robot path planning with avoiding obstacles in known
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Research ArticleRobot Path Planning with AvoidingObstacles in Known Environment Using FreeSegments and Turning Points Algorithm
Imen Hassani , ImenMaalej , and Chokri Rekik
Control and Energy Management Laboratory (CEM Lab), National Engineering School of Sfax, University of Sfax, Sfax, Tunisia
Correspondence should be addressed to Imen Hassani; imenhassani@yahoo.fr
Received 23 February 2018; Revised 24 April 2018; Accepted 13 May 2018; Published 11 June 2018
Academic Editor: Luis Gracia
Copyright ยฉ 2018 Imen Hassani et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Currently, the path planning problem is one of the most researched topics in autonomous robotics. That is why finding a safe pathin a cluttered environment for a mobile robot is an important requirement for the success of any such mobile robot project. In thiswork, a developed algorithm based on free segments and a turning point strategy for solving the problem of robot path planningin a static environment is presented. The aim of the turning point approach is to search a safe path for the mobile robot, to makethe robot moving from a starting position to a destination position without hitting obstacles.This proposed algorithm handles twodifferent objectives which are the path safety and the path length. In addition, a robust control law which is called sliding modecontrol is proposed to control the stabilization of an autonomous mobile robot to track a desired trajectory. Finally, simulationresults show that the developed approach is a good alternative to obtain the adequate path and demonstrate the efficiency of theproposed control law for robust tracking of the mobile robot.
1. Introduction
Nowadays, robots are considered as an important element insociety. This is due to the replacement of humans by robotsin basic and dangerous activities. However, designing anefficient navigation strategy for mobile robots and ensuringtheir securities are the most important issues in autonomousrobotics.
Therefore, the path planning problem is one of the mostinteresting and researched topics. The aim of the robot pathplanning is to search a safe path for the mobile robot. Alsothe path is required to be optimal. In this sense, severalresearch works tackling the path planning problem have beenproposed in the literature [1โ4]. Until now, many methodshave been used for path planning of mobile robots. Amongthese strategies, the geometry space method such as ArtificialPotential Field [5, 6], Agoraphobic Algorithm [7], and VectorField Histogram [8, 9].These methods give the heading anglefor avoiding obstacles. The strategy of dynamic windows hasbeen used in [10, 11]. This approach is a velocity-based localplanner that calculates the optimal collision-free velocity
for a mobile robot. Another method used in [12] is namedturning point searching algorithm which consists of findinga point around which the mobile robot turns without hittingobstacles.
In the other side, several research works for trackingcontrol of a wheeled mobile robot have gained attentionin the literature [13โ16]. The nonholonomic system suffersof nonlinearity and uncertainty problem. Because of thisuncertainty, the trajectory error for a wheeled mobile robothas always been produced and can not be eliminated. In thissense, many tracking methods are proposed in the literatureas Proportional Integral Derive (PID) controller [17] but thiscontroller becomes instable when it is affected by the sensorsensitivity [18]. Furthermore, a fuzzy logic controller is usedin [19] but this control law has a slow response time due tothe heavy computation [20]. Other works used sliding modecontroller in various applications [15, 16]. The aim advantageof this control system is its insurance for stability, robustness,fast response, and good transient [21].
The aim of the developed strategy is to solve the problemwhen the robot is located between two obstacles such as
HindawiMathematical Problems in EngineeringVolume 2018, Article ID 2163278, 13 pageshttps://doi.org/10.1155/2018/2163278
2 Mathematical Problems in Engineering
the following: how the robot can detect that the distancebetween the two obstacles is safe enough to reach the targetwithout collision and how to avoid obstacles and movebetween two obstacles in the shortest path. That is whythis work is based on selecting safe free segments in anenvironment encumbered by obstacles firstly. After that, adeveloped turning point searching algorithm is applied todeterminate the endpoint of the safe free segment which givesthe shortest path. This strategy is inspired from the approachgiven by Jinpyo and Kyihwan [12]. In fact, the strategypresented in [12] handles two fundamental objectives: thepath length and the path safety. This approach is focusedfirstly on searching the endpoint of a free segmentwhich givesthe shortest path. Hence, if the distance of the free segmentselected is larger than the robot diameter, the endpoint isconsidered as a turning point. If this is not the case, it mustreplay the algorithm to search a new endpoint of the freesegments. The disadvantages of this strategy are that it isfocused firstly on finding the shortest path without takinginto consideration the safety and, after that, it is focused onensuring a safe path navigation which leads to an extensiveand heavy computation andneedsmore time for planning theadequate path for a mobile robot. In order to overcome thesedisadvantages, our developed algorithm serves to ensure atfirst the path safety by selecting the safest free segments.Then,it searches the path length by determining the endpoint of thesafest free segments which gives the shortest path. Using thisstrategy, we can rapidly determine the safest and the shortestpath.Moreover, once the path is planned, a tracking lawbasedon sliding mode controller is used for the robot to follow thedesigned trajectory.
