different models of the driver, in order to verify the effects of the proposed assistance. The first test corresponds to the ideal case without the disturbance of the arm. The tracking controller takes in input the error between the reference trajectory and the current path and exerts the corresponding delayed torque on the steering wheel τ v = τ m . The second simulation corresponds to the case where the weight of the arm acts as a disturb on the tracking controller τ v = τ m + τ g , but the steering assistance is not activated. Finally, the last simulation corresponds to the same as in the second case, but the steering assistance is active τ v = τ m + τ g -U g . For sake of simplicity, the experimental validation has been done considering only the driving models of the second and the third simulations. For each scenario, the first test is done with the operator who drives with one hand, but without the steering assistance. The second test is done with the operator who drives with one hand and the assistance active. In all cases (simulations and tests), the LQR control is active, the observer provides the estimated muscular torque to the steering assistance and the friction torque is simulated. Further tests are in progress to validate the methodology with the NeCS-CAR.

Scenario

Parking manoeuvre

A ramp is given in input to the PID controller as reference trajectory for a speed of the vehicle at 0 km h -1 for the different models of the driver. Under these conditions, the main responsible of the road feeling feedback is the Sticking torque. This is the case, when the steering assistance system acts to diminish the driver's effort and it is active only when the driver's movement is in the opposite sense of the weight force. Let consider the case with the position of the ball on steering wheel is at 0 • . In Fig. 16(a) the reference trajectory is compared with the resulting trajectory at the output of each driver's model. It is evident the influence of the disturbance on the second model, when he tries to gain the straight position (at 25 s), while this disturbance is annealed by the steering assistance acting on the third model. In this scenario, it is possible to observe how the assistance helps to reduce the driver's strain (see Fig. 16(b)): during the first part of the simulation (at 10 s), the assistance is not active, because the weight helps the driver's movement, so the curve of driver's strain overlaps the curve corresponding to the second model, without assistance. During the second part, the compensation is active and the curve of strain overlaps that one corresponding to the first model, with a global improvement of the driving comfort. The experimental test concerning this scenario is shown in Fig. 17(a). To be able to compare the two experiences, a constraint in terms of timing and speed of the movement has been required to the operator. A fast movement entails the growth of the angular speed of the steering wheel, with a consequences on the measure of the mechanical power, because of a consequent augment of the dissipated energy on the tires. Experimental results shown in Figs. 17(a) and 17(b) validate the simulation results. The real movement is comparable with the simulated one and it is possible to verify the diminishing of the mechanical power. For both the situations, it is possible to show the drive's power consumption over the steering wheel plane (see Figs. 90 • the movement is in the same direction as the weight disturb, so the assistance does not compensate. When the movement is in the opposite sense despite the disturb, the assistance helps to diminish the steering power. It is possible to remark that the magnitude of the power in test is compatible with that one obtained in simulation.

θ v (rad) θ ref θ v for τ v = τ m θ v for τ v = τ m + τ g θ v for τ v = τ m + τ g -U g t 1 t 2 t 0 T (

Obstacle avoidance

To simulate the scenario of an obstacle avoidance, a step-wise reference trajectory is given in input to the tracking control. The vehicle is simulated with a cruise speed of 30 km h -1 , meaning that the Self-Alignment torque is the main responsible of the driver's steering feel. The driver's steering assistance is always active to compensate the disturbance and to improve the driving precision. The simulations results are shown in Fig. 20(a) with the 3 different driver's models.The influence of the disturb on the tracking control is clear, when it tries to change path direction, for instance, at the 5 second. On the experimental validation, this phenomenon is not evident, so as it happens in simulation. The requirement to reproduce exactly the same movement over a long period introduce many difficulties on the experiment and the behavior of the driver is probably different with respect to the model used in simulation. Anyway, it is possible to find some cases where the improvement is evident, as it is shown in Fig. 20(b). Zoom of the profile of the trajectory applied on the HIL setup, where the contribution of the steering assistance is evident.

Cruise control

The effect to maintain the same straight position for a long time may infer with driver's strain. In this case, the steering assistance helps to maintain the path following at high speeds and to improve the driving comfort. It is possible to verify and test the diminishing of the driver's strain to maintain for long time the same position.

