Page 74 - Human Inspired Dexterity in Robotic Manipulation

70 Human Inspired Dexterity in Robotic Manipulation

1 ð T 2
J¼ u ðtÞ dt, (5.38)
2 0
under the dynamic constraint (5.36) and the boundary conditions
T T
xð0Þ¼ 0,0,0,0,0,0,0½ Š , and xðTÞ¼ L,0,L,0,L,0,0Š . Proceeding along
½
the same line of reasoning as in the case of the minimum hand-jerk model,
one establishes the trajectory of the mass center in a form structurally similar
to Eq. (5.23). It can be shown then that the hand trajectory, established from
Eq. (5.31), will have the following representation:

!
5 2
X k X
x h ðtÞ¼ L α k t + β sinðω k t + φ Þ + γ tsinðω k t + ψ Þ , (5.39)
k
k
k
k
k¼0 k¼1
where the constant coefficients α k , β k , γ k , φ k , and ψ k are functions of ω 1 ,ω 2 ,
and T.
5.4 HAND MASS IDENTIFICATION
To use the minimum hand-force-change model, one needs to know the
mass of the hand. To estimate the parameter m h reliably, it is not enough
to know the mass of the hand predicted by anthropometric measures, that
can be done as, for instance, in [20]. This parameter actually accumulates
three factors: the mass of the hand itself, the configuration-depended inertia
of the human arm in multijoint movements reduced to the hand, and the
end-point inertia of the haptic device.
Typically the human hand is modeled as a mass-damper-spring system,
and this model holds for small movements and for a short period of time.
Based on this model, the hand impedance can be evaluated by perturbing,
by a robotic manipulandum, the hand during maintenance of a given pos-
ture, and measuring the hand displacement. This perturbation technique is
reported in [21–23]. However, its use for a lightweight haptic device with-
out additional hardware, such as the one proposed in [24], is problematic.
A simple method, based on the accommodation to forced vibrations and
applicable to the estimation of the hand mass only, can be constructed as fol-
lows. Assume that a human subject can follow the motion of a virtual object
of mass m o , connected to the hand by a spring of stiffness k and a damper of
viscosity b, without developing his or her own driving force (see Fig. 5.2).
Assume also that a periodic force of amplitude F and frequency Ω is applied
to the hand, and the total haptic force is defined as
f haptic ¼ bð_x o _x h Þ + kðx o x h Þ + F cosΩt: (5.40)