# Panoramic optic-flow profile guiding aerial exploration through turns in simulated environments

Xavier Daïni, Romain Raffin, Thibaut Raharijaona, Franck Ruffier

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Xavier Daïni, Romain Raffin, Thibaut Raharijaona, Franck Ruffier. Panoramic optic-flow profile guiding aerial exploration through turns in simulated environments. 2025. hal-05374840

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# Panoramic optic-flow profile guiding aerial exploration through turns in simulated environments

Xavier Daïni<sup>a</sup>, Romain Raffin<sup>b</sup>, Thibaut Raharijaona<sup>c</sup>, and Franck Ruffier<sup>a</sup>

<sup>a</sup>ISM, Aix Marseille Université, CNRS, Marseille, France; <sup>b</sup>Université Bourgogne Europe, LIB, UR 7534, Dijon, France; <sup>c</sup>Université de Lorraine, Arts et Métiers Institute of Technology, LCFC, Metz, France

## ARTICLE HISTORY

Compiled November 20, 2025

## ABSTRACT

Optic flow-based navigation, inspired by biological systems, differs from traditional vision-based techniques, reducing computational requirements and enabling a more reactive approach. So far, this optic flow-based approach suffers from certain limitations, including a reduced ability to make corridor exploration decisions, difficulty to detect frontal obstacle in advance and sensitivity to rotations. To overcome these challenges while preserving the benefits of direct optic-flow regulation, we propose to use the spatial profile of the optical flow field to (i) precisely control the UAV's orientation parallel to a corridor wall and (ii) head toward the aperture in a highly responsive manner. In addition, we showed that extracting cues from the spatial OF profile reconciles continuous trajectory and turns, without stopping the course of the aerial robot by specifying rules for visual inhibition during turns. We found that the specific optical flow signature created by the apertures in the environment facilitated decision-making and enabled the control architecture to maintain a certain pace for the UAV's navigation through the turns in corridors.

## KEYWORDS

UAV's, Optic Flow, Autonomous, Exploration, Multirotors

## 1. Introduction

Autonomous exploration of unknown environment is a typical problem in the field of robotics. Many solutions exist and have been proposed over the years. In the literature, there exist two main ways to achieve the exploration of a unknown environment:

* Creating an on-board map and planned navigation [deliberative control]

* Reacting to the environment as it evolves [reactive control]

While deliberative control strategies (such as relying on environment mapping and Simultaneous Localization and Mapping, SLAM) have been extensively studied and are effective for smooth navigation in unknown environments, they come with notable constraints [4]. These approaches typically require high-end hardware, including powerful onboard computing units and precise, often costly sensors to enable detailed environmental reconstruction. Furthermore, the computational cost necessary to ensure convergence [8] of such systems can induce latency, resulting in slower reaction times to sudden changes or unexpected obstacles. It is well established that deliberative architectures perform poorly in the face of unexpected or unmodeled conditions, as their decision-making relies on previous representations and pre-established models of the environment [29].

An alternative and widely studied visual cue in both insect-inspired and robotic navigation is the optic flow (OF). When an agent moves through a static environment, the resulting OF encodes rich spatiotemporal information [9], which can be leveraged to perform various navigational tasks. Note that the OF can also be used to create maps of the environment in a more deliberative approach [22], for example in aperture detection.
Several studies have demonstrated how OF can enable a robot to avoid obstacles by using insect-like saccadic motion strategies [12]. Single OF regulation techniques have been applied on a robot for altitude control [25], while more advanced dual OF regulations have experimentally enabled both wall-following and speed control [24]. The profile of the OF vectors field itself has also been used to perform in-flight adjustments in orientation relative to nearby surfaces [10]. In addition, the divergence of the OF vectors field has proven effective for tasks such as autonomous altitude control [15], landing [16], obstacle detection and avoidance [27,32] (on a real robot and in simulation, respectively). OF information can also support in-flight attitude estimation [7] as well as visual odometry [3].

In robotics, OF vector can also be directly measured using unitary directional OF sensor. However, relying on isolated local measurements often leads to reduced robustness, as such measurements are particularly sensitive to noise and inconsistencies. To mitigate this issue, [24] proposed increasing the number of sensors and adopted a more general equation framework to improve the reliability of the OF cue estimation. Other extraction methods rely on the use of cameras, which reduces the hardware complexity while benefiting from the richer and more structured visual data provided by image sequences. Various methods exist to estimate optic flow fields from camera images. These include sparse vector-based approaches, such as the Lucas-Kanade method [21], as well as dense estimation techniques [1,11], each suited to specific computational and application contexts.

However, Weber et al. [31] identified a fundamental limitation in OF-based navigation strategies: the use of OF for speed regulation can lead to instabilities, particularly in the presence of visual discontinuities such as apertures or gaps. More broadly, OF-based navigation often lacks anticipatory capabilities and tends to fail in predicting upcoming structural changes, such as turns or openings, thereby limiting its effectiveness in cluttered or unpredictable environments [24].

To address the challenge of lacking anticipations during reactive OF-based navigation in complex environments, pioneering studies [27,31] have proposed reducing translational speed during turns : authors used high speed reduction when encountering sharp turns or narrow openings (as soon as the angle of bending corridor exceeded 20°). In particular, [31]'s robot would come to a complete stop before a turn, while in [27]'s simulations, the agent's forward speed dropped by as much as 80% to prevent frontal collision. Nevertheless, this speed reduction strategy diverges from observed insect behavior: as shown in [17], insects tend to maintain relatively high speeds even when executing or approaching turns. Bio-inspired OF-based navigation strategies thus could benefit from further refinement to better mimic the insect's behavior.

As an example, studies led by Schilstra and van Hateren [14,26] on blowflies revealed that insect's flight includes rapid rotational saccades, during which the thorax and the head exhibit distinct dynamic responses. These findings challenge some simplifying assumptions made in previous models (e.g. [27,31]) and point to the potential benefits of accounting for such head/body decoupling.

Second, the compound eyes of insects consist of numerous ommatidia, each detecting local motion [20], giving a large number of cues in a large field of view. In addition, [13] has shown that bees have different landing performances depending on the landing-ground pattern and therefore the underlying OF profile generated. This suggests that insects likely also rely on the overall structure of the optic flow field, rather than isolated vectors, to guide their navigation. These findings are in accordance with neurophysiological studies that exhibited the presence of some neurons in the insect visual system which are specifically tuned to optic flow patterns [19], highlighting again the biological relevance of the OF profile. Incorporating such global information may thus enhance the robustness of an OF-based navigation strategies.

