# New interpretation of fractional potential fields for
robust path planning

Jean-Baptiste Receveur, Stéphane Victor, Pierre Melchior

► To cite this version:

Jean-Baptiste Receveur, Stéphane Victor, Pierre Melchior. New interpretation of fractional potential fields for robust path planning. Fractional Calculus and Applied Analysis, 2019, 22 (1), pp.113-127. 10.1515/fca-2019-0007. hal-02476698

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
NEW INTERPRETATION OF FRACTIONAL POTENTIAL
FIELDS FOR ROBUST PATH PLANNING
IN “FCAA” JOURNAL

Jean-Baptiste Receveur ¹, Stéphane Victor ², Pierre Melchior ³

Abstract

Trajectory planning for autonomous vehicles is a research topical subject.
In previous studies, optimal intermediate targets have been used in the Po-
tential Fields (PFs). PFs are only a path planning method, or a reactive
obstacle avoidance method and not a trajectory tracking method. In this
article, the PFs are interpreted as an on-line control method to follow an
optimal trajectory. An analysis and methodological approach to design the
attractive potential as a robust controller are proposed, and a new defi-
nition of a fractional repulsive potential to characterize the dangerousness
of obstacles is developed. Simulation results on autonomous vehicles are
given.

***Key Words and Phrases:*** Autonomous vehicles, Trajectory planning, Potential fields, Optimization, Fractional potentials, Robust control

1. Introduction

In previous studies [11], focus was set on finding an optimal trajectory
for an autonomous vehicle, which has been a growing issue in the recent
years. Motion planning or trajectory planning means finding a timed path,
or trajectory in a given environment. The Potential Field (PF) method is
a path planning method first developed by [6] and modified by [4] and [8].
It enables the avoidance of obstacles in a reactive manner, using the slope
of the gradient. It is much appreciated in an unstructured environment,
particularly for rovers [1], [2], and up to four forces to virtually move the
vehicle are considered.

In [11], optimal trajectories were obtained, and can be used as a virtual
optimal target in a PF. In this article the vehicle is called “Ego-Vehicle”
(EGV) to distinguish it from other vehicles. The PF-planned motion is used
on a tactic level (see Figure 1), faster than the strategic level (often handled
by the GPS in a car), and slower than the real-time operational control
level of a vehicle. Distinction is made between the attractive potential,
driving the EGV towards a goal, and the repulsive potential leading the
<figure><img src="image_2.png" /><figcaption>FIGURE 1. Control structure for an autonomous vehicle</figcaption></figure>

EGV away from obstacles. The contributions of this study are to propose an original trajectory control loop, as well as an analysis and a methodological approach to design the attractive potential field as a controller. The expected performance and implementation are described. Also, a new definition for the dangerousness of obstacles is defined with a fractional order so as to variate the potential slope function of the nature and speed of the obstacle. In this article, only two forces are considered but the fractional orders for the obstacle make the method safer. In section 2 the basic principles of the potential fields are introduced. In section 3, the different attractive PF methods are described. Then, in section 4 repulsive potentials are developed. Simulation results are presented in section 5, to finally conclude.

## 2. PF principles, control point of view and expected performances

2.1. Potential field principles. The EGV environment is filled with an artificial PF. It is attracted towards the point of lowest potential (called hereunder “target”), like a marble on a slope, and repelled away from zones of high potential (obstacles). An example of a PF map for one round obstacle is given in Figure 2. The attractive potential is the slope of the potential leading the vehicle towards a point (target), and the repulsive one is the positive slope around obstacles repelling the EGV.

This study follows the one proposed in [3] and in [11]. The EGV has to reach a known final point. In [11], an optimization method was developed, where a genetic algorithm puts intermediate targets in the PF (see Figures 1 and 3). In this article, a control method of an optimal trajectory is developed, whose structure is illustrated in Figure 3. From a control point of view, the attractive potential can be seen as a set-point to be followed, and the repulsive potential field as a disturbance on the control variable.

As the EGV is considered a mass point of mass *M*, the process itself is a double integrator. This control loop is used in section 5 for the tests, first without the repulsive PF, and then with it.
<figure><img src="image_3.png" /><figcaption>FIGURE 2. Example of one simple potential field</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIGURE 3. EGV control loop with attractive and repulsive PF acting as a controller and a perturbation</figcaption></figure>

Attractive and repulsive forces (respectively $\vec{F}^{att}$ and $\vec{F}^{rep}$) are negative gradients of a potential (respectively $U^{att}(\vec{p}, \vec{v})$ or $U^{rep}(\vec{p}, \vec{v})$).

