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Path planning with fractional potential fields for autonomous vehicles
Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World The International International Federation of Congress Automatic Control Control Toulouse, France,Federation July 9-14, 2017 The of Automatic Available online at www.sciencedirect.com The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 14533–14538
Path planning with fractional potential Path planning with potential Pathfields planning with fractional fractional potential for autonomous vehicles fields for autonomous vehicles fields for autonomous∗ vehicles ∗,∗∗∗ ∗ Julien Julien Julien Julien
Moreau ∗,∗∗∗ Pierre Melchior ∗ St´ ephane Victor ∗ ∗,∗∗∗ Pierre ∗ St´ ∗∗ Melchior ∗∗ Victor Moreau e phane Moreau Pierre Melchior e Victor ∗∗ Fran¸ c ois Aioun Franck Guillemard ∗,∗∗∗ ∗ St´ Moreau Pierre Melchior St´ ephane phane ∗∗ ∗∗ ∗∗ Franck ∗∗ Victor Fran¸ c ois Aioun Guillemard Fran¸ c Fran¸ cois ois Aioun Aioun ∗∗ Franck Franck Guillemard Guillemard ∗∗ ∗ IMS – UMR 5218 CNRS, Universit´ e de Bordeaux/Bordeaux INP ∗ ∗ IMS – UMR 5218 CNRS, Universit´ de Bordeaux/Bordeaux Bordeaux/Bordeaux INP – UMR Universit´ de cours 5218 de la CNRS, Lib´eration, 33405eee Talence cedex – France INP ∗ IMS IMS351 – UMR 5218 CNRS, Universit´ de Bordeaux/Bordeaux INP 351 cours de la Lib´ eeration, 33405 Talence cedex – France 351 cours de la Lib´ ration, 33405 Talence cedex – France Email: fi[email protected] 351 cours de la Lib´ e ration, 33405 Talence cedex – France ∗∗ Email: fi[email protected] Email: PSA Groupe Email: fi[email protected] fi[email protected] ∗∗ ∗∗ PSA Groupe PSA Groupe Chemin de Gisy, 78140 elizy-Villacoublay cedex – France ∗∗ V´ PSA Groupe Chemin de de Gisy, Gisy, 78140 V´ lizy-Villacoublay cedex – – France Chemin 78140 V´ eeelizy-Villacoublay cedex Email: [email protected] Chemin de Gisy, 78140 V´ lizy-Villacoublay cedex – France France ∗∗∗ Email: [email protected] Email: [email protected] OpenLab PSA Groupe-IMS Bordeaux Email: [email protected] ∗∗∗ ∗∗∗ OpenLab PSA Groupe-IMS Bordeaux ∗∗∗ OpenLab PSA Groupe-IMS Bordeaux OpenLab PSA Groupe-IMS Bordeaux Abstract: Path planning is an essential stage for mobile robot control. It is more newsworthy Abstract: planning is context an essential stage for mobile robot control. It is more newsworthy Abstract: Path planning an stage robot It more newsworthy than ever inPath the automotive and especially for autonomous vehicle. Also, path planning Abstract: planning is is context an essential essential stage for for mobile mobile robot control. control. It is is more newsworthy than ever need inPath thetoautomotive automotive and especially especially for autonomous autonomous vehicle. Also, path planning than ever in the context and for vehicle. Also, path planning methods be adaptive regarding life situations, traffic and obstacle crossing. In this than ever in the automotive context and especially for autonomous vehicle. Also, path planning methods need to be adaptive regarding life situations, traffic and obstacle crossing. In this methods need to be adaptive regarding life situations, traffic and obstacle crossing. In paper, potential field methods are proposed to cope with these constraints and autonomous methods need tofield be adaptive regarding life to situations, traffic and obstacle and crossing. In this this paper, potential methods are proposed cope with these constraints autonomous paper, field methods proposed to cope with these constraints and autonomous vehiclespotential are considered equippedare with all necessary sensors for obstacle detection. In this way, paper, potential field methods are proposed to cope with these constraints and autonomous vehicles are considered equipped with necessary sensors for obstacle detection. In this way, vehicles are considered equipped with allfractional necessary sensors obstacle detection. In way, Ge&Cui’s attractive potential field andall attractive potential field have been adapted vehicles areattractive considered equipped with necessaryattractive sensors for for obstaclefield detection. In this this way, Ge&Cui’s potential field andall fractional potential havebetter been adapted Ge&Cui’s attractive potential field and fractional attractive potential field have been adapted to the context of autonomous vehicles. In this way, this latter method ensures stability Ge&Cui’s attractive potential field and fractional attractive potential field have been adapted to the context of autonomous vehicles. In this way, this latter method ensures better stability to the of vehicles. this degree robustness with controlled vehicleIn acceleration. to the context context of autonomous autonomous vehicles. Inacceleration. this way, way, this this latter latter method method ensures ensures better better stability stability degree robustness with controlled vehicle degree robustness with controlled vehicle acceleration. degree robustness with controlled vehicle acceleration. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Path planning, automotive, autonomous vehicle, potential field, fractional potential Keywords: Path Keywords: Path planning, planning, automotive, automotive, autonomous autonomous vehicle, vehicle, potential potential field, field, fractional fractional potential potential field Keywords: Path planning, automotive, autonomous vehicle, potential field, fractional potential field field field 1. INTRODUCTION The main advantage is that the vehicle is able to navigate 1. INTRODUCTION The main main advantage is that that the vehicle is is able to navigate navigate 1. INTRODUCTION The advantage is vehicle to with no “a priori” on the the environment. However, this 1. INTRODUCTION The advantage is that vehicle is able able to navigate with main no could “a priori” on the the environment. However, this with no “a priori” on the environment. However, this method lead to behaviors unreadable by the driver. with no “a priori” on the environment. However, this Autonomous driving is going to be the heart of public method method could could lead lead to to behaviors behaviors unreadable unreadable by by the the driver. driver. Autonomous driving is going to be the heart of public method could lead to behaviors unreadable by the driver. Autonomous driving is going going to be be the thevehicles heart of of public transports in driving the future. Autonomous have the Potentials field methods are common path planning meAutonomous is to heart public Potentials field methods methods are(Inarn common path(2012)). planning metransports the Autonomous vehicles have the Potentials field common path planning mein mobile roboticsare et al. These transportstoin inimprove the future. future. Autonomous vehiclestravel havetime the thods potential passenger’s safety, reduce Potentials field methods are(Inarn common path(2012)). planning metransports in the future. Autonomous vehicles have the thods in in have mobile robotics et Unmanned al. These potential to improve passenger’s safety, reduce travel time thods mobile robotics (Inarn et al. (2012)). These methods been already used for Aircraft potential to improve passenger’s safety, reduce travel time and improve the comfort of the passengers. As a consethods in mobile robotics (Inarn et al. (2012)). These potential to improve passenger’s safety, reduce travel time methods(UAV) have been been already used for for Unmanned Unmanned Aircraft Aircraft and the the As consemethods have already used in Zhan et al. (2014), (2014) and improve improve the comfort comfort of ofinvest the passengers. passengers. As a consequence car manufacturers in the conception and System methods have been already used forAndrew-Fields Unmanned Aircraft and improve the comfort of the passengers. As aa conseSystem (UAV) in Zhan et al. (2014), Andrew-Fields (2014) quence car manufacturers invest in the conception and System (UAV) in Zhan et al. (2014), Andrew-Fields (2014) and Liu et al. (2016). Moreover, they have also been quence car manufacturers invest in the conception and the equipment of their vehicles with advance systems for System (UAV) Zhan etMoreover, al. (2014), Andrew-Fields quence car manufacturers invest in advance the conception and and Liu et al. al.in(2016). (2016). they have have also(2014) been the equipment of their vehicles with systems for and Liu et Moreover, they also been used in automotive especially for collision avoidance in the equipment of their vehicles with advance systems for assistance. In urban areas, intersections are considered as and Liu et al. (2016). Moreover, they have also been the equipment of their vehicles with advance systems for used in automotive especially for collision avoidance in assistance. In urban areas, intersections are considered as used in automotive especially for collision avoidance in Shibata et al. (2014) however no extension for a complete assistance. In urban areas, intersections are considered as bottlenecking; more than 44 % of all the accidents reported in et automotive especially for collisionfor avoidance in assistance. In urban areas, intersections are considered as used Shibata al. (2014) however no extension a complete bottlenecking; more than 44 % of all the accidents reported Shibata et al. (2014) however no extension for a complete navigation has been developed. bottlenecking; more than 44 % of all the accidents reported in the United States arise in intersection zones and are Shibata et al. however no extension for a complete bottlenecking; more than 44 % of all the accidents reported navigation has(2014) been in arise intersection zones are has been developed. developed. in the the United United States arise in inand intersection zones1and and are navigation responsible for States 8,500 deaths approximately million navigation has been developed. in the United States arise in intersection zones and are In order to use potential field for autonomous vehicle responsible for 8,500 deaths and approximately 11 million responsible foryear. 