Building on these biological insights, we propose a more robust and bio-inspired navigation strategy that exploits the profile of the OF vectors in a large field of view, to handle both small reorientations [10] and saccadic flight behavior [27]. Our approach focused on local deviations of the OF profile, known as outliers, in the field of OF vectors on either side, enabling the drone to identify navigable passages earlier, and thus adjust its behavior by maintaining its flight speed during the rotation phase. The paper is organized as follows: Section 2 introduces the optic flow (OF) equation allowing the study of the whole OF vectors field, known as spatial profile. Section 3 details the approach that allows extracting direct OF cues from this spatial profile. Section 4 describes the simulation environment, including the structure of the virtual scene, the parameters of the simulated camera,
and the drone's model. Section 5 defines the OF as a proprioceptive cues, and also discuss other important proprioceptives cues used in visual navigation. Section 6 details a two-state navigation strategy: *State 0* was specifically designed to handle translational flight in the presence of small body rotations ; To turn faster toward aperture, a *State 1* was introduced. Simulated results are given in Section 7 and Section 8 offers a comparative discussion between the proposed model and its biological counterparts, drawing parallels with visually guided behaviors observed in insects.

## 2. OF equation

The general expression of the OF vector $\omega$ was given in [18] and, expressed with the notation used in this study, becomes:

$$ \omega(\theta, \delta) = -\frac{1}{D_{\theta,\delta}}(\mathbf{T} - (\mathbf{T} \cdot \mathbf{d}_{\theta,\delta})\mathbf{d}_{\theta,\delta}) - \mathbf{R} \times \mathbf{d}_{\theta,\delta} \quad (1) $$

With:

* $(\theta, \delta)$ the angles (azimutal and elevation respectively) defining the 3D direction at which the OF vector is computed. The unitary vector defining this direction is $d_{\theta,\delta}$. In a 2D case, $\delta = 0$.

* $D_{\theta,\delta}$ the distance between the observer and the feature corresponding to the computed OF vector

* **T** and **R** translation vector and rotation vector respectively, defining the relative movement of the observer.

In this study, we focused on the norm of the optic flow, denoted as $\omega = \|\omega\|$, generated by a planar surface under conditions of pure translation (i.e., $\mathbf{R} = 0$). Furthermore, we only tackled the optic flow evolution at $\delta = 0$. This allowed for a major simplification in computation, while still allowing 2D navigation. The derivation of the corresponding expression for the flow norm in this context are detailed in [6], where the camera's optical axis is used as the angular reference frame (Figure 1). The formulation, in this frame of this study, was:

$$ \omega = \frac{\|V\|}{D_0 \cos(\alpha)} \sin(\theta - \psi_v) \sin(\theta + \alpha) \quad (2) $$

While interesting, Eq. (2) is not often used directly. In the literature, specific OF cues are used, such as:

* $\omega_{90^{\circ}}$ the translational OF at $90^{\circ}$ (Figure 1a). On the other side of the drone, $\omega_{-90^{\circ}}$ is defined the same way.

* $\theta_m^L$ and $\theta_m^R$ are the direction where the OF norms are maximum on the left (L) and right (R) side respectively.

## 3. Direct OF signature from spatial profile

### 3.1. Interesting cues in the spatial OF profile

Figure 1 schematically summarizes the OF cues extracted from the global OF profile. In particular, $\omega_{90}$ and $\theta_m$ which are directly linked to the observer's motion and the geometry of the surrounding environment (see Figure 1).

### 3.2. OF-Fitting spatial function

Several fitting function could be envisioned in order to model the 2D profile of the optic flow. Polynomial approximation approaches diverge in large Fields-of-View conditions, while the use of the theoretical function has the drawback of being sensible to small optical errors.
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Different UAV heading/wall configuration leading to different OF shapes and reference frame for the body. Representations deals only with the left (L) side, but the same definition apply for the right (R) side.</figcaption></figure>

(a) A camera moving along its retinal axis while being perpendicular to the faced plan, produces the OF profile in (b).

(c) A camera moving while being not being perpendicular to the faced plan, produces the OF profile in (d).

(e) A large difference in distance leads to a discontinuity in the OF field as shown in (f). In this type of cases, the value of $\theta_m^L$ becomes ill-defined. As such, the algorithm of this study is designed to disregard these configurations, as detailed in Appendix A, *Inhibition following aperture detection*.
To overcome these limitations, we opted for a Gaussian approximation (cf Eq.(3)). A Gaussian curve has the advantage of offering a window with one clearly defined maximum. Even if it approximates the theoretical function, a Gaussian has the advantage of being robust to errors in OF measurement. Furthermore, in case of two planes with different orientations, the curve would simply merge both, allowing for a smooth transition. Such characteristics was crucial when defining OF Aperture (cf Section 6).

Therefore, for each camera FoV (one per side), a Gaussian function was fitted to the OF norm acquired. On the Left (or Right) side, the expression of the function $f_{L,R}$ is:

$$f_{L,R}(\theta) = A_{L,R} \cdot e^{-(\frac{\theta-\theta_m^{L,R}}{\sqrt{2}\sigma_{L,R}})^2} \quad (3)$$

In Equation (3), $A_{L,R}$ represents the amplitude of the fitted profile (either $A_L$ or $A_R$), $\theta_m^{L,R}$ is the angle at which the fitted profile reaches its maximum, and $\sigma_{L,R}$ defines the spread (standard deviation) of the fitted Gaussian curve.

### 3.3. From OF vector field to OF cues

Starting with the OF profile defined by Eq.(3), the two main cues are accessible on each side. For the left (L) side $\omega_{90^\circ}$ and $\theta_m^L$, are defined as follows:

$$\omega_{90^\circ} : \omega_{90^\circ} = f_L(90^\circ) \quad (4)$$

$$\theta_m^L : f(\theta_m^L) = \max(f(\theta)) \quad (5)$$

Visual representation of these values can be seen on Figure 1. The same definitions are valid for the Right (R) side with $\omega_{-90^\circ}$ and $\theta_m^R$.

#### 3.3.1. From OF vector field to profile extraction

First and foremost, there was the need for OF profile to be acquired. In this study, we focused on the sparse vectorial representation, for its versatility, allowing for more or less vector, with better or worse quality to be extracted. However, for 2D navigation, only 2D OF information are needed, which simplifies the OF representation needed. In order to obtain a 2D OF-profile, the study of the sole central-band of the image is necessary. A potential extension could be to take into consideration the study along the vertical of the image, to add another 2D OF representation, unlocking full 3D navigation, but this consideration is outside the scope of the present study.

