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Four sample space launch missions were designed using rotating momentum transfer tethers (bolos) within low Earth orbit and a previously unknown phenomenon of “aerospinning” was identified and simulated. The momentum transfer tethers were found to be only marginally more efficient than the use of chemical rocket boosters. Insufficient power density of modern spacecrafts was identified as the principal inhibitory factor for tether usage as a means of launch-assistance, with power densities at least 10 W/kg required for effective bolos operation.

After 56 years of space exploration, the space launch industry remains reliant only on chemical rockets. This paper will investigate if any merit exists in implementing an alternative launch assist technology with functionality of existing upper stages—momentum transfer tethers.

For this purpose,

In 2003, Frisbee [

Experiments were performed in space to master tether deployment. The work of Gates, in 2001 [

Therefore, the author concludes that current technology is sufficient to deploy 10–20 km-long tethers in the LEO environment.

Although significant research has been conducted by the scientific community on hanging or librating tethers for momentum transfer, in 1888, Wiesel [

Among many rotating momentum-transfer tether configurations promoted on the Tether Unlimited site, the LEO bolo (rotating tether) [

The LEO bolo can even operate the propellant-free, extracting momentum necessary for orbital transfer from the momentum of bodies in higher orbits. Unfortunately, around the Earth, the only such body is the moon, and series of tethers able to handle at least 1.6 km/s tip speed are necessary for harnessing the orbital momentum of the moon rocks. An example of momentum-neutral tether design may be found in [

Orbital geometry of the LEO bolo (rotovator) launch assistance mission. (1) payload post-release orbit; (2) tether pre-capture orbit; (3) tether and payload post-capture orbit; (4) tether post-release orbit (circular); (5) payload pre-capture sub-orbital trajectory.

The problem of rotating bolo (rotovator) stability has been extensively discussed by the research community. In 1996, Luo [

Therefore, the author concludes that LEO bolo may rotate (but not liberate) predictably if excessive lengths (beyond 100 km) are not considered.

This section complements the technical designs of momentum transfer tethers in

Engineers accustomed to the LEO mission design are familiar with the phenomenon of air drag. Air drag impacts much of the lifetime and requirements for propulsion for all LEO spacecrafts. However, often overlooked, is the rotation momentum imparted on the LEO tethers by air drag.

The rotation momentum buildup (aerospinning) phenomenon is defined as self-sustained and self-limited prograde rotation of an object in orbit due to its interaction with the density gradient of the atmosphere. Intuitively, rotation torque appears as the lower portion of the tether encounters denser air. Therefore, the center of pressure is initially below the tether’s center of mass. The diagram of forces resulting in aerospinning is shown in

The diagram of forces resulting in tether aerospinning.

The role of atmosphere in tether angular movement did not go unnoticed previously. In 1949 and 1972, Philips [

Both, air drag and air-induced torque, are dependent on the ratio between spacecraft (tether) unit mass and cross-sectional area. For unity surface divided by mass, both, drag and momentum have an associated characteristic time. These are: time to reentry for air drag, and steady angular speed divided by initial angular acceleration for air-induced momentum (called a characteristic spin-up time in

The author has simulated normalized (1 kg weight and 1 m^{2} ram area tethers) reentry and spin-up times using the numerical integration in conjunction with the MSISE-90 atmospheric model. The simulation results of reentry time and aerospinning steady-state tip speed are presented in

The normalized reentry time for tethers on equatorial circular orbits.

As the numerical integration algorithm may suffer from high sensitivity to input parameter variations, the orbital lifetime model was verified against Table I-1 in [

Steady-state tip speed of the rotating tether on the equatorial circular orbit in average solar conditions (based on MSISE-90 atmospheric model).

The important feature visible on

The characteristic spin-up time of tethers on equatorial circular orbits.

Due to the gravity gradient stabilization, tether rotation cannot begin spontaneously as deployed. An initial tether angular velocity must be supplied by the momentum wheel or tip jet to start the rotation.

To calculate the rotating tether tip minimal speed at the nadir (or zenith) point, it is possible to equate the rotational energy of the tether at this point to the tidal energy of the tether stretched vertically (Equation (1a)). Equation (1a) states that to sustain rotation, the tether must have enough rotational energy to librate from a stable vertical orientation to an unstable horizontal orientation, with inertia of the tether acting against the tidal forces. Solving Equation (1a) with the gravity potential of Earth yields Equation (1b).

