Asian Journal of Control, Vol. 11, No. 2, pp. 214 225, March 2009
Published online in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/asjc.098
A NEW POTENTIAL FIELD METHOD FOR MOBILE ROBOT PATH
PLANNING IN THE DYNAMIC ENVIRONMENTS
Lu Yin, Yixin Yin, and Cheng-Jian Lin
ABSTRACT
A new potential field method for mobile robot path planning is proposed
in this paper. At present, most potential field methods are designed to be applied in the stationary environment, and several improved potential functions
have brought in the velocity factors in the dynamic circumstances. Based on
the consideration that the moving trend of the robot in the dynamic environments is also necessary to produce more reasonable path, this paper defines
new attractive potential function with respect to the relative position, velocity, and acceleration between the robot and the goal, as well as the repulsive
potential function with respect to the relative positions, velocities, and accelerations between the robot and the obstacles. The virtual forces are calculated
to make the robot plan its motion, not only with right positions, but also with
suitable velocities. Furthermore, the robot will keep a similar moving trend
with the goal and contrary trends with the obstacles. Finally, some methodic
simulations are carried out to validate and demonstrate the effectiveness of
the new potential field method.
Key Words: Path planning, mobile robot, potential field, dynamic environment.
I. INTRODUCTION
The problem of autonomous mobile robot path
planning has been researched extensively and deeply
for many years, and there are several classical methods showing considerable results. One of them is potential field method which has been studied a lot in past
decade [1–16]. The basic concept of the potential field
method is to fill the robot’s workspace with an artificial
potential field in which the robot is attracted to its goal
Manuscript received April 25, 2007; revised October 27,
2007; accepted March 3, 2008.
The authors are with the School of Information Engineering,
University of Science and Technology Beijing, 135# University
of Science and Technology Beijing, Beijing, China, 100083
(e-mail: [email protected]) and the Department of Computer Science and Information Engineering, Chaoyang University of Technology, No. 168, Jifong E. Rd., Wufong Township,
Taichung County 41349, Taiwan, China.
q
position and is repulsed away from the obstacles [3].
This method is particularly attractive because of its elegant mathematical analysis and simplicity. Most of the
previous studies use potential field methods to deal with
mobile robot path planning in stationary environments
where goals and obstacles are all stationary. However,
in many real-life implementations, the environments are
dynamic. Not only the obstacles are moving, so does
the goal [16].
There are several basic requirements for the conventional artificial potential field method: the robot is
considered as a particle in the workspace; the robot is
assumed to exist in an artificial potential field; the structure of the potential function reflects the structure of
free moving space.
The path planning process is iterative. First of
all, the force in the present position should be calculated, and then the robot moves one pace in the direction of this force. At that moment, the robot stops
if the goal is reached, otherwise, the iterative process
continues.
2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
L. Yin et al.: Mobile Robot Path Planning in the Dynamic Environments
1.1 Classical potential functions and virtual forces
The classical attractive potential function is:
Uatt (q) = 12 2g
(1)
where is a positive constant scaling factor, and
g = q − qg is the Euclidean distance between the
robot and the goal.
The attractive force can be obtained as:
Fatt (q) = −∇Uatt (q) = −(q − qg )
(2)
The classical repulsive potential function is:
⎧
1
1 2
⎪
⎨ 1
−
if (q) ≤ 0
(q) 0
(3)
Ur ep (q)= 2
⎪
⎩
0
if (q)>0
where is a positive constant scaling factor, (q) is the
minimum distance from the robot to the obstacle, and
0 is a positive constant, i.e. the influence distance of
the obstacles.
The repulsive force can be obtained as:
Fr ep (q) = −∇Ur ep (q)
⎧ 1
1
⎪
⎪
−
⎪
⎪
⎪
(q) 0
⎪
⎨
q−qc
1
=
×
⎪
⎪
2 (q) q−q ⎪
c
⎪
⎪
⎪
⎩
0
if (q)≤0
(4)
if (q)>0
where qc denotes the closest point to the robot in obstacle boundary.
1.2 Problem statement
In an effort to solve the problem of autonomous
mobile robot path planning in a dynamic environment,
a great many works have been done. Fujimura and
Samet [17] include time as one of the dimensions of
the model world and thus the moving obstacles can
be regarded as stationary in the extended world; Shih
and Lee [18] allow the planner to view the space-time
configuration of free space as disjoint polytopes that
represent a time-dependent environment consisting of
moving and stationary objects; Conn and Kam [19]
develop a motion planning algorithm for a point robot
traveling among moving obstacles in an N-dimensional
space; Ko and Lee [9] present a new solution approach
by defining a new concept of avoidance measure (AVM)
to drive the robot to avoid moving obstacles in real
q
215
time; Hussien [10] constructs repulsive potential functions by taking into account the velocity information.
