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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