8,500 deathsthe andtraffic approximately million wounds every Besides, delays generated by In order to use potential field for autonomous vehicle responsible for 8,500 deaths and approximately 1 million In order to use potential field vehicle navigation, a strategic schematic has autonomous been developed (see wounds every year. Besides, the traffic delays generated by order toa use potential field for for autonomous vehicle wounds every everylead year.toBesides, Besides, the losses traffic of delays generated by In intersections enormous resources. Therenavigation, strategic schematic has been developed developed (see wounds year. the traffic delays generated by navigation, strategic schematic been (see Fig. 1). Thisaa schematic has three has stages: intersections lead to enormous losses of resources. Therenavigation, strategic schematic has been developed (see intersections lead vehicles to enormous enormous losses of resources. resources. There- Fig. 1). This schematic has three stages: fore, autonomous answer to these issues. Thereintersections lead to losses of Fig. 1). This schematic has three stages: the strategic stage: data is collected and processed from fore, autonomous vehicles answer to these issues. Fig. 1). This schematic has three stages: fore, autonomous vehicles answer to these issues. the strategic strategic stage: data istocollected collected and processed from fore, answer these issues. --- the stage: processed from sensors in data orderis create anand immediate virtual The autonomous urban area vehicles represents the to sensitive case for au- perception the strategic stage: data istocollected and processed from perception sensors in order create an immediate virtual The urban area represents the sensitive case for auperception in order to create an immediate virtual of the sensors autonomous vehicle with obstacle informations The urban urbanvehicles. area represents represents the kind sensitive case for for auautonomous Indeed, this of environment is map perception sensors in order to create an immediate virtual The area the sensitive case map of of the the autonomous autonomous vehicle with obstacle obstacle informations tonomous vehicles. kind of environment is map vehicle with . . . ) and targets; tonomousautonomous vehicles. Indeed, Indeed, this kind of adaptive environment is (velocity, variable: vehiclesthis need to be regarmap of theposition, autonomous vehicle with obstacle informations informations tonomous vehicles. Indeed, this kind of environment is (velocity, position, . .. .. from ) and and targets; variable: autonomous vehicles need to be adaptive regar(velocity, position, . ) targets; the tactical stage: the previous informations, a variable: autonomous vehicles need to be adaptive regarding life situations, traffic and obstacle crossing. In this (velocity, position, . . . ) and targets; variable: autonomous vehicles need to be adaptive regarthe resulting tactical of stage: from the the previous previous informations, ding situations, traffic obstacle In this --- the tactical stage: from informations, an attractive repulsive potential fieldaa ding life life situations, vehicles traffic and and obstacle crossing. crossing. In with this force paper, autonomous are considered equipped the resulting tactical of stage: from theand previous informations, a ding life situations, traffic and obstacle crossing. In this force an attractive and repulsive field paper, autonomous vehicles are considered equipped with force resulting an attractive and repulsive potential field is of generated and according to apotential simple model paper, autonomous vehicles are considered considered equipped with with combination all necessary sensorsvehicles for obstacles detection.equipped force resulting of an attractive and repulsive potential field paper, autonomous are combination is generated generated and ideal according to aa simple simple model all combination is and according to model (punctual mass model), the position, velocity and all necessary necessary sensors sensors for for obstacles obstacles detection. detection. combination is generated and ideal according to a simple model all necessary for obstacles These recent sensors years, path planningdetection. for autonomous vehi- acceleration (punctual mass mass model), the the position, velocity and (punctual model), ideal position, velocity and are generated by simulating the motion of a These recent planning for vehi(punctual mass model), the position, velocity and Thesehas recent years, path planning for autonomous autonomous vehi- charged cles been years, widelypath studied in literature. In Makarem acceleration are generated generated byideal simulating theforce; motion of of a These recent years, path planning for autonomous vehiacceleration are by simulating the motion particle subjected to this generated cles has widely studied literature. Makarem acceleration are generated by this simulating theforce; motion of aa cles Gillet has been been widely studied in infunction literature. In Makarem -charged and (2011), a navigation for In autonomous charged particle subjected to generated cles has been widely studied in literature. In Makarem particle to this the operational stage: the position, force; velocity and and aa navigation function for charged particle subjected subjected to ideal this generated generated force; and Gillet Gillet (2011), navigation function for autonomous autonomous vehicles has(2011), been developed but no pedestrians have been acceleration the operational operational stage: the ideal position, velocity and and Gillet (2011), a navigation function for autonomous --- the stage: the ideal and from the previous stageposition, are used velocity as references vehicles has been developed but no pedestrians have been the operational stage: the ideal position, velocity and vehicles has hasinbeen been developed but no pedestrians pedestrians have been as considered thisdeveloped method. but Another interesting method acceleration from the the previous stage are areusing usedthe as references references vehicles no have been acceleration from previous stage used as a classic control loop, the controller deviation considered in this method. Another interesting method acceleration from the previous stage are used as references considered in this this method. Another interesting method called “tentacles method” hasAnother been developed in A.method Chebly between as aa classic classic control loop, loop, the controller using to thegenerate deviationa considered in method. interesting as control controller using the deviation references and the measures in order called “tentacles method” has in as a classic control loop, the controller using to thegenerate deviationa called “tentacles method” has been been developed in A. A. Chebly and Charara (2015). The idea is todeveloped deploy a set of Chebly virtual steering between references and measures in order called “tentacles method” has been developed in A. Chebly between references and measures in order to generate aa of wheel torque T w and a wheel w. and Charara (2015). The idea is to deploy aa set of virtual between angle references andβ measures in order to generate and Charara (2015). The idea is to deploy set of virtual tentacles with different curves in front of the autonomous steering angle of wheel β and a wheel torque T . w w and Charara (2015). The idea is to deploy a set of virtual steering angle of wheel β and a wheel torque T . tentacles with different curves of autonomous angle only of wheel βw a wheel torque Tw w and stage w .Indeed, In this paper, the tactical is presented. tentacleseach withtentacle different curves in in front front of the the vehicle autonomous vehicle, representing a possible path. steering tentacles with different curves in front of the autonomous In this paper, only the tactical stage is presented. Indeed, vehicle, each tentacle representing a possible vehicle path. In this paper, only the tactical stage is presented. Indeed, strategic is atactical data processing problem whereas vehicle, each tentacle representing possible vehicle path. the Each path is tentacle then evaluated according to a set of criteria. In paper,stage only the stage is presented. Indeed, vehicle, each representing aa possible vehicle path. thethis strategic stage is aa data data processing processing problem whereas whereas Each path is then evaluated according to aa set of criteria. the strategic stage is problem Each path is then evaluated according to set of criteria. Each path is then evaluated according to a set of criteria. the strategic stage is a data processing problem whereas
Copyright © 2017, 2017 IFAC 15098 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 15098 Copyright ©under 2017 responsibility IFAC 15098 Peer review of International Federation of Automatic Control. Copyright © 2017 IFAC 15098 10.1016/j.ifacol.2017.08.2076
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 14534 Julien Moreau et al. / IFAC PapersOnLine 50-1 (2017) 14533–14538
Vehicle measures STRATEGIC STAGE
Obstacles
Frep
Repulsive force +
Simple model +
Target
Attractive force
βw
+
Navigation
Vehicle
Controller
Ref.
