**Nanomedicine,
Volume I: Basic Capabilities**

**©
1999 Robert A. Freitas Jr. All Rights
Reserved.**

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

**4.3.4.1 Gimballed Nanogyroscopes**

Pivoted gyroscopes of the type described in Section
4.3.3.4 are impractical for orientational sensing in all but the largest
medical nanodevices. Setting w_{max} (Eqn.
4.17) equal to w_{min} (Eqn.
4.18), taking a_{min} = g = 9.81 m/sec^{2} (gravitational
acceleration), and optimizing for minimum sensor size gives a critical nonwobbling
nonbursting spin velocity w_{crit} ~ 6.8
x 10^{7} radians/sec at a minimum device radius of ~35 microns. If a
pivoted gyroscope is smaller than this minimum radius, then spinning it slower
than w_{crit} causes the gyro to tumble helplessly,
while spinning it faster than w_{crit} tears
the gyro apart.

However, a triaxially gimballed nanogyroscope may be regarded
as having its pivot point near its center of mass, so L_{p} becomes
very small and w_{min} becomes very slow,
making nanogyroscopes feasible in medical nanorobots. Taking h = 1 micron, r
= 0.5 micron, r = 3510 kg/m^{3} and a_{min}
= g in Eqn. 4.18, then if the gimbals are
aligned such that L_{p} = 1 nm between the pivot point and the effective
center of gravity along the rotation axis, then w_{min}
~ 400 rad/sec. If gimbal tolerances are improved to L_{p} = 0.1 nm,
then w_{min} ~ 125 rad/sec.

How stable is a nanogyroscopic orientation standard? There are two principal considerations.

First, Brownian thermal rotation in all three angular degrees
of freedom (q, j and y)
gives rise to a small nutation around the nanogyroscopic precession axis of
magnitude Dq_{nutate} ~ p_{j}
/ p_{y}, where p_{j}
and p_{y} are angular momenta of the gyro.^{448}
The minimum value for p_{j} is approximated
by (1/2) kT ~ (1/2) p_{j}^{2} / I_{1},
where I_{1} = (1/4) m r^{2} + (1/12) m h^{2}, the moment
of inertia around j, and m = p
r^{2} h r. The maximum value for p_{y}
is given by the burst-strength condition w < w_{max}
(Eqn. 4.17) as p_{y}
~ I_{3} w, where I_{3} = (1/2) m
r^{2}, the moment of inertia around y. Hence:

For T = 310 K, r = 0.5 micron, h = 1 micron, s_{w}
~ 10^{10} N/m^{2}, Dq_{nutate}
~ 0.8 microradian.

Second, the gimbal bearings have small frictional losses,
exerting small torques on the gyro and causing a small precession away from
the original orientation. The angular velocity of precession w_{precess}
~ T_{gimbal} / (I_{3} w) (rad/sec)
where T_{gimbal} is the frictional torque caused by an imperfect gimbal
bearing. Assume that the gimbal is driven by external forces to oscillate at
some frequency n_{gimbal}, and that a gimbal
bearing dissipates power at the rate of P_{gimbal} (watts), so that
each oscillatory motion of the gimbal dissipates E_{gimbal} = P_{gimbal}/n_{gimbal}
(joules). If the angular amplitude of the gimbal motion is a_{gimbal}
(radians) and the gimbal radius is r_{gimbal}, then the gimbal bearing
travels X_{gimbal} = a_{gimbal} r_{gimbal}
(meters) per gimbal oscillation and the force applied in dissipating an energy
E_{gimbal} is F_{gimbal} = E_{gimbal}/X_{gimbal}.
Thus, the torque applied by each gimbal oscillation event is T_{gimbal}
= F_{gimbal} r_{gimbal} = P_{gimbal} / (n_{gimbal}
a_{gimbal}); P_{gimbal} = k_{p}
v_{gimbal}^{2} (see Drexler^{10}),
where v_{gimbal} = a_{gimbal} n_{gimbal}
r_{gimbal} is the sliding speed of the gimbal bearing surfaces during
movement and k_{p} is a constant that depends solely on the geometry
of the bearing.

The change in orientation angle caused by each gimbal oscillation
event is Dq_{osc} = w_{precess}
/ n_{gimbal}. The number of gimbal oscillations
N = t n_{gimbal}, where t is elapsed time
since the gyro was last calibrated. If gimbal oscillations are independent and
randomly distributed, then the total change in orientation angle is very roughly
Dq_{precess} ~ Dq_{osc}
N^{1/2}; assuming r_{gimbal} ~ r, then:

From Drexler^{10}, k_{p}
= 2.7 x 10^{-14} watt-sec^{2}/m^{2} as a conservative
value for a small (~2 nm) stiff bearing. Random hydrodynamic flows induced by
thermal fluctuations inside biological cells have a characteristic duration
t_{fluct} ~ 10 millisec,^{1069}
which suggests a_{gimbal} ~ 0.06 radians
(~3°) from Eqn. 3.2 for an R ~ 1 micron in cyto
nanorobot. Taking r = 0.5 micron, h = 1 micron, r
= 3510 kg/m^{3} for diamond (giving m ~ 3 x 10^{-15} kg), w
= 1 x 10^{9} rad/sec (just below the w_{max}
~ 4.8 x 10^{9} rad/sec bursting speed), and assuming n_{gimbal}
~ t_{fluct}^{1} = 100 Hz (to which the results are not
terribly sensitive), then Dq_{precess} ~
10^{-8} t^{1/2} (radians). Thus a Dq_{precess}
~ 1 microradian orientation shift takes ~1 hour; a Dq_{precess}
~ 4 microradian shift takes ~1 day; and a Dq_{precess}
~ 70 microradian (~10 arcsec) shift takes ~1 year. (Total angular error Dq_{total}
= Dq_{precess} + Dq_{nutate}.)
Thus a gimballed nanogyroscope may be sufficiently stable to serve as a readily
storable, easily transportable, and highly accurate onboard orientation standard.

Last updated on 17 February 2003