Proceedings of the 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems

Maui, Hawaii, USA, Oct. 29 - Nov. 03, 2001

The D Linear Inverted Pendulum Mo de A simple mo deling for a

bip ed walking pattern generation

Shuuji Ka jita Fumio Kanehiro Kenji Kaneko Kazuhito Yok oiand Hirohisa Hirukawa

National Institute of Advanced Industrial Science and Technology AIST

Umezono Tsukuba Ibaraki Japan

Email ska jitaaistgojp

Abstract The mo deling the ThreeDimensional Linear Inv erted

P endulum Mo de DLIPM is derived from a general

threedimensional inv erted p endulum whose motion is

F or D walking c ontr ol of a bipedrobot we analyze

constrained to mov e along an arbitrarily dened plane

the dynamics of a thr e edimensional inverted p endu

It allows a separate controller design for the sagit

lum in which motion is constr aine d to move along an

tal xz and the lateral yz motion and simplies

arbitr arily dened plane This analysis leads us a sim

awalking pattern generation a great deal

ple line ar dynamics the Thre eDimensional Linear In

verte d Pendulum Mode DLIPM Geometric nature

of trajectories under the DLIPM and a method for

Derivation of D Linear Inverted

walking p attern gener ation ar e discusse d A simula

tion r esult of a walking contr ol using a dof bip e d

Pendulum Mo de

robot model is also shown

Motion equation of a D in verted

p endulum

Intro duction

When a bip ed rob ot is supp orting its b o dy on one

leg its dominant dynamics can be represented by a

Research on h umanoid rob ots and bip ed lo como

single inv erted p endulum which connects the supp ort

tion is currently one of the most exciting topics in the

ing fo ot and the center of mass of the whole rob ot

eld of rob otics and there exist many ongoing pro jects

Figure depicts suc h an inverted p endulum consist

Although some of those w orks hav e already demon

ing of a p oint mass and a massless telescopic leg The

strated very reliable dynamic bip ed walking we

p osition of the point mass p x y z is uniquely

believe it is still imp ortant to understand the theoret

sp ecied by a set of state variables q r

r p

ical background of bip ed lo comotion

A lot of researc hes dedicated to the bip ed walk

x rS

ing pattern generation can b e classied into t w o cate

p

gories The rst group uses precise knowledge of dy

y
rS

r

namic parameter of a rob ot eg mass lo cation of cen

z rD

ter of mass and inertia of each link to prepare walking

q

patterns Therefore it mainly relies on the accuracy

S
S

S  sin S  sin D 

p r

r r p p

of the mo del data

Let f b e the actuator torque and force asso

r p

Contrary the second group uses limited knowledge

ciated with the state variables r With these

r p

of dynamics eg lo cation of total center of mass total

inputs the equation of motion of the D inverted p en

angular momentum etc Since the con troller knows

dulum in Cartesian co ordinates is given as follows

little ab out the system structure this approachmuch

relies on a feedbackcontrol

In this pap er w e tak e a standp oint of the second

x

r

T

A A A

approach and intro duce a new mo deling which rep

y

m J

p

resents a rob ot dynamics with limited parameters

z f
mg

0-7803-6612-3/01/$10.00  2001 IEEE 239

D Linear In v ertedPendulum Mo de

Although the moving pattern of the p endulum has

v ast p ossibilities wewant to select a class of motion

that would b e suitable for walking For this reason we

apply constraints to limit the motion of the p endulum

The rst constraint limits the motion in a plane with

given normal vector k k
and z intersection z

x y c

z k x k y z

x y c

Forarobotwalking on a rugged terrain the normal

v ector should match the slop e of the ground and the

zintersection should b e the exp ected average distance

of the center of the rob ots mass from the ground For

further calculation we prepare the second derivatives

of

Figure D Pendulum

z k x k y

x y

Substituting these constraints into equations and

where m is the mass of the p endulum and g is gravit y

w e obtain the dynamics of the p endulum under

acceleration The structure of the Jacobian J is

the constraints F rom straightforward calculations we

get

rC S

p p

p

A

g

k

rC
S

J

r r

y

y
xy
xy
u

q

r

rC S D
rC S D D

z z mz

r r p p

c c c

k g

x xy
xy u x

p

z z mz

c c c

C  cos C  cos

r r p p

where u u are new virtual inputs which are intro

r p

T o erase the inv ersed Jacobian that app ears in

duced to comp ensate input nonlinearity

T

let us multiply the matrix J from the left

C

r

u

r r

rC
rC S D x

r r r

D

A A

rC
rC S D y

m

p p p

C

p

u

p p

S
S D z

p r

D

rC S

r r r

In the case of the walkingona atplane wecan

A A

rC S D

mg

p p p

set the horizontal constraint plane k k

x y

f D

and we obtain

Using the rst ro w of this equation and multiplying

g

y
u y

r

DC weget

r

z mz

c c

g

D

x u x

rS mg

m
rDy
rS z

p

r r

r

z mz

C

c c

r

In the case of the walking on a slop e or stairs

By substituting kinematic relationship of equations

and w e get a go o dlo oking equation that de where k k  weneed another constraint From

