fourpend2015_v05

1 Copyright © 2015 by ASME
DRAFT
Proceedings of the ASME 2015 International Mechanical Engineering Congress & Exposition
IMECE2015
November 13-19, 2015, Houston, Texas
IMECE2015-51551
A FOUR-PENDULUM OMNIDIRECTIONAL SPHERICAL ROBOT: DESIGN
ANALYSIS AND COMPARISON
Brian P. DeJong
Central Michigan University
Mount Pleasant, MI, USA
Kumar Yelamarthi
Brad Bloxsom
ABSTRACT
This paper presents a four-pendulum design for an
omnidirectional spherical robot. The robot moves by the
rotation of tetrahedrally-located pendulums to change the center
of mass, and can do so instantaneously in any direction. In that
respect, it is omnidirectional unlike the more common single-
pendulum designs. In the paper, the theoretical design is
discussed and contrasted to existing theoretical center-of-mass
designs in terms of directionality, torque arm, eccentricity, and
radius of gyration. The torque arm comparison is based on the
envelope of the dynamically-controlled mass (i.e., how far in
any direction can the center of mass be moved), while the
eccentricity and gyration criteria are based on homogeneity and
size of the mass moment of inertia (i.e., how much does the
design “wobble” while rolling; how much inertia does it have).
In these regards, the four-pendulum architecture is a strong
design – better than some in one respect, better than the others
in other respects. A prototype has been built and tested that
successfully implements the four-pendulum propulsion, and its
results are compared to theoretical ones.
INTRODUCTION
Spherical mobile robots are an interesting research area
with subtly complex dynamics and a variety of mechanisms [1-
3]. Various propulsion schemes have been used, and since one
scheme has not emerged clearly better than the rest, new
designs are still being presented. The ball-shaped robots are
intriguing because they have often have no externally-visible
sources of propulsion and yet can navigate the non-holonomic
rolling-ball-on-plane space. They have application in patrol
and surveillance [4], extraterrestrial exploration [5],
environmental monitoring [6], underwater [7], and even child
development [8].
This paper presents one such new design (Fig.1), but in doing
so, also attempts to quantify its performance to similar designs.
The design presented here is an omnidirectional tetrahedral
four-pendulum design that uses the position of the pendulums
to change the center of mass of the robot and thus create a
rolling torque. The robot is omnidirectional in that it can roll in
any direction instantly, unlike the more common single- or
double-pendulum designs. Using pendulums, the robot is
Figure 1. Sketch of four-pendulum mechanism
R
-R
-R
R
0
0
R
-R
0
+z
x
y
z
xz
+y
-y
rp
rm
θϕ

shown to be theoretically more agile than other existing
tetrahedral designs but more eccentric in its rotational inertia,
which may lead to more wobbling without proper control
schemes. Specifically, the new design is compared to similar
designs in terms of directionality, torque arm, inertia
eccentricity, and inertia magnitude.
RELATED WORK
Spherical robots can be roughly categorized by their
propulsion mechanism. One type of propulsion mechanism
uses conservation of angular momentum to induce rolling of a
rigid outer shell. These robots spin a heavy internal flywheel
(or flywheels), and then tilt it to produce shell rolling. For
example, Brown and Xu [9] built disk-on-edge robot that
moved via an internal gyroscope. Bhattacharya and Agrawal
[10] present a robot using two perpendicular rotors, while Shu
et al. [11] present a single rotor also acting as a liftable
pendulum. Joshi and Banavar [12] use four rotors that allow
for rolling in any direction. The main advantage of this type of
design is that it can produce torques equivalent to moving the
robot’s center of mass outside of its shell – more torque than
center-of-mass mechanism discussed below. One complication
of this design is that the resulting precession torque can move
the robot in unintended directions. The use of a rotor pair has
also been used by Schroll [13] to enhance a center-of-mass
design.
