full Paper - Nguyen Dang Binh

contact point, c we know that all the forces which are exertable
(Fset) belong to the vector subspace generated by the
columns of Jc:
Fset Im Jc
The forces outside Fset are “controlled” by the joint
reaction-forces. They cannot be programmed or controlled
from the virtual environment and are outside of the interfacemobility
space. In such directions the control is completely
left to the operator. The force feedback information produced
are such that they cancel the user efforts in order to
produce no movements along that directions.
The following result due to the force mapping can be
fixed:
1. not all forces can be replicated by means of the haptic
contacts;
2. when the rank of the Jacobian matrix is equal to 6, the
force feedback presented to the operator can be an approximation
of the virtual force computed into the VE.
For example with reference to figure 14 the force textbfF
is not exactly reproducible.
Virtual
Force
Massimo Bergamasco / Haptic Interfaces
Replicated
Force
Figure 14: Not all forces are exactly reproducible
If we decompose the force components along the two orthogonal
spaces: Fb Im Jc and Fa Im Jc Ý only the
Fb component can be reproduced by the contact. Let consider
the force mapping rule which determines the effective
component at joints (see next paragraph):
M j J T c F J T c Fa Fb J T c Fb
3. the cases in which an approximation is present can be
distinguished into:
¯ kinematic singularities due to the posture. Such kind
of singularities are generally avoided when designing
an exoskeletric structure;
¯ due to the mechanical structure (DOF at the contact
6).
c­ The Eurographics Association 2005.
13
4.4. Mapping Forces into Contacts
In virtual environment forces can be generated between any
couple of movable objects. With reference to the virtual representation
of the operator body, the contact point usually
does not match with the correspondent points of the haptic
contacts except that in the case of a simple interaction context
(one point attachment and metaphoric interfaces).
The strategy of generating force on the operator hand
should consider that the system can never exactly replicate
the avatar sensations whenever they have been generated by
forces applied in points which are different from the haptic
contacts.
In these cases the mapping strategy will introduce an error
of replication.
The interface is in a
singular configuration
The set of
possible FF
is limited
The contact Jacobian is
singular
Figure 15: Singularity position in the interface can reduce
the set of replicable and controllable forces
When the contact force should be applicated exactly on
the mechanical contact between the operator and the interface,
if no contact singularities are present, the required force
at joints can be computed as:
M J T c Fc
which produces on the contact the exact reaction force, in
fact multiplying both sides by a small joint-angular change,
we obtain:
δωM δL δωJcFc δxFc L
that satisfies the virtual work principle for the reaction force.
When the contact force is generated outside the haptic
contact (C) of the operator with the mechanical interface,
in an external point (A) there are two different strategies for
computing the proper contribution to the motors:
1. the simplest strategy does not take care of the contact
constraints and assumes that the attachment rigidly holds
the link by allowing the force and torque exchange along