IDETC/CIE2021 the ASME 2021
International Design Engineering Technical Conferences and
Computers and Information in Engineering Conference August 17-20 2021 Virtual
Online DETC2021-68093

# Design And Analysis Of Three-Output Open Differential with 3-DOF

Rama Vadapalli  Nagamanikandan Govindan  K Madhava Krishna  International
Institute of Information Technology Hyderabad, Hyderabad, India

###### Abstract

This paper presents a novel passive three-output differential with three
degrees of freedom (3DOF), that translates motion and torque from a single
input to three outputs. The proposed Three-Output Open Differential is
designed such that its functioning is analogous to the functioning of a
traditional two-output open differential. That is, the differential translates
equal motion and torque to all its three outputs when the outputs are
unconstrained or are subjected to equivalent load conditions. The introduced
design is the first differential with three outputs to realise this outcome.
The differential action between the three outputs is realised passively by a
symmetric arrangement of three two-output open differentials and three two-
input open differentials. The resulting differential mechanism achieves the
novel result of equivalent input to output angular velocity and torque
relations for all its three outputs. Furthermore, Three-Output Open
Differential achieves the novel result for differentials with more than two
outputs where each of its outputs shares equivalent angular velocity and
torque relations with all the other outputs. The kinematics and dynamics of
the Three-Output Open Differential are derived using the bond graph method. In
addition, the merits of the differential mechanism along with its current and
potential applications are presented.

## INTRODUCTION

A differential is a mechanism that translates motion and torque from an input
to its outputs in accordance with the load conditions the outputs experience
[1]. The traditional two-output open differential $(2-OD)$ has a single input
which passively operates two outputs. It has two degrees-of-freedom (2DOF) and
when in operation, translates equal motion and torque from its single input to
its two outputs if they are subjected equal loads [2]. However, when the load
conditions of the outputs differ, the motion translated to each output is
different. The $2-OD$ plays a vital role in automobiles. During a turn, the
$2-OD$ enables the automobile to rotate the wheel that travels a longer
distance faster than the other wheel and potentially eliminates the slip and
drag of the wheels [3]. By doing so, the $2-OD$ considerably reduces the
stresses induced. Three-output differentials play a similar role in delivering
motion to three outputs.

Three-output differentials have existed for over a decade and have
predominantly been used in pipe climbing robots with three tracks or wheels
[4, 5, 6, 7]. One such differential, the multi-axle differential gear was
implemented in the in-pipe robot MRINSPECT VI (Multifunctional Robotic crawler
for In-pipe inspection VI) [5]. The multi-axle differential gear has three
sets of planetary gears arranged in a series where its three outputs are the
ring gears of each planetary gear. The series arrangement of the planetary
gears permit the ring gears to rotate at different speeds, however; it
encounters a limitation [7]. The input does not translate equal motion to the
outputs when the outputs are subjected to equal loads. This limitation is
transpired because all three outputs do not have equivalent dynamics with the
input. The schematic of the mechanism is as shown in Fig. 1$(a)$ where one of
the outputs $($Output $Z)$ has a different link to the input when compared to
the other two outputs $($Output $X$ and Output $Y)$. This causes the three
outputs of the differential to behave differently under equal loads [7].

Another solution for three-output differential was developed by S. Kota and S.
Bidare $(1997)$ using standard bevel gear setups and epicyclic gear trains
$($planetary gear$)$ [8]. A similar approach is proposed by Diego Ospina and
Alejandro Ramirez-Serrano $(2020)$ for the use of differentials with more than
2DOF in underactuated robotic arms [9]. The differentials proposed by S. Kota
and S. Bidare $(1997)$ and by Diego Ospina and Alejandro Ramirez-Serrano
$(2020)$ have schematics similar to those shown in Fig. 1$(a)$ and carry
similar limitations [8, 9].

Figure 1: CURRENT DIFFERENTIALS WITH (a) THREE OUTPUTS (b) FOUR OUTPUTS.

There exist four-output open differentials such as the 2-2D differential gear
mechanism that translates equal motion to all its outputs when the outputs are
under equivalent load conditions [6, 7]. The 2-2D differential has four sets
of planetary gears arranged in a series where its four outputs are the ring
gears of each planetary gear. Its schematic Fig. 1$(b)$ is such as connecting
the inputs of two $2-OD$ to the outputs of another $2-OD$ [7]. The 2-2D
differential gear mechanism has a limitation, that is the effect an output has
on the other three outputs is not equivalent even when the other outputs are
under the equal loads. This transpires because the dynamics an output shares
with the other three outputs is not equivalent. This limitation becomes more
apparent when one of outputs’ (Output A) motion is ceased and the other three
outputs (Output B, Output C and Output D) are unconstrained Fig. 1$(b)$. In
such cases the Output B will rotate faster than the Output C and Output D.

