a friction constraint method for the force distribution of quadruped robots

7/25/2019 A Friction Constraint Method for the Force Distribution of Quadruped Robots

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Proceedings of the 1999 IEEYASME

International Conference on Advanced Intelligent Mechatronics

September 19-23, 1999 Atlanta,

USA

A

Friction Constraint Method for the Force

Distribution

of

Quadruped Robots

Debao

Zhou

and

K. H. Low

School

of

Mechanical and Production Engineering

Nanyang

Technological

University,

Singapore

639798

Abstract--One of the important issues of walking machine

active force control is a successful distribution of the body

force to the feet to prevent leg slippage. In this article, a new

force distribution method, the Friction Constraint Method

FriCoM), is introduced. The force distribution during the

walking

of

a typical quadruped crawl gait is analyzed by using

the FriCoM. Computation results show that the distributed

forces of the feet are continuous during the walking. This

reflects the change of the force distribution during actual

conditions. The comparison with a pseudo-inverse method

shows that the FriCoM is more practical. The FriCoM also

requires less computation time than that by an incremental

optimization method. Some problems, such as the singularity,

in the application of the FriCoM are discussed. The FriCoM

will be used in the active force control of a quadruped robot

that is taken as the platform for the research on the study of

terrain adaptation.

Index

terms-- quadruped walking robots, dynamic equations,

force

distribution friction constraints

the Friction Constraint Method FriCoM), for the control

of

quadruped walking robot is introduced. Further discussion

on the method can be found in reference

3.

This paper is organized as follows: Section I1 provides the

problem statement and reviews of the previous force

distribution optimization methods for walking machines.

The proposed method FriCoM) is introduced in Section

111. Section

IV

presents the force distribution results

obtained by the FriCoM during one walking period

of

a

quadruped crawl gait whose parameters are similar to that

given by Liu and Wen

[4].

he results given by the FriCoM

are similar to those given in reference 4. By using the

physical parameters

of

the proposed quadruped robot,

results of the FriCoM are compared with those obtained

from the pseudo-inverse method

[2, 5,

6,

71

and an

incremental method

[8].

The effectiveness and efficiency of

the proposed approach are also demonstrated. Certain

problems with the FriCoM are discussed in Section

V.

Some concluding remarks are

given in Section

VI.

I. INTRODUCTION

11.

PROBLEM ONSIDEREDNDPREVIOUS

ORKS

An important problem for a walking machine with active

force control is how to successfully distribute the body

force to the feet. These forces are usually indeterminate

during the walking motion because of the closed-chain

system [

11

The optimization of the leg-end force values is

to select a solution of a reference force for the active force

control to prevent foot slippage and to support the body. It

is also defined as “the leg force distribution problem”

[2].

The active force control of a walking machine requires an

efficient approach to optimize the foot force distribution in

real time. In the control system of the walking machine,

much time is required for the motion planning, the

computation of the inverse kinematics of the legs, the

processing signal coming from sensors, and the motion

control of the body and legs, etc. All these works must be

done on-line,

so

it is necessary for the control system to

quickly obtain a suitable force distribution and the force

setpoint in the active force control of legs.

The objective of this work is to develop an effective

method to obtain quickly a suitable force distribution.

To

achieve such an objective, a new force distribution method,

0-7803-5038-3/99/ 10.00 1999

IEEE 866

There are two assumptions for the force distribution: 1) the

foot contacts with the ground by a point or a small area,

and 2) the walking machine walks in a statically stable

way. Based on these assumptions, the equilibrium

equations can be expressed in terms of the coordinate

system

( o b Y b z b )

see Fig. 1):

.

2

f,

+ F x

= 0 ,

=I

i= l

=I

Authorized licensed use limited to: Khajeh Nasir Toosi University of Technology. Downloaded on December 21, 2009 at 05:46 from IEEE Xplore. Restrictions apply.



where

n

is the total number of the supporting legs;f k J , and

fl, are the forces acting at the i-th foot along x , y, and

z

axis

of the body-fixed frame 0,

xb

Yb Zb), respectively;

F,,

F,,

F,,

T,,T,, are the forces and torques acting at the frame

coordinates of i-th leg-end in the body-fixed fram e;

x,, yc,

and z are the coo rdinates of the c enter of the gravity of the

walking machine in the frame

(ob

Xb

Yb

Zb .

For

simplicity,

the frame

(obxb

Yb

zb

will always keep its orientation the

same as that of the terrain-fixed frame

(0,X, Z,)

in the

following discussion.

