Game Development Reference
In-Depth 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 proportional-integrator equaliser and whose dis-
cretised transfer function is:
+
Z c ( z )
=
( B c +
K c /s ) s
(9.13)
z
1
T z