Method of estimating contact force of bone-conducted sound transducer with a two-degrees-of-freedom vibrating model

Satoki Ogiso; Koichi Mizutani; Keiichi Zempo; Naoto Wakatsuki; Yuka Maeda

doi:10.7567/1347-4065/ab1fd9

1. Introduction

In bone conduction, sound propagates through skin and bone into the inner ear.¹⁾ Bone-conducted sound does not travel through the outer ear or the middle ear.^2–9) This feature of bone-conducted sound can be useful to help compensate for conductive hearing loss^10–12) or for the clinical assessment of hearing loss. Also, bone conduction shows a unique phenomenon in which ultrasound above 20 kHz can be heard.^13–15) This phenomenon is called bone-conducted ultrasound, and it enables even some patients with severe to profound hearing loss to hear.¹⁶⁾ Bone-conducted ultrasound is also applied to hearing aids.¹⁷⁾

For these applications, bone-conducted sound is presented from a sound transducer attached directly to the bone or placed on the skin. A directly attached transducer is called a bone-anchored hearing aid (BAHA).¹⁰⁾ A BAHA provides a stationary connection between the transducer and the bone. However, BAHA is not always preferable, as its placement is invasive and the tissue around the transducer must be cared for. A non-invasive method to present bone-conducted sound is to place a transducer on the skin instead. The transducer is typically placed on the mastoid to aid hearing and for clinical assessment. Alternatively, the transducer can be placed on different points depending on the purpose. For radio communication with higher speech intelligibility, for example, the transducer is placed on the forehead¹⁸⁾ or on the tragus.¹⁹⁾ From each point, the sound propagates in the head by the vibration of the transducer. In this paper, we focus on a transducer placed on the tragus.

Several factors affect bone-conducted sound by non-invasive transducers. One of them is the contact force of the transducer, shown in Fig. 1. With different contact forces, the absolute threshold of hearing can be affected by more than 10 dB.²⁰⁾ Unlike other transducers for mechanical systems using ultrasound^19–32), a bone-conducted transducer is not used with matching layers or with rigid mounts. International standards on bone-conducted sound transducer calibration reflect the effect of contact force on bone-conducted sound. One of the current international standards for the calibration of bone-conducted sound transducers requires the contact force to be fixed at 5.4 ± 0.5 N (ISO 389-3:2016) for reproducible calibration and fitting. To avoid the problem of the contact force changing the sound, mainly two approaches are used. One of the most common methods is to use a headband to stabilize the bone-conducted sound transducer.³³⁾ Although the headband is very easy to use, it prevents the measurement of the contact force. The wearing of the headband is empirically mastered. Also, there are reports that sound can be changed by how the headband is worn.^34,35) For research purposes, the contact force of the headband is measured in advance with a force sensor. However, it is not practical for everyday users of the transducer to apply an external force sensor. Another method for dealing with the contact force is to locate a pressure sensor between the transducer and the skin.³⁵⁾ Although this method measures the contact force directly, the sensor itself affects sound propagation. If the pressure sensor is removed before the sound is emitted, the contact force is no longer monitored.

We propose a method that uses the electrical impedance of the transducer^36–38) to solve this problem of contact force estimation. The electrical impedance of the transducer shows a characteristic change according to the contact force. The contact force can be estimated without any extra sensor by measuring the change in electrical impedance. This estimation method was implemented with a neural network to estimate the nonlinear relationship between electrical impedance and contact force.³⁷⁾ However, the method depends entirely on the measured data, since the structure of the neural network does not reflect the physical vibration. If the variation of the data is limited, the neural network fails to learn the correct relationship between impedance and force. To solve this problem, we previously proposed a vibration model with two-degrees-of-freedom and verified its characteristics by measuring the electrical impedance.³⁹⁾ However, the model was verified only with the measured data and has not been applied to the estimation.

In this paper we present a method of estimating the contact force of a bone-conducted sound transducer with a two-degrees-of-freedom vibrating model. This method can guarantee the physical validity of the estimation regardless of the data since it explicitly utilizes the physical model. Also, the amount of data required for the calibration process would be reduced compared to the neural network.