However, there are still some inevitable problems in
these improved methods to baffle their applications.
Ge and Cui [16] present an effective potential
function to allow the robot to do a good job in dynamic
environment. The velocities of the robot, the goal, and
the obstacles are all taken into account. Nevertheless,
the robot still has a trend to go apart with the goal if
their accelerations are different in the final goal position. As a result, it is a reasonable idea to have consideration for the accelerations of the robot, the goal, and
the obstacles, then bring these factors into the potential
function.
In this paper, a new potential function conformation with consideration for both velocity and acceleration is proposed. In a dynamic environment where the
goal and the obstacles are moving, the attractive potential function is defined with respect to the relative position, velocity, and acceleration of the robot to the goal;
and the repulsive potential function is defined with respect to the relative position and velocity of the robot
to the obstacle, what’s more, the relative acceleration
of the robot to the obstacle is also a key factor in the
function. In this new potential function conformation,
the on-line measurement information is essential and
important to the path planning process.
This paper is organized as follows: In Section
II, the new potential function conformation is given;
meanwhile, the attractive and repulsive forces are described and analyzed respectively. Some simulation results which prove the effectiveness of the new function
are shown in Section III. At last, Section IV presents
the conclusion of the research work in this paper.
II. NEW POTENTIAL FUNCTIONS AND
VIRTUAL FORCES
2.1 Preconditions for the conformation of potential
function
In the new potential field this paper presents, the
robot can land on the goal without any requirement for
its velocity and acceleration; it can also land on the
goal where it shares the same velocity with the goal;
furthermore, the moving trend of the robot can be kept
accordant with the goal because of the acceleration consideration. These behaviours of the robot will be sorted
later.
For the simplification of the analysis, as conventional, we give the assumptions and statement as follows.
2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
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Asian Journal of Control, Vol. 11, No. 2, pp. 214 225, March 2009
the relative velocity between the robot and the goal to
the function have been done. In this attractive potential function conformation, the relative acceleration between the robot and the goal is brought in skillfully and
effectively as follows:
i 1
ai
Fi
mr
Uatt (q, v, a) = q q − qg i + v v − vg j
+a a − ag k
Fi
1
i
i 1
Fig. 1. A simple flow chart to show the iterative process to calculate
the acceleration a.
Assumption 1. The mass m r , position q and velocity
v of the robot are known. And in the process of calculation, the virtual force Flast in the last moving interval
is memorized, by which we can get the acceleration a
of the robot, i.e.
a=
(8)
Fatt (a i ) Frep (a i )
Flast
mr
(5)
So we can use this a to calculate F in the next
moving interval by the method in this paper, and then
the acceleration in the next moving interval will be obtained. The rest may be deduced by analogy. A simple flow chart is given to show the iterative process in
Fig. 1.
Assumption 2. The position qg , velocity Vg and acceleration ag of the goal are known.
Assumption 3. The obstacles are convex polygons
whose shapes, positions, velocities vobs and acceleration aobs can be accurately measured or calculated
on-line.
Based on these assumptions, the potential function
and force can be calculated as
U = Uatt + Ur ep
(6)
F = Fatt + Fr ep
(7)
and
2.2 Attractive potential function and virtual force
The attractive potential function is often defined
as a function with respect to the distance from the goal
to the robot previously, and some attempts to bring in
q
where q , v and a are some positive constant scaling
factors; i, j and k are scalar positive parameters; q, qg ,
v, vg , a and ag are all the functions of time t, and they
denote the positions of the robot and the goal, the velocities of the robot and the goal, and the accelerations
of the robot and the goal respectively, in n-dimensional
workspace, q = (x1 , . . . , xn )T , qg = (x g1 , . . . , x gn )T ,
v=(v1 , . . . , vn )T , vg =(vg1 , . . . , vgn )T , a = (a1 , . . . ,
an )T , ag = (ag1 , . . . , agn )T ; q − qg is the Euclidean
distance between the robot and the goal; v − vg is the
magnitude of the relative velocity between the robot
and the goal; a − ag is the magnitude of the relative
acceleration of the robot and the goal.
We can conclude in (8) that the attractive potential
Uatt (q, v, a) decreases as the relative distance, velocity, and/or acceleration between the robot and the goal
decrease, and it becomes minimum zero when and only
when the relative distance, velocity, and acceleration are
all zero at the same time.
Apparently, this new attractive potential function
will degenerate to the conventional quadratic function
when q = 12 , v = a = 0 and i = 2, i.e.
Uatt (q, v, a)=Uatt (q)=q q−qg i = 12 2g
(9)
When a = 0, the attractive potential function will
degenerate to the potential function form proposed in
paper [16], i.e.
