# Dynamic Modeling of an Out-Pipe Inspection Robot and Experimental Validation of the Proposed Model using Image Processing Technique

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## Abstract

Gas and liquid pipelines surround us. To ensure reliable product delivery and to maintain pipeline integrity, asset managers should consider routine pipeline inspection and holistic management programs to extend pipeline life and prevent risk. Therefore, pipe inspection robots are of special interest to industries. In this paper, we present a new and simple locomotion strategy for an out-pipe inspection robot which can provide adjustable tractive force and can also be utilized to support active diameter adaptability. The advantages proposed by this design include simplicity, low manufacturing costs, online inspection capability and short operational time. Here a dynamic model of the robot is presented with the required assumptions. The mathematical model of 2-DOF robot is obtained using the well-known Lagrange equation. Modeling and simulations were conducted to test the validity and practicality of the proposed design and strategies. The prototype has successfully traveled along a pipe of 20 cm diameter. The results obtained from our dynamic model are then validated by experimental data.

## Keywords

Inspection robots Image processing Out-pipe robots Dynamic modeling of out-pipe robots## List of symbols

- \( q_{1} \)
Center height of driving wheels

- \( q_{2} \)
Angle of rotation of rigid guides with respect to the horizon

*R*Radius of pipe

- \( d_{\text{wheel}} \)
Distance between closer sides of driving wheels

- \( \alpha_{\text{w}} \)
Included angle between the vertical axis and the line from the supporting point of driving wheels to the pipe center

- \( \alpha_{\text{wi}} \)
Included angle between the vertical axis and the line from the supporting point of idler wheels to the pipe center

- \( d_{\text{wheelidler}} \)
Distance between centers of idler wheels

- \( d_{p1} \)
Distance between center of mass of sliding plate and four rigid guides and axis of driving wheels

- \( d_{p2} \)
Distance between axis of driving wheels and mass center of sliding plate

- \( l_{\text{hor}} \)
Projected distance between axis of driving wheels and idler wheels on horizontal axis

- \( l_{{{\text{base}} \times {\text{w}}}} \)
Normal distance between center of the idler wheels and base plate

*P*Potential energy

*Q*Vector of generalized forces

*q*Vector of generalized coordinates

- \( \dot{q} \)
Vector of generalized velocities

- \( \ddot{q} \)
Vector of generalized accelerations

- \( M(q) \)
Inertia matrix

- \( C(q,\dot{q}) \)
Coriolis and centrifugal force vector

- \( g(t) \)
Gravitational force vector

- \( \tau (t) \)
Applied torque/force vector

- \( l_{0} \)
Initial length of springs

- \( d_{\text{spring}} \)
Deflection of the springs

- \( y_{c1} \)
Center of mass height of the frame

- \( y_{c2} \)
Center of mass height of base plate with respect to reference frame

- \( v_{c1} \)
Center of mass velocity of the frame

- \( v_{c2} \)
Center of mass velocity of the sliding plate

*K*Kinetic energy

- \( l_{\text{tot}} \)
Total length of guides

- \( tp1 \)
Thickness of sliding plate

- \( r_{\text{iw}} \)
Radius of idler wheels

- \( r_{\text{w}} \)
Radius of driving wheels

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