A small-scale hyperacute compound eye featuring active eye tremor: application to visual stabilization, target tracking, and short-range odometry

In this study, visual hyperacuity results from an active process whereby periodic micro-movements are continuously applied to an artificial compound eye. This approach was inspired by the retinal micro-movements observed in the eye of the blowfly Calliphora (see figure 1(a)). Unlike the fly's retinal scanning movements, which result from the translation of the photoreceptors (see figure 1(b)) in the focal plane of each individual facet lens (for a review on the fly's retinal micro-movements see [10]), the eye tremor applied here to the active CurvACE by means of a micro-stepper motor (figure 1(c)) results from a periodic rotation of the whole artificial compound eye. Section 2.3 shows in detail that both scanning processes lead to a rotation of the visual axis.

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Here we describe in detail the active version of the CurvACE sensor and establish that this artificial compound eye is endowed with hyperacuity, thanks to the active periodic micro-scanning movements applied to the whole eye.

As described in [3], the CurvACE photosensors array has similar characteristics to the fruitfly eye in terms of the number of ommatidia (630), light adaptation, the interommatidial angle (Δφ = 4.2° on average), and a similar, Gaussian–shaped angular sensitivity function (ASF). This specific ASF removes 'insignificant' contrasts at high spatial frequencies.

To replicate the characteristics of its natural counterpart (see figure 2(a)), CurvACE was designed with a specific optical layer based on the assembly consisting of a chirped microlens array (lenslet diameter: 172 μm) and two chirped aperture arrays. In the case of active CurvACE, the optical characteristics remain constant during the scanning process.

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Each ASF of active CurvACE (figures 2(b) and (c)) can be characterized by the acceptance angle $\Delta \rho$, which is defined as the angular width at half of the maximum ASF. The ASF along 1D $s(\psi )$ of one CurvACE ommatidium can therefore be written as follows:

$$\begin{eqnarray}&&s(\psi )=\frac{2\sqrt{\pi {\rm ln} (2)}}{\pi \Delta \rho }{{e}^{-4{\rm ln} (2)\frac{{{\psi }^{2}}}{\Delta {{\rho }^{2}}}}}\end{eqnarray} \tag{ 1 }$$

where ψ is the angle between the pixel's optical axis and the angular position of a point light source.

Figures 3(a) and (b) compare the rotation of the visual axes resulting from the translation of the retina behind a fixed lens (e.g., in the case of the fly's compound eye) with the rotation of the visual axes resulting from the rotation of the whole eye (e.g., like the mechanism underlying the micro-saccades in the human's camerular eye [41]), respectively. In the active CurvACE, we adopted the second strategy by subjecting the whole eye to an active micro-scanning movement that makes the eye rotate back and forth.

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As shown in figure 3, the retinal micro-scanning movements are performed by a miniature eccentric mechanism based on a small stepper motor (Faulhaber ADM 0620-2R-V6-01, 1.7 grams in weight, 6 mm in diameter) connected to an off-centered shaft [42]. This vibrating mechanism makes it possible to control the scanning frequency by just setting a suitable motor speed.

A general expression for the pixel's output signal is given by the convolution of the ASF $s(\psi )$ with the light intensity I of the 1D scene as follows:

$$\begin{eqnarray}&&Ph({{\psi }_{c}})=\int _{-\infty }^{+\infty }s(\psi )\cdot I(\psi -{{\psi }_{c}})\ d\psi \end{eqnarray} \tag{ 2 }$$

where ${{\psi }_{c}}$ is the angular position of a contrasting feature (edge or bar) placed in the sensor's visual field. For example, $I(\psi )$ can be expressed for an edge as follows:

$$\begin{eqnarray}I(\psi )=\left\{ \begin{array}{ccccccccccccccc} {{I}_{1}} & \;{\rm for}\;\psi \lt 0 \\ {{I}_{2}} & \;{\rm for}\;\psi \geqslant 0 \\ \end{array} \right.\end{eqnarray} \tag{ 3 }$$

and for a bar:

$$\begin{eqnarray}I(\psi )=\left\{ \begin{array}{ccccccccccccccc} {{I}_{1}} & \;{\rm for}\;|\psi |\lt L/2 \\ {{I}_{2}} & \;{\rm for}\;|\psi |\geqslant L/2 \\ \end{array} \right.\end{eqnarray} \tag{ 4 }$$

with L the width of the bar (expressed in rad).

