Touchless underwater wall-distance sensing via active proprioception of a robotic flapper

The flapping plate is actuated by a 3DoF robotic mechanism (figures 2(a) and (b)) consisted of three servos with encoder resolutions of 360/4096° (XM450-W350-R, Robotis). Only 1 DoF from the topmost servo was used in this study for generating flapping motion. The plate was thus assumed to move only within the horizontal plane. The other two servos remained fixed in the positions shown in figure 2(b), keeping the plate's span horizontal and with an angle of attack of 90°. The servos had integrated PID controllers to reach the desired angles. The PID gains were manually tuned to be 200, 10, 10, respectively, to avoid oscillations around the set point and to reduce jerkiness. The same robotic mechanism was also used in our previous work [35–37].

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A six-component force–torque load cell (Nano17-IP65, ATI) was mounted on the output shaft of the bottom servo and attached to the plate, thereby measuring the internal forces and moments between the plate and the robotic mechanism. The sensor had resolutions of 1/320 N for forces and 1/64 N mm for moments. The robot and the load cell were immersed in a mineral oil tank (White Mineral Oil, Tulco, density $\rho = {826}$ kg m⁻³, viscosity $\nu = {6e-6}$ m s⁻² at 22 °C) of size ${81}~\textrm{cm}\times{81}~\textrm{cm}\times{81}~\textrm{cm}$. T-slotted beams installed over the center of the tank allowed the plate to be positioned at different wall-distances as shown in figure 2(c).

The servo control and force–torque data collection were performed using a Speedgoat real-time computer (Performance Real-Time Target Machine, Speedgoat GmbH). Code for the target machine was generated using a Simulink® model. The force–torque readings were collected at 200 Hz, while the reading of the encoder angle and the sending of the angle commands were alternated over serial (RS-480) at 100 Hz each.

The rectangular plates were made of 3 mm thick acrylic and was attached at the mid-chord to the bottom servo's shaft. Three aspect ratios of the plates were used for varying the Rossby number (Ro) (figure 2(d)). Ro measures the degree of three-dimensional effects in the fluid flow, and determines the stability of leading-edge vortices (LEV) observed in flapping wings of birds and insects [38]. For rectangular plates, Ro is equivalent to the aspect ratio, and is defined as [39]:

$$\begin{align} Ro = \frac{R}{c} \end{align} \tag{ 1 }$$

where $R = s+o$ is the plate tip radius, s is the span (or plate) length, $o = {5.1}~\textrm{cm}$ is the offset between the plate root and the rotation axis, and c is the chord length. Equation (1) was used to create three sets of s and c for the rectangular plates (figure 2(d), table 1) while keeping the surface area ($S = s\cdot c$) constant at 108 cm². The designed values of Ro (2, 3.5, 5) fell in the range observed in biological creatures [39].

Table 1. Plate shape and motion parameters for each Ro and $A^*$. s is the span length, c is the chord length, φ₀ is the flapping amplitude, and f is the flapping frequency.

Ro	$A^*$	s (cm)	c (cm)	φ0 (°)	f (Hz)
2	2	12.4	8.7	57.3	0.116
	3			84.9	0.077
	4			114.6	0.058
3.5	2	17.1	6.3	32.7	0.229
	3			49.1	0.153
	4			65.5	0.114
5	2	20.8	5.2	22.9	0.348
	3			34.4	0.232
	4			45.8	0.174

Dimensionless amplitude of the flapping motion ($A^*$) measures the unsteadiness of the fluid flow generated by the flapping plate, as a lower amplitude at the same average flapping speed (determined by the Reynolds number) leads to more frequent back-and-forth. $A^*$ affects the integrity of the LEV [39]. Although the lower stroke amplitude of mosquito wings has been speculated to lead to better signals in a fluid-based active sensing scenario as compared to bird wings [33], the effects of dimensionless amplitude on perception are yet to be extensively studied. $A^*$ was defined as:

$$\begin{align} A^* = \frac{\phi_0 R}{c} \end{align} \tag{ 2 }$$

where φ₀ is the flapping amplitude. In addition, the Reynolds number at the radius of gyration ($Re_{\mathrm{g}}$) was defined as [35, 36, 39, 40]:

$$\begin{align} Re_{\mathrm{g}} = \frac{c U_{\mathrm{g}}}{\nu} = \frac{4 \phi_0 f R_{\mathrm{g}} c}{\nu} \end{align} \tag{ 3 }$$

where $U_{\mathrm{g}}$ is the average velocity at the radius of gyration $R_{\mathrm{g}} = \sqrt{I/S}$ (with $I = \frac{s^3c}{12}+S (o+\frac{s}{2})^2$ being the second moment of plate area about the rotation axis) and f is the flapping frequency. We designed three values of $A^*$ (2, 3, 4), of order 1 like insect flyers [39], at a fixed $Re_{\mathrm{g}}$ of 800, which were used to determine φ₀ and f for the each Ro (table 1) at which the plate flapped sinusoidally.

The plates were placed at 14 distances from a wall of the tank, along a straight line in the middle of the tank (figure 2(c)). The distance was measured from the plate tip to the wall ($d_{\mathrm{tip}}$), which went from 1 to 40 cm in increments of 3 cm. This range corresponded to 1.9 plate lengths for Ro = 5 and 3.1 plate lengths for Ro = 2. While the distance from the force–torque sensor to the wall in the above distance measure was dependent on the plate length, an alternative wall-distance measure from plate root (or the force–torque sensor) to the wall ($d_{\mathrm{root}}$) was also calculated, which was independent of the plate length. The depth of the center of the plate was set to be 40 cm, which was more than 4 times the chord length, thereby leading to minimal effects of surface waves on the force–torques.

For each of the 3 Ro and at each of the 14 $d_{\mathrm{tip}}$, time series data including the force–torque and the flapping angle were collected in 5 separate experimental trials. In each trial, 14 flapping cycles worth of data was collected at all 3 $A^*$ in the order of $A^*$ = 2, 3, 4. To remove the transient force–torque behavior when switching between $A^*$, the first 3–4 cycles of data after setting the flapping motion for each $A^*$ were discarded, after which the measurements did not appear to vary from cycle to cycle upon visual inspection. Thus, data for a total of 70 flapping cycles was collected at each distance for each combination of Ro and $A^*$.

We used supervised learning to train a LSTM architecture for wall-distance prediction (figure 3). The inputs to the LSTM included the measured six-component force and torque and the flapping angle within one flapping cycle, i.e. a 7-dimensional time-series data, which were mapped by the LSTM to its output as the predicted wall-distance—a single number. LSTM was chosen because of its ability to process time-series data with long-term dependencies [41]. Multiple LSTM units were stacked in multiple layers (table 2). The output of the last LSTM layer at only the last time-step was propagated into the next-stage hidden and output dense layers. Dropout was also used between all layers for regularization.

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The force–torque, flapping angle, and wall-distance data were pre-processed (figure 3), to improve the learning performance. These include the following:

• (i)  
  Dividing into cycles: The neural network was trained to output the wall-distance given the time-series data from one flapping cycle (starting at 0 phase).
• (ii)  
  Dividing into training/validation/testing sets: To avoid bias towards data from any experimental trial or wall-distance in training or in testing, the 14 input-output samples in each trial at each wall-distance were randomly distributed into the three sets in a 64.3%/14.3%/21.4% split, leading to 630/140/210 samples in the respective sets. Additionally, the training process was repeated 5 times to mitigate biases introduced by the stochasticity in the splitting and in the gradient descent. Thus, our results are based on $5\times210 = 1050$ testing samples.
• (iii)  
  Downsampling: In a preliminary test and in [42], it was found that the learning performance greatly increased when the temporal resolution of the input force–torque and angle data were reduced by downsampling. Here, downsampling was done by averaging a number of consecutive time-steps into a single time-step.
• (iv)  
  Normalization: For each Ro and $A^*$, the input time-series and the output wall-distances were normalized by having their means subtracted and then divided by their standard deviations.

The neural network was trained with gradient descent with Keras in TensorFlow. The loss function was the mean absolute error (MAE). Adam was used as the optimizer to update the neural network weights because of its adaptive learning rate [43], which was initially set at 0.0005. Exponential learning rate decay (with a rate of 0.9 and a step of 5000) further reduced the learning rate with the number of training epochs to help the weights converge. The weights were updated with every mini-batch of 32 samples from the training set, and this process was repeated for 5000 epochs. Finally, early stopping was used to combat overfitting—the weights from the epoch that had the lowest validation set loss were restored at the end of training.

