Mechanism and control of a quadruped walking robot - IEEE Control ...

Mechanism and Control of a Quadruped Walking Robot Hironori Adachi, Noriho Koyachi, and Eiji Nakano ABSTRACT: This paper provides a description of the quadruped walking robot called “TURTLE-1.’’ A new link mechanism named ASTBALLEM is used for the legs of the robot, and highly rigid and easily controllable legs are constructed using this mechanism. Each leg has two degrees of freedom and is driven by two dc servomotors. The motion of the legs is controlled by a microcomputer, and various gaits are generated so that the robot walks not only statically but also quasidynamically. When the walking mode is static, the center of gravity of the robot is kept statically stable. However, when its walking is quasidynamic, a two-legged supporting period is required.

Introduction Interest in robot locomotion has been increasing recently. The research in legged locomotion is especially active because of the potential to traverse irregular terrain. However, legged locomotion has demerits, such as the complexity of mechanisms and control. Many multilegged robots have been developed using various leg mechanisms. Each of these leg mechanisms has certain advantages as well as disadvantages. To overcome the disadvantages, the authors propose a new mechanism for legs. This paper reports on the construction of a walking robot called “TURTLE-1’’ (Fig. l), which employs this mechanism. With the exception of bipeds, most legged robots maintain their body in a statically stable position while walking. This means that, at any particular moment, three or more legs must be on the ground. In the case of quadrupeds, only one leg at a time can be moved. If a quadruped is assumed to walk with constant speed, this means that the foot velocity Presented at the 1987 International Conference on Industrial Electronics, Control, and Instmmentation, Cambridge, Massachusetts, November 3-6, 1987. Hironori Adachi and Noriho Koyachi are with the Robotics Department of the Mechanical Engineering Laboratory, Ministry of International Trade and Industry, Namiki 1-2, Tsukuba-shi, Ibaraki, 305 Japan. Eiji Nakano is a Professor in the Mechanical Engineering Department of the University of Tohoku, Aoba, Sendai-shi, Miyagi, 980 Japan.

Fig. 1. Photograph of TURTLE-1. of the returning period is required to be more than three times that of the supporting period. Therefore, static walking limits walking speed. In order to walk faster, quasidynamic walking is proposed in this paper. There has been extensive research on legged vehicles. Frank [ l ] considers the quadruped robot’s gait and its control, and McGhee and Iswandhi [2] and Klein and Briggs [3] show the potential for legged locomotion over rough terrain. Research on leg mechanisms includes Hirose [4],who proposes a gravitationally decoupled actuator system for energy-efficient legged locomotion, and Kaneko et al. [SI, who develop a four-bar linkage mechanism for leg motion. Lee and Shih [6] report the gait control of a quadruped in static walking. Dynamic walking is studied by Miura et al. [7], who investigate quadruped dynamic walking, and Kajita and Kobayashi [8], who report on biped dynamic walking and introduce orbit energy. Recently, a six-legged vehicle was developed that cames an operator who provides supervisory-level commands [9], [101.

(1) Simple leg structure;

(2) High leg rigidity;

Z

I

B

0

New Leg Mechanism In general, the trajectory of foot motion with respect to the body forms a closed loop. The trajectory is considered desirable if it includes a partial straight line because a straight line trajectory reduces up-and-down movement of the robot’s center of gravity. This results in reduced energy consumption and increased convenience for transporting some loads. From a practical point of view, straight (or approximately straight) line motion is simpler to control. In particular, in an articulated-type leg, it is necessary to control 0272-1708/88i1000-0014 $01 00

14

and coordinate several actuators for straight line motion, although the link mechanism presented here [SI generates straight line motion using only one actuator. The authors have developed a new link mechanism (Fig. 2) for approximately straight line motion. This link mechanism has two degrees of freedom: one is the rotation of bar OA around point 0; the other, a variation in the length of bar OA. The two independent variables are rotation angle 0 and relative length s, while L is the maximum length of bar OA. Point B is the passive slide mechanism along the z axis. The authors named this mechanism ASTBALLEM (Approximately Straight Line 7hree Bar Link Mechanism for Leg Motion). The characteristics of legs using the ASTBALLEM mechanism are as follows:

9 1988

Fig. 2 . Principle of ASTBALLEM.

