MICROGRAVITY EXPERIMENT OF ROCK CLIMBING
LOCOMOTION FOR EXPLORATION ROBOTON MINOR BODY
*Yudai Yuguchi1 , Wudom Tesshin1 , Kyohei Maruya1 , Kenji Nagaoka1 and Kazuya Yoshida1
1

Tohoku University, Aoba 6-6-01, Aramaki, Aoba-ku, Sendai, Japan, E-mail: {yuguchi, u-domu, maruya, nagaoka,
yoshida}@astro.mech.tohoku.ac.jp

ABSTRACTIn this study, we analyze the motion of a type ofrock-climbing robot used for minor body exploration.
The surface of a minor body features an irregular terrain and microgravity. Hence, it is effective for an
exploration robot to move by gripping the surface.
In the robot’s gait, the reaction force acting on thegripper of the supporting arm must be controlled toprevent the detachment of the gripper from the surface. Therefore, we formulate a control method tocontinue locomotion with reactionless motion by utilizing reaction null-space. Moreover, the control law
is experimentally verified using an air floating system
under microgravity. Furthermore, planar simulationshows that a reactionless axis affects continuous locomotion.

1

INTRODUCTION

With rapid advances in space development, the interest in minor body exploration has been constantly
increasing. An example of pioneering a spacecraft
for minor body exploration includes the NEARShoemaker, which successfully landed on asteroid
Eros and captured high-resolution photos of its surface [1]. The Japanese spacecraft Hayabusa2 waslaunched in 2014 and is expected to reach the asteroid Ryugu in 2018 [2]. Minor bodies are expected
to possess evidences revealing the evolutionary process of our solar system. Thus, implementing minor
body exploration is highly significant. In particular,
unmanned exploration using robots is effective for detailed investigations of minor body surfaces. Suchrobots are required to have the ability to reach ascientifically significant area; however, it is difficult
for them to move on such surfaces. This is because
these surfaces are unknown irregular terrains with extremely low gravity. In such environments, it is difficult for a robot to maintain its position on the surface, and it can easily float because of surface contact
forces. Therefore, wheeled and legged mechanisms
aimed at planetary exploration are unsuited for minor

body exploration [3]. Matsuno et al. proposed the useof a legged robot to control the contact with an asteroid surface [4]. However, a locomotive strategy was
not mentioned. To address this issues, Yoshimitsu
et al. proposed a hopping mobility system in MINERVA that was carried by the Japanese spacecraft
Hayabusa [5]. This mechanism facilitated effective
movement about an asteroid surface, and was implemented in other hopping robots [6], [7]. However, it
is difficult for these robots to reach their specific location because of their repetitive bouncing locomotion.
Furthermore, a robot that moves by ciliary vibrations
has been studied; however, it is unsuitable for roughterrains [8].
For locomotive problems, Yoshida et al. proposed a type of rock-climbing robot [9]. In this paper,
we refer to this robot as a “ground-gripping robot.”
By gripping the surface of a minor body, the robot
can prevent its flotation and implement secure exploration activities. The robot gait can be divided into
three phases, as shown in Fig. 1. Phase 1 comprises
the motion of surface gripping. Thus far, we formulated a gripping condition for the case in which thegripper and surface are symmetric, as in [10]. After this phase, the arm gripping the surface becomesthe renewed supporting arm. Phase 2 consists of the
motion of detachment from the surface, and Phase 3
comprises the movement of the idling arm. By repeating this gait, the robot can continuously moveon the surface. In this paper, we focus on Phase 3,
in which motion control is required considering thereaction force acting on the gripper of the supporting arm. When the idling arm is moved, all reaction
forces act on the gripper of the supporting arm. If
the reaction force exceeds the gripping force, it detaches from the surface, and the robot’s locomotionceases. As an example of reaction control for space
robots, Nenchev et al. proposed reactionless control
that utilizes the reaction null-space [11]. As anotherapproach, Wu et al. proposed the path planning of amanipulator using a genetic algorithm [12]. However,this method is unsuitable for the precise control of thetip position because the method is designed in joint

