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Omnidirectional Static Walking of a Quadruped Robot on a

Slope

Paper:

Omnidirectional Static Walking of a Quadruped Roboton a Slope

Lei Zhang , Shugen Ma , Yoshinori Honda , and Kousuke Inoue

Department of Systems Engineering, Faculty of Engineering, Ibaraki University4-12-1 Nakanarusawa-Cho, Hitachi-Shi 316-8511, Japan

E-mail: shugen@dse.ibaraki.ac.jp, zhang@roboken.dse.ibaraki.ac.jp COE Research Institute, Ritsumeikan University

1-1-1 Noji Higashi, Kusatsu-Shi 525-8577, Japan Computron Corporation

1-11-2 Tonnya-Cho, Maebashi-Shi 371-0855, Japan

[Received May 17, 2005; accepted July 25, 2005]

We propose successive gait transition with arbitrary

body posture to enable a quadruped robot to walk

statically and omnidirectionally on a slope. Body pos-

ture is determined by rotation around 3 axes, roll,pitch, and yaw. Successive gait transition with a min-imum number of steps on a slope is realizable using

common foot position before and after gait transition.

The time required to transit between gaits is reduced

by carefully designing foot position in crawling and ro-

tating while limiting foot reachable region on a slope.

The robot thus walks into any direction with arbi-trary body postures. In this study, we also verify a

tradeoff relation between motion speed and body pos-

ture. Computer simulation and experiments verified

the feasibility of our proposed method and the stabil-

ity of gait transition based on static stability margin.

Keywords: quadruped robot, omnidirectional walking,

body posture, successive gait transition, walking on a

slope

1. Introduction

Quadruped robots have advantages over wheeled robots

in their ability to walk on irregular terrain in any direc-

tion. Researches on quadruped robots have been widely

carried out [13]. Fukuoka et al. also realized dynamicwalking of a quadruped robot on irregular terrain using a

neural system method [4]. Since stability is more impor-

tant than speed in conveyance tasks, we focused on static

walking for its stability. To increase stability, Tsukakoshi

et al. proposed an intermittent crawl gait [5]. More re-cently, Konno et al. presented an adaptive intermittent

crawl gait [6]. Since the zigzag trajectory or 4-leg sup-

port period used by a quadruped robot on a slope, its

movement is generally slow. Semiautonomous walking

based on leg transition at the perimeter of leg movementhas been proposed [7], and gait transitions forward and

backward, turning left and right, and rotation have been

discussed in cooperative sideway movement [8], but om-

nidirectional static walking with arbitrary body posture on

a slope has not progressed as far as other types.

We discussed successive gait transition [9] for aquadruped robot to walk omnidirectionally and staticallyon a horizontal plane using crawl and rotation gaits. It

transited between gaits successively, stably, and continu-

ously with the least number of steps using the common

foot position (CFP), a leg position common to two gaits

before and after gait transition. Gait transition chooses

the crawl or rotation from the turning center and trans-fers from one gait to another continuously when changing

the turning center. The applicable environment was re-

stricted to a horizontal plane, however, making it difficult

for the quadruped robot to adapt well to different environ-

ments. We have considered expanding omnidirectionalstatic walking to a slope [10], keeping the robot horizon-

tal for conveyance tasks. But in such a case, there will

occur a problem that the leg reachable region becomes

narrow and motion speed becomes slow. To improve mo-

tion speed of robot, we considered body posture with the

largest leg reachable region, i.e., parallel to the slope, al-though this required shifting the CFP for successive gait

transition. Leg reachable region also becomes narrow,

slowing the robot. So we considered the relationship be-

tween motion speed and body posture, determining what

would enable the fastest motion speed on a slope.In this paper, we propose integrating (1) static walk-

ing with high environmental adaptability, (2) omnidirec-tional movement, and (3) high motion speed. Section 2

discusses how to realizing omnidirectional static walking

on a slope using specified body posture. Section 3 de-tails successive gait transition on a slope with this body

posture, so that the robot moves into all direction with a

minimum number of steps through CFP design. Section

4 explains computer simulation and experiments demon-

strating the feasibility of our proposed method and results,

clarifying the tradeoff between motion speed and bodyposture.

Journal of Robotics and Mechatronics Vol.18 No.1, 2006 51

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Zhang, L. et al.

Fig. 1. Quadruped robot TITAN-VIII.

leg 1

leg 2

leg 3

leg 4

Z

c2

1

c4

Approximatedleg movement area

3

Z

Y

Leg reachable regionRi

Approximatedleg movement areaAi

Slope

X

(a) Robot model (b) Leg movement area Aiandreachable region Rion a slope

Y

c

Ri

Ri

X

Fig. 2. Robot model, leg movement area A i and reachable

region Ri.

2 . Omnidirectional Static Walking on a Slope

The quadruped robot we used was TITAN-VIII (Fig.1)

[11], assuming mechanical leg movement area Ai to bean octahedron centering on the CFP ci for standard pos-

ture on a horizontal plane (Fig.2). The basal plane ofAi

is a rectangle. The height from ci to the vertex of Ai inpositive and negative directions was made the same as the

mechanical movement limit for the legs. Leg reachableregion Ri on a slope is a polygon resulting from the in-

tersection of leg movement area Ai and the slope. This

can be formulated easily and enables gait planning in real

time. Our proposed method is, of course, applicable to

any leg movement area.

2.1. Coordinates

In considering omnidirectional static walking of aquadruped robot on a slope, we set the right-hand coor-

dinates shown in Fig.3. We assume the slope with fixedinclination angle as the known parameter and the heightfrom the center of gravity (COG) to the slope as a fixed

value, h. Body posture is determined by rotations around3 axes of body coordinate, roll, pitch and yaw, (bR, bP,bY).Body coordinate : Origin G is located at the COG,

its X-axis is at the front of the robot, its Y-axis is in the

left direction and Z-axis is at the top.

Horizontal plane coordinate : Origin G is an inter-

section of the gravity direction from the COG and the

slope that results from oblique parallel projection fromG to the slope. Its Z-axis is normal to the horizontal

Z

bY

Y

bP

bR

GG ,

G

Z

Y

X

GG ,

Z

X

Y

Vertical projection of COG

Hcog(height of COG)

COG G

Fig. 3. Coordinates , and .

X X

X X

X

X X

Y

Y

Y

Y

Y

Y

Y

Y

Y

X-Crawl

Y-Crawl

Rotation

CFP: leg placement common to two gaits before and after gaittransition

Fig. 4. CFP for each gait on a slope.

plane and its X

-axis is in the direction of the orthogonal

projection of the X-axis to the horizontal plane.

Slope coordinate : Origin G is common to G. The

directions of

s axes result from rotating

around Y

-

axis at angle and around Z-axis at angle bY. Each

rotating matrix is set to E j and EkbY.

Homogeneous transformation matrices between coor-

dinates are described as

T

EjbP EibR EjbP EibR p

0 0 0 1

. . . . (1)

T

E jEkbY 0

0 0 0 1

. . . . . . . . (2)

T T

T . . . . . . . . . . . . (3)

where p (p

0 0 h T

) is the vector from the ori-

gin of to that of. Rotating matrix EjbP EibR results

after rotating

around Y

-axis at angle bP and around

X

-axis at angle bR. Gaits are first planned on slope co-

ordinate , then transformed to those on body coordinate

, to control the robot.

2.2. Common Foot Position (CFP) on a Slope

To minimize steps in gait transition among crawl androtation gaits, foot trajectories before and after gait tran-

sition use the CFP shown in Fig.4 [9]. Setting this

52 Journal of Robotics and Mechatronics Vol.18 No.1, 2006