Destination Accessibility of a Hopping Robot on Small Celestial Bodies in
Microgravity*Keita Kobashi1 , Toshiyasu Kaneko1 , Kenji Nagaoka1 , and Kazuya Yoshida1
1
Tohoku University, Aoba 6-6-01, Aramaki, Aoba-ku, Sendai, Japan
E-mail: {kobashi, kaneko, nagaoka, yoshida}@astro.mech.tohoku.ac.jp

ABSTRACT
Hopping mobility is effective for the use of an exploration robot on small celestial bodies such as asteroids andcomets. However, the rebounding dynamics in such a caseis influenced by non-uniform gravity and the surface shapeof the body. Therefore, it is challenging for the robot toaccess its destination. This paper presents various movements of robots using hopping mobility. Moreover, thispaper also presents the destination accessibility of suchrobots. In order to analyze the various movements of therobots, we focused on the hopping and rebounding motionof the robot. First, we modeled the hopping motion andrebounding motion, respectively, using the contact force
model and linear springs and dampers. Second, we verified the hopping model through experiments in a 2D microgravity environment. Finally, we performed a dynamic
simulation of the rebounding. The results showed that thecharacteristics of the rebounding motion are related to the
hopping motion and that the characteristics can be significantly adjusted by altering the rebounding attitude andangular velocity of the robot. Based on this result, thedestination accessibility is evaluated by using the following four characteristics, (1) moving accuracy, (2) travelingtime, (3) energy efficiency, and (4) ability to leap over obstacles. This paper presents the conditions of hopping andrebounding motion that these four evaluation elements ofdestination accessibility emerge noticeably.

1 INTRODUCTION
There are several small bodies in our solar system. Smallbodies such as asteroids and comets contain information
regarding the origin of the solar system. However, the detailed composition of asteroids is unknown. Therefore, theexploration of asteroids for clarifying their composition isof great importance for revealing information regarding
the origin of the solar system. Increasing attention has
been focused on the exploration of small celestial bodies
in recent years.
In 2010, Hayabusa [1, 2], an asteroid explorer developed by The Japan Aerospace Exploration Agency(JAXA), succeeded in its mission of collecting a sample
from an asteroid called ”Itokawa”. Asteroids can be classified into several types based on their emission spectrum

and their orbit, and Itokawa is classified as an S-type asteroid. This mission revealed the composition of S-type asteroids. To obtain more information regarding asteroids,
asteroid-surface exploration is required. In general, exploration robots that can traverse an asteroid surface playa crucial role in asteroid-surface exploration. In the caseof Mars exploration, the dynamic frictional force acting
on contact points of exploration robots is sufficiently large
owing to the existence of higher gravity. Hence, the utilization of friction as the driving force, such as in thecase of wheel mobility, is an appropriate method of facilitating movement. However, an asteroid-surface environment is very different from that of Mars. The characteristics of the environment include the existence of microgravity (10−6 ∼10−4 G) and uneven surfaces. On asteroid surfaces, if exploration robots do not actively push
the surface, the frictional force acting on the robots isvery small due to the microgravity environment of asteroids. Therefore, it is necessary for asteroid-exploration
robots to use appropriate moving methods. Some examples of exploring robots include MINERVA [3], which
was developed by JAXA, MASCOT [4] developed by European Space Agency (ESA) and German Aerospace Center (DLR) and Hedgehog [5] developed by Stanford University and the Jet Propulsion Laboratory at the National
Aeronautics and Space Administration (NASA/JPL). All
the aforementioned examples adopted the use of hopping
mobility. Robots that use hopping mobility actively push
the surface of the ground. Therefore, even if the robots
are in a microgravity environment, a large frictional force
acts on the robots. Therefore, hopping movement is anappropriate moving method for this application. These
three robots have similar hopping mechanisms albeit withminor differences The hopping mobility allows for a highperformance in leaping over obstacles and high travelingtime efficiency. Moreover, the three robots have the advantage of a simplified structure in which their actuator is
attached inside their body. However, the implementationof a soft-landing technique for such a robot is still challenging, and thus, the robot repeatedly rebounds on an asteroid surface until it comes to a stop. Therefore, factors
affecting destination accessibility, such as the locomotion
accuracy of the hopping robot involves uncertainties.
In this study, an analysis of the hopping and rebound-• Case2: The contact point slips

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
















Figure 1 : Process of Hopping and Rebounding
ing characteristics of a hopping robot such as MINERVA
that is equipped with an internal torquer is performed using numerical simulations. Section 2 presents the dynamics models of hopping and rebounding. Section 3 presents
the experiments for verifying the accuracy of these dynamics models. Section 4 presents the numerical simulations of the rebounding motion and the argument of thedestination accessibility. In this study, as part of the destination accessibility analysis, the effects of the attitude andangular velocity of the robot at collision with the asteroid
surface on the locomotion accuracy, traveling time, energy
efficiency, and characteristics of leaping over obstacles areclarified.