Uatt (q, v, a) = Uatt (q, v) = q q − qg i
+v v − v(g) j
(10)
As the negative gradient of the attractive potential
function, the corresponding virtual attractive force Fatt
acting on the robot can be calculated.
Fatt (q, v, a) = −∇Uatt (q, v, a)
= −∇q Uatt (q, v, a)−∇v Uatt (q, v, a)
−∇a Uatt (q, v, a)
(11)
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L. Yin et al.: Mobile Robot Path Planning in the Dynamic Environments
and
qg
∇q Uatt (q, v, a) =
*Uatt (q, v, a)
*q
(12)
∇v Uatt (q, v, a) =
*Uatt (q, v, a)
*v
(13)
∇a Uatt (q, v, a) =
*Uatt (q, v, a)
*a
(14)
where the subscripts q, v, and a denote the gradients
in terms of position, velocity and acceleration respectively.
An important remark should be represented here:
When 0<i ≤ 1, Uatt (q, v, a) is not differential to q at
q = qg ; When 0< j ≤ 1, Uatt (q, v, a) is not differential
to v at v = vg ; and When 0<k ≤ 1, Uatt (q, v, a) is
not differential to a at a = ag . So the values of i, j
and k may bring in some problems on some but not all
occasions.
The behaviour of the robot are divided to three
sorts: we call it hard-landing when the robot gets the
goal and just share the same position without the consideration for its velocity or acceleration; we call the situation as semi-soft-landing when the robot gets the goal
and share the same position and velocity at that moment
irregardless of its moving trend factor, i.e. the acceleration a; and another behaviour is called soft-landing
when the robot reaches the goal with sharing not only
the same position and velocity, but also the same moving trend. Obviously, the soft-landing mode is the one
we expect the robot to implement.
Let
Fatt (q, v, a) = Fattq (q) + Fattv (v) + Fatta (a)
(15)
we can get
Fattq (q) = −
=−
=−
*Uatt (q,v,a)
*q
*(qq−qgi+vv−vg j+a a−agk )
*q
*(q q − qg i )
*q
= −q iq − qg i−1 ∇(q − qg )
= −q iq − qg i−1 eqgr
= q iq−qg i−1 eqrg
(16)
where eqrg is a unit vector pointing toward qg from q.
q
vg
x2
goal
Fattq
v vg
Fattv
Fattq
v
Fattv
ag
Fatt
a
a ag
q
robot
Fatta
o
x1
Fig. 2. The process to get the attractive force Fatt in 2-D workspace.
In the same way, we can also get that
Fattv (v)=−
*Uatt (q, v, a)
=v jv−vg j−1 evrg
*v
(17)
where evrg denotes the unit vector whose direction is
the same as the direction of the relative velocity of the
goal with respect to the robot, and
Fatta (a)=−
*Uatt (q, v, a)
=a ka−ag k−1 earg
*a
(18)
where earg denotes the unit vector whose direction is
the same as the direction of the relative acceleration of
the goal with respect to the robot.
The process to get the attractive force Fatt from the
relative position, velocity, and acceleration of the robot
with respect to the goal is shown explicitly in Fig. 2.
In this example, the workspace is set to be 2dimensional, and the positions, velocities, and accelerations are all marked in the coordinates. First of all, the
vector Fattq whose direction is from the robot to the
goal can be easily gotten by (16) as the conventional
method; and the vector Fattv whose direction is just
opposite to the direction of (v − vg ) can be gotten by
(17). Then we can get the vector (Fattq + Fattv ) by
superposition principle as marked in Fig. 2. The vector
(a − ag ) is also knowable in the similar way where
the vector (v − vg ) is obtained, so the vector Fatta
can be calculated in (18). At last, the resultant force
Fatt = Fattq + Fattv + Fatta is known by superposition principle and marked as the thick real line with an
arrowhead in Fig. 2.
From Fig. 2, it can be shown apparently that the
force Fattq devotes itself to make the robot reach the
present position of the goal, as well as the force Fattv
to make the robot move with the same velocity as the
2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
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Asian Journal of Control, Vol. 11, No. 2, pp. 214 225, March 2009
goal, and the force Fatta to make the robot keep the
same moving trend as the goal. As the resultant force
of those forces, Fatt is possessed of all properties they
have.
2.3 Repulsive potential function and virtual force
In some improved attempts for the conventional
repulsive potential function, the relative velocities between the robot and the obstacles have been added into
the function form in variable modes, but still, the factor
of moving trend has not been considered in the process
of obstacle avoidance.
In the paper, the traditional motion by which the
figures of the robot and obstacles are treated is used, i.e.
the robot is considered as round whose diameter is the
biggest dimension of the robot itself, and the distance,
velocity, acceleration between the robot and obstacle is
defined as those between the center of the round robot
and the nearest point to the robot in the obstacle boundary. So these closest points are considered in every moving step instead of the whole figures of the obstacles,
which allows us to ignore the rotation of the obstacle
with respect to the robot.
