2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)
October 25-29, 2020, Las Vegas, NV, USA (Virtual)Tumbling and Hopping Locomotion Control fora Minor Body Exploration Robot
Keita Kobashi1 , Ayumu Bando1 , Kenji Nagaoka2 , and Kazuya Yoshida1
Abstract— This paper presents the modeling and analysis of anovel moving mechanism “tumbling” for asteroid exploration.
The system actuation is provided by an internal motor andtorque wheel; elastic spring-mounted spikes are attached tothe perimeter of a circular-shaped robot, protruding normal
to the surface and distributed uniformly. Compared with theconventional motion mechanisms, this simple layout enhances
the capability of the robot to traverse a diverse microgravity
environment. Technical challenges involved in conventional
moving mechanisms, such as uncertainty of moving direction
and inability to traverse uneven asteroid surfaces, can now
be solved. A tumbling locomotion approach demonstrates two
beneficial characteristics in this environment. First, tumbling
locomotion maintains contact between the rover spikes and theground. This enables the robot to continually apply control
adjustments to realize precise and controlled motion. Second,owing to the nature of the mechanical interaction of the
spikes and potential uneven surface protrusions, the robot can
traverse uneven surfaces. In this paper, we present the dynamics
modeling of the robot and analyze the motion of the robot
experimentally and via numerical simulations. The results ofthis study help establish a moving strategy to approach thedesired locations on asteroid surfaces.

I. I NTRODUCTION
The exploration of asteroids is an interesting and scientifically lucrative endeavor owing to the wealth of information
asteroids may contain about the origin and evolution of oursolar system. The gravitational acceleration of asteroids has
been estimated to be between 1.0×10−5 m/s2 and 1.0×10−2
m/s2 . Owing to the microgravity environment on asteroid
surfaces, friction-based locomotion such as wheel mobility, which significantly depends on gravity, cannot achievesufficient locomotion [1]. Moreover, robots utilizing such
locomotion are likely to experience significant bumps andground reaction forces resulting in periods of unintentional
floating. [2].
To overcome friction-based locomotion limitations, several
moving mechanisms have been invented for adapting to
the microgravity environment; examples include hopping
mobility and the ciliary vibration drive [3] – [5]. A samplingof hopping and ciliary locomotion robots includes MINERVA, MASCOT, and MINERVA-II as developed by Japan
Aerospace Exploration Agency (JAXA) and Centre National
1 Keita Kobashi, Ayumu Bando, and Kazuya Yoshida with
Department of Aerospace Engineering, Graduate School ofEngineering, Tohoku University, Sendai, 980-8579, Japan {kobashi,

yoshida}@astro.mech.tohoku.ac.jp
2 KenjiNagaokawithDepartmentofMechanicalandControlEngineering,
Graduate
SchoolofEngineering,
Kyushu Institute of Technology, Kitakyushu, 804-8550, Japannagaoka.kenji572@mail.kyutech.jp
978-1-7281-6211-9/20/$31.00 ©2020 IEEE

Fig. 1.

3D Image of Our Robotd’etudes Spatiales (CNES), and JAXA, respectively. However, the conventional mechanisms have vital problems toaccess the desired locations on asteroid surfaces. Therefore,
these problems should be addressed for the success of future
missions.
Hopping mobility is induced by an impulsive force thatis derived from robots’ internal actuator. Various actuators
have been explored in previous studies, such as motors
with a flywheel [6], elastic energy derived from internal
springs [7], and the deformation of a bimetal exposed toa temperature gradient [8]. Although the effectiveness of the
hopping mobility was demonstrated by MINERVA-II during
the landing on the surface of Ryugu, the robots utilizing
this hopping mobility may experience repeatedly reboundon asteroid surfaces and lose their moving directivity. Whilesome space probes have observed the surfaces of asteroids [9]and [10], the characteristics of asteroids’ surfaces are varied,and hence, further investigation is required. Consequently, forpractical use, the soft-landing techniques for such hopping
rovers that can adapt to unknown asteroid surfaces, arestill difficult to achieve. Hence, such hopping rovers cannot
explore their desired locations.
Alternatively, ciliary vibration-type motion is generated
by the oscillation of the robot [11] – [13]. For ciliary-type
motion robots, elastic bodies attached to the body of therobot are bend, and the elastic forces generated from thetransformation of its cilia work as the propulsion forces.
Owing to this mobility, the motion is predictable [14];however, most of the sequence of this mobility is crawling on
the ground. Therefore, ciliary-type motion robots experiencegreat difficulty of traversing uneven surfaces, even if the
obstacles are small.
Moreover, legged robots utilizing their limb to grasp
boulders on asteroid surfaces have been invented by the National Aeronautics and Space Administration (NASA) [15].
This mechanism overcomes the challenges of the moving
directivity and the traversability. However, such robots have

