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# Design and performance measurement of an ultrasonic rangefinder with signal correlation in the context of autonomous urban drone navigation

Jean-Claude Rembert-Baudet

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Jean-Claude Rembert-Baudet. Design and performance measurement of an ultrasonic rangefinder with signal correlation in the context of autonomous urban drone navigation. 2024. hal-04394025v2

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Submitted on 15 Jan 2024

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# Design and performance measurement of an ultrasonic rangefinder with signal correlation in the context of autonomous urban drone navigation

ASTEK Group

**Abstract-** Drones represent new types of robots that allow the integration of different sensors that are inexpensive, easy to use, and provide users with many new options regarding image and geospatial data collection. Technological advances in electronics, big data and artificial intelligence have enabled the development of new fields of application for these devices as well as their robustness and utility. To ensure autonomous flight, each drone must have a 3D perception of the environmental obstacles present as well as an understanding of their respective trajectories and their impact on its own navigation. For this, UAVs use several sensors to be more reactive in an environment full of obstacles. The objectives of the work are to model the performance of an acoustic telemetry device, with an ultrasonic transmitter and receiver, intended to be mounted in an urban drone with a wingspan of a few tens of centimeters. The innovative solution proposed in this work to make telemetry from acoustic wave is the use of a device, integrable on an urban drone flying at a maximum speed of about 15 Km/h and at a height of about 7 meters, based on acoustic sensors whose concept is based on the calculation of the autocorrelation between the transmitted and received signals to measure the distances between the drone and the obstacles. The work showed that the wind speed or the mobility of the obstacles have little impact on the telemetric performances of the device. They also showed that the nature of the material of the obstacle as well as its surface state or its size influence the accuracy of the telemetric measurement.

**Index Terms**- Urban drone, Ultrasonic sensor, Doppler effect, Wind, Obstacle mobility, Autocorrelation of analog and digital signals, Amplification electronics, Signal digitization, Telemetry measurement.

## I. INTRODUCTION

Detecting obstacles in the drone's environment is a key function to ensure a collision-free flight. One of the methods used for this purpose is acoustic telemetry. This technology has several advantages.

Compared to other types of sensors (infrared or laser for example), the ultrasonic sensor is relatively inexpensive. The measurement is quite accurate. However, one of the main drawbacks of this technology is its sensitivity to temperature and pressure. It is also sensitive to other acoustic sensors in the vicinity since interference between the waves emitted by the different sensors is possible. This is due to the fact that the information we get on obstacles with the time-of-flight technology is not always reliable since the technology does not differentiate the echoes received. However, it is still a very common technology.

In the current work, another technology of acoustic telemetry is proposed; the telemetry by autocorrelation of transmitted and received signals. The calculation of this function allows the differentiation of the echoes emitted by the device from those coming from the external environment through the similarity between the echoes and the emitted signals. It is then potentially possible to extract more information and to improve the reliability as well as the telemetric performances. Autocorrelation is a characteristic of data that shows the degree of similarity between values of the same variables over successive time intervals. In signal processing, we can use this mathematical tool to measure the similarity of the received echo to the transmitted signal. For a given signal $f(t)$ the continuous autocorrelation $R_f(\tau)$ is the continuous cross-correlation of $f(t)$ with itself, at time interval $\tau$, and is defined as :

$$R_f(\tau) = f^*(\cdot - \tau) \circ f(\cdot) = \int_{-\infty}^{\infty} f(t + \tau)f^*(t)dt = \int_{-\infty}^{\infty} f(t)f^*(t - \tau)dt,$$

Where $f^*$ represents the complex conjugate of $f$ and $\circ$ represents the convolution operator ($f = f^*, \forall f(t) \in \mathbb{R}$).

The objective of this paper is to present the performance modeling of such a telemetric device based on the emission of an acoustic wave by a piezoelectric ultrasound transmitter and received by a single acoustic receiver also piezoelectric after the
<figure><img src="image_4.png" /></figure>

wave has been reflected by a potential obstacle. More specifically, it aims to quantify the impacts of wind, moving objects and obstacle material on the telemetric performance of the device. After a state of the art of acoustic telemetry technologies including the correlation of transmitted and received signals, the work will quantify the impacts of wind speed and obstacle on the telemetry performance of the device. Then, the work will identify some uncertainties of measurements related to the electronics of reception of the return wave and to the digitization of the transmitted and received signals for the needs of numerical correlation of these last ones. Finally, the work will quantify the impacts of these uncertainties on the telemetry performances of this new technological device of acoustic telemetry.

