Game Development Reference
InDepth Information
v
c
*
v
h
*
v
e
Haptic
tool
Virtual
coupling
Virtual
world
Operator
F
c
*
F
*
F
h
Figure 9.7
Haptic system in network form
Let us consider a stiff manipulator with one degree of freedom, mass
M
m
and a viscous
friction
B
m
, where index
a
refers to the robot actuator (Figure 9.6). The hybrid matrix
of such a system is given by:
F
h
−
M
m
s
v
h
F
a
+
B
m
1
=
(9.8)
v
a
−
1
0
A simple implementation in the virtual world will be to fix
ν
a
=
F
e
on the
virtual object that one is stiffly linked to. Such a coupling in continuous time verifies
both the condition of passivity and unconditional stability. Problems occur as soon as
the previous relation is discretised. We shall later note
x
∗
the discretised
x
variable. By
using the Tustin method that conserves passivity, we obtain for
P
11
:
ν
e
and
F
a
=
P
11
(
z
)
=
Z
m
(
z
)
=
(
M
m
s
+
B
m
)
s
→
(9.9)
2
T
z
−
1
z
+
1
Moreover, we can consider the zero blocking function, concerning
F
c
:
1
2
z
+
1
P
12
(
z
)
=
ZOH
(
z
)
=
(9.10)
z
We thus obtain the following discrete hybrid matrix:
F
h
−
Z
m
(
z
)
ZOH
(
z
)
−
v
h
F
c
=
(9.11)
v
c
10
By applying the unconditional stability criterion (9.6) to this system, we find:
R
e
(
ZOH
(
z
)
/

ZOH
(
z
)

)
≥
1
⇐⇒
cos(
ZOH
(
z
))
≥
1
(9.12)
∠
With this inequality (virtually) never being fulfilled, the system as implemented is not
unconditionally stable. This problem is resolved by addition of a virtual coupling
element in the loop (Figure 9.7).
This element is generally the equivalent of a spring
viscous friction system
(Figure 9.8), that corresponds to a proportionalintegrator equaliser and whose dis
cretised transfer function is:
+
Z
c
(
z
)
=
(
B
c
+
K
c
/s
)
s
→
(9.13)
z
−
1
T
z
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