A climb-glide routine can potentially yield a useful increase in range compared with the same airframe/propeller/motor combination operating straight-and-level at best range speed point or best endurance point respectively.
Let me explain briefly with graphs of the results how I have come to that conclusion, at least theoretically so far, hopefully including some useful detail on the method of my calculations and models. I intend to experimentally validate this concept at some point with my APM-powered Skyfun, so I have tailored this example with measured and guestimate data as best I can for this style and size of UAV. The propeller is oversized compared with what is normally run on the Skyfun (8" vs 5-6") due to the non availability of data at the smaller sizes.
Best Endurance Steady-state Flight
Firstly, we look at the airframe drag power curves as a function of airspeed. For this model a simple Prandtl model of an aircraft is used. The form drag power and induced drag power combine to form a "U"-shaped total drag power curve with a minima representing the best endurance point (lowest power consumption). The point where a tangent line to this drag power curve passing through the origin touches the curve, (or, in the case of head or tailwind components, with the origin point offset in the speed axis by the appropriate offset) yields the minimum energy consumption per unit distance - i.e. best range speed. This is all valid for steady-state, straight-and-level operation and is only considering aerodynamic power absorbed.
Adding in the propeller and electric motor models to the calculation, we can estimate the shaft power and electrical power consumed to overcome this aerodynamic drag. The propeller model is a regressed 4th order polynomial surface model based on UIUC propeller data The efficiency curve for an 8x6" APC-E propeller shown here shows how the polynomial model compares with the UUIC data. This model is actually two models - one for thrust coefficient and one for power coefficient. It isn't a perfect model by any stretch, but it allows an analytical solution for the overall modelling process.
The electric motor is modelled as a first-order model as described by Mark Drela
The motor Kv was estimated to give a voltage requirement for a 25% climb gradient at 28.5m/s of approximately 14.8V (4S battery pack assumption). This is somewhat arbitrary and the assumption does influence the motor efficiency somewhat, but it also gives a realistically useful top speed estimate of 33-34m/s. It is possible to optimise the motor efficiency in cruise a fraction further by manipulating the Kv value, but only by choosing to compromise climb and max speed performance. The motor model used yields a peak efficiency of a realistic 81%
Having modeled the propeller and the motor, these two power curves sit respectively above the aero drag curve by an amount representing the successive inefficiency of each of these processes. The minima of the electric power curve now yields the best electrical endurance speed and, likewise, the tangent point for best electrical range speed. Note that both points are at somewhat increased airspeeds - something to bear in mind even for straight and level operations!
Just for interest, here below are the propeller and motor efficiency curves for steady-state flight. It shows clearly some of the compromises of steady-state operation where the efficiency points of the electric motor and propeller are obviously not co-located. Notice too that increasing speed increases the total propulsion efficiency, excepting that the gain here is subsequently lost in airframe drag.
System Efficiency during Climb
If we now add a climb (vertical) velocity, then knowing the airframe mass (in this example specified at 1kg), we can quite easily calculate the additional power required to climb and add this to the net thrust power. Not forgetting to correct for the increased hypotenuse speed of the climb gradient, this generates a new thrust value the propeller needs to generate, hence a higher propeller speed, more shaft power and therefore more electrical power. This is shown in the following graph, where the best range climb point (curve minima) is clearly visible. A key point to understand is that much of the extra power expended is being stored in a potential energy "battery". (PEB). There are some subtleties here that I have not modeled, since increasing climb angle transfers some of the airframe lift duties to the propeller thrust. I initially attempted to do the vector math, but quickly threw it in the too-hard-for-now basket, however I may revisit this. I do suspect the effects are small and potentially favourable.
Note how the best range speed during climb is faster than at straight-and-level. This is useful because it compensates for the climb angle reducing the horizontal speed and also helps offset the slower best L/D speed during the glide period. In this example, as we will see in the calculations below, the net horizontal speed (but not necessarily ground speed) is within a bull's roar of the best range speed when straight and level.
Glide Performance
For this modelling exercise, I have assumed, reasonably enough, that the glide mode will be at best glide ratio, which in this case is about 8.5:1. The second graph above shows the peak L/D to be coincident with the best range speed from the airframe drag power curve - no voodoo in that, really.
It's worth noting that optimal glide performance will only be achieved with a stopped propeller (or, ideally a folded propeller!). With a fixed propeller this will probably require the continuous use of the ESC brake function which may consume some power. Such are the differences between theory and practice.