## II. STATE OF THE ART

In this section, a review of the existing literature, where the papers most related to the problem of acoustic telemetry using autocorrelation of transmitted and received signals, will be made. Acoustic time-of-flight telemetry is the technology of choice for several research works and a lot of knowledge concerning the performance of this technology has been developed. Therefore, papers dealing with acoustic time-of-flight telemetry, and addressing the notions of impact of physical parameters of the urban environment on the telemetric performances, will also be presented.
The paper [1] explores the possibility of recognizing hand gestures using ultrasound imaging based on distance calculation between different points of the hand and the device. Deep learning models were used to recognize gestures from ultrasound images. However, the hand remains motionless, at a close distance (less than 20cm) from the device and the work does not conclude on the impact of the material on the telemetric measurement, as the hand is composed of organic tissue. The paper [2] presents a high-resolution sensor composed of three ultrasonic sensors: a transceiver and two additional receivers, which allows a significant improvement of the information extraction process compared to a single ultrasonic sensor. At each of the three receivers, the precise arrival times of the echoes are extracted from the raw signals of the sensors through a filter based on the calculation of the autocorrelation of the received signal. Concerning the properties of the objects in the environment, this paper is only interested in the shape of the objects. The influence of the object's material and its orientation on the sensor's performance is missing. The paper [3] presents a measurement setup consisting of several ultrasonic sensors used in parallel, to perform telemetric measurements. The sensor is based on the

calculation of the autocorrelation (calculated by matched filter) of the received signal to detect obstacles. However, the paper does not address the properties of the obstacles and their influences on the telemetric performance of the studied method. Paper [4] presents work on the statistical properties of acoustic emission through rock samples as a function of applied uniaxial loading. It was found that the parameters of the autocorrelation function of the acoustic emission event series change significantly near failure. However, the acoustic waves used for the calculation of the autocorrelation function are waves produced by the rock sample itself and, therefore, all the factors influencing the performance of the telemetry device in our work have not been treated, by what in our operational context we are interested in the calculation of the autocorrelation of the reflected signal. The paper [5] proposes a method of Doppler estimation by multi-branch autocorrelation. This method measures not only the temporal compression of the acoustic waves over the estimation interval, but also the gradient of the temporal compression by calculating several autocorrelation functions and introducing a frequency offset to each function. This being said, the paper is concerned with underwater acoustic communication, so the reflection of acoustic waves is not part of this work and therefore the paper cannot inform us about the influence of obstacle characteristics and the reflection properties of acoustic waves in relation to any telemetry performance. The paper [6] presents a technology for exploiting the Doppler effect to recognize people by their gait. An acoustic wave is incident on the walking subject. The motion of the walker induces a Doppler shift in the frequency of the reflected sound, which is detected by the sensor. However, the paper is not interested in quantifying the impact of object motion on the telemetry performance of the device nor that of wind or obstacle material since the wave reflects off people. The paper [7] proposes to design a device emitting and receiving acoustic echoes through a microphone array and then allowing the calculation of 3D coordinates. The calculation is based on the principle of triangulation telemetry. In addition, the article presents the impact of the intrinsic and extrinsic characteristics of the obstacle on the reflection of the wave and notions of acoustic wave physics. This paper is a theoretical study explaining the interaction between an obstacle and an acoustic wave and does not present any experiments of acoustic wave reflection on physical obstacles, nor any numerical results of telemetry performance. Finally, the paper [8] investigates the shape identification capabilities of some simple obstacles such as outer corner (convex), inner corner (concave), flat wall, door with open door, round shaped object,
<figure><img src="image_5.png" /></figure>

and their orientation in a coordinate system using an ultrasonic sensor system for a modular mobile robot. The technology used in this article to measure the distance is time-of-flight measurement. However, the paper uses the classical method of calculating the time of flight of the acoustic wave to determine the distance. The autocorrelation technology of the transmitted signal and the received signal was not used. Furthermore, the experiments performed in this study are all static, i.e., the object and sensors are fixed. Furthermore, neither the influence of the obstacle size nor its orientation on the telemetry performance has been studied. To conclude, there is no study in the scientific literature that addresses our problem. Indeed, even if the autocorrelation technology is used, either the operational context is different, or the impacts of the wind, the mobility of the obstacles or the influence of the intrinsic properties of the materials have not been simultaneously evoked and quantified. Direct time-of-flight measurement is the most common method of acoustic telemetry.

