This paper reports on the design, modeling, analysis, and evaluation of a micro-electromechanical systems acoustic sensor and the novel design of an acoustic vector sensor array (AVS) which utilized this acoustic sensor. This research builds upon previous work conducted to develop a small, lightweight, portable system for the detection and location of quiet or distant acoustic sources of interest. This study also reports on the underwater operation of this sensor and AVS. Studies were conducted in the lab and in the field utilizing multiple acoustic sources . The sensor operates at resonance, providing for high acoustic sensitivity and a high signal-to-noise ratio (SNR). The sensor demonstrated a maximum SNR of 88 dB with an associated sensitivity of −84.6 dB re 1 V/μPa (59 V/Pa). The sensor design can be adjusted to set a specified resonant frequency to align with a known acoustic signature of interest. The AVS demonstrated an unambiguous, 360-degree, in-plane, azimuthal coverage and was able to provide an acoustic direction of arrival to an average error of within 3.5° during field experiments. The results of this research demonstrate the potential usefulness of this sensor and AVS design for specific applications. The design, modeling, analysis, and evaluation of a micro-electromechanical systems (MEMS) directional acoustic sensor operating at resonance and an associated acoustic vector sensor (AVS) is presented. Determining the direction of arrival (DOA) of sound has long been an active field of study in acoustics. The DOA can be determined via many different devices and techniques.In the field of MEMS vector sensors, significant inspiration has been drawn from biology. Many MEMS acoustic sensors have been designed based on the hearing organs of humans as well as certain lizards, mosquitos, locusts, and flies . The acoOrmia ochracea, and how the fly is able to determine the DOA of sound. This hearing organ consists of two mechanically coupled tympana (eardrums). In 2006 Arthur and Hoy [Ormia ochracea, have been investigated [In 1995 Miles et al.  describe and Hoy  demonstr and Hoy  analyzedstigated ,26,27,28Ormia ochracea inspired sensors is that these sensors typically exhibited a cosine-like directionality, where a maximum response is detected with the DOA normal to the sensor surface. This response drops sinusoidally to zero as the DOA is rotated by ninety degrees [The most important factor to  degrees .Ormia ochracea-inspired sensor utilizing an intermembrane bridge. The sensor consisted of two separate circular membranes mechanically coupled by a structure that pivoted between the two membranes. They reported a theoretical acoustic DOA resolution of two degrees with an optimal frequency range of 300 Hz to 1500 Hz. In 2022 Rahaman and Kim [Ishfaque et al. , in 2019Another key aspect of the sensor presented in this paper is that it is designed to operate at, or near, resonance. Typically, microphones are designed to have a relatively constant sensitivity over a large frequency range. Sensors that are designed to operate near resonance trade increased performance at the cost of a reduced frequency range. In applications where tonal detection or signature-based harmonic detection is desired, resonant acoustic sensors can be designed with the advantage of mechanically filtering noise outside of the frequency range of interest ,34,35,36In 1998 Schoess et al.  demonstrIn 2017, Kusano et al.  demonstrAlthough not a MEMS sensor, in 2020 Lee et al.  presenteThe sensor presented in this paper follows years of research at the Naval Postgraduate School. Touse et al. ,21, in 2Rabelo et al. , in 2020A common problem with the AVSs discussed above is the limitation in azimuthal range. Single sensors or a combination of them were not able to resolve 360 degrees of acoustic DOA without ambiguity or a priori knowledge of some parameters. The AVS design presented in this paper addresses these shortfalls. In this paper, we describe the design and characterization of a double-wing MEMS sensor, similar to the ones demonstrated in ,27. UsinDistant and quiet acoustic sources can be difficult to detect and track with conventional acoustic detectors. High SNR, directionality, small size, light weight, and low power consumption are sensor characteristics often required by many applications, particularly those associated with national defense. AVSs meeting these requirements can be made using two MEMS directional acoustic sensors operating at resonance ,26,27,28Ormia ochracea, there are significant differences between the fly and this sensor. The fly utilizes multiple vibration modes of its hearing organ to determine acoustic DOA with a single pair of mechanically coupled membranes. The sensor presented here utilizes only one vibration mode and calculates acoustic DOA using two MEMS sensors and an omnidirectional microphone.While the sensor described in this paper is inspired by the The directional sensor presented in this paper is a MEMS device consisting of a pair of identical wings connected by a bridge, as shown in When exposed to acoustic waves, the sensor is subject to a rocking mode (the wings move opposite to each other) and a bending mode (wings move with each other), as shown in The sensor is microfabricated by a commercial foundry (MEMSCAP)  on a 400WC represents the interdigitated comb finger capacitors of the MEMS directional sound sensor. Assuming that WC is biased under a constant voltage REFV, when the wings are at rest, an equilibrium position of the comb fingers’ overlap is achieved and the capacitance between them is constant. Consequently, the accumulated charge on its plates, WQ, is constant as well. Under these conditions, no electric current  flows to, or from, the op amp nor does it flow through the feedback network; the output voltage CAV follows the inverting input voltage, which in this case is virtual ground. When an acoustic pressure wave impinges on the sensor’s wings, their equilibrium position at rest is changed along with the overlap of comb fingers, resulting in a change in the wing capacitance to a new value, WC’. If the biasing voltage is kept constant, the accumulated charge in the comb finger capacitors must readjust to a new value, WQ’. The charge difference, ∆WQ, is pushed to, or pulled from, the remainder of the circuit. Since the op amp input impedance is much larger than that of the feedback capacitor, FC, all the charge variations from the sense capacitor, WC, move through FC, developing an output voltage, CAV, given byTwo Sallen–Key filter stages are added to limit the passband between 100 and 3500 Hz.The sensor is intended to be used both in air and underwater. For underwater operation, the sensor is installed in a smaller PCB with a similar readout circuit. The sensor and PCB are enclosed in an air-filled, watertight housing, shown in The sensor operation can be approximated as a driven, damped harmonic oscillator. The complexities in the shape of the sensor and various sources of damping make this design appropriate for finite element modeling (FEM). However, some simple assumptions and approximations allow for analytical methods to be used for the sensor design.w is the width of the beam, t is the thickness of the beam, and L is the length of the beam. The stiffness of the wing is much greater than the stiffness of the bridge, therefore the wing does not bend significantly compared to the bridge. Therefore, the sensor can be further simplified to a simple mass-loaded beam. In this model, the wing is treated as a point mass (effm) located at the end of the bridge with an equivalent moment of inertia in the wing. The natural resonant frequency can then be modeled byIn the bending mode, the wings bend up and down together, causing no torque to be applied to the torsional arms. Analytical models of the sensor can be simplified to a single wing in a clamped-free configuration. The stiffness of the micro-scale sensor bridge matches that of larger-scale beams .