Abstract—A haptic interface is a kinesthetic link between a
human operator and a virtual environment. This paper addresses
fundamental stability and performance issues associated with
haptic interaction. It generalizes and extends the concept of a
virtual coupling network, an artificial link between the haptic
display and a virtual world, to include both the impedance and
admittance models of haptic interaction. A benchmark example
exposes an important duality between these two cases. Linear
circuit theory is used to develop necessary and sufficient conditions
for the stability of a haptic simulation, assuming the human
operator and virtual environment are passive. These equations
lead to an explicit design procedure for virtual coupling networks
which give maximum performance while guaranteeing stability.
By decoupling the haptic display control problem from the design
of virtual environments, the use of a virtual coupling network
frees the developer of haptic-enabled virtual reality models from
issues of mechanical stability.
Index Terms—Absolute stability, force feedback, haptic interface,
impedance control, two-port network, unconditional stability,
virtual reality.
I. INTRODUCTION
Ahaptic interface conveys a kinesthetic sense of presence
to a human operator interacting with a computer generated
environment. Historically, human-computer interaction
has taken place through one-directional channels of information.
Visual and audio information is sent from the computer
to the operator. Keyboard, mouse, and joystick inputs transfer
human inputs to the machine. Human neuromuscular and
decision responses close this information loop. Oscillatory
behavior is possible in this configuration, for example, when
attempting to track a moving target with the mouse in the
presence of delay in the rendering of graphics. Since there
is no kinesthetic energy flow to the operator, such an event
is at worst annoying, but never physically threatening. Haptic
interaction is fundamentally different in that physical energy
flows bi-directionally, from and to the human operator. The
haptic display, typically some form of robotic manipulator,
creates a feedback loop which includes not only the human
neuromuscular and decision responses, but also the biomechanical
impedance characteristics of the operator’s contact
with the device. The human grasp may stabilize an otherwise
unstable system by absorbing mechanical energy. Conversely,
the human grasp may destabilize an otherwise stable system by
reflecting energy back into the system. Since the haptic device
actively generates physical energy, instabilities can damage
hardware and even pose a physical threat to the human.
A number of authors have considered issues of stability
in haptic simulation. Minsky et al. [1] explored stability
problems in the haptic display of simple virtual environments.
They noted a critical tradeoff between simulation rate, virtual
wall stiffness, and device viscosity and provided insights
into the role of the human operator in stability concerns.
A more rigorous examination of the stability problem was
performed by Colgate et al. [2]. They used a simple benchmark
problem to derive conditions under which a haptic display
would exhibit passive behavior. Salcudean and Vlaar [3]
studied the stability properties of a discrete, proportionalplus-
derivative, virtual wall implementation for a magnetically
levitated force feedback joystick. They found very low device
friction significantly limited the achievable stiffness of the
virtual environment. A much higher perceived stiffness was
achieved using a braking pulse at the moment of impact with
the virtual surface. While each of these works are significant
contributions to the field, their analyzes are limited to specific
assumptions about the type of haptic display used and the
type of virtual environment being simulated. The problem lies
in the fact that no distinction is made between the virtual
environment and the control law for the haptic device. In fact,
in the above examples, the virtual environment is the control
law. It is encumbered with twin roles of creating realistic force
feedback cues to render a virtual scene and ensuring the haptic
device remains stable.
One way of decoupling the haptic device control problem
from virtual scene generation is the introduction of an artificial
coupling between the haptic display and the virtual
environment. Colgate et al. [4] introduced the idea of a virtual
coupling for haptic displays which guarantees stability for
arbitrary passive human operators and environments. Zilles
and Salisbury [5] presented a heuristically motivated “godobject”
approach which greatly simplifies control law design.
Ruspini et al. [6] use a virtual “proxy” extension of the
god-object to couple a Phantom device to a three degree-offreedom
constraint based simulation. These implementations
can be grouped together as special cases of a virtual coupling
network, a two-port interface between the haptic display and
the virtual environment. This network can play the important
role of making the stability of the haptic simulation independent
of both human grasp impedance and the details of virtual
environment design. All of the above-mentioned work focuses
on one particular class of haptic display, those which render
impedance. No similar work on virtual couplings has appeared
for the complementary case of haptic displays which render
admittance and very little exists in explicit criteria for the
design of virtual coupling networks.
This paper extends the concept of a virtual coupling to
admittance displays and attempts to treat the problem of
stable haptic interaction in a more general framework which
encompasses any combination of haptic display and virtual
environment causality. Llewelyn’s criteria for “unconditional
stability” is introduced as a tool in the design and evaluation
of virtual coupling networks. A benchmark example illustrates
some fundamental stability and performance tradeoffs and
brings to light an important duality between the impedance
and admittance models of haptic interaction.
II. PRELIMINARIES
A. Terminology
The following terms are used throughout this paper.
Haptic display mechanical device configured to convey
kinesthetic cues to a human operator.
Haptic displays vary greatly in kinematic structure, workspace,
and force output. They can be broadly classified into two
categories, those which “measure motion and display force”
and those which “measure force and display motion” [7]. The
former will be referred to as impedance displays, the latter
as admittance displays. Impedance displays typically have
low inertia and are highly back-drivable. The well known
Phantom [8] family of haptic displays, the McGill University
Pantograph [9], and the University of Washington Pen-Based
Force Display [10] fall into this class, along with many
others. Admittance displays are often high-inertia, non backdrivable
manipulators fitted with force sensors and driven by a
position or velocity control loop. Examples include Carnegie
Mellon University’s WYSIWYF Display [11] and the Iowa
State/Boeing virtual aircraft control column [12], both of
which are based upon PUMA 560 industrial robots.
Haptic interface includes everything that comes between
the human operator and the virtual environment.
This always includes the haptic device, control software,
and analog-to-digital/digital-to-analog conversion. It may also
include a virtual coupling network which links the haptic
display to the virtual world. The haptic interface characterizes
the exchange of energy between the operator and the virtual
world and thus is important for both stability and performance
analysis.
Virtual environment computer generated model of some
physically motivated scene.
The virtual world may be as elaborate as a high-fidelity walkthrough
simulation of a new aircraft design, or as simple as
a computer air hockey game. Regardless of its complexity,
there are two fundamentally different ways in which a physically
based model can interact with the haptic interface. The
environment can act as an impedance, accepting velocities (or
positions) and generating forces according to some physical
model. This class includes all so-called penalty based approaches
and to-date has been the most prevalent [1]–[3], [8],