### III. STUDY STRATEGY

Some impacts, such as wind, drone mobility or moving obstacles, will be studied in a purely theoretical way. It is also necessary to define the function of the autocorrelation of the signals and its link with the telemetry and to quantify the physico-geometric parameters which would influence this function. To do this, in a first step, a theoretical model will be established. The accuracy of the measurement of a telemetry device represents the ability of the sensor to give a measurement close to the exact value of the distance. Several geometrical-physical factors can influence the value of this precision. The intuitive parameters influencing the precision of the telemetric measurement are multiple but we can quote as example the size, the material of the obstacle, its shape as well as its orientation. For this reason, we will, in a second step, try to measure the impact of the size and the material of the obstacle on the accuracy of the telemetry device. The current work will only use flat obstacles, because we consider that the obstacles in an urban environment are generally flat. To do this, the indoor environment is privileged in order to minimize the influence of external elements (temperature, humidity, wind...) whose characteristics are not controlled. The obstacle will be unique, stable, close to the device and with a surface orthogonal to the incident wave vector.

### IV. MODELING

#### A. Impacts of wind speed and obstacle mobility speed

The wind corresponds to the displacement of the material medium in which an acoustic wave of any frequency propagates. If the material medium is the ambient air, assimilated as a perfect gas, the acoustic velocity can be related to the temperature $θ$ (in Kelvin) such that:

$$c = \sqrt{\frac{\gamma R \theta}{M}}$$

Where $γ$ is the Laplace coefficient, $R$ the constant of perfect gases and $M$ the molar mass of air. For air at 20°C, $c ≈ 343m.s^{-1}$.

Noting by $v_1$ the speed of the drone with respect to the ground; $v_2$ the speed of the obstacle and $v_3$ the wind speed with respect to the ground and in the direction of the vector $\vec{v}_1$; this celerity $c$ is measured in relation to the material medium. In the case of wind blowing at the algebraic speed $v_3$, the velocity of the acoustic wave with respect to the ground becomes simply $c + v_3$ if the wave is carried and $c - v_3$ if the wave is against the direction. If the wind is transverse, the acoustic wave is deviated from its initial axis of emission. We will not study this particular case which remains possible. However, the navigation of the urban drone with a transverse wind can be even more problematic than in the case of a weak wind against the direction considering that it can make deviate the drone from its trajectory.

Measuring a distance using the autocorrelation of transmitted and received signals is equivalent to measuring the time it takes the acoustic wave to make a round trip between the device and the obstacle. Noting by $d$ the distance between the device and the obstacle at the time of transmission, this round trip time of the acoustic wave is then:

$$t_{a-r} = \frac{2c}{(c - v_3 + v_1)(c + v_3 - v_2)} d$$

The drone could know its speed of movement $v_1$ in relation to the ground. If it takes into account the hypothesis of the immobile obstacle compared to the ground and in the absence of wind, it will then measure a time of round trip of the wave equivalent to a distance $d' \neq d$ and worth:

$$d' = \frac{c + v_1}{2} t_{a-r} = \frac{c(c + v_1)}{(c - v_3 + v_1)(c + v_3 - v_2)} d$$

The relative error is therefore a function of the speeds:

$$e_r = \frac{d' - d}{d} = \frac{v_2 c + (v_1 - v_3)(v_2 - v_3)}{(c - v_3 + v_1)(c + v_3 - v_2)}$$

For a stationary obstacle relative to the ground, for a wind blowing at 50km/h (about 14m/s) and for a
<figure><img src="image_6.png" /></figure>

drone flying at 15km/h, the maximum relative error would be 0.21%. In the absence of wind and for an obstacle moving at 50km/h (speed of cars in town), we would have a maximum relative error of 4.3%.

Thus, wind has very little effect on the range error, unlike the speeds of the fastest obstacles in the city, which are motor vehicles. In the following, we will neglect the impact of wind and simply consider that $v_3 = 0$.

## B. Autocorrelation function and telemetric measurement uncertainty

The acoustic sensors are made of piezoelectric materials capable of emitting an acoustic wave controllable by an input voltage $e(t)$ of frequency $f$ or to generate an output voltage $r(t)$ on reception of an acoustic wave centered on the frequency $f$. The control voltage of the acoustic transmitter is thus a sinusoidal voltage modulated in amplitude. If we consider the amplitude of the envelope of the acoustic emission, modulated in amplitude, as a square signal of amplitude $e_0$ of period $T$ and duty cycle $\chi$ the received signal is theoretically, neglecting the possible deformations of the carrier of the wave, a square signal of the same period and with the same cyclic ratio but with an attenuated amplitude that we note $r_0$ with a delay $t_{a-r}$. To avoid that the return wave in amplitude is superimposed on 2 consecutive emissions in amplitude, it is necessary to choose a value of $\chi$ slightly lower than 0.5. Moreover, to avoid that the return wave is superimposed on the following consecutive emission, it is necessary to choose $T$ such that:

$$T(1 - \chi) \geq 2P/c$$

With $P$ is the detection range which represents the distance beyond which the return acoustic wave will not be detected by the ultrasonic receiver. We note by $G$ the quantity related to the product of correlation between the envelopes of the emission signals noted $e_a(t)$ and of reception $r_a(t)$ at a given time $t_0$ and such that by definition:

$$G = \frac{1}{T} \int_{t_0}^{t_0+T} e_a(t) r_a(t) dt$$

By normalizing to 1 the amplitudes of $e_a$ and $r_a$ that is, in the case of $e_a(t)$:

$$e_a(t) = \begin{cases} 1 & \text{pour } t_0 < t \leq t_0 + \chi T \\ 0 & \text{pour } t_0 + \chi T < t \leq t_0 + T \end{cases}$$

And in the case of $r(t)$:

$$r_a(t) = \begin{cases} 1 & \text{pour } t_0 + t_{a-r} < t \leq t_0 + t_{a-r} + \chi T \\ 0 & \text{pour } t_0 + t_{a-r} + \chi T < t \leq t_0 + t_{a-r} + T \end{cases}$$

We have then considering that, to recall, $v_3 = 0$:

$$G = \chi - \frac{t_{a-r}}{T} = \chi - \frac{2cd}{T(c + v_1)(c - v_2)}$$

By measuring $G$ and considering that the obstacle is immobile, the drone is then able to measure a distance with the obstacle $r'$ such that:

$$d' = \frac{T}{2}(\chi - G)(c + v_1)$$

The relative error is therefore written in absolute terms:

$$e_r = \left| \frac{d' - d}{d} \right| = \frac{1}{(\chi - G)(1 - \frac{v_2}{c})} \left( \Delta G + \frac{|v_2|}{c}(\chi - G) \right)$$

Where $\Delta G$ is the absolute error on the estimate of $G \neq \chi$. If $\Delta G = 0$ there is indeed an error of 4.3% when considering $|v_2| = 50\text{km. h}^{-1}$, the maximum travel speed in the city. Next, we will quantify the sources of uncertainty for the estimation of $G$ and thus quantify the quantity $\Delta G = \Delta(\chi - G)$.

## C. Sources of uncertainty in the estimation of the quantity $G$

### D. Uncertainty related to the phase shift of the receiving electronics

At the receiving sensor, it is necessary to amplify the acoustic wave reflected on the obstacle. For this, the amplification electronics must have a maximum gain around the frequency of the carrier $f = 40\text{kHz}$. This electronics can cause a phase shift equivalent to a delay or an advance of arrival of return wave. In the worst case, the phase shift is $\pi$ rad and the time shift would then be $\frac{1}{2f}$ which would correspond to a telemetric error of $\frac{c}{4f} = 2,1\text{mm}$ which would be negligible. Thus, the receiving electronics do not contribute significantly to the uncertainty on the estimation of the value of $G$.

1) Uncertainty related to the digitization of signals

The transmission and reception signals are necessarily digitized before calculating the quantity $G$ defined such that for $k$ integer representing the k-th analog to digital conversion (with $k = 1$ representing the first analog to digital conversion) in the time interval $I = [t_0 + t_{a-r}, t_0 + \chi T]$ we have:

$$G = \frac{1}{T F_e} \sum_{k=1}^{M} e[k] \times r[k]$$

Where $e[k]$ and $r[k]$ are the digitized values of the transmission $e(t)$ and reception voltages $r(t)$. With $F_e$ representing the sampling frequency of the analog-to-digital converter and with $M$ a natural number equal to the integer part of the quantity $(\chi T - t_{a-r})F_e$. The parameter $M$ then represents the number of samples digitized on a duration $\chi T - t_{a-r}$ of the interval $I$. Moreover, we have the following conditions:

$$f \gg F_e \gg \frac{1}{T}$$

On the temporal level, we can write the voltages $e$ and $r$ such as:

$$\begin{aligned} e(t) &= e_p(t)e_a(t) \\ r(t) &= r_p(t)r_a(t) \end{aligned}$$
<figure><img src="image_7.png" /></figure>

With $e_p(t)$ and $r_p(t)$ the voltages of the carriers (perfectly sinusoidal voltages of frequency $f$) respectively of emission and reception.