(2)kbriω is the vibrational frequency of the beam, ρ is the fluid density, μ is the fluid dynamic viscosity, and b is the characteristic length (width of the wing for our purposes). The Reynold’s number is used to calculate a hydrodynamic function (Γ), which is used to determine the resonant frequency and quality factor. Multiple equations are needed to calculate Γ, and are beyond the scope of this paper. Suffice to say, we used the hydrodynamic function to calculate the resonant frequency and quality factor usingω1 and Q1 are the resonant frequency and quality factor modeled by Sader’s method. Γr and Γi are the real and imaginary parts of the hydrodynamic function detailed in [Resonating MEMS devices are subject to multiple damping effects. The significance of these damping sources depends strongly on the design of the device . The damailed in . When thmR is the mechanical resistance. wA, cA, wg, cg, and N are the surface area on the sides of the wing, surface area on the side of a single comb finger, the gap distance between the wing and substrate, gap distance between comb fingers, and number of comb fingers on the wing, respectively. The Couette flow contribution to the quality factor of the sensor is thenSader’s model assumes that the beam vibrates alone in the fluid. Our sensor is surrounded by a substrate. There is a narrow gap between the edges of the wing and substrate, and an even more narrow gap between the interdigitated comb fingers of the wing and substrate. Couette flow in these gaps was considered. The drag force from the Couette flow can be related to the mechanical resistance of a simple harmonic oscillator system ,45 as = Q1. tQ is evaluated against this best fit curve to determine the resonant frequency, tω.A best fit curve for To date, the sensor described in this paper (model 7-1) has been operated only in air (the sensor has been used in underwater applications while encased in an air-filled housing). However, a similar sensor (model 7-3) has been operated in air and in a low-viscosity silicone oil. FEM analysis of the sensor designs was conducted using COMSOL Multiphysics software version 6.1. The FEM included the wings, substrate, and a sphere of surrounding fluid. To reduce computing requirements, the model was bisected along the center of the length of the sensor bridge with symmetry constraints applied along this bisection.The material properties of the substrate and sensor components were modeled as anisotropic, single-crystal silicon based on the properties of silicon used in the sensor fabrication . The elaThe FEM included pressure acoustics and solid mechanics. Acoustic wave radiation conditions were applied to the outside edge of the sphere. Boundary loads were applied between the wings and substrate to simulate the resistance from the flow through the gaps between the sensor and substrate. Separate boundary loads were applied to the top surface of the wings to account for form friction. The acoustic wave was modeled as a background pressure field plane wave. The acoustic DOA was set using an adjustable parameter to control the wave direction. Two studies were conducted as part of the FEM analysis: a frequency sweep and a directionality analysis, where the displacement of the wing and the associated phase shift from the pressure wave were modeled. The frequency sweep was performed for a constant angle of incidence of 45 degrees, as seen in Electro-static effects were not modeled in the FEM analysis. Data collection on earlier generations of similar sensors showed that electrostatic forces do not appreciably affect the performance of the sensors. For this sensor, measurements were taken with the sensor electrically isolated and electrically connected to a powered PCB. No significant changes to the resonant frequency or quality factor were noted.The FEM resonant frequency was 699 Hz with a quality factor of 33. Laser vibrometry measurements of multiple sensors showed an average bending mode frequency of 671 Hz and an average quality factor of 27. The sensor resonant frequencies ranged from 662 Hz to 679 Hz. These results indicate that there are physical effects which impact the sensor response that were not fully accounted for in the FEM, such as dimensional variation due to fabrication tolerances and actual material properties. However, the FEM, as it stands, is a useful tool for sensor design.SV, CV, and OV are the sine, cosine, and omnidirectional sensor voltage Fourier transforms. SM is a correction function that accounts for the differences in frequency response of the two MEMS directional sensors. The term sgn(.) is the sign function. Since the omnidirectional microphone is not ideal, only the phase contribution was used (sgn(num) and sgn(den) terms) to determine the quadrant.The MEMS acoustic sensors described in this paper were used to create an AVS array by placing two collocated MEMS sensors perpendicular to each other with an omnidirectional microphone (Knowles MEMS microphones model: SPM0687LR5H ) or hydrMany underwater acoustic localization applications require the knowledge of azimuth and elevation, which was not possible with the single AVS demonstrated in this manuscript. The use of two devices orthogonally placed would solve this problem. A more elegant option being studied by our group is to use the rocking vibrational mode of the MEMS sensors, which has a sine dependence on the incoming sound, in combination with the bending motion. This would provide 3D unambiguous coverage with the same arrangement demonstrated in this paper.All in-air laboratory measurements were taken inside an anechoic chamber. The chamber is made with 12-inch-thick concrete walls and is mechanically and acoustically isolated from the rest of the building housing the chamber. It is surrounded on the walls, ceiling, and floor with fiberglass wedges which absorb 99% of incident sound for frequencies greater than 100 Hz . Mechanical sensitivity measurements were taken using a laser vibrometer setup consisting of a Polytech data management system (DMS), OFV-534 laser unit, and OFV-5000 controller, as shown in Electrical sensitivity and directionality measurements were taken in an anechoic chamber with the sensor electrically connected to a PCB. The MEMS sensors  were mounted on a B&K Model 5960 turntable, operated by a B&K Type 5997 turntable controller. A rubber dampening device was installed between the sensor and turntable to reduce mechanical coupling. An acoustic signal generated by a Zurich Instruments Multifunction Lock-in Amplifier (MFLI) was sent to a Techron 5507 amplifier to a JBL 2450H speaker with a 2380A cone, which was pointed toward the AVS. Parallel signals from each sensor were sent to individual MFLIs and to a microprocessor . An Agilent 33220A waveform generator was used in conjunction with the MFLI to generate broadband white noise. The experimental setup is shown in Electrical sensitivity measurements were made with the sensor facing the speaker (DOA = 0 degrees), while the speaker produced a frequency sweep generated by the MFLI. The output of the sensor was sent to the MFLI. A Piezotronics Model 378A21 calibrated reference microphone was positioned near the MEMS sensor during the frequency sweeps. The microphone signal was sent through a Piezotronics Model 482C sensor signal conditioner  to a separate MFLI. These signals were divided to generate the electrical acoustic sensitivity of the sensor, as discussed in Sensor directionality was measured with single tones (near sensor resonance) generated by the MFLI. The turntable was rotated at a constant angular velocity through a 360-degree circle. The sensor output was sent to the MFLI, which continuously recorded sensor output as a function of time during the rotation. The time corresponded to the DOA angle and a radial plot of sensor directionality was generated, as in Figure 11b.SM. This correction function was ingested in the arctangent estimator algorithm, which was run by a Teensy 4.0 microprocessor in the control box.Prior to an AVS being used to calculate DOA, it must be calibrated. To calibrate the AVS, the turntable was rotated such that the DOA was set to 45 degrees. The signal generator produced a frequency sweep. The outputs of each MEMS sensor and omnidirectional microphone were sent through a microprocessor control box to a computer. A computer program received the output from the three sensors and computed the correction function, To characterize the AVS performance, the AVS was exposed to various acoustic signals . The turntable was adjusted to a known DOA value, and the microprocessor was triggered. The microprocessor calculated the DOA. The actual and calculated DOA were recorded. The turntable was then rotated to a new DOA value and the process was repeated.AVSs were operated in the field to measure DOA accuracy when exposed to actual gunshots. A node consisting of the AVS (encased in a protective housing and dust cover) and microprocessor control box were placed on an Edelkrone Pan Pro motorized single-axis rotation device mounted on a tripod, as shown schematically in Individual underwater MEMS sensors were measured in the SWT, which is an aluminum