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complex systems that require intricate controllers. Therefore,
the practical use of legged robots in real missions is stillaffected by various obstacles.
In contrast to asteroid exploration, the proposed tumbling
locomotion exhibits advantages in overcoming these significant challenges. Tumbling locomotion is induced by aninternal actuator and the spring-mounted spikes attached tothe perimeter of the robot. The reaction torque generated by
the internal actuator makes the robot rotate, and the contacts
between the spikes and the ground have the spikes transform.
Thus, the robot obtains its motion through the frictional andelastic forces of the spikes interacting with the ground.
During tumbling, the robot maintains contact with theground; thus, contributing to the controllability of the robot
and consequently, the predictability of the robot’s motion.
Moreover, the tumbling locomotion enables the robot to traverse uneven surfaces because the spikes are able to grapple
steps and small obstacles on asteroid surfaces. Accordingly,
the tumbling motion improves the chances of the robot
approaching its desired locations and expands the scope of
its action on asteroid surfaces.
In this paper, we mention the dynamics modeling of thesmall tumbling robot and analyze the characteristics of the
tumbling motion based on numerical simulations. Moreover,some physical experiments are conducted to validate theproposed dynamics model.

(a) Hopping(b) Ciliary Vibration
Fig. 2.

Notions of Conventional Movement

Fig. 3.

II. L OCOMOTION M ECHANISM

Tumbling Mechanism

III. D EFINITION OF T UMBLING AND H OPPINGIn this section, we address the mechanism of the proposed
tumbling locomotion. The notions of conventional hoppingand ciliary vibration drive are described in Fig. 2 (a) and
Fig. 2 (b). As shown in Fig. 2 (a), the internal actuatorof the robot realizes the hopping mobility. Robots utilizing
this mobility have a significant drawback that the robots lose
their moving directivity because of the repeated rebounding.
Conversely, the ciliary vibration drive described in Fig. 2(b) has high directivity. While traversing asteroid surfaces,
the robots assuming ciliary vibration driving mobility havehigh chances to touch the ground because of the microhopping, which enables them to have high moving directivity.
However, the robots cannot traverse uneven surfaces byassuming this moving mode.
On the contrary, the proposed moving mechanism, tumbling, can overcome these drawbacks involved in the conventional mobilities while holding the advantages of these twomoving modes. Moreover, the tumbling movement can begenerated by an internal torque, which can also generate thehopping movement. Therefore, robots utilizing the tumbling
mobility are able to switch two moving modes using an
actuator.
The sequence of motion is shown in Fig. 3. The robot
receives the reaction torque derived from the rotation of
its internal motor. Based on the magnitude of the reaction
torque, the robot selects one of the two moving modes,
tumbling and hopping.

For clarity, we define the two significant modes of therobot’s motion, tumbling and hopping.
Tumbling: As shown in Fig, 4 (a), the spikes of therobot contact the ground, sequentially. Periods of nocontact with the ground are possible; however, less thana revolution in the air is required.
• Hopping: As shown in Fig. 4 (b), contact is not maintained with the ground. The spikes of the robot are notrequired to contact the ground sequentially. A single
revolution or more is necessary while floating in theair.
•

IV. DYNAMICS M ODELINGIn this study, we address the robot in a 2D environment.
The robot exhibits a circular shape and has eight spikes connected by linear springs and dampers. The internal actuatoris located at the geometric center of the robot.

