Vol. 3, No. 1 | January - June 2020 

 

 

SJET | ISSN: 2616-7069 | Vol. 3 | No. 1 | © 2020 Sukkur IBA University 

55 

 

Static Analysis of Six Bar Tensegrity Ball Structure Robot 

Muhammad Basit Chandio1, Ani Luo1, Asif Raza1, Sanaullah Khushak1, 

Muhammad Nazim Lakhan2, Altaf Hussain Shar2, Kishore Chand2, Fahad 

Hussain3, Zubair Ali Shah4 

Abstract: 

All pre-stressed structures called as tensegrity structures have been introducing into 

robotics, modern architectural designs, and medical necessities called bio-tensegrity, space 

structure alternates, and many other emerging technologies because of its numerous useful 

properties.  Six bar tensegrity ball robot structure has an essential importance in the field of 

robotics due to its deployable, movable, deformable, and easily controllable capabilities. This 

structure has 6 rigid bars (in compression) and 24 flexible strings (in tension) that are connected 

in such a 12 nodes arrangement that provides the highest spatial symmetrical shape to this 

structure among most of the tensegrity structure. In this paper, the internal forces of bars and 

strings on each node have been studied, and the force equation on every node has been 

developed. Furthermore, the balance of all bars and strings forces on the whole structure has 

been analytically verified to assure the structure remains statically pre-stressed under the zero 

net effect of all internal forces applied by bars and strings. 

Keywords: static analysis; force equations; equilibrium 

1. Introduction 

Tensegrity structures are formed by a 

combination of rigid elements (the bars) and 

elastic elements (the strings) [1-4]. The bars 

are always in compression and the strings in 

tension. The entire structure stands by itself 

and maintains its form solely because of the 

internal arrangement of the strings and the 

bars [5-8]. Tensegrity structures were used 

only in arts and architecture in the early1960s 

                                                 

1
College of Mechanical & Electrical Engineering, Harbin Engineering University, Harbin, 

China 
2College of Material Science and Chemical Engineering, Harbin Engineering University, 

Harbin, China 
3Department of Mechanical Engineering MUET SZAB Campus Khairpur Mir's, Pakistan 
4College of Power & Energy Engineering, Harbin Engineering University, Harbin, China 

Corresponding Author: basitchandio47@gmail.com 

when fuller coined this tensegrity idea [9-12]. 

Orthopedic surgeon Stephen Levin found a 

tensegrity structure could be a new 

biomechanical approach and coined a bio-

tensegrity term in the medical field in the 

1970s[13], later these structures have been 

remaining a vital part in medical 

researches[14-17]. Tensegrity structures are 

also used in robotics because of implement 

flexibility, dynamism, and light-weighting 



Chandio M.B (et al.), Static analysis of six bar tensegrity ball structure robot                                              (pp. 55 – 67) 

SJET | ISSN: 2616-7069 | Vol. 3 | No. 1 | © 2020 Sukkur IBA University 

56 

  

[18-21]. These structures are also used in 

space applications due to super deplorability 

[22, 23].  

Louani modeled the mathematics of the 

six-bar tensegrity ball structure robot[24] and 

analyzed the possible and feasible ways to 

drive this structure [25-27]. NASA proposed 

a six-bar tensegrity ball robot structure as 

workable in landing on different surfaces for 

planetary exploration missions and easy to 

move and control the structure [28, 29]. 

       It is vital to recognize the all internal 

forces exerted by all bars and strings on each 

node of six-bar tensegrity ball structure robot 

in a static position in order to move it by 

servomotors or to land it for planetary 

exploring mission to protect the payload. 

Skelton in his book and Julio in his paper have 

mentioned basic general methods to analyze 

the static tensegrity structures [30, 31]. But, 

there is a gap of static analysis of the six-bar 

tensegrity ball structure robot in research. 

Here, a static study of the six-bar tensegrity 

ball structure has been done to analyze the 

position and forces exerted by all bars and 

strings on each node and the overall net effect 

of all internal forces of the structure has been 

observed to verify the static mode of structure.  