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space. Furthermore, a calculated load for reactionlesscontrol is lesser than that for this method. We applied the reactionless control to the ground-gripping
robot and clarified that the initial base height affects
stable locomotion through planar simulation [13].
In this study, we verified reactionless locomotion
in microgravity through an experiment and simulations. The equation of motion of the ground-gripping
robot is first formulated, followed by the formulationof the simultaneous control law, combining the reactionless control and the control of the idling arm’s
tip position. Moreover, the effectiveness of the control
method is confirmed through a single-step locomotion
experiment using an air floating system. Finally, weshow that the reactionless axis affects stable locomotion on irregular terrains through planar simulationand discuss singularities.

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DYNAMICS OF
GROUND-GRIPPING ROBOTIn this section, we first present a dynamic model ofthe ground-gripping robot, followed by its equationof motion.

2.1

Nomenclature

Fig. 2 shows the dynamic model of a dual-arm robot.
The symbols used in this paper are defined below.
F s ∈ R6
τ ∈ Rn: Vector of force and momentum on thegripper of the supporting arm.
: Vector of torque on manipulator joints.

H s ∈ R6×6 : Inertial matrix of the gripper of the
supporting arm.
n×n
Hm ∈ R: Inertial matrix of the manipulator.
H sm ∈ R6×n: Inertial matrix of coupling between thegripper of the supporting arm andmanipulator.
6
xs ∈ R: Vector of the position and orientationof the gripper of the supporting arm.
nφ∈R
: Vector of the joint angle of themanipulator.
c s ∈ R6
: Nonlinear velocity-dependent term of
the gripper of the supporting arm.

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: Nonlinear velocity-dependent term of
the manipulator.
J s ∈ R6×6 : Jacobian matrix of the gripper of the
supporting arm.
6×n
Jm ∈ R: Jacobian matrix of the manipulator.
F h ∈ R6
: Vector of force and momentum on the
manipulator’s tip.
H ∗ ∈ Rn×n : Generalized inertial matrix.
c ∗ ∈ Rn
: Generalized nonlinear
velocity-dependent term.
The “manipulator” represents links other than thegripper of the supporting arm.

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Equation of Motion

For modeling the equation of motion, we made thefollowing assumptions:• The gravitational force is negligible.
• Contact forces do not act on the robot.
• At least one arm grips the surface.
By dividing all the links between the gripper of the
supporting arm and the manipulator, the equation of
motion is given as


 
 
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ẍs
H s H sm
Fs
cs
=
+
τ
cm
H Tsm H m
φ̈


J sT
−
Fh
(1)
J mT
The upper and lower parts of (1) denote the equationsof motion about the gripper of the supporting arm
and the manipulator, respectively. By assuming that
contact forces do not act on the tip of the manipulator(F h = 0), and by eliminating the acceleration of thegripper of the supporting arm ẍs from (1), the control
torque τ can be written as
τ = H ∗ φ̈ + c∗ + H sm T H s −1 F s(2)

H m − H sm T H s −1 H sm
cm − H sm T H s −1 cs

(3)
(4)

where
H∗
c∗

≡
≡

3

CONTROL LAW

Substituting (8) into (10), vector ξ is calculated as

In this section, we formulate the desired joint angularvelocity. We first describe the reactionless constraint
by utilizing the reaction null-space. Next, we present
the simultaneous control law combining the tip position control and constraint using the task-priority
method. Moreover, we explain the generation of thedesired tip trajectory.

