Additional Actuations for Obstacle Overcoming by a Leg Mechanism

Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011

Additional Actuations for Obstacle Overcoming by a Leg Mechanism Tao Li. Marco Ceccarelli LARM: Laboratory of Robotics and Mechatronics, DiMSAT University of Cassino, Via Di Biasio 43, 03043, Cassino (Fr) Italy, (e-mail: {taoli, ceccarelli} @unicas.it) Abstract: This paper presents a solution for improving the obstacle overcoming ability of LARM leg mechanism. In order to guide the solution, an analysis on gait adjustment is discussed with design guidelines. Kinematic equations of the proposed leg mechanism are formulated for a computer oriented simulation of operation overcoming obstacles. Numerical simulations are made to indicate that the solution is feasible. Keywords: Leg Mechanisms, Biped robots, Gait Analysis, Simulation walking gait, which lead to low walking capability especially for obstacles overcoming.

1. INTRODUCTION In recent years, research works on biped robots have been addressed great passion by corporations, institutes and universities. This is largely because legged locomotion has many advantages, such as high efficiency and excellent suitability in people’s day life environment, like ascending and descending stairs and obstacle overcoming.

At LARM, Laboratory of Robotics and Mechatronics in the University of Cassino, a research line is devoted to the low-cost easy-operation leg mechanism design. A single DOF leg mechanism with linkage architecture has been proposed with low-cost easy-operation features (Liang et al., 2008). It can realize human-like walking gait but cannot realize obstacle overcoming. A further study is proposed here on improving the obstacle overcoming ability of this leg mechanism.

A leg mechanism will determine not only the degrees of freedom of a robot, but also actuation system efficiency and its control strategy. Therefore, leg mechanisms are fundamental for design and operation issues of a biped walking robot (Carbone et al., 2005).

2. GAIT ADJUSTMENT FOR OBSTACLES OVERCOMING

Most of the existing biped robots have leg mechanisms with three actuating motors at the hip, knee and ankle joints. These kinds of leg mechanisms show anthropomorphic motion capability. But they have also drawbacks, such as the control system design is very complex and difficult, the cost is large, and electronics hardware and sophisticated control algorithms are also needed at the same time. In addition, they are not energy efficient because of the “Back-driven” effect and heavy masses of motors with gear boxes (Yoshiaki et al., 2002; Kaneko et al., 2004; Omer et al., 2005), but all the robots mentioned in these papers have the ability of obstacle overcoming.

The foot point of a human-like leg mechanism traces an ovoid curve while walking. The dimension of the ovoid curve is characterized by the length w and height h as shown in Fig. 1. A leg can overcome obstacles with suitable size, like A with the width w1 and height h1 as shown in Fig. 1(a), at its normal gait. But when the size of the obstacle changes into a larger one named B with a larger width w2 and a larger height h2, the length and the height of the gait should be changed immediately from w and h to w’ and h’, respectively, as shown in Fig. 1(b), i.e. adjustments should be made of the leg swinging motion.

On the other hand, reduced DOF (degree of freedom) leg mechanisms have advantages such as low-cost and easy-operation mainly because fewer motors are used (Song et al., 1989; Funabashi et al., 1991; Shieh et al., 1997). Such kinds of biped robots are similar to their costly counterparts in the sense that they can offer the capacities to develop and improve new biped walking algorithms, and they are more affordable. But meanwhile, fewer motors make them not as flexible as their counterparts, and what’s more, most of the reduced DOF leg mechanisms cannot realize adjustment of its 978-3-902661-93-7/11/$20.00 © 2011 IFAC

Fig. 1. Obstacles with different sizes: (a) for standard walking; (b) for augmented DOF motor 6898

10.3182/20110828-6-IT-1002.00351

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

There are two aspects of adjustments during obstacle overcoming. The first one is walking adjustment in approaching obstacles, and the second one is walking adjustment in overpassing obstacles.

robot body. This paper is focused on the motion strategy, by assuming that the robot has been properly equipped. 3. A KINEMATIC ANALYSIS OF A PROPOSED SOLUTION

In Fig. 2, normal gaits are represented by long dashed dotted lines, while the adjusted gaits are represented by continuous lines. As shown in Fig. 2(a), the leg mechanism walks with a normal gait. There is an obstacle A which is not on a suitable position. Thus, if the leg mechanism makes no adjustment but still walks with the normal gait, it will collide with obstacle A. In this situation the first kind of adjustment is essential.