We bring both the velocity and acceleration factors
into the conformation of the repulsive potential function
as follows.
Ur ep (q, v, a)
⎧ 1
1
⎪
⎪ 1
+2 vr o +3 ar o ,
−
⎪
⎪
obs −Rr ob 0
⎪
⎪
⎪
⎪
⎪
if (obs −Robs ) ≤ 0 and vr o >0 and
⎪
⎪
⎪
⎪
⎪
⎪ ar o >0
⎨
1
1
=
1
+2 vr o ,
−
⎪
⎪
⎪
obs −Rr ob 0
⎪
⎪
⎪
⎪
⎪
if (obs −Robs )≤0 and vr o >0 and
⎪
⎪
⎪
⎪
⎪
ar o ≤0
⎪
⎪
⎩
0, if (obs − Robs )>0 or vr o ≤ 0
vr o = (v − vobs )T er o
(20)
ar o = (a − aobs )T er o
(21)
and
where er o is a unit vector pointing from the robot to
the obstacle. When vr o ≤ 0, the robot is moving away
from the obstacle, no avoidance behaviour is needed; on
the contrary, when vr o >0, the robot is moving toward
the obstacle, and an avoidance consideration should be
taken.
From (20) and (21), we can get that
∇v vr o = er o
(22)
∇q vr o = ∇q [(v − vobs ) er o ]
(qobs − q)
= ∇q (v − vobs )T
qobs − q
T
=
1
[vr o er o − (v − vobs )]
qobs − q
∇a ar o = er o
∇q ar o =
1
[ar o er o − (a − aobs )]
qobs − q
(23)
(24)
(25)
Let vr o⊥ er o⊥ be the velocity component perpendicular to vr o er o , and ar o⊥ er o⊥ be the acceleration component perpendicular to ar o er o , we can get that
(19)
In the repulsive potential function definition, Rr ob is
the radius of the robot; 0 is a positive constant reflecting the influence distance of the obstacles. The obstacle avoidance behaviour may need to be considered just
when the robot is inside the influence range of the obstacle; 1 , 2 and 3 are all positive scaling factors.
In the basic assumption that the position qobs ,
velocity vobs and acceleration aobs of the closest
point on the obstacle can be obtained on-line, obs =
q − qobs denotes the distance between the robot centre to the nearest point on the obstacle boundary. The
relative velocity and acceleration between the robot
q
and the obstacle in the direction from the robot to the
obstacle are given by
vr o⊥ er o⊥ = v − vobs − vr o er o
(26)
and
(27)
ar o⊥ er o⊥ = a − aobs − ar o er o
2 ,
ar o⊥
=
where
vr o⊥ = v−vobs 2 −vor
T
2 , and e
a−aobs 2 −aor
e
=
1.
So
(23)
and
(25)
r o⊥ r o
can be also expressed as
∇q vr o = −
1
vr o⊥ er o⊥
q − qobs (28)
∇q ar o = −
1
ar o⊥ er o⊥
q − qobs (29)
and
The detailed relationship among the vectors mentioned in above equations is shown explicitly in Fig. 3.
2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
219
L. Yin et al.: Mobile Robot Path Planning in the Dynamic Environments
x2
x2
obstacle
obstacle
v vobs a aobs
qobs
v vobs a aobs
vro ero
Frepq 2
vobs
aro ero
vro ero
aro ero
robot
v
Frepq 3
vobs
Frepq 3
aobs
Frepq 2 q
Frepq
a
q
qobs
Frepa
Frep
Frepv
o
aobs
v
a
Frepv
Frepq1
Frepa
robot
x1
x1
o
Fig. 4. The process to get the repulsive force Fr ep in 2-D workspace.
Fig. 3. The presentation of the vector relationship in 2-D workspace.
