research article combination of the dvz method, particle...

Research ArticleCombination of the DVZ Method, Particle Filter, and FuzzyController for Electric Wheelchair Navigation

Malek Njah and Mohamed Jallouli

Control and Energy Management Laboratory (CEMLab), National School of Engineering of Sfax, University of Sfax, BP W,3038 Sfax, Tunisia

Correspondence should be addressed to Malek Njah; [email protected]

Received 30 July 2014; Revised 31 October 2014; Accepted 3 November 2014; Published 18 November 2014

Academic Editor: Sandro M. Radicella

Copyright © 2014 M. Njah and M. Jallouli. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Electric wheelchair is one of the many engines used for the movement of aged and disabled people. This paper introduces anobstacle avoidance using deformable virtual zone (DVZ), particle filter to improve localization and fuzzy controller to join desiredtarget.This controller is developed to increase the independence of disabled and aged people, specifically those who suffer not onlydisability in the lower limbs but also visual disturbances. To overcome these problems, different perceptive abilities or sensors wereintroduced. In this context, we developed a control system composed by fuzzy controller to join a target, DVZmethod for obstacleavoidance, and particle filter for localization. Also, we present the simulation results of the wheelchair navigation system.

1. Introduction

Several works have been developed in this context. Thepublication of Lapierre et al. [1] threatened the DVZ methodcombined with the path followed, and the work of Gil Pintoet al. [2, 3] provides an active approach to planning basedon the DVZ method. This control strategy is based on theGPS sensor and ultrasonic sensors for generating an optimaltrajectory to ensure cooperation between nonholonomicrobots in an unknown environment. The particulate filter isa probabilistic method allowing the multisensor data fusionand localization. The coordination of these strategies resultsin an automatic obstacle avoidance navigation to reach atarget point [4, 5]. This paper presents control system basedon the DVZ method for obstacle avoidance. The principle ofthis method is to define the robot/environment interaction asa risk zone surrounding the wheelchair (Figure 2). The DVZcharacterizes the deformable zone geometry and depends onthe wheelchair speed. When the risk zone is disturbed by anobstacle, it can be reformed bymodifying the velocities of thewheelchair [1, 6].

In addition we will detail the use of the particle filtermethod for localization. This method uses a probabilistic

approach [4, 7]. It generates a set of possible positions in eachsampling period.This data fusionmethod gives the estimatedposition of the chair using distance between wheelchair andobstacles.

Also, we use a fuzzy controller already developed inprevious work that allows the wheelchair to reach the desiredposition [8, 9].

In the following we will detail the used methods and thesimulation of the control system.

2. Using Fuzzy Controller for ElectricWheelchair to Join Target

In this paper the fuzzy controller permits reaching the targetposition (𝑋

𝑇, 𝑌𝑇) from any position (𝑋, 𝑌).

The considered wheelchair is composed of two indepen-dent coaxial driving wheels and two free-rotating wheels.The configuration of this type of robot is characterized bythe position on the 𝑋-axis, 𝑌-axis, and orientation 𝜃 in aCartesian coordinate system illustrated by Figure 1. Considerthe following.

Hindawi Publishing CorporationInternational Journal of Navigation and ObservationVolume 2014, Article ID 137891, 7 pageshttp://dx.doi.org/10.1155/2014/137891

2 International Journal of Navigation and Observation

XO XTXk

YT

Yk

Y

Y1 VL

O1

V VR

X1

𝜃

𝜑

Target

d

𝜃T

Figure 1: Wheelchair and target position.

(𝑋𝑘, 𝑌𝑘) is wheelchair position at the sampling period

𝑘.𝑉𝐿and 𝑉

𝑅are the velocities of right and left wheels.

The objective is to move the electric wheelchair andreach the desired position.𝑉 is linear speed of the electric wheelchair.𝜃 is the angle of wheelchair orientation.𝐿 is the distance separating the driving wheels ofelectric wheelchair.