The LSTM design parameters and pre-processing parameters were designed separately for each Ro and $A^*$. Based our preliminary tests and also inspired by the approach in [42], parameters that had large impacts on the LSTM model's performance and their possible options were identified (table 2). The downsampling rate affected the smoothing of the time-series and reduced the input's dimensionality, the number of LSTM units and layers affected the model's complexity, and the dropout rate affected the strength of the regularization. All combinations of these parameters were used to train the LSTM model using the process described above, and the combination yielding the lowest validation set loss was chosen (table 3). Regarding the large model sizes in proportion to the number of training samples: the larger models still yielded better validation set losses than the smaller models, and were thus used in this work. For example, only models with 128 and 192 LSTM units per layer were selected for all Ro and $A^*$ cases as they had lower validation losses than the 64-unit models, even though the latter would have resulted in the smallest model size. Nonetheless, the smaller models we used in preliminary work (e.g. 8 LSTM units with only 1 layer) still achieved notable prediction performance.

Table 3. Parameters and hyperparameters found best for each Ro and $A^*$ among the choices in table 2 for the neural network in figure 3. The time steps per cycle is a result of sampling rate, flapping frequency and downsampling rate at each Ro and $A^*$.

Ro	$A^*$	LSTM units per layer (C)	LSTM layers (L)	Dropout (D)	Downsampling rate	Time steps per cycle
2	2	192	2	0.5	15	114
	3	128	2	0.2	15	172
	4	192	2	0.2	15	229
3.5	2	192	3	0.5	15	58
	3	192	3	0.5	15	87
	4	192	3	0.5	15	116
5	2	192	3	0.5	5	115
	3	192	3	0.5	5	172
	4	192	2	0.2	15	76

We also evaluated the effects of Ro and $A^*$ on predicting the wall-distances using the above LSTM framework. We compared the 9 combinations of Ro and $A^*$ based on the error statistics of the test set predictions. The medians of the prediction errors for all samples in the test set were used as a measure of the expected prediction error, and the lower (Q₁) and upper (Q₃) quartiles were used as measures of variability in the prediction quality. To give a more comprehensive picture of the variability in the prediction errors, the expected lower ($W_{\mathrm{L}} = Q_1 - 1.5 IQR$) and upper ($W_{\mathrm{U}} = Q_3 + 1.5 IQR$) extremes of the errors were also obtained, with $IQR = Q_3-Q_1$ being the inter-quartile range. In the sections that follow, the lower and upper whiskers in the box plots represent $W_{\mathrm{L}}$ and $W_{\mathrm{U}}$, the lower and upper limits of the box represent Q₁ and Q₃, and the line crossing the box represents the median, respectively [44].

SHAP [34] values signify the contribution of each input feature, quantifying how much each feature increases or decreases the model's output relative to a baseline output. Because of its additive properties, SHAP values for individual input features can be calculated separately and then be summed up to explain the deviation of a particular prediction from the baseline. Additionally, SHAP values for individual predictions can be averaged over multiple predictions to explain, in general, what input features are important to the model [45]. Here, SHAP was used to identify the temporal phases of flapping during which the input had the most pronounced influence on the predictions. Since the inputs here are multi-dimensional time-series, a SHAP value was first assigned to every time step of every force–torque component and flapping angle. Then, the SHAP values were summed over the 7 input dimensions, resulting in a one-dimensional time-series containing the contributions of each time step to the LSTM model's output. The absolute value was taken to get only the magnitude of the time step's importance. Finally, the SHAP time-series for all samples at the same wall-distance in the testing set were averaged, yielding a single SHAP time-series for each combination of Ro and $A^*$ at a wall-distance.

The cycle-averaged proprioceptive forces and torques at all $d_{\mathrm{tip}}$ are plotted in figure 4 for one case (Ro = 5, $A^* = 2$). Overall, the presence of the wall at different distances had subtle, but identifiable effects on force and torque components. The most noticeable changes occurred close to when the flapping plate reversed its direction (i.e. stroke reversals). The changes in the forces and torques were relatively small compared to their respective absolute values, except for components with smaller magnitudes, especially the spanwise torque.