Our contribution is to develop a new algorithm forsolving the problem of robot path planning with static obsta-cles avoidances. This planning, also called static path plan,presents the advantage of ensuring safety and shortness ofthe path. Moreover, the proposed algorithm is characterizedby a reactive behavior to find a collision-free trajectoryand smooth path. On the other side, the mobile robotshould track the trajectory without collision with obstacles.So, a sliding mode control is proposed for guaranteeingrobustness, stability, and reactivity.
The rest of this paper is organized as follows. Section 2presents the mobile robot model used in this work. Thedifferent steps of the proposed algorithm for path planningpurpose are described in detail in Section 3. In Section 4, aslidingmode controller is used for trajectory tracking. Finally,simulation results and conclusion are presented and analyzedin Sections 5 and 6, respectively.
2. Mobile Robot Model
Several research works for autonomous navigation have beenapplied to different types of mobile robots [22, 23]. In thiswork, we consider the Khepera IV mobile robot whichhas two independent driving wheels that are responsiblefor orienting and commanding the platform by acting onthe speed of each wheel. Thus, the schematic model of thewheeled mobile robot Khepera IV is shown in Figure 1.
y
x
L
X
Y
o
o1
VLX1
Y1
VR
Figure 1: Schematic representation of Khepera IV.
The kinematic model of a nonholonomic mobile robot isgiven as follows:
๐๐ฅ๐๐ก =๐๐ + ๐๐ฟ2 cos๐ผ
๐๐ฆ๐๐ก =
๐๐ + ๐๐ฟ2 sin๐ผ๐๐ผ๐๐ก =
๐๐ โ ๐๐ฟ๐ฟ
(1)
where (๐ฅ, ๐ฆ) are the robotโs Cartesian coordinates, ๐ผ is theangle between the robot direction and ๐ axis, ๐๐ and ๐๐ฟ are,respectively, the robot right and left wheel velocities, and ๐ฟ isthe distance between the two wheels.
3. Path Planning Algorithm
In order to solve the path planning problem, an algorithmbased on finding the turning point of a free segment isproposed.
3.1. Principle of the Proposed Algorithm. A free segment isconsidered as the distance between two endpoints of twodifferent obstacles (see Figure 2). It searches the endpoint of asafe segment where the mobile robot turns around this pointwithout hitting obstacles.
When there are no obstacles, the path planning problemdoes not arise. In fact, the robot moves from an initialposition ๐๐ to a goal position ๐๐ in a straight line which willbe considered as the shortest path. However, when themobilerobot encounters with ๐ obstacles as shown in Figure 2 (๐ =3), the robot should be turning without collision withobstacles. So, the major problem is how to determinate asuitable path from a starting point to a target point in a staticenvironment. To solve this problem our developed algorithmis proposed to search for a turning point of a safe free segmentwhich gives the shortest path and allows the robot to avoidobstacles. Once the turning point is located, a dangerouscircle with radius ๐ ๐ is fixed in this point. In this case, ourproposed strategy aims to search for the turning point ๐๐ก๐of the safe free segment around which the robot turns safely.
Mathematical Problems in Engineering 3
Path1Path2
Obstacle
Obstacle
Obstacle
Dangerous circle
Safe segmentDanger segment == Free segmentFree segment
Turning point, PtpTurning point, Ptp
SP1SP2
RD RD
pi
pg
Figure 2: Framework of navigation.
