44-59
Al-Khwarizmi Engineering Journal,Vol. 11, No. 4, P.P.
Hexapod Robot Static Stability Enhancement using Genetic
Firas A. Raheem
*,**Department of
(Received
Abstract
Hexapod robot is a flexible mechanical robot with six legs. It has the ability to walk over terrain. The hexapod robot
look likes the insect so it has the same gaits. These gaits are
statically stable at all the times during each gait in order not to fall with three or more legs continuously contacts with
the ground. The safety static stability walking
kinematics are derived for each hexapod
R2010a for all gaits and the geometry in order to derive the equations of the sub
hexapod’s leg. They are defined as the sub
with each other and they are useful to keep the legs stable from falling during each gait. A smooth gait was analyzed
and enhanced for each hexapod’s leg in two phases, stance phase and swing phase.
two approaches first, the modified classical stability margins. In this approach, the range of stability margins is
evaluated for all gaits. The second method is called
static stability by getting the best stability margins for hexapod robot
planning of hexapod robot with smaller error than the first approach and with better new stable coordinates of legs tips
than the first method. In addition, the second approach is useful for getting the better new stable center body
coordinates than center body coordinates in the first approach of hexapod robot.
Keywords: Kinematics, Stability Margin,
1. Introduction
The hexapod robots are mechanical vehicles
that walk with six legs; they have attracted
considerable attention in recent decades. There are
several benefits for hexapods rover
efficient one to maintain for statically stable static
on three or more legs, it has a great deal of
flexibility in how it can move [1].
difficult problem of generation and control of the
sequences of placing and lifting legs such that
any instant body should be stable and moving
from one position to other. The gait is defined as
generation and sequences of legs during the
hexapod motion [2]. Hexapod robot looks like
insect so it has the same gaits [1]. The
of hexapod are: Wave, Ripple, and Tripod gait
[3]. The GA is determined the optimal movement
Khwarizmi Engineering Journal,Vol. 11, No. 4, P.P. 44- 59 (2015)
Hexapod Robot Static Stability Enhancement using Genetic
Algorithm
Firas A. Raheem* Hind Z. Khaleel**
of Control and Systems Engineering / University of Technology
*Email: dr.firas7010@yahoo.com
**Email: hhindzuhair@yahoo.com
(Received 22 March 2015 ; accepted 30 June 2015)
is a flexible mechanical robot with six legs. It has the ability to walk over terrain. The hexapod robot
look likes the insect so it has the same gaits. These gaits are tripod, wave and ripple gaits. Hexapod robot needs to stay
times during each gait in order not to fall with three or more legs continuously contacts with
static stability walking is called (the stability margin). In this paper,
for each hexapod’s leg in order to simulate the hexapod robot model walking
for all gaits and the geometry in order to derive the equations of the sub-constraint workspaces for each
hexapod’s leg. They are defined as the sub-constraint workspaces volumes when the legs are moving without collision
with each other and they are useful to keep the legs stable from falling during each gait. A smooth gait was analyzed
and enhanced for each hexapod’s leg in two phases, stance phase and swing phase. The propose
the modified classical stability margins. In this approach, the range of stability margins is
evaluated for all gaits. The second method is called stability margins using Genetic Algorithm (GA)
the best stability margins for hexapod robot and these results are useful to get best stable path
planning of hexapod robot with smaller error than the first approach and with better new stable coordinates of legs tips
first method. In addition, the second approach is useful for getting the better new stable center body
coordinates than center body coordinates in the first approach of hexapod robot.
argin, Workspace, Genetic Algorithm and Hexapod Robot.
robots are mechanical vehicles
that walk with six legs; they have attracted
considerable attention in recent decades. There are
s rover such as:
efficient one to maintain for statically stable static
on three or more legs, it has a great deal of
flexibility in how it can move [1]. There is a
generation and control of the
sequences of placing and lifting legs such that at
any instant body should be stable and moving
from one position to other. The gait is defined as
generation and sequences of legs during the
robot looks like
The three gaits
e, Ripple, and Tripod gait
is determined the optimal movement
by the hexapod leg for walking robot with three
degrees of freedom, with higher positioning
precision By using an Analytical Hierarchy
Process [4]. The stability of
problem to maintain it from fall during its walking
with using the (GA) in order to get the optimal
movement. Hexapod simulation with a
used to determine the robot's movements for
generating chromosomes which are created from
repeated sequence of static leg positions, the
fitness function equation using the main factor
stability and efficiency [5]. The objective function
is used as the stability margin to find optimal
walking gaits for an 8-legged robot as in [6].