A. Equation of Motion
From Fig. 5, when one of the robot spikes is in contact
with the ground surfaces, the forces obtained using the1872

B. Motor Equation
The robot adopts a DC motor as its internal actuator. The
centrifugal force produced by an eccentric motor can be
expressed as follows:



dθM 2


cos(θM + θ )FMx = me re
dt
2

(7)

dθM

FMy = me resin(θM + θ )dt(a) Tumbling

Moreover, we discuss the centrifugal torque derived fromthe DC motor. This robot system has a brushed DC motorthat rotates the eccentric masses. Generally, the equations of
a DC motor can be expressed as follows:(b) Hopping
Fig. 4.

dθM
d 2 θM
+ νM
+ Tr
dt 2
dt
dIM
dθM
VM = LM+ RM IM + KEdt
dt

Definition of Motion

TM = KT IM = JM

(8)
(9)

In the case that the motor does not have a gear head, νM
is ideally equal to 0. In addition, since the center of gravityof the eccentric masses is not collocated with the rotation
axis of the motor, the following disturbing torque affects the
motor.
Tr = −me re cos(θM + θ )Fig. 5.

However, in this case, Tr is virtually equal to 0 because
g is significantly smaller than 1 owing to the microgravity
environment.
Moreover, we assume VM ≈ RM IM + KE dθdtM in the subsequent analysis model since an electrical time constant ofa motor LM /RM is significantly smaller than a mechanical2
time constant. For these approximations, ddtθ2M and TM are

Reaction Forces

following equations act on the robot.
m

n
d2x
=
∑ fi
dt 2 i=1

(1)

n
d2y
m 2 = mg + ∑ Ni
dt
i=1

I

Variables
m
I
i
n
fi
Ni
TM
x
y
θ
ri
φi

(2)

n
n
d2θ
=
N
r
cos
φ
−
fi ri sin φ + TM
i
∑
∑
dt 2
i=1
i=1

(3)

Since the spikes are connected by linear springs anddampers, Ni and fi are described as follows:Ni =TM
cos φ + FKi sin φ
ri


TM


 r sin φ − FKi cos φifi = sgn(v )µN
xi
i



sgn(vxi )µd Ni

(10)

(4)

Name
Total mass of the robot
Moment of inertia of the robot
Spike indice
Number of spikes contacting surface
Frictional force acting on ith spike
Normal force acting on ith spikeCentrifugal torque of the motor
Horizontal position of the robot
Vertical position of the robot
Attitude of the robot
Rotational radius
Angle between ith spike

TABLE IVARIABLES IN E QUATIONS (1)–(3)(Static Friction)(Maximum Static Friction)(Dynamic Friction)(5)

FKi = k(h0 − h) − d

dh
dt

(6)

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Variables
FKi
vxi
µ
µd

Name
Elastic-damping force of the ith spike
Horizontal portion of translational velocity of the ith spike
Static friction coefficient
Dynamic friction coefficient
TABLE II
VARIABLES IN E QUATIONS (4)–(6)described as follows:d 2 θMdθM
KT=
VM − KE
dt 2
JM RM
dt


dθM
KT
VM − KE
TM =
RM
dt

(11)
(12)

TM is the exerted torque from the motor, hence the robot
receives TM as its reaction torque.

(a) TM = 0.005 N· m(b) TM = 0.006 N· m(c) TM = 0.007 N· m(d) TM = 0.008 N· m(e) TM = 0.009 N· m(f) TM = 0.01 N· m

