Design and Suction Cup Analysis of Wall Cli

Pergamon Compuk~s Elect. Engng Vol. 22, No. 3, 193-209, 1996 pp.

Copyright 0 1996 Elsevier Science Ltd

PII: 0045-7906(95)ooo39-9 Printed in Great Britain. All rights reserved

0045-7906/96 $15.00 + 0.00

DESIGN AND SUCTION CUP ANALYSIS CLIMBING ROBOT

B. BAHR,’ Y. LI’ and M. NAJAFI’

OF A WALL

‘Mechanical Engineering Department, Wichita State University, Wichita, KS 67260, U.S.A. and 2Mathematics and Computer Science, Kent State University, Ashtabula, OH 44004, U.S.A.

(Received for publication I3 December 1995)

Abstract-This paper describes the design, kineostatic and safety analysis of a two legged mobile robot that can climb from one surface to another surface by using suction cups. Three artificial constraint equations are derived to create the working Jacobin matrix, and then a progam is written to calculate the reaction and actuator force (moment), based on which, the most dangerous position of the robot is found. Also, we study the amount of the stress distribution on the deformable suction cup due to the exertion of the gravitational force, as well as load distribution of the robot. Copyright 0 1996 Elsevier Science Ltd

Key words: Wall climbing, mobile, robot, suction cup, legged, safety, design, analysis, elasticity.

0. INTRODUCTION

Robotics systems, in principle, present a new technical means for complex automation and the manufacturing process, and in hazardous environments, human labor can be eliminated by their application. These operations are usually monotonous, primitive and often strenuous and hazardous to human health. Nevertheless, a large part of human work effort is still spent on those operations. Experience has shown that many manual work operations cannot be automated by traditional automation, and robotization is the only solution. It appears that industrial robots are replacing the human counterparts for loading, unloading, inspection, maintenance, welding, painting, and precision manufacturing.

On the other hand, the industrial robots are not designed to work in the unstructured environments such as sky scrapers, nuclear power plants, large storage tanks and aircraft structures. Therefore, in recent years a number of researchers have developed robots that can climb on vertical walls and have addressed the need for robots in the unstructured environment [l-lo]. The area of mobile climbing robots is a new and fast-growing interdisciplinary field. Many researchers are challanged to create smarter mechanisms for the locomotion and climbing of robots. To perform tasks that are not within reach of the conventional robotics manipulators, a new generation of robotics systems should be developed. These climbing robots use vacuums, magnetic force, or propeller systems for adhesion to surfaces.

Wichita State University (WSU) developed a climbing robot named Robotics System for Total Aircraft Maintenance (ROSTAM) in 1990 [lo]. It consists of four legs and a central body. The four legs have the same design and each leg has two degrees of freedom. The first degree is a rotation one, which enables the leg to rotate around the gear shaft. This rotational motion is transmitted from a gear head DC motor to a worm gear, then to the rotational linkage that is part of the leg. The second is a longitudinal one, which enables the leg to extend along the cylinder axis. The longitudinal motion of the leg is created by sending compressed air to the ports of a pneumatic cylinder that causes a rod to extend or retract. Each leg has two suction cups that are 11.53 cm in diameter, and each is equipped with an ejector. A newer version of ROSTAM-III (Fig. 1) was also designed at WSU, with the rotation mechanism in the center and only two legs. The robot can change its moving direction at any desired angle.

In this paper, the following sections are discussed: in Section 1, the design of a new light-weight robot is described along with the description of its movement. In Section 2, the kinetostatic analysis of the robot is developed. Also in this section, the kinematic analysis of the robot is developed along

B. Bahr er crl

Fig. I. ROSTAM-III on the wing of an aircraft.

with inverse dynamic analysis for the force and moment analysis that are required for the design section. Since the robot uses suction cups for adhering to surfaces, the theoretical safety analysis of the suction cup is discussed in Section 3, along with the experimental verification. Section 4 develops the safety analysis for the robot based on the results of the Section 2 and 3. In Section 5, we study the deformation of the suction cup due to external froces. There, the set of partial differential equations (which are yet to be determined), along with imposed boundary conditions, are introducued. The solutions to this system represent the amount of the displacements from the equilibrium position due to pressure distribution on the suction cup.

