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Towards long-distance inspection for in-pipe robots in water distribution systems with smart motion facilitated by particle filter and multi-phase motion controller

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Abstract

Incidents in water distribution systems cause water loss and water contamination that requires the utility managers to assess the condition of pipelines frequently. However, pipelines are long and access to all parts of it is a challenging task; current in-pipe robots have the limitation of short-distance inspection and inability to operate in-service networks. In this work, we improve the design of our previously developed in-pipe robot as reported by Kazeminasab et al. (IEEE 5th international conference on robotics and automation engineering (ICRAE), 2020) and a multi-phase motion controller is proposed that ensures reliable motion for the robot during operation. The controller provides stabilized configuration with zero velocity at junctions (i.e. phase 1), stabilized configuration with velocity tracking at straight paths (i.e. phase 2), and facilitates desired amount of rotation around two axes for the robot (i.e. phase 3). We propose a localization technique with a particle filter that receives information about the surrounding environment with a rangefinder sensor. The map of the operation that is needed for the particle filter is an array that comprises non-straight paths in the operation that the robot passes through. We also develop a method that facilitates smart navigation for the robot in different configurations of pipelines. In this method, the localizer (i.e. particle filter) is synchronized with the multi-phase motion controller, and the robot switches between different phases of the controller based on its location in the network. We validated the functionality of the controller in the three phases with simulations and also the localization and navigation methods with experimental results. The results show that the robot has smart navigation with the synchronized motion controller and the localizer inside pipelines.

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Acknowledgements

This research is supported by Texas A&M Engineering Experiment Station (TEES) internal financial source by Dr. M Katherine Banks.

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  1. Texas A&M University, College Station, USA

    Saber Kazeminasab & M. Katherine Banks

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  1. Saber Kazeminasab
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Correspondence to Saber Kazeminasab.

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Appendices

Appendix A

1.1 Robot modeling

Figure 

Fig. 18
figure 18

Free body diagram of the robot in a pipe with an inclination angle, \(\alpha\)

Full size image

18 shows the free body diagram of the robot inside the pipeline. \(F_{i}\), \(i = 1,2,3\) are traction forces that the motors generate. We have

$$ F_{i} = \frac{{\tau_{i} }}{R}\quad i = 1,2,3 $$
(3)

In Eq. (3), \(\tau_{i}\), \(R\), and \(i\) are motor torque, wheel radius, and the wheels numbers, respectively. The motor torque (\(\tau\)) is linearly computed with current passing through it. We can express the governing equation of the gear motors with the following set of equations:

$$ \begin{array}{*{20}c} {\frac{{v_{{{\text{co}}}} }}{{L_{m} }} - \frac{{v_{e} }}{{L_{m} }} - \frac{{R_{m} }}{{L_{m} }}i_{m} = \frac{{{\text{d}}i_{m} }}{{{\text{d}}t}}} \\ {v_{e} = K_{v} \dot{\vartheta }} \\ {\tau_{m} = K_{v} i_{m} } \\ {\frac{{n^{2} }}{{I_{l} + n^{2} I_{R} }}\tau_{m} = \ddot{\vartheta }} \\ \end{array} $$
(4)

The parameters in Eq. (4) are described in Table

Table 4 Parameters in gear-motors dynamical equations
Full size table

4.

$$ \sum F_{x} = {\text{ma}} \to F_{1} + F_{1} + F_{1} - {\text{mg}}\sin \left( \alpha \right) - F_{d} = {\text{ma}} = m\ddot{x} $$
(5)
$$ \sum M_{Y} = I_{{{\text{yy}}}} \ddot{\phi } \to \frac{\sqrt 3 }{2}F_{3} L\cos \left( {\theta_{3} } \right) - \frac{\sqrt 3 }{2}F_{2} L\cos \left( {\theta_{2} } \right) = I_{{{\text{yy}}}} \ddot{\phi } $$
(6)
$$ \sum M_{z} = I_{{{\text{zz}}}} \ddot{\psi } \to \frac{1}{2}F_{3} L\cos \left( {\theta_{3} } \right) + \frac{1}{2}F_{2} L\cos \left( {\theta_{2} } \right) - F_{1} L\cos \left( {\theta_{1} } \right) - {\text{mg}}\cos \left( \alpha \right)L\sin \left( {\theta_{1} } \right) = I_{{{\text{zz}}}} \ddot{\psi } $$
(7)

The parameters in Eqs.)5–7( are listed in Table

Table 5 Parameters in robot dynamical equations
Full size table

5. The central processor is desired to remain at the center of the pipe, hence, the arm angles (\(\theta_{i} , i = 1,2,3\)) are equal to each other and we have

$$ \cos \left( {{\uptheta }_{{\text{i}}} } \right) = \frac{{\text{D}}}{{2{\text{L}}}} $$
(8)

where \(D\) is the pipe diameter and is variable.

