# Design and Control of a Large-Range Nil-Stiffness Electro-Magnetic Active Force Sensor

Jonathan Cailliez, Antoine Weill-Duflos, Mokrane Boudaoud, Stephane Regnier, Sinan Haliyo

► To cite this version:

Jonathan Cailliez, Antoine Weill-Duflos, Mokrane Boudaoud, Stephane Regnier, Sinan Haliyo. Design and Control of a Large-Range Nil-Stiffness Electro-Magnetic Active Force Sensor. 2020 IEEE International Conference on Robotics and Automation (ICRA), May 2020, Paris, France. pp.9244-9250, 10.1109/ICRA40945.2020.9197096. hal-05208150

HAL Id: hal-05208150

https://hal.sorbonne-universite.fr/hal-05208150v1

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# Design and Control of a Large-Range Nil-Stiffness Electro-Magnetic Active Force Sensor

Jonathan Cailliez *, Antoine Weill-Duflos*, Mokrane Boudaoud, Stéphane Régnier and Sinan Haliyo

**Abstract**— Active force sensors are key instruments to get around the tradeoff between the sensitivity and the measurement range of conventional passive force sensors. Thanks to their quasi-infinite stiffness in closed loop, active sensors can be applied for force measurements on samples with a wide range of stiffness without interference with the mechanical parameters of the sensor. MEMS (Micro-Electro Mechanical Systems) active force sensors have been wildly developed in the literature but they are ill adapted for force measurements at the Newton level needed in meso-scale robotics. In this article, a novel structure for a meso-scale active force sensor is proposed for the measurement of forces from the milli-newton to the newton. This novel meso-scale sensor is based on a nil-stiffness guidance and an electromagnetic actuation. This paper deals with its design, identification, calibration and closed loop control. The sensor exhibits nil-stiffness characteristic in open loop and an almost infinite stiffness in closed loop. This allows measuring forces with a large range of gradients. First experiments shows the ability of this new sensor architecture to measure low frequency forces up to 0.8 N with a precision of 0.03 N and a closed loop –20 dB cutoff frequency of 73.9 Hz.

## I. INTRODUCTION

Force sensing and control is a key capability in robotics for a myriad of applications requiring a direct interaction between the end effectors and the external environment. The most common force sensing technique relies on the measurement of the deformation of a flexible structure, attached to the force sensor probe, with a known stiffness [1]. Such sensors are referred as *passive* sensors. For high precision position measurements of the probe, one can use laser [2], optical encoder [3] [4], piezoresistive [5], [6], capacitive [7], [8] or more exotic strategies such as the current flowing through a MOSFET transistor [9] or an optical trap principle [10].

The performances of passive sensors in terms of sensitivity and measurement range depend on the mechanical properties of the sensor flexible structure. The stiffness is one of the most influent parameters. A high sensitivity can be obtained with low stiffness sensors but at the cost of a low measurement range while high stiffness passive sensors allow for a higher measurement range but with a reduced sensitivity [11]. In addition, the stiffness of passive sensors interferes with the force measurement when it is close to

that of the object in contact with the probe. To avoid such an interference, the sensor stiffness must be very high compared to that of the external environment. This condition seriously restrict the usability of passive force sensor for the measurement of forces on objects with a large stiffness variation [12].

The alternative *active* sensor working principle is based on force balancing between an unknown force and a known quantity, as in Roberval balance [13]. Such sensors incorporate an actuator [14] whose objective is to keep the position of the probe at a fixed value when an external force is applied on it. This is achieved thanks to a closed loop control feedback. Therefore, using this principle, the external force can be measured from the control signal of the actuator. This measurement principle overcomes the previously cited drawbacks of passive sensors. The sensitivity and the measurement range of the sensor do not become longer dependent on the properties of the suspensions but to that of the actuator, which offers much more possibilities to reduce the measurement range to sensitivity tradeoff. Moreover, such sensors have a quasi-infinite closed loop stiffness which makes them suitable for the measurement of forces on objects with a large stiffness variation without interference with the mechanical parameters of the sensor. Another important benefit of the active sensing is to avoid the displacement of the probe during the measurement. This links precisely a measured force to a given position, and allows for instance a precise observation of a distance-dependent force field. With a passive sensor, the probe displacement has to be taken into account [15], [6]. This approach works also equally well regardless attractive or repulsive forces, or even in the case of time-varying fields.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Schematic view of the active force sensor working principle.</figcaption></figure>

