Diagnosis of Mechanical Rotor Faults in Drones Using Functional Gaussian Mixture Classifier

Abstract

The article presents the topic of propeller damage detection on unmanned multirotor drones. Propeller damage is dangerous as it can negatively affect the flight of a drone or lead to hazardous situations. The article proposes a non-invasive method for detecting damage within the drone’s hardware, which utilizes existing sensors in the Internal Measuring Unit (IMU) to classify propeller damage. The classification is performed by using the Bayesian Gaussian Mixture Model (BGMM). In the field of drone propeller damage detection, there is a significant issue of data scarcity due to traditional methods often involving invasive and destructive testing, which can lead to the loss of valuable equipment and high costs. Bayesian methods, such as BGMM, are particularly well-suited to address this issue by effectively handling limited data through incorporating prior knowledge and probabilistic reasoning. Moreover, using the IMU for damage detection is highly advantageous as it eliminates the need for additional sensors, reducing overall costs and preventing added weight that could compromise the drone’s performance. IMUs do not require specific environmental conditions to function properly, making them more versatile and practical for real-world applications.

1. Introduction

In today’s world, drones play an increasingly important role in various fields, from environmental monitoring and transporting blood to hospitals, to border inspections. With the rise in their popularity and applications, the risk associated with operating unmanned aerial vehicles (UAV) also increases. One of these risks is propeller damage. This is one of the most common and critical threats to their operation, as the result of such an incident is increased energy consumption to compensate for the damaged propeller, and in the worst-case scenario, it can lead to a sudden destabilization of the drone. Additionally, damaged propellers can generate significant noise, further indicating the need for prompt detection and maintenance. Unmanned aerial vehicles use inertial measurement units (IMU) to stabilize the drone. The same sensors can provide the data necessary for real-time diagnostics, without human intervention, expand the infrastructure or increase the production costs of the drones.

Various factors, such as collisions, material wear, or atmospheric conditions, can cause damage to propellers. These damages can significantly impact the drone’s performance, ranging from decreased flight stability to complete loss of control over the device. Therefore, detecting propeller damage at the earliest possible stage is crucial to ensure the safe and efficient operation of drones.

To enhance flight safety and minimize operational risk [1], standards increasingly incorporate the use of UAVs. These standards involve the addition of redundancy in propulsion systems. By employing this approach, drones are capable of continuing flight even in emergency situations, such as the loss of an engine or propeller [2]. Consequently, in the conducted studies, an octocopter was selected due to its design and propulsion system redundancy, which provide additional operational safety, especially in missions requiring a higher level of safety.

One of the main challenges in drone visual diagnostics is accessing data regarding the device’s technical condition. Inspectors often require costly and time-consuming inspection procedures, which are not always feasible in field conditions. In the context of real-time damage detection during flight, applying data analysis methods from the IMU, such as Gaussian mixture models, offers a solution to this problem.

Current field of propeller diagnosis operates in both dynamic and static environments. By processing a series of images taken while the drone is in a stable position with Yolo v5 convolutional neural network (CNN), we were able to detect three distinct categories of propeller condition: completely damaged, partially damaged and healthy [3]. This method used machine learning on two levels: passive and active. Passive learning involved training the model on labeled data, while active learning involved selecting the most significant data from the validation set, labeling it, and adding it to the training set. The results achieved were at the level of 85.8%, with no significant difference observed between passive and active learning. It is worth noting that this type of analysis could inspect only the upper half of a propeller due to the nature of the data, while damages on the lower half go unnoticed.

The acoustic-based approach is also widely used. When combined with an artificial neural network, it was able to detect imbalances in the blades of the UAV propeller with a high precision of 97.63% [4]; however, measurements had to be conducted indoors. A more robust method that could be performed both on the ground and during flight involved analysis of spectrograms using CNN. While an accuracy of 96.67% was achieved, this model struggled with generalization, with accuracy dropping to 55% when a different model was used [5]. For each type of drone, transfer learning was required to maintain high accuracy. Aside from a neural network, Discrete Wavelet Transform decomposition and Fourier Transform were used together to detect certain frequencies that correspond with rotational speed. These are used for detecting an artificially unbalanced blade during takeoff and hovering [6].

