The effect of inherent mechanism dynamics and sampling on Z-width

2.3 The effect of inherent mechanism dynamics and sampling on Z-width

Figure 5 shows the general layout of a haptic display, with a strong focus on the manipulandum. This representation allows us to appreciate the effects of manipulandum dynamics, sensor and actuator performance and sample-and-hold on the performance of the whole system. We can also see that these components must be highly robust, since they have to form a stable connection between two largely unspecified systems (i.e. the human operator and the real-time simulation).

Figure 5. Components of a haptic display. v(t) and

are the velocity and force at the endpoint of the manipulandum;

is a measurement of x(t), the state of the manipulandum;

is the actuators' implementation of

, the force output of the virtual environment simulation.

We are particularly interested in the effect of inherent manipulandum dynamics and sample-and-hold on the achievable dynamic range of haptic displays, so we have developed the simplest model possible that still captures the salient features of these components (Figure 6). This model has only one degree of freedom and uses an ideal actuator, position sensor, sampler and zero order hold. The model of the mechanism dynamics approximates the behavior of the 1 DOF device in our laboratory.

Figure 6. Model of a one degree-of-freedom haptic interface. m is the inherent mass of the display, b is inherent damping, v is velocity, x is position,

is the sampled position, T is the sampling rate, u is the control effort, and f is the force applied by the operator. H(z) is the (linear) pulse transfer function of the virtual environment.

As a point of departure, an impedance will be considered achievable if it can be implemented passively. The following theorem, proven in [9], is useful:

Theorem -- A necessary and sufficient condition for passivity of the haptic interface model in Figure 6 is:

(1)

Here, b is the inherent damping of the display, T is the sampling time, H(z) a pulse transfer function representing the virtual environment, and .

This result shows that indeed the implementation of passive elements in a haptic display can be non-passive. Furthermore, we can see that two other factors play a role as well : 1) inherent manipulandum dynamics (in this case, the viscous damping of the mechanism) and 2) sampling time.

As stated above, we have chosen the "virtual wall" as a benchmark problem, since it utilizes a unilateral constraint and a wide range of impedances. We will consider a common implementation composed of a virtual spring and damper in mechanical parallel, together with a unilateral constraint operator (see Figure 7).

Figure 7. Model of a virtual wall as a spring and damper in parallel. K defines the virtual stiffness, B the virtual damping, and

the location of the wall.

The total force experienced by the operator is given by (2):

(2)

A velocity estimate is obtained via backward difference differentiation of position, giving the following pulse transfer function within the wall:

(3)

where K > 0 is a virtual stiffness, and B is a virtual damping coefficient (we will allow B to be positive or negative). A condition for passivity is found by inserting (3) into (1). After some manipulation:

(4)

which can be further reduced to:

(5)

The following conclusions may be drawn from this analysis:

To achieve passivity, some physical dissipation is essential.
With other variables (b and B) fixed, the maximum achievable virtual stiffness is proportional to the sampling rate.
The maximum achievable virtual damping for zero stiffness is independent of the sampling rate.
With other variables (b and T) fixed, higher virtual stiffnesses can be achieved at lower values of virtual damping.

While these are potentially useful guidelines for design, it is important to recognize that passivity is a conservative design requirement. This is because, even if an interface is active, a human operator may not be able to destabilize it. As an example, we rarely find instances in which the instability caused by coupling to a human operator exceeds 150-200 Hz, but examination of (4) shows that, for positive B, passivity is most readily violated at the Nyquist frequency, which may be 500 Hz or greater. At the Nyquist frequency, the factor

is negative, but at less than half the Nyquist frequency, it is positive. If it is positive at the frequency of instability, then the implications of the last bullet point above are completely reversed: higher virtual stiffnesses can be achieved at higher values of virtual damping.

With the above caveat in mind, the implications for haptic interface design can be discussed. For instance, to implement stiff, dissipative walls (high K, B), it is apparently helpful to maximize b and minimize T. Fast sampling is a standard objective, but maximizing damping goes against conventional wisdom [14]. It is generally argued that the dynamics of a haptic interface should be dominated by the virtual environment (which is, after all, the programmed behavior we wish to display) rather than any inherent dynamics (which is considered parasitic). In other words, the interface hardware should be "transparent." Unfortunately, the notion of transparency places focus on mimicking the governing equations (e.g., state equations) of physical systems, but not on obeying underlying physical laws (such as conservation of energy). Adding physical damping helps the sampled-data system to behave as physical law would dictate.

But is there a cost to additional damping? Is the behavior inside the wall improved at the cost of the behavior outside the wall? The answer is no. The reason for this answer is that, as seen in equation x, one may introduce negative virtual damping outside the wall. In fact, since K = 0, one may select B = -b, resulting in zero net damping (although this is borderline passive, and perfect cancellation is difficult to achieve in practice). This importance of physical damping will be further elaborated in Section 4.

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