1. DESIGN OF THE TWO LEGGED ROBOT

The objective of this current research work was to design a robot that has a universal property so that it can move and climb on any material surface; i.e. a climbing robot that can move on flat surfaces, right angle surfaces or parallel surfaces. Therefore, the two legged robot was designed as shown in Fig. 2, which consists of two structures (right and left), and a two leg assembly connected to the right and left structures respectively. At the end of each leg there is one suction cup with a diameter of 11.35 cm. Both legs are connected with a right and left structure by a worm system, respectively. In addition, a rotating mechanism was designed on the right leg with a pair of bevel gears so that the robot can change with moving direction without any disturbance to the vacuum line. This mechanism enables the robot to rotate without any limitations. The linear motion of the mechanisms consists of one ball lead screw and two linear bearings which connect the right and left side together. The robot is very light weight (about 4 kg) and portable to any location. Its weight-carrying capacity is at least 8 kg so that the robot can carry a load such as a video camera, lights, and other inspection tools.

The robot has a total of four degrees of freedom controlled by four DC motors. Three motors produce the rotating motion and one produces linear motion. When the suction cup on the right leg is attached to a surface and the suction cup on the left leg is not attached on the surface, the relative motion between the couple of bevel forces the robot body to rotate around the right leg. The linear motion of the robot is produced by one DC motor, one ball lead screw and two linear bearings. When the suction cup on the right leg is attached to a surface and the left leg’s suction

cup is not attached, the ball lead screw rotates counter clockwise to move the left structure that is connected to the left leg. After the cycle is finished, the robot is in the expanded position and

Design and suction cup analysis of a wall climbing robot 195

the left leg will attach itself to the surface. Then the right leg releases from the surface and the ball screw rotates clockwise which means the robot moves one step.

If it is desired for the robot to climb a right angle surface, first the right suction cup is attached on the surface (Step 1 in Fig. 3), second it raises the left structure and left leg, and third it moves the left structure and left leg toward the wall so that the left suction cup touches and is adhered to the side wall (Step 2 in Fig. 3). After the left suction cup is stuck firmly on the side wall, the right body and right leg moves toward the left body (Step 3 in Fig. 3). Finally, the left leg is rotated around the center of the left worm gear, and at the same time the right leg rotates around the center of the right worm gear so that both legs are touching the wall and are firmly attached to the surface (Step 4 in Fig. 3). The robot is able to climb from one surface to another at 180 degrees, which means the two surfaces are parallel.

2. KINETOSTATIC ANALYSIS

In this paper, a method of Kinestatic (Inverse dynamic) analysis is introduced to obtain the minimum torque iV, or force Fi that will be required for each joint. In addition, the maximum N and F that will be exerted to the suction cup will be used for safety analysis in Section 3.

2.1. Notation

xi, Y,: the global coordinates of the mass center of link i, i = 14 A, E: are revolute joints c: is a prismatic joint B, D, F: are the mass centers of link 2, 3 and 4, respectively x+, Y{: the global coordinates of the point j at link i, where j = A, B, C, D, E, P

::’ zi,; the local coordinates axis of the mass center of link i I the local coordinates of point j at link i

the length of the link i

the angle between axe 5, and axe Xi constraint function, where n = l-8

Fig. 2. ROSTAM-IV climbing ceiling to a side vertical wall.

Step I

B. Bahr CI (I/.

Step 4

////l/J// Fig, : I-he repr~~~~tat~~n of the robot movmg from floor to the side wall.

constant in the ith constraint equatron the distance between the mass center of link 2 and the mass center of link 3 vector of coordinates for a system the mass matrix the Jacobian matrrx. whrch IS the partial derivates of @, with respect to X,, Y, and Cp, contains the known forces such as gravity contains forces of actuatot vector of Lagrange multipliers vector of Lagrange multipliers including consideration of artificial constraint equations nonzero elements reaction forces of joint 2 in X and Y-direction reaction forces of joint 3, in X and Y-direction reaction torque of joint 4 equal to actuator torque NZ of motor 2 equal to actuator force F3 associated with motor 3 equal to actuator torques N4 of motor 4

Consider the following case: the robot moving from the floor to an inclined surface (as shown in Fig. 4), no rotation of the first joint is needed. So, the robot manipulator can be considered as a four bar, open loop mechanism with three degrees of freedom. A set of Cartesian coordinates are assigned to the robot links so that the method in reference [1 l] can be used along with a dynamical analysis software to perform the forward or inverse dynamic analysis. Since the center line of the suction cup needs to be perpendicular to the surface, the 44 keeps constant during the motion. The two constraint equations for revolute joint 2 are

~:=X,+5~coscp,-~I\sincp,-XXZ-5:sincp,coscpz+~rsin~,=O

(bz*= Y,+<fsincp,+~~coscp,- Y?-tfsinrp -2-~7:coscp~=O.

