GAIT ANALYSIS OF A FREE-CLIMBING ROBOT ON SLOPED TERRAIN
FOR LUNAR AND PLANETARY EXPLORATION*Yuki Shirai1 , Hayato Minote1 , Kenji Nagaoka1 , and Kazuya Yoshida1
1

Tohoku University, 6-6-01 Aoba, Aramaki, Aoba-ku, Sendai, Miyagi, Japan
E-mail: {shirai, minote, nagaoka, yoshida}@astro.mech.tohoku.ac.jp

ABSTRACTIn this paper, we discuss free-climbing gaits on sloped terrains based on tumble stability margin and manipulability
in a quantitative way. The tumble stability margin considers a tumbling moment according to forces and moments
acting on a robot with legs, and the manipulability considers the movements of the legs. Considering these parameters, we propose a more suitable gait called hybrid gaitfor slope walking. Hybrid gait increases the tumble stability margin by shifting the main body of the robot to improve stability during slope walking. To validate the proposed gait, we conducted numerical simulations and experiments using a prototype robot. The result shows that
although the robot overturns during diﬀerent gaits, suchas adaptive gait and sway compensation gait, it does not
tumble when using the hybrid gait on sloped terrain. Ourfindings confirm that the proposed gait achieves the moststable locomotion on slopes.

1 INTRODUCTION
Recently, the use of mobile robots for lunar and planetary
exploration has received attention because these robots
provide a possible means for conducting more challenging
missions. In particular, lunar lava tubes, including steep
cliﬀ walls, have attracted a great deal of attention fromscientific interests and as a future habitat for humans [1].
Although NASA’s wheeled rovers have achieved robotic
exploration on the Martian surface, it is extremely diﬃcult
for them to access areas like rock outcrops. For example,
the Mars Exploration Rover, Opportunity, tried to examine
layering in the rock outcrops of Victoria Crater, for thesestratified layers help us to understand the geologic historyof the site. Despite NASA’s eﬀorts, Opportunity could
not access the layering and gave up deploying its scientific instruments as the slopes were too steep [2]. Hence,in order for a robot to access these kinds of complicatedenvironments, free-climbing capabilities on cliﬀ walls are
required. As shown in Figure 1, we have proposed a freeclimbing robot moving on a cliﬀ by gripping its surface forrobotic exploration [3]. NASA has also developed freeclimbing robots, and the latest model, LEMUR 3 [4], is aquadruped robot with a microspine gripper on each leg. It
demonstrated free-climbing motion on natural cliﬀ walls.

Figure 1 : Conceptual drawing of a free-climbing robot
In order for a free-climbing robot to maintain climbing motion on a cliﬀ face, gait control is necessary forbetter stability. It is also important to evaluate the stabilityof the robot on irregular terrains based on suitable stability criteria. In the past, research on walking robot was
the standard for evaluating static stability and used the relation between the support leg polygon, which is createdby connecting the points of the support leg touching the
ground, and the projected point of the center of gravity on
the ground. However, this standard does not consider external forces and moments acting on the robot, such as theforce of the gripper. In addition, the zero moment point(ZMP) has been used for evaluating dynamic stability criteria [5]. However, the concept of the ZMP targets walking on flat terrain; thus, it cannot be applied as a criterion of dynamic stability on rough rocky terrain. To overcome these issues, previous research has proposed tumble stability, which considers the tumble direction of therobot when all ground contact points, except two remaining contact points, are hypothetically lost [6]. If a given
point among the ground contact points that have been hypothetically lost can produce a support force to suppress
that tumble, the walking robot will not tumble. Although
the tumble stability margin is used as a criterion of tumble stability and it can be used on rocky terrains, limited
research studies have taken advantage of the tumble stability margin in real robot experiments. Moreover, when
the legged robot walks, whether the robot can realize theintended walking motion depends on the reachable area ofthe leg. Hence, manipulability as a criterion of the reachable area should be taken into consideration.
In this study, we compare and analyze the tumble stability of various gaits based on the tumble stability marginഥࡹand manipulability. We then propose a more suitable gaitfor slopped terrains. To validate the proposed gait, we
conducted numerical simulations and experiments using a
prototyped robot. Furthermore, appropriate leg link parameters for free-climbing are discussed, which will be aneﬀective benchmark for design optimization.