IEEE

I € € € Control Systems Mogorrne

(3) Approximate straight line motion realized with one degree of freedom;

0 . 3 y

/ 2.4

(4) Reaction force of robot weight has little influence on driving force. Now consider the kinematics of ASTBALLEM. In the coordinate system shown in Fig. 2, the length of bar OA is equal to s times L (with s greater than 1) and the length of bar BC is equal to a. Furthermore, when the length of AB is equal to ka and the length of AC is equal to (1 - k ) (with k between 0 and l), we obtain the kinematics

x = (sL/k) sin 8 Z

= SL cos 8

-

/

(1)

L(k-’

-

1)

+I

The inverse kinematics of ASTBALLEM is obtained directly as

o = tan-’ s

=

-0.3

Fig. 4. Gradient of the trajectory.

Robot‘s body

(3)

(p/q)

(1/L)(p2 + q’”’

Sensors and Control System

(4)

where (5 1

p=kx =z

+ (1

- k)(d

- x’)’’’

(6)

Figure 3 shows the trajectory of point C under the conditions that a = 300 mm, L = 25 mm, k = 0.25, s increases from 1.0 to 2.4, and 8 is given in steps of 10 deg. As seen in Fig. 3 in the range where 1.0 < s < 1.2 and -60 < O < 60 deg, the trajectory is approximately a straight line. Moreover, from the gradient of the trajectory in Fig. 4, it is found that the trajectory is nearly horizontal when s is smaller than 1.4 and the absolute value o f x is relatively small. Therefore, it is convenient to require that (1) a trajectory in this range is used for leg motion in the supporting period, and (2) a trajectory with a relatively large value of s is used for the returning period. This means that the leg motion of the supporting period can be realized by exercising only one degree of freedom. Consider a further effect due to the reaction force of the robot’s weight. The diagram

-100

walking robot. This robot is 500 mm long, 330 mm wide, 380 mm high, and weighs 17 kg. The link dimensions used in this robot are a = 300 mm, L = 25 mm, and k = 0.25. Each leg has two degrees of freedom driven by dc servomotors. The length of the link s is between 1.O and 2.5, and it is vaned by a ballscrew driven by a worm gear. The range of the link rotation 8 is between +90 deg, and it is varied by a gear mechanism as shown in Fig. 7. A leg retraction mechanism is installed at the end of each leg. With this mechanism, the length of the leg can be shortened about 30 mm by on/off control. (However, the retraction mechanism was not used in the walking experiment because the mechanism lacked sufficient power to accomplish the task.) The robot is separated from the control computer and the motor driver, and they are connected with cables.

Fig. 5. Reaction force effect. Some part of the reaction force of the robot’s weight F, becomes the force F2 that rotates bar OA .

1.o

0

esl -1.0

Fig. 6 . Reaction force variation. in Fig. 5 shows that when a leg supports the robot body, the reaction force of the robot’s weight F , acts at point C, and some part of this force produces the force F2 that rotates bar OA. The magnitude of this reaction force effect is indicated in Fig. 6, where the axis of the ordinate indicates the ratio of the force acting on bar OA F2 and reaction force F l . From this figure, it is shown that the reaction force has little influence on the rotation of bar OA during the time of the supporting period.

In general, two types of sensors-internal and external-are necessary for an intelligent robot. In the case of TURTLE-1, the internal sensors include the following: potentiometers (8 channels) to detect the link length and the link angle; tachometer generator (8 channels) to detect the angular velocity of the motor; and two attitude sensors. For external sensors, microswitches for detecting contact with the ground and force sensors are attached in each foot. Figure 8 shows the overall control system including the sensors mentioned earlier. The robot is controlled by a microcomputer. The motor speed command signal from the computer is transmitted to the motor driver through the digital-to-analog (DIA) converter. The motor driver has a velocity control circuit that sends electric current corresponding to the difference between the speed command signal and the feedback signal from the tachometer generator. Position control of the motor is achieved with software. In the computer, the desired position and the present position signal from the potentiometer are compared, and a motor speed command signal is sent to the motor driver based

Construction of Quadruped Robot Mechanism 1.o

Fig. 3. Trajectory of point C.

