6 EXPERIMENTS

233

(z~(t~,+~)) ~

V2(Z 2i +

2) =

D +

c~

2

I z3(t2i+2 )

+ z3(t2i+

2) aa 2

where D = V2(~ 1, 0, 0) and

7"1 =

Yc,sw-

Yd"

Since

-Jr- O~CZ2(t 2i +

2) 1

lim

z2(t 2i+

1) = 0

i--~ oo

lim

z2(t 2i +

2) = 0

i---~ oo

lim

z3(t

2i +

2) -- 0

i--* oo

we obtain that, for any e > O, there is i* such that for i > i*,

I V2(Z 2i + ,) - DI < e

I V2(Z 2, + 2) - DI <

Therefore, since

Vz(Z(t))

is decreasing for

t2i +

1

< t

<

t2i+2 , we have proved that for any

e > 0, there exists i* such that, for all i > i* and all t e [t2i + 1,

tzi+

2],

D

+ e >

V2(Z(t))

>

D

-- e,

In other words, for all i> i* and all

t6[t2i+a, t2i+2],

the trajectory

Z(t)

is in an e-

neighborhood of the level set

d = {Z ~

R": V2(Z )

=

V2(51,

0, 0)}

This, together with the observation that the trajectory in the free space is itself in an

e-neighborhood of the point {z 1 = 21, z 2 = 0}, completes the proof that for any e > 0, there

is T such that for all t > T the trajectory is contained in an e-neighborhood of a~/.

Since d is entirely contained in the set B, by Theorem 1, d cannot contain any nontrivial

invariant set. This proves that there cannot be infinitely many switchings. In view of Lemma

1, the last switching is from the free space to the constrained space.

6 EXPERIMENTS

The intensive simulation and experimental studies were conducted at the Center for Robotics

and Automation in order to verify the theoretical results. Experimental comparisons with

other typical contact control schemes were also studied. The trajectory tracked consists of a

straight line in free space and a straight line on the constrained surface.

A 6-DOF dual-arm PUMA 560 manipulator [41] manufactured by Unimation Inc. was

used for the experiments. The manipulator is hardware interfaced to a Motion Tek universal

motion controller [42], which is an electronic system used to control a number of motors for

the robot system. It is highly modular in configuration and is capable of up to 1000 Hz servo

rate on every axis. Each controller stack consists of five modules. The power supply module

provides the logic and servo power for the entire control stack. The joint processor module

contains a 10-MHz, 32-bit NS 32016 microprocessor that runs both the configuration set up

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