The two constraint equations for translational joint 3 are

45:=(x:-xE)(Y:- Y;>-(Yr:- Y;)(X,B-X:)=o

4: = ‘p? - cp? - c, = 0.

Design and suction cup analysis of a wall climbing robot 197

The two constraint equations for revolute joint 4 as

46*= Y,+{:sincp,+r]Fcoscp,- Y,-54Esincp,--~cos~4=0.

The equations of motion of the robot manipulator can be written as

(7-l)

Since q, Q and ii are calculated kinematically, and A?, 4, @q’, g(‘) are known, equation (2.1) can be rewritten as

(2.2)

In order to find the actuator torque N2 of motor 2, actuator force F3 associated with motor 3 and actuator torque N4 of motor 4, the following artificial constrain equations are used

$7*=(p*-(p,-c,=o (2.3)

where

$8’ = -[(x3 - x2)2 + (Y, - Y,)’ - &]/2d, = 0, (2.4)

d2 = ,/(X3@*) -X2(2*))’ + (Y3(t*) - Y,(t*)) 2. (2.5)

So, the equations of motion can be rewritten as

(2.6)

E Link 4

Fig. 4. The robot moving on an inclined surface.

198 B. Bahr ei al.

Using equations (2.3)-(2.5) the new Jacobian [1 1] matrix is

@: =

-1 0

0 -1

0 0 0 0

0 0

0 0

0 0

(X,-X*) (y?- Y,)

4 4

0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -10 0 0

0 1 0 @ -1 0 0

0 0 1 0 0 -1 0

I 0 0 0 0 0 0

0 -(X,-X,) -(y?- Y*)

4 4

0 o 0 0

0 0 0 -1 0 0 1

and thus, the Lagrange multiplier can be calculated as

i. * = [ii, &. ii. I.,, j.5. &. A,, &, i.,]r,

(2.7)

(2.8)

which is used for selection of the motors and the linkage design of the robot. However, for joint 3 the reaction forces are not directly known. Therefore, the following equations should be used

,fi,, = ( Yi - YF)i?

I;,.= -(XS - Y;)&

For the safety analysis of the suction cup the forces i., and 1, and the moment 1, will be used in Section 3.

3. SAFETY ANALYSIS

The safety analysis is the most important consideration in designing the climbing robot, which should be secured firmly on the surface while climbing. There are two dangerous circumstances that could occur. One is slipping of the robot, and the other is falling of the robot. The following theory section will describe how the equations were derived and formulated. The procedure section will discuss how the experiments were conducted to measure the breaking loads, shear, and moment on the suction cups.

3.1. Theoretical analysis qf‘ the suction cup

In this section the theoretical analysis is developed to find P’,,, for suction cups with moment and force applied on them. If the kinematic requirements of the robot are known, the minimum vacuum V,,, inside the suction cup needs to be determined for selection of the suction cups.

Notation.

R: Lo:

N,, Nb, No: Fx, I;,:

Fp:

Fn, F,: M,M,:

Radius of the suction cup Length from the small arc at 6, to the rotation axe (Z-axe) Force intensity at right side, left side and at the angle (3 Total force acting on the suction in X-direction (shear force), Y-direction (pull force) Total force acting on the suction in y direction due to vacuum pressure Total normal and friction forces acting on the suction cup Total moment acting on the suction cup from outside and due to normal force, respectively Equivalent working pressure Vacuum pressure, critical vacuum pressure for suction cup’s falling down, sliding and the minimum vacuum pressure for avoiding both falling down and sliding.

