Game Development Reference

In-Depth Information

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

(9.1)

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

chapter.

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

property.

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:

=

xT

·

∂

∂t
E
store

=

+

P
in

P
diss

(9.2)

The considered system is called passive if and only if we obtain the inequation (9.3).

t

t

=

−

+

≥−

∀

≥

P
in
dτ

E
store
(
t
)

E
store
(0)

P
diss
dτ

E
store
(0)

t

0

(9.3)

0

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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