# Mobile Synchronization Recovery for Ultrasonic Indoor Positioning

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Architecture and Synchronization Recovery

_{SILENCE}, 1, 2, 3, 4…) starting from the time t

_{0BEACONS}, and the time interval between emissions is T

_{REPETITION}(see Figure 2). Each emission duration is T

_{EMISSION}< T

_{REPETITION}. The beacons belong to the same circuit and are intrinsically synchronized with each other. The sequence of the four signals is repeated in identical frames emitted at regular time intervals of duration T

_{FRAME}(frame repetition time). Chirps are the simplest and more suitable ultrasonic signals that allow to take full advantage of correlation-based ranging techniques. Anyhow, the following reasoning applies to any type of signal that is able to provide a ranging with the required accuracy.

_{FRAME}, according to its internal timer.

_{0BEACONS}; (2) the listening window which starts at t

_{0MD}(see Figure 2). They are repeated with equal periods but with different starting times. There is, therefore, a lag or time offset between the two processes T

_{OFFSET =}t

_{0BEACONS}- t

_{0MD}that is unknown.

_{OFFSET}, for example Reference Broadcasting [37,38], which however require additional hardware (wires, RF, etc.) and protocols. On the contrary, here an innovative solution completely devoid of synchronization hardware that allows the recovery of T

_{OFFSET}is proposed.

_{1}, TOA

_{2}, TOA

_{3}, TOA

_{4}, respectively. From these, three TDOA dt

_{j}(j = 1, 2, 3) are obtained:

_{1}= TOA

_{2}− TOA

_{1}

_{2}= TOA

_{3}− TOA

_{1}

_{3}= TOA

_{4}− TOA

_{1}

_{air}:

_{j}(j = 1, 2, 3) from the time differences are calculated as:

_{1}= dt

_{1}⋅c

_{ai}

_{r}= l

_{2}− l

_{1}

_{2}= dt

_{2}⋅c

_{air}= l

_{3}− l

_{1}

_{3}= dt

_{3}⋅c

_{air}= l

_{4}− l

_{1},

_{1}, l

_{2},…l

_{4}are the distances between the MD and the four beacons. In what follows, it is assumed that, as usual, in a room the air flows and temperature gradient are kept under control for reasons of well-being of the occupants of the room. Residual fluctuations have very small effects on positioning accuracy. Alternatively, methods based on direct or indirect measurement of the speed of sound along the propagation path (see e.g., [40]) can be adopted.

_{RPi}= (x

_{RPi}, y

_{RPi}, z

_{RPi}) is the reference position of the i

^{th}(i = 1, 2,…4) beacon.

_{RP1}= (0, 0, 0), X

_{RP2}= (a, 0, 0), X

_{RP3}= (a, b, 0), X

_{RP4}= (0, b, 0), respectively, the (4) can be rewritten as follows:

_{1}, finally is obtained:

_{1}and from this, through the (2), of l

_{2}, l

_{3}, and l

_{4}. From l

_{1}, TOF

_{1}= l

_{1}/c

_{air}is obtained, and from this, knowing TOA

_{1}= TOF

_{1}+ T

_{OFFSET}, T

_{OFFSET}is finally estimated.

_{OFFSET}from (9) strongly depends on the position of the MD with respect to the four beacons. The (9) produces bad estimates in some positions, where the denominator becomes very small or almost zero. In the space points set where ${d}_{1}+{d}_{3}-{d}_{2}=0$ there is an unlimited error and (9) is “blind”. Each T

_{OFFSET}estimated directly from (9) is therefore affected by a considerable uncertainty or noise (see Figure 6).

_{OFFSET}is the same for all the beacons, which belong to the same synchronous circuit, and, above all, T

_{OFFSET}is the same for any position of the MD.

_{OFFSET}can be used to refine the estimate of the true value of T

_{OFFSET}.

_{OFFSET}is the only unknown factor.