For ensuring safety, we select the segment whose distance ๐๐๐(๐ = 1, . . . , ๐ โ 1) is larger than the robot diameter ๐ท๐ with amargin for security ๐ฟ (๐๐๐ โฅ ๐ท๐ + ๐ฟ). On the other hand, thesegmentwhose distance ๐๐๐ is smaller than the robot diameteris considered as a danger segment (see Figure 2). In this work,we take into account only safe segments and danger segmentsare ignored. Furthermore, and to determinate the shortestpath, we have determined the point ๐๐ก๐ of the safest segmentwhich gives the shortest path.Then a dangerous circle is fixedat this point and the robot turns and moves towards thetangential direction to this circle. Even when there is a dangerproblem, our proposed algorithm will be reactive to allowthe robot to avoid obstacles and reach the goal. In this case,the robot reserves the determined turning point and searchesfor a new turning point to avoid collision with obstacles.To more clarify our strategy, the different notions of thealgorithm are incorporated in Figure 2 and the basic principleis summarized in a flowchart presented in Figure 3.
3.2. Static Path Planning Steps. The aim of this section is tofind a safe path as short as possible. In this approach, it isdefined as the path having the tangential direction to thecircle located on the searched turning point.
3.2.1. Selection of the Safe Path. The safe path aims to finda free path that helps the robot to reach the target withouthitting obstacles of the environment. The selection of a safesegment needs to follow the next steps:
(i) Step 1: Find out all free segments of the environment(see Figure 4). Equations (2) and (3) show how todeterminate the value of the distance ๐๐1 that con-nects points ๐2 and ๐3 and the distance ๐๐2 thatconnects points ๐4 and ๐5:
๐๐1 = โ(๐ฅ๐3 โ ๐ฅ๐2)2 + (๐ฆ๐3 โ ๐ฆ๐2)2 (2)
No
Solve problemYes
Planning trajectory
Placement of the dangerous circle
Danger problem
Determination of the shortest path
Determination of the turning point
Calculate SPi
Selection of safe free segments (SPi โฅ Dr + )
Figure 3: The proposed algorithm.
Obstacle
Obstacle
Obstacle Safe free segment
Danger free segment
pi
p6
p5
p2
p1
p3p4
pg
SP1
SP2
Figure 4: Determination of free segments (safe-danger).
๐๐2 = โ(๐ฅ๐5 โ ๐ฅ๐4)2 + (๐ฆ๐5 โ ๐ฆ๐4)2 (3)
where (๐ฅ๐๐,๐ฆ๐๐) (๐=2..5) corresponds to the coordinateof endpoints of free segments.
(ii) Step 2:The segment whose distance ๐๐๐ (๐ = 1, . . . , ๐ โ1) is larger than ๐ท๐ is considered as a safe segment.
4 Mathematical Problems in Engineering
Obstacle
Obstacle
Obstacle
pi
p6
p5
p2
p1
p3p4
pg
SP2 SP1Turning point (Ptp)
SD1
SD2
Figure 5: Determination of the shortest path.
However, the segment whose distance ๐๐๐ is smallerthan ๐ท๐ is considered as a danger segment. Only safesegments are taken into consideration for the rest ofthis work. Danger segments whose number is ๐๐ areignored. In this step, we define the number of safesegments as
๐๐ = {{{๐ โ 1 if ๐๐๐ โฅ (๐ท๐ + ๐ฟ)๐ โ 1 โ ๐๐ elsewhere.
(4)
Once the safety criteria are handled, in the next section weare interested to determinate the shortest path.
3.2.2. Determination of the Shortest Path. When the robotgoes to reach the target position, it is important to do it inthe shortest path as possible.The objective of determining theshortest path can be divided into three steps:
(i) Step 1: Calculate distances ๐๐ท1 and ๐๐ท2 between therobot and the target with consideration of the safefree segment (see Figure 5).These distances should becalculated as follows:
๐๐ท1 = โ(๐ฅ๐2 โ ๐ฅ)2 + (๐ฆ๐2 โ ๐ฆ)2
+ โ(๐ฅ๐ก โ ๐ฅ๐2)2 + (๐ฆ๐ก โ ๐ฆ๐2)2(5)
๐๐ท2 = โ(๐ฅ๐3 โ ๐ฅ)2 + (๐ฆ๐3 โ ๐ฆ)2
+ โ(๐ฅ๐ก โ ๐ฅ๐3)2 + (๐ฆ๐ก โ ๐ฆ๐3)2(6)
(ii) Step 2: It concerns the determination of the turningpoint which is defined as the point around which the
Obstacle
Obstacle
Obstacle
pi
pg
SP2
SP1
RD
p1
p6
p5
p2
p3p4
Ptp
Figure 6: Dangerous circle placement.