-
Fatt
Tw
Vehicle measures
OPERATIONAL STAGE
TACTICAL STAGE
Vehicle measures
Fig. 1. Plan of regulation at three levels: strategic (blue), tactical (green) and operational (orange)) the operational is a control problem. The fractional differentiation has been already used in automotive (Morand et al. (2016)). This present work is based on the extension of the work developed in Melchior et al. (2001),Melchior et al. (2003) and Metoui et al. (2009) on fractional potential field. Firstly, the fractional potential field method is presented in section 2. After that, section 3 shows extensions on fractional potential field method to autonomous vehicle. A set of simulation results is presented in section 4, to finally conclude in section 5. 2. FRACTIONAL POTENTIAL FIELD In motion control, a charged particle is subject to: -an attractive potential field tied to the target -and a repulsive one tied to obstacles for avoidance. Compared to a classic potential field, a fractional definition of the potential field allows to create a less constrained potential field curve.
Fig. 2. Fractional repulsive potential field created around an obstacle 1
2.1 Fractional repulsive potential field
Fractional potential field
n−2 r n−2 −rmax n−2 n−2 rmin −rmax
f or rmin < r ≤ rmax ,
where r is the distance of the particle to the considered obstacle, rmin and rmax the boundaries of the repulsive potential field and n the fractional order. The potential field can be shaped by using the parameter n (see Fig. 3). For example, when a charged particle is on a direct course to an obstacle, the higher the parameter n, the farther from the obstacle the move of the particle. Thus, the parameter n can be interpreted as a danger parameter. The potential field is applied to each obstacle and free road is obtained by the truncation of the potential map (Melchior et al. (2003)). The resulting force applied to the particle is given by: → − → − F = −grad( U ).
n=4
0.8
Let us consider the fractional repulsive potential field U (see Fig. 2) defined by : U (r) =
n=5
0.9
n=3
0.7
n=2
0.6 0.5
n=1
0.4 0.3 0.2 0.1 0
0
5
10
15 Distance (m)
20
25
30
Fig. 3. Influence of parameter n on the potential field 2.2 Fractional attractive potential field The resulting force expressed by Ge and Cui (2002) is given by:
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− → → → → → F = αp (− x target − − x ) + αv ( − v target − − v)
(1)
Proceedings of the 20th IFAC World Congress Julien Moreau et al. / IFAC PapersOnLine 50-1 (2017) 14533–14538 Toulouse, France, July 9-14, 2017
where αp and αv are weighting coefficients, the particle → → position − x and the target position − x target , the particle → − → − velocity v and the target velocity v target . For the fractional attractive potential field, an extension of equation (1) gives: − → dn → → → → F = αp (− x target − − x ) + αv n ( − v target − − v ), dt
(2)
where the fractional derivative of the position is considered instead of the velocity.
1 X(s) = F (s) M s2
3.1 Limits of classic Ge&Cui’s attractive potential field in autonomous vehicle context
Γ(s) =
According to equations (2) and (3), the fractional attractive potential field can be interpreted as a position controller. This interpretation leads to the elementary control loop schematic (see Fig. 4). xtarget+ F x ε 1 α + α sn v
In this section, the drawbacks of classic attractive potential field when applied to the autonomous vehicle are presented. A solution using the concept of way-point is proposed. Finally, an upgrade of this solution using the fractional attractive potential field is explained.
(3)
where s is the Laplace variable, X(s) the Laplace transform of the vehicle position and F(s) the Laplace transform of the resulting force (repulsive or attractive).
p
3. APPLICATION TO THE AUTONOMOUS VEHICLE
According to Fig. 4, the Laplace transform of the vehicle acceleration can be expressed (with n = 1) according to:
Let us consider a punctual mass model: G(s) =
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M s2
− x
αv 3 αp s M 2 αp s
αv αp s
+
+1
Xtarget (s).
(8)
Consequently, two main problems may occur with this definition. First, this transfer function is not proper (numerator degree higher than denominator one) leading to implementation problems. The second problem concerns the target position: the farther the target, the greater the attractive force magnitude and the greater the vehicle acceleration magnitude. Thus, if the vehicle acceleration is too high, it could be uncomfortable for passengers and in a worst case, the system limits may be reached. 3.2 Classic attractive potential field with limitation of vehicle acceleration
Fig. 4. Elementary control loop schematic with Ge&Cui’s attractive potential field when n = 1 and fractional attractive potential field in other cases The open-loop expression associated to this schematic is expressed as β(jω) =
αv (jω)n +αp M (jω)2
=
jω n 1+( ω ) c M αp
(jω)2
First of all, the causality problem can be fixed by using the vehicle velocity. In this way, vehicle velocity is used instead of the derivative of the vehicle position. However, the target velocity needs to be added as an input signal as illustrated on Fig. 5.
. In high
vtarget
+
−
frequencies, it then becomes: β(jω) ≈
+ s2
εv
αv αp , f or ω >> ωc = M (jω)2−n αv
1 n
αv
(4) xtarget+ ε p
or else,
β(jω) ≈
ωug jω
n
α v
M
1 n
.
αp
+
F
+
−
1 M
γ
1 s
1 s
x
x
(5)
where n� = 2 − n and the unit gain frequency: ωug =
v
Fig. 5. Control loop schematic using attractive potential field and velocity (6)
In this way, the vehicle acceleration becomes:
Such an open-loop is expressed by a fractional order n� and the damping ratio is then given by (Oustaloup et al. (2014)): π ξ = −cos( � ). (7) n In this case, the damping ratio only depends on the parameter n� and it is independent of the model parameters. In this way, robustness of stability degree relative to the vehicle mass variation is ensured.
Γ(s) =
αv 2 αp s αv M 2 α p s + αp s
+1
Vtarget (s)
+
M 2 αp s
s2 Xtarget (s) v +α αp s + 1
(9)
In order to limit the acceleration to a γmax , equation (9) becomes with s = jω:
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Proceedings of the 20th IFAC World Congress 14536 Julien Moreau et al. / IFAC PapersOnLine 50-1 (2017) 14533–14538 Toulouse, France, July 9-14, 2017
Considering a stationary target, Vtarget = 0 and equation αv 2 (6), the expression of the attractive potential field coefαp (jω) max |Γ(jω)| ≤ max M |Vtarget (jω)| ficient is expressed as: v α (jω)2 + α (jω) + 1 αp p 2 (jω) γmax M n� + max M |Xtarget (jω)| (10) αv = M ωug . (14) and αp ≤ v α (jω)2 + α (jω) + 1 |X α target | p
p
with a stationary target, Vtarget = 0 and thus, the attractive potential field coefficients can be expressed as:
4. SIMULATION
4.1 Acceleration constraint simulation γmax M αv = 2ξ αp M and αp ≤ . |Xtarget |
(11)
These equations show that the target needs to be stationary and not too far from the vehicle in order to limit the acceleration. In the light of these constraints, a set of targets may be added throughout the vehicle path. These targets are called waypoints. The vehicle has to only consider the closest waypoint as attractive point, after reaching it, the following waypoint is used as the following attractive point, and until the end. 3.3 Fractional attractive potential field with limitation of vehicle acceleration
The autonomous vehicle is attracted by a single waypoint according to the attractive potential field theory. The attractive potential field coefficients are chosen in accordance with expressions (11) where ξ = 1 (without damping), M = 750kg (light vehicle) and the waypoint is placed 100m farther than the vehicle. An acceleration limit is fixed to 0.5g where g is the gravity acceleration. The Bode Γ(s) diagram of the sensitivity function Xtarget (s) (blue curve) and the acceleration constraint (green curve) are plotted on Fig 7). As expected, this choice of coefficient respects the acceleration constraint in the useful frequency band. Sensitivity function Bode gain diagram −20
In the same way, a similar schematic can be established in the fractional case (see Fig 6). +
−
εv
v
−60 Magnitude (dB)
vtarget
−40
αv s1−n
xtarget+ ε p
−80
−100
+
αp
+
F
1 M
−
γ
1 s
1 s
x −120
Sensitivity function Acceleration constraint
x
−140 −3 10
Fig. 6. Fractional attractive potential field with causality In that case, the vehicle acceleration becomes:
Γ(s) =
αv n+1 αp s αv n M 2 αp s + αp s
+1
+
Vtarget (s) s2
M 2 αp s
+
αv n αp s
+1
Xtarget (s).