3.1

Reactionless Motion Control

We assume that no external force acts on the robot,
and the initial linear and angular momenta with respect to the center of mass of the supporting arm’s
gripper are zero. In this case, the following conservation laws of linear momentum P ∈ R3 and angularmomentum L0 ∈ R3 can be established:


P
(5)
= H s ẋs + H sm φ̇ = 0
L0From (5), the condition whereby the momenta dueto the gripper of the supporting arm are both zero isgiven as(6)
H sm φ̇ = 0
As long as the manipulator moves under this constraint, the reaction force does not act on the gripperof the supporting arm. (6) represents the constraint
on six dimensions with respect to the inertial frame.
However, in this study, reactionless control is applied
to a limited number of dimensions. Therefore, we
formulated a revised condition by rewriting H sm asĤ sm φ̇ = 0(7)

The desired joint angular velocity that satisfies (7)can be expressed as
φ̇d
RRN S

=

RRN S ξ(8)≡+
I − Ĥ sm Ĥ sm(9)where ξ and RRN S respectively denote an arbitrary
vector and the projector on the null-space of the inertial coupling matrix Ĥ sm . RRN S is called the reaction null-space.
Subsequently, the simultaneous control method
combined with the tip position control is expressed
through task-priority redundancy resolution. To enable the robot to continue locomotion, it is importantthat the gripper of the supporting arm does not detach from the surface. Therefore, we set reactionlessmotion control as the primary task.
The relationship between the manipulator’s tip
velocity ẋh and the joint angular velocity φ̇ is generally expressed asẋh = J m φ̇

(10)

+

ξ = J̃ m K (xd − xh )(11)

where J̃ m is expressed asJ̃ m ≡ J m RRN S(12)In (11), ẋh is replaced by a control gain matrix K
and the desired tip position and orientation xd forthe tip position control. Moreover, the superscript“+” represents the pseudo inverse. Furthermore, thedesired joint angular velocity for simultaneous control
is obtained by substituting (11) into (8) as+

φ̇d = RRN S J̃ m K (xd − xh )

3.2

(13)

Measurement of Manipulability

A reactionless motion control utilizing the reaction
null-space comprises dynamic and algorithmic singularities [14]. Thus, we devised three types of equations for measuring each manipulability. The manipulability measures u and v regarding dynamic singularity are expressed asqT(14)|Ĥ sm Ĥ sm |
u ≡
q
v ≡
(15)
|J m J m T |Moreover, the manipulability measure w regarding algorithmic singularity is presented asqw ≡ |J q J q T |(16)
where J q is expressed as
J q ≡ J̃ m RRN S +

3.3

(17)

Tip Trajectory Generation

As the gripper of the idling arm moves away from andthen close to the minor body’s surface, unintended
contact and friction between the gripper and surface
must be avoided. When the contact or frictional force
exceeds the gripping force of the supporting arm’s
gripper, the gripper detaches from the surface. Thus,
the robot cannot maintain stable locomotion. In response to this demand, the trajectory whereby thegripper of the idling arm is vertically moved awayfrom and then close to the surface is generated in two
dimensions. To satisfy this condition, the trajectoryis made elliptical so that the trajectory from the initial tip position in Phase 3 to its desired position isconnected smoothly. Fig. 3 shows a model of the elliptical trajectory, in which x0 denotes the vector of

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the initial tip position. The coordinates of the initialand desired tip positions can be written asx0xx0y= a cos θ1 cos ψ − b sin θ1 sin ψ + rx (18)
= a cos θ1 sin ψ + b sin θ1 cos ψ + ry (19)

xdx
xdy

= a cos θ2 cos ψ − b sin θ2 sin ψ + rx (20)= a cos θ2 sin ψ + b sin θ2 cos ψ + ry (21)where a, b, θ, ψ, and r(= [rx , ry ]T ) denote the lengthof the long axis of an ellipse, the length of its shortaxis, the angle between the long axis and tip position,
the inclination angle of the ellipse, and the vectorof the central coordinates of the ellipse, respectively.
Furthermore, conditional equations whereby the tipof the idling arm moves away from and then close
to the surface in a direction perpendicular to it are
expressed as


dy
a tan
(22)
|θ=θ1 − α ≤ ǫ0
dx


dy
|θ=θ2 − β ≤ ǫd
a tan
(23)
dx
where α and β denote the tangent angles of the ellipse
at the initial and desired tip positions, respectively.
Moreover, the acceptable error ranges ǫ0 and ǫd are
set to address cases in which a perfect elliptical trajectory cannot be generated vertical to the surface. Bysolving the constrained non-linear optimization problem using (18) - (23), the desired elliptical trajectoryis generated.