mobile robot turns for avoiding obstacles; the processis achieved after comparing the distances ๐๐ท1 and ๐๐ท2.The endpoint of the safe free segment which givesthe shortest path corresponds to the searched turningpoint ๐๐ก๐ as shown in Figure 5.
(iii) Step 3: It concerns the placement of the dangerouscircle. Once the turning point is determined, a dan-gerous circle with radius ๐ ๐ท is fixed at this point asshown in Figure 6.
3.3. Problems Examination. Even the adequate path is deter-mined, some problems can persist whose results make therobot damaged and can not avoid obstacles. Some problemcases are highlighted in this work.
3.3.1. Collision Danger Problem. Path planning problemmeans that the path should be safe enough to go throughwithout collision. However, a collision danger problem canpersist in some cases:
(i) Case 1: If there is an intersection between the robotand the obstacle. To better concretize the problem,Figure 7 is given: path 1 presents an example of amobile robot where it is entrapped by the obstacle andit can not avoid it. To remove the collision betweenthe robot path and obstacle, path 2 is presented andturned around a second dangerous circle with radius๐ ๐ท. So, we can conclude that path 2 is safe enoughfor the robot to go to the destination point withoutcollision.
(ii) Case 2: If the distance between the line tangent of thedangerous circle and the endpoint of an obstacle ๐(see Figure 8) is less than the robot radius (๐ โค ๐ ๐),
Mathematical Problems in Engineering 5
Obstacle
Obstacle
Obstacle
Path 2
Path 1
Dangerous circle
Second dangerous circle
SP2SP1
RD
RD
pi
p6
p5
p2
p1
p3p4
pg
Figure 7: Collision danger problem-case 1.
Obstacle
Obstacle
Obstacle
Dangerous circle
d
Obstacle
SP1
SP2
SP3 RD
pi
p6
p5
p2
p1
p3p4
pg
Figure 8: Collision danger problem-case 2.
a turning point algorithm is applied and a dangerouscircle is centered at the adequate turning point (seeFigure 9).
3.3.2. Problem of Local Minima. A local minima problem canexist when all segments are danger or the robot is entrappedwith obstacles. To escape from such a situation, the robot goesfar away from those obstacles until reaching the target (seeFigure 10).
4. Sliding Mode Control
After planning the path of the robot Khepera IV, a slidingmode controller is proposed for robust tracking trajectory
Obstacle
Obstacle
Obstacle
Dangerous circle
Obstacle
Dangerous circle
SP1
SP2
SP3
RD
RD
pi
p6
p5
p2
p1
p3p4
pg
Figure 9: Placement of the turning point.
Obstacle
Obstacle
Obstacle
Robot path
pi
p5
p1
p6
p3
p2p4
pg
Figure 10: Local minima problem resolution.
([15, 16]). In this strategy, two positions are needed tobe known as shown in Figure 11: the desired position๐๐=(๐ฅ๐, ๐ฆ๐, ๐ผ๐) which is defined as the desired position to bereached and the current robot position ๐=(x, y, ๐ผ) which isdefined as its real position at this moment. Furthermore, thedifference between the reference position ๐๐ and the currentposition ๐ is called the tracking error position๐๐=(๐ฅ๐, ๐ฆ๐, ๐ผ๐).The expression of ๐๐ is defined in equation (7) as follows:
๐๐ = [[[
๐ฅ๐๐ฆ๐๐ผ๐]]]= [[[
cos๐ผ sin๐ผ 0โ sin๐ผ cos๐ผ 00 0 1
]]][[[
๐ฅ๐ โ ๐ฅ๐ฆ๐ โ ๐ฅ๐ผ๐ โ ๐ผ
]]]. (7)
Tracking trajectory can be introduced as finding theadequate control vector ๐ = (V, ๐ค)๐ (V is the linear velocity
6 Mathematical Problems in Engineering
Y
X
yr
xr
y
x0
Xe
Ye
e
r
Figure 11: Tracking error.
of the wheeled mobile robot and๐ค is its angular velocity). Sothat the error position ๐๐ converges asymptotically to zero.The autonomous mobile robot is controlled according to
๐๐ = V + ๐ฟ.๐ค2๐๐ฟ = V โ ๐ฟ.๐ค2 .