J. Cailliez, M. Boudaoud, S. Haliyo and S. Régnier are with Sorbonne Université, UMR 7222, ISIR, F-75005 Paris, France. A. Weill-Duflos is with McGill University, Centre for Intelligent Machines, 3480 Rue University, Montréal, Québec H3A0E9, Canada. E-mail: cailliez(at)isir.upmc.fr, mokrane.boudaoud(at)sorbonne-universite.fr
antoine.weill-duflos(at)mcgill.ca, sinan.haliyo(at)sorbonne-universite.fr,
stephane.regnier(at)sorbonne-universite.fr

<footnote>* Both authors contributed equally to this work</footnote>
Most of active force sensors reported in the literature are based on the MEMS (Micro Electro Mechanical Systems) technology [11], [12], [16], [17]. The comb drive actuator is the standard actuation mechanism for active MEMS force sensors. They allow force measurements at the kHz frequency range but their measurement range is limited to the µN range. Therefore, such sensors are ill adapted for meso-scale robotics requiring force measurements at the Newton level. Note that here, the term meso-scale is referred to the scale 1 mm to 100 mm in contrast to micro-scale (1 µm - 1 mm) and nano-scale (1 nm - 1 µm).

This paper introduces a novel prototype of meso-scale active force sensor. The challenge is to design an active force sensor using an actuation principle allowing high amplitude force measurement (Newton level), a resolution of few tens of µN with a quasi-infinite stiffness in closed loop and a wide linear operating range. The design strives to eliminate parasitic forces by getting rid of the internal stiffness of the sensor in open loop as well as avoiding most losses like dry friction. The sensor (Fig. 1) is based on an electro-magnetic actuation, an air bushing system with a guiding system to hold the sensor probe and an optical sensor for the probe position measurement. This sensor offers the advantages of active sensing and can be embedded on robotic systems requiring precise low frequency force measurements at the Newton level.

The paper is structured as follows. Section II introduces the sensor architecture and its main features as well as its theoretical sensitivity calculation. Modeling and identification of the sensor are presented in section III. The closed loop control and calibration methodologies as well as the experimental dynamic force measurements are presented in section IV. Section V ends the paper.

## II. DESCRIPTION OF THE SENSOR

The sensor is built around a mobile probe carried on a cylindrical air bushing, with a magnetic voice coil actuator and an optical position sensor. A controller is designed to balance the external force applied to the probe and to keep it at a fixed position in steady state. A schematic view of the active force sensor working principle is shown in Fig. 1. A 3D CAD view of the sensor is presented in Fig. 2. The

<figure><img src="image_3.png" /><figcaption>Fig. 2: 3D CAD representation of the active sensor.</figcaption></figure>

designed sensor can be observed in Fig. 10.

### A. Sensor's components

1) Guiding mechanism: In order to minimize the perturbations on the motion of the probe due to the air movements and to achieve a smooth gliding, a porous air bushing is included in the sensor architecture. The model S300601 from Newway Air bearing<fnref n="1" /> is chosen for its relatively small size. A stainless steel shaft is machined according to the specifications of the bearing, with a diameter of 6.337 mm and a tolerance of -5 µm to -14 µm. When the bushing provides a 60 PSI pressure input, a 2 µm to 10 µm air gap is established around the shaft as long as a 12 N radial load is not exceeded on the sensor probe.

2) Actuation: A voice coil actuator is selected. Such an actuator can provide very high accelerations and a high positioning accuracy when suitably controlled. It can achieve a settling time of two milliseconds or less. It has also linear electro-mechanic characteristics. In the sensor, the moving (i.e. probe) and fixed parts of the voice-coil are not in contact and no wiring is required. The force output is independent of the probe position hence it is a pure force generator. The selected actuator is a NCC01-04-001-1X from H2W technologies<fnref n="2" />. It provides a maximum force of 0.8 N in short bursts and 0.27 N continuously. The force constant of the voice coil is 0.45 N A⁻<fnref n="1" />. It is driven by a OPA548 power amplifier from Texas instruments. It has been chosen because of its tunable current limitation and thermal shutdown protection.

3) Position sensor: In order to measure the position of the probe, a micro-epsilon ILD1420-10 optical sensor<fnref n="3" /> is chosen. It is based on an optical triangulation with a resolution of 0.5 µm at 4 kHz sampling rate.

4) Data acquisition and control system: Characterization, identification and control algorithms are programmed using Matlab/Simulink software. The implementation of the control algorithms and the data communication with the sensor are made through a DSpace DS1004 controller board at 10 kHz sampling frequency.