_{OFFSET}were really constant, in order to obtain an accurate estimate of the T

_{OFFSET}, it would be enough to make an average of the noisy values of the rough T

_{OFFSET}estimates coming from (9), provided that the noise has zero mean value. This is in fact the approach followed by [33], which used a moving average. In practice, however, this approach does not work well.

_{OFFSET}. Therefore, T

_{OFFSET}(k) has to be estimated as a function of time, where k is a time index (see below).

_{FRAME}duration, i.e., the repetition time of the positioning operation, ref(k) is the noisy ramp to be followed, i.e., the current value of T

_{OFFSET}(k), y(k) = T

^{*}

_{OFFSET}(k) is the estimate of the true T

_{OFFSET}(k) after the noise rejection, x

_{1}(k) and x

_{2}(k) are the internal states of the ramp follower, and K is a constant parameter tuned by trial and error procedure, the smaller its value, the greater the noise rejection and convergence time of the ramp follower. Note that (10) is only one of the possible ways to estimate a ramp shaped T

^{*}

_{OFFSET}(k), however a thorough discussion on this issue is beyond the scope of this work.

_{OFFSET}which varies very slowly, obtaining the T

^{*}

_{OFFSET}. It is worth noting that (10) converges recovering the synchronization without having any prior information on the T

_{OFFSET}and starting from the first value provided by (9). It is also possible to observe that Equations (9)–(10) do not introduce any constraint on the trajectory of the measured MD positions to estimate l

_{1}and T*

_{OFFSET}. In fact, (10) converges regardless of the trajectory followed by the MD.

^{*}

_{OFFSET}(k) sequence thus obtained, although heavily filtered over time, allows to obtain the TOF(k)

_{i}= TOA(k)

_{i}+ T

^{*}

_{OFFSET}(k) sequence and therefore to use the intersection of spheres (12)–(13), which produces much more accurate results than the intersection of hyperboloids (4), therefore overcoming the issue of noisy results from (4). Ultimately, this approach allows trajectories to be tracked with reduced noise even in the presence of abrupt variations.

_{OFFSET}(k) estimate allows finding the estimates l

^{*}

_{i}(k) through the following:

^{*}

_{1}(k) = [TOA

_{1}(k) + T

^{*}

_{OFFSET}(k)]·c

_{air}

^{*}

_{2}(k) = [TOA

_{2}(k) + T

^{*}

_{OFFSET}(k)]·c

_{air}

^{*}

_{3}(k) = [TOA

_{3}(k) + T

^{*}

_{OFFSET}(k)]·c

_{air}

^{*}

_{4}(k) = [TOA

_{4}(k) + T

^{*}

_{OFFSET}(k)]·c

_{air}.

_{i}≥ 0 (i = 1, 2...4).

_{OFFSET}(0) and T*

_{OFFSET}(0) are set to zero. Subsequently, the algorithm works as an infinite loop. During the signal acquisition phase, the ultrasonic signal is received, sampled, and recorded for a period of time suitable for surely acquiring all four ultrasonic emissions considering the duration of the twitter used. The recording duration is therefore longer than the T

_{FRAME}used. The details are described in Section 3. The successive step includes the cross-correlation between the recorded signal and a copy of the expected signal, previously stored in the MD memory. In this way, four sharp consecutive peaks are obtained. The positions, or lags, of the peaks in the computed cross-correlation vector are proportional to the four TOAs of the four ultrasonic signals travelling from the emitter to the MD.

_{OFFSET}frame by frame, this order may appear altered in reception. For example, due to the clock drift, after a certain operation time, the sequence “4, silence window, 1, 2, 3” can be received instead of the expected sequence “1, 2, 3, 4, silence window”. Furthermore, it is clear that T

_{OFFSET}has a cyclic behavior, since every time it exceeds the T

_{FRAME}value, it returns to zero, wrapping around the value T

_{FRAME}as in the modulus operation. The correct sequence of events is restored at the receiver periodically in the third step of the algorithm by considering each intervened T

_{OFFSET}reset along time.

