Vol. 3 No. 2 Jun. 1999

JOURNAL OF SHANGHAI UNIVERSITY

On Maximal Weight Moment and Overturn Tendency of Multi-Sucker Wall-Climbing Mechanism Qian Jinwu Gong Zhenbang (School o f Mechanical and Electronic Engineering, Shanghai University) Zhang Qixian ( BeOing University o f Aeronautics and Astronautics ) For a multi-legged robot climbing on a vertical wall, how to increase its anti-overturn capacity (AOC)

Abstract

and to detect overturn tendency are most important. They form the basis of off/on-line gait programming and safety monitoring system. In the paper, after investigating the indeterminate statics of an insect-like six-legged wall-climbing mechanism in various support patterns, the authors obtained the analytical expressions of maximal weight moment with respect to different support patterns, and the overturn tendency while payload was increased. As a side product, the authors find an unusual phenomenon governing the relationship between the robot's AOC and the number of functioning suckers, which is explained theoretically by distributed mass-spring statics model.

Key words robot, wall climbing mechanism, load capacity, statics

I

The safety of wall-climbing robot is of great importance in robot design and gait programming. Safety requirements include both anti-overturn capability (AOC) and anti-slippery capability. When the wall-climbing mechanism with several vacuum suckers walks in a specific gait, the support pattern and the number of suckers in function change continually in a definite way. For a specific support pattern, AOC is expressed by the maximal allowable weight moment, which can be calculated as follows[13 :

-Fs Mm~x -

min (R01)'

in body's coordinate system,

(xi,Yi),

Introduction

(1)

1 Xl

[A] =

fjl

=

1 X2

1 X3

. . . . . . . ° °

, ° °

. . . . . . o°o

1 ~n

Yl

Y2

Y3

. . . . . .

Yn

f41

f42

f43

f44

0

f51 •"

f52

f53

2

:

-fnl

fn2

fn3

A5 0

,~tl~

l,lXn

(3C2 -- X j ) ( y 3 -- y j ) -- (X3 -- X j ) ( y 2 -- y ) ) ,

zJ)(Yl - YJ) - ( z l - x J ) ( Y 3 - Yi), .~3 = (Xl - x i ) ( y 2 - yi) - (x2 - x j ) ( Y l - y j ) , fj2 = (X3 --

f,, = - (h~ + fi2 + f.), j = 4,5,'",n

i=O~n

where n is the number of suckers in function, Fs is the suction force of a single sucker, R0i is the i-th element of vector {R 0 } : [R0},×l = [ A ] ~ , I Q 0 } , × I ,

(2)

where [A] is a matrix related with sucker's positions

Received Jun. 18,1998 Qian Jinwu, Prof., Schoolof Mechanicaland ElectronicEngineering, Shanghai University, 20 Chengzhong Road, Shanghai 201800

and { Q0 } is the unit load vector specified as { Oo} = ( 0 , - 1 , 0 , " - , 0 ) r l .

In the process of optimal gait programming using AOC as the goal function, repeated calculation of M ~ x is needed, which is time-consuming as formula (2) requires matrix inverse calculation. This leads to difficulty for real-time gait programming on wall surfaces with obstacles. To overcome this difficulty, it is necessary to calculate the analytical solutions for different support patterns in advance. When the analytical expressions of AOC are obtained, the great enhancement of gait programming

Vol. 3 No. 2 Jun. 1999

Qian J. W. : On Maximal Weight Moment and Overturn Tendency of ...

efficiency as well as real-time safety monitoring system become possible. In the paper, the authors consider a six-legged wallclimbing robot walking up/down along a vertical wall

Step 1 Write static force equilibrium equation for a specific support pattern, ~~

surface. Besides, the detaching tendency of functioning suckers from the wall is also obtained as side products. All the results are of great importance for the safety-monitoring system to take samples or monitor the conditions of critical suckers.

2

Statics Model and Analytical Procedure of AOC

The statics model of the wall-climbing robot in a specific support pattern is shown in Fig. 1, where zo is the gravitational force, Fs is the suction force of a single sucker. The reaction force on a sucker from the wall surface is decomposed into normal component N~ and tangent component Fmi which is not shown in the figure.

1 x2

Nn-l_ ~ N,

x y

Fig. 1 Robot's statics model Imagining that the actual gravitational moment of the wall-climbing mechanism

M=WH increase gradually, the minimal normal wall reaction force component Nmin becomes smaller and smaller. At the critical moment when Nmin equals zero, M comes to the maximal allowable gravitation moment M ~ x , which is defined as AOC for a specific pattern. If the actual gravitational moment M exceeds M r ~ , the functioning sucker with the minimal normal force component would detach from the wall first and the robot' s tipping-over follows. Based on the above analysis, we can draft the deduction procedures to obtain analytical solution of AOC.

-" "-"

11

N2

XnJ

0

Fs --

_-- _

Y2 "'" Y, Ii

$

where ( x i , Yi ) is the Cartesian coordinates of the i-th functioning sucker in body coordinate system oxyz. If the number of functioning suckers n >~4, the statics problem becomes indeterminate. In order to solve N1, N 2 , ' " , N , , we must supplement a number of deformation coordination requirements. Under the consumption that the rigidity of the overall robot is much greater than that of the sucker's rubber brim, the displacements of foot tips in Z direction under gravitational force are supposed to be distributed in a planar fashion. With standard four-point co-plane equation and the Hooke' s law considered, we have

xl - xj Yl - Yj N 1 - Nj x 2 - xj

%

149

Yz-Yi

N 2 - Nj

x3-xj Ya-Yi j =4,"',n.

