EXTENDED GRIPPING CONDITIONS OF ROCK CLIMBER-LIKE
ROBOT FOR ASYMMETRIC GRIPPING CONFIGURATIONIN MICROGRAVITY*Kyohei Maruya1 , Yudai Yuguchi2 , Wudom Tesshin3 ,Kenji Nagaoka4 , and Kazuya Yoshida5
1

Tohoku University, Aoba 6-6-01, Aramaki, Aoba-ku, Sendai, Japan, E-mail: maruya@astro.mech.tohoku.ac.jp
Tohoku University, Aoba 6-6-01, Aramaki, Aoba-ku, Sendai, Japan, E-mail: yuguchi@astro.mech.tohoku.ac.jp
3
Tohoku University, Aoba 6-6-01, Aramaki, Aoba-ku, Sendai, Japan, E-mail: u-dom@astro.mech.tohoku.ac.jp
4
Tohoku University, Aoba 6-6-01, Aramaki, Aoba-ku, Sendai, Japan, E-mail: nagaoka@astro.mech.tohoku.ac.jp
5
Tohoku University, Aoba 6-6-01, Aramaki, Aoba-ku, Sendai, Japan, E-mail: yoshida@astro.mech.tohoku.ac.jp
2

ABSTRACTAn advanced robotic locomotion mechanism is required
for exploring a minor body which has irregular surfaces
in a microgravity environment. In this paper, we propose
a rock climber-like robot that can perform with accurate
mobility on an unknown minor body. One of the technological requirements for the robot is a gripping mechanismthat can grip tightly on an uneven surface. To meet this requirement, the gripping conditions must be theoreticallyderived. This paper presents the conditional equations forthe grippable range on an irregular surface based on friction theory, and the validation of the developed equations
experimentally using an air-floating test bed system.

1 INTRODUCTION
Exploration of minor bodies such as asteroids or comets
has received a lot of attention in recent years. Minor bodies are expected to hold clues that can reveal the stateof the primary solar system and its evolutionary process.
They are also regarded as possible targets for manned explorations in future, and hence, it is useful to investigate
them. To enhance the achievements of the explorationof minor bodies, using small robots that can move on
their surfaces is an eﬀective exploration method. Furthermore, from a viewpoint of scientific interest, the robots are
required to reach destinations where scientific attention
would be high. However, there are technical diﬃculties in
the accomplishment of robotic locomotion on minor bodies because of two characteristic conditions: the gravity
on the minor body is extremely small, and the minor bodyhas an unknown irregular terrain. In such conditions, it isdiﬃcult for a robot to maintain contact with the surface. In
particular, traditional wheeled or tracked robots can easily
hop by the reaction force from the surface.
For an asteroid exploration robot, hopping mobility has been studied as an eﬀective method of exploration. MINERVA is a representative asteroid explo-

ration robot, which was carried in the Japanese spacecraft
Hayabusa [1]. Following MINERVA, MINERVA-II1 [2]and MINERVA-II2 [3] were developed and equipped in
the Hayabusa 2 spacecraft that is cruising on its target
path. These robots can move by the reaction forces or
torques generated by built-in actuators. However, the hopping robots have diﬃculties in reaching the desired destination precisely, because they do not have soft-landing
mechanisms, and thus the destination depends very muchon the surface conditions. In contrast, Yoshida et al. [4]proposed a rock climber-like robot that can move on a minor body by gripping its surface, as shown in Figure 1.
This locomotion method enables the robot to reach thedestination by continuous stable locomotion. For the realization of this robot system, two technical challenges
should be addressed: One is robust gait control for stable
locomotion on a rough terrain in a microgravity environment, and the other is reliable gripping control for holding
on an irregular surface. This paper presents information
on how the latter is addressed.
To date, some researchers have studied robotic
gripping mechanisms in a microgravity environment.
The NASA/JPL’s robotics group proposed the robot
LEMUR IIB and conducted gripping experiments using a
prototyped robot [5]. Its gripping mechanism is composedof 256 sharp hooks and flexible elements to grip the surface of a rock. This is a super-redundant mechanism, andhence the friction characteristics of the contact points canbe ignored during its gripping phase. However, the gripper
was not getting released easily from the surface, and thehooks broke frequently [6]. Research on gripping mechanisms for base-fixed manipulators has been conducted forground applications. The unidirectional nature of the constraints by using fingers was addressed in the analysis ofform closure [7] and force closure [8]. The contact dynamics between the fingers and the objects was also analyzed [9, 10]. Furthermore, stability analysis of gripping against a disturbance applied to the gripped object
was performed [11]. Yuguchi et al. [12] have applied the(a) Overall view of the robot