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Nevertheless, these relatively small changes in the forces and torques were sufficient for the trained LSTM to predict the wall distance with high accuracy. This can be shown by comparing the predictions and ground truth for the testing cases (figure 5), which showed excellent match for all combinations of Ro and $A^*$. In figure 6(a), even $W_{\mathrm{U}}$—an indicator of the worst expected prediction error—is less than half the plate lengths of each Ro, further confirming the LSTM's ability to successfully decode the wall-distance from the proprioceptive force–torque signals.

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The prediction errors for each Ro and $A^*$ combination, including the predictions at all wall-distances, are summarized in figure 6(a), which shows that the predictions were more accurate with lower Ro or plate aspect ratio. Going from Ro of 2 to 3.5, the median of the errors increased slightly by less than 0.25 cm. However, Q₃ and $W_{\mathrm{U}}$ increased more substantially, from 1 to 2 cm and from 3 to 7 cm, respectively. Thus, this increase in Ro increased the possibility of having large prediction errors that were greater than the median errors, even though the medians themselves were comparable between the two Ro at 0.25 cm. Going from Ro of 3.5 to 5, the median errors increased more significantly from 0.25 to 1 cm, while the Q₃ and $W_{\mathrm{U}}$ also increased. Despite the increase, the median prediction error in all cases were below 5% of the plate length.

Using $d_{\mathrm{tip}}$ as the distance measure, the lower Ro or aspect ratio plates had their force–torque sensor placed closer to the wall due to the shorter plate length. Alternatively, we also used $d_{\mathrm{root}}$ as the distance measure, which made the force–torque sensor in all Ro cases placed at an almost identical distance from the wall. A separate set of LSTM models were thus trained with only the force–torque data having comparable $d_{\mathrm{root}}$ between the different Ro. The prediction errors using this approach is shown in figure 6(b). Compared to figure 6(a), Q₃ and $W_{\mathrm{U}}$ of Ro = 2, $A^*$ = 2 are significantly higher, making its performance almost identical to Ro = 3.5, $A^*$ = 2. This may be expected as the sensor on the Ro = 2 plate starts 12 cm—almost an entire plate length—further from the wall than when using $d_{\mathrm{tip}}$. Except this case, the lower Ro had lower prediction error in all other cases, agreeing with the earlier conclusions based on $d_{\mathrm{tip}}$ as the distance measure.

The prediction accuracy also increased moderately with the flapping motions of higher $A^*$ (figure 6(a)). At Ro = 2, Q₃ and $W_{\mathrm{U}}$ are clearly higher for $A^*$ = 2. At Ro = 3.5, Q₃ and $W_{\mathrm{U}}$ at $A^*$ = 2 were higher than at $A^*$ = 4. However, $A^*$ = 3 at Ro = 3.5 has only slightly higher errors than $A^*$ = 2, but significantly higher errors than $A^*$ = 4, breaking the trend of better predictions being at higher $A^*$. A similar trend is observed at Ro = 5, where the distribution of prediction errors between the three $A^*$ are almost the same. Thus, while the increase in the prediction quality with $A^*$ was more pronounced at Ro = 2, the impact of $A^*$ at higher Ro was less substantial compared to the effect of Ro.

The prediction accuracy also depended heavily on the distance to the wall (figure 7). The errors were lowest closest to the wall, then increased with distance, and either plateaued or lowered again at the furthest distances. The predictions being best nearest to the wall was expected since the wall would be able to influence the flow generated by the plate more significantly. The lower prediction errors at the furthest distances from the wall may also be explained similarly, but due to the influence of the opposite wall of the oil tank. Because the length of the tank is only between 4 to 7 plate lengths, being further from one wall of the tank meant getting closer to the opposite wall. The decrease in prediction errors at the furthest wall-distances was most pronounced for the Ro = 5 plate, which—having the longest plate length—reached closest to the opposite wall, and possibly had its flow altered more than the shorter plates. Thus, larger alterations in the self-generated flow by both the front and back walls could have improved the wall-distance predictions.

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Figure 8 shows the magnitude of the SHAP at each phase of a flapping cycle for all combinations of Ro and $A^*$ and at all wall-distances. The largest peaks of SHAP values occurred during or after stroke reversals, while the SHAP values during the middle flapping portion were relatively low (except Ro = 2 and $A^*$ = 2). Therefore, these results suggest that the proprioceptive forces and torques near the stroke reversals were the most important for wall-distance predictions, and the unsteadiness in the flow may facilitate the underwater wall perception.

