Toward Guaranteed Stability in the Haptic Display of Virtual Environments
1
Jui-Chang Tsai
Ph.D. Dissertation, Northwestern University, 1996
1996 NWU
http://lims.mech.northwestern.edu/jctsai-lib/dissertation/jctdiss0.ps
http://lims.mech.northwestern.edu/jctsai-lib/dissertation/jctdiss0.pdf
A haptic display is a device that allows human operators to manipulate
and feel virtual environments.  An important issue in the haptic
display of virtual environments is stability.  Often, unintended
oscillations of the human/display system are found to occur.  This
dissertation addresses the problem of guaranteed stability in the
haptic display of virtual environments. The approach taken to this
problem is based upon ensuring not only discrete time stability, but
also discrete time passivity.  The introduction and background of this
work are included in Part I.

In Part II, we consider the effect of unilateral nonlinearities on the
stability of discrete-time control systems, a problem of some
importance in haptic display.  For example, a virtual wall is a simple
form of unilateral constraint.  Sufficient conditions for the
stability of linear, shift-invariant systems in feedback with
unilateral nonlinearities are derived.  The derivation follows a
method originally presented by Mitra in which the existence of
periodic oscillations is first assumed, then conditions leading to a
contradiction are found.  The results developed here are presented
graphically in the Nyquist plane, allowing direct comparison to other
well-known criteria, such as Tsypkin's Condition.  It is shown that
the new criterion is much less conservative.

In Part III, we show that A-stability relates to the passivity of the
discrete time simulation.  If A-stable methods are used to map linear
time-invariant systems to discrete time, the results will be discrete
time passive.  Furthermore, the need for real-time simulations of
complex dynamic systems in haptic display suggests that simulations
may need to be distributed across a network of processors.  A parallel
simulation structured in a physically meaningful way allows the user
to apply the powerful insights and methods of physical network theory.
We show that, by modeling junctions in the simulation network using
transmission lines, the inherent delays are accommodated, and discrete
time passivity is guaranteed.

In conclusion, we are able to construct A-stable simulation networks
through the transmission line model and A-stable integration
technique.  We believe that this is a promising approach to the
real-time, A-stable simulation of multibody systems.