# Enhancing IMU Accuracy in MRAVs: A Theoretical and Experimental Approach to Vibration Damping

Lorenzo Balandi, Paolo Robuffo Giordano, Marco Tognon

► To cite this version:

Lorenzo Balandi, Paolo Robuffo Giordano, Marco Tognon. Enhancing IMU Accuracy in MRAVs: A Theoretical and Experimental Approach to Vibration Damping. ICUAS 2025 - International Conference on Unmanned Aircraft Systems, May 2025, Charlotte (North Carolina), United States. pp.1-7, 10.1109/icuas65942.2025.11007828 . hal-05025884

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Submitted on 8 Apr 2025

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<figure><img src="image_1.png" /></figure>
# Enhancing IMU Accuracy in MRAVs: A Theoretical and Experimental Approach to Vibration Damping

Lorenzo Balandi<fnref n="1" />, Paolo Robuffo Giordano<fnref n="1" />, and Marco Tognon<fnref n="1" />

*Abstract*— This paper analyses the problem of mechanical vibrations on Flight Controllers (FCs) of Multi-Rotor Aerial Vehicles (MRAVs) and proposes solutions based on vibration theory. First, we analyze the raw Inertial Measurement Unit (IMU) data obtained from real flights to understand the dynamical characteristics of a baseline damping configuration. We confirm that the motor-propeller units are the main source of vibration in these systems. We then develop two models used to understand how to effectively damp vibrations on IMU. We improve the baseline configuration using commonly available components placed according to a theoretical analysis and we discuss the experimental results. The new damping configuration greatly decreases the amplitude of vibrations on acceleration and angular velocity measurements.

<figure><img src="image_2.png" /><figcaption>Fig. 1: The fully-actuated hexarotor used in this work (left) with IMU frame, the Paparazzi Chimera flight controller with IMU in the geometrical center (right).</figcaption></figure>

## I. INTRODUCTION

Manned and unmanned Multi-Rotor Aerial Vehicles (MRAVs) are widespread in research and industry. Thanks to their versatility and performance, they are used in different sizes for many different tasks. MRAVs are equipped with onboard electronics for sensing and actuation. In particular, an Inertial Measurement Unit (IMU) is usually present and embedded in the Flight Controller (FC) mounted on the vehicle frame [1]. The IMU includes accelerometers and gyroscopes providing linear acceleration and angular velocity, respectively. These devices are often based on Micro Electro Mechanical Systems (MEMS) technology, which is sensitive to electronic noise, vibrations from the environment and measurement offsets. While electronic noise and offsets can be dealt with by IMU calibration, mechanical vibrations from the environment represent a critical issue in MRAV and one needs to perform a pre-filtering of the raw data for making it usable in any subsequent step. Data from IMUs are then used for state estimation and control algorithms and allow the deployment in real world scenarios.

There are two main solutions to filter noise: software and mechanical filtering. Software filtering consists in applying digital low-pass filters to reduce signal components with frequencies higher than the cutoff frequency of the filter. A mechanical filter is a mechanical component equivalent to a spring-damper system used to mechanically isolate the sensitive electronics from the source of noise. These solutions are usually not sufficient alone and are thus used together. Furthermore, the cutoff frequency represents a balance between the vehicle's motion frequencies we want

to detect and the noise frequencies we want to eliminate. It is important to note that a lower cutoff frequency results in a higher delay in the output signal, with obvious negative effects for the closed-loop stability.

In this work, we focus on the mechanical damping of vibrations affecting the FC on a MRAV. This problem is crucial for safe and efficient operations. Noisy measurements produce noisy state estimation and thus control inputs, which increase instability and energy consumption. Moreover, some recently developed acceleration-based controllers [2]–[4] have proven to be very effective for MRAV trajectory tracking, but they are extremely sensitive to acceleration noise and usually require heavy software filtering. Another extreme phenomenon caused by vibrations in MRAVs is IMU saturation, which can lead to artifacts in the sensed data, bad state estimations and potentially crashes. Indeed, during our experimental campaigns we encountered IMU saturation which resulted in undesired behaviors as shown in the attached video. Many custom platforms are proposed for contactless and contact-based [5]–[9] operations, but little work is dedicated to the problem of vibrations identification and damping. There are no general procedures motivated by theory to tackle this problem.