There are also examples of spherical robots that move by
distorting or transforming the shell itself. These designs are all
distinct and this vein of research is relatively unexplored. For
example, separate designs by Artusi et al. [14] and Wait et al.
[15] deform panels on the robot’s shell by dielectric actuators
and air bladders, respectively. Sugiyama et al. [16] propose a
wheel or shell made out of shape-memory alloys.
The most common propulsion scheme, and perhaps the
simplest to implement, is to change the robots internal center of
mass to induce a torque and thus rolling of a rigid outer shell.
Within this scheme, there are various methods changing that
center of mass, as discussed below. The main advantage of this
design is its actuation simplicity. However, the main limitation
is the torque is restricted by the envelope of possible center of
masses, as discussed below.
The first center-of-mass method in the literature is that of
an internal wheel or car mechanisms – often explained as a
robotic hamster in its wheel. The wheel or car rolls internally in
one direction, changing the robots center of mass, and inducing
rolling of the ball in that direction. For example, Halme et al.
[17] use a spring-loaded internal unicycle drive unit to shift the
center of mass. Bicchi and colleagues [18], and later Alves and
Dias [19], use a small car as the drive unit. From the external
(visible) point of view, these robots are omnidirectional in that
the shell can roll in any direction. However, they are not
instantaneously omnidirectional – the wheel or car must steer
first before being able to move in a sideways direction. They
also suffer from the typical grip-wear tradeoff of friction drives:
increasing the normal force on the drive wheels reduces drive
slip but also greatly increases steering friction and wear.
The most common method is to use a single two-degree-
of-freedom pendulum, here called the “single-pendulum”
design. Raising the pendulum causes the robot to roll forward
or backward, and tilting the pendulum while raising it causes
the robot to steer left or right while rolling. For example
Michaud and Caron [20] introduce Roball that is re-prototyped
as a toy for child development studies. Seeman et al [4] apply
the GroundBot robot to security tasks. Schroll [13] analyzes
the dynamics of a single-pendulum spherical robot that also
includes conservation of angular momentum enhancement.
Zheng et al. [21] analyze the BHQ-1 robot. Park et al [22]
proposes a control scheme for a single-pendulum robot.
Likewise, Pokhrel et al. [23] introduces SpheRobot, and Hogan
et al.i [5] discuss a wind-blown spherical robot that also uses a
hanging payload as a single pendulum. Recently, several
designs have emerged with two smaller hanging pendulums
that allow for stick-slip spin-in-place about the vertical axis.
Example designs are proposed and analyzed by Ghanbari et al.
[24], Yoon et al. [25], and Wang and Zhao [26].
These single-pendulum designs are more “wheel” than
“ball” because they all have a main axis/axle that the robot
must roll about. Thus they are not omnidirectional – when
facing forwards, they cannot purely roll sideways. However,
the single pendulum means that all of the dynamic mass of the
robot can be moved very close to the outer shell, maximizing
the torque created for a center-of-mass design.
A third way of moving the center of mass is by distributed
masses sliding along radial, such as tetrahedral, spokes. By
sliding the (four) masses as much to one side as possible, the
robot rolls in that direction – here, this type of design is called
“four-sliders”. Tomik et al. [27] first presented this design
along with a geometric motion planner. Javadi and Mojabi [28]
present a similar design with a simulation. The main advantage
of this design is that it is omnidirectional – the robot can move
in any direction instantly. It does not need to steer either via
spin or via rolling in a small arc. However, the distributed
masses means that some of the dynamic mass is not ideally
located for torque generation. For example, if attempting to
roll forward, one mass may be located on a spoke aimed
backwards and thus not able to apply a positive torque. (Even
worse, it may be applying a negative torque.)
A proposed but understudied design [29] uses cables and
pulleys to move a single mass about the interior of the robot.
This design allows for omnidirectional travel and maximized
center-of-mass torque, and thus has a lot of potential, but is not
heavily explored yet. Here, this design is called the “cable
mass” design.
