Content uploaded by Ahmad Ataka Awwalur Rizqi
Author content
All content in this area was uploaded by Ahmad Ataka Awwalur Rizqi on Jul 09, 2018
Content may be subject to copyright.
King’s Research Portal
Link to publication record in King's Research Portal
Citation for published version (APA):
Ataka, A., Lam, H-K., & Althoefer, K. A. (2018). Reactive Magnetic-field-inspired Navigation Method for Robots
in Unknown Convex 3D Environments. IEEE Robotics and Automation Letters.
Citing this paper
Please note that where the full-text provided on King's Research Portal is the Author Accepted Manuscript or Post-Print version this may
differ from the final Published version. If citing, it is advised that you check and use the publisher's definitive version for pagination,
volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are
again advised to check the publisher's website for any subsequent corrections.
General rights
Copyright and moral rights for the publications made accessible in the Research Portal are retained by the authors and/or other copyright
owners and it is a condition of accessing publications that users recognize and abide by the legal requirements associated with these rights.
•Users may download and print one copy of any publication from the Research Portal for the purpose of private study or research.
•You may not further distribute the material or use it for any profit-making activity or commercial gain
•You may freely distribute the URL identifying the publication in the Research Portal
Take down policy
If you believe that this document breaches copyright please contact librarypure@kcl.ac.uk providing details, and we will remove access to
the work immediately and investigate your claim.
Download date: 09. Jul. 2018
IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JUNE, 2018 1
Reactive Magnetic-field-inspired Navigation Method
for Robots in Unknown Convex 3D Environments
Ahmad Ataka1,2, Hak-Keung Lam1, and Kaspar Althoefer2
Abstract—With a shift in current robotics application from
known, well-defined environments towards unknown environ-
ments, the robot’s ability to avoid unknown obstacles in real-time
whilst relying on limited information about spatial constraints
in its path becomes essential. Taking inspiration from the laws
of electromagnetism, we present a novel navigation method,
whereby the moving robot induces an artificial electric current
onto the obstacle surface generating, in turn, a magnetic field
guiding the robot along the obstacle’s boundary without affecting
its kinetic energy. Our method has several advantages over
existing methods: 1) it guides point-like robots towards the
goal without suffering from local minima in 3D environments
populated with convex obstacles, 2) it does not need any prior
knowledge of obstacle positions and geometries, 3) it only requires
environmental sensor information that is spatially and temporally
local to generate motion commands iteratively. Our navigation
method is tested in simulations and experiments, showing that
a point-to-point navigation of point-like robots and the end
effector of the Baxter’s arm has been successfully achieved in a
collision-free manner towards a goal position in a 3D environment
populated with unknown convex obstacles.
Index Terms—Reactive and Sensor-Based Planning, Collision
Avoidance, Motion and Path Planning.
I. INTRODUCTION
THE fields of robot navigation and path planning have
been extensively studied in the robotics community over
the past thirty years [1], [2]. Developed navigation methods
range from classic geometrical planning, which tries to pro-
duce a series of collision-free configurations from an initial
configuration to the final configuration, to motion planning,
which produces the necessary control signals needed to move
the robot towards the goal. Achieving collision-free navigation
becomes more challenging as the robot applications move from
well-defined environments towards unknown environments
where the obstacle’s position and geometry are not known
beforehand. Thus, the robot’s ability to reach the goal whilst
Manuscript received: February, 24, 2018; Revised May, 22, 2018; Accepted
June, 20, 2018.
This paper was recommended for publication by Editor Tamim Asfour upon
evaluation of the Associate Editor and Reviewers’ comments. This work was
supported by King’s College London, the EPSRC in the framework of the
NCNR (National Centre for Nuclear Robotics) project (EP/R02572X/1), q-
bot led project WormBot (2308/104059), and the Indonesia Endowment Fund
for Education, Ministry of Finance Republic of Indonesia.
1A. Ataka and H.K. Lam are with the Centre for Robotics Research (CoRe),
Department of Informatics, King’s College London, WC2R 2LS, United
Kingdom ahmad_atak_awwalur.rizqi@kcl.ac.uk
2A. Ataka and K. Althoefer are with the Centre for Advanced Robotics
@ Queen Mary (ARQ), Faculty of Science and Engineering, Queen Mary
University of London, Mile End Road, London E1 4NS, United Kingdom
k.althoefer@qmul.ac.uk
Digital Object Identifier (DOI): see top of this page.
avoiding obstacles on-line with limited information of the
environment becomes important.
Early robot navigation and planning techniques emerged
from the well-known piano mover’s problem, where the goal is
to move a piano through a cluttered environment, often using
the configuration space of the robot for planning [3], [1]. Ap-
proaches based on the configuration space, though, are plagued
by the costly numerical computation, especially when dealing
with high-degree-of-freedom robots [4]. Many variants of the
original configuration space approach have been developed,
with an aim to create practical solutions that can achieve real-
time path planning and navigation. For example, sampling-
based planning does not require a complete description of
the C-Space and can produce planning results more quickly.
Popular examples of this approach are probabilistic roadmaps
(PRMs) [5] and rapidly-exploring random trees (RRTs) [6]
which have triggered a lot of variances as reported in [7].
However, most of these methods assume to have perfect prior
knowledge of the environment. Other approaches which aim
to generate paths for static environments and modify the path
as the environment changes also suffer from the same problem
[8], [9].