(12)
In order to limit the acceleration to a γmax , equation (12) then becomes: max |Γ(jω)� ≤
αv n+1 αp (jω) αv 2 n αp (jω) + αp (jω) 2
max | M
+ max | M
αp
(jω)2
+1
−1 Frequency 10 (rad/s)
0
10
1
10
Fig. 7. Sensitivity function Bode gain diagram To make sure that this constraint is well and truly respected, a simulation in a straight line is made. The attractive crossing point is placed in the point of coordinates (100m, 100m) while the vehicle is placed at the origin of the mark (0m, 0m). Figures 8, 9 and 10 represent respectively the position, the speed and the acceleration of the vehicle according to X-axis. The simulation being in a straight line and the crossing point being equidistant to the origin, only the temporal answers according to the X-axis are presented. Note that the vehicle reaches its destination in 30s with a maximum speed of 30km/h and maximum acceleration of 0.5g according to the acceleration constraint. If an obstacle appears during this motion, the trajectory will be locally adjusted in order to avoid it. 4.2 Stability degree robustness simulation
||Vtarget (jω)|
(jω) ||Xtarget (jω)|. v n +α αp (jω) + 1
−2
10
(13)
Now, the vehicle mass varies (single driver, full loaded vehicle,...). The classic and the fractional Ge&Cui’s attractive potential field are designed for a maximum mass
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Proceedings of the 20th IFAC World Congress Julien Moreau et al. / IFAC PapersOnLine 50-1 (2017) 14533–14538 Toulouse, France, July 9-14, 2017
in the fractional case, the phase margin is held, ensuring the stability degree.
100 X
90
14537
Distance (m)
80
Nichols Chart
70
50
60
40
50
30
40
20
0.25 dB
Open−Loop Gain (dB)
0.5 dB
30 20 10 0
0
20
40
60
80
1 dB 3 dB 6 dB
10 0
Classical Ge&Cui M=900 kg
−10
Classical Ge&Cui M=750 kg −20
100
Time (s)
Classical Ge&Cui M=600 kg Fractional Ge&Cui M=900 kg
−30
Fractional Ge&Cui M=750 kg Fractional Ge&Cui M=600 kg
−40
Fig. 8. Evolution of vehicle position
−50 −180
−150 Open−Loop Phase (deg)
−120
30
Fig. 11. Nichols chart of fractional and classic open-loops
20
15
10
Step Response (position)
Position (m)
Step Response (position)
Position (m)
Velocity (km/h)
25
1
0.5
Classic M=900 kg Classic M=750 kg
0
20
40
Time (s)
60
80
Fractional M=750 kg
10 20 30 Time (seconds)
Fractional M=600 kg
0
40
10 20 30 Time (seconds)
Step Response (force)
100
Fig. 9. Evolution of vehicle velocity
20000
15000
15000
10000 5000
10000 5000
0
0.6
40
Step Response (force)
20000 Force (N)
0
Force (N)
0
Fractional M=900 kg
0.5
Classic M=600 kg
0
5
1
0 0
10 20 30 Time (seconds)
40
0
10 20 30 Time (seconds)
40
0.5
Fig. 12. Illustration of robustness with fractional potential field (right) compared to classic potential field (left)
Acceleration (g)
0.4 0.3 0.2
Step Response (position) 1.2 Amplitude (m)
0.1 0 −0.1
0
20
40
Time (s)
60
80
1.1 Classical M=900kg
1
Classical M=750kg Classical M=600kg
0.9
100
2
4
6
8
10 12 14 Time (s) (seconds)
16
18
20
Step Response (position)
Fig. 10. Evolution of vehicle acceleration and same unit gain frequency ωug = 0.39rad/s. Then, the impact of mass variations on the stability degree is tested. Fig. 11 represents the Nichols chart of the open-loops for the classic cases (dotted line) and fractional ones (full line) for a maximum mass M = 900 kg (red), a medium mass M = 750 kg (green) and a minimum mass M = 600 kg (blue). In the classic case, mass variations lead to phase variations around the unit gain frequency ωug (the phase margin varies between 65 and 70 degrees) contrary to the fractional case where there is no phase variations around ωug (the phase margin is kept equal to 60 degree). Thus,
Amplitude (m)
1.15 1.1 1.05 1
Fractional M=900kg
0.95
Fractional M=750kg
0.9
Fractional M=600kg 2
4
6
8
10 12 14 Time (s) (seconds)
16
18
20
Fig. 13. Illustration of robustness with fractional potential field (down) compared to classic potential field (up) Fig. 12 and Fig. 13 presents the step responses of the position for classic (dotted line) and fractional cases (full
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4
Amplitude (N)
2
ACKNOWLEDGEMENTS
Step Response (Attractive force)
x 10
This work took place in the framework of the OpenLab ‘Electronics and Systems for Automotive’ combining IMS laboratory and PSA Groupe company.
1
0
−1
0
0.05
0.1
REFERENCES
0.15
Time (s) (seconds) 4
Amplitude (N)
2
Step Response (Attractive force)
x 10
1.5 1 0.5 0
0
0.05
0.1
0.15
Time (s) (seconds)
Fig. 14. Illustration of robustness with fractional potential field (down) compared to classic potential field (up) with corresponding attractive force (blue) and force constraint (red) line) for a maximum mass (red), a medium mass (green) and a minimum mass (blue). In the fractional case, the damping is kept constant when the mass varies as opposed to the classic case. Finally, Fig. 14 is the corresponding attractive force (blue) with the force constraint (red). Both of them respects the force constraint however, in the fractional case, the curve is far below the limit as compared to the classic case. Thus, the fractional attractive potential field response time can be improved while it is not possible in the classic case.
5. CONCLUSION AND PERSPECTIVES Research works on autonomous vehicle lead car manufacturers to focus on the urban area. This environment establishes a critical point for the autonomous vehicle because it is very variable due to the presence of numerous mobile obstacles (pedestrians) or due to the diversity of the road infrastructures (intersections, gyrating crossroads, . . . ). It is thus essential to develop adaptive path planning towards the environment of the autonomous vehicle. In this way, a non exhaustive state of the art of these methods allows to put the potential field method in relation to the existing. The main advantage of this method is its adaptability which is a real need for autonomous vehicles. However, this method requires some modifications to become completely applicable to the autonomous vehicle. From this perspective, an adaptation of the attractive potential has been proposed, enabling acceleration limits and stability degree robustness guarantee. A set of attractive points called waypoints are also introduced to better answer to the constraints of the autonomous vehicle.