## 2.2. Expected performances.
The set-point trajectory has been found using a genetic algorithm [11] and is optimal according to several criteria, particularly the time criterion. As the objective is to follow this optimal trajectory in a closed-loop configuration, it is necessary that the time criterion value corresponding to the response of the closed-loop system be very close to the optimal criterion value. In this study, a time relative error of less than 1%, between the optimal trajectory and the closed-loop output is sought, namely

$$ \left\| \frac{J_{\text{time opt}} - J_{\text{time cp}}}{J_{\text{time opt}}} \right\| < 1\% \qquad (2.1) $$

where $J_{\text{time opt}}$ is the optimal value of the time criterion, and $J_{\text{time cp}}$ the value of the criterion for the real-time trajectory. If the static error is null,
then the time when the EGV meets the virtual target must be such as:

$$ \frac{t_f - t_{cp}}{t_f} < 1\%, \Rightarrow tr_{5\%} = 0.01 \times 20 = 0.2s, \qquad (2.2) $$

for an example optimal time of 20s. It is exactly the same inequality as before once simplified. $t_f - t_{cp}$ can be seen as the 5% response time. Variations of the EGV mass or other parameters directly imply performance variations, unless the controller is robust such as proposed in section 3.2.

### 3. Attractive potential fields

3.1. Speed dependent attractive potential field. The attractive potential $U^{att}$ was first defined as a bowl-shaped function so as to attract the EGV towards a point. But [4] made $U^{att}$ dependent of the relative position and velocity of the target and EGV such as:

$$ U^{att}(\vec{p}, \vec{v}) = \alpha_p \|\vec{p}_{target} - \vec{p}\| + \alpha_v \|\vec{v}_{target} - \vec{v}\|, \qquad (3.1) $$

where $\vec{p}$ and $\vec{v}$ are, respectively, the EGV position and speed, and $\vec{p}_{target}$ and $\vec{v}_{target}$ those of the target (or the reference). $\alpha_p$ and $\alpha_v$ define weight coefficients, however authors in [8] have interpreted them as proportional derivative controller parameters (see equation (3.5)).

It then becomes:

$$ \vec{F}^{att}(\vec{p}, \vec{v}) = \alpha_p(\vec{p}_{target} - \vec{p}) + \alpha_v(\vec{v}_{target} - \vec{v}), \qquad (3.2) $$

which, denoting $\epsilon_z = z_{target} - z$ with $z$ being either $x$ or $y$, comes down to:

$$ F_z^{att} = Ma_z = \alpha_p \epsilon_z + \alpha_v \frac{d\epsilon_z}{dt}. \qquad (3.3) $$

If $\vec{a}_{target} = \vec{0}$, then in the Laplace domain:

$$ F_z^{att}(s) = (\alpha_p + \alpha_v s)E_z(s), \qquad (3.4) $$

where $E_z(s)$ is the Laplace transform of $\epsilon_z$. For causality reasons, the proportional derivative controller here-above is transformed into a lead phase controller:

$$ F_z^{att}(s) = \frac{(\alpha_p + \alpha_v s)}{1 + \frac{s}{\omega_c}} E_z(s), \qquad (3.5) $$

where $\omega_c$ is the filter cut-off frequency to make the controller causal.

If the mass $M$ vary between 600kg up to 900kg for the considered EGV, a robust controller is needed. Therefore, a fractional order $n_{att}$ has been introduced in [3]. Two more intuitive approaches of the fractional attractive potential are proposed in the next sections.
3.2. Implicit form of a lead-phase controller. A previous study [3] presents a fractional order controller in an explicit form. Here put under, an implicit form, as proposed by [5], and the controller becomes:

$$C(s) = \left( \frac{\alpha_p + \alpha_v s}{1 + \frac{s}{\omega_c}} \right)^{n_{att}} = \alpha_p^{n_{att}} \left( \frac{1 + \frac{\alpha_v}{\alpha_p} s}{1 + \frac{s}{\omega_c}} \right)^{n_{att}} \quad (3.6)$$

a form that reveals a non integer lead-phase controller, or more explicitly a robust CRONE controller of second generation, whose expression is:

$$C(s) = C_0 \left( \frac{1 + \frac{s}{\omega_b}}{1 + \frac{s}{\omega_h}} \right)^{n_{att}} . \quad (3.7)$$

The identification gives $C_0 = \alpha_p^{n_{att}}$, $\omega_b = \frac{\alpha_p}{\alpha_v}$ and $\omega_h = \omega_c$. In terms of settings, this method is easier, as the unit gain frequency $\omega_u$ is directly deduced from $tr_{5\%} \simeq \frac{3}{\omega_u}$.