x y

scrib es the dynamics along the yaxis xy  wesee

D

mg y m
z y y z

u x u y xy
xy

r

r p

C

mz

r

A similar pro cedure for the second row of yields

Therefore wehav e the same dynamics of and

the equation for the dynamics along the xaxis

in the case of an inclined constraint plane byintro duc

ing the following new constraint ab out the inputs

D

mg x mz x
xz

p

C

p

u x u y

r p

240

A unit mass is driven by a force vector of magnitude

Equations and are indep endent linear

f that is prop ortional to the distance r betw een the

equations The only parameter which go verns those

mass and the origin The force magnitude can b e dis

dynamics is z ie the z intersection of the constraint

c

tributed into x and y elementasfollows

plane and the inclination of the plane never aects the

horizontal motion Note that the original dynamics

w ere nonlinear and we derived linear dynamics with

y

g

f f

y

out using any approximation

y

r z

c

Let us call this the ThreeDimensional Linear In

g x

x f f

verted Pendulum Mo de DLIPM The rst author

x

r z

c

and Tani intro duced a tw odimensional version of this

dynamics mo de in and Hara Yokokaw a and

Equation reminds us the celestial mechanics

Sadao extended it to three dimensions in the case of

under the gravityeld In this case the force magni

zero input torque in

tude is

k

G

f

G

r

Nature of the D Linear Inv erted

where k is a parameter determined from the gravity

G

Pendulum Mo de

constant and the mass of the gra vit y source Both

in the equations and the force vectors are

In this section w e examine nature of tra jectories

parallel to the p osition vectors from the origin to the

under the DLIPM with zero input torques u

r

mass This results wellknown Keplers second lawof

u

p

planetary motion areal velo cit y conservation

g

Let us see howtheforceofgravit y can b e separated

y

y

z

into x and y direction

c

g

x x

xk k x

G G z

c

f

Gx



r r

x y

With a given initial condition these equations deter

k yk y

G G

mine a tra jectory in D space Figure sho ws two

f

Gy



r r

x y

examples

Since x aects f and y aects f wemust always

Gy Gx

wo equations as a set Contrarilythe x

1 treat these t

and y element of the DLIPM can be treated inde

0.8

p endently at all times This gives us great advan tages

alking patterns as we 0.6 in analyzing and in designing w

z

will see in the following sections 0.4

0.2

Geometry of the tra jectory

0 0.5 0.5 0 0 -0.5 -0.5 y x Y

Y'

D Linear Inverted Pendulum Mode Figure (v , v ) x y

(x, y)

