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Journal of Intelligent & RoboticSystemswith a special section on UnmannedSystems ISSN 0921-0296 J Intell Robot SystDOI 10.1007/s10846-016-0414-4

Design and Analysis of a Four-PendulumOmnidirectional Spherical Robot

Brian P. DeJong, Ernur Karadogan,Kumar Yelamarthi & James Hasbany

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J Intell Robot SystDOI 10.1007/s10846-016-0414-4

Design and Analysis of a Four-Pendulum OmnidirectionalSpherical Robot

Brian P. DeJong · Ernur Karadogan ·Kumar Yelamarthi · James Hasbany

Received: 26 April 2016 / Accepted: 21 August 2016© Springer Science+Business Media Dordrecht 2016

Abstract This paper presents the design, analysis,and comparison of a novel four-pendulum sphericalrobot. The proposed mechanism rolls omnidirection-ally via four tetrahedrally-located pendulums that shiftthe robot’s center of mass to create rolling torque.The nine dynamic equations of motion are derivedvia the Lagrangian and nonholonomic constraint equa-tions, and then simulated numerically; results showsuccessful propulsion with expected behaviors. Themechanism is then compared to existing center-of-mass designs in terms of directionality, drive torquearm, and inertia eccentricity. In these regards, the four-pendulum design is a balance of existing designs: itis omnidirectional with eccentricity and torque capa-bility that are in the middle of the range exhibited byexisting designs. In addition, the new four-pendulummechanism has been built and tested as a successfulproof-of-concept prototype.

B. P. DeJong (�) · E. Karadogan · K. Yelamarthi ·J. HasbanyCentral Michigan University, Mount Pleasant,MI 48859, USAe-mail: b.dejong@cmich.edu

Ernur Karadogane-mail: ernur.karadogan@cmich.edu

Kumar Yelamarthie-mail: kumar.yelamarthi@cmich.edu

James Hasbanye-mail: hasba1jh@cmich.edu

Keywords Spherical robot · Dynamics ·Center-of-mass · Nonholonomic

1 Introduction and Related Work

Remotely controlled mobile robots continue tobecome more popular in society and in the researchliterature. One interesting vein of research is in theuse of spherical robots – mobile robots with an outerspherical shell that traverse an environment by rolling.These ball-shaped robots are intriguing because theymove without any externally visible mode of propul-sion and must navigate the nonholonomic ball-and-plane workspace. Thus, they require relatively novelmotion mechanisms, subtly complex dynamic analy-sis, and advanced control algorithms [1–3]. No singlepropulsion scheme has emerged clearly better than theothers, and so new mechanisms continue to be pre-sented. Because of their outer shell, spherical robotshave the advantages of being durable, non-tippable,and capable of being fully sealed from the outsideenvironment. They have found broad applications inpatrol and surveillance [4], extraterrestrial exploration[5], environmental monitoring [6], underwater [7], andeven child-development [8].

This paper presents a novel spherical robot designthat uses four internal pendulums for propulsion. Assuch, it is omnidirectional (it can roll in any direc-tion instantly) unlike most of the existing sphericalrobot designs, and yet comparably agile. Here, the

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design is described and implemented on a physicalprototype as a proof of concept. The mechanism’sdynamics are analyzed from first principles, and itsmotion is numerically simulated. The design is alsoanalyzed and contrasted with existing designs in termsof directionality, torque arm, and inertia eccentricity.

Spherical robots can be roughly categorized bytheir drive mechanism. One technique uses conserva-tion of angular momentum by having one or moreinternal flywheels that are tilted to induce rolling ofthe sphere. For example, Bhattacharya and Agrawal[9] use two perpendicular rotors, while Joshi andBanavar [10] use four rotors to achieve omnidirec-tionality. Gajamohan et al. [11] use three rotors in acube robot to rotate and balance omnidirectionally.These robots have the advantage of being able toproduce drive torques higher than pendulum designs(i.e., torque arms are greater than the sphere’s radius)but the disadvantages of large internal energy andundesired precession torques.