## 3. Algorithm Simulation and Experimental Results

#### 3.1. System Simulation Setup and Results

_{OFFSET}at the accuracy level that can be achieved with the given sampling frequency F

_{S}. The beacon system is fixed at the center of the ceiling of a 4 × 4 × 3 m

^{3}room. The beacons are fixed at the corners of a square having side a = 50 cm, so as to constitute an element that can easily be integrated into a typical room ceiling panel.

_{TOA}[18,38]. In the following simulations, F

_{S}= 1/T

_{S}= 192 kS/s (F

_{S}= 192 kHz) and Δ

_{TOA}= ± T

_{S}/2 have been set.

_{S}corresponds to a space interval of 1.8 mm. A random value for the starting T

_{OFFSET}(0) and a Beacon Set-MD relative clock drift of 200 ppm has been assumed.

_{OFFSET}(k) = T

_{OFFSET}(0) + Δ

_{CLOCK}⋅k,

_{CLOCK}= 25 μs is computed taking into account that a 200 ppm clock drift applies to an external 32768 Hz crystal, from which the microprocessor clock is derived, and T

_{FRAME}is 0.125 s.

^{2}rectangle, with 12 positioning measures or frames every 1 m along the way (se Figure 4) and the T

_{FRAME}is 0.125 s (i.e., frame rate 8 Hz). The rectangular trajectory is repeated for a total of 200 positioning frames to show the convergent behavior of the T

_{OFFSET}recovery process. Figure 4 shows the estimated trajectory compared to the true one indicated by the last 12 position estimates, i.e., after algorithm convergence. The overall 3D positioning error at coordinates (x’, y’, z’) is given by the Euclidean distance ${e}_{ED}=\sqrt[2]{{\left({x}^{\prime}-x\right)}^{2}+{\left({y}^{\prime}-y\right)}^{2}+{\left({z}^{\prime}-z\right)}^{2}}$ between each ground truth point at coordinates (x’, y’, z’) and the estimated one at coordinates (x, y, z). Figure 5 reports the e

_{ED}behavior along the 200 positioning frames. The last 12 values refer to the positioning shown in Figure 4, after the convergence of the process from successive application of (9) and (10) starting from no prior information on the T

_{OFFSET}value.

_{ED}is shown in Figure 5: The error obtained using the proposed method decreases over time as ramp follower converges (thick solid line) approaching the small error achievable by the synchronized TOF positioning (dash-dot line), while the error obtained using TDOA remains large (dotted line).

_{OFFSET}, and its over time estimate T

^{*}

_{OFFSET}using the ramp follower (10) with K = 2.5.

#### 3.2. Experimental System Realization

_{FRAME}= 125 ms, T

_{EMISSION}= 2.66 ms, T

_{REPETITION}= 5 ms, and T

_{SILENCE}= 109.34 ms (see Figure 2).

^{2}panel. The four transducers are placed at the corners of a 50 × 50 cm

^{2}square, face up toward the room volume (see Figure 9).

_{S}= 192 kHz). The microphone is an FG-6163 (Knowles Acoustics, Itasca, Illinois, USA), a micromachined condenser microphone encapsulated in a cylindrical package, 2.6 mm length and diameter, 0.79 mm acoustical receiver window diameter, and 80 mg weight (see Figure 8). This microphone is the same one that equips the MD described in [18].

#### 3.3. Experimental Results

_{FRAME}). This positioning rate is actually limited by the computational power of the employed PC.

_{OFFSET}along the successive positioning frames. After the convergence of the synchronization recovery process, the microphone coordinates measured by the system are in good agreement with the ones measured using the synchronism information (see Figure 10).

_{OFFSET}estimation. Microphone absolute positioning uncertainty is instead mainly due to the TOA time quantization error propagation, deeply discussed in our previous work [18].

_{OFFSET}, and its estimate over time T

^{*}

_{OFFSET}using the ramp follower (10) with K = 2.5.