The distance between the electric wheelchair and thetarget is defined by the following expression:

𝑑 = √(𝑋𝑇− 𝑋)2+ (𝑌𝑇− 𝑌)2,

𝜑 = 𝜃𝑇− 𝜃,

(1)

where

𝜃𝑇= tan−1

(𝑌𝑇− 𝑌𝑘)

(𝑋𝑇− 𝑋𝑘). (2)

The electric wheelchair is represented with the discreteform of the kinematic model:

𝑋(𝑘+1)

= 𝑋(𝑘)

+ 𝑇𝑉𝑅(𝑘)

+ 𝑉𝐿(𝑘)

2cos 𝜃𝑘,

𝑌(𝑘+1)

= 𝑌(𝑘)


+ 𝑉𝐿(𝑘)

2sin 𝜃𝑘,

𝜃(𝑘+1)

= 𝜃(𝑘)


− 𝑉𝐿(𝑘)

𝐿,

(3)

where 𝑇 is the sampling period.The used fuzzy controller is developed in our previous

work [8, 9]. When we use the fuzzy controller of the type

Risk zoneObstacle

Obstacle

X

Y

X1

Y1

x

y

𝜃

𝛼dh(𝛼)

c(𝛼)

cy

ax

cx

d(𝛼)

Figure 2: Obstacle avoidance with DVZ method.

Takagi-Sugeno of the order 0, the distance “𝑑” and the angle“𝜑” are the input. The left and the right velocity (𝑉

𝐿, 𝑉𝑅) of

two driving wheels are the output. The input membership ofvariables “𝑑” and “𝜑,” linguistic inference table of velocity, andfuzzy repartition of output variables are detailed in previouspapers [8, 9]. In this paper we combined the fuzzy controllerto join defined target with obstacle avoidance using DVZ.Weimproved the localization using the particle filter. In the nextsection we describe the DVZ method for obstacle avoidance.

3. Obstacle Avoidance by the DVZ Method

The DVZ method, based on a reflex behavior, uses theconcept of deformable virtual zone developed by Zapataet al. [10, 11]. This method associates a risk zone to theelectric wheelchair. This zone depends on the kinematics ofthe wheelchair. The deformation of the risk zone is due tointrusion of proximity information (obstacles within the riskzone).The system response ismade to reform the danger zoneto its nominal shape.Themovement of the platform dependson the area surrounding the robot. If an obstacle is detected,the areawill deform.Thismethod treats all forms of obstacles.We can apply this type of control with the use of the kinematicmodel of the robot, which facilitates its application to a realsystem that navigates in an unknown environment [1, 12].Therepresentation of the risk zone is shown in Figure 2.

In the following we will describe the principle of the algo-rithm DVZ applied to the electric wheelchair. It develops inan unknown environment. It is surrounded by a deformablevirtual zone; the geometry of this area depends on the linearand angular velocities and the wheelchair coordinates. Thedeformations of the risk zone are due to the interaction withthe environment through the sensors. The calculation of thedeformationmeasurements from the sensors is used to definethe response of the platform.

3.1. The Undeformed Risk Zone. In order to obtain an ana-lytic expression for the algorithm DVZ, it is considered as

International Journal of Navigation and Observation 3

an elliptical shape risk zone. Either𝑃 = [𝑥 𝑦]𝑇 a point on the

ellipse with the axes 𝑐𝑥and 𝑐𝑦. It is assumed that the reference

position by the considered DVZ is translated from the centerby a vector 𝑎 = [𝑎

𝑥𝑎𝑦]𝑇. Hence the equation is given by

(𝑥 − 𝑎𝑥

𝑐𝑥

)

2

+ (𝑦 − 𝑎𝑦

𝑐𝑦

)

2

= 1. (4)

Coefficients 𝑎𝑥, 𝑎𝑦, 𝑐𝑥, and 𝑐

𝑦are selected heuristically

[1, 6]. They depend on the speed of the wheelchair. We canmodify the coefficients “√5/3” and “−2/3” of the expression𝑐𝑦and 𝑎𝑥according to themaximum speed of the wheelchair,

the range sensors, and response time of the system

𝑐𝑥= 𝜆𝑐𝑥𝑢2+ 𝑐

min𝑥

,

𝑐𝑦=

√5

3𝑐𝑥,

𝑎𝑥=

−2

3𝑐𝑥,

𝑎𝑦= 0.