V. N UMERICAL S IMULATIONIn this section, we discuss the numerical simulations
conducted to evaluate the characteristics of the tumbling andhopping motions. The dynamics model discussed in Section
IV is used in these simulations. Parameters of the robot usedin these simulations are listed in Table V. Here, rbody andlspike denote the radius of the body and the length of a spike,
respectively.
A. Tumbling Characteristics
1) Analysis of Translational Velocity: To begin with,
we analyze the translational velocity of the robot during
tumbling. We assume a flat surface and assign TM in therange between 0.01 N·m and 0.015 N·m until the end of thesimulation. In this simulation, we set g 2.5 × 10−3 m/s2 .
The x and y portions of translational velocity vx and vy
are shown in Fig. 6 (a) – (f). In these figures, the profilesof the velocity appear as almost two piecewise linear lines.
The slope of the linear line indicates the acceleration of
the robot. We assign α1 and α2 as the representatives ofthese acceleration, which α1 > α2 . When the spikes of
the robot receive static frictional forces, the acceleration isequal to α1 ; conversely, when the spikes receive dynamic
frictional forces, the acceleration of the robot is α2 . Hence,
the acceleration of the robot appears to experience one
significant discrete change, while tumbling and the friction
coefficient changes at this discrete change. As mentioned
in Section IV, the static and dynamic frictional forces
Variables
me
re
θM

Name
Eccentric mass
Effective radius between the rotational axis andthe center of gravity of an eccentric mass
Rotation angle of the motor
TABLE III
VARIABLES IN E QUATIONS (7)Variables
JM
νM
Tr
KT
IM
LM
RM
KE
VM

Fig. 6.

Time Profile of Tumbling Velocity

are TM /r sin φ − FKi cos φ (Static Friction) and TM /r cos φ +FKi sin φ , respectively. Generally, the static friction forces
are larger than the dynamic friction forces; therefore, α1
corresponds to the static friction forces and α2 corresponds
to the dynamic friction forces. The friction forces depend on
TM . Thus, the large torque TM enlarges α1 and α2 . However,
α1 and α2 are not proportional to TM since the energy that the
robot receives from TM are partially converted to the verticalcomponent of the robot’s momentum.
2) Climbing Ability: To evaluate the traversability on
uneven asteroid surfaces with the tumbling motion, analyzing
the ability to climb a step plays a key role. We assume
the initial state of the robot as one of its spikes is on the
ground and the other is about to locate on a step. Using thissimulation, we evaluate the ability of the robot to climb astep. The result of this simulation is represented in Table.
VI. In these simulations, we set g 2.5 × 10−3 m/s2 and TM
0.015 N·m until the end of simulation. However, g and TM
are irrelevant to the climbing ability. Even if g is large, theName
Sum of the moment of inertia including all eccentric masses
Motor’s damping coefficient
Disturbing torqueMotor’s torque constantMotor’s current
Inductance of the motor
Inner resistance of the motor
Inverse electromotive force constant
Input voltage
TABLE IV
VARIABLES IN E QUATIONS (8)–(9)Parameters
m
I
me
µ
µd
k
d
h0
rbody
lspikeValue
4.45 kg
0.0154 kgm2
0.0012 kg
2.25
1.8
15 N/m0.3 Ns/m
0.015 m0.10607 m
0.015 mTABLE V
PARAMETERS FOR N UMERICAL S IMULATION

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(a) Success Case (0.08 m)

Fig. 8.

(b) Failure Case (0.12 m)
Fig. 7.

Climbing Simulation

robot can climb by receiving the large TM . This result showsthat the threshold is located between the heights of 0.10 m
and 0.12 m. When the spike of the robot is on top of the step,then the robot has finally climbed the step. However, whenno spike is on the step, one of the spikes initially touches thelateral side of the step. This contact occurs when the robot
and its repulsive force are not into contacts with the step.
Therefore, the threshold can be determined by the sum of
the radius of the body and spike length of the robot because
this length determines whether the spike is on the step ornot.

spike must maintain the contact with the ground until φireaches 90◦p
. To meet this requirement, the natural frequency( fnatural = k/m) of the robot should be low. When thelength of the springs attached to the spikes exceeds theoriginal length, the springs start shrinking. This makes therobot to commence floating. Hence, m should be large toprevent the robot from floating. In particular, when g is
below 10−5 m/s2 order, the amplitude of vibration extends
compared to 10−3 m/s2 or larger case. Consequently, the importance of the heavy mass of the robot increases especiallyin a low gravity environment.
Second, we address the second condition. In this discussion, the robot must be under the first condition. If the spiketouching the ground slips before φi reaches 90◦ , the robot
starts floating in the air. Hence, the spike of the robot must
maintain the static friction until φi reaches 90◦ . To maintainthe static friction, the horizontal component of the robot’s
pushing force should satisfy the following equation.