In this section, it is assumed that no elastic deformation takes place. The suction cup is a rigid body and the intensity N, of normal force F, between the suction cup and the contact surface

Design and suction cup analysis of a wall climbing robot 199

- b

A A-View

Fig. 5. The schematic of the suction cup with force distribution.

distribute linearly along the X-direction as shown in Fig. 5. Based on the above assumptions for normal force as a function of the suction cup size and the extreme normal reaction forces intensity

N, - N, R - R sin 9 -----= Nb-N, 2R ’

Equation (3.1) can be rewritten as

The

The

and

No=--- %+Na Nb-Nixsine 2 2 ’

total normal force acting on the suction cup is

dF,, = 2N,(R de)

F,, = dO=rcR(N,,+Na).

force acting on the suction cup due to vacuum pressure is

I;, = nR2V

the force balance in the y-direction is

Fy + F,, - F,, = 0

which can be expressed as

where

F =nR2P* ” 9

P’=V--$.

Now, by setting (3.3a) equal to (3.3b), the following equation is obtained

Nb + N, = RP*.

The total moment about point “0” can be calculated as follows

dM, = 2(R sin 6)(N,R de)

(3.1)

(3.2)

(3.3a)

(3.3b)

(3.4)

(3.5)

s n/2 IV,= (2R sin 0)

Nb+N, Nb-N, - -

2 ___ sin 8 (R de)

-n/2 2

200 B. Bahr et al.

or

M, = --zR”(N, - N,).

The moment balance for the suction cup is

M+M,=O

(3.6)

that is.

TKR’(N, - N,) = hf.

From equations (3.5) and (3.7) the normal force at “a” and “b” can be written as

(3.7)

(3.8)

(3.9)

Therefore, to avoid the suction cup (robot) falling down. the normal force intensity at “a” should satisfy the following condition

,Y, 3 0. (3.10)

By substituting equations (3.4) and (3.8) into (3. lo), the equation relating vacuum pressure of the suction cup to the moment and the pulling force is obtained as

Thus, the critical vacuum pressure for falling down is

. (3.11)

To avoid suction cup slippage from the surface, the following condition should be satisfied

IF,1 < Fr. (3.12)

Thus, Fr can be calculated as

Fl- = F&f:

By substituting (3.3b) and (3.13) into (3.12). the following

(nR2V - F,)f’> IF,/.

relationship is obtained

(3.13)

Thus, the critical vacuum pressure for sliding can be calculated by

(3.14)

Considering equations (3.1 I) and (3.14). the following minimum vacuum pressure is needed to avoid both falling and slippage of the suction cup from the surface

C’,,, = max[ VS, Vf]. (3.15)

3.2. Experimental procedure for suction cup testing

The purpose of this part is to verify the results of the suction cup analysis experimentally. This is done by performing three different types of experiments. Due to the weight and the payload of the robot, there exist shear force, pulling force and the tipping moment at the suction cups (see Fig. 4). The first group of experiments measures the validation of (3.15) when moment and pulling force are applied to the suction cup from the robot. The second group of experiments is when the shear and pulling forces are applied. The third group uses a combination of shear, moment and pulling force on the suction cup.

Design and suction cup analysis of a wall climbing robot 201

PULLING

r-4

Load cell -t_$C=i

Testing surface Suction cup

1

I-

Fig. 6. Suction cup experiment setup.

Roller

PIABSO Nitoplastic suction cups were selected for these experiments and a flat, aluminium plate was used for the contact surface to simulate the vertical wall (aircraft body). The history of shear load, pull load and the suction cup vacuum for every experiment was collected and stored by the computer. Figure 6 shows a schematic of the experimental setup. A sliding mechanism generated

Fig. 7. Setup for moment measurement.

202 B. Bahr <‘I ui

SAMPLE GRAPH 1: PULL AND SHEAR

PULL,SHEAR,&VACUUM vs. CYCLE

0 10 20 30 40 50 60 CYCLE[SECOND]

8 PULL -dACUUM-- SHEAR

Fig. 8. The pulling. shear. and moment for ;I typical suction cup

by two-sided rollers allows one degree of freedom in sliding motion. A shear load cell is attached to the sliding mechanism measuring the horizontal shear force. A hydraulic cylinder (not shown in Fig. 6) was used to generate a vertical. upward pulling force.

