#  Design and Analysis of Modular Pipe Climber-III with a Multi-Output
Differential Mechanism ††thanks: All the authors are with Robotics Research
Center, International Institute of Information Technology, Hyderabad-500032,
India. (e-mail<EMAIL_ADDRESS>
Vishnu Kumar1, Saharsh Agarwal1, Rama Vadapalli1, Nagamanikandan Govindan1, K
Madhava Krishna1

###### Abstract

This paper presents the design of an in-pipe climbing robot that operates
using a novel ‘Three-output open differential’(3-OOD) mechanism to traverse
complex networks of pipes. Conventional wheeled/tracked in-pipe climbing
robots are prone to slip and drag while traversing in pipe bends. The 3-OOD
mechanism helps in achieving the novel result of eliminating slip and drag in
the robot tracks during motion. The proposed differential realizes the
functional abilities of the traditional two-output differential, which is
achieved the first time for a differential with three outputs. The 3-OOD
mechanism mechanically modulates the track speeds of the robot based on the
forces exerted on each track inside the pipe network, by eliminating the need
for any active control. The simulation of the robot traversing in the pipe
network in different orientations and in pipe-bends without slip shows the
proposed design’s effectiveness.

###### Index Terms:

Three-Output Open Differential (3-OOD), Mechanism, Design, Pipe climber

##  I Introduction

Pipe networks are omnipresent, primarily used to transport liquids and gases
in industries and urban cities. Most often, the pipes are concealed to comply
with the safety guidelines and to avoid risks. This makes inspection and
maintenance of pipes very difficult. Buried pipes are highly prone to
clogging, corrosion, scale formation, and crack initiation, resulting in leaks
or damages that may lead to catastrophic incidents.

Various In-Pipe Inspection Robots (IPIRs) [1] were proposed in the past to
conduct regular preventive inspections to avoid accidents. Wall-pressed IPIRs
with single, multiple and hybrid locomotion systems [2], Pipe Inspection
Gauges (PIGs) [3], actively controlled IPIRs with articulated joints and
differential drive units [4] were also extensively studied. Furthermore, bio-
inspired robots with crawler, inchworm, walking mechanisms[5], and screw-
drive[6] mechanisms were also shown to be suitable for different requirements.
However, most of them use active controlling methods to steer and manoeuvre
inside the pipe. Dependence upon the robot’s orientation inside the pipe added
to the challenges, also leaving the robot vulnerable to slip if traction
control methods are not involved. The Theseus [7], PipeTron [8], and PIRATE
[9] robot series use separate segments for driving and driven modules that are
interconnected by different linkage types. Each segments align or change the
orientation for negotiating turns. Additionally, robust active steering makes
such robots reliant on sensor data and heavy computation.

Pipe climbing robots with three symmetrical modules are more stable and
provide better mobility. Our earlier proposed modular pipe climbers [10, 11]
have used three driving tracks arranged symmetrically to each other, similar
to MRINSPECT series of robots [12, 13, 14]. In such robots, to actively
control the three tracks, their velocities were pre-defined for the pipe-
bends. This posed a limitation for the robot to negotiate pipe-bends only at a
particular orientation corresponding to the pre-defined velocities[10, 11, 12,
13, 14]. In real applications, the robot’s orientation change if it
experiences slip in the tracks during motion. This limitation can be solved by
using a passively operated differential mechanism to control the robot.
MRINSPECT-VI [15, 16] uses a multi-axial differential gear mechanism to
control the speeds of the three modules. However, for the division of the
driving torque and speed to the three modules, the layout of the differential
shown in Fig. 1(a) is used. This strategy made the first output (Z) to rotate
faster than the other two outputs (X and Y), making output Z easily affected
by slip [15]. This is caused because the outputs of the differential does not
share equivalent kinetics with the input. Other previously proposed solutions
for Three-output differentials (3-OD’s) [17, 18] also followed a similar
differential layout, as shown in Fig. 1(a).

(a) (b)

Figure 1: (a) Layout of the previous Three-output differentials(3-OD’s) (b)
Layout for Three-Output Open Differential (3-OOD)

Our ‘Three-Output Open Differential’ eliminates the mentioned limitation by
realizing equivalent output to input kinetic relations [19]. In the devised
layout all the three outputs are equally affected by the input, shown in Fig.
1(b). This contributes for the robot to eliminate slip and drag in any
orientation of the robot during its motion. Additionally, the differential
mechanism in the pipe climber enhances the ease of use by reducing the
dependency on the active controls to manoeuvre through the pipe networks.

