RESEARCH ON THE PIPELINE TRANSPORT ROBOTS ELEMENTS

1. Introduction

Robots of various types and designs are widely used for effective control and repair of gas pipelines, water pipelines or communication lines [1, 2, 5, 6]. One kind namely, the pipeline transport robots those with elastic elements is discussed here. The object of the present project is the development of an original ro- bot with elastic elements and the investigation of its movement.

2. Designs of Robots Elements

Usually a robot consists of several transporting blocks connected with elastic elements [3] (Fig 1). The transporting blocks are equipped with elements that contact with the internal surface of the pipe in several points and ensure fixing of the block in the pipe. Be- cause of a simple structure and sufficiently good fixa- tion in the pipe robots with elastic elements consi- sting of rings, half-rings and similar easily deformable elements may be usable as well.

A transporting block of such robots consists of a drive, deformable elastic elements that pay a role of block fixing supports and block connecting elements.

When one block is fixed, other blocks may turn and move along the fixed block. This enables the equip-

Bronislovas SPRUOGIS

RESEARCH ON THE PIPELINE TRANSPORT ROBOTS ELEMENTS

In the article are reviewed constructions of the pipeline transport robots elements and scheme of new original construction is presented. The mathematical model of the pipeline transport robot is formed and its motion equations are presented.

Keywords: a pipeline transport robot, robots elements, vibration exciter.

ment to go round obstacles on a movement in the pipe.

However, some robots with transporting blocks of this type have some imperfections (such as large si- zes, limited power supply possibilities) that cause li- mited possibilities of their application.

The most frequent imperfection – such robots may be used only in straight fragments of pipelines (Fig. 1) and their capability to move in sections of different diameters of pipes is limited.

So, one of newer designs providing extended possi- bilities of application is a robot with ring elements, consi- sting of transporting blocks connected with elastic ele- ments (Figs 2, 3). Such robots consist of rings or their elements with the ends of the latter connected with ela- stic link elements and with each other forming blocks.

The interconnected blocks form a chain of the blocks.

In such a way an increased capability of the robot to manoeuvre and permeability of the robot on its movement in pipes is ensured.

Fig. 2. The schemes of the robot block with ring ele- ments a) and its connection b): 1 - ring-sha- ped elastic element, 2 - the element connec- ting blocks and their ring elastic elements, 3 – damper, 4 - spring

Fig. 1. The scheme of the transport robot with ring- shaped elements: 1 - pipe, 2 - transporting blocks, 3 - elastic support elements

3. The Mathematical Model of the Trans- port Robot with Ring Elements Located Perpendicularly to the Direction of its Movement

The mathematical model of a robot is developed for the determination of its characteristics and at the beginning it may be usable in a simplified form. So, a robot with elastic ring elements situated perpendi- cularly to the direction of its motion (Fig 3) may be simulated by one-support dynamic model and exami- ned as a dynamic model of one-dimensional vibration exciter with two degrees of freedom (Fig 4).

The coordinates of the point A (x₀; y₀) of the excitable mass m in the coordinate system X₀0₀Y₀ and the point A(x, y) of the same mass in the coordinate system X0Y are expressed by the following equations [3, 4]:

; sin y cos x z

x= + ₀ α− ₀ α (1)

; cos y sin x

y= ₀ α+ ₀ α (2)

(

^{x z}

)

^{cos y sin ;}

x₀ = − α+ ₀ α (3)

(

^{x z}

)

^{sin ycos .}

y₀ =− − α+ α (4)

The kinetic and potential energy of the system as well as its dissipative function shall be expressed as follows:

[

^{m z m(x y )}

]

^;

5 , 0

T = ₀² + ²+ ² (5)

[

^{c x c y c (y y )}

]

^;

5 ,

0 _{x 0}²+ _{y 0}²+ _a − _s ²

Π = (6)

[

^{H x H y H y}

]

^;

5 , 0

D= _x₀²+ _y₀²+ _a² (7) where m – mass; x, x₀, y, y₀– shifts of the robot and its mass; y_s – distance between support and the coordinate axel X, c_x, c_y, c_a – coefficients of stiffness;

H_x, H_y, H_a – coefficients of resistance of a linear shift;

f₀, f₁ – the coefficients of dry and liquid friction.