In addition to using differential mechanisms, differential speed can be
realised by electronically controlling the speeds of each wheel. Our
previously developed robots, the Modular Pipe Climber I and II predefined the
speeds for each track in accordance to the bends of the pipe [10, 11, 12].
Other pipe climbing robots such as MRINSPECT $IV$ and LS-$01$ have also used
similar methods in modulating speeds [13, 14]. The need to pre-define the
speeds of each track is a substantial limitation which restricts the robots’
success to a highly predictable environment.

Figure 2: THREE-OUTPUT OPEN DIFFERENTIAL $(3-OOD)$.

Contribution: The Three-Output Open Differential $(3-OOD)$ as shown in Fig. 2
is designed such that it realises all the functional abilities of a
traditional $2-OD$, but with three outputs, i.e., $(i)$ the three outputs
operate with equal speed when the outputs are unconstrained or under equal
loads, $(ii)$ all three outputs have equivalent input to output kinematics and
dynamics and $(iii)$ the kinematics and dynamics between any two of its
outputs is equivalent.

The design of the $3-OOD$ along with its components are presented in the next
section. The section (Kinematics and Dynamics), establishes the kinematics and
dynamics of the $3-OOD$ by means of the bond graph method. Finally, the
current and potential applications of the mechanism along with our future work
are presented.

## DESIGN OF THE 3-OOD

The design intent for the $3-OOD$ was to develop a functional architecture
that could achieve all the results achieved by the traditional $2-OD$, but
with three outputs.

### Design Synthesis

The functioning of the $2-OD$ and the principles it operates are studied for
the purpose of incorporating these principles into the design of the $3-OOD$.
$2-OD$ chiefly operates on two principles. (i) The effect the input has on its
outputs is variable with the load conditions the outputs experience [15]. (ii)
The input shares equivalent dynamics with all its outputs [15]. However, there
is a third principle that needs to be satisfied when the differential has more
number than two outputs, that is, (iii) the dynamics an output shares with all
the other outputs is equivalent. The currently existing open differentials
with three outputs as represented in Fig. 1$(a)$ only satisfy the first
principle. The currently existing open differentials with four outputs as
represented in Fig. 1$(b)$ satisfy the first principle and second principle,
but do not operate under the third principle.

Figure 3: INITIAL DESIGN (a) TOP VIEW (b) SCHEMATIC.

To integrate all the three principles determined into a design, the initial
synthesis was to have a single central input transfer motion to three $2-OD$’s
$(D_{1}$, $D_{2}$ and $D_{3})$ as shown in Fig. 3$(a)$. Each side gear
$(SG_{1}$, $SG_{2}$, $SG_{3}$, $SG_{4}$, $SG_{5}$ and $SG_{6})$ of the three
$2-OD$’s is connected with the adjacent side gear of its neighbouring $2-OD$
to combine and form an output $(X_{1}$, $X_{2}$ and $X_{3})$ of a three-output
differential Fig. 3$(a)$. Figure 3$(b)$ illustrates the schematic of the
initial design. On analysing the initial design, it was observed that though
the initial design could achieve differential speed for its three outputs, the
two adjacent side gears $(SG_{2}$, $SG_{3})$ or $(SG_{4}$, $SG_{5})$ or
$(SG_{1}$, $SG_{6})$ at the respective outputs $(X_{1}$, $X_{2}$ and $X_{3})$
are forced to move with equal speeds. As a consequence, at any given moment,
at least two of the three outputs will always have equal speeds regardless of
the load conditions the outputs experiences.

Figure 4: (a) $3-OOD$ (b) TWO-OUTPUT DIFFERENTIAL $(2-OD_{1-3}$) (C) TWO-INPUT
DIFFERENTIAL $(2-ID_{1-3})$.

To eliminate the limitation present in the initial design three two-input open
differentials $(2-ID_{1}$, $2-ID_{2}$ and $2-ID_{3})$ are used so that they
facilitate differential speed between the adjacent side gears at the outputs
as shown in Fig. 4. Each two-input differential receives motion from the
adjacent side gears of its neighbouring two-output differentials $(2-OD_{1}$,
$2-OD_{2}$ and $2-OD_{3})$ and combines the motion to a single output
$(O_{1}$, $O_{2}$ and $O_{3})$ as shown in Fig. 4.

Figure 5: SCHEMATIC OF THE $3-OOD$.

Novelty of 3-OOD: The proposed design of the $3-OOD$ satisfies all the three
aforementioned principles of operation of the traditional $2-OD_{2}$. The
schematic as shown in Fig. 5 illustrates that each output shares the same
dynamics with the other outputs. As a result, the change of loads for one of
the outputs will have the equal effect on the remaining outputs.