In this paper the main problem is when
hexapod robot walking and may be fall down if
the legs are not constraints so the
are analyzed, one that called the
Al-Khwarizmi
Engineering
Journal
(2015)
Hexapod Robot Static Stability Enhancement using Genetic
Hind Z. Khaleel**
Technology
is a flexible mechanical robot with six legs. It has the ability to walk over terrain. The hexapod robot
. Hexapod robot needs to stay
times during each gait in order not to fall with three or more legs continuously contacts with
. In this paper, the forward and inverse
simulate the hexapod robot model walking using MATLAB
constraint workspaces for each
es when the legs are moving without collision
with each other and they are useful to keep the legs stable from falling during each gait. A smooth gait was analyzed
The proposed work focused on the
the modified classical stability margins. In this approach, the range of stability margins is
stability margins using Genetic Algorithm (GA) that enhanced the
and these results are useful to get best stable path
planning of hexapod robot with smaller error than the first approach and with better new stable coordinates of legs tips
first method. In addition, the second approach is useful for getting the better new stable center body
obot.
leg for walking robot with three
degrees of freedom, with higher positioning
precision By using an Analytical Hierarchy
The stability of hexapod is a main
problem to maintain it from fall during its walking
the (GA) in order to get the optimal
simulation with a (GA) are
used to determine the robot's movements for
generating chromosomes which are created from a
repeated sequence of static leg positions, the
fitness function equation using the main factor
stability and efficiency [5]. The objective function
is used as the stability margin to find optimal
legged robot as in [6].
er the main problem is when
robot walking and may be fall down if
the legs are not constraints so the two approaches
one that called the modified
Firas A. Raheem
classical stability margins is depended on
according to constraints of each leg in order
fall and the other approach is called
margins enhancement using Genetic Algorithm is
based on the stable ranges values that getting from
first approach. The Genetic Algorithm is used to
get the best stability margins and these results
useful to get best stable path planning
robot.
2. Modeling of Hexapod Robot
The legged locomotion verities by verity of
usual terrain and it presents a set of difficult
problems (foot placement, obstacle avoidance,
load distribution, common stability) which must
be taken into account both in mechanical
construction of vehicles and in development of
control strategies [7]. Besides that, these issues
are using models that mathematically explain the
verities of situations and for that; th
modeling becomes a practical tool in
understanding systems complexity and for testing
and simulating diverse control approaches
The robot structure considered has (6) identical
legs and each leg has (3) degree of freedom, in
addition to that, all the related points for each
joint have been put on the model, the legs
numbering as shown in Figure 1, robot’s
coordinate o (xo, yo, zo).
Fig. 1. Hexapod robot structure
The z-axis pointing up, the x
forward and the y-axis pointing left
to right hand rule. Hexapod robot
consisting of two types, one is forward kinematic
and its inverse, below will discuss in deta
each type of kinematic.
Al-Khwarizmi Engineering Journal, Vol. 11, No.
45
is depended on
f each leg in order not to
fall and the other approach is called stability
using Genetic Algorithm is
that getting from
The Genetic Algorithm is used to
these results are
path planning of hexapod
Modeling of Hexapod Robot
The legged locomotion verities by verity of
usual terrain and it presents a set of difficult
problems (foot placement, obstacle avoidance,
common stability) which must
be taken into account both in mechanical
construction of vehicles and in development of
. Besides that, these issues
are using models that mathematically explain the
verities of situations and for that; the robot
modeling becomes a practical tool in
understanding systems complexity and for testing
ting diverse control approaches [8].
The robot structure considered has (6) identical
legs and each leg has (3) degree of freedom, in
all the related points for each
joint have been put on the model, the legs
1, robot’s center
robot structure.
axis pointing up, the x-axis pointing
and according
robot modeling
consisting of two types, one is forward kinematic
and its inverse, below will discuss in details for
2.1. Forward kinematics for One Leg
hexapod Robot
The successful design of a legged robot
depends to a large amount on the leg design
chosen. Since all aspects of walking are ultimately
governed by the physical limitations of the leg, it
is important to select a leg that will allow a
maximum range of motion a
unnecessary constraints on the walking.
three-revolute kinematical chain
been chosen for each leg mechanism in order to
imitate the leg structure as shown in Fig
direct geometrical model for each leg mechanism
is formulated between the moving frame
oi(xi,yi,zi) of the leg base, where i=1…
fixed frame o (xo,yo,zo) [9].
Fig. 2 .Model and coordinates frame for leg
kinematics.
