**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

**4.3.3.2 Displacement
Accelerometers**

Formed elements in the human bloodstream (e.g., red cells,
white cells, platelets) typically are buffeted by much smaller accelerations.
For example at 310 K, instantaneous random thermal accelerations of a_{thermal}
~ 500 g's are experienced by a 0.2-micron virus particle, ~0.05 g's by a 2-micron
platelet, ~10^{-4} g's by a 10-micron neutrophil -- and ~0.4 g's by
a 1 micron^{3} spherical nanorobot. Force-based sensors cannot easily
detect such low accelerations, since the smaller the sensor, the larger the
a_{min} for a given SNR (Eqn. 4.11).
An alternative approach uses displacement sensors to determine an object's velocity
twice in rapid sequence, allowing acceleration to be computed from the difference.

Consider a constantly accelerating object that triggers three
clocking latches at positions x_{1}, x_{2}, and x_{3},
in sequence. Time and position of the object are measured as (x_{1},
t_{1}), (x_{2}, t_{2}), and (x_{3}, t_{3}).
The measured constant acceleration is then given by

** _{}**
{Eqn. 4.12}

where v_{32} = (x_{3} - x_{2}) / (t_{3}
- t_{2}), v_{21} = (x_{2} - x_{1}) / (t_{2}
- t_{1}), t_{32} = (t_{3} + t_{2}) / 2 and t_{21}
= (t_{2} + t_{1}) / 2. If adjacent latches are an equal distance
x_{latch} apart so that x_{latch} = (x_{3} - x_{2})
= (x_{2} - x_{1}), and taking t_{1} = 0, then Eqn.
4.12 reduces to

Assume that a slowly accelerating external object passes three
clocking latches that are fixed to the exterior of a nanorobot of radius R.
Measurements of the object's velocity incur an unavoidable error due to nanorobot
thermal motion. The object is observed to pass between latch pairs during a
measurement time ~t_{meas}. During this same interval, from Eqn.
3.1 the nanorobot translates a distance DX =
(kT t_{meas}/ 3 p h R)^{1/2}, incurring
a maximum velocity measurement error of ~DX / t_{meas},
and thus from Eqn. 4.12 a maximum acceleration
measurement error of ~2 DX / t_{meas}^{2}
if (t_{3} - t_{2}) ~ (t_{2} - t_{1}) ~ t_{meas}.
Thus the minimum detectable acceleration is

Taking T = 310 K, h = 1.1 x 10^{-3}
kg/m-sec and R = 1 micron, then a_{min} ~ 10^{-7} g's for t_{meas}
= 1 sec; a_{min} ~ 0.004 g's for a more reasonable t_{meas}
= 10^{-3} sec measurement time in the in vivo environment. Onboard reference
oscillators with the requisite accuracy are described in Section
10.1.

Last updated on 17 February 2003