2 DYNAMICS MODELINGThe following two main dynamics models are used for
simulating the motion of the robot: hopping and rebounding dynamics models. In this research, hopping means
the torque applied by the robot torquer actively pushes theground and the robot is pushed up by the reaction force.
In contrast, rebounding means that the robot is pushed up
owing to a collision between the robot and the ground after hopping. The process of hopping and rebounding isshown in Fig. 1.

2.1 Hopping
The dynamics model of the hopping robot is shown in
Fig. 2. The robot can hop by using the reaction torque ofthe internal torquer that is attached to the center of gravity(CoG) of the robot. This research considers the case of aflat surface and a cubic robot. The robot applies a pushingforce in the direction perpendicular to the line A shown in
Fig. 2. The equation of the hopping motion of the robot isgiven as follows:• Case1: The contact point does not slip
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
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



I

m

d2 x T= cosβ
r
dt2

d2 x
T
= µd sinβ
2
r
dt

m

d2 y T= sinβ
r
dt2

(
d2 α
π) T
× sin β= − T + r cos α +
2
4
r
dt
(
π) T
+ µd r sin α +× sin β
4
r

(1)

(2)

(3)

(4)

where m is the mass of the robot, I is the moment of inertia, g is the gravitational acceleration, µd is the dynamic
friction coefficient, r is the length from the robot CoG
to the contact point, T is the reaction torque exerted by
the torquer, α is the attitude of the robot, and β is the angle between the pushing force vector of the robot and theground. If the static friction coefficient µ = ∞, i.e., theedge of the robot comes into contact with the ground anddoes not slip, the equation of hopping motion can be described using Eq. (1). In contrast, if µ , ∞, the equationof hopping motion can be described using Eq. (1) or Eq.
(3). In particular, in the case of µ < 1, the equation canbe described by Eq. (3). This paper targets only the caseof µ < 1. Hence, the hopping angle θ can be expressed asfollows:vx(5)
θ = tan−1 = tan−1 µdvyMoreover, from Eq. (3), it is clear that the contact pointof the robot always slips while the torquer of the robot applies a torque. It is also clear that the pushing force of therobot becomes 0 when T = 0 or sin β = 0. Therefore, thecondition for which hopping is realized is either the torqueis cancelled or α = 45◦ . Here, a relationship between the
angular acceleration and the torque obtained from Eq. (4)can be expressed as follows:
d2 α
∝T
(6)
dt2
Therefore, the time for hopping has the following relationship.
1
t∝ √
T

d2 y Tm 2 = sinβ
r
dt

(
π) T
d2 α
=
−
T
+
r
cos
α
+
× sin β
4
r
dt2
)
(
T
π
× cos β+ r sin α +
4
r

I

m

(7)

Moreover, the reaction force experienced by the robot during hopping is proportional to T from Eq. 3. Thus, thetranslational impulse experienced by the robot during hopping can be expressed as follows.
√F x ∆t ∝ T
(8)
√
(9)
Fy ∆t ∝ T

Figure 2 : Dynamics Model of Hopping
Therefore, the initial velocity can be expressed as follows.
√vx ∝ T(10)√vy ∝ T(11)

2.2 Rebounding
The ground is modeled as comprising linear springs anddampers as shown in Fig. 3. In order to simulate the characteristics of the ground, the elastic constants used arekx = 5000 [N/m] and ky = 10000 [N/m] and the viscosity constants are dx = 5 [Ns/m] and dy = 10 [Ns/m]. ∆xand ∆y are the virtual penetration or displacement of thecontact point along the x and y axis, respectively. The
horizontal and normal contact forces Nx and Ny can be expressed as follows:
k1 ∆x + d1 v x , if v x = 0
Nx = 
(12) µd N y ,otherwise
Ny = k2 ∆y + d2 vy(13)

Figure 3 : Dynamics Model of Rebounding
2D. The users of this environment can change the acceleration of gravity by tilting the stone plate. The verticaland horizontal dimensions testbed were 15 times 15 cm,
and its weight was 4.3 kg. The edges of this testbed were
fabricated using acrylonitrile butadiene styrene copolymer(ABS) in a 3D printer. The testbed also had a microcontroller Arduino, which is connected operator’s PC byusing XBee. Using this connecting system, the operators
can control the rotational frequency of the motor. The position of the testbed is recorded using a motion tracking
system.