The micro-scanning movements of the pixels can be modelled in the form of an angular vibration ${{\psi }_{{\rm mod} }}$ of the optical axes added to the static angular position ${{\psi }_{c}}$ of the contrast object (an edge or a bar). The equations for the two pixels ($P{{h}_{1}}$ and $P{{h}_{2}}$) are therefore:

$$\begin{eqnarray}&&P{{h}_{1}}(\psi (t))=Ph\left( {{\psi }_{c}}(t)+{{\psi }_{{\rm mod} }}(t)-\frac{\Delta \varphi }{2} \right)\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&P{{h}_{2}}(\psi (t))=Ph\left( {{\psi }_{c}}(t)+{{\psi }_{{\rm mod} }}(t)+\frac{\Delta \varphi }{2} \right)\end{eqnarray} \tag{ 6 }$$

with ${{\psi }_{{\rm mod} }}$ obeying the following sinusoidal scanning law:

$$\begin{eqnarray}&&{{\psi }_{{\rm mod} }}(t)=A\cdot {\rm sin} (2\pi {{f}_{{\rm mod} }}\cdot t)\end{eqnarray} \tag{ 7 }$$

With A and f_mod describing the amplitude and the frequency of the vibration, respectively. In the case of a whole rotation of the eye, this scanning law is easily achievable by a continuous rotation of the motor with an off-centered shaft. In the case of a translation of the pixels behind a lens, the law should be weighted with the tangent of the ratio between the retinal displacement and the focal length f of the lens.

At the end, the photosensor response is a modulated convolution of the light intensity with the Gaussian sensitivity function.

The LPU defined in figure 4 is the application of algorithms presented in [13] and [14]. The first paper ([13]) leads to the signal Output_Pos resulting from the difference-to-sum ratio of the demodulated pixel output signals described by equation (8). The demodulation is realized by means of a peak filter that acts as both a differentiator and a selective filter centered at the scanning frequency (${{f}_{p}}={{f}_{{\rm mod} }}$ ). Then, an absolute value function cascaded with a low-pass filter smooths out the pixel's output signals. The second paper ([14]) explains in detail the edge/bar detection based on the observed phenomena that the two pixels' output signals are in phase in the presence of an edge and in opposite phase in the presence of a bar. At the output of the LPU, the signal ${{\theta }_{i}}$ represents the position of the contrasting feature in the FOV (see equation (9)).

$$\begin{eqnarray}&&Outpu{{t}_{Pos}}=\frac{P{{h}_{{{1}_{demod}}}}-P{{h}_{{{2}_{demod}}}}}{P{{h}_{{{1}_{demod}}}}+P{{h}_{{{2}_{demod}}}}}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{{\theta }_{i}}({{\psi }_{c}})=Outpu{{t}_{Detector}}.Outpu{{t}_{Pos}}\end{eqnarray} \tag{ 9 }$$

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With Output_Detector equal to (−1) or (1) and ${{\theta }_{i}}$ the output signal of an LPU (see figure 4).

The characteristic static curves of the active CurvACE obtained with a contrasting edge and a black bar 2.5 cm in width subtending an angle of $2.86{}^\circ$ are presented in figure 5.

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The curve in figure 5(a) has a tangent hyperbolic profile with respect to the angular position of the edge. It can be clearly seen by comparing the two curves plotted in figure 5 that the slopes of the characteristic static curves obtained with a bar and an edge are inverted. A theoretical explanation for the inversion of the slopes is given in [14]. This inversion justifies the use of an edge/bar detector in the LPU (see figure 4) to compensate for it and still be able to distinguish the direction of the movement.

Moreover, the characteristic curves are independent of the ambient lighting condition. Figure 6 shows that the Output_Pos signal remains constant even if the ambient light varies over about one decade (from 180 to 1280Lux). A peak is visible at each light change and corresponds to the transient phase during light adaptation of the CurvACE photosensors, which lasts only 250 ms. In addition, figure 6 shows that active CurvACE is a genuine angular position sensing device able to provide the angular position of a contrasting object placed in its FOV. Indeed, when the scanning is turned off and on again, the value of the measurement remained the same. This experiment shows that the micro-scanning movement allows us to measure the position of a contrasting object without any drift. A limitation comes here from the contrast, as it cannot be theoretically higher than 81.8% for an edge because the auto-adaptative pixel output signal is no longer log linear for changes of illuminance (in $W/{{m}^{2}}$) greater than one decade (see [3]).