#### 3.3.2. OF cues signatures in the spatial profile

From the OF profile and knowing the fitting function Eq.(3), $\omega_{\pm 90^\circ}$ and $\theta_m^{R/L}$ are easily accessible. The advantage of a fitting function is that it is based on a large number of measurements, which directly enhances the quality of the estimate.

**OF divergence** Another important OF cues is the OF divergence $\omega_{div}$. The definition of the divergence is $\omega_{div} = \frac{V_h}{h}$ (with $V_h$ the speed along the Z-axis and h the height).

In this paper we computed an adapted divergence, relying on the direct sum of both the norms $\omega_{-20^\circ}$ and $\omega_{20^\circ}$:

$$\tilde{\omega}_{div} = |\omega_{20^\circ}| + |\omega_{-20^\circ}| \quad (6)$$

This adapted divergence Eq.(6) allowed to compute adaptive thresholds, based on the lateral optic flow, indicating an incoming obstacle and allowing to switch states as defined Figure 5b. Computation of the thresholds is given in Appendix A.
Although it is theoretically possible to compute $ω_{-20°}$ and $ω_{20°}$ directly from the fitted optic flow profile, doing so would be inappropriate in practice. Indeed the fitting function assumes a lateral planar surface, and is thus no longer valid in the presence of frontal obstacles, which alter the geometry of the environment and invalidate the model assumptions. Notably, divergence-based cues are primarily intended to detect frontal obstacles.

Rather we rely on a local, data-driven approach. Specifically, $ω_{-20°}$ and $ω_{20°}$ are estimated using the median OF magnitude computed over the angular intervals [-25°, -15°] and [15°, 25°], respectively. This method ensures robustness to local irregularities without relying on geometric assumptions that may no longer hold.

# 4. Simulation Setup

## 4.1. Simulated Environments

The simulated environment was made on Gazebo, based on a series of oriented planes as shown in Figure 2. The images were acquired by a simulated camera (defined below) sending its images to a Python script by a ROS-bridge. The Python scripts dealt with the Optic flow, computed the cues and the motor commands which were sent to a Matlab model (defined below as well). The model performed the simulation during a time step and loop with Gazebo by exporting the position at $t + Δt$. In our experiments, $Δt = 0.02s$ while dynamic simulation ran at 1000 Hz.

## 4.2. Simulated Cameras

The 2 simulated cameras, each featured a wide field of view (FoV) of 160°, consistent with panoramic bio-inspired vision systems. The generated images had a resolution of 1280×100 pixels and were assumed to be free from optical distortions. The physical pixel size was set to $7.8 \cdot 10^{-7}$ by $7.8 \cdot 10^{-7}$ meters along the horizontal axis, enabling high angular resolution across the image width. This configuration ensured sufficient spatial sampling to support optic flow computation over a broad visual field. We also used one camera per drone side (Left and Right) to ensure simultaneous detection of left and right features.

Finally, it has to be noted that, contrary to OF measurement from dedicated sensors, OF acquired by a camera with a planar retina is the projection of a 3D vector along a sphere onto a 2D plane. The back-projection of the retinal-OF is possible, and already used in [6] (and the detailed computations in [30]) allowing a return to the usual 3D spherical OF.

**Image resolution** Finally, since computing optic flow from full-resolution images with a vertical dimensions of 1-pixel would result in unreliable estimates, we opted to use band-like images of 100 pixels height, as discussed in sections 3.3.1 and 4.2. A vertical resolution of 100 pixels provides sufficient information to reliably compute optic flow. This trade-off ensured a balance between computational efficiency and quality of the motion cues extracted.

## 4.3. Fully Actuated Drone simulated model

The drone used in this study is a fully-actuated hexarotor. The model itself was the one defined in [5] (with simulated parameters given in Appendix B, Table 1). As the regulation is based on OF, a movement is needed to generate OF. For this purpose, an initial non null speed was defined, resulting in $V_0 = 0.5$ m/s along the X-axis.

**Altitude regulation** to constraint the UAV exploration in the horizontal plane. The simulated drone model was fully actuated and integrated the complete 3D dynamics of the platform, including non-linear effects. As a result, variations in altitude, roll and pitch occurred during flight maneuvers, particularly during rotational motions. To mitigate this, these 3 flight variables were assumed to be perfectly measured and were tightly regulated throughout all scenario, ensuring that the drone
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Simulation setup, upper view. (a) is the texture patch present on each walls to create OF-usable textures. (b) is the Bending corridor, with a corridor of width 2 m and two angles of +30° and -30°. (c) is the Square-shaped circuit, with corridors of width of 2 m. (d) is the S-turn, with corridor width of 2 m but corridor aperture of 1.5 m.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>**Figure 3.** (a) Example of Optic flow with OF acquired in a bending corridor, with specific OF cues appearing. (b) Bending corridor with corresponding drone position and movement resulting in OF shown in (a).</figcaption></figure>

exploration remained effectively constrained to a 2D plane.

## 4.4. Simulation parameters

In the simulation workspace, several parameters were defined: all were not described in the control loop. Notably, time-based low-pass filters, on the measured OF cues are used, and inhibition on measures are triggered when the drone is rotating. Detailed information on such parameters are presented in Appendix A.

# 5. Proprioceptive cues

## 5.1. Optic Flow as a functional proprioceptive cue

In robotics, the concept of proprioception is traditionally defined as the sensing of the agent's internal state — such as joint angles, body orientation, gyrometer, or internal accelerations — using sensors like IMUs or encoders. However, drawing from biological systems, this definition can be broadened. In neuroscience, proprioceptive information is not limited to signals from within the body; it encompasses any sensory input that contributes to the perception of the body's own motion or orientation, even if it originates from external senses like vision [23].

In this context of usage, Optic Flow is considered functionally proprioceptive. In our robotic application, we adopted this biologically grounded perspective: since Optic Flow is used here directly to infer self-motion – rather than to compute absolute distances to external objects – we treated it as a functional proprioceptive cue, aligning with its role in biological systems.

## 5.2. Other proprioceptive sensors

In the present study, only the direction of the speed vector, denoted by $\psi_v$, was used. Such speed direction can be directly derived from a single OF vector measurement downwards. Combined with a perfectly regulated altitude, this enabled the simulation to operate within a consistent 2D framework while not making use of the speed vector.