Some works focus on vibrations characterization and damping for MRAVs. [10]–[12] recognized the onboard motors as the main cause of vibration noise on acceleration signals, and identified the blades of the propellers cutting the air as the main source. These vibrations are transmitted to the FC through the arms and the frame. As a *n*-Degrees of Freedom (DoF) system has *n* natural frequencies, the resonance of MRAVs frames has been analyzed [10]–[13] and some relatively low resonance frequencies were found. Although these results depend on the specific platform, they suggest that the problem of vibrations is non negligible

<footnote n="1">¹ Univ Rennes, CNRS, Inria, IRISA, Rennes, 35042, France, lorenzo.balandi@inria.fr, marco.tognon@inria.fr, prg@irisa.fr. This work was supported by the project ANR-23-CE33-0013 "FlyHandyBot". Experiments presented in this paper were carried out thanks to a platform of the Robotex 2.0 French research infrastructure.</footnote>
with the materials generally used to build frames. General practices to move low resonance frequencies higher in the spectrum are to adopt stiff materials (e.g., carbon fiber) and securely tighten the screws. An important result from [13] is that the electronic card of the FC is an extremely rigid body which transmits all the vibrations in the frequency range of interest (< 1000 Hz), and hence the motor-propeller units are definitely the main vibration source. In [12] the authors experimentally verified that long carbon fiber bars damp vibrations transmitted from the motors to the MRAV frame, depending on length and shape of the bars.

However, these works do not provide general rules for designing mechanical vibration isolators or for improving existing systems. [13] identifies performance targets for a damping system for FCs in terms of desired cutoff frequency, but it is unclear how to physically realize devices with such characteristics and no general guidelines are provided. Moreover, none of the above mentioned works analyzes the angular velocity signal from the IMU, which is crucial for attitude control of MRAVs. In general, there is a disconnection between the Mechanical Engineering and the Aerial Robotics communities. The latter conceived novel platforms and suitable control algorithms, but the mechanical design lacks proper considerations for the isolation of the FC from vibrations. Usually adopted solutions are copied from commercial products without critical investigations.

Given this gap, the goal of this paper is to analyze raw IMU signals from the FC of a fully-actuated hexarotor MRAV with a baseline damping configuration to characterize the vibration problem. We discuss how to effectively damp vibrations providing the theoretical basis from which general considerations can be drawn. Leveraging the developed theory, we improve the baseline damping using common dampers placed in an optimized way considering the geometry of the FC. Experimental data confirms the theoretical results.

We believe that this paper can help researchers and practitioners in tackling the problem of vibrations on sensitive electronics components of MRAVs, providing theoretical foundations validated by practical results. We report the theory to ease the generalization from the platform used in this work to other MRAVs, without the need for complex analyses and instruments.

Finally, note that the problem of vibrations is relevant also for other types of aerial vehicles. For example, in [14] the authors study the vibration introduced by the flapping wings to the body in a flapping wings flying robot.

## II. MULTIROTOR VIBRATION ANALYSIS

In order to study the IMU damping, it is necessary to identify and characterize the problem of vibrations. We analyze real flight data from the onboard IMU in time and frequency domains to understand the amplitude of vibrations and their main frequency components.