(5)

The distance 𝑑ℎis determined from the equality given by

the following defined in [1, 6]:

𝐴𝑑ℎ (𝛼)2+ 𝐵𝑑ℎ (𝛼) + 𝐶 = 0. (6)

Hence the expression 𝑑ℎis written as

𝑑ℎ (𝛼) =

−𝐵 + √(𝐵)2− 4𝐴𝐶

2𝐴,

(7)

along with

𝐴 = (𝑐𝑦cos (𝛼 − 𝛾))

2

+ (𝑐𝑥sin (𝛼 − 𝛾))

2,

𝐵 = 2 ⋅ (𝑎𝑥cos (𝛼 − 𝛾) 𝑐

2

𝑦+ (𝑎𝑦sin (𝛼 − 𝛾) 𝑐

2

𝑥)) ,

𝐶 = (𝑎𝑥𝑐𝑦)2

+ (𝑎𝑦𝑐𝑥)2

− (𝑐2

𝑥𝑐2

𝑦) ;

(8)

𝛾 is the angle between axes 𝑐𝑥and 𝑋

1. The risk zone is

rigidly attached to the electric wheelchair and follows theposition and direction during movement therefore (𝛾 = 0).The detection range of the sensors used covers the front ofthe mobile platform.

3.2. The Deformed Risk Zone. The deformation of the area atrisk is due to the existence of an obstacle. This deformation iscalculated from the measurements of distance by the sensors.Distances measured by the ultrasonic sensors are rated 𝑐(𝛼).The distance 𝑑(𝛼) between the sensor and the risk zoneis distorted by saturation obtained 𝑐(𝛼) according to thefollwoing algorithm:

𝑑 (𝛼) = 𝑐 (𝛼) 𝑠𝑖 𝑐 (𝛼) < 𝑑ℎ (𝛼)

= 𝑑ℎ (𝛼) 𝑠𝑖𝑛𝑜𝑛.

(9)

3.3. The Deformation. The intrusion report necessary forthe development of response of the control algorithm upondetection of an obstacle is present. “𝐼” is the expression ofintrusion given by equation report (10). When the distancebetween the robot and the obstacle approaches zero, it isfound that “𝐼” tends to infinite.Then, a limitationwas appliedon the values of “𝐼” for a limited intrusion. This intrusiongives a good efficiency of obstacles avoidance system:

𝐼 = ∫𝜋

𝛼=0

𝑑ℎ (𝛼, 𝑢) − 𝑑 (𝛼)

𝑑 (𝛼)𝑑𝛼. (10)

The derivation of the expression of the intrusion versustime gives the following result [1]:

𝐼 = 𝐽𝑢

𝐼�� + 𝐽𝜃

𝐼𝑟 + 𝐹

Rob.vel𝑢, (11)

with

𝐹Rob.vel

= ∫𝜋

𝛼=0

𝑑ℎ (𝛼)

𝑑2 (𝛼)cos (𝛼) 𝑑𝛼. (12)

The derived expressions are given in [12].The inverse kinematicmodel is used to generate velocities

necessary for controlling the twodrivingwheels of the electricwheelchair. The displacement allows obstacle avoidance. Wedenote the expression𝑉

𝐼= 𝐼2/2which satisfies the Lyapunov

stability theorem [1, 6]. The equation of the derivative of 𝑉𝐼

relative to the intrusion “𝐼” gives

��𝐼= 𝐼 (𝐽

𝑢

𝐼�� + 𝐽𝜃

𝐼𝑟 + 𝐹

Rob.vel𝑢) . (13)

On our application, we will set the control system of theelectric wheelchair given by [1]

�� = − 𝐾𝑢𝐽𝑢

𝐼𝐼 −

𝐹Rob.vel

𝐽𝑢𝐼

𝑢,

𝑟 = − 𝐾𝑟𝐽𝜃

𝐼𝐼.

(14)

Parameters𝐾𝑢and𝐾

𝑟are positive gains resulting in ��

𝐼≤

0 ∀𝑡.

3.4. Application of the DVZ Method for Obstacle Avoidance.We tested the algorithm of the obstacle avoidance usinga developed simulator for electric wheelchair. The controlparameters are selected according to constraints such asspeed limits, the response time of the system, and the sam-pling period.The distances between obstacles and wheelchairare calculated in the simulation. We will discuss severalcases in the simulation. In the following we will detail thesimulation of the obstacle avoidanceDVZmethod algorithm.We will consider a case for the validation of the algorithmcombining the fuzzy controller to reach a target point andobstacle avoidance.Then we will consider a complicated casewith many obstacles.

