Member, IEEE Fr Éd Éric Giraud
email: frederic.giraud@polytech-lille.fr.

Paul Sandulescu
email: paul.sandulescu-6@etudiants.ensam.eu.

Michel Amberg
email: michel.amberg@univ-lille1.fr.

Member, IEEE Betty Lemaire-Semail

Florin Ionescu

B Lemaire

Modeling and Compensation of the Internal Friction Torque of a Travelling Wave Ultrasonic Motor

Keywords: Force Feedback, Real Time Control, Haptic Display, Ultrasonic Motor. 3

This paper deals with the control and experimentation of a one degree of freedom haptic stick, actuated by a travelling wave ultrasonic motor. This type of actuator has many interesting properties such as low speed operation capabilities and a high torque-to-weight ratio, making it appropriate for haptic applications. However, the motor used in this application displays non linear behaviour due to the necessary contact between its rotor and stator. Moreover, due to its energy conversion process, the torque applied to the end-effector is not a straightforward function of the supply current or voltage. This is why a force-feedback control strategy is presented, which includes an online parameter estimator. Experimental runs are then presented to examine the fidelity of the interface.

INTRODUCTION

U LTRASONIC motors output high torque at low rotational speeds, and can thus be directly mounted on the lever of a 1 degree of freedom force feedback stick, without any speed reducer or cable driven system. Compared to classical electromagnetic solutions, which use a DC motor mounted with a speed reducer and cables, we can expect a reduction in the weight, backlash and bulk size of the kinematic chain. Moreover, they are built with non magnetic materials, providing a solution for force-feedback devices used in a Magnetic Resonance Imaging (MRI) environment [START_REF] Flueckiger | fmri compatible haptic interface actuated with traveling wave ultrasonic motor[END_REF][2] [START_REF] Izawa | Mr compatible manipulandum with ultrasonic motor for fmri studies[END_REF]. Moreover the rotor is firmly pressed on the stator as a consequence of the design of the motors; thus, the shaft is locked if no power is applied to the motor. This property can be used in order to simulate stiff walls very easily. These advantages make Ultrasonic Motors a good alternative to other technologies in force-feedback applications.

However, these motors present many disadvantages due to their energy conversion process. In fact, the output torque is not a straightforward function of an electrical value. For DC motors, however, for which the output torque T is proportional to the flowing current I such that T = KI, the relationship cannot be applied for Ultrasonic Motors. Dry friction occurring at the stator-rotor interface is invoked to explain the torque production. But it is an extremely non linear process, which is difficult to model at very low speed. Torque production is thus a particular issue in these motors.

To cope with this problem, a specific control has to be designed in order to tune the output torque T of the motor. For example, Schulte et al. built a haptic stick driven by a non-commercial Travelling Wave Ultrasonic Motor (TWUM) [START_REF] Schulte | Active control stick driven by a piezo electric actuator[END_REF]. Results have shown good fidelity. But the control used a mapping of the torque versus speed characteristics of the motor. If the motor is replaced, a new identification process needs to be implemented. By mounting additional strain gauges that allow the measurement of the motor's torque, it is possible to propose a closed loop control which does not need a precise identification of the motor's torque-speed characteristics. Previous work has shown force feedback operations using such a control [START_REF] Dai | A force feedback device actuated by piezoelectric travelling wave ultrasonic motors[END_REF]. However, the loop corrector used was a simple PI which did not produce satisfactory control of the torque, even at low rotational speeds.

This article proposes a new control algorithm. It is based on the inversion of an accurate modelling which takes into account the internal friction torque. Accuracy of the loop corrector is thus improved by compensating this friction. To achieve this, it is necessary to identify the relationship between the friction and the operating conditions. A mean square algorithm is then used in order to identify on-line parameters of the friction modelling.

In section 2 of this article, we present the modelling which includes friction, and the control scheme deduced from its inversion. This specific control includes a torque estimator which is also explained. Experimental runs are presented in section 3.

MODELLING AND CONTROL

Modelling of a Travelling Wave Ultrasonic Motor

The travelling wave ultrasonic motor is driven by two sinusoidal voltages with a constant amplitude and variable pulsation. Two stationary waves are created by each voltage, one is combined with the other to produce a travelling wave. This wave is able to propel the rotor which is pressed on the stator. The speed of the rotor is a function of the vibration amplitude W of each flexural wave, the phase shift ϕ between the two supply voltages, and also depends on the operating conditions and on some parameters of the motor's design, such as the stator's thickness and the motor's diameter. This function is not straightforward, but under "ideal" contact conditions between the stator and rotor (no sliding and punctual contact conditions), the "ideal" rotational speed of the motor, called ω id , would be:

R1 → ω id = λW sin(ϕ) (1)
where λ is a coefficient taking into account the stator's geometry, and -π 2 ≤ ϕ ≤ π 2 . We note relation 1 as R1 because it is used later in a graphical representation.

Equation 1 is useful to obtain the motor's operating principle, but does not precisely describe its behaviour. In a previous study [START_REF] Giraud | Analysis and phase control of a piezoelectric traveling-wave ultrasonic motor for haptic stick application[END_REF], we presented a modelling which takes into account the effect of the actual speed of the motor on the motor's output torque, and allowing us to write relation R2 as:

R2 → T m = f 0 (ω id -ω) (2)
where

• ω is the revolving speed of the rotor • f 0 is a parameter which is identified from the torque-speed characteristics of the motor Relation 2 shows how the actual speed of the motor can be modified by the load torque. In fact, we can write ω = ω id -Tm f0 : if ω id is constant, ω decreases if T m increases. Under no load conditions, T m = 0 leads to ω = ω id : using (1), the no load revolving speed of the motor should be a sinusoidal function of ϕ for a constant vibration amplitude W as depicted in figure 2, and a linear function of the vibration amplitude W for constant ϕ. However, this behaviour is not verified by experimental measurements, such as those presented in figure 3. In fact, the actual motor presents a dead zone, which is a vibration amplitude below which the rotor sticks on the stator. In what follows, we call W T H the vibration threshold below which no revolution of the rotor is experienced.

In order to take into account these phenomena, one may use a non linear relationship instead of (1), as in [START_REF] Giraud | Precise position control of a traveling-wave ultrasonic motor[END_REF]. But in this paper, we choose a linear relationship, and to include non linearities in an additional friction torque T f instead. Then T f is considered as a perturbation, and there exist online methods to identify and compensate it. The idea is to employ methods already used in force-feedback applications, and to synthesize torque controllers more easily for TWUM. So, the output torque of the motor T is given by relation R3:

R3 → T = T m -T f (3)
In this modelling, T f is calculated in order to simulate the dead zone effect. To achieve this, we write:

T f = f 0 λW T H sin(ϕ) if W ≥ W T H (4)
For the no load condition (T = 0), and using ( 1), ( 2), ( 3) and ( 4), we can write:

f 0 (ω id -ω) = f 0 λW T H sin(ϕ) (5)
which yields:

ω = λ(W -W T H )sin(ϕ) (6)
For small vibration amplitudes below the threshold, W ≤ W T H , dry friction occurs and the rotor sticks to the rotor, leading to ω = 0. This behaviour is not modelled by equation 4: the method proposed in the paper implies that we work above the threshold, and that we keep W > W T H .

These relationships are consistent with experimental runs achieved on TWUM with a wave amplitude control [START_REF] Giraud | Analysis and phase control of a piezoelectric traveling-wave ultrasonic motor for haptic stick application[END_REF]. In fact, if W > W T H and ϕ = π 2 , ω is a linear function of W and ω = 0 is reached for W = W T H , as shown in figure 3. Moreover, for W constant, ω varies with sin(ϕ) as depicted in figure 2. Thus, this is proof for equation [START_REF] Schulte | Active control stick driven by a piezo electric actuator[END_REF].

Finally, the dynamic of the angular velocity ω of the rotor is given by the dynamic mechanical equation, leading to relation R4:

R4 → J dω dt = T m -T r -T f ( 7
)
where J is the inertia of the motor and T r the load torque. We represented this modelling in a Causal Ordering Graph [START_REF] Barre | Inversion-based control of electromechanical systems using causal graphical description[END_REF] in order to deduce the torque control structure by inversion. In this graph, equations ( 1), ( 2), ( 3) and ( 7) are denoted by their causal relation R1..4.

Of course, this is a very simplified modelling of the motor. But it can be verified by the experimental runs of figures 2 and 3 that the modelling matches the actual behaviour of the motor for steady state operations.

It is now possible to deduce, by inversion, the torque control, as is shown in the next section.

Torque Control

We can see from (2) that the output torque of the motor not only depends on supply conditions -leading to a certain vibration amplitude W -but also on the actual speed of the motor ω. A consequence of this is that to control the motor's output torque ω id , and thus the supply conditions of the motor, should always be adapted to the actual speed ω. Moreover, if the motor is not powered up, ω id = 0: there exists a torque which blocks the motor's shaft. The energy supplied to the motor is thus necessary to overcome this high blocking torque. Hence, this modelling shows how a TWUM behaves differently to electromagnetic motors, where the current controls the torque independently of the rotor's velocity. It also points out how performances of a control can be different depending on wether we consider a force feedback application -for which the output torque should be controlled as precisely as possible -or a position control -for which ω = 0 is a mandatory condition at rest.