### A. Experimental setup

The aerial platform used in this work is the fully-actuated hexarotor called Tilthex [6] depicted in Fig. 1. The Tilthex

<figure><img src="image_3.png" /><figcaption>Fig. 2: Baseline damping configuration for the FC.</figcaption></figure>

is composed of six brushless motors and 13" propellers mounted on carbon fiber bars and a central frame made of several discs of glass fiber to host electronics and batteries. The onboard computer is mounted at the bottom of the central frame, while the FC is mounted on its top, as visible in Fig. 1. The motors can be controlled to spin in the range [16, 110] Hz. The Tilthex mounts a Paparazzi Chimera<fnref n="1" /> open-source FC (Fig. 1) whose position has never been optimized. It embeds an IMU (3 axis Accelerometer + 3 axis Gyroscope) providing raw measurements for the linear acceleration $a \in \mathbb{R}^3$ and angular velocity $\omega \in \mathbb{R}^3$, both expressed in the IMU frame depicted in Fig. 1. This sensor is relatively cheap and comparable to IMUs mounted on other commercial FCs. The accelerometer is set to the range $\pm 8$ g, while the gyroscope to $\pm 1000°/s$. The baseline damping configuration considered in this work consists of 4 cylindrical rubber dampers between the FC and the robot frame, as depicted in Fig. 2. This solution is cheap and a standard practice in aerial robotics. However, the performance in attenuating the vibration noise can be improved using simple analysis and mechanical vibration theory as shown in this work. Raw IMU data are sampled and logged at 1 kHz, while a nonlinear state estimator running at the same frequency fuses them with motion capture data. Without loss of generality, we can define the body frame coincident with the IMU frame with the x- and y- axes defining a plane parallel to the one of the motors. In all the experiments, the Tilthex is controlled to hover with zero attitude, namely, with the z-axis parallel and opposite to the gravity vector.

### B. Data analysis

From a complete flight we extract 10 seconds of raw data and we apply the Fast Fourier Transform (FFT) algorithm (without the gravity bias on $a$) to extract frequency information. Linear acceleration, $a$, and angular velocity $\omega$ obtained with the baseline damping configuration are plotted in red in Fig. 3, both in time and frequency domains.

From the time domain plots we can see that all signals are noisy, and in particular the linear acceleration along $z$ has high amplitude vibrations. Considering the accelerometer upper limit set to $+8$ g, this represents a potential risk of IMU saturation. Regarding the frequency analysis, for both acceleration and angular velocity along $x$ and $y$ axes, the

<footnote n="1">¹https://wiki.paparazziuav.org/wiki/Chimera/v1.00</footnote>
<figure><img src="image_4.png" /><figcaption>(a) Accelerations in time domain.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(c) FFT of accelerations.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(b) Angular velocities in time domain.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(d) FFT of angular velocities.</figcaption></figure>

<caption>Fig. 3: IMU data with baseline and improved damping systems in hovering.</caption>

frequency components are concentrated around the average motors spinning frequency in hovering (66 Hz), its double, and multiple harmonics. The average motors speed and its multiples are indicated as black vertical lines in Fig. 3. This result suggests the presence of low resonance frequencies of the MRAV structure similarly to [10], [12]. Along the z axis, the spectral content is concentrated around double the motors frequency, because each motor mounts a two-blades propeller. This result is similar to what found in [11]. Note that these frequencies are relatively low and cannot be fully canceled by low pass filters with cutoff frequencies in the order of the Hz (frequencies of interest of the MRAV motion), as opposed to, e.g., electrical high-frequency noise. As stated in [10]–[12], we conclude that the motors-propellers are the main source of vibrations in the system and the number of blades influences their frequency. Note that with more dynamic reference trajectories in which the vehicle frame is required to move and tilt, the sharp peaks observed in the FFT are expected to be more spread out and less pronounced, because motor speeds differ and vary over time.

Next, we provide the theoretical foundations to effectively damp the identified vibrations in a more principled way.

### III. VIBRATION DAMPING MODELING

In this section, we develop two models for the FC damping system. The first is used to quantify the force transmitted from the MRAV frame to the FC and is then exploited to practically enhance the baseline damping configuration. The second is used to study how the geometrical properties of the FC and the dampers placement influence the IMU data, and in particular the angular velocity.

