Implementing low impedances with a damped haptic display

SECTION 4 : Implementing low impedances with a damped haptic display

We've shown in previous sections that increasing the inherent physical damping of a haptic display increases the maximum impedance that the device can emulate. The downside to the increased physical damping is that it makes the unpowered haptic interface feel highly viscous. Thus, in order to display extremely low impedances, the motor actually has to help the user move the handle through the fluid. This section will review two approaches to damping cancellation, focusing on each method's ability to increase the Z-width of a haptic display.

4.1 Velocity-based damping cancellation

The first approach to damping cancellation is negative virtual damping (i.e. an assistive force that is a function of velocity). This technique relies on an accurate model of the damper, since we are inferring the forces that the damper applies to the motor shaft solely from its measured rotational velocity. The simplest such model that one might expect to behave reasonably is :

(8)

where B is the magnitude of the negative virtual damping and v is the rotational velocity of the motor shaft. Therefore, the output impedance can be approximated by :

(9)

where b is the inherent physical damping, m is the inertia of the motor shaft and handle, and s is the Laplace variable. The negative virtual damping, B, is increased until the device is displaying near-zero net damping (i.e., the physical damping is nearly canceled).

Unfortunately, a small change in the physical characteristics of the damper can make the output impedance of the display less than zero (unstable at DC, due to the net negative damping). To get an idea of how much the physical characteristics of the damper changed from trial to trial, we measured the physical damping as a function of angular velocity (without an operator grasping the handle). Figure 14 shows the results of several trials within the same device configuration.

Figure 14. Inherent physical damping of the 1 DOF haptic display vs. velocity for several different trials. Note the non-linearity and the time-variance of the damping from trial to trial.

Figure 14 demonstrates two important characteristics of the inherent damping of the haptic display : 1) the damping is non-linear, and 2) the damping is time-varying. The first feature isn't a problem in terms of performance, as we simply need to use nonlinear negative damping to match the shape of the damping characteristic. However, unless we update this curve in real time, our estimate of the inherent damping of the mechanism can be off by as much as 25% at any given time. We theorized that because the fluid was dissipating mechanical energy in the form of heat, the fluid was heating up, causing its viscosity to vary with use.

To test this hypothesis, we constructed a simplified heat transfer model of the damper (see Figure 15). It assumes pure radial conduction across the fluid and the cup, that the effect of the aluminum vane is negligible as far as heat transfer is concerned, and that the heat is generated along the surface of the vane. These assumptions, together with the simplification of geometry, make the model a poor representation of the actual heat transfer in the damper. However, an analysis of this model gives us insight into why the damping changes from trial to trial.

Figure 15. Heat transfer model of the damper.

and

are the coefficients of thermal conductivity for aluminum and silicone oil, h is an approximate coefficient for convection from aluminum to unforced air, and L is the length of the cylinder (into the page).

is the temperature of the air both outside and inside the cup and

is the temperature of the vane.

Dampers absorb mechanical energy, and dissipate that energy in the form of heat. The steady state heat flux, , out of the damper fluid is given by :

(10)

where b is the damping of the fluid (assumed linear for this analysis), is the rotational velocity, and is the torque applied by the damper to the motor shaft. Assuming nominal parameters of b=0.75 Nm-sec/rad and =5 rad/sec, this leads to a heat flux of about 18 watts. Since we have specified the temperature of the air both inside and outside the cup, and the rate of heat transfer, we should be able to solve for the temperature difference between the vane and the air outside the damper . The first step is to break the problem into two domains : from the outside edge of the vane outward and from inside edge of the vane inward (see Figure 16) :

Figure 16. Heat transfer of the damper spilt into two domains. The boundary conditions for the models are that both and must be in agreement for each domain.

For both domains, the temperature difference between the vane and the outside air is given by :

(11)

where and are the shape factors for the fluid and the cup, respectively, and A is the surface area of the cup. The shape factor for a cylinder is given by :

(12)

where is the outer radius, is the inner radius, and L is the length of the cylinder. Obviously, the shape factors for the two domains are different, as is the surface area, since the radii for the outer domain will be larger than for the inner one. Table 1 is a catalog of the shape factors and surface areas relevant to this analysis.

Parameter	Outer domain	Inner domain
Shape factor of cup	163	113
Shape factor of fluid	2500	2100
Surface area of cup	28	8

Table 1. Catalog of shape factors and surface areas for the damper heat transfer model.

Substituting the geometric factors from Table 1 and assuming the temperature of the vane is the same for both domains, we obtain :

(13)

Since and must add up to the total energy dissipation of the damper, (10), we find that

This calculation cannot be relied upon too heavily, since most of the temperature difference is accounted for in the convection term, which varies greatly with air circulation. However, the results for the other two terms are important. Conduction through aluminum is so efficient in this particular case that the temperature gradients across it can be ignored. Thus, the temperature on one side of an aluminum wall should approximately equal the temperature on the other side. At this point, one might theorize that regulating the temperature of the outside of the cup would have the effect of regulating the temperature of the fluid as well. Unfortunately this approach will not work, as the fluid itself appears capable of maintaining a temperature gradient, making temperature regulation an unattractive option. Informal experiments back up the results of our model. Damping cancellation achieved in this manner often becomes unstable after a few minutes of continuous use, accompanied by temperature rises of a few degrees Fahrenheit on the outside of the damper.

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