Aircraft trajectory from point a to point b

[badescuga]

Hello!
I've started working on a project that involves some light weight aircrafts. I am not a physicist(not at all) at my origin, I'm more the computer science and mathematics kind of fellow . I need to determine an aircraft trajectory form a point ( longitude and latitude ) to another taking of course in consideration the forces applied (like wind,Centrifugal force etc.) . Can anyone point to me from where to start, a short introduction or even a final answer? Any information is well recieved!

Kind regards,
Alexandru Badescu

 

[Filip Larsen]

Welcome to PF!

Even though the basics are fairly simple, the topic has potentially many complicated details that you may or may not be required to know for your project, so it is a bit difficult to provide you with help without knowing more.

If you are completely new to both the piloting and engineering aspect of this topic, you may want to search your references for keywords like "aircraft navigation dead reckoning great circle" for a pilots perspective on the matter, perhaps spiced up with "dynamics" if you also want information that would be of interest to engineers.

I will recommend that you try dig out some of that basic information, try to apply it to your project, and then return here with specific questions if this gives you problems.

 

[Bob S]

Look up haversines at

http://en.wikipedia.org/wiki/Haversine_formula

Use havesines if you want to follow Great Circle routes not including Earth's rotation. I don't know about programs that give Great Circle routes between specific lats and longs however. I don't know of anything that takes into account the Earth's rotation, however. For example, the Great Circle route from London to Singapore is different than the route from Singapore to London.

Bob S

 

[Cleonis]

I need to determine an aircraft trajectory form a point ( longitude and latitude ) to another taking of course in consideration the forces applied (like wind,Centrifugal force etc.) .

I gather you want to take the Earth's rotation into account.As a starting point for thinking about that, consider the case of an airship. (Airships are also referred to as 'Zeppelins'.)

Take the case of a neutrally buoyant airship, no wind, and the airship floats above a fixed point on the ground.
Notice that in that case you don't have to consider centrifugal effect. The airship stays above the same latitude, just as the waters of the oceans all remain at their latitude.
The solid Earth has an equatorial bulge. If the solid Earth would not have an equatorial bulge then the waters of the oceans would gather around the equator. The Earth's equatorial bulge is precisely enough to keep all water at its own latitude. This is not by coincidence, of course; the Earth's rotation has an effect on the solid Earth, the solid Earth has an equilbrium shape that matches its rotation rate.

With this geophysical background established I return to the case of an airship. Let's say you're at a latitude of 60 degrees,

- When you are flying in eastward direction then you are circumnavigating the Earth's axis faster than the Earth's own rotation: you will tend to swing wide.
- When you are flying in westward direction then you are circumnavigating the Earth's axis slower than the Earth's own rotation: the airship will tend to move away from the equator, closer to the nearest pole.

My point is: aircraft motion is not ballistics.
A projectile is not supported; a projectile is subject only to gravity. Airships and aircrafts, however, are supported. Unlike a projectile an aircraft creates lift to keep itself in the air. Therefore formulas that work for projectiles won't work for airplanes.

I don't know whether in aircraft flight planning the above described effects are taken into account at all. Possibly these effects are swamped by unpredictable wind effects.

If you do want to take the above described effects into account then check out the code of the Java applet http://www.cleonis.nl/physics/ejs/inertial_oscillation_simulation.php" a Java environment for creating simulations.

As I said, I don't know whether the flight planning software in, say, jet liners take the Earth's rotation into account. Jet liners follow designated corridors. All kinds of factors may cause the aircraft to drift away from the planned direction; you have to adjust all the time. Usually the flight crew will switch on the autopilot.
So I don't know about calculating the flight path in advance; maybe they just wing it.

 

[rcgldr]

The effect on a flight from Earth rotation is not significant compared to prevailing winds. Although a light aircraft won't be flying in a jet stream, there are still moderate winds to take into account. If you can find stronger tail wind or weaker head wind by going a bit off course or flying at a diferent altitude, it ends up being a shorter (time wise) trip.

