To ensure robust interactive behavior, for example with a wrench tightening a bolt, the physical world relies heavily upon the property of passivity. The wrench and bolt are obvious examples of passive systems, neither being able to provide energy to the other. It is well known that the coupling of passive systems is guaranteed to be stable. We feel that this stable coupling is utilized in humans' adept manipulation of passive physical systems. To preserve this skill for the manipulation of virtual environments and ensure stability of interaction, we need to address the behavior of the dynamic system as a whole (i.e. the simulation, the manipulandum and the human operator).
The difficulty in any traditional stability analysis of this system is the unmodeled dynamics of the human operator. Even though the virtual environment itself might be stable, interaction with a human operator via a manipulandum often causes instability. In our studies of virtual environments, we have had many experiences with human operators adjusting their own behavior until oscillations resulted (Figure 3).
However, if the haptic display is truly passive, then it seems reasonable that human operators cannot destabilize the system, since the environment will not be a continual source of energy. This type of analysis gives us a way to assess the performance a haptic display. As mentioned above, any given device will have a finite range of impedances which it can successfully emulate. We can further hone this definition to include only impedances that can be implemented passively. This range of impedances is termed the "Z-width" of the device, and reflects the types of virtual environments it can successfully implement.
Before proceeding with a mathematical discussion of passivity, a simple example will illustrate how apparently passive virtual environments can become non-passive. In this case, the source of the non-passivity is the sample-and-hold. Other possible sources are sensor inaccuracies, pipeline delay, errors in A/D-D/A, and actuator limitations. Consider the simulation of a simple Hookian spring, where we try to enforce the constitutive relation F = kx. Let us assume we sample position, and output force through a zero-order hold. Figure 4 shows how the force changes with position for both a real spring and a sample-and-hold virtual spring. Notice how the force updates whenever the position is sampled, leading to a "staircase" effect. This staircase can lead to net energy generation by the virtual spring, as compared to its passive physical counterpart which can only release as much energy as was stored in it. Unfortunately, this energy generation by normally passive environments opens the possibility of instabilities driven by the haptic display.
To assess passivity of a specific virtual environment mathematically (i.e. determine the Z-width), we must first choose a model for the manipulandum and virtual environment. Careful analysis of this model will give us insight into how various factors affect the Z-width of a haptic display.