A few spherical robots move by distorting or trans-forming their outer shell, or by simply using theenvironment’s wind similar to tumbleweed. Sugiyamaet al. [12] use shape-memory alloy to deform and rollwheels and spheres, while Wait et al. [13] and Artusiet al. [14] deform panels on a robot’s shell via airbladders and dielectric actuators, respectively.

The most common propulsion scheme, however,is to internally change the robot’s center of mass toinduce rolling. This scheme has the advantage of actu-ation simplicity, but the disadvantage of limiting themaximum torque arm to less than the sphere’s radius.There are several approaches to modifying the centerof mass, as described below.

One proposed but under-studied design [15] uses asingle internal mass that is hung and moved by severalcables and pulleys. This design has the advantage ofminimal static mass (i.e., almost all of the mass can beclustered in the movable mass), but its torque arm islimited by the locations of the cables – the mass cannotmove past the bisection line from one cable mount toanother.

A more common center-of-mass technique is to usean internal car or drive unit that rolls on the interiorof the shell, similar to a hamster in a hamster wheel.For example Halme et al. [16] present a spring-loadedunicycle drive unit, while Bicchi and colleagues [17],and later Alves and Dias [18], use a small internalcar. These designs have the advantage of well-studied

drive mechanisms and result in a sphere that can rollin any direction, but are usually not instantaneouslyomnidirectional, as the drive unit must turn beforeclimbing the shell. They also suffer from the typicalgrip-wear tradeoff of friction drives – to get good trac-tion, the drive wheels need large normal forces, butthis increases steering friction and wear.

The most common center-of-mass design is a sin-gle internal pendulum that can rotate about a mainrolling axis and tilt to induce steering when rolling.For example, Seeman et al. [4] and Michaud andCaron [19] present single pendulum designs andapplications. Schroll [20] analyzes and simulates thesingle-pendulum dynamics, while Hogan et al. [5]discusses a robot that uses a hanging payload asthe pendulum. Designs also exist with two hangingpendulums and stick-slip spinning, such as Ghanbariet al. [21] and Yoon et al. [22]. The single-pendulumdesign has the advantage of a simple (typically two-degree-of-freedom) drive mechanism; its primary dis-advantage is that it is not omnidirectional – single-pendulum designs are more “steered wheel” than“rolling sphere” because of their main drive axis.When facing forwards, for instance, these designscannot roll purely sideways.

Alternatively, some spherical robots move the cen-ter of mass by sliding multiple masses along internaltetrahedral spokes. As the masses slide along theirrail, the center of mass changes and the sphere rolls.Javadi and Mojabi [23], Sang et al. [24], and Tomiket al. [25] present such designs. These robots are omni-directional instantaneously, but suffer from reducedtorque arms since the masses cannot all converge onthe ideal center-of-mass location. For example, whenrolling forward, one or more distributed masses maybe restricted to a backwards spoke and thus reducerolling torque.

2 Four Pendulum Design

The spherical robot presented here uses fourtetrahedrally-located pendulums to change the robot’scenter of mass and thus roll the sphere. The mecha-nism, shown in Fig. 1, includes four pendulums thatrotate about the four tetrahedral axes. Their configu-ration (rotating about tetrahedral axes versus inline)and location (radial distance to motor, rm and pen-dulum rod length, rp) are optimized for maximum

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Fig. 1 CAD representation of four-pendulum mechanism

center-of-mass envelope without risking pendulumoverlap. Conceptually, this location is such that thefour pendulum’s arcs touch at the outer shell. Fromrest, the mechanism can apply rolling torque in anydirection instantly, and in practice, the pendulums arerotating in full circles as the robot rolls.

The mechanism has been implemented in a proof-of-concept prototype, shown in Fig. 2. The electronicsand batteries are housed in a rapid-prototyped tetra-hedral box in the center of a 33-cm-diameter plasticshell, with motors mounted on the box facing out-wards. The robot uses a Maple r5 microprocessor,Futaba 75 MHz RF transmitter and receiver, and LiPobatteries for the motors (11 V) and electronics (3.7 V).The motors are Polulu 12V brushed DC with built-in50:1 gear reduction and 3200 CPR-output encoders.