 

[Filip Larsen]

I don't know whether in aircraft flight planning the above described effects are taken into account at all. Possibly these effects are swamped by unpredictable wind effects.

The Coriolis force is indeed insignificant for normal aircraft operations. Even for a trans-sonic aircraft the effect accounts for a maximum around 5 milli-g, with aerodynamic, gravitational and propulsive forces in the 1-10 g area. The effect is completely ignored both during flight planning and actual flying.

Jet liners follow designated corridors. All kinds of factors may cause the aircraft to drift away from the planned direction; you have to adjust all the time. Usually the flight crew will switch on the autopilot.
So I don't know about calculating the flight path in advance; maybe they just wing it.

Jetliner pilots plan their flight very accurately referring to routes passing through well-known navigation waypoints. Such a flight plan is uploaded or entered into the planes Flight Management System (FMS) and the autopilot/director is then able to follow the planned route laterally using input from inertial systems (INS/IRS), satellites (GPS) and radio-navigation (VOR, NDB) and vertically using pressure and radar altimeters. In many controlled airspaces aircrafts are required to be able to stay within fairly narrow distance from the assigned route and altitude.

 

[Cleonis]

Jetliner pilots plan their flight very accurately [...]

Yes, the flight is plotted in detail. What is unknown, of course, is precise direction and velocity of wind encountered along the way.

An unsophisticated autopilot would keep the nose of the aircraft pointing to the destination, but that would give a systematic error.

I surmise that the autopilot must be sophisticated in the following way: to keep the airplane very close to the plotted route the autopilot must point the nose away from the destination, just enough to counteract the effect of a sideways wind.

 

[Filip Larsen]

I surmise that the autopilot must be sophisticated in the following way: to keep the airplane very close to the plotted route the autopilot must point the nose away from the destination, just enough to counteract the effect of a sideways wind.

Yes, all autopilots found in civilian aircrafts do a closed-loop correction in order to follow inputs like gyro compass heading, VOR radials, ILS localizers, or general ground tracks in case of a full GPS/FMS. However, none of these tracking modes correct for wind in an open-loop maner (like a pilot does when he calculates a correction angle for a known wind), instead the wind can derived by computer or pilot from knowledge of air/ground speed, track and heading. Sophisticated autopilots that have good access to air data may utilize some form of open-loop correction for changes or gusts in wind, especially during the landing phase where track error margins may be very low when performing automatic landing.

 

[badescuga]

I gather you want to take the Earth's rotation into account.

Thanks for the great reply!

I should have given more details : My aircraft in theory is going to be a patrol aircraft and the surface in which it's going to navigate won't be larger then 10 km x 10 km .. so i think that the Earth rotation isn't quite an issue ( if it is, please correct me, i may be wrong ) .

I will calculate every one second the position of the aircraft and generate a trajectory taking in account the current new position. The aircraft is fairly light ( 1,5 - 3 kg for now ) , must i take the centrifugal force in consideration if i recalculate anyway my trajectory every 1 second?

The thing i am most concerned about is the wind, especially for my lightweight aircraft!

 

[Filip Larsen]

Thanks for the great reply!

I should have given more details : My aircraft in theory is going to be a patrol aircraft and the surface in which it's going to navigate won't be larger then 10 km x 10 km .. so i think that the Earth rotation isn't quite an issue ( if it is, please correct me, i may be wrong ) .

I will calculate every one second the position of the aircraft and generate a trajectory taking in account the current new position. The aircraft is fairly light ( 1,5 - 3 kg for now ) , must i take the centrifugal force in consideration if i recalculate anyway my trajectory every 1 second?

The thing i am most concerned about is the wind, especially for my lightweight aircraft!

You are correct that wind will be the most dominant "unknown" to affect your aircraft, especially for a slow moving RC craft (not that an RC craft has to be slow). As mentioned earlier, the effects of a rotating Earth can safely be ignored.