Figure 1 : Conceptual illustration of rock climber-likerobotgripping analysis to the rock climber-like robot moving
on a minor body. They derived the static ground gripping
characteristics in a microgravity environment based on the
manipulation theory established with the theory of polyhedral convex cones [13]. In addition to the theoreticalanalysis, they evaluated the characteristics experimentally,
where the relative positions of the ground surface and thegripping points were assumed to have symmetrical configuration. However, for practical applications, the gripping
characteristics must be extended to asymmetric gripping
conditions.
This paper presents an extended theory of the groundgripping conditions in asymmetrical configuration. The
extended theory is a generalized framework for evaluating
the gripping capability of the rock climber-like robot. The
theory is validated based on experiments with planar motion in a microgravity environment using an air-floating
test bed system.

2 GRIPPING CAPABILITY ANALYSIS
While gripping the ground surface, the inclination angleof the surface and the coeﬃcient of friction at the contact point have a large influence on the gripping capability.
We define gripping capability in terms of an equation thatgives the conditions under which the gripping mechanism
can generate a constant gripping force without slippage.
This section presents the theory of static gripping conditions, including asymmetric configuration of the relative
positions in a microgravity environment.

2.1 Formulation of Gripping Conditions
Given that the robot is in a static condition, the reaction
force f and gravitational force W exerted on the robot, are
as shown in Figure 2. Here, fiB (i = 1, 2, · · · , n) is the reaction force on the i-th finger and the subscript B indicatesthat the variable is defined in the coordinate system fixed
on the robot base ΣB {xB , yB }. Furthermore, xiB is the position vector from the center of gravity of the robot to thetip of the i-th finger. The equilibrium equations of the re-(b) Gripping state of the hand
Figure 2 : Asymmetric gripping state of ground grip locomotive robot in ΣB
action force and the moment around the center of gravity
can be expressed as follows.
n∑fiB = −W
(1)
i=1
n
∑

xiB × fiB = 0

(2)

i=1

To satisfy the equilibrium conditions, the gripping force− fiB on the ground surface must be generated inside of
the frictional cone without slip on the ground. Thus, weapproximate the frictional cone as a polyhedral cone. We
define the edge vectors of the polyhedral cone for the imth finger as d1iB , d2iB , ... , diB, where m is the number ofthe edge line of the polyhedral cone. Consequently, thecondition that − fiB works inside the polyhedral cone isgiven as follows.
m
∑
j
j
j
(3)
∃RiB
≥ 0 s.t.
RiB
diB = − fiB
j=1

Accordingly, Eq. (3) provides the no-slip condition
for the i-th finger. In addition, substituting Eq. (3) into
Eqs. (1) and (2), the gripping conditions are re-written as
follows.
n ∑
m
∑
j
j
j
∃RiB
≥ 0 s.t.
RiB
diB = W(4)
i=1 j=1n ∑m∑
i=1 j=1

jj
RiB
diB × xiB = 0(5)

2.2 Analysis of Possible Gripping Range theGroundIn this study, we assume the robot to be a dual-arm robot
model in two dimensions for simplification, i.e., n = m =2. Moreover, µ is the coeﬃcient of static friction between
the ground and the tip of the robot, and the gravity force
W is assumed to be zero because of microgravity. θ1 andθ2 (where −π ≤ θ1 ≤ π and −π ≤ θ2 ≤ π) are the inclined angles of the gripping surface with respect to theyB axis, as shown in Figure 2. A counter-clockwise direction indicates a positive angle, and thus in Figure 2, θ1 ispositive and θ2 is negative. For simplification, we define
2ϕ ≡ θ1 − θ2 , and rotate the base coordinate system by(θ1 + θ2 )/2, as shown in Figure 3. In the rotated coordinate system Σϕ {xϕ , yϕ }, the yϕ axis is in the same direction
as the bisector of the inclination angle of the ground. From
this definition, Eqs. (4) and (5) can be re-written with respect to the yϕ axis as follows.
∃Riϕj ≥ 0 s.t.