### Components Of The Differential

The design of the $3-OOD$ comprises a single input $(I)$, three two-output
open differentials $(2-OD_{1-3})$, three two-input open differentials
$(2-ID_{1-3})$ and three outputs $(O_{1-3})$ as shown in Fig. 4 and Fig. 6.
The input of the $3-OOD$ is located at the centre of the mechanism and the
three two-output differentials $(2-OD_{1-3})$ are arranged symmetrically
around the input in a circular pattern at $120^{o}$ apart from each other.
Each of the three two-input differential $(2-ID_{1-3})$ is arranged radially
in between the two-output differentials. The three two-input differentials
form a circular pattern with $120^{o}$ between each other and bisect the two-
output differentials with an angle of $60^{o}$ as shown in Fig. 2. The outputs
of the three two-input differentials $(2-ID_{1-3})$ form to be the three
outputs of the $3-OOD$.

Each of the three $2-OD_{1-3}$ have six moving parts, an input $(I)$ which is
shared by all three, a ring gear $(R_{1-3})$ that is connected to the input,
two bevel gears $(B_{1-6})$ which are arranged inside the ring gear and two
side gears $(S_{1-6})$, which are connected to its bevel gears Fig. 4 and Fig.
6. Similarly, each of the three $2-ID_{1-3}$ have six moving parts, an output
$(O_{1-3})$, a ring gear $(R_{4-6})$ that is connected to the output, two
bevel gears $(B_{7-12})$ which are arranged inside the ring gear and two side
gears $(S_{7-12})$, which are connected to its bevel gears Fig. 4.

Figure 6: $3-OOD$ (EXPLODED VIEW).

The input $(I)$ provides motion to the three ring gears $(R_{1-3})$ of
$2-OD_{1-3}$. $2-OD_{1-3}$ translate the motion received by their respective
ring gears $(R_{1-3})$ to their respective side gears $(S_{1-6})$ depending on
the load conditions the side gears experience. The side gears $(S_{1-6})$
transfer their motion to the adjoining side gears $(S_{7-12})$ of
$2-ID_{1-3}$, which is translated to the three outputs $(O_{1-3})$ via their
respective ring gears $(R_{4-6})$. When the two side gears of a two-input
differentials ($2-ID_{1-3}$) move with equal speeds, their respective bevel
gears do not engage, and the ring gear will move with the same speed as the
side gears. When the two side gears of a two-input differential ($2-ID_{1-3}$)
have different speeds, $2-ID_{1-3}$ translates the differential speed to a
single output. Under equal loads, all the twelve side gears $(S_{1-12})$ and
the six ring gear $(R_{1-6})$ move at equal speeds. Overall, the design of the
$3-OOD$ comprises six open differentials symmetrically arranged radially at
$60^{o}$ to each other. The six differentials work together to translate the
motion from the ring gears of the three two-output differentials
$(2-OD_{1-3})$ to the ring gears of the three two-input differentials
$(2-ID_{1-3})$. The three ring gears $(R_{4-6})$ of the three two-input
differentials independently power the three outputs $(O_{1-3})$. The $3-OOD$
is symmetric in design with the centre of mass of the mechanism coinciding
with its geometric centre, which ensures that the mechanism is well balanced
around its geometric centre in terms of weight and vibrations that can occur
during its operation.

In total, the $3-OOD$ has thirty-four moving parts, a single input $(I)$,
three outputs $(O_{1-3})$, six ring gears $(R_{1-6})$, twelve bevel gears
$(B_{1-12})$ and twelve side gears $(S_{1-12})$ as shown in Fig. 6. The gear
arrangement between the central input and the ring gear of each of
$2-OD_{1-3}$ is a worm gear arrangement. However; the central input and the
ring gears of $2-OD_{1-3}$ can also be configured to a crown gear arrangement.
Each of the twelve bevel gears have a crown gear and each of the twelve side
gears have a crown gear at one end of the shaft which connect to the bevel
gears of its respective differential. The side gears have another crown gear
at the other end of the shaft that is designed to operate at an angle of
$120^{o}$ and is connected to side gear of the adjacent differential as shown
in Fig. 4 and Fig. 6. The gear arrangement between the outputs and the ring
gears of $2-ID_{1-3}$ is a crown gear arrangement.

## KINEMATICS AND DYNAMICS

The kinematics and the dynamics of the $3-OOD$ are derived by means of the
bond graph modelling technique. The bond graph method is used as it enables a
direct correlation for the mathematical model of the system with its physical
framework [16]. The kinematic scheme $($kinematic diagram$)$ as shown in Fig.
7 illustrates the structure and the links between the components of the
$3-OOD$. The bond graph model as shown in Fig. 8 is developed based on its
corresponding kinematic scheme. All the parameters used in formulating the
kinematic scheme and the bond graph model are listed in Table 1.