In [START_REF] Giraud | Analysis and phase control of a piezoelectric traveling-wave ultrasonic motor for haptic stick application[END_REF], the torque controller directly adjusts W from the error between the torque reference T ref and T . The control scheme is simple, but the torque controller has to be finely tuned in order to exhibit the required performances. In this paper, we find the control scheme by inverting the causal ordering graph as in figure 4. Starting from the torque T , we have first to invert R3 to calculate a reference for T m , namely T mreg , which is directly deduced from relation R3, leading to equation 8 and relation Rc3:

Rc3 → T mreg = T ref + T f (8)
Using the same approach, relations R2 and R1 are inverted, leading to references for ω id , W and ϕ ( [START_REF] Giraud | Precise position control of a traveling-wave ultrasonic motor[END_REF]). In this control scheme, no torque control is required. However, the control is accurate only if the friction torque T f is perfectly compensated. But large errors are experienced if parameters of equation ( 4) are not well identified. This is why this article deals with a torque estimator which is implemented on-line in a torque control of the travelling Wave Ultrasonic Motor.

Parameter Estimator

Many methods exist for implementing an estimation of the parameters of equation (3) [START_REF] Janot | Analysis and application of a robust identification method[END_REF], [START_REF] Kwon | Control of the haptic interface with friction compensation and its performance evaluation[END_REF]. We have chosen to implement a least squares estimation method. Moreover, we will consider a recursive method because we are interested in identifying those parameters on-line.

In the context of TWUM, a measurement of the motor's output torque T is achieved by strain gauges glued on the haptic stick. Then, equations ( 2),( 3) and (4) lead to equation ( 9):

T = f 0 λW sin(ϕ) -f 0 ω -f 0 λW T H sin(ϕ) (9)
The algorithm should estimate on-line parameters f 0 , W T H and λ, from inputs T , ω, W and ϕ, which can be measured or regulated. We can build a vector matrix which contains the estimated values:

X n = (f 0 λ, -f 0 , -f 0 λW T H ) T ( 10
)
where n defines the number of sampling periods. We also define a vector matrix with the measured data:

A n = (W sin(ϕ), ω, sin(ϕ)) (11)
The estimator should minimize |A n X n -T |. To achieve this, we define a square matrix P which is updated upon the arrival of new data, as described in [START_REF] Yang | Applied Numerical Methods Using MATLAB[END_REF]:

P n+1 = P n -P n .A n [A T n P n A n + 1] -1 A T n P n (12)
It is shown in [START_REF] Yang | Applied Numerical Methods Using MATLAB[END_REF] that A T P n A is scalar, and the inverse is thus easily computable. It is useful to set:

K n+1 = P n .A n [A T n P n A n + 1] -1 (13) yielding:
P n+1 = P n -K n+1 A T n P n (14)
The recursive method allows the calculation of the new vector of estimated values:

X n+1 = X n + P n+1 A n (T -A T n X n ) ( 15
)
where T is the measured torque at the n-th sampling period.

Simulation tests were carried out in order to check how long the algorithm takes to converge. We conclude that 2.5 sec are necessary to converge at 100µsec of the sampling period. This is why this torque estimator adapts to slow drifts of the parameters, due to temperature for example. Thus, a faster torque controller is necessary to compensate for the modelling errors.

Torque Controller

Despite the online friction torque estimator, errors may still occur during transient operation; a torque controller is then added in parallel in order to reduce torque error. In this study, we used a simple PI controller, whose parameters are tuned to obtain a fast torque response. The final torque control scheme is detailed in figure 5.

The block R e is used to compensate the friction torque estimation error. The output of R e , which is called T f ε is then equal to the difference between the estimated friction torque Tf and the actual one T f . Because a perfect compensation of T f would result in Fig. 5. Torque Control with a torque controller added. The torque estimator is not represented. a perfect control of T , T f ε is then calculated from the difference between the actual output torque T and its reference T ref , leading to:

Re → T f ε = (K p + K i s )(T ref -T ) (16)
The two blocks R c1 , R c2 describe the inversion of the relations R 1 and R 2 ; they are not detailed in this paper, but information about their synthesis can be found in [START_REF] Giraud | Precise position control of a traveling-wave ultrasonic motor[END_REF]. Finally, this control scheme produced an experimentally measured response time of about 3ms for the closed loop torque control.

IMPEDANCE FIDELITY TEST

Experimental Setup

A one-degree-of-freedom force feedback lever was built to verify the control laws. The experiment involves use of the lever of the digitracker which is free to rotate about the horizontal axis, and is presented in figure 6. On the shaft, we attached a variable resistor which outputs a voltage proportional to the angular position of the lever (a 2kΩ low friction resistor from Vishay, N 0 1 in figure 6). On the right hand side of the figure, the TWUM is fixed to the shaft (N 0 2 in figure 6) in order to achieve force feedback operations. The motor used is a Shinsei USR30, its technical specifications are presented in table 1. In order to measure the torque of the motor, two strain gauges (CEA 13015 UW, 120Ω from Vishay) were bonded on the lever, yielding a torque sensor. The measured torque differs from T by a small amount due to the small shaft and the position sensor friction, and shaft inertia, which are negligible compared to the lever's inertial torque. The motor is supplied by a linear power amplifier. Typical voltage frequency is 50kHz, but this value may vary from 55kHz to 49kHz. On the motor, a deformation sensor is bonded on the stator. This sensor is used for the deformation amplitude control. The control laws are computed on a DSP SBC6711 board; W is regulated by a closed loop control whose response time is estimated to be constant and equal to 1ms. More details about this control can be found in [START_REF] Giraud | Analysis and phase control of a piezoelectric traveling-wave ultrasonic motor for haptic stick application[END_REF]. We chose a sampling period of 100µsec. Two electronic boards were built to adapt analog input and output to the motor. First, a rectifying amplifier converts the voltage from the sensor to the DSP. A second one is based on a Voltage Controlled Oscillator, and converts a voltage from the DSP to a frequency of two sinusoidal voltages with the same amplitude. Finally, a digital output of the DSP is used to modify the voltages' phase differences. This control hardware is shown in figure 7.

Experimental results and discussion

If no power is applied to the stator, the motor is locked in its position because the rotor is pressed on the stator. Consequently, simulating a zero torque on the lever is something which is not natural, and which demonstrates how well friction compensation is achieved. This is why, during this test session, the torque reference is set to 0. The user should then be free to manipulate the lever at any speed condition. Results are presented in figure 8, in the torque-speed plane. As can be seen, a zero force is achieved. However, when the motor's speed reverses, a torque spike evaluated at 5mN m appears. This spike is due to the difficulty which exists in removing the motor's static friction. These spikes are small compared to the motor's rated torque. But this figure can increase if the user's rotational speed becomes faster, and can increase up to 15mN m.

Experimental data from controlled viscous friction tests are presented in figure 9. During this test, we set T ref = f v ω where f v is the parameter used to obtain the desired friction. Low to high friction levels were programmed, leading to several values of f v = such as f v = 0.0001, 0.001, 0.01, 0.1 N m.sec. In the torquespeed plane, characteristics of the measured torque T should then be straight lines. This is actually the case except around ω = 0 and for low friction. In fact, in that case, the parasitic torque leading to the spike is relevant. These spikes decrease the fidelity of the haptic interface.

Finally, we created an elastic virtual environment, by imposing T ref = K v θ + f v ω. Some friction was added in order to stabilize the virtual stiffness, however its value was sufficiently low not to change the elastic behavior. Results are presented in figure 10. They show that the measured torque is close to its reference; no hysteresis appears in the torqueposition plane, showing that motor's static friction is well compensated. For the K v = 5.7N m/rad case, we required the user to be stronger than the forcefeedback device. By doing so, the motor saturates and the output torque attains its limit. This results in horizontal lines in the torque-position plane. One may remark that the torque limit is smaller than the motor's rated torque (100mN m). During these experiments, the friction torque estimator was running. We have drawn its output as a function of time to verify its accuracy. We compare in figure 11 the calculated friction torque from ( 4) to the measured one. In fact, from (3), we obtain T f = T m -T . This is not a direct measurement of T f , but it should be equal to Tf if the ω(rad.s

ref = K v θ + f v ω.
estimation is perfect. Thus, the friction estimator is accurate because the estimated value of T f is very close to the measured one. Moreover, the parameter estimator is helpful to follow the slow drifts of T f and to estimate these parameters without a specific measurement process. In the end, the error between T ref and T is small at the input of the torque controller Re. If we remove the friction torque estimation, then fidelity of the forcefeedback device only relies on the torque controller Re. In that case, large torque errors are measured at each change in direction of the lever, as those shown in figure 12. Consequently, in the T -θ plane, the spikes that appear are those presented in [START_REF] Giraud | Analysis and phase control of a piezoelectric traveling-wave ultrasonic motor for haptic stick application[END_REF]. This test shows the experimental performance of the estimator because we obtain a satisfactory torque control of the motor even without a high-performance torque controller.

CONCLUSION

This paper presents a control algorithm for a travelling wave ultrasonic motor used during force feedback operations. This control compensates internal friction of the motor, and uses an online friction estimator. An experimental test bench was designed and has shown that the motor could be directly mounted on the haptic stick to allow force-feedback. The control presented good performances at low rotational speed.

Despite not being able to avoid the spike that arises when the rotational speed reverses, we measured torque spikes of 5mN m which is low compared to the maximum output torque. We then simulated a variable stiffness spring. The largest simulated stiffness was set to 1.92N m/rad and we imposed torque limitation of the motor at 0.05N m which is lower than its rated torque. These performances were obtained with a commercial product. They may encourage use of Travelling Wave Ultrasonic Motors in force-feedback applications in order to reduce the weight or bulk size of the system. To attain better performances, work needs to be carried undertaken in order to improve the motor's control. To reach the same performance level of today's force-feedback device, a specific design of the motors may however also be necessary.