The setup for group 2, which is used for measuring the moment, is shown in Fig. 7. Therefore, a rigid rod with a length of 101.6mm (4 inches) is added to the experiment. A moment would generate by the pull force multiplied by the tipping distance between the pulling axis and the center of the suction cup.

During the group 1 experiments, the suction cup would be attached along the center line of the pulling load. A constant pull force would be set. then followed by an increasing shear load until the suction cup failed to stick. The shear load in this case increased from zero to a maximum value and then dropped to a constant value. At this time shear load has overcome the static friction between the suction cup and the aluminium plate. The failure condition is considered when the

PULL

1.

4! Horizontal length -i /

r

I !

5-p’ ’ SHEAR

& \ : \\ Vertical length

\: I

SHEAR _lI!?Y I

_-.

Fig. 9. The schematic of the deformed suction cup under different loads.

Design and suction cup analysis of a wall climbing robot

Table I. The length and masses of various linkages

1, (m) 0.3 m, (kg) 0.0 Mm) 0.6 mz (kg) 0.72 Mm) 0.6 m, (kg) 0.72 L(m) 0.3 Mkg) 0.36

203

shear froce reached its maximum value. The experimental results of the pull and shear experiment are shown in Fig. 8. Suction cup vacuum force was found by both theoretical and experiment method. The comparison of the data show that group 1 experiment is 4.0%, group 2 is about 3.0% and group 3 is 5.0% less than theoretical value, respectively. These errors can be contributed to the following factors: (a) The pulling and the moment applied to the suction cup case the cup to deform as seen in Fig. 9. This deformation reduces the vacuum area of the suction cup; (b) dust particles on the aluminium surface permits some air leakage.

4. CALCULATION EXAMPLE FOR THE ROBOT

The equations in Sections 2 and 3 are used to calculate the forces and the moments for the robot. Three driving conditions [l l] are assumed for the kinematic calculation, and the kineostatic analysis. Since the path of point “P” in Fig. 4, is a straight line, the path of the mass center C, is also a straight line. The three driving constraint equations can be written as

@,=X,-X,(O)- V,t=O (4.1)

@*= Y4- Y,(O)- v,t=o (4.2)

@j = (p4 - (P4m = 0, (4.3)

where: I’,, V, are the velocities of link 4 in the X and Y-direction, respectively. The kinematic and kinetostatic calculation based on the following conditions and assumptions:

(4 0-N

;; 63 0-l

no elastic deformation takes place on the suction cup; friction coefficient between suction and surface was assumed to be f = 0.05, and the radius of suction cup was R = 0.1134 m; inclined surface angle a = 30”; V, = 0.0866 m/s, and V, = 0.05 m/s; time r = 0.0-l .O set; parameters of links are listed in Table 1.

The robot has two revolute joints defined by points “A”, and “E”; and a prismatic joint which is defined by points “B”, “C”, and “D” as shown in Fig. 4. Because, we are interested in the kinematics of the point “P”, the local coordinates are assigned as shown in Table 2. The robot is initially estimated to be at the following global coordinates listed in Table 3.

After solving (2.6), the results are which is equation (2.8) that are plotted in Fig. 10-12. Figure 10 shows thatf,, andf& are constant for the duration of the motion of the link 2, 3 and 4. Therefore, the forces acting from the robot to the suction cup are constant. Figure 11 shows that the absolute value of the F2 increases with time. This shows that the maximum value of the force F2 should be used for selection of the motor as well as the part design for joint three. Figure 12 shows the actuator torques for Nz and N4. The N4 stays constant during the motion of the robot, because there is no angular acceleration of link 4. The IN2 1 increases with increasing time and it is much greater than N4. This means that the motor 2 should supply more power to the robot than motor 4 for this movement.