#### A. 1D damping system modeling

The FC can be modeled as a mass *m*, while its connection to the MRAV frame can be represented as a spring-damper system (Fig. 4). The displacement of *m*, expressed in the axis with positive direction as in Fig. 4, is measured from static equilibrium and denoted with *w*(*t*). The axis can correspond to any of the IMU frame axes, as the equivalent stiffness along an axis can be determined, allowing for the decoupling of multi-DoFs systems. [15]. The frame is modeled as a vibrating base whose motion *h*(*t*) is assumed to be known
<figure><img src="image_8.png" /><figcaption>Fig. 4: 1 DoF mass-spring-damper system with exciting base (left), free body force diagram (right).</figcaption></figure>

(i.e., it does not add degrees of freedom to the system) and harmonic, so that it can be expressed as $h(t) = H_0e^{i\omega t}$, where $H_0$ is the intensity, $i$ is the complex unit and $\omega$ is the frequency of vibration. The relative displacement is thus $w(t) - h(t)$. Analyzing the forces acting on the mass as depicted in the free body diagram in Fig. 4, we obtain

$$m\ddot{w}(t) + c(\dot{w}(t) - \dot{h}(t)) + k(w(t) - h(t)) = 0, \quad (1)$$

where $k$ and $c$ are the stiffness and damping coefficients, respectively. We can rewrite (1) as

$$\begin{aligned} m\ddot{w}(t) + c\dot{w}(t) + kw(t) &= c\dot{h}(t) + kh(t) \\ m\ddot{w}(t) + c\dot{w}(t) + kw(t) &= H_0(k + ic\omega)e^{i\omega t}, \end{aligned} \quad (2)$$

where we replaced $h(t) = H_0e^{i\omega t}$ and its derivative. Dividing both sides by $m$ and using the definition of natural frequency $\omega_n = \sqrt{k/m}$ and damping factor $\xi = \frac{c}{2m\omega_n}$, we obtain

$$\ddot{w}(t) + 2\xi\omega_n\dot{w}(t) + \omega_n^2 w(t) = H_0(\omega_n^2 + 2i\xi\omega\omega_n)e^{i\omega t}. \quad (3)$$

If $\xi > 0$, the system is damped and will reach the steady state condition. We are interested in the steady state response of $w(t)$, which will be in the form $w_{ss}(t) = W_{ss}e^{i\omega t}$. Plugging $w_{ss}(t)$ and its derivatives in (3) we finally obtain

$$\begin{aligned} W_{ss}(-\omega^2 + 2i\omega\xi\omega_n + \omega_n^2)e^{i\omega t} &= H_0(\omega_n^2 + 2i\xi\omega\omega_n)e^{i\omega t} \\ \implies W_{ss} &= H_0 \frac{1 + 2i\xi\bar{\omega}}{1 - \bar{\omega}^2 + 2i\xi\bar{\omega}} = H_0 T(i\bar{\omega}), \end{aligned} \quad (4)$$

where $\bar{\omega} = \omega/\omega_n$ and $T(i\bar{\omega})$ is the transmissibility, namely a function describing the steady state response of $w(t)$ given a base excitation $h(t)$ with a certain frequency. In the last derivation we exploited known results for the response of a second order system subject to harmonic excitation, the interested reader is referred to [15] for a complete discussion. This approach is well-known in the mechanical community and is used to study systems such as cars suspensions on imperfect roads or buildings subject to seismic excitations.

It is interesting to analyze the magnitude of the transmissibility, also called displacement transmissibility [15]

$$|T(i\bar{\omega})| = \frac{\sqrt{1 + (2\xi\bar{\omega})^2}}{\sqrt{(1 - (\bar{\omega})^2)^2 + (2\xi\bar{\omega})^2}}, \quad (5)$$

which is a quantitative indication of how the base excitation is transmitted to the vibrating mass through the spring-damper, which can act as a damping system or as an

<figure><img src="image_9.png" /><figcaption>Fig. 5: Transmissibility magnitude for varying $\xi$.</figcaption></figure>

amplifier. Fig. 5 depicts the transmissibility magnitude for various $\xi$. This plot shows several areas of interest:

* $\bar{\omega} = 0$ ($\omega = 0$, constant excitation): the force at steady state is fully transmitted from the base to the mass;

* $\bar{\omega} \approx 1$ ($\bar{\omega} \approx \omega_n$): the force from the base is greatly amplified as we are exciting the system natural frequency;

* $\bar{\omega} > \sqrt{2}$: the spring-damper system is effective as the magnitude of the transmitted force is decreased depending on $\xi$.