##  II Design of the Modular Pipe Climber-III

###  II-A Structure of the Robot

The CAD model of the proposed robot, Modular Pipe Climber-III, is shown in
Fig. 2(a). The differential mechanism (3-OOD) is positioned inside the robot’s
central chassis and it drives each track individually through their respective
driving sprockets. The output velocity of the 3-OOD mechanism is translated to
the driving sprockets through a bevel gear setup. The detailed view of the
3-OOD mechanism is shown in Fig.3 (a). An irregular nonagon central chassis of
the robot houses three modules (A, B, C) circumferentially 120∘ apart from
each other, illustrated in Fig. 2(b). Each module houses a track to encompass
the driving and driven sprockets, and a spring-loaded mechanism to attach with
the chassis. The linear springs supported with a guiding rod (Linkage) between
the modules and the chassis push the modules radially outwards when they are
compressed. The stoppers restrict the radial extension of the modules within
the permissible limits. When the robot is deployed inside the pipe, the
spring-loaded tracks undergo deflection and press against the pipe’s inner
walls, providing them the necessary traction to move or hold position. Each
module in the robot can also compress asymmetrically, as detailed in section
III.

Figure 2: (a) Robot structure (b) Top view of the robot

###  II-B Three-Output Open Differential (3-ODD)

The Three-Output Open Differential (3-OOD) is the principal constituent of the
proposed robot, Fig. 3(a). The 3-OOD mechanism comprises a single input $(U)$,
three two-output open differentials $(2-OD_{1-3})$, three two-input open
differentials $(2-ID_{1-3})$ and three outputs $(O_{1-3})$. The differential’s
input $(U)$ is located at the central axis of the robot chassis. The three
$2-OD_{1-3}$, shown in Fig. 3(c) are arranged symmetrically around the input
$(U)$, with an angle of $120{{}^{\circ}}$ between any two. The $2-ID_{1-3}$,
shown in Fig. 3(b) are fitted radially in-between $2-OD_{1-3}$. The single
output of $2-ID_{1-3}$ form the three outputs $(O_{1-3})$ of the 3-OOD.
$2-OD_{1-3}$ comprises of gear elements such as ring gears $(R_{1-3})$, bevel
gears $(B_{1-6})$ and side gears $(S_{1-6})$, while $2-ID_{1-3}$ include ring
gears $(R_{4-6})$, bevel gears $(B_{7-12})$ and side gears $(S_{7-12})$. The
side gears $(S_{1-6})$ of $2-OD_{1-3}$ is meshed with their adjacent side
gears $(S_{7-12})$ of $2-ID_{1-3}$, to transfer the torque and speed from
$2-OD_{1-3}$ to it’s adjacent $2-ID_{1-3}$. The input $(U)$ of the worm gear
provide motion to $2-OD_{1-3}$ simuntaneously. Each two-output differential
$(2-OD_{1-3})$ then transfer received motion to its adjacent two-input
differentials $(2-ID_{1-3})$, depending on the load conditions experienced by
its respective side gears $(S_{1-6})$. The motion received by the side gears
$(S_{7-12})$ of $2-ID_{1-3}$ further translates them to the three outputs
($O_{1-3})$. The six differentials ($2-OD_{1-3}$ and $2-ID_{1-3})$ work
together to translate motion from the input to the three outputs ($O_{1-3})$.

Figure 3: (a) 3-OOD mechanism (Isometric view) (b) Two-output differential
$(2-OD_{1-3})$ (c) Two-input differential $(2-ID_{1-3})$

When $O_{1-3}$ experience different loads, the side gears ($S_{7-12})$ in
$2-ID_{1-3}$ transfers different loads to the side gears $(S_{1-6})$ of
$2-OD_{1-3}$. Under this condition, $2-OD_{1-3}$ translates differential speed
to its adjacent $2-ID_{1-3}$. When $O_{1-3}$ experience equal load or no load,
all the side gears experience the same load making them rotate at the same
speed and torque.

