Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 8, No. 2, 2023 

 

121 

Design	and	Optimization	Analysis	of	Downhole	Robot	
Support	Mechanism	Based	on	Finite	Element	Method	
Wei Wu, Xiaolin Ren, Yijun Liu, Pengpeng Zhang, Yu Li 

School of Mechanical Engineering, Xi 'an Shiyou University, Xi 'an 710065, China 

 

Abstract: To increase the traction force of the telescopic downhole robot, the support mechanism of the downhole robot was 
optimized, and a slider-spring transmission mechanism was introduced to solve the problem of jamming. A mathematical model 
of the support mechanism was established, covering the relationship between the contact normal pressure and the hydraulic 
cylinder load. The finite element simulation model of the friction block of the support mechanism was established based on 
Ansys software. Stress analysis of the casing under different contact normal pressures was carried out, and the frictional 
characteristics of the support mechanism friction block were obtained. The results show that the support mechanism composed 
of four linkages and friction locking blocks proposed in this paper can adapt well to casing of different diameters. The slider 
spring, as a transmission mechanism, has the elasticity and buffering effect, which makes the support mechanism have a certain 
flexibility, alleviating the vibration impact during obstacle crossing and reducing the impact on logging instruments. It enables 
the robot to passively adjust during obstacle crossing, effectively improving the traction capability of the downhole robot. 
Through finite element simulation, it was found that the depth of the plow groove produced by the friction block under normal 
pressure is basically proportional to the normal pressure. The greater the normal pressure, the deeper the plow groove, and the 
plow groove friction coefficient increases continuously with the increase of normal pressure, but the change in the plow groove 
friction coefficient gradually decreases as the pressure increases. 

Keywords: Downhole robot; Finite element simulation; Optimization design; Support mechanism. 
 

1. Introduction 
The telescopic downhole robot is a robot used for 

downhole operations such as horizontal well drilling and 
logging. It has a large traction force and traction distance, 
which can solve the problem of insufficient traction force of 
traditional wheeled downhole robots in conveying continuous 
oil pipes and downhole operation instruments. However, the 
control system of the telescopic downhole robot is one of the 
key technologies for its underground operation. Currently, 
existing telescopic downhole robots face challenges such as 
complex support structure, jamming problems, and low 
control accuracy, which restrict their response speed and 
control precision. However, due to the confidentiality 
measures and high R&D costs of hydraulic telescopic 
downhole robot technology, its development in China is 
relatively lagging behind. Currently, research institutions and 
companies at home and abroad are increasing their research 
and application efforts on hydraulic telescopic downhole 
robots to promote further breakthroughs and applications of 
this technology. 

This paper proposes a support mechanism composed of 
four linkages and friction locking blocks, which can adapt 
well to casing of different diameters. The slider spring, as a 
transmission mechanism, has the elasticity and buffering 
effect, which makes the support mechanism have a certain 
flexibility, alleviating the vibration impact during obstacle 
crossing and reducing the impact on logging instruments. It 
enables the robot to passively adjust during obstacle crossing. 
The frictional characteristics of the friction block support 
mechanism were obtained through finite element simulation. 

2. Working Principle of Downhole 
Robot and Supporting Mechanism 

2.1. Working principle of retractable 
underground robot 

The telescopic downhole robot consists of two parts: the 
front working section and the rear working section. These two 
parts work independently and in coordination with each other. 
The working principle is as follows: 

 

 
1. Left support cylinder; 2. Left support mechanism; 3. Left 

traction cylinder; 4. Control system; 5. Right traction 
cylinder; 6. Right support mechanism; 7. Right support 

cylinder; 8. Directional casing; 9. Power drilling tool; 10. 
Drill bit. 

Figure 1. Working principle of telescopic downhole robot 
 
The motion principle of the telescopic downhole robot is 

shown in Fig. 1. When crawling inside the wellbore or 
pipeline, the left and right working sections alternate to 
achieve telescopic crawling forward. One motion cycle can 
be divided into 7 motion steps. After completing one cycle of 
motion, the robot moves forward by the stroke of 2 telescopic 



 

122 

cylinders. 

2.2. Working principle of the support 
mechanism 

The friction locking block is designed based on the support 
structure of multi-link and four-link mechanisms with the 
casing wall. According to the driving principle of the motor-
driven screw nut in existing downhole robots, a slider-spring 
transmission mechanism driven by a hydraulic cylinder is 
proposed. The support mechanism is shown in Figure 2, and 
it includes two steps: 

(1) Support action: When the servo valve receives the 
control signal, it adjusts the spool displacement, causing the 
hydraulic cylinder piston rod to exert an output force F1, 
compressing the pressure spring and pushing the slider to 
slide in the piston sleeve. At the same time, the return spring 
is compressed, driving the right push rod to move to the left. 
This continues until the locking block completely contacts the 
casing wall, and the return spring is no longer compressed. As 
the piston rod continues to move, the pressure spring 
continues to compress, providing sufficient contact normal 
pressure for traction until the hydraulic cylinder piston stroke 
ends. 