A. Chebly, G. Tagne, R.T. and Charara, A. (2015). Local trajectory planning and tracking for autonomous vehicle navigation using clothoid tentacles method. In proceedings of International IEEE Conference on Intelligent Vehicles Symposium (IV). Andrew-Fields, R. (2014). Continuous Control Artificial Potential Function Methods and Optimal Control. Ph.D. thesis, Air Force Institute of Technology. Ge, S. and Cui, Y. (2002). Dynamic motion planning for mobile robots using potential field method. Autonomous Robots, 13, 207–222. Inarn, C., Melchior, P., Metoui, B., and Oustaloup, A. (2012). Robust path planning for 3D dynamic environment based on fractional attractive force. 5th IFAC Symposium on Fractional Differentiation and Its Applications (IFAC FDA’12). Liu, Y., Zhang, X., Guan, X., and Delahayel, D. (2016). Potential odor intensity grid based UAV path planning algorithm with particle swarm optimization approach. Mathematical Problems in Engineering, 2016, 1–16. Makarem, L. and Gillet, D. (2011). Decentralized coordination of autonomous vehicles. In proceedings of the 18th IFAC World Congress, 44, 13046–13051. Melchior, P., Orsoni, B., Lavialle, O., Poty, A., and Oustaloup, A. (2003). Consideration of obstacle danger level in path planning using A∗ and fast-marching optimisation: comparative study. Signal processing, 83, 2387–2396. Melchior, P., Orsoni, B., and Oustaloup, A. (2001). Weyl fractional potential in path planning. Proceedings of the Sixth IEEE ECC’2001. Metoui, B., Melchior, P., Najar, S., Abdelkrim, M., and Oustaloup, A. (2009). Robust path planning for dynamic environment based on fractional attractive force. Sixth IEEE International Multi-Conference on Systems, Signals and Devices (IEEE SSD’09). Morand, A., Moreau, X., Melchior, P., Moze, M., and Guillemard, F. (2016). Crone cruise control system. IEEE Transactions on Vehicular Technology, 65, 15–28. Oustaloup, A., Sabatier, J., Lanusse, P., and Melchior, P. (2014). Fractional Order Differentiation and Robust Control Design: CRONE, H-infinity and Motion Control. Wiley-ISTE, Paris. Shibata, N., Sugiyama, S., and Wada, T. (2014). Collision avoidance control with steering using velocity potential field. IEEE Intelligent Vehicles Symposium. Zhan, W., Wang, W., Chen, N., and Wang, C. (2014). Efficient UAV path planning with multiconstraints in a 3D battlefield environnement. Mathematical Problems in Engineering, 2014, 1–12.
The adaptation of the attractive potential field method is solved in this paper, the repulsive one needs to be scaled regarding the attractive potential field. Moreover, life situation scenarii need to be simulated and validated before testing this strategy on a real autonomous vehicle. 15103
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 14533–14538
Path planning with fractional potential Path planning with potential Pathfields planning with fractional fractional potential for autonomous vehicles fields for autonomous vehicles fields for autonomous∗ vehicles ∗,∗∗∗ ∗ Julien Julien Julien Julien
Moreau ∗,∗∗∗ Pierre Melchior ∗ St´ ephane Victor ∗ ∗,∗∗∗ Pierre ∗ St´ ∗∗ Melchior ∗∗ Victor Moreau e phane Moreau Pierre Melchior e Victor ∗∗ Fran¸ c ois Aioun Franck Guillemard ∗,∗∗∗ ∗ St´ Moreau Pierre Melchior St´ ephane phane ∗∗ ∗∗ ∗∗ Franck ∗∗ Victor Fran¸ c ois Aioun Guillemard Fran¸ c Fran¸ cois ois Aioun Aioun ∗∗ Franck Franck Guillemard Guillemard ∗∗ ∗ IMS – UMR 5218 CNRS, Universit´ e de Bordeaux/Bordeaux INP ∗ ∗ IMS – UMR 5218 CNRS, Universit´ de Bordeaux/Bordeaux Bordeaux/Bordeaux INP – UMR Universit´ de cours 5218 de la CNRS, Lib´eration, 33405eee Talence cedex – France INP ∗ IMS IMS351 – UMR 5218 CNRS, Universit´ de Bordeaux/Bordeaux INP 351 cours de la Lib´ eeration, 33405 Talence cedex – France 351 cours de la Lib´ ration, 33405 Talence cedex – France Email: fi[email protected] 351 cours de la Lib´ e ration, 33405 Talence cedex – France ∗∗ Email: fi[email protected] Email: PSA Groupe Email: fi[email protected] fi[email protected] ∗∗ ∗∗ PSA Groupe PSA Groupe Chemin de Gisy, 78140 elizy-Villacoublay cedex – France ∗∗ V´ PSA Groupe Chemin de de Gisy, Gisy, 78140 V´ lizy-Villacoublay cedex – – France Chemin 78140 V´ eeelizy-Villacoublay cedex Email: [email protected] Chemin de Gisy, 78140 V´ lizy-Villacoublay cedex – France France ∗∗∗ Email: [email protected] Email: [email protected] OpenLab PSA Groupe-IMS Bordeaux Email: [email protected] ∗∗∗ ∗∗∗ OpenLab PSA Groupe-IMS Bordeaux ∗∗∗ OpenLab PSA Groupe-IMS Bordeaux OpenLab PSA Groupe-IMS Bordeaux Abstract: Path planning is an essential stage for mobile robot control. It is more newsworthy Abstract: planning is context an essential stage for mobile robot control. It is more newsworthy Abstract: Path planning an stage robot It more newsworthy than ever inPath the automotive and especially for autonomous vehicle. Also, path planning Abstract: planning is is context an essential essential stage for for mobile mobile robot control. control. It is is more newsworthy than ever need inPath thetoautomotive automotive and especially especially for autonomous autonomous vehicle. Also, path planning than ever in the context and for vehicle. Also, path planning methods be adaptive regarding life situations, traffic and obstacle crossing. In this than ever in the automotive context and especially for autonomous vehicle. Also, path planning methods need to be adaptive regarding life situations, traffic and obstacle crossing. In this methods need to be adaptive regarding life situations, traffic and obstacle crossing. In paper, potential field methods are proposed to cope with these constraints and autonomous methods need tofield be adaptive regarding life to situations, traffic and obstacle and crossing. In this this paper, potential methods are proposed cope with these constraints autonomous paper, field methods proposed to cope with these constraints and autonomous vehiclespotential are considered equippedare with all necessary sensors for obstacle detection. In this way, paper, potential field methods are proposed to cope with these constraints and autonomous vehicles are considered equipped with necessary sensors for obstacle detection. In this way, vehicles are considered equipped with allfractional necessary sensors obstacle detection. In way, Ge&Cui’s attractive potential field andall attractive potential field have been adapted vehicles areattractive considered equipped with necessaryattractive sensors for for obstaclefield detection. In this this way, Ge&Cui’s potential field andall fractional potential havebetter been adapted Ge&Cui’s attractive potential field and fractional attractive potential field have been adapted to the context of autonomous vehicles. In this way, this latter method ensures stability Ge&Cui’s attractive potential field and fractional attractive potential field have been adapted to the context of autonomous vehicles. In this way, this latter method ensures better stability to the of vehicles. this degree robustness with controlled vehicleIn acceleration. to the context context of autonomous autonomous vehicles. Inacceleration. this way, way, this this latter latter method method ensures ensures better better stability stability degree robustness with controlled vehicle degree robustness with controlled vehicle acceleration. degree robustness with controlled vehicle acceleration. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Path planning, automotive, autonomous vehicle, potential field, fractional potential Keywords: Path Keywords: Path planning, planning, automotive, automotive, autonomous autonomous vehicle, vehicle, potential potential field, field, fractional fractional potential potential field Keywords: Path planning, automotive, autonomous vehicle, potential field, fractional potential field field field 1. INTRODUCTION The main advantage is that the vehicle is able to navigate 1. INTRODUCTION The main main advantage is that that the vehicle is is able to navigate navigate 1. INTRODUCTION The advantage is vehicle to with no “a priori” on the the environment. However, this 1. INTRODUCTION The advantage is that vehicle is able able to navigate with main no could “a priori” on the the environment. However, this with no “a priori” on the environment. However, this method lead to behaviors unreadable by the driver. with no “a priori” on the environment. However, this Autonomous driving is going to be the heart of public method method could could lead lead to to behaviors behaviors unreadable unreadable by by the the driver. driver. Autonomous driving is going to be the heart of public method could lead to behaviors unreadable by the driver. Autonomous driving is going going to be be the thevehicles heart of of public transports in driving the future. Autonomous have the Potentials field methods are common path planning meAutonomous is to heart public Potentials field methods methods are(Inarn common path(2012)). planning metransports the Autonomous vehicles have the Potentials field