As outlined in Fig. 1(b), all the outputs ($O_{1}$, $O_{2}$, $O_{3}$) share
equal kinetics with the input. In addition, the outputs also share identical
kinetics with each other. This results in the change of loads in one of the
outputs imposing an equal effect on the other two outputs that are
undisturbed. The outputs operate with equal speeds when there is no load or
equal load acting on all the outputs. The 3-OOD mechanism operates its outputs
with differential speed when the outputs are under varied loads. When one of
the outputs is operating at a different speed while the other two outputs are
experiencing the same load, then the two outputs with the same loads will
operate with equal speeds.

The 3-OOD mechanism designed for the Modular Pipe Climber III, advances its
the three tracks with equivalent speeds when moving inside a straight pipe.
But while manoeuvring inside the pipe-bends, the differential modulates the
track speed of the robot such that the track travelling the longer distance
rotates faster than the track travelling the shorter distance.

##  III Kinematics and Dynamics

###  III-A Kinematics and Dynamics of the 3-OOD mechanism

Figure 4: Three-Output Open Differential: (a) Power flow diagram (b) Kinematic
scheme (c) Bond Graph Model

Fig. 4(a) represents an illustration of the direction of speed and torque flow
in the differential. The kinematic scheme shown in Fig. 4(b) illustrates the
connectivity of the links and joints of the 3-OOD mechanism. The Three-Output
Open Differential’s kinematic and dynamic equations are formulated using the
bond graph model shown in Fig. 4(c).

The input angular velocity ($\omega_{u}$) from the motor is equally
distributed to the three ring gears of the two-output differentials as
$\omega_{r1}$, $\omega_{r2}$ and $\omega_{r3}$. They rotate at equal angular
velocities and with equal torque i.e. $1/k$ times $\omega_{u}$ and $k/3$ times
$\tau_{u}$, where $1/k$ is the gear ratio of the input to the ring gears
($1/k$ = $1/20$). Moreover, a two-output differential dictates that the
angular velocity of its ring gear is always the average of the angular
velocities of its two side gears. These two side gears can rotate at different
speeds while maintaining equal torque [20].

$\omega_{u}=\frac{k(\omega_{1}+\omega_{2})}{2}=\frac{k(\omega_{3}+\omega_{4})}{2}=\frac{k(\omega_{5}+\omega_{6})}{2},$
(1)

The angular velocities $\omega_{r1}$, $\omega_{r2}$ and $\omega_{r3}$ are then
translated to their side gears. Side gears $(S_{1}$, $S_{7}$), $(S_{3}$,
$S_{8}$) and $(S_{5}$, $S_{11}$) are connected such that they do not have any
relative motion between the gears of the same pair, i.e., angular velocities
of side gears of the same pair is always equal $(\omega_{1}$$=$$\omega_{7})$,
$(\omega_{3}$$=$$\omega_{8})$ and $(\omega_{5}$$=$$\omega_{11})$. Substituting
in (1)

$\begin{split}\omega_{7}=\omega_{1}=\frac{2(\omega_{i})}{k}-\omega_{2},\hskip
2.168pt\omega_{8}=\omega_{3}=\frac{2(\omega_{i})}{k}-\omega_{4},\hskip
2.168pt&\omega_{11}=\omega_{5}=\frac{2(\omega_{i})}{k}-\omega_{6}\end{split}$
(2)

Similarly, the output angular velocities $\omega_{O1}$, $\omega_{O2}$,
$\omega_{O3}$ are derived from the velocities $\omega_{r4}$, $\omega_{r5}$,
$\omega_{r6}$ of the respective ring gears $R_{4}$, $R_{5}$, and $R_{6}$. The
output to side gear relation is equated with (2), to attain the angular
velocity equation for the input to the outputs.