Conclusions

A methodology to adapt the generic steering assistance law was developed and validated experimentally. This methodology can be used in case of a driving with 1 or 2 arms. The improvements due to this methodology can be shown in Table 7. It is not exhaustive, because it is not possible to obtain the percentage of improvements for all the cases, but it provides a preliminary idea, except for the precision that it is not possible to verify during the test because there is not a reference. For the parking, all criteria are improved in simulation and tests. The obstacle avoidance cannot be verified in test, because we did not study a trajectory that allows us to have the equivalence of the movement. The cruise control always find an improvement from the introduction of the assistance, because it is active all the time. On the other hand, the muscular contraction and co-contraction to maintain the same position over a long time (cruise control scenario) can be shown only with the electromyography (EMG). Further EMG tests are planned to validate this metrics. The driver's model can be improved to keep into account more pathologies, that can deal with the project VolHand. Finally, the study concerning the asymmetry abilities of the driver is under development.

% improvement

Figure 1 :

 1 Figure 1: Examples of (a) ball mounted on the steering wheel for total/partial invalidity; (b) radio command used to show the visibility of the vehicle.

Figure 2 :

 2 Figure 2: Examples of (a) forks in case of good mobility of the wrist, but bad mobility of the fingers; (b) tripod in case of partial/complete invalidity of the wrist and the fingers.

Figure 3 :

 3 Figure 3: Model of the general architecture of the EPAS system

Figure 4 :

 4 Figure 4: Mechanical model of the EPS system

Figure 5 :

 5 Figure 5: Bode diagram of the system in open loop (blue) and in closed loop.

Figure 6 :

 6 Figure 6: Behavior of the assistance system at 0 km/h

V

  . Ciarla, C. Canudas de Wit, J. Dumon, F. Quaine, V. Cahouet-GIPSA-Lab -Report on Experimental Validation 11

Figure 8 :

 8 Figure 8: NeCS-CAR teleoperation setup (NeCS Team GIPSA-lab/INRIA)

Figure 9 :

 9 Figure 9: (a) Steering wheel trajectory applied from the operator on the system. (b) Response of the EPS system in terms of steering wheel rate. Note the oscillations of the systems between 5 and 8 s.

Figure 10 :

 10 Figure 10: Angular position (a) and angular speed (b) of the steering wheel. The oscillations of the steering column are annealed by the LQR control.

Figure 11 :

 11 Figure 11: Observer validation: (a) Trajectory imposed on the test, with two hands and oscillation annealing control. (b) Profile of the angular speed of the electrical motor and comparison of the estimated torque with the driver's torque measure C mmeas

Figure 12 :

 12 Figure 12: From the top to the bottom: comparison of the estimated torque with the load torque, obtained from the LuGre model, and estimation error.

Figure 13 :

 13 Figure 13: (a) Comparison between the measured friction torque and the simulated one (b) Observer performance into the estimation of the friction torque sensed from the NeCS-CAR.

  ) and 14(b) for the range of speeds of interest. The corresponding wheel torques are shown in Figs. 15(a) and 15(b). These results validate what it has been shown in

Figure 14 :

 14 Figure 14: (a) Steering wheel trajectory and (b) corresponding angle of wheels of the vehicle.

Figure 15 :

 15 Figure 15: (a) Sticking and (b) Self-Alignment torque for different speeds of the vehicle.

gFigure 16 :

 16 Figure 16: (a) Profile of the trajectory of the steering wheel used to simulate parking manoeuvre. (b) Profile of the mechanical power dissipated during the simulation.

Figure 17 :

 17 Figure 17: (a) Profile of the trajectory of the steering wheel used to test the parking manoeuvre on the HIL setup. (b) Profile of the mechanical power dissipated during the test.

Figure 18 :

 18 Figure 18: Power consumption during the simulation

Figure 19 :

 19 Figure 19: Power consumption during the experimental test

Figure 20 :

 20 Figure 20: (a) Profile of the trajectory of the steering wheel used to simulate obstacle avoidance. (b) Zoom of the profile of the trajectory applied on the HIL setup, where the contribution of the steering assistance is evident.

