dynamical motion control for quadruped walking with autonomous distributed system

7/25/2019 Dynamical Motion Control for Quadruped Walking With Autonomous Distributed System

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Dynamical Motion Control for Quadruped Walking

with Autonomous Distributed System

Katsuhiko INAGAK I and Hisato KOBAYASHI

Department

of

Electrical Engineering, Hosei University

Koganei, Tokyo

184,

Japan

Abstract his paper discusses dynamical mo-

tion control

for

quadruped walking machine.

Quadruped walking machine requires dynamical

cc..trol to keep its stability in high speed walk-

ing. We make the dynamical compensator by

using a weight oscillator built in the center

of

the body. First we calculate the exact oscil-

lators motion that can neutralize the moment

con.pletely. Next we try to use

a

harmonic os-

cillation for the oscillators motion instead of the

accurate motion. Finally we use a notion of au-

tonomous distributed control to our motion con-

trol and expand it to quasi-dynamical walking.

I. INTRODUCTION

Quadruped walking machine requires dynamic walking

in high speed walking, because the number of standing

legs is less than three. In tr ot gait t hat is known as a

typical dynamic walking pattern, two pairs of diagonal

legs make standing phase, respectively. Thus, there ex-

ists a moment when the body

is

falling down around the

supporting axis. If we leave this moment out of consid-

eration, walking machine may walk with rolling motion,

repeatedly. This fact influences the accuracy of the t ra-

jectory, because each leg makes landing at unexpected

timing. Therefore , we should restrain the moment for

stable walking. For this problem, a method which uses a

notion of ZMP(Zero Moment Point) has been proposed.

In this method, the moment by the gravity can be neu-

tralized by acceleration generated by the trajectory of

the body. However, this method requires vibration

of

the body, and such motion is not desired for some pur-

poses s x h as transportation.

In th is paper, we utilize an oscillation of a small weight

instead of oscillating the body. Our method makes it

possible to neutralize the moment without oscillation of

the body. Usually, horses running accompanies a head

shaking. We think this motion expects similar effect to

1004

the oscillator. By the way, we must not ignore energy

cost problem to discuss the walking machine, because it

has poor energy efficiency compared with other mobile

machines. On this point of view, utilization

of

the os-

cillator causes increase of the body weight. But, we can

settle this problem easily by using mechanical elements

as the oscillator. More important point is to reduce the

energy consumption to drive the oscillator.

In th e first part of this pape r, we make the motion of

the oscillator clear, and show that its motion is similar

to sine wave. Then, we analyze the moment and energy

consumption in case that a simple sine wave is applied

to motion of the oscillator instead of the regular motion.

Finally, we apply this method into our autonomous dis-

tributed control system that has been already proposed.

11.

MOTION L A N N I N G O F O S C I L L A T E R

A .

Standard Gait Rule

First, let us consider gait rules to apply the oscillator

system. In our former report, we have already proposed

a method of gait transition from low speed walking to

high speed walking. Three typical gaits (crawl gait , tro t

gait , gallop gait) are connected smoothly in our method.

The crawl gait is known

as

a typical static gait in slow

speed walking. And, as increase

of

the walking speed,

walking manner changes into the trot gait and the gallop

gait in order.

In this study, we consider only from the crawl gait to

the trot gai t, because the gallop gait requires a jumping

motion. It is difficult to make the gallop gait by only

the oscillator system.

Fig.1

shows a walking pattern of

four legs from the crawl gait to th e trot gait. Black part

means that leg is in the standing phase, and white part

is in the swing phase. We can see that the period of the

standing legs is reduced from three to two

as

increase

of the walking speed . And, during the period of two

standing legs, the standing legs are placed in diagonal

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R-F

R-H

L-F

L-H

0.2

0.1

0.0

Fig. 1 . Gait transition from crawl

gait

to trot gait

Pa1

Pa

2

D i r e c t i o n

Fig.

2 .

Required accelerative direction

position of the body.

B.

M o t i o n

Planning

First of all, we consider the motion planning of the

os-

cillator during the trot gait, because the dynamical con-

trol is always required in this gait . Next, we consider

the motion of the oscillator. There are some directions

of accelerat,ion to generate

a

moment to neutralize the

moment by the gravity. In the trot gait, the pair of

the standing legs is changed at an ins tant. Thus, re-

quired direction of the acceleration is also changed at

an instant as shown in

Fig.2.

At this time, compo-

nent of sideways direction is kept in the same direction,

but that of lengthwise direction is changed into oppo-

site direction. Therefore, it is not wise to generate the

acceleration to the lengthwise direction. Accordingly,

we give the oscillator

a

prismatic motion in sideways di-

rection as degrees of freedom.

Fig.3

shows a walking

situation for our study . The body of the walking ma-

chine goes straight in the y axis direction at

a

constant

speed. Points Pl q,

l),P2 22,~2)

are on the support-

ing diagonal line. At

t

= 0 ,

y

coordinate of t.he body is

/

/

@ C o n t a c t

W a l k i n g

D i r e c t i o n ,/

I

Fig.

3.

Walking situation

on y1, and that is on y2 at

t

=

T .

We can describe the

moment M1 by the gravity around the supporting axis

as follows.

where,

In these equations,

E , )

means

x

coordinate

of

the

point

C

in

Fig.3.