$\begin{split}&\omega_{O1}=\frac{2j(\omega_{u})}{k}-\frac{j(\omega_{2}+\omega_{4})}{2},\hskip
1.8063pt\omega_{O2}=\frac{2j(\omega_{u})}{k}-\frac{j(\omega_{3}+\omega_{5})}{2},\\\
&\hskip
50.58878pt\omega_{O3}=\frac{2j(\omega_{u})}{k}-\frac{j(\omega_{1}+\omega_{6})}{2}\end{split}$
(3)

where $\omega_{u}$ is the angular velocity of the input $(U)$, $j$ is the gear
ratio of the ring gears $(R_{4-6})$ to the outputs ($j$ = 2:1), while
$\omega_{O1}$, $\omega_{O2}$ and $\omega_{O3}$ are angular velocities of the
output. $\omega_{1}$, $\omega_{2}$, $\omega_{3}$, $\omega_{4}$, $\omega_{5}$
and $\omega_{6}$ are the respective angular velocities of the side gears
$(S_{1}$, $S_{2}$, $S_{3}$, $S_{4}$, $S_{5}$, $S_{6})$. Therefore, the output
angular velocities ($\omega_{O1}$, $\omega_{O2}$, $\omega_{O3}$) are dependent
on the input angular velocity ($\omega_{u}$) and the side gear outputs
($\omega_{2}$, $\omega_{4}$), ($\omega_{3}$, $\omega_{5}$) and ($\omega_{1}$,
$\omega_{6}$).

Meanwhile, the torques $\tau_{R4}$, $\tau_{R5}$ and $\tau_{R6}$ of the ring
gears $(R_{4}$, $R_{5}$ and $R_{6})$ is the sum of the torques of their
corresponding side gears $S_{7}$, $S_{8}$, $S_{9}$, $S_{10}$, $S_{11}$ and
$S_{12}$. Similar to angular velocity relation for input to outputs, by
equating the ring gears to side gears relation with the output to the ring
gear relation, a relationship between output torques ($\tau_{O1}$,
$\tau_{O2}$, $\tau_{O3}$) to the input torque ($\tau_{u}$) is obtained.

$\begin{split}&\tau_{O1}=\frac{k(\tau_{u})}{3j}-\frac{(I_{1}\dot{\omega}_{7}+I_{3}\dot{\omega}_{8})}{j},\
\hskip 14.45377pt\tau_{O2}=\frac{k(\tau_{u})}{3j}\hskip 1.8063pt-\\\
&\frac{(I_{4}\dot{\omega}_{9}+I_{6}\dot{\omega}_{10})}{j},\ \hskip
14.45377pt\tau_{O3}=\frac{k(\tau_{u})}{3j}-\frac{(I_{2}\dot{\omega}_{12}+I_{5}\dot{\omega}_{11})}{j}\end{split}$
(4)

where $I_{1}$,$I_{2}$,$I_{3}$,$I_{4}$,$I_{5}$,$I_{6}$ are the inertia
exhibited by the side gears, $\dot{\omega}_{7}$, $\dot{\omega}_{8}$,
$\dot{\omega}_{9}$, $\dot{\omega}_{10}$, $\dot{\omega}_{11}$,
$\dot{\omega}_{12}$ are the angular acceleration of the respective side gears
and $\tau_{u}$ is the input torque.

The 3-OOD mechanism has three degrees of freedom in its output. Equations (3)
and (4) validates that the behavior of each output is impacted by the input
($U$) as well as the other two outputs.

Equal speeds and torque: Identical side gears $(S_{7-12})$ exhibit equal
inertia $(I_{1}=I_{2}=I_{3}=I_{4}=I_{5}=I_{6})$. The side gears $(S_{7-12})$
operate with equal angular velocity $(\omega_{n}=\frac{\omega_{i}}{k}$, where
$n$ ranges from $1$ to $12)$ and angular acceleration when all three outputs
$(O_{1-3})$ experience equal loads. Substituting these relations in (3) and
(4) we get,

${\bf\omega_{O1}}={\bf\omega_{O2}}={\bf\omega_{O3}}=\frac{j(\omega_{u})}{k}$
(5)
${\bf\tau_{O1}}={\bf\tau_{O2}}={\bf\tau_{O3}}=\frac{k(\tau_{u})}{3j}-\frac{2(I_{1}\dot{\omega}_{1})}{j}$
(6)

Equations (5) and (6) presents the novelty of the differential to provide
equal motion characteristics in all three outputs that experiences equal loads
or when left unconstrained. But when the gear components experience a
resistance across a junction, the angular velocity and the torque changes
depending on the external resistive force.