Table 1 :

 1 Parameters of the EPS system and the bicycle model• Small slip angles (β f and β r ):

	Symbol	Description	Value
	Jv	Steering wheel inertia	0.05 kgm 2
	Jm	Motor inertia	0.0004 kgm 2
	Jc	Column inertia	0.04 kgm 2
	Jw	Rack inertia	0.000784 kgm 2
	k	Column stiffness	100 N m rad -1
	N1	Steering column-wheels gear ratio	13.67
	N2	Motor-steering column gear ratio	17
	Bv	Steering wheel viscosity	0.06 N m rad -1 s -1
	Bm	Motor shaft viscosity	0.0032 N m rad -1 s -1
	J0	Wheel real inertia of the HIL setup	0.0303 kgm 2
	B0	Wheel real viscosity of the HIL setup	0.039 N m rad -1 s -1
	m	Vehicle total mass	1621 kg
	l f	Distance from the center of gravity to the front axle	1.15 m
	lr	Distance from the center of gravity to the rear axle	1.38 m
	Iz	Vehicle inertia	1975 kgm 2
	C yf	Front axle tire cornering	28559 N rad -1
	Cyr	Rear axle tire cornering	40698 N rad -1

Table 2 :

 2 Constant parameters of the LuGre model

Table 3 :

 3 Summarizing table with simulation/test scenarios and corresponding improvement critera

		Vehicle speed Ball position	Metrics
	Parking manoeuvre	0 km/h	0 • / 90 •	Driver's energy E d
	Obstacle avoidance	30 km/h	0 • / 60 •	Driving precision D p
	Cruise control	30 km/h	0 • / 60 •	Driver's Strength S d

V. Ciarla, C. Canudas de Wit, J. Dumon, F. Quaine, V. Cahouet-GIPSA-Lab -Report on Experimental Validation

To estimate the torque exerted by the mass of the healthy arm of the driver, an off-line estimation procedure is adopted. The theory about this procedure is under study and concerns the workpackage Gipsa-Tuning


M Breuilly

G Malandain ⋆ , N

J Guglielmi

T Pourcher

P R Franken

J Darcourt

IMAGE-BASED MOTION DETECTION IN 4D IMAGES AND APPLICATION TO RESPIRATORY MOTION SUPPRESSION

Keywords: small animal imaging, single photon emission computed tomography, respiratory motion detection

Respiratory motion may blur the tomographic reconstruction of PET or SPECT images, which subsequently impairs quantitative measurements, e.g. in the upper abdomen area. Respiratory phase-based gated reconstruction addresses this problem for CT images, but deteriorates the signal-to-noise ratio and other intensity-based quality measures for ET images.

This article describes an image-based motion detection method for dynamic images, which works for both 4D-CT and 4D-SPECT images. This method allowed us to identify the motionless phases from the image-based motion signal. Interestingly, we found that the peak of motion according to the pressure signal, recorded during the acquisition and considered as a reference, is temporally shifted compared to the motion-phases identified by the proposed method. Moreover, these observations permit to reconstruct a breath hold like 3D SPECT image improving the quality of the 3D reconstruction compared to both 4D reconstruction and standard 3D reconstruction.

INTRODUCTION

Emission tomography (ET) is an efficient imaging tool for physiological and functional processes. It provides in vivo information and allows longitudinal studies. For such studies, qualitative analysis may not be sufficient to reveal subtle differences between groups (for instance, in oncological studies where different therapy strategies have to be assessed), and quantitative analysis is thus necessary. ET quantification may be impaired by physical (e.g. attenuation, etc) or physiological (e.g. motion) causes. Methods exist to take care of the former both for Positron Emission Tomography (PET) or Single-Photon Emission Computed Tomography (SPECT) [START_REF] Ritt | Absolute quantification in SPECT[END_REF]. Motion compensation, however, is still an active research topic.

In ET imaging, motion artefacts can be handled by gating protocols. An additional signal, which is considered representative of the motion of interest, is recorded and synchronised with the data acquisition. Reconstruction is then based on a selection of the acquired data. Such signals can be acquired by an external device and include electrocardiogram [START_REF] Livieratos | Respiratory gating of cardiac PET data in list-mode acquisition[END_REF], Real-time Position Management [START_REF] Grotus | Fully 4D list-mode reconstruction applied to respiratory-gated PET scans[END_REF] or Multidimensional Respiratory Gating [START_REF] Sa Nehmeh | A novel respiratory tracking system for smart-gated PET acquisition[END_REF], and pressure sensor (chest belt) [START_REF] Chang | Implementation of an automated respiratory amplitude gating technique for PET/CT: clinical evaluation[END_REF][START_REF] Van Elmpt | Optimal gating compared to 3D and 4D PET reconstruction for characterization of lung tumours[END_REF].

