© 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 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 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 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 (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