We numerically investigate the basin of attraction of chaotic behavior on the ... the trajectories are attracted to is referred to as a "strange attractor." In Section 2 ...
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CHARACTERIZATION OF CHAOS IN THE ZERO DYNAMICS OF KINEMATICALLY REDUNDANT ROBOTS
Matthew Varghese, Andreas Fuchs and Rangaswamy Mukundan Department of Electrical and Computer Engineering
Clwrkson University Potsdam, NY 13699-5720
Abstract
In Section 2, we outline techniques of feedback linearization when the decoupling matrix is nonsquare and has more columns than rows. In Section 3, simulation results are presented. Poincare maps are shown, as well as the power spectra of the chaotic time trace of one of the coordinates of the robot arm's end point. The slice of a basin of attraction of the chaotic attractor is also displayed.
Chaos is a ubiquitous and robust phenomenon that is now
wel documented in nonlinear dynamical systems. The recent explosion of research in the feedback linearization of nonlinear control problems has uncovered interesting internal behavior in partially linearizable system. The intemal dynamics subject
to the constraint that the output is zero is defined as the zero dynamics of the partially linearized system. We numerically investigate the basin of attraction of chaotic behavior on the zero dynamics submanifold. The theoretical ramifications of such studies could lead to new avenues in the global feedback stabilization of nonlinear systems when the zero dynamics converge to a compact but strange attractor.
1
2 Feedback linearization for nonsquare systems In state space form, we use results from Isidori [11] to formulate a nonlinear feedback law for nonsquare systems. Multivariable nonlinear systems can be written as i
Introduction
Yl
We examine complicated behavior in the zero dyanamics of kinematically redundant manipulators. The theory of inputoutput linearization via nonlinear feedback has received a lot of attention recently. Work by Isidori, Byrnes, Dayawansa et al., [4, 5, 7, 8,10,12, 13] have put the subject on a level so that one can see applications to a host of practical problems. In cases where the linearized system has less dimensions than the original nonlinear system, the notion of zero dynamics, which is the nonlinear analog of transmission zeros, plays an important role.
m
=
=
(2.1)
hl(z).
(2.2)
i=1
=hp(x3
(2.3) where f(x), g1(x),. . ,gm(x) are smooth vector fields and hi (x), ... , hp(x) are smooth functions defined on an open set of aW. yp
The vector relative degree {'YlvY2,. . .-,} at a point x0 is defined if the following conditions are met in the neighborhood of xo:
{V
(i)
L92Lkhi(z)
Earlier work on feedback linearization relied on assumptions that the zero dynamics subsystem is exponentially stable. Sastry and Isidori 116] have addressed the subject of bounded tracking when the zero dynamics is exponentially stable. By using a version of the converse theorem of Lyapunov, it is possible to show that the zero dynamics is bounded input bounded output stable.
f(X) +Egi(x)ui,
=
I '-1 hi
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dLf-1h2
R, = row (dh, . , dL-" -1 h, . , dh,, . . .,y dLfP--hp), (2.10) (2.11) R2 = C01(91, . ,m, ... aP l . .aQ gm)-
dL7 h'-
.
I
..
This is imposible because
-
RjR2 is a t x mz, matrix, which has a block triangular
do', dt
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O' (X), * ' * t
OY(z)-i O.+8
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(2.14) .,+,(x), ...,
(X),
=
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(2.12) (2.13)
0
col [O' (m)
A(x°) has rank p.
In the new coordinate system,
Proposition 2.2 Consider a nonsquare system that has relative degree {1yi, . .., y} at o and 71 + 72 + *+ Let t = 71+ .+yp and
#4,(x) = Lf-h . It is always possible to find n - r more functions 0,(x) such that the transformation given by
(2.23)
Since the distribution C has dimension y, it is possible to find n - y functions in the set { 17 02, -,tn-mi with the prop. erty that CU {t+ 1, qg+2, ... , .} are linearly independent and L,,Oi=0, + 1 < i < n and 1