For a robot with no prior knowledge of the environment, a
more suitable method is sensor-based navigation, in which the
planning is seen as a control problem. In this approach, the
environment is sensed by the robot and the information is used
to generate a trajectory on-line by producing a required motion
signal considering the dynamics or kinematics model of the
robot. It is also possible to use only the current sensor data,
hence, forgetting past data, as, for example done in reactive
navigation [10]. One popular example of a reactive navigation
method is the Artificial Potential Field (APF) method inspired
by electrostatic phenomena where the robot is repelled by
obstacles and attracted towards the goal [11]. Its simplicity and
elegance triggered a lot of research and led to the development
of many APF variances such as the one reported in [12].
However, its drawbacks are also well documented, mainly
the local minima problem whereby the robot can get stuck
in undesired configurations even for fairly simple obstacle
configurations even though a solution (a collision-free path
to the goal) exists [2].
The inherent local minima in the artificial potential field
method motivated the incorporation of a vortex field, where
the field produced by an obstacle is designed to circulate the
obstacle surface [13]. However, the authors do not provide any
stability or collision avoidance guarantee. Taking inspiration
from the phenomena of wave expansion, the fast-marching
method generates a local-minima-free potential by starting
2 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JUNE, 2018
a ”wave” from the goal configuration until it reaches the
initial configuration [14]. This method, though, relies on the
knowledge of the environment prior to the robot’s movement.
Another method exploited the power diagram and convex
optimization to design a globally stable feedback planner
with unique attractor in the goal position [15]. Unlike the
classical navigation function method, this method only needs
to know the location of the nearby obstacle obtained from an
onboard sensor. Nonetheless, this approach is only suitable
for a topologically simple planar environment consisting of
only spherical obstacles and robot. Other studies suggested
applying the properties of an artificially-induced magnetic
field for robot navigation, such as [16]-[17], which claim
to resolve the problem of entrapment in local minima. This
method though also requires the geometry and location of
obstacles to be known beforehand. In [18], the concept of
circular fields was introduced and applied to partially unknown
environments. This method, however, requires prior knowledge
of the location of the centre of the obstacles.
The concept of ”circular fields”, where the field pushes
the robot onto a circular path rather than repelling the robot
away from the obstacle, as described in [16]-[18], was further
extended. Similar approaches under the heading of ”gyro-
scopic force” were used for planar point-like robots to do
boundary following [19], obstacle avoidance [20] and, in the
context of multi-agent systems, formation control [20]. A
recent work explored the possibility of applying the sliding-
mode-based navigation to handle maze-like and even dynamic
environment [21]. However, all these methods are designed
specifically for planar mobile robot applications. There were
also recent efforts to apply the concept of gyroscopic forces in
3D for fully-actuated and under-actuated system [22] but the
method is limited to cylindrical and spherical obstacles with
perfectly-known geometry. Gyroscopic-force-based algorithms
were also used for multi-robot formation in 3D [23]-[25].
However, none of these works deals with the problem beyond
the scope of collision avoidance among point-like agents for
formation control.
In this paper, we present a magnetic-field-inspired algorithm
for reactive robot navigation. An artificial electric current,
induced by the robot on the obstacle surface, will generate
a magnetic field which alters the robot’s movement without
affecting its energy. The algorithm is able to guide point-
like robots towards the goal despite using only local sensory
information of the environment and without the need for prior
knowledge of obstacle position and geometry. Our algorithm
goes beyond the standard APF method, being capable of
generating paths in environments with convex obstacle without
experiencing local minima. Our approach also outperforms
previous magnetic-field-inspired navigation methods [16]-[18]
since it does not rely on prior knowledge of the obstacles’
geometrical properties. Our approach also provides a clear
improvement over gyroscopic force based approaches [19]-
[25] as our method is designed to be generic so that it can be
used not only for planar mobile robots but also for robots in
3D environments with an unknown arbitrarily shaped convex
obstacle. This work extends our previous work which describes
the preliminary formulation of the magnetic-field-inspired
(a) (b)
dl0r
dB
l0v
BF
Fig. 1. (a) An electric current flows in the direction of dlowill produce a
magnetic field dB, directed into the page. (b) A charged particle under the
influence of the magnetic field Bproduced by current lochanges it motion
direction due to the force F. For robot navigation, this particle can be seen
as a robot with the current-carrying wire serves as an obstacle surface.
navigation method applied to mobile robot operating in planar
environments only [26]. To the best of our knowledge, this is
the first time a general reactive navigation method exploiting
magnetic field properties is applied to the problem of robot
navigation in 3D environments without prior knowledge of
obstacles and capable of guiding the robot to the goal in a
globally stable way.
II. IN SPI RATIO N
The inspiration for our method comes from the laws of
electromagnetism. A wire segment of length dlocarrying an
electrical current ioas depicted in Fig. 1a will produce a
magnetic field dB as described in the following [27]
dB =µ0
4π
iodlo×r
|r|3,(1)
where rdenotes the position vector of a point in space with
respect to the wire segment, µ0is a permeability constant, and
×denotes the vector cross product operation.