Combining Climb and Glide Modes
For the horizontal speed calculations, we need to correct the airspeed to consider only the horizontal component to make valid comparisons with straight-and-level flight. Using the example of this study, this takes the best range climb speed of 18m/s at 15% gradient down to 17.46m/s horizontal speed and for the 8.5:1 glide, 15m/s to 14.89m/s. To compare directly with straight-and-level operation, if we take a climb-glide cycle, the average speed is 15.7m/s - fractionally slower than the 16m/s straight-and-level cruise. The electrical energy consumption is assumed to only occur during the climb phase, so factoring this over the whole climb-glide cycle (working in energy units/unit distance or J/m) gives an average energy consumption of 2.18J/km compared with 2.50J/km when travelling straight-and-level - a 12.7% decrease in energy consumption or a 14.6% increase in range.
This isn't actually the optimal case as calculated by this model. The 25% climb gradient yields a slightly better 15.2% increase in range, but since this model doesn't include the battery discharge efficiency, we can't be sure this really is optimal.
Note that it is now possible to trade this range gain off in the glide phase to recover or even increase the average speed over the straight-and-level condition. The choice is yours!
Where do the Efficiency Gains come from?
If we plot efficiency curves for propeller and motor together it becomes immediately clear that the vast majority of the efficiency gain is found in operating the electric motor closer to its peak efficiency. It must be emphasised that this is for this case only! Other airframe, motor and propeller combinations will move this around.
Indeed it opens the possibility of re-sizing the motor in lieu of climb-glide operation, however this will most likely limit the maximum power output of the motor, limiting climb rate and making take-off difficult without applying additional measures.
Summary
This was prepared for my own curiosity, but I hope that it can shed some light on the possibilities that relatively simple modelling can offer in helping to optimize a fixed-wing UAV. All this was done using MS Excel, including the regression analysis of the propeller model.
I am in no way claiming that climb-glide is the best solution for everyone's application. Merely that it is a way to improve the energy efficiency of the relatively simple hardware set that most people adopt for their UAV's.
very interesting!
I did exactly the same math using an optimization routine to choose the "best" climb and glide profile based on some requirements. I was actually trying to optimize aircraft weight (via battery weight reduction due to lowered power requirements) and flight time for a given mission (see here)
As you said, this can work well for non-optimized designs (like most of us are flying).
Very nice writeup! I'm eager to see if your results bear this out!
Nice concept. But I suspect that some of this gain you are seeing is from autopilots currently wasting energy maintain level flight at fixed airspeed.
The reason I say this is that any airplane will have a natural cruise speed where it will maintain a level flight without any trim. If you increase the airspeed above this natural state, the plane will start to climb and below it will decline.
But autopilots (that I know of) does not take this into account. The airspeed is set more or less arbitrarily, and if the plane naturally wants to climb the autopilot will force it level again by applying elevator and vice versa. And this process bleeds a lot or energy.
So in principle if you goal is energy efficient flight, it would be better to maintain altitude using the throttle to vary the airspeed instead of elevator and fixed airspeed. And the normal approach in full size to get around the motor peak efficiency problem, is to use variable pitch propellers so that the motor can run more efficiently at any airspeed.
Or to look at it from another perspective. Your climb-glide is essentially a very slow acting PID. And usually the goal of a PID is to find the optimal (stable) state for a system, since this is where the system is most efficient.
So while is have no doubt your approach works, I suspect as you mention in your conclusion that this it is more of a workaround to the real problem(s) causing the system to be inefficient in the first place. It would be very interesting to see how this would compare to a variable airspeed - level flight approach since this is also something that should be also possible to implement in existing systems.
I agree that this is covering up sub-optimized level flight solutions. I also wonder what is the best way to figure out the optimized flight speed of the air-frame and optimized weight vs. battery size. And finally, what prop size vs. pitch to meet these two needs since we can't change pitch of the prop.
If you can nail those two things you can probably get quite a bit more flight time out of our current systems.
Clearly for various real world air frames / motors / propellers / weights, this method can often provide an improvement in efficiency.
However, when looked at from the other end, an airframe / propeller / motor / weight combination designed to be fully optimized at a particular cruising speed, I would think the best efficiency would always be realized in level flight at that relative air speed.
Of course a case can be also made for soaring planes that can take advantage of various sources of lift and wind assist and this has already been done to a small degree with thermal seeking autopilots.
It seems as soon as you depart from the planes optimal flight conditions a climb and glide (or possibly just operate a reduced thrust while descending) method might potentially provide some efficiency gain.
