i-SAIRAS2020-Papers (2020)

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MOBILITY ANALYSIS OF A TUMBLING AND HOPPINGROVER WITH AN INNER ACTUATOR AND SEVERAL
ELASTIC SPIKES IN MICROGRAVITY*Keita Kobashi1 , Kenji Nagaoka2 , Kazuya Yoshida11Department of Aerospace Engineering, Tohoku University, Aoba 6-6-01, Aramaki, Aoba-ku, Sendai, Japan,
E-mail: {kobashi, yoshida}@astro.mech.tohoku.ac.jp
2
Department of Mechanical and Control Engineering, Kyushu Institute of Technology, Sensuicho 1-1, Tobata-ku,
Kitakyushu, Japan, E-mail: nagaoka.kenji572@mail.kyutech.jp

ABSTRACTThis paper presents the analyses of the tumbling andhopping mobility of a novel moving mechanism on
small celestial bodies in microgravity. The robot consists of an inner motor with a flywheel and eight elastic spikes connected to the perimeter of the robot.
The tumbling and hopping motion of the robot canbe switched by controlling the torque of the motor.
Hence, the robot can traverse a large region with highmoving accuracy. In this paper, we conduct several
numerical simulations to analyze the characteristicsof the mobility by assigning various values of elastic
and damping coeﬃcient of spikes, and the torque ofthe motor. The results are useful to construct thefeasible motion planning for real missions.

1

INTRODUCTION

In recent years, asteroid exploration has attracted scientists due to the wealth of clues which reveal the origin of our solar system asteroids are expected to have.
For the detailed investigation of asteroids, utilization of locomotive robots is an eﬀective exploration
method. The gravitational acceleration of asteroid
surfaces is estimated to be between 1.0 × 10−5 m/s2
and 1.0 × 10−2 m/s2 . Several locomotion mechanisms
have been invented so far, however, due to the microgravity environment, the candidates of the feasible
mechanisms for locomotion on asteroid surfaces arelimited. For instance, wheeled locomotion, which is
the most prevailing mechanism for space exploration,is less likely to achieve suﬃcient locomotion [1] because this mechanism utilizes the frictional forces between a robot and the surfaces and the magnitudeof the frictional forces depends on the acceleration ofgravity. Moreover, robots adopting the wheeled mobility are prone to experience unintentional detachment from the surfaces [2].
To cope with the technical challenges involved inwheeled mobility, hopping [3], [4], [5], ciliary vibra-

Figure 1 : 3D image of the proposed systemtion [6], and limbed locomotion [7] were considered.
To begin with, we mention the hopping mobility.
Inner-actuated hopping mobility is a promising
method to move on asteroid surfaces because it realizes a simple mechanical system and the capability totraverse uneven surfaces. This mobility is induced byinternal impulsive forces derived from motors withflywheels [8], the deformation of a bimetal exposed
to a temperature gradient [9], and the elastic energy
stored in internal buckling springs [10]. However,robots utilizing this mobility rebound repeatedly after hopping. For this reason, it is diﬃcult to achievea desired initial velocity and realize the soft landing.
Ciliary vibration locomotion generated by an internal motor with eccentric masses [11] also has asimple mechanical system and this mobility enables
robots to move accurately. Although this mechanism
enables accurate locomotion, the motion is limited to
crawling and consequently has great diﬃculty traversing uneven surfaces.
Limbed locomotion [7] demonstrates its high capability to traverse uneven surfaces by grasping boulders. However, to realize this locomotion, robots require several legs with some joints and this leads toincreasing the cost for development and the failurerisks for operation. Moreover, to utilize such complex
limbed systems, intricate gait control is required.
The tumbling locomotion displayed in Fig. 1 ex-

i-SAIRAS2020-Papers (2020)

hibits a simple mechanical system with several elastic spikes through linear springs and dampers andan inner actuator. This locomotion is able to traverse uneven surfaces by utilizing the elastic damping
forces derived from the deformation of its spikes andthis contributes to increasing the predictability of themoving direction. Moreover, this simple mechanicalsystem also realizes hopping mobility by changing themagnitude of the motor’s torque. Therefore, by applying the hopping mobility when obstacles are toolarge to traverse by the tumbling motion, the robot
can move across a vast scope of asteroid surfaces withhigh moving accuracy. For these reasons, analyses of
the characteristics of tumbling and hopping motion
are quite important for the application of this mechanical system to real exploration missions.
In this paper, we evaluate the characteristics ofthe tumbling and hopping motion by 2-dimensional
numerical simulations. In chapter 2, the locomotion
mechanism and the dynamics model of the robot ismentioned. In chapter 3, the capability of the proposed system to realize moving modes is addressed.
In chapter 4, the analyses of hopping mobility are
conducted and the characteristics of the hopping mobility is discussed. In chapter 5, the analyses of tumbling mobility are conducted and the characteristicsof the tumbling mobility is discussed. The dynamics
model mentioned and utilized in this study has beenvalidated by my previous work [12].