Table 1: PARAMETERS OF COMPONENTS OF $3-OOD$ Components | $\omega$ | $\alpha$ | Torque | Inertia
---|---|---|---|---
Source $(S_{e})$ | $\omega_{e}$ | $\alpha_{e}$ | $\tau_{e}$ |
Input $(I)$ | $\omega_{i}$ | $\alpha_{i}$ | $\tau_{i}$ | $I_{i}$
Output $(O_{1})$ | $\omega_{O1}$ |  | $\tau_{O1}$ |
Output $(O_{2})$ | $\omega_{O2}$ |  | $\tau_{O2}$ |
Output $(O_{3})$ | $\omega_{O3}$ |  | $\tau_{O3}$ |
Ring Gear $(R1)$ | $\omega_{R1}$ | $\alpha_{R1}$ | $\tau_{R1}$ |
Ring Gear $(R2)$ | $\omega_{R2}$ | $\alpha_{R2}$ | $\tau_{R2}$ |
Ring Gear $(R3)$ | $\omega_{R3}$ | $\alpha_{R3}$ | $\tau_{R3}$ |
Ring Gear $(R4)$ | $\omega_{R4}$ | $\alpha_{R4}$ | $\tau_{R4}$ |
Ring Gear $(R5)$ | $\omega_{R5}$ | $\alpha_{R5}$ | $\tau_{R5}$ |
Ring Gear $(R6)$ | $\omega_{R6}$ | $\alpha_{R6}$ | $\tau_{R6}$ |
Side Gear $(S_{1})$ | $\omega_{1}$ | $\alpha_{1}$ | $\tau_{1}$ |
Side Gear $(S_{2})$ | $\omega_{2}$ | $\alpha_{2}$ | $\tau_{2}$ |
Side Gear $(S_{3})$ | $\omega_{3}$ | $\alpha_{3}$ | $\tau_{3}$ |
Side Gear $(S_{4})$ | $\omega_{4}$ | $\alpha_{4}$ | $\tau_{4}$ |
Side Gear $(S_{5})$ | $\omega_{5}$ | $\alpha_{5}$ | $\tau_{5}$ |
Side Gear $(S_{6})$ | $\omega_{6}$ | $\alpha_{6}$ | $\tau_{6}$ |
Side Gear $(S_{7})$ | $\omega_{7}$ | $\alpha_{7}$ | $\tau_{7}$ | $I_{1}$
Side Gear $(S_{8})$ | $\omega_{8}$ | $\alpha_{8}$ | $\tau_{8}$ | $I_{3}$
Side Gear $(S_{9})$ | $\omega_{9}$ | $\alpha_{9}$ | $\tau_{9}$ | $I_{4}$
Side Gear $(S_{10})$ | $\omega_{10}$ | $\alpha_{10}$ | $\tau_{10}$ | $I_{6}$
Side Gear $(S_{11})$ | $\omega_{11}$ | $\alpha_{11}$ | $\tau_{11}$ | $I_{5}$
Side Gear $(S_{12})$ | $\omega_{12}$ | $\alpha_{12}$ | $\tau_{12}$ | $I_{2}$

Figure 7: KINEMATIC SCHEME OF THE DIFFERENTIAL.

Figure 8: BONG GRAPH MODEL OF THE DIFFERENTIAL.

It can be inferred from the bond graph model that the ring gears $(R_{1-3})$
of $(2-OD_{1-3})$ rotate with equal angular velocities that is $(\frac{1}{k})$
times the angular velocity of the input $(I)$ and with equal torque that is
$(\frac{k}{3})$ times torque at the input, where $(k^{-1})$ is the gear ratio
of the input to the ring gears. For the current design it is chosen that
$(k^{-1}=\frac{1}{20})$ because the worm gear arrangement at the input
requires a low speed ratios for smooth transmission.

$\omega_{R1}=\omega_{R2}=\omega_{R3}=\frac{\omega_{i}}{k},$ (1)
$\tau_{R1}=\tau_{R2}=\tau_{R3}=\frac{k(\tau_{i})}{3},$ (2)

where $\omega_{R1}$, $\omega_{R2}$ and $\omega_{R3}$ and $\tau_{R1}$,
$\tau_{R2}$ and $\tau_{R3}$ are the respective angular velocities and torques
of the ring gears ($R_{1}$, $R_{2}$ and $R_{3}$), $\omega_{i}$ and $\tau_{i}$
are the angular velocity and torque of the input $(I)$. The side gears
$(S_{1}$ and $S_{2})$ of $2-OD_{1}$ can operate at different speeds
$\omega_{1}$ and $\omega_{2}$, while maintaining equal torques
$(\tau_{1}=\tau_{2})$ [17]. Similarly, the torques of the corresponding side
gears $(S_{3}$ and $S_{4})$ and $(S_{5}$ and $S_{6})$ of $2-OD_{2}$ and
$2-OD_{3}$ are equal $(\tau_{3}=\tau_{4})$,$(\tau_{5}=\tau_{6})$. The angular
velocity $\omega R_{1}$ of the ring gear $(R_{1})$ is always the average of
the angular velocities $\omega_{1}$ and $\omega_{2}$ of the side gears
$(S_{1}$ and $S_{2})$ and the sum of the torques of the side gears $\tau_{1}$
and $\tau_{2}$ is equal to the torque $\tau_{R1}$ of the ring gear [17].