N 3 - Ni

= 0,

(3)

Step 2 Calculate the normal component of reaction force from the wall, Ni, i = 1 , 2 , n. Step 3 Determine the minimum of Ni, i.e. Nmi n, which is related with suction force F , , actual weight moment M and features of the support pattern. Step 4 In the equation to calculate Nmin, substitute M with Mn~x, and let Nmi n equals to zero, thus obtaining the expression of Mmax.

3

Analytical Results of AOC

According to the above proposed procedures, the AOC and sucker detaching tendency for various support patterns are obtained, which are listed in Table 1. Analyzing Table 1, we can have the following discussions. Firstly, the AOC is proportional to suction force Fs. It is also related with some specific parameters of a support pattern, such as b and d, etc. Secondly, the specific parameters of a support pattern, that affect AOC, are the dimension in the same direction as gravity force. Thirdly, the AOC is proportional to the number of functioning suckers in general . But there existes such

150

Journal of Shanghai University Table 1 Analyticalsolution of AOC and sucker detaching tendency support pattern

---

AOC ( M ~ )

/

3-leg support

bF,

i

~C)

sucker i detaches first

j

b2+d2 F max( b, d)

k 2-2 4-leg support

sucker detaching tendency

(1) sucker i detaches first if d > b (2) sucker j detaches first if d < b (3) sucker i and j detach simultaneously ifd=b

arrangement i

~-"@ J

~ ~/. / / / 2
sucker i detaches first

1-3 arrangement

(1) sucker j detaches first if 3d>2(2bl + b2) 5-leg support

6 leg support

3b~+2b~ +2btb2 +2d 2 max[3d,2(2bl + 62)] F,

(cx + c2)F, max[ (2bl + 62),(2dl + d2)] ct =2(b~ + b 2 + bib2) c2=2(d21 + d 2 + did2)

(2) sucker i detaches first if 3d<2(2bt + b 2) (3) sucker i and j detach simultaneously if 3d = 2(2bx + b2)

(1) ~ k e r i c~ches first if 261 + b2>2dl + d2 (2) suckerj &~_hes first if 261 + bz<2dl + d2 (3) sucker i and j detachsimultamously if 2ba + b2 =2dr + d2

exception that more suckers can cause the reduction of

shown in Fig. 3.

AOC, as in the situation of a 4-sucker support pattern with three suckers in one side and one in the other, also

By comparison with AOC of 3-sucker support pattern, we can see that only when 0 ~ x ~ 0 . 5 b , the mid-

referring to the Fig. 2.

dle suction cup of 4-sucker support pattern is doing

Let x and (b - x ) to substitute bl and b2 in the relevant formula of AOC written in Table 1, we get:

goodness. While 0 . 5 b ~ x ~ b , the middle sucker does not help anymore, it only brings harms to the contrary. The reduction of AOC caused by the middle sucker

2 ( x 2 - bx + b2).~,s., Mmax = x+b The function of Mmax with respect of x is graphically

reaches its worst if x = ( r ~ _ 1)b. To explain such an unusual phenomenon, a distributed mass-spring model is introduced for further analysis.

Vol. 3 No. 2 Jun. 1999

Qian J. W. : On Maximal Weight Moment and Overturn Tendency of ... i

4

------/-~)J / /

k

Fig.2 Four-leg support pattern bFs

side of each mass is connected to a spring respectively, thus forming the mass-spring system. When the mass-

2bFs

spring system is put on the level ground, each spring

°73:1U 0.5b

has a initial deformation, causing that the ground reaction forces Nio linearly distributed. When the moment M exerting on the frame increas-

X

Fig.3

es, the mass-spring system will tip over. At the critical situation as shown in Fig. 4 ( b ) , the ground reaction forces N i will he linearly distributed as follows,

M , ~ - x function

t~

b

Explanation of Unusual Features for AOC

The wall-climbing mechanism can be analogized by a distributed mass-spring system shown in Fig. 4. Three centralized masses whose individual gravity force is Fs are placed in a line. They are connected with each other by a rigid weightless frame along one side. The other

/

0

151

.~l

f~ i

/// 1

(a)

(a) initial status

(b)

(c)

Fig. 4 Distributed mass-spring model for analogy (b) critical overturn status (c)two-mass model for compai'ison

N 1 =0,

N2 - N1 N3 - N1 x b A~ all the forces acted on the system satisfy force equilibrium condition N2 + N3 = 3F3, we can see the following results from the overturn tendency with respect to the acting point of the 3rd mass with the ground. (1) If 0~ F , , meaning that mass 2 makes the AOC of the system decrease, as corn-

pared with two-mass system shown in Fig. 4 ( c ) . In brief, the effect of mass 2 on AOC of the system depends on its relative position between mass 1 and mass 3. This explains why 4-sucker support is sometimes no better than 3-sucker support from the viewpoint of AOC of the wall-climbing mechanism.

5

Conclusion

The analytical solutions of AOC and the sucker-detaching tendency in various support patterns are obtained for the multi-sucker mechanism crawling on a vertical wall. Although the results are deduced from sixlegged robotic mechanism like insect leg arrangement, it is also applicable to 4-legged wall-climbing mechanism. The analytical solutions of AOC obtained are proved effective in both gait programming and safety-monitor-

152

Journal of Shanghai University

ing computation because matrix inverse calculations are

or programming gait on line.

avoided in on-line obtaining of AOC.

References

AOC of wall-climbing mechanism is not always proportional to the number of functioning suckers. In a 4sucker support pattern there existed abnormal situation. Special attention must be paid while selecting foothold

1

Oian Jinwu, Gong Zhenbang, Zhang Qixian, et al. , Antioverturn distance and its applications in multi-legged wallclimbing robots, Proc. 2 nd Asian Conf. on Robotics and Its Applications, Beijing, Oct. 1994 : 318 - 322.