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2 STABILITY MODEL OF QUADRUPEDROBOT
A typical gait stability analysis of a walking robot is basedon the ZMP. This standard can be basically applied to awalking robot on a horizontal surface. However, as the
ZMP does not target walking on irregular terrain, one isrequired to use a standard for evaluating dynamic stability
on rocky terrain. In addition, the free-climbing robot requires a specialized gripper to tightly hold the terrain surface, whereas most walking robots simply equip frictional
spheres or circular plates on their leg tips. The walking
criterion including the gripper eﬀects is therefore necessary for the free-climbing.
This section first introduces the concept of tumble stability and its standard called tumble stability margin. We
also describe the manipulability of a legged robot.

2.1 Tumble Stability Margin
In this study, we focus on the static walking of a legged
robot. From the view of lunar and planetary exploration,a reliable robot is necessary in unknown or uncertain environments. Hence, we discuss static walking when thestatic stability of a robot is always maintained. In thisregard, however, an inertial force acts on the robot evenwhen the robot walks statically. Accordingly, in this study,
we evaluate the dynamic stability of the robot during static
walking. Furthermore, a four-legged robot has the minimum number of legs required for static walking. In termsof mission cost for lunar and planetary exploration, it isdesirable for the payload of the rocket to be as light as
possible. We therefore analyze static walking for a fourlegged robot.
This study adopts the tumble stability margin ( T S M )as the criterion of tumble stability, which can be used inrough terrain and includes dynamic eﬀects and the gripping force of the gripper. Here, we introduce T S M . First,
we calculate the sums of forces Fa and moments Ma required for acceleration and deceleration of all parts of therobot. Assuming that the walking robot is a mass concentration system of the L links, with each mass being mi andeach position vector being qi , Fa and Ma are expressed asfollows.
Fa =

L
∑
i=1

mi q̈i(1)

Figure 2 : Conceptual diagram of tumble stability margin

Ma =

L
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mi qi × q̈i

(2)

i=1

The forces acting on the walking robot are gravity, reaction forces from the ground surface, and other external
forces. Let FG and MG be a force and moment totaling
the gravity acting on each link, and F0 , M0 be a force
and moment totaling the other external forces except thecontact force from the ground. As shown in Figure 2, fromthe equilibrium of the forces, the reaction force F and moment M that should be obtained from the ground surface
can be expressed as follows.
F = Fa − FG − F0

(3)

M = M a − MG − M 0(4)Here, if the coordinate system is the inertia coordinate
system, we define the ground touching point as Pa , Pb ，
· · · , Pn , and the position vector to the ground touching
point as p1 , p2 , · · · , pn . Among these, the moment M abaround the line segment connecting two arbitrary ground
touching points Pa , Pb can be calculated from F and M.
First, F is the translational force working on the origin of the coordinate system. The moment MbF aroundthe ground touching point Pb , which is given by this translational force, is expressed as follows.
MbF = F × pb

(5)