And. other variables are as follows.

mb : massofthe body

m, : mass

of

the oscillator

x,

:

I coordinate

of

the oscillator

g

: acceleration of gravity

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On the other hand, we can make

a

moment to neu-

tralize

M1

by reactive force of the oscillators motion.

This moment

M z

can be described as follows.

Mz

=

-m,hx,

sin 8

(6)

where, h is height of the oscillator

Therefore, the total moment Ma is described as fol-

lows.

To keep the stability of the body, we should solve

following differential equation, because

k f b

must always

keep zero.

where,

a l

= -ag m,+mb) (9)

mbg z l k -

bg m,

+

mb) (10)

2

l =

The solution of this equation is:

where, C1 and

C2

are unknown constants.

C. Boundary Condation

Next, let us consider boundary condition t o find the con-

stants C1,Cz. As stated before, pair

of

the supporting

legs is changed at an instant in the trot gait. Thus, the

motion of the oscillator should be connected smoothly

in this time. Moreover, the total motion of the oscilla-

tor should make gentle oscillatory motion.

From these

reasons, the boundary condition should be described

as

follows.

Sm 0)

=

S , T )

=

0 (12)

By using this condition, the parameters C1,Cz are

derived

as

follows.

(14)

a1

m

1

c2 = --

% I

1+ exp

-m

)

Fig.4 shows a result of a generated trajectory of the

oscillator in the trot gait. Fig.5 shows acceleration of

the oscillator in this time. We can see that the trajectory

is similar to sine wave.

D.

Energy

Consumption

Problem

In the former section, we solved the motion of the oscil-

lator that can neutralize the moment. But, this method

has some problems. First , required acceleration shown

in Fig.5 has discontinuous points at the changing point

of the supporting legs. Such motion may be difficult

to realize and have a bad influence on energy consump-

tion. Even if we can realize

thc

intended motion, walking

motion commonly accompanies other kinds of undesired

acceleration such as vibration of the body. For example,

position error of a standing leg causes such vibration,

because closed mechanical linkage is formed by plural

stand ing legs on the ground. After all, we think

it

is

impossible to neutralize the moment completely. W hat

should be noticed is energy consumption problem ra ther

than precise motion control.

Then, we examine an effect when a single harmonic

oscillation is used insbead of the exact motion in the

former section. It is easy to generate the single harmonic

oscillation. And, th is motion is suitable for economy of

the energy consumption. The motion of the oscillator

can be described as follows.

15)

- = l o cos

ut

+ a )+ l +

2

In thi s equat ion, we must decide the angular veloc-

ity and the phase a suited for the characteristic of

the regular trajectory shown in

Eq.11.

Therefore, we

decided them

as

follows.

x

-

T

w

a = o

(17)

Then, let

us

think about remained undecided value,

amplitude

4.

he amplitude

A

should be decided as the

moment has minimum value. The moment of the body

can be calculated by substituting Eq.15 to Eq.7.

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0 .

U .

x 0 .

- 0 .

- 0 .

4 0 - I

1

I I

e e

e 0

4 0 -

I

1 I

-

0 . 0

0 . 5

1 . 0

1 . 5 2 . 0 2 . 5

y [ml

Fig. 4 . Oscillatory trajectory in trot gait

I I

I I 1

0 . 0 0 . 5

1 . 0

1

?5 2 . 0 2 . 5

y m l

Fig. 5 . Acceleration of oscillation

our former report, we generated a periodic signal based

2 1 + 2 2

E = m A o ( h w 2 + g )

COS

( U t ) + a i t + b i + m g - 2 18)

Therefore, 4 hould satisfy following condition.

on

a

basic motion of a leg. Thus , this method is very

compatible with the oscillator s motion. First p art of

this section, we show basic notion of the autonomous

distributed system and apply it t o the dynamical motion

control

(19)

The solution of this equation is: A . Phase

Szgnal

4a

1

m w 2 h w 2 +

g)T

(20)

Fig.6

shows time response of the moment. The solid

line is the moment when the oscillator makes single har-

monic oscillation with amplitude

A

in Eq.20. And, the

dotted line is the moment when the oscillator is fixed on

the moment compared with the case that the dynamical

motion is not taken.

Our control system consists of one central controller

and four distributed controllers for each leg, as shown

in Fig.7. The central controller makes a basic motion

model of leg by solving following equations.

4 0

=

the center of the body. Our proposed method can reduce

; =

h k1

Ust,

Us,

p,n){q +

p

m ( l , p , 21)

4

= h ( k ,

U s t . h u , P , q ) { - P + q

m ( l , p , q ) } (22)

111. A U T O N O M O U S D I S T R I B U T E D C O N T R O L where

In this section, we adopt the notion of the autonomous

distributed control to our dynamical motion control. In

m ( l , p ,

=

l 2 -

p 2 + q 2 )

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Fig. 6. Time response of moment

Central

Controller

Rhythm

Generator

Left-Fore Left-Hind

Right-Fore Right-Hind

Leg

Distributed Controllers

Fig. 7. Autonomous distributed control system

h ( k , % t ~ ' s W > P ? q )

q+p-:{:p,q)

{ q

+

P 4 l . P Q)l v sw

' .

-usW I

{ q

+ p m l .p . }

I

U,t

u s t

dynamical motion control for quadruped walking with autonomous distributed system

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