Fig. 1 .

 1 Fig. 1. Causal Odering Graph of the travelling Wave Ultrasonic Motor.

Fig. 2 .

 2 Fig. 2. No-load revolving speed as a function of ϕ with W constant. Comparison between modelling (solid lines) and experimental runs (crosses).

Fig. 4 .

 4 Fig. 4. Torque control (bottom part) and modelling (top part) of a TWUM.

Fig. 6 .

 6 Fig. 6. The experimental test setup. 1: position sensor 2: TWUM 3: Strain gauges.

Fig. 7 .

 7 Fig. 7. The hardware setup.

Fig. 8 .

 8 Fig. 8. T as a function of ω when T ref = 0.

- 1 )Fig. 9 .Fig. 10 .

 1910 Fig. 9. T as a function of ω for T ref = f v ω.

Fig. 11 .

 11 Fig. 11. Transitory operation of the force-feedback device. A and B: comparison between T and T ref ; C and D: comparison between Tf and T ; E, F and G: estimation of f 0 , λ and W T H of equations 3 and 4, and H: T as a function of θ.

Fig. 12 .

 12 Fig. 12. Output torque T compared to T ref when the friction torque estimator is removed. The ellipse highlights errors when the direction changes.

TABLE 1

 1

	technical specifications of a Shinsei USR 30
	rated torque	0.1Nm
	rated speed	200 rev/min
	supply voltage	200 volts
	supply frequency	50kHz
	outer diameter	30mm


Frédéric Giraud

Betty Lemaire-Semail

Julien Aragones

Jacques Robineau

Jean-Thierry Audren

Precise Position Control of a Travelling Wave Ultrasonic Motor

Precise position control of a Travelling Wave Ultrasonic Motor is achieved, avoiding the traditional drawbacks attributable to non-linear torque generation: overshoot or slow response time. For that purpose, a behavior model control is proposed and presented. With this control law, a quick and precise response is obtained. In this article, we present a position control scheme of an inertial load. The guideline used for this control was a rotation of 90 0 in a response time of about 200ms with a position error of 0.6mrad, targeting a typical application for avionics. In the paper, a shinsei USR30 is used, but the method can be applied to other Ultrasonic Motors.

I. INTRODUCTION

Travelling Wave Ultrasonic Motors (TWUM) exploit a piezoelectrically generated flexural wave which propagates at the surface of a stator. This wave is able to propel by contact a rotor strongly pressed on it. The friction that the contact mechanism produces is at the origin of numerous advantages, among which we can find: holding torque without supply, a high torque to mass ratio and high torque -low speed characteristics. Thus, while a speed reducer is often needed with an electromagnetic motor, it becomes unnecessary in an application using TWUM, leading to lightweight and compact applications. These features increase significantly the interest of avionics in these motors; they can be used for example for finely positioning optical lenses in embedded optronic devices.

Unfortunately, the torque generated by a TWUM is non linear: it depends on the rotational speed, and a large static friction torque appears at low speed; in that context, position controls are not straightforward and often fail in applications where precise position and robustness are both required. Simple position correctors are not sufficiently robust compared to external load torque or parameters variations, since they are better suited to roughly linear process.

On the one hand, it has been shown, for example in [START_REF] Maas | Characteristics of inverter-fed ultrasonic motors[END_REF], that to obtain torque at low rotational speed, and thus attain good position error, it is useful to change the temporal phase shift between the two supply voltages. On the other hand, some authors experimented with many techniques to achieve robustness in position controls, for example adaptive controls [START_REF] Senjyu | Quick and precise positilon control of ultrasonic motors taking account of frictional force control[END_REF], fuzzy-logic [START_REF] Lin | Fuzzy neural network position controller for ultrasonic motor drive using push-pull dc-dc converter[END_REF] and neural networks [START_REF] Senjyu | Position control of ultrasonic motors using neural network[END_REF] [START_REF] Tanaka | Precise position control of an ultrasonic motor using the pid controller combined with nn[END_REF] are used to make up for lack of knowledge about the motor and its load. Robustness and precision are thus obtained, but at the expense of tuning complexity, while transient performancesovershoot and response time -are not compared.

In the present paper we propose a position control of an inertial load actuated by a TWUM which requires robustness, precision and short response time. It is based on a coarse but causal modelling of the motor which is presented in section II. The controller presented doesn't belong to adaptive controllers family, but the non-linearities and the parameters changes are compensated thanks to a behavioral model control which will be described in section III. At last, the actuator's behavior will be checked by experimental results.

II. MODELLING OF THE TORQUE GENERATION

A. Modelling overview.

Each sinusoidal voltage, named v α and v β , supplying the motor creates a stationary flexural waves, w α and w β , which bends the stator. Superimposing both standing waves creates an elliptical motion of the stator's particles. This motion creates frictional forces which are at the origin of the torque. A detailed description of the principles of the propagation of the wave can be found in [START_REF] Sashida | An introduction to Ultrasonic Motors[END_REF]. As for [START_REF] Hagood | Modeling of a piezoelectric rotary ultrasonic motor[END_REF], we developed an analytical modelling of the stator's vibration from which a causal modelling is derived [START_REF] Giraud | Analysis and phase control of a piezoelectric traveling-wave ultrasonic motor for haptic stick application[END_REF]. However, in this article, we focus our attention on torque generation. This is why it is supposed that control of the two stationary waves is obtained by an external loop. Considering that no cross couplings between the two phases exists, the following equations will be admitted:

w α (t) = W sin(2πf t) w β (t) = W sin(2πf t -ϕ) (1) v α (t) = V sin(2πf t+Ψ) v β (t) = V sin(2πf t+Ψ-ϕ) (2)
with W the stator's deformation amplitude, V the supply voltage amplitude and f the voltage's frequency. In normal operating mode, ϕ is set to π 2 for a positive rotational speed, and to -π 2 for a negative one. But in order to improve performances at low rotational speed, variations of ϕ in the range of [-π 2 ; π 2 ] will be applied. To describe the torque generation process, we previously introduced the ideal rotational speed ω id [START_REF] Giraud | Analysis and phase control of a piezoelectric traveling-wave ultrasonic motor for haptic stick application[END_REF] [START_REF] Giraud | Causal modeling and identification of a traveling wave ultrasonic motor[END_REF]; ω id is assumed to be the rotational speed of the rotor under ideal contact position: no slippage between stator and rotor and a punctual contact. A kinematic study of the stator [START_REF] Giraud | Causal modeling and identification of a traveling wave ultrasonic motor[END_REF] helps to express the value of ω id which is given by:

ω id = (2πf )k h b 2 W sin(ϕ) (3)
with the following constant parameters • k : wavenumber of the stator • b : radius of the stator,

• h : thickness of the stator. It should be noted here that the stator is a mechanical resonator, and the vibration amplitude is named W . So, the voltages of frequency f are tuned in close vicinity of the eigenfrequency: in equation 3, f can be considered constant. Modification of ω id is achieved by tuning f , but for the sole purpose of tuning W thanks to the resonant process.

However, due to the stator/rotor contact mechanism, the actual speed of the rotor ω is different from ω id . First, because at low wave amplitude, stick-slip effect causes tribological uncertainties which stop the motor. This is illustrated in figure 1(a) on the measured curve of the no load speed of the motor as a function of W where a dead zone around W = 0 appears. One way to reduce this dead-zone effect is to change the temporal phase shift of the supply voltages. For a constant wave amplitude W above a threshold -named W T H -, and a varying ϕ, ω id decreases, and the dead-zone disappears, as depicted figure 1(b), on which we recognize the sinusoidal shape of equation 3. Second, because the motor decelerates when it is loaded by an external load torque T r . On the figure 2, we have drawn the torque-speed characteristics of the motor for a constant ω id (either by keeping ϕ constant or W constant).

So, these experimental runs highlight many nonlinearities in the torque generation. Modelling exists for describing the stator/rotor contact mechanism; for example, by using Coulomb's friction law [START_REF] Maas | Simulation model for ultrasonic motors powered by resonant converters[END_REF] which can be extended to take into account many nonlinear mechanisms, such as the tangential deformation of the contact layer [START_REF] Wallaschek | Contact mechanics of piezoelectric ultrasonic motors[END_REF]. However, this modelling is suited to simulation or prototyping of the motors, and becomes very complex when we attempt to build a control scheme from it. This is why we aim to define a simpler modelling.

