On the Stabilization of a Biped Walking Robot Model with Circular Arced Soles

Copyright (£) IF AC Advances in Control Education, Gold Coast. Queensland, Australia, 2000

ON THE STABILIZATION OF A BIPED WALKING ROBOT MODEL WITH CIRCULAR ARCED SOLES Yoshifumi OKUYAMA, Fumiaki TAKEMORI * and Atsuo YABU **

• Faculty of Enginee'ring. Tottori University, 4-101, Koyama-cho Minami, Tottori, 680-8552 Japan Phone: +81-857-31-5211. Fax: +81-857-31-0879, E-mail.·[email protected] **

Kagawa Polytechnic College, Japan

Abstract: 'Ne made a biped walking robot model with circular arced soles which corresponds to a tumbler system with rotational joints and legs. The stabilization of a standing posture of the tumbler system was realized by using a small gyroscope. The stabilizing control system is composed of a one-input two-output (SIMO) feedback system. This paper describes the analysis and design of the feedback system. which controls the posture by the support of one leg and the drive of the crotch joints of the free leg. In addition. the dynamic stabilization of the locomotion by changing the support leg will be studied in this paper. The results of our experiment and computer simulation show that the posture of the robot model can be maintained and the locomotion will be stably sustained. Copyright ~ 2000 IFAC Keywords: \Valking robot. tumbler system, stabilizing control, dynamic stability

1.

INTRODUCTIO~

the support of one leg and the drive of the crotch joints of the free leg. In addition, the dynamic stabilization of the locomotion by changing the support leg will be studied in this paper. The results of our experiment and computer simulation show that the posture of the robot model can be maintained and the locomotion will be stably sustained.

In this paper, we will analyze a walking robot model as a controlled system. The robot model, which we made as a trial, is a tumbler system with rotational joints and legs with feet whose soles are circular arcs (Yabu et al., 1998a: Okuyama et al., 1998a: Okuyama et al., 1998b). It is intentionally designed to fall down when supported by one leg. On uneven surfaces. it is difficult to detect the posture (attitude) of the robot model between the supporting leg and the floor surface (Pannu et al., 1993; Kajita et al., 1995; Yamaguchi et al., 1995). Therefore, we used a small gyroscope as a sensor for attitude detection. First, we tried to control the posture of the tumbler system by fixing the knee joint of the supporting leg in a straight stretched position and driving the actuator of the crotch joint. By simplifying the system in this way, we made the control system as a one-input twooutput system.

2. BIPED WALKIf"C ROBOT MODEL 2.1 Outline of Tumbler System

Our walking robot model is a tumbler system composed of joints and legs with feet whose soles are circular arcs. as shown in Photo 1. The tumbler system consists of two circular arc areas at the bottom (soles) which consist of feet. three joints which consist of knee joints and a crotch joint, and the legs which connect to these parts. The prototype modeL made as a trial, was composed of the following parts. A square aluminum timber was used for the leg parts, and a thick acrylic board was used for the motor brackets. In

This paper describes the analysis and design of the feedback system, which controls the posture by

299

As for the robot model, kinetic energy T. potential energy U and dissipation energy V are expressed as follows:

T =

1

.2

1

'2

2"C1 9a + 2"C1 9 b "?

,

..

+ C3
V =

COS9b

+ C 7 cos 9a + Cs .

:2 2"1 R CPa'

(3) (4) (5)

where

+ l~) + mb(r 2 + l~) + la mbl~ + h marll + mb rl 2

Cl = m a (r2 C2 = Photo 1 Biped walking robot model.

C3

=

C 4 = -mb rl 3

addition , each of the three joints is driven by a DC motor with harmonic gear, and the angle of each joint is detected by a potentiometer.

C 5 = -mb 12 13

C6

This robot model has a wide structure so that the feet will not fall to the right or left; however, it is intentionally designed to fall back or forth regardless of the posture of the free leg when feedback control is not taking place. The physical parameters of the tumbler system are shown in Table 1.

= magI I

+ mbg l 2

C 7 = -mbg l 3 Cs

=

(ma

+ mb)gr.

Here, the parameters of each part of the system is defined as: r: Radius of the curvature of the circular arced

part [m]

Table 1. Physical parameters.

