Physics of Drones

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Thanks Talisker, that was a good read. Also the part on prop balancing helps validate what I think about static balancing.
 
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The discussion suggests DJI’s RTH speed is set to optimum- we know this not to be true. At least with regards to maximum distance travelled for a given power consumption.
 
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What do DJI say about the optimisation of RTH speed? Given that the battery running out is by far the biggest hazard to a drone, there’s not much else they could have set it to other than using the minimum energy to travel a given distance. It’s not that easy to check in reality, since fluctuating wind conditions mean that it’s hard to repeat a test under the same conditions. A couple of days ago my P3S activated RTH after losing signal in pretty calm conditions and the flight log shows it returned at a steady 9.7 metres per second, which is near enough what the piece says.
 
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I have never seen any comment from DJI as to why the RTH speed was set to 10ms.... I can tell you that optimum speed for maximum distance for a given battery charge is close to flat stick. Well above 10ms. Have a look at the distance threads and/or try it yourself. It’s well demonstrated.
 

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I have never seen any comment from DJI as to why the RTH speed was set to 10ms.... I can tell you that optimum speed for maximum distance for a given battery charge is close to flat stick. Well above 10ms. Have a look at the distance threads and/or try it yourself. It’s well demonstrated.

Is it completely clear that is true? While the author doesn't actually present the supporting calculation for the assertion that the RTH speed is set to maximize distance, the calculations are relatively trivial. I posted a similar calculation on the Mavic forum a while back that suggested a pitch of around 30° was optimal, but I didn't use model specific data and that does change things.

The motor thrust (force) vector F, which aligns approximately with the aircraft z-axis, decomposed into vertical and horizontal components at a pitch θ, solves for equilibrium in constant-velocity flight. Ignoring body aerodynamic lift, which is not large for a quadcopter:

F cosθ = Mg [1]​

F sinθ = -D = -ku² [2]​

where M is the mass of the aircraft and D is the horizontal drag, which goes roughly with the square of the airspeed, u.

Eliminating θ ( since cos²θ + sin²θ = 1):

F = √(M²g² + k²u⁴) [3]
Power, P, is related to thrust by P² = /k, which allows us to express power as a function of airspeed:

P = (M²g² + k²u⁴)^(3/4)/√k₂ [4]​

and the ratio of airspeed / power, which is equal to energy per unit distance, as

u/P = √k₂u/(M²g² + k²u⁴)^(3/4) [5]​

We don't need to know k₂ to calculate a normalized u/P, but we do need k₁. That can be obtained from the published specifications on pitch angle vs. speed and combining equations [1] and [3].

Mg/cosθ = √(M²g² + k²u⁴) [6]
Giving k₁ = 0.033 for the P4P and 0.016 for the MP.

That gives the following relationship between the airspeed/power ratio and airspeed:

airspeed_power_ratio.png


That suggests maximum range for the P4P at 16.4 mph and for the MP at 14.4 mph. The author calculates a slightly higher optimal airspeed, but has a couple of errors in his estimates.

There are a few uncertainties and simplifications in those calculations, but I find it difficult, given the shapes of those curves, to see how the real peak u/P values could be at much higher airspeeds. Are there actual solid data to support much greater speeds for maximum distance, or is it mostly anecdotal?
 
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I always thought since a component of battery usage goes into defying gravify and keeping the craft aloft, that getting it back as quickly as possible would rule the day.
 
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Is it completely clear that is true? While the author doesn't actually present the supporting calculation for the assertion that the RTH speed is set to maximize distance, the calculations are relatively trivial. I posted a similar calculation on the Mavic forum a while back that suggested a pitch of around 30° was optimal, but I didn't use model specific data and that does change things.

The motor thrust (force) vector F, which aligns approximately with the aircraft z-axis, decomposed into vertical and horizontal components at a pitch θ, solves for equilibrium in constant-velocity flight. Ignoring body aerodynamic lift, which is not large for a quadcopter:

F cosθ = Mg [1]​

F sinθ = -D = -ku² [2]​

where M is the mass of the aircraft and D is the horizontal drag, which goes roughly with the square of the airspeed, u.