_{c}_{earth}_{earth}

A plot of minimal initial tip speed of a hanging tether is shown in

In 2005, Pelaez [

The minimal nadir speed of the tether tip relative to the tether center of mass to sustain tether rotation.

Micrometeoroids and orbital debris (MMOD) hazards play an important role in tether design.

The colliding objects frequency on LEO can be extracted from Figures 7–19 in [

_{year}^{-10}^{-2.56}

Collision frequencies for various debris sizes.

Particle size | Collision frequency for 1 m^{2} single-side target |
---|---|

100 µm | 2.8 collisions/year |

200 µm | 0.48 collisions/year |

500 µm | 0.045 collisions/year |

1 mm | 0.0077 collisions/year |

2 mm | 0.0013 collisions/year |

2.5 mm | 0.00074 collisions/year |

3 mm | 0.00046 collisions/year |

Although, in 1995, Forward and Hoyt [

An alternative, an advanced form of a Whipple shield (stopping 6.3 mm particles for 10.8 kg/m^{2} weight already flown [^{2} per layer was integrated into software used to generate the sample mission budgets in

This Section is intended to outline the conditions when the momentum transfer tether with tip mass or counterweight may be useful. In addition, the related problem of matching rotation phase of the bolo with the payload trajectory will be discussed with a resulting conclusion on the inconvenience of tip masses or counterweights for launch-assist missions using LEO bolos.

In 2001, Ruiz [

The bolo must not only be deployed to LEO, but also rotated to its operational tip velocity if aero-spinning alone cannot be relied upon. Such a spin-up requirement is non-trivial, and requires, most realistically, a hydrazine-powered (or electric-ion) low-thrust rocket pod on either end of the tether. Therefore, minimization of the energy required for spin-up will allow reduction of bolo launch mass. Such minimization of energy spent generally requires minimization of the moment of inertia of the entire bolo. Therefore, placement of any massive components on the bolo tip (as counterweight) will result in larger energy requirements for bolo spin-up, as illustrated in

The energy (normalized) spent to spin-up a tether with a point-mass counterweight on one end, from standstill to fixed free tip velocity.

In addition, it is advisable to reduce the bolo moment of inertia to allow easy adjustment of the rotation phase. Such an adjustment is necessary to catch payloads launched from the fixed Earth’s location to the lowest point of tether.

Relying on a booster rocket to provide cross-time to meet the bolo tip in its lower point will result in the booster flying long segments in non-ballistic trajectory, with resulting delta-v losses of 400–800 m/s.

The phase of the bolo can be adjusted either by tip rockets (very expensive, in terms of propellant mass and added weight) or by reeling/unreeling the tether to the central hub. In 2006, Williams [

Given the lesser role of aerospinning (described in

This Section links the momentum transfer tether system architecture and environment discussion in

The primary component of the bolo is the high-tension rope (tether), characterized by the critical tip velocity, dependent on strength to weight ratio. In 2001, Ziegler [

However, using only ultimate tensile strength and density of material to calculate breaking force may lead to over-optimistic calculations of critical tip velocity and tether mass, as happened to Jokic [

The following derating from the ultimate tensile strength should be used:

If loads are periodic or constant (as opposed to impulsive), use maximal yield strength;

If brittle failure is possible in metal elements by a factor of 3 [

For time-dependent degradation (especially for Zylon);

For a maximal operating temperature (or up-rate, if operating temperature is low);

For ultraviolet light and radiation-induced degradation;

For the bundling of yarn (not all yarns are tensioned equally)—this penalty increases with tether diameter;

For the weaving of rope (bundles are twisted and pre-stressed);

For atomic oxygen erosion (for tethers on LEO) or decomposition catalysts present (copper on polyethylene);

For the creep during service life (most important for polymer ropes, especially UHMWPE) [

For the extra length of the tether compared to length of test specimens (variability of strength increases logarithmically with an increase in sample length).

All derating coefficients may be cumulative.

