In this appendix, we derive the passivity condition for a specific implementation of a virtual wall. This implementation is similar to that of [9], except that this time the differentiator is cascaded with a first order low pass filter of time constant . Thus the control law we are attempting to implement is :

(A1)

If a backwards difference mapping is used, the Z-transform is :

(A2)

Recall the general passivity result from Section 2.3 :

(1)

Substituting (A2) into (1), we obtain the following for a passivity condition :

(A3)

After some manipulation, (A3) reduces to (A4) :

(A4)

It is important to recall at this time that can vary from 0 to , allowing to vary from -1 to +1. Differentiating the right hand side with respect to and solving it when set to zero allows us to find the worst case value of :

(A5)

which simplifies to :

(A6)

and

(A7)

Substituting (A7) into (A4) :

(A8a)

and

(A8b)

Both of these conditions must be met for the haptic display to be passive. By looking at the coefficients of B, we can see that when B is positive, (A8a) is more likely to be violated, and that when B is negative, (A8b) is more likely to be violated. Thus, the final result is shown in (7).

(7a)

(7b)