(Microsoft Word - \335\321\307\323 \346\307\323\343\307\301124-136)


 

   

Al-Khwarizmi 

  Engineering 

  Journal 
         Al-Khwarizmi Engineering Journal,Vol. 13, No. 4, P.P. 124- 136 (2017) 

 

Applying A* Path Planning Algorithm Based on Modified 

C-Space Analysis 
 

Firas A. Raheem*                 Asmaa A. Hussain** 
*,* *Department of Control and Systems Engineering/ University of Technology 

   *Email: 60124@uotechnology.edu.iq 

**Email: asmaa.alshawi@yahoo.com 

  
(Received 25 October 2016; accepted 7 March 2017) 

https://doi.org/10.22153/kej.2017.03.007 
 

 

Abstract  
 

In this paper, a modified derivation has been introduced to analyze the construction of C-space. The profit from using 

C-space is to make the process of path planning more safety and easer. After getting the C-space construction and map 

for two-link planar robot arm, which include all the possible situations of collision between robot parts and obstacle(s), 

the A* algorithm, which is usually used to find a heuristic path on Cartesian W-space, has been used to find a heuristic 

path on C-space map. Several modifications are needed to apply the methodology for a manipulator with degrees of 

freedom more than two. The results of C-space map, which are derived by the modified analysis, prove the accuracy of 

the overall C-space mapping and construction, and then a successful and guaranteed path from a start to goal configuration 

has been obtained without any collision probability. The results had been achieved by (Matlab R2015a) software, which 

run on Intel (R) Core (TM) i3-3120M CPU.  

Keywords: Configuration Space, Degrees of Freedom, Topology, Path Planning. 
 

 

1. Introduction 

 
Due to the importance of path planning and 

obstacle avoidance problems in practical areas, 

there are more researches and studies about it in the 

last decades. In the classical mechanics, the 
configuration space (C-space) represent very 

important tool to describe and analyze the motion 

planning for many important systems, especially in 

robotics [1]. To understand the C-space, first it is 
important to know what the term of configuration 

means in robotic systems. It means a complete 

description, which determines every point position 
on the robot uniquely. So, the C-space of a robot 

typifies the space of all configurations, which are 

processed by the robot. As the robot has n degrees 

of freedom, the C-space for this robot is n-
dimensional [2]. 

C-space deals with static obstacles, or dynamic 

obstacles with slow motion. Also, the shape and 

dimensions of each obstacle must be known before 
finding C-space construction if the path planning 

that will be used is off-line. 

From the first time the C-space was defined [3], 
it played very important role in robotic systems 

especially in path finding problem. Then, the usage 

of C-space continue in many researches and 

applications. Until today, more analysis and 
modifications with many type of robots are applied 

on C-space [4- 7]. 

Path planning for robot arm requires optimized 
global path generation, in which the robot arm can 

avoid any obstacle in the Cartesian W-space. The 

feasibility of path planning depends on some 

factors: map accuracy, number of obstacles and 
current configuration of the robot. There are many 

approaches and algorithms have been proposed to 

solve the path planning problem. This problem is 
related with finding the shortest path between the 

start and goal configurations. The most known 



Firas A. Raheem                                            Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 124- 136 (2017) 
 

125 
 

algorithms that are being used to find shortest path 
are graph search algorithms. These algorithms are 

based on node-edge marking [8]. Graph search 

methods speculate the cost and generate a sequence 
of segments that extent from the start point to the 

goal point through vertices [9]. One of the 

graphical search methods is the A* algorithm. It is 

a graph, heuristic and greedy search algorithm, that 
means the determined solution is approximated not 

optimal (it is Best-First Search) [10]. The main 

advantage of A* that it can be modified easily and 
by change the weights of the traversal costs, it can 

be extended [11]. Usually A* algorithm is applied 

to find the path on Cartesian W-space, while in this 

paper, a different approach adopted by the 
application of A* algorithm on C-space analysis, 

which includes a modified derivation based on 

geometrical analysis to find the construction of C-
space for two revolute joints planar robot arm. 

