1. Introduction
During the last decade, robots, in particular those with servomotors, have gained momentum in diverse areas such as rehabilitation systems [
1,
2,
3], human interaction systems [
4], bio-inspired systems [
5,
6,
7,
8], and in surgery [
9]. In general, their functionality depends on their performance trajectory; in this regard, an essential aspect to be considered in each programmed task is the value of the natural frequencies (NFs) of the robot, which can magnify the robot vibrations, affecting its performance [
10,
11,
12,
13,
14]. Hence, it is of paramount importance to implement a method to accurately identify the NFs in order to obtain the best performance of the robot.
In literature, different strategies based on FEM and experimental procedures to estimate NFs in robots have been presented [
15,
16,
17,
18]. FEM is the most used experimental procedure for vibration measurements integrated with signal processing techniques because it can estimate the real behavior of robots [
19,
20,
21,
22,
23,
24]. Although FEM provides results that are very close-to-real [
25], it is necessary to perform an experimental validation of the behavior of the robot to identify the NFs with accuracy [
26]. An important step of the experimental procedure is the use of a signal processing technique to estimate the NFs of robots [
22]. In this regard, Fourier-based methods [
19,
20,
21,
22,
23] and wavelet transform (WT) [
24,
27] have been the most commonly employed methods for performing this task. For example, Min et al. employed the Fourier transform method to identify the NFs of an STR6-05 robotic arm, a 6-DOF heavy load manipulator, to develop a collision detection method based on vibration signals [
22]. Similar work was presented by Yuan et al. [
23], who applied Fourier transform to estimate the NFs of a robotic manipulator of 6 DOF in order to reduce its vibrations in a specific frequency range using a magnetorheological elastomer absorber on a spindle. Moreover, a swept sinusoidal signal for measuring the vibration response of the robot was used.
On the other hand, Chen et al. used the WT method for NFs identification of a simulated 4-DOF system subjected to forced excitations [
28]. Klepka et al. applied the WT method for estimating the NFs of a simulated 2-DOF system [
29]. The works presented above [
19,
20,
21,
22,
23,
24,
27] have presented promising results for estimating the NFs of robots [
19,
20,
21,
22,
23,
24,
27]; nevertheless, when a system or robot presents closed NFs, the methods presented are not efficient for identifying them [
19]. Besides, Fourier-based methods are affected by noisy signals in complex systems and require a large number of samples [
30], and an appropriate selection of the decomposition level and wavelet mother is necessary for WT to estimate the NFs appropriately [
28]. For these reasons, it is essential to implement a signal processing technique for estimating the NFs of a robot under noisy signals with accuracy, especially closed NFs, using a small amount of data to avoid wear and tear on the robot during testing.
In recent years, a technique called the MUltiple Signal Classification (MUSIC) algorithm has provided promising results for analyzing induction motors [
31,
32], evaluating the behavior of civil structures [
33,
34,
35], and impact-source-localization in composite structures under deformation conditions [
36], among other applications. This technique presents diverse advantages such as noise immunity and high resolution and does not require a large amount of experimental information to estimate the frequencies contained in the analyzed signal with high accuracy [
33]. It is also important to mention that the MUSIC algorithm provides an increased detectability of frequencies with a low amplitude as measured in robots [
19,
22,
29], which is a significant advantage in this task. Hence, the MUSIC algorithm could provide an excellent alternative for estimating the NFs of robots.
This paper identifies the NFs of two different experimental platforms of a 2-DOF planar robot by FEM and its validation through the MUSIC algorithm. The FEM for two experimental platforms is presented. The experiment consists of an impulse-based trajectory applied to the end effector of the robot in order to excite it and obtain the response; then, the measured vibration signals are processed by the MUSIC algorithm and the fast Fourier transform (FFT) method [
37] to identify the NFs. The experimental results show that the MUSIC algorithm can identify the NFs of the 2-DOF planar robot with a short sample data set and higher accuracy than the FFT method. Furthermore, the error between the MUSIC and the FEM results is lower than that obtained with FFT and the FEM results.