4

LOCOMOTION EXPERIMENTIn this section, we describe our experiment of singlestep locomotion according to the control law statedin Section 3. In this experiment, we used an air floating system to emulate motion in microgravity. We
first present the configuration of the air floating system, followed by the experimental conditions. Next,
we verify gait control by comparing motion with andwithout the reactionless control law.

Air Floating System

To validate the gait control method, we developed
an air floating system, as shown in Fig. 4. Table 1
lists the link parameters. Link 1 denotes the gripperof the supporting arm, and the gripping state wassimulated by setting the inertial parameters to large
virtual values. The gripper of the supporting arm
was fixed on a flat plate mounted on the side of thesurface plate. This system can float on the surface ofthe plate by emitting compressed air stored in an air
tank by using air bearings mounted at the bottom of
the system. There was almost no friction between thesystem and surface plate, and the system simulated
the motion in planar microgravity. Furthermore, thissystem had four degrees-of-freedom (DOF) and usedmotors (RH-8D-3006-E100AL) manufactured by Harmonic Drive Systems Inc. Each tip was equipped witha force/torque sensor (WDF-6M200-3) produced by
WACOH-TECH Inc.

4.2

Experimental Conditions

In this experiment, the reactionless axis was set parallel to the minor body’s surface gripped by the gripperof the supporting arm. This is to prevent the sideslipof the rock being gripped due to the motion of theidling arm. Moreover, the initial angles of each joint
were set as φ = [35◦ , −35◦ , −35◦ , 35◦ ]. The desired
tip position of the idling arm was set at +0.08 m along
the I x-axis and +0.04 m along the I y-axis from theinitial tip position, and the desired tip posture wasset at −60◦ , relative to the I x-axis. The tip postureangle was controlled at a constant rate during movement. These desired values simulated the motion of
the robot climbing up a slope. Further, the control
time, measurement time, and control gain were set at
16 s, 18 s, and K = diag(2, 2, 2) in both cases.

4.3

Experimental Results

Fig. 5 shows the motion sequences of the air floating system when reactionless control is incorporated.
Fig. 6 shows the reaction forces with and without
reactionless control. Fig. 6 (a) shows that approximately 0.5 N of Fy was exerted on the tip of the
supporting arm. Moreover, both forces in Fig. 6 (a)show that the robot moved while continuously vibrating because of joint stiffness. In contrast, Fig. 6 (b)shows that Fy remains close to zero and the vibrationof the robot was suppressed. Thus, reactionless control law reduces the risk of sideslipping of the rock
being gripped by the supporting arm’s gripper andsuppresses the vibration of the robot during motion.
Moreover, Fig. 7 shows the tip position and postureof the system with the incorporation of reactionlesscontrol. Fig. 7 (a) shows that the control value is con-

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sistent with the desired value, and the elliptical trajectory is realized. Furthermore, the tip posture angle
γ in Fig. 7 (b) followed the desired value. Therefore,
the validity of the gait control method was confirmed.

5

MOBILITY ANALYSIS

In this section, we apply two types of reactionless axes
to the gait control law and verify the continuous locomotion of a ground-gripping robot on a rough terrain
through planar simulation. We first present the simulation conditions, including the simulation model ofthe robot and the generation method of the roughterrain. In the simulation, the adaptability of eachgait to the rough terrain is discussed in terms of themanipulability measure.