(obXb

Yb

Z b ; Mg

iS the weight Of the body;

X i ,

y ; ,

Z;

are the

Fig. 1. Frame sy stems and parameter definit ionof a 4-legged robot.

In ca se of three supporting legs (n

=

3) , the forces on these

feet are indeterm inate because there are

9

unknown forces

but only six e quations. In orde r to determine the reference

force in an active force control, the solution of the force

values

must

be optimized. Many me thods had been applied

to solve this problem. To control the 6-legged robot,

AMBLER, Nagy et al. [9] used a method which was

combined with the Hooke’s aw for the compliant contact of

the feetlground. They did not, howeve r, consider ny

‘horizontal’ force com ponent. The pseudo-inverse method

[7] is one of the most comm on-used methods. Lehtinen

[2]

and Alexandre [6] use this method to obtain the force

distribution for 6-legged walking robots. Klein and Chung

[5] also proposed this m ethod, but the lateral forces were

not considered. A stiffness matrix method was developed

by

Gao

et

a1

[

10, 1 ]. In this method, the force distribution

is obtained by mod eling the stiffness in the legs and their

effects on the leg-foot contacts. Howe ver the stiffness of

the whole body is muc h more difficult to determine because

of the complex structure of the legs, body and actuators.

The incremental methods include the compact-dual LP

method by Cheng and Orin [12], a method by Klein and

Kittivatcharapong [13] and a m ethod by Gardner [14].

Some optimization methods such as Simplex [I21 are used

to obtain an optimal force distribution solution. A

hypothesis of a zero interaction force field wa s introduced

by Waldron [15]. The components of each pair of contact

forces projected on the joining line of their corresponding

contact points are equal. The structure of this force field

equilibrating force field) is analogous

to

a helicoidal

vector field and is the planar components of the contact

forces. This assumption provides additional equations

desired for a complete solution. Liu and W en [4] c ombined

867

the forcelm oment equilibrium equations with some optimal

relations and obtained a set of linear equations. To solve the

discontinuity problem of optimal solution, they used the

principle of the convex combination [4].

A new force distribution for the motion of the leg-ends of a

quadruped-walking machine is proposed and introduced

here. The a im of this m ethod is to find a force distribution

that is quickly satisfied by the friction constraints.

111.

FRICTION

ONSTRAINTETHODFRICOM)

For the statically stable walk of a qua drupe d robot, there

are

two

kinds of supporting phases: 3-leg supporting phase

and 4-leg supporting phase [16]. A m ethod is proposed for

the force distribution problem in these

two

phases by using

the friction constraints. The method is called the Friction

Constraint Method, sho rtly the FriCoM.

A .

Force Distribution fo r a 3-leg Supporting Phase

For the force distribution during a 3-leg supporting phase,

both the vertical

(zb

direction) and horizontal

(

or

Yb

direction) forces are considered together with the friction

constraints.

Step

I:

Vertical o rce s without considering the horizontal

forces

By assuming that all the horizontal forces are zero, the

vertical forces acting at the three leg-ends satisfy the

following equations:

s,

+Az fm - Mg

=

0

Lap aq d r = M a c

T,,

,lc, hi.,k

f d r

= Mgxc

+ T.,

2 4

2b)

2c)

where p , q, r are the numb ers of the three suppo rting legs

(p,

q , r

=

1,

2 ,

3

or 4,

and p

f

q

f

r ) . Equations

(2)

can be

used to solve for the three vertical forces acting at the leg-

ends.

Step

2:

Forces acting at onefo ot

The vertical forces obtained from Equations 2) are only

approximate values and the horizontal forces must satisfy

the friction constraints in order to keep a statically stable

walk. Acco rding to the slippage constraints, the horizontal

forces acting at one leg-end can be estimated by f k

= pKJk

and A, = pKiJz ( i

=

p ,

q or r) , where p is the friction

coefficient. The parameters Kk?

Kiy

1

2

IKkl 2 0 and 1 L

I&

1

0) are used to estimate the horizontal forces. The

forces acting at the i-th leg-end, f k ,

A

and

Az

can then be

obtained. The i-th foot here is defined as the

reference oot.

Step

3:

Forces acting at other eet

The forces acting at the reference foot can be estimated

during the second step of the force distribution method.

These force values will then be used to co mpute the forces

acting at the other two feet.




