Urban Aerial Defence / Paper 01 / Interactive edition
The next front
The low sky over our cities, and how to defend it
Provisional build. The full sweep is still solving some aircraft; only those it has finished are shown, and numbers for the others read as a question mark.
TriangleTinyTriangle Engineering LtdOctober 2026Triangle Gym, studies/agility
Abstract
Counter drone defence has so far been reactive. Cheap quadcopters spread along the front in Ukraine, and jamming followed1; then fibre optic drones arrived that jamming cannot touch2,3. Defences were built around the propeller driven Shahed, and jet powered versions came that outrun them4,5. Each time, the defence is designed for the attacker of the day, and is behind by the time it is fielded. The next attacker is already visible. Autonomous racing drones from Zurich and Delft have beaten world class human pilots6,7, and the same autonomy will fly an attack drone low, fast and evasive through the streets of a city, at the limit of what its airframe can physically do. Nothing in service today is built for that, and waiting for it to appear before designing a defence repeats the same mistake. This paper starts from the other end. It assumes autonomy will push every aircraft, attacker and interceptor alike, to its physical limit, and asks which defence then works best. Six airframes are modelled from first principles at equal mass and power and flown to their limits by a minimum time optimiser, and their agility then defends a site in a city scenario, attacks per case. Two things decide the outcome. The first is sensing. A sensor on the site itself sees only so far, and buildings hide anything flying low until it is close, so many drones are seen too late to reach. With only that sensor, the defence never catches much more than half of them, however many interceptors it has. Sensors on the tall buildings around the site see further out and down the streets, and lift that limit. The second is the airframe: once the site can see, helicopters defend it successfully in almost every attack, where it takes fixed pitch quadrotors.
- Attacks per case
- Drones per attack
- Helicopters to defend
- 95% OF ATTACKS
- Quadrotors to defend
- FIXED PITCH
1The mission
Take a site in the middle of a city: a data centre, a substation, a stadium. Five waves of small drones come at it from every side, 50 in all. Most fly high over the roofs, where any sensor in reach sees them. The rest, of them, fly low along the streets, hidden by the buildings until they turn a corner. A handful of interceptors wait on the site's roof, all flown by the same autonomy. Each one accelerates and responds exactly as fast as its airframe physically can, taken from the study in Sections 4 to 7; nothing is tuned by hand.
The 3D view needs WebGL, which this browser has turned off. The comparison below still runs.
Fig. 1The city in three dimensions, 480 m across, buildings 15 m to 60 m tall. Solid diamonds in a box are drones the autonomy is tracking; ghosted diamonds are drones nobody can see yet. Ink cubes are interceptors, with their trails and a line to where guidance is aiming; every aircraft has a hairline down to the ground to show its height. Drag to turn the city (on a phone, drag sideways). Tap a rooftop to put a sensor on it (up to four), and tap the sensor to remove it; a tall building sees further down the streets. The site's own sensor sits on a 20 m mast on its roof, and the mission starts with two more on the tallest nearby towers. Switch to Site mast only to see what the site sees by itself. Five waves, 50 drones. Each drone that gets through costs the site 5% of its integrity; at zero it is overwhelmed.
| Airframe | Intercepted | Got through | Site | Mean time to intercept |
|---|
| Press Compare all airframes to fly this one attack against each of them. Section 2 flies a hundred. |
2What the mission shows
Every configuration is flown against the same set of attacks. An attack is generated from a seed that fixes each drone's entry point, route, altitude and timing, so every airframe faces identical attacks and a difference between airframes cannot come from the draw. From a baseline of interceptors, ms autonomy latency, an interceptor top speed of m/s, threats at m/s with of them in the streets, and sensors on the site's mast and two towers with m reach, one factor is varied at a time. Fig. 3 varies interceptor count and sensing together. Two measures are reported: the share of the 50 drones intercepted, and the defence rate, the share of attacks in which fewer than 20 drones reach the site. Section 9 gives the full method.
Six airframes are compared as interceptors, each at 3.1 kg and 3 kW. They differ in how quickly they can change the magnitude and the direction of their thrust (Section 3), and are grouped below by how they change its magnitude. Sections 4 and 5 describe each in full.