tube with an outer diameter of 30.5 cm (12 in), an inner diameter of 25.4 cm (10 in) and is 61.0 cm (24 in) tall. An Electro Voice UW30 underwater loudspeaker was placed at the bottom of the SWT on vibration-damping material. The SWT was filled with water such that the water surface was approximately 33 cm (13 in) above the top of the speaker. The sensor, contained in a watertight housing, was held on a suspended rod with a rotating mechanism, which allowed for the DOA of the sensor to be adjusted.An acoustic signal was generated by an MFLI and sent to a Hewlett Packard 467A amplifier, which powered the speaker. Acoustic measurements were taken by a B&K Type 8103 hydrophone. The hydrophone output was sent to a Stanford Research Systems Model SR560 low-noise preamp and then to an MFLI. The MEMS sensor output was sent directly to an MFLI.Electric acoustic sensitivity measurements were made with the sensor facing the speaker (DOA = 0 degrees). A frequency sweep was generated by the MFLI. Outputs from the hydrophone and MEMS sensor were sent to MFLIs. The output of the hydrophone was used to calculate the acoustic pressure, which was then used with the MEMS sensor output to determine sensitivity.The directionality measurements for the underwater sensor were taken in the similar manner to the in-air directionality data. Directional measurements were made by connecting the Edelkrone Pan Pro rotation device to the rotator at the end of the suspension rod. The MFLI generated a single tone while the sensor was rotated at a constant angular velocity through 360 degrees. Underwater AVS measurements were taken at the TRANSDEC anechoic pool. The AVS was suspended on a rotating pole 2 m away from a Lubell Labs VC2C underwater speaker. Both the sensor and speaker were 6 m from the surface and bottom of the pool. As with in-air AVS data collection, the MEMS sensor outputs were sent to the microprocessor control box. Instead of a microphone, the underwater AVS used a B&K Type 8103 hydrophone as the omnidirectional sensor. The hydrophone output was sent to a Stanford Research Systems Model SR560 low-noise preamp and then to the microprocessor. The rotating pole was adjusted to set the AVS to a known DOA while a constant tone was played by the underwater speaker. The microprocessor was triggered, and the DOA was calculated. The actual and calculated DOAs were recorded. Then, the pole was rotated to set a new DOA and the process was repeated.SM, in (10) and (11).Laser vibrometry measurements were taken for all MEMS sensors. Wing displacement was measured and compared against the acoustic pressure of a frequency sweep to determine the mechanical sensitivity (μm/Pa). These measurements were conducted in an anechoic chamber and the response of a typical pair of sensors is shown in Electrical sensitivity (V/Pa), as well as directionality, were measured in an anechoic chamber. Noise measurements were performed in an anechoic chamber with the sensor readout electronically powered and with no acoustic stimuli in two different configurations. First, the sensor wings were free. Then, the sensor wings were cemented to the substrate to prevent their natural vibration. CV and SV) with the DOA set to 45 degrees. Ideally, the frequency responses of both sensors at 45 degrees should be the same, however, since they are not, SM = CV/SV is applied to the sine signal. The signal processing electronics calculate the DOA, using (10), as the AVS is rotated.Prior to use in the field, the AVSs were calibrated in an anechoic chamber. This process is described in Measurements were taken in an anechoic chamber and in the field for various acoustic sources, particularly gunfire and small multi-rotor aircraft (drones). The anechoic chamber gunshot data and drone data show a systematic error offset to the positive side. This indicates a misalignment of the turntable used to rotate the AVS when setting the DOA. Field data typically have larger errors due to reverberations, acoustic reflections, and background noise. In field experiments, the AVS occasionally collects data on gunshot sounds that reflect off nearby structures, which is the case with the outlier points in The DOA errors for both drones and gunshots (for data taken in an anechoic chamber) fall within a 0-to-5-degree range . It should be noted that the drone sound used to characterize the AVS . A diagram of the experimental setup is shown in The sensitivity and DOA response of the underwater sensor were measured in a similar manner to the in-air measurements. Underwater AVS measurements were taken at the Transducer Evaluation Center (TRANSDEC) which is a six-million-gallon Navy facility designed to test underwater acoustic devices. The MEMS sensors were individually evaluated and the AVSs were calibrated at TRANSEC prior to conducting the DOA measurements. It is noticeable that, even though the estimator used underwater was the same as that used for in-air operation, the error is higher. Possible sources of this error included surface reflections and effects from the AVS mounting apparatus. Surface reflections were observed for the frequency ranges investigated with this AVS. As noted above, the back lobes of the MEMS sensors are smaller than the front lobes when operated underwater. This phenomenon was seen both in the SWT and the anechoic pool using different mounting schemes. Therefore, the underwater housing is likely affecting the sensor response.CV is near zero and the algorithm is less sensitive to fluctuations. The maximum error magnitudes are near the center of each quadrant , where the amplitude mismatch is most significant. Phase mismatches  lead to a different sinusoid-like DOA error centered at approximately half of the mismatch in the phase. The correction factor, SM, compensates for the differences between vs. and CV. However, vs. and CV are determined by applying a Fourier transform to the sine and cosine sensor signals, respectively. Consequently, there are limitations on the frequency bin size of vs. and CV. Broadband signals  are naturally integrated across these frequency bins and the error is minimized. Single-tone sound sources do not benefit from this phenomenon and consequently show larger errors.A sinusoid-like shape to the DOA error is observable for the underwater sensor. There are two primary causes of a sinusoid-like DOA error: amplitude and phase mismatches between the two MEMS sensors. The minimum error amplitudes are seen at cardinal angles , where either vs. or Ormia-inspired MEMS acoustic sensors to determine acoustic DOA.This paper describes the design and experimental characterization of directional acoustic MEMS sensors and an AVS comprised of two collocated and orthogonally aligned sensors in combination with a commercial omnidirectional microphone (or hydrophone). The AVS can determine in-plane acoustic DOA over a non-ambiguous 360 degrees. This AVS is meant to operate near resonance and to be used in the air and underwater. The configuration of the MEMS sensors, AVS design, and algorithms used make this a novel approach to using The small size, light weight, and low power requirements of the MEMS acoustic sensor and associated AVS show potential for their use as a man-portable sensor for detecting and locating acoustic contacts of interest. The MEMS acoustic sensor demonstrates very high SNR near resonance. This makes the sensor ideal for detecting quiet and distant acoustic targets of interest. The resonant frequency of the MEMS sensor is based on the physical characteristics of the sensor  allowing for bespoke sensors to be designed for acoustic targets that emit specific acoustic tones.The maximum SNR for this MEMS sensor was determined to be 88 dB with a corresponding maximum sensitivity of −84.6 dB re 1 V/μPa (59 V/Pa). The sensor demonstrates a cosine-like acoustic directionality for the bending eigenmode. This sensitivity is significantly larger than comparable MEMS acoustic sensors, as seen in The accuracy of the AVS was measured both in the lab and in the field for use in both the air and underwater. While calculating the DOA of gunshots in the field, the average best case AVS accuracy was approximately 3.5 degrees. AVS accuracy, measured in the lab, was determined to be less than an average of 2 degrees over a 360-degree arc. This performance is an improvement on previous laboratory measurements reported by this group of 3.4-degree accuracy over a ±60-degree arc . The perThese results indicate the great potential of this type of MEMS sensor for DOA determination in multiple domains. The specific characteristics and figures of performance can be modified by design, according to the application demands. Future work includes the development of multi-resonance MEMS sensors for use in detecting very quiet broadband acoustic sources.