     In this paper, a six-bar tensegrity ball 

structure robot has been defined and a node 

matrix has been developed. Later, Individual 

force equations for all nodes have been 

established. In the last, all individual force 

equations on each node have been examined 

to validate the static position of the structure 

by balancing the bars forces and strings 

forces. 

2. Introduction of the Six Bar 
Tensegrity Ball Robot Structure 

Six bar tensegrity ball robot structure has 

6 rigid bars in compression and 24 flexible 

strings in tension.  All bars are shown by blue 

color and strings are shown by red color and 

are connected by 12 nodes. Each node is 

connected with one bar and four strings. In 

this structure, all six bars are lying along 

three different planes. Each plane has two 

parallel bars. Bars b1 & b2 lay along XY 

plane, bars b3 & b4 lay along YZ plane and 

bars b5 & b6 lay along XZ plane. Similarly 

nodes1,2,3 & 4 are in XY plane, nodes 5,6,7 

& 8 are in YZ plane and nodes 9,10,11 & 12 

are in XZ plane as illustrated in Fig.1. 

 

 
Fig. 1.  Six bar tensegrity ball structure robot 

illustrated (a) in MATLAB & (b) in ADAMS 

All members of six-bar tensegrity 

structure are connected by 12 nodes and a 

node ni can be represented in Cartesian 

coordinates as 

n = n n n ,i 1,12i ix iy iz 
     

  

(a) 

Group I 

strings 

Group II 

Strings 

Group 

III 

strings 

b1 & 

b2 in 
XOY 

b3 & b4 
in YOZ 

b5& b6 
in ZOX 

(b) 



Chandio M.B (et al.), Static analysis of six bar tensegrity ball structure robot                                              (pp. 55 – 67) 

SJET | ISSN: 2616-7069 | Vol. 3 | No. 1 | © 2020 Sukkur IBA University 

57 

  

Node matrix (N) can be obtained by putting 

all nodes in a column of the matrix. 

= n , n , n ...n
1 2 3 12 3×12

 
 

N        

Coordinates of all 12 nodes along three axes 

can be rearranged as displayed in Eq. (1) [24]. 

 

5 71 2 3 4 6 8 9 10 11 12

0.5 0.5 0.5 0.5 0 0 0 0 0.5 0.5 0.5 0.5

3 12
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0 0 0 0 R

0 0 0 0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

n n n n n n n n n n n n

L L L L L L L L
b b b b d d d d

N L L L L L L L L
d d d d b b b b

L L L L L L L L
d d d d b b b b

   


     

   

 
 
 
 
 

        (1) 

 

                

3. Internal Forces on Structure 
Every node of the six-bar tensegrity ball 

structure is acted by four strings forces and 

one bar force. We have modeled separate 

equations of forces for each node in all three 

axes. Force density in strings is the tension 

applied string per unit its length (i-e: γ=Ts/Ls) 

and force density by the bar is the force 

applied by bar per unit its length (i-e: 

λ=Fb/Lb) [24, 30]. Six bar tensegrity structure 

is spatial structure and lies along three space 

axes. Each plane contains four nodes.  

3.1. Internal Forces on Nodes laying 
along  XY Plane 

Plane XY contains four nodes 1, 2, 3 & 4. 

We find the force equations of each node. 

Here, node 1 is connected with strings s1, s2, 

s3, s4 & bar b1 in such an arrangement that 

strings pull the node inwards and bar pushes 

it back by the forces Ts1, Ts2, Ts3, Ts4 & Fb1 

respectively, but the node does not move in 

any direction as shown in Fig.2. 