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To endow a robot with the capability to sense its linear speed and position, a novel sensory fusion algorithm was developed using several LPUs in parallel. In this article, a 2D region of interest (ROI) composed of 8 × 5 artificial ommatidia in the central visual field was used in order to expose several pixels to the same kind of movements (figures 9(a) and (c)). In other words, the pattern seen by the sensor during a translation of the robot is a succession of edges and bars. The algorithm used here and depicted in figure 7 implements the connection between the 8 × 5 photosensors' output signals to an array of 7 × 5 LPUs in order to provide local measurements of edge and bar angular positions. Then a selection is performed by computing the local sum S of two demodulated signals $P{{h}_{demod}}$ obtained from two adjacent photosensors:

$$\begin{eqnarray}&&{{S}_{n,n+1}}=P{{h}_{{{(n)}_{demod}}}}+P{{h}_{{{(n+1)}_{demod}}}}\end{eqnarray} \tag{ 10 }$$

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Indeed, as a signal Output_Pos is pure noise when no feature is in the FOV, an indicator of the presence of a contrast was required. The sum of the demodulated signals was used here to give this feedback, because we observed that the contrast is positively correlated with the sum and the signal-to-noise ratio. Therefore, at each sampling step, each local sum S is the thresholded in order to select the best LPU's outputs to use. All sums above the threshold value are kept and the others are rejected. The threshold is then increased or decreased by a certain amount until 10 local sums have been selected. The threshold therefore evolves dynamically at each sampling time step. Lastly, the index i of each selected sum S gives the index of the pixel pair to process. Thus, the computational burden is dramatically reduced. Moreover, this selection helps reduce the data processing because only the data provided by the 10 selected LPUs are actually processed by the micro-controller.

In a nutshell, the sensory fusion algorithm presented here selects the 10 highest contrasts available in the FOV. As a result, the active CurvACE is able to assess its relative linear position regarding its initial one and its speed with respect to the visual environment.

It is worth noting that the selection process acts like a strong non-linearity. The output signal ${{\theta }_{fused}}$ is therefore not directly equal to the sum of all the local angular positions ${{\theta }_{i}}$. The parallel differentiation coupled to a single integrator via a non-linear selecting function merges all the local angular positions ${{\theta }_{i}}$, giving a reliable measurement of the angular orientation of the visual sensor within an infinite range. The active CurvACE can therefore be said to serve as a visual odometer once it has been subjected to a purely translational movement (see section 5.1). Mathematically, the position is given through the three equations as follows:

$$\begin{eqnarray}\left\{ \begin{array}{ccccccccccccccc} \Delta {{P}_{{{i}_{sel}}}} & = & {{\theta }_{{{i}_{sel}}}}(t)-{{\theta }_{{{i}_{sel}}}}(t-1) \\ {{\theta }_{fused}}(t) & = & {{\theta }_{fused}}(t-1)+\frac{1}{10}\mathop{\sum }\limits_{i=1}^{10}\Delta {{P}_{{{i}_{sel}}}} \\ {{V}_{x}}(t) & = & \frac{K}{Ts}({{\theta }_{fused}}(t)-{{\theta }_{fused}}(t-1)) \\ \end{array} \right.\end{eqnarray} \tag{ 11 }$$

As shown in figure 7, the robot's speed is determined by applying a low-pass filter to the fused output signal S_fused (which is the normalized sum of the local displacement error $\Delta {{P}_{isel}}$), whereas the robot's position is determined in the same way as ${{\theta }_{fused}}$, with the gain K.

To sum up, the algorithm developed here sums the local variation of contrast angular positions in the sensor FOV to be able to give the distance flown with the assumption that the ground height is known.