common path planning mein mobile roboticsare et al. These transportstoin inimprove the future. future. Autonomous vehiclestravel havetime the thods potential passenger’s safety, reduce Potentials field methods are(Inarn common path(2012)). planning metransports in the future. Autonomous vehicles have the thods in in have mobile robotics et Unmanned al. These potential to improve passenger’s safety, reduce travel time thods mobile robotics (Inarn et al. (2012)). These methods been already used for Aircraft potential to improve passenger’s safety, reduce travel time and improve the comfort of the passengers. As a consethods in mobile robotics (Inarn et al. (2012)). These potential to improve passenger’s safety, reduce travel time methods(UAV) have been been already used for for Unmanned Unmanned Aircraft Aircraft and the the As consemethods have already used in Zhan et al. (2014), (2014) and improve improve the comfort comfort of ofinvest the passengers. passengers. As a consequence car manufacturers in the conception and System methods have been already used forAndrew-Fields Unmanned Aircraft and improve the comfort of the passengers. As aa conseSystem (UAV) in Zhan et al. (2014), Andrew-Fields (2014) quence car manufacturers invest in the conception and System (UAV) in Zhan et al. (2014), Andrew-Fields (2014) and Liu et al. (2016). Moreover, they have also been quence car manufacturers invest in the conception and the equipment of their vehicles with advance systems for System (UAV) Zhan etMoreover, al. (2014), Andrew-Fields quence car manufacturers invest in advance the conception and and Liu et al. al.in(2016). (2016). they have have also(2014) been the equipment of their vehicles with systems for and Liu et Moreover, they also been used in automotive especially for collision avoidance in the equipment of their vehicles with advance systems for assistance. In urban areas, intersections are considered as and Liu et al. (2016). Moreover, they have also been the equipment of their vehicles with advance systems for used in automotive especially for collision avoidance in assistance. In urban areas, intersections are considered as used in automotive especially for collision avoidance in Shibata et al. (2014) however no extension for a complete assistance. In urban areas, intersections are considered as bottlenecking; more than 44 % of all the accidents reported in et automotive especially for collisionfor avoidance in assistance. In urban areas, intersections are considered as used Shibata al. (2014) however no extension a complete bottlenecking; more than 44 % of all the accidents reported Shibata et al. (2014) however no extension for a complete navigation has been developed. bottlenecking; more than 44 % of all the accidents reported in the United States arise in intersection zones and are Shibata et al. however no extension for a complete bottlenecking; more than 44 % of all the accidents reported navigation has(2014) been in arise intersection zones are has been developed. developed. in the the United United States arise in inand intersection zones1and and are navigation responsible for States 8,500 deaths approximately million navigation has been developed. in the United States arise in intersection zones and are In order to use potential field for autonomous vehicle responsible for 8,500 deaths and approximately 11 million responsible foryear. 8,500 deathsthe andtraffic approximately million wounds every Besides, delays generated by In order to use potential field for autonomous vehicle responsible for 8,500 deaths and approximately 1 million In order to use potential field vehicle navigation, a strategic schematic has autonomous been developed (see wounds every year. Besides, the traffic delays generated by order toa use potential field for for autonomous vehicle wounds every everylead year.toBesides, Besides, the losses traffic of delays generated by In intersections enormous resources. Therenavigation, strategic schematic has been developed developed (see wounds year. the traffic delays generated by navigation, strategic schematic been (see Fig. 1). Thisaa schematic has three has stages: intersections lead to enormous losses of resources. Therenavigation, strategic schematic has been developed (see intersections lead vehicles to enormous enormous losses of resources. resources. There- Fig. 1). This schematic has three stages: fore, autonomous answer to these issues. Thereintersections lead to losses of Fig. 1). This schematic has three stages: the strategic stage: data is collected and processed from fore, autonomous vehicles answer to these issues. Fig. 1). This schematic has three stages: fore, autonomous vehicles answer to these issues. the strategic strategic stage: data istocollected collected and processed from fore, answer these issues. --- the stage: processed from sensors in data orderis create anand immediate virtual The autonomous urban area vehicles represents the to sensitive case for au- perception the strategic stage: data istocollected and processed from perception sensors in order create an immediate virtual The urban area represents the sensitive case for auperception in order to create an immediate virtual of the sensors autonomous vehicle with obstacle informations The urban urbanvehicles. area represents represents the kind sensitive case for for auautonomous Indeed, this of environment is map perception sensors in order to create an immediate virtual The area the sensitive case map of of the the autonomous autonomous vehicle with obstacle obstacle informations tonomous vehicles. kind of environment is map vehicle with . . . ) and targets; tonomousautonomous vehicles. Indeed, Indeed, this kind of adaptive environment is (velocity, variable: vehiclesthis need to be regarmap of theposition, autonomous vehicle with obstacle informations informations tonomous vehicles. Indeed, this kind of environment is (velocity, position, . .. .. from ) and and targets; variable: autonomous vehicles need to be adaptive regar(velocity, position, . ) targets; the tactical stage: the previous informations, a variable: autonomous vehicles need to be adaptive regarding life situations, traffic and obstacle crossing. In this (velocity, position, . . . ) and targets; variable: autonomous vehicles need to be adaptive regarthe resulting tactical of stage: from the the previous previous informations, ding situations, traffic obstacle In this --- the tactical stage: from informations, an attractive repulsive potential fieldaa ding life life situations, vehicles traffic and and obstacle crossing. crossing. In with this force paper, autonomous are considered equipped the resulting tactical of stage: from theand previous informations, a ding life situations, traffic and obstacle crossing. In this force an attractive and repulsive field paper, autonomous vehicles are considered equipped with force resulting an attractive and repulsive potential field is of generated and according to apotential simple model paper, autonomous vehicles are considered considered equipped with with combination all necessary sensorsvehicles for obstacles detection.equipped force resulting of an attractive and repulsive potential field paper, autonomous are combination is generated generated and ideal according to aa simple simple model all combination is and according to model (punctual mass model), the position, velocity and all necessary necessary sensors sensors for for obstacles obstacles detection. detection. combination is generated and ideal according to a simple model all necessary for obstacles These recent sensors years, path planningdetection. for autonomous vehi- acceleration (punctual mass mass model), the the position, velocity and (punctual model), ideal position, velocity and are generated by simulating the motion of a These recent planning for vehi(punctual mass model), the position, velocity and Thesehas recent years, path planning for autonomous autonomous vehi- charged cles been years, widelypath studied in literature. In Makarem acceleration are generated generated byideal simulating theforce; motion of of a These recent years, path planning for autonomous vehiacceleration are by simulating the motion particle subjected to this generated cles has widely studied literature. Makarem acceleration are generated by this simulating theforce; motion of aa cles Gillet has been been widely studied in infunction literature. In Makarem -charged and (2011), a navigation for In autonomous charged particle subjected to generated cles has been widely studied in literature. In Makarem particle to this the operational stage: the position, force; velocity and and aa navigation function for charged particle subjected subjected to ideal this generated generated force; and Gillet Gillet (2011), navigation function for autonomous autonomous vehicles has(2011), been developed but no pedestrians have been acceleration the operational operational stage: the ideal