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I lack some 3 dimensional understanding.

What are these angles referenced to? For example, R, P, Y all start from 0, plane is flying horizontal, then it inclines its nose by 90 degrees, moving vertical like a rocket.
Now pitch is 90 degrees, but what now determines the roll?
While moving horizontal, roll may be the angle between the normal vector of the earth's surface and the airplane's up vector.
But when pitch = 90 degrees, what is roll?

Update: Can any (yaw, pitch, roll) triple set exactly describe the attitude/orientation of an aircraft?

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    $\begingroup$ Regarding your update: the question about sensors doesn't have too much to do with the rest of the question, you should probably ask a new one for that. Also have look at this question: What's the minimum number of sensors for a hobby GPS waypoint following UAV?. $\endgroup$ – Bianfable Jul 25 at 15:12
  • $\begingroup$ @Bianfable thanks for all your efforts. $\endgroup$ – muyustan Jul 25 at 15:38
  • $\begingroup$ @quietflyer Well, I am a total starter in the aviation area, so I probably could not express myself as I wanted. Let me try again, suppose I want to design a flight controller for an aircraft. I want to know that can I get meaningful roll,pitch,yaw values via an IMU(accelerometer+gyroscope) and determine exactly in which position is my aircraft is. I think I mix the motions and attitudes. Suppose uav takes off from the runway and all the angles are taken as 0 now. With the following order, pitch 90 deg, roll 180 deg, pitch -90 deg. All these movements resulted in a -180 deg yaw? $\endgroup$ – muyustan Jul 25 at 19:44
  • $\begingroup$ @quietflyer I am just too confused that should I use a tracker like that or just read the values of pitch,roll and yaw by the help of accelerometer (I have no idea how to get yaw by the way) and use them to relate a specific orientation. $\endgroup$ – muyustan Jul 25 at 19:45
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    $\begingroup$ Re, "my main aim is actually to be able to define the orientation for a flying vehicle" I would either use quaternions if I wanted to represent just the orientation, or I would use a matrix representation of a rigid coordinate transformation if I wanted to represent both orientation and position. Using angles to represent 3D orientations is messy, it has singularities (google "gimbal lock"), and it's too easy to make mistakes. $\endgroup$ – Solomon Slow Jul 26 at 22:04
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What are the exact meanings of roll, pitch and yaw?

It depends somewhat on whether you are speaking from a pilot's point of view or from an engineer's point of view. Your reference to yaw, pitch, and roll ANGLES is indicates you are looking at the situation from an engineering point of view. A pilot would tend to think more in terms of yaw, pitch, and roll as expressing RATES of rotation, or in some cases as expressing angular (rotational) changes from the aircraft's PREVIOUS POSITION rather than from a prescribed starting reference position. If the desire is to express the aircraft's 3-dimensional position in space relative to the plane of the ground and relative to north, a pilot would tend to speak of heading, pitch ATTITUDE, and roll ATTITUDE (or bank angle). But read on for more.

But in the pitch = 90 degrees case I cannot easily understand how can I think the roll.

When the pitch attitude is 90 degrees nose-up or nose-down, the yaw attitude (heading) and roll attitude (bank angle) become undefined, from the pilot's point of view.

(I've now found that this was also pointed out in this related answer which also has some other content that pertains to your question: What is the relation between roll angle and pitch angle? : )

This describes almost every possible attitude uniquely, unless the pitch angle is +/- 90 degrees. Then roll and yaw will become ambiguous.

Note that if you are banked (i.e. nonzero bank attitude or roll attitude, which ever you prefer to call it), then a motion that purely involves a nose-up pitch rotation and no yaw rotation or roll rotation will also increase the bank angle, and will also change the heading. Note also that a constant-banked climbing or descending turn (i.e. nose-up or nose-down pitch attitude) at a constant pitch attitude involves a roll rotation as well as pitch and yaw rotations. So, a pitch rotation is not exactly the same as a change in pitch attitude...and a lack of roll rotation does not always imply that the bank attitude (roll attitude) is not changing-- it's all a little complicated isn't it. Generally speaking, the rate of rotation about any given axis (pitch, yaw, or roll) is not the same as the rate of change of the pitch attitude, yaw attitude (heading), or roll attitude (bank angle), respectively. For example an aircraft pointing straight up can YAW through 180 degrees and end up pointing straight down-- it has changed pitch attitude with no pitch rotation.

