{Eqn. 9.45}
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
9.3.1.3 Pneumatic Manipulators
Differential tensions also can be applied to a nanomanipulator structure using gases (e.g., pneumatic actuators) or liquids (e.g., hydraulic actuators). Nanoscale pneumatic actuator elements must be of size L >> lgas (Eqn. 9.23) to ensure continuum flow. Thus for L ~ 1 micron, minimum pneumatic operating pressure pgas >> 0.2 atm; for L = 10 nm, minimum pgas >> 24 atm.
There are many examples of pressurized hydraulic tissue inflation in macroscale biology, including sea anemones, nematodes, echinoderm tube feet, spider legs, pupal butterfly wing extension, piscine swim bladders, and human sex organs. The pressurized pumping of DNA through a nanotube in the T4 bacteriophage has already been described (Section 9.2.4). As another possible example in nanobiology, the sperm of the sea cucumber Thyone rapidly elongates its ~700-nm wide spherical acrosomal vesicle into a 50-nm wide, 90-micron long tubular acrosomal process in less than 10 sec after a rapid influx of water doubles the volume of the 0.5 micron3 acrosomal cup in which the vesicle is stored in 50-70 millisec (Fig. 9.4). This has led to the proposal that the elongation is at least partly hydraulically driven by a profilactin osmotic pressure of ~0.5 atm, because the speed of actin polymerization alone appears to be about an order of magnitude too slow.1230
Nanoscale manipulators may be constructed of simple orthotropic tubes with flexible ribbing and an elastic but nonextensible spine that allows manipulator flexure in only one direction when pressurized (Fig. 9.5). Experiments using compound manipulators constructed using centimeter-sized pneumatic orthotropic segments that were pressurized to 3-6 atm produced 90° flexures in 0.3-1 sec with 2-5 mm placement accuracy (0.4%-0.9%) and load-bearing strengths up to ~20 N at the tip.1231
Suzumori et al1232 have constructed, tested and analyzed a simple but elegant pneumatic manipulator ~4 mm in diameter that should be readily adaptable to the microscopic scale. As shown in Figure 9.6, the manipulator consists of a flexible cylinder made of three separate 120°-sector longitudinal chambers that are permanently bonded together. Each chamber may be independently pressurized or depressurized through flexible tubes connected to pressure control valves. In the first configuration, purely circumferential reinforcing fibers (winding angle a = 0°) allow the tri-chambered manipulator to easily deform in the axial direction, while resisting deformation in the radial direction. Pressure applied equally to the three chambers thus causes axial stretching. Pressure applied to only one chamber causes the manipulator to bend away from that chamber, so the device can be flexed in any direction by controlling the pressure in the three chambers. In a second configuration, spiral reinforcing fibers (a = 5°-20°) force a circumaxial rotation of the cylinder when chamber pressure is applied, allowing useful screwing motions to be generated at the tip. Thus the manipulator can achieve pitch, yaw, stretch, and rotation, although these four movements are not fully independent because there are only three independent control parameters.
Consider the tri-chambered pneumatic manipulator of Figure 9.6 with one end fixed, a = 0°, tube radius rtube, tube wall thickness twall, and chambers of equal relaxed length L0 along the center axis that are then pressurized to p1, p2, and p3, respectively. The fiber-reinforced tube wall material has Young's modulus Etube in the transverse direction. Then the x-y projection of manipulator rotation around the vertical axis (q) is given by:
For each of the three chambers, q1 = q, q2 = q + (2 p / 3), and q3 = q - (2 p / 3). The angular deflection of the manipulator down from the vertical axis (j) is the same for all three chambers, given as:
Lmanip is the new stretched length of the manipulator at the center axis, Rcurve is the radius of curvature of the manipulator, and the linear deflection from the vertical axis is:
and Eqn. 9.48 is readily evaluated using the quadratic formula. Making a few simplifying assumptions, the bending stiffness of the manipulator is given approximately as:
and thus the maximum force that can be exerted at the tip is:
Finally, the lowest resonant frequency nres of a tri-chambered pneumatic manipulator of mean density rmanip, mass Mmanip ~ p rtube2 L0 rmanip, and carrying a tip load of Mload is given by:1232
where the second moment of area Itube ~ (1/4) p rtube4 + (1/2) Lmanip rtube3. Manipulator power requirement in the absence of a viscous operating medium is:
where nmanip is manipulator operating frequency.
