External mechanical disturbances compensation with a passive differential measurement principle in nanoforce sensing using diamagnetic levitation

Sensors and Actuators A 238 (2016) 266–275

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

External mechanical disturbances compensation with a passive differential measurement principle in nanoforce sensing using diamagnetic levitation Margot Billot, Emmanuel Piat ∗∗ , Joël Abadie ∗ , Joël Agnus, Philippe Stempflé Femto-ST Institute, Univ. Bourgogne Franche-Comté, Univ. de Franche-Comté/CNRS/ENSMM/UTBM, Besanc¸on, France

a r t i c l e

i n f o

Article history: Received 28 July 2015 Received in revised form 27 October 2015 Accepted 23 November 2015 Available online 3 January 2016 Keywords: Micro and nano force sensor Magnetic spring Diamagnetic levitation Kalman filtering Unknown input estimation Passive disturbances compensation

a b s t r a c t Nanoforce sensors using passive magnetic springs associated to a macroscopic seismic mass are known to be a possible alternative to force sensors based on elastic microstructures like Atomic Force Microscopes if the nanoforces that have to be measured are characterized by a bandwidth limited to a few Hertz. The estimation of the unknown force applied to the seismic mass is based on the deconvolution of the noisy measurement of the mass displacement which has an under-damped dynamic. Despite their high performances in terms of linearity, resolution and measurement range, such force sensors are extremely sensitive to low frequency environmental mechanical disturbances, like the angular variations of the anti-vibration table supporting the device or the residual seismic vibrations that are not filtered by the table. They are also sensitive to the temperature evolution of the ambient air. The evaluation, modeling and compensation of such environmental disturbances have to be specifically studied in the context of magnetic springs associated to a macroscopic seismic mass because of their important negative effects in terms of low frequency drifts and oscillatory disturbances. This article presents an estimation and a passive compensation strategy of the low frequency and non-stationary mechanical disturbances that is based on a differential principle. This approach is applied to a nanoforce sensor based on diamagnetic levitation developed in the last decade. It does not necessitate to add new types of sensors in the measurement chain such as very high resolution and low frequency inclinometers or accelerometers in order to estimate the mechanical disturbances. In term of performances, the force estimation error remains in the nanonewton level over periods of time of several minutes when external temperature remains constant. © 2015 Elsevier B.V. All rights reserved.

1. Addressed problem 1.1. Introduction Achieving progresses in the measurement of micro and nanoforce remains a necessity in domains like material sciences at micro and nano scales or micro and nanorobotics. For instance, in the first case, it is often necessary to characterize, in terms of adhesion and friction, functionalized and/or structured surfaces at micro and nanoscales. In the second case, performances of micro and nano-objects manipulation during micro-assembling should

∗∗ Corresponding author. ∗ Principal corresponding author. E-mail addresses: [email protected] (M. Billot), [email protected] (E. Piat), [email protected] (J. Abadie), [email protected] (J. Agnus), philippe.stempfl[email protected] (P. Stempflé). http://dx.doi.org/10.1016/j.sna.2015.11.032 0924-4247/© 2015 Elsevier B.V. All rights reserved.

be improved if the forces applied on the objects are directly taken into account in the control loops driving the micro-actuators. Such progresses concern both force sensors embedded in microrobotics devices and force sensors embedded in dedicated measurement platforms. Whatever is their final use, all micro or nanoforce sensors use a transducer to convert the force into a measurable effect. In many force sensors, this effect is related to the displacement x of a force–displacement transducer. Most classical designs are based on elastic microstructures with one or several degrees of freedom which have a high resonant frequency: AFM based microforce sensors [1,2], piezoresistive microforce sensors [3,4], capacitive microforce sensors [5], piezoelectric microforce sensors [6]. Another important challenge that leads to original development of micro and nano force sensors is present in the field of metrology of small forces (below the mN). The strategies used by National Metrology Institutes (NMIs) to try to establish traceable measurements of small forces are varied. The simplest consists in calibrating the sensor using a dead weigh which is possible when the