Update to the question: I wonder whether any (yaw, pitch, roll) triple set exactly describes the attitude/orientation of an aircraft or not.

So could you please tell me that can any orientation of a flying object described with a unique y,p,r triplet?

In ALMOST all cases the answer is "yes". The exception, from the pilot's point of view, is when the nose is pointing straight up or straight down (i.e. the pitch attitude is plus or minus 90 degrees). In that case the yaw and roll attitude (heading and bank angle) become undefined and there is no easy way to distinguish between the aircraft's belly pointing north, south, east, or west, at least from the pilot's point of view. But read on for more on how an engineer would see the situation.

It has been noted that this can be resolved through the use of "quaternions". To learn more about that, you should probably ask another question.

Actually there is some need for refinement here. Taking off our "pilot's hat" and donning our engineer's "pocket protector"-- If we follow the CONVENTION given in this Wikipedia article on Euler angles https://en.wikipedia.org/wiki/Flight_dynamics_(fixed-wing_aircraft) , we note that if we START with an imaginary aircraft (this is a thought experiment, not an actual flight maneuver) in level flight with the nose pointing north, and THEN we rotate around the yaw axis by some prescribed yaw angle, FOLLOWED by a rotation around the pitch axis by some prescribed pitch angle, FOLLOWED by a rotation around the roll axis by some prescribed roll angle, we DO end up with the aircraft's position in space fully described. With this particular sequence of rotations, starting from wings-level flight pointing north, the angles that we've rotated about the yaw, pitch, and roll axes DO end up being exactly the same as the aircraft's heading, pitch attitude, and bank angle (roll attitude). Following this convention, the aircraft's three dimensional position in space CAN be fully described at all times with a yaw, pitch, roll "triplet"-- even including the directions that the canopy and the belly are facing when the aircraft is pointing straight up or straight down (pitch angles of 90 or -90 degrees). Note however that unless we arbitrarily REQUIRE the roll angle to be zero in the straight-up or straight-down cases, we notice that the straight-up and straight-down cases are not described by a UNIQUE set of yaw, pitch, and roll angles. For example, yaw, pitch, and roll angles of 90, 90, and 0 will give the same attitude in space as angles of 0, 90, and minus 90 -- in either case the aircraft is pointing straight up with its belly pointing east and its canopy pointing west. As another answer noted, the common thread is that in the nose straight-up or straight-down cases, the direction that the belly and canopy are pointing is tied to ( yaw angle minus roll angle) in the nose-straight-up case and ( roll angle minus yaw angle) in the nose-straight-down case. Therefore, any yaw-pitch-roll triplets that have a pitch angle of plus 90 degrees, and roll-minus-yaw equating to some given value, are describing the same orientation of the aircraft in space, including the direction that the canopy and belly are pointing.

For most pilots it is more intuitive to simply say that the heading and bank angle become undefined when the nose is pointing straight up or straight down, but the other method obviously has applications for defining the aircraft's attitude in space for engineering purposes.