#### 3.4. Error Propagation Remarks

_{1}, x

_{2},…x

_{k}), small changes Δx

_{1}in x

_{1}, Δx

_{2}in x

_{2},…, Δx

_{k}in x

_{k}, all propagate to produce a worst-case small change Δy in y in the following manner:

_{1})

^{2}have been set to zero, however, it is sufficiently accurate for most purposes.

_{j}is the quantization error of the measurement of the Beacon-MD distance. In the worst case, Δl

_{j}is equal to the half quantization interval, i.e., Δl

_{j}= vT

_{s}/2, where v is the speed of the sound in air and T

_{s}the sampling period. In (17), the resulting values must be considered values that cannot under any circumstances be exceeded, and not as standard deviations. However, in practice, the typical values may be smaller but reasonably of the same order of magnitude.

_{i}≤ 0.9 mm (i.e., half range quantum Δl

_{j}= vT

_{s}/2, with F

_{S}= 192 kHz) and considering that the geometrical center of the system is (0.25, 0.25, 3), at the furthest and less favorable point in the 4 × 4 × 3 m

^{3}room P = (−1.75, 2.25, 0) using (17) $\Delta {P}_{MAX}=\sqrt{\Delta {x}^{2}+\Delta {y}^{2}+\Delta {z}^{2}}=2.94\text{}cm.$

_{MAX}must be regarded as upper bound of the real error, which occurs only in the worst case when all the Δl

_{i}(i = 1, 2, 3) assume their maximum value at the same time. Equation (17) shows that to obtain a positioning accuracy of the order of centimeters, measurement accuracy of the order of millimeters is required. Moreover, it shows that the error magnitude is approximately inversely proportional to the length of the sides a and b of the rectangle formed by the Beacons.

_{OFFSET}has been estimated.

_{1}by (9) following the variation of a and b, which (9) does not explicitly show. Figure 15 shows the simulation estimation error trend of (9) due to the quantization error on the estimate of d

_{1}, d

_{2}, and d

_{3}, for a grid of points belonging to the plane z = 1.5 m (Beacon Set at z = 3), and by considering in sequence a = b = 1.5, 0.5, 0.25 m, respectively. Due to the very large dynamic of the error values, they are shown in decibels (dB).

_{1}, and consequently of T

_{OFFSET}, from (9) strongly depends on the position of the MD with respect to the four beacons, but also on the beacon separation, as shown by Figure 15. The (9) produces bad estimates in some positions, where the denominator becomes very small or almost zero, as in the neighborhood of the “central cross” artifact visible in Figure 15. On the “central cross”, or in its neighborhood, due to the combined effect of the quantization error on the d

_{i}s, it results ${d}_{1}+{d}_{3}-{d}_{2}=0$ in many points with an unlimited error. In practice, the values of (9) are suitably limited with an adequate threshold tuned by trial and error procedure before feeding them to the ramp follower (10). Larger Beacon Sets considerably reduce the size of the “bad” region, as shown by Figure 15. On the other hand, smaller Beacon Sets are more easily integrated into room ceiling panels.

#### 3.5. Discussion

_{OFFSET}and that, after a certain transient time from the beginning of the operations, the synchronism is recovered, thus avoiding any RF reference signals. The clock drift problem afflicting every real system is solved with a suitable output estimator. The accuracy of the T