Thrust by blade pitch
Helicopter
One large rotor at constant speed. Blade pitch changes its thrust almost at once.
Reverses thrustHovers
Variable pitch quadrotor
Four small rotors at constant speed, each changing thrust by blade pitch, like a helicopter.
Reverses thrustHovers
Thrust by rotor speed
Fixed pitch quadrotor
The common drone. It changes thrust by spinning its rotors up and down.
One wayHovers
VTOL wing
A fixed pitch quadrotor with a wing, so it can hover and also fly on the wing.
One wayHovers
Thrust by turbine or propeller
Thrust vectoring jet
A small turbine with a steerable nozzle. The turbine is slow to change its thrust.
One wayHovers
Fixed wing
A small aeroplane. It cannot hover, so it circles while it waits.
One wayCannot hover
Airframe, with few interceptors
With three interceptors, the helicopter intercepts of drones on average, the variable pitch quadrotor , the fixed pitch quadrotor and the jet (Fig. 2). The two airframes that change thrust by blade pitch lead. The ranking holds across attacks, not only in the mean: the helicopter's 10th percentile () equals the fixed pitch quadrotor's 90th (). No airframe defends the site at this count; the best defence rate is .
Bars / Share interceptedThree interceptors, defaults
| Airframe | Mean | 10th to 90th | | Defended |
|---|
Sensing sets the ceiling
Fig. 3 shows the defence as interceptor count and sensing vary together. With the site's mast as the only sensor, the intercept share saturates: helicopters intercept with three interceptors and with ten, and the site falls in almost every attack. More interceptors cannot engage drones that are detected too late to reach. More sensing raises the ceiling: with ten helicopters, two tower sensors give , four give , and four with double the reach, 320 m, give .
Lines / Interceptors against sensing
Fig. 3Each line is one way of sensing the city: the site's mast alone, the mast and two towers (the default), the mast and four towers, and the four towers seeing 320 m instead of 160 m. Along each line, the number of interceptors grows from two to ten. Defended is the share of attacks the site survived; Intercepted is the share of drones caught. Pick an airframe to see how many of it the site needs.
With adequate sensing, the airframe sets the count
With four tower sensors at 320 m reach, helicopters defend the site in of attacks. Fixed pitch quadrotors need to reach the same rate, as many, and the jet does not reach it with ten.
Sensing sets the ceiling. The airframe sets how many interceptors it takes to reach it.
The two are complements, not substitutes. An agile airframe cannot make up for late detection, and with adequate detection a site still needs as many fixed pitch quadrotors as helicopters. Sections 3 to 7 explain where the difference between airframes comes from.
Latency and street share
Against the baseline threat, cutting autonomy latency from 200 ms to 2 ms raises the helicopter's intercept share by percentage points, and moving drones from above the roofs into the streets, from none to 80%, lowers it by . Both results depend on the threat. The baseline drones weave slowly and do not fly at their physical limit, so a track extrapolated at constant velocity stays accurate. Latency is expected to matter against targets that manoeuvre hard at close range (Section 8), which is the next run of this study. The small effect of street share is partly a property of this city's long street routes.
3Why the airframe matters
Learned controllers already outperform expert humans in agile vehicles. Autonomous quadrotors have beaten world champion drone racing pilots6,7, and a learned driver has beaten the best human drivers in simulated motor racing8. In each case the vehicle was unchanged; the controller simply operated closer to the vehicle's physical limit than a human could.
If such controllers become common, controller quality stops separating competitors, and the vehicle's limit decides the outcome: how quickly it can change its velocity, in any direction, from any state. In a close engagement over a city, that is the quantity that matters.
Autonomy will maximise any airframe. So the most agile airframe wins.
The problem therefore has two parts: which airframe is the most agile, and how fast the loop from sensing to command must run for the airframe, not the software, to be the limit. Sections 4 to 7 address the first and Section 8 the second.
The hypothesis tested is that a helicopter is the most agile airframe of this size. Its rotor turns at near constant speed and stores substantial kinetic energy. Thrust magnitude is set by blade pitch, which a servo changes within a few hundredths of a second, through zero to negative thrust. Thrust direction is set by cyclic, which tilts the rotor disc through blade flapping before the fuselage rotates. A fixed pitch quadrotor must instead change rotor speed to change its thrust, and rotate its whole body to redirect it.