5. Results and Discussion
Table 6 shows the values of the first five identified NFs by the FFT method and the MUSIC algorithm, as well as their corresponding analytical values obtained by FEM for the two cases of study. As it is shown, the MUSIC algorithm can identify the NFs at a very similar level when compared with the analytical calculation from FEM. Besides, the MUSIC algorithm does not require a long time window to provide a high-frequency resolution, avoiding wear on the actuators during testing through a short robot testing time. On the other hand, the FFT method presents a significant difference compared to FEM due to the noise present in the signal; this is a significant advantage of the proposed methodology because most of the real signals have a considerable noise level.
Figure 10 shows the percentages of similarity between the results of the MUSIC algorithm and FFT method against FEM. These values are obtained by:
where
is the analytical calculation results,
is the experimental results (MUSIC or FFT), and
is the error percentage for the corresponding modes.
Observing
Figure 10, the results of the FEM analysis by ANSYS™ using the two cases of study provide an acceptable accuracy compared with experimental results obtained with the MUSIC algorithm, i.e., the results of FEM are close-to-real, with a maximum error of
for the case of study 1 (denoted by the solid blue line) and
error for the case of study 2 (denoted by the dotted blue line). However, if the FEM results are validated using the experimental results of the FFT method, the percentage error is more significant, at
(denoted by the solid black line) and
(denoted by the dotted black line) for the cases of study 1 and 2, respectively. In this regard, the proposed methodology can contribute to the planning of the trajectories of the robot. It will also be useful for the selection of controller gains to avoid exciting the robot in the NFs and assist in the correct selection of notch filters at the output of a controller.
Notice that the MUSIC algorithm requires the previous selection of the algorithm order, which is chosen according to the number of frequencies or components found in the analyzed signal. In this regard, the number of frequencies contained in the signal is unknown, but it is known that the trajectories used in this type of robot are in the range from
to
[
20,
23]. In this sense, the FEM results allow the selection of the MUSIC algorithm order because they provide the number of natural frequencies contained in the range of interest (
to
), showing that an order of 10 is the most reliable for identifying the main NFs contained in the signal. On the other hand, the MUSIC algorithm presents more computational complexity than the FFT; however, it is superior to the FFT method for identifying the NFs of both robots under noisy signals. The numbers of operations for computing the FFT (
) [
48] and MUSIC (
) [
49] are given by the following equations:
where
represents the data length.
6. Conclusions
This paper presents a novel methodology to identify NFs in 2-DOF planar robots, a FEM is applied in two cases of study, and the results are validated with a vibration signal analysis through the MUSIC algorithm. Two cases of study are modeled in ANSYS™ software, and a FEM is applied to estimate the NFs of the robots. Also, a simulated signal is analyzed to show the effectiveness of the MUSIC algorithm. The experimentation consists of an impulse-based trajectory applied to the end effector of the robot to excite the mechanism; the vibration signals are measured by a triaxial accelerometer and processed by the proposed methodology and the FFT method to show the advantages of the MUSIC algorithm compared with the traditional method. The experimental results show that the MUSIC algorithm is closer to the FEM results, and it is a useful methodology for NF identification in 2-DOF planar robots because it is not affected when the signal is contaminated with a high-level noise. In this context, the MUSIC algorithm is advantageous and has a higher resolution than traditional Fourier-based methods. The experimental cases are carried out in an time window. Hence, the simulated and experimental results show that the MUSIC algorithm does not need a long sample time window to obtain a high-frequency resolution. The proposed methodology has been developed in a 2-DOF planar robot without loss of generality; that is, the instrumentation and signal acquisition process is the same for an n-DOF robot arm. The MUSIC algorithm can also contribute to path planning of the robot, the selection of gains of a controller to avoid exciting the robot in the NFs, and the correct selection of notch filters at the output of a controller.