11 : Distance from the center of r to the center of T

12 ma la J

D

0.27 [m] 0.33 [m] 23 [kg] 0.13 [kgm"] 0.021 [kgm"] 4.2 [Nms/rad]

h 13 mb

h K R

gravity of the support leg [m] Distance from the cent er of r to the crotch joint [m] 13: Distance from crotch joint to the center of gravity of the free leg [m]
0.125 [m] 0.28 [m] 1.5 [kg] 0.10 [kgm"] 1.08 [Nm/V] 0.4 [Nrns/rad]

[2:

2.2 Mathematical Model and Control In the first step of our research on stabilizing control of the robot model, each knee joint of the tumbler system was adjusted to a certain angle, as is shown in Fig. 2.1. The attitude of the entire tumbler system was controlled by only using the actuator of the crotch joint. Then, in order to verify the robustness of the attitude control system, the knee joint of the free leg was bent and stretched similar to a walking motion.

On the other hand, driving torque by

The equation of motion for this robot model can be given by using the following Lagrange type equations:

300

T

can be given

T=Ku-J~-D~,

(6)

where dJ = 9b - dJ a, K and u are the torque constant and the input voltage of the motor; furthermore J and D are moment of inertia and viscosity coefficient converted into load side, respectively. Thus, the following equations of motion are obtained: .....

CIOa -r- 2C3 9a COSO a

u

. . .2 - C 3 ifJ a sinifJa

+ JO a

Fig. 1. SIMO feedback control system.

+D~a

+ R~a + C4 0b - JOb + C 56bCOS(
b2

-Dd; -

+ C5~a cos( Oa - rj)b) - J Oa

c 5 6a sin(Oa -

+C7 sin Ob

=

=

F =

C 2 9b + J~b + D~b +C4 6b cos rj)b

+ C4 + C 5 )K (Cl + 2C3 + C 4 + C 5 )K (Cl + 2C3 + J)(C2 + J) - (C4 + Cs - J)2

bl = -(C2

(7)

From these equations, we can obtain the transfer function of the controlled system as:

9b)

Ku,

(8)

P(s) In order to analyze and design this type of control system. Eqs. (7) and (8) were linearized near the origin of variables Oa and Ob. and can be rewritten as follows:

+ 2C3 + J)~a + (D + R)~a + C6 0a +(C4 + C 5 - J)~b - D~b = -Ku, (9) (C4 + C 5 - J)~a - D~a + (C2 + J)~b +D
The angle detection of the crotch joint of the tumbler by the potentiometer indicates the angle of the free leg in relation to the supporting leg as shown in Photo 1, and not the angle of the free leg in the direction of the center of gravity (Yabu et al., 1998b). Moreover, because the position of the center of gravity also changes according to the attitude of the free leg, the following angle correction must be performed.

(11) the following state equation can be obtained from Eqs. (9) and (lO):

+ Bu,

y = Cx

Therefore, we will present the following definition:

14: Distance from the crotch joint to the center

(12)

of gravity of the upper free leg [m] Length of link of the upper free leg [m] 16: Distance from knee joint of the free leg to the center of gravity of the lower free leg [m] mbu: 11ass of the upper free leg [kg] mbl: Mass of the lower free leg [kg]

Is:

where

A

~ F-' [~f ~~: :~:

Then. corrected angle B=F-I[b l b2 OO]T

C= [0o 00 10 0]1 (C2

a12 =

(C2 + C 4 + Cs)D

al3

= -(C2

al4

= (C4

a22

=

a23

=

a24 =

9corr =

+ C 4 + Cs)D + (C z + J)R

all =

a21 =

(13)

2.3 Angle Correction for the Center of Gravity

Assuming that the state variable is written as

= Ax

C(sI - A)-l B.

Therefore, the controlled system can be treated as a SIMO (Single-Input and Multi-Output) feedback system, as shown in Fig. l.

(Cl

:i;

=

tan

!/JCOTr

can be given as

m bl sin Oa -l( 14 m bu + mbl(/S + 16

16

. ) .(14) cos 0 c )

Thus,
( 1.5)

+ J)C6

+ Cs - J)C7 (Cl + 2C3 + C4 + C 5 )D + (C4 + Cs - J)R -(Cl + 2C3 + C 4 + C5 )D (C4 + C 5 - J)C6 -(Cl + 2C3 + J)C7

3. CONTROL SYSTEM DESIGN BY THE LOOP-SHAPING METHOD The feedback control system must be robust against many nonlinearities, disturbances due to outside circumstances, and uncertainties such as

301

the difficulty of finding accurate system parameters for the controlled system, such as the position of the center of gravity and the moment of inertia of the leg, or changes in leg positions and system parameters by driving the knee joint as well as friction.