In this paper, the BH3-
is taken as a case study
lengths of the hexapod’s leg
= (5.7 cm), L3 = (10.8 cm
frame starts with link (0) which is the
robot body where the leg is jointed to; link (1) is
the coxa, link (2) is the femur and link (3) is the
tibia. Legs are distributed symmetrically around
the axis in the direction of motion (
The general form for the transformation matrix
from link (i) to link (i-1) using Denavit Hartenberg
parameters is given in the
d1
R1
θ1
z1
L1
x0
yo
zo
xc
leg base
θ3
Khwarizmi Engineering Journal, Vol. 11, No. 4, P.P. 44- 59 (2015)
rd kinematics for One Leg of
successful design of a legged robot
depends to a large amount on the leg design
chosen. Since all aspects of walking are ultimately
governed by the physical limitations of the leg, it
is important to select a leg that will allow a
maximum range of motion and not inflict
onstraints on the walking. The
revolute kinematical chain (R1, R2, R3) has
been chosen for each leg mechanism in order to
imitate the leg structure as shown in Figure 2. A
direct geometrical model for each leg mechanism
formulated between the moving frame
) of the leg base, where i=1…6, and the
[9].
l and coordinates frame for leg
-R hexapod robot model
case study of hexapod robot. The
’s leg are: L1 = (2.9 cm), L2
10.8 cm) [10]. The robot leg
frame starts with link (0) which is the point on the
robot body where the leg is jointed to; link (1) is
the coxa, link (2) is the femur and link (3) is the
tibia. Legs are distributed symmetrically around
direction of motion (x in this case).
The general form for the transformation matrix
) using Denavit Hartenberg
Eq. 1 [9,11]:
R2
x1
L2
R3
L3
x3
z3
x2
z2
y0
θ2
Firas A. Raheem Al-Khwarizmi Engineering Journal, Vol. 11, No. 4, P.P. 44- 59 (2015)
46
T���� =
�cos θ� − sin θ� cos α�sin θ� cos θ� cos α� sin θ� sin α� a� cos θ�−cos θ� sin α� a� sin θ�0 sin α�0 0 cos α� d�0 1 �
…(1)
The transformation matrix is a series of
transformations:
1. Translate di along zi-1 axis.
2. Rotate θ�about zi-1 axis.
3. Translate a� about xi-1 axis (a�= Li for i=1...3).
4. Rotate α� about xi-1 axis.
The overall transformation is obtained as a
product between three transformation matrixes:
T�������� = T��������� T����� ���� … "2$
Considering Figure 2 and using Eq. 2 the
coordinates of the leg tip are: x = cosθ� ∗ "L� + L) ∗ cosθ) + L*∗ cos"θ) − θ*$$, y = sinθ� ∗ "L� + L) ∗ cosθ)+ L* ∗ cos"θ) − θ*$$, z = d� + L) ∗ sinθ)+ L* ∗ sin"θ) − θ*$ … "3$
Where: d1 is the distance from the ground to the
coxa joint. Li are the lengths of the leg links.
2.2. Inverse Kinematics using Geometric
Approach
The inverse kinematics problem consists of
formative the joint angles from a given position
and orientation of the end frame. The solution of
this problem is significant in order to transform
the motion assigned to the end frame into the joint
angle motions matching to the desired end frame
motion. The goal is to find the three joint
variables θ1, θ2, and θ3 corresponding to the
desired end frame position. The end frames
orientation is not a matter, where only paying
attention in its position [9].
Fig. 3 . Illustrations for solving inverse kinematics.
Using Eq. 3 and considering the following
constraints: all joints allow rotation only about
one axis, femur and tibia always rotate on parallel
axes, and the physical limitation of each joint can
determine the joint angle. The coxa joint angle
can be found using the following function as can
be seen from Figure 3 A. θ� = tan�� 0y�x�1 … "4$
In order to determine the other two angles a
geometrical approach is considered. The leg tip
coordinates were transformed to coxa frame using
the transformation matrix below:
T��������� = 0"R��������� $ −"R��������� $4 ∗ d���������0 1 1 … "5$
The angle θ) of the femur servo position can be
derived directly from the triangle Figure 3 B. θ) = φ − φ� … "6$
The angle φ� is the angle between the x-axis and
line a and can be calculated with the following
function: φ� = tan�� 0y*x*1 … "7$
Where x3 and y3 are the leg tip coordinates in coxa
frame. The law of cosine results is applied: φ = cos�� 9L)) + a) − L*)2 ∗ L) ∗ a : … "8$
Where: a = <=*) + >*) … "9$
From the Eq. 6, the femur angle can be found
from[9]:
L
Leg tip
x1, y1
y
Femur
joint
Coxa joint
Tibia
joint
L2
θ*
θ)
φ*
φ�
x3,y3
a
A B
y
x x
φ
Firas A. Raheem Al-Khwarizmi Engineering Journal, Vol. 11, No. 4, P.P. 44- 59 (2015)
47
θ) = cos�� 9L)) + a) − L*)2 ∗ L) ∗ a :− tan��"y*, x*$ … "10$
By applying the law of cosines, the φ* angle is
found:
φ* = cos�� 9L)) + L*) − a)2 L) ∗ L* : … "11$
Considering Figure 3 B, the @* can be found as
follows [9]: θ* = π − φ3 … "12$
3. Workspace of Hexapod’s Leg
In this paper the hexapod's leg workspace has
been computed and analyzed. Hexapod's leg
workspace can defined as the set of reachable
points by the end-effector for each foot. These
points (positions) depended on the leg orientation
(the mechanical limits of the joints). The
mechanical limits of the joints restrict leg motion
and are a major factor to consider when
developing walking algorithm for a hexapod
module. The working volumes for each leg are
identical because each leg of hexapod has the
same geometrical configuration and joint limits;
the analyzed of the two approaches to evaluate the
constraint workspace for BH3-R hexapod robot
[10]. The limits of the joint variables for a
representative one leg are shown in Table 1 [10].