B. Simplified Modelling

The proposed simplified modelling is obtained by making a linear approximation of the torque-speed characteristics. The torque T produced by the motor can be thus written as

R 1 -→ T = f 0 (ω id -ω) (4)
f 0 can be identified by measuring the average slope of the curves of the figures 2, while W T H is a threshold measured for ϕ = ± π 2 . The dead-zone effect can then be modelled by revising the equation 3:

R 2 -→ ω id with (5)
ω id = (2πf )k h b 2 (W -W T H ) sin(ϕ) if W > W T H ω id = 0 if 0 < W < W T H (6)
Finally, the dynamic of the rotor in the tangential direction can be given by the equation 7:

J dω dt = T -T r (7)
This set of equations, giving a simplified description of the torque generation, can be represented using the Causal Ordering Graph (COG - [START_REF] Guillaud | Synthesis of control laws with the causal ordering graph: Application to elimination of vibration in drive of mechanical laws[END_REF]). This graphical description tool of physical systems first defines two kinds of model for elementary objects: a causal processor is associated with an energy storage object and is represented by an oriented arrow symbol. Symbols with a bidirectional arrow are associated with dissipative objects as well as instantaneous relations. To match the requirement of causality in the representation, equation 7 must be revised to:

R 3 -→ ω = 1 J (T -T r ) dt (8)
The COG of the TWUM is then represented in figure 3. In this modelling, the relations which are not linear are highlighted by a double circle. The parameters, whose values are given table I, are identified as follow:

• f 0 is directly given by the average slope of figure 2, • k h b 2 is given by the slope of figure 1(a) (whose actual slope is in fact (2πf )k h b 2 according to 3) • W T H is given by the dead-zone of figure 1(a) The response of this modelling is compared to experimental measurements (figure 4) for a step of W from 0, 77µm to 0, 94µm. Both responses are consistent, checking the validity of the modelling, where f 0 is almost constant at the most.

This modelling is the basis of the position controller; the next section describes the calculation of the regulator's parameters.

f 0 J k h b 2 W T H 0.0224N.m.s 10 -4 kg.m 2 70m -1 0, 28µm

III. ROBUST AND PRECISE POSITION CONTROL

A. Introduction

When the modelling is represented by a COG, the control laws can be deduced by inversion of the graph [START_REF] Guillaud | Synthesis of control laws with the causal ordering graph: Application to elimination of vibration in drive of mechanical laws[END_REF]. Since we have separated causal and instantaneous relations in the modelling, we operate differently for each case (figure 5). For instance, an instantaneous relation can be directly inverted :

if R → y = R(u) then R c → u REG = C(y REF ) (9) and y = y REF if C = R -1 .
A causal relation can't be directly inverted. So, a control loop is achieved, which then requires a feedback:

if R → y = R(u) then R c → u REG = C(y REF -y) (10) and y = y REF if C → ∞.
Thus, it is now possible to deduce the position control law by inverting the COG of figure 3.

Among the possible strategies, the one depicted on figure 6 is a position control with a speed control in serial. The output of the speed controller (R c3 ) is a reference for the torque; however, on TWUM, the torque cannot be directly controlled by an electrical variable -a current or a voltageas in the case of electromagnetic motors. Thus, R 1 has to be inverted, and this is the function of R c1 . Since f 0 varies with the operating point, R 1 and R 2 are strongly non-linear; their inversion, based on a linear expression, leads to slow dynamic responses and static error. This is why, in [START_REF] Maas | High performance speed control for inverter-fed ultrasonic motors optimized by a neural network[END_REF] a neural network is used to inverse the contact model and compensate the nonlinear torque generation. Good results are thus obtained at the expense of control complexity. The control presented in the next section uses linear controller only. But the overall scheme helps to compensate the parameters variations.

B. Behavior model control

Behavior Model Control (BMC) has been successfully applied to electromagnetic motors control to solve the unknown parameters variations or to remove the effect nonlinearity This is why we experimented with scheme in our application.

The basic principle of BMC relies on a linear modelling of the process with constant parameters (the behavior model) . Together with the reference modelling output is used online to compute the error. This error is minimized using a main controller RM . If the actual process is equal to its modelling, outputs of RM should be sufficient for controlling it with respect to the reference. But because discrepancy between the actual process and the simplified modelling exists, there may be some difference between their outputs.So, behavior controller, RB is used to remove the difference between the actual process and its modelling. Controllers output are both applied to the tuning input of the process. To sum up, the behavior controller has to enforce the actual process to follow the simplified modelling behavior. As for the main controller, it is able to impose the closed-loop dynamic. The actual outputs of the motor behave like if linear at the condition that the behavior controller is faster than the main one [START_REF] Vulturescu | Behaviour model control for cascaded processes: application to an electrical drive[END_REF].

Figure 7 depicts the overall scheme of the BMC designed for a state space control. For a better comprehension, in this modelling the input of the Behavior Model is ω idM , the inversion of R 2 is achieved for the actual motor and will be detailed in section III-B.3. We are now detailing the calculation of the controller R M and R B .

1) Calculation of the Main Controller:

The corrector is based on a linearized modelling of the TWUM established in section II. For that purpose, the non linearities are removed and a state space modelling is used yielding:

ẊM = A M X M + B M ω idM (11)
with the state vector defined as

X M = θ M ω M (12)
A M = 0 1 0 -f0 J (13) B M = 0 f0 J (14)
The control of the TWUM is deduced from a state space control:

R M → ω idM = K(X M ref -X M ) = (K 1 ; K 2 )(X M ref -X M ) (15)
with

X ref = θ ref 0 (16)
The characteristic polynomial P M (s) for this closed-loop system is the determinant of sI -(A M -B M K). This yields

P M (s) = 1 + 2 ξ ω 0 s + s 2 ω 0 (17)
with

ω 2 0 = f 0 J K 1 ξ = 1 2ω 0 (1 + K 2 ) f 0 J (18)
The parameters K 1 and K 2 are then calculated so as to match the requirements of the guidelines, i.e. ξ = 1 and ω 0 = 38rad/sec, which corresponds to a step response time of 200ms with no overshoot. The values of K 1 and K 2 are given in table II.

K 1 K 2 7, 1s -1 -0, 71
2) calculation of the behavior corrector: The method described in the previous section is applied to the calculation of the behavior corrector; however, because no static error between the outputs of the simulated and actual motor is allowed, the state vector is upgraded to include the integrative value of θ:

ẊB = A B X M + B B ω idB (19)
with the state vector defined as

X B =   θ θ ω   (20)
A B =   0 1 0 0 0 1 0 0 -f0 J   (21)
B B =   0 0 f0 J   (22)
Once again, the control of the simulated TWUM is deduced from a state space control:

R → ω idB = G(X Bref -X B ) = (G 1 ; G 2 ; G 3 )(X Bref -X B ) (23)
with

X Bref =   θ M θ M ω M   (24)
And the characteristic polynomial P B (s) for this closedloop system is also the determinant of sI -(A B -B B G):

P B (s) = G 1 f 0 J + G 2 f 0 J s + f 0 J (1 + G 3 )s 2 + s 3 (25)
Since P B is a third order type polynomial, we can chose the location of the roots in order to fulfill the requirements of the guideline. This can be achieved using Naslin's empirical procedure [START_REF] Kim | Transient response control via characteristic ratio assignment[END_REF]. Given a polynomial with real positive coefficients: P (s) = a 0 +a 1 s+a 2 2 +...+a n-1 s n-1 +a n s n a i > we introduce the following characteristic pulsatances:

0 = a 0 a 1 , β 1 = a 1 a 2 , ..., β n-1 a n-1 a n-2 (27)
and the ratios:

α 1 = a 2 1 a 0 2 , α 2 = a 2 2 a 1 a 3 ... α n-1 = a 2 n-1 a n-2 a n ( 28
)
It can be shown that if those ratios have the same value, named α the principal characteristic ratio, and are bigger than 2, the transient response is equivalent to a second order response with a "good" damping and the rising time of the step response is given by:

t D ≈ 2, 2 β 0 ( 29
)
In this article, good results have been obtained for t D = 60ms and α = 2.8ms. So, considering equations 25, 27 and 28, leads to the expressions and the values of G 1 , G 2 et G 3 summarized at table III: We have designed the controllers for the behavior model control R M and R B . They are calculated for a specific value of J. This suits well for our application where the load is supposed to be a pure and constant inertial load. But the controller still remains stable with good performances in the case of extra load torque and small variations (100%) of the parameter J [START_REF] Giraud | Position control of a small travelling wave ultrasonic motor[END_REF]. If larger variations are experienced, we should estimate on line its value and adapt the controllers parameters; or design a non linear adaptive controller, such as [START_REF] Senjyu | Position control of ultrasonic motors using mrac and dead-zone compensation with fuzzy inference[END_REF]. Now we still have to inverse relation R 2 to obtain R c2 (see figure 7). This is dealt with in the next section.

3) phase difference control: For the very low values of ω id , it is known that W should not be decreased below a threshold (W T H , figure 1(a)). Because in this condition the rotor sticks on the stator, we would rather keep W constant and adjust ϕ so as to attain the right value for ω id . We then define W M in , the lowest value for W , with W M in > W T H and we have chosen the following strategy for R c2 :

R c2 → (W, ϕ) = f (ω idREG ) with W = sup(W M in , 1 k h b 2 (2πf ) |ω idREG | + W T H ) and 0 0.1 0.2 0.3 time (sec) θ (rad) θ θ M θ REF π /8 π /4
=      ω id |ω id | . π 2 if |ω idREG | > (2πf )k h b 2 (W M in -W T H ) else arcsin( ω idREG (2πf )k h b 2 (WMin-WT H ) )
(30) This strategy is summarized by figure 10. Finally, the reference value of W has been saturated to avoid stator damage.

IV. EXPERIMENTAL RUNS

A. Experimental setup

Experimental runs have been carried out to verify the performances of this control. The test bench is made using a shinsei USR30[19] and the supply voltages are produced by a linear power amplifier. The photograph of figure 11 shows the motor and its inertial load, attached to an optical encoder. This mechanical load provides the same behavior on the motor's shaft as the real optronic system the control is designed for. One will notice the size of the motor, which is small compared to the load (the motor inertia can in fact be neglected).