Eliminating θ ( since cos²θ + sin²θ = 1):

F = √(M²g² + k²u⁴) [3]
Power, P, is related to thrust by P² = /k, which allows us to express power as a function of airspeed:

P = (M²g² + k²u⁴)^(3/4)/k₂ [4]​

and the ratio of airspeed / power, which is equal to energy per unit distance, as

u/P = k₂u/(M²g² + k²u⁴)^(3/4) [5]​

We don't need to know k₂ to calculate a normalized u/P, but we do need k₁. That can be obtained from the published specifications on pitch angle vs. speed and combining equations [1] and [3].

Mg/cosθ = √(M²g² + k²u⁴) [6]
Giving k₁ = 0.033 for the P4P and 0.016 for the MP.

That gives the following relationship between the airspeed/power ratio and airspeed:

View attachment 99664

That suggests maximum range for the P4P at 16.4 mph and for the MP at 14.4 mph. The author calculates a slightly higher optimal airspeed, but has a couple of errors in his estimates.

There are a few uncertainties and simplifications in those calculations, but I find it difficult, given the shapes of those curves, to see how the real peak u/P values could be at much higher airspeeds. Are there actual solid data to support much greater speeds for maximum distance, or is it mostly anecdotal?
It might be the OP did not perform any calculations to arrive at his assumption that 10m/s might provide greatest range for a given power consumption. In the absence of any further clarification from him it might be that he simply relied on an assumption that DJI had chosed the most efficient speed for RTH- we are entitled to believe this to be true from his narrative.

In any case your calculations are wrong. The most efficient speed for max range is demonstrated for both P3/4 as around 31mph. Many users, myself included, have confirmed this in actually flying the AC.

The following article has some good numbers which reflect actual performance. 14m/s crusie speed for P3 will provide for a distance or 14km in 17min on a fully charged pack. Would be interested to learn from you what you missed in your calcs.

long range drone DJI Phantom 3 professional
 

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The most efficient speed for max range is demonstrated for both P3/4 as around 31mph. Many users, myself included, have confirmed this in actually flying the AC.
I am not a physicist by any stretch of the imagination. But I disagree with this statement. From my flying time, and this is based on that alone. I have achieved much better range at a slower speed. This of coarse would depend on an individual specific battery's use, maintenance and the individuals flying preference. Based on my aircraft alone, with no other variables other than the obvious, ( Wind, RH and so forth), overall the range I achieve is greater when flown much less then the 31mph. Normally, between 10-20mph maximum. I have compared these, granted over not at max range, but the levels indicate to me that the slower speed is more efficient, as far as the range goes. Like I said, there are no technical calculations involved here, it is only from my own personal flights. That said, it can vary, from aircraft, to battery and flying conditions.
 

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Quick addendum: This might be considered "flight time" vs "max range". One could make the assumption ( Yes I said assumption) that these two entities are related. I have no calculations to verify this, yet if the criteria as posted above are "assumed" correct there are many more variables than can be sufficiently accounted for.
 

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It might be the OP did not perform any calculations to arrive at his assumption that 10m/s might provide greatest range for a given power consumption. In the absence of any further clarification from him it might be that he simply relied on an assumption that DJI had chosed the most efficient speed for RTH- we are entitled to believe this to be true from his narrative.

In any case your calculations are wrong. The most efficient speed for max range is demonstrated for both P3/4 as around 31mph. Many users, myself included, have confirmed this in actually flying the AC.

The following article has some good numbers which reflect actual performance. 14m/s crusie speed for P3 will provide for a distance or 14km in 17min on a fully charged pack. Would be interested to learn from you what you missed in your calcs.

long range drone DJI Phantom 3 professional

The author of the article linked in this thread clearly did perform calculations. He had the necessary derivations to do so - he simply didn't explicitly report the form used.