  

Fig. 2.  Forces on node 1 

0
1 2 3 41

F T T T Ts s s sb
       (2) 

As we already mentioned λ=Fb/Lb & 

γ=Ts/Ls in section 3, therefore, by putting 

substitutes of bar force and all string forces in 

terms of force densities in eq. (2), we achieve 

 

 

n9 

n5 

n2 n
7
 

n10 



Chandio M.B (et al.), Static analysis of six bar tensegrity ball structure robot                                              (pp. 55 – 67) 

SJET | ISSN: 2616-7069 | Vol. 3 | No. 1 | © 2020 Sukkur IBA University 

58 

  

(n ) (n ) (n ) (n ) (n ) 0
5 71 1 2 1 1 2 1 3 9 1 4 10 1

n n n n n                  

All bars have the same force density as λ, and all strings have the same force density as γ when 

no external force acts on the structure.  

 

We get Eq. (4) by putting coordinates of nodes in Eq. (3) 

 

Force equations along the x-axis and y-axis 

on node 1 are Eq. (5) & Eq. (6) respectively 

and force equation along the z-axis is zero. 

 (L ) ( 2 ) 0L L
b d b

      (5)  

 ( 2 ) 0L L
b d

                 (6)  

Eq. (5) & Eq. (6) are force equations in 

terms of force densities, bars length, and the 

perpendicular distance between two parallel 

bars.  

Similarly, we find force equations on 

node 2. this node is connected with strings s5, 

s6, s7, s8  & bar b1 in such an arrangement that 

strings pull the node inwards and bar pushes 

it back by forces Ts5, Ts6, Ts7, Ts8 & Fb1 

respectively, but this node does not move in 

any direction also as illustrated in Fig. 3. 

 
Fig 3.  Forces on node 2 

(n ) (n ) (n ) (n ) (n ) 0
5 5 7 71 2 1 2 6 2 11 2 8 12 2

n n n n n              

4
5 7 9 10 11 2

4 0
5 71 2 9 10 1

41 2 5 7 9 10 1

n n n n nn nx x x x x x x

n n n n n n ny y y y y y y

n n n n n n nz z z z z z z



   

       

    

  
  
  
  

   
 

        

 
0.5 ( 0.5 ) 0 0 0.5 0.5 4 0.5

0.5 0.5 0.5 0.5 0 0 4 0.5 0

0 0 0.5 ( 0.5 ) 0.5 ( 0.5 ) 4 0

L L L L L
b b d d b

L L L L L
d d b b d

L L L L
d d b b



      

        

       

  
  
  
     

 
                             

 

n1 

n
5
 

n
7
 

n
11

 

n
12

 

(3) 

(4) 



Chandio M.B (et al.), Static analysis of six bar tensegrity ball structure robot                                              (pp. 55 – 67) 

SJET | ISSN: 2616-7069 | Vol. 3 | No. 1 | © 2020 Sukkur IBA University 

59 

  

 0
5 76 81

F T T T Ts s s sb
      (7) 

Similarly, by putting substitutes of bar force and all string forces in terms of force densities eq. 

(7), we achieve 

(n ) (n ) (n ) (n ) (n ) 0
5 5 7 71 2 1 2 6 2 11 2 8 12 2

n n n n n                

All bars have the same force density as λ, and all strings have the same force density as γ 

when no external force acts on structure, therefore 

 ( ) ( 4 ) 0
5 72 1 11 12 2

n n n n n n n           

 
4

5 7 11 12 22 1

4 0
5 72 1 11 12 2

42 1 5 7 11 12 2

n n n n nn nx x x x x x x

n n n n n n ny y y y y y y

n n n n n n nz z z z z z z



   

       

    

  
  
  
  

   

 (8) 

We get Eq. (9) by putting coordinates of nodes in Eq. (8) 

 
( 0.5 ) 0.5 0 0 ( 0.5 ) ( 0.5 ) 4 ( 0.5 )

0.5 0.5 0.5 0.5 0 0 4 0.5 0

0 0 0.5 ( 0.5 ) 0.5 ( 0.5 ) 4 0

L L L L L
b b d d b

L L L L L
d d b b d

L L L L
d d b b



         

        

       

  
  
  
     

 
(9) 

Force equations along the x-axis and y-

axis on node 2 are in Eq. (10) & Eq. (11) 

respectively and force equation along the z-

axis is zero 

 ( ) ( 2 ) 0L L L
b d b

        (10) 

 ( 2 ) 0L L
b d

    (11) 

Eq. (10) & Eq. (11) are force equations in 

terms of force densities, bars length, and the 

perpendicular distance between two parallel 

bars. Again, node 3 is connected with strings 
s9, s10, s11, s12 & bar b2 in such an arrangement 

that strings pull the node inwards and bar 

pushes it back by forces Ts9, Ts10, Ts11, Ts12 & 

Fb2 respectively, but this node also does not 

move in any direction as shown in Fig. 4.  