### B. Theoretical sensitivity

When an external force $F_{ext}$ is applied on the sensor probe in the horizontal direction (Fig. 1), the dynamic equation of the probe motion can be expressed as follows:

$$m \times \ddot{X}_{probe} = F_{coil} - F_{ext} - F_{f} \quad (1)$$

Where: $m$ and $X_{probe}$ are the mass and the position of the probe respectively. $F_{coil}$ is the force generated by the actuator. $F_{ext}$ is the external force applied on the probe. $F_{f}$ is a friction force. In equation (1), the damping factor of the probe is neglected.

The choice of the guiding mechanism is made so that $F_{f} \ll F_{ext}$.

<footnote n="1">¹https://www.newwayairbearings.com</footnote>

<footnote n="2">²https://www.h2wtech.com</footnote>

<footnote n="3">³https://www.micro-epsilon.fr</footnote>
In static mode (i.e. $t \to \infty$), equation (1) becomes:

$$F_{\text{coil}} = F_{\text{ext}} \qquad (2)$$

In dynamic mode, the equation (1) can be described in the frequency domain as follows:

$$m\omega^2 X_{\text{probe}} = F_{\text{ext}} - F_{\text{coil}} \qquad (3)$$

Hence:

$$|m\omega^2 X_{\text{probe}}| = |F_{\text{ext}} - F_{\text{coil}}| \qquad (4)$$

The objective in the sequel is to define the relationship between the relative force measurement error $\Delta_F$, the pulsation $\omega$ and the actual force $F_{\text{ext}}$. This analysis can be used as a basis for the sensor design specifications.

$$\Delta_F = \left| \frac{F_{\text{ext}} - F_{\text{coil}}}{F_{\text{ext}}} \right| \qquad (5)$$

Let us consider $\epsilon_x$ the smallest displacement value provided by the position sensor. For $X_{\text{probe}} = \epsilon_x$, equation (4) allows considering:

$$|F_{\text{ext}} - F_{\text{coil}}| \le |m\omega^2 \epsilon_x| \qquad (6)$$

Therefore:

$$\Delta_F \le \left| \frac{m\omega^2 \epsilon_x}{F_{\text{ext}}} \right| \qquad (7)$$

We consider in this analysis that $\Delta_F \le 1$

Taking into account the values $m = 18.3$ g and $\epsilon_x = 0.5\mu m$, the map representing the relative force measurement error $\Delta_F$ w.r.t. the actual force $F_{\text{ext}}$ and the frequency is shown in Fig. 3.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Relative force measurement error $\Delta_F$ w.r.t. the actual force $F_{\text{ext}}$ and the frequency.</figcaption></figure>

### III. MODELING AND IDENTIFICATION

The goal of this section is to propose a knowledge model of the active force sensor using the scheme of Fig. 4. For control purposes, we consider here $U_{\text{controller}}$ as the input of the system and $U_{\text{pos}}$ as its output.

<figure><img src="image_5.png" /><figcaption>Fig. 4: Block diagram of the closed loop system.</figcaption></figure>

#### A. Knowledge model

The controller output will be used as a measurement of the applied force.

Let us consider the following model for the voice coil actuator:

$$F_{\text{coil}} = k_e \times U_{\text{controller}} \qquad (8)$$

With $F_{\text{coil}}$ the force exerted by the coil, $U_{\text{controller}}$ the controller output and $k_e$ the static gain of the actuator.

Let us now consider the dynamic equation of the probe modeled as a mass-damper system:

$$m \times \ddot{X}_{\text{probe}} = F_{\text{coil}} - F_{\text{ext}} - \nu \dot{X}_{\text{probe}} \qquad (9)$$

with $m$ the mass of the sensor probe, $F_{\text{ext}}$ the force to be measured by the sensor, $X_{\text{probe}}$ the probe position, and $\nu$ a damping coefficient.

The transfer function between $X_{\text{probe}}$ and $F_{\text{coil}}$ can then be expressed as follows (with $p$ the Laplace variable):

$$H(p) = \frac{X_{\text{probe}}}{F_{\text{coil}}} = \frac{1}{m \times p^2 + \nu p} \qquad (10)$$

Finally, for the optical position sensor, the relationship between the output voltage $U_{\text{pos}}$ and the position of the probe $X_{\text{probe}}$ is modeled by a static gain as follows:

$$U_{\text{pos}} = k_{\text{pos}} \times X_{\text{probe}} \qquad (11)$$

with $k_{\text{pos}}$ the sensor gain.