The mathematics expression of the dynamic mo- del of the system shall be the following equations:

•

when y≤ys, there is no contact with the support at the point A:

( )

[

^{− +}

]

⁺

+^{c x z cos}α ^ysinα ^cosα x

m _x

( )

[

^{− −}

]

⁺

+c_y x z sinα ycosα sinα

( )

[

^{− +}

]

⁺

+^Hx ^x ^z ^cosα ^y^sinα ^cosα

( )

[

^{x z sin ycos}

]

^{sin F ;}

H_y − − = _x

+   α  α α (8)

( )

[

^{− +}

]

⁻

+^{c x z cos}α ^ysinα ^sinα y

m _x

( )

[

^{− −}

]

⁺

−^cy ^{x z sin}α ^ycosα ^cosα

( )

[

^{− +}

]

⁻

+^Hx ^x ^z ^cosα ^y^sinα ^sinα

( )

[

^{x z sin ycos}

]

^{cos F ;}

H_y − − = _y

− ^{ } α ^ α α (9)

( )

[

^{− +}

]

⁻

−^{c x z cos}α ^ysinα ^cosα z

m₀ _x

( )

[

− −

]

−

−c_y x z sinα ycosα sinα

( )

[

^{− +}

]

⁻

−^Hx ^x ^z ^cosα ^y^sinα ^cosα

( )

[

^{x z sin ycos}

]

^{sin F ;}

H_y − − = _z

−   α  α α (10)

•

when y≥ ys, there is a contact with the support at the point A and the following forces appear.

Along the axis 0Y:

(

^{y y}

)

^.

c y H

F_so= _a+ _a − _s (11) Along the axis 0X:

. x f x Nsign f

F_f = ₀ + ₁ (12) The normalized force of pressure:

; F

N= _so (13)

Fig. 3. The scheme of a robot blocks with ring ele- ments located along the direction of its move- ment: 1 - elastic ring element, 2 - supports with excitable masses, 3,5 - electrostatic exci- ting elements, 4 - elastic element, 6 - hinge block jointing element

Fig. 4. The scheme of the dynamic model of the sys- tem of the one-dimensional vibration exciter with two degrees of freedom

( )

[

^{− +}

]

⁺

+^{c x z cos}α ^ysinα ^cosα x

m _x

( )

[

− −

]

+

+c_y x z sinα ycosα sinα

( )

[

^{− +}

]

⁺

+^Hx ^x ^z ^cosα ^y^sinα ^cosα

[ ( )

^{x z sin ycos}

]

^{sin F F ;}

H_y − − + _f = _x

+   α  α α (14)

( )

[

^{− +}

]

⁻

+^{c x z cos}α ^ysinα ^sinα y

m _x

( )

[

^{− −}

]

⁺

+^cy ^{x z sin}α ^ycosα ^cosα

( )

[

^{− +}

]

⁻

+^Hx ^x ^z ^cosα ^y^sinα ^sinα

( )

[

^{x z sin ycos}

]

^{cos F F ;}

H_y − − + _so = _y

− ^{ } α ^ α α (15)

( )

[

^{− +}

]

⁻

−^{c x z cos}α ^ysinα ^cosα z

m₀ _x

( )

[

^{− −}

]

⁻

−c_y x z sinα ycosα sinα

( )

[

^{− +}

]

⁻

−H_x x z cosα ysinα cosα

[ ( )

^{x z sin ycos}

]

^{sin F .}

H_y − − = _z

−   α  α α (16)

After the solution of the above-presented equ- ations we find the characteristics of the dynamic mo- del of the one-dimensional vibration exciter with two degrees of freedom (Figs 5, 6).

On the further analysis of the obtained solution and the optimization of the parameters of the dynamic models it was concluded that the design of the robot requires improvement. The obtained shifts of the vi- bration exciter were very small increasing the exciting force. So in order to ensure more effective dynamic characteristics of the robot it was decided to change its design for the one where deformable rings are de- formed in the direction of the movement of the robot (Figs 2, 7).

4. The Mathematical Model of the Trans- port Robot with Ring Elements Situated Along the Direction of its Movement Developing a mathematical model of a robot with elastic ring-shaped elements (Figs 2, 7) some changes are introduced to simplify the calculations. Points of a ring element may be singled out and their coordina- tes may be used for the description of the model or the ring elements may be replaced with solid members.