$\tau_{1}=\tau_{2}=\tau_{3}=\tau_{4}=\tau_{5}=\tau_{6}=\frac{\tau_{R1}}{2},$
(3) $\omega_{R1}=\frac{(\omega_{1}+\omega_{2})}{2}.\vspace{-0.25in}$ (4)

Similarly,

$\begin{split}&\omega_{R2}=\frac{(\omega_{3}+\omega_{4})}{2},\quad\omega_{R3}=\frac{(\omega_{5}+\omega_{6})}{2},\
\omega_{R4}=\frac{(\omega_{7}+\omega_{8})}{2},\\\
&\omega_{R5}=\frac{(\omega_{9}+\omega_{10})}{2},\quad\omega_{R6}=\frac{(\omega_{11}+\omega_{12})}{2},\end{split}\vspace{-0.15in}$
(5)

where $\omega_{R2}$, $\omega_{R3}$, $\omega_{R4}$, $\omega_{R5}$ and
$\omega_{R6}$ are the respective angular velocities of the ring gears
$(R_{2}$, $R_{3}$, $R_{4}$, $R_{5}$ and $R_{6})$ and $\omega_{3}$,
$\omega_{4}$, $\omega_{5}$, $\omega_{6}$, $\omega_{7}$, $\omega_{8}$,
$\omega_{9}$, $\omega_{10}$, $\omega_{11}$ and $\omega_{12}$ are the angular
velocities of the respective side gears $(S_{3}$, $S_{4}$, $S_{5}$, $S_{6}$,
$S_{7}$, $S_{8}$, $S_{9}$, $S_{10}$, $S_{11}$ and $S_{12})$. Substituting Eqn.
(4) and Eqn. (5) in Eqn. (1) and Eqn. (3) in Eqn. (2),

$\omega_{i}=\frac{k(\omega_{1}+\omega_{2})}{2}=\frac{k(\omega_{3}+\omega_{4})}{2}=\frac{k(\omega_{5}+\omega_{6})}{2},\vspace{-0.4in}$
(6) $\tau_{i}=\frac{6(\tau_{1})}{k}.$ (7)

It can be inferred from the bond graph model in Fig. 8 that the inertia
$I_{i}$ of the input $(I)$ has a derived causality.

$I_{i}\alpha_{i}=\tau_{e}-\tau_{i},\vspace{-0.15in}$ (8)

where $\tau_{e}$ and $\tau_{i}$ are the respective torque at the source
$S_{e}$ and the input $(I)$ and $\alpha_{i}$ is the angular acceleration of
the input Fig. 8 similarly,

$\begin{split}&I_{1}\alpha_{7}=\tau_{1}-\tau_{7},\quad
I_{2}\alpha_{12}=\tau_{2}-\tau_{12},\quad
I_{3}\alpha_{8}=\tau_{3}-\tau_{8},\\\ &I_{4}\alpha_{9}=\tau_{4}-\tau_{9},\quad
I_{5}\alpha_{11}=\tau_{5}-\tau_{11},\quad
I_{6}\alpha_{10}=\tau_{6}-\tau_{10},\end{split}\vspace{-0.25in}$ (9)

where $I_{1}$,$I_{2}$,$I_{3}$,$I_{4}$,$I_{5}$ and $I_{6}$ are the respective
inertia exhibited by the side gear $(S_{7}$, $S_{12}$, $S_{8}$, $S_{9}$,
$S_{11}$ and $S_{10})$ and $\alpha_{7}$, $\alpha_{12}$, $\alpha_{8}$,
$\alpha_{9}$, $\alpha_{11}$ and $\alpha_{10}$ are their respective angular
accelerations. The side gears $(S_{1}$ and $S_{7})$ are connected such that
there exists no relative motion between them.