 

 

2. Modified C-Space Derivation 
   

For a manipulator, the space that represented by 
the variable parameters of joints, either slide 

position or rotation angles, is called C-space. It 

includes two main regions: C-free and C-obstacle, 
in which the second region is a prohibitive because 

it represents collision between the robot arm and 

obstacles and between the links of the robot arm 

[4]. The geometrical representation of the obstacle 

(s) can be denoted as �, so the configuration space 
obstacles will be as below [1]: 

���� = {	: ��	
 ∩  � ≠ ∅}                                … �1
 

While C-free can be represented as below [1]: 

����� = {	: ��	
 ∩ � = ∅}                              … �2
  

Where: 

����  : C-space obstacles 
	: Robot configuration 
��	
 : The set of points of the space confined by 
the robot at configuration 	 
The C-space as overall can be defined as [1]: 

������ = ���� ∪ �����                                       …  �3
 

The mathematical study of the surface 

properties, which are do not change if this surface 
has been applied for certain deformations, such as 

bending or stretching, is called topology of space. 

The topology for robot arm with two revolute joints 
can be described as [12]: 

�� ∗  �� =   !                                                      …  �4
 

       The above equation means that the C-space for 
two revolute joints robot arm is a torus (with no 

joint limits for the joints). Where ��  is the circle 
description, and  ! is the 2-dimensional torus 
surface [12]. Each joint angles pair is corresponded 

by a unique point on the torus. 

 

(a) 2R Robot Arm  (b)2Torus  (c)Sample 

representation 
 

Fig. 1. Topologically representation of the 2-

dimensional C-space [12]. 

To find the mapping of C-space, there are some 
basic shapes of the obstacle(s) will be analyzed.  

 

2.1. C-Space Construction for Point 

Obstacle 
        

Mathematically, the simplest obstacle may be 

found in the workspace is the point. In this paper, a 
modified derivation for C-space analysis will be 

discussed, in which the collision checking based on 

geometrical analysis. 

 

1) First Link-Point Obstacle 
 

A collision between the first link and the point 

obstacle will acquire if the following condition 

achieve: 
#��� = #�     $%&    '� ≥ )    $%&    *� ≥ % …  �5
        
Where: 

d��� =   
-./
0./

 ;         #��� =tan
.� -

0
 ;               …  �6
                                                 

'�, *�: First link end effector coordinate 
), %: Point obstacle coordinate 



Firas A. Raheem                                            Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 124- 136 (2017) 
 

126 
 

   2ᶿ
Point Obstacle

First Link 

Reachability 

Circle

   1ᶿ
obsᶿ

 
 

Fig. 2. The first link with the point obstacle 

geometrical analysis. 

 
 

For Fig. 2, the coordinate of point obstacle is 

inside the first link reachability circle.  If the slope 

of the first link equal to the slope of the point 
obstacle, the collision is acquire. 

 

 

2. Second Link-Point Obstacle 
     

The following mathematical derivation explains 

the condition of collision between the second link 

and point obstacle: 

7 =   8  −   #!                                                 …  �7
 
; =   #���    − #�                                            …  �8
 
By sine's law: 

=>
?@A B

=   
CDEF
?@A G

               H =  sin.�
=>∗?@A G

CDEF
    … �9
                                   

While, 

L7 +  ; +  HN =  8                                      …  �10
 
There is a collision between the robot and the point 

obstacle, where (7,  ; and H) are the angles used for 
the analysis as shown in Fig. 3. 

Point Obstacle

 

Fig. 3. The second link with the point obstacle 

geometrical analysis. 

These collision conditions are applied for any 
position of the point obstacle inside first or second 

link, not only at the end effector for each of them. 

 

2.2. C-Space Construction for Line Obstacle 
 

The robot links can be handled as two lines, 

with regardless of their width. In our research, the 

modified derivation of C-space analysis includes 

two analyses for checking collision. 
 