and
Now the virtual repulsive force generated by the
obstacle can be calculate as
Fr ep (q, v, a)
⎧
Fr epq + Fr epv + Fr epa ,
⎪
⎪
⎪
⎪
⎪
if (obs −Robs )≤0 and vr o >0 and ar o >0
⎪
⎪
⎨
= Fr epq + Fr epv ,
⎪
⎪
⎪
⎪ if (obs −Robs )≤0 and vr o >0 and ar o ≤0
⎪
⎪
⎪
⎩
0, if (obs −Robs )>0 or vr o ≤0
(30)
−er o
= −1 −
(obs − Rr ob )2
1
vr o⊥ er o⊥
−2 −
q − qobs 1
ar o⊥ er o⊥
−3 −
q − qobs q
ar o⊥
er o⊥
q − qobs +2 vr o +3 ar o
= −2 er o
= 2 eor
(32)
Fr epa = −∇a 1
1
1
−
obs −Rr ob 0
+2 vr o +3 ar o
= −3 er o
= 3 eor
(33)
2.4 Local minimum problem
eor
vr o⊥
er o⊥
+ 2
q − qobs (obs − Rr ob )2
+3
1
1
−
obs −Rr ob 0
Fig. 4 shows the process to get the repulsive force
Fr ep marked as the thick real line with an arrowhead
in 2-D workspace. According to the superposition
principle, we have Fr epq1 = (1 eor )/(obs − Robs )2 ,
Fr epq2 = (2 vr o⊥ er 0⊥ )/q − qobs and Fr epq3 =
(3 ar o⊥ er 0⊥ )/q − qobs . The virtual force component
Fr epq1 which will keep the robot away from the obstacle is in the opposite direction of er o . The components
Fr epq2 and Fr epq3 will act as the steering forces for
detouring. Fr epv will strengthen the repulsive effect
according to the parameter 2 , and Fr epa will do the
same job according to 3 only when ar o >0.
In (30), we can get that
1
1
−
Fr epq = −∇q 1
obs − Rr ob
0
+2 vr o +3 ar o
= 1
Fr epv = −∇v 1
(31)
When employing the new potential functions for
dynamic motion planning, local minimum problem does
exist and should be taken care of. For example, consider the case when the robot, the obstacle and the goal
move in the same direction along the same line and the
obstacle is in between, as shown in Fig. 5. Assuming
that the goal moves outward or synchronously with the
2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
220
Robot
Asian Journal of Control, Vol. 11, No. 2, pp. 214 225, March 2009
Obstacle
Goal
Fig. 5. Local minimum problem.
obstacle (this assumption ensures that the obstacle is
between the robot and the goal all the time), the robot
is obstructed by the obstacle and cannot reach the goal.
To solve the problem, the simplest method is to
keep the robot moving according to the total virtual
force as usual and waiting for the goal or obstacles to
change their motion. Since the environment is highly
dynamic where the goal and obstacles are moving, the
situation where the configuration of the obstacles and
goal keeps static is rare. Thus, the waiting method is
often adopted.
However, if after a certain periods waiting, the
configuration of the goal and obstacles is still unchanged
and the robot is still trapped, it can then be assumed
that the configuration will not change temporarily, and
the robot will have to take other approaches to escape
from the trap situation. Since the configuration of the
robot, obstacles and goal is relatively stationary, the
conventional local minimum recovery approaches such
as wall following method, which are designed for the
stationary environment cases, can be applied.
III. SIMULATION RESULTS
Comprehensive simulation studies are carried out
to validate the effectiveness and efficiency of the new
potential field conformation.
In the real situation, the electric motor of the robot
drives it to move in every moving interval according to
each velocity that the algorithm gives. The calculation
and execution procedure compose an iterative discrete
process to follow the goal and avoid the obstacles. So
a discrete mode is applied to simulate the behavior of
the robot in the real world.
The simulation is conducted in a MATLAB environment to demonstrate the whole planning process in
a clearly visible way.
3.1 Planning behaviour to goal without obstacle
For simplicity, let’s consider the environment
where no obstacle exists firstly. In this circumstance,
the attractive potential is the only one to influence the
behaviour of the robot.
q
Before the simulation is carried on, The analysis
of the choice of the parameters is necessary. The new
attractive potential function contains six parameters, q ,
v , a , and i, j, k. By choosing different parameter
settings, different performances can be obtained. Here,
consider the case where i = j = k = 2 according to
the most familiar choice in the path planning problem.
Then from the equation (15), (16), (17) and (18), The
total attractive force is given by
Fatt = q iq−qg i−1 eqrg +v jv−vg j−1 evrg
+a ka − ag k−1 earg
= 2q q − qg eqrg + 2v v − vg evrg
+2a a − ag earg
= 2q (qg − q) + 2v (vg − v)
+2a (ag − a)
(34)
The ultimate purpose of the new method is to make
the robot maintain the same position, velocity and acceleration with the goal, so the desired situation is that
q = qg , v = vg and a = ag , from which we can obtain
the desired force Fdes = 0. Once that situation occurs,
the robot will keep the present moving state until the
state of the goal changes.
Let Fdes = Fatt , we have
2q (qg −q)(t)+2v (vg −v)(t)+2a (ag −a)(t)=0 (35)
Define e(t) = qg (t) − q(t), the equation (35) can
be written as
ë(t) + (v /a )ė(t) + (q /a )e(t) = 0
(36)
Because all the parameters in (36) are positive,
system (36) is stable but with different performances
for different choices of parameters. The characteristic
equation is given by
s 2 + 2εs + 2 = 0
(37)
where ε and are the damping ratio and
natural frequency of the system, and ε = v /(2 q a ), =
q / a . It is obvious that a = 0.