(2) Retraction action: When the servo valve receives the 
control signal, the solenoid valve switches, causing the 
hydraulic cylinder piston rod to move in the opposite 
direction, driving the slider to move the right push rod to the 
right. The return spring and pressure spring return to their 
initial state, and the hydraulic cylinder piston returns to its 
initial position. 

 

 
Figure 2. Working principle of hydraulic telescopic 

downhole robot support mechanism 
 
The support structure proposed in this article, composed of 

a four-link mechanism and a friction locking block, is able to 
adapt well to different diameter casings. The slider spring, as 
a transmission mechanism, provides the support structure 
with elasticity and a cushioning effect, allowing it to have a 
certain degree of flexibility. This helps to alleviate vibration 
and impact during obstacle traversal, reducing the impact on 
logging instruments. It enables the robot to passively adjust 
during the obstacle traversal process. 

3. Mechanical Modeling of Support 
Mechanism 

3.1. Mechanical model of friction block 
support mechanism 

According to the working principle of the support structure, 
the model is simplified to reduce its complexity, and a 
mechanics analysis is established as shown in Fig. 3. A 
coordinate system, xOy, is established with the fixed hinge of 
the left push rod as the origin, and the principle of virtual work 
is applied for the mechanics analysis. 

 

 
Figure 3. Mechanical analysis of friction block support 

mechanism 
 
Where  𝐿   is the length of the left push rod,  𝐿   is the 

length of the support rod,  𝐿  is the length of the right push 
rod,  𝐿   is the length of the parallel rod, 𝛼  is the angle 
between the left push rod and the horizontal direction, 𝛽 is 
the angle between the right push rod and the horizontal 
direction,𝛿  is the thickness of the locking block, D is the 
inner diameter of the casing, d is the diameter of the support 
rod, h is the offset between the right push rod and the 
horizontal direction at the hinge connecting the slider,  𝑘  is 
the spring stiffness of the pressure spring,  𝑘  is the spring 
stiffness of the reset spring, N is the contact normal pressure 
between the support structure and the casing wall,  𝐹  is the 
load force of the hydraulic cylinder,  𝐹   is the horizontal 
pushing force of the right push rod, and f is the traction force 
of the support structure. 

The geometric relationships established according to 
Figure 3 are as follows: (followed by a description of the 
geometric relationships) 

 
𝑆 𝐿 cos𝛼 𝐿 𝐿 𝑐𝑜𝑠 𝛽
𝑆 𝐿 𝑠𝑖𝑛 𝛼 𝑑 2⁄ 𝛿
𝐿 𝑠𝑖𝑛 𝛼 ℎ 𝐿 𝑠𝑖𝑛 𝛽

            (1) 

 
In the formula (1), 
 

𝑠𝑖𝑛𝛼

𝑠𝑖𝑛𝛽
                 (2) 

 
Total differentiation of the equation (1) is obtained: 
 

𝑑𝑆 𝐿 𝑠𝑖𝑛 𝛼 𝑑𝛼 𝐿 𝑠𝑖𝑛 𝛽 𝑑𝛽
𝑑𝑆 𝐿 cos 𝛼 𝑑𝛼
𝐿 𝑐𝑜𝑠 𝛼 𝑑𝛼 𝐿 𝑐𝑜𝑠 𝛽 𝑑𝛽

        (3) 

 
According to the virtual work principle: 
 

𝑁𝑑𝑆 𝐹 𝑑𝑆 0                 (4) 
 
Simultaneous (1), (3), (4) can be obtained: 
 

𝑁 𝐹 𝑡𝑎𝑛𝛼 𝑡𝑎𝑛𝛽                (5) 
 
In the formula, 
 

⎩
⎪
⎨

⎪
⎧𝑡𝑎𝑛𝛼

𝑡𝑎𝑛𝛽
             (6) 

 
The thrust of the right push rod of the support mechanism 

is: 



 

123 

𝐹
            (7) 

 
Therefore, the tractive force is: 
 

𝑓 𝜇𝐹 𝑡𝑎𝑛𝛼 𝑡𝑎𝑛𝛽                (8) 
 
According to equation (8), it can be concluded that, under 

the condition of determining the parameters of the linkage, 
the main factor affecting the traction performance of the 
underground robot is the coefficient of friction, μ. 