Analysis One 
 

To implement the checking, there are several 

constraints can be put: 

1)  First Link – Line Collision 
In Fig. 4, if the first joint angle, at specific 

configurations, has values locate at the range of 

angles of end points for line obstacle, this may 
cause collision between the first link and the 

obstacle: 

#P�  ≤  #�  ≤ #P!                                               …  �11
 
Where: 

#PR  : The angles of line obstacle end points, i = 1, 
2. 

It is important to take in consideration that the 
length of the first link equal or more than the 

distance between the robot's base and the line 

obstacle, as in the following equation: 

 S'�, *�T ≥ U'=V , *=VW                                        …  �12
                                                                 
S'�, *�T : The coordinate of first link end effector 
S'=V , *=VT : Matrix of line obstacle points 
 j=1, 2, …, number of line obstacle points 

 

(a) Horizontal Line (b) Vertical Line (c) Sloping Line 

Fig. 4. Cases of first link – line obstacle collision. 

 

 

1) Second Link – Line Collision 
As it was done in checking collision with point 

obstacle, the geometrical analysis can be used to 

find the collision angle between the second link 
and the line obstacle. Equations (5, 6, 7, 8, and 9) 

can be used but it is applied for the intersection 

point between the second link and the line obstacle. 



Firas A. Raheem                                            Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 124- 136 (2017) 
 

127 
 

Fig. 5 illustrate the situations of collision between 
the second link and the line obstacle. 

   

obsᶿ   obsᶿ

   

obsᶿ    obsd

   obsd

 
 

(a) Horizontal Line (b) Vertical Line (c) Sloping Line 

Fig. 5. Cases of second link – line obstacle collision. 
 

 

Analysis Two 
 

Another method for checking collision by 

checking the equality of each link slop with the 
slop of each point in the line obstacle: 

For the first link: 

  
X>./
Y>./

=  
XZ[ ./

YZ[./
                                                  …  �13
  

For the second link: 

 
X\.X>
Y\.Y>

=  
XZ[.X>
YZ[.Y>

                                                …  �14
 

An important condition must be considered for 

equations (13) and (14): the length of each link 

should be more than or equal to the distance of the 
robot's base to that point (for equation 13) or from 

the first link end effector to the point (for equation 

14). 

 
2.3. C-Space Construction for Polygonal Shapes 

Obstacle 
 

Any two dimensions shape formed by straight 

lines, is called polygon. It has three lines (such as 

triangle), four lines (such as square, rectangle, 
rhombic… etc.), or more [13]. So the same 

methods that have been used for line obstacle can 

be used here. 
 

2.4. C-Space Construction for Circle Obstacle 

 

For a circle that is known in center ('� , *� ) and 
radius (rad), the collision can be checked by the 

following modified derivation of C-space 

analysis: 
The distance from the center of the circle to each 

point on the first and second link will be calculated: 

For the first link: 

&��]
 = ^�'� − ] ∗ '�
! + �*� − ] ∗ *�
!  … (15) 
For the second link: 

 &!�]
 = ^�'� − _!
! + �*� − !̀
!           …  �16
 

_! = '� + ] ∗ a! ∗ cos�#� + #! 
               …  �17
 
!̀ = *� + ] ∗ a! ∗ sin�#� + #!
                …  �18
 

d� = minS&��]
T                                        …  �19
 
d! = minS&!�]
T                                        …  �20
 
Where: 

] = 0.1, 0.2, … , �  
It is important to denote that ] is the resolution 

step of deviation of each link, while � is the total 
length of link. Theoretically, the step ] may be 
taken 0.1, smaller than 0.1, or biggest. If the 
following condition verified, that means there is a 

collision between the circle obstacle and the robot 

arm: 

dR  ≤ g$&                                                    …  �21
 
h = 1, 2. 

The distances &R can be implemented by the 
form of deviation as below: 

&R� = S&R�, &R!, … , &Ri T                               …  �22
 
j: Number of steps of deviation (]). 
Fig .6 show the distances that used in the analysis 

for checking collision with circle obstacle.  