For conventional attractive potential function,
v = 0, which leads to a zero damping ratio, the relative
position between the robot and the target will oscillate
with constant amplitude at the natural frequency , i.e.
the robot will be oscillating around the target as will
be verified by simulation later.
For the new attractive potential function, v >0,
and the damping ratio ε>0. The position error reduces asymptotically. For 0<ε<1, the system is under
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L. Yin et al.: Mobile Robot Path Planning in the Dynamic Environments
30
30
Original position of the goal
25
25
20
20
Trajectory of the goal
x2 15
x2 15
The trajectory of the goal
Trajectory of the robot
10
10
Goal' position at t=12s
Original position of the robot
5
5
0
0
The position of the robot at t=12s
0
5
10
15
20
25
30
35
40
45
50
0
5
10
30
25
The trajectory of the goal
20
x2 15
The trajectory of the robot
The original position of the robot
5
The position of the robot at t=12s
0
0
5
10
15
20
25
30
35
40
45
50
x1
Fig. 7. The panning result at t = 12s for i = j = k = 2, and q = 3,
v = a = 0.5.
damped-the tracking error converges to zero with oscillation. For ε>1, the system is over damped-there is
no overshoot but the responses are very sluggish.
If there is no requirement of soft-landing, any positive q , v and a may be used to allow the robot to
hit the goal. However, if soft-landing is required, the
velocity and acceleration error should be zero once the
robot reaches the goal. When ε ≥ 1, the position, velocity and acceleration of the robot approach those of
the goal as the time approaches infinity. That means the
robot will soft-land on the goal as time increases.
For other choices of i, j, and k, different performances can be obtained. As the analysis becomes difficult to obtain, numerical simulation investigations are
presented next.
First of all, a trajectory of moving goal is given
in Fig. 6 whose original position qg = (10, 27.5)T ,
original velocity vg = (0.4, −0.4)T , and acceleration
is a constant vector ag = (0.4, −0.2)T . For being
q
20
25
30
35
40
45
50
Fig. 8. The panning result at t = 12s for i = j = k = 2, and q = 0.5,
v = a = 3.
Fig. 6. The trajectory of the goal.
10
15
x1
x1
convenient to compare the path planning results, we
present the simulation results at the same time t = 12s.
Now we give the parameters of the robot: the original position q = (2.5, 25)T , the original velocity
v = (0, 0)T , and the original acceleration a = (0, 0)T ,
so the robot is stationary in the original position. At
last, the mass of the robot m r = 10, and the moving
interval of the robot is set to be ti = 0.3s.
In Fig. 7, the parameters are chosen as i= j=k=2,
and q = 3, v = a = 0.5. It’s clear that the robot can
follow the goal very quickly, but the trajectory is oscillating a lot. The reason to explain this phenomenon is
that the parameter q to decide the influence degree of
relative position between the robot and the goal is set
big enough, whereas the ones v and a to decide the
influence degrees of relative velocity and acceleration
is so small. In this circumstance, the robot plans its
behaviour according to the position factor mainly with
little consideration of velocity and acceleration factor
so that the velocity direction and moving trend look so
different with the goal. For that reason, let’s adjust the
parameters to increase the influence degree of the relative velocity and acceleration, meanwhile, decrease the
influence degree of the relative position. We set q =
0.5 and v = a = 3 without i, j, k changed. Then the
moving trajectory of the robot can be obtained as shown
in Fig. 8. From the planning result, it’s clear that the
velocity direction and moving trend are very similar to
the goal, however, it takes too many steps for the robot
to follow the position of the goal, so the robot follows
the goal too slowly. The reason of this phenomenon is
that the parameter q to decide the influence degree of
the relative position between the robot and the goal is
too small, whereas the ones v and a to decide the
influence degrees of the relative velocity and acceleration are big, so that the robot plans its behaviour
2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
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Asian Journal of Control, Vol. 11, No. 2, pp. 214 225, March 2009
30
30
The trajectory of the robot
25
x2
20
20
15
x2 15
The trajectory of the goal
10
i=2
j=1.5
k=1.5
t=12s
25
The trajectory of the goal
10
Original position of the robot
5
5
The trajectory of the robot
The position of the robot at t=12s
0
0
5
10
15
20
25
30
35
40
45
0
50
0
5
10
15
20
25
30
35
40
45
50
x1
x1
Fig. 9. The panning result at t = 12s for i = j = k = 2, and q = 2,
v = 4, a = 3.
Fig. 11. The panning result at t = 12s for q = 2, v = 4, a = 3, and
i = 2, j = k = 1.5.