The coordinates of the points of the simplified model of a robot (Fig 8) [4] may be written as follows:

0,0E+00 1,0E- 05 2,0E- 05 3,0E- 05 4,0E- 05 5,0E- 05

0 20 40 60 80 100

Time, s

The shift according to X axis, m

Fig. 5. The shifts of the exciter according to X axis dependence upon the time

0,0E+00 1,0E- 07 2,0E- 07 3,0E- 07 4,0E- 07 5,0E- 07

0 20 40 60 80 100

Time, s

The shift according to Y axis, m

Fig. 6. The shifts of the exciter according to Y axis dependence upon the time

Fig. 7. The scheme of a robot with ring elements 1-pipe, 2, 3, 4 - the blocks formed of elastic elements and united into the chain

Fig. 8. The simplified scheme of a robot calculation

(

^{i,j 1, 0 i,j 1}

)

1 j ,

i X L Y

A _{+ +} + ₊ ; (17)

(

i i 1

)

1 i,

i X X

2

X ₊ = 1 + ₊ ; (18)

(

i i 1

)

²

1 2 i,

i X X

4 R 1

Y ₊ = − − ₊ ; (19)

here R-the length of a section;

(

^{i i 1}

)

1 i,

i X X

2

X ₊ =1  +  ₊ ; (20)

(

^{i i 1}

)

1 i, i

1 i 1 i

i,

i X X

Y 4

X

Y X ₊

+

+ =− − +  − 

 ; (21)

here dt

= d

⋅

.

The kinetic energy of the system:

( )

^{+ ∑}

( )

∑ +

= ⁺

=

= + +

1 q

1 i

i2 0 q

1 i

2i, 1 2 i

1 i,

i m X

2 Y 1 X 2m

T 1    ; (22)

here q=1,2,…n;

where

(

⁺

)

⁺

=

+ _{+ +}

+ 2

1 i 2 i

1 i, 2 i

1 i,

i X X

4 Y 1

X   

(

^{i i 1}

)

²

1 i,

i X X

4K

1 + − +

+   ; (23)

(

i i 1

)

1 i, i 2

1 i, i

1 i 1 i

i,

i K X X

Y X X 4

K 1 _{+ +}

+

+ + ⎟⎟ = −

⎠

⎞

⎜⎜

⎝

⎛ −

= ; (24)

( ) ( ) ^{( )}

⎢⎣⎡ + + + − +

= 1 K₁₂ X₁² X₂^{2 2} 21 K₁₂ X₁X₂ 8

T m    

(

⁺

) (

⁺

)

⁺

⁽

⁻

⁾

⁺

+ 1 K₂₃ X₂² X₃^{2 2} 21 K₂₃ X₂X₃

(

⁺

) (

⁺

)

⁺

⁽

⁻

⁾

⁺

+ 1 K₃₄ X₃² X₄^{2 2} 21 K₃₄ X₃X₄

(

^1+K45

) (

^X42+X52

)

²⁺²

⁽

^1−K45

⁾

^X4X5_⎥⎦^⎤+

(

^{12 22 32 42 52}

)

0 X X X X X

m  +  +  +  + 

+ . (25)

The forces of inertia are found:

( )

i i i

X X I

T X

T dt

d =

∂

− ∂

∂

∂

 ; (26)

( )

^{X1 = m}₄

[ (

^1+K12

)

^X1+

(

^1−K12

)

^X2+

I  

( )

K12^'X

(

X¹ X²

)

² m⁰X¹ 2

1

1







 − ⎥⎦⎤+

+ ;

( ) (

j ⁼

[

1⁺Kj−1,j

) (

Xj⁺1⁻Kj−1,j

)

Xj−1⁺

(

1⁺Kj,j+1

)

Xj⁻

4 X m

I   

(

⁻

)

⁻

( ) (

⁻

)

⁺

− _{+ + − −}

−

2 j 1 j ' j X , 1 j 1 j 1 j ,

j K X X

2 X 1 K 1

1 j









(

Kj,j 1

) (

^'X X^j X^{j 1}

)

² m⁰X^j 2

1

j







 − ⎥⎦⎤+

+ _{+ +} ; (27)

here j=2,…,q-1

( ) (

q ⁼

[

1⁺Kq−1,q

) (

Xq⁺ 1⁻Kq−1,q

)

Xq−1⁻ 4

X m

I  

(

Kq 1,q

) (

^'X X^{q 1} X^q

)

² m⁰X^q 2

1

1 q







 − ⎥⎦⎤+

− _{− −}

− . (28)

The potential energy of the system:

(

^{X X L}

)

C _{1 2}

'

X1 = − +

Π ; (29)

(

^{j j 1 j 1}

)

'

Xj=C2X −X ₋ −X ₊

Π ; (30)

(

^{X X L}

)

C _{q q 1}

'X_q = − ₋ −

Π . (31)

The dissipative function:

( ) ( )

⎢⎣⎡ − + − +

= 2 3 ²

2 2

1 X X X

2 X

D H    

(

⁻

)