# 6. Exploration through turns

To enable exploration through turns, we designed a two-state architecture. The first state manages standard OF-based regulation, including minor reorientations to maintain alignment with nearby walls. The second state is dedicated to executing larger exploratory rotations, during which visual feedback becomes unreliable.
<figure><img src="image_5.png" /><figcaption>Figure 4. a. Drone position and movement in the square-shaped circuit top-right corner, with 3 Regions on interest marked by ①,② and ③, being the left wall, the aperture and the right wall respectively.</figcaption></figure>

b. Corresponding Optic Flow field generated by conditions in (a), with OF in regions ①,② and ③ highlighted.

c. Fitting of OF field and detection of an outlier cluster (green).

d. Fitting of OF field and position of $\theta_{OFAperture}$ used to detect aperture in the environment.

## 6.1. State 0: Wall Following

The dual lateral regulation was handled in State 0: Wall Following (see Figure 5).

* The first loop (shown in black) maintains the system within the 2D framework by relying on perfect measurements of the drone's geometry and high-gain controllers.

* The Dual direct OF regulation (shown in blue) are control loops that governs both lateral positioning and speed control. It ensures stable navigation along the corridor based on optic flow measurements on each side.

* The Reorientation (shown in red) control loop computes the desired orientation $\psi_d = \psi + \alpha$ to correct the measured angular deviation.

**Dual direct Optic Flow regulation** Lateral regulation based on optic flow cues is a well-established approach in bio-inspired navigation. The objective is to maintain a lateral positioning by controlling the optic flow measured at $\pm 90^{\circ}$ relative to the heading direction — that is, between $\omega_{90^{\circ}}$ and $\omega_{-90^{\circ}}$. Fundamentally, the side with the higher optic flow magnitude corresponds to the closer obstacle, as described by Equation (4), where the optic flow incé $D_0$. With a regulation based on this higher side, the drone maintains adjustments toward the side where the risk of collision is greater.

Equation (4) shows the link between distance $D_0$, forward speed $V_x$ and lateral optic flow $\omega_{90^{\circ}}$:

$$ \omega_{\pm 90^{\circ}} = \frac{V_x}{D_{\pm 90^{\circ}}} \qquad (7) $$

With:

* $V_x$ the projection of the speed vector $V$ on the retina $X$-axis

* $D_0$ the distance camera-surface in the direction of the optic $Y$-axis

At the same time, lateral regulation also tackle speed control. Indeed, the regulation of the sum $\omega_{90^{\circ}} + \omega_{-90^{\circ}}$ allows to control the forward speed $V_x$.
**Reorientation parallel to a corridor wall** Reorientation aim at maintaining the robot parallel to the closer planar surface followed (case shown Figure 1a). In order to measure the relative orientation error to be corrected, Eq.(8) is of help, by introducing $\alpha$.

The angle $\alpha$ is the relative orientation difference between the observer and the planar surface, as shown Figure 1c. Eq.(8) make the link between $\alpha$, the Left optic flow profile ($\theta_m^{L/R}$) and the speed vector direction $\psi_v$ using the relation:

$$ \alpha = 2\theta_m^{L/R} - \psi_v \quad (8) $$

With:

* $\alpha$ the relative angle between the surface and the retina plane.

* $\psi_v$ the angle between the speed vector $V$ and the retina $X$-axis.

* $\theta_m^{L/R}$ is the observation angle at which the fitted curve is at its maximum (taking the camera's optical axis as 0-reference) for either the Left or Right side, depending on $max(\omega_{90^{\circ}}, \omega_{-90^{\circ}})$.

Once again, Equation (8) holds also for the Right side.

The simulated drone measures its orientation $\psi$ (heading). The link between $\alpha$ and $\psi$ (the drone heading) is thus quite simple. If a non null $\alpha$ was detected; hence the drone was not parallel to the followed wall, the desired heading became $\psi_d = \psi + \alpha$.

In addition, to ensure the reliability of the measurement, $\alpha$ was constrained within $\pm 15^{\circ}$, as larger values cannot be guaranteed to remain accurate under the current estimation method. Larger rotations were handled in section 6.

Finally, since the drone is equipped with a left (L) and right (R) camera, both relative angles $\theta_m^L$ and $\theta_m^R$ can be extracted (see Figure 3). However, regulating both simultaneously is impractical. To simplify, only the side with the stronger translational optic flow (i.e., $max(\omega_{90^{\circ}}, \omega_{-90^{\circ}})$) is regulated by Eq.(8). This choice naturally prioritizes the closest wall, which is both more critical for navigation and typically provides a more reliable orientation estimate due to stronger and better-fitted optic flow.

## 6.2. State 1: Heading toward the aperture

In State 0 direct optic flow regulation has been employed in a purely reactive manner, responding to immediate environmental cues. However, in more complex situations, the ability to anticipate and exploit openings—such as side apertures—becomes advantageous, especially to avoid abrupt maneuvers like wall-following rebounds. Achieving such anticipatory behavior requires here a preliminary step: detecting the presence of apertures within the visual field.

Optic flow proved to be useful, by enabling the identification of local deviations in the vector field — referred to as optic flow outliers — which are indicative of environmental apertures.

**OF outliers as a signature of aperture** An optic flow outlier is defined as a local deviation between the measured optic flow field and the expected planar OF function, here a Gaussain curve for planar wall. When one of these planes ended abruptly within the Field of View, being replaced by a more distant surface (as illustrated in Figure 1e), the geometric discontinuity was reflected in a corresponding discontinuity in the optic flow magnitude (Figure 1f).

From a modeling perspective, Figure 4c shows that the fitted function (using a least-squares algorithm) attempted to minimize the overall error, thereby smoothing over the discontinuity. We exploit this property by identifying the 10 highest fitting errors within a $\pm 30^{\circ}$ angular range around the optical axis. Apertures located further in the periphery of the field of view were considered less relevant for immediate navigation.
However, fitting error alone was insufficient to reliably detect an aperture. To reinforce detection, we required that these errors be significant — specifically, their median value must exceed 50% of the fitted Gaussian amplitude. Then we had the guarantee that the OF outliers did not come from a singular OF error, and that these outliers were significant enough to impact the OF profile.