October I988

The mechanism ASTBALLEM was applied to construct the TURTLE-1 quadruped

Fig. 7. Leg driving mechanism.

15

( 2 ) Existence of a three-legged supporting period,

I

I

4

CPU 8086187 Force sensor 4 ch

M

I10 Fig. 8. Control system of TURTLE-1 on the computed difference. In this system, 8 channels of D/A and 22 channels of A/D are needed. Moreover, 4 bits of digital input for microswitches and 4 bits of digital output for control of the leg retraction mechanisms are also needed.

Quasidynamic Walking First, TURTLE-1 was tested for static walking. In this manner of walking, three legs are always on the ground and only one leg is in the air. The projection of the center of gravity on the ground is always within the support polygon, which is made by connecting the footpoints of the supporting legs. In other words, the duty factor (fraction of a locomotion cycle that each leg spends in contact with the ground) must be larger than 0.75. Because of this duty factor, the foot velocity of the returning period is required to be more than three times that of the supporting period. Fast walking can be defined by a duty factor of less than 0.75, but the robot is not capable of fast walking in the static walking mode. Reducing the duty factor requires an increase in the foot velocity of the supporting period. As the duty factor gets smaller, it becomes necessary to support the body of the robot with only two legs. The manner of walking that has both the twolegged supporting period and the threelegged (or more) supporting period is named quasidynamic walking. Now consider a gait analysis of a quasidynamic walking robot. To simplify the analysis, the following assumptions are made: (1) The one-legged supporting period does not exist (implying that the value of the duty factor 0 is between 0.5 and 0.75),

16

( 2 ) The gait has symmetry, i.e., the phase difference between the right front leg and the left front leg is a half-cycle, and the same relation exists for the right rear and left rear legs.

The phase difference q5 is defined to be the phase difference between the right front leg and the left rear leg, as shown in Fig. 9. In this figure, the x axis indicates foot position in the direction of motion, d is the foot stroke, and the axis of the abscissas indicates the time normalized by the gait cycle. The right-up line expresses the motion of the returning period, the right-down line expresses the supporting period, and vertical motion is not considered. Imposing these assumptions, we can classify the gait with duty factor 0 and phase difference as parameters. Varying q5 from 0 to I , the gait is classified into the 12 types listed in the Table. In the Table, each type of gait is characterized by the following points: (1) Existence of a four-legged supporting period, X

(3) Frequency of a two-legged supporting period in a cycle, (4)Which pair supports the body during a two-legged supporting period, and (5) Order of returning during a two-legged supporting period. Now consider a suitable gait using these characteristics. First, a gait that has a lateral pair, two-legged supporting period is undesirable since the robot loses horizontal freedom of motion in the frontal plane. In the gait whose order of returning is rear-front, the support triangle moves from front to rear and the body is not maintained statically stable even when three feet are on the ground. Moreover, from the standpoint of stability, it is better to have a shorter, two-legged supporting period. For the constant value of duty factor 0,the gait that does not have a fourlegged supporting period has a shorter twolegged supporting period. Judging from these factors, gait type 3 is the most suitable for quasidynamic walking.

Robot Motion During Two-Legged Supporting Period As an example of a quasidynamic walking gait, the feet position-time trajectory with the conditions of 0 = 0.7 and q5 = 0.2 is shown in Fig. 10. In this figure, the twolegged supporting period is seen at A-B and C-D. Next, we analyze the robot motion of this period. Figure 11 shows the top view and the skew view of the robot at time A . Time A is a beginning time of the two-legged supporting period. The skew view is seen from the extended line QR, here Q and R are constant points of the supporting legs with the ground. Now we consider the motion of the robot during the two-legged supporting period as that of an inverted pendulum whose length varies. In the skew view of Fig. 1I , when the robot’s center of gravity G moves along the orbit that is parallel to the U axis, the equation of motion is as follows, where m is the mass of the robot, h the height of G, and g the acceleration of gravity. mu

0.5

3 !