For orientation sensing, the robot uses a CHRoboticsCHR-UM6 six-axis orientation sensor that incorpo-rates accelerometer, rate gyro, and magnetic sensorinformation.

The prototype successfully rolls in any direction,as commanded by the user. As an initial prototype, itis not optimized for minimum static mass nor doesit include position feedback. It has a static mass of2.6 kg with a dynamic mass of 1.6 kg; the dynamic-to-static mass ratio could be greatly improved infuture prototypes. The four-pendulum configurationallows for ideal (point-mass) pendulum rod lengths of13.5 cm in a 33 cm shell; the actual pendulums haveeffective lengths of 10.0 cm. Initial tests show that theprototype can accelerate with 0.6 m/s2 and achieve amaximum speed of 0.7 m/s. The prototype proves fea-sibility of the four-pendulum design and allows forfuture testing.

3 Mechanism Dynamics

3.1 Kinematics

To describe the dynamics of the four-pendulum mech-anism, we define six inertial frames: a static Worldframe, a Body frame centered on the sphere’s centerof mass, and four pendulum frames (numbered 1 to4) centered on the corresponding pendulum’s centerof mass. The mechanism and frames are sketched inFig. 3.

The Body frame rotates in the World frame using z-x’-z” intrinsic Euler notation with angles φ, θ , and ψ ,and translates with x and y. For simplicity, the frameis assumed to also be at the sphere’s geometric centersuch that it does not translate in the vertical direction.

Fig. 2 Four-pendulumrobot prototype

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1

2

3

4

rp

rm

xBody

yBody

Body

World

x1y1

1

Fig. 3 Sketch of four-pendulum coordinate frames

The homogeneous standard rotation and translationmatrices for the Body are Rφ , Rθ , Rψ , and Dxy , withtotal transformation from Body to World of

WBT = DxyRφRθRψ. (1)

The sphere’s position in the world is

WpS = WBT

[0 0 0 1

]T = [x y 0 1

]T. (2)

The ith pendulum frame is defined such that the xi-axis is aligned with the pendulum arm and the zi-axisis parallel to its rotation axis. As the pendulum rotateswith angle αi , the transformation from i to the motor’sposition is (using notation c and s for cosine and sine)

Ti =

⎡

⎢⎢⎣

cαi −sαi 0 rpcαi

sαi cαi 0 rpsαi

0 0 1 rm0 0 0 1

⎤

⎥⎥⎦ . (3)

The motors are mounted in a tetrahedral pattern

defined by the tetrahedral angleε = cos−1(−1/

3

)≈

109.5◦ and 120◦ rotations about the z-axis, repre-sented by rotation matrices

Rε =

⎡

⎢⎢⎣

cε 0 sε 00 1 0 0

−sε 0 cε 00 0 0 1

⎤

⎥⎥⎦ ,

R120 =

⎡

⎢⎢⎣

c120◦ −s120◦ 0 0s120◦ c120◦ 0 00 0 1 00 0 0 1

⎤

⎥⎥⎦ . (4)

Therefore, the transformations for the four pendu-lums areB1T = Ti=1,

B2T = RεTi=2,

B3T = R120RεTi=3,

B4T = RT

120RεTi=4. (5)

Pendulum i located at

Wpi = WBT

Bi T

[0 0 0 1

]T = [xi yi zi 1

]T. (6)

The system therefore has 9 generalized coordi-nates: the three Euler angles, the two translations, andthe four pendulum angles:

q = [φ, θ, ψ, x, y, α1, α2, α3, α4]T . (7)

The translational velocities of the sphere and pen-dulums are the time derivatives of the p’s above:

Wps = [x y 0 0

]T(8)

Wpi = d

dt

(WBT

Bi T

[0 0 0 1

]T )

= [xi yi zi 0

]T. (9)

The mechanism is to be simulated in MATLAB, soit is advantageous to rewrite the Wpi equation to elim-inate time derivatives of functions. This can be doneby using the chain rule:

Wpi =[

∂W pi

∂q1

∂W pi

∂q2· · · ∂W pi

∂q9

]q. (10)