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In this work, we showed that the fluid flows generated by a flapping plate did encode distance information relative to an underwater wall, and it could be successfully decoded by an LSTM receiving the proprioceptive forces and torques measured at the base of the plate. Notably, the perception of the wall was made possible by the plates' active flapping movements, thus making the sensing 'active' [33]. Without the active movements, there would be no flow in the tank and no hydrodynamic force or torque for the sensor to measure. Existing works using potential flow theory have shown that the presence of a wall can alter the pressure gradient near a moving airfoil [46], which could in turn result in changes in the hydrodynamic forces and torques and be reflected in the proprioception. Unlike most methods in the literature, an external flow, such as an incoming freestream [47], was not needed to carry the information about the surroundings to a stationary pressure sensor array. Thus, this study substantiates the potential roles of active sensing using self-generated flows in facilitating underwater perception.

While some of the features extracted from the forces and torques may be specific to our experimental setup due to the minimalistic flapping kinematics and the unavoidable imperfections and uncertainties inherent in it, the overall approach is expected to be applicable to other conditions. The specific decoding methodologies and the design of the neural network will likely need to be tailored to the application at hand, for which the kinematics and morphology may be designed to optimize the thrust, lift, or other metrics [48]. However, as long as the repeatability and resolution of the sensing apparatus is adequate, it should be possible to distinguish the effects of walls or other underwater objects on the proprioception, like it was here as evidenced by the visible changes in the forces and torques with wall-distance (figure 4). Moreover, detection of a wall parallel to or in front of a swimming robot may be more useful practically. Our results suggest that the flapper can also potentially detect the opposite wall that is behind the flapper (or in front of a swimming robot). As we moved the flapper further from the wall that its tip is facing, and towards the opposite wall (figure 7 by the secondary x-axes), the prediction error in most cases decreased. Thus, we expect the prediction of wall-distance in other robotic applications to also be feasible.

It is also worth mentioning that the airflows generated by rotating blades in rotary-wing fliers can also enable detection of walls or ground. For example, the presence of surfaces (of various orientations) within the airflow field generated by quadcopters can alter its pressure distribution, which can be measured directly via an array of pressure sensors, and used for surface detection [33]. The presence of a wall can also indirectly alter the body states of a quadcopter via aerodynamic interaction, which can be measured using an Inertial Measurement Unit (IMU), and decoded for wall detection [49]. Our work differs from the above methods in the sense that the perception is enabled by measuring the internal forces and torques between the plate and the robotic mechanism, i.e. proprioception instead of exteroception.

The most critical temporal phase for wall-distance prediction coincided with the stroke reversal, where the plate experiences largest acceleration or deceleration, likely corresponding with highest unsteadiness in the self-generated flow, including rapid vortex formation and shedding. Thus, this suggests that the unsteady excitation in the fluid flow can facilitate its interaction with the wall and generate rich information regarding the wall distance. The role of active flow generated by reciprocal motion in prompting the underwater perception was also reported in the literature. For example, in the work of Meyer et al [50] where a vibrating sphere was used to stimulate a living fish's lateral line, the response of the fish's neurons during the phase at which the sphere reverses its direction could be used to discern the 2D orientation of its vibration direction. However, active sensing with self-generated flow is possible even without reciprocal motion. For instance, the increase in pressure measurement (generated by the movement of the sensor array on underwater vehicle towards the wall) was used to detect the wall [26], and the stagnation pressure at the nose of blind Mexican cave fish has been observed to increase when moving towards a wall [14]. Interestingly, the rate of collision of these fish with the walls increased when they were beating their tails compared to when they were only gliding [13], suggesting that some self-generated flows may be detrimental to perception. Therefore, further investigation of the unsteady flow features that enable perception is needed, which were promoted by reciprocal motion in our work and by body translation in some other works in literature.