Similarity and dierence with gravity

X' eld θ

X e can regard that equations and repre

W O

sen t a force eld for a unit mass

g

r f

Figure DLIPM proje ctedontoXYplane

z c

241

following condition must be satised Since B

Figure shows a DLIPM tra jectory whichispro

rarely happ ens we do not consider eq and

jected on to XY plane Motions along Y and X are

go v erned by the equations and resp ectively

gz xy  x y

c

By integrating each equation we obtain a time invari

an t parameter named the orbital energy

Using this equation wecan calculate the geometric

g

shap e of the DLIPM By substituting eq and

y y E

y

z

c

g

x x E

x

gz x y x y

c

z

c

E gz x E gz y

x c y c

In Figure it is sho wn another co ordinate frame

X Y which rotates from the original frame XY

The nal form is a simple quadratic equation

Since the DLIPM is a dynamics under the central

g

g

force eld as discussed in the last section the new

x y

z E z E

c x c y

frame X Y also gives a prop er representation of the

DLIPM The new orbital energy is calculated as

Since E andE eq forms a hyp erb olic

x y

g

curve

cx sy cx sy
E

x

Hyp erb olic curves also app ear in Kepler motion

z

c

and one of its examples is a swing by ight of the

g

sx cy
sx cy E

y

V o y ager spacecraft approached Jupiter in

z

c

It seems in teresting that w e obtained same shap e of

where c
cos s
sin By simple calculations we

tra jectory from a totally dierent p otential eld

can verify that the total energy of the system do es not

c hange by the wa y of the co ordinate setting

D walking pattern generation

E E E E constant

x y

x y

When the Y axis corresp onds to the axis of sym

Outline

metry like in Figure E and E b ecome maximum

x y

and minimum resp ectively Therefore w e can cal

Figure sho ws an example of a w alking pattern

culate the axis of symmetry b y solving the following

based on the DLIPM In this pap er we assume sup

equation

port leg exc hange of constant pace Tochange the

w alking sp eed and direction the rob ot mo dies fo ot

E

x

placements sho wn as small circles in Figure

As  c Bsc

When we pro ject the walking motion onto X and Y

axis we observe decoupled motions gov erned byequa

where

tions and Figure Each motion follo ws

A
gz xy  x y

c

the D version of the Linear Inv erted Pendulum Mo de

that we describ ed in the former pap er

B
gz x  y  x  y

c

The solution is when B 

Pattern generation along a lo cal axis



tan AB

Now the problem b ecomes a control of the motion

along X or Y axis for eachstep Let us assume that

when B A 

the rob ot is rep eating single supp ort phase of duration

T and double supp ort phase of duration T

s dbl

Figure illustrates successive steps in X direction

n n

The initial b o dy state x v and thenal body

when B A

i i

n n

state x v hav e the following relationship

f f

atany x





 

n

n

x

C T S x

T c T If the Y axis happ en to b e already the axis of sym

f

i

n n

S T C

T c T metry should b e zero F rom equations and

v v

i f

242

C
coshT T S
sinhT T

T s c T s c

T o control the walking sp eed wemust c hange the

0.15

fo othold p oint E to mo dify the initial condition of

the supp ort phase DF When the desired status

0.1

at the end of supp ort p ointFisgiven as x v we

d d

can dene error norm with certain weight a b as

0.05

Y [m]




 

N
ax  x bv  v

d d

f f

0

By substituting eq into this equation and calcu


-0.05

which minimizes N weob lating the fo othold of x

i

tain a prop er control la w 0 0.05 0.1 0.15 0.2 0.25 0.3

X [m]



aC x  S T v bS T v  C v D x

T d T c d T c d T f T

i

Figure Walking p attern gener ate dfrom the D

 

D
aC bS T

T T c

T

LIPM A robot takes seven steps from left to right

Motion of the tip of the inverte d p endulum is shown

as pieces of hyperb olic curves solid lines We as

sume that the robot is in double support and moves

(1) (1) (2) v(2)

a straight line betwe ene ach support phase dotte d on v v = v

z i f i f

lines Smal l cir cles ar e fo ot places and dashe d lines

D D' F erb olic curve indic ate primary axes of hyp B

0.1 A C E 0.05 x

(2) (2) 0 x(1) x(1) d x x x,y [m] i f i f

-0.05

-0.1

0 0.5 1 1.5 2 2.5 3 3.5 4

Figure Two succ essive steps in the sagittal plane

e il lustr ated The body travels from B to D in the

0.2 ar

singlele g support phase then moves from D to D in

0.1



the doubleleg support phase with c onstant speed v f

0

and then travels D to F in the sec ond singlesupport

vx,vy [m/s]