The article that you linked has some serious issues. At least one of the basic results is demonstrably incorrect - that the aerodynamic force on a moving Phantom is downwards and increases monotonically with airspeed. Tilting a wing forwards reduces lift - it doesn't make it negative. If that were true then maximum flight time would be achieved at hover, which even DJI specifications show is incorrect. That article also suffers from the problem that the author posits a few equations, some trivially correct, others not, but then just asserts that they solve to give the results he displays. His equations are incomplete in that case, because they do not, and he shows none of his method.

As for why my calculations might be wrong - I do ignore aerodynamic lift. We know that it is positive (not negative as your linked article asserts), and so increasing airspeed reduces the component of motor thrust required to keep the aircraft aloft. Most simple calculations ignore this but it will shift the u/P curves to the left. For the Mavic, DJI quotes the flight time in hover as 24 minutes, and the maximum achievable flight time as 27 minutes at 15.5 mph. That's already faster than my calculation for maximum range and so, if correct, implies that maximum range speed will be higher. Maximum range speed trivially has to be higher than maximum flight time speed for any reasonable functions for lift and drag.

It's also possible that something in my assumptions or derivation is incorrect, which is why I included it in full - hoping that one of you might check it for errors. I'll go back over it and also see if I can add a defensible term for lift to see how much difference that makes.
 
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sar104

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I am not a physicist by any stretch of the imagination. But I disagree with this statement. From my flying time, and this is based on that alone. I have achieved much better range at a slower speed. This of coarse would depend on an individual specific battery's use, maintenance and the individuals flying preference. Based on my aircraft alone, with no other variables other than the obvious, ( Wind, RH and so forth), overall the range I achieve is greater when flown much less then the 31mph. Normally, between 10-20mph maximum. I have compared these, granted over not at max range, but the levels indicate to me that the slower speed is more efficient, as far as the range goes. Like I said, there are no technical calculations involved here, it is only from my own personal flights. That said, it can vary, from aircraft, to battery and flying conditions.

This is my concern - lack of hard data, at least that I have seen. Easy enough experiments to perform I guess - one could simply fly a large diameter circle at various speeds to generate distance travelled.
 

sar104

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It might be the OP did not perform any calculations to arrive at his assumption that 10m/s might provide greatest range for a given power consumption. In the absence of any further clarification from him it might be that he simply relied on an assumption that DJI had chosed the most efficient speed for RTH- we are entitled to believe this to be true from his narrative.

In any case your calculations are wrong. The most efficient speed for max range is demonstrated for both P3/4 as around 31mph. Many users, myself included, have confirmed this in actually flying the AC.

The following article has some good numbers which reflect actual performance. 14m/s crusie speed for P3 will provide for a distance or 14km in 17min on a fully charged pack. Would be interested to learn from you what you missed in your calcs.

long range drone DJI Phantom 3 professional

OK - resolved. It was a simple error in my u/P calculation - I neglected to square the k₁ term in equation [5] when performing the actual numerical calculation. Not surprisingly that makes a significant difference:

airspeed_power_ratio_fixed.png


That looks much more in line with your observation that almost flat out (or maybe absolutely flat out for the Mavic Pro) maximizes distance:

P4P: 38.3 mph
MP: 40.4 mph
 
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OK - resolved. It was a simple error in my u/P calculation - I neglected to square the k₁ term in equation [5] when performing the actual numerical calculation. Not surprisingly that makes a significant difference:

View attachment 99679

That looks much more in line with your observation that almost flat out (or maybe absolutely flat out for the Mavic Pro) maximizes distance:

P4P: 38.3 mph
MP: 40.4 mph
It will be interesting to learn if the OP did any calcs. I’m not surprised you came through in the end, the numbers don’t lie. Direct observations don’t either- that’s why I was convinced something was off. When I first got a phantom 3 I did numerous runs, some one way to the extent to the battery (in the desert over private property) so I was pretty clear on the practical considerations. It would seem DJI has set the hard limit on max tilt angle as the best compromise to get reasonable battery performance.
 