2
2.1

LOCOMOTION MECHANISM
AND DYNAMICS MODELING
Locomotion Mechanism

In this section, we address the locomotion mechanismof the tumbling motion. Fig. 2 (a) displays the 2dimensional schematic figure of the proposed mechanical system. As shown in Fig. 2 (a), the robot has an
inner motor with a flywheel at the geometric centerand eight elastic spikes around the perimeter of its
body. When the internal motor starts rotating, thereaction torque is induced and this leads the robot
to rotate. After rotating, the robot utilizes the elastic and damping forces derived from the deformationof the spikes to propel and to continue its tumbling
motion. The feasibility of this mechanism has been
demonstrated by physical experiments [12].

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equations of motion can be described as follows:mnd2 x ∑
=
fi
dt2
i=1

(1)

m

n
∑
d2 y
=
−mg
+
Ni
dt2
i=1

(2)

I

n
n
∑
d2 θ ∑
=
N
r
cos
ϕ
−
fi ri sin ϕi + TM
i i
i
dt2
i=1
i=1

(3)

The dynamical interaction between the body and the
spikes is modeled by elastic and damping forces since
the spikes are connected through linear springs anddampers.
Ni =Table 1 : Variables in Equations
Variables
m
x
y
n
i
fi
g
Ni
I
θ
ri
ϕi
TM
vxi
µ
µd
FKi
K
h0
h
D
θhop
vxhop

2.2

Dynamics ModelingIn this study, we consider the 2-dimensional dynamics
of the proposed mechanical system. Fig. 3 displays
the forces acting on the robot. From Fig. 3, theTM
cos ϕi + FKi sin ϕi
ri

vyhop
F
∆t

Name
Total mass of the robot
Horizontal position of the robot
Vertical position of the robot
Number of spikes
in contact with the groundIndex of spikes
Frictional forces
acting on the i th spike
Acceleration of gravity
Normal forces
acting on the i th spike
Moment of inertia of the robot
Attitude of the robot
Rotational radius between
the body and the spike
Angle between the i th spikeand the groundCentrifugal torque of the motor
Horizontal portion of translational
velocity of the i th spike
Static friction coeﬃcient
Dynamic friction coeﬃcient
Elastic and damping forceof the i th spike
Elastic coeﬃcient of a linear spring
Initial length of a linear spring
Length of a linear spring
Damping coeﬃcient of a damper
Hop angle of the robot
Horizontal portion of
hop velocity of the robot
Vertical portion of
hop velocity of the robot
Sum of the reaction forces
Period of maintaining contact
with the ground(4)

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5083.pdf(a) Mechanical system and moving mechanism

Figure 3 : Dynamics model(b) Validation of the feasibility(a) Tumbling

Figure 2 : Notion and feasibiity of the mechanical system and the moving mechanismTM(Static Friction)
 r sin ϕi − FKi cos ϕiifi = −sgn(v )µN(Maximum Static Friction)
xi
i



−sgn(vxi )µd Ni(Dynamic Friction)(5)
FKi = K(h0 − h) − D

dh
dt

(6)

Besides, in the context of evaluating the hopping
mobility, the hop angle θhop is an important parameter. In this paper, we define θhop as the following
equation.
θhop = tan−1

vxhop
vyhop

(7)

As described in Eq. 7, the hop angle is the ratio of
the horizontal and vertical portion of the translational
velocity of the robot at the time of detaching fromthe surface. We define that the initial translational
velocity at the time of detaching from the surface is
hop velocity.
Here, the variables used in these equations arelisted in Tab. 1.

3

MOVING MODES

In this section, we mention the three moving modes
with their definitions.

(b) Hopping

Figure 4 : Definition of motion
1. Tumbling: As shown in Fig, 4 (a), the spikes of
the robot contact the ground, sequentially. Periods of no contact with the ground are possible;
however, less than a revolution in the air is required.
2. Hopping: As shown in Fig. 4 (b), contact is
not maintained with the ground. The spikes of
the robot are not required to contact the ground
sequentially. A single revolution or more is necessary while floating in the air.
An example of a motion sequence of tumbling andhopping is shown in Fig. 5. We apply the torque tothe robot for 0.3 seconds. Besides, the time profiles ofthe position of the geometric center of the robot whiletumbling and hopping are shown in Fig. 6. Fig. 6
indicates the capability that the proposed mechanicalsystem realizes the tumbling and hopping motion ina microgravity environment by changing the magnitude of the torque. If the acceleration of gravity is10−6 m/s2 or less, the robot tends to realize the hopping motion. The reason is that the pushing force ofthe robot becomes small in the case of desiring the
tumbling locomotion and the center of the vibration

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2.5 × 10−3 m/s2 environment.