Monitoring damage using micro-electromechanical accelerometers placed next to each of the drone’s motors also shows potential. Based on the collected vibrations in the time domain, Fourier Transform is performed, which is the base for the propeller damage detection analysis. However, the conclusion is that using this method in real-time during the flight of a UAV is difficult due to the interference of motor vibrations, which transfer to other motors [7].

The methods described in the literature review utilize external sensors. The installation and integration of external sensors involve additional costs. Moreover, increasing the weight of the drone can negatively affect its performance during flight and increase power consumption. We decided to instead utilize data from the Internal Measurement Unit (IMU), with a focus on accelerometer measurements.

Classical machine learning models based on data analysis have hidden layers, which disrupt the transparency of the network. As a result, analyzing the network’s operation becomes more challenging. Machine learning models also require a large amount of accumulated data to function effectively once trained. Using Bayesian methods overcomes the lack of relatively rare labeled real data from faulty appliances. This field shows potential, as demonstrated in several cases, such as fault detection in diesel engines, where, despite having data from fixed rotational speed, the model could be applied to a much wider spectrum [8]. Additionally, there is work utilizing Bayesian methods to develop models that estimate the remaining useful time of a system by incorporating historical data from sensors, addressing challenges related to data scarcity [9]. Another example is a model designed to recognize faults in the early stages in permanent magnet synchronous motors [10]. The Gaussian Mixture Model is a promising method, as it applies to various classification problems. One such problem is cable fault detection [11]; however, the model is not only limited to this, and was tested across many common classification tasks [12]. In the authors’ other work, they detected faults in the frequency domain, where Bayesian methods created a generative model to supplement a lack of data [13]. This approach reduces inspection costs, time, and potential material losses. Introducing analytical methods into the operational practice of drones can contribute to their increased reliability and safety, which is crucial given their growing popularity and commercial applications.

Functional data analysis (FDA) is a set of methods for analyzing data in the form of functions, with a particular emphasis on time series data. FDA turns a complicated signal into a number of simpler basis functions with coefficients. FDA is a well-established field in statistics, with a focus on bases in functional spaces such as polynomials and wavelets. This maturity can be seen in review papers [14] and special issues in prestigious statistics journals [15]. Combining FDA with Bayesian methods allows analysts to obtain probability distributions of basis coefficients, create generative models, and model uncertainty. Analysts can apply FDA both in time and frequency domains, the latter of which can also be used in motor diagnostics [16].

In this paper, we focus on detecting damages on propellers using data from the accelerometer acquired by the IMU in the time domain. We propose two Bayesian Gaussian Mixture Model classifiers for recognizing damage of the propeller. Our main contributions are as follows:

• Construction of Bayesian functional model using splines,
• Binary and three class classifier that recognizes the type of damage the propeller has suffered

The rest of the paper is organized as follows. First we describe the experimental setup, available measurements, their initial analysis and reasoning for proposed analysis. Then we move to the Bayesian framework, where we show how we modeled the signals to compensate for data scarcity, then we propose the classifier and describe how it works. Then we move to the analysis of the results. We show how classification is performed for individual signals, and then provide detailed testing over 100 randomized training runs.

2. Materials

In this paper, we focus on diagnostics of the octocopter Figure 1. Due to its design and motor arrangement, this type of drone can withstand more demanding environmental conditions. In case of the loss of even three propellers, it can continue its mission [17]. In the unmanned multirotor, a flight controller was used, which has a triple-redundant system and, in case of failure of one of the inertial navigation systems, can still complete the mission. The components used in the drone are presented in

Table 1. Components used in the drone and ground station.

Component	Details	Location
Flight Controller	Orange Cube-CubePilot-Australia	Drone
Satellite Navigation System	GNSS HERE 3+-CubePilot -Australia	Drone
Communication Module	RFD900x-RFDesign-Australia	Drone
Camera and Video Transmitter	DJI Air Unit 03 Set-DJI-China	Drone
Two Electronic Speed Controllers (ESCs)	T-Motor Velox 45 A-T-Motor-China	Drone
Eight Motors	T-MOTOR 4006 380 kV-T-Motor-China	Drone
Converter	D24V50F5-Pololu-United States of America	Drone
Capacitor	Capacitor 50 V 1500-Jamicon-Poland	Drone
Battery	Tattu 6S 10,000 mah-Grepow-China	Drone
Ground Computer		Ground Station
Transmitter	FrSky Taranis Q X7-FrSky-China	Ground Station
Communication Module	RFD900x-RFDesign-Australia	Ground Station
Video Receiver	DJI Goggles V2-DJI-China	Ground Station

.