position, velocity and and Gillet (2011), a navigation function for autonomous --- the stage: the ideal and from the previous stageposition, are used velocity as references vehicles has been developed but no pedestrians have been the operational stage: the ideal position, velocity and vehicles has hasinbeen been developed but no pedestrians pedestrians have been as considered thisdeveloped method. but Another interesting method acceleration from the the previous stage are areusing usedthe as references references vehicles no have been acceleration from previous stage used as a classic control loop, the controller deviation considered in this method. Another interesting method acceleration from the previous stage are used as references considered in this this method. Another interesting method called “tentacles method” hasAnother been developed in A.method Chebly between as aa classic classic control loop, loop, the controller using to thegenerate deviationa considered in method. interesting as control controller using the deviation references and the measures in order called “tentacles method” has in as a classic control loop, the controller using to thegenerate deviationa called “tentacles method” has been been developed in A. A. Chebly and Charara (2015). The idea is todeveloped deploy a set of Chebly virtual steering between references and measures in order called “tentacles method” has been developed in A. Chebly between references and measures in order to generate aa of wheel torque T w and a wheel w. and Charara (2015). The idea is to deploy aa set of virtual between angle references andβ measures in order to generate and Charara (2015). The idea is to deploy set of virtual tentacles with different curves in front of the autonomous steering angle of wheel β and a wheel torque T . w w and Charara (2015). The idea is to deploy a set of virtual steering angle of wheel β and a wheel torque T . tentacles with different curves of autonomous angle only of wheel βw a wheel torque Tw w and stage w .Indeed, In this paper, the tactical is presented. tentacleseach withtentacle different curves in in front front of the the vehicle autonomous vehicle, representing a possible path. steering tentacles with different curves in front of the autonomous In this paper, only the tactical stage is presented. Indeed, vehicle, each tentacle representing a possible vehicle path. In this paper, only the tactical stage is presented. Indeed, strategic is atactical data processing problem whereas vehicle, each tentacle representing possible vehicle path. the Each path is tentacle then evaluated according to a set of criteria. In paper,stage only the stage is presented. Indeed, vehicle, each representing aa possible vehicle path. thethis strategic stage is aa data data processing processing problem whereas whereas Each path is then evaluated according to aa set of criteria. the strategic stage is problem Each path is then evaluated according to set of criteria. Each path is then evaluated according to a set of criteria. the strategic stage is a data processing problem whereas
Copyright © 2017, 2017 IFAC 15098 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 15098 Copyright ©under 2017 responsibility IFAC 15098 Peer review of International Federation of Automatic Control. Copyright © 2017 IFAC 15098 10.1016/j.ifacol.2017.08.2076
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 14534 Julien Moreau et al. / IFAC PapersOnLine 50-1 (2017) 14533–14538
Vehicle measures STRATEGIC STAGE
Obstacles
Frep
Repulsive force +
Simple model +
Target
Attractive force
βw
+
Navigation
Vehicle
Controller
Ref.
-
Fatt
Tw
Vehicle measures
OPERATIONAL STAGE
TACTICAL STAGE
Vehicle measures
Fig. 1. Plan of regulation at three levels: strategic (blue), tactical (green) and operational (orange)) the operational is a control problem. The fractional differentiation has been already used in automotive (Morand et al. (2016)). This present work is based on the extension of the work developed in Melchior et al. (2001),Melchior et al. (2003) and Metoui et al. (2009) on fractional potential field. Firstly, the fractional potential field method is presented in section 2. After that, section 3 shows extensions on fractional potential field method to autonomous vehicle. A set of simulation results is presented in section 4, to finally conclude in section 5. 2. FRACTIONAL POTENTIAL FIELD In motion control, a charged particle is subject to: -an attractive potential field tied to the target -and a repulsive one tied to obstacles for avoidance. Compared to a classic potential field, a fractional definition of the potential field allows to create a less constrained potential field curve.
Fig. 2. Fractional repulsive potential field created around an obstacle 1
2.1 Fractional repulsive potential field
Fractional potential field
n−2 r n−2 −rmax n−2 n−2 rmin −rmax
f or rmin < r ≤ rmax ,
where r is the distance of the particle to the considered obstacle, rmin and rmax the boundaries of the repulsive potential field and n the fractional order. The potential field can be shaped by using the parameter n (see Fig. 3). For example, when a charged particle is on a direct course to an obstacle, the higher the parameter n, the farther from the obstacle the move of the particle. Thus, the parameter n can be interpreted as a danger parameter. The potential field is applied to each obstacle and free road is obtained by the truncation of the potential map (Melchior et al. (2003)). The resulting force applied to the particle is given by: → − → − F = −grad( U ).
n=4
0.8
Let us consider the fractional repulsive potential field U (see Fig. 2) defined by : U (r) =
n=5
0.9
n=3
0.7
n=2
0.6 0.5
n=1
0.4 0.3 0.2 0.1 0
0
5
10
15 Distance (m)
20
25
30
Fig. 3. Influence of parameter n on the potential field 2.2 Fractional attractive potential field The resulting force expressed by Ge and Cui (2002) is given by:
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− → → → → → F = αp (− x target − − x ) + αv ( − v target − − v)
(1)
Proceedings of the 20th IFAC World Congress Julien Moreau et al. / IFAC PapersOnLine 50-1 (2017) 14533–14538 Toulouse, France, July 9-14, 2017
where αp and αv are weighting coefficients, the particle → → position − x and the target position − x target , the particle → − → − velocity v and the target velocity v target . For the fractional attractive potential field, an extension of equation (1) gives: − → dn → → → → F = αp (− x target − − x ) + αv n ( − v target − − v ), dt
(2)
where the fractional derivative of the position is considered instead of the velocity.
1 X(s) = F (s) M s2
3.1 Limits of classic Ge&Cui’s attractive potential field in autonomous vehicle context
Γ(s) =
According to equations (2) and (3), the fractional attractive potential field can be interpreted as a position controller. This interpretation leads to the elementary control loop schematic (see Fig. 4). xtarget+ F x ε 1 α + α sn v
In this section, the drawbacks of classic attractive potential field when applied to the autonomous vehicle are presented. A solution using the concept of way-point is proposed. Finally, an upgrade of this solution using the fractional attractive potential field is explained.
(3)
where s is the Laplace variable, X(s) the Laplace transform of the vehicle position and F(s) the Laplace transform of the resulting force (repulsive or attractive).
p
3. APPLICATION TO THE AUTONOMOUS VEHICLE
According to Fig. 4, the Laplace transform of the vehicle acceleration can be expressed (with n = 1) according to:
Let us consider a punctual mass model: G(s) =
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M s2
− x
αv 3 αp s M 2 αp s
αv αp s
+
+1
Xtarget (s).
(8)
Consequently, two main problems may occur with this definition. First, this transfer function is not proper (numerator degree higher than denominator one) leading to implementation problems. The second problem concerns the target position: the farther the target, the greater the attractive force magnitude and the greater the vehicle acceleration magnitude. Thus, if the vehicle acceleration is too high, it could be uncomfortable for passengers and in a worst case, the system limits may be reached. 3.2 Classic attractive potential field with limitation of vehicle acceleration
Fig. 4. Elementary control loop schematic with Ge&Cui’s attractive potential field when n = 1 and fractional attractive potential field in other cases The open-loop expression associated to this schematic is expressed as β(jω) =
αv (jω)n +αp M (jω)2
=
jω n 1+( ω ) c M αp
(jω)2
First of all, the causality problem can be fixed by using the vehicle velocity. In this way, vehicle velocity is used instead of the derivative of the vehicle position. However, the target velocity needs to be added as an input signal as illustrated on Fig. 5.
. In high
vtarget
+
−
frequencies, it then becomes: β(jω) ≈
+ s2
εv
αv αp , f or ω >> ωc = M (jω)2−n αv
1 n
αv
(4) xtarget+ ε p
or else,
β(jω) ≈
ωug jω
n
α v
M
1 n
.