(i) Figure 3 indicates a case of an obstacle avoidance. Inthis simulation the wheelchair must reach a targetpoint with coordinates (𝑋

𝑇= 4m, 𝑌

𝑇= 4m) with a


Obstacle

−1−1

0

0

1

1

2

2

3

3

4

4

5

5

6

6

Obstacle

Target (XT = 4m, YT = 4m)

Y(m

)

X (m)

Starting point (X = 0m, Y = 0m)

Figure 3: Application of the DVZ method.

starting point (𝑋 = 0m, 𝑌 = 0m) and a departureangle 𝜋/4. The obstacle is in the middle of the path.The trajectory traveled by the electric wheelchairpresented in Figure 3 is obtained by applying thefuzzy navigation controller combined with the DVZmethod for obstacle avoidance.

(ii) In the last simulation, the navigation algorithm hasprovided obstacle avoidance using the left sensors.We will consider now a second simulation for com-plicated case. The navigation is performed in anenvironment containing a hallway and obstacles.

Figure 4 represents the case where the starting point ofthe wheelchair is (𝑋 = −8m, 𝑌 = 4m) and a wheelchairorientation angle 𝜃 = 0. The target is (𝑋

𝑇= 5m, 𝑌

𝑇= 3m).

Figure 4 shows the trajectory with obstacle avoidance. Asthe tracking system is based on odometry information,the actual path may be different because of measurementerrors and the statement of the initial position. The electricwheelchair detects and avoids obstacles without collision.Hysteresis function ensures switching between nonoptimizedfuzzy controller to achieve the target location and the DVZalgorithm. We limit the wheel speeds during obstacle avoid-ance to ensure system stability during navigation.

In the next section, we will add a method of localizationusing particle filter to improve the localization and perfor-mance of the control algorithm.

4. Particle Filter

Particle filter, also known as sequential Monte Carlo (SMC)methods estimation techniques, is based on simulation.Theyare used in mobile robotics for localization, path planning,and vision [13]. In the following we will describe the use of

10

8

6

4

2

0

0 42 6

Target

Starting point

Obstacle

Obs

tacle

Obstacle

Y(m

)

−2

−2−6

−6

−4

−4−8

X (m)

(X = −6m, Y = 4m)

(XT = 5m,

YT = 3m)

TargetgObstacle

Obs

tacle

(XT = 5m,

YTYY = 3m)

Figure 4: Wheelchair navigation in an environment with manyobstacles.

the particle filter and fusion data used for locating an electricwheelchair for the disabled.

4.1. Particle Filter Algorithm for Locating an ElectricWheelchair. Particle filtering is a sequential version of theMonte Carlo methods to solve filtering problems. The con-ditional distribution of the state is represented by a finitenumber of weighted systems called Dirac particles. Eachparticle represents a state of probable weighted system. Itsweight is based on the confidence level of the particle. Itrepresents the probable state of the system. Historically thefirst filter contains a step redistribution presented as particlefilter algorithm with interaction or condensation [4].

For the simulation, we use a discretemodel defined by thesystem of (3).

Each particle is a configuration of the electric wheelchair.This particle is associated with a weight 𝑝𝑖 representing theconsistency between the configuration of the particle and theconfiguration of the electric wheelchair in its environmentfrom observations.

First Step (initialization of particle). Consider

𝑝𝑖= (𝑥𝑖, 𝑤𝑖) , (15)

where 𝑤𝑖 is associated with the weight of the 𝑖 particle con-figuration 𝑥𝑖 and 𝜀𝑖

𝑥, 𝜀𝑖𝑦, 𝜀𝑖𝜃are random variables:

𝑃𝑖= (

[[

[

𝑥init + 𝜀𝑖𝑥

𝑦init + 𝜀𝑖𝑦

𝜃init + 𝜀𝑖𝜃

]]

]

,1

𝑁) . (16)


Second Step (prediction). During this step, data encoders willbe used to move the particles. The evolution of the particle isgiven by

