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Ch06-I044963.fm Page 24 Tuesday, July 25, 2006 11:50 AM
Ch06-I044963.fm
24 24 Page 24 Tuesday, July 25,2006 11:50 AM

change the superficial moment of inertia of a rod. This means that the rod length perceived by
grasping arid wielding a rod can be changed, because the perceived length of a rod is proportional
to its moment of inertia(Turvey, 1996).

SUPERFICIAL MOMENT OF INERTIA

One can perceive the length of a rod by simply grasping one end of the rod and gently wielding it,
without even seeing the rod. This kind of touch is referred to as "dynamic touch" (Turvey, 1996).
It has been reported that the perceived length of a rod by dynamic touch increases exponentially
with the moment of inertia of the rod(Chan, 1994). These results imply that perceived length can
be changed freely by varying the moment of inertia of the rod. In the following, we demonstrate
that it is possible to vary the moment of inertia of a rod superficially, without actually varying
its shape or mass distribution.

Consider a rod that is grasped by one end and wielded around the wrist joint, which is assumed
as the origin. If the external torque T gX is applied to the rod in addition to the torque applied
by the wrist, Tx, as shown in Figure l(a), then the equation of motion of the rod is given as

Tx - mgh cos <p + T gX = I(Ax- (1)
where m is the mass of the rod. g is the acceleration due to gravity, h is the center of the mass
of the rod, 7 0 is the moment of inertia of the rod, and <t> and Qx(= 4>) a r e the rotation angle and
angular velocity, respectively, of the rod rotating around the wrist joint.
Next, the external torque T tJx is generated so as to be proportional to the angular acceleration
Q x as

T gX = -ISlx (2)
where 7, is an arbitrary constant. Equation 1 reduces to

~ mgh cos 6 = (7 0 + I S)Q X- (3)
T x
Comparing Eqn. 3 and Eqn. 1, the moment of inertia changes from 7o to 7n + I s by applying the
external torque, T I]X- according to Eqn. 2. We refer to the moment of inertia, 7 S, as the superficial
moment of inertia. Thus, the external torque T gx, satisfying Eqn. 2, can be generated.
It is desirable to design the virtual cane as a non-installing and non-equipping device in order to
make it possible to use the virtual cane in various locations. This requirements of the design can
be fulfilled using a gyroscope. Thus, external torque T,,x is generated by a haptic force generator
consisting of a gyroscope.

Figure l(b) shows the haptic force generator represented as a simplified model consisting of a rod
and a rotor. The rotor is attached firmly to the rod, but can be tilted in the plane perpendicular
to the rod axis. The external torque, T ax, generated by the spinning rotor can be derived using
the coordinate systems having a common origin and the Euler angles (<p. 0,%b) shown in the figure.
The XYZ-coordinate system is the global coordinate system, in which the external torque T gX
is represented as the component along the X-axis. It is assumed without any loss of generality
that the device is wielded in the YZ-plane of the global coordinate system. The x'yV-coordinate
system is obtained by a rotation of the XYZ-coordinate system about the X-axis through an
angle 4>. Consequently, the y'-axis of the x'y'z'-coordinate system coincides with the central axis
of the rod. The x"y"z"-coordinate system, in which the z"-axis coincides with the rotor axle.