While the body moves from B to D the tip of

-0.1 phase

the swing legtravels fr om A to E dashed curved line -0.2

0 0.5 1 1.5 2 2.5 3 3.5 4

osition of E we can control the nal

time [s] By changing the p



body speed v at the point F Except for our inserted

f

doublesupport phase this is the same ide a pr op ose d

Figure XY position and velocity in a walk of the g

by Miura and Shimoyama

ure The tick line shows x motion and the thin line

shows y motion The p osition graphs jump the dis

T o determine the fo othold E we also need the dis

tanc e of the step length at each support foot exchange

tance that body tra v els in the double supp ort The

sinc e we are taking an origin at a fo ot place of e ach

distance is

support The vertical dotte d lines indicate the time of



d v T

dbl

f

the supporting mode change In this walking p attern

the robot is taking s for single support and s for

The motion of the swing leg is planned to arriveat

double support

the p oint E at the exp ected touchdown time dashed

curve from A to E in Figure

243

Control of walking direction

T o sp ecify the walking direction we rotates the ref

erence XY frame for step to step See Figure The

con trol of eq automatically shap es the rob ot mo

tion to follo w the giv en frame of reference Figure

illustrates a w alking along a circle In this walking

pattern the reference frame is rotated rad so

that the rob ot returns the stating p oint with steps

0.5

0.4

0.3 Y [m]

0.2

Figure A Humanoid Robot model T otal mass of

robot is kgs and the c enter of mass is lo c ate d

0.1 the

m in the height from the oor level The rob ot

for each leg but the arms and the head do 0 has dof

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

deled as one block with the

X [m] not c ontain any joints mo

body

Figure Walk on a circle

T able Important link parameters

Simulation of h umanoid walking

length m

The DLIPM has b een already applied to a non

y

width of p elvis

h umanoid typ e bip ed rob ot that has telescopic legs

thigh link

Though the rob ot could p erform D dynamic bip ed

sc hank link

w alking without using prepared tra jectorywalking di

anklesole distance

rection control was not yet considered

fo ot length

In this pap er w e examine a simulated humanoid

fo ot width

rob ot whose motion is generated b y the DLIPM

weightkg

Figure sho ws the outlo ok of the rob ot used in the

body

simulation The rob ot has dof for eac h leg but

crotchyaw

the arms and the head are mo deled as one blo ckwith

crotchroll

the body T able shows imp ortant physical param

thigh

eters that was determined b y considering an actual

shank

mechanical design The size of the fo ot is m

ankle pitch

m length width T otal mass of the rob ot is

fo ot

kgs and the center of mass is lo cated m in the

moment of inertia of the b o dy kgm

height from the o or level

I

xx

F or the dynamic simulation we used the OpenHRP

I

yy

simulator which w as develop ed in METIs h umanoid

I

zz

rob otics pro ject The walking pattern of Figure

w as used as a reference input to the rob ot and the cen

y distance b etw een the right and left hip joints

ter of mass of the rob ot b o dy was controlled to follow

this Towalk on a at o or the b o dy heightwas kept

244

constant The reference joint angles and sp eeds were

calculated byinv erse kinematics so thatthe p osition

and the velo city of the feet with resp ect to the b o dy

corresp ondto the sp ecied motion Finally the ref

erence joint angles and sp eeds were realized byaPD

feedback controller

F urthermore w e must consider the body and the

fo ot rotation around z axis Although the walking

pattern of Figure is assuming an ideal rob ot that

can step tow ards any direction at all time the rob ot

of Figure has the limit of join t angles and it must

a v oid collision b etw een the left and the right legs For

this reason we designed additional pattern for the fo ot

orien tation with resp ect to the b o dy so that the body

faces instantaneous walking direction in the middle of

eac h supp ort

Figure shows the snapshots of the simulation and

Figure sho ws the b o dy tra jectory and fo ot place

ments We see that the tra jectory of the h umanoid

rob ot did not close a circle at the th steps This

causes from a small fo ot slip o ccurred in eachsupport

the friction co ecientbetween the fo ot and the o or

In Figure we can see the slip as blur of

the fo otholds that should b e xed p oints At the same

time the fo ot slips around z axis and that diverted the

rob ot from the planned walking direction

Despite of the error caused by slip of the feet it is

imp ortant that a stable dynamic walk was realized by

the DLIPM tra jectories We b eliev e the error can

b e reduced byintro ducing additional feedbackcontrol

Figure Snapshots of humanoid simulator

to the DLIPM system F or example F ujimoto and

Kawamura prop osed a feedback comp ensation of yaw

axis rotation by arm swing motion By intro ducing

suc h metho ds wecan exp ect more accurate walking

direction control

0.7

0.6

Summary and Conclusions

0.5

In this pap er weintro duced the ThreeDimensional

0.4

Linear Inv erted P endulum Mo de DLIPM that is

y [m]

alking control in a D space We discussed

useful for w 0.3

a nature of the DLIPM and prop osed a simple walk

0.2

ing pattern generation that can sp ecify walking sp eed

The walking pattern was tested on a

and direction 0.1

dof h umanoid rob ot in a dynamic simulator and a

0

dynamically stable walk along a circle was successfully

−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 ulated

sim x [m]

Acknowledgments

Figure Hip motion and foot plac e

This researc h was supp orted by the Humanoid

Rob otics Pro ject of the Ministry of Economy Trade

245

Sano A and Furusho J Realization of Natural and Industry through the Manufacturing Science and

T ec hnology Center Dynamic Walking Using The Angular Momentum

Information Pro c of ICRA Cincinnati

pp

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