$\omega_{1}=\omega_{7},$ (10) $\alpha_{1}=\alpha_{7}.\vspace{-0.15in}$ (11)

Similarly,

$\omega_{2}=\omega_{12},\quad\omega_{3}=\omega_{8},\quad\omega_{4}=\omega_{9},\quad\omega_{5}=\omega_{11},\quad\omega_{6}=\omega_{10},$
(12)
$\alpha_{2}=\alpha_{12},\quad\alpha_{3}=\alpha_{8},\quad\alpha_{4}=\alpha_{9},\quad\alpha_{5}=\alpha_{11},\quad\alpha_{6}=\alpha_{10}.$
(13)

Substituting Eqn. (10) and Eqn. (12) in the Eqn. (6),

$\begin{split}&\omega_{7}=\omega_{1}=\frac{2(\omega_{i})}{k}-\omega_{2},\qquad\omega_{8}=\omega_{3}=\frac{2(\omega_{i})}{k}-\omega_{4},\\\
&\omega_{11}=\omega_{5}=\frac{2(\omega_{i})}{k}-\omega_{6}.\end{split}\vspace{-0.15in}$
(14)

### The Input To Outputs Kinematics

Similar to Eqn. (1) the angular velocities $\omega_{O1}$, $\omega_{O2}$ and
$\omega_{O3}$ of the respective outputs $(O_{1}$, $O_{2}$ and $O_{3})$ is $j$
times the respective velocities $\omega_{R4}$, $\omega_{R5}$ and $\omega_{R5}$
of the corresponding ring gears $(R_{4}$, $R_{5}$ and $R_{6})$, where $j$ is
the gear ratio of the ring gears $(R_{4-6})$ to the outputs $(O_{1-3})$. For
the current design it is chosen that $(j=\frac{2}{1})$.

$\omega_{O1}=j(\omega_{R4}),\quad\omega_{O2}=j(\omega_{R5}),\quad\omega_{O3}=j(\omega_{R6}).\vspace{-0.15in}$
(15)

Substituting Eqn. (5) in Eqn. (15),

$\begin{split}&\omega_{O1}=\frac{j(\omega_{7}+\omega_{8})}{2},\qquad\omega_{O2}=\frac{j(\omega_{9}+\omega_{10})}{2},\\\
&\omega_{O3}=\frac{j(\omega_{11}+\omega_{12})}{2}.\end{split}\vspace{-0.3in}$
(16)

Further substituting Eqn. (12) and Eqn. (14) in Eqn. (16),

$\omega_{O1}=\frac{2j(\omega_{i})}{k}-\frac{j(\omega_{2}+\omega_{4})}{2},\vspace{-0.3in}$
(17)
$\omega_{O2}=\frac{2j(\omega_{i})}{k}-\frac{j(\omega_{3}+\omega_{5})}{2},\vspace{-0.3in}$
(18)
$\omega_{O3}=\frac{2j(\omega_{i})}{k}-\frac{j(\omega_{1}+\omega_{6})}{2}.\vspace{-0.15in}$
(19)

Equivalent kinematics of the outputs: Equations (17), (18) and (19) show that
all the three outputs have equivalent kinematics with the input. The outputs
$(O_{1-3})$ relate to the input $(I)$ with additional dependence on the side
gears $(S_{1-6})$. This is because the $3-OOD$ has three degrees of freedom
(3DOF) and the behaviour of an output is impacted by the input $(I)$ and the
other two outputs. In Eqn. (17) the influence of the outputs $(O_{2}$ and
$O_{3})$ on the output $(O_{1})$ is seen in the form of $\omega_{2}$ and
$\omega_{4}$.

### The Input To Outputs Dynamics

Similar to Eqn. (2), the torque $\tau_{O1}$, $\tau_{O2}$ and $\tau_{O3}$ of
the respective outputs $(O_{1}$, $O_{2}$ and $O_{3})$ is ($j^{-1}$) times the
respective torque $\tau_{R4}$, $\tau_{R5}$ and $\tau_{R6}$ of the
corresponding ring gears $(R_{4}$, $R_{5}$ and $R_{6})$.

$\tau_{O1}=\frac{\tau_{R4}}{j},\quad\tau_{O2}=\frac{\tau_{R5}}{j},\quad\tau_{O3}=\frac{\tau_{R6}}{j}.\vspace{-0.15in}$
(20)

It can be inferred from the bond graph model in Fig. 8 that the torque
$\tau_{R4}$ is the sum of the torques $\tau_{7}$ and $\tau_{8}$ of the side
gears $S_{7}$ and $S_{8}$ of $2-ID_{1}$ and similarly for $\tau_{R5}$ and
$\tau_{R6}$

$\tau_{R4}=\tau_{7}+\tau_{8},\quad\tau_{R5}=\tau_{9}+\tau_{10},\quad\tau_{R6}=\tau_{11}+\tau_{12}.\vspace{-0.15in}$
(21)

Substituting Eqn. (21) in Eqn. (20),

$\tau_{O1}=\frac{(\tau_{7}+\tau_{8})}{j},\quad\tau_{O2}=\frac{(\tau_{9}+\tau_{10})}{j},\quad\tau_{O3}=\frac{(\tau_{11}+\tau_{12})}{j}.$
(22)