To reduce operational risk and avoid increasing costs associated with expanding the system with additional safety features, the flight controller uses a built-in IMU. Automated missions standardized the data and reduced random factors such as unforeseen maneuvers and induced oscillations in the drone. This standardization of the data aims to make analysis easier. Based on the data collected from automated flights Figure 2 and after preprocessing, researchers created two classifiers: binary and three-class. The statistical model based on the Gaussian Mixture Classifier performs the analysis and detection. The binary classifier predicts whether the drone has a damaged propeller, while the three-class classifier estimates whether the drone has all propellers intact or whether a propeller has a 15 mm or 35 mm cut Figure 3.

During the configuration of the Ardupilot autopilot, the system set IMU data logging to a microSD card. After the flight, one could read and analyze the recorded data.

2.1. Data Collection

To collect high-quality data that can be easily compared, researchers conducted a series of automated flights. Utilizing the capabilities of the open-source autopilot “Ardupilot” [18], they logged data from the start of the mission to its end every 10 ms. This method allowed them to read and analyze the data from the MicroSD card. They carried out each automated flight on a pre-programmed route, which standardized the test conditions shown in Figure 2.

During various flights, we conducted a propeller cutting operation to collect data. Initially, we performed flights with all fully intact propellers to obtain baseline reference data. Then, in the next series of flights, we shortened one of the propellers by 15 mm (Figure 3). We conducted automated flights again in this configuration to record changes in the drone’s behavior and flight parameters.

In the final series of automated flights, we shortened the same previously damaged propeller by another 20 mm, resulting in a total shortening length of 35 mm (Figure 3). Our purpose was to understand how gradual propeller damage affects the drone’s stability, maneuverability, and performance.

In each series of flights, we collected data using the IMU, accelerometer, and gyroscope. This allowed us to precisely monitor flight parameters such as vibrations, acceleration, and changes in the drone’s orientation.

2.2. Preprocessing

The first step we took was to divide the collected data into smaller segments, facilitating data management and allowing for more detailed analysis. Next, we applied Euclidean normalization to the accelerometer and gyroscope data. This normalization reduced the number of dimensions to one, making the data more homogeneous and easier to analyze, and allowing for better comparison of different data sets. We present an example of preprocessed data in the Figure 4.

The following step was to remove unit jumps that resulted from the landing and takeoff of the UAV, as these jumps could introduce noise into the analysis and affect the selection of statistical methods. After these steps, the accelerometer data were ready for further analysis. We exported the accelerometer data in the time domain.

We applied two sampling methods to the data analysis: a sliding window and an aggregating window with data from the flight. The sliding window allowed us to analyze the signal in short, overlapping segments. Each segment consisted of 175 samples with 125 samples overlap, effectively yielding new diagnoses. The aggregating window summed the previous samples and the current one, where the duration of the current sample was 1750 ms.

3. Methods

In this section we introduce the main concepts on which our work is built. Firstly, we introduce the idea for construction of Bayesian functional data models with splines. Then we present the algorithm for classification Bayesian Gaussian Mixture Model and briefly explain how it works. Finally, we introduce Stan, a computational platform, primarily designed for Bayesian inference.

3.1. Bayesian Functional Spline Models

We represent given measurement, y, as B-spline basis that can be drawn from one of M data generating processes (which represents our fault classes), each described by a unique set of parameters. To create this model, we need to build the likelihood function and then use it to develop the posterior distribution. The data generating process is given by

$$\begin{array}{c}y\sim \mathrm{Normal}(\mu ,\sigma )\\ \mu ={\displaystyle \sum _{m=1}^M}{\beta }_m{\varphi }_m\left(t\right)\end{array}$$