The LARM single DOF leg mechanism is composed of a Chebyshev four-bar linkage LCEDB and a pantograph mechanism PGBHIA, as shown in Fig. 3. The Chebyshev mechanism LCEDB is used to generate a suitable ovoid curve for the point B, in which LE is a crank, CD is a rocker, and EDB is a coupler. Joint at pivot point C is fixed on the frame of the mechanism. The pantograph mechanism PGBHIK is used to amplify the input trajectory of point B into output trajectory with the same shape at point A. The amplify ratio of the pantograph mechanism depends on the length of bar HI and bar IA or the ratio of PA and PB.

There are several kinds of adjustments which can be made toward a successful obstacle overcoming target. One is to increase the value of the step length and another is to make an adjustment to the touchdown point. But in general, a wider step length will make the leg mechanism more unstable, so making an adjustment to the touchdown point is a better way. In this situation, the leg mechanism needs to adjust the step length to a smaller one wM to reach a desirable touchdown point M as shown in Fig. 2(a). In the next step, it will overcome obstacle A still at its normal gait, so the second adjustment is not needed.

A parametric study has been proposed to characterize the operation performance of the proposed leg mechanism as function of its design parameters (Liang et al., 2008). Actually, the lengths of the linkages only determine a proper shape and size of the generated ovoid curve produced by the Chebyshev linkage and the amplification ration of the pantograph mechanism. Therefore, only three parameters a, p, and s can be considered as significant design variables.

However, the leg mechanism could not overcome an obstacle B, which has both a larger width and a larger height than A, at its normal gait. At this moment, both the first and the second adjustments are essential. In the first adjustment, step length is adjusted into a smaller one wN, in order to reach a desirable touchdown point N; and then the second adjustment changes both the length and the height of the gait from w and h to w’ and h’, respectively, as shown in Fig. 2(b). After doing these two adjustments, the leg mechanism can overcome obstacle B successfully.

Y X

Fig. 2. Walking adjustments for overpassing different kinds of obstacles: (a) for a normal size; (b) for a wider and higher size Thus, for a successful obstacle overcoming target, the leg mechanism should have the ability to adjust both the step length and the height of the gait. The size of an obstacle and distance between foot and the obstacle can be known as measured by additional devices that will be installed on the

W’

Fig. 3. A kinematic scheme of the LARM leg mechanism with capability in overpassing obstacles by two additional sliding joints 6899

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

The parametric study have analyzed the shape of the generated ovoid curve as function of three parameters a, p, and s. Main results of the parametric study are:

in which

- The size of the walking step, i.e. the step length and the step height, can be modified by changing the parameter a.

k1  (x B  p) (yB  s)

- By varying parameter p, the ovoid curve generated at point A has only displacements along x-axis without change of the step length and the step height.

k2 

2  2 tan 1[(1  1  k 32  k 24 ) (k 3  k 4 )]

(7)

(8)

b12  yB2  x B2  (l2  b2 )2  p2  s 2  2px B  2sy B 2b1 (y B  s)

(9)

k3  (x B  p) (yB  s)

(10)

- By varying parameter s, the ovoid curve generated at point A has only displacements along y-axis without change of the step length and the step height.

k4 

Although the step length and the step height can be modified by changing the parameter a, the step height only increase when the step length decrease. Thus, if the step length and the step height need to be increased at the same time, it is not enough to change the parameter a only.