A particle with positive charge qmoving close to the
current-carrying wire will be affected by the presence of
the magnetic field as depicted in Fig. 1b. A force Fwhose
direction is perpendicular to both particle’s velocity vand
magnetic field Bwill be applied to the particle as a result
of this interaction and is given by [27]
F=qv×B.(2)
Substituting Bfrom eq. (1) and dropping the infinitesimal
notation, the force acting on the particle is given by
F=µ0qio
4π
v×(lo×r)
|r|3.(3)
The magnetic field interacts with the electric charge in a way
that the produced force is always perpendicular to the moving
charge velocity direction and, thus, changing its movement
direction. Inspired by this physical phenomenon, we can think
of a robot as a charged particle with velocity vand the obstacle
surface as a current-carrying wire. The artificial current lo
flowing on the obstacle surface located at position rowith
respect to the robot is designed in such a way that generated
force Fwill guide the robot away from a head-on collision
and follow obstacle boundary instead.
It is noted that, by definition, ro=−r, so eq. (3) can be
rewritten in a more general form as follows
F=cla×(ro×lo)f(|ro|,|˙
p|),(4)
ATAKA et al.: REACTIVE MAGNETIC-FIELD-INSPIRED NAVIGATION METHOD FOR ROBOTS IN UNKNOWN CONVEX 3D ENVIRONMENTS 3
(a) (b)
lo
la
ro
y
x
lo
rola
θ
Fig. 2. (a) The workspace of a point-like robot moves towards the goal
(red circle) in the vicinity of the polygonal obstacle (black). loand ladenote
the artificial current on the obstacle surface and the robot’s velocity. (b) The
scenario of a robot moving in the vicinity of a flat obstacle surface.
where c>0 is a scalar constant, lastands for the robot’s
velocity direction, lostands for the artificial current on the
obstacle surface, and f(|ro|,|˙
p|)≥0 is a scalar function
which depends on the robot-to-obstacle distance |ro|and/or
robot’s speed |˙
p|. We introduce a skew-symmetric matrix ˆ
l
to replace the vector cross product operation l×of a vector
l=lxlylzTas follows
ˆ
l=
0−lzly
lz0−lx
−lylx0
.(5)
III. PROP OSE D ALGORITHM
To explain the idea of artificial current generation on the
obstacle surface, it is better to analyze a point-mass robot
moving in R2, as depicted in Fig 2a. When the robot first
senses the obstacle surface in front of it, there are 2 choices of
movement: either following the surface to the left or right. To
minimize unwanted oscillation, the induced current direction
loon the obstacle surface (which will be followed by the robot)
is designed to be the projection of the robot’s velocity direction
laon to the obstacle surface.
With no knowledge of the obstacle geometry, we can create
a reasonable assumption regarding the obstacle surface relying
only on range sensor information. For an obstacle whose
closest point sensed by the robot located in position rowith
respect to the robot, we assume that the obstacle surface at this
point is perpendicular to the direction of ro. This simplification
is reasonable since we want the robot to move perpendicular to
the vector connecting the robot and the closest obstacle point
at any time. Using geometry, the current direction locan then
be written as
lo=la−(lT
aro)ro
|ro|2.(6)
Using this formulation, there could be a special case where the
artificial current becomes zero, i.e. when the robot’s direction
lais in line with vector ro. The solution to this problem will
be explained at the end of this section, where the current will
be slightly modified into (29).
To make the robot move towards the direction of lo, the
force equation in (4) is modified as follows
F=cla×(lo×la)f(|ro|,|˙
p|).(7)
For a robot with position vector p, the robot’s velocity direc-
tion lais defined as la=˙
p
|˙
p|.
Lemma 1. The obstacle avoidance force Fin (7) will not
change the magnitude of the robot’s velocity |˙
p|.
Proof. Assuming c f (|ro|,|˙
p|) = 1 and lo×la=a, using the
skew-symmetric definition in (5), we get
lT
aF=lT
a(ˆ
laa) = 0.(8)
Using Newton’s law of motion, for a robot with mass m, speed
v=|˙
p|, and velocity ˙
p=vla, we get
F=md˙
p
dt =m(dv
dt la+vdla
dt ).(9)
Combining with (8), we get
lT
aF=m(dv
dt +vlT
a
dla
dt ) = 0.(10)
Due to the fact that lais a unit vector with a constant magni-
tude |la|=1, dla
dt refers to the change of the vector’s direction
only. From geometry, the change in a vector’s direction is
perpendicular to the vector direction itself. Hence, we can
conclude that lT
a
dla
dt =0, and then we can simplify (10) into
lT
aF=mdv
dt =0. Since the mass is not zero, the only solution
is dv
dt =0, i.e. the speed is constant.
Lemma 2. The force Fwhose direction is described in (7)
created by induced current loof the obstacle will tend to move
the robot in the direction of induced current lo.
Proof. Suppose we define d=(lT
aro)
|ro|2and c f (|ro|,|˙
p|) = 1,
using definition in (6) and the fact that ˆ
lala=0, the force
Fwill be F=d(ro×la)×la. The product of this force Fand
obstacle current locan be written as
FTlo=d((ro×la)×la)T(la−dro) = g−h,(11)
where g=d((ro×la)×la)Tlaand h=d2((ro×la)×la)Tro.