2 ∑
2
∑

j

Riϕj diϕ = 0(a) Overall view of the robot(6)

i=1 j=1
2 ∑2∑

j

Riϕj diϕ × xiϕ = 0(7)(b) Gripping state of the handi=1 j=1

On the basis of the geometric relationship shown in Figj
ure 3, the edge vectors of the polyhedral cone diϕ can begiven as follows.
[][]−cos(ϕ − α)−cos(ϕ + α)
1
1
d1ϕ =
, d2ϕ =
−sin(ϕ − α)−sin(ϕ + α)
(8)
[
]
[
]
cos(ϕ
−
α)
cos(ϕ
+
α)
2
2
d1ϕ
=
, d2ϕ
=
−sin(ϕ − α)−sin(ϕ + α)
where α = tan−1 µ represents the half-angle of the friction cone. Substituting Eq. (8) into Eqs. (6) and (7), thefollowing simultaneous inequalities are obtained.
−x21ϕ sin(ϕ + α) + y21ϕ cos(ϕ + α) ≤ 0







 x21ϕ sin(ϕ − α) − y21ϕ cos(ϕ − α) ≤ 0
x21ϕ sin(ϕ + α) + y21ϕ cos(ϕ + α) ≥ 0 x sin(ϕ − α) + y cos(ϕ − α) ≤ 0
21ϕ
21ϕ

From Eqs. (10) and (11), it can be said that there exist a maximum inclination angle of the surface and arange( of gripping points) for positive gripping. Here,√2sin−1 y21ϕ / x21ϕ
+ y221ϕ represents the slope between thetwo fingertips in Σϕ . Therefore, once x1ϕ is decided, thepossible range of x2ϕ for gripping the ground surface islimited, as shown in Figure 4.
Additionally, the expression Eq. (11) can be obtainedin ΣB by rotating Σϕ again through −(θ1 + θ2 )/2 as follows.

• θ1 ≥ θ2 :
(9)

(
)T
(
)T
where x1ϕ ≡ x1ϕ y1ϕ , x2ϕ ≡ x2ϕ y2ϕ , x21ϕ ≡ x1ϕ −x2ϕ , and y21ϕ ≡ y1ϕ − y2ϕ . By solving Eq. (9), the gripping
capability can be finally determined as follows.
α−ϕ≥0






y21ϕ
−1 

 ≤ α − ϕ−(α − ϕ) ≤ sin  √


2
x21ϕ
+ y221ϕ 

Figure 3 : Asymmetric gripping state of ground grip locomotive robot in Σϕ

(10)

(11)







y21B

 ≤ α − θ1
−θ2 − α ≤ sin−1  √ x2 + y2 
21B
21B

(12)

• θ1 ≤ θ2 :






y21B

 ≤ α−θ1 −π (13)−θ2 −α+π ≤ sin−1  √ x2 + y2 
21B
21B
(
)T
(
where x1B ≡ x1B y1B , x2B ≡ x2B
x2B , and y21B ≡ y1B − y2B .

)T
y2B , x21B ≡ x1B −Figure 4 : Schematic view of the grippable range
Figure 5 : Air-floating test bed system

3 GRIPPING EXPERIMENTThis section presents the gripping experiments conductedon the simulated ground using an air-floating test bed system. The range of grippable points is determined by the
point at which the fingers slip on the surface. The theoretical results are verified through comparison with theexperimental results.

3.1 Air-Floating Test Bed System
To evaluate the gripping performance in a microgravity
environment, we developed the air-floating test bed system, which is shown in Figure 5. This system performs
planar motion in a microgravity environment using air
bearings and air tanks. Frictionless motion can be simulated between the system and the surface plate. Moreover,this system has a dual-arm which has two motor-driven
joints and a six-axis force/torque sensor in each arm. The
driving motors (RH-8D-3006-E100AL) are produced by
Harmonic Drive Systems Inc., and the force/torque sensors (WDF-6M200-3) are produced by WACOH-TECH
Inc. On each force/torque sensor, a specific finger mechanism is attached. The system can measure the data of thereaction force while gripping the surface. The motion of
the system is measured by a motion capture system (OptiTrack FLEX:V100R2) produced by NaturalPoint Inc., andthe measured data is used as the basic information.

3.2 Simulated GroundFigure 6 shows a view of the simulated ground. The simulated ground is composed of two flat plates and these areconnected by two passive hinges. The inclination angle
between the two plates can be changed arbitrarily. Furthermore, sandpapers are attached on the surface of theplates to simulate various frictional surfaces. In the experiments, the simulated ground was fixed on a flat plane
using clamps.