2. Estimation of model parameter change by contact force

2.1. Proposed model of bone-conducted sound transducer

The proposed model of the mechanical part of the bone-conducted sound transducer is shown in Fig. 2. The model consists of two mass-spring-damper models. Parameter m_h represents the mass of the housing and magnet of the transducer. k_h and c_h in the model are the skin's spring and damper components, which form a Kelvin–Voigt model and which connect the housing and the diaphragm. m_d represents the mass of the diaphragm and skin. From a previous examination,³⁷⁾ it is known that the proposed model can reproduce the change in electrical impedance by the contact force. The model is used to estimate c_h by the least-squares method in this paper. The relationship between contact force and c_h is discussed, as are the limitations of the proposed model.

2.2. Experimental setup

The experiment was conducted with 12 human subjects (ages 21–27 years, normal hearing). Figure 3 shows the setup of the experiment. The bone-conducted sound transducer (AS400, Aftershockz) was placed anterior to the acoustic ear opening. The electrical impedance of the transducer was measured by an impedance analyzer (E5061B, Keysight) at a frequency range of 10 Hz-60 kHz. The contact force was changed at 0.0, 0.1, 0.3, 0.5, 1.0, 3.0, and 5.0 N. From preliminary experiments, the parameters of the proposed model were determined as m_h = 3 g, k_h = 350 N m⁻¹, m_d = 3 g, k_s = 1000 N m⁻¹, c_s = 0.001 N s m⁻¹. c_h was estimated for the frequency range of 10 Hz to 1 kHz, where the impedance changes drastically. Impedance was measured 10 times for each subject at each contact force.

**Fig. 3.** (Color online) Experimental setup for measuring the electrical impedance of a bone-conducted sound transducer on a human head.
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2.3. Results and discussion

An example of the measured electrical impedance and fitting result of the proposed model is shown in Fig. 4. Figures 4(a), 4(b) show two peaks in the electrical impedance. These peaks can be understood as the resonance of the housing and the diaphragm in Fig. 2. The damping coefficients c_h at these contact forces approximately 0.2 N s m⁻¹ in both cases. For greater contact force, only one peak can be found in the electrical impedance. The peak indicates that the housing and the diaphragm are vibrating in-phase. The damping coefficients for these higher contact forces are from 2.1 to 5.1 N s m⁻¹. With these results, we can confirm the relationship between c_h and contact force.

The relationship between c_h and contact force F for all subjects is shown in Fig. 5. The relationship between c_h and contact force F was confirmed for all subjects. The relationship between c_h and F can be confirmed as an exponential function if the contact force was in the range of 0.3–3.0 N. The relationship between c_h and F above 3.0 N is not an extension of that below 3.0 N. The reason for this difference can be found in Figs. 4(f), 4(g). The fitting results of the model in Figs. 4(f), 4(g) have a larger error than those of the model in Figs. 4(c)–4(e). If F is larger than 3.0 N, the measured electrical impedances have backbone curves. This implies that a nonlinear spring is in the system. The nonlinearity of the skin was omitted in the proposed model. The lack of a nonlinear component is considered a limitation of the proposed model. For this reason, the proposed contact force estimation method was tested within the range of F = 0.0–1.0 N. In the future, to estimate higher contact forces, the model should include a nonlinear component.

3. Estimation of the contact force with the proposed model

3.1. Proposed method of estimating contact force

An overview of the proposed model is shown in Fig. 6. The proposed method consists of a calibration component and an estimation component. The calibration component estimates the relationship between damping coefficient c_h and contact force F with known contact forces [Figs. 6(i)–6(1)]. As discussed in the previous section, the relationship between the damping coefficient and the contact force can be modeled as an exponential function in the 0.0–1.0 N range. For this reason, we assume that the damping coefficient and the contact force have the following relationship.

$\begin{eqnarray}&&F=\exp \left(\displaystyle \frac{{c}_{h}-a}{b}\,\right),\end{eqnarray} \tag{ 1 }$

where a and b are the parameters of the line, c_h is the damping coefficient, and F is the contact force. The best-fit parameters a and b [Figs. 6(i)-(5)] are estimated in the calibration component. Then, the contact force F can be estimated by this equation with damping coefficient c_h [Fig. 6(ii)-(4)]. c_h is estimated from the electrical impedance. The fitting of Eq. (1) was achieved with the least-squares method.