Finally, the decision to reorient toward a detected aperture required the divergence of the optic flow $\tilde{\omega}_{div}$, computed at $\pm 20^\circ$, exceeded a threshold. A low divergence value suggested the absence of an imminent frontal obstacle, and the current trajectory could be maintained. Conversely, a significant divergence (thresholds detailed in Appendix A) indicated a risk of frontal collision, prompting the need to explore alternative paths. Figure 4 presents the steps used for the detection of such OF Aperture. The *State 1* was activated only when two conditions were met: an aperture was detected with a frontal obstacle appearing (*OF Aperture ≠ 0* and $\tilde{\omega}_{div} > T$), and the drone was flying strait and not currently undergoing visual inhibition (i.e. $|\dot{\psi}| < \eta$). During this **Heading toward the aperture** state, the parameters of the Sliding Mode Controller (SMC) were also modified, as detailed in Table 2, in order to accommodate the aggressive reorientation dynamics.

It is important to note that precise alignment was not required to end *State 1*, as minor corrections can be effectively handled in *State 0*. Instead, the main criterion for ending the saccadic maneuver was achieving both an orientation error manageable by *State 0* (*err<sub>θ</sub> < 20°*) and a sufficiently low angular velocity ($|\dot{\psi}| < 4η$), allowing a return to optic flow-based regulation in *State 0*.

The *Heading toward the aperture* (shown in green in Figure 5) method thus identified and processed optic flow outliers together with divergence cues. Its role was to determine the most likely direction of an aperture within the visual field. The output, $\theta_{Aperture}$, represented the angular direction the drone should turn to in order to pass through the detected opening.

## 6.3. Inhibiting vision during turns

Direct dual lateral mechanism assumed a purely translational motion. However, when an orientation error was detected, the drone initiated a corrective maneuver (called reorientation) that introduced a non-negligible rotational component. In such cases, the optic flow became distorted, and lateral regulation was adversely affected.

Interestingly, insects exhibited a flight strategy alternating between straight flight segments and rapid saccadic rotations. Inspired by this biological behavior, our approach implemented an inhibition mechanism: when the rotational rate exceeded a given threshold $\eta$ (taken at 0.05rad/s in the simulations), optic flow-based regulation was temporarily suspended (visual inhibition) to preserve control robustness.

During the rotations, an additional cue was exploited to regulate the drone's motion: the direction of the speed vector $\psi_v$. This cue can be measured from the ventral optic flow (i.e., the optic flow generated when looking below the drone). Regulating $\psi_v$ contributes to two complementary objectives: i. linking $err_y$ to $\sin(\psi_v)$ naturally mitigates lateral drift during rotations by driving $\psi_v$ toward zero; ii. linking $err_x$ to $\cos(\psi_v)$ enables a gradual reduction of the forward speed.

## 6.4. Other proprioceptive control loops

In both states, it was essential to control various dynamic parameters to ensure flight safety. Two parameters are particularly important in the visual regulation loops: the heading, denoted as $\psi$, and the speed vector direction, denoted as $\psi_v$. The heading is crucial during rotational movements and can be obtained from an onboard Inertial Measurement Unit (IMU). Meanwhile, the speed vector direction causes a shift in the optical flow (OF) distributions from Equation (8) and can be estimated using the ventral optical flow direction.

Additionally, since the drone model incorporates 3D roll and pitch (denoted as $\beta$ and $\gamma$), these factors also needed to be considered during regulation. Both roll and pitch can be measured by an onboard IMU.
<figure><img src="image_6.png" /><figcaption>**Figure 5.** (a) Improved control scheme with the addition of an exploration state in order to allow for exploration and obstacle avoidance. The classical OF based regulation (in blue) is obtained by the lateral regulation loops [24] and the orientation regulation loop [10]. A new regulation is added (in green) to deal with OF outliers and add an anticipatory exploration trigger. In this figure, the 3D position of the drone is noted by X, Y, Z and the orientation is noted by the roll $\beta$, pitch $\gamma$ and yaw $\psi$, also called heading.</figcaption></figure>

(b) State machine creating the saccadic behavior during state transitions. Note that to transition from *State 0* to *State 1*, both an aperture needs to be detected and the divergence triggers, more information in Appendix A.

In the simulations $\omega_{sum}^{des} = 1.5 \text{ rad.s}^{-1}$, $\omega_{max}^{des} = 1 \text{ rad.s}^{-1}$ and $\eta = 0.05 \text{ rad.s}^{-1}$. In addition, in *State 0*, during visual inhibition, $err_Y$ is computed using a sign function as it allows for a strict return to a low lateral movement. In *State 0*, the speed is still regulated by $err_X = err_{sum}$. Indeed, with a low rotational OF, its addition or retraction from a translational OF would be compensated between measurements on the Left and Right.

During higher rotations however, like in *State 1*, the sum could no longer be used as the rotational OF part would exceed the translational one on both sides. Thus we used cos and sin function to regulate the speed based on its direction, to force the lateral speed to decrease promptly, and maintain a velocity. Sliding Mode Controller (SMC) parameters can be found in Appendix C.
<figure><img src="image_7.png" /><figcaption>**Figure 6.** (a) Navigation results with the exploration algorithm implemented in a bending corridor. (b) Time between plots: $\Delta t = 4$ s. (c) Focus on variable evolution during the flight shown in (b).</figcaption></figure>

i) Represents the current state of the algorithm.

ii) Is the drone's rotation speed, linked to visual inhibition when $|\dot{\psi}| > \eta$.

iii) Is the evolution of the speed norm.

iv) Is the evolution of the orientation with respect to the closest wall of the setup.

v) Is the evolution of the lateral optic flow at +90° and -90°.

# 7. Simulated Results

We successfully tested our navigation algorithm on several simulated environments, along different initial conditions.

Figure 11 illustrate the algorithm's robustness to various environment complexity and starting positions with focus given on specifics ones in Figures 6, 7, 8, and 9. In particular, Figure 6 highlights the improved reorientation mechanism, which enables smooth transitions combining translations and small corrective rotations, thus improving on the state of the art. Meanwhile, Figures 7 and 8 highlight the key role of the new state machine in exploration control, which enables a drone to detect and turn decisively towards lateral openings, thus supporting OF-based decision-making strategies for the first time. Finally, Figure 10 illustrates the reliability of our method over time, with successful autonomous navigation over several loops in a closed, square-shaped environment.