Assuming that the p-th leg-end is the reference foot, we can

rewrite Equations 1) as:

in which

B F = Q , 3 4

Q = bqX

f q y

f ,

f q z f n T

3b)

-T

+ f ,Y ,

-

,.,

k 3 C )

F = { - 4 - L z

-Fy-Ly

M g - t -

.

M a c-

T

-

, Y p +

fpyz p

- Mg4 - T + f P P P P JP

1 1 0 0 0

0 0 1 1 0 0

- Y , - Y , x , x r

0 0

The forces acting at other two leg-ends can then be

obtained by using Equations (3).

Step 4 : Searching fo r the suitable,result which is satisfied

by the frictio n constraints

From the pervious three steps, the forces acting at the three

leg-ends are obtained. It is still unknown w hether the forces

satisfy the friction constraints: rJ p , V; l/JbS p nd fu>O.

If the friction constraints are not satisfied, the first three

steps must be re-computed by changing some parameters

until the friction constraints are satisfied. Note that the

horizontal forces equal to zero

Change the magnitude o

Fig. 2 . Evaluation procedures

of

the FriCoM.

following parameters will affect the results: a) Friction

factors between the leg-end and ground,

p,

K , and

Kfy;

b)

Magnitude of external generalized forces Fx,

Fy, Fz,

I; Ty

and

Tz;

c) Direction of the horizontal force f n and J,,.

Furthermore, the following steps are specified in the

searching algorithm: a) Increasing

K ,

from 0 to 1; b)

Increasing

K ,

from 0 to 1 ; c) Changing the direction off,;

d) Changing the direction ofJy. W henJz > 0,

If

I

z

nd

xyl I

dZ suitable result which is satisfied by the friction

constraints is obtained. The main steps of the force

distribution method are illustrated in Fig. 2.

B. Force Distribution fo r a 4-leg Supporting Phase

For the statically stable walking of a quadruped machine,

there is one 4-leg supporting phase between two 3-leg

supporting phases. As shown in Fig. 3, from time

t ,

to

t,,

there is a 3-leg supporting phase and the supporting legs are

1, 2 and 3. From time t2 to t , , there is a 4-leg supporting

phase with the first leg being lifted and the fourth leg being

lowered. From time t , to

t4,

there

is

another 3-leg supporting

phase and the supporting legs are 2, 3 and 4. In a 4-leg

supporting phase, the forces at each foot should be

smoothly changed from the end of the previous 3-leg

supporting phase to the beginning of the next 3-leg

supporting phase. Therefore in a 4-leg supporting phase,

the distribution of the forces can be any desired continuous

scalar function varying during the supporting-leg changing

phase

[4,

171.

For simplicity, it is assumed that the forces at

each leg-end will change linearly with time during one 4-

leg supporting phase, which can be seen

in

Fig.

3.

3- leg

i

4- leg i 3-leg

5

s u ppor t in g ph a s e . s u ppor tin g ph a s e ; s u pport in g ph as e

Leg 1

t2

t4

Time s)

-_-_

ifting leg upporting leg

Fig.

3.

Foot force dis tribution for a support-leg-changing phase.

IV. SIMULATION

OF

FORCEDISTRIBUTION

SIN

A

QUADRUPEDRAWL AIT

A

periodic nonsingular forwardhackward quadruped

creeping gait is used for the simulation. The gait

parameters, including the duty factor A, leg motion phase

p nd the initial support position xi,

y i ,

z f ) ,determine the

statically stable margin of the gait. They should be chosen

correctly, so that the gait is statically stable. Different

values of the parameters of a statically stable gait were

assumed in different sets of simulations. One simulation

result is shown here to compare with the results given by

868




Liu and Wen [4]. The gait diagram is show n in Fig 4.

The data used in the simulation are as follow s:

1. Foot initial positions: x,=x2=25/24 m, x,=x4=-23/24 m;

y1=y ,=0 .5m,y2=y4=-0 .5m;z l=-0 .6m,z ,= -0 .7

m, z

= -0.5

m,

z4 = -0.6

m;

2.

Duty factors:

A

= 11/12

i

= 1,2,3,4);

3.

Leg motion phase:

p4= 4/12,

=

6/12, fi

=

10/12, p

= 12/12;

4. The x-y plane of the body-fixed frame is parallel to the

horizontal ground;

5 .

The suppo rt surfaces are parallel to each other;

6.

The total body forces are

F, = -50

N,

F,

=

-25

N,

F, = 0

N,

M g = 1000

N and moments are all zero. The

acceleration forces of the legs are ignored;

The locom otion cycle time is T

=

10 s; the stride length

of the gait is A = 1 m; and the velocity of the vehicle is

constant.