)cy,cx(

 
 

Fig. 6. Geometrical analysis of circle obstacle. 
 

3. A* Algorithm 
 

A* algorithm is most familiar search algorithm 

that is used to find the shortest path. It represent an 

extension of Dijkstra's algorithm. The usage of 
heuristic is what make the A* different from 

another graph search algorithms. The heuristic 

provides an estimated distance from a current node 

to the goal node, it denoted by h(n), also the A* 
take in account the cost from the start node to the 

current node, and it is denoted as g(n). The cost 

function is determined as below [9]: 

  f(n) = g(n) + h(n)                                     …  �23
 



Firas A. Raheem                                            Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 124- 136 (2017) 
 

128 
 

For C-space map, each node represented by a 

pair of joint angles, n: (#�, #! ). 
The A* algorithm includes two lists: OPEN list 

(O_List) and CLOSED list (C_List). It choose the 

nodes that have minimum cost functions to create 
a path from start node to goal node. The fulfillment 

of this algorithm on C-space map can be explained 

briefly by the following flowchart: 
 

C-Space Construction

Discretizing C-space map

Initialization of Start 

Configuration

End

Remove n_best at which 

f(n_best)<=f(n) from O-List to 

C_List

End

Expand all n_best neighbor nodes that not 

in C_List

Add current node 

to O_List

If g(n_best)+Path cost<g(current 

node):update back pointer(current 

node)=n_best

Yes

No

Yes

No

Yes
No

Start

Is O_List 

empty?

n_best=goal 

configuration?

Current node 

is in O_List?

 
 

 Fig. 7. A* algorithm on C-space map flowchart. 
 

 
 

4. Results and Discussion 
 

The results of finding C-space construction and 

applying A* algorithm on it, will be displayed and 
discussed in section 4.1 and 4.2 respectively. 

 

 4.1. C-Space Construction Results 
 

For the point obstacle, it can be observed that 
the C-space map of the collision between the first 

link and the point obstacle has the straight-line 

shape, Fig. 9, while the collision between the 
second link and the point obstacle has a curved 

shape. The upper curvature is the result of elbow 

down configuration, Fig. 8, and elbow up 

configuration form the lower curvature as in Fig. 
10. 

 

 
(a) 

ra
d

, 
2

Ө
  
  
 

rad, 1Ө      
(b) 

 

Fig. 8. (a) Second link-point collision in elbow down 

in Cartesian W-space. (b) Corresponding C-space 

map. 

 
 



Firas A. Raheem                                            Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 124- 136 (2017) 
 

129 
 

 
(a) 

ra
d

, 
2

Ө
  

  
 

rad, 1Ө      
(b) 

 

Fig. 9. (a) First link-point collision. (b) 

Corresponding C-space map. 

 
 

 
(a) 

 

ra
d

, 
2

Ө
  
  

 

rad, 1Ө     
 

(b) 
 

Fig. 10. (a) Second link-point collision in elbow up. 

(b) Corresponding C-space map. 

 

 

As the point obstacle approaching to the end 

effector of the robot arm, the C-space obstacle 
points will decrease, as shown in Fig. 11. 

 
 

 
(a) 

ra
d

, 
2

Ө
  
  
 

rad, 1Ө      
(b) 

 

Fig. 11. (a) Point obstacle collides with second link 

only. (b) Curve points decrease as the point be 

nearest to the end effector. 
 

 

 



Firas A. Raheem                                            Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 124- 136 (2017) 
 

130 
 

The C-space map of line obstacle and polygonal 
shape obstacle is achieved by the analysis 

introduced previously in section 2.2 and 2.3. The 

same shape of the curve of C-space map for point 
obstacle applied for each point on the line(s). Here, 

these curves in compact form will represent the C-

space line or polygonal obstacles, as illustrated in 

Fig. 12 through Fig.15.      
 

 
(a) 

ra
d

, 
2

Ө
  

  
 

rad, 1Ө      
(b) 

 

Fig. 12. (a) Robot arm with line obstacle in 

workspace. (b) Corresponding C-space map. 