1.o

1: Right front leg 2:Left front leg 3:Right rear leg 4: Left rear leg

Fig. 9. Feet position-time trajectory. The phase difference between 1 and 2 and between 3 and 4 is 0.5. I#J is a phase difference between 1 and 4 (or 2 and 3).

-

(mg/h)u = 0

(7)

The orbit energy [8] E is expressed as follows:

E

=

mu2/2 - (mg/2h)u2

(8)

From Eq. (8), it is found that when the kinetic energy m u 2 / 2 is larger than the potential energy (mg/2h)u2at time A , the center of gravity G is moved into the opposite side;

IEEE Conlrol Systems Magazine

Table Classification of Gait

4

Type 1 2 3 4

5 6 7 8 9 10

ThreeLegged Supporting

E* E DNE DNE

DNET E E E

DNE E E E DNE DNE

E E DNE E E E

DNE E

E E

0

0

-p

- 0.5 p - 0.5 p-0.5-1-p

1 - 0 1 -p 0.5 0.5 0.5 /3

-

-

-

p

11

12

FourLegged Supporting

1.5

______________

P 1.5

-

1.5 -

p

-

-

p

1.0

Two-Legged Supporting Frequency

Order of Returning

Supporting Pair

Simultaneous Front-rear'F Front-rear Front-rear Rear-front Rear-front Rear-front Simultaneous Front-rear Front-rear Front-rear Rear-front Rear-front Rear-front

Diagonal Diagonal Diagonal Diagonal Lateral Lateral Lateral Lateral Lateral Lateral Lateral Diagonal Diagonal Diagonal

2 2

~

*E = Exists. tDNE = Does not exist. $Front-rear means that first the front leg returns and next the rear leg returns in a two-legged supporting period

i.e., the robot is able to continue walking. This condition is expressed as follows, where uo i s the value of U at time A .

i

U0

>

lull1 =Pluol

Thus, when the condition of v, is met, the robot continues walking. In the gait 4 = p - 0.5, and d, and d2 are expressed as follows:

(9)

where

p

r A

B

C D 1 : Right front leg 2: Left front leg 3: Right rear leg 4: Left rear leg

Fig. 10. Feet position-time trajectory of type 3 (0 = 0.7, 4 = 0.2).

=

(g/h)"2

(10)

Equation (9) is about the U axis, so we reform it about the x axis, where zjr is the velocity along the x axis at time A , and d, and d2 are the foot positions of supporting legs at time A :

(12)

dz

(13)

=

(1 - PW(2P)

Using these equations, Eq. (11) now becomes V,

> (3

-

4P)dp/(4P)

(14)

After solving Eq. (7) with the initial condition of U = uo and U = vi at t = 0, we obtain the following solution: U

Top view

d, = (2 - 3P)d/(20)

= [(ug

+ vl/p)eP' + (uo - ~ ~ / p ) e - ~ ' ] / 2 (15)

This solution is rewritten to obtain the solution about the x axis: x = [{2v, -

-

{2v,

p(dl

+ d2)}eP'

+ p(dI + d2)WP'1/(4p) (16)

Here v, is the velocity along the x axis at t = 0. During the two-legged supporting period, as the body moves, as expressed in Eq. (16), the supporting legs must be driven contrariwise. The time of the two-legged supporting period t2 is calculated as follows:

/

t2 =

(W) In [(Vi

-

~OP)/(VI+

U0P)l

Skewview

Fig. 11. Robot state at time A .

October 1988

17

In Fig. 10, the velocity of the supporting legs during the two-legged supporting period is shown to have the same constant velocity as that of the three-legged supporting period. However, since the body moves as expressed in Eq. (16), this is not correct. Therefore, strictly speaking, the duty factor must be recalculated using t2. In general, the value of this duty factor is smaller than the duty factor used for analysis of the gait.