###  III-B Track Velocity

TABLE I: Theoretical velocities of $v_{A}$,$v_{B}$ and $v_{C}$ in Elbow and
U-section for various robot orientations ($\mu$)

Robot Orientation |  $\alpha$ = $0^{\circ}$ |  $\alpha$ = $30^{\circ}$ |  $\alpha$ = $60^{\circ}$
---|---|---|---
Velocity ratio’s ($v_{A}$:$v_{B}$:$v_{C}$) |  0.670 : 1.165 : 1.165 |  0.715 : 1.285 : 1 |  0.835: 1.329: 0.835
Inner-module velocity (in mm/sec) |  $v_{tA}$ = 33.69 |  $v_{tA}$ = 35.90, $v_{tC}$ = 50.24 (Center module) |  $v_{tA}$ = $v_{tC}$ = 41.96
Outer-module velocity (in mm/sec) |  $v_{tB}$ = $v_{tC}$ = 58.51 |  $v_{tB}$ = 64.57 |  $v_{tB}$ = 66.78

Figure 5: Two-dimensional representation of robot’s orientation in pipe bends

The output velocity of the differential outputs are equal to the speed of the
driving module sprockets.Therefore, the output velocities ${\omega_{O1}}$,
${\omega_{O2}}$ and ${\omega_{O3}}$ of the differential are translated into
track speeds $v_{tA}$, $v_{tB}$ and $v_{tC}$. The input speed for the robot is
120 rpm, thereby translating 12 rpm to the outputs under equal loading
conditions. The sprocket diameter is constant ($D_{s}$= 80 mm) for all the
three tracks.

When the robot travels in straight pipes, the tracks experience equal loads in
all three tracks i.e., $\omega_{t1}$=$\omega_{t2}$=$\omega_{t3}$

$v_{tA}=v_{tB}=v_{tC}=\frac{\pi\times{D_{s}}\times
12}{60}=50.24\textrm{mm/sec}$ (7)

The average speed of the three tracks are always equal to the speed of the
robot, $v_{R}$.

$v_{R}=\frac{v_{tA}+v_{tB}+v_{tC}}{3}=50.24\textrm{mm/sec}$ (8)

The speed of the tracks required to negotiate a pipe-bend depends on the
orientation of the robot with respect to the bend direction. Ho Moon Kim et
al. [16], in their paper suggested a method for calculating the respective
velocities of the three tracks inside pipe bends. Assuming that the robot
enters the pipeline with the configuration shown in Fig. 5, $\Theta$ is the
angle of the pipe bend, R is the radius of curvature of the pipe bend and r is
the radius of the pipe. The velocity of the track A is derived from Fig. 5

$v_{tA}=v_{R}(\frac{R_{A}-r\cos({\mu})}{R_{A}})\vspace{-0.05in}$ (9)

In a similar way, the velocities $v_{tB}$ and $v_{tC}$ for their respective
tracks B and C are obtained. The robot is inserted at different orientation of
the modules with respect to OD. The modulated speeds for the tracks are
calculated for bend-pipes in orientations $\mu$ = $0^{\circ}$, $\mu$ =
$30^{\circ}$, $\mu$ = $60^{\circ}$, shown in table I.

###  III-C Asymmetrical Compression

The linear springs in the module provides robot the flexibility to negotiate
bends easily. The maximum compression possible in each module is $16mm$. There
are additional tolerances in the module holes, so that asymmetrical
compression is possible. This helps the robot to overcome obstacles and
irregularities in the pipe network that it may face in real world
applications, shown in Fig. 6(a). In Fig. 6(b), the front end of the module is
compressed completely whereas the rear end is in its maximum extended state
possible inside the pipe. The maximum asymmetrical compression allowed in a
single module of the robot is shown in Fig. 6(b). Thus, $\phi$ is the maximum
angle the module can compress asymmetrically.

In $\bigtriangleup$$XYZ$, $YZ$ is perpendicular to $XZ$ and $\angle$ $YXZ$ is
$\phi$, Thus, the maximum permissible asymmetrical bend is

$\phi=\tan^{-1}(\frac{YZ}{XZ})=\tan^{-1}(\frac{12}{150})=4.574^{\circ}$ (10)