Therefore, the relationship between $U_{\text{pos}}$ and $U_{\text{controller}}$ can be modeled by the following transfer function:

$$H_{\text{ol}}(p) = \frac{U_{\text{pos}}}{U_{\text{controller}}} = \frac{k_e \times k_{\text{pos}}}{m \times p^2 + \nu p} \qquad (12)$$

The experiments have shown that despite the natural integrator in the model of equation (10), i.e. pole at zero, the system still presents a non zero static error. This error is attributed here to a static friction which is thus included in the model. When including static friction, the model becomes non-linear. The system model is modified to take into account static friction, noted $\Delta_{\text{friction}}$, as in Fig. 4.
## B. Parametric identification

By its design and working principle, the sensor probe has no recall force (nil stiffness in open loop). Therefore, the parametric identification of the sensor has to be done in closed loop. The parameters of the model of Fig. 4 that have to be identified are: $v$, $k_e$, and the static friction $\Delta_{friction}$. The parameters $k_{pos}$ is taken from the optical sensor specifications and the mass $m$ is estimated to be equal to 18.3 g.

For the identification, the following protocol is used.

* The sensor is fed back with a proportional controller,

* the position of the probe is recorded for a step input reference,

* the knowledge model is implemented in simulink,

* a grid search is performed to identify the knowledge model parameters, trying to minimize the model error with experimental data.

For the identification, a proportional controller with a gain $k_p = 0.2$ has been used. Simulation and experimental step responses of the closed loop system are shown in Fig. 5a.

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5: Experimental and simulation step responses of the sensor probe position in closed loop with (a) $k_p = 0.2$ and (b) $k_p = 0.1$</figcaption></figure>

It can be observed that despite the open loop integrator, the system exhibits a non-zero static error, caused by static friction. The model fits the experimental results well in terms of resonant frequency and fits reasonably well in terms of damping. The modeling error is assumed to be acceptable for

<table><caption>TABLE I: Parameters of the sensor's non linear model identified through a grid search.</caption><thead><tr><th>Physical parameter</th><th>symbol</th><th>value</th></tr></thead><tbody><tr><td>Gain of the optical position sensor</td><td>$k_{pos}$</td><td>1981 V m<sup>-1</sup></td></tr><tr><td>Mass of the sensor</td><td>$m$</td><td>18.3 g</td></tr><tr><td>Gain of voice coil</td><td>$k_e$</td><td>0.2467 N V<sup>-1</sup></td></tr><tr><td>Damping coefficient</td><td>$\nu$</td><td>0.3570 N s m<sup>-1</sup></td></tr><tr><td>Static friction</td><td>$\Delta_{friction}$</td><td>0.0025 N</td></tr></tbody></table>

further control design purposes. The identified parameters are summarized in table I.

Furthermore, simulation and experimental closed loop data have been compared when modifying the controller gain at a value that has not been used during the identification. For $k_p = 0.1$, results are shown in Fig. 5b. They show that even if the controller gain is modified, the model fits well the experimental data. So the parameters of table I are kept for the development of a controller.

## IV. CLOSED LOOP CONTROL, CALIBRATION AND FORCE MEASUREMENT

### A. Closed loop state feedback control

A state feedback control strategy is chosen for the controller design. It allows a precise closed loop dynamic control by a wise pole assignment. Equation (12) can be described in the state space leading to the following state space representation:

$$ \begin{cases} \dot{Z} = A \times Z + B \times U_{\text{controller}} \\ U_{\text{pos}} = C \times Z \end{cases} \quad (13) $$

with

$$ Z = \begin{Bmatrix} \dot{X} \\ X \end{Bmatrix}, \quad A = \begin{bmatrix} \frac{-\nu}{M} & 0 \\ 1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} \frac{k_e}{M} \\ 0 \end{bmatrix}, \quad C = \begin{bmatrix} 0 & K_{\text{pos}} \end{bmatrix} \quad (14) $$

The system is of order 2 but because of the static friction, an integrator is added into the control loop. The number of the states becomes equal to 3. The system is fully observable. A Luenberger observer is designed for the states observation and feedback control. The poles are assigned in closed loop at -100, -400, -500, leading to a closed loop -20 dB cut-off frequency of 73.9 Hz. Experimental and simulated step responses are shown in Fig. 6. This result demonstrates (i) firstly that the identified model is well adapted for the control design and (ii) secondly the ability to perform a precise closed loop positioning of the sensor probe, which is of primary importance for further active force measurement.

