Torque-based damping cancellation

4.2 Torque-based damping cancellation

The basic problem with velocity-based negative virtual damping is that we're relying on a model of the damper to ensure system performance. While the nonlinear characteristic of the physical damping doesn't pose any problems, the time-varying characteristic eliminates it as a good technique for implementing low impedances. A better approach is to measure the torque applied by the fluid to the motor shaft directly, and then use that information to cancel the effect of the damping. With this technique, the damping cancellation torque varies in real time with the damping torque. The typical approach to measuring torque is to measure strain in a flexure element using strain gauges, and from that signal calculating torque. Figure 17 shows a block diagram for this implementation.

Figure 17. Block diagram of torque-based damping cancellation. b1 and b2 are the damping coefficients of the manipulandum before and after the addition of the damper. Note that

and

are the transfer functions of the continuous and discrete damping filters, respectively.

An important clarification should be made at this point : we don't want to entirely cancel out the effects of the damper (that would just take us back to where we started!). We want the stability provided by inherent physical damping with the backdriveability of the original design (no damper, low friction, etc.). In the language of Z-width, damping extends the high end of the Z-width by absorbing extra energy that can be produced by high virtual impedances, making them more stable. However, we still want the low end of the Z-width to be near-zero impedance. This brings about the concept of frequency-dependent damping. The situation described above corresponds to a physical damper that works only at frequencies above the human motion bandwidth (less than 10 Hz). For a device to feel backdriveable to a human only requires near-zero impedances in the 0-10 Hz range, since the human operator has no way to explore higher frequencies. Appropriate low-pass analog and digital filtering of the damper cancellation torque can achieve this effect. In Figure 17, if we assume the cutoff frequency of the digital filter is much less than the sampling time, then we can model the discrete time filter as its continuous time counterpart. This assumption allows us to combine and into a single continuous time filter, . The output impedance of this system for H(z) set to zero is given by :

(14)

where m is the inertia of the motor shaft/handle, b1 and b2 are the damping coefficients of the manipulandum before and after the damper was added, respectively. is the transfer function of the damper filter. This system is passive iff

(15)

Any filter satisfying this constraint will be guaranteed to produce a passive output impedance. Thus, if is chosen to have a lowpass characteristic, we can achieve the effect of a frequency-dependent damper (i.e. one that works only at frequencies outside the human actuation bandwidth). Details of the design of these filters can be found in [7].

The remainder of Section 4 will be spent dealing with the practical issues of constructing a force sensor from scratch. In particular, both damper and force sensor design play critical roles in the proper implementation of torque-based damping cancellation. In 4.2.1, we will use the Navier-Stokes' equations to obtain expressions for the torque applied to the motor shaft by the fluid in the damper. In 4.2.2, we will show how these torques can be measured using strain gauge circuitry. In 4.2.3, we will show that, in addition to the tangential shear force on the damper cup, the vane moving through the fluid causes a pressure buildup, which in turn results in radial forces on the cup of the damper. In 4.2.4, we will show how these forces can show up as unwanted signals in the strain gauge circuitry, resulting in a poor signal to noise ratio. In 4.2.5, we will summarize how we redesigned the damper based on the these results.

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