αp
+
F
+
−
1 M
γ
1 s
1 s
x
x
(5)
where n� = 2 − n and the unit gain frequency: ωug =
v
Fig. 5. Control loop schematic using attractive potential field and velocity (6)
In this way, the vehicle acceleration becomes:
Such an open-loop is expressed by a fractional order n� and the damping ratio is then given by (Oustaloup et al. (2014)): π ξ = −cos( � ). (7) n In this case, the damping ratio only depends on the parameter n� and it is independent of the model parameters. In this way, robustness of stability degree relative to the vehicle mass variation is ensured.
Γ(s) =
αv 2 αp s αv M 2 α p s + αp s
+1
Vtarget (s)
+
M 2 αp s
s2 Xtarget (s) v +α αp s + 1
(9)
In order to limit the acceleration to a γmax , equation (9) becomes with s = jω:
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Proceedings of the 20th IFAC World Congress 14536 Julien Moreau et al. / IFAC PapersOnLine 50-1 (2017) 14533–14538 Toulouse, France, July 9-14, 2017
Considering a stationary target, Vtarget = 0 and equation αv 2 (6), the expression of the attractive potential field coefαp (jω) max |Γ(jω)| ≤ max M |Vtarget (jω)| ficient is expressed as: v α (jω)2 + α (jω) + 1 αp p 2 (jω) γmax M n� + max M |Xtarget (jω)| (10) αv = M ωug . (14) and αp ≤ v α (jω)2 + α (jω) + 1 |X α target | p
p
with a stationary target, Vtarget = 0 and thus, the attractive potential field coefficients can be expressed as:
4. SIMULATION
4.1 Acceleration constraint simulation γmax M αv = 2ξ αp M and αp ≤ . |Xtarget |
(11)
These equations show that the target needs to be stationary and not too far from the vehicle in order to limit the acceleration. In the light of these constraints, a set of targets may be added throughout the vehicle path. These targets are called waypoints. The vehicle has to only consider the closest waypoint as attractive point, after reaching it, the following waypoint is used as the following attractive point, and until the end. 3.3 Fractional attractive potential field with limitation of vehicle acceleration
The autonomous vehicle is attracted by a single waypoint according to the attractive potential field theory. The attractive potential field coefficients are chosen in accordance with expressions (11) where ξ = 1 (without damping), M = 750kg (light vehicle) and the waypoint is placed 100m farther than the vehicle. An acceleration limit is fixed to 0.5g where g is the gravity acceleration. The Bode Γ(s) diagram of the sensitivity function Xtarget (s) (blue curve) and the acceleration constraint (green curve) are plotted on Fig 7). As expected, this choice of coefficient respects the acceleration constraint in the useful frequency band. Sensitivity function Bode gain diagram −20
In the same way, a similar schematic can be established in the fractional case (see Fig 6). +
−
εv
v
−60 Magnitude (dB)
vtarget
−40
αv s1−n
xtarget+ ε p
−80
−100
+
αp
+
F
1 M
−
γ
1 s
1 s
x −120
Sensitivity function Acceleration constraint
x
−140 −3 10
Fig. 6. Fractional attractive potential field with causality In that case, the vehicle acceleration becomes:
Γ(s) =
αv n+1 αp s αv n M 2 αp s + αp s
+1
+
Vtarget (s) s2
M 2 αp s
+
αv n αp s
+1
Xtarget (s).
(12)
In order to limit the acceleration to a γmax , equation (12) then becomes: max |Γ(jω)� ≤
αv n+1 αp (jω) αv 2 n αp (jω) + αp (jω) 2
max | M
+ max | M
αp
(jω)2
+1
−1 Frequency 10 (rad/s)
0
10
1
10
Fig. 7. Sensitivity function Bode gain diagram To make sure that this constraint is well and truly respected, a simulation in a straight line is made. The attractive crossing point is placed in the point of coordinates (100m, 100m) while the vehicle is placed at the origin of the mark (0m, 0m). Figures 8, 9 and 10 represent respectively the position, the speed and the acceleration of the vehicle according to X-axis. The simulation being in a straight line and the crossing point being equidistant to the origin, only the temporal answers according to the X-axis are presented. Note that the vehicle reaches its destination in 30s with a maximum speed of 30km/h and maximum acceleration of 0.5g according to the acceleration constraint. If an obstacle appears during this motion, the trajectory will be locally adjusted in order to avoid it. 4.2 Stability degree robustness simulation
||Vtarget (jω)|
(jω) ||Xtarget (jω)|. v n +α αp (jω) + 1
−2
10
(13)
Now, the vehicle mass varies (single driver, full loaded vehicle,...). The classic and the fractional Ge&Cui’s attractive potential field are designed for a maximum mass
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Proceedings of the 20th IFAC World Congress Julien Moreau et al. / IFAC PapersOnLine 50-1 (2017) 14533–14538 Toulouse, France, July 9-14, 2017
in the fractional case, the phase margin is held, ensuring the stability degree.
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Fig. 10. Evolution of vehicle acceleration and same unit gain frequency ωug = 0.39rad/s. Then, the impact of mass variations on the stability degree is tested. Fig. 11 represents the Nichols chart of the open-loops for the classic cases (dotted line) and fractional ones (full line) for a maximum mass M = 900 kg (red), a medium mass M = 750 kg (green) and a minimum mass M = 600 kg (blue). In the classic case, mass variations lead to phase variations around the unit gain frequency ωug (the phase margin varies between 65 and 70 degrees) contrary to the fractional case where there is no phase variations around ωug (the phase margin is kept equal to 60 degree). Thus,
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Fig. 13. Illustration of robustness with fractional potential field (down) compared to classic potential field (up) Fig. 12 and Fig. 13 presents the step responses of the position for classic (dotted line) and fractional cases (full
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Proceedings of the 20th IFAC World Congress 14538 Julien Moreau et al. / IFAC PapersOnLine 50-1 (2017) 14533–14538 Toulouse, France, July 9-14, 2017
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ACKNOWLEDGEMENTS
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This work took place in the framework of the OpenLab ‘Electronics and Systems for Automotive’ combining IMS laboratory and PSA Groupe company.
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REFERENCES
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Fig. 14. Illustration of robustness with fractional potential field (down) compared to classic potential field (up) with corresponding attractive force (blue) and force constraint (red) line) for a maximum mass (red), a medium mass (green) and a minimum mass (blue). In the fractional case, the damping is kept constant when the mass varies as opposed to the classic case. Finally, Fig. 14 is the corresponding attractive force (blue) with the force constraint (red). Both of them respects the force constraint however, in the fractional case, the curve is far below the limit as compared to the classic case. Thus, the fractional attractive potential field response time can be improved while it is not possible in the classic case.
5. CONCLUSION AND PERSPECTIVES Research works on autonomous vehicle lead car manufacturers to focus on the urban area. This environment establishes a critical point for the autonomous vehicle because it is very variable due to the presence of numerous mobile obstacles (pedestrians) or due to the diversity of the road infrastructures (intersections, gyrating crossroads, . . . ). It is thus essential to develop adaptive path planning towards the environment of the autonomous vehicle. In this way, a non exhaustive state of the art of these methods allows to put the potential field method in relation to the existing. The main advantage of this method is its adaptability which is a real need for autonomous vehicles. However, this method requires some modifications to become completely applicable to the autonomous vehicle. From this perspective, an adaptation of the attractive potential has been proposed, enabling acceleration limits and stability degree robustness guarantee. A set of attractive points called waypoints are also introduced to better answer to the constraints of the autonomous vehicle.
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The adaptation of the attractive potential field method is solved in this paper, the repulsive one needs to be scaled regarding the attractive potential field. Moreover, life situation scenarii need to be simulated and validated before testing this strategy on a real autonomous vehicle. 15103