 
Fig. 4. Forces on node 3 

n
8
 

n
1

n
6
 

n
9
 

n
4
 



Chandio M.B (et al.), Static analysis of six bar tensegrity ball structure robot                                              (pp. 55 – 67) 

SJET | ISSN: 2616-7069 | Vol. 3 | No. 1 | © 2020 Sukkur IBA University 

60 

  

 0
9 10 11 122

F T T T Ts s s sb
      (12) 

Similarly, by putting substitutes of bar force and all string forces in terms of force densities 

eq. (12), we achieve

 

 

 

All bars have the same force density as λ and all strings have the same force density as γ when 

no external force acts on structure, therefore 

 0(n - n ) + (n + n + n + n - 4n ) =
43 6 8 9 10 3

    

 
4

3 4 6 8 9 10 3

4 0
3 4 6 8 9 10 3

43 4 6 8 9 10 3

n n n n n n nx x x x x x x

n n n n n n ny y y y y y y

n n n n n n nz z z z z z z



    

       

    

  
  
  
  

   

 (13) 

We get Eq. (14) by putting coordinates of nodes in Eq. (13) 

 
0.5 ( 0.5 ) 0 0 0.5 0.5 4 (0.5 )

0.5 ( 0.5 ) 0.5 ( 0.5 ) 0 0 4 ( 0.5 ) 0

0 0 0.5 ( 0.5 ) 0.5 ( 0.5 ) 4 0

L L L L L
b b d d b

L L L L L
d d b b d

L L L L
d d b b



      

             

       

  
  
  
     

 
(14) 

Force equations along the x-axis and y-

axis on node 3 are in Eq. (15) & Eq. (16) 

respectively and force equation along the z-

axis is zero 

 ( ) ( 2 ) 0L L L
b d b

      (15) 

 ( 2 ) 0L L
b d

     (16) 

Eq. (15) & Eq. (16) are force equations in 

terms of force densities, bars length, and the 

perpendicular distance between two parallel 

bars. 

Likewise, node 4 is connected with 

strings s13, s14, s15, s16 & bar b2 in such an 
arrangement that strings pull the node 

inwards and bar pushes it back by forces Ts13, 

Ts14, Ts15, Ts16 & Fb2 respectively, but the 

node does not move in any direction as 

illustrated in Fig. 5. 

 

Fig 5.  Forces on node 4 

0
2 9 10 11 12

(n - n ) + (n - n ) + (n - n ) + (n - n ) + (n - n ) =
43 6 3 8 3 9 3 10 3

      

n
3
 n

6
 

n
1

 

n
1

 

n
8
 



Chandio M.B (et al.), Static analysis of six bar tensegrity ball structure robot                                              (pp. 55 – 67) 

SJET | ISSN: 2616-7069 | Vol. 3 | No. 1 | © 2020 Sukkur IBA University 

61 

  

 0
13 14 15 162

F T T T Ts s s sb
      (17) Similarly, by putting substitutes of bar 

force and all string forces in terms of force 

densities eq. (17), we achieve 

 

 

 

All bars have the same force density as λ, and all strings have the same force density as γ when 

no external force acts on structure, therefore 

 
4 3 11 4

(n - n ) + (n + n + n + n - 4n ) = 0
6 8 12

    

 
4

4 3 6 8 11 12 4

4 0
4 3 6 8 11 12 4

44 3 6 8 11 12 4

n n n n n n nx x x x x x x

n n n n n n ny y y y y y y

n n n n n n nz z z z z z z



    

       

    