MOTOR

Inertial load Position enc. In the next section, the experimental trials are presented.

B. experimental results

For the first trial, the reference for θ REF is small. The evolutions of θ, θ M and θ REF are depicted on figure 8(a) On this figure, θ M respects the guidelines, because the response time is less than 200ms and there is no overshoot. On the same figure the actual output of the motor is depicted; as one can see, θ follows θ M and thus also respects the guideline. According to figure 8(b), the actual revolving speed and the model's revolving speed are similar, excepting at the starting of the motor: this is due to the time response that exists between the stator's deformation amplitude W and the actual deformation amplitude measured on the motor. This point is detailed on figure 8(c) where both measurement and reference of W are depicted. On figure 8(d), the evolution of ϕ is depicted. At t = 0, ϕ = π 2 in order to have the most important torque. Then, as ω M decreases, ϕ also decreases according to figure 1(b) . This strategy has been chosen to avoid small wave amplitude W , and thus avoid the dead-zone of figure 1(a). A fine position control is achieved since no stick-slip appears between stator and rotor. For steady state, ϕ is set to a constant value (near 0) to compensate residual static friction on the shaft. Moreover, ϕ increases gradually because the electronic device controlling ϕ doesn't let continuous variation.

For the second trial, larger step responses are experimented with. Now, a bigger error is observed between θ and θ M (figure 9(a). This is due to the saturation of W which has been set in R c2 . So, the speed of the motor is limited and can't be equal to the speed required to follow the behavior of model 9(b). But it is interesting to note that the motor catches up the trajectory of the model as soon as possible, avoiding any overshoot.

V. CHECKING ROBUSTNESS BY SIMULATION

During operation of the device, an external torque can be applied on the load; a position error can then appear if the control is not sufficiently robust. With a BMC, the robustness against perturbations is obtained by the behavior controller R B , which is supposed to be faster than the main controller. This is why, rejection of perturbation is "good" with such a control scheme. This is confirmed by the simulations results of the figure 12 which shows the position error to a step variation of the external torque of 0.05N m. The maximum error observed is 10mrad, which is acceptable, compared to the resolution -0.6mrad-of the guidelines. These results agree with the experiment, as detailed in [START_REF] Giraud | Position control of a small travelling wave ultrasonic motor[END_REF].

This approach may also be applied for other typical load conditions. In particular, for large disturbance torque, or coulomb friction torque. Moreover, as far as a linear process modelling can be developed, the proposed BMC method can be used for other type of actuators. In the case of ultrasonic motors -which are typically non-linear -the main issue is then to deduce a modelling sufficiently simplified to come to a straightforward main controller, but also accurate enough to make the work of the behavior controller easier [START_REF] Giraud | Analysis and phase control of a piezoelectric traveling-wave ultrasonic motor for haptic stick application[END_REF] [START_REF] Pigache | Modelling and identification of a planar standing wave ultrasonic motor[END_REF].

VI. CONCLUSION

The aim of this paper was to achieve a precise position control of a TWUM for typical avionic applications in the field of optical lense positioning. Although the dynamic and mechanical properties of the actuator are interesting, its control is not straightforward, which accounts for the many non linearities of the contact phenomena at the stator/rotor interface.

To achieve a precise control, we propose both a modelling approach well adapted to control issue, and a behavior model control to cope with the modelling uncertainties. For that purpose, an experimental identification of the TWUM's mechanical characteristics has been achieved, allowing the building of a simplified modelling. The causal structure of position control was then built and the regulator's parameters determination was explained. After implementation, the control law was checked experimentally and showed dynamic responses which respect the guidelines, as well as good robustness behavior.

  (a) ω id as a function of W .(b) ω id as a function of ϕ; experimental and modeling (W = 0, 65µm).

Fig. 1 . 2 (

 12 Fig. 1. Identification of ω id

Fig. 2 .

 2 Fig. 2. Characteristics of the loaded motor.

Fig. 3 .

 3 Fig. 3. COG of the TWUM

Fig. 5 .Fig. 6 .

 56 Fig.5. Inversion of elementary processors[START_REF] Guillaud | Synthesis of control laws with the causal ordering graph: Application to elimination of vibration in drive of mechanical laws[END_REF].

Fig. 7 .

 7 Fig. 7. COG of a the Behavior Model Control.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Small step response of the motor -first run.

  ϕ

Fig. 10 .

 10 Fig. 10. Computation of W and ϕ. Inversion of R 2

Fig. 11 .

 11 Fig. 11. Experimental setup with the shinsei φ30mm motor.

Fig. 12 .

 12 Fig. 12. Position error for a step torque variation. Simulation result; at t=0, Tr = 0.05N m.

Betty

  Semail was born in France, in 1964. She received her Ph. D degree in 1990 from University of Paris XI, Orsay and habilitation degree in 1997 from University of science and technologies of Lille. She is now professor in university of Lille 1. She is member of the electrical engineering and power electronics laboratory of Lille (L2EP) and responsible of the research axis upon control of electrical systems. She has been working motors and her main field of interest now deals with the modelling and control of piezo-electric actuators, for positioning and force feedback applications. Julien Aragones was born in France in 1975, is graduated of University of Montpellier where he achived his PhD in automation and signal processing in 2002. He joined Sagem défense sécurité in 2003 where he works on helicopter sight heads. He is in charge of turret control for advanced studies in airborne optronic.

TABLE I PARAMETERS

 I IDENTIFICATION FOR USR30 AND ITS LOAD

Jean-Thierry Audren was born in Paris, France on March 1st , 1951. He was graduated Engineer from Ecole Nationale Supérieure d'Electronique et d'Electromécanique de Caen in 1974. Since 1975 he has been in charge of RD in guidance, avionics and optronics systems. Since 1995 he has been leading research in the field of high power piezo motors in SAGEM. He is currently Deputy Director of the Air Land Warfare Business Unit of SAGEM, in charge of business Development.

Jacques P. Robineau is gratuated of "Ecole Supérieure d'Electricité" (french top engineering school) in 1990 and achieved his PhD in electromagnetism. He has been working in electromagnetism numerical simulation (RF and optics) for 10 years, dealing with many aircraft and optronic laser stealth projects. He joined Sagem défense séecurité where he is in charge of advanced studies in airborne optronics.


Frédéric Giraud

Member IEEE Betty Semail

A Torque Estimator for a Travelling Wave Ultrasonic Motor -Application to an Active Claw

Depending on its electrical to mechanical energy conversion process, the torque on a Travelling Wave Ultrasonic Motor(TWUM)'s shaft is not directly proportional to a measurable electrical variable, such as current or voltage, but is derived from a complicated process at the stator/rotor interface. The load torque is thus quite unknown, and this can be a disadvantage in applications where a torque limitation is required or a torque measurement is needed. The aim of this article is to come up with a straightforward torque estimator on a TWUM. For that purpose, the motor is modelled, this modelling leads to different estimator strategies. More specifically, we chose a strategy for which a speed sensor is useless, relying only on the stator's resonant behavior. The parameters of the motor needed for the estimator are measured afterwards while some non linearities are identified and taken into account. Several experimental trials are then carried out to check the performance of the estimator. Finally, a claw actuated by a TWUM is presented, because this is a typical application where the knowledge of the torque helps guarantee the safety of the device.

I. INTRODUCTION

Travelling wave Ultrasonic Motors (TWUM ) exploits a piezoelectrically generated flexural wave which propagates at the surface of a stator. This wave is able to propel by contact a rotor strongly pressed on the stator. The friction caused by the contact mechanism provides us with numerous advantages, including: breaking without supply, a high torque to mass ratio and high torque/low speed characteristics. Thus, while a speed reducer is often needed with an electromagnetic motor, it becomes useless in applications using TWUM , therefore leading to lightweight and compact applications.

Unfortunately, unlike classical electromagnetic motors, it is very difficult to exactly obtain the torque operating on a TWUM shaft. This is due to the different energy conversions and more specifically to the complications caused by the contact mechanism between the stator and the rotor. However, for many applications, limiting or controlling the actuator's torque is necessary; in fact, when a fault appears in a system, the torque on the motor's shaft increases (or decreases) dramatically, which should be detected by the controller. While this can be easily done with a DC-motor, TWUM doesn't offer this simplicity naturally.

A torque estimator for a TWUM is nonetheless possible to achieve. For instance, in [START_REF] Maas | Model based control for ultrasonic motors[END_REF], a possible method is proposed: a neural network learns off line all the torque-speed characteristics of the motor, while during operation, the torque is estimated on line by inverting the network. This method is based on the characterization of the stator/rotor contact mechanism; a speed measurement is necessary and a large number of trials have to be carried out because torquespeed characteristics are not linear.

In this article, a different torque estimator is proposed. This estimator doesn't rely on the mechanical load characteristics, nor on the stator/rotor contact mechanism, but on the stator parameters. This estimator relies on a modeling which is described in the first section of this paper. Then, the torque estimator's principle is then developed as well as an experimental approach which is useful when identifing the modelling parameters. Finally, the accuracy of the torque estimator is checked on an active claw application, comparing the estimated torque to the measured one.