3.3 Experimental Condition
Figure 7 shows an overview of the experimental setup.
In the experiments, the range of the relative position of

Figure 6 : Simulated groundthe fingertips for stable gripping was investigated by examining the boundary at which the fingertips begin toslip. According to the experimental results by Yuguchi
et al. [12], a cone-shaped finger is more eﬀective for gripping. Therefore, we also used cone-shaped fingers in theexperiments, as shown in Figure 8. The material of thefingers is SUS304. The fingers were contacted in a direction perpendicular to the ground. Moreover, the fingertips
were controlled to move with a constant velocity, and the
motion was stopped when the reaction forces were greaterthan 1.5 N for safe operation. For the simulated ground,#100 sandpaper was used. The coeﬃcient of static friction
between the sandpaper and the tip of cone-shaped fingers
was obtained from the measured data as follows [12].
µ = 3.08
To verify the derived theory, we conducted experiments in five diﬀerent conditions of the inclination anglesof the simulated ground: ϕ = 55◦ , 60◦ , 65◦ , 70◦ and 75◦ .
From Eq. (10) and with µ = 3.08, the maximum inclination angle for gripping was calculated as 72◦ . Here, we
define η as the slope between the fingertips. In the experiments, the left finger was fixed at a position 7 cm fromthe top of the simulated ground. Then, the contact position of the right finger was changed so that η changes withincrements of 1◦ . The experiments in each condition wereconducted five times. The acceleration due to gravity was
set to 0.016 G.

80
60
40
20

Gripping capability [%]Experiment

TheoryExperiment

100
80
60
40
20

0
-20
-10
0
10
20
Slope between fingertips [°]

0
-20
-10
0
10
20
Slope between fingertips [°](a) ϕ = 55◦(b) ϕ = 60◦

TheoryExperiment

100
80
60
40
20

Gripping capability [%]Gripping capability [%]Gripping capability [%]Figure 7 : Experimental setup of the robot and the simulated ground

Theory100

TheoryExperiment

100
80
60
40
20

0
-20
-10
0
10
20
Slope between fingertips [°]

0
-20
-10
0
10
20
Slope between fingertips [°](c) ϕ = 65◦(d) ϕ = 70◦

Figure 9 : Comparison of theoretical and experimental
gripping capabilities

Figure 8 : Cone-shaped finger

3.4 Experimental Results
Figure 9 shows the theoretical and experimental results on
gripping capability for several slopes between the fingertips. The blue lines and the red cross marks indicate thetheoretical values and experimental results, respectively.
In the case of ϕ = 55◦ in Figure 9(a), the theoretical grippable range can be calculated from Eq. (11) as follows.
−17◦ ≤ η ≤ 17◦
It may be noted that η = 0 means that the arrangementof the system and the simulated ground is symmetrical.
Furthermore, Figures 10 and 11 show the time history of
the reaction forces. These graphs show that the force increased and became constant at a value greater than 1.5Nin the case of successful gripping; on the other hand, theforce was not generated or was reduced to zero in the caseof failure to grip. Therefore, the gripping capability wasdetermined by the reaction force.
From Figures 9(a)∼(d), it can be seen that the grippable range is narrower in the experiments than in the theoretical results. In addition, we conducted the test for thecase of ϕ = 75◦ and η = 0, a case in which it is impossible to grip the surface. The result of this case was thatthe robot was not able to grip the surface during any of thefive attempts. Considering the practical applications, wedefine the grippable boundary as the angle η at which therobot was able to grip during all the five attempts. With
this definition, the theoretical and experimental values of

the grippable range are shown in Table 1. A comparison of
the theoretical and experimental grippable ranges for various inclination angles of the simulated ground is shown in
Figure 12. From Table 1 and Figure 12, it can be seen thatin case when the inclination angle is between 55◦ and 65◦ ,the value of the absolute error in each grippable boundary
angle was within 4◦ . In the case of ϕ = 70, there was no
grippable point in the experiment, though the theoreticalgrippable boundary angle is ±4◦ . Moreover, the deviationsin the experimental and the theoretical values increased
with increase in the inclination angle. From these results,
it can be seen that regardless of the magnitude of the inclination angle, the test bed system has an error of 6◦ ∼ 7◦ inthe grippable range η. These errors occurred based on theaccuracy of the friction coeﬃcients measured and the positioning accuracy of the manipulator joints including themechanical parts and motors. However, these errors are
small, and thus the validity of the derived theory was confirmed. The rock climber-like robot can achieve gripping
locomotion on a minor body in practical applications, if
suﬃcient margin is considered for the gripping boundary.