30
30
Trajectory of the goal
25
20
The trajectory of the goal at t=12s
i=1.5
j=2
k=2
t=12s
25
The trajectory of obstacle1 at t=12s
20
x2
15
x2 15
10
10
Trajectory of the robot
5
5
The trajectory of obstacle2 at t=12s
0
0
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
x1
x1
Fig. 10. The panning result at t = 12s for q = 2, v = 4, a = 3, and
i = 1.5, j = k = 2.
according to the relative velocity and acceleration
mainly with little consideration of the position factor.
To obtain excellent path planning trajectory, a reasonable choice of the parameters are very important. Now
we choose parameters as i = j = k = 2, q = 2,
v = 4, and q = 3. The planning result is shown in
Fig. 9 which presents a considerably ideal following
path. Similarly, to present the influence of the parameters i, j and k, we keep that q = 2, v = 4 and
a = 3, and choose different values of i, j and k, the
planning results are shown in Figs 10 and 11 respectively. It’s clear that the bigger the value i and smaller
the values j and k are, the faster the robot follows the
goal and the worse the oscillation phenomenon is; on
the contrary, the smaller the value i and bigger the values j and k are, the slowlier the robot follows the goal
and the slighter the oscillation phenomenon is.
q
Fig. 12. The trajectories of the goal and obstacles between t = 0s and
t = 12s.
3.2 Planning behaviour to goal with obstacles
After testing the planning effectiveness in the environment without obstacle, a dynamic multi-obstacle
environment is assumed for the robot to follow its goal
by the new potential function. In the first instance, trajectories of the goal and obstacles at t = 12s are shown
in Fig. 12, where the moving state of the goal is still the
same as the above-mentioned situation. For obstacle1,
its original position and velocity are qobs1 = (8, 17)T ,
vobs1 = (2.5, −2)T , respectively, and its acceleration
is constant vector aobs1 = (−0.1, 0.15)T ; and for obstacle2, its original position and velocity are qobs2 =
(17, 22)T , vobs2 = (−3, −1)T , and its acceleration is
aobs2 = (0.5, 0.05)T . Based on this situation, the parameters of the potential function are chosen as i =
j = k = 2, q = 3, v = 5, a = 4, 1 = 12, 2 =
15, 3 = 15, 0 = 4, and the radius of the robot is
2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
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L. Yin et al.: Mobile Robot Path Planning in the Dynamic Environments
30
30
The trajectory of the goal at t=0.9s
The trajectory of the goal at t=1.8s
25
25
The trajectory obstacle2 at t=0.9s
20
q3
x2 15
The trajectory of obstacle2 at t=1.8s
20
The trajectory of obstacle1 at t=0.9s
x2 15
q3
q2
q1
q2
10
10
The trajectory of the robot at t=0.9s
q1
5
5
q0
0
5
10
The trajectory of obstacle1 at t=1.8s
q0
0
0
q6
q5
q4
15
20
25
30
35
40
45
50
0
5
10
15
20
x1
Fig. 13. The trajectories of the robot, goal and obstacles between
t = 0s and t = 0.9s.
35
40
45
50
Fig. 15. The trajectories of the robot, goal and obstacles between
t = 0s and t = 1.8s.
The trajectory of the goal at t=2.1s
The trajectory of the goal at t=1.2s
25
25
x2 15
q3
q2
10
The trajectory of obstacle2 at t=2.1s
qobs27
20
The trajectory of obstacle2 at t=1.2s
20
The trajectory of obstacle1 at t=1.2s
q4
x2 15
The trajectory of the robot at t=1.2s
10
q1
q7
q6
q5
The trajectory of obstacle1 at t=2.1s
q0
0
0
5
10
qobs20
q0
5
5
15
20
25
30
35
40
45
50
0
5
Fig. 14. The trajectories of the robot, goal and obstacles between
t = 0s and t = 1.2s.
Rr ob = 0.7. Figure 13 shows the states of the robot, goal
and obstacles at t = 0.9s, where the original position
of the robot is presented as q0 , and its current position
is q3 , so the robot is just in the third moving interval.
It’s apparently to see that the robot has already entered
into the influence range of obstacle1, which means the
robot is going to collide with obstacle1 in next moving
interval if no obstacle avoidance behaviour is taken. So
the repulsive force Fr ep by obstacle1 begins to work.
It makes the robot adjust its own velocity to avoid the
obstacle. The obstacle avoidance result in the next interval at t = 1.2s is shown in Fig. 14, where q4 denotes
the current position of the robot, and it can be seen that
the robot has avoided the collision with obstacle1 successfully. Then the robot continues to follow the dynamic goal without any repulsive force acting until it
moves to the position q6 at time t = 1.8s as Fig. 15
shows. The robot has entered into the influence range of
10
15
20
25
30
35
40
45
50
x1
x1
q
30
30
30
0
25
x1
Fig. 16. The trajectories of the robot, goal and obstacles between
t = 0s and t = 2.1s.
obstacle2 by which the repulsive force has been produced. In the effect of this force, the velocity of the robot
is adjusted again to avoid the collision with obstacle2.