The latter is the case of interest and can be used to identify performance targets as done in [13]. Given a desired attenuation for the disturbance frequency components (e.g., the propeller's frequency identified in Sec. II), we can identify $\xi$ and the target natural frequency for the damping system (in practice, this corresponds to the desired stiffness coefficient $k$, because the FC mass is not a degree of freedom).

In practice, it is difficult to realize a passive system with a desired stiffness coefficient, there are no general rules and parameters identification may not be easy without specific machines. For example, the half-power method [15], used to estimate the damping in a mechanical system, requires machines to excite the system at variable frequencies. Nevertheless, we can use the results above to draw general considerations: the transmissibility plot tells us that a 'good' damping system should have low stiffness $k$ and damping $c$. These hints will be applied in the next Section to improve the damping of vibrations at the IMU by just using cheap components placed in a proper way.

Note that the results obtained are useful also for multi DoFs systems as they can often be treated as multiple decoupled 1 DoF systems [15]. For MRAV, for example, one could consider an equivalent stiffness for the translational and rotational DoFs, treating them as independent, as shown in the next Section for a 3 DoFs model.

Finally, note that the natural frequency of the damping system cannot be set arbitrarily low, as this could also filter out the actual motion of the MRAV captured by the IMU signals. The appropriate frequency depends on both the task and the platform. Small quadrotors, for example, have fast motion, but also higher frequency noise from propellers than the one identified in this work.
<figure><img src="image_10.png" /><figcaption>Fig. 6: 3 DoF model for the FC.</figcaption></figure>

## B. 3D damping system modeling

To investigate the influence of FC geometry and dampers placement on IMU readings, we develop a 3 DoFs model. The Chimera FC has different moments of inertia along x and y axes because of its non-square geometry. If not properly considered, this can lead to different performance along different axes. The 3 DoFs model described in the following is one of the simplest models which allows to consider the influence of system parameters on the natural frequencies corresponding to the rotational DoFs.

We model the FC as a rigid planar body with mass m and inertia $I_x$ and $I_y$ along x and y axes respectively, as depicted in Fig. 6. The system has 3 DoFs: the rotation $\theta_x$ (roll) around x, the rotation $\theta_y$ (pitch) around y and the translation z. The reference frame is as in Fig. 1, with origin in the geometrical center<fnref n="2" /> (as the Chimera FC mounts the IMU in that position). The dimensions of the board are $l_x$ along x and $l_y$ along y. The system is connected to a vibrating base through four dampers placed at distances a and b from the center along the y-axis, and c and d along the x-axis. Each connection is modeled as a spring-damper system with coefficients $k_i$ and $c_i$, $i = 1, ..., 4$. This model is a modified version of the well-known full-car model, but here it may be $l_x \neq c+d$ and $l_y \neq a+b$ since the dampers can be placed freely using custom plates such as those described in the following.

With the assumptions of small angles and displacements measured from static equilibrium, the total displacements in the z direction at points 1 to 4 are

$$l_1 = -a\theta_x - c\theta_y + z \quad (6)$$

$$l_2 = -a\theta_x + d\theta_y + z \quad (7)$$

$$l_3 = b\theta_x + d\theta_y + z \quad (8)$$

$$l_4 = b\theta_x - c\theta_y + z, \quad (9)$$

while the elastic and viscous forces acting at those points are

$$F_i = k_i l_i + c_i \dot{l}_i. \quad (10)$$

The motion equations for $z$, $\theta_x$ and $\theta_y$ are

$$m\ddot{z} = \sum_{i=1}^{4} F_i \quad (11)$$

$$I_x \ddot{\theta}_x = bF_3 + bF_4 - aF_1 - aF_2 \quad (12)$$

$$I_y \ddot{\theta}_y = dF_2 + dF_3 - cF_1 - cF_4. \quad (13)$$

With the assumptions $k_i = \bar{k}$, $c_i = \bar{c}$ (i.e., same dampers used in all four points) and $a = b$, $c = d$ (i.e., dampers mounted symmetrically wrt the origin), the motion equations can be simplified obtaining three decoupled systems