In order to determine the robot's speed and position accurately, the gaze direction should be orthogonal to the terrain. But as a simplification, we chose to align it with the vertical, assuming that the ground is mostly horizontal. Therefore, the eye has to compensate for the robot's roll angle. To this end, the eye is decoupled from the robot's body by means of a servo motor with a rotational axis aligned with the robot's roll axis. The gaze control system, composed of an inertial feedforward control, makes the eye looking always in a perpendicular direction to the movement during flight (figure 10). The rotational component introduced by the rotating arm supporting the robot can be neglected in this study.

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HyperRob is an aerial robot with two propellers and a carbon fiber frame, which includes two DC motors. Each motor transmits its power to the corresponding propeller (diameter 13 cm) via a 8 cm long steel shaft rotating on microball bearings in the hollow beam, ending in a crown gear (with a reduction ratio of 1/5). Two Hall effect sensors were used to measure the rotational speed of each propeller, and hence its thrust, via four magnets glued to each crown gear. Based on the differential thrust setpoints adopted, HyperRob can control its attitude around the roll axis, which is sensed by a six-axis inertial sensor (here an InvenSense MPU 6000). The robot's position in the azimuthal plane is controlled by adjusting the roll angle. In the robot, where only the roll rotation is free, only one axis of the accelerometer and one axis of the rate gyro are used. As shown in figure 11, active CurvACE is mounted on a fast micro-servomotor (MKS DS 92A+) which makes it possible to control the gaze with great accuracy ($0.1{}^\circ$) and fast dynamics (60° within 70 ms, i.e. 860°/s). This configuration enables the visual sensor to be mechanically decoupled from the body (see section 4.1). The robot is fully autonomous in terms of its computational resources and its power supply (both of which are provided onboard). The robot alone weighs about 145 g and the robot plus the arm weigh about 390 g.

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All the computational resources required for the visual processing and the autopilot are implemented on two power lean micro-controllers embedded onboard the robot. The first micro-controller (Microchip dsPIC 33FJ128GP802) deals with the visual processing, whereas the second one (Microchip dsPIC 33FJ128GP804) is responsible for stabilizing the robot. The two micro-controllers have a sampling frequency of 500 Hz. The robot's hardware architecture is presented in detail in figure 12.

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The micro-controller (μC) Vision communicates with CurvACE via a serial peripheral interface (SPI) bus and collects the pixel values of the ROI. The signal S_fused is computed and sent to the μC Control via another SPI bus. The latter completes the computation of the position $\bar{X}$ and the speed ${{\bar{V}}_{x}}$. This solution was chosen in order to keep the number of data sent via the SPI bus to a minimum. The μC Control estimates the robot's attitude on the roll axis via a reduced complementary filter (inspired by [43]) and controls the robot's linear position on the basis of the two visual measurements (X and V_x see figure 13). The μC Control then sends the propeller speed setpoints to a custom-made driver controlling the rotational speed of each propeller in a closed-loop mode.

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Assuming the robot to be a rigid body and simplifying the dynamic model for a quad-rotor presented in [44] in the case of a single roll axis, the robot's dynamics can be written as follows:

$$\begin{eqnarray}\left\{ \begin{array}{ccccccccccccccc} {\dot{X}} & = & {{V}_{x}} \\ {{{\dot{V}}}_{x}} & = & -\frac{{{T}_{nom}}}{m}{\rm sin} ({{\theta }_{r}}) \\ {{{\dot{\theta }}}_{r}} & = & {{\Omega }_{r}} \\ {{{\dot{\Omega }}}_{r}} & = & {{\Gamma }_{\theta }}=\frac{2\;l}{I}\delta \\ \end{array} \right.\end{eqnarray} \tag{ 12 }$$

where X is the robot's lateral position, V_x is its lateral speed, ${{\theta }_{r}}$ is the roll angle, ${{\Omega }_{r}}$ is the rotational roll speed, l is the robot's half span, I is the moment of inertia, δ is the differential thrust, and T_nom is the nominal thrust.

