Common-form statistical relations for conceptual sizing of electric and hybrid VTOL unmanned aircraft

Designing an aircraft that can hover like a helicopter and then fly forward like a plane is one of the most demanding puzzles in modern engineering. These machines, known as vertical take-off and landing or VTOL drones, must satisfy two very different sets of physical rules at the same time. To hover, they need powerful rotors pushing air straight down to hold their weight. To fly forward efficiently, they need wings that catch the air to generate lift while the rotors spin much slower to save energy. The challenge for engineers is that these two requirements often pull in opposite directions. A design with large rotors for hovering might have heavy, draggy wings that ruin its forward speed, while a sleek, fast plane might not have enough power to lift off vertically. For decades, creating these hybrid machines has required building expensive prototypes, testing them, and then starting over when the math didn't work out.

A team of researchers at the Technical University of Sofia has proposed a new way to solve this puzzle before a single piece of metal is cut. Instead of treating every new drone design as a unique mystery, they have gathered data from dozens of existing electric and hybrid drones built between 2011 and 2026 to find a common pattern. They looked at four main ways these machines are built: some have separate motors for lifting and flying, some tilt their entire engines, some tilt their wings, and some fly like a plane that lands on its tail. By analyzing the real-world numbers from these different designs, the researchers found that despite their different shapes, they all follow a similar set of statistical rules. This discovery allows engineers to make a reliable first guess about how big the wings should be, how large the rotors need to be, and how much battery power is required, simply by knowing the drone's weight and its target speed.

The researchers began by assembling a massive collection of data from peer-reviewed studies, manufacturer specifications, and research demonstrators. They focused on electric and hybrid drones, separating them into their four architectural families. For each family, they looked for the relationship between the aircraft's weight and the size of its wings, as well as the relationship between the weight and the total area of the spinning rotors. They discovered that for a given cruising speed, the weight of the aircraft divided by the wing area follows a predictable pattern for each design type. For example, a drone that uses separate motors for lifting and cruising tends to have a different wing size relative to its weight compared to a drone that tilts its wings to fly. By calculating these patterns, the team created a set of coefficients, or scaling factors, that act as a quick reference guide. These factors tell an engineer that if they choose a specific design style, they can immediately estimate the necessary wing area and rotor size without running complex computer simulations.

The study also mapped out the power requirements for these machines. The researchers found that the energy needed to hover is directly linked to how heavily the rotors are loaded, which is a measure of how much weight each square meter of rotor area must support. Heavier loading requires more power, but the relationship follows a consistent curve across all the different drone types. Similarly, the power needed to fly forward at a steady speed was found to be proportional to the speed itself, with a specific multiplier for each design style. This means that once an engineer decides on a cruising speed, they can instantly calculate how much power the motors will need to sustain that flight. The team also looked at battery storage, determining how much energy per kilogram of aircraft weight is typically installed in successful designs. This helps designers avoid the common pitfall of building a drone that is too heavy with batteries to lift off, or one that runs out of power before reaching its destination.

To prove their method works, the researchers applied their formulas to a specific concept called TERES-02, a hybrid drone designed to carry a payload of 50 kilograms. They started with just three basic inputs: the total weight, the desired cruising speed of 33 meters per second, and the number of lift rotors, which was four. Using their statistical rules, they calculated that the drone would need a wing area of 1.75 square meters and rotors with a diameter of roughly 0.88 meters. The math then revealed that the motors would need to produce about 12.87 kilowatts of power to hover and 2.23 kilowatts to cruise. Finally, they determined that the battery would need a minimum capacity of 1.34 kilowatt-hours to handle a five-minute vertical flight segment. This entire process, which usually takes weeks of detailed modeling, was completed in a single step, providing a solid starting point for further development.

The value of this work lies in its ability to reduce uncertainty at the very beginning of the design process. By using a single mathematical structure that adapts to different shapes through simple coefficients, the researchers have created a tool that works for almost any fixed-wing VTOL drone. They are careful to note that these formulas are for the initial screening phase and are not a replacement for detailed engineering later on. The data they used comes from a limited number of existing aircraft, so the confidence in the results is highest for designs that closely resemble the ones in their database. For example, the data for tilt-rotor designs was based on only five aircraft, making those specific numbers more tentative than the data for lift-and-cruise designs, which had eight examples. The researchers also identified that a major limitation of their current database is the lack of consistent, real-world power measurements, as many sources only report theoretical or rated power rather than what the motors actually use in flight.

Despite these limitations, the study offers a clear path forward for the next generation of unmanned aircraft. It separates the geometry of the aircraft from the energy system, allowing designers to mix and match different power sources, such as batteries or fuel cells, with the most efficient airframe shape. The researchers emphasize that their method is not a magic solution that eliminates the need for testing, but rather a way to ensure that the first design attempt is grounded in the reality of what has already been built. By relying on the collective experience of dozens of previous projects, engineers can avoid the most common mistakes and focus their resources on refining the design rather than guessing its basic dimensions. This approach turns the early stages of aircraft design from a game of chance into a more predictable science, helping to bring these versatile machines from the drawing board to the sky more quickly.