**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

**8.3.1 Dead Reckoning**

Perhaps the simplest but least accurate method of positional navigation is dead reckoning -- the determination of position by keeping an account of the distance and direction traveled, without reference to any exogenous sources of information other than the beginning point. For example, starting from a well-defined initial location, a legged nanorobot counts the exact number of footfalls taken in all directions; the measured length of each footfall gives the distance traveled. However, anchor points on biological membranes are positionally unstable (Section 9.4.3.1), so a micron-sized bipedal walker with a 100-nm leg stride that achieves even a very optimistic 1% footpad positional stability (~1 nm) during each leg placement cycle will find that the accumulated error in its computed position has reached one body length (~1 micron) after just 1000 strides, or ~100 microns (~100 body-lengths) of locomotion. This level of accuracy is equivalent to ~one cell-width of error per ~2 mm (~20,000 strides) of travel.

Accuracy may be at most 2 orders of magnitude better when
negotiating hard or very firm surfaces such as tooth enamel or bone. The classical
positional variance of a telescoping nanomanipulator capable of 100 nm of horizontal
travel is at best 0.01 nm (Section 9.3.1.4); a device
stiffness of ~10 nN/nm gives a limb deflection of ~0.1 nm if ~nN forces are
applied, resulting in a measurement error of 0.01%-0.1% at each footfall. Additional
measurement errors due to varying strains caused by fluctuating normal bone
loads are of similar magnitude. For example, strain ~ m g / E A ~ 0.01% for
the ~2.5-cm diameter human femur, taking a human mass of m ~ 10^{2}
kg supported by two femurs of total cross-sectional area A ~ 10 cm^{2},
with Young's modulus E ~ 10^{10} N/m^{2} for wet compact bone
(Table
9.3) and acceleration of gravity g = 9.81 m/sec^{2}. Pressures exerted
during chewing may reach 10-100 atm (Chapter 28),
giving a maximum natural strain on tooth enamel of (10^{7} N/m^{2})
/ E ~ 0.01%, taking E = 7.5 x 10^{10} N/m^{2} for enamel (Table
9.3).

Navigational accuracy via dead reckoning may be up to 1-2 orders of magnitude poorer in other cases, especially during nanorobot swimming (Section 9.4.2) which requires additional corrections for fluid motions relative to surfaces.

Sequential monitoring of accelerations and rotations alone
produces even less accurate results. Assuming typical environmental accelerations
of ~0.4 g's experienced by a 1-micron nanorobot, then data sampling at ~10 KHz
to an accuracy of ~4 x 10^{-5} g per measurement (Section
4.3.3.2) yields a cumulative error of ~0.4 g after ~10^{4} measurements,
a mere 1 second of travel. Pendular orientation sensors accurate to ~2 milliradian
(Section 4.3.4.2) produce ~1 radian of accumulated
error after a non-recalibrated chain of just 500 measurements.

Dead reckoning is most useful in two special circumstances:

1. hard-frozen tissues presenting immobile and highly anchorable surfaces; and

2. localized navigation requiring only very short distances to be traversed, such as locomotion between organelles in cyto or where abundant functional or positional cues allow frequent recalibration of the estimated current location.

Last updated on 19 February 2003