Experimental Results An experiment was made on both static and quasidynamic walking. Before the robot begins to walk, the control computer calculates the foot trajectory and makes the reference table. Figures 12 and 13 show the output of the force sensors attached to the feet during walking. The higher level indi-

cates that a leg receives the robot’s weight, i.e., the leg is on the ground. Figure 12 shows the data obtained for static walking (6 = 0.75),and Fig. 13 shows the data obtained for quasidynamic walking ( p = 0.7) at a walking speed of 30 mmlsec for static walking and 50 mm/sec for quasidynamic walking. In Fig. 12, it is found that there are four types of supporting patterns. In part 1, the left rear leg is returning; in part 2, the left front leg is returning. In the case of quasidynamic walking, although there is no clear supporting pattern like that shown in Fig. 12, it is found that there are two two-legged supporting periods for each cycle. Theoretically, a two-legged supporting pattern and a three-legged supporting pattern should be seen. The difference between an actual robot and the inverted pendulum model, along with the effect of returning leg motion, account

for the disappearance of a clear supporting pattern in quasidynamic walking.

Conclusion In this paper, a new leg mechanism is proposed, and the construction of a quadruped walking robot that employs this mechanism is described. The legs of the robot have merits, such as rigidity and controllability. Moreover, quasidynamic walking is proposed as a means to attain faster walking. The gait and motion of the robot are analyzed, and a walking experiment is conducted. Even with the advances described in this paper, TURTLE-1 and its walking have not been perfected. In the future, it is planned to increase the degrees of freedom and to perfect the dynamic walking.

Acknowledgments

r

The authors would like to express appreciation to Mr. Abe and Dr. Nakamura of the Mechanical Engineering Laboratory for their advice and encouragement. Thanks also to Mr. Tsuji and Mr. Nakahara of Hitachi Zosen Technical Research Laboratory, Inc., for their kind help in designing the hardware.

I 1 1 2 , 3, 4 ,

References

Time (1 seddiv) Walking speed: 30 m d s e c Stroke: 80 mm

Fig. 12. Output of foot force sensor ( p = 0.75).

w

A+

-

Right front leg

Leftfront leg

VLPLJL

U

I

I

I

I

I

I

Time (1 seddiv) Walking speed: 50 mm/sec Stroke: 80 mm

Fig. 13. Output of foot force sensor (0 = 0.7).

18

I

A. A. Frank, “Automatic Control System for Legged Locomotion Machines,” USC Rept., p. 273, 1968. R. B. McGhee and G. I. Iswandhi, “Adaptive Locomotion of a Multilegged Robot Over Rough Terrain,” IEEE Trans. Syst., Man, Cybern., vol. SMC-9, no. 4, pp. 176182, 1979. C. A. Klein and R. L. Briggs, “Use of Active Compliance in the Control of Legged Vehicles,” IEEE Trans. Sysr., Man, Cybern., vol. SMC-10, no. 7, 1980. S . Hirose, “A Study of Design and Control of a Quadruped Walking Vehicle.” Inr. J . Robotics Res., vol. 3, no. 2, pp. 113-133, summer 1984. M. Kaneko, M. Abe, and K. Tanie, “A Hexapod Walking Machine with Decoupled Freedoms,” IEEE J . Roborics Autom., vol. RA-1, no. 4, pp. 183-190, 1985. T.-T. Lee and C.-L. Shih, “A Study of the Gait Control of a Quadruped Walking Vehicle,” IEEE J . Roborics Autom., vol. RA2, no. 2, pp. 61-69, 1986. H. Miura, I. Shimoyama, M. Mitsuishi, and H. Kimura, ‘‘Dynamic Walking of Quadruped Robot (COLLIE-1),” Proc. 2nd ISRR, pp. 317-324, 1984. S . Kajita and A. Kobayashi, “Dynamic Walk Control of a Biped Robot with Potential Energy Conserving Orbit,” Trans. SICE (in Japanese), vol. 23, no. 3, pp. 75-81, 1987. K. J. Waldron and R. B. McGhee, “The

IEEE Control Systems Magazim

Adaptive Suspension Vehicle,” IEEE Contr. Sysr. M a g . , vol. 6, no. 6, pp. 7-12, 1986. [lo] C.-K. Tsai, H.-C. Wong, and D. E. Orin, “Modified Hybrid Control for an Electrohydraulic Robot Leg,” IEEE Contr. Syst. M a g . , vol. 7, no. 4,pp. 12-18, 1987.