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sar104

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It will be interesting to learn if the OP did any calcs. I’m not surprised you came through in the end, the numbers don’t lie. Direct observations don’t either- that’s why I was convinced something was off. When I first got a phantom 3 I did numerous runs, some one way to the extent to the battery (in the desert over private property) so I was pretty clear on the practical considerations. It would seem DJI has set the hard limit on max tilt angle as the best compromise to get reasonable battery performance.

It's definitely reassuring when observation and prediction broadly agree. Interestingly, for the P4P, DJI's specifications for pitch angle vs. airspeed are not quite represented by a constant drag coefficient relationship. Specifically, at the three data points that they quote (31 mph @ 25°, 36 mph @ 35° and 45 mph @ 42°), the implied drag coefficients vary from 0.0310 to 0.0375. I used an average for the calculation.
 
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Quick addendum: This might be considered "flight time" vs "max range". One could make the assumption ( Yes I said assumption) that these two entities are related. I have no calculations to verify this, yet if the criteria as posted above are "assumed" correct there are many more variables than can be sufficiently accounted for.
Max flight time is realised at significantly reduced speeds to that which provide for max distance on a given battery charge. This is also well demonstrated in practice...
 
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Interesting discussion. The original article does give the function for the energy used to travel a given distance that was minimized to produce the answer of about 10 metres per second as the optimum speed. There is now a supplementary paragraph showing the energy usage graph and a discussion of the difference between the mechanical energy needed (the one that was originally minimized) and the battery energy used. The answer for the battery usage is still near 10 metres per second. Travelling as fast as 14 metres per second isn't the most efficient because the drag increases as the square of the air speed. The energy required to overcome this drag is more than that saved by needing to keep aloft for a shorter time. As with a car on the open road, making a journey at the speed limit takes more fuel than at a lower speed, because of the high drag at high speed.
 

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Interesting discussion. The original article does give the function for the energy used to travel a given distance that was minimized to produce the answer of about 10 metres per second as the optimum speed. There is now a supplementary paragraph showing the energy usage graph and a discussion of the difference between the mechanical energy needed (the one that was originally minimized) and the battery energy used. The answer for the battery usage is still near 10 metres per second. Travelling as fast as 14 metres per second isn't the most efficient because the drag increases as the square of the air speed. The energy required to overcome this drag is more than that saved by needing to keep aloft for a shorter time. As with a car on the open road, making a journey at the speed limit takes more fuel than at a lower speed, because of the high drag at high speed.

That was never clear at all, at least to me, since he never expressed total power as a function of horizontal velocity. He did calculate additional power needed for horizontal motion, but if he minimized just that function then he will obviously have got the wrong answer. In any case, I think it's clear from both empirical data and my analysis that his conclusion is incorrect - the total power per unit distance function has a minimum at a much higher speed.
 
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Looking back at sar104's comment that "The article that you linked has some serious issues. At least one of the basic results is demonstrably incorrect - that the aerodynamic force on a moving Phantom is downwards and increases monotonically with airspeed" it seems to me the article did not say this. The only case mentioned of a downward only force was the case of the drone hovering stationary or rising vertically. The section on forward motion shows the drone applying a force that is both upward (to counteract the weight) and horizontal (to counteract the drag). The horizontal force must increase as the square of the horizontal speed, as stated in the article, so it seems to me the forces mentioned are the ones needed.
 

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Looking back at sar104's comment that "The article that you linked has some serious issues. At least one of the basic results is demonstrably incorrect - that the aerodynamic force on a moving Phantom is downwards and increases monotonically with airspeed" it seems to me the article did not say this. The only case mentioned of a downward only force was the case of the drone hovering stationary or rising vertically. The section on forward motion shows the drone applying a force that is both upward (to counteract the weight) and horizontal (to counteract the drag). The horizontal force must increase as the square of the horizontal speed, as stated in the article, so it seems to me the forces mentioned are the ones needed.

That was a reference to the article linked in post #8, not the one that you linked. Your one explicitly ignored aerodynamic lift as second order, as did my approach.
 

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