Neglecting the frictional force of the slider, the load force 
of the support cylinder is given by: 

 
𝐹 𝑘 ⋅ ∆𝑥 𝐹 𝐹 𝑘 ⋅ 𝛥𝑥 𝐹       (9) 

 
Using formula (8), we can obtain: 
 

𝐹                  (10) 

 
Therefore, the hydraulic cylinder load is: 
 

𝐹 𝑘 ⋅ 𝛥𝑥           (11) 

 
From the above equation, it can be seen that the locking of 

the support structure is related to the length of the support 
structure push rod, the thickness of the locking block, and the 
diameter of the pipeline. Increasing the spring force can 
increase the load force of the hydraulic cylinder. Under the 
same traction conditions and with a constant spring force, the 
larger the pipe diameter, the smaller the required load force of 
the hydraulic cylinder. If it is necessary to maintain a constant 
load force, the hydraulic cylinder needs to adjust the 
compression amount of the spring to increase the spring force. 

4. Optimal Design of Support 
Mechanism 

According to the API standard for oil casing, the material 
selected for the study is Q235 5"~7" casing. This means that 
it must satisfy the mechanical requirements of the minimum 
casing inner diameter for the structure, as well as the size 
requirements of the maximum casing inner diameter for the 
structure. Therefore, the support structure designed in this 
study has an adaptive adjustment function for the casing inner 
diameter. 

For the 5"~7" casing inner diameter, the maximum traction 
force f_max for the telescopic downhole machine is 5000N. 
The friction angle φ between the locking block and the casing 
wall is 35°, and the friction coefficient is μ=0.7. Taking the 
example of a 110mm pipeline inner diameter, the maximum 
positive pressure of the support structure is approximately 
7143N. Therefore, the positive pressure N generated between 
a single support foot and the pipeline should be greater than 
3571.5N (assuming two support feet are symmetrically 
distributed). 

4.1. Support Structure Optimization Design 
To use a traction robot to perform horizontal well testing, it 

is necessary to design a structure with outer diameter 
dimensions and traction force that meet the requirements of 

horizontal well testing. In order to meet the special pipeline 
environment requirements for the robot in horizontal wells, 
the maximum traction force is achieved with the shortest axial 
length and the smallest outer diameter size. The structure of 
the traction force adjustment mechanism is complex, and in 
order to obtain the best combination of performance and 
structural parameters, the key parameters need to be 
optimized. 

Optimization design variables for the multi-variable 
function with nonlinear constraints: [Not provided in the 
original text] 

 
𝑥 𝑥 , 𝑥 , 𝑥 , 𝑥 𝐿 , 𝐿 , 𝐿 , ℎ          (12) 

 
Objective function: 
 

𝑓 𝑥 𝑚𝑖𝑛 𝑓           (13) 

 
Constraints: 
 

 
2

2 2 2 2 2
2 3 1

2

2 2 2 2 2
2 3 1

2

2

2 2 2 2 2
2 3 1

2

1

2 2
2 3

1 2 3

2

2
2

3

( ) ( )
2 2

( ) ( ) 0
2 2

( ) ( ) 0
2 2

(
2

0
D d D d

h L L L h

D d D d
h L L L h

D d D d
h L L L h

D d
h

L L L

L

L

L L L

 

 

 

  
         
 

  
          
 

  
  


 
    
  

 



 
  
 
 
  



      
 


  

2

2 2 2
1

1 3 2 4

) ( ) 0
2

10
2

D d
L h

D
h L L L L

 














  

          

      


 



 
  

(14) 

 
In this paper, the Angle 𝛼  between the support arm and 

the initial position of the horizontal axis is set to 15°. 
"Through MATLAB programming, fmincon() function 
toolbox is used to optimize the solution, and the results are as 
follows: 

 
Table 1. Optimized parameters of support mechanism 

Serial number Main parameter Value 
1 L1/mm 54 
2 L2/mm 85 
3 L3/mm 80 
4 h/mm 10 
5 D/mm 110-150 
6 d/mm 10 
7 𝛿/mm 6 
8 k1/kN/m 10 
9 k2/kN/m 5 

4.2. Mechanical analysis of the drive 
mechanism after optimization 

After the parameter values of the drive mechanism are 
determined, k values of the optimized drive unit under 
different pipe diameters are calculated, and cftool toolbox in 
MATLAB is used to fit them. The fitting results are shown in 
Fig.4. 