 
 

 
(a) 

 
 

 

ra
d

, 
2

Ө
  

  
 

rad, 1Ө      
(b) 

Fig. 13. (a) Robot arm with square obstacle in 

Cartesian W-space. (b) Corresponding C-space map. 
 
 

 
(a) 
 

 
(b) 

Fig. 14. (a) Robot arm with rectangular obstacle in 
Cartesian W-space. (b) Corresponding C-space map. 

 



Firas A. Raheem                                            Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 124- 136 (2017) 
 

131 
 

 
(a) 

r
a
d

, 
2

Ө
  

  
 

rad, 1Ө      
(b) 

Fig. 15. (a) Robot arm with octagonal obstacle in 

Cartesian W-space. (b) Corresponding C-space map. 

 

                      

The figures (12-15), represent the case of 

collision with both links. The number of points on 

C-space map are increase as the obstacle in the 
Cartesian W-space be near to the first link. In 

addition, it can be observed that the  �����  is on the 
both sides of the ���� . Therefore, to find the path 
from one side to other, the range of changing #� is 
chosen in somehow for unification the region of 

����� which make the path planning easier. 
For circle obstacle, Fig. 16, there are two main 

parts of edges of the C-space map. First, the 
collision between the first link and the circle 

obstacle circumference, which represented by the 

vertical straight lines with all values of #!. Second, 
the outer upper and lower curved shapes represents 

the collision between the second link and the circle 

obstacle circumference. 

 

 
(a) 

r
a
d

, 
2

Ө
  
  
 

rad, 1Ө      
(b) 

Fig. 16. (a) Robot arm with circular obstacle in 

Cartesian W-space. (b) Corresponding C-space map. 
 
 

For any C-space obstacle shape, the map of C-

space is formed by matched angles. Each pair of 

matched points in the obstacle in Cartesian W-

space has a pair of matched angles in C-space. For 
example: 

 
Table 1,  

Two Matched Points in Cartesian W-Space and 
Their Symmetry Angles in C-space 

X(m) Y(m) kl�mno. 
 kp(deg.) 
0.5808 0.2111 -52 144 

0.211 0.5808 142 -144 

 
 

This pair of points represent the first point and 

last point on the C-space map for circle obstacle 
with center of (0.5, 0.5) and radius equal to 0.3 m, 

which is illustrated in Fig. 17. It can be seen that 

the symmetry points on C-space have the same #! 
value with different sign, which is verifies the two 

solutions of elbow up and elbow down of the 

second joint. 

 



Firas A. Raheem                                            Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 124- 136 (2017) 
 

132 
 

 
 

Fig. 17. The matched points analyzing for circle 

obstacle. 

 

 

For any obstacle shape, if there is a collision 

between the robot links and the circumference of 
that obstacle, by the way any inner point of the 

obstacle has a corresponding point on the C-space 

map. 

 
4.2. Results of Applying A* Algorithm on 

Modified C-Space 
 

The following two environments have different 

obstacle shapes and tasks. First, we do all the steps 

of C-space analysis and then the construction to get 
the exact map, which includes the collision and free 

areas. Second, the start and goal configuration, 

which are represented by the dot and the star shape 
respectively, have been computed by the inverse 

kinematic equations. 

For the first and second environment, the start 
and goal points represented by the dot and the star 

shape respectively. 
 

 
 

Fig. 18. Cartesian W-space of the first environment. 

 

 
 

 
 

Fig. 19. C-space map of the first environment. 

 

d
e
g

, 
2

Ө
  

  
 

deg, 1Ө     

Start 

Point

Goal Point

 
 

Fig. 20. A* path in C-space map for the first 
environment. 

 

Start 

Point

Goal Point

 
 

Fig. 21. The movement of robot arm from start to 

goal point for the first environment. 