Typically, after all deratings, loading all of the potential tether materials beyond 30% of their respective ultimate tensile strength is dangerous, and some synthetic fibers (UHMWPE, polyester, Kevlar in marine environments) may be usable only up to 5%–20% of their ultimate tensile strength [

One is sometimes misled by the stated structural factors of 1.5 to 2.0 (corresponding to loads 67% to 50% of ultimate tensile strength) of the structural components (wings, for example) of aircraft. Such low structural factors are possible only if component is available for defectoscopy or is flying a very short time. The impressive reliability of modern aircrafts is the function not only of advanced materials, but also of inspection infrastructure. If one cannot monitor (and as necessary, repair) small cracks or scars before they grow to a dangerous size, it is better to double the structural factors of component in question.

In addition, the geometry of the rotating tether is very unfavorable to timely crack detection, even if defectoscopy equipment is available during normal tether operation. Critical cracks may be only 1–2 mm long in steel fibers (necking and then brittle failure) and 10–20 mm long in UHMWPE ropes (creep, fibrillar, and then brittle failure).

For example, let us estimate usable 10 km tether strength for two materials: 2 GPa New Japan Steel wire for suspension bridge cable [

Due to the complex nature of derating, as much data as possible was derived from the actual testing of the ropes rather than from material specifications.

Overall, although UHMWPE provide 10× better average strength-to-weight ratio compared to steel filament, the performance of the UHMWPE is more derated because of strength variability, insufficient testing, and creep.

The usable strength of the example high-tension materials.

Materials | Akashi bridge cable wire | Dyneeema SK78 multi-filament yarn |
---|---|---|

Ultimate tensile strength | 2.0 GPa | 2.34 GPa (for a fill factor of 0.78) |

Diameter | 5.23 mm | 0.54 mm × 10 |

Effective density | 7700 kg/m^{3} |
756 kg/m^{3} (for a fill factor of 0.78) |

Length to implement | 10 km | 10 km |

(1) Yield derating | IIMD * | IIMD * |

(2) Brittleness derating | 0% (below fatigue limit) | 0% (below creep limit) |

(3) Degradation derating | IIMD * | n/a |

(4) Temperature derating | IIMD * | IIMD * |

(5) Radiation derating | 0% (steel is radiation hard) | 30% (estimated) |

(6) Bundling derating | n/a (single wire) | IIMD * for 0.54 mm, 12% for 5.4 mm |

(7) Weaving derating | n/a (single wire) | IIMD * |

(8) Erosion derating | 0.0007%/year (274 km orbit) | 0.26%/year (274 km orbit) |

(9) Creep derating | IIMD * | IIMD * for 1 year lifetime (−70%) ^{#} |

(10) Length derating | 0% (tested in 2 km pieces) | −70% (7% strength variability in 0.5 m test samples up-rated to 10 km length) |

Total IIMD * derating | −68.75% | 70% |

Total derating (1 year) | −68.75% | 81.6% |

1/(Structural factor) | IIMD * | 33.3% |

Usable tether strength | 625 MPa ^{&} |
287 MPa |

Derated tip velocity | 284 m/s | 616 m/s |

Tether usable strength | 13.4 kH | 15.0 kH |

^{#} If length derating exceeds creep derating, it is safe to apply creep derating only; * Included in manufacturer data; ^{&} Similar stress was applied to steel hanging wires at the Pylons of Messina (608 MPa at 3.6 km span).

Tapered tethers, although theoretically more mass-effective, were not considered. Although the tapered fibers are known to naturally occur in sea urchins and sea cucumbers, which condones the superior strength of their ligaments, as noted in 2000 by Trotter [

This Section defines the interface between highly speculative momentum transfer architecture and hardware outlined in

Catch of payload by LEO bolo has very strict requirements for speed, location, and time-matching between the payload and bolo tip. A space-docking experience is not applicable for rendezvous with momentum transfer tethers. In 2006, Williams [

Historically, most resembling the bolo capture was the air recovery of Corona satellites, Stardust sample recovery, or attempted recovery of the Genesis sample capsule. A satellite being recovered closely resembles the bolo tip, while recovery aircraft mostly resemble the suborbital, maneuvering payload. The Corona satellite was captured at high relative velocities (up to 60 m/s) using a combination of hooks hanging from aircrafts and loops created by parachute ropes attached to the Corona satellite. It is a flight-proven solution.

Research is currently in progress, specifically for bolo by Newton [

Requirements for the orbital booster (or payload only) to have an increased agility in order to be caught will surely result in payload-bearing modules having too much deadweight, out-weighing any benefit from any ∆v savings provided by the bolo.