In this work, we present a diagnosis system for rolling bearings that leverages simultaneous measurements of vibrations and machine rotation speed. Our approach combines the robustness of simple time domain methods for fault detection with the potential of machine learning techniques for fault location. This research is based on a neural network classifier, which exploits a simple and novel preprocessing algorithm specifically designed for minimizing the dependency of the classifier performance on the machine working conditions, on the bearing model and on the acquisition system set-up. The overall diagnosis system is based on light algorithms with reduced complexity and hardware resource demand and is designed to be deployed in embedded electronics. The fault diagnosis system was trained using emulated data, exploiting an ad-hoc test bench thus avoiding the problem of generating enough data, achieving an overall classifier accuracy larger than 98%. Its noteworthy ability to generalize was proven by using data emulating different working conditions and acquisition set-ups and noise levels, obtaining in all the cases accuracies greater than 97%, thereby proving in this way that the proposed system can be applied in a wide spectrum of different applications. Finally, real data from an on-line database containing vibration signals obtained in a completely different scenario are used to demonstrate the distinctive capability of the proposed system to generalize. In industrial systems, rotating machines play a vital role. Rolling element bearings are often one of the most vulnerable components in machinery. Their health condition significantly impacts the machine performance, efficiency, stability, and life cycle , moreoveWhen one of the elements of the bearing is damaged, its defect impacts on the other components during the rotation and each impact starts a transient vibration, whose intensity is determined by the severity of the fault. As such, vibration signals caused by faults can be seen as trains of pulses repeated with a periodicity dictated by the rotation speed of the damaged components.The time-domain analysis is the most direct approach for detecting faults in bearings since it is based on the evaluation of scalar indices to measure the energy and to spot the presence of pulses . This anOn the other hand, frequency-domain or spectral analysis is the most employed method for fault recognition and location. This method exploits the transformation of the vibration signals into the frequency domain using discrete Fourier transform (DFT), allowing thus for detecting the characteristic periodicity  of each type of fault.Actually, real fault vibration signals are complex and can be the combination of periodic components and non-stationary ones, and this behavior suggests the application of time-frequency analysis. Many time-frequency techniques have been applied for bearing diagnosis, such as short time Fourier transform (STFT), Wigner-Ville distribution, and wavelet transform.There are some well-known drawbacks of the above-mentioned traditional methods: time domain methods are not suitable for fault locations, frequency domain methods are often disturbed by noise or temporary non stationarity due to sliding, whereas time-frequency approaches suffer from the selection of the right time window length which changes with respect to the operating conditions, or for Wavelet from selecting the appropriate mother wavelet and decomposition level.As a result, many researchers have sought in machine learning (ML) techniques solutions able to overcome some of these problems. During the last decades a huge quantity of research work on this topic has led to a variety of different possible solutions, such as artificial neural networks (ANN), support vector machines (SVM), neural fuzzy network ,13 BayesThe main drawback in all these approaches is the need of a very large quantity of labelled data to train and properly verify the diagnosis system, and this is especially true for DL techniques. In fact, when using small datasets, classical ML algorithms can compete with or even outperform deep learning networks . The neeAdditionally, often experimental signals stored in these databases  obtained with simple artificially injected defects, and labelled as examples of single fault signatures, contain portions affected by sliding, or non-stationarity of the machine rotation regime or of the faulty condition, which are not isolated and labelled as such. These signal parts should be discarded or isolated when training the diagnosis system, reducing further and dramatically the size of the exploitable datasets.In this work we proposed a diagnosis system for rolling bearings, based on the simultaneous measurement of vibration and machine rotation speed, that combines the robustness of simple time domain methods for fault detection, with the potential of ML methods for fault location. The fault classifier leverages a simple preprocessing algorithm that allows for minimizing the dependency of the classifier performance on the machine working conditions, on the bearing model, and on the acquisition system set-up. The overall diagnosis system is based on light algorithms with reduced complexity and hardware resource demands and is designed to be deployed in embedded electronics. The fault diagnosis system is trained using emulated data, exploiting an ad-hoc test bench thus avoiding the problem of generating enough data.The novelty of the proposed approach resides mainly in the tailored preprocessing algorithm, which leverages the transformation of a time signal into a gray scale image, thus permitting the usage of many available neural models for the following classification task. But beyond this benefit, which has already been exploited in many different works ,31,32, tThe approach presented in this paper holds significant promise as it capitalizes on the growing availability of low-cost accelerometers with extended bandwidth capabilities of up to 20 kHz. These accelerometers, developed using state-of-the-art Micro-Electro-Mechanical Systems (MEMS) technologies, accompanied by embedded processing, offer a tangible opportunity to deploy widespread distributed low-cost diagnosis systems.A rolling bearing comprises three main components: the outer race, the inner race, and the rolling elements , as shown in Wear typically damages one of these three components. In the event of a defect in the bearing components, vibrations are generated. The magnitude of these vibrations is directly linked to the severity of the damage. Moreover, the specific damaged bearing element, the type of bearing, and the rotational speed of the machine all contribute to determining the frequency content of the vibration signal.When one of the elements of the bearing is damaged, its defect periodically impacts on the other components during the rotation. Each impact excites the mechanical resonance of the metallic structure near the bearing, whose fundamental frequency is usually much higher than the frequencies of the impacts, resulting in a transient vibration that can be described by a decaying sine wave. Therefore, the overall vibration signal caused by a fault, in the absence of sliding of the rolling elements (pure rolling), is a train of transient vibrations repeated with a fundamental frequency dictated by the rotation speed of the faulty component. In The vibration signals are sensed by an accelerometer fixed to the stator of the machinery as close as possible to the roller bearing, therefore the amplitudes of the sensed transient vibrations depend on the distance between the impact locations and the sensor, which can be fixed as for outer race faults or variable in the other two cases. The received signal is therefore a transient train that can be amplitude modulated.faultf represents characteristic frequency related to the particular failure , resf is the machine component resonance frequency, and Ff and FK represent the amplitude modulation frequency and modulation index, respectively. Finally, cA1 and cA2 represent the amplitudes of the transient vibrations.Summarizing, a widely accepted mathematical model for the resulting bearing failure vibration signal, if no sliding occurs, is the following :(1)t=1+KFK = 0, Ff = 0 and cA1 = cA2. For an inner race defect, the vibration signal is