Amid and in addition to the mechanism literature, there are
several veins of ball-on-plane dynamics applied to spherical
robots, motion planning, and spherical robot control research.
Besides such work included in the already cited papers – or by
similar papers by cited authors – is work done by Der et al.
[30], Qiang et al. [31], Peng et al. [32], Tao et al. [33], Ivanova
and Pivovarova [34], and Karavaev and Kilin [35]. The

dynamics and control of the four-pendulum mechanism
presented here are complicated (as they are for other spherical
robots), but they are not discussed in detail in this paper.
FOUR PENDULUM DESIGN
The spherical robot presented here is a center-of-mass
design that uses four pendulum masses in a tetrahedral
arrangement. Figure 1 shows a sketch of the (ideal) mechanism,
where R is the sphere’s radius. Each pendulum rotates about a
tetrahedral direction – changing the angle of the pendulums
changes the robot’s center of mass. This mechanism can create
rolling torque in any direction instantly, and in practice, the
pendulums are rotating in full circles as the robot rolls in any
one direction.
The main benefit of this design is its omnidirectionality,
with larger torque possible than previous tetrahedral designs.
The four pendulum design can roll in any direction
instantaneously, and can shift its center of mass farther forward
than the four-slider design, as will be shown here.
Figure 2 shows pictures of a physical prototype. The
robot’s electronics and battery are housed in and around central
rapid-prototyped tetrahedral box, and the four motors are
mounted facing out from that box. The box is centered in the
shell using four spokes that reach through gaps between
pendulum arcs, and the shell is a 33-cm-diameter plastic ball.
Figure 2. Pictures of the prototype four-pendulum
spherical robot.
The robot uses the Maple r5 as its processor. It is remote
controlled using a Futaba 75 MHz RF transmitter and receiver,
and carries one battery for the motors (11V 1500mAh LiPo)
and a smaller one for the electronics (3.7V 850mAh LiPo). The
motors are Pololu 12V brushed DC with a 50:1 reduction and
64 CPR encoders for position feedback. The robot senses
orientation via an onboard orientation sensor (CHR-UM6) that
processes accelerometer, rate gyro, and magnetic sensor data.
Given the desired torque magnitude and direction from the RF
data, the robot computes the desired center of mass location in
world coordinates, converts that position to robot and then
pendulum coordinates, and moves the pendulums to achieve
that center.
The tetrahedral transformations are as follows (see labels
in Fig. 1). The direct angle between axes is the standard
tetrahedral angle,
φ = cos−1
−
1
3
"
#
$
%
&
' ≈109.5° , (1)
and the z-rotation from “xz” pendulum to “+y” pendulum is
ϕ =120° . (2)
Let the pendulums be identical with length rp rotating on motor
axes at tetrahedral distances of rm. Given pendulum i's rotation
angle, θi, the position of that pendulum in robot coordinates, xi,
is
xi = Ri
rp cosθ
rp sinθ
rm
!
"
#
#
#
#
$
%
&
&
&
&
, (3)
where Ri is the rotation matrix for that pendulum. Here,
R+z =
1 0 0
0 1 0
0 0 1
!
"
#
#
#
$
%
&
&
&
, (4)
Rxz =
cosφ 0 sinφ
0 1 0
−sinφ 0 cosφ
"
#
$
$
$
$
%
&
'
'
'
'
, (5)
R+y =
cosϕ −sinϕ 0
sinϕ cosϕ 0
0 0 1
"
#
$
$
$
$
%
&
'
'
'
'
• Rxz , (6)
R−y =
cosϕ sinϕ 0
−sinϕ cosϕ 0
0 0 1
"
#
$
$
$
$
%
&
'
'
'
'
• Rxz . (7)
For maximum rolling torque, the pendulum length should
be as large as possible without collisions between pendulums.
That is, larger rp results in larger torque. For this design, the
maximum rp without collision is when two pendulum arcs
touch – by trigonometry, this occurs when
rp = Rsin
φ
2
!