(8)
The process of designing a sliding mode controller is dividedinto two steps:
(i) Step 1:The choice of the sliding surface: ๐ is defined asthe switching function because the control switchesits sign on the sides of the switching ๐ = 0. Therefore,๐ฅ๐=0 is chosen at the first switching function. When๐ฅ๐=0, the Lyapunov candidate function is definedas ๐ = (1/2)๐ฆ2๐ . Then, we determinate the timederivative of V:
๏ฟฝ๏ฟฝ = ๐ฆ๐ ๐ฆ๐ = ๐ฆ๐ (โ๐ฅ๐๐ค + V๐ sin (๐ผ๐))= โ๐ฅ๐๐ฆ๐๐ค โ V๐๐ฆ๐ sin (arctan (V๐๐ฆ๐)) . (9)
We notice that ๏ฟฝ๏ฟฝ โค 0 because V๐๐ฆ๐sin(arctan(V๐๐ฆ๐)) โฅ0. We define ๐ผ๐ = โarctan(V๐๐ฆ๐) as a switchingcandidate function.Then, the expression of the vectorof sliding surfaces is given as follows:
๐ = [๐ 1๐ 2] = [๐ฅ๐
๐ผ๐ + arctan (V๐๐ฆ๐)] . (10)
(ii) Step 2: The determination of the control law: thedesigning of a sliding mode controller needs firstlyto establish an analytic expression of the adequatecondition under which the state moves towards andreaches a sliding mode. However, a chattering phe-nomenon can be caused by the finite time delaysfor computations and limitations of control. That is
Obs4
Obs1
Free segment 2
Obs2
Free segment 3Obs3
Free segment 4
Obs5
Free segment 5
Obs6
Free segment 6
Obs7
Free segment 1
0 100 200 300 400 500 600 700 800 900โ1000
100
200
300
400
500
600
700
800
900
Figure 12: Environment mapping.
why the switching function is defined as a saturationfunction. The control law is defined then as
๐ = โ๐๐ ๐๐ก (๐ ) . (11)
It is noted that the reaching control system is notonly able to establish the reaching condition but alsoable to specify the dynamic of the switching function.By differentiating the vector of the sliding surfacesdefined in equation (10), we obtain
๐ = [ ๐ 1๐ 2] = [
โ๐1๐ ๐๐ก (๐ 1)โ๐2๐ ๐๐ก (๐ 2)] = [[[
๐ฅ๐๐ผ๐ + ๐๐พV๐ V๐ + ๐๐พ๐ฆ๐ ๐ฆ๐
]]]
= [[[
๐ฆ๐๐ค + V๐ cos๐ผ๐ โ V
๐ค๐ + ๐๐พ๐V๐ V๐ +
๐๐พ๐๐ฆ๐ (V๐ sin๐ผ๐ โ ๐ฅ๐๐ค) โ ๐ค
]]]
(12)
where๐๐พ๐V๐ =
๐ฆ๐1 + (V๐๐ฆ๐)2
and๐๐พ๐๐ฆ๐ =
V๐1 + (V๐๐ฆ๐)2 .
(13)
5. Simulation Results
In mobile robot navigation, the building of the environmentis considered an essential issue to carry out motion planningoperations. In this section, to demonstrate the basic abilityof the proposed algorithm, we present some simulationresults. In all simulations, we will present results of anenvironment including seven obstacles which are placed withan arbitrary way (see Figure 12). Table 1 presents the initialcenter coordinates of static obstacles.
The simulations are performed for the cases where thetarget coordinate (๐ฅ๐ก, ๐ฆ๐ก) is fixed while the robot positionchanged.
Mathematical Problems in Engineering 7
Obs4
Obs1
Obs2
Obs3Obs5
Obs6
Obs7
200 400 600 8000X [mm]
0
100
200
300
400
500
600
700
800
900Y
[mm
]
(a) Navigation with safe segments ((๐ฅ๐, ๐ฆ๐)=(0, 0)).