That said it is not clear to me what your setup and goals are. Do you want to predict the ground track over the next T seconds each second based on actual position and velocity measured aboard the aircraft? Or do you want to plan the track second by second before you launch the aircraft? Is the calculation going to be used to control the aircraft, present to a human operator during flight, or just as some kind of data logging or mapping thing?

In case you are aiming to predict the track before launch and then hope you can set your aircraft to follow this track autonomously (i.e. using open-loop control) then you are going to be disappointed. If you want any kind of automatic control of your aircraft to follow a track you have to have closed-loop control and if it also has to be autonomous it will most likely require a significant sophisticated device, like a small computer interfaced with GPS, accelerometers and actuators. On the other hand, I would not be surprised if someone already has made an iPhone or Android application that could make an RC aircraft capable of autonomous flight with a minimum of hardware :)

 

[Cleonis]

[...] must i take the centrifugal force in consideration if i recalculate anyway my trajectory every 1 second?

Well, the information/reasoning that allows to you decide whether to take some centrifugal force in consideration is in post #4 of this thread. So I guess I have to refer you back to post #4.

 

[badescuga]

You are correct that wind will be the most dominant "unknown" to affect your aircraft, especially for a slow moving RC craft (not that an RC craft has to be slow). As mentioned earlier, the effects of a rotating Earth can safely be ignored.

That said it is not clear to me what your setup and goals are. Do you want to predict the ground track over the next T seconds each second based on actual position and velocity measured aboard the aircraft? Or do you want to plan the track second by second before you launch the aircraft? Is the calculation going to be used to control the aircraft, present to a human operator during flight, or just as some kind of data logging or mapping thing?

In case you are aiming to predict the track before launch and then hope you can set your aircraft to follow this track autonomously (i.e. using open-loop control) then you are going to be disappointed. If you want any kind of automatic control of your aircraft to follow a track you have to have closed-loop control and if it also has to be autonomous it will most likely require a significant sophisticated device, like a small computer interfaced with GPS, accelerometers and actuators. On the other hand, I would not be surprised if someone already has made an iPhone or Android application that could make an RC aircraft capable of autonomous flight with a minimum of hardware :)

My goal is to have a software in which i incorporate a map that shows the current position of the airplane and constantly reloads the current position . when i click on a point on the map, i want to calculate the trajectory to that point and send to the aircraft new data ( rotate to the left 15 degrees etc. ) . I will recalculate the trajectory every second and send the airplane the new data.

 

[Filip Larsen]

Well, the information/reasoning that allows to you decide whether to take some centrifugal force in consideration is in post #4 of this thread. So I guess I have to refer you back to post #4.

I would like to emphasize once again, that the rotation of Earth has no practical significance for the dynamical behavior of aircrafts and is almost alway ignored.

Anyone is of course free to include any such effect in any analysis they may be making, but it corresponds more or less to include the gravity of the moon in the dynamical description of a roller coaster. If you are designing such a roller coaster, you for sure have more pressing modelling needs than making sure you get the moons gravity right.

 

[Filip Larsen]

My goal is to have a software in which i incorporate a map that shows the current position of the airplane and constantly reloads the current position . when i click on a point on the map, i want to calculate the trajectory to that point and send to the aircraft new data ( rotate to the left 15 degrees etc. ) . I will recalculate the trajectory every second and send the airplane the new data.

What kind of sensors, actuators and controllers do you have on the aircraft? Is is already capable of autonomous flight or is it only remote controlled?

Calculating a course to track from one point to another on a map is very simple, but getting the aircraft to follow that track is potentially rather complicated and depend a lot on what equipment you have. Typically you would have a controller on the aircraft controlling ailerons and elevator (and perhaps power) with several control laws governing level flight, turns and heading and you can then build a tracking capability on top of that, but it is not a trivial matter to design such a tracking computer that behaves well.