 



 

124 

 
Figure 4. Polynomial curve fitting results 

 
The fitting degree is 99.67%, and the function expression 

of 𝐹 /𝑁 and pipe diameter is  
 

0.0001623𝐷 0.05599𝐷 5.048        (15) 
   
The relationship between the friction block spring force 

and the positive pressure can be expressed as 
 
𝐹 0.0001623𝐷 0.05599𝐷 5.048 ∙ 𝑁     (16) 

4.3. Optimization Feasibility Verification 
In order to achieve a sufficiently large traction force, it is 

feasible to increase the normal pressure between the friction 
block and the pipe wall, thereby generating sufficient friction 
force between the support structure and the inner wall of the 
oil pipe. By combining multiple sets of driving short sections 
in a certain way, the load can be distributed to multiple 
support structures, achieving the goal of making the robot 
structure more compact. For downhole robots, during formal 
downhole operations, the required instrument string weight 
can be achieved by increasing the number of active short 
sections. 

5. Finite Element Analysis of Support 
Mechanism Friction Blocks 

5.1. Ploughing Effect 
The traction force of underground robots mainly depends 

on the coefficient of friction between the friction blocks and 
the casing. There are many factors that affect the friction 
coefficient between the friction blocks and the casing. In 
order to increase the contact friction and improve the 
obstacle-surmounting ability, the friction blocks are often 
designed in a gear shape. Under a large positive pressure, the 
gear tips produce slight plastic deformation on the casing, 
resulting in very small grooves on the pipe arm. Therefore, 
traditional methods are not suitable for studying friction. The 
ploughing effect refers to the process in which rough peaks of 
hard metal are embedded into soft metal, causing the soft 
metal to flow and produce a groove. The research process in 
this paper uses a hexahedral tooth shape, which has better 
durability and stronger shear resistance compared to pointed 
teeth, resulting in less damage to the casing. 

 

 
Figure 5. Schematic diagram of friction performance 

analysis of hexahedral drive wheel 
 
The horizontal plane projection of the trapezoidal tooth 

contact surface of the hexahedron is a trapezoid, with the 
length of the lower side of the trapezoid being 𝐿  , the 
indentation depth being h, the inclination angle of the friction 
block tooth being α, the top angle of the friction block tooth 
being β, and the friction force of the ploughing effect being 
𝐹 , then the horizontal plane projection area is: [The rest of 
the text seems to be a mathematical equation, which I cannot 
translate accurately without the specific symbols and context.] 

 
𝑆 ℎ 𝑡𝑎𝑛 𝛽 90∘ 𝑙 ℎ           (17) 

 
The contact area is projected as: 
 

𝑆 𝑙 2ℎ 𝑡𝑎𝑛 𝛽 90∘ ℎ 𝑡𝑎𝑛     (18) 

 
Suppose the casing is an isotropic material and the positive 

pressure is: 
 

𝑁 𝑆 𝜏                     (19) 
 
In the formula: 𝜏 —Embedded shear force, Pa, the value is 

approximately equal to the yield limit of the casing material. 
The friction required for the furrow effect is: 
 

𝐹 𝑆 𝜎                       (20) 
 
In the formula: 𝐹 —Furrow effect friction, N; 
𝜎 —Yield strength of casing material, Pa. 
Therefore, the friction coefficient of the furrow effect is: 
 

𝜇                        (21) 

 
Coupling (17) ~ (21) can obtain friction factors: 
 

𝜇
∘

∘
           (22) 

 
The equation shows that under the ploughing effect, the 

friction coefficient between the friction block and the casing 
is related to the top angle and inclination angle of the friction 
block tooth, the length of the lower side, and the embedding 
depth. 



 

125 

5.2. Finite Element Analysis 
(1) Simulation Model Establishment and Import 
Based on the previous section and referring to the 

simulation conclusions of other scholars, this paper 
establishes a simulation analysis of a friction block with 
𝑙  =4mm, 𝛼 =60° , 𝛽 =120° , 𝑠 =0.02mm. The friction block 
and pipeline models are created in Solidworks and saved in 

Parasolid (*.x-t) format, then imported into the Workbench 
finite element software. In Figure 6, (a) shows the overall 
model of the friction block, and (b) shows the finite element 
simulation model. The finite element simulation model 
simplifies the friction block to a single tooth, allowing for 
improved simulation accuracy by increasing mesh density 
while reducing computational simulation time to enhance 
efficiency. 

 

     
(a) Friction block model           (b) Finite element simulation model 

Figure 6. Friction block model and simulation modelDefine material propertiesDefine material properties 
 
The material properties are set in the static analysis module. 

