Nanomedicine, Volume I: Basic Capabilities

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

9.4.3.5 Legged Ambulation

Consider a 1 micron3 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 106 nm2 underside area of the device. To allow tenfold redundancy, at any one time only Nleg = 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 Nleg working legs is:

{Eqn. 9.82}

The maximum dislodgement or "headwind" force normally encountered along blood vessel walls is Fdis ~ 40 pN (Section 9.4.3.3). The viscous force on each leg is approximated by Eqn. 9.75 as Fleg ~ FnanoN ~ 6 pN, taking Lleg = 100 nm, Rleg = 15 nm, and vleg = 1 cm/sec. From Eqn. 9.73, Fnano ~ 100 pN, taking h = 1.1 x 10-3 kg/m-sec for plasma at 310 K, Rnano ~ 0.5 micron, and vnano = 1 cm/sec. Ftotal = 200 pN, easily within the capacity of a single leg (Section 9.3.1.4), giving an allocation of Ftotal/Nleg = 20 pN per leg. Maximum safe towing force is <~300 pN/leg, assuming 100 nm2 footpads and a 3 x 106 N/m2 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 (Nleg -1) legs stay anchored at their footpads. Given a full center-to-center working arc of Xarc ~ 80 nm, each leg must travel Xswing = Xarc/Nleg = 8 nm in a time tswing = Xswing / vleg = 0.8 microsec at a velocity vleg = 1 cm/sec, with a per-leg duty cycle of fduty = Nleg-1 = 10% and an operating frequency of nleg = fduty/tswing ~ 100 KHz. From Eqn. 9.74, nanorobot motive power is Pnano = Ftotal vnano / e% ~ 10 pW, taking e% ~ 0.20 (20%). Conservatively taking each footpad binding event as costing Ebind ~ 100 zJ (Section 4.2.1), then footpad binding power requirement is Pbind = Nleg nleg Ebind ~ 0.1 pW, a negligible contribution.

For the least conservative gait, only 1 leg stays anchored while the remaining (Nleg -1) legs are in motion. In this case, Xswing = Xarc = 80 nm, giving tswing = 8 microsec and nleg ~ 10 KHz, but motive power, nanorobot velocity, and leg duty cycle are unchanged. Doubling leg length to Lleg = 200 nm while holding Rleg, vleg, and vnano unchanged decreases operating frequency to nleg ~ 5 KHz while increasing Ftotal to 230 pN and Pnano 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