Parameters in these equations are indicated in Fig. 3. In addition, in order to avoid collision between the bar HA and obstacle, as indicated in Fig. 3, the angle ankle joint angle φ1 must always be larger than arctan[2h / (w ' w)] .

Based on these parametric study results, for the first adjustments analyzed in part 2, an actuator can be adopted to change the values of the parameter a by making L moveable in a slider guide within the frame of the mechanism. An increase of a will decrease the step length while decrease of a will increase the step length.

By using (1) to (11), numerical simulations can be computed to check the motion properties. 4. NUMERICAL EXAMPLE WITH CONSTANT a & s Simulation programs have been developed in the MATLAB environment to study the kinematic performance of the proposed leg mechanism. Design parameters for normal walking without obstacle are listed in Table 1, in which a is 0.05 and s is 0.03.

Meanwhile, suitable change of s can be obtained by an actuator to increase the step height. An added actuator changes the value of the parameter s as a function of α by making P moveable in a guide within the frame of the mechanism. However, for the purpose of increasing the step height, a suitable function of s is needed to coordinate its motion with crank motion.

Table 1.

When the crank LE rotates around the point L, an output curve can be generated at point B. When a reference frame XY is fixed at point L with X-axis laying along the direction of straight line LC in Fig. 1, coordinates of point B can be formulated as a function of input crank angle α in the form

 x B  m cos   (c  f ) cos    yB  msin   (c  f )sin 

b12  yB2  x B2  (l2  b2 )2  p2  s 2  2px B  2syB (11) 2(l2  b2 )(yB  s)

Design parameters for the first adjustment (expressed in meters)

a

b

c

d

s

m

0.05

0.02

0.0625

0.0625

0.03

0.025

f

p

l1

l2

b1

b2

0.0625

0.03

0.3

0.2

0.075

0.15

A simulation result is shown in Fig. 4. The step length is approximate to 0.48m and the step height is approximate to 0.09m, which are the normal step sizes.

(1)

  2 tan 1[(sin   sin 2   p12  p22 ) (p1  p 2 )]

(2)

p1  cos   a m

(3)

p2  a cos  c  (a 2  m2  c2  d 2 ) 2mc

(4)

y(m)

in which

Kinematic equations of the pantograph mechanism can be formulated in the form

 x A  x B  b 2 cos 2  (l1  b1 ) cos 1   yA  yB  b2 sin 2  (l1  b1 )sin 1

(5)

in which x(m)

1  2 tan 1[(1  1  k12  k 22 ) (k1  k 2 )]

(6)

Fig. 4. Motion output with fixed leg body with a=0.05m and s=0.03m 6900

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

In order to compute the trajectory of point L, which is fixed on the body of the leg mechanism, the foot point A is fixed on terrain with a reference frame xy of which x-axis is parallel to X-axis and y-axis is parallel to Y-axis, as shown in Fig. 3. Coordinates of point A in (5) can be written in matrix form as

The generated ovoid curve is composed of an approximate straight-line segment and a curved segment with a symmetrical shape. The straight-line segment starts at the actuation angle α=π/2 and ends at α=3π/2, which is called the swinging phase. During the next π interval the actuation angle goes from α=3π/2 to α=π/2, corresponding to the coupler curve segment, which is called the supporting phase.

 m    x cos  cos  cos   cos   A   cf  2 1   y    sin   sin   sin   sin    b  (12)  A  2 1 2   l1  b1 

Fig. 6 shows the sequence positions of the leg mechanism with each position of point L on the trajectory of the supporting phase. 5. NUMERICAL SIMULATION OF AN OBSTACLE OVERCOMING PROCESS

Coordinates of point L can be computed in matrix form as  m    x  cos   cos   cos  cos   L  2 1  c  f    y   sin  sin  sin  sin    b  (13)  L  2 1  2   l   1 b1 

Based on the previous analysis, a method has been proposed for the leg mechanism to overcome the abovementioned obstacle B in Fig. 1. Steps of this method are: (1) Decreasing the step length to reach a desirable touchdown point;

A simulation has been computed in MATLAB environment with point A fixed on terrain with values of a and s as shown in Table 1. The result is shown in Fig. 5.