Figure 13 shows several positions of the robot from time 0 to 1 s. The calculation for the joint force and motor torques at t = 1 .O second are as follows: for joint 2, Lx = 0.0, _& = - 18 N, for

Table 2. The local coordinates of the mass center and selected points at various linkages

c: = 0.0 q: = 0.3 e: = -0.3 q:=o.o 5:=0.3 ?/:=o.o r:= -0.15 q: = 0.0 s: = 0.0 q; = 0.0 cf = 0.3 q; = 0.0

cp=o.o qp = 0.0 c:= -0.15 11: = 0.0

204 B. Bahr rr (I/.

Table 3. The tmtial coorxhnate system of the robot

X, = 0.0 Y, = 0.0 ‘p, = 0.0 X, = 0.2913 Y: = 0.3hlh ‘P? = 0.241 I x, = 0.3476 Y, = 0 1716 9, = 0.241 I X, = 0.72 Y4 = 0.3 ‘Pi= -1.0472

motor 2, IN21 = 6.3 Nm, for motor 4, N4 = 0.3 Nm. These forces and moments were used in (3.15) to calculate the minimum required vacuum pressure to hold the robot at that position as

c’nl,” = max[ b’(. I’.] = 1 1447 N/m’.

5. DEFORMATION ANALYSIS OF THE SUCTION CUP

Due to the elasticity and the geometrical configuration of the suction cup, assumed to have circular boundary surface, we use the following Navier equation (in spherical coordinate system [12]) which represents the displacement vector 6 in a homogeneous, isotropic, linearly elastic medium:

where

pV++(j. +p)VV.G+F=O. (5.1)

U = displacement vector, p = constant of modules of rigidity.

VE

;i= (1 +v)(l -25

where

E = Young’s modulus, v = Poisson’s ratio, P = body force/volume,

where E. and p are Lame’s elastic constants. The vector operator V in spherical polar coordinates is defined

4 -9

2 z -11

Y -13

!z -16

-17

-19

c: : : : ’ : : : :: :: :: -

0 0.1 0.2 0.3 04 0.5 0.6 0.7 0.8 0.9 1.0

TIME T (set)

- Jzx - f2, Fig. IO. The reaction force of the joint 2 vs time.

Design and suction cup analysis of a wall climbing robot 205

-4’35 I -4.40

-4.10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

TIME T (set)

Fig. Il. The actuator force of the motor 2.

0.8 0.9 1.0

Unit vectors associated with the two coordinate systems are related by the (see Fig. 14):

t, = sin 4 cos 06, + sin 4 cos t& + cos &?, ,

62, = cos f$C, + cos f$ sin ec, - sin f#G3,

ts = - sin &, + cos BP,.

In the present work we shall restrict ourselves to the formulation and solution of problems within the framework of classical, linearized elasticity theory. In particular, we consider a solid body occupying the volume u with surface r = T,uT, . The body force field P is prescribed in u while the surface traction vector T and displacement vector U are prescribed on r, and r, , respectively. For a unique solution of the problem to be posed, it is sufficient that

P is specified in v, T is specified on To = {0]f9~[0,27r]), zi is specified on r, = {c#I]~E[~,$c]}.

-10 ! , I

6 0.1 6.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

TIME T (set)

+ N2 - Nq

Fig. 12. The actuator torque of the motor 2 and 4.

206 B. Bahr er al.

//I/////// t

Fig. 13. The robot motion from zero to one second.

The typical problem of classical elasticity theory is usually stated in the following manner: given the shape (geometry) of the solid in the reference configuration, and given the body force field P in U, the surface traction T on To, and the displacement vector ~2 on r, , we are to find the displacement field ti in v. Once the displacement vector field is found, the strain and stress tensor fields can be readily calculated as follows: let

Fig. 14. Spherical polar coordinates.

Design and suction cup analysis of a wall climbing robot 207

Then the stress-strain relations can be expressed as

t,, = Ad + 2pc,,

t,,, = Ad -I- ~/AC,,

tgl = Ad + 2/+

where d = E,, + coo -t cb6 = V . k.

The strain-displacement relations are au,

c,, = &- 1

1 CM = za 1

aa, zj + (sin 0)u, + (cos 0)~ 1 , 1

'k+ = 'W = 5 [

i au, au, ~- sin8 a4 +x -(cos8)u, 1 )

The equilibrium equations (5.1) in spherical coordinates can be expressed in the form

(1 + 2p) g -F [

2 + (cot 8)0, - A%$ 1 +sI=o, (if2P)f&2Zlr -&$$+

[ 1 +fs = 0, l aA

@+2/A--- [ am0 O@ i aa,

r sin C#I a4 2P Y$+y;z +f+=O, 1 where

(5.2a)

(52b)

(5.2c)

(5.3)

(5.4a)

(5.4b)

(54c)