I still say that the chance of success probably would increase a lot if you can find an existing RC aircraft controller capable of autonomous flight.

 

[badescuga]

What kind of sensors, actuators and controllers do you have on the aircraft? Is is already capable of autonomous flight or is it only remote controlled?

Calculating a course to track from one point to another on a map is very simple, but getting the aircraft to follow that track is potentially rather complicated and depend a lot on what equipment you have. Typically you would have a controller on the aircraft controlling ailerons and elevator (and perhaps power) with several control laws governing level flight, turns and heading and you can then build a tracking capability on top of that, but it is not a trivial matter to design such a tracking computer that behaves well.

I still say that the chance of success probably would increase a lot if you can find an existing RC aircraft controller capable of autonomous flight.

The sensor included is GPS for now but I'm sure that there are going to be more installed , any suggestions ? The plane can perform autonomous flight given a trajectory on the ground but the user is unable to comunicate with the plane while it's in flight . It has a micro-controller that helps it fly. Due to the fact that the trajectory isn't necessarily optimal, if i continuously calculate the trajectory, the only thing that can modify abruptly my flight is the wind and i am worried that due to the light mass, the airplane could in fact be deviated quite a lot.

 

[rcgldr]

It appears that what you're trying to make is a UAV (unmanned aircraft vehicle). There's been some issue with the FAA regarding these, but not sure if it's just a clasification (stating they aren't remote control models), or additional regulations. A UAV managed to cross the Atlantic ocean, but a radio control pilot took over in order to land the aircraft. If I remember correctly, it took several attempts (so lost models) before one of them completed the crossing and landed.

There are already systems for radio control models that can restore level flight if a model is disturbed (or bad pilot inputs). Some of these were visual, but now I think they rely on gyros.

A typical UAV will use feedback from GPS to yaw the model to compensate for any crosswinds. There are universities that do research with UAV's, but I'm not sure if there are commercially available packages for models.

 

[Filip Larsen]

The sensor included is GPS for now but I'm sure that there are going to be more installed , any suggestions ? The plane can perform autonomous flight given a trajectory on the ground but the user is unable to comunicate with the plane while it's in flight . It has a micro-controller that helps it fly. Due to the fact that the trajectory isn't necessarily optimal, if i continuously calculate the trajectory, the only thing that can modify abruptly my flight is the wind and i am worried that due to the light mass, the airplane could in fact be deviated quite a lot.

If your aircraft already is able to perform autonomous straight and level flight, and the controller for this can be commanded in bank angle or heading, and you have a GPS provide you with position and velocity, then in principle it should be possible to make a "guidance" module that enables the aircraft to follow a track. However, it is quite likely a complicated exercise to get all the details right, especially if you try without having any control engineering knowledge to help you through the process, as both analog and digital control systems have a fair deal of "gotchas" to overcome.

Regarding the "guidance law" you may be able to do something like the following, which mimics how a "big" guidance computer operates: The computer use the notion of a leg consisting of the track from one waypoint to the next. These waypoints usually comes from a route (like your patrol route), but may also come from current position when you want to make initial intercept of the route. Knowing the leg track and your current position and velocity, the computer can then, based on one or more guidance laws, calculate the desired velocity vector, compare this with the actual velocity and provide the control computer with a guidance signal (turn/bank left, fly straigth, etc).

The guidance laws are a prioritized set of conditions that when triggered give rise to a particular relationship between current position and velocity and the desired velocity. For instance, far from the leg the relationship may be adjusted to intercept the leg in minimum time, but once we get close to the leg the desired velocity shift towards the end waypoint and avoid (dampen out) any tendency to oscillate back and forth over the track line. Near the end waypoint you may want to guide velocity towards an upcoming turn. Depending on your requirements you may be needing several more such guidance laws, and you probably need a simulation/test environment so you can fine-tune any parameters associated with these.