Appendix B

2.1 Linear quadratic regulator (LQR) controller

To design the LQR controller, considering the stabilizing states, \(x_{s} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} \phi & {\dot{\phi }} \\ \end{array} } & {\begin{array}{*{20}c} \psi & {\dot{\psi }} \\ \end{array} } \\ \end{array} } \right]^{T}\), Eqs. (6) and (7) are linearized around the equilibrium point, \(x_{s}^{e} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } \\ \end{array} } \right]^{T}\) with Taylor expansion and neglect higher-order terms. The resulting equations and construct the system’s auxiliary matrices. We have:

$$ x_{s} = \left[ {\begin{array}{*{20}r} \hfill {x_{1} = \phi } \\ \hfill {x_{2} = \dot{\phi }} \\ \hfill {x_{3} = \psi } \\ \hfill {x_{4} = \dot{\psi }} \\ \end{array} } \right] $$
(9)
$$ u = \left[ {\begin{array}{*{20}c} {\tau_{1} = u_{1} } \\ {\tau_{2} = u_{2} } \\ {\tau_{3} = u_{3} } \\ \end{array} } \right]{ } $$
(10)

Hence, we can write

$$ \dot{x}_{{\varvec{s}}} = \left[ {\begin{array}{*{20}c} {x_{2} } \\ {\zeta_{1} } \\ {x_{4} } \\ {\zeta_{2} } \\ \end{array} } \right] = {\varvec{F}} = \left[ {\begin{array}{*{20}c} {F_{1} } \\ {F_{2} } \\ {F_{3} } \\ {F_{4} } \\ \end{array} } \right] $$
(11)

where \(\zeta_{1} = \frac{1}{{{\text{RI}}_{{{\text{yy}}}} }}\left[ {\frac{\sqrt 3 }{2}\tau_{3} L\cos \left( {\theta_{3} } \right) - \frac{\sqrt 3 }{2}\tau_{2} L\cos \left( {\theta_{2} } \right)} \right]\) and \(\zeta_{2} = \frac{1}{{I_{{{\text{zz}}}} }}\Big[ \frac{1}{2R}\tau_{3} L\cos \left( {\theta_{3} } \right) + \frac{1}{2R}\tau_{2} L\cos \left( {\theta_{2} } \right) - \frac{1}{R}\tau_{1} L\cos \left( {\theta_{1} } \right) - {\text{mg}}\sin \alpha \sin \theta_{1} \Big]\).