Obs4
Obs1
Obs2
Obs3Obs5
Obs6
Obs7
200 400 600 8000X [mm]
0
100
200
300
400
500
600
700
800
900
Y [m
m]
(b) Navigation with safe segments ((๐ฅ๐, ๐ฆ๐)=(400, 0)).
Obs4
Obs1
Obs2
Obs3Obs5
Obs6
Obs7
200 400 600 8000X [mm]
0
100
200
300
400
500
600
700
800
900
Y [m
m]
(c) Navigation with safe and danger segments ((๐ฅ๐, ๐ฆ๐)=(0, 0)).
0
Obs4
Obs1
Obs2
Obs3Obs5
Obs6
Obs7
200 400 600 8000X [mm]
100
200
300
400
500
600
700
800
900Y
[mm
]
(d) Navigation with safe and danger segments ((๐ฅ๐, ๐ฆ๐)=(400, 0)).
Figure 13: Path planning ((๐ฅ๐ก, ๐ฆ๐ก)=(250, 750)).Table 1: Center coordinates of obstacles.
Obstacles ๐ฅ๐๐๐ ๐ฆ๐๐๐ Obstacle 1 550 100Obstacle 2 640 400Obstacle 3 640 600Obstacle 4 400 500Obstacle 5 70 680Obstacle 6 100 400Obstacle 7 150 130
In this section, we present the case when the robot startsfrom the initial positions (๐ฅ๐, ๐ฆ๐)=(0, 0) and (๐ฅ๐, ๐ฆ๐)=(400, 0)as shown in Figures 13(a) and 13(b), where all free segmentsare safe. We notice that the robot turns around circles which
Table 2: Center coordinates of obstacles ((๐ฅ๐ก, ๐ฆ๐ก)=(250, 750)).Obstacles ๐ฅ๐๐๐ ๐ฆ๐๐๐ Obstacle 1 550 100Obstacle 2 640 400Obstacle 3 640 600Obstacle 4 400 500Obstacle 5 200 550Obstacle 6 150 420Obstacle 7 150 300
are located in the adequate turning points and reaches thetarget for each modification of the robot position.
Even the obstacle centers changed their positions asshown in Table 2, and the path navigation changes are shown
8 Mathematical Problems in Engineering
Obs4
Obs1
Obs2
Obs3Obs5
Obs6
Obs7
200 400 600 8000X [mm]
0
100
200
300
400
500
600
700
800
900Y
[mm
]
(a) Navigation with safe segments ((๐ฅ๐, ๐ฆ๐)=(0, 0)).
Obs4
Obs1
Obs2
Obs3Obs5
Obs6
Obs7
200 400 600 8000X [mm]
0
100
200
300
400
500
600
700
800
900
Y [m
m]
(b) Navigation with safe segments ((๐ฅ๐, ๐ฆ๐)=(400, 0)).
Obs4
Obs1
Obs2
Obs3
Obs5
Obs6
Obs7
0
100
200
300
400
500
600
700
800
900
Y [m
m]
200 400 600 8000X [mm]
(c) Navigation with safe and danger segments ((๐ฅ๐, ๐ฆ๐)=(0, 0)).
Obs4
Obs1
Obs2
Obs3
Obs5
Obs6
Obs7
0
100
200
300
400
500
600
700
800
900Y
[mm
]
200 400 600 8000X [mm]
(d) Navigation with safe and danger segments ((๐ฅ๐, ๐ฆ๐)=(400, 0)).
Figure 14: Path planning ((๐ฅ๐ก, ๐ฆ๐ก)=(500, 750)).
in Figures 13(c) and 13(d) because of the appearance of dangersegments.
Figure 16 illustrates the navigation of the mobile robotwith safe segments and danger segments. That robot startsfrom different initial positions (๐ฅ๐, ๐ฆ๐)=(0, 0) (see Figures14(a) and 14(c)) and (๐ฅ๐, ๐ฆ๐)=(400, 0) (see Figures 14(b)and 14(d)). The obstacle center coordinates are addressed inTable 3.