_{OFFSET}estimate depends on the kind of estimator employed, in our case a ramp follower. As a drawback, while the ramp follower converges to the true value, it provides a sufficiently small error only after many estimation cycles. Estimation could be improved with a more refined estimator, which is however beyond the scope of this work.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Zhang, D.; Xia, F.; Yang, Z.; Yao, L.; Zhao, W. Localization Technologies for Indoor Human Tracking. In Proceedings of the 2010 5th International Conference on Future Information Technology, Busan, South Korea, 21–23 May 2010; pp. 1–6. [Google Scholar]
- Przybyla, R.J.; Tang, H.Y.; Shelton, S.E.; Horsley, D.A. 3D ultrasonic gesture recognition. In Proceedings of the 2014 IEEE International Solid-State Circuits Conference Digest of Technical Papers (ISSCC), San Francisco, CA, USA, 9–13 February 2014; pp. 210–211. [Google Scholar]
- Ionescu, R.; Carotenuto, R. 3D Localization and Tracking of Objects Using Miniature Microphones. Wirel. Sens. Netw.
**2011**, 3, 147–157. [Google Scholar] [CrossRef][Green Version] - Carotenuto, R. Touchless 3D gestural interface using coded ultrasounds. In Proceedings of the 2012 IEEE International Ultrasonics Symposium, Dresden, Germany, 7–10 October 2012; pp. 146–149. [Google Scholar]
- Ebisawa, Y. A pilot study on ultrasonic sensor-based measurement of head movement. IEEE Trans. Instrum. Meas.
**2002**, 51, 1109–1115. [Google Scholar] [CrossRef][Green Version] - Kasprzak, H.T. “Ultrasonic measurement of fine head movements in a standard ophthalmic headrest”. IEEE Trans. Instrum. Meas.
**2010**, 59, 164–170. [Google Scholar] [CrossRef] - Torres-Solis, J.; Falk, T.H. A review of indoor localization technologies: Towards navigational assistance for topographical disorientation. Ambient Intell.
**2010**, 51–84. [Google Scholar] [CrossRef][Green Version] - Marco, A.; Casas, R.; Falco, J.; Gracia, H.; Artigas, J.I. Location-based services for elderly and disabled people. Comput. Commun.
**2008**, 31, 1055–1066. [Google Scholar] [CrossRef] - Mainetti, L.; Patrono, L. A survey on indoor positioning systems. In Proceedings of the 22nd International Conference on Software, Telecommunications and Computer Networks (SoftCOM), Split, Croatia, 7–19 September 2014; pp. 111–120. [Google Scholar]
- Whitehouse, K.; Karlof, C. A practical evaluation of radio signal strength for ranging-based localization. Mob. Comput. Commun. Rev.
**2007**, 11, 41–52. [Google Scholar] [CrossRef] - Amann, M.C.; Bosch, T.M.; Lescure, M.; Myllylae, R.A. Laser ranging: A critical review of unusual techniques for distance measurement. Opt. Eng.
**2001**, 40, 10–19. [Google Scholar] - Yüzbaşioğlu, Ç. Improved range estimation using simple infrared sensors without prior knowledge of surface characteristics. Meas. Sci. Technol.
**2005**, 16, 1395. [Google Scholar] [CrossRef] - GiPS tech Srl. Available online: http://www.gipstech.com/it/indoor-localization-and-navigation-technology/ (accessed on 20 March 2019).
- Ijaz, F.; Yang, H.K.; Ahmad, A.W. Indoor Positioning: A Review of Indoor Ultrasonic Positioning systems. In Proceedings of the 15th International Conference on Advanced Communications Technology (ICACT), PyeongChang, South Korea, 27–30 January 2013; pp. 1146–1150. [Google Scholar]
- Ureña, J.; Hernández, Á.; García, J.J.; Villadangos, J.M. Acoustic Local Positioning with Encoded Emission Beacons. Proc. IEEE
**2018**, 106, 1042–1062. [Google Scholar] [CrossRef] - Kino, G.S. Acoustic Waves: Devices, Imaging, and Analog Signal Processing; Prentice-Hall: Upper Saddle River, NJ, USA, 1987; Volume 100. [Google Scholar]
- Saad, M.M.; Bleakley, C.J. Robust high-accuracy ultrasonic range measurement system. IEEE Trans. Instrum. Meas.
**2011**, 60, 3334–3341. [Google Scholar] [CrossRef][Green Version] - Carotenuto, R.; Merenda, M.; Iero, D. An indoor ultrasonic system for autonomous 3D positioning. IEEE Trans. Instrum. Meas.
**2019**, 68, 2507–2518. [Google Scholar] [CrossRef] - Seco, F.; Jiménez, A.R.; Prieto, C.; Roa, J.; Koutsou, K. A survey of mathematical methods for indoor localization. In Proceedings of the 2009 IEEE International Symposium on Intelligent Signal Processing, Budapest, Hungary, 26–28 August 2009; pp. 9–14. [Google Scholar] [CrossRef]
- Nardone, S.C. A closed-form solution to bearings-only target motion analysis. IEEE J. Oceanic Eng.