What agility means here
Agility here is the rate at which an aircraft can change its velocity vector, not its top speed or its thrust to weight ratio. Equation (1) shows the two limits: how quickly thrust magnitude T and thrust direction n can change. Each airframe in this paper is a different mechanism for changing them.
m dv/dt = T(t) n(t) + L + D + m g(1)
T thrust magnitude: set by rotor speed, blade pitch, or a turbine's spool
n thrust direction: set by tilting the body, the rotor disc, or a nozzle
L, D lift and drag of a wing or fuselage, if there is one
The study therefore measures time directly. Each airframe performs four manoeuvres from hover or steady flight, and its score is the minimum time in which it can complete each one.
| Manoeuvre | Start | Goal | What it stands for |
|---|
| Change velocity | Hover | 10 m/s in a chosen direction | Taking off after a target that has just been seen |
| Reach | Hover | A point 3 m away, in a chosen direction | A dodge, or the last few metres of a chase |
| Twitch | Hover | 2 m/s in a chosen direction | How quickly it starts to respond at all |
| Turn around | Level, 20 m/s | 20 m/s the other way, by any path | A target that doubles back |
Directions are sampled every 45 degrees in the vertical plane, from straight down through forward to straight up and back. Final attitude is free: the aircraft may arrive in whatever orientation is fastest.
4Six aircraft, one budget
Every airframe has a mass of 3.1 kg, and every electric airframe has 3 kW of peak shaft power. Parameters are taken from datasheets where available; those that had to be assumed are varied in Section 7.
| Aircraft | Changes thrust by | Points thrust by | Key numbers |
|---|
| Helicopter | Collective blade pitch, at constant headspeed; reverses | Cyclic tilts the rotor disc through flapping, then the body follows | 1.29 m rotor, 2500 rpm, flapping lag , inflow lag , kJ in the rotor |
| Quadrotor, fixed pitch | Rotor speed: motor torque against prop inertia; cannot reverse | Differential thrust rotates the whole body | Four 13 in props, thrust to weight |
| Quadrotor, variable pitch | Blade pitch per rotor, at governed speed; reverses | Differential thrust rotates the whole body | Same geometry, 9000 rpm governed |
| Thrust vectoring jet | Turbine spool | Vectoring nozzle, fast | 100 N turbine, thrust to weight , spool lag s |
| Fixed wing | Propeller | Lift, bounded by wing area, speed and stall | 0.6 m2, stalls at m/s, cannot hover |
| VTOL wing | As the quadrotor | As the quadrotor, plus a wing through any angle of attack | 0.35 m2 wing |
Rotor thrust is computed with blade element momentum theory, including the axial flow through the disc, so thrust falls in a fast climb as it does on a real rotor. The speed of a governed rotor is a state: power demanded above the motor's limit is drawn from the rotor's kinetic energy, and the headspeed droops to a floor. Wings use attached flow up to stall and flat plate flow beyond it, so the VTOL wing can also use its wing as an airbrake.
Flying them
Each manoeuvre is posed as a minimum time optimal control problem (2): find the actuator commands, within every rate and range limit, that reach the goal soonest. Trajectories are divided into 50 intervals and solved by direct multiple shooting with CasADi and IPOPT. The result is the best any controller could achieve with that airframe, which is the limit the hypothesis assumes autonomy will reach.
minimise T
over u(t), 0 <= t <= T
subject to dx/dt = f(x, u) the aircraft, actuators included
x(0) = hover or level flight
goal(x(T)) = 0 velocity, or position, reached
actuator rates and ranges, stall, power(2)
Problems of this kind have local optima, and a local optimum can make an airframe appear slower than it is. Each problem is therefore solved from several initial guesses, then each direction is solved again from its neighbours' solutions until none improves. The procedure is identical for every airframe. The VTOL wing is also started from the quadrotor's solution, so its wing can cost it time only through physics.
Trajectories / Minimum time, side viewx forward, z up
Subject
Fig. 4Each aircraft starts in a hover (or, for the turn around, level at 20 m/s) and flies the fastest path the optimiser can find to the goal. Every path is a solved trajectory, played back slowed down; the board on the right is the finishing order. The fixed wing cannot hover, so it only appears in the turn around. Pick a subject to draw it in ink.
