With the increasing interest in the use of multi-rotor UAVs for long range applications such as pipeline inspection, large-area mapping, transporting packages or even people, the question arises: “how far can they fly?”.

In this post I’ll present a theoretical approach, an experiment and a conclusion to answer this question. Let’s start with some basic theory that can be used for all flying things. The distance *d* that a UAV can cover, flying at a constant mass and velocity *v* is presented by:

With *t* the flight time calculated as the available energy in the battery divided by the total required electrical power:

Assuming the power required for propulsion is some orders of magnitude larger then power for payload or avionics, the total required electrical power can be calculated as the power required for propulsion divided by the overall efficiency of the propulsion system (losses in cables, ESC, motors, propellers, air).

For the same battery chemistry and manufacturing process, we can assume that the energy available from the battery linearly depends on its mass:

For Li-Po batteries, a value of 180Wh/kg is possible or in SI units:

The mass of the battery is a fraction of the total mass of the UAV. For example a UAV with a total mass of 4kg, including 1kg of battery has a 25% battery fraction:

The theoretical power that is needed for propulsion is calculated as the flight speed multiplied by the drag:

For fixed-wing aircraft this formula makes sense but multi-rotors also consume propulsion power to create lift, right? The propellers create both lift AND propulsion. This is analog to an airplane which uses wings and a propulsion unit (propeller or jet) to create its lift and propulsion respectively. So just consider the propellers of a multi-rotor acting as a wing and a propulsion unit simultaneously.

We do not know the drag of the UAV and we cannot measure it. Why not? For a fixed wing aircraft you can put the aircraft in the wind tunnel and measure the drag with a force sensor with the propulsion unit producing zero thrust. Or we can measure the thrust of the propulsion unit in flight with a force cell. We can do this because the propulsion unit and the ‘lift unit’ (=wings) are separated systems. Because the propellers of a multi-rotor act both as wing and propulsion unit, we cannot simply remove them and measure the drag in a wind tunnel. This would be the same as measuring the drag of a fixed-wing aircraft without the wings. Because we cannot separate lift and thrust (= drag in regime flight), we continue with a generally accepted ratio in aviation: the maximum lift-to-drag ratio, also known as the aerodynamic efficiency or glide ratio. A glide ratio of 10 for example means, if you shut down the propulsion unit, that you can travel maximum 10 meters while descending 1 meter.

The glide ratio of a fixed-wing aircraft is easily measured this way and are in the order of 10 to 40. For helicopters, it can also be defined by gliding down in auto-rotation. Therefore the blades are given a negative pitch and they act like a wing of a glider, but then rotating. Typical ratios are around 4 or lower. Because we cannot change the pitch of the blades of multi-rotor propellers, it is not possible to determine the lift-to-drag ratio by gliding down in auto-rotation.

We’ll have to find a way to determine exact the lift-to-drag ratio for multi-rotors later on, but let’s assume for now we know the ratio and it is less than a full-size helicopter in auto-rotation (4) and more than a wingsuit (2.5).

We also know the lift, which is equal to the weight:

Rearranging former equations allows us to calculate the maximum range:

Since the propellers act as both wings and propulsion unit, it is also not possible to determine solely the efficiency of the propulsion unit. Therefore, we have to consider the two factors marked in yellow as one unknown value. We can make an educated guess however: the efficiency will always be <1 and the lift-to-drag ratio will be probably <4. Therefore, we can calculate the maximum range of a multi-rotor with 180Wh/kg batteries and 25% battery as:

The energy density of the battery and the fraction of the battery are easily measured and should be maximized to get the highest range.

We assumed that

but we need a way to know this number for a multi-rotor. This number is a combination of aerodynamic and electric/mechanical efficiency of a multi-rotor and would be a good benchmark to validate the capability of a multi-rotor to cover large distances.

Doing some more math with former equations allows us to determine the highest value of this number by performing some test flights at different flight speeds (yay! No wind tunnel test or force cells required, only a speed and a power measurement!)

If we want to consider only the aerodynamic performance of the multi-rotor, including the propellers, we can define a ‘theoretical glide ratio’, which we cannot achieve because auto-rotation is not possible:

With P_mech the total mechanical power delivered to the propeller shafts during cruise flight. This can be measured or calculated if the efficiency of wires, ESC and motor is known for all possible operating points.

Now the big question: what is the theoretical glide ratio of a multi-rotor? During my PhD, I performed many test flights with different multi-rotors and here are the results for a popular setup:

The best total flight efficiency of this setup was:

occurring at a total mass of 1.6 kg and a speed of 16.3 m/s. With a 25% battery fraction and 180Wh/kg batteries, this results in a range of about 20km. Because I measured the efficiency of the motor and ESC in all possible operating regimes on a test bench, I was able to also compute the theoretical glide ratio:

So that is somewhere in between a flying squirrel and a wingsuit (not too good!).

Conclusion: The range of multi-rotor can be easily computed if you know the ‘flight efficiency’, which is a combination of the efficiency and aerodynamics of the ESCs, motors, propellers and the body/arms/wings. The best flight efficiency of a conventional quadcopter was calculated from flight data as 1.2 and the accompanying ‘theoretical glide ratio’ was 2.13. People working on VTOL hybrid solutions should be able to increase this number towards glide ratios of fixed-wings (> 6). However… battery fraction is of equal importance if you want to maximize your range.

Question: what is your flight efficiency and battery fraction? I’d love to see some comparison between multi-rotors, helicopters, fixed wings and hybrids!

If you like to learn more about the performance of multi-rotors for high speeds, long range, endurance, payload capacity, you want access to all the data on different vehicles or for more specific questions contact me on bart.theys@kuleuven.be