Table 1.
The range of angles for one hexapod’s leg [10].
Link Name The range of one robot’s leg
angle in degree
Coxa −90 < @� < 90
Femur −45 < @) < 90
Tibia 0 < @* < 135
These joint variable limits, then, separate the
reachable area from the unreachable area.
Reachable areas move with the body. The region
included within the reachable area is known as the
unconstrained working volume (UWV). The
constrained working volume (CWV) is defined as
a subset of the original working volume, for each
leg, that ensures static stability. Therefore, the
(CWV) sets soft limits for each leg so as to
exclude points from the working volume that may
lead to instability. In our case, the working
volume is also constrained to prevent leg
collisions. An excluded area for hexapod's legs,
then, is that part of the reachable area where, if a
foot were placed there, instability or leg collision
might result.
Fig. 4. Flowchart Workspace of hexapod’s leg.
Figure 4 shows that the flowchart of
workspace. The workspace of robot leg is
computed from kinematics and geometry as
follow:
3.1. Unconstrained Workspace
The unconstrained horizontal workspace of
hexapod leg is the reachable areas include the
sections in the xy plane around the individual
coxas and within the mechanical joint limits, the y
plane equal (30 cm). The unconstrained vertical
workspace, or z-plane reachable area, depends on
the height of the hexapod’s center-of-body above
the terrain is (5.5 cm).
To define the maximum unconstrained vertical
workspace, if a leg were extended to its fullest,
the added lengths of radius body (13.75 cm), coxa
(2.9 cm), femur (5.7 cm), and tibia (10.8 cm) y-
plane would equal (30 cm). The minimum
unconstrained vertical workspace, If a leg were
lack extent to its fullest, (i.e θ* = 1350), the y-
plane would equal (14.7 cm) while the z-plane
equals to (9 cm).
3.2. Constrained Workspace
The calculations of CWV results are used for
six basic constraints:
• the height (z = plane) from the ground to the CB
(center body of robot) = (10.8 cm) is fixed for
minimum, maximum reach, the vertical maximum
Firas A. Raheem Al-Khwarizmi Engineering Journal, Vol. 11, No. 4, P.P. 44- 59 (2015)
48
reach equal (24.25 cm) if θ* = 800 and minimum
reach equal (21 cm) if θ* = 97o,
• the suitable posture of robot = (22.35 cm) if θ*
= 90o,
• The terrain is flat,
• The legs are not allowed to collide or overlap,
and
• The horizontal workspace of hexapod leg is the
reachable areas include the sections in the xy
plane around the individual coxas, y-plane =
24.25 cm and within the mechanical joint limits
but in this case limit joint (half range of coxa
angle is taken in order not to the legs collide) so
the range is "−45 < θ� < 45$ degree.
Another approach for constrained workspace is
derived, the more details in [12], [13]. For the
hexagonal model the mathematically Eq. 13 for the
radius of the annulus is :
r���) = "r��D + Q$) + F�) ∗ PH) … (13)
Where: r��� , r��D, Q, P defined by Figure 5.
Fig. 5. The relationship between the reachable area
and annuls.
And the rectangular area is the reachable area
of each leg of robot, for our hexapod robot rmax =
(10.5 cm) from coxa joint. Added the length of
center robot (13.75 cm), rmax = (24.25 cm). The
center of leg tip point is (22.35 cm) that it is equal
to the posture robot in method1 above comparing
between two constraints workspaces methods are
found that the maximum reaches of the leg are
equal for our hexapod robot.
4. Modified Classical Stability Margins
Analysis Approach of Hexapod Robot
The first gait of the hexapod robot is the tripod
gait. In this gait the three legs stay on the ground
(support pattern) while the other legs are on the
air. The analysis of static stability depended on
the Eq. 14 (the triple equation in the Figure 6)
[14] that only computes the S1 where S2 and S3 are
evaluated from the same pervious equation but
with changing the coordinates of legs for each
S1, S2 as in the Figure 6 of three triangles and
there are two conditions to set the robot stable
first if S1, S2 and S3 are >= zero, the tripod is
considered stable other, the tripod is considered
unstable more details in [14].