M. Billot et al. / Sensors and Actuators A 238 (2016) 266–275

measurement direction is vertical. This restrictive approach has the major disadvantage that the uncertainty associated with the weight, in mass metrology, increases gradually as the latter decreases [9]. In terms of alternatives, many micro and nanoforce sensors are based on elastic microstructures. Some works have been done by NMIs to develop traceable low forces sensors using piezoresistive MEMS [10] or capacitive cells (comb-drives) [9]. Another measurement principle is the mass comparators. Schematically, a comparison of mass acts via an elastic structure (micro or macro) which is moved along an axis to mechanically load an artefact with an unknown force. This artefact is itself in contact with an electromagnetic balance which generates an opposite electromagnetic force adjusted by a current. This current is controlled in order to generate a controlled displacement of the loaded artefact. When a null displacement is reached, the electromagnetic force is then the opposite of the unknown loading force. The electromagnetic balance is previously calibrated with dead weights, which allows to know the electromagnetic force versus the current and thus the force applied to the loaded artefact. In such designs, the electromagnetic balance is an active force sensor. The measured output value that has to be controlled is a displacement, the associated reference is a null displacement, the controller output is a current and the known generated force is a known function of the current. Different mass comparators have been developed to measure vertical forces using masses of 1200 g [11], 210 g [12], 41 g [13] (PTB, Germany) and 5 g [14] (KRISS, Korea). These comparators reach different working ranges (from a few hundred nanoNewton to Newton) and resolutions (minimum 1 nN) [10]. An original electromagnetic balance using a reference death weight of 1000 g is also developed to measure the horizontal forces that exceed the micronewton [15]. Controlling the force orientation during micronano force measurement is particularly difficult. Thus, if a sensor is designed to measure the force in only one direction, an error in the orientation of the applied force leads to a measurement error. This error can be either estimated or not. If it is estimated, the force measurement can be corrected. This is for instance the case with the previously referenced mass comparator which has a tilting sensitivity. Tilting angle must be inferior to 0.1 ␮rad to guarantee an 1 ␮N accuracy. This tilting error is measured with a high-resolution inclinometer. If the error is not corrected it should be bounded and included in the uncertainty calculation. For instance, the uncertainty associated to the vertical force sensor described in [9] needs a cosine correction factor as the force applied is never strictly vertical. The temperature is also known to be a source of disturbance in the field of metrology and the force sensors developed by NMIs are always controlled in temperature (generally with 0.01◦ accuracy) over long periods of time [13,16] in order to establish a traceability of micro-nano-force to International System of Units standards. To conclude this short review of micro and nanoforce sensors, all sensors are subject to external disturbances that affect their dynamic and that should be estimated and compensated if their bandwidth and amplitude corrupt the force that has to be estimated. A possible categorization of such environmental disturbances and their domains of influence can be found in [7,8]. Their estimation and active or passive compensation in micro and nanoforce sensors remains an open research problem.

of the macroscopic size of the force–displacement transducer, these force sensors, developed in the last decade, are commonly used in force measurement macroscopic platforms with one [17] or several degrees of freedom [18]. Indeed, they have been used for instance with success to characterize the mechanical behavior of human ovocytes whose stiffness is commonly below 0.01 N/m [19]. Contrary to sensors based on micro elastic structures, very long range of force amplitude can be measured without any risk of breaking the force–displacement transducer, linearity also remains excellent over classically ±1 mm displacements of the transducer and a nanonewton resolution can be reach thanks to the low stiffness of the sensor that can easily be adjusted by varying the magnets configuration of the device. Moreover, the mechanical part manufacturing is a low cost processing and does not necessitate any complex machining. As the transducer mass can be measured with a micro balance, its second-order behavior can be identified (including the stiffness corresponding to the steady state behavior) using a zero-input response [17]. These is no parametric indetermination during calibration like for micro-nano force sensors whose mass is hard to determine. When the mass is unknown, the calibration requires thermal noise measurement or indirect methods with additional hypotheses that are generally difficult to verify. If only one Degree of Freedom is considered and under the assumption that the environmental disturbances are completely negligible, their force–displacement dynamic is a second-order under-damped transfer because of the inertia of the macroscopic mass, the very small viscous friction applied on the mass and the absence of dry friction. When an adequate design of the magnetic spring is used, if a force Fx (t) is applied along an axis x , the linear displacement x(t) obtained corresponds to the following dynamic in a global inertial reference frame R0 attached to the laboratory: x m¨x + Kvx x˙ + Km x = Fx

(1)

where m, Kv and Km are the mass, the viscous damping coefficient of the mass and the magnetic stiffness of the transducer. Thus, classical open-loop steady-state equation: x F x = Km x,

x Km >0

(2)

cannot be used to determine the force Fx (t) because the evolution of the successive derivatives of the mass displacement x is absolutely not negligible over long periods of time. That is why, the force estimation becomes in this context an open-loop deconvolution problem of a noisy output signal whose principle is shown in Fig. 1. The force F(t) applied on the transducer is corrupted by environmental disturbances. The resulting displacement dynamic is measured with a sensor corrupted by measurement noises and gives the measurement signal xkm , sampled at a period Ts . Knowing this signal, it is necessary to compute at each sampling time the estimation Fˆk of the real force F(kTs ) applied on the transducer, thanks to a real-time deconvolution.

1.2. Nanoforce sensors using magnetic springs Nanoforce sensors with low resonant frequency that are based on low stiffness magnetic springs associated to a macroscopic seismic mass (from milligramme to gramme) are a possible alternative to classical designs based on elastic microstructures to measure low frequency forces or quasi static forces. The unknown force is applied on a macroscopic seismic mass connected to a passive magnetic spring and induces a displacement response of the mass. Because

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Fig. 1. Force estimation using a deconvolution approach.

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Fig. 2. Macroscopic seismic mass used as force–displacement transducer (maglevtube).

This problem has been addressed in the past for a nanoforce sensor based on diamagnetic levitation [17,20,21] but without taking into account the environmental disturbances acting on the transducer (see Fig. 1). The seismic mass, called maglevtube, is a ten centimeter-long capillary tube stuck on two small magnets M2 (see Fig. 2). It is levitating passively around a given equilibrium state thanks to the diamagnetic levitation principle. Typical maglevtube mass is around 20 mg. As illustrated in Fig. 3, damping ratio , overshoot and 5% settling time of the displacement response xkm of the maglevtube to a force step are around 6 × 10−3 , x is between 97% and 20 s, respectively. Typical magnetic stiffness Km 0.005 N/m and 0.03 N/m. Resonant frequency is around 4 Hz. The measurement xkm of x is done with a confocal chromatic sensor. It is aimed at the deflector located at the rear of the maglevtube (see Fig. 4). Coils located at the rear of the sensor are only used during the sensor calibration to generate a zero input response. The a priori information on F(t) mentioned in Fig. 1 consists in modeling the deterministic second-order dynamic of the maglevtube and merging it with an uncertain modeling of the input force in order to obtain an extended state-space model that will be used to estimate the input force with a time-varying discrete Kalman filter. The input of the Kalman filter (that corresponds to the deconvolution block in Fig. 1) is the noisy measurement xkm of the maglevtube displacement xk : xkm = xk + nk ,