amplitude modulated since the defect moves with respect to the accelerometer, being integral with the rotating shaft. The modulation frequency Ff is equal to the machine rotating frequency RPM/60, and the two amplitudes cA1 and cA2 are equal.More in detail, for an outer race fault the defect is in a fixed position with respect to the sensor so cA1 and cA2 are different, since they are associated with the defect hitting the inner race and the outer race alternately, during the spinning of the ball; in this case there is also the modulation effect given by the movement of the rolling element, integrally mounted in the cage. The modulating frequency is related to the cage speed and the frequency of the modulation is called fundamental train frequency (FTF). In this case, the failure frequency ffault is equal to two times the characteristic frequency BSF.Finally, in the rolling element defect, the two amplitudes Note that with the above-described model, the vibrations’ signals are always periodic.Considering the mathematical representation of the three different bearing failure classes, it is possible to distinguish the type of failure as well as the presence or absence of the failure, analyzing the signals either in time or in the frequency domain. Obviously, the pulse train nature of fault vibration signals is associated with wide band spectra, due to the high frequency content of the transients, while the features meaningful for diagnosis reside in the low frequency content.cf) are needed together with long observation windows. Constraints on the large analog bandwidth and high sampling rates are posed by the transient frequency band, whereas the observation window length must be selected on the basis of the fault frequencies, which in turn change with the RPM and are usually less than 10 RPM. Many preprocessing techniques or classifier architectures need to be adapted, when changing the ratio cf/RPM, even if this problem is often disregarded in the literature [Therefore, to acquire fault vibration signals, large band sensors and readout electronics and high sampling frequencies , is a simplification of a complex scenario. In real cases, the behavior of the vibration signals is more complex. Apart from sliding, there are effects of attenuation in the signals given by the machine structure as well as related to the type of failure.This is given by the fact that especially for ball defects, the fault is present in a small part of the bearing and the impacts with the other bearing components create little amounts of energy with respect to the other failures. Moreover, the amount of energy created by a certain fault during the rotation of the shaft is also related to the amount of mechanical load and to the rotation speed of the shaft.Besides the amount of energy generated by a fault impact, there is the aspect of the sensor position with respect to the impact location. In real case scenarios, for instance in harsh environmental conditions scenarios, such as in gas turbines, the vibration sensor placement is always a problem. For the best detection of the bearing vibration behavior, the sensor should be placed as near as possible to the bearing to be monitored or on the bearing itself; however, it is not always possible to place the sensor in the best position. High temperatures, mechanical constraints, and geometries usually determine placements far from the very bearing that must be monitored, increasing the distance between the sensor and the vibration source, which leads to the attenuation of the vibration signal and to the deteriorating of its information content.A placement far from the bearing to be monitored could also cause other problems related to interferences from other vibration sources, for instances vibrations created by gear transmissions, chains, gearmotors and so on. In this context it is not possible to recover the characteristic behavior of the bearing failure.It is important to also stress that the signal vibration behavior shown in As previously stated, bearing failures progress gradually over time, often originating in a single bearing component like a race but then escalating to multiple components, leading to a more complex vibration signal with new harmonics which deviates from the conventional model. Moreover, in demanding environments, like high temperatures, suitable accelerometers might be lacking for wideband measurements. In this case, the accelerometer acts as a resonant low pass filter, heavily distorting the vibration signals.The information that can be recovered from the signal in all these critical but realistic cases presented in this subsection, is only related to the vibration amplitude, and the sensor output signal can be exploited only for alarm, i.e., for fault detection, and not for fault location.From a failure detection perspective, classifying the vibration signals according to the characteristics related to the mathematical representation of the three bearing failure classes is surely viable, as discussed above. Nevertheless, as discussed in What is always true is that when a failure occurs there will be a change of the vibration signal behavior with respect to the one recorded in ‘normal’ conditions.Typical approaches for the bearing early failure detection, implemented in condition monitoring setups, such as in oil and gas rigs and in wind turbines, for instance, rely on the analysis of the vibration signal during the machine operations, by evaluating the magnitude of its deviation from the ‘normal operation baseline’ exploiting several different metrics, e.g., in the time domain such as the root mean square value (rms) or the kurtosis over a time window lasting some rotation periods. An approach of this kind allows for detecting the presence of the failure but not for discriminating among the various types of failures.Therefore, in this paper we propose the approach shown in Both the preprocessing and classifier are simple, low-computational burden techniques. They demand only a small amount of hardware resources, making them well-suited for deployment in low-complexity microcontrollers. Moreover, their design facilitates edge data processing in distributed sensor nodes.The proposed feature extraction algorithm operates in the time domain and aims at transforming the time domain signal into an image, i.e., an arbitrarily long-time sequence into a fixed sized matrix.However, processing the signals in the time domain has some obvious drawbacks. As the speed varies, considering the sampling frequency fixed, the vibration signal periods will be composed of a variable number of samples related to the ratio between the sampling frequency and the rotating speed. Several works exploit time-domain signals to realize 2D features of the different bearing failures, used as input of various type of classifiers ,31,32. ITo have a representation independent from the rotating speed, resampling the signal, with a sampling period related to the speed might be a feasible solution requiring some additional computational burden.K × K square matrices In this paper, by exploiting the a priori knowledge of the rotation speed, a 2D representation of the time-domain signal is proposed which is related only to the type of failure if present and if no sliding occurs. In particular, three fault,f defined in faultf = BPFO, BPFI, 2BSF), the three fault periods in terms of number of samples, fault,N are found as follows:At first, for all the possible fault frequency wT = cNT is considered, where Then, a time window of the sampled signal of length Subsequently, the sampled signal is normalized as follows:Finally, each wT value. Gray scale images corresponding to the 2D Each matrix K can be arbitrarily chosen and represents the signal amplitude dynamics. A K value in the range  is more than reasonable, given the typical signals to noise ratios for vibration signals.Notice that the matrix size In the presence of a specific fault (I or O or B), in ideal cases, when the vibration signals are described by Equation (1) and when the noise and the discretization errors both in amplitude and the time domain are negligible, the matrices Therefore, the simultaneous analysis of the three images Obviously in real cases the presence of noise and the sampling of the vibration signal causes these images to distort, obtaining features as those shown in F, is a K × 3 K image built as follows:Finally, the three images derived from With this feature extraction procedure, the extracted features lose their dependency on the ratio between sampling frequency and machine rotating