(1)

where y is a sampled signal with uncertainty of normal distribution given by $\sigma$. Functions ${\varphi }_m\left(t\right)$ are B-splines on the assigned M knot grid. $\mu$ represents the transformed parameters of the model, as it corresponds to the mean of the fitted distribution of each class. ${\beta }_m$ are terms regulating each spline value. All are described by normal distribution:

$${\beta }_m\sim \mathrm{Normal}({\mu }_0,{\sigma }_0)$$

(2)

Parameters ${\mu }_0,{\sigma }_0$ are not known and are obtained through inference, making the proposed model a hierarchical one. Relations of the entire model are presented using Bayesian network plate notation in Figure 5. In order to ensure we chose the right amount of splines, we conducted experiments consisting of calculating the Watanabe–Akaike information criterion (WAIC), and cross referencing the obtained values with statistics of 10 classification runs from the model. Lower WAIC values were obtained as more splines were used, while model performance slightly declined, which was interpreted as a warning of excessive model complexity. Therefore, the number of splines should be chosen in such a way that the model is sufficiently informative but not overfitted. As a result, 15 splines were chosen for the binary classifier and 20 for the three-class classifier. The basis for the binary classifier is shown in Figure 6.

Hyper-parameters were given uniform priors, as we have no justification for others. $\sigma$ was given an exponential prior, as it has a heavier tail than the half-normal, allowing extra flexibility. The proposed model allows sampling of parameters from posterior and sampling from posterior predictive distribution. Posterior predictive distribution is a useful tool for generating predicted data from inferred parameters, as can be seen in Figure 7.

3.2. Gaussian Mixture Model Classifier

Gaussian Mixture Models (GMMs) are a powerful statistical method, commonly used for data classification because of their effectiveness in modeling complex distributions. GMMs work on the assumption that data are generated from a mixture of several Gaussian distributions, each defined by its own set of parameters. This approach is especially beneficial for handling data that exhibit multiple local phenomena and non-stationarity. By utilizing multiple components, GMMs can capture diverse statistical characteristics within the data, offering a flexibility that single Gaussian models cannot achieve [19].

Using GMMs as classifiers provides certain advantages. They assign probabilities of belonging to each class instead of just labelling them, which is beneficial in real-world problems, since it can lead to detecting faults in early stages where repair could be much cheaper and easier compared to more serious damages that occurs over longer periods of time. Moreover, the parameters of the GMM (means, variances, and mixture coefficients) are fitted through the model based on the observed data [19].

In the context of this paper, we consider that each mixture of components is a signal reconstruction using FDA with coefficients as distributions of hyperparameters. In such cases, each representation of signals, whether faulty or healthy, and the class probability can be recovered.

3.3. Stan

Stan is a powerful computational platform, primarily designed for Bayesian inference, developed to handle complex models and enhance the precision and efficiency of statistical analyses. It is built on advanced algorithms like Hamiltonian Monte Carlo (HMC) and the No-U-Turn Sampler (NUTS), which are sophisticated variations of Markov Chain Monte Carlo (MCMC) methods [20]. These algorithms excel at exploring the intricate landscapes of high-dimensional parameter spaces often found in statistical models, ensuring comprehensive and efficient sampling. In our application, we utilized the HMC approach for sampling. The creation of Stan was motivated by the statistical community’s need for a more accessible, scalable, and efficient tool for applying Bayesian methods to real-world challenges. It features a user-friendly, high-level programming language that simplifies the specification of complex models, making Bayesian analysis more approachable for non-statisticians. Stan’s language enables users to define their models once and then apply various available algorithms, facilitating a wide range of statistical analyses without requiring algorithm-specific coding [21]. Stan also offers various interfaces, or ’wrappers’, catering to users familiar with different programming environments. We employed CmdStanPy, which allows Python users to execute Stan models by interacting directly with Stan. These interfaces enable seamless integration into data analysis workflows across diverse platforms.

4. Results

GMM was implemented for classification using the Stan framework. Each classification is produced by calculating statistics from 4000 samples generated by stan. The dataset consisted of a total of 158 samples produced by a sliding window, using data from an accelerometer from three different classes. Two types of models were considered binary and three-class classifier.

4.1. Binary Classifier

For binary classifier, samples from 15 mm and 35 mm cavities were merged together as “damaged” class. Performance was calculated as a statistic of 100 independent classifications. Each time, the model was trained on seven randomly chosen samples from both classes. An example of classification results can be seen in Figure 8. This model achieved $97.57\%$ accuracy. It is worth noting that most of its mistakes come from classifying healthy data as damaged.