The magnitude of this moment around the line segment
connecting the points touching the ground Pa , Pb is given
as follows.
)(p − pb
M abF = MbF · a(6)| pa − pb |Next, M is the moment around the origin of the coordinate system. The magnitude of this moment around theline segment connecting the points touching the ground
Pa , Pb is given as follows.
()
pa − pb(7)
M abM = M ·| pa − pb |Consequently, from Eq. (6) and Eq. (7), the moment M abaround the line segment connecting two arbitrary points
touching the ground Pa , Pb is expressed as follows.
()()
p − pbp × paM ab = M · a
+F· b
(8)
| pa − pb || pa − pb |Here, it is assumed that the ground-touching points Pa , Pb
have not lifted up and that the moment M ab is not zero
when more than two points are touching the ground. In
this condition, if the direction in which the tumbling occurs is actually supported by that point, then the walking
robot does not tumble so long as the moment is in the direction of one of the points touching the ground.
Here, we introduce tumble stability, which takes into
consideration the gripping force of the gripper. Let thegripping force of the gripper holding the contact point j
be F j . Then, the moment generated from the gripping
force that acts around the tumble axis can be expressed asfollows.
() ()
pb − p j × pa − p j(9)M jab = F j ·|pa − pb |By considering this moment, we can express the moment
working on the robot with a gripper (the free-climbing
robot) around the line segment connecting two arbitrary
ground-touching points Pa and Pb .
(())
p − pbp × pa+F· b− M jab(10)
M ab = M · a| pa − pb || pa − pb |By evaluating the moment M ab on Eq. (10), it is possible
to discriminate the tumble stability considering the gripping performance.
Furthermore, let n be the number of legs, and calculate the moment M ab for all combinations of a = 1 to n,b = 1 to n, where a , b. When all the determined tumble moments are cancelled by the remaining support legs,
the minimum absolute value of the moment M ab can becalculated. Thus, the tumble stability margin is definedas the value at which this value is further divided by theweight of the walking robot [6]. The tumble stability margin ( T S M ) can be expressed as follows.
TSM =min| M ab |
mg

(11)

If the tumble stability is not satisfied, the tumble stability margin is set to zero. By dividing min| M ab | by theweight of the robot, the unit of the tumble stability margin
becomes the dimension of length. Moreover, when the
robot walks on a horizontal surface, this standard shows
the same value as the ZMP. Thus, T S M has continuity
with the conventional standard for evaluating dynamic stability using the ZMP.
Using the tumble stability margin, it is possible toevaluate the dynamic stability of the walking robot onrough surfaces in a quantitative way.

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2.2

Manipulability

Manipulability is a criterion of the reachable area. It is
important to evaluate manipulability as well as the tumble stability margin when a legged robot walks on roughground surfaces. In addition, we can regard the legged
robot as manipulators connected to a body. Once the manipulator falls into a singular point, we cannot operate it
anymore. This is a particularly important problem when
the robot climbs a slope. When the legged robot walks ona slope, remaining within the stable tumble stability margin is desirable. Therefore, to execute this motion, the leg
sometimes falls into a singular point.
Here, we introduce manipulability. We consider ann degree of freedom leg and define a joint vector θ =[θ1 , θ2 , · · · , θn ]T that represents the posture of the leg.
Then, the manipulability w can be expressed using Jacobian J(θ) as follows [7].
√w = det(J(θ)J T (θ))(12)
In this study, we analyze gaits based on the leg shownin Figure 3. In Figure 3, θ1 , θ2 , and θ3 represent the joint
angles of the joints 1, 2, and 3, respectively, and L1 , L2 ,and L3 represent the lengths of the links 1, 2, and 3, respectively. The manipulability w of the target robot canbe expressed as follows.
w = L2 L3 |(L1 + L2C2 + L3C23 )S 3 |(13)where, Cn = cos θn , S Y = sin θn , and Cnm = cos(θn + θm ).

3

HYBRID GAITIn this section, we describe multiple gaits for slopes andpropose a stable gait called hybrid gait for slope walking.
Table 1 shows parameters for each gait. Each gait analyzed in this study will be explained below with reference
to its gait diagram. The gait diagrams show the time-series
data of the leg tip position with respect to the center of therobot’s main body.

3.1

Standard Gait

Here, we consider a robot walking on a horizontal planein the x direction. Figure 4 shows the gait diagram ofTable 1 : Definition of gait parameters
ParameterT [s]T ′ [s]T ′′ [s]T ′′′ [s]β [-]∆x0 [m]x0∗ [m]y∗0 [m]h [m]h0 [m]
θ [rad]µ [-]Definition
Time of a gait cycle
Time of a swing phase
Time of a stance phase
Time of a quadruped support phase
Duty ratio: Ratio of T ′′ to T
Stridex-directional stroke
y-directional stroke
Height of center of gravity from slope
Height of center of gravity from horizontal plane
Slope angle
Coeﬃcient of static friction

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Figure 4 : Gait diagram for x-directional standard gaitstandard gait. In Figure 4, the blue rectangle is the reachable area of a leg. Standard gait is a gait in which theduty ratios of all the legs are equal in planar walking, thereference positions of the respective legs are symmetrically arranged with respect to the traveling direction, andthe phase diﬀerence between the left and right legs is
0.5. When the robot realizes standard gait on a horizontal
plane, the robot always realizes walking while maintaining static stability. However, with duty ratio β being 0.75,
T S M is zero at the moment when the free leg is switched.