E[n2k ] = R

(3)

The zero-mean white Gaussian noise nk with the known variance R is due to the confocal chromatic sensor. The output of the Kalman filter is an extended state estimation Xˆ ke that include the force estimation Fˆk at each sampled time tk = kTs :



Xˆ ke = xˆ k

xˆ˙ k

Fˆk

T

(4)

Fig. 4. Force sensor prototype.

developed in [21]. Fig. 5 shows the measurements performed with the maglevtube for a test of adhesion breaking between two microobjects. At the beginning of this test, the force is gradually increased in order to separate the objects. At t = 3s, the contact is lost and the real interaction force is quasi instantaneously decreasing to zero. The force estimation Fˆk using the time-varying Kalman filter is compared to the direct and simplest way to estimate the force given by Eq. (2): x m Fˆsimple = Km xk

(5)

It can be observed that, using the time-varying Kalman filter, a high performance gain can be obtained in terms of dynamic improvement and noise reduction of the force estimation.

1.3. Environmental disturbances acting on the mass

and its associated 3 × 3 covariance matrix Pk conditioned by all the measurements done. The complete estimation process is fully

The previous estimation process was done under the assumption that the environmental disturbances are totally negligible, which in practice is not the case.

Fig. 3. Example of a step response of the maglevtube.

Fig. 5. Experimental force measurement.

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1.3.1. Initialisation of the estimation process An initialization of the estimation process is done before each force measurement campaign: no force is applied on the seismic mass over a long period of time. The noisy and disturbed residual displacement of the seismic mass is recorded and an average displacement is computed over a window of one second (1000 measurements). This average value is then subtracted to the measurement of the displacement in order to obtain an estimation of the force that is close to zero. Obviously, this initialization process is tedious because it has to be repeated before each force measurement and it does not compensate the environmental disturbances during the measurement campaign. 1.3.2. Measurement of environmental disturbances After this initialization process, the measurement of the external force immediately begins and the environmental disturbances add over time an unknown additive error on the force estimation. Fig. 6 shows the force estimation Fˆk over a long period of time (5 min) if no external force is applied on the maglevtube tip (i.e. Fx (t) = 0). Thus a correct force estimation process should give an estimation equals to zero. The visible low frequency drift (15 nN in 5 min) and the variations with higher frequency of the estimation Fˆk in the nanonewton scale (see zoomed view) show clearly that some nonstationary environmental physical parameters have a significant impact on the dynamic of the force sensor and make the force estimation erroneous. The evaluation, modeling and compensation of such environmental disturbances have to be specifically studied in the context of magnetic springs associated to a macroscopic seismic mass because of their important negative effects in terms of low frequency drifts and oscillatory disturbances. To improve the estimation process a differential approach is developed in this article using a set of two force sensors based on diamagnetic levitation. The first one is used classically for force measurement, while the second one estimates the mechanical disturbances in order to reject them in the first one. Both sensors are placed side-by-side on an anti-vibration table in order to decrease the influence of the vibrations due to people working and walking near the experiment. Two main low frequency external mechanical disturbances have been identified and are modeled in Section 2. These two disturbances, that have a significant influence on the maglevtube dynamic, are the small angular variations of the anti-vibration table supporting the device and the low-frequency seismic vibrations of the ground that are insufficiently filtered by the anti-vibration table (denoted in Section 2, (t) and xa (t),

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respectively). The air displacement due to infra-sounds is supposed negligible in our quiet environment because the device is enclosed in a chamber. Section 3 explains how the identified environmental mechanical disturbances influence the deconvolution process. Section 4 presents the passive differential approach that is used on the force sensor based on diamagnetic levitation in order to compensate the external mechanical disturbances induced by angular variations (t) and vibration acceleration x¨ a (t). The advantage of such an approach is that it does not necessitate to add new types of sensors in the measurement chain like inclinometers or accelerometers with high resolution at low frequency. Section 5 presents simulated results and the final section deals with experimental results introducing thermal influence. 2. Maglevtube modeling with low-frequency mechanical disturbances Let G be the center of gravity of the maglevtube and x its position in the frame Rt (shown in Fig. 7) attached to the tabletop of the anti-vibration table (base of the sensor). Coordinate x is therefore the longitudinal displacement of the maglevtube (along er direction in Fig. 7) measured with the confocal chromatic sensor located on the tabletop. The value x is set to zero when the maglevtube is in steady state without any external excitation. The frame Rt is a noninertial reference frame because of the low frequency vibrations transmitted by the ground to the tabletop. These vibrations are only considered in the direction of the measurement. As shown in Fig. 7, let note (t) the rotation angle between R0 (attached to the laboratory) and Rt . The anti-vibration table oscillates with a very low frequency. Therefore the angular acceleration of Rt respect to R0 is ¨ ˙ = 0, (t) =0 closed to zero and is supposed to be negligible ((t) and (t) = ). Let note xa (t) the position of the table expressed in R0 which is an unmeasured quantity contrary to x(t). The fundamental principle of dynamic applied in the noninertial reference frame Rt (O , er , e , ez ) is given by: G = ma /R



t

F + Ftrans + Fcent + FCor + FEul

G is the transducer mass in which m is the maglevtube mass, a /R



t

 F is the sum of all the external forces acceleration along er, applied to the maglevtube and Ftrans , Fcent , FCor and FEul are the translation, centrifugal, Coriolis and Euler force, respectively, defined by:  0 Ftrans = −ma

(7)

Fcent = −m˙ ez ∧ (˙ ez ∧ O G)

(8)

FCor = −2m˙ ez ∧ vG/R

(9)

/Rt

t

Fig. 6. Force estimation versus time for an external force Fx (t) = 0 with environmental disturbances.