As an example of a nanomedically useful pneumatic manipulator, consider a tri-chamber design with L0 = 1200 nm, rtube = 200 nm, twall = 50 nm, Etube = 108 N/m2, and maximum operating pressures up to ~1000 atm (pressure/volume relationships for air calculated using the ideal gas law are ~5% in error at 100 atm, ~50% in error at 1000 atm; Table 10.2). Application of a p1 = 6 atm pressure pulse to one chamber of the manipulator produces an Ftip ~ 0.1 nN lateral force and a d ~ 1 nm tip displacement at zero load, costing ~3 kT/cycle at T = 310 K. Application of a 60 atm pulse produces ~1 nN and a 10 nm displacement (~300 kT/cycle); a 600 atm pressure step produces a ~10 nN force and a 100 nm no-load tip displacement at an energy cost of ~30,000 kT/cycle or Pmanip ~ 100 pW continuous power draw at nmanip = 1 MHz. Minimum driving pressure increments of ~6 atm (~3 kT/cycle) allow the manipulator tip to be reliably positioned in repeatable step sizes of ~1 nm, well above the classical positional variance of the tip due to thermal noise10 which is Dx = (kT / kshaft)1/2 ~ 0.2 nm, given a bending stiffness of ktube ~ 0.1 N/m for this design.
Manipulator volume is ~0.15 micron3 and manipulator mass Mmanip ~ 0.1 picogram. At zero load in vacuo, nres ~ 30 MHz up to ~1000 atm; nres falls to ~5 MHz for a 1 picogram load (roughly the mass of a 1 micron3 nanorobot) and to ~1 MHz for a ~30 picogram load, so up to ~MHz operating frequencies seem reasonable at modest loads. (For comparison, a 1 nN lifting force supports a 100,000 picogram mass against gravity.) At maximum cyclical tip displacement d = 100 nm and nmanip = 1 MHz, mean tip velocity is vtip ~ d nmanip = 10 cm/sec. Maximum viscous drag power in vivo is Pdrag ~ 140 pW using the formula for Stokes frictional dissipation (Eqn. 9.74) at a 10 cm/sec tip velocity and taking r ~ (2 rtube L0)1/2 for the manipulator and h = 1.1 x 10-3 kg/m-sec in tissue plasma at 310 K.
The pneumatic design outlined above is quite versatile. Manipulator tip displacement scales as d ~ L0 p / Etube. Manipulator tip force scales as Ftip ~ rtube3 / L02; tip force is nearly independent of Etube down to Etube ~ 108 N/m2, below which Ftip falls rapidly. Pneumatic nanomanipulators using diamondoid wall materials (e.g., Etube ~1011 N/m2) can have a stiffness on the order of kmanip ~ 10-100 nN/nm (comparable to the telescoping manipulator described in Section 9.3.1.4), although in such cases the maximum pneumatic tip displacement even at maximum operating pressures may be limited to ~1 nm or less. Longitudinal cables can be added to actively control the helical pitch (e.g., a) of the reinforcing fibers;529 adding both circumferential and longitudinal reinforcing fibers allows deformation of compliant tube structures into multiply-coiled shapes.1255 Adding tensioned control cables and additional layers of pneumatic chambers can give a full six degrees-of-freedom (DOF) manipulator, albeit with some increased complexity of the mechanism. Control of individual axial segments (e.g., selective locking/unlocking or differing Young's moduli of adjacent segments) allows arbitrary serpentine flexures through a three-dimensional work volume (Fig. 9.7). End effector control lines may be routed along the central axis.
Many interesting designs for "tentacle" or "snake" manipulators, also known as highly-redundant or hyper-redundant manipulators, have appeared in the literature.1231,1233,1270-1273,1625,2387 These may be fabricated as stacks of multiple pneumatic segments connected coaxially. A biological snake may have ~200 separate jointed segments, though each joint has a pitch range of only ± 10°; snake robots having artificial joints with pitch ranges up to 135° have been built.1625 Such designs are useful in nanomedical robotic systems because hyper-redundant manipulators can be made extremely fail-safe -- if one joint fails (e.g., by locking tight, not falling limp, by design), the accessible work volume is not significantly diminished.
Last updated on 20 February 2003
{Eqn. 9.46}
{Eqn. 9.47}
{Eqn. 9.48}
{Eqn. 9.49}
{Eqn. 9.50}
{Eqn. 9.51}
{Eqn. 9.52}
{Eqn. 9.53}