# 8. Discussion

**Toward real world transfer** The choice of an environment made of corridors is plausible in many real-life contexts such as urban canyons, indoor corridors, tunnels, or structured natural passages (e.g., forest trails). This type of environment is particularly relevant for optic flow-based navigation strategies. However, transferring the proposed approach to more complex or open environments raises several challenges. In natural outdoor scenes, visual texture, illumination, and motion dynamics can vary significantly, leading to less reliable optic flow estimates. Similarly, in cluttered or dynamic settings, the presence of moving objects or irregular geometries may distort motion cues and challenge the stability of the control laws.

**Range of the Exploration** The simulations presented in this study were conducted within a defined range of initial conditions. Specifically, in the bending corridor setup, which measures 2 meters in width, the drone was initially positioned at regular intervals of 25 cm across the corridor's width. Similarly,
<figure><img src="image_8.png" /><figcaption>**Figure 7.** (a) Navigation results with the exploration algorithm implemented in a S-turn. (b) Time between plots: $\Delta t = 4$ s. (c) Focus on variable evolution during the flight shown in (b).</figcaption></figure>

i) Represents the current state of the algorithm.

ii) Is the drone's rotation speed, linked to visual inhibition when $|\dot{\psi}| > \eta$.

iii) Is the evolution of the speed norm.

iv) Is the evolution of the orientation with respect to the closest wall of the setup.

v) Is the evolution of the lateral optic flow at +90° and -90°.

Note that *State 1: Heading toward the aperture* triggered very shortly before visual inhibition was activated. It is a perfectly normal behavior as the drone detected an aperture while in translation movement, triggered a state change, before rotating in the desired direction a time step later.

<figure><img src="image_9.png" /><figcaption>**Figure 8.** (a) Navigation results with the exploration algorithm implemented in a square shaped circuit. (b) Time between plots: $\Delta t = 4$ s. (c) Focus on variable evolution during the flight shown in (b).</figcaption></figure>

i) Represents the current state of the algorithm.

ii) Is the drone's rotation speed, linked to visual inhibition when $|\dot{\psi}| > \eta$.

iii) Is the evolution of the speed norm.

iv) Is the evolution of the orientation with respect to the closest wall of the setup.

v) Is the evolution of the lateral optic flow at +90° and -90°.
<figure><img src="image_10.png" /><figcaption>**Figure 9.** (a) Navigation results with the exploration algorithm implemented in a square shaped circuit but with a starting relative angle of -45°. (b) Time between plots: Δt = 4 s. (c) Focus on variable evolution during the flight shown in (b).</figcaption></figure>

i) Represents the current state of the algorithm.

ii) Is the drone's rotation speed, linked to visual inhibition when $|\dot{\psi}| > \eta$.

iii) Is the evolution of the speed norm.

iv) Is the evolution of the orientation with respect to the closest wall of the setup.

v) Is the evolution of the lateral optic flow at +90° and -90°.

in the square and S-shaped environments, the same spacing was applied, while ensuring that the drone never started too close to the side walls. This precaution was taken to account for two key limitations:

* First, the visual regulation algorithm is based on the camera's perspective, which is located at the center of the drone. Although our model is point-like, we have deliberately limited the initial conditions by avoiding positioning the drone too close to the walls in the simulations (see Figure 6). This precaution aims to maintain a safety margin consistent with what a real drone would require, while allowing us a minimum distance slightly smaller than that strictly imposed by its geometry, in order to maintain a good diversity of test cases.

* Second, when the camera is positioned too close to a textured surface, the apparent motion reaches extreme values, leading to unreliable optical flow estimation. Since the perceived motion depends on both the drone's speed and its distance from the surface, operating too close to walls compromises the robustness of the visual cues used for navigation.

**Intertwined translational and rotational optic flow** This study demonstrates the effectiveness of leveraging optic flow (OF) cues extracted from the entire Field-of-View for an improved navigation in complex environments. However, as motion complexity increased, challenges arose from the coupling between translational and rotational components of the OF. To address this, we adopted a bio-inspired saccadic flight strategy, with visual inhibition during rotations, effectively decoupling pure translations and complex motion (rotation with translations). Yet, this approach temporarily still suspends the OF-based regulation during saccades.

This lack of visual regulation can prove to be an issue as illustrated in Figure 7. Initial conditions that place the drone too close to the upper boundary can result in crashes, particularly when a rapid 180° turn is required within a limited space and timeframe. Such situations are referred to here as *aggressive maneuvers*.

Finally, the choice of a 3D model for the drone also adds the possibilityies of movement along a *z* axis improve the realisme of the study. Indeed the roll, pitch and *z* movements do highly impact the optic flow aswell. Having them as perturbations thus increases the realisme.
<figure><img src="image_11.png" /><figcaption>**Figure 10.** (a) Navigation results within the square shaped circuit for 3 consecutive turns (b) evolution of the parameters during the simulation.</figcaption></figure>

i) Represents the current state of the algorithm.

ii) Is the drone's rotation speed, linked to visual inhibition when $|\dot{\psi}| > \eta$.

iii) Is the evolution of the speed norm.

iv) Is the evolution of the orientation with respect to the closest wall of the setup.

v) Is the evolution of the lateral optic flow at +90° and -90°.
<figure><img src="image_12.png" /><figcaption>**Figure 11.** Drone trajectories in different environments (a,b,d) and starting conditions (c). In (c) the starting angles range from -45° to +45° by steps of 15°, excluding 0 already shown in (a).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 12.** (a) Simulated UAV response with its camera fixed to its body. It was necessary to inhibit vision during the whole body rotation, resulting in a longer time without the OF-based wall-aware control.
(b) Experiments in [14] in which the dynamical response of the blow fly's head was observed to be much shorter than its body response, resulting in a longer stabilised vision during saccadic flight.</figcaption></figure>
**Decoupling camera from body to enable aggressive maneuver**

Successfully executing demanding, aggressive, maneuvers requires maintaining visual control even during body rotations, thereby minimizing the duration of saccadic phases or control inhibition.

One promising direction is to incorporate an actuated Degree of Freedom (DoF) for the camera. This would enable independent head-like rotations, as observed in flying insects (see Figure 12). Indeed, we make the hypothesis that such difference in dynamics allows visual stabilization and continued OF-based control even during body rotations of the insect.

Such DoF, achievable on drones through the use of a decoupled or gimbal-mounted camera, could significantly impact the OF-based navigation strategy by enabling the use of visual feedback even during rotational maneuvers.