Figure 6: (a) Robot overcoming obstacle (b) Asymmetrical module compression

##  IV Dynamic Simulations of the Pipe Climber

Simulations were conducted to analyze and validate the motion capabilities of
the robot in various pipe networks. The same will give us more insights into
the dynamics and behaviour of the developed Modular Pipe Climber-III in real
testing environments. Hence, multi-body dynamic simulations was performed in
MSC Adams by converting the design parameters into a simplified simulation
model. To reduce the number of moving components in the model and to decrease
the computational load, the tracks were simplified into roller wheels. Each
module houses three roller wheels in the simplified model. Therefore, the
contact patch provided by the tracks to the pipe walls are reduced from 10
contact regions to 3 contact regions per module. The simulation parameters
such as the track velocities ($v_{tA}$, $v_{tB}$, and $v_{tC}$) and the module
compression for each track A,B and C were analyzed. Simulations were performed
by inserting the robot in three different orientations of the module ($\mu$ =
0∘, $\mu$ = 30∘, $\mu$ = 60∘) in both the straight pipes and pipe-bends.

Figure 7: (a) Pipe network; Robot orientations: (b) $\mu$=0∘, (c) $\mu$=30∘,
(d) $\mu$=60∘

The robot is simulated inside a pipe-network designed according to ASME B16.9
standard NPS 11 and schedule 40. The simulations were conducted for four test
case scenarios in the pipe network consisting of Vertical section, Elbow
section (90∘ bend), Horizontal section and the U-section (180∘) for different
orientations ($\mu$ = 0∘, $\mu$ = 30∘, $\mu$ = 60∘) of the robot, refer Fig.
7. The total distance of the pipe structure is $D_{pipe}$ = 3,023.49mm. The
distance travelled by the robot ($D_{R}$) in pipe is calculated from center of
the robot body and the track’s individual distance travelled is calculated
from the center roller wheel mounted in each modules. Therefore, we get the
total robot’s path, by subtracting the robot’s length from $D_{pipe}$ (i.e.,
$D_{R}$ =$D_{pipe}$ \- $(L_{R})$ = 2,823.49mm, where $L_{R}$= 200mm).The
robot’s path in vertical climbing and the last horizontal section is measured
by subtracting $L_{R}/2$ from their respective section length. The input ($U$)
of the 3-OOD is given a constant angular velocity of 120rpm ($\omega_{u}$=
120rpm) and motion of the robot including the track velocities ($v_{tA}$,
$v_{tB}$, and $v_{tC}$) are studied in the simulation.

Figure 8: Simulation of the robot

###  IV-A Vertical section and Horizontal section

In the vertical section and horizontal section, the robot follows a straight
path. Therefore, the tracks experience equal loads on all three modules in
both the test cases. As a result, the differential provides equal velocities
for all the three tracks, equivalent to the robot’s average velocity $v_{R}$.
The observed track velocities in the simulation for the orientation $\mu$ =
0∘, is $v_{R}$= $v_{tA}$= $v_{tB}$= $v_{tC}$= 50.03 mm/sec. Similarly, for
$\mu$ = 30∘, the velocities are $v_{R}$= $v_{tA}$= $v_{tB}$= $v_{tC}$= 50.22
mm/sec and for $\mu$ = 60∘, the velocities are $v_{R}$= $v_{tA}$= $v_{tB}$=
$v_{tC}$= 51.36 mm/sec. Therefore, all the values correspond to the
theoretical results shown in (8), with an absolute percentage error (APE)
lesser than 2.2%. This error quantifies the actual amount of deviation from
the theoretical value [21].

$\small{APE=(\frac{Simulation\;velocity-
Theoretical\;velocity}{Theoretical\;velocity})\times 100\%}$ (11)

To negotiate an initial length of 550 mm ($D_{R}$ = 550 - $(L_{R}/2)$= 450 mm)
in vertical climbing, the robot takes 0 to 9 seconds. Figures 8(a) and 8(b)
represents the robot’s ability to climb the pipe vertically against gravity.
Starting from 24 seconds till 31 seconds, the robot moves a distance of 350 mm
in the first horizontal section, shown in Fig. 8(d). In the smaller horizontal
section of distance 150mm ($D_{R}$ = 150 - $(L_{R}/2)$= 50 mm), the robot
moves from 59 to 60 seconds, shown in Fig. 8(f).