𝑃𝑖

𝑘+1=((

(

[[[[[[[[

[

𝑥𝑖

𝑘+ 𝑇

𝑉𝑑𝑘

+ 𝑉𝑔𝑘

2cos 𝜃𝑘

𝑦𝑖𝑘+ 𝑇

𝑉𝑑𝑘

+ 𝑉𝑔𝑘

2sin 𝜃𝑘

𝜃𝑖𝑘+ 𝑇

𝑉𝑑𝑘

− 𝑉𝑔𝑘

𝐿

]]]]]]]]

]

,𝑤𝑖

𝑘

))

)

. (17)

Step Three (estimation). In this step the information of thedistance sensors is used. For each particle 𝑖 and 𝑗 for eachsensor, we evaluate error 𝑒𝑖

𝑦

𝑒𝑖

𝑦= 𝑦𝑗− 𝑦𝑗

𝑚(𝑥𝑖) , (18)

where 𝑦𝑗 is the distance given by jth sensors and 𝑦𝑗

𝑚(𝑥𝑖) is

calculated by the developed simulator of sensors distanceassociated with the expression 𝑥𝑖.

We consider a Gaussian error 𝑁(0, 𝜎𝑑); that is to say, the

probability of measuring 𝑦𝑗 configuration 𝑥𝑖 is given by

𝑝 (𝑦𝑗| 𝑥𝑖) = exp(−1/2)((𝑒

𝑖

𝑦)2/𝜎2

𝑦). (19)

The new coefficient of the particle is calculated using the fol-lowing relation:

𝑤𝑖

𝑘+1= 𝑤𝑖

𝑘

𝑁

∑𝑗=1

𝑝 (𝑦𝑗| 𝑥𝑖) . (20)

The weight of the particles is adjusted using

𝑊𝑖=

𝑤𝑖

∑𝑁

𝑗=1𝑤𝑗

. (21)

Step Four (resampling). Resampling is performed when theinequality 𝑁eff < 𝑁th is checked [7, 14]. Initially, we create[𝑁𝑤𝑖] copies particles 𝑥𝑖. [𝑁𝑤𝑖] is the integer part of 𝑁𝑤𝑖.With this approximation, the number of generated particlesis less than𝑁. Therefore, it is necessary to add more particlesobserved by𝑁 and given by the following expression:

𝑁 =

𝑁

∑𝑖=1

(𝑁𝑤𝑖− [𝑁𝑤

𝑖]) . (22)

Resampling is to generatemultinomial𝑁newparticles.Theseparticles are weighted by

𝑤𝑖=

(𝑁𝑤𝑖 − [𝑁𝑤𝑖])

𝑁. (23)

In the next section we will describe the results obtainedby combining the fuzzy controller, the DVZ method, andthe particle filter for navigation and localization of electricwheelchair for disabled persons.

10

8

6

4

2

0

0

Target

Obstacle

2 6

Obs

tacle

4 8 10

Hallway

Y(m

)X (m)

−2

−2

−4

−4

−6

−6−8−8

Targeg

Obstacle

Obs

tacle

Hallwayl

Position obtained with odometry dataPosition obtained with particle filter

Hal

lway

(XT = 8.5m,

YT = 9.5m)

Starting position (X = −5m, Y = −5m, 𝜃 = −𝜋/2)

Figure 5: Simulation result of an electric wheelchair obtained by thecombination of DVZ method, particle filter, and fuzzy controller.

5. Result of the Electric Wheelchair Navigation

The navigation of the electric wheelchair is provided by thefuzzy controller. The controller generates the wheels velocityto reach the target point. The DVZ method is used toavoid the obstacles in the way. The location of the electricwheelchair is calculated by the kinematic model by using theinformation received from encoders. We will introduce thelocalization using the particle filter.The sampling period usedis 𝑇 = 1 second.

(i) Figure 5 shows the simulation result of a case ofnavigationwith obstacle.Thenavigation environmentpresents obstacles to the right and left and a hallway.

The starting position is considered (𝑋 = −5m, 𝑌 = −5m,and 𝜃 = −𝜋/2) and the target position is (𝑋

𝑇= 8.5m,

𝑌𝑇= 9.5m). An error in the measurements of the sensors

is added to test the localization algorithm by particle filter.These errors are due to the geometric errors parametersof the wheelchair (wheel diameter, length, and width), theslipping ofwheels, errors of speedmeasurement, and errors ininitial values of the orientation and position.The localizationerror increases when the wheelchair does not detect anobstacle. The localization with particle filter is used when thewheelchair avoids an obstacle with a known position. It isfound that the algorithm of particle filter allows data fusionobtained by the simulation of the encoders and the distancemeasured by the distance sensors. Initially 1000 particles aregenerated. Particles evolved during navigation using (17).