## 9. Conclusion

In this study, we presented a novel model for bio-inspired navigation based on the optic flow profile. This improved approach combines two classical strategies: one dedicated to lateral positioning and speed control, and the other to orientation regulation. By drawing inspiration from the saccadic flight behavior of insects, our model enables a drone to navigate safely through corridors while maintaining relatively high speeds even during rotations by anticipating apertures.

By exploiting sparse optic flow measurements obtained via the Lucas-Kanade method from cameras with large FoV, we introduced the concept of optic flow outlier—visual signatures associated with environmental openings. Leveraging the optic flow vector field across the entire visual field, we demonstrated the drone's ability to navigate in complex environments, including square circuit, S-turns, and bending corridors. Notably, the drone was capable of performing multiple consecutive turns within the square-shaped circuits.

Finally, the limitations observed in certain scenarios — particularly in S-turn configurations — highlight the potential benefit of an additional degree of freedom between the sensor and the drone's body. Inspired by the head-body decoupling found in insects, we hypothesize that this degree of freedom could enhance visual regulation during rotational phases by mitigating the effects of rotational optic flow, which remains a major challenge in optic flow-based navigation.

Building on the current results, two main research directions are envisioned for future work:

First, now that we have characterized the key optic flow cues and visual inhibition rules required to identify and navigate through openings, the next step will be to integrate learning-based methods to enhance their OF cues detection. In particular, supervised (or even unsupervised) learning approaches could be trained to robustly extract these visual features even in more complex environments. Such an approach follows the path initiated by works like [28] (see also [2] for review), which demonstrate the ability of convolutional neural networks (CNN) to estimate optical flow directly from raw images. Extending these methods to detect OF signature cues relevant for navigation could significantly improve robustness and generalization. Another addition could be to use unsupervised learning to find new optic flow cues, that the current method may not take into consideration.

Second, we plan to extend our current exploration framework to more natural and irregular environments, such as caves, mines, or other confined spaces with non-planar surfaces. These scenarios pose additional challenges for both perception and control, especially in the presence of varying textures, irregular geometries, or low-light conditions.

## Open Data

The simulated flight, as well as optic flow vectors acquired at each steps, can be founded openly at:

https://doi.org/10.57745/MEV4FF
## Acknowledgment

We thank J. de Jesús Castillo Zamora for his kind help with the theoretical and modeling framework used in this study as well as C. Demonceaux for our fruitful discussions.

Financial support was provided via a ProxiLearn project grant (ANR-19-ASTR-0009) to F.R. and via SpotReturn project grant (ANR-21-ASRO-0001-02) to T.R. and F.R. from the ANR (Astrid Program). The participation of X.D. in this research was made possible by the joint PhD grant from the Agence Innovation Défense (AID) and Aix-Marseille University. X.D., T.R. and F.R. were also supported by Aix Marseille University and the CNRS (Life Science, Information Science and Engineering and Science & technology Institutes).

During the preparation of this work the authors used ChatGPT (OpenAI) in order to improve the readability and rephrasing. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.

## Biographical note

**Xavier Daini** is a PhD candidate at ISM (Aix Marseille Université), Marseille, France. His research focuses on bio-inspired robotics and visual navigation using optic flow and drones.

**Romain Raffin** is a Full Professor at Université Bourgogne Europe (LIB, UR 7534), Dijon, France. His research focuses on geometric processing, including curves and surfaces, subdivision surfaces, and point clouds, as well as geometry standardization.

**Thibaut Raharijaona** is a full Professor Université de Lorraine and Arts et Métiers Institute of Technology (LCFC), Metz, France. His research interests include control systems, robotics, and visual perception.

**Franck Ruffier** is a CNRS Research Director at Lab-STICC UMR 6285, ENSTA, Brest, France. He specializes in bio-inspired control systems, visual guidance, and insect flight modeling.

## A. Simulation parameters

Several simulation parameters were implemented to ensure realistic and stable drone behavior:

**Optic Flow Filtering** The optic flow cues $ω_{90°}$ and $ω_{-90°}$, as well as the desired orientation to stay parallel to the walls $ψ_d$ were processed using a five-sample median sliding filter (in the scheme Figure 5). This temporal smoothing reduces the influence of noise and outliers on the decision-making process. The same sliding median filter is applied on OF Aperture detected, ensuring a consistent detection before triggering exploration.

**Gaussian parameters boundaries** Fitting parameters are by default any number between $-∞$ and $+∞$. However, the fitting function usually allows for specific boundaries to be selected to ease the process of fitting. During our simulations, taking the definition of a gaussian of Eq. (3) we defined the following boundaries for either sides (Left or Right):

* $A_{L/R} \in [0, 1.4]$ (rad/s)

* $θ_{m}^{L/R} \in [-50, 50]$ (degrees)

* $σ_{L/R} \in [0, 50]$ (degrees)

**Inter Saccadic Interval (ISI)** Once a small rotational saccade is triggered, the drone is constrained to wait until the rotation is completed before initiating a new one. A rotation is considered complete when the remaining orientation error falls below 5 degrees, the angular velocity is lower than 0.1
rad/s and the time since the rotation triggered is superior to the ISI (of 0.5s in our simulations). This mechanism prevents the accumulation of successive, unstable orientation corrections, but also increases the time needed to correct high orientation error (cf Figure 7).

**Inhibition Following Aperture Detection** When an opening is detected—through the identification of an OF aperture, a 0.5 second inhibition period is enforced during which no small rotations can be triggered. This constraint addresses the observation that orientation regulation is less reliable in the presence of openings (as shown in theory Figure 1c and from simulated OF Figure 4d), and helps to ensure that visual feedback is based only on undisturbed data.

**Divergence trigger** In order to initiate exploration behaviors, we defined a condition based on a adapted divergence of the optic flow, which we defined as the sum of $\omega_{20^{\circ}}$ and $\omega_{-20^{\circ}}$. A high divergence indicates the presence of a frontal obstacle, and thus the necessity to reorient the trajectory toward a potential aperture in the environment.

Since the system relies on the optic flow magnitude (norm), both values are treated as positive, and their sum provides a compact indicator of frontal obstruction.