II. PRINCIPLE OF THE TORQUE ESTIMATOR

A. Modeling overview

Travelling wave ultrasonic motors are often described using an electromagnetic equivalent circuit. The resonant behavior is taken into account by a R-L-C serial branch, while the rotational effect of a mechanical load attached to the motor is modeled by an equivalent parallel resistor [START_REF] Sashida | An introduction to Ultrasonic Motors[END_REF] [START_REF] Giraud | Analysis and phase control of a piezo electric travelling wave ultrasonic motor for haptic stick application[END_REF]; sometimes, the value of the capacitance of the R-L-C branch varies in order to take into account the effect of the axial force which presses the stator against the rotor. This modeling is very useful in designing the power electronics in front of the motor, but is not satisfactory when describing the energy conversion process; in fact, this modeling is valid for steady state operations only, and both the normal and tangential forces are mixed in the circuit.

The next section deals with a modeling we proposed for taking into account transitory operations [3][4]. This modelling relies on a two stages decomposition of the stator-rotor contact mechanism. The first stage, named interaction between stator and an "ideal" rotor relies on the assumption of a punctual contact. Punctual means that the stator is in contact with the ideal rotor at one point only along a wavelength. Moreover, no sliding is considered between the ideal rotor and the stator, so both have the same instantaneous speed. The second stage, named interaction between the real and the "ideal" rotor aims at taking into account in a global approach the real contact mechanism so as to fit the experimental torque-speed characteristics. Although the real contact mechanism isn't well described, this modelling has the advantage to lead to simple equations of which the validity domain will be checked in chapter III through experimental trials.

On the figure 1, we have represented the travelling wave which propels the ideal rotor by only one point along a wavelength. In the fame fixed to the contact point ( -→ u r , -→ u θ , -→ u z ) the travelling wave propagates toward the right (along -→ u θ ) although the ideal rotor moves toward the left (along --→ u θ ). We also define the z-axis about which the motor and the contact point turn. The location of the contact point is defined by the rotation angle θc, measured from a fixed frame (O, x, y).

1) Causal modeling in (α, β) frame: The TWUM is composed of a ring shaped stator on which small piezo-electric elements are bonded. These elements have alternate polling directions, and are divided into two groups, named phase α and phase β. These two phases are supplied by two independent voltages, vα and v β . Because of the piezo-electric effect, the first energy conversion stage converts these voltages into the forces Fα and F β inside the stator with:

Fα = N vα F β = N v β (1)
and N the force factor. These forces bend the stator. And if the excitation frequency of the two supply voltages is sufficiently close to the resonant frequency of the stator, two stationary waves appear at the stator surface: the bending waves, whose height are named wα and w β . This second stage of the energy conversion can be described by the equation ( 2)

m ẅα + ds ẇα + cwα = Fα -Frα m ẅβ + ds ẇβ + cw β = F β -F rβ (2)
with

• m the vibrating mass of the stator,

• c the vibrating stiffness of the stator,

• ds the damping coefficient. In this equation, Frα and F rβ have been added to take into account the effect of external forces on the wave propagation. Their expression which depends on FN (the force pressing the rotor against the stator along the axis of the motor) and F ′ T (the tangential force deriving from the load torque T ) is quite difficult to determine, and strongly depends on the contact condition at the stator-rotor interface [START_REF] Hagood | Modeling of a piezoelectric rotary ultrasonic motor[END_REF] [START_REF] Maas | Simulation model for ultrasonic motors powered by resonant converters[END_REF]. However, in the ideal case of a punctual contact condition, these forces are simply expressed using the location of the contact point θc, and the well known rotational matrix, usually used in the study of electromagnetic machines. This calculation has been developed in [START_REF] Giraud | Causal modeling and identification of a travelling wave ultrasonic motor[END_REF], and derives from power balance of the system:

Frα F rβ = R(kθc) FN F ′ T R(θ) = cos(θ) -sin(θ) sin(θ) cos(θ) (3)
with

F ′ T = k h b 2 T (4)

And

• h the thickness of the stator • b the radius of the rotor • k = 2πb λ with λ the wavelength of the excited mode. The contact points are located at the travelling wave's crest. It can be shown that they are situated at an angle given by wα and w β using the expression shown in [START_REF] Hagood | Modeling of a piezoelectric rotary ultrasonic motor[END_REF]

kθc = atan w β wα (5)
This can be summarized by figure 2; in this figure, we define W , a time varying vector, which co-ordinates are wα and w β . And in the same way, vα and v β are projections of a time varying vector V on axis α and β. In this figure, a (d, q) frame appears, but will be explained further.

Moreover, as it can be seen in figure [START_REF] Maas | Model based control for ultrasonic motors[END_REF], the contact point speed which propels the rotor, has two components:

a b q ✁ q ✁ a b ✂ ✄ ☎ ✆ a b W W ✝ V V
Fig. 2. Voltage vector V and Wave vector W in (α, β) frame.

• V N id along the axis of the motor, lifts up the rotor to enable the propagation of the travelling wave. During steady state, this speed is null.

• V ′ T id in the tangential direction , from which the rotational speed of the rotor ω id derives. A kinetic study of the vibrating stator shows then that the expression of those speeds can be written using ẇαβ and R(kθc) once again:

V N id V ′ T id = R(-kθc) ẇβ ẇα (6)
The rotational speed of the ideal rotor ω id and V ′ T id is given by equation ( 7) [START_REF] Sashida | An introduction to Ultrasonic Motors[END_REF]:

ω id = k h b 2 V ′ T id (7)
However, the speed ω id does not actually correspond to the rotor speed, because the contact conditions observed at the statorrotor interface are very different from those of the ideal contact conditions: in fact, because of friction, the actual rotational speed ω of the rotor decreases with the load torque T . Several models ( [START_REF] Maas | Simulation model for ultrasonic motors powered by resonant converters[END_REF][7] [START_REF] Maeno | Finite element analysis of the rotor/stator contact on e ring-type ultrasonic motor[END_REF]) exist to describe this behavior, but they are too complex to be taken into account in a control scheme. In order to achieve a straightforward modelling which may be useful for actuator control, a linear modelling for the global behavior of the motor can be proposed based on experimental results. This is the third energy conversion, given by equation ( 8) and [START_REF] Maas | Optimized drive control for inverterfed ultrasonic motors[END_REF].

T = f0 (ω id -ω) (8)
FN = KN (V N id -VN ) (9)
Of course, this process is very non linear, and the value of the parameters f0 and KN should vary as a function of the operating point. However, we find here the key variables governing the evolution of the motor torque: the amplitude of the travelling wave, because ω id derives using it, and the rotational speed of the rotor itself. This roughly outlines the torque estimator described in [START_REF] Maas | Optimized drive control for inverterfed ultrasonic motors[END_REF] from the speed and the stator's deformation measurements.

In addition, the dynamic of the rotor, along the two degrees of freedom, leads to lay down two other equations:

mr dV N id dt = FN -Fτ (10)
Jr dω dt = T -Tr (11)
Where:

• mr and Jr are respectively the mass and rotational inertia of the rotor,

• Tr is the external load torque applied on the rotor,

• Fτ is the normal force pressing the rotor against the stator. Finally, all these equations yield a modeling which can be represented by way of a Causal Ordering Graph as shown in figure 3. In this diagram we see the internal variables of the system, the ellipses represent the equations using integral relations (single arrow) or independent time relations (double arrow) only. The relations named Rx are refereed to the equation number (x) in the paper. This method is helpful in deducing the control scheme all it requires is a simple inversion of the graph. The graph in figure ( 3) is divided into four parts, each one represents one energy conversion process or the dynamic of the rotor. Moreover, it shows the couplings of the motors:

• the two independent voltage lines α and β are coupled via the rotational matrix,

• the two external forces FN and F ′ T are mixed and Frα and F rβ act on the travelling wave. These couplings make global comprehension of the system difficult. This is why in the next section we try to remove them by taking advantage of the experience of the modelling of the electromagnetic AC machines.

2) Causal modelling in a rotating reference frame: the two rotational matrices, on the stationary waves (eq. 6) and on the forces (eq. 3), lead to setting down two new variables V d and Vq. As it is for the electromagnetic machines where current or voltages are expressed in a rotating reference frame fixed to the rotating electromagnetic field, V d and Vq represent the supply voltages in a rotating reference frame fixed to the rotating travelling wave. So, this leads us to set down:

vα v β = R(kθc) V d Vq (12)
So, according to the figure 2, V d and Vq are the coordinates of the voltage vector V in the rotating frame (d, q). Using V d and Vq, equations 2, 3, 6, are revised to produce two new equations respectively on d axis and q axis respectively, where the couplings are removed:

m V ′ N id + dsV ′ N id + (c -mk( θc) 2 ) V ′ N id dt = N V d -F ′ N (13) 2m V ′ T id + dsV ′ T id = N Vq -F ′ T (14)
In fact, in equation ( 14), the variables of the tangential axis appear only on the q axis, though equation (13) shows that the normal variables only appear on the d axis. This equation is of second order type, and completely describes the resonant behavior of the motor.

Figure [START_REF] Giraud | Causal modeling and identification of a travelling wave ultrasonic motor[END_REF] shows the Causal Ordering Graph of the actuator in the rotating reference frame, reduced to the q axis as it deals with the torque.

Thanks to this graph's inversion, it is easy to deduce the torque control of the piezo-actuator [START_REF] Giraud | Analysis and phase control of a piezo electric travelling wave ultrasonic motor for haptic stick application[END_REF]; however, in this paper, we focus our attention on the torque estimator, which is what the next section deals with.