$$\begin{eqnarray}\left\{ \begin{array}{ccccccccccccccc} 2\delta & = & {{T}_{1}}-{{T}_{2}}={{c}_{T}}(\omega _{r1}^{2}-\omega _{r2}^{2}) \\ {{T}_{1}} & = & {{T}_{nom}}+{{\delta }^{\star }} \\ {{T}_{2}} & = & {{T}_{nom}}-{{\delta }^{\star }} \\ \end{array} \right.\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray*}\Leftrightarrow \left\{ \begin{array}{ccccccccccccccc} {{\omega }_{r1}} & = & \sqrt{\frac{\left( {{T}_{nom}}+{{\delta }^{\star }} \right)}{{{C}_{T}}}} \\ {{\omega }_{r2}} & = & \sqrt{\frac{\left( {{T}_{nom}}-{{\delta }^{\star }} \right)}{{{C}_{T}}}} \\ \end{array} \right.\end{eqnarray*}$$

where c_T is the thrust coefficient, and ${{\omega }_{r1}}$ and ${{\omega }_{r2}}$ are the right and left propeller speeds, respectively, and $\delta ={{f}_{prop}}(s){{\delta }^{\star }}$, where f_prop(s) and δ^⋆ correspond to the closed-loop transfer function of the propellers' speed and the differential thrust reference, respectively.

As described in figure 13, the autopilot controlling both the robot's roll and its position is composed of four nested feedback loops:

• the first feedback loop controls the robot's rotational speed by directly adjusting the differential thrust, and hence the roll torque.
• the second feedback loop yields setpoints on the previous one for tracking the robot's reference roll angle.
• the third feedback loop adjusts the robot's linear speed by providing roll angle setpoints.
• the fourth feedback loop controls the robot's linear position and yields the reference speed.

In the first and second feedback loops, the roll angle's estimation is obtained by means of a reduced version of a complementary filter described in ([43]). In the case of HyperRob, since only a 1D filtering method is required, the attitude estimator becomes:

$$\begin{eqnarray}\left\{ \begin{array}{ccccccccccccccc} {{{\bar{\theta }}}_{r}} & = & {\rm arcsin} \left( \frac{{{Y}_{acc}}}{g} \right) \\ \dot{{\hat{b}}} & = & -{{k}_{b}}({{{\bar{\theta }}}_{r}}-{{{\hat{\theta }}}_{r}}) \\ {{{\hat{\Omega }}}_{r}} & = & {{{\bar{\Omega }}}_{r}}-\hat{b} \\ {{\dot{{\hat{\theta }}}}_{r}} & = & {{{\hat{\Omega }}}_{r}}+{{k}_{a}}({{{\bar{\theta }}}_{r}}-{{{\hat{\theta }}}_{r}}) \\ \end{array} \right.\end{eqnarray} \tag{ 14 }$$

where ${{\bar{\theta }}_{r}}$ is the roll angle calculated from the accelerometer measurement Y_acc, $\hat{b}$ is the estimated rate gyro's bias, ${{\bar{\Omega }}_{r}}$ is the rate gyro's output measurement, ${{\hat{\theta }}_{r}}$ is the estimated roll angle, and ${{\hat{\Omega }}_{r}}$ is the unbiased rotational speed. Here k_a and k_b are positive gains which were selected so as to obtain a convergence time of 3 s and 30 s for the estimated angle and the estimated rate gyro's bias, respectively.

The complementary filter therefore yields the values of the rate gyro bias, the unbiased roll rotational speed, and the roll angle.

The fused visual output signal S_fused provided by the active CurvACE depends on the visual environment: if there are no contrasts, the sensor will not detect any visual cues and will therefore not be able to specify the robot's position accurately. The richer the visual environment is (in terms of contrasts), the better the position measurement. In order to compare the output of the sensor with the ground-truth measurements, three experiments were conducted under different visual conditions.

In these experiments, the robot was moved manually over two different panels under two different ambient lighting conditions from right to left and back to the initial position with its gaze stabilization activated. The results obtained, which are presented in figure 14, are quite similar for each trip, giving a maximum error of 174 mm. Figure 14 shows that the output in response to a textured panel and one composed of a single 5 cm wide black bar was similar. Therefore, assuming the distance to the ground to be known, the active CurvACE was able to serve as a visual odometer by measuring the robot's position accurately in the neighbourhood of its initial position.