4

CONCLUSION

This study evaluated the extended static gripping conditions of a rock climber-like robot in asymmetric gripping
configuration of the relative positions of the ground andthe gripping points in a microgravity environment. The
gripping capability was analyzed based on the experimental results and theoretical analysis. As a result, it was
found that there is a relationship between the inclination
angle of the ground and the range of gripping points forForce [N]Force [N]

2
0
total

-2
0

Table 1 : Theoretical and experimental values of the grippable range

4

4

2.5

ye

5
7.5
Time [s]

xe

10

2

ϕ

TheoryExperiment

55◦
60◦
65◦
70◦
75◦

−17◦ ≤ η ≤ 17◦−12◦ ≤ η ≤ 12◦−7◦ ≤ η ≤ 7◦−2◦ ≤ η ≤ 2◦
Failure−13◦ ≤ η ≤ 14◦−9◦ ≤ η ≤ 8◦−4◦ ≤ η ≤ 4◦
FailureFailure

0
total

-2
0

(a) Left fingertip

2.5

ye

5
7.5
Time [s]xe10(b) Right fingertip

Figure 10 : Reaction forces in the case of successful gripping (ϕ = 55◦ , η = −13◦ )Absoluteerror
3◦ ∼ 4◦3◦ ∼ 4◦
3◦
-

2
0
-2
0

4

xe

Force [N]Force [N]

total

ye

2.5

5
7.5
Time [s](a) Left fingertip

10

total

ye

xe

2
0
-2
0

2.5

5
7.5
Time [s]10(b) Right fingertip

Figure 11 : Reaction forces in the case of failure to grip
(ϕ = 75◦ , η = 0◦ )positive gripping. Then, in the gripping experiments using an air-floating test bed system, the validity of the conditional equations for the grippable range was verified. In
future research, it is necessary to evaluate the gripping capability on a real rough terrain.

References[1] T. Yoshimitsu, T. Kubota, et al. (2003), “MicroHopping Robot for Asteroid Exploration”, Acta Astronautica, vol. 52, pp. 441–446.
[2] T. Yoshimitsu, T. Kubota, , et al. (2014), “Development of Hopping Rovers for a New Challenging Asteroid”, Proceedings of the 12th International Symposium on Artificial Intelligence, Robotics and Automation in Space, #5C-01.
[3] K. Nagaoka, K. Yoshida, et al. (2015), “Development of MINERVA-II2, a Micro-Robot for Asteroid Surface Exploration with Innovative Mobility”,
The 11th Low-Cost Planetary Missions Conference,
Berlin, Germany.
[4] K. Yoshida, T. Maruki, et al. (2002), “A Novel Strategy for Asteroid Exploration with a Surface Robot”,
Proceedings of the 34th COSPAR Scientific Assembly, pp. 281–286.
[5] A. Parness, M. Frost, et al. (2012), “Gravityindependent Mobility and Drilling on Natural RockUsing Microspines”, Proceedings of 2012 IEEE International Conference on Robotics and Automation,
pp. 3437–3442.
[6] A. Parness, M. Frost, et al. (2013), “Gravityindependent Rock-climbing Robot and a Sample Ac-

Grippable range, 2(α -φ) [ °]60
4TheoryExperiment

50
40
30
20
10
0
45

50

55
60
65
70
Inclination angle, φ [ °]7580

Figure 12 : Comparison of the theoretical and experimental grippable ranges for various inclination angles
quisition Tool with Microspine Grippers”, Journal of
Field Robotics, vol. 30, no. 6, pp. 897–915.
[7] K. Lakshminarayana (1978), “Mechanics of Form
Closure”, ASME Technical report, no. 78-DET-32.
[8] V. Nguyen (1988), “Constructing Force- Closure
Grasps”, The International Journal of Robotics Research, vol. 7, no. 3, pp. 3–16.
[9] J. Kerr and B. Roth (1986), “Analysis of Multifingered Hands”, The International Journal of Robotics
Research, vol. 4, no. 4, pp. 3–17.
[10] P. Akella and M. Cutkosky (1989), “Manipulatingwith Soft Fingers: Modeling Contacts and Dynamics”, Proceedings of the 1989 IEEE International
Conference on Robotics and Automation, vol. 2,
pp. 764–769.
[11] A. Nakanishi and Y. Hayakawa (2011), “Stability
Analysis of Multi-Fingered Grasp under Destabilizing Gravity Eﬀect”, Proceedings of the 18th World
Congress The International Federation of AutomaticControl, vol. 12, pp. 14667–14674.
[12] Y. Yuguchi, W. F. R. Ribeiro, et al. (2015), “Experimental Evaluation of Gripping Characteristics Basedon Frictional Theory for Ground Grip Locomotive
Robot on an Asteroid”, Proceedings of the 2015
IEEE International Conference on Robotics and Automation, pp. 2822–2827.
[13] S. Hirai and H. Asada (1993), “Kinematics and Statics of Manipulation Using the Theory of Polyhedral Convex Cones”, The International Journal of
Robotics Research, vol. 12, no. 5, pp. 434–447.