Phase-based gating divides the period into several phases of equal duration, with the assumption that the motion can be neglected within each phase. This method is used for both cardiac [START_REF] Büther | Detection of respiratory tumour motion using intrinsic list mode-driven gating in positron emission tomography[END_REF] and respiratory [START_REF] Bundschuh | Local motion correction for lung tumours in pet/ct-first results[END_REF] motions and allows the reconstruction of dynamic or 4D (3D+t) images. However, using only a fraction of the acquired data deteriorates the signal-tonoise ratio (SNR) of each 3D image, especially in ET. Therefore, Dawood et al. propose to co-register each 3D image and then to sum them up [START_REF] Dawood | Lung Motion Correction on Respiratory Gated 3-D PET/CT Images[END_REF]. This method permits to obtain a 3D image with minimum motion and improved SNR, but assumes that image co-registration can be achieved with minimal error. Another gating approach, amplitude-based gating, assumes that the signal amplitude is representative of the motion of interest. A range of amplitude values is selected, and the associated acquired data are used for reconstruction. For instance, images at end of exhalation or inhalation can be reconstructed by selecting either the lowest or the highest amplitude values of a respiratory representative signal. In human imaging, such a method has been used for respiratory motion in PET [START_REF] Chang | Implementation of an automated respiratory amplitude gating technique for PET/CT: clinical evaluation[END_REF][START_REF] Sa Nehmeh | A novel respiratory tracking system for smart-gated PET acquisition[END_REF][START_REF] Van Elmpt | Optimal gating compared to 3D and 4D PET reconstruction for characterization of lung tumours[END_REF]. In the case when a motion model is available, some authors proposed to incorporate the model into the tomographic reconstruction procedure in order to take into account all the acquired data to reconstruct a 3D still image. In human imaging, cardiac [START_REF] Blondel | 3-d tomographic reconstruction of coronary arteries using a precomputed 4-d motion field[END_REF] and respiratory [START_REF] Lamare | List-mode-based reconstruction for respiratory motion correction in PET using non-rigid body transformations[END_REF][START_REF] Reyes | Model-based respiratory motion compensation for emission tomography image reconstruction[END_REF] motion models have been developed. In addition to the difficulty in defining an accurate motion model, such methods also imply a huge computational cost.

The motivation of our work is the oncological studies in a molecular pre-clinical imaging context, for tumours in the upper abdominal area. Therefore, motion artefacts have to be reduced before conducting any quantitative studies. While motion compensation has been widely studied in human imaging, the literature is less abundant for pre-clinical imaging and mostly concerns µ-CT [START_REF] Guo | 4D micro-CT using fast prospective gating[END_REF][START_REF] Martiniova | In vivo micro-ct imaging of liver lesions in small animal models[END_REF]. The proposed method will take advantage of the particular respiratory pattern of the anaesthetized mice (still periods separated by gasping breaths [START_REF] Cavanaugh | In vivo respiratory-gated micro-ct imaging in small-animal oncology models[END_REF]). Detecting the gasping breaths allows in turn to select the still periods and to use the associated data to reconstruct a motion-free 3D image. However, it appeared that the acquired respiratory representative signal (here a pressure signal) is not representative of the respiratory motion. Therefore it motivates an ad-hoc image based motion detection.

IMAGING PROTOCOL

We acquired experimental animal images (both SPECT and CT) using a dedicated small animal SPECT/CT scanner (eXplore speCZT CT 120, GE Healthcare Bioscience, London ON, CA). SPECT and CT camera are located on two different gantry. Thus the system does not allow synchronous SPECT and CT images acquisition and SPECT and CT images have to be registered with a known global rigid transformation (translation). The output of the SPECT camera is a list-mode of all events detected on the detectors: every millisecond, the time, the gating index, and the properties of all detected photons are recorded.