B. The torque estimator

The principle of a torque estimator is outlined by inverting the Causal ordering graph of the figure 4. There are thus three different paths, as summarized figure 5:

• 1-from relation [START_REF] Ghouty | Hybrid modeling of a traveling wave piezo electric motor[END_REF]. This is difficult because both the load inertia and the load torque Tr should be known,

• 2-from relation [START_REF] Maeno | Finite element analysis of the rotor/stator contact on e ring-type ultrasonic motor[END_REF] and the measurements of both ω and ω id as proposed by [START_REF] Maas | Optimized drive control for inverterfed ultrasonic motors[END_REF]; this estimator needs the identification of the parameter f0,

• 3-from relation ( 14) and the measurements of Vq and V ′ T id . In order to avoid a speed sensor, the torque estimator will rely on equation 14. However, inverting this equation is not straightforward, because of the derivative term 2m V ′ T id , which should be approximated. A closed loop estimator could then be proposed, whose output would converge toward the solution. But considering a high performing estimator is required especially in the case of steady state operations -or slow variations of V ′ T id , the problematic term 2m V ′ T id is deleted in equation ( 14):

R14 R1 R7 R4 R8 R11 Vq Tr VTid T F'T Ν F k h b² ωid ω R1 q Sol. 1 Sol. 2 Sol. 3 Measurement required
dsV ′ T id = N Vq -F ′ T (15)
Finally, using equations 4 in 15 leads to

T = b 2 kh N Vq -dsV ′ T id (16)
Achieving the torque estimator.

III. EXPERIMENTAL SETUP

A. The experimental test bench 1) Measuring Vq: The TWUM used during the test is a Shinsei USR30 [START_REF] Shinsei | Internet web site[END_REF]. This motor has a 30mm diameter and can provide a rated torque of 0.1N m for a rated speed of 200rev/min. It is a commercial product and thus is readily available. On the stator of this motor, an extra electrode is glued to the ring of piezo material. Due to the direct piezo-electrical conversion, the voltage measured on this sensor, named vEA, is directly proportional to the deformation of the stator at this point. Unfortunately, this deformation is a combination of wα and w β : thus it is difficult to build θc from the equation ( 5) because wα and w β are not measured independently. In that case, measuring Vq from Vα, V β and ( 12) is impossible. This leads us to set down the assumption that the amplitude of the two stationary waves are sinusoidal functions of time in quadrature, so:

wα(t) = W cos(2πf t) w β (t) = W sin(2πf t) (17)
With f the frequency of the supply voltages and W the travelling wave's amplitude.

In order for this assumption to be fulfilled the construction of the must be sufficiently careful as to ensure a good positioning of the stator's electrodes on the piezo ring, thus avoiding dissymmetry between the two phases. However, this condition is often achieved in the Shinsei USR30 when it is supplied by two sinusoidal voltages in quadrature with the same amplitude:

vα(t) = V cos(2πf t + Ψ) v β (t) = V sin(2πf t + Ψ) (18)
with V the amplitude of supply voltages, and Ψ, the phase difference between vα(t) and wα(t). On the experimental test bench, V is kept constant, though f varies.

On the one hand, the equations ( 5) and ( 17) yield to:

θc = atan W sin(2πf t) W cos(2πf t) = 2πf t (19)
On the other hand, if we consider the location of the sensor, vEA is in fact proportional to K(wα + w β ), with K the conversion factor of the auxiliary electrode. This consideration yields to:

vEA = K(W sin(2πf t) + W cos(2πf t)) = K √ 2 2 W cos(2πf t + π 4 ) (20)
In conclusion, from equations (19), ( 12) and (20), θc is equal to the instantaneous phase of the sensor's voltage vEA minus π 4 . The considerations of equations ( 19) and ( 12) yields:

V d Vq = R(-2πf t) V cos(2πf t + Ψ) V sin(2πf t + Ψ) = V cos(Ψ) V sin(Ψ) (21)
So, Vq may be deduced by:

Vq = V sin(Ψ) (22)
and Ψ can be measured from the phase difference between vα and vEA, which is experimentally achieved thanks to an integrated chip.

2) Measuring V ′ T id : to complete the identification of T from equation ( 16), it is necessary to know V ′ T id , but however this is an internal variable. It may be calculated thanks to equation [START_REF] Maas | Simulation model for ultrasonic motors powered by resonant converters[END_REF], and from the measurement of ẇα and ẇβ . Once again, to reach these values, we use the previous assumption which leads to equation ( 17)

ẇα(t) = Ẇ cos(2πf t) + (2πf )W sin(2πf t) ẇβ (t) = Ẇ sin(2πf t) + (2πf )W cos(2πf t) (23)
Of course, in order to obtain the equation ( 23) we need to consider that the excitation frequency f is kept constant. This is not rigorously correct during transitory operations. But the stator of a TWUM is a mechanical resonator, f is chosen very close to its resonant frequency and thus considered constant.

Using equations ( 23), ( 19) and the rotational matrix (6) now yield:

V ′ T id = -ẇαsin(2πf t) + ẇβ cos(2πf t) = -Ẇ cos(2πf t) + (2πf )W sin(2πf t) sin(2πf t) + Ẇ sin(2πf t) + (2πf )W cos(2πf t) cos(2πf t) = (2πf )W (sin 2 (2πf t) + cos 2 (2πf t)) = (2πf )W (24)
Therefore, V ′ T id is proportional to the travelling wave's amplitude. If we now take equation 20 into account, W can be deduced by extracting from vEA its own amplitude named VEA. V ′ T id is given by:

V ′ T id = (2πf ) 1 K √ 2 2 VEA ( 25
)
Since the different state variables of the torque estimator can be measured, the next section deals with the identification of the parameters appearing in the equation 16.

B. Measurement of the stator parameters

Several methods have been proposed to identify stator parameters [START_REF] Ghouty | Hybrid modeling of a traveling wave piezo electric motor[END_REF]. Most of them rely on the electrical equivalent circuit, and the parameters are found by fitting the curve of the variation in the frequency of the current into a phase with a 2 nd order type equation. However, this approach has several drawbacks among which there is the difficulty, firstly of precisely detecting the stator's resonance -because a small variation of the frequency leads to a greater variation of the current when approaching the resonance frequency -and secondly of carrying out an identification of the effect of the tangential forces. This is why we proposed an other method [START_REF] Giraud | Causal modeling and identification of a travelling wave ultrasonic motor[END_REF]. This approach relies on a Vq control detailed in [START_REF] Giraud | Analysis and phase control of a piezo electric travelling wave ultrasonic motor for haptic stick application[END_REF] which allows easy characterisation of the resonant frequency and avoidance of couplings between the normal and tangential axes. The identification process is divided into two steps:

• identification of ds • identification of k h b 2
1) Identification of ds: For an unloaded motor which has reached its steady state, the travelling wave's amplitude W derives from the equation 16 where T is set to 0. In these conditions:

ds2πf W = N Vq (26)
Due to the experimental test bench, and equation ( 22), Ψ is in fact tuned instead of Vq, and this equation is transformed into:

ds2πf W = N V sin(Ψ) (27)
Identifying ds can then be achieved by keeping V constant; plotting the evolution of W against Ψ should result in a sinusoidal shape, whose crest gives the ratio N ds2πf . However, this method is quite sensitive to non linearities which induce variations of ds with W [START_REF] Maas | Averaged model of inverter-fed ultrasonic motors[END_REF]. This is why, instead of V , W is kept constant. So, for each value of Ψ, V is tuned in order to have W equal to a value we fixed before.

A closed loop including a Voltage Controlled oscillator controls the voltage frequency f so as to have the required value for Ψ. This angle is measured by using a phase detector (figure 6); the controlled scheme has been previously described in [START_REF] Giraud | Analysis and phase control of a piezo electric travelling wave ultrasonic motor for haptic stick application[END_REF].

We then plot the evolution of W V against Ψ. The experimental trials, for each rotational direction and several wave amplitude, are given in figure 7. Table I summarizes the values of ds identified from this method. It appears that ds depends more on the operating wave amplitude W and less on the rotational direction. In order to take into account this non linearity in the torque estimator, we are led to representing in figure 8 the evolution of ds in W , or rather in 1 W because this results in a linear shape.

Equation 28 then fits this curve:

ds = -ds0 + ρ W ( 28
)
The values of ds0 and ρ are obtained from the figure 8: So, taking into account the non-linearity of ds, the torque estimator can be achieved by merging the equations 16, 23 and 28:

(-ds0 + ρ W )2πf W = N Vq -k h b 2 T -ds02πf W + 2πf ρ = N Vq -k h b 2 T ds02πf W = 2πf ρ + k h b 2 T -N Vq (29)
In this equation, the global parameter k h b 2 is still to be calculated. This next section deals with this calculation.

2) Identification of k h b 2 : The identification of this parameter derives once again from equation ( 16). But now, T will vary and Vq is adjusted to keep W constant. So, the evolution of Vq in T should be a straight line whose slope is k h N b 2 . We have represented in figure 9 the result curves for several values of W and both rotational directions.

The results show that the experimental trials and the modelling are consistent, since the curves are nearly straight. However, the slope of these lines depends on the level of the stator's deformation W and also on the direction of rotation. These variations reveal nonlinearities in the modelling; in fact, they express the limits of this modelling, and more specifically the limit imposed by the assumption that there is only one contact point. This is why, an average value of k h b 2 is then calculated from the trials, and the slope of the curves drawn on figure 9.