3.2 Hopping Experiment
The hopping experiment was executed using the aforementioned environment. The process of this experimentis as follows:
1. Setting the attitude of the testbed α to 0 and being
settle on the simulated surface
2. The motor in the testbed is rotated for a certain
amount of time.
3. The testbed hops.

3 EXPERIMENTS
In this section, some experiments conducted for the verification of the dynamics models are presented.

3.1 Environment of Experiment
The environment considered for the experiment in orderto verify the validity of the dynamics model of hopping isshown in Fig. 4. Moreover, the testbed is shown in Fig. 5.
The testbed was fitted with a motor, air tanks, and air bearings under its body. The air tanks contained compressed
air, and the robot blows air through its air bearings. Using these instruments, the robot can float against the stone
plate and can move in the microgravity environment in

The conditions of the hopping experiment are as follows:• The acceleration of gravity is 5.0 × 10−4 G .
• The time of applied torque is 0.3 s (limit of the motor).
• The materials used for contact point of the robot are
ABS and rubber.
• The frequency of rotation of the motor is 30 Hz.
In the case of the hopping motion, the appearance ofthe contact point comparing two different types of materials is shown in Figs. 7 and 8 (t = 0 is the time at which thetorque is applied). These figures ilustrate the view from

Figure 4 : Environment of Experiment
Figure 6 : Recording Position(a) 0 s(b) 0.1 s(c) 0.2 s(d) 0.3 s(e) 0.4 s(f) 0.5 s

Figure 5 : Air-floating Testbed
above the testbed, and the schematic is shown in Fig. 6.
Figures 7 and 8 illustrate the cases of ABS and rubber,respectively. Even if the material changes, i.e., the friction coefficient changes, the edge of the testbed always
slips between the time of torque application and that of
hopping. In general, the friction coefficient of rubber is
1 or less. Thus, this result is suitable for the dynamics
model. The attitude of hopping is almost 40◦ and 30◦ , respectively, when the material is ABS and rubber. This isbecause the CoG of the testbed is not located at its geometric center, and the motor is not located at the CoG;
moreover, the applying time of torque reaches 0.3 s before the testbed hops.
The position of the geometric center of the testbed
when the frequency of rotation is 30 Hz and the material
is ABS is shown in Fig. 9. The position of the geometriccenter of the testbed when the frequency of rotation is 30
Hz and the material is rubber is shown in Fig. 10. Here,the position of the geometric center is almost that sameas that of the CoG. From these figures, the initial velocity of the testbed during hopping is derived. As shown in
Fig. 9, the translational velocities (v x , vy ) are 0.013851
m/s and 0.060575 m/s, respectively. As shown in Fig. 10,
the translational velocities are 0.036832 m/s and 0.051758
m/s, respectively. Therefore, the ratio of the translational
velocities v x /vy is 0.23 and 0.714 for ABS and rubber, respectively. Here, on considering the dynamic friction coefficient of ABS and rubber as 0.2∼0.3 and 0.6∼0.7, respectively, the resultant initial velocity is suitable for the
dynamic model.

Figure 7 : State of Contact Point During Hopping (Material of Contact Point: ABS)

4

SIMULATION ANALYSIS

4.1 Rebounding Simulation
In this study, the dynamics calculation was performed using MATLAB, and the simulation results are visualizedusing Gazebo to obtain more reliable simulation results.

4.1.1 Effect of attitude angle at collision
At first, an effect of the attitude angle α of the robot when
focusing on the translational velocity after the rebound
was analyzed. The analysis conditions are as follows:• α = 10, 20, 30, and 40 [◦]• µd = 0.1, 0.2, · · · , 0.8, and 0.9 [-]• vx /vy = 0.1, 0.2, · · · , 0.8, and 0.9 [-]This simulation was performed with a total of 324 conditions. The image of this simulation is shown in Fig. 11.