The angular velocities of the sphere in World andBody coordinates are

Wωs = φ + θ + ψ =

⎡

⎢⎢⎣

00φ

0

⎤

⎥⎥⎦ + Rφ

⎡

⎢⎢⎣

θ

000

⎤

⎥⎥⎦

+RφRθ

⎡

⎢⎢⎣

00ψ

0

⎤

⎥⎥⎦

=

⎡

⎢⎢⎣

θ cφ + ψsφsθ

θsφ − ψcφsθ

φ + ψcθ

0

⎤

⎥⎥⎦

=

⎡

⎢⎢⎣

0 cφ sφsθ 00 sφ −cφsθ 01 0 cθ 00 0 0 1

⎤

⎥⎥⎦

⎡

⎢⎢⎣

φ

θ

ψ

0

⎤

⎥⎥⎦ , (11)

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and

Bωs = R−1ψ R−1

θ

⎡

⎢⎢⎣

00φ

0

⎤

⎥⎥⎦ + R−1

ψ

⎡

⎢⎢⎣

θ

000

⎤

⎥⎥⎦ +

⎡

⎢⎢⎣

00ψ

0

⎤

⎥⎥⎦

=

⎡

⎢⎢⎣

sψsθ cψ 0 0cψsθ −sψ 0 0cθ 0 1 00 0 0 1

⎤

⎥⎥⎦

⎡

⎢⎢⎣

φ

θ

ψ

0

⎤

⎥⎥⎦ . (12)

It is advantageous to define both – as shown later,the World form is needed for the constraint equations,while the Body form is used with the (Body-frame)inertia for the kinetic energy.

The relative angular velocity of pendulum i withrespect to the sphere is

Bωi/s =Bi T

iωi/s =Bi T

[0 0 αi 0

]T. (13)

Therefore, the pendulum’s total angular velocity is

Bωi =Bωs + Bωi/s . (14)

3.2 Equations of Motion: Lagrange Equations

The Lagrange equations of motion and generalizedcoordinates are

L = Ktotal − Utotal

d

dt

(∂L

∂qk

)− ∂L

∂qk

=n∑

i=1

λiaik + Fk (15)

where Lis the Lagrangian, Ktotaland Utotal are thesystem’s kinetic and potential energies, k correspondsto the generalized coordinate (1 to 9), n is the num-ber of constraint equations, the λ’s are the Lagrangemultipliers for the constraints, the a’s are the coeffi-cient in the constraint equations for each generalizedcoordinate, and the F ’s are the nonconservative forces(friction, motor torques, etc.) on the generalized coor-dinates. Using the chain rule and rewriting followingthe method described in [26], the equations become

9∑

j=1

⎛

⎝d

(∂K

/∂qk

)

dqj

qj +d

(∂K

/∂qk

)

dqj

qj

⎞

⎠

− ∂K

∂qk

+ ∂U

∂qk

=n∑

i=1

λiaik + Fk (16)

where j sums through the generalized coordinates. Athird version is useful for control applications:

M (q) q + V (q, q) + G(q) =n∑

j=1

λiaj + F (17)

where M is the inertia matrix, V is the Coriolisand centripetal vector, and G is the gravity vector. Thematrix and vectors have terms

Mkj (q) =d

(∂K

/∂qk

)

dqj

,

Vk (q, q) =∑9

j=1

⎛

⎝d

(∂K

/∂qk

)

dqj

qj

⎞

⎠ − ∂K

∂qk

,

Gk (q) = ∂U

∂qk

. (18)

The mechanism has static mass M(shell, frame,electronics, motors, etc.) and moveable pendulummasses of m each. The translational (‘ t’) kineticenergies are

Ks t = 1

2W pT

s MW ps = 1

2M

(x2 + y2

)(19)

Ki t = 1

2W pT

i mW pi = 1

2M

(x2i + y2

i + z2i

). (20)

The sphere has moment of inertia tensor Is androtational kinetic energy

Ks r = 1

2BωT

s IsBωs

= 1

2

(Ixxs

Bω2x + Iyys

Bω2y + Izzs

Bω2z

). (21)