A monitoring system (BioVet, m2m Imaging Corporation, Newark, USA) was used with a pneumatic pressure sensor placed under the animal abdomen. The pressure signal was recorded (1 ms-sampled rate) and used to monitor the animal anaesthesia. Body temperature was maintained at 37 • Celsius and anaesthetized gas rate was regularly controlled to keep the respiratory rate between 60 and 100 breaths per minute. Once a stabilized respiratory rate was reached, the image acquisition started.

At the beginning of each acquisition, an amplitude threshold was arbitrarily fixed in order to detect the rising edge of the pressure signal variation. Each time the signal passes this threshold, a trigger is sent to the camera. In the SPECT mode, this trigger is recorded in the list-mode to be used for latter gated reconstruction. We assume that this trigger corresponds to the same time in the cycle for each respiratory cycle. A respiratory cycle is then defined by the interval between two consecutive triggers in the SPECT list-mode. In the gated CT mode, every trigger initiated the acquisition of a projection.

MOTION DETECTION

The principle of the proposed method is to detect, from a first 4D reconstruction, the phases exhibiting some motion, and then to discard the data associated with those phases in the final reconstruction. For CT images, the 4D sequence reconstructed by the constructor software is sufficient to detect these phases. However, some extra work has to be conducted for the reconstruction of 4D SPECT images.

4D SPECT image reconstruction

From the recorded triggers in the SPECT list-mode data, we compute the mean C and the standard deviation σ of the res-piratory cycle length. The data associated with cycles outside ±3σ are discarded because we suspect that they are associated with some respiratory artefacts. This concerns around 3% of the data.

Then, since the respiratory cycles exhibit some length variation, we build cycles of uniform length C s by redistributing the data of each cycle into a cycle of length C s . C s is chosen as the closest value to C, that is a multiple of the number N of phases being reconstructed. This way we ensure that each respiratory cycle will be correctly binned in the right number of phases. The i th cycle is defined by the interval [t i , t i+1 [ where t i is the time of the i th trigger. The events in the list mode data associated to time t in this interval will now be associated to time t ′ defined by

t ′ = (i -1)C s + (t -t i ) C s t i+1 -t i ( 1
)
Then it is straightforward to reconstruct a 4D image of N phases from this new list-mode. However, the number of detected events differs from one reconstructed phase to the other. Thus, we normalized all the reconstructed phases with respect to one reference phase by multiplying the voxel intensities by the number of events used in the reference phase divided by the number of events used in the considered image.

4D image analysis

The principle of the method is that the voxels reconstructed at the position x will exhibit some intensity variation if they are located in an area containing a moving edge. Given a 4D image made of N 3D phase images I n , n ∈ [1, N ], we compute an amplitude image A such that :

∀x : A(x) = max{I n (x)} n∈[1,N ] -min{I n (x)} n∈[1,N ] (2)
This amplitude image will have high intensity values at the location of the moving edges. This image is then thresholded to keep only the regions of potential motion, generating a mask M . A threshold at 15% (resp. 30%) of the maximum amplitude was arbitrarily chosen for SPECT (resp. CT) images.

According to the pressure signal, we observed that around 2/3 of the respiratory cycles do not exhibit significant motion. Since N phases are reconstructed, with N = 15, this corresponds to around 10 phases without motion; the 5 other phases are the cause of the intensity variations due to motion. Therefore, for each voxel x ∈ M , we compute the amplitude after discarding every possible sequence of 5 consecutive phases. We define S m as a sequence of 5 consecutive phases:

S m = {k ∈ [1, N ]|∃j ∈ [0, 4] : k -m ≡ j[N ]} (3)
The discarded sequence yielding the smallest amplitude is associated to a possible motion under the considered voxel. This sequence is defined by its first phase m(x) computed by:

m(x) = arg min m∈[1,N ] (max{I n (x)} [1,N ]\Sm -min{I n (x)} [1,N ]\Sm ) (4)
We cumulated the obtained sequences S m(x) into a histogram:

∀n ∈ [1, N ], h(n) = card{x ∈ M/n ∈ S m(x) } (5)
The resulting histogram h is automatically thresholded [START_REF] Otsu | A threshold selection method from gray-level histograms[END_REF], yielding two groups of phases. The smallest group (typ. 5 to 6 phases) is associated with motion state and the largest group (typ. 10 or 9 phases) is associated with the motionless state.

In the latter group, p denotes the first phase and L denotes the number of phases of the largest group.