From the definition of “stability margin,” (sm),
is the shortest distance from the vertical projection
of the center of robot to the boundaries of the
support pattern in the horizontal plane [15] the
proposed method is explained below:
In Figure 6 L1 is derived in the Eq. 16 (the
distance line between two points) and the same
thing of L2 is computed, L3 for other legs, each L
is considered as a base of the one triangle while
the areas of the S1, S2 and S3 are previously
computed so that the stability margin is the
shortest perpendicular distances from L1, L2 and
L3 to the center of robot (H1, H2, H3 respectively).
H1 is computed as Eq. 17 as well as the H2, H3 are
computed in the same manner.
the stability margins are computed and analyzed
for all cases of the hexapod legs motion for three
gaits (tripod, ripple and wave) for example the
support pattern of tripod when robot lifts legs (1,
3 and 5) :
Fig. 6. The stability analysis for the tripod gait
when legs (2, 6 and 4) on the ground and legs (1, 3
and 5) on the air.
The triple equation of the Figure 6 is:
S� = �) J 1 1 1xKL xM x)yKL yM y)J . . . "14$
The
support
pattern
CB
L1
L2
L3
S1
S2
S3
Leg 2
Leg 6
Leg 4
H1
H3
H2
Firas A. Raheem Al-Khwarizmi Engineering Journal, Vol. 11, No. 4, P.P. 44- 59 (2015)
49
Where (xKL, yKL) the coordinate of center body,
(x), y)) the coordinate of Leg 2, (xM, yM) the
coordinate of Leg 4.
The simplification of the Eq. 14 : S� = �) O"xM − xKL$"y) − yKL$ − "x) −xKL$P"yM − yKL$QR … "15$
L� = sqrt ""x) − xM$) + "y) − yM$)$ … "16$
Where L1 is the distance between two points H� = 2 ∗ 0S�L�1 … "17$
From Figure 6 the stability margin are computed
as: sm� = min"H� , H), H*$ … "18$ sm� is the stability margin of the support pattern
of legs (2, 4 and 6) similarly, the sm) is derived
for other three legs when (1, 3 and 5). Besides
that, Eq. 18 is derived for the other gaits of robot
(wave and ripple).
5. Walking, Mechanism of Leg Motion and
Path Planning of Hexapod Robot
The mechanism of leg motion is very
complex problem that each leg is forward and
back motion. It derived from insect motion that
has two phases: swing (the leg in the air) and
stance (the leg in the ground) phases [3]. The
equations of motion in [16] are derived for two
phases. The walking of hexapod robot is
developed by combing the stance phase [16]
explained by Eq. 19 and the swing phase [17] as
in Eq. 20 to get our modified smooth gait for one
hexapod’s leg as below: x�W� = x� − v ∗ T ∗ cos"ϕ$, y�W� = y� − v ∗ T ∗ sin"ϕ$ …(19)
Where x�, y� the coordinates of leg tip derived
from the forward kinematic, ϕ is the direction of
motion, T is the period required to complete one
cycle and changes during type gait and v
describes how many centimeters per gait cycle the
hexapod robot should move.
The equations in swing phase are: x�W� = 2 ∗ xZ[ ∗ dt� �\"1 − cos 0 ] ^ _`ab1$, y�W� = 2 ∗ yZ[ ∗ dt� �\"1 − cos 0 ] ^ _`ab1$, z�W� = h ∗ "1 − cos 0 ] ^ _`ab1$ …(20)
xZ[ , yZ[ are the speed of the hexapod robot’s truck in
x and y directions, dt� �\ is the time duration for
each step and h is the height of each step. The
movement of the center of body that moves from
start point to the goal point so the new center
point [14] is calculated as: x��W� = x�� + d ∗ cos"ϕ$, y��W� = y�� + d ∗ sin"ϕ$ …(21)
Where d is the step size. The path planning start from point (0,0) cm and
end with goal point (500,0)cm for the straight
line.
6. Stability Margins Enhancement using
Genetic Algorithm
The results of sequences of three main gaits for
hexapod robot are found the range values for the
stability margins as mentioned above. Each gait of
hexapod robot, has the sequence of gait such as
the tripod gait has two sequences, the first
sequence lifts legs (1, 3 and 5) where the legs (2,
4 and 6) on the ground. The second sequence lifts
legs (2, 4 and 6) where the legs (1, 3 and 5) are on
the ground.
In the proposed method, an intelligent method
which is (GA) has been applied to find the best
stability margin during each sequence in each
gait. While modified classical stability margins
analysis method which used to find the range
values for stability margins is considered as
constraints for Genetic Algorithm to find the best
stability margins based on these constraints. Also,
the genetic algorithm has been applied to find the
best positions of legs tips as described below:
A. Parameters Initialization of Genetic
Algorithm
The initial populations for genetic algorithm
are the coordinates of each leg’s tip (xi, yi) and the
coordinates of center body (xCBi, yCBi) while the
(zi) is constant (the motion of hexapod robot on
the flat plane). Chromosomes (individuals)
represent the solutions for optimization problem.