Each pendulum’s inertia must be shifted to themotor via the Parallel Axis Theorem to define tensoriIi and rotational (‘ r’) kinetic energy

iIi = m

⎡

⎢⎢⎣

Y 2i + Z2

i −XiYi −XiZi 0−XiYi X2

i + Z2i −YiZi 0

−XiZi −YiZi X2i + Y 2

i 00 0 0 0

⎤

⎥⎥⎦ (22)

Ki r = 1

2BωT

i

(Bi T·iIi ·Bi TT

)Bωi (23)

where Xi = rpcαi , Yi = rpsαi , and Zi = rm. Thetotal kinetic energy is

Ktotal = Ks t + Ks r +4∑

i=1

(Ki t + Ki r) (24)

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The potential energies are

Us = [0 0 Mg 0

] · WpS = 0, (25)

Ui = [0 0 mg 0

] · Wpi =mgzi, (26)

with total

Utotal = Us +4∑

i=1

Ui. (27)

The R-radius sphere is assumed to roll withoutslipping, implementing two nonholonomic constraintequations (n = 2):

f1 = x − R · Wωys = 0, (28)

f2 = y + R · Wωxs = 0. (29)

Defining rolling damping as broll , motor dampingas bmotor , and motor torques as τi , the constraint coef-ficients and nonconservative forces can be modeledas

a1 =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂f1/

∂φ∂f1

/∂θ

∂f1/

∂ψ

100000

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, a2 =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂f2/

∂φ∂f2

/∂θ

∂f2/

∂ψ

010000

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

F =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

000

−broll

−broll

τ1 − bmotor

τ2 − bmotor

τ3 − bmotor

τ4 − bmotor

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (30)

The four-pendulum equations of motion have beensimulated in MATLAB. To simulate the nine coordi-nate and two constraint equations, the multipliers aresolved for using the x and y equations:

λ1 = M4 (q) q + V4 (q, q) + G4 (q) − F4

λ2 = M5 (q) q + V5 (q, q) + G5 (q) − F5 (31)

where the 4 and 5 subscripts signify rows of the matri-ces and vectors. The multipliers are then substituted

back into the remaining seven Lagrange equations (ksignifies row, from 1 to 3 and 6 to 9):

(Mk − a1kM4 − a2kM5) q

+ (Vk−a1kV4−a2kV5) + (Gk−a1kG4 − a2kG5)

= (Fk − a1kF4 − a2kF5) . (32)

The two constraint equations are differentiated(again using the chain rule):

df 1

dt= d

dt

(aT1 q

)= q +

[∂aT

1

∂qq

]T

q

= aT1 q + qT ∂a1

∂qq = 0 (33)

df 2

dt= aT

2 q + qT ∂a2∂q

q = 0

The differentiation is a standard method (e.g., [27])and is needed to make the new inertia matrix full rank.These equations are in Eq. 18 form — for example,

[aT1

]q +

(qT ∂a1

∂qq

)+ (0) = 0 + 0 (34)

— and can be combined with the seven remainingLagrange equations. With these modifications, thesimulation takes the resulting nine equations, invertsthe new inertia matrix, and solves for q.

4 Simulation Results

The mechanism dynamics have been simulated inMATLAB for various initial conditions and torquesusing normalized values of M = 1.0 (modeled as ahomogenous sphere), m = 0.25 (modeled as a pointmass), and R=1. In the results shown here, the rollingdamping and motor damping are assumed zero. Dueto the equations’ nonholonomic nature, the simulationis susceptible to ill-conditioned configurations thatresult in singularities in the inverted inertia matrix, andresults show slight numerical errors in energy conser-vation; both limitations are similar to those seen byother authors (e.g., [5, 20]). The following three exam-ples, however, demonstrate the validity of the equa-tions and mechanism, and the simulation’s usefulnessfor future analysis and control schemes testing.