5.1

Simulation Conditions

In the simulation, a dual-arm robot with each arm
having three DOF was used to improve manipulability over the air floating system. Table 2 lists the linkparameters. Locomotion was simulated by switching
the state quantities of the links corresponding to eacharm when the tip of the idling arm reached the desired tip position and posture angle. Moreover, thesimulation was executed by setting two reactionlessaxes. One was set parallel to the surface (case 1),
and other was set vertical to the surface (case 2). In
addition, case 2 is useful for preventing detachmentof the gripper from the surface because the allowable
reaction force in this direction depends on the gripping force. Furthermore, the reaction force acting on
the gripper of the supporting arm F s was calculated
Table 1 : Link parameters of air floating system
mass [kg] inertia [kg·m2 ] length [m]link 11000
1 × 1090.13597
link 20.565
0.00216655
0.14997Base
9.690
0.15481713
0.15800
0.565
0.00216655
0.14997
link 40.350
0.00085437
0.13597
link 5Figure 5 : Motion sequence of air floating system.
by applying the following equation:
F s = −kp ∆xs − kc ẋs(24)where the parameters of stiffness kp and viscosity kc
were set at 3000.
A minor body surface was simulated by randomly
setting inclinations and heights of lines 9 mm apartalong the X-axis, and connecting them with lines 1
mm apart along the X-axis. The range of the inclination was randomly set from −30◦ to ∼ 30◦ , relative to the X-axis. On the contrary, the range of theheight was randomly set from −10 mm to ∼ 10 mm
according to the height of the previous line’s center.
Moreover, the initial angles of each joint were set at
φ = [−15◦ , −105◦, 30◦ , 30◦ , −105◦ , −15◦]. Further,
the stride length along the X-axis, the control time ina single step, the number of motion cycles, the control
gain, and the number of times the simulation was run
were set at 0.04 m, 8 s, 8 cycles, K = diag(12, 12, 12),and 50 times, respectively.

5.2

Simulation Results

Fig. 8 illustrates the motion sequences of the robot
for case 1. Fig. 9 and Fig. 10 show the reaction forces
acting on the gripper of the supporting arm and the

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length [m]
0.050
0.080
0.080
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On the contrary, Fig. 11 illustrates the motion
sequences of case 2. Fig. 12 and Fig. 13 show the reaction forces acting on the gripper of the supportingarm and the manipulability measures, respectively.
In case 2, there are many situations in which the robot
was able to move across the surface for all motion cycles, and the number of divergences was 8. The divergences were caused by ingression to the algorithmic
singularity such as in case 1. Therefore, it is clarified
that the reactionless axis affects the stable locomotion of a ground-gripping robot. Furthermore, it isnecessary to construct a control law for avoiding or
passing through algorithmic singularity.

inertia [kg·m2 ]1 × 109
0.0001675
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manipulability measures, respectively. Fig. 9 depicts
that almost no reaction force acted in the direction
parallel to the surface. However, Fig. 10 (b) showsthat the calculation entered algorithmic singularity
while the robot was moving. The calculations for not
only this surface type but also for all surface types
were diverged in case 1. This is because the base
moves in the opposite direction of locomotion to accomplish the reactionless gait.

link 1link 2link 3Baselink 5link 6link 7Figure 7 : Tip position and posture of the idling arm.

mass [kg]
1000
0.30
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6

CONCLUSIONSIn this paper, we focused on the reaction forces acting
on the tip of the supporting arm of a ground-gripping
robot, and proposed a reactionless gait for stable
locomotion. Moreover, we verified the effectivenessthrough an experiment and simulations. We first described the equation of motion of the ground-gripping
robot, followed by the simultaneous control law using
the reaction null-space. In addition, the manipulability measures and generation of the elliptical tip trajectory were formulated. Furthermore, we developed
an air floating system, and confirmed a single-step
locomotion through the experiment. We then conducted two types of simulations of continuous locomotion on irregular terrain. The result showed thereactionless axis affects the feasibility of the robot.
In future research, we plan to develop a gripper
suitable for the environment of a minor body and ver-

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Figure 11 : Robot motion sequence in case 2.
ify the continuous locomotion using our air floating
system. Moreover, we need to address algorithmic
singularity to accomplish stable locomotion on any
irregular terrain.

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