**1997**, 22, 168–178. [Google Scholar] [CrossRef] - Navidi, W.; Murphy Jr, W.S. Statistical methods in surveying by trilateration. Comput. Stat. Data Anal.
**1998**, 27, 209–227. [Google Scholar] [CrossRef] - J Abreu, J.M.; Ceres, R.; Calderon, L.; Jiménez, M.A. Measuring the 3D-position of a walking vehicle using ultrasonic and electromagnetic waves. Sens. Actuators
**1999**, 75, 131–138. [Google Scholar] [CrossRef] - Ho, K.C. Solution and performance analysis of Geolocation by TDOA. IEEE Trans. Aerosp. Electron. Syst.
**1993**, 29, 1311–1322. [Google Scholar] [CrossRef] - Ruiz, D.; Ureña, J.; Gude, I.; Villadangos, J.M.; García, J.C.; Pérez, C. New iterative algorithm for hyperbolic positioning used in an Ultrasonic Local Positioning System. In Proceedings of the 2009 IEEE Conference on Emerging Technologies & Factory Automation, Mallorca, Spain, 22–25 September 2009; pp. 1–4. [Google Scholar] [CrossRef]
- Filonenko, V.; Cullen, C. Indoor Positioning for Smartphones Using Asynchronous Ultrasound Trilateration. ISPRS Int. J. Geo-Inf.
**2013**, 2, 598–620. [Google Scholar] [CrossRef][Green Version] - Bordoy, J.; Hornecker, P.; Höflinger, F.; Wendeberg, J.; Zhang, R.; Schindelhauer, C. Robust tracking of a mobile receiver using unsynchronized time differences of arrival. In Proceedings of the International Conference on Indoor Positioning and Indoor Navigation, Montbeliard-Belfort, France, 28–31 October 2013; pp. 1–10. [Google Scholar] [CrossRef]
- Wendeberg, J.; Höflinger, F.; Schindelhauer, C. Calibration-free TDOA self-localisation. J. Loc. Based Serv.
**2013**, 7, 121. [Google Scholar] [CrossRef] - Bancroft, S. An Algebraic Solution of the GPS Pseudorange Equations. IEEE Trans. Aerosp. Electron. Syst.
**1985**, 21, 56–59. [Google Scholar] [CrossRef] - Geyer, M. Solving passive multilateration equations using Bancroft’s algorithm. In Proceedings of the 17th DASC AIAA/IEEE/SAE Digital Avionics Systems Conference, Bellevue, WA, USA, 31 October–7 November 1998; p. F41-1. [Google Scholar] [CrossRef]
- Villadangos, J.M.; Ureña, J.; Mazo, M.; Hernández, A.; De Marziani, C.; Pérez, M.C. Ultrasonic Local Positioning System with Large Covered Area. In Proceedings of the 2007 IEEE International Symposium on Intelligent Signal Processing, Alcala de Henares, Spain, 3–5 October 2007; pp. 1–6. [Google Scholar] [CrossRef]
- Ureña, J.; Hernández, A.; Jiménez, A.; Villadangos, J.M.; Mazo, M.; García, J.C. Advanced sensorial system for an acoustic LPS. Microprocess. Microsyst.
**2007**, 31, 393–401. [Google Scholar] [CrossRef] - Yayan, U. A Low Cost Ultrasonic Based Positioning System for the Indoor Navigation of Mobile Robots. J. Intell. Rob. Syst.
**2014**, 78, 541–552. [Google Scholar] [CrossRef] - Saad, M.M.; Bleakley, C.J.; Ballal, T. High Accuracy Reference-free Ultrasonic Location Estimation. IEEE Trans. Instrum. Meas.
**2012**, 61, 1561–1570. [Google Scholar] [CrossRef] - Choi, H.H.; Jin, M.H.; Lim, D.W.; Lee, S.J. Dilution of Precision Relationship between Time Difference of Arrival and Time of Arrival Techniques with No Receiver Clock Bias
User Positioning with Particle Swarm Optimization. J. Electr. Eng. Technol.
**2016**, 11, 709–718. [Google Scholar] [CrossRef][Green Version] - Li, X.; Deng, Z.D.; Rauchenstein, L.T. Contributed Review: Source-localization algorithms and applications using time of arrival and time difference of arrival measurements. Rev. Sci. Instrum.
**2016**, 87, 041502. [Google Scholar] [CrossRef] [PubMed] - Cobos, M.; Antonacci, F.; Alexandridis, A.; Mouchtaris, A. A Survey of Sound Source Localization Methods in Wireless Acoustic Sensor Networks. Int. J. Wireless Mobile Comput.
**2017**, 2017, 1–24. [Google Scholar] [CrossRef] - Elson, J.; Girod, L. Fine-grained time synchronization using reference broadcasts. ACM SIGOPS Operating Syst. Rev.
**2002**, 36, 147–163. [Google Scholar] [CrossRef] - Carotenuto, R.; Merenda, M.; Iero, D. Using ANT communications for node synchronization and timing in a wireless ultrasonic ranging system. IEEE Sens. Lett.
**2017**, 1, 1–4. [Google Scholar] [CrossRef] - Jackson, J.C.; Summan, R.; Dobie, G.I.; Whiteley, S.M. Time-of-flight measurement techniques for airborne ultrasonic ranging. IEEE Trans. Ultrason. Ferroelect. Freq. Control
**2013**, 60, 343–355. [Google Scholar] [CrossRef] - Figueroa, J.F.; Barbieri, E. Position detecting system and method. US Patent 5,280,457, 1992. [Google Scholar]
- Doyle, J.C.; Francis, B.A.; Tannenbaum, A.R. Feedback Control Theory; Courier Corporation: North Chelmsford, MA, USA, 2013. [Google Scholar]
- SensComp, Inc. Available online: http://www.senscomp.com/pdfs/Series-7000-Ultrasonic-Sensor-spec.pdf (accessed on 20 March 2019).