Further substituting Eqn. (3), Eqn. (7) and Eqn. (9) in Eqn. (22),

$\tau_{O1}=\frac{k(\tau_{i})}{3j}-\frac{(I_{1}\alpha_{7}+I_{3}\alpha_{8})}{j},\vspace{-0.4in}$
(23)
$\tau_{O2}=\frac{k(\tau_{i})}{3j}-\frac{(I_{4}\alpha_{9}+I_{6}\alpha_{10})}{j},\vspace{-0.4in}$
(24)
$\tau_{O3}=\frac{k(\tau_{i})}{3j}-\frac{(I_{2}\alpha_{12}+I_{5}\alpha_{11})}{j}.\vspace{-0.15in}$
(25)

Equivalent dynamics of the outputs: Equations (23), (24) and (25) show that
all the three outputs of the differential have equivalent dynamics with the
input.

### When Three Outputs Are Under Equal Loads

Since the side gears $(S_{7-12})$ are all identical, they exhibit equal
inertia. Furthermore, when all three outputs $(O_{1-3})$ are under equal loads
(resistive torque ‘$\tau_{R}$’), the side gears $(S_{7-12})$ operate with
equal angular velocity and angular acceleration.

$\begin{split}&\omega_{1}=\omega_{2}=\omega_{3}=\omega_{4}=\omega_{5}=\omega_{6}=\\\
&\omega_{7}=\omega_{8}=\omega_{9}=\omega_{10}=\omega_{11}=\omega_{12}=\frac{\omega_{i}}{k},\end{split}$
(26)
$\begin{split}&\alpha_{1}=\alpha_{2}=\alpha_{3}=\alpha_{4}=\alpha_{5}=\alpha_{6}=\\\
&\alpha_{7}=\alpha_{8}=\alpha_{9}=\alpha_{10}=\alpha_{11}=\alpha_{12},\end{split}$
(27) $I_{1}=I_{2}=I_{3}=I_{4}=I_{5}=I_{6}.\vspace{-0.15in}$ (28)

Substituting Eqn. (6), Eqn. (26) in Eqn. (17), Eqn. (18) and Eqn. (19),

${\bf\omega_{O1}}={\bf\omega_{O2}}={\bf\omega_{O3}}=\frac{j(\omega_{i})}{k}.\vspace{-0.15in}$
(29)

Substituting Eqn. (27) and Eqn. (28) in Eqn. (23), Eqn. (24) and Eqn. (25) and
subtracting the resistive torque $(\tau_{R})$ from the torque of each output

${\bf\tau_{O1}}={\bf\tau_{O2}}={\bf\tau_{O3}}=\frac{k(\tau_{i})}{3j}-\frac{2(I_{1}\dot{\omega}_{1})}{j}-\tau_{R},\vspace{-0.15in}$
(30)

Equivalent angular velocities and torque : Equations (29) and (30) illustrate
the novel ability of the differential to translate equal motion
$(\omega_{O1}=\omega_{O2}=\omega_{O3})$ and equal torque
$(\tau_{O1}=\tau_{O2}=\tau_{O3})$ to all its three outputs that are
unconstrained $(\tau_{R}=0)$ or under equal loads.

### Kinematics Between The Outputs

The kinematics an output of the $3-OOD$ shares with the other two outputs is
determined by substituting Eqn. (12) and Eqn. (16) into the previously
established input to outputs kinematics in Eqn. (17).

$\omega_{O1}=\frac{2j(\omega_{i})}{k}+\frac{j(\omega_{10}+\omega_{11})}{2}-(\omega_{O2}+\omega_{O3}),\vspace{-0.15in}$
(31)

Further substituting Eqn. (12) and Eqn. (6) in Eqn. (31),

$\omega_{O1}+\omega_{O2}+\omega_{O3}=\frac{3j(\omega_{i})}{k}.\vspace{-0.15in}$
(32)

### Dynamics Between The Outputs

The dynamics an output of the $3-OOD$ shares with the other two outputs is
determined by substituting Eqn. (9), Eqn. (11) and Eqn. (13) into the
previously determined input to outputs dynamics in Eqn. (23).