In the proposed work, the better results of new
stable population are evaluated. These results are
considered as initial coordinate (xi, yi) for walking
of the hexapod robot. These stable positions are
repeated till hexapod robot reaches to the goal
point.
The size of genes in each chromosome is
various according to the type of gait (support
pattern for each sequence) for example the size of
Firas A. Raheem Al-Khwarizmi Engineering Journal, Vol. 11, No. 4, P.P. 44- 59 (2015)
50
chromosomes equals to eight variables in tripod
gait. The first sequence in tripod gait when lifting
legs (1, 3 and 5) where legs (2, 4 and 6) on the
ground have been calculated with center body
coordinates as the following formula: chromosome = gx� y� x) y) xh yh xM yMi.
The other sequences of hexapod gaits types are
formulated as in the above example and according
to each gait.
B. Fitness Function:
The fitness function of GA is used as
minimization of the cost function (stability
margins) as derived previously, Where each sm =min"H�, H), … , Hh$ and the Hi defined in section
(4). For example ,the fitness function of tripod
gait equals to stability margin (sm) is selected
according to the each sequence of each gait such
as the Eq. 18
The idea of proposed approach is different
from the approach in [17]. The difference is that
the researchers in [17] that the minimum fitness
stability margin is squared value (fitness function
=sm2) in order to get the high values of stability
margin (sm), while in proposed work, the
minimum fitness stability margin is (fitness
function =sm) and all values are high and within
the constraints of legs tips without need to square
the stability margin (sm). All cases of our
approach results of the stability margin are within
the ranges of stability margins from modified
classical analysis method.
In the GA, the best stability margins are
evaluated while in the modified classical approach
the ranges of the stability margins are obtained.
The all cases of the fitness equations for three
gaits are derived from the modified classical
stability method where in each sequence there is
fitness function as an example for Eq. 18 is the
fitness function of tripod gait for the first
sequence (where the robot lifting legs (1, 3 and 5)
and legs (2, 4 and 6) are on the ground).
C. Crossover Operation
The main processing of crossover is to get two
parents of coordinates of leg’s tips and the center
body coordinates and producing from them the
children. Crossover operator is applied to produce
a better offspring (children).
D. Mutation Operation
After the crossover operation, the output string
from crossover is subjected to mutation process.
Fig. 7. Proposed flowchart of GA.
7. Flowchart of Proposed Genetic
Algorithm
After complete the modified classical method,
below will describe the developed method using
Genetic Algorithm. In the Figure 7 shows that
flowchart of proposed Genetic Algorithm. It has
consists of steps where described as below:
• Defined the variables which determined the
leg’s tip of hexapod robot (chromosomes).
• Generate the coordinates of leg’s tip and center
point of body (chromosomes).
• Find the cost function (stability margins
equations as in section (4) to get the best
stability margin for each sequence for each
gait.
• Select mates, mating, and mutation.
• Convergences condition, if the generation <=
nG (100 generations).
• Done.
8. Simulation Results
The simulation results consist of two
approaches where illustrated as below:
Firas A. Raheem Al-Khwarizmi Engineering Journal, Vol. 11, No. 4, P.P. 44- 59 (2015)
51
8.1. The Modified Classical Stability
Margins
Analysis of the stability margins above for
three gaits (ripple, wave and tripod) are simulated
for each gait within the steps as below:
a- The wave gait cases
1
2
3
4
5
6
Fig. 8. The configuration of hexapod robot’s leg of
sequences of wave gait.
Firas A. Raheem Al-Khwarizmi Engineering Journal, Vol. 11, No. 4, P.P. 44- 59 (2015)
52
b- The ripple gait cases
1
2
3
4
Fig. 9. The configuration of hexapod robot’s leg of
sequences of ripple gait.
c- The tripod gait cases
1
2
Fig. 10. The configuration of hexapod robot’s leg of
sequences of tripod gait.
The figures (8-10) show that the configurations
of hexapod robot’s legs for three gaits. The yellow
lines indicated to the support patterns. From
support patterns the (sm) is evaluated for each
sequence in every gait and the simulation results
of all stability margins are described in
simulations below:
Firas A. Raheem Al-Khwarizmi Engineering Journal, Vol. 11, No. 4, P.P. 44- 59 (2015)
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(a)
(b)
Firas A. Raheem Al-Khwarizmi Engineering Journal, Vol. 11, No. 4, P.P. 44- 59 (2015)
54
(c)
Fig. 11. Minimum and maximum stability margins
for, (a) ripple gait, (b) tripod gait, and (c) wave gait.
