Drone Ground Effect: Hover Thrust Gain and Pitch Stability

Hover thrust and power. Cheeseman and Bennett, ARC R&M 3021 (September 1955), equation (2), gives the constant-power gain $G_{CB}=[1-(R/4h)^2]^{-1}$. Hayden's 1976 fit is implemented as reproduced in NASA NDARC Theory, release 1.12, §12-4.1.2, PDF p. 122: $f_H=\min\{1,[0.9926+0.03794/(h/2R)^2]^{-1}\}$ and $G_H=f_H^{-2/3}$. A gain $G$ corresponds to the ideal induced-power factor $G^{-3/2}$, rather than its reciprocal.

Multirotor interaction. Sanchez-Cuevas et al. (2017), equations (3)–(4), are implemented in full for a level square quadrotor. With adjacent spacing $d$ and diagonal $b=\sqrt2d$, the denominator is

The total gain is $1/D$. The first three correction terms are the self, adjacent and diagonal image-source contributions; the last is the empirical fountain/body term.

Edge and pitch extension. Circular-segment geometry gives the fraction $c_i$ of each projected rotor disc above the pad. Following the area-weighting approximation described in the paper's §3.1, isolated excess thrust is proportional to $c_i$. For the multirotor option, each neighbour's image contribution is weighted by its own $c_j$, and the resulting excess gain is weighted by $c_i$. For small pitch angles, image-source velocities are projected onto the common tilted rotor axis, including the different rotor heights. This is a derived finite-edge extension of the published full-plane model, not an edge-flow solution from that paper.

The body correction is multiplied by the mean of the six pair products $c_ic_j$. Its additional force acts at the aircraft centre, so it contributes no direct pitching moment. This recovers equation (4) exactly for complete level coverage and suppresses the fountain term when only one rotor is supported. Rotor thrust moments are summed using body-frame lever arms. A centred ±0.01° difference gives $K_\theta=-\partial M/\partial\theta$ in N·m/rad; positive means locally restoring.

Impingement pressure. Ideal actuator-disc momentum gives $v_i=\sqrt{T_i/(2\rho A)}$, incident jet speed $w_i=2v_i$ and stagnation scale $q_i=\rho w_i^2/2=T_i/A$. The chosen envelope is $p_i=q_i\exp[-r_i^2/R^2]$; over an infinite plane its integral is $T_i$. The footprints follow the tilted jet axes, are superposed, and are clipped at the pad edge without renormalization. This diagnostic is separate from the thrust correlation.

Numerical resolution. The 241 edge-sweep points concentrate 100 intervals on each rotor crossing, with additional samples across the intervening gap and beyond the outer tips. Pressure is evaluated on four disjoint 43 × 43 local tiles, each extending at most 3.5R from its jet centre in either coordinate and split at the intervening midplanes. Thus sample spacing never exceeds R/6, regardless of arm length. The square layout and equal left/right thrust make the Gaussian envelope separable in x and y. Its continuous maximum is obtained by 45 bisection iterations on each one-dimensional Gaussian-pair derivative, independently of the plotted samples. Trapezoidal integration of the same local axes checks recovery of the unclipped rotor-thrust integral.

Four equal, nonoverlapping rotors on a square X layout; centre of mass at the square centre; hover in still, incompressible air above a horizontal, rigid pad. The pad is infinite laterally and extends to negative x, with no nearby lower floor beyond its straight edge.

The baseline power is the out-of-ground-effect hover power for the entered mass, divided equally between the four rotors. Geometry-based gains do not change with load in this model. Ideal induced power excludes profile drag, motor/ESC loss and auxiliary loads.

The coverage-weighted image interactions, pair-weighted body term, central application of body lift and Gaussian pressure footprint are explicit modelling choices derived here. The empirical coefficient 2 comes from the 2017 tests and is not universal.

Use for quasi-static hover and small pitch (±10°). The lowest blade tip must remain at least 0.6R above the pad; this conservative working limit follows the approximate lower extent of conventional ground-effect data discussed by NASA NDARC, rather than the mathematical singularity at h/R = 0.25. Small-angle edge extension and local pitch stiffness are not a flight-dynamics or controller-stability solution.

Cheeseman–Bennett and Hayden are isolated-rotor correlations. The 2017 full-plane interaction/body model is empirical; body shape and rotor spacing affect its accuracy. Its partial-coverage extension is not experimentally calibrated for arbitrary pads. No vortex-ring state, wake memory, edge suction, fountain pressure solution, pad thickness, wind, ground roughness or porous-surface loss is resolved.

The 241-point height and edge sweeps are evaluated directly; edge samples are concentrated near the projected rotor tips. The pressure picture blends four rotor-scaled 43 × 43 tiles. Very small Gaussian tails beyond 3.5R are omitted between widely separated tiles. Mesh spacing ≤R/6 limits peak undersampling to less than 1.4%; the colour limit uses the continuous envelope maximum. Checks require at least 99.999% recovery of the unclipped pressure integral across close and widely spaced layouts. The pressure map is a momentum-based explanatory envelope, not CFD or a pad structural load. The infinite-plane envelope conserves vertical jet momentum only for level jets; clipping an edge removes support without redistributing pressure.

Power saving refers to ideal induced power and equal reference commands scaled together to obtain weight-equivalent axial thrust. With pitch, vertical thrust also includes cos(θ); at an edge, actual balanced hover needs differential rotor commands. Do not infer battery endurance, fixed-RPM thrust, required flight-control gains or safe landing clearance from these results. Validate aircraft-specific predictions with controlled measurements.