(6)

Fig. 7. Reference frames and coordinates.

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Fx (t) generates a dynamic of the seismic mass that is observed through its displacement x in the reference frame Rt attached to the tabletop. As this displacement is the output of the following second-order linear system: x m¨x + Kvx x˙ + Km x = Fx

(17)

a real-time linear deconvolution can be used to estimate the input Fx (t) with the noisy observed output xkm given by (3). Because both (1) and (17) have an equivalent structure, this deconvolution can be based on the same approach as for the estimation of the force Fx (t) mentioned in Section 1.1, substituting Fx (t) by Fx (t). Such approach used an uncertain a priori input model based on a stochastic Wiener process [20]: ˙ F(t) = ω(t) Fig. 8. Forces applied to the maglevtube.

FEul = −m¨ ez ∧ O G  0 with a

/Rt

(10)

the acceleration of Rt respect to R0 . In our case, as ¨ and

 and a  G = x¨ .er.  ˙ are assumed to be zero, Fcent = FCor = FEul = 0 /Rt Moreover,

⎡

 0 Ftrans = −ma

/Rt

−m x¨ a cos

⎤

⎢ ⎥ = ⎣ −m x¨ a sin ⎦ 0

(11)

/Rt

Then,



F = F + Fw + Fmag + Fvisc

(12)

 Fw , Fmag and Fvisc the unknown external force to be meawith F, sured, the weight, the return magnetic force (magnetic spring) and the viscous friction force (mainly due to the air friction against the rear deflector), respectively. Assuming that the unknown force and the tube displacement are collinear to er , that the latter displacement remains in the linear domain and that its speed is small, the forces are defined by:

⎡

Fx

F = ⎣ 0 0

⎤ ⎦

⎡

m g sin 

⎤

Fw = ⎣ −m g cos  ⎦

/Rt

Fmag = −Km x er ,

0

(13)

(14)

where g is the gravity acceleration, Km is the magnetic stiffness, Kv the viscous damping coefficient. All the non-zero forces applied to the maglevtube are represented in Fig. 8. The fundamental principle of the dynamics (6), in projection along er , becomes: m¨x + Kv x˙ + Km x = F x + m(g sin  − x¨ a cos )

(15)

Eq. (15) shows that the dynamic of the maglevtube in the reference frame Rt depends on the force Fx (t) and the disturbances  and x¨ a (t). For instance, with m equal to 20 mg, a tiny angle  equal to 5 ␮rad leads to a disturbance force m g sin equal to 1 nN because of the high value of the seismic mass. Therefore, this type of force sensor based on a macroscopic seismic mass is particularly sensitive to very small angular variation of the anti-vibration table. 3. Input estimation including the environmental disturbances Let Fx (t) be the unknown input in (15) that includes the force to be measured and the environmental disturbances:

Fx (t)

Fx = F x + m(g sin  − x¨ a cos )

F(t) in (18) is the modeled unknown input Fx (t). The process ω(t) is a stationary zero-mean infinite-variance white Gaussian stochastic process representing the fact that the evolution of the input derivative is not known accurately. The autocorrelation function ω,ω of this process is characterized by its power spectral density WF˙ :

∀ ∈ IR

ω,ω () = WF˙ ı()

(19)

WF˙ is a scalar parameter that has to be set by the end-user. It will influence the input estimation in a given way explained later. The extended state-space model combining the maglevtube dynamic (17) and the input model (18) is: X˙ e (t) = A X e (t) + Mω(t)

(20)

x(t) = CX e (t)

(21)



with X e (t) = x

⎡

0

x˙

F

1

0 1 m

⎢ 1 1 A=⎢ − x − x ⎣ Km Kv 0

0

T

and:

⎤

⎡ ⎤ 0 ⎥   ⎥ M=⎣ 0 ⎦ C= 1 0 0 ⎦

(22)

1

0

The discretization of (20) with a sampling period Ts and a zero-order hold (zoh) on ω(t) leads to:

/Rt

Fvisc = −Kv x˙ er

(18)

(16)

e Xk+1 = F Xke + ˝k

(23)

xk = C Xke

(24)

with F = Xke



eA Ts

and:

x˙k

Fk

= xk

T

,



˝k = ωkx

ωkx˙

ωkF

T

(25)

The zero-mean white Gaussian process noise ˝k characterizes the uncertainties added on xk , x˙ k and Fk due to both the uncertain input model used and the discretization of the maglevtube dynamic. Its covariance matrix Q depends on Ts and WF˙ and is equal to:





Q = E ˝k ˝kT = WF˙ (Ts )



Ts

(Ts ) =

T

eAt MMT eA t dt

(26) (27)