B. Switch Tumbling and Hopping
Although the tumbling motion enables the robot to approach the desired locations accurately, some large obstacles
on asteroid surfaces can inhibit its movement owing to thelimitation of the climbing ability. Hence, the robot should
achieve the hopping motion to leap over obstacles, and wehave to assess the condition for switching the tumbling andhopping motion. Two conditions are considered for switchingthese modes.
1) The spike touching the ground must maintain contact
with the ground until φ exceeds 90◦ .
2) To maintain static friction, the torque TM should be
below µrFKi .
First, we deal with the first condition. If the spike ceases
a contact with the ground before φi reaches 90◦ , the robot
starts floating and stops the tumbling motion. Therefore, theAbility to
Climb

0.02

0.04

X

X

Height of Step [m]
0.06
0.08
0.10
X

X

X

TABLE VIR ESULTS OF C LIMBING S IMULATION

0.12

Conditions of Switching Conditions

fi < µNi
(13)


TM
TM
sin φ − FKi cos φ < µcos φ + FKi sin φ
(14)
i.e.
r
r
This equation claims that the horizontal component of its
pushing force should be below the maximum static friction
force. We can control the torque actively and it mostly
affects the horizontal component of its pushing force whenφ becomes 90◦ . Therefore, we can obtain the following
condition by substituting φ = 90◦ .
TM < µrFKi(15)

VI. E XPERIMENTS
To verify the dynamics modeling proposed in Section
IV, we conduct some experiments. The experiment environment and the test bed are shown in Fig. 9 (a) and (b),respectively. This test bed comprises electric circuits witha power battery, an air tank, a brushed DC motor with an
eccentric mass, air bearings attached to the bottom of thetest bed, and eight spikes made from acrylonitrile-butadienestyrene (ABS) polymer. Owing to the compressed air in
its air tank and the flowing of air from the bottom air
bearings, the test bed can float on the granite table. This
enables the robot to move in a two-dimensional emulated

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Fig. 10.

Sequence of Tumbling Locomotion

in experiments and numerical simulations. Moreover, Fig. 11(b) shows the x component of the translational velocity of therobot in experiments and numerical simulations. This figure
indicates the validity of the dynamics model mentioned in
Section IV. Moreover, the translational velocity of the robot
increases as the torque proliferates, and the friction status
affects the acceleration of the robot. Owing to the applying
time of 3 seconds, the velocity decreases suddenly approximately 4 or 5 seconds unlike the numerical simulations.
Contacts between a spike and the simulated surface cause
the sudden reduction since the contacts disperse the energyof the robot.

(a) Experiment Environment

B. Climbing Ability(b) Test Bed
Fig. 9. Overview of the Microgravity Simulating Experiment Environment
and Test Bed

microgravity environment. The DC motor is attached to thetest bed perpendicularly to the granite table. The weight ofthis test bed is 4.45 kg. The DC motor is controlled remotely
via XBEE, wireless communication modules. The rotational
speed of the motor is controlled by a proportional-integraldifferential (PID) controller that is implemented on Arduino,
a micro controller. The position of the test bed is recordedby an opti-track, motion capture cameras. The accelerationof gravity is generated by slightly tilting the granite table.
On the surface that the robot moves, sand paper on a steel
plate is attached. This sand paper contributes to simulating
the friction coefficient of uneven asteroid surfaces.
A. Tumbling Locomotion
First, we verify the feasibility of the tumbling locomotion.
Fig. 10 shows the tumbling locomotion. Fig. 10 indicatesthat the proposed method actually works in a microgravity
environment and its spikes can keep its contact with thesimulated surface while tumbling.
Subsequently, to confirm the dynamics model, we apply
the motor’s torque listed in Table. VII to the robot for 3s (owing to the specification of the motor) and analyze
the tumbling motion. In Table. VII, ftarget and ω̇ denotethe target motor frequency and the angular acceleration of
the motor, respectively. In this experiment, we arrange g
2.5 × 10−3 m/s2 . Fig. 11 (a) shows the x position of the robot