Algorithmically, it is possible to define $e[k]$ and $r[k]$ in order to normalize them if they exceed a certain value. Thus, we can define them such that for any $k \in \mathbb{N}$:

$$e[k] = \begin{cases} 1 & \text{si } e \ge n_e s \\ 0 & \text{sinon} \end{cases}$$

$$r[k] = \begin{cases} 1 & \text{si } r \ge n_r s \\ 0 & \text{sinon} \end{cases}$$

Where $n_e$ and $n_r$ are natural numbers greater than or equal to 1 and $s$ represents the discretization step (in $V$). On an amplitude of 5V, a sampling on 10bits gives $s \approx 4,9mV$. Here, $e$ and $r$ represent the voltage values of the transmission $e(t)$ and reception signals $r(t)$ that need to be sampled. To avoid considering the electronic noise as part of the useful signal, it is necessary to choose $n_e$ and $n_r$ such that:

$$n_e > \frac{3\sigma_{b_e}}{s}, \quad n_r > \frac{3\sigma_{b_r}}{s}$$

With $\sigma_{b_e}$ and $\sigma_{b_r}$ are the standard deviations of the electronic noise of the transmitting circuit and the receiving circuit respectively. When the standard deviation of the electronic noise of the receiving circuit is important and when $r(t)$ has a small amplitude (far obstacle) even after amplification, the digitization may be problematic by not detecting the useful reception signal. In the time interval $t \in I = [t_0 + t_{a-r}, t_0 + \chi T]$ we have $e_a = r_a = 1$ and ideally it is necessary to have for all $k \in [1, M]$, $e[k] = r[k] = 1$ except that this is not the case at each sampling, which can generate an uncertainty in the measurement of $G$. In this time interval $I$ we can then write:

$$e(t) = e_p(t) = \frac{e_0}{2}(1 + \cos(2\pi ft + \varphi))$$

$$r(t) = r_p(t) = \frac{r_0}{2}(1 + \cos(2\pi f(t - t_{a-r}) + \varphi))$$

With $\varphi$ any phase shift and $e_0, r_0$ the amplitudes of the transmission and reception voltages.

In probabilistic terms, the probabilities of sampling $e$ and $r$ when they are lower than their threshold values $n_e s$ and $n_r s$ are also respectively $p_e$ and $p_r$ such that:

$$p_e = \frac{1}{\pi}\arccos(1 - \eta_e)$$

$$p_r = \frac{1}{\pi}\arccos(1 - \eta_r)$$

$$\text{With } \eta_e = \frac{2n_e s}{e_0} \text{ and } \eta_r = \frac{2n_r s}{r_0}$$

However, this is a perfectly deterministic problem. Indeed, once the first sampling is done, it is possible to predict if the following samplings will give 1's or 0's. However, as the sampling frequency is low compared to the frequency of the carrier which is 40kHz, any

random fluctuation of the sampling frequency (electronic jitter) will make the following measurement unpredictable. For this reason, the prediction of the detection of the wanted signal will be treated in probabilistic form. Thus, for a $k \in [1, M]$ the product $e[k] \times r[k]$ is zero with a probability of $p$ such that:

$$p = p_e + p_r - p_e p_r$$

Thus, the median number of times that the scan provides the value 0 to the product $e[k] \times r[k]$ for $k \in [1, M]$ is $pM$. If we approximate $M$ à $(\chi T - t_{a-r})F_e$ the average uncertainty on $G$ would then be $\Delta G$ such that:

$$\Delta G = pG$$

One would then tend to find a value of $G$ lower than it should be and one could then consider the obstacle as more distant than it really is. By considering $v_2 = 0$ and $v_1$ negligible compared to the error on the estimation of the distance $d$ taking into account this problem of digitization would then be worth:

$$\Delta d \approx \frac{1}{2}pcTG$$

Depending on the value of $p$ the uncertainty on the depth estimate $d$ can be very important. It is therefore necessary to make the algorithm more complex for an abusive setting to zero for $e[k]$ or/and $r[k]$ when they should not be. To do this, the proposed algorithm is based on the simple principle of reducing an abusive 0 at a given time to a 1 if the A/D converter subsequently detects another 1 within less than $\chi T$. This program is functional from the first 1 detected. For a $k \in [1, M]$ given, a 0 abusive will certainly become a 1 if $k \gg 1$ and $k \ll M$ but this program will have flaws for the first conversions when the envelopes of the transmission and reception voltages are superimposed on the temporal plane but also for the last conversions. Indeed, for the first conversions, it is necessary to detect a first 1 for the program to be functional. It is thus possible to have a certain number of 0 before detecting the first 1. As for the last conversions, if we detect an abusive 0, there is less probability to detect a 1 thereafter because there will be less sampling to be done. We will then quantify the performance of this algorithm. There is very little chance that the noise exceeds the threshold of digitization to 1 outside the moments when the envelopes of the transmission and reception overlap. We will therefore consider this untimely phenomenon as non-existent. Thereafter, we will call by $\Delta G_-$ the contribution of the reduction of the value of $G$ at the beginning of conversion (Obtaining abusive 0 before having the first 1) and by $\Delta G_+$ that at the end of conversion (Obtaining abusive 0's without any other occurrence of 1 during the temporal superposition of the 2 envelopes). Thus, we have with this new algorithm an uncertainty on $G$ such that:

$$\Delta G = \Delta G_- + \Delta G_+$$
<figure><img src="image_8.png" /></figure>