$\tau_{O1}=\frac{2k(\tau_{i})}{3j}-\frac{(I_{1}\alpha_{7}+I_{3}\alpha_{8})}{j}-\frac{(\tau_{1}+\tau_{3})}{j}.\vspace{-0.15in}$
(33)

Substituting Eqn. (3) and Eqn. (9) in Eqn. (33),

$\begin{split}\tau_{O1}=\frac{2k(\tau_{i})}{3j}-\frac{(I_{1}\alpha_{7}+I_{2}\alpha_{12}+I_{3}\alpha_{8}+I_{4}\alpha_{9})}{j}-\frac{(\tau_{9}+\tau_{12})}{j}.\end{split}\vspace{-0.15in}$
(34)

Further substituting Eqn. (9), Eqn. (13) and Eqn. (22) in Eqn. (34),

$\begin{split}\tau_{O1}+\tau_{O2}+&\tau_{O3}=\frac{k(\tau_{i})}{j}\\\
&-\frac{(I_{1}\alpha_{7}+I_{2}\alpha_{12}+I_{3}\alpha_{8}+I_{4}\alpha_{9}+I_{5}\alpha_{11}+I_{6}\alpha_{10})}{j}.\end{split}$
(35)

When motion and torque are provided to one of the outputs, the input receives
no motion $(\omega_{i}=0$, $\tau_{i}=0)$ because the worm gear arrangement at
the input only allows one-way motion from the input to the ring gears. The
motion provided to the outputs is shared between the other two outputs which
will rotate in the opposite direction. Substituting $(\omega_{i}=0)$ in Eqn.
(32), $(\tau_{i}=0)$ in Eqn. (35),

$\omega_{O1}=-(\omega_{O2}+\omega_{O3}),$ (36)
$\begin{split}\tau_{O1}=&-\tau_{O2}-\tau_{O3}\\\
&-\frac{(I_{1}\alpha_{7}+I_{2}\alpha_{12}+I_{3}\alpha_{8}+I_{4}\alpha_{9}+I_{5}\alpha_{11}+I_{6}\alpha_{10})}{j}.\end{split}$
(37)

The negative sign indicates that $(O_{2})$ and $(O_{3})$ move in the opposite
direction of $(O_{1})$.

Novelty of 3-OOD: Equations (29) and (30) illustrate the novel ability of the
differential to translate equal motion $(\omega_{O1}=\omega_{O2}=\omega_{O3})$
and equal torque $(\tau_{O1}=\tau_{O2}=\tau_{O3})$ to all its three outputs
that are unconstrained $(\tau_{R}=0)$ or under equal loads. Equations (32) and
(35) show the equivalent kinematic and dynamics relations that the outputs of
the differential share with each other. Furthermore, Eqn. (36) and Eqn. (37)
illustrate that when the input is ceased from motion and motion is provided to
one of the outputs, the two other outputs receive equal motion when under
equal loads. In such cases the $3-OOD$ behaves as a $2-OD$ where the output
which receives motion acts as the input and the other two outputs will act as
the outputs of a $2-OD$.

## CURRENT AND POTENTIAL APPLICATIONS

Figure 9: $3-OOD$ INSIDE A PIPE CLIMBER.

The $3-OOD$ is designed to be used inside a pipe climbing robot as shown in
Fig. 9 [18, 19]. It can equip the three tracks of the robot with differential
speed so that, during a turn it can rotate the track travelling the longer
distance faster than the track that travels the shorter distance. But when
moving in a straight path the differential can rotate the thee tracks with
equal speeds. Using the differential in a pipe climber will potentially
eliminate slip and drag of its tracks and reduce the stresses the robot
experiences during turns and thus providing a smoother motion of the robot
[18]. Other current applications can be found in robotic arms such as the
SARAH (Self-Adapting Robotic Auxilary Hand) $($Self-Adapting Robotic Auxilary
Hand$)$ and sensor less in-hand manipulation by underactuated robot hand [9,
20, 21]. The differential can also find applications in single-input multiple-
spindle wrench $($nut-runner$)$ as suggested by S. Kota and S. Bidare (1997)
[8]. Other potential applications for the differential include but not limited
manipulators, legged robots, and wheeled robots [22]. In the near future, a
prototype of the differential will be made to further research the functioning
of the robot under different load cases and also study the transmission
efficiency and energy flow of the differential. In addition, we will be
implementing the differential in a pipe climbing robot.

## CONCLUSION

The novel design of the $3-OOD$ is presented. $3-OOD$ is designed such that
its functioning ability is analogous to the traditional $2-OD$. The kinematics
and dynamics of the differential are established using the bond graph model.
From the analytical and theoretical study conducted it is observed that the
$3-OOD$ is the first differential with three-outputs to realise all the
functioning abilities of a traditional $2-OD$. That is, $(i)$ its outputs
operate with equal speed when the outputs are unconstrained or under equal
loads, $(ii)$ its input to output kinematics and dynamics is equivalent for
all three outputs and $(iii)$ its kinematics and dynamics between any two of
its outputs is equivalent. It is also observed that when one of the outputs is
ceased from motion and the other two outputs are left unconstrained, they
operate with equal angular velocities and equal torques. The need for such a
mechanism is discussed with relevant current and potential applications in
fields such as robotics and automobile engineering.

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