  
  
  
  

   

                (18) 

We get Eq. (19) by putting coordinates of nodes in Eq. (18) 

 )

( ) ) )

0.5 0.5 0 0 0.5 0.5 4 ( 0.5

0.5 0.5 0.5 ( 0.5 0 0 4 ( 0.5 0

0 0 0.5 ( 0.5 ) 0.5 ( 0.5 ) 4 0

d d

d d d

d d

b b b

b b

b b

L L L L L

L L L L L

L L L L



   
  

 
  
     

     

           

       
 (19) 

Force equations along the x-axis and y-

axis on node 4 are in Eq. (20) & Eq. (21) 

respectively and force equation along the z-

axis is zero 

 ( ) ( 2 ) 0L L L
b d b

            (20) (20) 

 ( 2 ) 0L L
b d

                               (21) (21) 

Eq. (20) & Eq. (21) are force equations in 

terms of force densities, bars length, and the 

perpendicular distance between two parallel 

bars. 

3.2. Internal Forces on Nodes laying 
along YZ Plane 

Plane YZ also contains four nodes 5, 6, 7 

& 8. Here in this section, we just put the force 

equations on each node lying in the YZ plane, 

and these equations can be found with the 

same method used in section 3.1. 

Here, node 5 is connected with four 

strings s1, s5, s17, s18 & one bar b3 in such an 
arrangement that strings pull the node 

inwards and bar pushes it back by forces Ts1, 

Ts5, Ts17, Ts18 & Fb3 respectively, but the 
node does not move in any direction as 

shown in Fig. 6 

 0
51 17 183

F T T T Ts s s sb
      

(22) 

We found force equations in terms of 

force densities, bars length, and the 

perpendicular distance between two parallel 

bars along the y-axis and z-axis on node 5 in 

2 4 3 13 4 14 4 14 11 4 15 4
0

6 8 12
(n - n )+ (n - n )+ (n - n )+ (n - n )+ (n - n )=      



Chandio M.B (et al.), Static analysis of six bar tensegrity ball structure robot                                              (pp. 55 – 67) 

SJET | ISSN: 2616-7069 | Vol. 3 | No. 1 | © 2020 Sukkur IBA University 

62 

  

Eq. (23) & Eq. (24) respectively and force 

equation along the x-axis as zero. 

 ( ) ( 2 ) 0L L L
b d b

      (23) 

 ( 2 ) 0L L
b d

    (24) 

 

 

Fig. 6.  Forces on node 5 

Similarly, Ts9, Ts13, Ts19, Ts20 & Fb3 are 
forces exerted by four strings s9, s13, s19, s20  

& one bar b3 on node 6 in such manner that 
strings pull the node inwards and bar pushes 

it back and the node does not move in any 

direction as shown in Fig. 7. 

 0
9 13 19 203

F T T T Ts s s sb
      (25) 

We found force equations in terms of 

force densities, bars length, and the 

perpendicular distance between two parallel 

bars along the y-axis and z-axis on node 6 as 

in Eq. (26) & Eq. (27) respectively and force 

equation along the x-axis as zero. 

 ( ) ( 2 ) 0L L L
b d b

        (26) 

 ( 2 ) 0L L
b d

    (27) 

Four strings s2, s6, s21, s22, & one bar b4 
exert forces Ts2, Ts6, Ts21, Ts22 & Fb4 

respectively one node 7 in such manner that 

this node has no movement in any direction 

as shown in Fig. 8. 

 

 

Fig. 7.   Forces on node 6 

 

Fig. 8. Forces on node 7 

 0
2 6 21 224

F T T T Ts s s sb
      (28) 

n
9
 

n
1
 

n
6
 

n
1

 

n
2
 

n
3
 

n
5
 

n
4
 

n
9
 

n
1

 

n
1
 

n
2
 

n
10

 

n
12

 

n
8
 



Chandio M.B (et al.), Static analysis of six bar tensegrity ball structure robot                                              (pp. 55 – 67) 

SJET | ISSN: 2616-7069 | Vol. 3 | No. 1 | © 2020 Sukkur IBA University 

63 

  

We found force equations in terms of 

force densities, bars length, and the 

perpendicular distance between two parallel 

bars along the y-axis and z-axis on node 7 as 

in Eq. (29) & Eq. (30). Moreover, the force 

equation along the x-axis is zero. 