(2) Increasing the step length and the step height at the same time to complete the obstacle overcoming process. Decreasing the step length of the leg mechanism in step (1) can be solved by increasing the value of the parameter a, while increasing of the step length in step (2) can be solved by decreasing the value of the parameter a. Furthermore, a suitable function need to be found for s to solve the problem of increasing the step height as coordinated with α input motion.

0

( 2)



/2

y(m)

 3 / 2

In order to make the first adjustment, parameter a is set as 0.06 to decrease the step length to reach the desirable touchdown point. Simulation result is shown in Fig. 7. The step length is approximate to 0.39m; it is smaller than the normal step length, which is 0.48m as shown in Fig. 4. Simulation result with point A fixed on terrain is shown in Fig. 8.

x(m)

y(m)

y(m)

Fig. 5. Simulation result of trajectory of point L with foot point A fixed on terrain

x(m) x(m)

Fig. 7. Simulation with a=0.06m, s=0.03m with body fixed

Fig. 6. Sequence positions of the leg mechanism during the supporting phase, for a leg with a=0.05m and s=0.03m

In order to make the second adjustment, s is set as a function of α as 6901

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

s  h(1  sin 2 T) , with   [0, 2]

(14)

y(m)

y(m)

in which α is the input angle of crank LE. This function is obtained by repeated simulation for optimal solution.

x(m)

Fig. 9. Simulation result with parameter a=0.04m, and s=-0.03[1+sin(0.5α)] with body fixed

x(m)

Fig. 8. Simulation result with a=0.06m, s=0.03m with point A fixed on terrain Design parameters for the second adjustment (expressed in meters)

a

b

c

d

0.04

0.02

0.0625

0.0625

f

p

l1

l2

0.0625

0.03

0.3

0.2

b1

b2

m

s

0.075

0.15

0.025

s=-0.03[1+sin(0.5α)]

y(m)

Table 2.

x(m)

Fig. 10. Simulation result with parameter a=0.04m, and s=-0.03[1+sin(0.5α)] with point A fixed on terrain

Design parameters of the simulated leg mechanism for the second adjustment are listed in Table 2.

6. MECHANICAL DESIGN AND ITS SIMULATION The proposed obstacle overcoming leg mechanism is shown in Fig. 11 as seen from above. Two additional motors (motor 1 and motor 3) are assembled to change the parameters a and s as discussed, respectively.

Simulation result with body fixed is shown in Fig. 9. The step length is approximate to 0.61m and the step height is approximate to 0.16m; both of them are larger than the normal step length and height, which are 0.48m and 0.09m, respectively, as shown in Fig. 4. Simulation result with point A fixed on terrain is shown in Fig. 10.

As shown in Fig. 11, crank 1 and crank 2 are the input cranks which are actuated by motor 2 through a drive shaft a (motor 2 and motor 3 are under the body frame, thus they can’t be seen in Fig. 11). G1a and G1b are two bevel gears, which are used to transform the motion from motor 2 to drive shaft a. Motor 2, G1a, G2b, drive shaft a, crank 1, and crank 2 are assembled on a movable platform, which is actuated by motor 1. Parameter a changes while the platform moving forward or backward under the actuation by motor 1, which is a screw motor. The platform is assembled on four sliders which can make sliding motion on guide 1 and guide 2.

Thus from the reported simulation results, both the first and the second adjustments can be early achieved for an obstacle overcoming task by using two additional sliding joints. However, sizes of the obstacle are not specified, so the values of a and s are not computed as function of general sizes of obstacles. They just show that the step length and the step height can be adjusted to desirable values for the leg mechanism to overcome an obstacle larger than normal. Actually, once the sizes of an obstacle are specified, values of a and s should be computed specifically for them.