In Figure 11 shows that the stable range of
stability margins for three gaits and these values
are considered the constraints to the genetic
algorithm of our proposed work.
8.2. Best Stability Margins using Genetic
Algorithm
From the study in the section (4), the results of
best stability margins are shown in below:
Fig. 12. The best fitness value for tripod gait when
the legs (2, 6, and 4) are on the ground and lifting
legs (1, 3 and 5).
Fig. 13. The best fitness value for tripod gait when
the legs (1, 3 and 5) are on the ground and lifting
legs (2, 6 and 4).
0 10 20 30 40 50 60 70 80 90 100
3.935
3.936
3.937
3.938
3.939
Generation
F
it
n
e
s
s
v
a
lu
e
Best fitness: 3.9356
Average Distance Between Individuals
Best f itness
0 10 20 30 40 50 60 70 80 90 100
5.1
5.2
5.3
5.4
Generation
F
it
n
e
s
s
v
a
lu
e
Best fitness: 5.1304
Average Distance Between Individuals
Best fitness
Firas A. Raheem Al-Khwarizmi Engineering Journal, Vol. 11, No. 4, P.P. 44- 59 (2015)
55
Fig. 14. The best fitness value for ripple gait when
the legs (2, 3, 5, and 4) are on the ground and lifting
legs (1, and 6).
Fig. 15. The best fitness value for ripple gait when
the legs (4, 1, 2, 3 and 6) on the ground and lifting
leg (5).
Fig. 16. Best fitness value for ripple gait when the
legs (1,2,3,6 and 5) on the ground and lifting leg (3
and 4).
Fig. 17. The best fitness value for ripple gait when
the legs (1, 3, 6, 5 and 4) on the ground and lifting
leg (2).
Fig. 18. The best fitness value for wave gait when
the legs (2, 3, 6, 5 and 4) on the ground and lifting
leg (1).
Fig. 19. The best fitness value for wave gait when
the legs (1, 3, 6, 5 and 4) on the ground and lifting
(2).
0 10 20 30 40 50 60 70 80 90 100
6.26
6.265
6.27
6.275
6.28
Generation
F
it
n
e
s
s
v
a
lu
e
Best fitness: 6.2614
Best f itness
0 10 20 30 40 50 60 70 80 90 100
12
13
14
15
Generation
F
it
n
e
s
s
v
a
lu
e
Best fitness: 13.1925
Average Distance Between Individuals
Best fitness
0 10 20 30 40 50 60 70 80 90 100
8.48
8.5
8.52
8.54
Generation
F
it
n
e
s
s
v
a
lu
e
Best fitness: 8.4736
Best f itness
0 10 20 30 40 50 60 70 80 90 100
12.55
12.6
12.65
12.7
12.75
Generation
F
it
n
e
s
s
v
a
lu
e
Best fitness: 12.5673
Best fitness
0 10 20 30 40 50 60 70 80 90 100
6.6
6.62
6.64
6.66
Generation
F
it
n
e
s
s
v
a
lu
e
Best fitness: 6.6121
Best fitness
0 10 20 30 40 50 60 70 80 90 100
13.98
13.99
14
14.01
14.02
Generation
F
it
n
e
s
s
v
a
lu
e
Best fitness: 13.9895
Average Distance Between Individuals
Best f itness
Firas A. Raheem Al-Khwarizmi Engineering Journal, Vol. 11, No. 4, P.P. 44- 59 (2015)
56
Fig. 20. The best fitness value for wave gait when
the legs (1, 2, 6, 5 and 4) on the ground and lifting
leg (3).
Fig. 21. The best fitness value for wave gait when
the legs (1, 2, 3, 6 and 5) on the ground and lifting
leg (4).
Fig. 22. The best fitness value for wave gait when
the legs (1, 2, 3, 6 and 4) on the ground and lifting
leg (5).
Fig. 23. The best fitness value for wave gait when
the legs (1, 2, 3, 5 and 4) on the ground and lifting
leg (6).
The above Figures (12 - 23) show that the best
stability margins for each sequence in each gait
and all the stability margins are within the range
of constraints that described in classical stability
margins analyses method.
The analysis equations of Eq. 21 used of find
the next points of the hexapod center body and the
errors center body of the path planning for two
approaches is shown in Table 2, below:
Table 2,
Shows that the errors of two approaches modified
classical stability margins and stability margins
Enhanement using GA.
Errors path planning
of modified classical
stability margins
Errors path planning of
stability margins
Enhancement using GA
1- Tripod gait has less
error value equals
(8.3630e-004) cm.
2- Wave gait has less
error value equals
(0.0023) cm.
3- Ripple gait has great
error value equals
(0.0520) cm.
1- Tripod gait has less error
value equals (2.3590e-004)
cm.
2- Wave gait has error value
equals (0.0012) cm.