 ( ) ( 2 ) 0L L L
b d b

      (29) 

 ( 2 ) 0L L
b d

     (30) 

Similarly, node 8 is balanced by forces 

Ts10 ,Ts14 ,Ts23 , Ts24 & Fb4 applied by four 

strings s10, s14, s23, s24 & one bar b4 as shown 
in Fig.9.  

 

Fig. 9. Forces on node 8 

 0
10 14 23 244

F T T T Ts s s sb
      

(31) 

We found force equations in terms of force 

densities, bars length, and the perpendicular 

distance between two parallel bars along the 

y-axis and z-axis on node 8 as in Eq. (32) & 

Eq. (33) respectively and force equation 

along the x-axis is zero. 

 ( ) ( 2 ) 0L L L
b d b

        (32) 

 ( 2 ) 0L L
b d

     (33) 

3.3. Internal Forces on Nodes laying 
along XZ Plane 

Plane XZ also contains four nodes 9, 10, 

11 & 12. Here in this section, we just put the 

force equations on each node lying in the XZ 

plane, and these equations can be found with 

the same method used in section 3.1. 

So, node 9 is connected with four strings 
s10, s14, s23, s24 & one bar b5 in such an 
arrangement that strings pull the node 

inwards and bar pushes it back by the forces 
Ts10, Ts14, Ts23 , Ts24 & Fb5     respectively but 

node remains in the same position as shown 

in Fig.10.  

 

Fig 10.  Forces on node 9 

 0
3 11 17 195

F T T T Ts s s sb
      

(34) 

We found force equations in terms of 

force densities, bars length, and the 

perpendicular distance between two parallel 

bars along the x-axis and z-axis on node 9 as 

n
7
 

n
10

 

n
3
 

n
4
 

n
12

 

n
5
 

n
10

 

n
1
 

n
6
 

n
3
 



Chandio M.B (et al.), Static analysis of six bar tensegrity ball structure robot                                              (pp. 55 – 67) 

SJET | ISSN: 2616-7069 | Vol. 3 | No. 1 | © 2020 Sukkur IBA University 

64 

  

in Eq. (35) & Eq. (36) respectively and force 

equation along the y-axis as zero. 

 ( 2 ) 0L L
b d

    (35) 

 ) ( 2 ) 0L L L
b d b

      (36) 

Similarly, four strings s4, s12, s21, s23, & 
one bar b5 apply forces Ts4, Ts12, Ts21, Ts23 & 
Fb5    respectively on node 10 in such manner 

that node does not move in any direction as 

shown in Fig. 11. 

 

 

Fig. 11.  Forces on node 10 

 0
4 12 21 235

F T T T Ts s s sb
      (37) 

We found force equations in terms of 

force densities, bars length, and the 

perpendicular distance between two parallel 

bars along the x-axis and z-axis on node 10 

as in Eq. (38) & Eq. (39) respectively and 

force equation along the y-axis as zero. 

 ( 2 ) 0L L
b d

    (38) 

 ) ( 2 ) 0L L L
b d b

       (39) 

Node 11 is connected with strings s7, s15, 

s18, s20, & b6 in such an arrangement that 

strings pull the node inwards and bar pushes 

it back Ts7, Ts15, Ts18 , Ts20 & Fb6 respectively, 

but the node does not move in any direction 

as illustrated in Fig.12. 

 

Fig. 12.  Forces on node 11 

We found force equations in terms of 

force densities, bars length, and the 

perpendicular distance between two parallel 

bars along the x-axis and z-axis on node 11 

as in Eq. (41) & Eq. (42) respectively and 
force equation along the y-axis as zero. 