On the right part of Fig. 11 is the driving mechanism used to change the parameter s. It is composed of two clutches (C1 and C2), three bevel gears (G2a, G2b and G2c), and 6902

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

two drive shafts (Drive shaft b and c). C1 and C2 can work together to change parameter s of each leg to the opposite directions at the same time, and also they can work separately to change parameter s of each leg to the same direction, respectively.

based on the analysis of gait adjustment. This method contains two kinds of adjustments toward the step length and the step height. Suitable values of a and suitable function of s can achieve these two aspects of adjustments. A suitable mechanism for the proposed method is designed. Simulation results both in the MATLAB and Solidworks environment showed that the proposed method is effective and feasible.

Crank 1

Frame L1

P1

Drive shaft a

Guide 1

(Motor 2)

C1

Drive shaft b

ACKNOWLEDGMENT The first author likes to acknowledge Chinese Scholarship Council (CSC) for supporting his PhD study and research at LARM in the University of Cassino in Italy for the years 2010-2012.

G2a G1b

G2c G2b

G1a

(Motor 3)

Motor 1 C2

Guide 2 L2

Movable platform

REFERENCES

Drive P2 shaft c

CARBONE, G., CECCARELLI, M. (2005). Legged robotic systems. cutting edge robotics ARS scientific

Crank 2

book. pp. 553-576, Wien.

Fig. 11. Mechanical design of the proposed leg mechanism

YOSHIAKI, S., RYUJIN, W., and CHIAKI, A. (2002).

Fig. 12 and Fig. 13 show the simulation results of the proposed obstacle overcoming leg mechanism. Fig. 12 is the simulation of overcoming an obstacle A at the normal gait of the leg mechanism, while Fig. 13 is the simulation of overcoming an obstacle B at the adjusted gait. A has the maximum size that the leg mechanism can overcome at its normal gait. B is bigger than A, which verifies that the proposed method of obstacle overcoming is effective.

Intelligent ASIMO: system overview and integration. Proceeding of the IEEE/RSJ International Conference on Intelligent Robots and Systems, EPFL, Switzerland September 30-October 4: 2478-2483. KANEKO, K., KANECHIRO, F., KAJITA,S., et al. (2004). Humanoid robot HRP-2. Proceeding of the 2004 IEEE International Conference on Robotics and Automation, New Orleans, USA, April 26-May 1: 1083-1090. OMER, A.M.M., OGURA, Y., KONDO, H., et al. (2005). Development of a humanoid robot having 2-DOF waist and 2-DOF trunk. Proceeding of the 2005 5th IEEE-RAS International Conference on Humanoid Robots, Tsukuba, Japan, December 5-7: 333-338. SONG, S.M., WALDRON, K.J. (1989). Machines that

Fig. 12. Simulation result of overcoming obstacle A (width: 75mm; height: 40mm)

walk-the adaptive suspension vehicle, Cambridge MA: The MIT Press. FUNABASHI, H., HORIE, M., TACHIYA, H., et al. (1991). A synthesis of robotic pantograph mechanisms based on working spaces and static characteristics charts, JSME International journal Series III, 34(2). SHIEH, W.B., TSAI, L.W., and AZARM, S. (1997). Design and optimization of a one-degree-of-freedom six-bar leg mechanism for a walking machine. Journal of Robotic systems, 14(12): 871-880.

Fig. 13. Simulation result of overcoming obstacle B (width: 100mm; height: 50mm)

LIANG, C., CECCARELLI, M., and TAKEDA, Y. (2008).

7. CONCLUSIONS

Operation analysis of a one-DOF pantograph leg

In this paper, in order to make the proposed LARM Chebyshev-Pantograph leg mechanism with ability to overcome an obstacle, a method has been proposed as

Workshop on Robotics in Alpe-Adria-Danube Region,

mechanism. CD Proceedings of the 17th International RAAD'2008, Ancona, n. 50. 6903