In all the above, the wind does not appear explicitly. If your aircraft fly with known or constant speed and is going to fly high enough that mechanical wind turbulence from the surface is only expected to be minor, it may make sense to try build and maintain statistical (e.g. Kalman filtered) state concerning wind speed and direction based on your actual track. Having access to accurate wind data will allow for more accurate guidance laws, e.g. by correcting the desired velocity for wind effects.

All this said, I must say it is difficult to provide you with help that is more specific than the above, as the more specific it is the more likely it is not to be applicable to your situation. It may also be that there are other (less classic) approaches that will be more suitable for you to achieve your goal. For instance, studies in insectile robotics show that you can get quite sophisticated (but perhaps not very flexible) behavior from a so-called subsumption architecture and its not unthinkable that you could get something usable by following that road.

 

[badescuga]

If your aircraft already is able to perform autonomous straight and level flight, and the controller for this can be commanded in bank angle or heading, and you have a GPS provide you with position and velocity, then in principle it should be possible to make a "guidance" module that enables the aircraft to follow a track. However, it is quite likely a complicated exercise to get all the details right, especially if you try without having any control engineering knowledge to help you through the process, as both analog and digital control systems have a fair deal of "gotchas" to overcome.

Regarding the "guidance law" you may be able to do something like the following, which mimics how a "big" guidance computer operates: The computer use the notion of a leg consisting of the track from one waypoint to the next. These waypoints usually comes from a route (like your patrol route), but may also come from current position when you want to make initial intercept of the route. Knowing the leg track and your current position and velocity, the computer can then, based on one or more guidance laws, calculate the desired velocity vector, compare this with the actual velocity and provide the control computer with a guidance signal (turn/bank left, fly straigth, etc).

The guidance laws are a prioritized set of conditions that when triggered give rise to a particular relationship between current position and velocity and the desired velocity. For instance, far from the leg the relationship may be adjusted to intercept the leg in minimum time, but once we get close to the leg the desired velocity shift towards the end waypoint and avoid (dampen out) any tendency to oscillate back and forth over the track line. Near the end waypoint you may want to guide velocity towards an upcoming turn. Depending on your requirements you may be needing several more such guidance laws, and you probably need a simulation/test environment so you can fine-tune any parameters associated with these.

In all the above, the wind does not appear explicitly. If your aircraft fly with known or constant speed and is going to fly high enough that mechanical wind turbulence from the surface is only expected to be minor, it may make sense to try build and maintain statistical (e.g. Kalman filtered) state concerning wind speed and direction based on your actual track. Having access to accurate wind data will allow for more accurate guidance laws, e.g. by correcting the desired velocity for wind effects.

All this said, I must say it is difficult to provide you with help that is more specific than the above, as the more specific it is the more likely it is not to be applicable to your situation. It may also be that there are other (less classic) approaches that will be more suitable for you to achieve your goal. For instance, studies in insectile robotics show that you can get quite sophisticated (but perhaps not very flexible) behavior from a so-called subsumption architecture and its not unthinkable that you could get something usable by following that road.

i'll start to work on something and see where things start to get messy and i'll come back with more question. You have been most helpful! thanks for your time!

Regards,
Alexandru Badescu

 

[Buzzworks]

The sensor included is GPS for now but I'm sure that there are going to be more installed , any suggestions ?

I would suggest you visit more appropriate forums than PF. There are forums out there that is dedicated to RC hobbies. I've heard some from the RC, especially on RC helicopters that uses GPS to hover at a predetermined location. The equipment is typically used for RC helicopters/planes with onboard video equipment, usually for videography/photography purposes.

The only thing I could suggest you for aircraft to be less affected by wind is to increase speed and/or increase wing loading - higher weight per wing/lifting surface area - simply by using smaller wings. However, the trade off for using higher wing loading is reduced loitering efficiency or endurance. Flying wings with electronic stabilization tend to give both speed and loitering efficiency (that's why maybe modern UAV's typically feature flying wing designs).