3.2 Adaptive Gait
Figure 5 shows the robot standing on a slope. In Figure 5,
the distance d between the point projecting the center ofgravity on the slope and the point where the center of gravity is perpendicular to the slope is expressed as follows.
d = h tan θ(14)When the robot walks in Figure 5, a large load acts on
the legs in a low position; when the leg becomes free, thetumble stability margin decreases. As a result, there isa possibility that the robot may tumble. Thus, adaptive
gait prevents the robot from tumbling by maintaining the
tumble stability margin by shifting the main body to theupper side of the slope by d (Figure 6) [8].
Figure 7 shows the gait diagram of adaptive gait, inwhich the main body is translated ∆x in the x direction.
By realizing this gait, the legged robot can walk stablyon the slope. Here, a blue rectangle in Figure 7 is thereachable area. However, the legs are outside the reachable areas as the body is translated by ∆x. It is for thisreason that adaptive gait is not always possible. It is impossible to move the leg outside of this reachable region,
where manipulability becomes zero. To solve this problem, we apply adaptive gait that considers the reachable
area as shown in Figure 8. In Figure 8, we can see that thestroke of the leg decreases. Hence, adaptive gait, by taking into consideration the reachable area, keeps the tumble stability margin larger than that of standard gait, while
preventing the leg from moving out of the reachable areaby reducing the stride.

3.3

Sway Compensation Gait

Previously, sway compensation gait was proposed [9].
This gait enables the robot to walk more stably during dynamic motion by adjusting the acceleration of the body inthe y axis direction. In this study, we apply this gait andmake the robot more stable during static walking. In otherwords, by moving the main body semi-statically, the projected point of the center of gravity of the robot remains
inside the support polygon. Figure 9 shows the gait di-DŽǀŝŶŐƌĞĂZĞĂĐŚĂďůĞƌĞĂ

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agram of sway compensation gait. The gait diagram ofsway compensation gait in the x axis direction is the sameas Figure 4. Then, Figure 9 shows the gait diagram in they axis direction. In sway compensation gait, the body is
moved left and right beforehand in the quadruped support
phase. Therefore, sway compensation gait is able to prevent the tumble stability margin from being zero in theswing phase.

3.4 Hybrid Gait
In adaptive gait and sway compensation gait, by moving
the body of the robot with respect to the slope, the tumble
stability of the robot is maintained during slope walking.
However, in both gaits the reachable area depends on the
hardware of the walking robot, and this area determines
the limits of each gait. For that reason, it is necessary tohave a gait that can maximize utilization of the leg’s reachable area and movement on a steeper slope. In this study,
we therefore propose a gait called hybrid gait that combines both adaptive and sway compensation gaits. This isa gait in which the body translates in the x axis direction,
which is the walking direction with respect to the slope,and the body swings to the right and left in the quadruped
support phase. This prevents the tumble stability margin
from becoming zero. It is for this reason that hybrid gait
allows for slope walking.

4 SIMULATION ANALYSIS OF SLOPEWALKINGIn this section, we examine the eﬀectiveness of each gaitby simulation. We performed gait analysis on the robot asshown in Figure 10. The joint arrangement of the robot
prototype is the same as Figure 3 and L1 = 2.9 [cm], L2 =

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considering manipulability
5.6 [cm], L3 = 14.1 [cm]. Furthermore, we define themass of the body, link 1, link 2, and link 3 as m0 , m1 ,m2 , and m3 , respectively, where m0 = 676 [g], m1 = 95[g], m2 = 110 [g], m3 = 9 [g]. In addition, we set gaitparameters as T = 11 [s], β = 0.8, ∆x0 = 5.0 [cm], andh = 11.3 [cm]. In adaptive gait and sway compensation
gait, the swinging displacement of the body is set to 4.0
cm. In hybrid gait, the body is swung forward by 4.0 cm
and left and right by 4.0 cm.