"
#
$
%
& ≈ 0.82R
rm = Rcos
φ
2
!
"
#
$
%
& ≈ 0.58R
. (8)
Simulations confirm that these dimensions result in the largest
possible torque.
The following sections compare this mechanism to similar
designs with a summary shown in Table 1. The comparisons
are made for the ideal, theoretical designs (no static mass, all
point-mass dynamic mass). Such a comparison is useful
separate from the pragmatic comparison of costs, realistic
masses and dimensions, and ease of construction or control.

Design Single
Pendulum
Four
Sliders
Cable
Mass
Four
Pendulums
Directionality Steered Omni Omni Omni
Torque arm, rT <100% <29% <100% <67%
Eccentricity, e 100% 0-20% 0 or
100%
36-100%
Radius of
Gyration, k
100% 0-178% 0-100% 87-216%
Table 1. Comparison of Several Theoretical Center-of-
Mass Designs
DIRECTIONALITY COMPARISON
This section compares the four-pendulum design to
existing center-of-mass designs in terms of what is called
“directionality”. This term is not meant as a technical term and
applies to the robot’s shell on the (floor) plane.
A sphere on a plane has five degrees-of-freedom: two
translational degrees corresponding to the location of the
sphere’s center, and three rotational degrees corresponding to
the sphere’s orientation at the point. The ideal spherical robot
would be able to change any of these degrees instantaneously
(and thus be in the technical sense, holonomic).
However, a rolling sphere is non-holonomic in that it has
no-slip conditions and must roll to change its orientation (and
position). An ideal rolling spherical robot would be able to roll
in any direction instantaneously without steering – similar to a
sports ball rolling on a floor or field. Such omnidirectionality
has obvious advantages when navigating complex
environments. In this sense, the four-pendulum, sliding-
masses, and cable-mass designs are omnidirectional, while the
more prevalent single-pendulum designs are not.
TORQUE ARM COMPARISON
The second comparison to be made between the four-
pendulum design and existing center-of-mass designs is in
terms of its theoretical normalized torque arm. Here, the
torque arm, rT is defined as the effective torque arm of the
dynamic center of mass envelope, as a percentage of the sphere
radius:
rT =
horizontal distance to dynamic CoM
sphere radius
•100%
=
xCoM
2
+ yCoM
2
R
•100%
. (8)
Thus, normalized torque can be 0 to 100% for center-of-mass
designs, but larger than that for conservation-of-angular-
momentum designs.
Each robot implementation has some “static” mass
consisting of its components that do not move to change the
center of mass and thus the drive torque – its shell, internal
framework, and possibly electronics, sensors, battery, and
motors. It also has some “dynamic” mass that can be moved to
change its center of mass – the pendulums or sliding masses,
which may include some or all of the electronics, sensors,
battery, and motors. A robot-building design goal is to
minimize the static mass while maximizing the dynamic mass,
to allow for the maximum possible torque.
A theoretical-mechanism design goal is to create as large a
torque arm as possible from the dynamic mass, assuming the
static mass is negligible. In this sense, the single-pendulum and
cable-mass designs have the largest torque arm possible of
100% – if modeled as a point mass, the center of mass can be
moved anywhere in the sphere, out to the sphere radius. (In
application, the static mass decreases this to a more typical 50-
67% [13].)
The four-sliding-mass design, while omnidirectional, has a
much smaller torque arm. If the masses are equal, then
simulation shows that its center of mass envelope for all
possible slider configurations is a polyhedron with a maximum
torque arm of 29%. The maximum torque arm also varies
based on the direction – in some directions, it is only 20%.