**Figure 1.**System architecture. The Beacon Set Unit emits the ultrasonic chirp signals through the four beacons B

_{1}, B

_{2,}, ..., B

_{4}; the microphone onboard the MD receives four ultrasonic signals and calculates its own position. The beacons belong to the same circuit and are intrinsically synchronized with each other.

**Figure 2.**Time diagram for the Beacon Set and the mobile device (not in scale): The four beacons emit ultrasonic signals in a predefined sequence (i.e., 1, 2, 3, 4, T

_{SILENCE}, 1, 2, 3, 4…) starting from the time t

_{0BEACONS}, and the time interval between emissions is T

_{REPETITION}(see Figure 2). The duration of each ultrasonic emission is T

_{EMISSION}(not displayed) < T

_{REPETITION}.

**Figure 3.**Block diagram of the proposed positioning algorithm that operates in an infinite loop after initializing T

_{OFFSET}(0) = 0 and T*

_{OFFSET}(0) = 0. Peak position sequence restoration at the third step of the algorithm is required when the incoming arrival instants (i.e., the detected peaks) do not appear in the natural sequence that occurs for certain values of T

_{OFFSET}.

**Figure 4.**Simulated trajectory of the moving MD (diamonds) and estimated positioning (dots): Last 12 positioning frames. Beacons are indicated by triangles. The positioning error, expressed as Euclidean distance between reference and estimated points e

_{ED}, is shown by the last 12 values of Figure 5.

**Figure 5.**Decreasing positioning error e

_{ED}over 200 successive positioning frames: TDOA (dotted line), synchronized TOF (dash-dot line), and proposed method (thick solid line).