5The race
The helicopter reaches 10 m/s fastest in every direction tested: straight down, straight up and forward. The fixed pitch quadrotor of equal mass and power takes , and . The jet, limited by the time its turbine takes to spool, takes forward.
Polar / Time by directionHovering aircraft only
Fig. 5Each closed line is one aircraft. Its distance from the centre in each direction is the time it takes to do the manoeuvre that way, so a smaller shape is a more agile aircraft, and a round shape is one that is equally agile in every direction. The subject picked above is drawn in ink.
| Aircraft | 10 m/s, mean | 10 m/s, worst | Reach 3 m, mean | Twitch, mean | Turn around |
|---|
A wing does not improve agility from hover. The VTOL wing is slower than the quadrotor it is built on in every manoeuvre, because the wing adds mass and drag before there is airspeed for it to use. Even turning around at 20 m/s, where a wing should help, it takes against the quadrotor's . The fixed wing takes , and the helicopter, which tilts its rotor back and pulls, .
6Why the helicopter wins, and where it does not
Three mechanisms account for the result. Only the first is the one the hypothesis named.
Blade pitch, as predicted
The twitch, a 2 m/s change from hover, isolates how quickly an airframe begins to move. Vertically, where only collective changes, the helicopter takes . The fixed pitch quadrotor takes upward and downward: it must accelerate four rotors against their own inertia, and it cannot push downward at all, only reduce thrust and fall. The helicopter's collective passes through zero to negative thrust and drives it down faster than free fall.
But it is the pitch, not the helicopter
The variable pitch quadrotor is the control case: the fixed pitch quadrotor's geometry, with each rotor governed at constant speed and its thrust set by collective. Its vertical twitch, up and down, matches the helicopter's, and its horizontal twitch is shorter, against , because a small body rotates sooner than a large disc tilts. The advantage the hypothesis named therefore belongs to blade pitch control, not to helicopters as such. The helicopter's remaining lead, 10 m/s downward in against and forward in against , comes from the next two mechanisms.
Stored energy
The helicopter's rotor stores about kJ at headspeed. For a fraction of a second it can deliver more power than its motor, repaid as the headspeed droops. Small quadrotor props store almost none. The advantage is physical, but it is not the mechanism the hypothesis named, and it lasts only as long as the allowed droop.
Disc area
The helicopter's 1.29 m rotor sweeps m2; four 13 inch props sweep m2, about a quarter. At equal power a larger disc produces more thrust, so part of the helicopter's lead is its size. Both are realistic designs at 3.1 kg, but an urban interceptor must also fit through gaps, and the helicopter's span is about one and a half times the quadrotor's.
Where it loses
With four times the power, a thrust to weight ratio near 12, typical of racing quadrotors, the fixed pitch quadrotor reaches 10 m/s forward in and upward in , both faster than the helicopter. Downward it remains slower, against , because a fixed pitch rotor cannot produce negative thrust at any power.
Verdict, as the study stands
At equal mass and power, the helicopter is the most agile airframe tested: the fastest to 10 m/s in every direction and the fastest to turn around at speed. The mechanism the hypothesis named, rapid blade pitch, is real but not unique to the helicopter; a variable pitch quadrotor starts to move as quickly, and slightly faster sideways. The helicopter's further lead comes from stored rotor energy and disc area. A fixed pitch quadrotor with several times the power can beat it in some directions, never downward, and only by carrying the battery and motors that power requires. The design that follows is a pitch controlled airframe with as much rotor energy and disc area as the mission's space allows, of which the helicopter is the purest form.
7Does it survive the assumptions?
Several helicopter parameters were assumed rather than taken from datasheets. Each that could change the result is varied below, together with the quadrotor's power and the jet's spool time. Each value is the mean time to reach 10 m/s across four directions.
Slowing blade flapping threefold, the most pessimistic reading of the rotor head, raises the helicopter's forward time from to ; it remains faster than the quadrotor at baseline power. Halving the collective servo speed adds ms on average, because the servo is not the limiting element. Allowing 5% headspeed droop instead of 15% has no measurable effect on this manoeuvre. None of these changes alters the ranking against the baseline quadrotor, the jet or the fixed wing.
8How fast must the autonomy be?
