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.

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Stable haptic interaction with virtual environments

It is shown that the stability of a linear twoport is invariant under arbitrary lossless terminations, under interchange of input and output, and under "immittance substitution," a transformation group involving the arbitrary interchanging of impedance and admittance formulations at both ports. The quantity k = \frac {2 Re (\gamma_{11}) Re (\gamma_{22}) - Re(\gamma_{12} \gamma_{21})} {|\gamme_{12} \gamma_{21}|} (where the \gamma may be any of the conventional immittance z, y, or hybrid h, g matrix parameters) is the simplest invariant under these transformations, and describes uniquely the degree of stability, provided Re(\gamma_{11}), Re(\gamma_{22}) \geq 0; the larger k is, the greater the stability, and in particular k = 1 defines the boundary between unconditional and conditional stability. The quantity k is thus the basic invariant stability factor. Its definition is also extended to include the effect of terminating immittances, which may be padding resistances or source and load immittances, or both. Certain power-gain functions, including the maximum available power gain, are shown to be invariant under immittance substitution, and k is identified as a function of ratios between them, where they exist. This provides a fundamental way of determining k, apart from calculating it from matrix parameters, and indicates that it is a measure of an inherent physical property.

Stability and Power-Gain Invariants of Linear Twoports

The realisation of negative-impedance convertors (n.i.c.s) and invertors (n.i.i.s) using the bridge-type circuit is briefly surveyed. An equivalence relating the nullor to infinite-gain controlled sources is first proved, and is then used for the derivation of nullator-norator equivalent circuits for n.i.c.s and n.i.i.s. Some properties of networks containing singular elements, which are used in the realisation of gyrators, are investigated. Nullator-norator equivalent circuits for gyrators are derived by using the n.i.c.s and n.i.i.s. They are converted into physical networks by using the proved equivalence. Gyrator circuits are obtained by replacing nullors by operational amplifiers. A stability analysis of the gyrator circuits is produced and a relevant theorem concerning passivity is proved. The feasible Qfactors and the accuracy of the gyrator circuits are indicated by some experimental results. A generalised-impedance convertor (g.i.c.) is defined by generalising the n.i.c. theory, and it is shown that the gyrator circuits described can be used as g.i.c.s. The application of the gyrator and g.i.c. circuits in the synthesis of RC-active networks is considered. Finally, a highpass filter using gyrators and a bandpass filter using g.i.c.s are designed, and the experimental results are given.

Realisation of gyrators using operational amplifiers, and their use in RC-active-network synthesis

A bilateral teleoperation experiment with Engineering Test Satellite 7 (ETS-VII) was conducted on November 22, 1999. Round-trip time for communication between the National Space Development Agency of Japan ground station and the ETS-VII was approximately seven seconds. We constructed a bilateral teleoperator that is stable, even under such a long time delay. Several experiments, such as slope-tracing task and peg-in-hole task, were carried out. Task performance was compared between the bilateral mode and the unilateral mode with force telemetry data visually displayed on a screen. All tasks were possible by bilateral control without any visual information. Experimental results showed that kinesthetic force feedback to the operator is helpful even under such a long time delay, and improves the performance of the task.

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Ground-space bilateral teleoperation of ETS-VII robot arm by direct bilateral coupling under 7-s time delay condition

The goal of control law design for haptic displays is to provide a safe and stable user interface while maximizing the operator's sense of kinesthetic immersion in a virtual environment. This paper outlines a control design approach which stabilizes a haptic interface when coupled to a broad class of human operators and virtual environments. Two-port absolute stability criteria are used to develop explicit control law design bounds for two different haptic display implementations: the impedance display and admittance display. The strengths and weaknesses of each approach are illustrated through numerical and experimental results for a three degree-of-freedom device. The example highlights the ability of the proposed design procedure to handle some of the more difficult problems in control law synthesis for haptics, including structural flexibility and noncollocation of sensors and actuators.

Control law design for haptic interfaces to virtual reality

The problem of the minimum loss in relation to the singing point is investigated for generalized transmission systems that must be stable for any combination of passive terminating impedances. It is concluded that the loss may approach zero db only in those cases where the image impedances seen at the ends of the system are purely resistive. Moreover, in such cases, the method of overcoming the transmission loss, whether by conventional repeaters or by series and shunt negative impedance loading, or otherwise, is quite immaterial to the external behavior of the system as long as the image impedances are not changed. The use of impedance-correcting networks provides one means of insuring that phase of the image impedance of the over-all system approaches zero. General relations are derived which connect the image impedance and the image gain of an active system with its over-all performance properties.

F. B. Llewellyn

Papers

Some Fundamental Properties of Transmission Systems