0

Like for the force estimation without mechanical disturbances in Section 1.1, the extended state Xke can be estimated with a time-varying discrete Kalman filter whose input is the noisy measurement xkm . Parameters needed by the Kalman filter are the matrices defined in Eq. (22), the covariance matrix Q and the scalar variance R of the noise provided by the confocal chromatic sensor. Knowing the estimation Xˆ ke provided by the filter, the input ˆ k of Fx (t) at each time tk = kTe is given by: estimation F ˆ k = CF Xˆ e F k

(28)

M. Billot et al. / Sensors and Actuators A 238 (2016) 266–275

with



CF = 0

0 1



271

(29)

The time-varying Kalman filter must be initialized before any force measurement (i.e. choice of Xˆ 0e and P0 ) and a numerical computation of the covariance matrix Q must be done each time the power spectral density WF˙ is changed by the end-user during the force estimation process. This scalar WF˙ enables the end-user to adjust a trade-off between the bandwidth of the sensing of Fx (t) and the resˆ k . This trade-off is the same than the one olution of the estimation F analyzed in [21] for the force estimation without any mechanical disturbances. 4. Force estimation by differential deconvolution The previous section gives a way to compute an estimation of Fx = F x + m(g sin  − x¨ a cos ).

(30) Fx

In order to obtain an estimation of it is necessary to compensate the effects of the disturbances induced by  and x¨ a in the previous equation. A possible solution could be the direct measurement of these disturbances using adequate sensors like inclinometers and accelerometers with sufficient resolution at low frequency. Instead of this, a differential principle is developed in this article. This principle avoids to use of new types of sensors in the measurement chain by using two distinct maglevtubes and enables to compensate all the common mechanical disturbance modes that are applied on these two maglevtubes (even if not modelized). The two force sensors are placed side-by-side on the same anti-vibration table. Therefore, because the table rigidity is supposed to be infinite at low frequency, it is assumed that the two sensors are subject at the same time to the same perturbations due to table movements ( and x¨ a (t)). The first maglevtube is dedicated to the measurement of the external force applied on its tip, whereas the second one is only excited by the perturbations. These perturbations can be estimated and used to compensate the ones applied on the first maglevtube. Fig. 9 shows the experimental configuration with the two sensors enclosed in a chamber and stand on the anti-vibration table. Using Eq. (15) from Section 2, the dynamic of the two maglevtubes are described by the following equations: m1 x¨ + Kv1 x˙ + Km1 x = F x + m1 (g sin − x¨ a cos)

(31)

Fig. 9. Differential force sensor prototype using two maglevtubes.

m2 x¨ + Kv2 x˙ + Km2 x = m2 (g sin − x¨ a cos)

(32)

where mi , Kvi and Kmi are the mass, the viscous damping coefficient and the magnetic stiffness of the maglevtube i, respectively (i = 1, 2). The diagram from Fig. 10 allows to estimate the force applied to the maglevtube 1 compensating the mechanical disturbances. The compensation is deduced from (31) and (32). Indeed, the timeˆ 1,k varying Kalman filter corresponding to the maglevtube 1 gives F which is an estimation at time tk = kTs of the right-hand-side in (31), i.e. F x + m1 (g sin − x¨ a cos).

(33)

Moreover, the time-varying Kalman filter corresponding to the ˆ 2,k of maglevtube 2 gives the estimation F m2 (g sin − x¨ a cos).

(34)

ˆ 2,k by m1 /m2 and to subTherefore, it is just necessary to multiply F ˆ ˆ tract the result from F1,k to obtain Fk the estimation of Fx at time kTs . This rejection approach of the perturbation due to  and x¨ a (t) using a mechanical differential design is quite similar in its principle to a common mode rejection usually used in electronics and, as previously mentioned, does not necessitate the measurement

Fig. 10. Diagram of differential principle.

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of  and x¨ a (t) with dedicated sensors with high resolution at low frequency. 5. Simulated results In order to illustrate the influence of the modeled disturbances  and x¨ a (t), the complete prototype presented in Section 4 is simulated using Matlab-Simulink with the following parameters: • zero force or force step Fx set to an amplitude of 10 nN; • quasi-static sinusoidal variation of the tabletop angle  with a frequency of 0.0005 Hz; • mono-modal variation of the table position xa (t) with a frequency of 0.8 Hz; • sampling rate Ts of 0.001 s; • R1 and R2 the variance of the displacement sensors equal to 4 × 10−16 m2 ; • Characteristics of the maglevtube 1: m1 = 21.40 mg, Km1 = 0.0120 N/m, K1 = 8.5026 × 10−6 ; • Characteristics of the maglevtube 2: m2 = 21.58 mg, Km2 = 0.0149 N/m, K2 = 1.1848 × 10−5 ; • Kalman filters adjustments: WF˙ = 1.10−18 N2 /Hz and WF˙ = 1.10−18 N2 /Hz.