To quantify ΔG_it is necessary, on the probabilistic level, to calculate the probability of obtaining a first 1 from k = 1 on the n-th conversion. That is to consider that the A/D converter gives abusive 0 for the first conversions and a 1 at the n − 1 conversions and a 1 at the n-th conversion. This probability noted p_(n) is worth for 1 ≤ n ≤ M :

$p_{-}(n) = p^{n-1}(1 - p)$

The median value of the number of possible abusive 0's at the beginning of the conversion is then 〈n〉_− such that :

$$
\langle n \rangle_{-} = \sum_{n=1}^{M} (n-1) \times p_{-}(n)
$$

Which can be written explicitly for p ≠ 1 :

$$
\langle n \rangle_{-} = \frac{p + (M - 1)p^{M+1} - Mp^{M}}{1 - p}
$$

We deduce that on average for p ≠ 1 :

$$
\Delta G_{-} = \frac{\langle n \rangle_{-}}{TF_{e}} = \frac{p + (M - 1)p^{M+1} - Mp^{M}}{(1 - p)TF_{e}}
$$

To quantify ΔG+, we will define an integer n ∈ [1, M] for which the A/D converter obtains an abusive 0 and we wish to quantify the probability for which, we would not obtain any more 1 (whereas we should) from this integer n to the integer M. We note this probability p+(n) such that :

$p_{+}(n) = p^{M-n+1}$

The median value of the number of possible abusive 0 at the end of the conversion is then 〈n〉_+ such that

$$
\langle n \rangle_{+} = \sum_{n=1}^{M} (M - n + 1) \times p_{+}(n)
$$

We deduce for p ≠ 1 :

$$
\langle n \rangle_{+} = p \frac{Mp^{M+1} - (M + 1)p^{M} + 1}{(1 - p)^{2}}
$$

We can then deduce that for p ≠ 1 :

$$
\Delta G_{+} = \frac{\langle n \rangle_{+}}{TF_{e}} = p \frac{Mp^{M+1} - (M + 1)p^{M} + 1}{(1 - p)^{2}TF_{e}}
$$

Thus, we can deduce that for p ≠ 1 :

$$
\Delta G = \frac{(2 - p)(p - p^{M+1}) - M(1 - p)p^M}{(1 - p)^2 TF_e}
$$

Note that if p_e may be low, p_r is all the closer to 1 as t_a-r is important. Considering that the voltage at the terminals of a piezoelectric acoustic receiver is sensitive to the acoustic overpressure on the sensor, a simplistic model would be to write simply by considering p_e ≪ p_r :

$$
p \sim p_r = \frac{1}{\pi} \arccos(1 - K t_{a-r}^2)
$$

With K a physical quantity specific to the acoustic device but also to the nature of the obstacle and defined as :

$$
K = \frac{2}{t_{a-r}^2 |_{\max}}
$$

Where $t_{a-r}|_{max}$ is the maximum time of flight for an acoustic wave to be detected.

On the other hand, we can write approximately M = (χT − t<sub>a-r</sub>)F<sub>e</sub>. Therefore, we can estimate the value of ΔG as a function of t<sub>a-r</sub> and thus of d ≈ <sup>1</sup>&frasl;<sub>2</sub> ct<sub>a-r</sub> knowing t<sub>a-r</sub>|<sub>max</sub> (and thus the maximum detection range of the return wave : t<sub>a-r</sub>|<sub>max</sub> = 2P/c if v<sub>1</sub> = v<sub>2</sub> = v<sub>3</sub> = 0 and, in this case we can write K = <sup>c<sup>2</sup></sup>&frasl;<sub>2P<sup>2</sup></sub>). Thus, for a detection range of 5m (it is difficult to envisage higher ranges for acoustic transmitters and receivers), it will be necessary to take in the precise case where χ = 0,5 at least T = 58,3ms. Here, we can see that t<sub>a-r</sub>|<sub>max</sub> = T(1 − χ). Furthermore, if the acquisition is done on Arduino, the maximum sampling frequency without modifying the default configuration of the ATmega 328 microcontroller registers, is F<sub>e</sub> = 3,77kHz. We can then plot the relative error of G, ΔG/G as a function of the distance r between the device and the obstacle in the case where v<sub>1</sub> = v<sub>2</sub> = v<sub>3</sub> = 0.