_⎥⎦^⎤

+

+... X_q−1 X_q ² ; (32)

(

^{1 2}

)

'

X H X X

D 1



 =  − ; (33)

(

^{j j 1 j 1}

)

'

X H 2X X X

D_j1 =  −  − −  + ; (34)

(

^{q q 1}

)

'

X H X X

D_q =  −  − . (35)

The differential equations of the movement of the system

( )

' f 12

X 'X

1 D F F

X

I +Π 1+ 1+ 12 = ; (36)

( )

⁺ j⁺ ' j⁺ fj,j₋1⁺ fj,j₊1⁼

X ' X

j D F F

X

I Π 

1 j , j 1 j ,

j F

F ₋ + ₊

−

= ; (37)

( )

^'X ^{f q,q 1 n}

' X

q D F F F

X

I +Π q+ q+ q,q₋1=− ₋ − ; (38) here F_n – the useful resistance force.

The friction force in the point A_j,j+1 is equal to

(

^{j j 1}

)

0 1 j , j

f N f sign X X

2 F 1

1 j ,

j ₋ = −  +  − ; (39)

here N_j,j-1 – the normalised force at the point A_j,j-1 ac- ting between the immovable surface of support and the robot’s ring element point A_j,j-1; f₀– dry friction coefficient.

At the point A_j, the external exciting force is equal to:

1 j , j 1 j , j

s F F

F =− ₋ + ₊ ; (40)

Prof. dr hab. Alvydas PIKUNAS Automobile Transport Department Vilnius Gediminas Technical University,

Basanaviciaus g. 28, LT-2009 Vilnius, Lithuania, Tel.: 370-5-2744789; Fax: 370-5-2745068 E-mail: vaida@ti.vtu.lt

Prof. dr hab. Bronislovas SPRUOGIS Mgr inÝ. Arvydas MATULIAUSKAS

Department of Transport Technologic Equipment, Vilnius Gediminas Technical University,

Plytins g. 27, LT-2040 Vilnius, Lithuania, Tel.: 370-5-2744783; Fax: 370-5-2745060 E-mail: tti@ti.vtu.lt

( )

( )

⎪⎪

⎩

⎪⎪⎨

⎧

= + +

−

<

+

−

>

+ +

= +

−

−

−

−

. 0 X X kai , 1 , 1

, 0 X X kai , 1

, 0 X X kai , 1 X

X sign

1 j j

1 j j

1 j j 1

j j

















(41)

When Y_j,j-1=Y_a, i.e. to the distance to the support, in the equations F_{f j,j-1}ą0. And when Y_j,j-1<Y_a, then F_{f j,j-1}=0.

The solution of these equations and a further ana- lysis of the obtained results as well as the optimiza- tion of parameters of the mathematical model allow to improve the design of the robot and to develop its optimal variant.

5. Conclusions

Designs of pipeline transport robots elements are reviewed and the scheme of a new original construc- tion is presented. Mathematical models of robots with elastic elements are developed and their dynamic equ- ations are written.

6. References

[1] Komori M., Suyama K.: Inspection robots for gas pipelines of Tokyo Gas. Advanced Robotics, Vol 15, No 3, 2001, p. 365 - 370.

[2] Choi H.R., Ryew S.M.: Robotic system with active steering capability for internal inspection of urban gas pipelines.

Mechatronics, Vol 12, Issue 5. 2002, p. 713-736.

[3] Kulvietis G., Mištinas V., Matuliauskas A., Ragulskis K., Spruogis B.: Dynamic Parameter Investigation of Pipe Robots.

Journal of Vibroengineering No 2 (7). 2001, p. 55 - 58.

[4] Ɍimoshenko S.P., Yang D.Kh., Weaver W.: Vibrations in engineering (Ʉɨɥɟɛɚɧɢɟ ɜ ɢɧɠɟɧɟɪɧɨɦ ɞɟɥɟ). Mashinostrojenije, 1985. 472 p. (in Russian).

[5] Matuliauskas A., Spruogis B.: Pipeline Robots With Elastic Elements. TRANSPORT. Vilnius: Technika, 2002, XVII t.

Nr.5, p. 177-181.

[6] Spruogis B., Ragulskis K., Bogdeviþius M., Ragulskis M., Matuliauskas A., Mištinas V.: In pipeline walking robot.

Patent of Lithuania (Vamzdyno viduje žingsniuojantis robotas. Lietuvos patentas) LT4968B.-Vilnius, 2000. (in Lithuanian).