Given these constraints, the only currently achievable flight profile leading to successful bolo rendezvous is reuse of the main engine in the last stage of the orbital booster to roughly match the acceleration of the bolo tip for a few seconds before the catch moment.

Typical upper stages have linear accelerations up to 30 m/s^{2}, throttleable down to 60% of nominal thrust, and have angular accelerations up to 4 rad/s^{2} (typical for Falcon-1e upper stage). To equalize longitudinal and transverse agility, these parameters result in free acceleration of 6 m/s^{2} with acceleration lagging about 0.2 s after controls. An approximate correctable course deviation as a function of the rendezvous window duration can be approximated as Equation (4):

If _{corrections}_{lag}_{max}_{min}^{2}, _{rendezvous}

Such a ∆v loss for terminal guidance would be in the order of 100 m/s for the typical upper-stages and a rendezvous window duration of 3 s.

Payloads with attachment hardware must be designed to disconnect from last stage immediately on successful attachment with the tether tip, and at the same moment the last-stage engine must start the shutdown to prevent re-contact with the payload. Engine shutdown speed requirements are expected to be less severe compared to normal stage separation due to the payload acceleration provided by the tether.

In

A tentative hook-and-loop catch mechanism for LEO bolos. (1) tether; (2) stiffening rods; (3) load redistribution ring; (4) shock absorbing ropes; (5) saddle for hook; (6) hook release (pyrotechnic-actuated hinge); (7) payload hook; (8) payload; (9) payload adapter to rocket last stage.

Tether design (1) is covered in

The space launch mission budgets using momentum transfer tethers (bolo) on LEO.

Mode | Existing steel tether | Existing PE tether | 10 W/kg hub and PE | Double-strength PE |
---|---|---|---|---|

Catch mass | 246 kg | 247 kg | 247 kg | 243 kg |

Tether rotation period | 52 s | 73 s. | 73 s | 152 s |

Tether length | 4 km | 9 km | 9 km | 32 km |

Tether tip speed | 242 m/s | 387 m/s | 387 m/s | 661 m/s |

Tether tip acceleration | 29.2 m/s^{2} |
33.3 m/s^{2} |
33.3 m/s^{2} |
27.3 m/s^{2} |

Tether tension without payload | 6.2 kH | 6.8 kH | 6.8 kH | 23.3 kH |

Tether tension with payload | 13.4 kH | 15 kH | 15 kH | 30 kH |

∆v for payload | 474 m/s | 759 m/s | 691 m/s | 1310 m/s |

∆v for tether ^{&} |
13.8 m/s | 16.4 m/s | 88.7 m/s | 15.3 m/s |

Tether mass (with shield) | 852 kg | 810 kg | 810 kg | 3416 kg |

Tether MMOD shield weight | 190 kg | 651 kg | 651 kg | 3257 kg |

Tether MMOD shield blocking class | 2 mm | 2.5 mm | 2.5 mm | 3 mm |

Tether effective drag area | 20.6 m^{2} |
56.7 m^{2} |
56.7 m^{2} |
238 m^{2} |

Total effective drag area (night glider mode) | 30.3 m^{2} |
70.3 m^{2} |
70.1 m^{2} |
260.5 m^{2} |

Tether hub power | 7.6 kW | 10.7 kW | 11.2 kW | 17.4 kW |

Tether hub mass | 7.6 ton | 10.7 ton | 1.1 ton | 17.4 ton |

Tether catch orbit, perigee × apogee (km) | 299 × 321 | 357 × 383 | 322 × 458 | 420 × 444 |

Post-release (circular) tether orbit altitude | 299 km* | 357 km* | 322 km* | 420 km* |

Pre-capture payload orbit, perigee × apogee (km) | 111 × 299 | −218 × 365 | −214 × 317 | −591 × 404 |

Post-release payload orbit, perigee × apogee (km) | 301 × 487 | 375 × 958 | 327 × 858 | 448 × 1455 |

Payload increase ^{#} |
25 kg | 42 kg | 34 kg | 79 kg |

Base payload | 246 kg | 247 kg | 247 kg | 243 kg |

Tether hub thrust for post-release orbit departure | 60 mH | 45 mH | 90 mH | 53 mH |

Thrust necessary for tether 1 week-long re-boost | 193 mH | 311 mH | 283 mH | 527 mH |

Refueling frequency, every Nth payload | N = 24 | N = 17 | N = 17 | N = 10 |

Lifetime required to break-even mass on LEO | 10.89 years | 7.95 years | 1.91 years | 7.32 years |

Average time to break due micrometeorites | 115 years | 97 years | 97 years | 44 years |

^{&} Not taking into account terminal guidance ∆v loss (about 100 m/s); * Polar orbit was assumed for all cases; ^{#} Compared orbital delivery to 185 × 185 km parking orbit without momentum transfer tethers.