An interceptor responds to a target's manoeuvre only after its autonomy has sensed it, estimated it and commanded a new path. Its response time is that latency plus the airframe's own twitch time (3).
t_response = t_latency + t_twitch(3)
Lines / Response timeTwitch averaged over four directions
Fig. 6Response time against end to end latency, on a log axis. Where a line is flat, the airframe is the limit; where it climbs, the software is.
The variable pitch quadrotor has the shortest mean twitch, ; the helicopter's is , and its fastest actuator, the flapping rotor, settles in about . A typical camera based tracking pipeline at 30 frames per second has an end to end latency near 100 ms. At that latency the helicopter responds in , half of it waiting on software. At 10 ms it responds in , almost all of it airframe.
To use the airframe, the loop has to run faster than the airframe moves: sense to command in about 10 ms, inner loops at 500 Hz or more.
9How the mission is built
- Airframes from the study. An interceptor's acceleration from hover depends on direction, as measured: 10 m/s divided by its time to reach 10 m/s straight up, straight down or level, interpolated between them. Its turning acceleration at speed is 40 m/s divided by its turn around time at 20 m/s. Its response lag is the part of its twitch time not explained by the acceleration limit. The fixed wing has no hover results, so its lag is assumed and it orbits while it waits. The variable pitch quadrotor's turn around did not converge, so it turns at its mean hover acceleration.
- Top speed is a setting, not a result. The study does not measure it, so all airframes share one value.
- Sensing. Sensors update at 100 Hz. A drone is detected by a sensor in range only if the line between them clears every roof, so a drone above the roofs is seen from far away and one in the streets only along clear lines. Interceptors fly over the roofs, never through them. An interceptor's own seeker holds its target within 60 m.
- Autonomy. Guidance uses the newest track at least one latency old, extrapolated at constant velocity, and steers for the predicted intercept point. Latency costs nothing against a target flying straight and most against one that turns.
- An intercept ends the engagement. The mission counts an intercept when the interceptor reaches its target.
- A hundred attacks per case. Section 2's results come from running the same model without its display: every airframe against the same seeded attacks, one setting varied at a time from the baseline in Section 2. For Fig. 3, interceptors and sensing are varied together.
- Defending the site. Each drone that reaches the site costs 5% of its integrity, so an attack is defended if fewer than 20 of the 50 drones get through. The 5% is a modelling choice, not a measurement: it sets how many drones a site can absorb, and so how many interceptors it needs.
- Simplified. Point masses, no wind, no downwash, unlimited endurance. The mission shows what the study's numbers imply for a defended site; it does not predict a real engagement.
10Limits of this study
- Planar. Airframes are modelled in the vertical plane. Roll, yaw and the tail rotor are not modelled, and a helicopter's lateral agility differs from its longitudinal agility.
- No ground effect, wake interaction or vortex ring state. Momentum theory is not valid in a steep descent, and the helicopter's best results are downward. A real rotor descending into its own wake may lose part of that advantage.
- Idealised actuators. Servos, motors and turbines are modelled as rate and range limits with first order lags, not identified on hardware.
- Not size matched. Airframes share mass and power but not span. A size matched comparison is the next run.
- Local optima remain possible. Solutions are not guaranteed to be global. The variable pitch quadrotor's reach is not mirror symmetric ( one way, the other), so its true best is probably slightly faster than shown.
- One city, one kind of attacker. The mission uses one generated city, and its drones fly to the site rather than search or coordinate. A different layout or a more capable attacker could change the numbers in Section 2, though on this evidence not the ranking of airframes.
- The physics limit, not a controller. The times reported are physical limits, not the performance of a real controller. Closing that gap is the other half of the problem.
11Conclusion
Counter drone defence falls behind for as long as it is designed around the attacker of the day. This paper starts instead from the physical limits of the aircraft, and finds that defending a site in a city depends on two things, in order. Sensing sets the ceiling: a site that relies on its own sensor sees too little, too late, and no number of interceptors makes up for it. Once the site can see, the airframe sets the cost: aircraft that change thrust by blade pitch, the helicopter above all, defend the site with far fewer interceptors than a fixed pitch quadrotor, against in this study. Physics, not the attacker of the day, is the basis on which the defence should be designed.