**Fig. 6.** (Color online) Method of estimating contact force using a physical model. (i) Calibration by pairs of electrical impedance and contact force, (ii) estimation of the contact force with calibrated parameters.
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The proposed method was evaluated by the experiment described in Sect. 2.2. The parameters of the model were set as m_h = 3 g, k_h = 350 N m⁻¹, m_d = 3 g, k_s = 1000 N m⁻¹, and c_s = 0.001 N s m⁻¹. For the calibration step, the results from only 0.3 N and 0.5 N were used to estimate parameters a and b.

3.2. Results and discussion

The results of the calibration process are shown in Fig. 7. The black line shows the calibrated relationship between c_h and F. By using the c_h values at F = 0.3 and 0.5 N, the parameters were determined as a = 4.69 and b = 2.92. The proposed method estimates the contact force from c_h of the measured electrical impedance with these parameters. The results of contact force estimation at contact forces of 0.0–1.0 N are shown in Fig. 8. From this figure, the estimation error was mostly within the ±0.3 N range. The estimation error had a slight bias to −0.2 N. This bias is attributable to the calibrated result. In Fig. 7, the calibrated line is far lower than the measured c_h at F = 0.1 N. This error is attributable to the lack of nonlinearity of the skin in the proposed model. The error of c_h estimation can be confirmed in Figs. 4(a), 4(b). Employment of nonlinearity is required in order to estimate a contact force lower than 0.1 N or higher than 3.0 N. The mean and standard deviation of the error were 0.2 N and 0.2 N, respectively. The 90th percentile of the estimation error was 0.4 N. This estimation error is reasonably low compared to the 90th percentile error of 0.4 N by the previously proposed method using a neural network.³⁷⁾ The variation in the error was attributable to the variation of c_h shown in Fig. 5. The variation of c_h maps directly to the estimation error with Eq. (1) and causes a variation in contact force estimation. The problem here is that the proposed model is considered a fixed parameter model in this research. The stiffness of the skin may change according to the skin condition or interpersonal variation. These changes were all estimated as the change in c_h, which makes apparent variation of c_h regardless of actual damping. According to the international standard for the calibration of bone-conducted sound transducers, the contact force should be within an error of time ±0.5 N (ISO 389-3:2016). The results showed that 97% of the estimation error was within this range. Although the standard is for the mastoid process and cannot be applied directly to a tragus-applied transducer, the proposed method can achieve this error range only with the electrical impedance. The feasibility of applying the proposed method to the mastoid should be tested.

**Fig. 8.** Estimation results of the contact force with the proposed method.
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Note that this result was achieved with calibration at only 0.3 N and 0.5 N. With such a small amount of data, the previous method using a neural network³⁷⁾ is not applicable, as thousands of parameters are required before learning converges. The 90th percentile of the estimation error in the previous method was 0.4 N with 770 sets of calibration data. The number of the calibration dataset was determined as a product of conditions: seven different contact forces, 10 measurements for each contact force, 11 subjects for the experiment. On the other hand, the 90th percentile of the estimation error shown in Fig. 8 was 0.4 N for the proposed method, which uses only 240 sets of c_h values. In the previous study,³⁷⁾ when the number of calibration data was decreased to 280 (four subjects were used for calibration), the 90th percentile error increased to 1.0 N. This error means that the previous method is not practical if the number of data sets is reduced to only 280. Although the contact force range was more limited than in the previous method, the comparison above might indicate accurate estimation with fewer data.

4. Conclusions

This paper proposes a method of estimating the contact force of a bone-conducted sound transducer with a two-degrees-of-freedom vibrating model. First, the damping coefficient of the vibrating model was estimated with electrical impedance measured in 12 human subjects. The impedance was measured in the frequency range of 10–60 kHz. The damping coefficient of the vibrating model was estimated with the least-squares method basing on the measured impedance. The results showed that the damping coefficient of the model was related to the contact force. Also, the relationship between the damping coefficient and the contact force was fitted with an exponential function when the contact force was lower than 3.0 N. Second, a method of estimating the contact force was proposed. The contact force was estimated by estimating the damping coefficient from the measured impedance and calculating the corresponding contact force from the damping-contact force curve. The contact force estimation was evaluated in the range of 0.0–1.0 N. The estimation results showed that the proposed method achieves reasonable accuracy compared to the previous method using a neural network. Also, the estimation required only two calibration conditions, at the contact forces of 0.3 N and 0.5 N, by employing a physical model. This drastically reduced the calibration process compared to the previous method using a neural network, which required the determination of over 1000 parameters. In future studies, the nonlinearity of human skin stiffness should be modeled to estimate greater contact forces.