(a) 0 s(b) 0.1 s

Position of the geometric center [mm]

90
X
Y

80

70

60

50

40

30

20
0

0.5

1

1.5

Time [s](c) 0.2 s(d) 0.3 s(e) 0.4 s(f) 0.5 s

Figure 10 : Position of Geometric Center of Testbed (30Hz, Rubber)

Figure 8 : State of Contact Point During Hopping (Material of Contact Point: Rubber)
Figure 11 : Image of Collision without Angular Velocity
If 0◦ ≦ α ≦ 45◦ ,Position of the geometric center [mm]

100
X
Y

90

(
(
π)
π)
M = −N x r cos α ++ Ny r sin α +
4
4

80

(14)

70

If 45◦ ≦ α ≦ 90◦ ,
60

(
(
π)
π)
− Ny r sin α +M = −N x r cos α +
4
4

50

40

30
0

0.5

1

1.5

Time [s]Figure 9 : Position of Geometric Center of Testbed (30 Hz,
ABS)

(15)

4.1.2 Effect of rotational velocity at collision
Next, the effect of a constant ω on the translation speed
after the collision was analyzed. The analysis conditions
are as follows:• ω = -1.0, -1.1, · · · , -1.8, and -1.9 [rad/s]

The resulting relationship between the ratio of thehorizontal velocity before and after the collision and µd
is shown in Figs. 12, 13, and 14, where the velocity ratios
are v x /vy = 0.1, v x /vy = 0.5, and v x /vy = 0.9 respectively.
The ratio in these figures are negative because of a calculation error in this simulation. This result confirms thatthe rate of decrease of the post-collision velocity becomes
smaller with an increase in α. This is because the robot’s
angular velocity ω after the collision decreases in proportion to the increase in the moment M of the frictional force
as α approaches 40◦ . Therefore, the translation momentum after the collision increases with an increase in α,
where the given impulse is constant without a collision
attitude change. This characteristic was observed for thevarious values of vx /vy . Here, M is given by the following
equations.

• α = 10, 20, 30, and 40 [◦]• vx /vy = 0.3, 0.5, and 0.7 [-]This simulation was performed with a total of 120 conditions. The image of this simulation is shown in Fig. 15.
The resulting relationship between the vertical velocity ratio before and after the collision and v x /vy are shownin Figs. 17, 18, and 19, where ω is -1.3 rad/s, -1.6 rad/s,
and -1.9 rad/s, respectively. From this result, it is foundthat the vertical velocity ratio becomes larger than 1 in thecase that α is 30◦ or 40◦ at the collision. Here, the normal
force Ny can be expressed as follows:{}
Ny = k2 ∆y + d2 vy + rω sin (β){(π)}= k2 ∆y + d2 vy + rω sin
−α
4

(16)

Ratio of velocity before and after collision

0

-2

-4

-6

-8

α = 10 deg
α = 20 deg
α = 30 deg
α = 40 deg

-10
0.2
0.4
0.6
0.8
Dynamic friction coefficient of the ground,µdFigure 12 : Ratio of Horizontal Velocity Before and After
Collision (Without Angular Velocity, v x /vy = 0.1)

Figure 14 : Ratio of Horizontal Velocity Before and After
Collision (Without Angular Velocity, v x /vy = 0.9)

Ratio of velocity before and after collision

1

0.5

0

Figure 15 : Image of Collision with Angular Velocity
-0.5

-1

α = 10 deg
α = 20 deg
α = 30 deg
α = 40 deg

4.2.1 Moving mode (i)

-1.5
0.2
0.4
0.6
0.8
Dynamic friction coefficient of the ground,µdFigure 13 : Ratio of Horizontal Velocity Before and After
Collision (Without Angular Velocity, v x /vy = 0.5)
As shown in Eq. (16), Ny reaches a maximum value when
α is almost 0◦ . In addition, as shown in Eq. (15), M
reaches a minimum value when α is 45◦ . Therefore, inorder to maximize the translational velocity after the collision, it is necessary to increase Ny and reduce M. In thiscase, the most appropriate value of α is within 30◦ ∼40◦ .
This appropriate range also does not depend on ω.

4.2 Destination Accessibility
The vertical and horizontal impulse, Fy ∆t and Fx ∆t, experienced by the robot at the collision are mvy ≦ Fy ∆t ≦2mvy and −2µd mvy ≦ Fx ∆t ≦ −µd mvy in accordancewith the linear spring–damper model. The horizontal impulse during hopping is µd mvy , and thus, the velocity of
the contact point becomes 0 when the robot collides withthe ground surface. The image of this phenomenon isshown in Fig. 16. The experiment for verifying this phenomenon is conducted with the environment shown in Fig.
4. The experiment results confirmed that the aforementioned phenomenon occurred. In addition, the testbed rotated because the testbed experienced the negative torqueof the dynamic friction force at the collision.