The scalar gcan be simplified as
g=−dlT
a(la×(ro×la)) = −dlT
a(ˆ
laˆ
rola)).(12)
By definition of the skew-symmetric matrix in (5), we can
see that for any vector a, the term lT
a(ˆ
laa) = 0, so we can
conclude that g=0. To simplify the term h, without losing
generality, we can write down laand roas la=lax layT
and ro=rox royT. This will lead to
h=−d2(−layrax +lax ray)2≤0,(13)
FTlo=d2(−layrax +lax ray)2≥0.(14)
This concludes that the component of force Fin the direction
of lowill always have the same direction of lo, i.e. tend to
move the robot towards the direction of lo.
Lemma 3. Assuming the obstacle surface to be flat and the
robot does not collide with an obstacle, the final direction of
the robot due to the force Fwhose direction described in (7)
will be parallel to the obstacle surface.
Proof. Without loss of generality, we assume that the obstacle
surface lies on x-plane as depicted in Fig. 2b, so the closest
point on the obstacle is located at ro=0roTwith respect
4 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JUNE, 2018
to the robot. Defining angle between laand loas θ, we write
down the robot’s velocity direction as la=cosθsin θT.
From (6), the obstacle current is given by lo=cosθ0T.
Assuming f(|ro|,|˙
p|) = 1, the force Ffrom (7) will be
F=cˆ
la(ˆ
lola) = csin2θcosθ−sin θcos2θT.(15)
From Lemma 1, a robot with a constant speed vwill have the
following equation of motion
F=mv dla
dt =mv ˙
θ−sinθcos θT.(16)
Without loss of generality, suppose we analyze the equation
of motion in ydirection and combine (15) and (16) to obtain
the equation as follows
mv dθ
dt cos θ=−csin θcos2θ.(17)
After simplification, we will get the following equation
Z1
cosθsin θdθ=−Zc
mv dt.(18)
The final equation will have the following form
θ(t) = arctan(Ae−c
mv t),(19)
where Ais a constant which depends on the initial angle when
the robot senses the obstacle for the first time. As we can see,
as t→∞,θ→0, meaning that the robot’s final direction is
parallel to the robot surface.
To complete the description of the algorithm, we choose the
function f(|ro|,|˙
p|)to have the following form
f(|ro|,|˙
p|) = |˙
p|
|ro|.(20)
The reason for this is that we want the force to be larger when
the robot comes closer to the obstacle and when the robot has
a higher speed.
Lemma 4. For a robot with velocity direction lalocated at
initial distance |ro|from the flat obstacle with induced current
lodescribed in (6), the robot’s distance to the obstacle surface
will never be zero when the initial direction of lais not in the
direction of ro.
Proof. Since we do not use assumption f(|ro|,|˙
p|) = 1 any
longer, eq. (17) can be rewritten as
mv dθ
dt cos θ=−csin θcos2θv
r⇔˙
θ=−c
mr cosθsin θ,(21)
where rstands for the distance to the obstacle surface. For the
scenario in Fig. 2b, the velocity component in ydirection is
the same as −˙ras follows
˙r=−vsinθ⇔dr
dθ
˙
θ=−vsinθ.(22)
Substituting (21) into (22) and simplifying, we obtain
Z1
rdr =mv
cZ1
cosθdθ.(23)
The final equations has the following form
r(θ) = B|secθ+tan θ|C,(24)
B=r0
|secθ0+tan θ0|C,(25)
where r0stands for initial distance to the obstacle when the
angle between the robot’s velocity to the obstacle surface is
θ0and C=mv
c. The closest distance between the robot and
obstacle occurs when ˙
r=−vsinθ=0, i.e. θ=0. Substituting
the value θ=0 to (24), we get that the robot’s closest distance
to the obstacle is given by rf=B. As we can see, the value
of Bnever reaches zero except when θ0=π/2, i.e. when the
robot’s initial direction lais parallel to ro.
Remark 1. Lemma 4 guarantees that the robot’s distance to
the flat obstacle surface will never be zero. Hence, by definition
of the convex set, this property will also be true for any convex-
shaped obstacle.
Lemma 5. The final equilibrium direction of the robot de-
scribed in Lemma 3 is globally asymptotically stable.
Proof. Suppose we define the Lyapunov function candidate as
follows
V=−ln(lT
alo).(26)
This form is chosen since it reflects how much the direction
of the robot ladeviates from the obstacle surface lo. For this
Lyapunov function candidate, we get
˙
V=−˙
lT
alo+lT
a˙
lo
lT
alo
.(27)
Using the scenario in Fig. 2b, we can write down the
rate of each vector as ˙
la=˙
θ−sinθcos θTand ˙
lo=
˙
θ−sinθ0T. Substituting (21) into the equation, we get
˙
V=−2c
mr sin2θ.(28)
By definition, both cand mare always positive, while,
according to Lemma 4, except at special condition θ0=π
2,
rwill always be greater than zero. Hence, ˙
Vwill always
be negative except when θ=0 or θ=π, which are the
equilibrium points in which V=0 and ˙
V=0. Both cases
happen when the robot is parallel to the obstacle surface and
hence, conclude the proof.
Lemma 4 shows that the robot will never touch the obstacle
surface except when the initial robot direction lais parallel to
the obstacle position with respect to the robot ro. In reality, this
special condition is not likely to happen, as the imperfection
of sensor or actuator will make the vector laand rostill have
an angle between them albeit small. However, to avoid this
problem, when the current magnitude on the obstacle surface
is smaller than some small positive constant ε, we take the
unit vector as the obstacle current. The current generation in
(6) can be modified as follows
lo=
lo,iif |lo,i|>ε
lo,i
|lo,i|if |lo,i| ≤ ε,(29)
where lo,iis obstacle current in (6).