Figure 9: Simulation results for (a) $\mu$ = 0∘, (b) $\mu$ = 30∘, (c) $\mu$ =
60∘

###  IV-B Elbow section and U-section

In the elbow section ($90^{\circ}$ bend) and U-section ($180^{\circ}$ bend),
the robot moves at a constant distance to the center of curvature of the pipe.
The 3-OOD mechanism modulates the output speeds of the tracks $v_{tA}$,
$v_{tB}$ and $v_{tC}$ according to the distance from the center of curvature
of the pipe. In all three orientations of the robot, the outer module tracks
rotates faster to travel a longer distance, while the inner module tracks
rotates slower to travel the shortest distance than the radius of curvature of
the pipe bend. In pipe bends, the simulation velocities of each track is
averaged seperately to approximate the observed track velocities without
fluctuations. The approximated velocities of each track is then compared with
their respective theoretical velocities to find the absolute percentage error
(APE), refer (11).

For the orientation $\mu$ = 0∘, the outer modules (B and C) move at an average
velocity of (58.7 mm/sec and 57.8 mm/sec), while the inner module (A) move at
an average velocity of 33.62mm/sec, refer Fig. 9(a). These values correspond
to the theoretical values $v_{tB}$ = $v_{tC}$ = 58.51 mm/sec and $v_{tA}$ =
33.69mm/sec with an APE of 1.2%. Similarly, the track velocities $v_{tA}$,
$v_{tB}$ and $v_{tC}$ for $\mu$ = 30∘ presented in Table I, matches with the
average value of the simulation results ($v_{tC}$= 50.3 mm/sec, $v_{tB}$= 63.8
mm/sec, $v_{tA}$= 37.3 mm/sec) with an APE of 3.8%, refer Fig. 9(b). Likewise,
the track velocity value for $\mu$ = 60∘, correspond to the simulation results
($v_{tB}$= 68.5 mm/sec, $v_{tA}$= 40.2 mm/sec and $v_{tC}$= 41.3 mm/sec) with
an APE of 2.5%, refer Fig. 9(c). In each orientations of both the straight and
bend sections, the error value is very minimal and they can be attributed due
to external factors in real-world conditions such as friction. Therefore,
minimal velocity fluctuations occurs in the simulation plot. From 9 to 24
seconds, the robot negotiates the elbow section (90∘ bend) of distance 657.83
mm, while it takes 31 to 59 seconds to travel 1315.66 mm in the U-section
(180∘ bend). Figures 8(c) and 8(e) shows the robot’s capability to traverse in
pipe-bends.

Eliminating slip and drag: The simulation results for the track velocities
$v_{tA}$, $v_{tB}$, $v_{tC}$ and the robot $v_{R}$ in different orientations
($\mu$ = 0∘, $\mu$ = 30∘, $\mu$ = 60∘), matches with the theoretical results
obtained in section III. It is observed from the simulation that in 60
seconds, the robot traverses throughout the pipe network uniformly at the
inserted orientation. The result correspond to our theoretical calculation
($D_{R}$/$v$= 3016.49/50.24 = 60.04 sec). This validates that the Three-Output
Open Differential eliminates slip and drag in the tracks of the pipe climber
in all orientations without any motion losses. In the simulation, the robot is
observed without any slip and drag in all orientations, which further impacts
in reduced stress influence on the robot and increased motion smoothness.

Figure 10: Linear spring deformation in the modules

Radial flexibility: The track in the modules clamp to the inner wall of the
pipe to provide traction during motion. The springs are initially pre-loaded
by a compression of 1.25 mm in all three modules equally when inserted in the
vertical pipe section, refer Fig. 10. In straight pipes, the robot moves at
the initial pre-loaded spring length. The deformation length increases by 1.5
mm for the inner and the outer modules when the robot is in motion near elbow
section and U-section. This deformation explains the radial flexibility
allowed for the modules to manoeuvre through the changing cross-section of the
pipe diameter in the bends during motion.

##  V Conclusion and Future works

The Modular Pipe Climber-III robot is presented with the novel Three-Output
Open Differential to control the robot mechanically without any active
controls. The differential has an equivalent output to input kinetics, whose
performance is completely analogous to the functionality of the traditional
two output differential. The simulation results validate successful traversal
of complex pipe networks with bends of up to 180∘ in different orientations
without slip. Adopting the differential mechanism in the robot achieves the
novel result of eliminating the slip and drag in all orientations of the robot
during the motion.

At the present, we are developing a prototype to perform experiments on the
proposed design [22]. In the future, we intend to extend the application of
the Three-Output Open Differential for other use cases.

## References

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