frequency, which is usually a scale factor in any other preprocessing and feature extraction technique either in the time or frequency domain. On the other hand, the type of roller bearing must be known a priori.N , This last parameter depends on the machine working conditions and can be adjusted based on the measured rotation speed as follows:pn equal, and also if the rotating speed varies.The pre-processing set up consists of selecting a threshold value in  and the Finally, and most importantly, the subsequent classifier design is independent from the machine working conditions, from the rolling bearing type, from the signal amplitude, from sampling time, and so on.K = 112 was selected for the ‘feature images’, therefore each fault feature image size is K × 3 K. In particular, the input layer was chosen according to the image size in pixels.The used neural network (NN) was tailored to the realized images representing the bearing failures. A value k = 4 for the classification of the three bearing failure classes  and for the class of faults associated with vibration signals characterized by no periodicity, which from now on is called ‘Noise’.An average pooling layer was then implemented, which performs an average pooling of 112 × 112 with a stride of (112 112). The stride dimensions were selected according to the dimension of the Using an FPGA for the entire evaluation would entail an extra microcontroller for signal acquisition and matrix image generation. However, this setup could potentially reduce computation time.Notice that the image processing task in this specific application is well defined because it consists of the recognition of few image types, with the need of rejecting the blurring effects of noise and of discretization (sampling). Therefore, available standard networks for image processing such as MobileNet and ResNet aimed at the solution of much more complex problems are not the best choice due to their complexity and to the operations such as rotations and striding performed to the image matrix (as shown later in this paper).Different datasets were exploited to train and test the proposed NN, based on emulated signals and on real signals available from online databases.Different datasets were obtained using emulated signals generated with a test bench, previously developed by the authors  and basesf, up to 10 kHz.The datasets were obtained considering different resonance frequencies, The fourth class of faults, representing faults with no evident periodicity (the related class is named ‘Noise’), was emulated by driving the shaker with white Gaussian pseudorandom noise signals generated by a waveform generator (AG33220A) added to sinusoidal waveforms at a frequency equal RPM/60.N used to form the fault feature images (image size 112 × 336) was chosen as per Equation (6) with pn = 8.Three different types of accelerometers were fixed to the shaker and used to sense the emulated fault vibration signatures. In particular, the used sensors were a piezoelectric accelerometer (Bruel & Kijer 4326) having a 10% bandwidth of 16 kHz and two embedded triaxial devices based on MEMS devices , previously realized and characterized in ,37,38 haThe training and validation dataset was obtained by preprocessing emulated signals relative to the four fault classes: ‘Noise’, and ‘O = Outer race Fault’, I = Inner Race fault’ and B = Ball defect’, by considering a machine resonant frequency of 10 kHz, and RPM = 1800 rpm, so that, BSF = 70.7 Hz, FTF = 11.9 Hz, BPFI = 162.5 Hz and BPFO = 107.5 Hz see also .All signals used to build the training and validation dataset were sampled at different sampling frequencies of 26 kHz, 53 kHz and 160 kHz.The training and validation dataset contains 588 images, 147 per class.Test datasets were obtained by injecting noise with different magnitudes, by varying the machine rotation speed, the sampling frequency, as well as the machine resonant frequency . These datasets were approximatively the same size of the training dataset with evenly distributed examples per class.cns(n) used to form the test datasets were obtained exploiting the following relationship:n, In particular, noisy signals, m > 1, were assumed fixed fractions of Standard deviations of the Gaussian noise with values up to 20% of the typical peak value of the vibration signals were used, so as for The proposed classifier was further tested with real data available in the online database of the Case Western Reserve University (CWRU).The signals available in this database are characterized by different rotation speed, sampling time, component vibration, natural frequency, and bearing geometry with respect to those used for signal emulation. The CWRU dataset is a popular, open-source, and easily accessible dataset. The CWRU bearing dataset is often used as a reference  dataset The bearing test rig arrangement used to obtain the CWRU data consists of a 2 hp Reliance electric induction motor, a torque transducer, a dynamometer, and control electronics. Two test bearings (fan end and drive end) support the motor shaft. Torque is applied to the shaft through a dynamometer and electronic control system. Some single point faults were induced in the rolling elements, the inner race and outer race, and each faulty bearing was reinstalled on the test rig. SKF  bearings were used. Acceleration signals were measured at locations near to and far-off the motor bearings, by multiple sensors. Accelerometers were attached to the housing with magnetic bases. For the drive-end bearing experiments, signals were sampled at 12 kHz and 48 kHz, while fan-end signals were sampled at 12 kHz.Drive end (DE) data were considered for the proposed system test. A dataset of 387 images, 129 for each class was used to test the NN network. The dataset was created with only three classes, corresponding to the three failures documented in the database . This differs from the datasets generated using emulated signals  and nIn the presence of the expected characteristic fault periodicity one of these spectral components has a value larger than others indicating the presence of a large spectral peak, whereas in its absence no large peak at the expected fault frequency is found and all the three normalized components are similarly small. It can be seen that in the case of O and I fault, the periodicity is present, whereas all the B faults in the database lack this feature  to match the dimensions of the images created for different datasets. Adjustments were also made to the output layers of both prebuilt NNs to align with the required number of classes for the classification task.Both prebuilt NNs underwent training and validation using the same dataset employed for the proposed network, achieving 100% accuracy in both training and validation. Testing of the trained networks used the datasets applied to test the proposed NN. The comparison reveals that the proposed approach demands fewer resources than standard prebuilt NNs designed for image classification. Regarding testing accuracies across three distinct datasets unrelated to the training dataset, it is evident that the proposed network achieves superior accuracy compared to the two prebuilt NNs, despite their high training accuracy.As discussed in In this study, we have introduced a machine learning methodology for diagnosing rolling bearing faults through vibration signal analysis. The proposed approach encompasses a lightweight preprocessing technique that leverages machine rotation frequency measurements. This enables the extraction of fault features without reliance on machine operational conditions, acquisition sampling frequency, or the responses of mechanical components and front-end electronics. The subsequent fault classification process is facilitated by a lightweight neural network (NN), facilitating the implementation of the entire diagnostic system using low-complexity, cost-effective hardware components. This design is tailored for embedded systems development.The effectiveness of our developed diagnostic system was validated using a sizable dataset created from diverse emulated data, each representing distinct characteristics. Subsequent testing included scenarios with varying rotation speeds of the shaft. Across all tested scenarios, the classification accuracy consistently exceeded 97%, even in the presence of substantial noise .Furthermore, our system performance was assessed using real-world data sourced from the CRWU database, demonstrating its capacity for generalization. The novelty of our approach, built upon classical fault diagnosis principles, is primarily attributed to the novel structure of the preprocessing algorithm and the generation of fault signature images. These images are solely dependent on the inherent periodicity intrinsic to the fault vibration signal. Importantly, they eliminate dependency on rotation speed, acquisition sampling rate, and the specific transient vibration responses of mechanical components.This transformative aspect simplifies the classification task, permitting the use of straightforward NN architectures. Moreover, this innovation facilitates training the network on a reference working condition database or its emulation, subsequently deploying the trained network in varying operational conditions. In essence, our methodology capitalizes on the intrinsic characteristics of fault vibration signals to create a versatile and efficient diagnostic system that can be confidently implemented across diverse settings.