RESULTS

Detected motion

Interestingly, the observed motion in gated 4D images appeared to be temporally shifted with respect to the pressure signal. This observation motivated the proposed ad-hoc image-based motion detection method. To exemplify this, we have compared the proposed motion detection method, applied to both SPECT and CT 4D images, to the lung volume (extracted from CT) and to the pressure signal. For this experiment1 , one healthy C57BL6 female mouse (weighing 26 g) was injected intra-peritoneally 100 MBq in 400 µL of 99m T c-pertechnetate. SPECT and 4D-CT images have been acquired one after the other. Both of the modalities acquisitions was synchronised to the pressure signal.

Motion was detected from 4D SPECT image using the proposed method. We observed that the peak of the detected motion (i.e. the phase that belongs to a majority of discarded sequences S m) is temporally shifted with respect to the pressure signal peak. As a consequence, the phases resulting from a threshold of the histogram h(n) differed from the ones that would have resulted from a threshold of the pressure signal. Changing the length of the discarded sequences (data non presented here) did not change the location of the peak. The same motion detection method has been applied to the 4D CT images, and the peak of the motion also appeared at the same phase. Last, we have computed the lung volume along the phases with an ad-hoc and simple method (Gaussian smoothing, threshold between -650 HU and -250 HU, and morphological closing). Again the peak of volume variation appeared at the same phase. The value of the temporal shift between the motion observed in images and the pressure variation did not seem to be predictable between animals or acquisitions. We suspect that are many contributing causes, including the placement of the pressure sensor.

Reconstruction of breath-hold like SPECT images

One NOD-SCID female mouse 1 (weighing 17.7 g) was injected intra-peritoneally 1.5 million cells of rat colonic adenocarcinoma expressing the Natrium Iodide Symporter (PROb-mNIS) in 50µL of Phosphate Buffered Saline. After 3 weeks of growth, this mouse was injected intra-peritoneally 180 MBq in 400 µL of 99m T c-pertechnetate and was scanned with both SPECT and 3D-CT. On the SPECT, PROb-mNIS peritoneal nodules of NIS-transfected cells were visible, some of them located just under the diaphragm.

We applied the proposed method on the reconstructed 4D-SPECT image. Extracting the motionless phases allowed the reconstruction of a breath-hold like 3D image (BH3D). For this image, only the data belonging to motionless phases are used. That is, for each cycle, all data from the phase p and for a time interval of length L × C s /N , this corresponds roughly to 2/3 of the original data. Figure 2 assesses the quality of the first results. First, BH3D image is less noisy than the 4D images at the end of exhalation or at the end of inhalation. Second, the zoomed-in view of the lesion is less blurred in BH3D than in the standard 3D image.

CONCLUSION

In this paper, we presented an image-based motion detection method for dynamic images. We extracted a discrete motion signal from the analysis of high intensity variation regions in dynamic images and compared it to the original pressure signal. This method has been tested on two imaging modalities, CT and SPECT. For both, we obtained similar imagebased motion signals. Moreover, a temporal shift between the peak of motion in the pressure signal and the peaks in the two image-based signals has been demonstrated, that justified posteriorly such an image-based motion detection approach.

This temporal shift has been confirmed by lung volume variation in the 4D-CT. Although the proposed method is quite simple, it achieved good results. Moreover, it allowed the reconstruction of breath-hold like 3D SPECT image.

For future work, the proposed method will be used on more SPECT images in order to reconstruct motionless 3D images. This will allow us to evaluate our method both qualitatively and quantitatively compared to other SPECT reconstruction schemes.

Figure 1

 1 presents the result of all image-based motion detection and illustrates the temporal shift between the normalized average pressure signal (continuous black curve) and a set of different image-based results: normalised histogram h 4D-SP ECT (n) computed on the 4D-SPECT image (blue staircase), normalised histogram h 4D-CT (n) of motion phase computed on the 4D-CT image (red curve with cross-shaped markers), and lung volume (mL) measured on the 4D-CT image (black dotted curve with circle-shaped markers).

Fig. 1 .

 1 Fig. 1. Image-based motion signal from 4D-SPECT and 4D-CT image compared to the pressure signal and lung volume variation.

Animal housing and procedures were conducted according to the guidelines of the French Agriculture Ministry and were approved by the local ethics committee.