Returning to a point already mentioned near the start of this answer-- sometimes with the words "pitch", "roll", and "yaw" it is not clear whether we are talking about a rotation rate, an attitude in space, or something else. For general pilot-speak, it seems a good rule of thumb to assume that these words refer either to rotation rates or to angles of rotation from the aircraft's previous position, unless it is very clear from context, or from additional attached words (e.g. "pitch ATTITUDE", or "pitched up TO 20 degrees") that they are being used to mean something else such as the aircraft's attitude with respect to the outside world. "Yaw" is the most squirrelly of the three-- we might sometimes say something like "we are going west but the nose is yawed 10 degrees right" to express a crab angle, or a sideslip angle-- i.e. the angle between the direction that the nose is pointing and something else, such as the direction of the ground track or the direction of the relative wind. (It wouldn't be clear which is meant from the above sentence, without further context.) It wouldn't be normal to say "the nose is yawed 40 degrees" to express a heading of 40 degrees, but in some particular contexts-- for example if we are working with "Euler angles" as described above-- a "yaw attitude" or "yaw angle" of 40 degrees could indeed be taken to mean a heading of 40 degrees. In many cases it is much clearer to just say "heading" rather than "yaw", "yaw attitude", or "yaw angle", if that's what we mean. Likewise for "slip angle" and "crab angle"-- both of these phrases are much clearer than any phrase involving the word "yaw".

But again, if we are using the system of Euler angles as described above for engineering purposes, then we might reasonably assume that "yaw", "pitch", and "roll" may be used to denote the corresponding Euler angles with no further elaboration. This wouldn't be normal pilot-speak.

As noted in another answer, it appears that "Tait-Brian angles" is a more accurate, or at least more specific, name for what I have called "Euler angles" in this answer. See the wikipedia link given above for more. Also see other answers for many well-presented, valid points.

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  • $\begingroup$ Thanks for the answer, my main aim is actually to be able to define the orientation for a flying vehicle, so could you please tell me that can any orientation of a flying object described with a unique y,p,r triplet? $\endgroup$ – muyustan Jul 25 at 15:46
  • $\begingroup$ Don't know why downwotes: perhaps TL;DR. You are generally correct, except that (90, 90, and 0) <=> (0, 90, minus 90). $\endgroup$ – Zeus Jul 26 at 2:15
  • $\begingroup$ My answer had some issues before recent edits. Also thanks for that last pointer; fixing. $\endgroup$ – quiet flyer Jul 26 at 2:17
  • $\begingroup$ thanks for your efforts, I can say that I am much more clear than the time when I asked the question. But still it is a very confusing situation. $\endgroup$ – muyustan Jul 26 at 6:41
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The key thing missing in your interpretation is that the order is important. Applying the same roll, pitch and yaw angles in a different order will result in a different orientation (mathematically this is because rotation matrices do not commute).

The correct order is:

  • Yaw (for an airplane this is typically called the heading)
  • Pitch
  • Roll

For any given attitude, there is a unique combination of yaw, pitch and roll angles that will (if applied in this order) give you the correct attitude.

These angles are only meaningful in a given coordinate system. We typically use a system where all angles zero means you are pointing North and wings and body are parallel to the ground. Based on this starting point you would apply the given angles.

Roll pitch and yaw (source: Wikimedia from this Wikipedia article: Aircraft principal axes)

See also the answer to this question: What is the relation between roll angle and pitch angle?


Let me expand a bit on your example:

We start will all angles equal to zero (level, pointing North) and pitch up 90 degrees such that our nose points straight into the sky. Yaw and roll have not changed. Now we start rolling. The roll angle would now tell us in which direction the belly of the aircraft is pointing (0 degrees means belly pointing North, 90 degrees means belly pointing West and right wing pointing North, and so on).

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    $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – Federico Jul 26 at 5:43
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    $\begingroup$ "Yaw (for an airplane this is typically called the heading)" - As far I understand, yaw is not the heading, but the angle between the heading and the front-to-back axis of the aeroplane. If the plane is moving in the same direction it's pointing, then the yaw is zero. If it's sideslipping a little, that's yaw. $\endgroup$ – Reversed Engineer Jul 26 at 7:08
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    $\begingroup$ @ReversedEngineer No, this is called the sideslip angle, it has nothing to do with yaw! $\endgroup$ – Bianfable Jul 26 at 7:10
  • $\begingroup$ Ah, thank you for clearing it up in my mind! $\endgroup$ – Reversed Engineer Jul 26 at 7:22
  • $\begingroup$ @ReversedEngineer in the context of the simulations used by aircraft manufacturers, yaw is the angle of the air-frame relative to the heading, the term 'sideslip angle' wasn't on I've heard used. For all i know, this usage may have been limited to within the operational analysis division one company (BAE). $\endgroup$ – Pete Kirkham Jul 26 at 13:32
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Roll, pitch, and yaw have two different meanings. They can refer to either

  • Euler angles, which describe the attitude (orientation) of an aircraft, or
  • ways that an aircraft can rotate.