 
 

 

 



Firas A. Raheem                                            Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 124- 136 (2017) 
 

133 
 

 

 
 

Fig. 22. Two-link robot arm end effector path in 

Cartesian W-space for the first environment. 

 

Start 

Point

Goal Point

 
 

Fig. 23. Cartesian W-space of the second 

environment. 

 

ra
d

, 
2

Ө
  
  

 

rad, 1Ө     

Start 

Point

Goal 

Point

 

Fig. 24. C-space map of the second environment. 

 

d
e
g

, 
2

Ө
  

  
 

deg, 1Ө     

Start 

Point

Goal 

Point

 

Fig. 25. A* path in C-space map for the second 

environment. 

Start 

Point

Goal 

Point

 
 

Fig. 26. The movement of robot arm from start to 

goal point for the second environment. 

 

Start Point

Goal Point

Region 

(b)

Region 

(c)

 
 

Fig. 27. Two-link robot arm end effector path in 

Cartesian W-space for the second environment. 

 

 



Firas A. Raheem                                            Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 124- 136 (2017) 
 

134 
 

From Fig.18 and Fig.23, both environments 
considered as a crowded environments because all 

obstacles are closed to the arm links with different 

distances. After getting the C-space construction of 
these environments, Fig.19 and Fig.24 show the 

overall C-space map, which include the ����� , ����  
and the start and goal configuration. According to 
the A* path on C-space map, as observed in Fig.20 

and Fig.25, the movement of the robot arm prove 

the success of A* algorithm on these maps, because 
in spite of the nearness of obstacles from the robot 

links, it can rescue itself from the crowded region 

and complete a smoothed path, as in Fig.21 and 
Fig.26. In addition, it is clear that if the A* path on 

C-space is optimal; it is not a condition to be 

optimal on Cartesian W-space for the same 

environment, and vice versa, but it represent a best 
solution for planning the path of the two revolute 

joints planar robot arm between obstacles. In the 

end effector path for both environments, Fig.22 and 
Fig.27, it is transparent there are several curved 

regions, denoted by region (a) in Fig.22 and 

regions (b) and (c) in Fig.27, represent a reflection 

of robot arm carful movement at these regions to 
avoid collision with obstacles. This sudden change 

insure the complete avoidance of the robot arm 

from obstacles.    
The resulted path and the changing of joint 

angles in C-space has the behavior shown in Fig. 

28 and Fig. 29 respectively: 
 

IterationsIterations

d
e

g
, 

p
a

th
 

_
1

Ө
  

  
 

d
e
g

, 
p

a
th

 
_

2
Ө

  
  

 

 

Fig. 28. The path of first and second joint angles 

along the iterations for the first environment. 

  
 

The change of joint angles can be calculated 

from the following equation: 

∆#R = #R �r
 − #R �r − 1
                             …  �24
 
Where: 

∆#R = Change of joint angles, i= 1, 2. 
t: Current iteration. 

 

IterationsIterations

p
a
th

 
_

2
Ө

∆ 

p
a
th

 
_

1
Ө

∆
  
  
 

 
 

Fig. 29. Changing of first and second joint angles 

along iterations for the first environment. 

 

d
e
g

, 
p
a
th

 
_

1
Ө

  
  

 

d
e
g

, 
p
a
th

 
_

2
Ө

  
  
 

 

Fig. 30. The path of first and second joint angles 
along the iterations for the second environment. 

 
p
a
th

 
_

2
Ө

∆ 

p
a
th

 
_

1
Ө

∆
  
  
 

 

Fig. 31. Changing of first and second joint angles 
along iterations for the second environment. 

 
 

From the above results, Fig. 30 and Fig. 31, the 
path and the behavior of changing each joint angle 

for the first environment is smooth. while the 

results of the second environment show there is a 
little of zigzag in the behavior of first joint angle 

because of the nearness of the triangular obstacle 

from the end effector path, so that was reflected on 
C-space by making the path take the form of the 

same obstacle border at particular region. 