**Figure 6.**Instantaneous T

_{OFFSET}(solid line) and estimated T

^{*}

_{OFFSET}(dash-dot line) while the MD is continuously moving on its trajectory. T

_{OFFSET}is affected by a relevant noise. T

^{*}

_{OFFSET}is the output of the ramp follower (10) and converges without initial guess or prior information.

**Figure 7.**(

**a**) Cumulative error distributions (percent of readings with error less than the value of a given abscissa) of the positioning over the 200 positioning frames (dotted line) and of the last 20 trajectory points (solid line) after convergence transient, compared to CDF of the TDOA technique (dash-dot line). (

**b**) X-axis zoomed portion. The positioning Euclidian error of the last 20 points obtained by the proposed method is below 5 cm.

**Figure 8.**Data emission/acquisition board, power amplifier, signal voltage multiplier, and 200 V DC-bias circuitry, and wired miniature microphone.

**Figure 9.**Beacon Set Unit: 52 × 52 cm

^{2}panel hosting four capacitive SensComp Series 7000 transducers. Their emission cone half angle (far field) is widened from 24.7 up to 80.95° at 50 kHz by reducing the aperture diameter down to 8.5 mm with an aperture mask made of white moldable material.

**Figure 10.**Experimental trajectory of the moving microphone obtained with synchronized range measurements (diamonds) and trajectory obtained using the proposed synchronization recovery method (dots): Last 20 positioning frames. Beacons are indicated by triangles.

**Figure 11.**Experimental decreasing positioning error e

_{ED}over 200 successive positioning frames: TDOA (dotted line) and proposed method (solid line).

**Figure 12.**Experimental instantaneous T

_{OFFSET}(solid line) and estimated T

^{*}

_{OFFSET}(dash-dot line) while the microphone is moved along its trajectory. T

_{OFFSET}is affected by a relevant noise. T

^{*}

_{OFFSET}is the output of the ramp follower (10).

**Figure 13.**Cumulative error distributions (percent of readings with error less than the value of a given abscissa) of the positioning over the 200 positioning frames (dotted line) and of the last 20 trajectory points (solid line) after convergence transient, compared to CDF of the TDOA technique (dash-dot line). The positioning Euclidian error of the last 20 points obtained by the proposed method is below 5 cm.

**Figure 14.**Simulation quantization error computed in a grid of points of plane z = 1.5 m, and by considering in sequence a = b = 0.5 m. The maximum error magnitude is 14.7 mm. Same error shape with a = b = 1.5 and 0.25 m, but with maximum error about 5.1 and 32.0 mm, respectively. Triangles show beacon positions. The error magnitude is of the same order but less than the value provided by (17), which is in fact the upper bound of the real error. The quadrangular symmetry of the Beacon Set produces an error shape with the same symmetry without blind or poor accuracy points.

**Figure 15.**Estimation error of l

_{1}from (9) due to the quantization error on the estimates of d

_{1}, d

_{2}, and d

_{3}computed in a grid of points of z = 1.5 m, with different beacon set sizes: a) a = b = 1.5 m, b) a = b = 0.5 m, c) a = b = 0.25 m. Triangles show beacon positions. The error magnitude is in decibels (dB). Larger Beacon Sets considerably reduce the size of the “bad” region.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Carotenuto, R.; Merenda, M.; Iero, D.; G. Della Corte, F. Mobile Synchronization Recovery for Ultrasonic Indoor Positioning. *Sensors* **2020**, *20*, 702.
https://doi.org/10.3390/s20030702

**AMA Style**

Carotenuto R, Merenda M, Iero D, G. Della Corte F. Mobile Synchronization Recovery for Ultrasonic Indoor Positioning. *Sensors*. 2020; 20(3):702.
https://doi.org/10.3390/s20030702

**Chicago/Turabian Style**

Carotenuto, Riccardo, Massimo Merenda, Demetrio Iero, and Francesco G. Della Corte. 2020. "Mobile Synchronization Recovery for Ultrasonic Indoor Positioning" *Sensors* 20, no. 3: 702.
https://doi.org/10.3390/s20030702