ATAKA et al.: REACTIVE MAGNETIC-FIELD-INSPIRED NAVIGATION METHOD FOR ROBOTS IN UNKNOWN CONVEX 3D ENVIRONMENTS 5
IV. IMPLEMENTATION
A point-like robot has a dynamic model as follows
¨
p=u(30)
where p∈R3stands for the robot’s position and u∈R3stands
for the acceleration input. The control law which will guide
the robot towards the goal and avoid obstacles is
u=Fg+Fo.(31)
Fgis an attractive PD control towards a goal pggiven by
Fg=−KP(p−pg)−KD˙
p,(32)
where KPand KDare positive constants. The obstacle avoid-
ance terms Fois a magnetic force described from (7), (20),
and (29) and will be activated once the distance between robot
and obstacle closer than a limit distance rlas follows
Fo=(cla×(lo×la)|˙
p|
|ro|if |ro|<rl
0if |ro| ≥ rl
.(33)
The robot in this case is assumed to have a 360◦laser sensor
so that it can sense the surrounding environments as far as
distance rlin all directions from the robot.
With the property of the field explained in Section III, for a
non-saturated input, it is shown that the robot will not collide
with an obstacle (Lemma 4). Assuming this is the case, we
can show the stability of control law in (31).
Lemma 6. The control law in (31)-(33) for a point-like robot
modeled in (30) is globally asymptotically stable.
Proof. Suppose we have the Lyapunov function candidate as
follows
V=1
2˙
pT˙
p+1
2KPeTe,(34)
where e= (p−pg)represents error vector. Differentiating with
respect to time, we get
˙
V=˙
pT¨
p+KPeT˙
e.(35)
Substituting (31)-(33) and the fact that ˙
e=˙
pand ˙
pTFo=0
according to Lemma 1, we get ˙
V=−KD˙
pT˙
p. We can see that
˙
Vis always negative except at equilibrium state p=pgand
˙
p=0where V=˙
V=0, and hence, this equilibrium is globally
asymptotically stable.
It is noted that when the obstacle is so large that it is
possible for the robot to be too far from the goal along
the obstacle’s boundary, the goal attraction component of our
algorithm could decrease the robot’s speed to zero. At this
point, the obstacle avoidance term will also be zero, and the
attraction to the goal could then force the robot to reverse its
travel direction, which could result in a condition where the
robot will oscillate without being able to reach the goal.
To avoid this problem, we add a goal relaxation (GR) term
which will decrease the contribution of the goal attraction as
the robot follows the boundary of the obstacle, reducing the
goal attraction when the robot gets closer to the obstacle as
described by
w1= (1−e−|ro|
αrl),(36)
TABLE I
LIS T OF PARAMETER VALUE S
Param Point-Like Robot Baxter
KP0.1 10
KD0.5 10
c5 20
rl3 m 0.1 m
ε0.05 0.05
α1.0 1.0
υ0.1 0.1
where αis a positive constant. The second property of the
GR term is that it will get smaller if the goal is still occluded
by an obstacle and will get bigger if the obstacle does not
obstruct the goal any longer as described by
w2=(1−rT
gro
|rg||ro|if |ro|<rl
1 if |ro| ≥ rl
.(37)
For a large obstacle, the attractive term should be even
smaller when the robot reaches a point where the distance
to the goal |rg|is larger than limit value rgl whilst, at the
same time, being in close proximity of the obstacle. One way
to limit distance rgl is using the initial distance to the goal rgi.
The weight is expressed as
w3=(e−|rg|−rgl
υif |ro|<rland |rg| ≥ rgl
1 otherwise
,(38)
where υis a positive constant. The overall GR term is then
expressed as γ(ro,rg) = w1w2w3, and the overall control signal
in (31) will be modified as follows
u=γ(ro,rg)Fg+Fo.(39)
V. RE SULTS AN D ANALYSIS
The magnetic-field-inspired navigation algorithm is imple-
mented to guide a point-like robot model in several simu-
lation scenarios. To better demonstrate the performance of
the algorithm, we also implement the algorithm to do point-
to-point navigation of the tip of a 7 DOFs Baxter arm
towards the goal. It is assumed that the robot only knows the
surrounding environment as far as rlin all directions from the
robot’s position. We compare the performance of the proposed
magnetic-field-inspired (MFI) algorithm with several reactive
algorithms (particularly the APF [11], variable speed force
field (VSFF) [12], circular field (CF) [18], and gyroscopic
force (GF) method [24], [25]). These reactive methods are
chosen since they do not rely on the availability of the
environmental map, do not need the information regarding
the obstacles shape or geometry, and can be applied in a 3D
environment with arbitrary-shaped obstacles. We also compare
the performance of the MFI algorithm to the other algorithms
when the goal relaxation (GR) term is added, except for the
APF method since it can be seen theoretically that the GR
terms will not stop the APF in avoiding the local minima.
All simulations and experiments are conducted in the Robot
Operating System (ROS) framework [28]. The constants used
6 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JUNE, 2018
Fig. 3. The trajectory of the robot is drawn in dashed lines moving towards
the goal (red mark) with a U-shaped obstacle drawn in black.