7.

Leg

Leg

2

Leg

3

Leg

4

- eg is in supporting phases

Leg is in swing phases

Fig. 4. Gait diagram of a quadruped crawl g ait (p= 11/12)

- .

-

1

-1

0

2 4

6

8

10

Time(s)

Fig. 5. Trajectory of the 1“ leg end.

According to the leg-end trajectory show n in Fig. 5 and the

external generalized forces, by using the computation a nd

control procedure shown in Fig.

6 ,

the force distributed to

the feet by the FriCoM is given in Fig.

7.

Using the body

parameters given by Liu and W en: the body length

L,

=

1.9

m,

the body width

L, = 1

m and the thigh

I = 0.4

m, the

shank link

1, =

0.4 m see Fig. l ,he torques acting at the

three joints can be evaluated by the Jacobian matrix [181 as:

- lS6PciS + 1 2 S ~ P S ( t S + , k ) ‘ l C , P c t S

+

1 2 C P S ( 6 + , k )

-

kC 8P s S

-

2c , P c ( , S + t k )

-

I s n P s S

-

2’,Pc(,S+2k)

- 2 C , P C ( , S + k )

-

P , P C ( , S + , k )

[g [ 0

- ’ l C ( & + 8 k ) + ‘ 2 s ( 6 + k ) ] [ z (4)

1 2 S ( , S + , k )

in which, s

=

sin;

c =

cos;

Ip =

eP;

s = &; Ik ek; ,s+lk) =

( +

ek . he subscriptsS and

k

represent the parameters

on the Joint-S, the Joint-P and the knee joint, respectively

Fig. 10). Results of the torques are shown in Fig.

8.

The

results ob tained from the FriCoM are continuous. Note that

the com putation time for one position is less than

0.01

s

in

a Pentium 200MHz computer.

In addition, we compared the FriCoM with the pseudo-

inverse method

[7]

and an incremental method [8]. It was

found that the FriCoM gave a better result than that

obtained from the pseudo-inverse method if a supporting

leg is near to the center of gravity.

By

employing the

Matlab software, the force distribution according to the

incremental method takes

2

s to

12

s depending

on

the

initial values, while the FriCoM takes less than

0.01s.

Because of the shorter computation time, the FriCoM w ill

be more efficient and suitable in the real-time control of

walking machines.

V.

DISCUSSION

A. Variation

of

the

Force

Distribution during CG Shift

It can be seen from Figs.

7

and 8 that the variation of the

distributed forces is continuous during the continuous

motion Fig. 5 ) of the walking machine. It reflects the

chang e of the fo rce distribution in practical conditions.

Leg-end trajectory Inverse kinematics

is changed to hip-

--*

(Incorporate the

joint angle)

fixed frame

trajectory into each

1

Motion

controller,

such as PID

controller

Motors

each joint

I

Encoders

Fig. 6 . Signal flow chart

of

the computation

of

the force distribution and torques

for

a quadruped robot.

869




-100

J

0 2

4

6 0

10

Time[@

(a)

0 2

6

8

10

Time(@

(b)

' - I

.......... .........

.................i..

- 5 4 I

0 2

4

. 6 8 10

Time(

(c)

Fig.

7.

Force distribution in a lo como tion cycle:

(a) Forces in Z direction;

b)Forces in

X b

direction;(c )Forces in Yb direction.

.................. ................... .................. ................... ..................

.............

?------.

0 2 4

6 10

Time(s)

Fig. 8:Joint torques of the 1 leg end in a locomotion cycle.

B A

Special Case: Flat Terrain

Assuming that the leg-end positions of three supporting

When zp

= z9 =

z the robot w alks on flat terrain. Under this

circumstance, according to Equations 3), the following

three e quatio ns can be solved forJ;Z ( i =

p ,

q or r ) :

legs are

(xp,

yp,

zp), x9,

y9,z9), and

(x,.,

yn

z,),

respectively.

f,. +f,

+f,

= M g 7

5 4

f p z Y p

+ q z Y ,+

f ,Y , =

M a ,

- FYZl- T, 9

t

5b)

f p x p + f

+

, x , = Ty

- F A

+ w v c . 5c)

However, there are six unknowns in the following

equations,

f p x

+ f , +

f

+

F,

= 0

3

6a)

f f , +f,

+ F y

= o 6b)

f p y x p

+fqyxq - f p x Y p s , Y ,

- L Y , +T =

09 6c)

so

it is still an undetermined system. The pseudo-inverse

method

[7]

is then used to solve Equa tions 6).