The four-pendulum design is also omnidirectional, and has
a larger torque arm than the four-slider, but less than the single-
pendulum and cable-mass designs. If the masses are equal,
then its center of mass can be calculated as the sum of the four
xi’s (Eqn. (3)) and the envelope of possible locations looks like
a sphere with eight truncations (see Fig. 3). The figure is
generated by simulating all possible configurations of the four
pendulums, and calculating the resulting dynamic center of
mass, but then plotting only the outer envelope. The maximum
torque arm in any direction ranges from 56% to 67%. That is,
the four-pendulum can create a torque equivalent to moving its
dynamic center of mass approximately two-thirds along its
sphere radius.
Figure 3. Four-pendulum theoretical torque-arm
envelope.
ECCENTRICITY COMPARISON
In addition, this section compares the four-pendulum
design to existing center-of-mass designs in terms of the
eccentricity, e, of the robot’s rotational moment of inertia, I,
about its center.
−0.5
0
0.5
−0.5
0
0.5
−0.5
0
0.5
z(normalized)
x (normalized)
y (normalized)

A general three-dimensional body has a moment of inertia
ellipsoid representing its angular inertia in all directions. The
axes of the inertia ellipsoid are called the principal axes, and
their magnitudes are the principal moments of inertia. The
largest inertia, Imajor, about any axis corresponds to the
ellipsoid’s semi-major axis, and the smallest, Iminor, corresponds
to the ellipsoid’s semi-minor axis.
Let Ixx, Iyy, and Izz be the moments of inertia about the x, y,
and z axes, and Ixy, Iyz, and Ixz be the corresponding products of
inertia. Then, the inertia matrix representing the inertia
ellipsoid is
I =
Ixx −Ixy −Ixz
−Ixy Iyy −Iyz
−Ixz −Iyz Izz
"
#
$
$
$
$
%
&
'
'
'
'
. (9)
The ellipsoid’s principal axes and values can be found from the
standard eigenvector problem
Av = λv (10)
where generic A is now the inertia matrix I; the eigenvalues, λ,
are the magnitude of the principal inertias/axes, and the
corresponding eigenvectors, v, are the cosines of the principal
axes with respect to the x, y, and z axes.
One of the well-known challenges in spherical robots is
that the robots tend to wobble as they roll. Angular momentum
( H = Iω ) is conserved, but because the robot’s moment of
inertia (I) is non-homogeneous (i.e., an ellipsoid and not a
sphere), the angular momentum (ω) also varies. The result is
similar to an American football that wobbles outside of a
perfect spiral.
Thus, having a homogeneous inertia (i.e., a spherical
inertia ellipsoid) is advantageous in spherical robots because it
reduces wobble. The more non-homogeneous, or “eccentric”,
the inertia ellipsoid is, the more the wobble. Here, eccentricity
is defined in the standard mathematical sense (also called first
eccentricity) but as a percentage:
e = 1−
Iminor
2
Imajor
2
•100%. (11)
In application, any static mass usually decreases eccentricity
because the overall mass is more distributed than the pendulum
masses.
A single-mass (pendulum or cable-controlled) design with
a point-mass bob has an inertia ellipse collapsed into a line – its
eccentricity is always 100% unless, in the cable-mass design,
the mass is in the center of the sphere and the eccentricity has
approached 0%. Thus, in any torque situation, these designs
have the potential for large wobble if control does not
compensate.
In terms of eccentricity, the four-slider design is much
better. Like the cable-mass design, when all four point masses
are at the center of the sphere, the eccentricity is 0%.
Otherwise, simulation shows it varies with a maximum of only
20% – the distributed-mass four-slider design is much less
prone to wobble than the single-pendulum or cable-mass
designs. This effect makes sense – distributing the dynamic
mass around the sphere homogenizing the ellipsoid, decreasing
the eccentricity.
The four-pendulum design is in-between the existing
designs. At no point can the four pendulums be moved into the
middle like the cable-mass or four-slider design, so the
eccentricity is never 0; it’s minimum is 36% (although while
still applying a torque). On the other hand, when the four
pendulums are paired up on opposite sides of the spheres, the
eccentricity is 100% – balanced about the center, but collapsed
into a line. A simulation shows that for all pendulum locations,
the eccentricity ranges for 36 to 100% with an average of 84%.