3- Ripple gait has great error
value equals (0.0196) cm.
The genetic algorithm here works as off-line
and getting the new stable coordinates of legs tips
of hexapod robot (new populations) before robot
start walking .From these coordinates are selected,
the best points of the center body in (x , y) plane
are calculated according to the best stability
margins values.
0 10 20 30 40 50 60 70 80 90 100
5.46
5.48
5.5
5.52
Generation
F
it
n
e
s
s
v
a
lu
e
Best fitness: 5.4661
Average Distance Between Individuals
Best f itness
0 10 20 30 40 50 60 70 80 90 100
12.69
12.695
12.7
12.705
12.71
Generation
F
it
n
e
s
s
v
a
lu
e
Best fitness: 12.6927
Best fitness
0 10 20 30 40 50 60 70 80 90 100
6.8
6.9
7
7.1
7.2
7.3
Generation
F
it
n
e
s
s
v
a
lu
e
Best fitness: 6.883
Best fitness
0 10 20 30 40 50 60 70 80 90 100
7.5
7.6
7.7
7.8
Generation
F
it
n
e
s
s
v
a
lu
e
Best fitness: 7.5258
Best f itness
Firas A. Raheem Al-Khwarizmi Engineering Journal, Vol. 11, No. 4, P.P. 44- 59 (2015)
57
9. Conclusion
In this paper the stability margins are
analyzed for all gaits of hexapod robot in two
approaches first is the modified classical approach
and second, is the stability margins enhancement
using genetic algorithm.
In first approach, the range of stability margins
values are evaluated for each sequence in every
gait where the hexapod walking from start point
(0,0) cm to the goal point (500,0) cm. For the
second approach, the best stability margins for
each sequence in every gait are calculated and the
results is useful to get best stable path planning
with smaller error than the first approach for three
gaits of hexapod robot as in Table 2, above .Also
in the second approach and according to the best
stability margins values.
The better new stable coordinates (positions of
legs) are gotten than stable coordinates of first
approach. In addition, the better new stable center
body coordinates are evaluated than center body
coordinates in the first approach
Notation
di translation along zi-1 axis a� = Li link length about xi-1 axis T���� forward kinematic matrix
x, y, z coordinates of hexapod's leg r��� maximum length of leg's workspace r��D minimum length of leg's workspace Q, P lengths of rectangular reachable area
S� triple equation
H� vertical line from the center point of
the robot’s body to the middle of the
base (L1)
sm stability margin
x�W�,
y�W�
backward smooth motion
x�W�,
y�W�,z�W�
forward smooth motion
ϕ direction of motion
T period required to complete one cycle
d step size
xZ[ ,yZ[ are the speed of the hexapod robot’s
truck in x and y directions
dt� �\ time duration for each step
h height of each step
x� �W�,
y� �W�
new center coordinates of hexapod's
body
xCBi,
yCBi
center coordinates of hexapod's body
v represents speed of how centimeter per
gait cycle the robot should move
GA Genetic Algorithm
Greek Letters
θ� rotation angle about zi-1 axis
α� rotation angle about xi-1 axis
φ� angle between the x-axis and line a
10. References
[1] X. Ding, Z. Wang, A. Rovetta and J.M. Zhu,
“Locomotion Analysis of Hexapod Robot,
Climbing and Walking Robots,” Behnam
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[2] M. I. Ahmad, D. K. Biswas and S. S Roy,
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[3] B. J., “Biologically Inspired Approaches for
Locomotion, Anomaly Detection and
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[4] D. A. Bucur, S. A. Dumitru, “Genetic
Algorithm for Walking Robots Motion
Optimization,” Institute of Solid Mechanics
of the Romanian Academy 15 C-tin Mille,
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[5] J. Currie, M. Beckerleg, J. Collins,
“Software Evolution Of A Hexapod Robot
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[6] B.L. Luk, S. Galt and S. Chen, “Using
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[7] K. W., D.B. and A. K., “Control and
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[8] J. Barreto, A. Trigo, P. Menezes, Dias J.,
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[10] http://www.lynxmotion.com/c-100-bh3-
r.aspx, Accessed on 17/11/2013.
[11] S. R. J., “Fundamentals of robotics: analysis
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[12] R. B. McGhee and G. I. Iswandhi, “Adaptive
locomotion of a multilegged robot over rough
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[13] Stanley Kwok-Kei Chu and Grantham
Kwok-Hung Pang, “Comparison between
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[14] Charles Andrew Schue, “Simulation of
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[15] S. M. Song and B. S. Choi, “The Optimally
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[16] S. Villumsen, “Modeling and Control of a six
legged mobile robot,” Master Thesis, 2010.
[17] M. A. Shahriari, K. G. Osguie, A. A. Akbar
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Science Journal, 2013.
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