### B. Calibration

The aim here is to refine the value of $k_e$ for the sensor calibration and force measurement. The calibration setup is described in Fig. 7. This setup includes the active force sensor and a reference force sensor (FS2050-0000-1500-G from TE connectivity). The reference force sensor is mounted on a micro-metric vernier positioner and compresses a spring between the reference force sensor and the active force
<figure><img src="image_8.png" /><figcaption>Fig. 6: Closed loop experimental and simulation step responses of the sensor probe position with the fully observed state feedback controller.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7: Schematic representation of the calibration setup.</figcaption></figure>

sensor probe. The probe is kept at a desired position thanks to the state feedback controller.

The following protocol is used to calibrate the actuator.

* The reference force sensor output at zero load is recorded,

* the vernier is used to compress the spring,

* output of the reference force sensor and controller output voltage are recorded,

* both values are plotted against each other,

* a linear fitting is performed to deduce the actuator force constant $k_e$.

Experimental results and fitting results are shown in Fig. 8. A reduced number of points is taken above 0.3 N in order to avoid damaging the voice-coil actuator by over-heating it. The static characteristic can be linearized with a static gain of 0.1429 N V⁻¹.

### C. Dynamic force measurement

The active sensor is in state-feedback closed loop. The setup of Fig. 7 is used to apply a variable force to the active sensor using the vernier positionner. Both the force reading of the reference force sensor and the output of the active force sensor are recorded. A comparison between the force reading of the active force sensor and that from the reference force sensor can be seen in Fig. 9a. Measurement error can be seen in Fig. 9b with a maximum error value around 0.03 N which is 4% of the sensor full range. This measurement error can be explained by the electrical noise as well as the bandwidth of the sensor. Indeed, when performing a spectrum analysis of the error, a strong 50 Hz component

<figure><img src="image_10.png" /><figcaption>Fig. 8: Static characteristic of the voice coil actuator, experimental results and linear fitting.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 9: Force measurement from the active force sensor and from the reference force sensor. Experimental filtered data (a) and measurement error (b).</figcaption></figure>
has been observed, denoting a pollution from the electrical circuitry. Further work will focus on reducing this error.

<figure><img src="image_12.png" /><figcaption>Fig. 10: Experimental setup for magnetic force measurement.</figcaption></figure>

Furthermore, dynamic force measurements at a fixed frequency are performed. To this end, the following setup is used. A magnet is fixed on a loudspeaker and the probe is fitted with a magnet so that an attractive force is applied on the active sensor. The position of the magnet fixed on the loudspeaker will be noted $X_{magnet}$. The loudspeaker is driven with a fixed frequency tone and is thus applying a force $F_{mag}$ that can be expressed as a function of the probe-magnet distance :

$$F_{mag} \propto \frac{1}{(X_{probe} - X_{magnet})^2} \quad (15)$$

Because of the power of 2 in the equation, the first frequency in the force spectrum will be the double of the tone. The goal here is to demonstrate the ability to measure forces with the adequate temporal shape at a fixed frequency. The measured force must have a temporal shape in accordance to equation (15). The force recorded by the sensor is normalized and compared with a function derived from equation (15). Experimental results for a tone of 30 Hz and 60 Hz are shown in Fig. 11a and 11b respectively demonstrating the efficiency of the proposed active force sensor for dynamic force measurements.

## V. CONCLUSION

In this article, a novel design for an active force sensor has been proposed. This design is based on a nil-stiffness guidance system and electro-magnetic actuation. The nil stiffness guidance system presents the advantage of having a theoretical resolution at the milli-newton range and the choice of the actuation allows the sensing of forces up to

<figure><img src="image_13.png" /><figcaption>Fig. 11: Magnetic force measurement from the active force sensor and comparison with the magnetic force model for a driving tone of 30 Hz (a) and 60 Hz (b).</figcaption></figure>

0.8 N. The design of the sensor has been presented along the reasons for the technological choices. A knowledge-based model of the sensor has been proposed, identified and validated. Using the knowledge model of the sensor, a controller has been developed for the sensor allowing for a closed loop –20 dB cut-off frequency of 73.9 Hz. A static characterization of the sensor has been performed and used for force measurement. This paper has presented the first results of force measurements with this new active force sensing technology.

## VI. ACKNOWLEDGEMENTS

This work has been partially sponsored by the research program Investissements d'avenir through the Robotex Equipment of Excellence (ANR-10-EQPX-44) and by the research program ANR JCJC Robine (S19JRAR086).

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