Drone Parachute: Deployment Height, Descent and Autorotation

The force–trajectory–time approach follows Knacke, NWC TP 6575 (1992), §§5.4.1–5.4.6 and §5.11. The ODE integrates downward distance, ground displacement, horizontal airspeed, downward speed and air-path length. Drag opposes the air-relative velocity; gravity remains active during inflation. Body and canopy drag areas add. Ordinary cases use adaptive Dormand–Prince integration. Stiff cases use Alexander's L-stable two-stage SDIRK method with step-doubling error control. Time-driven implicit drag stages have an exact quadratic solution; distance-driven stages use safeguarded Newton iteration for the coupled air path and drag area.

Before line stretch, canopy drag is zero. Filling progress ξ is elapsed inflation time divided by the specified filling time, or air path after line stretch divided by nD₀. The Knacke time option uses t_f=nD₀/V_s. The parametric dynamic drag-area fraction is clamp(ξ,0,1)^p multiplied by [1+(Cₓ−1)exp(−8 max(ξ−1,0))]. Thus the opening multiplier acts during filling and relaxes continuously toward steady drag after full inflation. This chosen profile requires calibration; it is not a digitized Knacke canopy curve. A separate adaptive opening solution and bounded scalar search refine the peak load. The instantaneous canopy force is ½ρV²(C_D A)_canopy times this fraction. Velocity decay and gravity determine the finite-mass opening load directly, without an independently fitted force-reduction factor.

The initial integration horizon includes acceleration from rest and is doubled automatically until the height range and full inflation are reached. Touchdown roots are searched only after the drop is bracketed. When impact precedes line stretch, touchdown time and velocity come from a separate body-only ODE solve, independent of the inflation sampling mesh. For ordinary cases, a bracketing sample interval is followed by a local ODE solve and root refinement, and touchdown velocity is taken from that solved state. Stiff-case event lookups use the compact interpolated trajectory. The minimum-height search starts at full inflation and finds the last downward crossing of the total ground-speed target; the entered reserve is then added to height lost. A sustained target requires the terminal ground speed to be below the target. Steady descent follows mg=½ρV_t²(C_D A)_total; impact energy uses both ground-velocity components.

For separate, non-overlapping rotor disks, v_h=√[mg/(2ρA)]. Axial momentum branches are shown for climb/hover and descent w≥2v_h; the intervening vortex-ring and turbulent-wake range is unresolved. A 64-station rigid, constant-chord, constant-pitch blade model integrates lift and drag. At each of 161 tip-speed/descent-speed ratios, 24 bisections solve the low-induction windmill branch T=2ρA v_i(w−v_i). Similarity scaling enforces aircraft weight balance including body drag. A bracketed zero of aerodynamic resisting torque plus motor torque is refined by 22 bisections. A candidate outside the entered linear-polar angle limit is rejected as an autorotation estimate. Brindejonc et al. (2007) motivates the torque sweep; their flapping hub and extrapolation into the unresolved regime are not applied to a rigid drone propeller.

Hybrid glide uses C_D=C_D0+k C_L², tan γ=C_D/C_L, L=mg cos γ and sink V sin γ. Best-range C_L is √(C_D0/k), capped at 95% of C_Lmax. Range uses height remaining after the specified transition and the horizontal wind.

Constant air density, gravity and horizontal wind; a point-mass aircraft with fixed isotropic body drag area; no wing lift in the parachute trajectory. Hover starts stationary relative to the ground. Cruise speed is relative to the air. Canopy mass is included in total mass; its motion relative to the aircraft is neglected.

The default canopy and deployment parameters describe a generic engineering example, not a tested commercial recovery system. The inflation profile, filling constant, dynamic opening multiplier, allowable load and reserve must come from the intended installation. The multiplier's post-opening relaxation is an assumed exponential with an e-folding progress interval of 1/8.

Rotor calculations assume identical, coplanar, non-overlapping disks, a 20% hub cutout and a 98% blade tip, uniform axial induction, linear lift slope and constant section drag. Motor resisting torque is taken constant at the candidate speed. The hybrid model uses an unpowered polar, retained control authority and an established straight glide.

Use as an engineering sizing and sensitivity model for low-altitude, subsonic recovery. Constant density over a large height change, large canopy/aircraft separation, tumbling, oscillation, line elasticity, reefing, canopy porosity changes and fluid–structure interaction are not resolved. The dynamic opening load omits line-stretch snatch and attachment stress concentrations; it is not a complete structural design load. Knacke's nD₀/V_s relation must not be extrapolated to deployment from zero airspeed; prefer measured low-speed inflation data. Integrated filling distance is a stated alternative model, not a universal low-speed correction.

The minimum modeled height is a speed/full-inflation threshold with a user-entered reserve, not a validated operational minimum. “Modeled criteria met” means only that this threshold and the entered load limit pass. It does not establish drone survival or permission to fly over people. A blank minimum height means that no sustained target is reached. Horizontal wind may make the selected total ground-speed target impossible regardless of canopy size.

The explicit solver uses relative tolerance 10⁻⁶. The stiff solver uses relative tolerance 2×10⁻⁷ and absolute tolerance 10⁻⁹ in the SI state variables, with step doubling and nonlinear implicit stages. Both solvers use the same drag, gravity and inflation laws. The explicit output mesh independently resolves initial flight, anticipated impact, canopy inflation and the long descent tail. For stiff cases the adaptive integration uses fewer requested output times, and event lookups and charts use PCHIP interpolation of fewer than 320 stored points concentrated around deployment and peak load. This compression can smooth short transients between stored points; stiff opening-load maxima are tracked at accepted half steps before compression. Ordinary opening-load maxima are additionally refined on a separate adaptive mesh with a bounded scalar maximum search. The animated drawing is schematic and vertical travel is rescaled to the canvas.

Momentum theory does not resolve the vortex-ring region. The fixed-pitch sweep searches tip-speed/descent-speed ratios 0.05–12 on the windmill branch only and does not extrapolate a torque root through the unresolved region. Linear-polar sweep points beyond the entered section angle limit are diagnostic only; a candidate there is not accepted. Neither spin-up time, rotor inertia, motor breakaway torque, reverse windmilling, attitude control nor flare is modeled. Even a valid steady torque balance does not show that autorotation can be entered after power loss.

Glide excludes transition dynamics, stall transients, turns, gusts and flare. Its distance is along the wind axis and is available only after the entered transition height loss. ASTM F3322-24a (DOI 10.1520/F3322-24A; edition status checked 31 October 2026) is named for system design and testing context; check against the standard in force. No standard's tests or approval criteria are reproduced.