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5.2.1. Above a horizontal textured panel

Lateral disturbances were applied by pushing the arm in both directions, simulating gusts of wind. In figure 15, it can be seen that all lateral disturbances were completely rejected within about 5 s, including even those as large as 40 cm. The dynamics of the robot could be largely improved by using a robot with a higher thrust or a lighter arm in order to reduce the oscillation. Figures 15(b) and (c) show that the robot was always able to return to its initial position. With its active eye, the robot can compensate for lateral disturbance as large as 359 mm applied to its reference position with a maximum error of only 25 mm, i.e., 3% of the flown distance. This error is presumably due to the selection process, which does not ensure the selection of the same features in the outward path and on the way back to the reference position; or maybe the assumption of a linear approximation of the inverse tangent function does not hold entirely within the entire FOV. As a consequence, thanks to the active visual sensor and its capability to measure the angular position of contrasting features, the robot HyperRob is highly sensitive to any motion and thus can compensate for very slow perturbation, ranging here from 0 to 391 mm. s⁻¹.

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5.2.2. Above an evenly sloping ground

In the previous experiment, it was assumed that the ground height must be known to be able to use the conversion gain in the fused output signal S_fused. The same experiment was repeated here above a sloping ground (see figure 16). The robot's height increased sharply in comparison with the calibration height. The robot's height varied in the range of +/−74 mm and starts with an offset of +96 mm compared to the calibration height. As shown in figure 16(b), the estimation of the traveled distance was always underestimated because the robot was always higher than the calibration height. However, the robot was still able to return to its starting position with a maximum error of only 45 mm (time $t=36.5$ s) for a disturbance of 210 mm (i.e., 10.5% of the flown distance).

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The robot's tracking performances are presented in this subsection. Three different tracking tasks were tested:

• tracking a moving textured panel (see section 5.3.1).
• tracking a moving textured panel with 3D objects placed on it (see section 5.3.2).
• tracking moving hands perceived above a stationary textured panel (see section 5.3.3).

5.3.1. Panel Tracking

In this experiment, the panel was moved manually and the robot's reference position setpoint ${{X}^{\star }}$ was kept at zero. The robot faithfully followed the movements imposed on the panel. The few oscillations which occurred were mainly due to the robot's dynamics rather than to visual measurement errors. Each of the panel's movements was clearly detected, as shown in figure 17(b), although a proportional error in the measurements was sometimes observed, as explained above.

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5.3.2. Tracking a moving rugged ground

In the second test, some objects were placed on the previously used panel to create an uneven surface. The robot's performances on this new surface were similar to those observed on the flat one, as depicted in figure 18. The visual error was not as accurately measured as previously over the flat terrain because of the changes in the height of the ground. But the robot's position in the steady state was very similar to that of the panel. The maximum steady-state error at $t=19$ s was only 32 mm .

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Table 1.  Controllers' parameters.

Controller	Transfer Functions	Parameters Value
Position controller	Kx	${{K}_{x}}=0.6\;{{s}^{-1}}$
Lateral speed controller	${{K}_{V}}.\frac{{{\tau }_{V}}s+1}{s}$	KV = 0.7
		${{\tau }_{V}}=\frac{1}{0.7}$
Roll controller	${{K}_{\theta }}$	${{K}_{\theta }}=0.6$
Robot rot. speed controller	${{K}_{\Omega }}.\frac{{{\tau }_{\Omega }}s+1}{s}$	${{K}_{\Omega }}=0.06$
		${{\tau }_{\Omega }}=0.45$
Motor rot. speed controller	${{K}_{\omega }}.\frac{{{\tau }_{\omega }}s+1}{s}$	Kv = 0.9
		${{\tau }_{\omega }}=0.05$

5.3.3. Toward figure-ground discrimination: hand tracking

The last experiment consisted of placing two moving hands between the robot and the panel. Markers were also placed on one of the hands in order to compare the hand and robot positions: the results of this experiment are presented in figure 19. As shown in the video in the Supplementary Data and in figure 19, the robot faithfully followed the hands when they were moving together in the same direction. By comparing the robot position error seen by the active CurvACE with the ground-truth error, it was established that the robot tracked the moving hands accurately with a maximum estimation error of 129 mm.

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Our visual algorithm selects the greatest contrasts in order to determine its linear position with respect to an arbitrary reference position. Therefore, when the hands were moving above the panel, some stronger contrasts than those of the hands were detected by the visual sensor, which decreased the accuracy of its tracking performances. However, this experiment showed that the robot is still able to track an object when a non-uniform background is moving differently, without having to change the control strategy or the visual algorithm. The robot simply continues to track the objects featuring the greatest contrasts.