4.1(a) Tumbling

Analyses of Hop Angle

In this section, we assess the hopping mobility of therobot by focusing on the hop angle mentioned in Eq.
7. To begin with, we assign several elastic and coeﬃcient of a linear spring K and a damper D, andobserve how the hopping behavior changes. The parameters used in this simulation are listed in Tab. 3.
In this case, the hop angle does not change significantly. In the case of the hopping motion, the robot
receives a large torque, hence, the period in contact
with the ground is quite short. For this reason, theelastic and damping forces do not significantly aﬀect
the impulse the robot receives before the detachmentfrom the ground.
Moreover, we consider enlarging(b) Hopping

Figure 5 : An example of a motion sequence of tumbling and hopping

Table 2 : Parameters of the robot
Variables [Unit]m [kg]
g [m/s2 ]I [kgm2 ]µµd
h0 [m]
rbody [m]lspike [m]Figure 6 : Time profile of the postion of the geometriccenter while tumbling and hoppingof a spike in contact with the ground shifts toward the
body of the robot and the length of the spike tends toexceed its natural length. Thus, the spike can easily
detach from the ground and finally, the robot starts
floating. For these reasons, the hopping motion is
likely to emerge in such environments.

4

SIMULATION ANALYSES OF
HOPPING MOBILITY

We discuss the hopping motion of the proposed mechanical system. Several numerical analyses are conducted to evaluate the characteristics of the hopping
motion. At first, the parameter of the robot is listed
in Tab. 2
In this section, all simulations are executed in g =

Value
4.45
2.5 × 10−30.0154
2.25
1.8
0.015
0.10607
0.015

Table 3 : Simulation conditions for analyzing hop angle against K and D
Variables [Unit]
TM [Nm]K [N/m]D [Ns/m]

Value
0.20
15 – 150
0.30 – 2.7

Table 4 : Simulation conditions for enlarging θhop
Variables [Unit]
TMsmall [Nm]TMlarge [Nm]K [N/m]D [Ns/m]tthreshold [s]Value
0.00120.20
150.30
1.6 – 2.8

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Hop Velocity, v [m/s]0.2vX
Fitting Curve of vX
vY
Fitting Curve of vY

0.15vX = 0.1354(TM)0.5

0.1

vY = 0.0751(TM)0.5
0.05

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Motor Torque, TM [Nm]Figure 7 : Hop angle against tthreshold

Figure 8 : Hop velocity against TM

the hop angle by utilizing the elastic and damping
forces derived from its spikes. By utilizing the forces,
when the torque of the motor is small, the robot cangain its horizontal portion of the translational velocity while holding the static friction of its spike in contact with the ground. We conduct some simulations
to analyze the hop angle in this case by applying asmall torque to the robot for a certain period (threshold time tthreshold ) and then applying a large enoughtorque to hop. The torque and threshold time usedin this simulation are listed in Tab. 4.
Fig. 7 shows the result of these simulations andplots the hop angle against the threshold time. In
Fig. 7, the longer threshold time enlarges the hop
angle. When the threshold time is long, the spike
can maintain static friction with the ground longer,
thus, the robot can obtain a relatively large horizontal
component of the translational velocity by the timeof transferring to the hopping. If the robot obtains
a large horizontal component of the translational velocity, the hop angle (θhop = tan−1 vxhop /vyhop ) increases and the robot can traverse a large distance ateach hop.

duced, therefore, a spike in contact with the groundslips on the ground and the large torque generates
much larger pushing forces than the elastic and damping forces generated by the robot’s spikes. Besides,in this case, the elastic and damping forces are irrelevant to the torque if K and D do not change because
the shrinkage of the spike depends on the attitude of
the robot before detaching from the ground. Moreover, in the case that the robot receives a large torqueand hops, the robot rotates pretty fast, therefore, theperiod of maintaining the contact with the groundbecomes too short. For these reasons, in Fig. 8, theelastic and damping forces induced by the external
compliances almost do not aﬀect the impulse which
the robot receives until starting the hopping motion
and vx and vy can be aﬀected by the magnitude of thetorque. Hence, the impulse which the robot receives
until the detachment is approximated as follows:

4.2

F ∝ TM
√
1
∆t ∝
TM
√
F ∆t ∝ TM

Analyses of Hop Velocity

In this section, we evaluate the hop velocity (vxhop ,vyhop ) of the robot. We assign several values of thetorque to the robot over each simulation and observe
the behavior of the robot. The conditions of this simulation are listed in Tab. 5. Fig. 8 displays the hop
velocity in x and y directions against the torque. In
the case of hopping, the relatively large torque is inTable 5 : Simulation conditions for analyzing hop velocity
Variables [Unit]
TM [Nm]K [N/m]D [Ns/m]Value
0.20 – 1.0
15
0.30

(8)
(9)
(10)

From these equations, the hop velocity is approximately proportional to the square root of TM . In thiscase, the characteristics of the hop velocity is almost
the same as hopping rovers which do not have external compliances (e.g. MINERVA [3]).

5ANALYSES OF TUMBLING
MOBILITY

We address the tumbling mobility of the proposedmechanical system in this section. Several numericalanalyses are conducted to evaluate the characteristics of the tumbling motion. In this simulation, thetorque of the motor is applied to the robot over thesimulation. Therefore, the proliferation rate of the

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5083.pdf(a) K = 15 N/m(b) K = 30 N/m(a) T = 0.005 Nm(b) T = 0.006 Nm(c) K = 45 N/m(d) K = 60 N/m(c) T = 0.007 Nm(d) T = 0.008 Nm(e) T = 0.009 Nm(f) T = 0.01 Nm

Figure 9 : Relationship between K and Translational
Velocity while Tumbling(a) D = 0.3 Ns/m(c) D = 1.5 Ns/m(b) D = 0.9 Ns/m(d) D = 2.1 Ns/m

Figure 10 : Relationship between D and Translational
Velocity while Tumblingtranslational velocity changes as the spikes’ state offriction transfers from static to dynamic. The parameter of the robot is listed in the same Tab. 2.
In this section, we analyze the translational velocity of the robot during tumbling locomotion. The
torque of the motor is applied to the robot over thesesimulations. At first, we assign several elastic coeﬃcient K to the spikes and mention the variationof the translational velocity during tumbling locomotion. The conditions for this simulation are listed
in Tab. 6. Figure 9 plots the translational velocity
against the time at each case. The x portion of theTable 6 : Simulation conditions for analyzing translational velocity assigning various K
Variables [Unit]
TM [Nm]K [N/m]D [Ns/m]Value
0.005015 – 60
0.30

Figure 11 : Relationship between Torque and Translational Velocity while Tumblingtranslational velocity does not tend to increase with
the increment of K. If K is large, it becomes diﬃcult
for the robot to maintain its contact with the grounddue to the increment of the natural frequency. Moreover, the period of floating becomes longer. Therefore, the x portion of the translational velocity does
not increase well when K is large.
Besides, we address the case that the various
damping coeﬃcient D are assigned to the spikes. The
conditions for this simulation are listed in Tab. 7.
Fig. 10 displays the translational velocity against thetime at each case. In this case, the translational velocity decreases since the spikes can easily dissipate
its reaction forces for propulsion. Finally, we menTable 7 : Simulation conditions for analyzing translational velocity assigning various D
Variables [Unit]
TM [Nm]K [N/m]D [Ns/m]Value
0.005015
0.30 – 2.1

Table 8 : Simulation conditions for analyzing translational velocity assigning various TM
Variables [Unit]
TM [Nm]K [N/m]D [Ns/m]Value
0.0050 – 0.0010
15
0.30

i-SAIRAS2020-Papers (2020)tion the case that we assign several torques TM to therobot. The conditions for this simulation are listed in
Tab. 8. Fig. 11 indicates that the translational velocity against the time at each case. In this case, thex component of the translational velocity increases
because as described in Eq. 4, the reaction forces of
the robot can be aﬀected by TM .

6

CONCLUSION

In this paper, we evaluated the characteristics of the
tumbling and the hopping mobility by numerical simulations. The hop angle is not aﬀected by the elastic
and damping coeﬃcient of the spikes and by utilizing
the elastic and damping forces of the spikes, we canenlarge the hop angle. The torque of the motor enlarges the hop velocity and the hop velocity is almost
proportional to the square root of the torque. On theother hand, the tumbling motion is aﬀected by theelastic coeﬃcient of the spikes and a large elastic coeﬃcient K leads the robot to float. The large torqueTM can also increase the translational velocity during
tumbling locomotion.

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