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Modeling and Human Performance in Manipulating Parallel Flexible Objects 81

λ 7 (T) ¼ 0), while the general structure of the optimal solution will
remain the same.
4. Minimum acceleration of the center of mass model. It would be instructive to
test also the minimum acceleration of the center of mass model [17]
because so far it has been tested only for a single mass-spring system,
and its applicability to reaching movements with parallel flexible
objects has been discussed only in theory [17]. However, the use of
the acceleration of the center of mass in the performance index does
not seem to be right for modeling of reaching movements in dynamic
environments. Indeed, one can easily create a simple environment
(e.g., a block of mass held by the hand over a vertical pulley, with
masses of the block and the hand being equal) where the center of mass
does not change. In this case the corresponding computational model
becomes inapplicable.
However, the idea of using the hand acceleration in the performance
index [32] can be insightful. As the model [32] implies bang-bang con-
trol, it is structurally similar to the minimum time control models. In this
connection, it would be reasonable to change the experimental protocol,
by removing the time constraint, and test the experimental data against
theoretical predictions by the minimum time [33] and the minimum
hand acceleration [32] models.

5.7 CONCLUSIONS
An analysis of human reaching movements in manipulation of parallel flex-
ible objects has been undertaken in this study. To predict the trajectory of
the human hand, the minimum hand-jerk and hand-force-change models,
claimed to be inapplicable to modeling of reaching movements in this spe-
cific dynamic environment [17], have been developed. It has been shown
that within these models, the optimal hand trajectory is composed of a
fifth-order polynomial (as in the classic-jerk model) and trigonometric terms
depending on the natural frequencies of the system and movement time.
A virtual reality-based experimental setup with a haptic simulator has been
designed, and the theoretical predictions have been tested against experi-
mental data. A method for the hand-mass identification, based on following
a periodic input force, has also been proposed. The experimental results
obtained confirmed the validity of both, the minimum hand-jerk and the
minimum hand-force-change models for modeling of human-like rest-
to-rest reaching movements.