4.1

Comparison of Manipulability

In this study, manipulability does not depend on slope
angle. Figure 11 shows time histories of minimum manipulability when each gait was executed for one cycle.
In Figure 11, the manipulability of standard gait is at a
maximum, and the manipulability of hybrid gait is relatively small. As a result, tumble stability is improvedat the expense of manipulability. However, hybrid gait
can make more use of the leg’s reachable area comparedwith adaptive gait and sway compensation gait. This has acorrelation with the displacement capable of swinging the
body of the robot, and the swingable displacement also increases as the reachable area increases. Consequently, although hybrid gait is considered to be an unfavorable gaitfrom the viewpoint of manipulability, it is more probablethat it will enable more stable walking than other gaits, as
its swingable displacement amount is large.

4.2

Comparison of Tumble Stability Margins

We simulate adaptive gait, sway compensation gait, andhybrid gait on a 23-degree slope for one cycle, and evaluate the time histories of tumble stability margin as shownin Figure 12. We can confirm that the tumble stability margin becomes a small value in the swing phase and a large

−4

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Figure 11 : Time histories of manipulability of gaits

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Standard Gait
Adaptive Gait
Sway Compensation Gait
Hybrid Gait

10

Tumble Stability Margin, TSM [m]

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Figure 9 : Gait diagram in an y-directional sway compensation gaitHybrid Gait
Adaptive Gait
Sway Compensation Gait

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10Figure 12 : Simulation results of time histories of the tumble stability margins of gaits on a 23-degree slope
in the simulation. In the experiment, the behavior of therobot was recorded using a motion capture system.

5.1
cFigure 10 : Quadruped robot ⃝Lynxmotion
Inc.
value in the support phase. In particular, we can confirm
that the values of the tumble stability margins of adaptive
gait and sway compensation gait were zero around 3.2 s
when the leg 3 becomes a free leg. Hence, the robot is
unstable at this time. On the other hand, for hybrid gait,
the tumble stability margin did not equal zero at one cycleof gait. Consequently, the robot could stably walk on theslope when using hybrid gait.

5 EXPERIMENTAL ANALYSIS OFSLOPE WALKING
In this section, we examine the eﬀectiveness of each gait
on a real robot system. We evaluate the tumble stability of
the robot when each gait is applied using the robot shownin Figure 10. Gait parameters are set to the same value as

Relation Between Gait and Tumble
Stability Margin

We determine the y axis in Figure 5 according to the righthanded system. We define the pitch angle as the angle thatthe direction of the right screw makes with the y axis when
the axis of rotation is negative. The pitch angular velocityof each gait was recorded on a 23-degree slope, and theresult is shown in Figure 13.
As shown in Figure 13, the pitch angular velocities
of adaptive gait and sway compensation gait have large
values, and the tumbling phenomenon of the robot starts
at around 3.2 s, at which time leg 3 becomes a free leg.
Figure 14 shows the state of adaptive gait when the robot
walks on a 23-degree slope. In Figure 14, we can seethat leg 1 loses contact with the ground at 4.0 s. Thus,
when the tumble stability margin became zero, a moment
causing the robot to overturn was generated, and it was
confirmed experimentally that the robot was dynamicallyunstable. On the other hand, the pitch angular velocityof the robot at the time of the hybrid gait showed a rela-Pitch Angular Velocity [deg/s]

25

Hybrid Gait
Adaptive Gait
Sway Compensation Gait

20
15
10
5

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0
−5
−10
−15
−20
−25

0

2

4

Time [s]6810

Figure 15 : Hybrid gait on a 23-degree slope

Figure 13 : Experimental results of time histories of pitch
angular velocity of gaits on a 23-degree slope−4‫ ݐ‬ൌ Ͳሾሿ‫ ݐ‬ൌ ʹሾሿTumble Stability Margin, TSM [m]5.1‫ ݐ‬ൌ Ͷሾሿ

Figure 14 : Adaptive gait on a 23-degree slope
tively small value with respect to adaptive gait and sway
compensation gait, and the robot walked steadily. Thisis confirmed in Figure 15. Accordingly, it was possible
to validate the usefulness of hybrid gait for slope walking
from the experiment.