(a) (b)
Fig. 4. The plot of (a) the trajectory covered by the robot l(t)and (b) the
position error e(t)for a U-shaped obstacle.
in simulations and experiments, retrieved from trial and error,
are listed in Table I. These parameters can be easily tuned
without radically disturbing the performance of the algorithm
since they only influence how strong the goal attraction is and
how close the robot will be to the obstacle surface. Simulation
and experimental results are shown in the video attachment.
A. Simulation Results
Fig. 3 shows the trajectory of the robot for an environment
consisting of a U-shaped obstacle. We can see that of all
navigation algorithms, the APF and VSFF fail to navigate
the robot passed the obstacle as the robot gets stuck in a
local minimum, while the rest of the algorithms without the
goal relaxation successfully reach the goal. This is due to the
repulsive nature of the field produced by the obstacle used in
the APF and VSFF methods. When we add the algorithms
with the goal relaxation, the CF method fails to guide the
robot towards the goal due to the fact that the GR term
decreases the influence of the goal attraction, failing to attract
the robot out of the obstacle influence. This is not the case for
the MFI algorithm since the artificial current depends on the
robot’s speed, and hence, the obstacle influence automatically
decreases when the robot slows down.
Among the 6 methods successfully navigating the robot
towards the goal, the MFI without the GR has the shortest
trajectory as shown in Fig. 4a. However, it requires the third
longest time to reach the goal (Fig. 4b) because the robot is
still being influenced by the obstacle, despite the fact that the
remaining path to the goal is free from obstacles. The same
problem occurs for CF which takes the longest time to reach
the goal (Fig. 4b). Here, with the introduction of the GR term,
as depicted in Fig. 4b, the MFI algorithm reaches the goal in
the shortest time when compared to the rest of the algorithms,
even when others also use the GR term. The GF algorithm
with GR, even though converge slightly faster compared to
when the GR term is not used, still converge in a slower time
Fig. 5. The trajectory of the robot is drawn in dashed lines moving towards
the goal (red mark) and avoiding multiple spherical obstacles drawn in black.
(a) (b)
Fig. 6. The plot of (a) the robot trajectory and (b) the position error for
multiple spherical obstacles.
to the MFI due to the repulsive field it employs to keep the
robot at a safe distance from the obstacle surface, which could
decrease the robot’s speed.
Fig. 5 shows the trajectory of the robot for an environment
consisting of multiple spherical obstacles. From Fig. 6b, we
can see that of all navigation algorithms, the APF and GF
fail to navigate the robot passed the obstacle while the rest of
the algorithms without the goal relaxation successfully reach
the goal. For the case of GF, the robot falls into a zero-speed
problem, where the robot’s speed decreases due to the goal
attraction in following the obstacle boundary, to the point
where its speed becomes zero, causing it to move back and
forth without being able to reach the goal.
Among the other methods which successfully navigate the
robot towards the goal, the MFI with and without the GR
converge in a short trajectory as shown in Fig. 6a, only second
to the VSFF with and without GR. In terms of the convergence
time, as depicted in Fig. 6b, the MFI algorithm reaches the
goal in the shortest time when compared to the rest of the
algorithms. The CF algorithm with GR also converges in a
very short time; however, unlike the MFI, it still relies on the
information of the obstacle’s central position in reaching this
performance.
The scenario in which the GR is even more crucial is one
where large obstacle areas are encountered as shown in Fig.
7. This environment consists of an obstacle so large that, in
order to circulate around the obstacle, the robot needs to go
through a critical point whose distance to the goal is farther
than the distance between the goal and the robot’s start point.
The problem of local minima occurs for the APF, VSFF
without GR, and even VSFF with GR, while the problem
of zero speed occurs for the case of the MFI, CF, and GF
methods without GR as shown in Fig. 7 and Fig. 8. In this case,
the attraction to the goal causes the robot’s speed to decrease
while following the obstacle’s boundary until it becomes zero.
Added with the GR, the MFI, CF, and GF successfully avoid
ATAKA et al.: REACTIVE MAGNETIC-FIELD-INSPIRED NAVIGATION METHOD FOR ROBOTS IN UNKNOWN CONVEX 3D ENVIRONMENTS 7
Fig. 7. The robot trajectory is drawn in dashed lines moving towards the goal
(red mark) with long planar obstacle drawn in black.
(a) (b)
Fig. 8. The plot of (a) the trajectory covered by the robot l(t)and (b) the
position error e(t)as a function of time for the case of large obstacles.
the obstacle and reach the goal. Among these 3 methods, the
MFI and CF have the fastest convergence time due to the
fact that the repulsive term in GF slows the robot down (see
Fig. 8b). However, it is noted that the MFI method achieves
this performance without any information of the environment,
unlike the CF which still needs the position of the obstacle’s
centre point.
B. Experimental Results
Next, we implement our algorithm to guide the tip of a 7
DOFs Baxter Arm (see Fig. 9). Here, the Baxter is opertated
using the joint torque control mode in which the joint torque
commands that we send is applied in addition to the gravity
compensation term and internal joint spring compensation. The
control signal described in (31), (39) is applied at the tip as
a task-space force, from which a joint torque is computed
via Jacobian transpose relation, and the joint acceleration is
computed using inverse dynamics.
In Fig. 10, we can see that the APF is the only method which
fails to guide the robot’s tip pass the obstacle. Besides, Fig.
