Drone–drone collisions over New York City via the kinetic theory of gases

Drones can deliver packages faster and at lower cost than street-level vehicles, especially for one vehicle trip per package.^1,2 But drone delivery faces numerous challenges: concerns about safety,^3–5 noise,⁶ a possibly unwelcoming public,⁷ and a several year pathway to achieving all the certifications required for full-scale operation.⁸ Given the speed and cost advantages of drone delivery, but still cognizant of the challenges, a number of companies have been developing and testing drone delivery systems. These include: Amazon Prime Air, Wing (a subsidiary of Alphabet), UPS Flight Forward (with Wingcopter), DHL, FedEx, Zipline, Flirtey, Matternet, and Uber Eats.^3,9–12 Gartner, a consulting firm, predicts that in 2026 more than one million drones will be engaged in retail deliveries worldwide.¹ For that same year, FedEx expects the U.S. parcel market alone to exceed 100 × 10⁶ packages per day, including drone deliveries as well as vehicle deliveries.¹³ Ninety percent of these could be deliveries by drone.^3,9

This envisioned future of drone delivery will only be realized if delivery by drone is safe. Collisions with buildings, towers, wires, flying taxis, passenger drones, helicopters, winged aircraft, and other delivery drones must be avoided. We will use the kinetic theory of gases to calculate rate of collisions between “dumb” drones that have no collision-avoidance capability, modeling the drones as large molecules. Such a capability is needed to reduce the number of collisions to zero or near-zero. We distinguish two strategies for achieving this:

1. Equipping each drone with a “sense and avoid” capability to detect an imminent collision and then swerve to avoid it. Amazon's Prime Air delivery drone^4,9 (see Fig. 1) utilizes this strategy.

2. Creating an Unmanned Aircraft System Traffic Management (UTM) system to ensure a safe separation between drones, using intelligence resident outside the drones themselves. A UTM is currently under development by the FAA and NASA, working in collaboration with other federal agencies and industry, but will only become well-defined over the next few years.^14,15 Because of the lack of specificity, we will not consider this strategy further in this paper.

If there are many drone flights over a city, but only one drone at a time is in the air, there will be no collisions. If many are in the air at one time, but all travel on radial paths from a central point, the outgoing drones at one altitude and the incoming at a lower, there will be no collisions, assuming care is taken to ensure that at the central point rising and descending drones do not conflict. However, if Amazon drones are launched from and return to one central point and Wing drones from a different central point, the Amazon and Wing drones, flying at the same altitude, could collide. We assume many such central points, roughly distributed uniformly over the area of New York City.

We refer to drones that have no detect and swerve capability as “dumb” drones, in contrast to “smart,” strategy 1 drones that do have that capability. Real-world delivery drones will all be smart. But will they be sufficiently smart so that p is small enough to satisfy $EPD×p≤ CPD acc$? We assume that the encounter rate EPD is the same for smart and dumb drones, and that, for dumb drones, each encounter results in a collision, so $CPD dmb=EPD.$

Note that 16 km is 2/3 of the maximum 24 km round trip distance claimed for the Prime Air drone and is the average delivery round trip distance assuming destinations have a uniform spatial distribution over a circle of radius 12 km. To see this, imagine a drone base station at the center of its circular delivery area, with radius R = the maximum one-way drone range, and destinations distributed uniformly throughout the delivery area. The average base-to-delivery distance is then $r ̂= 1 / π R 2 ∫ 0 R r 2 π rdr=( 2 / 3 ) R$⁠.

With 150 000 drone deliveries per day over New York City, and d = 16 km per delivery, the total distance traveled by the drone fleet per day will be $2.4× 10 6$ km, or 1400 λ_Cl, implying 1400 dumb drone collisions/day. However, this counts the collision of the ith drone with the jth, and the jth with the ith, as two distinct collisions. Dividing by two to correct for this double counting yields, for dumb drones, a rough estimate of CPD_dmb = 700 collisions/day. (When the author first arrived at this estimate, he was surprised that it was so large, having anticipated very few encounters, as there is much empty space above New York City.) Clearly, smart drones must be much better at avoiding collisions. If we arbitrary take CPD_acc = 10 collisions/year = 2.74 × 10⁻² collisions/day, then to meet the condition $EPD×p≤ CPD acc$⁠, p ≤ 0.39 × 10⁻⁴. The collision-avoidance capability of smart drones must reduce the collision rate by a factor of at least 1/p = 2.6 × 10⁴. Meeting this condition is the responsibility of drone manufacturers and will be a non-trivial challenge.