12

In order to achieve robust control, we used the loop-shaping method in the control system design (McFarlane and Glover, 1990; Okuyama et al. , 1998c) . The algorithm is written below.

'

, , -2 -3

Step 1: Introduce weighting functions \1, H' in series at the front and the rear of controlled system P and thereby shape the open loop frequency transfer function. Here, select V vV so that each of the unstable poles and zeros are not canceled out in It' PF. Step 2: Assuming that the transfer function of controlled system F( s) = W P\1, determine K(s), which minimizes maximum model error

r,me {s]

35

25 2 I 15 1 \

= \1 KW.

Step 3: The controller is given as K(s)

!

05 \

From the physical parameters as shown in Table 1 and Eq. (13), the transfer functions can be obtained as follows:

-0.5

Time

P ( ) 1 S

= ~a(s) = Nt{s)

P. ( ) 2 S

= ~b(S) = N 2(s)

U(S)

(16)

D(s) ,

U(S)

+ 81781.85 2 + 208656.85 + 408930 .0 K2n(S) = S4 + 25.61s 3 + 164.51s 2 + 442.02s + 623.5 K 2d( s) = S5 + 84.75 4 + 2502.0s 3 + 11281.5s 2 + 25374.2s + 81786.0

where

N1(s) = 0.211s2 - 25.262 N 2(5) = 4.768s 2 + 0.246s - 47.066 D(s) = s4

+ 17.775s 3 + 15.1145 2 -

Is]

Fig. 2. Computer simulation result.

(17)

D(s) ,

Support~

- . Free leg

These numerator and denominator polynomials may be simplified in a certain frequency range .

83.856s

-179.372. \\le chose the weighting functions as:

4. SHvIULATION AND EXPERHvIET'\TAL RESULTS

l'V (s) = [Wl (s), W 2(s)] = [195 ~s

Figure 2 shows a computer simulation result for stabilizing control of the tumbler system, when the above compensators were used. Figure 3 shows an experimental result for the inclination angle of the support leg where the knee joint of the free leg was fixed at -90[deg]. Although the attitude of the support leg vibrated in the vicinity of the equilibrium point within a range of 23[deg], the attitude of the entire tumbler system was maintained.

s

+ 4.2~ , -55] + 5.0

\1(s) = 1

(18)

Stabilizing compensators Kl(S) and K2(S) were determined as:

Kl(S)

=

K ( ) 2S

= 4 76

-1.69

.

X

10 4 Kln(S) Kld(S)

(19)

10 3 K2n(S) K2d(s) ,

(20)

X

Next, we show an experimental result in Fig. 4, where the knee joint of the free leg was bent and stretched in a range of -60 ~ -llO[deg]. Here , -60 '" -llO[deg] is a range where the free leg and the support leg can be crossed safely. In this experiment, the amount of time when the knee joint of the free leg was bent and stretched was set at random.

where

Kln(S) = s5 Kld(S)

= S6

+ 29.8s 4 + 272.1s 3 + 1132.95 2 + 2480.0s + 2618.6 + 89.7s 5 + 2925.5s 4 + 23791.5s 3

302

When choosing ka = 4.3 in Eq. (21 ), as a computer simulation for Eqs. (7) and (8 ), the behavior of the angle of the support leg
cb" [rad/s] -8 -1°0

10

20

30

40

SO

60

70

80

90

Time [5)

cjJa [rad]

Fig. 3. Experimental result for fixed knee joint (
-'TT'

7

I

I

//

-4

;

1-

SUppo<1leg

- - Knee ,IOInl

Fig. 5. Behavior of support leg in phase plane. r - .. t

t

1- ' I I

t

i

I

I

I

I

it

I

I ~

L

I

i

;, " "

- 1200

10

20

30

40

50

60

70

80

, 90

t •

Time [5) /

Fig. 4. Experimental results for bending and stretching movements (
-I

5. BIPED LOCOMOTION AND DYNAMIC STABILITY

Fig. 6. Time response of support leg.