The individual expected values, $T_+$ and $T_-$, are computed analytically based on the translational optic flow model for a planar surface. These values correspond to the expected optic flow magnitudes at $\pm 20^{\circ}$ when the distance to the forward obstacle is twice the one on the side. For the side distance, we use the maximum of the lateral optic flow (the maximum of both $\omega_{90^{\circ}}$ and $\omega_{-90^{\circ}}$ being noted $\omega_M$), adapted based on the current speed direction $\psi_v$:

$$T_{\pm} = \frac{\omega_M \cdot \cos(20^{\circ}) |\sin(\psi_v \pm 20^{\circ})|}{2 |\cos(\psi_v)|}$$

Exploration is triggered when the following conditions are jointly satisfied:

* The measured sum of optic flow exceeds the sum of the expected components: $\tilde{\omega}_{div} > (T_+ + T_-)$.

* The optic flow values at $\pm 20^{\circ}$ remain significantly non-zero: $\omega_{20^{\circ}} > 0.5 T_+$ and $\omega_{-20^{\circ}} > 0.5 T_-$,

* The current state is a regulated translational phase ($state = 0$),

* The drone is not already undergoing a significant rotation: $|\dot{\psi}| < \eta$, with $\eta$ set to 0.05 rad/s.

* An aperture is detected

Which is summarized by the following set of equations:

$$\left\{
\begin{array}{l}
\tilde{\omega}_{div} > (T_+ + T_-) \\
\omega_{20^{\circ}} > 0.5 \cdot T_+ \\
\omega_{-20^{\circ}} > 0.5 \cdot T_- \\
\text{state} = 0 \\
|\dot{\psi}| < \eta \quad \text{with } \eta = 0.05 \text{ rad/s} \\
\theta_{Aperture} \neq 0
\end{array}
\right.
\qquad (9)$$

These combined constraints ensure that exploration is only triggered when the system is in a stable translational phase, facing a potential frontal obstruction, and when the optic flow measurements provide sufficient evidence to support a reorientation maneuver.

In addition, when thresholds are exceeded, even if no OF aperture are detected, reorientations are nullified, as even a small rotation would risk blurring OF clues while a danger is approaching.

**Evasion Maneuver** In addition to reorientation driven by OF outlier, a final safety mechanism was implemented to prevent imminent collisions, inspired by the strategy proposed in [27]. This evasive behavior is triggered when the adapted divergence cue $\omega_{20^{\circ}} + \omega_{-20^{\circ}}$ reaches a higher threshold (twice the defined inbefore for OF-aperture based trigger) with no active rotation. To prevent collision, the robots turns about 60 degrees in the direction of the lower OF cues. This mechanism only triggers in case of imminent danger, hence only in the cases with a strong initial orientation error (Fig XX).
## B. Simulated drone parameters

The simulated hexarotor featured tilted propellers, making it fully actuated. All relevant parameters are listed in Table 1.

<table><caption>Table 1. Parameters of the simulated drone</caption><thead><tr><td>Parameter</td><td>Description</td><td>Value</td></tr></thead><tbody><tr><td>m</td><td>Mass</td><td>1.86 kg</td></tr><tr><td>a<sub>l</sub></td><td>Arm length</td><td>0.368 m</td></tr><tr><td>I<sub>x</sub></td><td>Inertia about x-axis</td><td>0.11 kg·m<sup>2</sup></td></tr><tr><td>I<sub>y</sub></td><td>Inertia about y-axis</td><td>0.11 kg·m<sup>2</sup></td></tr><tr><td>I<sub>z</sub></td><td>Inertia about z-axis</td><td>0.19 kg·m<sup>2</sup></td></tr><tr><td>g</td><td>Gravity</td><td>9.81 m/s<sup>2</sup></td></tr><tr><td>f<sub>tm</sub></td><td>Force-to-torque conversion factor</td><td>0.3</td></tr><tr><td>β</td><td>Propeller tilt angle (roll axis)</td><td>-25°</td></tr><tr><td>α</td><td>Propeller tilt angle (pitch axis)</td><td>35°</td></tr></tbody></table>

## C. Controller Parameters

In this study we have made use of a Sliding Mode Controller (SMC) to determine the control input U needed. This controller has its parameters defined in Table 2. As different types of maneuver are required from the system depending on the current state, each state has its own parameters.

<table><caption>Table 2. SMC parameters for each state</caption><thead><tr><th>Controlled variable</th><th>Parameter</th><th>State 0</th><th>State 1</th></tr></thead><tbody><tr><td rowspan="4">x (forward)</td><td>l<sub>x</sub></td><td>0.5</td><td>0.5</td></tr><tr><td>α<sub>gx</sub></td><td>1000</td><td>1000</td></tr><tr><td>K<sub>x</sub></td><td>0.5</td><td>0.5</td></tr><tr><td>K<sub>ix</sub></td><td>1</td><td>0.001</td></tr><tr><td rowspan="4">y (lateral)</td><td>l<sub>y</sub></td><td>5</td><td>0.3</td></tr><tr><td>α<sub>gy</sub></td><td>1000</td><td>1000</td></tr><tr><td>K<sub>y</sub></td><td>10</td><td>2</td></tr><tr><td>K<sub>iy</sub></td><td>0.001</td><td>0</td></tr><tr><td rowspan="4">z (vertical)</td><td>l<sub>z</sub></td><td>50</td><td>100</td></tr><tr><td>α<sub>gz</sub></td><td>1000</td><td>1000</td></tr><tr><td>K<sub>z</sub></td><td>1</td><td>0.1</td></tr><tr><td>K<sub>iz</sub></td><td>0.1</td><td>10</td></tr><tr><td rowspan="4">Roll (β)</td><td>l<sub>β</sub></td><td>50</td><td>40</td></tr><tr><td>α<sub>β</sub></td><td>2</td><td>2</td></tr><tr><td>K<sub>β</sub></td><td>1</td><td>0.1</td></tr><tr><td>K<sub>iβ</sub></td><td>1</td><td>1</td></tr><tr><td rowspan="4">Pitch (γ)</td><td>l<sub>γ</sub></td><td>50</td><td>40</td></tr><tr><td>α<sub>γ</sub></td><td>2</td><td>2</td></tr><tr><td>K<sub>γ</sub></td><td>1</td><td>0.1</td></tr><tr><td>K<sub>iγ</sub></td><td>1</td><td>1</td></tr><tr><td rowspan="4">Yaw (ψ)</td><td>l<sub>ψ</sub></td><td>50</td><td>0.5</td></tr><tr><td>α<sub>ψ</sub></td><td>1</td><td>0.03</td></tr><tr><td>K<sub>ψ</sub></td><td>5</td><td>0.05</td></tr><tr><td>K<sub>iψ</sub></td><td>0.01</td><td>0.005</td></tr></tbody></table>

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