As noted above, far more than 10%, up to 90%, of deliveries could be made by drone.^3,4 From Eq. (9), EPD varies as μ², because as μ increases, the cumulative distance traveled increases while the mean free path decreases. If 50% of packages are drone-delivered, EPD will be 25 times greater than for 10%. Table I shows the results of our analysis if 10%, 50%, and 90% of all deliveries are made by drone. For LPD = 150,000 over New York City, the drone density N = 35.4 drones/km³. The procedure for calculating the average separation between drones is presented in the  Appendix.

According to Ref. 13, the U.S. parcel market is expected to double in size, to more than 100 × 10⁶ packages per day by 2026. New York City's population is 2.6% of the U.S. population, so it can be expected to have roughly 2.6 × 10⁶ deliveries/day by 2026. Assuming again 10% of these NYC deliveries are by drone, there will be 2100 drone encounters/day in 2026, 53,500 at 50% drone delivery, and 173,000 at 90% drone delivery (upper bounds). Table II shows the same variables as Table I, but for 2.6 × 10⁶ deliveries per day in New York City.

The third and fourth rows from the top in Tables I and II display numbers based on the kinetic theory of gases and Eq. (6). Those in the bottom three rows do not depend on the kinetic theory, and so are valid for both strategies 1 and 2. All numbers in the two tables are valid for both smart and dumb drones, except for p_max, which applies only to smart drones. EPD estimates extend over a huge range: from a lower bound value of 4.7 (Table I) to an upper bound value of 170,000 (Table II). These numbers are encounters, not collisions. Even in the worst case, the number of collisions would be acceptable if the collision probability p ≤1.58 × 10⁻⁷ (assuming CPD_acc = 2.74 × 10⁻²). This implies a reliability, r = 1 − p ≥ 0.9999998. For comparison, ISO Standard 26262 implies a reliability of seven nines for future self-driving cars for each 30-mile round trip commute in the US.²⁰ 

One way to reduce EPD values is to fly drones in the same direction and speed but at different levels. If the 100 m thick drone fly zone is divided into four strata, with, say, drones in the 75–100 m stratum flying north at the same speed, those in the 50–75 m stratum flying east, etc., there will be no collisions except for drones ascending or descending through different strata. Let the density of those drones moving vertically be αN, where 0 ≤ α ≤ 1. In one round trip, the average time a drone spends moving vertically is $4l/ s v$⁠, where $l=50 m$ is the average distance the drone travels to reach the correct stratum, $s v$ is its vertical speed, and the factor of 4 accounts for the four vertical segments of each round trip. The average time spent in horizontal motion is $d / s$⁠, where d is the round-trip distance and s is the horizontal speed. For typical numbers (⁠ $s v=5 m/s$⁠, $s=48 km/h=13 m/s,$ and $d=16 km)$⁠, $α≈0.032≪1,$ so nearly all of the drones are traveling horizontally.

The probability of a vertically traveling drone suffering a collision with a horizontally moving one can be estimated with the help of kinetic theory. For a collision to occur, the center-to-center distance between drones must be less than 2a. In time $l / s v,$ a horizontal drone sweeps out a cylindrical “collision volume” equal to $4π a 2s l / s v$⁠, so the probability of a collision in a single round trip is $P ′=16π a 2s l / s vN$⁠. From Eq. (4), the probability of a collision within a cloud of drones moving with the same speed in random directions is $P= d / λ Cl=16π a 2dN/3$⁠, so $P ′ / P = 3 s l / d s v$⁠. Using the same numbers as above, $P ′ / P ≈ 0.025$⁠, a significant reduction in the encounter rate. Of course, travel distances will increase, offsetting some of this advantage: a drone traveling 1 km north and 1 km east will travel 2 km rather than 1.4 km “as the crow flies.”

While drone delivery promises many economic advantages, there are numerous quality-of-life issues to be addressed. Care must be taken to ensure pedestrians are not injured when deliveries are made to urban sidewalks or front steps, as opposed to private back yards. Table II suggests that, for a 50% drone delivery fraction, 2600–24,000 drones could simultaneously occupy the airspace over New York City. Combine the 31 drones/km² in Table II's upper bound 50% case with the sound of a single drone to imagine the noise level. The bottom three rows of each table give a sense of how drone-crowded the sky will be, with inter-drone separation ranging from 1600 m down to 140 m. Migrating geese beware!

Thanks to the two anonymous referees for their comments and insights, which were crucial to improving earlier versions of this paper. And thanks to Dr. Gastone Celesia, Frank Straka, and McLouis Robinet for very helpful discussions, advice, and encouragement.

Tables I and II include estimates for the average separation between nearest neighbor drones. We present here a procedure by which those estimates were calculated.