In the current world of hobby UAS and just starting in commercial UAS, the reality is efficiency is very poorly understood as it relates to the highly unoptimized current airframes, many of which have considerable excess / unnecessary drag simply resulting from bad airframe design and numerous appendages sticking into the wind messing up the laminar flow.
The RC gliders are clearly better in this regard, but our normal UAS airframes are about as aerodynamic as a Volkswagen Bus.
And most of them were designed by people who thought a lot more about how "cool" they looked than knowledgeably about how they actually work.
Climb and glide may be useful, but true aeronautical design is probably a lot more significant right now.
OK, there is, in my opinion, a lot of unnecessary opining going on here, if I may say so, when the data is right there in my post!
Firstly, in response to John - this is entirely theoretical and no autopilots are conisdered here. If you fly straight and level you will be somewhere on the drag curve shown in the second graph. The autopilot will indeed change the trim speed of the airframe to maintain straight-and-level flight but this simply allows it to run up and down the drag curve at varying speed. Nevertheless, there is an optimal speed for maximum range and another for maximum endurance. Airframe trim does not change these speed points. It's all there in this graph!
Secondly, actual airframe efficency isn't really the point of this exercise. It can be bad or really good, but the drag characteristic versus speed will follow a very similar pattern. What is important is wing loading which plays an important role in determining the actual maximum range speed. Noteworthy is that weight isn't necessarily the enemy here because higher wing loadings travel optimally faster. If you can keep your form drag under control, this can be advantageous for several reasons - Reynolds numbers being the first in my head!
Thirdly, of course you can attempt to optimise for cruise condition. The problem with this is you will be unable to climb or travel any faster than this cruise point. More importantly you won't be able to launch without supplementary thrust, which isn't the point of this exercise, although it might be something to consider adopting. Why is this so? Well, because maximum motor efficiency occurs at the maximum motor terminal voltage, if you are already there at cruise you have no option to increase power, at least with fixed pitch propellers. If you have any voltage headroom at all to climb, it will probably be more thrust efficient than the straight-and-level cruise condition.
Oddly enough, full-size self-launching and sustainer-engined sailplanes (retracting engine) use fixed climb pitch propellors, so climb-glide is used to optimize range. Touring motorgliders (fixed engine) usually have variable pitch propellors (or fixed cruise pitch propellors), so level cruise results in optimal range.
Hi Andrew,
I disagree with a single point regarding optimizing for cruise.
With brushless motors and many kinds of motors, they are definitely not most optimized at maximum output.
Most of them have a peak optimal output point which is greatest efficiency versus much higher potential maximum power but at reduced efficiency.
Most Multicopter motors are actually most efficient around 30 percent power or so and this is equally true for many kinds of fixed wing motors as well.
And this is also true for many ESCs.
Therefore you might have optimal cruise at approximately 30% but still have plenty of excess power for climbing, taking off, etc.
So in fact, for many cases of optimizing for cruise, this is simply not an issue.
Generally motors optimized for efficiency will weigh more than motors optimized for maximum performance, but as you said, weight is not necessarily a bad thing, and in fact, when designing an airframe for optimized cruise, it is just one more factor to be taken into account.
I was not in any way disputing your well thought out results, merely trying to point out that in this point in time many if not most of the UAS airframe designs are so poorly constructed and even more poorly understood that applying this concept in the real world is not yet likely to have the significance that it will have under more favorable circumstances.
I do believe that even with a plane constructed for optimal level flight efficiency at specified cruise, that any time you depart from the optimum taking into account the method you have described can be used to improve efficiency to at least some degree.
On a more interesting note, using the methods you have presented here, it seems likely you could even build that in as a semi autonomous response in an autopilot.
Variable pitch propellers, can also, of course be used to increase efficiency under varying operating conditions, singly or in concert with your climb and glide concept.
Best regards,
Gary
Gary,
my point regarding motor efficiency was that if peak efficiency is at 30% of rated power, the only way you have to achieve this 30% is by reducing terminal voltage (or voltage * PWM ratio). If you plot efficiency curves you will find motors become more efficient as you increase voltage. If you are not at full voltage (or 100% PWM) in the cruise then you can increase the motor efficiency (and concurrently, total propulsion efficiency) simply by increasing the voltage and beginning to climb...
Where the head wind and tail wind factor in the graph? I usually take advantages of mother nature to get me extra climb-glide efficiency on my tiny plane. Its works and flew more than hours on tiny pack. Small prop and lighter prop helps to reduce drag because folding prop are heavy.
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