Graph Percolation as Decision Threshold for Risk Management in Cross-Country Thermal Soaring

TL;DR

The paper reframes long-range, unpowered flight as a risk-management problem by modeling thermals as a random geometric graph and applying continuum percolation. It defines a percolation value $p = \mathbb{E}(n)/n_{critical}$ with $n_{critical} \approx 4.51$ and provides a practical in-flight estimator $p = \left( \frac{\lambda_1}{2} \right)^2 \pi \left( h \frac{L}{D} \right)^2 \frac{1}{4.51}$ to assess whether the thermal network percolates ($p>1$). Empirical analyses of Hobbs contest flights and a large WeGlide dataset show percolation values around 2.5 at thermalling onsets, and that pilots tend to exploit stronger thermals while gradually adjusting climbs to maintain connectivity, i.e., to stay in a percolating regime. The findings suggest percolation can unify risk management with speed-to-fly strategies and inform training and autonomous soaring systems for energy-efficient, reliable cross-country flight.

Abstract

Long range flight by fixed-wing aircraft without propulsion systems can be accomplished by "soaring" -- exploiting randomly located updrafts to gain altitude which is expended in gliding flight. As the location of updrafts is uncertain and cannot be determined except through in situ observation, aircraft exploiting this energy source are at risk of failing to find a subsequent updraft. Determining when an updraft must be exploited to continue flight is essential to managing risk and optimizing speed. Graph percolation offers a theoretical explanation for this risk, and a framework for evaluating it using information available to the operator of a soaring aircraft in flight. The utility of graph percolation as a risk measure is examined by analyzing flight logs from human soaring pilots. This analysis indicates that in sport soaring pilots rarely operate in a condition which does not satisfy graph percolation, identifies an apparent desired minimum node degree, and shows that pilots accept reduced climb rates in order to maintain percolation.

Figures (8)

• Figure 1: Cross-country soaring requires exploiting a sequence of updrafts to gain energy which is expended in gliding flight. Updrafts are stochastically distributed and cannot be reliably detected except in situ, putting an aircraft at risk of failing to find additional energy sources before reaching the ground.
• Figure 2: A collection of thermals which forms a random geometric graph whose connection degree is three-quarters of the critical threshold. The colored area surrounding each thermal indicates the glide range possible from the top of the thermal. Colors indicate clusters of thermals which can be traversed.
• Figure 3: A collection of thermal which forms a random geometric graph whose connection degree is at the critical threshold. The colored area surrounding each thermal indicates the glide range possible from the top of the thermal. Colors indicate clusters of thermals which can be traversed.
• Figure 4: Histogram of the percolation value at which pilots in the Hobbs contest begin thermalling.
• Figure 5: Normalized thermal strength distribution vs percolation at start of thermalling in Hobbs data.