Finally, we verify the robot’s ability to climb a step. In
this experiment, we prepare two steps whose heights are 0.1
m and 0.12 m, respectively. Based on the limitation of the
motor, we apply TM = 0.015 N·m to the robot for 3 seconds.
g is 2.5 × 10−3 m/s2 .
To begin with, we mention the success case of climbinga step. In this case, the height of the step is 0.1 m, which isapproximately equal to the sum of the radius of the body andspike length. Fig. 12 (a) shows the sequence of climbing a
step. From these figures, it is obvious that the robot actuallyclimbs a step.
Conversely, in the failure case, the height of the step is0.12 m, which exceeds the sum of the radius of the bodyand spike length. Additionally, for the other case, Fig. 12(b) shows the sequence of climbing a step. From Fig. 12(b), a spike of the robot has a contact with the lateral surfaceof the step before the robot completes its climbing motion.
The reaction force derived from this contact works as therepulsive force of the robot; therefore, the robot cannot climb
the step.
Thus, we can verify that the climbing ability of the robot
depends on the sum of the radius of the body and spike

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#1
#2
#3

ftarget [Hz]
20
25
30

ω̇ [rad/s2 ]
41.9
52.4
62.8

TM [N·m]
0.01000.01250.0150TABLE VII
T HE T ORQUE OF THE M OTOR A SSIGNED FOR
A NALYZING T RANSLATIONAL V ELOCITY

(a) Success Case (0.10 m)(a) x Position(b) Failure Case (0.12 m)
Fig. 12.

Climbing Experiment(b) x Velocity
Fig. 11.

x Position and Velocity of the Robot while Tumbling

length; specifically, the robot is able to climb only such steps
whose height is below the sum.

C. Switch Tumbling and Hopping(a) TM = 0.0100 N · m: Tumbling

Based on the switching condition mentioned in Section
IV, we validate the switching conditions of tumbling andhopping modes by experiments. Considering the condition
TM < µrFKi , we assign the motor’s torque listed in Table. VIII
to the robot. At the calculation of TM , we use FKi computedfrom the numerical simulation. In this experiment, we modify
g 1.4 × 10−3 m/s2 .
The results of this simulation are described in Fig. 13(a) and (b). Each of these figures shows the sequences oftumbling and hopping motion of the robot. These figuresindicate that the conditions function.

(b) TM = 0.0125 N · m: Hopping
Fig. 13.

#1#2

ftarget [Hz]20
25ω̇ [rad/s2 ]
41.9
52.4

TM [N·m]
0.01000.0125TABLE VIIIT HE T ORQUE OF THE M OTOR A SSIGNED FOR
VALIDATION OF S WITCHING C ONDITIONS

Switching Tumbling and HoppingVII. C ONCLUSION
In this study, we presented the dynamics modeling andanalysis of tumbling locomotion by numerical simulations
and physical experiments. The model considers the motor’s
torque and the elastic damping forces derived from the1877spikes. Thus, the robot’s motion can be expressed mathematically. Moreover, we analyzed the characteristics of its
tumbling motion by numerical simulations. The translational
velocity during tumbling locomotion positively depends onthe motor’s torque, and the robot can climb obstacles whose
height is below its radius. In addition, we performed somephysical experiments to verify the validity of the dynamics model. These experiments helped validate the proposed
dynamics model, and we obtained a deep insight into thecharacteristics of tumbling locomotion.
For future study, the following two issues should be
addressed: One is the mechanical approach, which addresses
the effects of the shape of the spikes. The other is the theoretical approach, which deals with the controller to generate
the motion of the robot to access the desired locations.

[15] A. Parness, N. Abcouwer, C. Fuller, N. Wiltsie, J. Nash and B.
Kennedy, “LEMUR 3: A limbed climbing robot for extreme terrain
mobility in space,” in Proc. 2017 IEEE Int. Conf. Robot. Autom.,Singapore, 2017, pp. 5467-5473.

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