Single-cell RNA-sequencing enables cell-level investigation of cell differentiation, which can be modelled using trajectory inference methods. While tremendous effort has been put into designing these methods, inferring accurate trajectories automatically remains difficult. Therefore, the standard approach involves testing different trajectory inference methods and picking the trajectory giving the most biologically sensible model. As the default parameters are often suboptimal, their tuning requires methodological expertise.We introduce Totem, an open-source, easy-to-use R package designed to facilitate inference of tree-shaped trajectories from single-cell data. Totem generates a large number of clustering results, estimates their topologies as minimum spanning trees, and uses them to measure the connectivity of the cells. Besides automatic selection of an appropriate trajectory, cell connectivity enables to visually pinpoint branching points and milestones relevant to the trajectory. Furthermore, testing different trajectories with Totem is fast, easy, and does not require in-depth methodological knowledge.https://github.com/elolab/Totem.Totem is available as an R package at  Single-cell RNA-sequencing (scRNA-seq) enables researchers to quantify the transcriptome of thousands or even millions of cells simultaneously at the single-cell level. The cells in a tissue undergo transcriptomic changes that are part of biological processes. If the changes happen gradually, the tissue contains cells derived from different stages of the process. If a correct trajectory that accurately models these different stages can be generated from the scRNA-seq data, this enables researchers to unravel the transcriptomic mechanisms of the processes. The modelling of trajectories from scRNA-seq data has formed a new branch in computational biology, known as trajectory inference.A vast number of trajectory inference methods has been developed  and the structure of the clustering results vary, generating vastly different trajectories when used as the basis for constructing the MST. To select a clustering from the large set of clustering results, Totem uses a new measure called cell connectivity, which is based on counting the number of edges (connections) between the clusters in an MST. The ratio of the number of edges and clusters (connectivity) is calculated for each cell and clustering, and the connectivity vectors are averaged to obtain the cell connectivity of the cells. The cell connectivity acts as a useful baseline for selecting an appropriate trajectory and deciding which milestones to include in the trajectory, while also helping to give a visual overview of the milestone network, i.e. how the milestones are connected and where the branching points and leaf nodes are located. Importantly, a key feature of Totem for its usability is that it allows to quickly and easily browse different MSTs, from which the user can select one or several MSTs for further analysis.To provide a system with better user experience for researchers who perform trajectory inference, we developed a new tool, Totem, for the inference of tree-shaped trajectories from single-cell data . Totem gTo benchmark Totem, we performed a comprehensive comparison using the dynverse benchmarking framework  and can In the following subsections, we go through the main steps of Totem trajectory inference workflow. Upstream of trajectory inference, Totem assumes that the input gene expression matrix has been normalized and quality control has been performed to remove bad-quality cells (t-distributed stochastic neighbour embedding (t-SNE)  embedding as input. The dimensionality reduction steps can be customized as the user sees best prior to Totem analysis, but Totem can also be used to transform gene counts into new features (feature extraction) with functions that utilize the dyndimred R package. In scRNA-seq data analysis, the most common way to perform feature extraction is the Principal Component Analysis (PCA) in which the number of principal components typically varies from 10 to 30. Alternatively, methods like Multi-Dimensional Scaling (MDS) or the more scalable landscape MDS (LMDS) can be used to generate an embedding with a smaller number of dimensions (e.g. from 2 to 5), which is usually not a sufficient number in PCA of scRNA-seq data.  (t-SNE)  and Unifk-medoids clustering algorithm  in the graph by dividing the degree of the vertex ijd, i.e. the number of edges that are connected to the vertex, by the number of vertices jk in the graph. We then create a connectivity vector cj that contains the connectivity values for all j, scale the connectivity values so that the maximum connectivity of each graph is always one, and calculate the cell connectivity of all the cells L′ connectivity vectors using the arithmetic mean:For each MSTThe cell connectivity is higher for cell populations that are farther from the leaf parts of the trajectory . We usedTo find a clustering that captures the tree structure of the data and gives clusters that are well defined, we use VRC, also known as the Calinski–Harabasz score , which mFor an MST corresponding to a selected clustering, Totem performs lineage smoothing using Slingshot  for eacht-SNE, UMAP, or MDS. To assist in biological interpretation, the user can also visualize expression levels of genes over the embedding.Totem includes visualization functions that aid trajectory inference, utilizing the dynplot R package  (k-medoids) clustering algorithm and ASW to select the optimal number of clusters, as well as TinGa, which is based on a GNG model  based metric or the Euclidean distance metric. If no starting cluster is specified by the user, Slingshot selects one so that the number of shared clusters between the lineages is maximized before the first branching point.In total, the GNG algorithm of TinGa has eight parameters in addition to the input embedding for which the trajectory is inferred. Of the eight GNG-related parameters, the maximum number of nodes  can be decreased if the trajectory has too many milestones  in the network. The maximum number of iterations  should not be tuned as the high iteration count ensures that the GNG algorithm converges. The six remaining parameters control how sensitively the graph nodes are linked and changed during the iteration, and their adjustment should be considered if tuning the maximum number of nodes does not improve the result. The original article of TinGa mentions that especially adjustment of the two epsilon parameters  10 000 times for each dataset, selected the top 100 clustering results with each method, performed the trajectory inference using Slingshot for each clustering, and ran the dynverse performance evaluation for the trajectories. For each benchmark dataset, we varied the number of evaluated trajectories from 1 to 100 and calculated the maximum and mean of the overall performance score. Finally, we calculated the mean of the overall scores across the datasets.To compare the clustering selection methods, we ran the k values  from which the number of clusters is sampled with replacement. We randomly selected 10 datasets from the top 50 datasets with the highest overall score in the benchmark . For each parameter configuration and dataset, we ran Totem 10 times using different random seeds. We calculated the dynverse overall score between the inferred and ground-truth trajectories  and the range of the To benchmark Totem, we used the dynverse benchmarking framework  that canP-value ≤ 0.001; Wilcoxon signed-rank test; Bonferroni-adjusted P-value ≤ 0.01) for datasets with a nonlinear trajectory , whereas TinGa was the