From $\omega_u$, $\omega_b$ and $\omega_h$ are deduced, to be far enough from the critical point so that the phase margin is almost constant, despite mass variations, in the desired frequency bandwidth: $\omega_b = \frac{\omega_u}{10\sqrt{10}}$ and $\omega_h = 10\sqrt{10}\omega_u$. If the phase margin needed is $M_\varphi$, then:

$$n_{att} = \frac{2}{\pi}(-\pi + M_{\varphi} - \arg(G(j\omega_u))) = \frac{2}{\pi}M_{\varphi}, \quad (3.8)$$

where $\arg(G(j\omega_u))$ is the process argument evaluated at the unit gain frequency $\omega_u$. In this study, the process phase is constant and equal to $-\pi$. Finally, $C_0$ is calculated such as:

$$C_0 = \frac{1}{M_{nom}\omega_u} \left( \frac{1 + \left( \frac{\omega_u}{\omega_h} \right)^2}{1 + \left( \frac{\omega_u}{\omega_b} \right)^2} \right)^{\frac{n_{att}}{2}}, \quad (3.9)$$

where $M_{nom}$ is the nominal mass of the EGV. An integer approximation of the controller can be easily calculated, with a number of pole and zero cells of $N$, see [5] for the details. This controller version provides an elegant methodology to compute the controller parameters. Nevertheless, $x$ and $y$ coordinates of the mass point are still controlled separately, and a physical interpretation is difficult to see from the final version of the controller. Also a physical interpretation is proposed in section 3.3, improving the methodology by combining $x$ and $y$ coordinates.

For example, from (2.2), $\omega_u \simeq 15rd/s$. $M_\varphi$ is chosen equal to 60°, then $n_{att} \approx 0.66$. $N = 6$ is often chosen for the approximation. The nominal mass $M_{nom}$ is equal to 750kg. See the open-loop design, plotted in Figure 6, to check the robustness of the stability degree.
### 3.3. Physical interpretation of the attractive potential fields and perspectives.
A more physical alternative approach to the attractive potential presented previously is proposed hereunder and illustrated in Figure 4. The EGV is linked to the target by a spring-damper system, but free in rotation around the target.

<figure><img src="image_5.png" /><figcaption>FIGURE 4. (A) Local referential (x-axis being oriented from the EGV to the target); (B) Model of the EGV-target link in a spring-damper-rope interpretation</figcaption></figure>

k, c and $d_{max}$ are respectively the spring stiffness, the damping ratio and the length of the rope. $\theta_t$, the angle between the x-axis and the EGV-target axis, is let free. In such manner, the EGV can navigate around the target depending on the forces exerted by the obstacles in the environment (repulsive PF). The corresponding force exerted by the target is then:

$$ \vec{F}^{att} = (kd_{ET} + c \frac{\partial d_{ET}}{\partial t})\vec{u}_t, \qquad (3.10) $$

where $d_{ET}$ is the distance between the EGV and the target, defined as $d_{ET} = \sqrt{(x_{EGV} - x_{tar})^2 + (y_{EGV} - y_{tar})^2}$.

Contrary to the section 3.2 conclusion where x and y were controlled separately, it is no more the case now. Plus, a physical meaning is much more visible here. The damper can also be expressed as a non-integer differentiation, which would lead to a controller such as explained in section 3.1. Furthermore, in an implicit fractional form, such a controller can also be related to the study of the CRONE suspension as initially introduced in [5].

The rope was added to model the maximum acceptable distance $d_{max}$ between the EGV and the target. This rope is necessary for two reasons:
* the further the target, the lesser the optimality;

* the saturation on the distance $d_{ET}$ makes the maximum force known.

This distance saturation provides two aspects:

* the further the EGV, the greater the acceleration to stay within the saturation;

* if the EGV is too far, the target will stop and wait for the EGV.

The first approach rises the issue of the force growing too much compared to the repulsive force, which might lead to collisions with obstacles (which could be solved by fixing $d_{max} = \max$ obstacle size), and also the issue of saturations in acceleration or speed which are limited. The second approach implies an attentive target, which stops and waits for the EGV to come closer if too far. For instance, in crossroads, an attentive target would wait for the EGV to come closer if separated by the crossing of a group of pedestrians or for the traffic lights to go green.