Thus, the four-pendulum design is more prone to wobble than
the four-slider design, but less than the single-mass designs.
GYRATION COMPARISON
The last comparison to be made between the designs is the
radius of gyration – a standard sense of the magnitude of a
moment of inertia. The normalized radius of gyration, k, is
defined here as percentage of the sphere’s radius:
k =
I
m
•
100%
R
. (12)
It can be visualized as the effective radius to a point mass that
results in the same moment of inertia. A larger radius of
gyration is equivalent to a larger moment of inertia, which can
be beneficial when maintaining angular speed, or detrimental
when trying to change angular speed. Usually, a design goal is
to minimize inertia to allow for a more nimble robot.
This is an important difference from minimizing
eccentricity. A homogenous sphere has an eccentricity of 0%
and a non-zero radius of gyration; the same mass as a point
mass can be placed at the same radius of gyration but will now
have an eccentricity of 100%.
For the single-pendulum design, the maximum theoretical
k is always 100% because there is only one point mass, at the
sphere’s radius. For the cable-mass design, the maximum k
ranges from 0 to 100%, linearly related to the radius to the
mass. For the four-slider design, the maximum k varies from 0
(when all masses are pulled in) to 178% (when all masses are
extended). For the four-pendulum design, the maximum k
varies from 81% to 218% – this design has a large moment of
inertia because of the masses cannot collapse into the center.
Thus the ranking of the four designs with regards to radius of
gyration depends on their configuration, but the cable-mass and
four-slider designs have the best possible configurations.
PROTOTYPE NUMBERS
The physical prototype was designed and built as a proof-
of-concept of the four-pendulum driving mechanism, and as an
initial testbed for control schemes. As such, it is a success.
The robot successfully rolls omnidirectionally as commanded
through the RF communication, and maintains (visually) proper
heading.
It should be noted that the comparisons above are for the
theoretical mechanism, and are dampened by the physical
application.

For example, the torque arm as calculated (56-67%) is for
the ideal mechanism; the physical prototype has a much smaller
torque arm. This prototype version is not optimized for
dynamic mass over static mass. It has a static mass of shell,
batteries, motors, and electronics totaling 2.6kg, with a
dynamic mass of 1.6kg – in Chase and Pandya’s [1] terms, its
power factor is 0.615. For a 33-cm-diameter sphere, the four
pendulums have ideal lengths of around 13.5 cm; because of
clearances and non-point-masses, the pendulum have effective
lengths of around 10 cm. The effective torque arm of the
prototype is closer to 20%. Initial tests with this prototype saw
it accelerate with 0.6 m/s2
up to a maximum speed of 0.7 m/s.
On the other hand, the eccentricity of the prototype is
better than theoretical. The inertia of the static masses means
that the inertia ellipse is less dependent on the pendulum
locations. The theoretical eccentricity ranges from 36% to
100% with an average of 84%; the prototype’s eccentricity
ranges from 30% to 92% with an average of 73%.
The radius of gyration also increases with the static mass.
For the prototype, it increases to 155-250%.
CONCLUSION
The four-pendulum omnidirectional spherical mobile robot
design has been presented and discussed. Compared to previous
center-of-mass designs, the four-pendulum design has large
directionality, medium torque arm, medium eccentricity, but
large radius of gyration. It is a viable spherical robot design.
A physical prototype has been built and tested. It
successfully rolls using the pendulums to change the center of
mass of the robot. The prototype is successful even with a
decreased torque arm and increased radius of gyration, and is
less prone to wobble than its theoretical equivalent.
Future work includes rigorous dynamical analysis, robust
control implementations, and a more optimized prototype. The
comparisons also suggest that the cable-mass mechanism is a
beneficial design and should be analyzed further.
ACKNOWLEDGMENTS
This work was funded in part by the Central Michigan
University FRCE Grant #48868.
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