4.2. Three-Class Classifier

For three-class classifier, both types of damage are distinguishable. Similarly to the previous model, training data consisted of seven randomly chosen samples, and itss performance is also a statistic of 100 independent runs. This model achieved lower accuracy $69.7\%$. An example of correct classification is presented in Figure 9, while wrong classification can be seen in Figure 10. It was noted that certain samples were especially problematic for the model, particularly for the propeller with the 15 mm cavity, as the model was unable to correctly classify them in most of runs.

5. Discussion

Based on the collected data from automatic octocopter flights, two types of classifications were performed using different methods, both based on the Gaussian Mixture Classifier. The first was a binary classifier, which used accelerometer data, achieving an accuracy of $97.57\%$.

Additionally, experiments were conducted with a three-class classifier, which aimed to predict the condition of the drone’s propellers. This classifier was able to determine whether the propellers were fully functional, if one had a 15 mm damaged area, or a 35 mm damaged area, thereby achieving an accuracy of $69.7\%$. These results demonstrate that it is possible to accurately monitor the technical condition of propellers using advanced classification methods. This model suffers from uncertainties when it tries to distinguish the type of damage. When its diagnosis still could be used as a warning, we aim to improve its precision. While current trials focus on increasing the number of splines used to represent the class or attempts to adjust priors were futile, we believe that incorporating data fusion methods will yield satisfactory results. Other types of experiments included changing samples from single vector containing a Euclidean norm to matrix containing data from each axis; however, results were similar at best. From our observation, the root of this problem is that, during certain movements, vibrations caused by the damaged propeller are greater along certain axes while being very similar across the remaining ones at the same timestamp. This confuses the model so that, on average, it performs worse. However, this topic has not been fully explored and we plan to test this approach further.

Future plans include, as mentioned earlier, the development of data fusion methods from the accelerometer and gyroscope, which can significantly increase the accuracy of damage detection [22]. Data fusion allows for a more comprehensive view of the analyzed parameters and can contribute to increased precision and reliability of forecasts. Although there were plans to develop a microcrack classifier, due to the lack of precise equipment and concerns regarding the safety of the testing procedures, it was decided not to address this issue in this article.

Furthermore, we plan on integrating the model with real-time data. This integration will require ground telemetry reading from the unmanned aerial vehicle and, using the MAVLINK protocol, reading the IMU data at the ground station [23]. At this station, a program for detecting damaged propellers will be run. Such infrastructure will allow for real-time monitoring of the drones’ technical condition and responding to any detected irregularities.

There are also plans to extend the Gaussian Mixture Classifier model to include anomaly detection functions. Such a modification would increase the classifier’s range of operation, making it more versatile and capable of detecting unexpected issues that may arise during the flight.

Another possible improvement is to include the Dirichlet Process to automatically detect the number of classes. It demonstrated promising results and also performed well in anomaly detection when combined with Gaussian Mixture Model [24].

The authors experimented with gyroscope data represented in the frequency domain after FFT; however, consistent results were not achieved. Choosing the right type of window and its size presented a set of challenges. Despite these setbacks, we still plan to further evaluate this data in the frequency domain.

The result of the implemented solutions and future considerations for the proposed algorithm’s development will be increased safety of the UAV. Improved accuracy in damage detection and rapid response to detected anomalies will contribute to reduce the maintenance costs of the drone infrastructure, minimizing the risk of failures, and extending the equipment’s lifespan. This, in turn, leads to more efficient and economical management of the drone fleet, which is crucial in the context of their commercial and industrial use.

6. Conclusions

The article presents the issue of detecting propeller damage on drones. Based on data collected during automated flights, which involved complex maneuvers such as turns, altitude changes, and forward motion, a statistical analysis was conducted using a Gaussian Mixture Classifier. The algorithm was designed to account for each type of malfunction occurring during the automated flights. The results of the experiments, comparing test data with validation data, showed that the prediction accuracy is promising. For practical and broader application, a series of studies in various environments and on different types of drones will be necessary. Nevertheless, the achieved performance of the implemented algorithms is sufficient to enhance the safety level of teleoperation.