C.

Singularity

During the solving procedures specified in Equations (3),

singularity would exist. In the following, the methods to

avoid different cases of singularity are de scribed. Again, p ,

q

and r denote the supporting leg number,

When

zp = zq,

the

Jh

leg-end cannot be selected as the

reference foot in order to avoid singularity. Similarly, when

xp

=

xq or yp =

yq,

he Jh eg-end cannot be selected as the

reference foot. Under the second condition, after the

computation of the plh leg-end and

qfh

leg-end forces, if

suitable result is still not available, the body frame can be

rotated an angle of

y

e.g. y= 45 ) along z b axis, then

By varying the angle of 3: the x and

y

coordinates of eac h

leg-end in the rotating kame

( o b

&'

yb' Z,?

will be

different to the previous coordinates in fram e ohxbYb

zh).

The singularity can be eliminated and the force com ponents

of fh , , 'nd

fk'

can then be obtained. For the forces in

frame ( o b Y b z b ) , the transformation can be written as:

When zp = z, and

((x,

=

xq and y q = y,) or (xq = x, and yp =

y J ) , none of the feet can be selected as the reference foot.

But this is a special case in wh ich the

CG

is on the edge of

the supporting polygon at the middle point of the line

through the pth leg-end and the J h leg-end). The force

distribution in this case can be obtained directly.

By

selecting any of the three feet as the reference foot, the

case of

xp =

xq

= x ,

or yp

= yq = y r

will c ause singularities in

the solving of Equations

3).

This is, in fact, an impossible

case as the three legs stand a long the same line.

VI.

CO NCLUDING

EMARKS

Force control is one of the ways to improve the terrain

adaptation of walking machine. The method of the force

distribution is to provide a reference force for the active

force control. Based on the equilibrium equations of a

quadruped-walking machine, a new efficient force

distribution method FriCoM) is introduced. This method is

used to evaluate the force distribution by considering the

870




friction constraints during the walking of a quadruped

machine. Th e following conclusions can be obtained:

1.

The forces distributed to the feet by the FriCoM are

continuous when the center of the gravity of the

walking machine shifts continuously. This enables a

smooth change of the force distribution in practical

conditions.

In some con ditions, the FriCoM yield s a result that

satisfies the friction constraints, whereas such a result

cannot be achieved by the pseudo -inverse method. The

FriCoM takes less computation time than that of an

incremental optimization method. The results o btained

from the FriCoM are quite similar to the results given

by Liu and Wen [4].

3. The com putation time in one posture using the

software Matlab by a Pentium 200MHz computer is

less than 0.01s. The force distribution algorithm is

more suitable in the real-time control of quadruped

walking machines.

2.

A quadruped robot shown in Fig. 9 is being built for future

implementation.

A

2-bar mechanism is selected as the leg

structure. There are

3

degrees of freedom in one leg. The

leg mechanism is shown in Fig.

10.

This quadruped-

walking machine is proposed to walking in various kinds of

terrain including hard-concave terrain, hard-conv ex terrain

and soft terrain. The FriCoM will be applied to the force

control of the proposed walking robot to improve the

terrain adaptation of the robot.

U

Fig. 9. The proposed walking machine.

gears

Fig. 10. Mechanical structure of one leg

VII. ACKNOWLEDGMENTS

The authors would like to thank Dr. Teresa Zielinska for

her various discussions and suggestions on the project.

Thanks are also due to the reviewers

for

their constructive

comme nts. The work is under the support of the Robotics

Research Ce nter, NTU.

VIII. REFERENCES

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Kinematic Chains,”

IEEE Journal of Robotics and Automation,

vol. 4,

no. 6, pp. 657-664, 1988.

[2] H. Lehtinen,

Force Rased Motion Control of a Walking Machine,

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179, Finland, 1994.

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World Congress

on

the Theory of Machines and Mechanisms

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[SI

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C.

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vol. 115, no.

3,

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E.

Orin, “Efficient Algorithm for Optimal Force

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vol. 6, no. 2, pp. 178-187, 1990.

[I31

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on

Robotics and Automation,

vol. 6, no. 1, pp. 73-85

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141 J. F. Gardner, “Efficient Computation of Force Distributions for

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vol. 10, pp. 427-433,

1992.

151

K. J. Waldron, “Force and Motion Management in Legged

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