$$ A = \left[ {\begin{array}{*{20}c} {\frac{{\partial F_{1} }}{{\partial x_{1} }}} & {\frac{{\partial F_{1} }}{{\partial x_{2} }}} & {\frac{{\partial F_{1} }}{{\partial x_{3} }}} & {\frac{{\partial F_{1} }}{{\partial x_{4} }}} \\ {\frac{{\partial F_{2} }}{{\partial x_{1} }}} & {\frac{{\partial F_{2} }}{{\partial x_{2} }}} & {\frac{{\partial F_{2} }}{{\partial x_{3} }}} & {\frac{{\partial F_{2} }}{{\partial x_{4} }}} \\ {\frac{{\partial F_{3} }}{{\partial x_{1} }}} & {\frac{{\partial F_{3} }}{{\partial x_{2} }}} & {\frac{{\partial F_{3} }}{{\partial x_{3} }}} & {\frac{{\partial F_{3} }}{{\partial x_{4} }}} \\ {\frac{{\partial F_{4} }}{{\partial x_{1} }}} & {\frac{{\partial F_{4} }}{{\partial x_{2} }}} & {\frac{{\partial F_{4} }}{{\partial x_{3} }}} & {\frac{{\partial F_{4} }}{{\partial x_{4} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]{ } $$
(12)
$$ B = \left[ {\begin{array}{*{20}c} {\frac{{\partial F_{1} }}{{\partial u_{1} }}} & {\frac{{\partial F_{1} }}{{\partial u_{2} }}} & {\frac{{\partial F_{1} }}{{\partial u_{3} }}} \\ {\frac{{\partial F_{2} }}{{\partial u_{1} }}} & {\frac{{\partial F_{2} }}{{\partial u_{2} }}} & {\frac{{\partial F_{2} }}{{\partial u_{3} }}} \\ {\frac{{\partial F_{3} }}{{\partial u_{1} }}} & {\frac{{\partial F_{3} }}{{\partial u_{2} }}} & {\frac{{\partial F_{3} }}{{\partial u_{3} }}} \\ {\frac{{\partial F_{4} }}{{\partial u_{1} }}} & {\frac{{\partial F_{4} }}{{\partial u_{2} }}} & {\frac{{\partial F_{4} }}{{\partial u_{3} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & { - \frac{\sqrt 3 D}{{4{\text{RI}}_{{{\text{yy}}}} }}} & {\frac{\sqrt 3 D}{{4{\text{RI}}_{{{\text{yy}}}} }}} \\ 0 & 0 & 0 \\ { - \frac{{\text{D}}}{{2{\text{RI}}_{{{\text{zz}}}} }}} & {\frac{D}{{4{\text{RI}}_{{{\text{zz}}}} }}} & {\frac{{\text{D}}}{{4{\text{RI}}_{{{\text{zz}}}} }}} \\ \end{array} } \right]{ } $$
(13)

Pipe diameter, \(D\), is generally variable and since in our experiments, the pipe diameter is \(\approx\) 18 cm, we have \(B = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & { - 123.72} & {123.72} \\ 0 & 0 & 0 \\ { - 193.55} & {96.77} & {96.77} \\ \end{array} } \right]\). Based on the motion sensors in the robot, output matrix in state-space representation,

\(C = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 1 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 1 & 0 \\ \end{array} } \\ \end{array} } \right].\) The state-space representation of the system in terms of stabilization is written as

$$ \begin{array}{*{20}c} {\dot{x}_{s} = {\text{Ax}}_{s} + {\text{Bu}}} \\ {y = {\text{Cx}}_{s} } \\ \end{array} $$
(14)

As for the LQR controller, a cost function, \(J\left( K \right)\), is defined that is written as

$$ J\left( K \right) = \frac{1}{2}\mathop \smallint \limits_{0}^{\infty } \left[ {x_{s}^{T} \left( t \right){\text{Qx}}_{s} \left( t \right) + u\left( t \right)^{T} {\text{Ru}}\left( t \right)} \right]{\text{d}}t $$
(15)

where \(Q\) is the nonnegative definite matrix that weights state vector,\(Q = \left[ {\begin{array}{*{20}c} {200} & 0 & 0 & 0 \\ 0 & {10} & 0 & 0 \\ 0 & 0 & {200} & 0 \\ 0 & 0 & 0 & {10} \\ \end{array} } \right]\), and \(R.\) The positive-definite matrix that weights the input vector and \(R = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\). To minimize the cost function, \(K\), gain matrix is computed with

$$ K = R^{ - 1} B^{T} P $$
(16)

\(P\) in Eq. (16) is computed with algebraic Ricatti equation:

$$ - {\text{PA}} - A^{T} P - Q + {\text{PBR}}^{ - 1} B^{T} P = 0 $$
(17)

We have \(K = \left[ {\begin{array}{*{20}c} { - 4.92} & { - 1.12} & { - 13.26} & { - 2.98} \\ { - 9.37} & { - 2.11} & {3.48} & {0.78} \\ { - 9.37} & { - 2.11} & {3.48} & {0.78} \\ \end{array} } \right]\) And the control input of the LQR controller is computed as

$$ u = - {\text{Kx}}_{s} $$
(18)

As for velocity controllers, we considered three PID controllers that each of them controls the velocity of one wheel. The parameters of the PID controllers are listed in Table

Table 6 PID Parameters for velocity tracking controller
Full size table

6.

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Kazeminasab, S., Banks, M.K. Towards long-distance inspection for in-pipe robots in water distribution systems with smart motion facilitated by particle filter and multi-phase motion controller. Intel Serv Robotics 15, 259–273 (2022). https://doi.org/10.1007/s11370-022-00410-0

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