**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

**9.4.3.5 Legged Ambulation**

Consider a 1 micron^{3} nanorobot cytoambulating using
legs tipped with appropriate footpads to traverse a vascular wall. Each leg
is assumed to be similar in size and function to the 100-nm long, 30-nm wide
cylindrical telescoping nanomanipulator described in Section
9.3.1.4. Each nanorobot has a total of 100 legs, occupying 7% of the 10^{6}
nm^{2} underside area of the device. To allow tenfold redundancy, at
any one time only N_{leg} = 10 legs are deployed and in use. The remainder
are stowed as spares.

Ignoring <~1% traction losses (Section
9.4.3.1), the total force that must be supplied by all N_{leg} working
legs is:

_{}
{Eqn. 9.82}

The maximum dislodgement or "headwind" force normally encountered
along blood vessel walls is F_{dis} ~ 40 pN (Section
9.4.3.3). The viscous force on each leg is approximated by Eqn.
9.75 as F_{leg} ~ F_{nanoN} ~ 6 pN, taking L_{leg}
= 100 nm, R_{leg} = 15 nm, and v_{leg} = 1 cm/sec. From Eqn.
9.73, F_{nano} ~ 100 pN, taking h = 1.1
x 10^{-3} kg/m-sec for plasma at 310 K, R_{nano} ~ 0.5 micron,
and v_{nano} = 1 cm/sec. F_{total }= 200 pN, easily within the
capacity of a single leg (Section 9.3.1.4), giving
an allocation of F_{total}/N_{leg} = 20 pN per leg. Maximum
safe towing force is <~300 pN/leg, assuming 100 nm^{2} footpads and
a 3 x 10^{6} N/m^{2} membranolytic limit for plasma membrane.^{1422}

Many N-podal gaits are possible.^{3499-3507}
In the most conservative gait, only 1 leg is moved at a time while the remaining
(N_{leg }-1) legs stay anchored at their footpads. Given a full center-to-center
working arc of X_{arc} ~ 80 nm, each leg must travel X_{swing}
= X_{arc}/N_{leg} = 8 nm in a time t_{swing} = X_{swing}
/ v_{leg} = 0.8 microsec at a velocity v_{leg} = 1 cm/sec, with
a per-leg duty cycle of f_{duty} = N_{leg}^{-1} = 10%
and an operating frequency of n_{leg} = f_{duty}/t_{swing}
~ 100 KHz. From Eqn. 9.74, nanorobot motive
power is P_{nano} = F_{total} v_{nano} / e% ~ 10 pW,
taking e% ~ 0.20 (20%). Conservatively taking each footpad binding event as
costing E_{bind} ~ 100 zJ (Section 4.2.1), then
footpad binding power requirement is P_{bind} = N_{leg} n_{leg}
E_{bind} ~ 0.1 pW, a negligible contribution.

For the least conservative gait, only 1 leg stays anchored
while the remaining (N_{leg }-1) legs are in motion. In this case, X_{swing}
= X_{arc} = 80 nm, giving t_{swing} = 8 microsec and n_{leg}
~ 10 KHz, but motive power, nanorobot velocity, and leg duty cycle are unchanged.
Doubling leg length to L_{leg} = 200 nm while holding R_{leg},
v_{leg}, and v_{nano} unchanged decreases operating frequency
to n_{leg} ~ 5 KHz while increasing F_{total}
to 230 pN and P_{nano} to 12 pW. Doubling the velocities doubles force
and operating frequency, and quadruples the power demand. Perhaps counterintuitively
from common macroscale experience,^{1486}
for small ambulators traveling in viscous-dominated media (e.g., low Reynolds
number ambulators), shorter legs may produce the highest motive velocity for
the lowest power requirement and applied force.

As a biological analog, tiny extensible hydraulic tube feet
specialized for burrowing and stepping locomotion, often with terminal suckers,
have been extensively described in echinoderms.^{1472-1475}

Last updated on 21 February 2003