Until the preceding section, we studied the stabilizing control which maintains the posture of the robot model (tumbler system) at the standing position, that is , the static stabilization of the tumbler system. However , in this section, we will consider biped walking (locomotion) of the above robot model, that is, the stabilization of an unstable tumbler system by the repetition of a change of the support leg for the free one. In other words, the dynamic stabilization of the robot model is studied.

On the other hand , when choosing ka = 4.5, the angle of support leg was maintained as shown in Figs. 7 and 8. Although a biped locomotion was realized by the control rule, the walking mode is not periodic because of the complexity (nonlinearity) of the robot model as shown in Eqs. (7) and (8). The schematic diagram of the walking mode is shown in Fig. 9. 6. CONCLUSIONS

Assume that the support leg is changed at angle [
[~a] i+l

In this paper, we presented a biped walking robot model with circular arced soles as a controlled object, i.e., a tumbler system with three rotational joints. First, we described the analysis and design of the feedback system, which controls the posture of the robot model by the support of one leg and the drive of the crotch joints of free leg. Then, we examined by computer simulaions the dynamic stabilization of the locomotion by changing the support leg.

= [Xl(tO)]i+l = ka[Xl(tj)]i , (i=0 , 1, 2 .. ·).

(21)

Here, ka is a kind of acceleration coefficient which is determined by the motion of the kicking leg.

303


Institute of Systems, Control and Information Engineers. 11 , pp. 145-153 (i n Japanese). Okuyama, Y et al. (199Sa). Stabilizing Control of a Biped Walking Robot ?-.Iodel Using a Gyroscope. Intelligent Robotic Systems (Ed. ~I. Vidyasagar, Tata ~IcGraw-Hill), pp. 97102. Okuyama, A. Yabu and F. Takemori (199Sb). Attitude Control of Biped Walking Robot t>.'1odel with Circular Arced Soles Using a Gyroscope. Proc. of IEEE International Conference on Robotics and Automation. Leuven, Belgium, pp. 1379-1384. Yabu, A.,F . Takemori and Y. Okuyama (199Sb). Attitude Control of Tumbler Systems with Joints Using a Gyroscope, Journal of the Robotic Society of Japan , 16, pp. 116-121 (in Japanese) . Pannu, S. , H. Kazerooni, G. Becker and A. Packerd (1993) . J.L-Synthesis Control for a Walking Robot. IEEE Control Systems, 16, pp. 543551. Kajita, S. and K. Tani (1995). Dynamic Biped Walking Control on Rugged Terrain Using the Linear Inverted Pendulum Mode. Tmns. of Soci ety of Instrument Control Engineers, 31 , pp. 1705-1714 (in Japanese). Yamaguchi , J., A. Takanishi and 1. Kato (1995). Biped Walking Control Method Adapting to an Unknown Uneven Surface - Realization of Biped Walking Adapting to a Horizontallv Unknown Uneven Surface -. Journal of th'e Robotics Society of Japan , 13, pp. 1030-1037. McFarlane, D, C. and K. Glover (1990). Robust Controller Design Using Normalized Coprime Factor Plant Descriptions, Lecture Notes in Control and Information Sciences, SpringerVerlag. Okuyama, Y, A. Yabu and F . Takemori (199Sc). Attitude Control of Biped Walking Robot Model with Circular Arced Soles Using a Gyroscope. Proc. of IEEE Conference on Conttrol Applications, Trieste, Italy, pp. 756-760.

[rad/s]

CPa

[radJ

o

-1T

1T

-4

Fig. 7. Behavior of support leg in phase plane.

rP.1

[r.d~

i

It

I,!! 1/", Of "

j I

Fig. 8. Time response of support leg. The results of our experiment and computer simulation show that the posture of the robot model could be maintained and also the locomotion could be stably sustained by some control tactics. Our research will be useful not only for the development of biped walking robots but also in regard to the verification of the static and the dynamic stabilization of unstable systems, that is, as a laboratory experiment for Education of Control Technology.

7. REFERENCES Yabu, A., Y Okuyama and F. Takemori (l998a). Attitude Control of Tumbler Svstems with Three Joints Using a Gyroscop~ . Tmns. of [m]

,

."

.

~.

o

';.:. 2

Fig. 9. Schematic diagram of walking mode.

304

3

[m]