<figure><img src="image_9.png" /><figcaption>Figure 1: Evolution of the relative error of G, ΔG/G as a function of the distance d between the device and the obstacle in meters.</figcaption></figure>

The more *d* is, the more important the error is. The last point for *d* = 4,95*m* is inconsistent in terms of measuring the relative error on *G* because for this point *M* = 1 only, which is not the case for the other measurement points of the curve. Obviously, by increasing the value of *F*<sub>e</sub> we could have better results. Hereafter, we have plotted for *d* = 4,1*m* the relative error on *G*, Δ*G*/*G* as a function of *F*<sub>e</sub> ≥ 5kHz.

<figure><img src="image_10.png" /><figcaption>Figure 2: Evolution of the relative error of G, ΔG/G as a function of the sampling frequency of the A/D converter for F<sub>e</sub> ≥ 5kHz for d = 4, 1m</figcaption></figure>

The evolution of this relative error is in 1/F<sub>e</sub> for
sampling frequencies higher than 5kHz.
<figure><img src="image_11.png" /></figure>

For low sampling frequencies $F_e \le 4,5kHz$ the evolution of the relative error on G does not evolve anymore in $1/F_e$. Thus, below, the evolution of $\Delta G/G$ as a function of $F_e \le 4,5kHz$ for $d = 4,1m$.

<figure><img src="image_12.png" /><figcaption>Figure 3: Evolution of the relative error of G, $\Delta G/G$ as a function of the sampling frequency of the A/D converter for $F_e \le 4,5kHz$ for $d = 4,1m$</figcaption></figure>

We can see that for a range of 5m and an average sampling frequency of 3770Hz, the relative error on G exceeds 10% beyond $d = 3,5m$ with a telemetric error of $\Delta d \approx \frac{1}{2}cT\Delta G = 14cm$. If we estimate the relative telemetric error $\Delta d/d$ in the case $v_1 = v_2 = v_3 = 0$ we see that we exceed only once the 10% relative error on the distances between 1m and P = 5m.

<figure><img src="image_13.png" /><figcaption>Figure 4: Evolution of the relative error of d, $\Delta d/d$ as a function of the distance d between the device and the obstacle in meters</figcaption></figure>

We therefore tend to think that the device is robust for the detection and characterization of the distance of the obstacle for the needs of navigation of a drone flying at low speed because it will be able to anticipate the obstacle and be able to bypass it especially that at the limit of the range, the relative error becomes less than 1% and this is verified even for ranges P less than 5m. In fact for $v_1 = v_2 = v_3 = 0$, $\frac{\Delta d}{d} = \frac{\Delta G}{\chi - G}$, beyond the fact that G decreases and therefore $\chi - G$ grows with d, $\Delta G$ is also a decreasing function of d beyond a certain distance as can be seen on the figure below for $F_e = 3770Hz$ and P = 5m. We notice then that its evolution is similar to that of $\frac{\Delta d}{d}$ as a function of d.

<figure><img src="image_14.png" /><figcaption>Figure 5: Evolution of the error on G, $\Delta G/G$ as a function of the distance d between the device and the obstacle in meters</figcaption></figure>

## E. Uncertainty related to the number of samples M

As the calculation of G is done according to a discrete sum, it is possible for small values of M to have a deviation with the integral calculation. This is mainly present when the temporal superposition of the two waves is not consistent and specifically at the limit of detection range of the reflected wave. Indeed, noting by $E(x)$ the integer part of a real $x \ge 0$ we have by definition $M = E((\chi T - t_{a-r})F_e)$ and thus the autocorrelation product in the form of a discrete sum that we call here $G_d$ which is:

$$G_d = \frac{E((\chi T - t_{a-r})F_e)}{E(F_e T)}$$

Thus, mathematically $G_d \neq G$ where G represents the correlation product in integral form of $e(t)$ and $r(t)$ over the time interval $I$. As a reminder:

$$G = \frac{1}{T} \int_{t_0}^{t_0+T} e_a(t)r_a(t)dt$$

We can approximate $E(F_e T) \approx F_e T$ thus, for $T = 58,3ms$ and $F_e = 3770Hz$ we have $F_e T = 219,8$ and thus the relative error on this approximation is less than 0.4%. Analytically, we recall that:

$$G = \chi - \frac{t_{a-r}}{T}$$

Now, we notice that:

$$(\chi T - t_{a-r})F_e \le E((\chi T - t_{a-r})F_e) + 1$$

And so:

$$\Delta G = G - G_d \le \frac{1}{F_e T}$$

We can then plot the relative error on G, $\Delta G/G$ as a function of the distance device - obstacle for a range of 5m, for $F_e = 3770Hz$, $\chi = 0,5$ and $T = 58,3ms$.
<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>Figure 6: Evolution of the relative error of G, ΔG/G as a function of the distance d between the device and the obstacle in meters.</figcaption></figure>

The telemetric error as a function of the distance device - obstacle varies in a decreasing way.