1

2

Fig. 11 shows the result of this simulation for a zero force and a step force. Without disturbances rejection, one can see the global and very low frequency drift due to the disturbance m1 gsin on which is added the disturbance m1 x¨ a cos  at a higher frequency. Despite the basic mono-modal disturbances that are simulated, one can see that the force estimation has an aspect that reminds the real response shown in Fig. 6. These disturbances are correctly rejected using the differential deconvolution. Fig. 11 also illustrates that this rejection is totally independant of the measured force Fx (t) (here equal to 0 or 10 nN) because, as shown by Eq. (31), the mechanical disturbances considered are added to Fx (t) and the deconvolution process presented in Section 3 is linear. 6. Experimental measurements 6.1. Influence of temperature Fig. 12 shows experimental results obtained with the prototype presented in Section 3. It gives the force estimation Fˆk when there is no external force applied on the maglevtube tip, thus Fx (t) is equal to 0. Due to the additivity of the mechanical disturbances over the external force emphasized at the end of the previous section, this choice enables to study the real disturbances estimation and compensation only without having to consider the additive dynamic due to an unknown external real force also applied to the maglevtube. In this case, the dynamic of both maglevtubes are given by: m1 x¨ + Kv1 x˙ + Km1 x = m1 (g sin  − x¨ a cos )

(35)

m2 x¨ + Kv2 x˙ + Km2 x = m2 (g sin  − x¨ a cos )

(36)

It can be seen that, contrary to what was expected with the simulation results (Fig. 11), the zero force estimated in the nanonewton scale using the differential principle does not stay close to zero. This is due to the fact that the two deconvolutions process (light blue and dark blue) do not give the same estimation of the input m1 (g sin − x¨ a cos), therefore the compensation using the differential principle does not work very well. An hypothesis to explain these differences between simulation and experimental results is that the temperature is not uniform in the chamber which contains the two sensors and induces news

Fig. 11. Simulation of force estimation versus time for an external force Fx (t) with environmental disturbances that are rejected or not.

disturbances that are different for each sensor (deformation of the mechanical structures supporting the external magnets and the confocal chromatic sensors). To verify this hypothesis, the temperature was monitored using two K-type wire thermocouple temperature sensors placed close to the maglevtube deflector of each force sensor. To be sure to only measure the very low frequency impact of the temperature on the low frequency drift, the anti-vibration table was blocked to suppress the low frequency influence of the slow variation of . Thus, the drift can not be the consequence of the terms mi gsin which remains constant in (31) and (32). Fig. 13 shows the results of an acquisition during four days. It can be observed that the temperature evolution is slow during the week-end and more important when heater is on because of the thermal exchange between the walls of the room where the sensors are placed and the big technical hall near it. It can also be clearly seen that there is an obvious correlation between the force sensor measurements and the temperature near each deflector. Nevertheless, this correlation is different for each force sensor because the two devices used in this experiment were made at two different periods in the

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past and thus are based on different mechanical design and materials and do not react in the same way to temperature changes. The observed correlations mean that the hypothesis made is verified: the measurement of the two maglevtubes is sensitive to temperature changes and, with the materials used, this sensitivity is unfortunately in the same order of magnitude than the one due to the mechanical disturbance associated to  (i.e. the term m1 gsin)). Indeed, it can be assumed that, in some special conditions, when the temperature is controlled and remains stable in the room and the chamber, the two sensors will give the same results and can be used for the compensation process. 6.2. Compensation when temperature is not varying

Fig. 12. Force estimation versus time for an external force Fx (t) = 0 nN with environmental disturbances.

Fig. 13. Force estimation and temperature versus time for an external force Fx (t) = 0 nN with environmental disturbances during 4 days.

The chamber in which the two sensors are enclosed is placed in an air-conditioned room in order to assure that the temperature can be uniform. After, a long period to reach the stabilization of the temperature inside the chamber, the experiments are done when the temperature is the same near the two sensors. This time, the antivibration table is not blocked any more and the same experiment as the one performed in Section 6.1 is performed again to obtain Fig. 12. Figs. 14 and 15 show two different experimental results obtained with the prototype placed in the airconditioned room. They give the force estimation Fˆk when there is no external force applied on the maglevtube tip (Fx (t) = 0) and illustrates, on two different situations, the typical improvement obtained for this passive compensation principle of environmental disturbances. Figs. 14 and 15 illustrate respectively the passive compensation of the disturbances due to the slow angular variation of the anti-vibration table (effect of ) and the vibrations of the ground insufficiently filtered by the anti-vibration table (effect of x¨ a (t)). It can be seen, in accordance with the simulation results (Fig. 11), that despite of the movements of the anti-vibration table, the force estimated using the differential principle stays close to zero. Standard deviation in Fig. 14 is equal to 250 pN. Moreover, comparison between Figs. 14 and 12 shows the importance of the temperature control to ensure a proper functioning of the compensation principle. As the mechanical structure of both sensors are different and as the materials used have a large thermal dilatation coefficient (100 × 10−6 K−1 ), the current design is not well-suited to perform differential compensation with a low sensitivity to

Fig. 14. Force estimation (using and not using differential principle) versus time for an external force Fx (t) = 0 nN with environmental disturbances ( influence).

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Fig. 15. Force estimation (using and not using differential principle) versus time for an external force Fx (t) = 0 nN with environmental disturbances (¨xa (t) influence).