(5.4d)

(54e)

(5.4f)

(5.5a)

(5.5b)

(5.5c)

(5.6)

The expression for V. Q and V x ti, the divergent and the curl of the vector quantities, in spherical coordinate are as follows:

v.a=$$+;ur+; f$+(cotB)u, [ 1 i au,

+----, r sin 8 a4

v x d =i!, ae +(cotB)u,-7- [ du, i au0

sm 8 a+ 1 + co [ 1 8% 8% %

r sin 0 a4 ar r 1

(5.7)

(5.8)

System of partial differential equations (5.5) can be solved by applying appropriate boundary conditions, i.e. for our problem, we can impose the following boundary conditions:

Crr = vmin 3 atr =a, O<tI <27r, O<I#J <in,

err = -g(r) at r = b, 0 < 8 < 2x, 0 < C#I < fn,

ti(r,t3,$)=0, a<r<b, 0 <6<2a, 4=0, (5.9)

208 B. Bahr el al.

The analysis, computational work, and comparison of the result of this section with previous section are still under consideration and will be subsequently published at a later date.

6. CONCLUSIONS

The model and kineostatic analysis of the climbing robot were under consideration. These analysis were used to determine the motor torques as well as the related forces at various joints. The results indicated that inner motor on the robot manipulator should have high torque in order to support the entire payload. In addition, the safety analysis of the robot was performed to determine when the robot will detach from the contact surface. To do this, the minimum vacuum, Vmin, in suction cup is calculated for different positions of the robot in this workspace. Suction cup was analysed both theoretically and experimentally for the various loads such as moment, shear and pulling forces. The results from experiments were lower than the theoretical about 5% partly because of the deformation of the suction cup under the above loading conditions. To address this issue an attempt was made to consider the elasticity of the suction cup material subjected to stress. Related formulation was developed.

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and Robotics, Vancover, Canada (1992). 9. B. Bahr and G. Li, Safety and survivability analysis of wall-climbing robot. NATO/AH, Vol. II, pp. 3499364 (1993).

10. B. Bahr and Y. Yin, Wall climbing robots for aircraft, ships nuclear power plants, sky scrapers, etc. Robotics and ManFfacturing Recent Trend in Research, Eduction and Application 5, 377-382 (1994).

11. P. E. Nikravesh, Computer-aided Analysis of Mechanical Systems. Prentice-Hall, New York (1988). 12. A. Cemal Eringen and S. Suhubi, Elustodynamics. Academic Press, New York (1975).

AUTHORS’ BIOGRAPHIES

Dr Bahr-received his B.S., M.S. and his Ph.D. from University of Wisconsin-Madison in 1980, 1983 and 1988, respectively. During his Ph.D. research he developed a seam tracking robot using computer vision and optical sensors. Dr Bahr, is currently teaching courses in robotics, control, instrumentation, and measurement in the Mechanical Engineering Department at Wichita State University. Dr Bahr is one of the pioneers in development of climbing robots for aircraft insepction. He has been invited to speak about the climbing robot at many organizations and conferences.

Design and suction cup analysis of a wall climbing robot 209

Yingjie Li was born in Sicuan, China, on April 3, 1963. He received the B.S. degree in mechanical engineering from Jiangxi Polytechnic University, China, in 1984, and the M.S. degree of mechanical engineering from Xian Jiaotong University, China, in 1987. He used to be a member of “863” High Technique Research Group of China in Robotics. Now, Mr Li is a Ph.D. student of Mechanical Engineering Department, Wichita State University, U.S.A. His research interests include robot, robot motion planning, constrained motion and coordination of robots, design and analysis of Mechanical System.

Dr Najti, who currently holds a faculty position in mathematics at Kent State University at the Ashtabula Campus, got his Bachelor’s degree in Mechanical Engineering, with emphasis on Design, Control, and Vibration, from the University of Texas at Austin. He received his Ph.D. in Applied Mathematics, with emphasis on Partial Differential Equation, Distributed Parameter Systems, and Dynamical Systems from Wichita State University. He has made conference presentations throughout the United States, such as AMS (American Mathematical Society), SIAM (Society for Industrial and Applied Mathematics), ACC (American Control Conference), and CDC (Conference on Decision and Control). He has numerous publications to his credit.