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4.1 Free Response

Figure 4 shows a sample free-response path and statevariables (i.e., no applied torques). For this simulation,the mechanism was given an initial rolling velocitytowards positive x:

q = [0, 0, −2π, 2π, 0, 0, 0, 0, 0]T . (35)

The results show the mechanism (and simulation)behaving as expected. The sphere rolls in the posi-tive x direction with wobbling due to the free-spinningpendulums and the inherent wobbliness of sphericalrobots (discussed later). The sphere positions, sphereangles, and pendulum angles are shown in Fig. 4b; thependulum angles are increasing as the sphere rotatesbut the pendulums do not.

4.2 Single-Pendulum Response

Figure 5 also shows the mechanism and simulationbehaving as expected. The figure shows a samplesingle-torque response path, with plots of state vari-ables and energy in the system. For this simulation,the initial velocities were zero but a constant (normal-ized) torque was applied to one pendulum (colored redin the figure):

F = [0, 0, 0, 0, 0, −1, 0, 0, 0]T . (36)

As expected, the mechanism rolls towards nega-tive x due to the applied torque and towards positiveydue to the unbalanced weight of the other three pen-dulums. The simulation did not apply any control

Fig. 4 Simulation resultsshowing a normalizedfree-response path due to aninitial rolling velocity. aInitial position andsubsequent path of sphere.b State variables

(a)

(b)

0 0.5 1 1.5 2 2.5 30

5

10

15

Pos

ition

s (n

orm

aliz

ed)

x

y

0 0.5 1 1.5 2 2.5 320

15

10

5

0

5

10

15

20

25

30

Time (sec)

Ang

les

(rad

)

1

2

3

4

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Fig. 5 Simulation resultsshowing a path and b statevariables, c energy whentorque is applied to onependulum

(a)

(b)

(c)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 52

0

2

4

Pos

ition

s (n

orm

aliz

ed)

xy

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 53

2

1

0

1

2

3

4

5

Time (sec)

Ang

les

(rad

)

1

2

3

4

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to the other pendulums or towards maintaining adesired rolling velocity. The energy plot shows thatthe total energy in the system is increasing due to theapplied torque, but once the work associated with thetorque (Wapplied) is removed, the remaining energiesremain relatively constant (within 3.2 %). As men-tioned previously, this slight fluctuation in the sim-ulation’s energy is expected and is due to numericalrounding and matrix-inversion approximations.

4.3 Double-Pendulum Response

Results from a third example simulation are shown inFig. 6. Here, the simulation applied constant torquesto two of the pendulums (colored blue and black in thefigure),

F = [0, 0, 0, 0, 0, 0, 0, −1, −1]T . (37)

Fig. 6 Simulation resultsshowing a path and b statevariables when torque isapplied to two pendulums

(a)

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 52

0

2

4

6

Pos

ition

s (n

orm

aliz

ed)

xy

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 54

3

2

1

0

1

2

3

4

Time (sec)

Ang

les

(rad

)

1

2

3

4

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causing the sphere to roll towards positive y. Becauseof the symmetry of both the actuated and the free-spinning pendulums, the path is a straight line. How-ever, the speed along the path is not constant, as shownby the slope of the y plot. The simulation did notattempt to control that speed.

5 Comparison to Existing Center-of-mass Designs

This section compares the new four-pendulum mech-anism to existing center-of-mass spherical robotdesigns, here called the single-pendulum, the four(tetrahedral) sliders, and the cable-mass designs. Thecomparison is made on three aspects of the mecha-nisms, as discussed below and summarized in Table 1.The terminology and comparisons used are benefi-cial for determining each mechanism’s strengths andweaknesses with regards to the others.

5.1 Directionality Comparison

The first comparison is on the directionality of themechanism. Here, directionality is meant as the num-ber of directions the sphere can roll instantly fromrest. It is meant as a general term rather than a tech-nical one. Ideally, a spherical robot should be able toroll in any direction instantly, as this provides navi-gational advantages in complex environments. In thatregard, the four-pendulum, four sliders, and cable-mass designs are omnidirectional, while the morecommon single-pendulum design is not.

