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active part, most often corresponding to a discretisation procedure (Colgate &
Shenkel, 1994);
For the second, we shall talk about the objective of “transparency''. The purpose
is to get close to the ideal system which is characterised as:
f h =
h f ·
f e
v h =
h v ·
v e
where h f and h v represent the force scaling and speed scaling respectively. Such an
objective implies good position tracking, i.e. x e
x h , at least in free mode;
The third criterion corresponds to the transient state of the system and helps to
define the relative values between stiffness and damping of the coupling equaliser
between the control element and the virtual manipulator. For example, one may
want a damping close to the critical damping when the arm and the virtual
environment are free.
We will deal with the stability problems related to haptic interfaces in the next
9.3.1 Passivity
We use the concept of passivity to manage the stability of the haptic systems. A system
is called passive if it can store, dissipate and restore energy, but without being able
to create any energy. Most of the standard physical elements have this mechanical
Definition 1: We call P in
y the power entering a system for which x is the input
vector and y the output vector. By noting E store as the energy stored by the system and
P diss the power that it dissipates, we obtain:
∂t E store
P in
P diss
The considered system is called passive if and only if we obtain the inequation (9.3).
P in
E store ( t )
E store (0)
P diss
E store (0)
This definition exactly corresponds to the intuitive concept of passivity. There are
numerous other ways to present this property of passivity (Micaelli, 2002). There are
a large number of results concerning passivity. Theorems 1 and 2 will be used for
quadripole applications.
Theorem 1: for Q ( jw ) admittance, impedance or hybrid quadripole (see annexe for
details regarding these matrices), the system is passive if and only if Q is real positive,
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