$$m\ddot{z} = -4\bar{k} - 4\bar{c}\dot{z} \quad (14)$$

$$I_x \ddot{\theta}_x = -4\bar{k}a^2\theta_x - 4\bar{c}a^2\dot{\theta}_x \quad (15)$$

$$I_y \ddot{\theta}_y = -4\bar{k}c^2\theta_y - 4\bar{c}c^2\dot{\theta}_y. \quad (16)$$

From (14) we recover a well known result for springs in parallel: the equivalent stiffness is the sum of the coefficients. From (15) and (16) we obtain the equivalent stiffnesses $k_{eq,\theta_x} = 4\bar{k}a^2$, $k_{eq,\theta_y} = 4\bar{k}c^2$, which can be used to compute the natural frequency of the system along the two rotational DoFs

$$\omega_{n,\theta_x} = \sqrt{\frac{k_{eq,\theta_x}}{I_x}} \quad \omega_{n,\theta_y} = \sqrt{\frac{k_{eq,\theta_y}}{I_y}}, \quad (17)$$

where the inertias are $I_x = \frac{1}{12}ml_y^2$ and $I_y = \frac{1}{12}ml_x^2$. It can be noticed that to decrease the natural frequency of the damping system (desirable to better damp vibrations as per Fig. 5) one could decrease the stiffness coefficient but also decrease the distances a and c. In practice, the latter solution may be limited by geometrical constraints depending on the design of the vehicle.

Depending on the dampers placement, one could modify the natural frequencies of the system and obtain different vibration damping around x and y. We can imagine this as a shift along the horizontal axis of the displacement transmissibility plot in Fig. 5, which can improve or worsen the damping. To obtain the same performance around x and y, we impose $\omega_{n,\theta_x} = \omega_{n,\theta_y}$ and simplifying computations, we obtain the following relation

$$\frac{a}{l_y} = \frac{c}{l_x}. \quad (18)$$

Considering $l_x$ and $l_y$ fixed (given by the FC geometry), we can fix a and compute b using (18) or vice versa.

# IV. EXPERIMENTAL VALIDATION

We describe here the hardware used to validate the theory and to improve the baseline damping configuration. Finally, we discuss the best results achieved.

## A. Damping structure

To verify the theoretical results of the previous Section while improving the baseline damping configuration of Sec. II, we designed two plates with eight holes each (Fig. 7a), to be used with the rubber dampers depicted in Fig. 7b. This design enables testing various configurations without modifying the plates. The dampers used are very cheap and

<footnote n="2">²This model can be easily extended to consider non-symmetrical placements for the IMU</footnote>
<figure><img src="image_11.png" /><figcaption>(a) Plates for T1 and T2 damping configurations (top view).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(b) Rubber dampers for T1, T2 and improved configurations.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(c) Improved damping system.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(d) Chimera FC with tensioned dampers.</figcaption></figure>

<caption>Fig. 7: Components used for T1, T2 and improved damping configurations.</caption>

commonly available in the market (which makes our method easily reproducible by the aerial robotics community) and can greatly reduce the vibration noise if properly placed. Indeed, even if their coefficients are not known, we can identify in which area of Fig. 5 the system is working (e.g., $\bar{\omega} > \sqrt{2}$) and apply what discussed in III-B and in IV-C to further lower $\omega_n$ and thus increase the damping effect. The FC is rigidly attached to the upper plate, while the lower plate is fixed to the MRAV frame. The dampers are mounted in the middle of the two plates, using the holes to place them. This structure allows to test different dampers placements and to validate the theory as discussed later. The offset between the holes of lower and upper plates is explained in the following.