An alternative solution may be to spin a secondary tether around the tip of the primary tether, nullifying tether tip acceleration at the moment of capture and therefore allowing low-acceleration rendezvous. That solution was first proposed in 2004 by Williams [

In this section, the conclusions on momentum transfer architecture (

A software package was developed in the Scilab for the semi-automatic calculation of the mass budget of the space launch mission using momentum transfer tethers. The Scilab scripts are provided on request or available on the author’s page at

The Tether_reentry.sce script works as follows:

The MSISE-90 atmospheric model is loaded and interpolated with a resolution of 1 km.

The distribution of atmospheric drag across the tether is calculated at five segments across the tether length for rotation angles with step 15 degrees.

The circular orbit altitude drop is calculated from energy and impulse conservation laws. Spacecraft assumed to collide inelastically with air molecules, therefore transferring the kinetic and potential energy of spacecraft to air, accelerating and ultimately heating it. The application of conservation laws is possible if the orbit decaying quasi-statically. Quasi-staticity mean drag acceleration several orders of magnitude below gravity acceleration, and gravity acceleration magnitude varying negligible per orbit. For this to happen, orbit must be near-circular and gravity field must be nearly-spherical. Real decaying low-earth orbits are slightly non-circular (orbital eccentricity is varying around 0.001 due non-spherical gravity fields). In addition, a few minutes before reentry, decay becomes not quasi-static due to increasing drag acceleration.

The time step is adapted to keep the orbit altitude drop rate in the 0.1–1 km/step range.

Simulation is terminated after the orbit is lowered to 100 km altitude.

The steps b–e are swept for various tether lengths and initial orbit altitudes. The tether rotation speed is derived from the Steady_tether_speed.sci script output.

The Steady_tether_speed.sci script works as follows:

The MSISE-90 atmospheric model is loaded and interpolated with a resolution of 1 km.

Initially assumed tip speed is 1 m/s and the speed adaptation coefficient is 1.4.

Torques arising from center of pressure and center of mass mismatch are calculated for nine tether segments and averaged over a rotation period with step 3 degrees.

If torque is found to be positive, estimated speed is multiplied by speed adaptation coefficient until torque becomes negative.

If torque is found to be negative, estimated speed is divided by speed adaptation coefficient until torque becomes positive.

When torque changes sign, the speed adaptation coefficient is set to the square root of itself.

The steps c–f are repeated until torque absolute value is reduced below the pre-determined accuracy.

The steps b–g are repeated while sweeping orbit altitude and tether mass.

The Tether_spinup_time.sci script works as follows:

The MSISE-90 atmospheric model is loaded and interpolated with a resolution of 1 km.

The output of Steady_tether_speed.sci is divided by torque calculated for zero rotation speed, but averaged for all possible tether orientations with step 3 degrees.

The tether_minspeed.sce script simply applies Equation (1) for the range of the tether lengths and circular orbit altitudes.

The tether_mmod.sce script calculates the total weight of MMOD shield for a given length and diameter of tether, using tabulated MMOD shield area density extracted from [

The tether.sce script uses laws of energy and momentum conservation to calculate the following parameters as a function of payload weight to tether weight ratio:

The ∆v imparted on the tether as a result of payload capture and release.

The ∆v imparted on the payload after being carried by the tether from a capture point directly below the tether hub to a release point directly above the tether hub.