temperature changes. Therefore the performance illustrated in Fig. 14 remains difficult to maintain over a significant period of time and one can see that a small raising difference between the two mechanical disturbances estimations still exists for instance for time superior to 300 s. A new design with a material having a much lower dilatation coefficient is necessary to conduct deeper analysis on the potential and performances of such passive compensation approach. Fig. 15 illustrates the compensation of seismic disturbances with a high amplitude (m1 x¨ a cos  = ±5 nN) in another experiment. In this case, the residual noise computed on a 30 s window has a standard deviation equal to 450 pN. This residual noise is due to two additive independent causes: (i) the residual seismic disturbances that are not totally canceled by the differential measurement principle and (ii) the additive residual electronic noise added by the two confocal chromatic sensors that remains after the filtering induced by the differential deconvolution process. This residual electronic noise amplitude is directly controlled by the choice of the values of the two parameters WF˙ and WF˙ (the lower 1 2 are these values, the lower is the residual electronic noise) and can be simulated if the variances R1 and R2 of both confocal chromatic sensors are determined. The result of such simulation without any mechanical disturbances and on the same time window is given in Fig. 16. The standard deviation of the residual electronic noise in this simulation is equal to 140 pN. Therefore the difference between 450 pN and 140 pN represents the possible margin of progress (for a given choice of parameters WF˙ and WF˙ ) to improve the global 1 2 noise amplitude associated to this passive differential compensation principle. The last Fig. 17 illustrates the estimation versus time of a real external force that is, this time, different from zero. This non-contact force is here a magnetic force applied on the maglevtube thanks to the coils normally used to calibrate the sensor. Such approach is useful to test the force sensor estimation with different temporal shape of external force by generating adequate current profiles in the coils. In order to better see the compensation, the anti-vibration table is touched at time t = 0 to amplify the dynamic of the angular variation of the tabletop ( disturbance). At time t = 50 s, the force step is generated (current in the coils has been adjusted to obtain an amplitude of 200 nN). Both force estimations using and not using the differential principle are shown. The disturbance estimation provided by the second force sensor is also provided. With the differential principle, the force estimation stays close to 200 nN, whereas it is not the case if the disturbances are not compensated. Nevertheless, even with the differential principle, a

Fig. 16. Simulated residual electronic noise due to the two confocal chromatic sensors with measured variances R1,2 = 1.75 × 10−15 m2 and with WF˙ 1 and WF˙ 2 set to 1 × 10−18 N2 /Hz.

Fig. 17. Force estimation (using and not using differential principle) versus time for an external force Fx (t) = 200 nN with environmental disturbances ( influence).

tiny drift remains due to the current temperature dynamic in the chamber. 7. Conclusion A novel passive differential measurement principle has been presented in this article to compensate the environmental mechanical disturbances that add a time-varying offset on the force estimation obtained with nanoforce sensors based on the diamagnetic levitation of a macroscopic seismic mass. Thanks to the use of two maglevtubes, this differential principle makes possible the estimation of the common low frequency and non-stationary mechanical disturbances that are applied on both transducers. The common mode rejection of these disturbances decreases the force estimation error by a significant amount on long period of time if the environmental temperature is not varying and should make possible to investigate, in the future, force estimations just below the nanonewton level. This approach also avoids the use of new types of sensors to estimate the disturbances effects that are added on the force (like inclinometers and accelerometers with high resolution at very low frequency). The current mechanical structure of

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the force sensors used is not optimized to minimize thermal dilatation. The choice of materials with a low sensitivity to temperature (low thermal dilatation coefficient) will be necessary for future high resolution nanoforce sensor designs using magnetic springs and a precise temperature control should increase the resolution. Nevertheless, these first results show that this differential approach of passive compensation is promising and is more efficient for the low-end of the frequency spectrum of the mechanical disturbances (very low frequency drift due to the angular variation of the antivibration table). Improvements remain possible to have a better seismic rejection at higher frequency. A new mechanical design is currently under development to reach this goal. Acknowledgment

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Conference on Intelligent Robots and Systems (IROS), San Francisco, Sept. 25–30, 2011, pp. 39–44. [21] E. Piat, J. Abadie, S. Oster, Nanoforce estimation based on Kalman filtering and applied to a force sensor using diamagnetic levitation, Sens. Actuators A: Phys. 179 (2012) 223–236.

Biographies Margot Billot is currently a third-year PhD student at the Femto-ST institute working under the supervision of Dr. Emmanuel Piat, Dr. Philippe Stempflé and Dr. Joël Agnus. In 2012, she received her Engineer’s degree from ENSMM (National Higher School of Mechanics and Microtechnology) and her Master’s degree from the University of Besanc¸on, France. Her current research topic is focused on micro force measurement using piezoresistive MEMS applied to nanotribological applications.