The virtual parameters $k$ and $c$ can be easily calculated using the method hereunder. Newton's second law applied to a mass point tied to a target with a spring and damper, presented in one dimension for simplicity sake, gives the following equation

$$M \frac{d^2 z}{dt^2} = kz_{tar} - kz + c \frac{dz_{tar}}{dt} - c \frac{dz}{dt} \quad (3.11)$$

with $z = x$ or $y$ or else, by applying the Laplace transform with null initial conditions, gives the following transfer function:

$$H(s) = \frac{Z(s)}{Z_{tar}(s)} = \frac{cs + k}{Ms^2 + cs + k} = \frac{1 + \frac{c}{k}s}{\frac{M}{k}s^2 + \frac{c}{k}s + 1} \quad (3.12)$$

which is a closed-loop second order transfer function, as the force depends on the measured error value. Its natural frequency and damping ratio are:

$$\omega_0 = \sqrt{\frac{k}{M}} \quad \text{and} \quad \xi = \frac{\omega_0 c}{2k} = \frac{c}{2} \sqrt{\frac{1}{Mk}} \quad (3.13)$$

According to (2.2), $k$ and $c$ can be determined such as:

$$k = M\omega_0^2 = \frac{M}{\tau^2} = \frac{25M}{\text{tr}_{5\%}^2} = \frac{2500M}{4} \quad (3.14)$$

As no overshoot is allowed for this system, the damping ratio is then fixed to $\xi = 1$, which means:

$$c = 2\sqrt{Mk}\xi = 2\sqrt{\frac{2500M^2}{4}} = 50M \quad (3.15)$$

Considering that the EGV-target distance $d_{ET}$ and its corresponding speed $v_{ET}$ are limited to
$$ \begin{cases} d_{ET} \in [0, d_{max}] \\ v_{ET} \in [-2v_{lim}, +2v_{lim}] \end{cases} \qquad (3.16) $$

The maximum attractive force $F_{max}^{att}$ can therefore be determined as:

$$ F_{max}^{att} = kd_{max} + 2cv_{lim}. \qquad (3.17) $$

$F_{max}^{att}$ is around $10^5$ for the values considered until now. Also, the maximum repulsive force $F_{max}^{rep}$ can be calculated with a weight factor $G_{rep} = \frac{F_{max}^{att}}{F_{max}^{rep}}$ to level the forces.

With this methodology, the parameters can be fixed such as presented in Table 1.

<table><caption>TABLE 1. Parameters table</caption><thead><tr><th>Parameter</th><th>Consequence on the system</th></tr></thead><tbody><tr><td>k: spring stiffness</td><td>tr<sub>5%</sub>: response time to reach 1% of the final value</td></tr><tr><td>c: damper viscous damping</td><td>ξ: damping ratio</td></tr><tr><td>G<sub>rep</sub>: repulsive potential gain</td><td>setting the distance to obstacles</td></tr><tr><td>d<sub>max</sub>: saturation between the EGV and the target</td><td>optimality of the trajectory</td></tr></tbody></table>

This method rises an interesting issue: in section 3.1 only the *x* and *y* position measures were used, now, in this new interpretation, the speed measure is also used to calculate $\frac{\partial d_{ET}}{\partial t}$. In a car, the speed norm is measured and not its *x* or *y* component, which means this interpretation is closer to the real application, which was studied in [9].

## 4. Repulsive Potential Fields

The repulsive potential field describes the obstacle environment around the EGV. The definition given hereunder in section 4.1 is used to shape obstacles function of the danger they represent.

4.1. **Fractional potential field.** In [10], a Weyl definition of the fractional differential was used to define a fractional repulsive potential field such as:

* if $r \in [r_{min}, r_{max}]$: $U_k^{rep}(r) = \frac{r^{n_k-2}-r_{max}^{n_k-2}}{r_{min}^{n_k-2}-r_{max}^{n_k-2}}$;

* if $r \notin [r_{min}, r_{max}]$: $U_k^{rep}(r) = 0$
<figure><img src="image_6.png" /><figcaption>FIGURE 5. Poty's fractional repulsive potential field shape</figcaption></figure>

where $r_{min}$ and $r_{max}$ are two chosen values, $r_{min}$ often being the obstacle radius, and $r_{max}$ the maximum obstacle influence radius. $n_k \in [0.1, 2[\cup]2, +\infty[$ is called order of the obstacle k, depending on the obstacle speed, nature, and vulnerability. This definition of the potential field is called Weyl's definition [7], and leads to potential values between 0 and 1.