6 BENCHMARK FOR DESIGNOPTIMIZATIONIn this section, we discuss the design guidelines of a freeclimbing robot having a better ability to walk on unevenground from the relation between the tumble stability margin and manipulability when changing the link ratio of
the leg. By evaluating manipulability and tumble stability
margin, it is possible to quantitatively evaluate the ratio of
the links for stable slope walking as well as not causing
gait failure.
For the simulation, the total of the lengths of link 2
and link 3 of the robot in Figure 3 was constant, and theregular gait was executed on the plane. We define the ratioof link length as r = L2 /L3 . Figure 16 shows the relation
between the ratio of link length and tumble stability margin. In Figure 16, as the link ratio increases, that is, L2
increases with respect to L3 , it can be confirmed that the
tumble stability margin increases. This is because link 2 ismore important for increasing the area of the support polygon compared to link 3. The length of link 3 projected on
the ground is shorter, owing to the trigonometric representation of the length between joint 2 and joint 3, whereas
link 2 is aﬀected only by joint 2. Therefore, in Figure 16,
when considering the tumble stability margin, it is shownthat the longer the length of link 2 is with respect to thelength of link 3, the better is the tumble stability.
Next, as shown in Figure 17, we discuss the relation

x 10

5.05

5

4.95

4.9

0

0.5

1

1.5

2

2.5

3

Length of Link2 / Length of Link3, r

Figure 16 : Relation between the ratio of link length andtumble stability margin
between the ratio of link length and manipulability. Figure 17 shows that the manipulability with respect to thelink ratio is a convex function with a peak at 1.33. Thisis because in Eq. (12), when L2 increases, L3 decreases.
From Eq. (12), it can be seen that the sum of the lengthsof link 2 and link 3 is constant. Hence, the peak is shownat a certain link ratio.
Here, from Figure 16 and Figure 17, the Pareto front,
which is a set of Pareto optimal solutions, is shown in Figure 18. We generated the Pareto front using the ranking
method. The Pareto front can be an optimal solution in
the multi-objective optimization problem. Accordingly, it
is necessary to select one of the Pareto optimal solutions
from the Pareto front based on the target robot. For example, because our target is reliable exploration in lunar andplanetary exploration, it is considered that, if the tumble
stability margin is as high as possible, it is diﬃcult for the
robot to tumble. In that case, we should select the Pareto
optimal solution at the top left with the highest tumble stability margin in Figure 18.

7

CONCLUSION

In this paper, we presented a new gait, hybrid gait, forfree-climbing on lunar planetary cliﬀ faces. In hybrid gait,
the main body of the robot translates in the walking direction and to the right and left. We analyzed walking stability of each gait during slope walking based on the tumble

−4

x 10

Manipulability, w

8
7.5
7
6.5
6
5.5
5
4.5
0

0.5

1

1.5

2

2.5

3

Length of Link2 / Length of Link3, r

Figure 17 : Relation between the ratio of link length andmanipulability
Tumble Stability Margin, TSM [m]

x 10

-4

5.06
r = 1.33
r = 1.5

5.04

r = 1.75
r = 2.0

5.02
r = 2.25

5

r = 2.5
r = 2.75

4.98
r = 3.0

7

7.5

Manipulability, w

8
x 10

-4

Figure 18 : Pareto front of manipulability and tumble stability margin
stability margin and manipulability with numerical simulations and experiments. We verified the tumble stabilityof hybrid gait and confirmed that hybrid gait is more stable than conventional gaits, including adaptive gait andsway compensation gait. Furthermore, we provided a design guideline for a quadruped robot for slope walking ona slope based on the tumble stability margin and manipulability.
In future work, we expect to generate gaits in uneven environments where grippable points are limited using model predictive control. Moreover, we expect to enhance the autonomy of the robot using SLAM in order toplan the path to the destination.

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