## B. 3 DoFs model validation

Using the setup described in Sec. II and the damping structure from Sec. IV-A, we performed two hovering flights with different dampers placements to validate the results obtained in Sec. III-B.

In the first experiment we placed four dampers using the holes indicated by the blue circles in Fig. 7a, while in the second we used the other four indicated by the red circles, resulting in two test configurations, called T1 and T2, respectively. Considering the FC dimensions, the dampers configurations and relation (18), we expect that: for T1 $\omega_{n,\theta_x} > \omega_{n,\theta_y}$, while for T2 $\omega_{n,\theta_x} > \omega_{n,\theta_y}$.

Fig. 8 shows the final results which verify the expected relations: for configuration T1 the vibrations around $x$ are less damped than that around $y$, while the opposite holds for configuration T2, despite with less difference in angular velocities amplitudes.

<figure><img src="image_15.png" /><figcaption>Fig. 8: Angular velocities with configuration T1 (top) and T2 (bottom).</figcaption></figure>

Finally, note that the baseline damping configuration described in Sec. II respects rule (18) because $a = l_y/2$ and $c = l_x/2$, and indeed the amplitudes of the angular velocity along the two axis, shown in Fig. 3b, are comparable. They are not exactly the same because of the model mismatch between the FC model (planar homogeneous body) and the real FC, whose geometry and mass distribution are more complex.

## C. Improved damping configuration and results

The baseline damping system already satisfies relation (18), ensuring similar damping around the two axes. To further reduce the stiffness—particularly along the z-axis—we introduce an additional damping stage. Mounting dampers in series effectively reduces the equivalent stiffness, following the relation $k_{eq} = \frac{k_1 k_2}{k_1 + k_2}$ and, according to the theory in Sec. III, this decreases the amplitude of vibrations transmitted from the frame to the FC. Therefore, we added the damping system used for configuration T2, placing it on top of the baseline configuration shown in Fig. 2. This combined setup allows us to minimally affect the balance provided by the baseline system (considering the geometrical constraints on the hexarotor) while reducing the stiffness to better attenuate vibrations transmitted to the FC, especially along the z-axis. As can be noticed from Fig. 7a, there is an offset between the axes of the lower and upper plates holes, so that the new dampers are mounted tensioned. Tensioning the dampers is a common approach for FCs of commercial platforms, but the reason for this particular choice is usually not explained. From III-A, we now know that this decreases the stiffness of the system, making the dampers more effective.

This damping system (see Fig. 7c and Fig. 7d), called improved damping configuration, led to the best filtering performance shown in Fig. 3. Comparing the results with the baseline configuration, it is clear that the improved damping configuration decreases the noise amplitude, especially on the z-axis, moving away the risk of IMU saturation. Moreover, the raw signals present less spikes with the improved damping system and the amplitudes of $\omega_x$ and $\omega_y$ are comparable as previously discussed. Regarding the FFT
of acceleration and angular velocity with the new system (Figures 3c and 3d), we can see that the frequency content is concentrated around the average propeller speed and its double for all plots. For acceleration, the amplitudes are much lower than the baseline case. For angular velocity, the lowest frequency peaks are slightly amplified and the amplitude of $ω_y$ is slightly greater than that of $ω_x$ (according to Sec. IV-B), however the higher frequency peaks have been almost completely removed and the amplitude of the signals in time domain is decreased with respect to the baseline configuration.

## V. CONCLUSION

We analyzed the vibration phenomenon on IMU signals of MRAVs. Using real-flight data, we characterized the problem and developed two models to analyze a damping system, considering the geometry of the FC. We then proposed an optimized placement of the dampers motivated by vibration theory which was then validated by experiments. Experimental results show that the improved dampers configuration achieves far better noise attenuation than the baseline configuration typically found in MRAVs, and this new damping configuration can be easily realized by means of a proper placement of commonly available components. In the future, we plan to further improve our damping system and to transfer it to other aerial platforms.

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