This work has been supported by the Labex ACTION project (contract “ANR-11-LABX-0001-01”). References [1] P. Rougeot, S. Régnier, N. Chaillet, Forces analysis for micro-manipulation, in: Proceedings 2005 IEEE International Symposium on Computational Intelligence in Robotics and Automation, Espoo, Finland, 2005, pp. 105–110. [2] B. Hoogenboom, P. Frederix, D. Fotiadis, H. Hug, A. Engel, Potential of interferometric cantilever detection and its application for SFM/AFM in liquids, Nanotechnology 19 (38) (2008) 384019. [3] P. Estevez, J.M. Bank, M. Porta, J. Wei, P.M. Sarro, M. Tichem, U. Staufer, 6 DOF force and torque sensor for micro-manipulation applications, Sens. Actuators A: Phys. 186 (2012) 86–93. [4] M. Billot, X. Xu, J. Agnus, E. Piat, P. Stempflé, Multi-axis MEMS force sensor for measuring friction components involved in dexterous micromanipulation: design and optimization, Int. J. Nanomanuf. 11 (3/4) (2015) 161–184. [5] F. Beyeler, S. Muntwyler, B.J. Nelson, A six-axis MEMS force-thorque sensor with micro-newton and nano-newtonmeter resolution, J. Microelectromech. Syst. 18 (2009) 433–441. [6] Y. Shen, N. Xi, W.J. li, Contact and force control in microassembly, in: IEEE 5th International Symposium on Assembly and Task Planning (ISATP), 2003, pp. 60–65. [7] M. Boudaoud, Y. Haddab, Y. Le Gorrec, P. Lutz, Effects of environmental noise on the accuracy of millimeter sized grippers in cantilever configuration and active stabilisation, in: Conference on Robotics and Automation, Shanghai, China, 2011, pp. 5240–5245. [8] M. Boudaoud, Y. Haddab, Y. Le Gorrec, P. Lutz, Noise characterization in millimeter sized micromanipulation systems, Mechatronics 21 (6) (2011) 1087–1097. [9] K.-H. Chung, S. Scholz, G.A. Shaw, J.A. Kramar, J.R. Pratt, Si traceable calibration of an instrumented indentation sensor spring constant using electrostatic force, Rev. Sci. Instrum. 79 (9) (2008) 095105. [10] M.-S. Kim, J.R. Pratt, Si traceability: current status and future trends for forces below 10 micronewtons, Measurement 43 (2) (2010) 169–182. [11] S. Niehe, A new force measuring facility for the range of 10 MN to 10 N, in: Proceedings of the XVII IMEKO World Congress, Dubrovnik, 2003, pp. 335–340. [12] J. Illemann, R. Kumme, The achievable uncertainty for balance-based force standard machines in the range from micronewton to newton, in: IMEKO 20th TC3, 3rd TC16 and 1st TC22 Merida, Mexico, 2007. [13] I. Behrens, L. Doering, E. Peiner, Piezoresistive cantilever as portable micro force calibration standard, J. Micromech. Microeng. 13 (4) (2003) S171. [14] M.-S. Kim, J.-H. Choi, J.-H. Kim, Y.-K. Park, Si-traceable determination of spring constants of various atomic force microscope cantilevers with a small uncertainty of 1%, Meas. Sci. Technol. 18 (11) (2007) 3351. [15] C. Diethold, F. Hilbrunner, Force measurement of low forces in combination with high dead loads by the use of electromagnetic force compensation, Meas. Sci. Technol. 23 (7) (2012) 074017. [16] N.-E. Khelifa, M. Himbert, Sensitivity of miniaturized photo-elastic transducer for small force sensing, Sens. Transducers 184 (1) (2015) 19–25. [17] J. Abadie, E. Piat, S. Oster, M. Boukallel, Modeling and experimentation of a passive low frequency nanoforce sensor based on diamagnetic levitation, Sens. Actuators: A Phys. 173 (2012) 227–237. [18] A. Cherry, E. Piat, J. Abadie, Analysis of a passive microforce sensor based on magnetic springs and upthrust bouoyancy, Sens. Actuators: A Phys. 169 (2011) 27–36. [19] J. Abadie, C. Roux, E. Piat, C. Filiatre, C. Amiot, Experimental measurement of human oocyte mechanical properties on a micro and nanoforce sensing platform based on magnetic springs, Sens. Actuators B: Chem. 190 (2014) 429–438. [20] E. Piat, J. Abadie, S. Oster, Nanoforce estimation with Kalman filtering applied to a force sensor based on diamagnetic levitation, in: IEEE International

Emmanuel Piat was born in 1968. He received his PhD degree in 1996 with a PhD dissertation on data fusion in a probabilistic context applied to mobile robotics (Université de Technologie de Compiègne, France). He was nominated as Associate Professor at the Top School ENSMM (Ecole Nationale Supérieure de Mécanique et des Microtechniques) in Besanc¸on at the end of 1998. At the same time, he joined the Laboratoire d’Automatique de Besanc¸on which became in 2008 the Automatic Control and Micro-Mechatronic Systems department of the FEMTO-ST institute. His current research topics are mostly focused on micro and nano force measurement. Joël Abadie was born in 1971. He received his Ph.D. degree on system control on November 2000 from the University of Franche-Comte, France. His PhD dissertation was on study and design of micro-actuators based on shape memory alloys for active endoscopy applications. He was nominated has research engineer in Besanc¸on at the CNRS (Centre National de la Recherche Scientifique) at the end of 2002. He joined the Laboratoire d’Automatique de Besanc¸on which became in 2008 the Automatic Control and Micro-Mechatronic Systems department of the FEMTO-ST institute. He has worked in 2006 on an international project in the field of miniinvasive surgical robotics. His research is now focused on micro and nanoforce measurements which is finding applications in the mechanical characterisation of artificial or biological micro-objets. He is also developing theoretical approach on mechanical modelling using energy. Joël Agnus received the Master of Science in Electrical Engineering in 1994 and the Ph.D degree in Automatic Control and Computer Sciences from the University of Besanc¸on, France, in 2003. He is a research engineer at ENSMM engineering school and FEMTO-ST/AS2M department. He is involved in microrobotics field, and more particular concerning microgrippers, piezoelectric material and piezoresistive force sensors within micromanipulation domain and/or for nano-tribology purpose.

Philippe Stempflé is an Assistant Professor in the MicroNano-Sciences & Systems Research Department at the FEMTO-ST Institute of Microtechnology. His research is focused on the tribological problems in the field of micro/nanoscale assembly and biomimicry. His main results are published in Biomaterials, Int. J. Nanotechnology, Tribology International, Tribology Letters, Wear, Materials Science & Engineering, Materials Characterization, Key Engineering Materials, and Int. J. of Surface Science and Engineering.