The auto-budget scripts, (Tether_autobudget_UHMWPE.sce and Tether_autobudgetSteel.sce), do automatically calculate power and propellant requirements for the catch-release-reboost momentum transfer cycle, given average orbit altitude and launch frequency. Optimization of these is left for the user. Other script outputs are used as input for auto-budget scripts. Average altitude of the tether capture orbit was swept to minimize orbital mass break-even time. Break-even time is the interval necessary for the bolo system to transfer, to orbit, additional (compared to purely rocket system) mass equal to the initial bolo weight (plus all ion motor fuel spent in the process). It was discovered that higher launch rates result in the reduction of orbital break-even time, up to a launch frequency of 180 per year. It was also found that a launch frequency above 52 launches/year (weekly) results in a bolo system that is undeliverable to orbit by medium-lift vehicles. Therefore, only bolo capture orbit average altitude was optimized, while launch frequency was fixed to 52 launches/year. For more robust optimization convergence, the gravity gradient on LEO was linearized. Therefore, all apogee and perigee data in ^{2}) were used for columns 1, 2, and 4. Solar panels are assumed to have an effective drag factor of 1 in sunlight and 0.2 in shadow, corresponding to the “night glider” operation mode.

The payload catch is assumed to happen near the perigee of the bolo orbit to avoid perigee lowering and excessive air drag during re-boost. The payload is released after a half-rotation of the bolo, still near the bolo perigee, to maximize momentum transfer (in line with [

In 2006, Ahedo [

Surprisingly, the concept that the re-boost engine specific impulse is a limiting factor, as stated in 2003 by Sorensen [

In addition, a small role of the material strength was found. Although bolo performance (pay-off time) moderately improves (by 27%) if steel tether is replaced for available high-strength UHMWPE rope (five times stronger for same weight), further doubling of tether strength results only in a 9% improvement in pay-off time. The main reason for such a performance stagnation is the limiting role of the payload catch specifications. If acceleration at the tether tip is kept at 30 m/s^{2}, as advocated in

However, power density of the bolo hub spacecraft was found to be very important, with 4.16 times the break-even time reduction if power density rises from 1 W/kg to 10 W/kg. In addition, although total ∆v delivered to payload is reduced because of the lower bolo mass, ∆v to reach catch point is actually decreased because of the bolo lower optimal orbit altitude. The entire ∆v penalty for the light (high power-density) hub is, thus, applied for reduction of the apogee of the post-release orbit. The result is more circular post-release orbits compared to missions using standard power density spacecrafts. These orbits are well-matching the launch-assistance purpose of the LEO bolo, outlined in

This section summarizes the factors critical for the efficacy of the LEO launch assistance mission using LEO bolos, calculated in

The development of a hypothetical space mission involving a satellite constellation delivered to orbit with the help of a LEO bolo has resulted in the following conclusions:

Specific impulse (~1500 s) of modern ion motors is sufficient to deliver a 3:1 reduction of fuel mass spent for bolo-assisted momentum transfer (compared to chemical rockets). Development of MXER tethers is not critical for the momentum-transfer applications, unless tethers with tip speeds approaching 1 km/s will be deployed.

Long-term strength of existing ropes is sufficient for LEO bolos. Tether maintenance crawler(s) may offer significant reduction of tether strength margin(s), but increasing tether strength does not significantly improve system performance, due to the added weight of the tether re-boost hardware and micrometeoroid shield.

The power density of modern spacecrafts (1 W/kg) is inadequate for LEO bolos deployment. Development of higher power-density (up to 10 W of electrical power per kg) spacecrafts is necessary if realistic (~1–40 launches/year) launch rates are expected. Otherwise, launch rates of 50–200 launches/year are required to pay-off the initial cost of LEO bolo deployment (even if not taking into account R&D costs).

The power-to-drag of modern solar arrays in “night glider” mode is inadequate for bolos operating below a 300 km altitude. Better power efficiency of solar cells or solar panels operating in “sun slicer” mode are desirable. However, operating in “sun slicer” mode still requires many advances in large structures and energy storage. A nuclear powered bolo for a 240–280 km catch orbit may be considered if politically acceptable.

Plain tethers are vulnerable to micrometeoroid damage. Either shielded or redundant tethers must be used. Currently, shielded tethers seem to be the most developed solution, and are weight-effective for tethers below 10 km long.

Payload capture requirements strongly favor high-agility upper stages of rocket boosters to reduce terminal guidance ∆v loss.

This work was partially supported by Japan Science and Technology Agency within the ERATO (Exploratory Research for Advanced Technology) project.

The author declares no conflict of interest.

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