<figure><img src="image_17.png" /><figcaption>Figure 7: Evolution of the relative error of d, Δd/d as a function of the distance d between the device and the obstacle in meters</figcaption></figure>

The relative telemetry error can be considered small if the error on the estimation of G is done in a discrete way. The interest of such a technological device is the fact that the relative error is lower at greater distances and thus makes it possible to measure the distance to an obstacle when that is at the limit of range of detection of the device. This is due to the fact that ΔG is increased by a constant value whatever the value of G which decreases and thus decreases the relative telemetric error.

## F. Summary of the total uncertainties on the accuracy of the telemetric measurement

Neglecting the uncertainty related to the electronic phase shift, the total relative telemetric error of the uncertainties related to the digitization of the signals and the number of samples *M* of the digitization varies for *F*<sub>e</sub> = 3770Hz, *P* = 5m according to the curve of the graph in the figure below.

<figure><img src="image_18.png" /><figcaption>Figure 8: Evolution of the total relative error of d, Δd/d as a function of the distance d between the device and the obstacle in meters</figcaption></figure>

The device therefore has the lowest relative error at the limit of the obstacle detection range, whose value is below 2%, which highlights the theoretical robustness of such a technological device based on an innovative measure for distance estimation. However, it should be noted that the detection range is specific to each type of obstacle (nature, shape, orientation etc.) and cannot be a guarantee for obtaining a reliable distance value in an environment with multiple types of obstacles. However, the telemetric accuracy remains satisfactory (mostly less than 10% error for any distance less than *P*).

## V. CONCLUSION

The acoustic devices based on the time of flight to realize distance measurements are technologically mature enough and allow to measure distances with a precision all the more weak as the distance of the obstacle to measure is important. The work carried out makes it possible to introduce a new concept of distance measurement based on the correlation of the emitted and received signals as long as they are superimposed temporally and this from ultrasonic acoustic wave. The work allowed to quantify the impacts of some measurement uncertainties on the telemetry performances of this type of technology. First of all, the theoretical measurements of the impacts of the wind speed or of the obstacle allowed to show that the wind contributes little to this relative telemetric error of the order of 0,21% only whereas the mobility of the obstacle contributes for a speed of 50km/h to induce a maximum relative telemetric error of 4,1%. Subsequently, the work has highlighted some measurement uncertainties related to the acquisition of the acoustic return wave and its digitization. The acquisition of the acoustic wave by an electronic amplification generates a temporal phase shift whose impact can be considered negligible on the telemetric measurement. Indeed, this phase shift delays the arrival of the acoustic wave and contributes to generate a constant telemetric error, whatever the distance of the device from the obstacle, of hardly a few millimeters.
<figure><img src="image_19.png" /></figure>

Finally, the digitization of the return signal made up of a carrier at 40kHz and a square envelope with a duty cycle of approximately 0.5 can generate errors in particular when the 2 envelopes of the transmitted wave and the received wave are superimposed temporally. A first type of digitization error at a given frequency is based on the capacity of an algorithm to detect a non-zero voltage from the value of the transmission and reception voltages when the value of each voltage exceeds a certain voltage threshold value and this when the envelope has a non-zero amplitude from a carrier whose amplitude is variable in time taking into account a given discretization step and a non-negligible standard deviation of noise which one does not necessarily wish to detect on the transmission and reception channels. A second source of uncertainty related to digitization is the fact that the calculation of the correlation product between the transmission and reception voltages is done discretely and not in integral form. The cumulations of these two digitization errors make it possible to reach typically for a device having a range of detection of 5m, relative telemetric errors at most of 11% when the distance to the obstacle is close to the range of detection but, paradoxically, the precision of the telemetric measurement is all the more important as the distance of the obstacle approaches the range of detection with an error lower than 1 to 2% at the range of detection. The technological device that we propose is singular in this atypical behavior. It would be able to measure at the detection range for a particular type of obstacle (example: concrete), the precise value of its distance. In a general framework, the telemetric error remains less than 10% for any distance below the detection range of the acoustic return wave, which makes this technological device robust in the complementarity of multiple sensor measurements for the needs of autonomous navigation of urban drone.

## ACKNOWLEDGMENT

This work was carried out by Astek Research Network part of Research and Development department of Astek Group.

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