So, the averaged slope of all those lines is 480N mV -1 yielding:

k h b 2 = 67.2m -1
The value of this parameter has to be compared to the one calculated from an analytical approach. In fact, considering the geometrical dimensions of the motor under study, kh/b can be evaluated to 107m-1. The gap between these two estimated values can be first explained by the difficulty to estimate the parameter h, that characterizes the neutral plane location. Secondly, this result may be induced by the modelling assumptions, in particular, the punctual contact one. Nevertheless, the approach leads to a modelling structure that we are able to identify and which agrees with the torque characteristics in a given validity domain. n

C. Experimental tries

Different trials have been carried out to illustrate the performance of this torque estimator. The first measurements shown in figure 10 depict the evolution of the estimated load torque and the measured 0 5 0 1 0 0 1 5 0 -0,1 -0,075 -0,05 -0,025 0 0,025 0,05 0,075 0,1

Vq(V) T(Nm) (1) (2) (3) (1) (2) (3)
W=0.65µm W=0.82µm W=1.16µm

CW -1 5 0 -1 0 0 -5 0 0 -0,1 -0,075 -0,05 -0,025 0 0,025 0,05 0,075 0,1 load torque for both rotational directions. During this trial, the stator wave's amplitude was kept constant while the load torque T was time varying.

T(Nm) Vq(V) (1) (2) (3) (1) (2) (3)
The second measurement of figure 11 shows the evolution of the measured load torque compared to its estimation when W was time varying. Both rotational directions are depicted, though motor operation (T > 0 and ω > 0, or T < 0 and ω < 0) and brake operation (T < 0 and ω > 0, or T > 0 and ω < 0) were here experienced.

Considering that the torque estimator is quite straightforward, the estimated load torque is found to be consistent with the measurements of the experimental trials of figures 11 and 10.

Estimation errors can however be measured on figure 11, revealing that the estimation depends on W . More obviously, errors are amplified for braking operations while they are acceptable for other conditions. This is mainly due to the averaged value of k h b 2 we had to use in equation 29 and which is a tradeoff between all the values identified for all the operating points of the chapter III-B. Performances of the estimator could then be improved by taking into account the variation of k h b 2 in W , but this would be achieved at the expense of complexity. Despite this problem, applications can take advantage of this torque estimator, and this is dealt with in the next section.

IV. APPLICATION: LIMITING THE TIGHTENING FORCE OF TWUM ACTUATED CLAW

A. Working principle of a claw

Because of torque vs speed characteristics, Travelling Wave Ultrasonic Motors are very suitable for applications which need a low speed but a high torque. In this field of applications, the electrically actuated claws are devices which are widely used, for example at the end of a robot arm to allow object prehension.

Different controls of the claw can be achieved. The first control type is a position control of the actuator of the claw between an opened position till the closed position. This is well suited for objects which all have the same dimensions. But if variable objects of random sizes have to be picked up, the larger objects are held extremely tight, while the smallest ones are not even caught (figure 12). This is why, instead of having a claw position control, a force limitation is preferred. In this operating mode, the claw closes up at a constant speed until the force at the claw 's tip reaches a limited value.

This last control method is quite easy to achieve with electromagnetic motors, because the torque is proportional to the supply current which is often controlled, and can thus be limited. But those motors have high speed characteristics, so they need a speed reducer in order to adapt the output speed of the shaft to the desired speed of the claw when closing. This increases the system's bulk and its cost.

In this application, a TWUM is interesting, because it can be directly mounted on the claw since its low-speed characteristic is well adapted to the system. The torque estimator developed in this article is implemented to ensure the limitation of the tightening force.

Open Closed

Open Closed Fig. 12. Effect of a controlled position claw when the object is sufficiently big, and in the case where it is too small.

The algorithm for the closing will be as follows:

• 1 wait for the starting signal • 2 close with a constant speed. During this step, W is kept constant, and the torque is estimated on line

• 3 stop the motor (set W to 0) when the torque estimator has reached a value previously fixed.

The next section presents the experimental runs of the claw actuated by a TWUM, with its torque limitation. from now on the limited value of the estimated torque will be referred to as T lim

B. Experimental runs

The claw fixed on the TWUM's shaft is depicted in figure 13. Several objects with different sizes can be positioned, and for each run, the internal variable of the motor can be recorded. Thus, figure 14 depicts the evolution of V q (measured with ψ), the motor rotational angle, the measured and estimated torque and finally the wave amplitude W -which is also controlled. These curves are obtained with T lim = 0.05N m. The cycle begins at t = 0 with the starting signal; a wave amplitude reference of about 1µm is ordered, and the claw closes. The estimated torque as well as the measured one are initially null, but at t = 50ms they both start to increase because the claw is in contact with the object to be grasped. Consequently, Vq increases in accordance equation ( 29), since W is kept constant. At t = 85ms, the estimated torque reaches the limitation: a reference value of W of 0µm is ordered which is obtained in almost 10ms. During this short time, the torque actually applied to the claw can be a little larger due to the time the wave amplitude needs to become null. Then the torque estimator is switched off because the very low wave amplitude leads to erroneous estimated torque values. So, the claw stops its grasping action when it meets an object. This contact is detected through an increasing in the motor's torque, which is estimated on line. Of course, the force with which the object is taken depends on the value of T lim . In figure 15, we have drawn the evolution of the estimated and the measured torque for several values of T lim . This run shows that it is possible to limit the tightening force of the claw by tuning the value of T lim . 1We must emphasize here that the control presented is not a torque regulation, but a torque limitation. In fact, when the motor is stopped, the force on the claw 's tips can change due to external conditions. A torque regulation implies that the torque keeps a constant value, even at null speed, this could be achieved with an improvement of the torque estimator at low wave amplitude. However, a limitation is enough in applications in which the objects are inanimate.

Motor Torque sensor Claw Pos. Encoder

V. CONCLUSION

In this article, we have presented a method to achieve a torque estimator for a TWUM. This estimator relies on the causal modelling of the motor which helps to deduce the estimator's equation. Compared to the torque estimators which rely on the torque/speed characteristics of the motor, we emphasize that the estimator presented here doesn't need the measurement of the rotational speed of the motor, this allows to removal of the speed sensor, and the measurements have confirmed the global structure of the modelling.

Experimental trials have been carried out to check the performances of the estimator along a wide range of operations, for time varying torque and for time varying speed of the motor. Results have been consistent with the measurements, but we highlighted that any non-linearities could reduce confidence in the estimation.

Although all the questions related to the precision of the estimated torque -especially for operations where W is varying -are still open, the estimator is tested on an application in the final section.

Fig. 1 .

 1 Fig. 1. View of the stator; under punctual contact assumption, the rotor is propelled by one point only along a wavelength.

Fig. 3 .

 3 Fig. 3. Causal ordering graph of the TWUM .

Fig. 4 .

 4 Fig. 4. Causal ordering graph of the TWUM in the rotating reference frame; axis q only.

Fig. 5 .

 5 Fig. 5. The three different ways to estimate the torque T can be deduced from the inversion of the COG.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Closed loop for the control of Ψ.

Fig. 8 .

 8 Fig. 8. Evolution of ds as a function of 1 W for both rotational direction and fitting.

CCWFig. 9 .

 9 Fig. 9. Vq as a function of T and comparison a straight line.

Fig. 10 .Fig. 11 .

 1011 Fig.10. Comparison between the actual load torque and the estimator's output for a constant travelling wave's amplitude.

Fig. 13 .

 13 Fig. 13. The experimental Setup.

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Internal variables of the motor for a closing sequence of a TWUM actuated claw , with limitation of the torque.

Frédéric

  Giraud was born in France in 1973. He is graduated from Ecole Normale Supérieure de Cachan, France in 1996 and received the B.S. degree in electrical engineering from Paris-XI University, Orsay, France in 1995, the M.S. Degree in electrical engineering in 1997 from the Institut National Polytechnique de Toulouse,Toulouse, France, and the phD degree from the Lille University, in 2002. He's a member of the electrical engineering and power electronics laboratory of Lille (L2EP) as an Assistant Professeur. His research deals with the modelling and the control of standing wave and travelling wave piezo-electric actuators, for positioning and force feedback applications. Betty Semail was born in Arras, France, on February 27, 1964. She received her Ph. D degree in 1990 from University of Paris XI, Orsay and habilitation degree in 1997 from University of science and technologies of Lille. Since 1990 she has been assistant professor in Ecole Centrale of Lille and she is now professor in university of Lille 1. She is member of the electrical engineering and power electronics laboratory of Lille (L2EP) and responsible of the research axis upon control of electrical systems. She has been working motors and her main field of interest now deals with the modelling and control of piezo-electric actuators, for positioning and force feedback applications.

The measured torque is actually about 0.04N m above the estimated one. This gap is due to the inertia of the torque sensor which creates a torque when the motor is breaking(Jt = 10 -5 kgm

). Better results should be obtained for a claw directly mounted on the motor's shaft.

We use a claw actuated by a TWUM. Several trials have shown that the device runs well, and we have shown that the tightening force of the claw can be tuned by adjusting the limited value of the estimated torque.

Future work should increase the precision of this torque estimator, more specifically at low rotational speeds, to allow sensorless torque regulators. This implies the control of voltages phases leading to the loss of quadrature. A new modelling should be introduced to take into account this voltage supply.

VI. ACKNOWLEDGMENT

This work has been carried out within the framework of the INRIA Alcove project and is supported by the IRCICA (Institut de Recherche sur les Composants logiciels et matériels pour l'-Information et la Communication Avancée).