second-best method for the nonlinear datasets. The violin plots also show that Totem’s distribution of the overall scores with the nonlinear trajectories was more concentrated to higher performance levels compared to TinGa and Slingshot. For linear trajectories, Slingshot achieved superior performance among the tested methods , while the overall scores of the linear datasets were, on average, similar for TinGa and Totem. The difference in the overall performance was mainly attributable to the differences in the topology accuracy (HIM) and the accuracy of the cell assignment onto the branches (F1 branches). The HIM score of Slingshot was close to perfect for most of the linear datasets but worse compared to TinGa and Totem for the nonlinear datasets.As shown in the rightmost panel of Selection of an appropriate clustering for constructing the MST is an important step in the trajectory inference process. Therefore, we took a closer look at the clustering selection step to investigate which clustering selection approach gives the best results when we consider multiple top-ranking trajectories simultaneously. We included two popular clustering selection methods, ASW and VRC, as well as the Totem method that uses cell connectivity and VRC, and also a method that ranks the clustering results into a random order (Random).When we tested multiple top-ranking clustering results and always selected the trajectory that gave the best performance in the evaluation, the result suggested that the random criterion gave the best performance . This isTo demonstrate the utility of Totem in practice, we considered two datasets from the dynbenchmark database. The datasets meet the prerequisites of Totem: they have a tree-shaped topology with a single starting point and no disconnected parts. Importantly, the cell types of the two datasets are connected in the correct way in their low-dimensional embeddings, in this case 5-dimensional LMDS, a property that generally hinders trajectory inference of many of the dynverse benchmark datasets irrespective of the trajectory method used.In the first example, we compared the trajectories  producedThe second example  shows siWe investigated the robustness of Totem to its parameters . The resWhile a large number of trajectory inference methods has been developed  with the In Slingshot, the between-cluster distances determine the MST and the milestone network. Our examples showed how even when the correct cell types were available, the MSTs predicted by Slingshot could still be inaccurate. One solution is to change the distance method, e.g. to a mutual neighbour-based method, but this will still not necessarily generate the correct network. Totem addresses this issue by providing a user-friendly interface to analyse MSTs generated based on different clustering results. With this interface the user can select an MST that best fits to the biological hypothesis and the cell connectivity profile without time-consuming and arbitrary parameter tuning. Totem then smoothens the MSTs of the selected clustering results using the simultaneous principal curves algorithm of Slingshot to obtain directed trajectories that include pseudotime. The trajectories can be exported as Slingshot or dynwrap objects to be used in downstream analysis, e.g. to perform differential expression analysis using tradeSeq  and performed comparably with TinGa for linear trajectories. The results suggested that while Slingshot was the best method for linear trajectories among the tested methods, it was significantly worse than TinGa and Totem for nonlinear trajectories. This happened because ASW is generally known to select a small number of clusters, which will more likely generate a linear trajectory than a nonlinear trajectory when used for MST estimation. In contrast, TinGa is more likely to overcomplicate the trajectories with redundant cycles and disconnected parts than Slingshot and Totem, both of which can only infer tree-shaped trajectories. Indeed, the performance differences between the methods were mainly attributable to differences in the accuracy of the topology (HIM) and branch assignment (F1 branches).k-medoids clustering results for each of the benchmark datasets and ranked them using different clustering evaluation methods, including ASW, VRC, and a Totem criterion that uses cell connectivity and VRC. When we investigated the performance scores of the top 100 clustering results, the results suggested that the Totem criterion and VRC provided the best average performance. However, the Totem criterion outperformed the VRC when we considered only the best trajectory from the set of trajectories. In other words, when testing multiple trajectories, it is more likely that we find a higher-performing trajectory with Totem than with the ASW and VRC methods, and the average performance of all the trajectories found by Totem will likely be at least as good as for the other methods.To investigate what is the most optimal way to select the clustering that is used to generate the MST, we generated 10 000 Totem has the same limitations as Slingshot. In particular, it cannot be used to infer trajectories that have cycles or both diverging  and converging  parts at the same time. In addition, unlike the updated Slingshot, Totem cannot currently handle disconnected trajectories. However, we do not consider this a major limitation because determining automatically which cell types belong to which disconnected sub-trajectories is not an easy task , and ElpAn important consideration is how to perform the upstream analysis steps prior to trajectory inference, including quality control, normalization, and dimensionality reduction. The choice of the normalization method has been shown to affect the cell types that can be identified from single-cell data , and conIn some cases, it is possible that the optimal trajectory, the one that is in line with the cell connectivity profile, is not readily obtainable from the top-ranking clustering results. This can happen when some cell types are extremely rare and some cell types are highly numerous. This cell number deviation makes it more difficult for the k-medoids to find a clustering result that creates the correct trajectory topology. To alleviate this issue, Totem’s vignette explains that in situations such as these, it is a good idea to increase the number of clusters so that the rare cell types are captured in the clustering. If some of the highly numerous cell types are overclustered, these cell types can be merged using a function implemented in the Totem R package. This approach requires slightly more manual work but will guarantee a clustering that matches the cell connectivity profile.As a potential future direction in Totem’s development, the discrete cluster-level connectivity values could be turned into more precise, cell-level connectivity values by measuring the distance of each cell from the central/mass point of the cluster. However, the current implementation of Totem can also model the cell connectivity at the single-cell level by aggregating the connectivity vectors of a large number of clustering results  using the arithmetic mean. In our analyses, the arithmetic mean provided a continuous connectivity gradient, as opposed to the median-based approach.To summarize, Totem is a tool designed to facilitate the inference of tree-shaped trajectories from single-cell data. It is built upon the popular Slingshot method, which uses a clustering to construct an MST and the simultaneous principal curves algorithm to obtain a directed trajectory along with pseudotime that quantifies cell differentiation at the single-cell level. The benefit of Totem over the available tools is that it is designed to provide a user-friendly interface for finding a clustering that is used as the basis of the trajectory. Although the clustering selection is highly critical for the success of the trajectory inference, performing it automatically in a way that will generate the correct milestone network in the trajectory remains challenging. To address this challenge, the analysis of different clustering results and their MSTs with Totem has been designed to be fast and easy without requiring complex parameter tuning and in-depth technical knowledge. As a notable difference compared to existing trajectory inference methods, Totem provides the cell connectivity measure, which aids the trajectory optimization by providing information about the location of the branching points and milestone transitions.btad515_Supplementary_DataClick here for additional data file.