In the case of moving mode (i), it is assumed that α canbe maintained at 0◦ at the collision through active controlof the torquer before the collision. By adopting this mode,the horizontal velocity of the robot s CoG after the collision is almost 0. Consequently, the robot can accurately
reach and stay near a destination point in a single hop (seeFig. 20(a)). In contrast, if the distance between the current
position of the robot and the destination is very large, thehopping velocity of the singe hop locomotion could exceed the escape velocity for the target body. Therefore, therobot should use multiple short-distance hops in a practical situation. Furthermore, moving mode (i) is inefficientfrom the viewpoint of energy consumption.

4.2.2 Moving mode (ii)In the case of moving mode (ii), it is assumed that α isapproximately 40◦ at the collision. In this mode, the horizontal velocity of the robot decays in small amounts owing to the collision. Thus, the robot can move more efficiently and faster than that using mode (i) (see Fig. 20(b)).

4.2.3 Moving mode (iii)
In the case of moving mode (iii), it is assumed that the rotating robot collides with the ground surface. In this mode,
the robot can possess a larger vertical velocity owing to
the collision than that before the collision. This suggeststhat the robot can leap over obstacles more efficiently byactively utilizing the rebound (see Fig. 20(c)).

Ratio of velocity before and after collision

1.3

Figure 16 : Image of Collision at 0◦
Ratio of velocity before and after collision

1.3

α = 10 deg
α = 20 deg
α = 30 deg
α = 40 deg

1.2
1.1

1.2
1.1

α = 10 deg
α = 20 deg
α = 30 deg
α = 40 deg

1
0.9
0.8
0.7
0.6
0.5
0.3

0.4
0.5
0.6
Ratio of translational velocity

0.7

Figure 18 : Ratio of Vertical Velocity Before and After
Collision (With Angular Velocity, ω = −1.6)

1
0.9
0.8
0.7
0.6
0.5
0.3

0.4
0.5
0.6
Ratio of translational velocity

0.7

Figure 17 : Ratio of Vertical Velocity Before and After
Collision (With Angular Velocity, ω = −1.3)

4.2.4 Comparison
Using the results of the three moving modes, the destination accessibility of each mode is summarized in Tab. 1.
Based on this comparison, it can be inferred that the hopping robot can adjust its mobility performance by simply
controlling the robot’s attitude and angular velocity at collision. This comprises the control of momentum transfer at the collision. In practical applications, the hopping
mode should be selected such that the robot s mission
requirements are satisfied.

5 CONCLUSION
In this study, the effects of various collision conditions ofa hopping robot on its rebounding performance are analyzed to obtain improved destination accessibility on an
asteroid with a microgravity environment. The results
confirmed that the robot can change the ratio of the translation and angular momentum after the rebound by controlling the robot’s attitude and angular velocity at collision. Therefore, this study suggests the possibility of theTable 1 : Destination Accessibility of Moving Modes
Mode
(i)
(ii)
(iii)

Accuracy✓

Time

Efficiency✓✓

Leaping✓

Figure 19 : Ratio of Vertical Velocity Before and After
Collision (With Angular Velocity, ω = −1.9)
achievement of a more adaptive hopping mobility to meetrequirements of accuracy, efficiency, or safety.

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Itokawa from Hayabusa, Science, Vol. 312, pp. 13411344, 2006.
[2] H. Kuninaka et al., Hayabusa Asteroid Explorer Powered by Ion Engines on the way to Earth, Proceedingsof the 31st International Electric Propulsion Conference, #267, 2009.
[3] T. Yoshimitsu et al., Micro-Hopping Robot for Asteroid Exploration, Acta Astronautica, Vol. 52, pp. 441446, 2003.
[4] M. Lange et al., MASCOT - Structures Design andQualification of an Organic Mobile Lander Platform for Low Gravity Bodies, Proceedings of the 13th
European Conference on Space Structures, Materials
and Environmental Testing, vol. SP-727, 2014.
[5] B. Hockman et al., Design, Control, and Experimentation of Internally-Actuated Rovers for the Exploration of Low-Gravity Planetary Bodies, Journal of
Field Robotics, vol. 34, pp. 5-24, 2016.

(a) Mode (i)(b) Mode (ii)(c) Mode (iii)

Figure 20 : Result of Bound Simulation