Hironori Adachi was born in Japan in 1958. He received the B.S. and M S . degrees in engineering from Keio University in 1981 and 1983, respeclively. He joined the Mechanical Engineenng Laboratory of the Agency of Industnal Science and Technology of the Uinistry of International Trade and Industry at Tsukuba Science City, where he is currently a Research Scientist i n the Autonomous Machinery Di-

vision of the Robotics Department. His current interests include control theory and mobile robots.

Noriho Koysehi was born in Kyoto, Japan, in 1957. He received the B.S. degree in mechanical engineering from Kyoto University in 1979. He joined the Mechanical Engineering Laboratory of the Agency of Industrial Science and Technology of the Ministry of Intemational Trade and Industry, Japan, in 1979, and is currently a Research Scientist in the Autonomous Machinery Division of the Robotics Department. His interests currently relate to mechanisms and control in robotics, rehabilitation engineering, and production engineering. He is a member of the Robotics Society of Japan, the Japan Society of Precision Engineering, and the Japan Association of Automatic Control Engineering.

Eiji Nakano received the B S , M S . and PhD degrees in engineenng from the University of Tokyo in 1965, 1967, and 1970, respectively. He had been working for the Mechanical Engineering Laboratory of the Agency of Industrial Science and Technology of the Ministry of International Trade and Industry from 1970 to 1987. During the time he has been at the Mechanical Engineering Laboratory, he has developed a number of new types of robots. Some of these are: a pair of anthropomorphic manipulators (MELARM), a patient care robot (MELKONG), an omnidirectional vehicle (ODV), a three-dimensional working wheelchair (MELCHAIR), a six-legged walking robot that can climb up and down stairs (MELCRAB),and so on. Since 1987, he has been a Professor at the Mechanical Engineering Department of the University of Tohoku .

Doctoral Dissertations The information about doctoral dissertations should be typed double-spaced using the following format and sent to: Prof. Bruce H. Krogh Dept. of Electrical and Computer Engrg. Carnegie-Mellon University Pittsburgh, PA 15213 University of Groningen Heij, Christiaan, “Deterministic Identification of Dynamical Systems.” Date: June 1988. Supervisor: Jan C. Willems. Current Address: Department of Mathernatics, University of Limburg, P.O. Box 616, 6200 MD Maastricht, the Netherlands. University of Illinois at Chicago Quinn, Stanley B., “Contributions to Controller Design in the Frequency Domain.” Date: April 1988. Supervisor: C. K. Sanathanan. Current Address: Borg-Warner Automotive Research Center, 1200 Wolf Rd., Des Plaines. IL 60018.

October I988

University of Adelaide Rogozinski, Maciej W., “Self-tuning f i e dictive Control.” Date: December 1987. Supervisors: M. J. Gibbard and A. P. Paplinski. Current Address: Department of Electrical and Electronic Engineering, University of Adelaide, G.P.O. Box 498, Adelaide, SA 5001, Australia.

Xi’an Jiaotong University Zuren, Feng, “High-speed and High-Precision Motion Control of Robotic Manipulators.” Date: March 1988. Supervisor: Baosheng Hu. Current Address: Institute of Systems Engineering, Xi’an Jiaotong University, Xi’an, China.

Xi’an Jiaotong University Naiqi, Wu, “Observability, Controllability, and Stabilization of Decentralized Control Systems-Theory and Design of Decentralized Control Systems.”

Date: March 1988. Supervisors: Baosheng Hu and Renhou Li. Current Address: Institute of Systems Engineering, Xi’an Jiaotong University, Xi’an, China.

Xi’an Jiaotong University Weiliang, Le, “Analysis, Modeling, and Control of Price System Reform in P.R. China.” Dare: October 1986. Supervisor: Baosheng Hu. Current Address: Institute of Systems Engineering, Xi’an Jiaotong University, Xi’an. China.

Xi’an Jiaotong University and City University (London) Jk,Lin, “Steady State Optimization and Control of Large Scale Systems.” Dare: September 1987. Supervisors: Baosheng Hu, Bai Wu Wan, and P. D. Roberts. Current Address: Institute of Systems Engineering, Xi’an Jiaotong University, Xi’an, China.

19