A new type of repulsive potential was defined by [4], but even though this method takes into account the relative speed between obstacles and the EGV, it has some major drawbacks such as a very complex setting of the distance of influence of an obstacle, or no way to differentiate types of obstacles. These issues are not present using the PF presented in this article.

4.2. **Order definition for the obstacles.** As was said in section 4.1, each obstacle has a different fractional differentiation order. In Figure 5 the shape of obstacles is presented, depending on $n_k$.

Knowing that the force exerted by the obstacle is directly linked to the slope of the potential field, and that the repulsive force should always increase when getting closer to the obstacle, the obstacle order should be less than 3, according to Figure 5. It then gives: $n_k \in [0.1; 3]$.

If the order $n_k$ of a given obstacle k is representative of the dangerousness of the obstacle, it should then depend on three elements:

$$n_k = f(Vuln_k, \Delta v_k, v_k) \quad (4.1)$$

where $Vuln_k$ is called the vulnerability of the obstacle, $\Delta v_k$ its speed variation potential, and $v_k$ its speed. Function f, to be determined, is increasing
with all three parameters, and should have a maximum of 3 and a minimum of 0.1. The minimum is arbitrary, so as not to be zero. Vulnerability can be defined as:

$$Vuln_k = Vuln_k^{EGV} + Vuln_k^{obst} \quad (4.2)$$

where $Vuln_k$ is chosen to be a number between 0 and 5 depending then on $Vuln_k^{EGV}$, $Vuln_k^{obst} \in [0; 2.5]$. Those are, respectively, the vulnerability of the EGV considering the obstacle, and the vulnerability of the obstacle considering the EGV, and where arbitrarily fixed. Both are considered of equal importance in this study, which explains why their influence on the total vulnerability $Vuln_k$ is the same. This relation could be explored further in later studies. For a road scenario, several types of obstacles are gathered in Table 2. From this table, where $v_k^{min}$ and $v_k^{max}$ are respectively the maximum and minimum speed of the obstacle, the potential speed variation is defined as $\Delta v_k = \max(|v_k - v_{min}|, |v_k - v_{max}|)$.

<table><caption>TABLE 2. Type of obstacles ($Vuln_k = Vuln_k^{EGV} + Vuln_k^{alter}$)</caption><thead><tr><th>Type of obstacle</th><th>[v<sub>k</sub><sup>min</sup>, v<sub>k</sub><sup>max</sup>] (km/h)</th><th>Vuln<sub>k</sub></th></tr></thead><tbody><tr><td>Other motorized vehicle</td><td>[0; v<sub>lim</sub>]</td><td>4 = 2 + 2</td></tr><tr><td>Bike, Rollerblades, ...</td><td>[0; 30]</td><td>3 = 0.5 + 2.5</td></tr><tr><td>Pedestrian</td><td>[0; 10]</td><td>2.5 = 0 + 2.5</td></tr><tr><td>Road side, sidewalk</td><td>0</td><td>0.5 = 0.5 + 0</td></tr><tr><td>Road sign, post, wall</td><td>0</td><td>2 = 2 + 0</td></tr></tbody></table>

Then let's shape $f$ as:

$$f(Vuln_k, \Delta v_k, v_k) = Vuln_k (A(v_k + \Delta v_k) + B) \quad (4.3)$$

where $A$ and $B$ are two coefficients to determine using $f(0.5, 0, 0) = 0.1 \Rightarrow B = 0.2$ and $f(5, v_{lim}, v_{lim}) = 3 \Rightarrow A = \frac{1}{5v_{lim}}$, which means:

$$f(Vuln_k, \Delta v_k, v_k) = Vuln_k \left( \frac{1}{5v_{lim}}(v_k + \Delta v_k) + 0.2 \right). \quad (4.4)$$

This formula enables the EGV to calculate an order for every obstacle appearing in the environment, depending on the speed they currently have and their type. Further amelioration could consider the direction of the obstacle, i.e. the relative speed between the EGV and the obstacle instead of the absolute speed.
<figure><img src="image_7.png" /><figcaption>FIGURE 6. Open-loop Nichols charts for different masses for controllers from [4] and [10] compared to the one in (3.6)</figcaption></figure>

