Page 76 - Human Inspired Dexterity in Robotic Manipulation

72 Human Inspired Dexterity in Robotic Manipulation

F
^ x o ðtÞ¼ 2 ð 1 bcosðΩt + βÞÞ, (5.48)
ðm h + m o ÞΩ
where

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2
fω ð1+ μÞΩ g +4n Ω 2
2
2
a ¼ , (5.49)
2 2
ðω Ω Þ +4n Ω 2
2
2
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ω +4n Ω 2
2
4
b ¼ , (5.50)
2 2
ðω Ω Þ +4n Ω 2
2
2
2nμΩ 3
tanα ¼ , (5.51)
2
2
ðω Ω Þfω ð1+ μÞΩ g +4n Ω 2
2
2
2
2nΩ 3
tanβ ¼ , (5.52)
ω ðω Ω Þ +4n Ω 2
2
2
2
2
and μ ¼ m o /m h .
By measuring the differences of the peak amplitudes for the hand and/or
the object, Δx h , Δx o , one can estimate the hand mass from the following
nonlinear (with respect to m h ) relationships:
2Faðm h Þ 2Fbðm h Þ
Δx h ¼ , Δx o ¼ : (5.53)
ðm h + m o ÞΩ 2 ðm h + m o ÞΩ 2
2
2
Note that for a relatively large natural frequency ω ≫ Ω and a sufficiently
small damping factor n, we have a 1, b 1, and the estimation equations
become linear and, therefore, easier to use. Either of them can be used for
the estimation of the hand mass m h . Practically, however, it is more prefer-
able to deal with the second one as the measurement of Δx o is less noisy
because Eq. (5.42) acts as a second-order low-pass filter.
5.5 EXPERIMENTAL RESULTS
To check the velocity profiles of reaching movements with multimass flex-
ible objects, we conducted an experiment. In the experimental setup, shown
in Fig. 5.3, a haptic device (PHANToM Premium 1.5 HF, maximum exer-
table force 37.5 N) is connected to the PC (Dell Optiplex 9020SFF, Intel
Core i7-4770, 3.4 GHz) through a parallel port card (LF811KB PCI-E).
Four naı ¨ve right-handed male subjects participated in the experiments.
The subjects were instructed to move a two-mass virtual flexible object (m 1