 

 



Firas A. Raheem                                            Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 124- 136 (2017) 
 

135 
 

5. Conclusion and Discussion 
 

The methodology of this work can be applied 

for any type of robot manipulator. Here, we tested 
it on 2R planar robot arm. In C-space, in general, 

the robot arm at specific configuration can be 

represented as a point in C-space map, which 
represent the values of first and second joint angles 

at this configuration. The step of changing joint 

angles had been chosen with particular value in 
order to achieve sufficient discretization resolution 

for getting accurate C-space map. The modification 

on C-space has been done by checking all the odds 

of collision by modified derivation depending on 
geometrical analysis. This analysis ease in 

application and gave accurate and exact C-space 

map. After applying A* algorithm on modified C-
space, we achieve a shortest, safe, and rather 

smooth path. (Matlab R2015a) was used to 

program the C-space construction and the 

algorithm of A*, that was by Intel (R) Core (TM) 
i3-3120M CPU. The key advantage of applying A* 

on C-space map not on Cartesian W-space is 

insuring the path in the safe area. For future work, 
the A* algorithm results can be enhanced by using 

classical interpolation equations. 

 
 

6. References 

 
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Configuration Space Construction and 

Optimization for Motion Planning", Published 
by Engineering Sciences Press, 2015. 

[2] Benjamin Wah, "Robot Motion Planning", 
Wiley Encyclopedia of Computer Science and 

Engineering, 2008. 
[3] T. Lozano- Pérez, "Spatial Planning: A 

Configuration Space Approach", IEEE 

Computer Society Washington, DC, USA, 
1983. 

[4] Kasper Althoefer, "Neuro-Fuzzy Motion 
Planning for Robotic Manipulators", 

Department of Electronic and Electrical 
Engineering, King's College London, 

University of London, 1996. 

[5] Wyatt S. Newman, Michael S. Branicky, "Real-
Time Configuration Space Transforms for 

Obstacle Avoidance", The International Journal 

of Robotics, 1991. 

[6] Debanik Roy, " Serial and Parallel Robot 
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(0975 – 8887) 

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  )2017( 124-136، صفحة 4د، العد13دجلة الخوارزمي الهندسية المجلم                          فراس عبد الرزاق رحيم                                          
 

136 
 

 
 

 

  المعدل  C-SPACEلتخطيط المسار على تحليل  *Aتطبيق خوارزمية 
  

  فراس عبد الرزاق رحيم*            أسماء عبد اللطيف حسين**
  الجامعة التكنولوجية /**قسم هندسة السيطرة والنظم،*

 uotechnology.edu.iq@60124*البريد االلكتروني: 
  asmaa.alshawi@yahoo.com**البريد االلكتروني: 

  

 

 

  الخالصة
  

وسهولة.  ) هي لجعل عملية تخطيط المسار أكثر امانC-space. الفائدة من استخدام ()C-spaceفي هذا البحث، تم تقديم اشتقاق مستحدث لتحليل بنية (
) لذراع الروبوت ثنائي الذراع، والتي تتضمن كل احتماالت االصطدام بين أجزاء الروبوت والعوائق المحيطة به، C-space( بعد الحصول على بنية وخارطة

-Cمسار المخمن على خارطة (، والتي عادة يتم تطبيقها في مجال العمل الكارتيزي للروبوت إليجاد المسار المخمن، تم تطبيقها إليجاد ال*Aخوارزمية 
spaceحريات حركة أكثر من اثنين. نتائج خارطة ( يريقتنا المطورة على ذراع روبوت ذ). بضع تعديالت الزمة لتطبيق طC-space والتي تم اشتقاقها ،(

ية الى نقطة الهدف تم الحصول عليه بدون أي ) وبنيته، ومن ثم مسار ناجح ومضمون من نقطة البداC-spaceبطريقتنا المطورة، تثبت دقة الخارطة الكلية (
 Intel (R) Core (TM) i3-3120M باستخدام حاسبة بالمواصفات التالية:  (Matlab R2015a)طة برنامجاستم الحصول على النتائج بو احتمالية اصطدام.

CPU.