Page 75 - Human Inspired Dexterity in Robotic Manipulation

Modeling and Human Performance in Manipulating Parallel Flexible Objects 71

Fig. 5.2 Hand mass estimation by following a haptic force.

The dynamics of the composite (the hand and the virtual object) system are
described as
m h €x h + bð_x h _x o Þ + kðx h x o Þ¼ F cosΩt, (5.41)
m o €x o + bð_x o _x h Þ + kðx o x h Þ¼ 0: (5.42)

The motion of the center of mass, x cm ¼ (m h x h + m o x o )/ (m h + m o ), is gov-
erned by ðm h + m o Þ€x cm ¼ F cosΩt and, under the zero initial conditions, one
obtains
F
x cm ðtÞ¼ ð 1 cosΩtÞ: (5.43)
ðm h + m o ÞΩ 2

Upon resolving for x o (with respect to x h and x cm ) and x h (with respect to x o
and x cm ), the system dynamics are decoupled as
F
2
2
€ x h +2n_x h + ω x h ¼ 2n_x cm + ω x cm + cosΩt, (5.44)
m 0
2
2
€ x o +2n_x o + ω x o ¼ 2n_x cm + ω x cm , (5.45)
where
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kðm h + m o Þ bðm h + m o Þ
ω ¼ , n ¼ (5.46)
m h m o 2m h m o
are, respectively, the natural frequency and the damping factor.
For the low resistance case (n < ω), the steady-state solution to the system
dynamics is developed as
F
^ x h ðtÞ¼ ð 1 acosðΩt + αÞÞ, (5.47)
ðm h + m o ÞΩ 2