Another simulation results present the case where all freesegments are safe (see Figures 15(a) and 15(b)). The robotturns around the dangerous circles until reaching the desiredtarget. By changing obstacle centers as shown in Table 4, weremark the appearance of dangerous segments. The robottakes into account just the free segments and moves in thesafe path (see Figures 15(c) and 15(d)).
Table 3: Center coordinates of obstacles ((๐ฅ๐ก, ๐ฆ๐ก)=(500, 750)).Obstacles ๐ฅ๐๐๐ ๐ฆ๐๐๐ Obstacle 1 550 100Obstacle 2 640 400Obstacle 3 640 600Obstacle 4 400 500Obstacle 5 240 550Obstacle 6 100 420Obstacle 7 150 250
Figures 16(a) and 16(b) show that the mobile robotensures reaching the destination with avoiding differentobstacles. Table 5 shows the center obstacle positions. In
Mathematical Problems in Engineering 9
Obs4
Obs1
Obs2
Obs3
Obs5
Obs6
Obs7
0
100
200
300
400
500
600
700
800
900Y
[mm
]
200 400 600 8000X [mm]
(a) Navigation in case safe segments ((๐ฅ๐, ๐ฆ๐)=(0, 0)).
Obs4
Obs1
Obs2
Obs3Obs5
Obs6
Obs7
200 400 600 8000X [mm]
0
100
200
300
400
500
600
700
800
900
Y [m
m]
(b) Navigation in case safe segments ((๐ฅ๐, ๐ฆ๐)=(400, 0)).
Obs4
Obs1
Obs2
Obs3
Obs5
Obs6
Obs7
200 400 600 8000X [mm]
0
100
200
300
400
500
600
700
800
900
Y [m
m]
(c) Navigation in case safe and danger segments ((๐ฅ๐, ๐ฆ๐)=(0, 0)).
Obs4
Obs1
Obs2
Obs3
Obs5
Obs6
Obs7
0
100
200
300
400
500
600
700
800
900Y
[mm
]
200 400 600 8000X [mm]
(d) Navigation in case safe and danger segments ((๐ฅ๐, ๐ฆ๐)=(400, 0)).
Figure 15: Path planning ((๐ฅ๐ก, ๐ฆ๐ก)=(750, 750)).
Table 4: Center coordinates of obstacles ((๐ฅ๐ก, ๐ฆ๐ก)=(750, 750)).Obstacles ๐ฅ๐๐๐ ๐ฆ๐๐๐ Obstacle 1 550 230Obstacle 2 630 450Obstacle 3 540 500Obstacle 4 400 500Obstacle 5 70 680Obstacle 6 160 400Obstacle 7 150 130
this case, we constate that there is a local minima problem.Therefore, the robot goes far away from obstacles and movesdirectly to the target (see Figures 16(c) and 16(d)).
Table 5: Center coordinates of obstacles ((๐ฅ๐ก, ๐ฆ๐ก)=(800, 500)).Obstacles ๐ฅ๐๐๐ ๐ฆ๐๐๐ Obstacle 1 550 240Obstacle 2 550 400Obstacle 3 540 510Obstacle 4 400 510Obstacle 5 200 560Obstacle 6 150 430Obstacle 7 150 260
From all simulation results, it is obvious to see that thedeveloped strategy is very reactive because the robot achievesthe obstacle avoidance in each modification of the robot
10 Mathematical Problems in Engineering
Obs4
Obs1
Obs2
Obs3
Obs5
Obs6
Obs7
200 400 600 8000X [mm]
0
100
200
300
400
500
600
700
800
900Y
[mm
]
(a) Navigation with safe segments ((๐ฅ๐, ๐ฆ๐)=(0, 0)).
Obs4
Obs1
Obs2
Obs3Obs5
Obs6
Obs7
200 400 600 8000X [mm]
0
100
200
300
400
500
600
700
800
900
Y [m
m]
(b) Navigation with safe segments ((๐ฅ๐, ๐ฆ๐)=(400, 0)).
Obs4
Obs1
Obs2
Obs3Obs5
Obs6
Obs7
0
100
200
300
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500
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800
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Y [m
m]
0 100 200 300 400 500 600 700 800 900โ100X [mm]
(c) Navigation with danger segments ((๐ฅ๐, ๐ฆ๐)=(0, 0)).