5.2 Torque Arm Comparison

The second comparison is on the theoretical normal-ized torque arm. When a mechanism shifts its centerof mass from its geometric center, the mass creates

Table 1 Comparison of several theoretical center-of-massdesigns

Design Single Four Cable Four

pendulum sliders mass pendulums

Directionality Steered Omni Omni Omni

Torque arm, rT <100 % <29 % <100 % <67 %

Eccentricity, e 100 % 0–20 % 0, 100 % 36–100 %

a rolling torque proportional to its distance. That dis-tance, or torque arm, can be modeled as the percentageof the sphere radius, assuming point masses and nostatic mass:

rτ = horizontal distance to CoM

R· 100 %. (38)

For center of mass designs, 0 < rτ < 100 %. Thisis an ideal mechanism calculation, since in actualitythe robot’s shell, frame, and electronics create a staticmass that reduces the effective torque arm (e.g., 50–67 % [20]).

Existing designs have either large or small max-imum torque arms. Using this metric, the single-pendulum design has a theoretical 100 % torque armsince the pendulum’s mass can be moved out to thesphere’s radius at any angle (<100 %). The cable massdesign has a similar maximum, although it is not atany angle (<100 %) – the mass cannot extend pastthe bisection line between any two cable mounts. Onthe other hand, the four-slider design has a much morerestricted torque arm – simulations show a center ofmass envelope of a polyhedron with maximum torquearm ranging from 20–29 %.

The proposed four-pendulum mechanism has amaximum torque arm length between those of theexisting designs. If the four pendulums are equal, thenits center of mass can be calculated as the average ofthe positions and the envelope of possible locationsis a truncated sphere. From the simulation, the maxi-mum torque arm ranges from 56–67 %, depending onthe angle. That is, the four-pendulum design can ide-ally generate torques equivalent to up to two-thirds ofits radius.

5.3 Eccentricity Comparison

The third comparison between designs is the eccen-tricity of the robot’s rotational moment of inertiaabout its center. A general three-dimensional bodyhas an inertia ellipsoid representing its rotational iner-tia in any direction. The eccentricity of the ellipse(mathematically, “first eccentricity”) is defined here as

e =√√√√1 − I 2minor

I 2major

· 100 %, (39)

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where Imajor and Iminor are the robot’s principalmoments of inertia about the ellipse’s major andminor axes.

A well-known challenge of spherical robots is thattheir inertia ellipses are not spherical (Imajor > Iminor).As the robot rolls, angular momentum is conserved.However, the robot’s angular inertia is not constant,and so the angular velocity varies resulting in wob-bling of the sphere. Thus, having a mechanism withlittle eccentricity (i.e., more homogeny) is advanta-geous since it results in less wobble.

Existing designs have either high or low eccen-tricity. A theoretical single-pendulum or cable-massdesign has one point mass, and thus an inertiaellipse collapsed into a line. Their eccentricity isalways 100 % (unless, in the cable-mass design, themass is at the center), resulting in wobbling. In thisregard, the four-slider design is better. Simulationshows that the four-slider design has an eccentric-ity of <20 %; the four-slider design is less prone towobbling.

Similar to the torque arm comparison, the new four-pendulum design’s eccentricity is between the existingdesigns’ eccentricities. The four-pendulum’s eccen-tricity varies depending on the configuration of thependulums, and ranges from 36 % to 100 %, with anaverage of 84 %. The 36 % occurs when the four pen-dulums are pointing in various directions, while the100 % occurs when they are paired at opposite sidesof the sphere. Thus, the four-pendulum design is lessprone to wobble than the single-pendulum and cable-mass designs, but more prone than the four-slidersdesign.

5.4 Comparison Summary

As noted, these comparisons are based on theoreticalmechanisms. A physical robot will have numbers thatare more conservative. The four-pendulum prototypeis a successful proof-of-concept robot that is not opti-mized for minimal static mass, but may be useful asan example. As mentioned previously, it has a staticand dynamic masses of 2.6 kg and 1.6 kg; in Chaseand Pandya’s [1] terminology, it has a power factor of0.615. The mechanism’s theoretical maximum torquearm is calculated as 56–67 %; the prototype’s effec-tive maximum is closer to 20 %. On the other hand,the prototype’s eccentricity is better because of thestatic mass. The mechanism’s theoretical eccentricity

is 36–100 %, with an average of 84 %; the prototype’seccentricity is 30–92 %, with an average of 73 %.