11b shows that the GF and VSFF methods fail to perfectly
guide the tip to reach the goal due to their repulsive term
causing the tip to be repelled too far and the Baxter arm stuck
in its joint constraint. In Fig. 11a, we can observe that the
final covered distance for the MFI with GR and without GR
are both smaller compared to the other methods. Fig. 11b
shows that the MFI and the CF, both with GR and without
GR, have the fastest error convergence rate compared to the
rest of the methods, shown as a very steep negative gradient at
the beginning of the movement. Once again, we argue that the
MFI still outperforms the CF due to the fact that it is able to
achieve a comparable performance in terms of the convergence
time to the CF even with less information.
Table II summarizes the results, comparing the APF, CF,
GF, and MFI methods both with GR and without GR. We
compare the ability of these methods to successfully reach the
Fig. 9. The experimental setup is shown in the left. The tip of 7 DOFs Baxter
arm is used as a point-like robot in 3D. The configuration of the robot, planar
obstacle, and desired tip position are shown in the right.
Fig. 10. The trajectory of the Baxter tip is drawn in dashed lines moving
towards the goal with obstacles drawn in black.
target and the path’s quality, characterized by the final path
l(t)and the time needed for the distance error e(t)to drop
below a specified value eb. Both variables are undefined when
the algorithm fails to guide the robot to reach the goal. The
value of ebis chosen to be eb=0.05rgi for the simulations,
where rgi stands for initial distance to the goal and eb=0.2
m for the experimental study due to the possible steady-state
positional error of the Baxter’s tip.
Table II demonstrates how in both aspects, simulation and
experiments, our reactive magnetic-inspired-field navigation
method outperforms the other methods in almost every sce-
nario, especially in terms of its ability to guide the robot
successfully to the goal and its low error convergence time. We
can see that the only algorithm which always works in every
scenario is the MFI with GR while the one without GR only
fails for the case of the large obstacle. The MFI without GR, in
every scenario except the multiple spheres always produce the
shortest trajectory. Despite taking longer trajectory, the MFI
with GR always has the best performance, or the second best
after the CF with GR for some cases, in terms of convergence
time despite only requires knowledge of the distance to the
closest obstacle and the robot’s speed.
VI. CONCLUSION
We present a reactive magnetic-field-inspired navigation
method capable of navigating the robot towards the desired
position. The algorithm takes inspiration from the magnetic
field laws and is used to guide the movements of a robot,
represented by a charged particle. The algorithm outperforms
the artificial potential field (APF) method because it is free
from local minima in convex environments. The algorithm
also improves on earlier magnetic-field-inspired navigation
methods found in the literature, as it does not rely on a priori
information of the obstacles’ geometries and locations. The
algorithm has been implemented successfully for a point-like
8 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JUNE, 2018
(a) (b)
Fig. 11. The plot of (a) the trajectory covered by Baxter’s tip l(t)and (b) the
position error e(t)as a function of time for the case of flat-surfaced obstacles.
TABLE II
SUMMARY OF RESU LTS
Robot Obstacle Algorithm Success Covered
Path (m)
Time
(s)
Simulation
U-Shaped APF 5- -
VSFF 5- -
VSFF+GR X49.84 87.28
GF X43.67 38.18
GF+GR X44.62 33.23
CF X43.46 192.88
CF+GR 5- -
MFI X42.38 82.00
MFI+GR X47.77 20.28
Multiple
Spheres
APF 5- -
VSFF X30.03 38.57
VSFF+GR X30.02 42.01
GF 5- -
GF+GR X51.68 28.50
CF X53.13 24.80
CF+GR X49.59 18.70
MFI X42.66 40.01
MFI+GR X44.88 19.08
Long
Plane
APF 5- -
VSFF 5- -
VSFF+GR 5- -
GF 5- -
GF+GR X87.25 122.17
CF 5- -
CF+GR X81.72 39.68
MFI 5- -
MFI+GR X93.25 40.45
Experiment
Plane APF 5- -
VSFF X1.86 6.74
VSFF+GR 5- -
GF 5- -
GF+GR 5- -
CF X1.24 3.95
CF+GR X1.11 4.00
MFI X1.07 5.10
MFI+GR X1.11 3.87
robot in R3, and also in an experimental setup using a 7DOF
Baxter arm. The results show that the algorithm outperforms
other reactive navigations. Our algorithm is also a promising
candidate to be used for the navigation of other type of robots,
such as flying robots, continuum manipulators, and swarming
multi-robot systems. Efforts to improve the algorithm by
considering the saturation of the robot’s actuators, uncertainty
in the sensor measurement, the orientation of the robot, and
environments with non-convex obstacles will be taken into
account in the future.
REFERENCES
[1] J.-C. Latombe, Robot Motion Planning. Norwell, MA, USA: Kluwer
Academic Publishers, 1991.
[2] H. Choset, K. M. Lynch, S. Hutchinson, G. A. Kantor, W. Burgard, L. E.
Kavraki, and S. Thrun, Principles of Robot Motion: Theory, Algorithms,
and Implementations. Cambridge, MA: MIT Press, Jun. 2005.
[3] S. M. LaValle, Planning algorithms. Cambridge university press, 2006.
[4] ——, “Motion Planning,” IEEE Robot. Autom. Mag., vol. 18, no. 1, pp.
79–89, Mar. 2011.