In previous works, it has been shown that spatial patterns in the aerodynamic pressure measured by insects' antenna or pressure sensors installed on robotic fliers can be decoded for touchless wall detection or aerodynamic imaging [33]; in addition, the spatial patterns in the lateral line sensory measurement also play a critical role in the underwater perception [14, 26, 51]. In this work, we showed that temporal patterns in the sensory inputs may also play a role in underwater perception, such as wall-distance estimation. While the forces and torques at the plate root do measure the net effect of the pressures acting on the plate, they are not able to measure the pressure gradient along the plate surface like a fish's lateral line or artificial sensor arrays [13, 21, 47, 51]. Thus, the success in wall-distance prediction using only measurements at one location, without knowing the spatial pressure distribution, demonstrates that wall-distance information is also encoded temporally, which also suggests that high unsteadiness in the flow can prompt underwater perception. Moreover, wall-distance could be predicted using only a single component of force or torque. For example, by utilizing changes with wall-distance at stroke reversal in only the normal force, as can be observed in figure 4, perception could be enabled solely by temporal patterns.

The Rossby number represents the ratio of inertial force and Coriolis force in the fluid flow, and therefore indicates the relative strength of rotational effects [38, 39]. A rotating plate has higher rotational effects (with a smaller local Ro) near its center of rotation, compared with those at its distal parts. Our results based on both $d_{\mathrm{tip}}$ and $d_{\mathrm{root}}$ show that lower Ro leads to higher prediction accuracy (figure 6), suggesting that the rotational effects may promote underwater perception. Note that, at the same $d_{\mathrm{root}}$, the tips of smaller Ro (or aspect ratio) plates were further away from the wall compared with those of higher Ro. As the prediction accuracy decreases with the distance within the majority of the distance range tested (figure 7), smaller Ro led to higher prediction accuracy regardless (figure 6(b)). It can be speculated that the stronger rotational effect in the flow may promote its interactions with the wall and generate richer temporal features (e.g. pressure gradient, vortex shedding, etc) that enhance the distance prediction. Additionally, at higher Ro, the diminishing prediction performance may be attributed to the fluid's added mass being positioned further from the transducer, thereby magnifying the inertial effects of the fluid during stroke reversals and causing the subtle variations in force–torque profiles from wall interactions to appear relatively smaller. However, detailed studies on the fluid field are needed to fully understanding the underlying physical processes.

In addition, $A^*$ provides a measure of the overall unsteadiness in the flow $A^*$ [39], and shows some moderate effect on the prediction accuracy. Our results show that higher $A^*$ (lower unsteadiness) led to either no significant change or slightly better prediction (figure 6(a)). This result does not align with our SHAP analysis which suggests that the instantaneous unsteadiness during the stroke reversal can enhance the prediction. As a speculation, the lower $A^*$ with smaller flapping amplitude may have generated more noises rather than meaningful temporal features, thereby, reducing the signal-to-noise ratio for the prediction. This result also seems to contradict the speculation made in [33] that lower stroke amplitudes in mosquitoes ($A^*\approx1.61$ [52]) could lead to better signals as the jets released from the wing could become more concentrated. Unlike in our study, these jets were directed towards the surface to be detected, while our flapping plate kinematics were unlikely to produce a strong concentrated jet towards the wall. While the wall-distance could still be predicted in our work, flow visualizations from Nakata's work [33] suggest possibility for improving prediction accuracy by optimizing the motion profile of the plates and, consequently, the direction of the self-generated flow, even when operating at the same $A^*$. However, as the effects of $A^*$ were not as strong as that of Ro (section 3.2), the effects of $A^*$ in perception could not be conclusively established in this study.

Finally, although we stayed at a low $Re_{\mathrm{g}}$ at which the flow is expected to be laminar, we suspect that turbulence may offer a tradeoff for perception. On one hand, turbulence has the potential of degrading the signal-to-noise ratio for wall-distance perception. In their behavioral and computational fluid dynamics studies, Windsor et al showed that the distance to a wall that can be detected by cave fish stays roughly the same and may even decrease slightly with increasing (body) Reynolds number (ranging from 1000 to 6000) [13, 14]. On the other hand, the dominance of inertial fluid effects over viscosity at higher $Re_{\mathrm{g}}$ may help in transmitting the encoded wall-distance signals to the proprioceptive sensors, rather than letting viscosity dissipate them. Additionally, the stronger flow excitation at high $Re_{\mathrm{g}}$ may possibly increase the signal-to-ratio of wall-distance in the presence of environmental noise, as hypothesized by Windsor et al [14]. The wall-distance prediction performance of this sensing mode in turbulent conditions compared to others, like vision, merits further study.