6 Conclusions and Future Work

The proposed four-pendulum center-of-mass spheri-cal robot is a novel and successful mechanism. Thenonholonomic equations of motion have been mathe-matically derived and numerically simulated, showingexpected behaviors. The mechanism is also a balanceof existing center-of-mass designs in that it is omni-directional, has torque arm around two-thirds of thesphere’s radius, and inertia eccentricity of an average84 %. The mechanism has been successfully imple-mented on a physical prototype that demonstrates thevalidity of the four-pendulum design.

The mechanism and equations show promise forfuture work. For example, the Euler-angle representa-tion used here is known for its ill-conditioning near“gimbal lock” orientations. One possible improve-ment is to derive and simulate the equations of motionusing quaternions rather than Euler-angles. Further-more, the simulation behaves as expected but theexamples show the sphere does not roll with con-trolled heading or speed. A next step is to applydrive-velocity and path-following control schemes tothe four-pendulum mechanism. In addition, the pro-totype is remote-controllable, but not autonomous;future work includes adding global position feedbackand autonomous control. Finally, the omnidirectional-ity of the four-pendulum design complicates the useof orientation-dependent sensors such as video cam-eras and ultrasonic position sensors since there is noconsistent roll axis. An unanswered research questionis how to apply, fuse, or simulate such sensors on anomnidirectional spherical robot.

Acknowledgments This work was funded in part by theCentral Michigan University FRCE Grant #48868.

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Brian P. DeJong is an Associate Professor of Mechanical Engi-neering in the School of Engineering and Technology at CentralMichigan University. He received a M.S. and Ph.D. in Mechan-ical Engineering from Northwestern University with researchin robotics. His current research is in mobile robots (spheri-cal, soundlocalization), teleoperation (improved interfaces), andengineering education.

Ernur Karadogan is an Assistant Professor of MechanicalEngineering in the School of Engineering & Technology at Cen-tral Michigan University, Mt. Pleasant, MI. He received hisB.S. degree in Aerospace Engineering in 2000 from MiddleEast Technical University (METU), Ankara, Turkey, and hisM.S. and Ph.D. degrees both in Mechanical Engineering fromOhio University (OU), Athens, Ohio in 2005 and 2011, respec-tively. His Ph.D. work was concentrated on the design of acable-actuated robotic lumbar spine to be used as a palpatorytraining and assessment device for medical students. Formerly,he worked as a research scientist in the Fluid Mechanics Lab-oratory at the National Metrology Institute (TUBITAK-UME),Gebze-Kocaeli, Turkey from 2001 to 2003; as a research asso-ciate in the Mechanical Engineering Department at OU from2005 to 2013; as an assistant professor in the Mechanical Engi-neering Department at the University of Texas Rio Grande Val-ley (formerly the University of Texas-Pan American), Edinburg,Texas between 2013 and 2015. His research interests include:Robotics, Haptics, Dynamic Systems & Control, Simulations inVirtual Environments, and Biomechanics.

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Kumar Yelamarthi received his Ph.D. and M.S degree fromWright State University in 2008 and 2004. He is currently anAssociate Professor of Electrical & Computer Engineering atCentral Michigan University. His research interest is in theareas of Wireless Sensor Networks, Internet of Things, mobilerobotics, computer vision, embedded systems, and engineeringeducation. He has published over 90 articles in archival jour-nals and conference proceedings in these areas. He served as PI,co-PI, and senior personnel in several externally funded grantsfrom organizations such as NSF, NASA, and the regional indus-try. He is an elected member of Tau Beta Pi engineering honorsociety, and Omicron Delta Kappa national leadership honorsociety, and a senior member of IEEE.

James Hasbany is a graduate student pursuing his Masters ofScience in Engineering at the School of Engineering and Tech-nology at Central Michigan University. He has served in bothResearch and Teaching assistantships. He received a B.S. inMechanical Engineering from Central Michigan University. Hiscurrent research area is in spherical robots.

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