Fig. 16 shows the avoidance result where the robot is
in the position q7 , as well as obstacle2 in the position
qobs27 .
By repeating the process to detect the environment
around, calculate the force acting, adjust the velocity to
follow the goal and avoid the obstacles continually, the
robot can accomplish the path planning task very well.
The planning result at t = 12s is given in Fig. 17.
IV. CONCLUSION
In this paper, a new potential field method is
proposed for the path planning of autonomous mobile
robots, by which the robot can follow the dynamic goal
2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
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Asian Journal of Control, Vol. 11, No. 2, pp. 214 225, March 2009
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The trajectory of the goal at t=12s
The trajectory of the robot at t=12s
25
20
x2 15
10
The trajectory of obstacle2
at t=12s
The trajectory of obstacle1 at t=12s
5
0
0
5
10
15
20
25
30
35
40
45
50
x1
Fig. 17. The trajectories of the robot, goal and obstacles between
t = 0s and t = 12s.
and avoid dynamic obstacles at the same time. The
new potential functions do not only take the relative
positions and velocities of the robot with respect to the
goal and the obstacles into account, but also bring in
the relative acceleration factors. As the negative gradient of the potential function, the virtual force exhibits
its effect to keep the robot to follow the goal with the
similar moving trend and avoid obstacles with contrary
moving trends. By choosing the suitable parameters
for the potential function, the reasonable planning path
can be produced successfully. At last, some methodic
simulations are carried out to verify and demonstrate
the effectiveness of the new potential field method for
mobile robot path planning.
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2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
L. Yin et al.: Mobile Robot Path Planning in the Dynamic Environments
Lu Yin received the B.S. and M.S.
degrees from the School of Information Engineering in University
of Science and Technology Beijing, China in 2003 and 2005. Now
she is a Ph.D. Student in Control
Theory and Control Engineering
of University of Science and Technology Beijing, China. Her main
research field includes robotics, artificial life, intelligent control, etc. The project she
joined in is National Natural Science Foundation of
China of the ‘Research and Application on Intelligent Control System Based on Artificial Life’ (NSFC:
60374032). Email: [email protected]
q
Yixin Yin was born in 1957. He
received the B.S. and M.S. degrees from the Department of Automation in Beijing University of
Iron and Steel Technology, China
in 1982 and 1984, and Ph.D. degree in the School of Information Engineering in University of
Science and Technology Beijing,
China. Now he is the Professor in
the School of Information Engineering of University
of Science and Technology Beijing, China, and council member of Chinese Association for Artificial Intelligence, council member of Chinese Association of
Automation, deputy director of Intelligent Automation
Committee of Chinese Automation Association, Committeeman of CSS&CAD committee of Chinese Association for System Simulation, etc. His main research
field includes artificial life, intelligent control, modeling
and control for complex systems, network control, flow
industry automation, etc. Email: [email protected]
225
Cheng-Jian Lin (S’93-M’95) received the B.S. degree in electrical
engineering from Ta-Tung University, Taiwan, in 1986 and the M.S.
and Ph.D. degrees in electrical
and control engineering from the
National Chiao-Tung University,
Taiwan, in 1991 and 1996. From
April 1996 to July 1999, he was
an Associate Professor in the Department of Electronic Engineering, Nan-Kai College,
Nantou, Taiwan. From August 1999 to January 2005, he
was an Associate Professor in the Department of Computer Science and Information Engineering, Chaoyang
University of Technology. From February 2005 to July
2007, he was a full Professor in the Department of Computer Science and Information Engineering, Chaoyang
University of Technology. Currently, he is a full Professor of Electrical Engineering Department, National University of Kaohsiung, Kaohsiung, Taiwan. He served as
the chairman of Computer Science and Information
Engineering Department, Chaoyang University of
Technology from 2001 to 2005. He served as the library
director of Poding Memorial Library, Chaoyang University of Technology from 2005 to 2007. Dr. Lin served as
the Associate Editor of International Journal of Applied
Science and Engineering from 2002 to 2005. His current
research interests are soft computing, pattern recognition, intelligent control, image processing, bioinformatics, and FPGA design. He has published more than
150 papers in the referred journals and conference proceedings. Dr. Lin is a member of the Phi Tau Phi. He
is also a member of the Chinese Fuzzy Systems Association (CFSA), the Chinese Automation Association,
the Taiwanese Association for Artificial Intelligence
(TAAI), the IEICE (The Institute of Electronics, Information and Communication Engineers), and the IEEE
Computational Intelligence Society. He is an executive
committee member of the Taiwanese Association for
Artificial Intelligence (TAAI). Email: [email protected]
or [email protected]
2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society