 ( 2 ) 0
b d

L L   
 

(41) 

 ) ( 2 ) 0L L L
b d b

      (42) 

Finally, node 12 is pulled by forces Ts8, 

Ts16, Ts22 , Ts24 applied by four strings s8, s16, 

s22, s24, and pushed back by force Fb6 applied 

 0
7 15 18 206

F T T T Ts s s sb
      (40) 

n
5
 

n
4
 

n
2
 

n
12

 

n
6
 

n
7
 

n
9
 

n
1
 

n
3
 

n
8
 

Ts4 



Chandio M.B (et al.), Static analysis of six bar tensegrity ball structure robot                                              (pp. 55 – 67) 

SJET | ISSN: 2616-7069 | Vol. 3 | No. 1 | © 2020 Sukkur IBA University 

65 

  

by bar b6, but the node does not experience 

any movement in any direction as shown in 

Fig.13. 

 

Fig. 13.  Forces on node 12 

 0
8 16 22 246

F T T T Ts s s sb
      (43) 

We found force equations in terms of 

force densities, bars length, and the 

perpendicular distance between two parallel 

bars along the x-axis and z-axis on node 12, 

as in Eq. (44) & Eq. (45) respectively and 

force equation along the y-axis as zero. 

 ( 2 ) 0L L
b d

     (44) 

 ) ( 2 ) 0L L L
b d b

       (45) 

3.4. Internal force on the whole 
structure 

We have found force equations of all 12 

nodes along three axes individually. Here, we 

add all these force equations to analyze 

overall force along each axis on the whole 

structure. We got net force as zero by adding 

all force components along the x-axis of all 

nodes as follows: 

(L ) ( 2 ) ( ) ( 2 ) ( ) ( 2 ) ( )

( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) 0

L L L L L L L L L
b d b b d b b d b b

L L L L L L L L L L
d b b d b d b d b d

                  

                 
 (46) 

Similarly, we get net force as zero by adding all force components along the y-axis of all 

nodes as follows 

( 2 ) ( 2 ) ( 2 ) ( 2 ) ( ) ( 2 )

( ) ( 2 ) ( ) ( 2 ) ( ) ( 2 ) 0

L L L L L L L L L L L
b d b d b d b d b d b

L L L L L L L L L
b d b b d b b d b

                  

                  
 

(47) 

Again, we get net force as zero by adding all force components along the z-axis of all nodes 

as follows 

n
2
 

n
4
 

n
7
 

n
8
 

n
11

 



Chandio M.B (et al.), Static analysis of six bar tensegrity ball structure robot                                              (pp. 55 – 67) 

SJET | ISSN: 2616-7069 | Vol. 3 | No. 1 | © 2020 Sukkur IBA University 

66 

  

( 2 ) ( 2 ) ( 2 ) ( 2 ) ) ( 2 )

) ( 2 ) ) ( 2 ) ) ( 2 ) 0

L L L L L L L L L L L
b d b d b d b d b d b

L L L L L L L L L
b d b b d b b d b



  

                  

                  
(48)

 

It can be analyzed by the above 

calculations that all internal forces exerted by 

rigid bars and flexible strings acting on the 

whole structure have no net effect along any 

axis, which results in the structure to remain 

in equilibrium, this null result of all internal 

forces of members on whole structure shows 

the all force equations developed for each 

node are valid. 

4. Conclusion 

We defined the structure of the six-bar 

tensegrity ball structure robot and developed 

its nodes matrix. Force equations for each 

node were developed individually. The 

overall net effect of forces applied by strings 

and bars of the structure was investigated, 

and it was observed that forces applied by all 

strings and bars on the whole structure 

balance each node and form a whole spatial 

balanced structure. 

ACKNOWLEDGEMENT 

This research has been sponsored by the 

National Natural Science Foundation of 

China (Grant No. 51605111, 51675114, 

51875111).  

NOMENCLATURE  

Lb Length of each bar 

Ld Distance between each couple of parallel 

bars 

Fb internal force of bars 

Ts Internal force of strings 

λ Bar force density 

γ String force density in group I strings after  

 

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