For measurement, a wireless sensor was chosen because of the need to walk freely without any constraints. We used MBIENTLAB METAMOTION which consists of a wireless Nordic Semiconductor chip that transfers data over a 2.4 GHz Bluetooth network. This sensor comprises of accelerometer and gyroscope. To enhance the precision and achieve additional detail in signal data, a 3-axis gyroscope with a sensitivity of 125 degrees/second was chosen. The sampling frequency of transmitted data was 200 Hz. The data were collected using a cellphone with Android 9.0 and a proprietary application used for transferring the data into the cloud for further processing.

For experimental part one, four volunteers were chosen: one healthy male and female and one male and female, both suffering from femoroacetabular impingement syndrome (on the same, right side). The sensor was placed on the right side of the pelvis, and each volunteer walked for 15 min on a treadmill.

For the second part of the experiment, 62 patients were selected. The inclusion criteria were indication to arthroscopy based on examination by physician and confirmed osteoarthritis stage based on X-ray imaging, and the exclusion criteria were morbid obesity (body mass index larger than 35) and disability to finish walking test because of the need of crutches to walk. Among them were 35 males and 27 females, the average age was 41 ± 12, and average weight was 74 ± 13 kg. In total, 19 patients had stage 1 osteoarthritis, 31 patients had stage 2 arthrosis, and 12 patients had stage 3 arthrosis on the grading scale for plain radiographs of the hip [20]. The sensor was placed on the right-hand side of the pelvis, and the patients walked approximately 10 steps. Three axes of the gyroscope signal were recorded as a time series, with a sampling frequency of 200 Hz.

A novel approach to signal processing was proposed to evaluate the risk of osteoarthritis. Osteoarthritis is often connected with crunching and grinding emerging from moving joints [13]. These mechanical disturbances are based on obstructions on mechanical movement and are often called crepitus. This grinding as a mechanical noise is then propagated through the body into the motion disturbances of the sensor which is sensitive enough to capture this mechanical noise while the patient is walking. To compare patients with different stages of osteoarthritis, it is first necessary to normalize the analyzed signal to eliminate any discrepancies, such as other types of pace. To achieve this, an approach using differential geometry was chosen. We treat three signals as time series of projections of the measured phenomena (pace) to three orthogonal axes. Euclidean differential geometry invariant, called arc length, was then used as a descriptor of the walking pace. Arc length is a mathematical object that is invariant under the action of the group ({text{SO}}left( 3 right) to {text{R}}^{3}), so it is invariant under translation and rotation in Euclidean space. This means that the resulting studied quantity (i.e., the arc length) amplifies small changes in the measured signals, which should be connected with crepitus, while maintaining and simplifying the information about individual steps. The arc length is calculated as follows:

$$sleft( t right) = mathop smallint limits_{0}^{t} sqrt {mathop sum limits_{i = 1}^{3} left( {frac{{{text{d}}G_{i} left( tau right)}}{{{text{d}}tau }}} right)^{2} } {text{d}}tau ,$$

where (G_{i}) is the i-th signal from gyroscope and (tau) is sampling time. To eliminate the natural linear trend of arc length, the derivative of (sleft( t right)) was used as an input to the next part of the signal processing. The derivative of arc length (DAL) was calculated as a difference between two adjacent samples in each sampling time. Arc length approach has already been proven to be a strong tool in biosignal processing [3, 18].

To evaluate the emergence of crepitus, spectral properties of the derivative of arc length were studied. We present the results of two hypotheses, which we propose in the presented paper.

The first hypothesis claims that with an increased stage of osteoarthritis, more parasitic mechanical noise should emerge in the frequency spectra of the processed signals. To further eliminate any disturbances, the derivative of arc length was decomposed to 2-s intervals, on which the Welch spectral analysis was performed. The resulting spectra are power spectral densities of the signals, where these spectral densities were calculated using Welch’s method. All of the measured spectra for each individual in these 2-s intervals were averaged. This ensures that minor local disturbances will not project into resulting spectral characteristics. After calculating the spectra for each patient, the resulting spectra for each stage of osteoarthritis were averaged. Consequently, the final result was three double-averaged spectral properties, each belonging to one stage of osteoarthritis.

The second hypothesis also takes the derivative of arc length as the input signal to represent the risk of osteoarthritis and the presence of crepitus. However, instead of Welch spectral analysis, continuous wavelet transformation (CWT) was used to analyze the spectral properties. CWT is effective at studying the low-frequency band of the spectra. We hypothesize that in the case of low risk of osteoarthritis, the pace and motion should be fluent without any discrepancies generated by crepitus. This means that the spectra in the higher part of the measured spectrum should contain all of the frequencies with various magnitudes without any significant frequency drops, similar to white noise. However, in the presence of crepitus, there should be visible drops in higher frequency spectra generated by spikes of mechanical noise generated by crepitus. This means that there are some resonances present. To analyze this, we first calculate the CWT of DAL. We then split the result of CWT into individual footsteps of measured subjects, and we average the CWT spectra over these footsteps. Finally, we take the average of these spectra over all subjects in each of the three groups with different stages of clinically confirmed osteoarthritis and compare them.

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