During that time detection of an obstacle, the particulatefilter gives the estimated actual position and removes particlesassuming a low weight. The particles removed are replacedby each other by the resampling. The navigation to achievethe target location is performed by the navigation fuzzycontroller associated with the obstacle avoidance using DVZmethod. The localization algorithm is using particle filter byusing the kinematics model for the evolution of the positionof the particles. The green curve is obtained by the particlefilter path.This curve is the connection between the particleshaving the highest weight. The simulation result showedimproved localization. The red curve (see Figure 5) gives theodometer position.The green curve obtained by fusion of thesensor datawith the particulate filter shows better positioningof the wheelchair in its environment. The positioning erroris considered in the simulation 400mm. In each position ofthe obstacle detection algorithm computes the actual positionlocation.

The simulations obtained (see Figure 5) show consider-able improvement in the precision of the localization by theparticle filter.Thismethod of localizationwill be used to reacha target location with better precision.

In the next section we will describe the proposed algo-rithm for applying the developed control system to a realwheelchair.

6. Proposed Algorithm for RealWheelchair Control

In this sectionwe present the proposed algorithm for applica-tion of the global control system.The algorithm used for realwheelchair is illustrated by Figure 6.

(i) In the case of navigation without obstacle, the func-tion “join target” is selected.The velocity generated bythe fuzzy controller is applied on two driving wheels.This controller steers the robot to reach the target.After a sampling period, themeasurement acquisitionsystem sends the information given by the encodersand ultrasonic sensors. This information is necessaryfor localization and obstacle avoidance.

(ii) In the case when obstacle is in a risk zone, thecondition (𝑑

ℎ(𝛼) > 𝑑(𝛼)) is verified. “𝑑(𝛼)” is

given by the measurement of each ultrasonic sensor(𝐷𝑆1, 𝐷𝑆2, 𝐷𝑆3, 𝐷𝑆4) and “𝑑

ℎ(𝛼)” is calculated by the

particle filter program. Each sensor admits an angle“𝛼” and range and zone of detection.The “switch bloc”selects the algorithm for localization and obstacleavoidance. The distance obtained by the ultrasonicsensors and the theoretical distance are used by theparticle filter for localization. After localization, thenew position is (𝑋

𝑘= 𝑋pf, 𝑌𝑘 = 𝑌pf) with (𝑋pf, 𝑌pf)

the coordinate of particle having the most importantweight calculated by (20). Next step is the obstacleavoidance using DVZ method. The DVZ functioncalculates the intrusion (equation (10)) and generatesthe linear and angular velocity using system of (14).We generate the velocity by the inverse kinematic

Switch between obstacle avoidance and join target

Join target using fuzzy controller

Calculation of theoretical distance between wheelchair and obstacle

Obstacle avoidance using DVZ method

Localization with particle filter

Encoders

Ultrasonic sensors

Speed left Speed right

End

Calculation of new positionusing kinematic model

No obstacle If obstacle is in a risk zone

Target achieved

S1S2S3

S4

S5S6S7

S8

DS1

DS1

DS2

DS2

DS3

DS3

DS4

DS4

VL VR

(XT, YT), obstacles position.Initial value (Xk = X0, Yk = Y0)

Xpf

Ypf

Xk = XpfXk = Ypf

Figure 6: Proposed algorithm for real wheelchair control.

model. Then, the generated velocities are applied ontwo driving wheels.

(iii) If the wheelchair achieved the target the algorithm isfinished.

7. Conclusion

In this paper we developed a control system for electricwheelchair navigation. The sensor data fusion is carried outby the particle filter for estimating the position of the electricwheelchair. It has been established with the kinematic modelof the wheelchair with the fuzzy controller which achievesa target point. For the obstacle avoidance we used a DVZmethod. With this control system we have improved thenavigation precision of the wheelchair. The particle filtermethod in the navigation gives more details on the actualposition and improves the quality of the control algorithm toachieve the target with minimum error.


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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