5. Simulation results

5.1. **Open-Loop without repulsive potential field.** Figure 6 presents the Nichols open-loops for three different controllers and three different masses. The first controller is the one described in section 3.1, the second one described in section 3.2 but with an integer order $n_{att} = 1$, thus representing a simple lead-phase controller, and the last one is an integer approximation of the CRONE controller described in section 3.2. It can be seen on this figure that the lead-phase controller and the CRONE controller bring almost the same stability degree with a phase margin around 60°. The CRONE controller is a fractional controller that provides a better stability degree robustness as the stability-precision dilemma is put in wrong as the damping ratio becomes independent of the mass variations. The phase margin is therefore kept for a CRONE controller contrary to a classic lead-phase controller.

The responses to a position ramp (or speed step responses) are given in Figure 7 to compare the three different controllers. The speed curves are plotted to better visualize the stability degree versus mass variations: the CRONE controller presents exactly the same overshoot despite mass variations which well illustrates the stability degree robustness.

The CRONE controller is held as it is the best controller among the three ones. The repulsive potential is added in the next section, to complete the tests.
<figure><img src="image_8.png" /><figcaption>FIGURE 7. Speed time response to a position ramp</figcaption></figure>

## 5.2. Scenario test with attractive and repulsive potential fields.

The scenario presented in Figure 8 is a classic 90° turn scenario on a one-way road. All obstacles are fixed here, and their type and resulting orders are given in Table 3. The resulting trajectory is shown in Figure 8, along with the different curves for the three main attractive methods presented: the lead-phase version, the CRONE version, and the Spring-Damper version. For these tests, the nominal mass of 750kg was used.

<table><caption>TABLE 3. Obtained orders for the different obstacles in the test scenario</caption><thead><tr><td>Obstacle Position(s)</td><td>Obstacle Type</td><td>Calculated order</td></tr></thead><tbody><tr><td>[29; 78], [94; 54], [60; 20]</td><td>Side walks</td><td>0.1</td></tr><tr><td>[45; 60]</td><td>Newsstand</td><td>0.4</td></tr><tr><td>[30; 50]</td><td>Posts</td><td>0.4</td></tr><tr><td>[57; 80]</td><td>Pedestrian</td><td>0.7</td></tr></tbody></table>

As can be seen in Figure 8, the three methods apparently lead to similar trajectories. The difference lies in the speed and acceleration curves, and consequently on the comfort for the passengers. When the loaded mass varies, which happens for a car carrying passengers or running low on fuel, the CRONE controller will ensure the same stability degree of the system. In this example, the spring stiffness and damper viscous damping
<figure><img src="image_9.png" /><figcaption>FIGURE 8. (a) Test scenario with obstacles, optimal trajectory (−) and EGV trajectory (−) with distance *d*, speed *v* and acceleration *a* curves, (b) for the lead-phase controller; (c) for the CRONE controller; (d) for the spring-damper system</figcaption></figure>

are not well calibrated, as the attraction gives rise to high oscillations on
the acceleration.
## 6. Conclusion

In this paper, the potential fields method, used in previous studies, was recalled and proposed in a closed-loop control configuration. A new definition has enabled to improve the methodology and consequently its conception. Physical interpretation has been given for the different coefficients of the controller leading to the definition of the controller damping ratio and unit gain frequency. Two new methods to calibrate the attractive potential have been presented, one using an implicit fractional lead-phase controller, and another one using a spring-damper system. For the repulsive potential, the methods from two articles have been recalled. Moreover, an obstacle order determination function has been proposed here, function of the obstacles vulnerability and their speed. Finally, some simulation tests have been pursued, illustrating the interest of the CRONE controller in terms of robustness. An example of trajectory has been shown, along with the different curves representing the position, speed and acceleration of the EGV.

For future works, the repulsive potential field method needs to be improved to reduce the EGV sensitivity to acceleration oscillations. Moreover, the calibration of the spring-damper method needs to be explored further, the two dimensions leading to more complex expressions of the damper viscous damping *c* and the spring stiffness *k*. The function calculating orders of obstacles could be modified so as to give more importance to the obstacle speed. This study could also be extended by considering the real dynamics of the vehicle instead of those of a mass-point limited by saturations.

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<footnote>1,2,3 Université de Bordeaux, IMS - UMR 5218 CNRS, Bordeaux INP, ENSEIRB-MATMECA, 351 Cours de la Libération, 33405 Talence CEDEX, France (e-mail: firstname.lastname@ims-bordeaux.fr).</footnote>