[5] L. E. Kavraki, P. vestka, J.-C. Latombe, and M. H. Overmars, “Prob-
abilistic roadmaps for path planning in high-dimensional configuration
spaces,” IEEE Trans. Robot. Autom., vol. 12, no. 4, pp. 566–580, 1996.
[6] S. M. LaValle, “Rapidly-Exploring Random Trees A New Tool for Path
Planning,” 1998.
[7] M. Elbanhawi and M. Simic, “Sampling-Based Robot Motion Planning:
A Review,” IEEE Access, vol. 2, pp. 56–77, 2014.
[8] O. Brock and O. Khatib, “Elastic strips: A framework for motion
generation in human environments,” Int. J. Robot. Res., vol. 21, no. 12,
pp. 1031–1052, 2002.
[9] J. Vannoy and J. Xiao, “Real-Time Adaptive Motion Planning (RAMP)
of Mobile Manipulators in Dynamic Environments With Unforeseen
Changes,” IEEE Trans. Robot., vol. 24, no. 5, pp. 1199–1212, Oct. 2008.
[10] S. M. L. Valle, “Motion Planning,” IEEE Robot. Autom. Mag., vol. 18,
no. 2, pp. 108–118, Jun. 2011.
[11] O. Khatib, “Real-time obstacle avoidance for manipulators and mobile
robots,” in Proc. IEEE Int. Conf. Robot. Autom., vol. 2, Mar. 1985, pp.
500–505.
[12] D. Wang, “A generic force field method for robot real-time motion
planning and coordination,” PhD Thesis, 2009.
[13] A. A. A. Rizqi, A. I. Cahyadi, and T. B. Adji, “Path planning and
formation control via potential function for UAV Quadrotor,” in Proc.
Int. Conf. Adv. Robot. Intell. Syst., Jun. 2014, pp. 165–170.
[14] A. Valero-Gomez, J. V. Gomez, S. Garrido, and L. Moreno, “The Path
to Efficiency: Fast Marching Method for Safer, More Efficient Mobile
Robot Trajectories,” IEEE Robotics Automation Magazine, vol. 20, no. 4,
pp. 111–120, dec 2013.
[15] O. Arslan and D. E. Koditschek, “Exact robot navigation using power
diagrams,” in Proc. IEEE Int. Conf. Robot. Autom., may 2016, pp. 1–8.
[16] L. Singh, H. Stephanou, and J. Wen, “Real-time robot motion control
with circulatory fields,” in Proc. IEEE Int. Conf. Robot. Autom., vol. 3,
Apr. 1996, pp. 2737–2742 vol.3.
[17] L. Singh, J. Wen, and H. Stephanou, “Motion planning and dynamic
control of a linked manipulator using modified magnetic fields,” in Proc.
IEEE Int. Conf. Control Applicat., Oct. 1997, pp. 9–15.
[18] S. Haddadin, R. Belder, and A. Albu-Schffer, “Dynamic motion planning
for robots in partially unknown environments,” in IFAC World Congr.,
vol. 18, 2011.
[19] F. Zhang, E. Justh, and P. Krishnaprasad, “Boundary following using
gyroscopic control,” in Proc. IEEE Conf. Decision Control, vol. 5.
IEEE, 2004, pp. 5204–5209.
[20] D. E. Chang and J. E. Marsden, “Gyroscopic forces and collision
avoidance with convex obstacles,” in New trends in nonlinear dynamics
and control and their applications. Springer, 2003, pp. 145–159.
[21] A. Savkin and C. Wang, Safe Robot Navigation Among Moving and
Steady Obstacles. Elsevier Science & Technology Books, 2015. [On-
line]. Available: https://books.google.co.uk/books?id=0lhorgEACAAJ
[22] G. Garimella, M. Sheckells, and M. Kobilarov, “A stabilizing gyroscopic
obstacle avoidance controller for underactuated systems,” in Proc. IEEE
Conf. Decision Control, Dec. 2016, pp. 5010–5016.
[23] E. W. Justh and P. Krishnaprasad, “Equilibria and steering laws for
planar formations,” Syst. & control Lett., vol. 52, no. 1, pp. 25–38,
2004.
[24] L. Sabattini, C. Secchi, and C. Fantuzzi, “Collision avoidance using
gyroscopic forces for cooperative Lagrangian dynamical systems,” in
Proc. IEEE Int. Conf. Robot. Autom., May 2013, pp. 953–958.
[25] ——, “Collision avoidance for multiple Lagrangian dynamical
systems with gyroscopic forces,” Int. J. Adv. Rob. Syst., vol. 14,
no. 1, p. 1729881416687109, 2017. [Online]. Available: http:
//dx.doi.org/10.1177/1729881416687109
[26] A. Ataka, H. K. Lam, and K. Althoefer, “Reactive Magnetic-field-
inspired Navigation for Non-holonomic Mobile Robots in Unknown
Environments,” in Proc. IEEE Int. Conf. Robot. Autom., May 2018, pp.
6983–6988.
[27] D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics
Extended, 10th Edition. Wiley, 2013. [Online]. Available: https:
//books.google.co.uk/books?id=DTccAAAAQBAJ
[28] M. Quigley, K. Conley, B. Gerkey, J. Faust, T. Foote, J. Leibs,
R. Wheeler, and A. Y. Ng, “ROS: an open-source Robot Operating
System,” in ICRA workshop on open source software, vol. 3, 2009, p. 5.