The friction block material is 40CrMoTi and the casing 
material is J55 (Q235). The corresponding material 
performance parameters [47] are shown in Tab.2: 

 
Table 2. Mechanical properties parameters of driving wheel and casing materials 

Name Materials 
Density 

/(kg/m3) 
Modulus of 

elasticity /GPa 
Poisson's 
ratio 

Yield strength 
/MPa 

Friction block 40GrMoTi 7850 210 0.3 835 
Cannula J55 (Q235) 7800 200 0.28 235 

 
(2) Mesh Division and Boundary Condition Settings 
In Workbench, a hexahedral mesh is used with increased 

mesh density on the contact surface of the tooth, the four 
lateral surfaces, and the pipe surface to improve accuracy 
while reducing computation time. The mesh size is 0.5mm, 
with a refinement level of 3. The friction block tooth is set as 
the contact surface, the pipe surface as the target surface, and 
a fixed constraint is applied to the lower surface of the casing. 
The friction block is set to move in the vertical direction and 
is fixed in other directions. A vertical downward pressure is 

applied to the friction block at values of 250N, 500N, 750N, 
1000N, 1250N, 1500N, 1750N, 2000N, 2250N, 2500N, 
2750N, and 3000N, respectively, to calculate the deformation 
of the pipe under the corresponding pressures. 

(3) Simulation Results and Friction Characteristics 
Analysis 

According to the simulation results, the cloud image of 
pipeline deformation under corresponding positive pressure is 
shown in Figure 7. 

 

 
(a)250N                          (b)500N 



 

126 

 
(c)750N                           (d)1000N 

 
(e)1250N                          (f)1500N 

 
(g)1750N                          (h)2000N 

 
(i)2250N                          (j)2500N 

 
(k)2750N                            (l)3000N 

Figure 7. Cloud image of pipeline deformation under positive pressure 
 

According to the pipe deformation cloud map and equation (22) in Figure 6, the furrow friction coefficient of the friction 



 

127 

block under corresponding positive pressure can be obtained, and the specific value is shown in Tab. 3. 
 

Table 3. Pipeline deformation and corresponding furrow friction coefficient under positive pressure of friction block 

Positive pressure 
/N 

Furrow depth 
/mm 

Friction 
coefficient of 

furrow 

Positive pressure 
/N 

Furrow depth 
/mm 

Friction 
coefficient of 

furrow 
250 0.00067373 0.2332 1750 0.0044571 0.8780 
500 0.0012735 0.3936 2000 0.0050938 0.9335 
750 0.0019102 0.5300 2250 0.0057306 0.9858 
1000 0.0025469 0.6412 2500 0.0063673 1.0300 
1250 0.0031836 0.7335 2750 0.0070040 1.0693 
1500 0.0035023 0.7741 3000 0.0076407 1.1043 

 
According to the results in Table 3, it can be concluded that 

when the friction block is subjected to pressure in the pipeline, 
the greater the pressure value, the deeper the furrow depth and 
the greater the friction coefficient of the furrow. The 
relationship between the positive pressure and the furrow 
depth-furrow friction coefficient is shown in Figure 8. 

 

 
Figure 8. Furrow depth and furrow friction coefficient 

generated by friction block under positive pressure 
 
Based on Figure 8, it can be concluded that the depth of the 

furrow produced by the friction block under positive pressure 
is proportional to the magnitude of the positive pressure. The 
greater the positive pressure, the deeper the furrow. The 
friction coefficient of the furrow increases continuously with 
the increase of the positive pressure. However, as the pressure 
increases, the magnitude of the change in the friction 
coefficient of the furrow gradually decreases. 

6. Conclusion 
A simplified mechanical model of the support mechanism 

for the telescopic downhole robot was established. Through 
optimized design, the best dimensions for the support 
mechanism within the 5" to 7" casing range were obtained, 
requiring minimal spring force and achieving maximum 
driving efficiency. The values for the linkages𝐿  and 𝐿 were 
determined to maximize transmission efficiency based on the 
minimum ratio of spring force to positive pressure. By 
providing the functional relationship between spring force 
and positive pressure, the mechanical analysis and calculation 

of the driving structure were simplified. Finally, the 
relationship between the depth of the furrow produced by the 
friction block under positive pressure and the magnitude of 
the positive pressure was verified. It was found that the 
greater the positive pressure, the deeper the furrow, and the 
friction coefficient of the furrow increases continuously with 
the increase of the positive pressure, but the magnitude of the 
change in the friction coefficient gradually decreases as the 
pressure increases. 

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500 1000 1500 2000 2500 3000
0

1

2

3

4

5

6

7

8
 犁沟深度
 犁沟摩擦系数

正压力/N

犁
沟
深
度

/m
m

×10-3

0.2

0.4

0.6

0.8

1.0

1.2

 犁
沟

摩
擦
系

数