Obs4
Obs1
Obs2
Obs3Obs5
Obs6
Obs7
0 100 200 300 400 500 600 700 800 900โ100X [mm]
0
100
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400
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600
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900Y
[mm
]
(d) Navigation with danger segments ((๐ฅ๐, ๐ฆ๐)=(400, 0)).
Figure 16: Path planning ((๐ฅ๐ก, ๐ฆ๐ก)=(800, 500)).
and the target positions and in presence of safe and dangersegments.
After planning the safest and the shortest path, it isrequired for the mobile robot to track reference trajectoriesbased on sliding mode controller. Figure 17 shows that themobile robot always follows the reference trajectory.
To more illustrate the performance of the sliding modecontroller, the error positions, and the two speeds (rightand left) of the wheels for the cases. Figures 15(a) and 16(b)were presented in Figures 18 and 19. Figure 18 shows that thetracking errors tend to zero which allows concluding thatthe proposed control law system provides a good trackingtrajectory.
In addition to this, Figure 19 presents the evolution oftwo speeds (right and left) of the wheels. For example,for Figure 19(b), initially the mobile robot advances with
the same speeds for both wheels. As soon as obstacle 1 isdetected, the control system provides a larger right wheelspeed compared to the leftwheel speed.After passing obstacle1, the two speeds are equal until the robot reaches the target.As soon as obstacle 2 is detected, the controller systemprovides a larger right wheel speed than the left wheel speed.After passing obstacle 2, we notice that the speed of theleft wheel is larger than the right wheel. This is to turn themobile robot to the target position. Once the robot is orientedtowards the target, the two speeds are equal until the robotreaches the target.
6. Conclusion
In this paper, an algorithm which searches for a turningpoint based on free segments is presented. It handles two
Mathematical Problems in Engineering 11
Obs4
Obs1
Obs2
Obs3
Obs5
Obs6
Obs7
Desired trajectory
Planned path
200 400 600 8000X [mm]
0
100
200
300
400
500
600
700
800
900Y
[mm
]
(a) Tracking planned path of Figure 15(a).
Obs4
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Obs2
Obs3Obs5
Obs6
Obs7
Desired Trajectory
Planned path
200 400 600 8000X [mm]
0
100
200
300
400
500
600
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Y [m
m]
(b) Tracking planned path of Figure 16(b).
Figure 17: Tracking planned path.
โ0.5
0
0.5
๏ผฒ ๏ผ
20 40 60 80 100 1200Time (s)
20 40 60 80 100 1200Time (s)
20 40 60 80 100 1200Time (s)
โ0.5
0
0.5
๏ผณ ๏ผ
โ2
0
2
๏ผ
(a) Case of Figure 15(a).
โ0.5
0
0.5
๏ผฒ ๏ผ
โ5
0
5
๏ผณ ๏ผ
โ5
0
5
๏ผ
10 20 30 40 50 600Time (s)
10 20 30 40 50 600Time (s)
10 20 30 40 50 600Time (s)
(b) Case of Figure 16(b).
Figure 18: Tracking Errors (๐ฅ๐, ๐ฆ๐, ๐ผ๐).
different objectives: the safe path and the path length. Theadvantage of the developed algorithm is that the robot alwayscan move from the initial position to the target position, notonly safely, but also on the shortest path regardless the shapeof the obstacles and the change of goal position in the knownenvironment. In the other side, the proposed sliding modecontrol is an important method to deal with the system. Thiscontroller demonstrates a good tracking performances suchas robustness, stability and fast response. Simulation resultsare performed on a platform Khepera IV to demonstrate thatthe proposed method is a good alternative to solve the pathplanning and trajectory tracking problems.
As a future work, it could be interesting to determinatepaths in dynamic environment.
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper.
12 Mathematical Problems in Engineering
ร104
VR
VL
5 10 150Time (s)
โ200
โ150
โ100
โ50
0
50
100
150
200V L
,VR
(mm
/s)
(a) Case of Figure 15(a).
ร105
VR
โ400
โ300
โ200
โ100
0
100
200
300
400
V L,V
R(m
m/s
)
0.5 1 1.5 2 2.5 30Time (s)
VL
(b) Case of Figure 16(b).
Figure 19: Evolution of the two speeds (right and left).
References
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