Euler angles

The attitude of an aircraft can be described by three angles: heading, pitch, and bank angle (sometimes called roll angle).

The heading $\psi$ (psi, sometimes called the yaw angle) of an aircraft is, essentially, the horizontal direction that the nose is pointing. It's expressed as a number from 0 to 360, where 90 is due east, 180 is due south, 270 is due west, and 360 is due north.

You can find the heading by pointing your finger in the direction the nose is pointing, then moving your finger straight up or down until it meets the horizon. The heading is the direction your finger is pointing.

The pitch $\theta$ (theta) of an aircraft, meanwhile, is essentially the vertical direction that the nose is pointing. The "lowest" possible pitch is 90 degrees down and the "highest" possible pitch is 90 degrees up. If the nose is pointed directly at the horizon, the pitch is 0.

The pitch is simply the angle between the horizon plane and the direction the nose is pointing.

Finally, the bank angle $\phi$ (phi, sometimes called the roll angle) is slightly more complicated. The range of possible bank angle values is a full circle, going from 0, to 90 degrees right, to 180 degrees, to 90 degrees left, and back to 0. (There's no difference between "180 degrees left" and "180 degrees right.")

The bank angle is the amount that the aircraft would have to roll in order to bring the wings level, with the top side of the aircraft facing up rather than down.

The singularities

The pitch is always defined. However, if the nose is pointing straight up or down, then the aircraft no longer has a heading or bank angle as described above. There's no heading because the nose isn't pointing horizontally at all; and there's no bank angle because it's not possible to roll the aircraft such that the top side is facing up.

That said, it would make sense to say that if the nose is pointing straight up or down, then the bank angle is always 0, and the heading is

  • the direction that the bottom of the aircraft is facing, if the nose is pointed up, or
  • the direction that the top of the aircraft is facing, if the nose is pointed down.

Rotations

The words roll, pitch and yaw refer to rotations about the aircraft's various axes. They refer to rotations from the aircraft's point of view (or the pilot's point of view).

Roll is a rotation about the longitudinal axis (the direction the nose is pointing). It's controlled using the ailerons. Pushing the control stick to the right, or turning the yoke to the right, causes the aircraft to roll right.

Pitch is a rotation about the lateral axis (a line parallel to the wings). It's controlled using the elevator. Pulling back on the stick or the yoke causes the aircraft to pitch up. Note that since these are rotations from the aircraft's point of view, if you're flying upside-down and you "pitch up," you're pointing the nose towards the ground, and if you "pitch down," you're pointing it towards the sky.

Yaw is a rotation about the vertical axis (a line perpendicular to the other two axes). Despite the name, the vertical axis isn't always oriented vertically. Yaw is controlled using the rudder. Stepping on the right rudder pedal causes the aircraft to yaw right.

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    $\begingroup$ +1 in general, but "heading" as a name is only used by pilots. Real "aviation experts" (i.e. engineers :) do use "yaw" (yaw angle) all the time. And its range is typically ±π. (The reference is often arbitrary; in most engineering problems it doesn't matter where "North" is). / The body vertical axis, to avoid confusion you mentioned in the last paragraph, is often called the normal axis. $\endgroup$ – Zeus Jul 26 at 1:50
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    $\begingroup$ This is the correct answer: two different axes frames, one relative to aircraft and one relative to earth. Not to be confused! $\endgroup$ – Koyovis Jul 26 at 2:13
  • $\begingroup$ Thanks for your answer, it helped. $\endgroup$ – muyustan Jul 26 at 7:18
  • $\begingroup$ @Zeus Huh, I didn't know that. All right, I've edited my answer to say it is called the yaw angle after all. $\endgroup$ – Terran Swett Jul 26 at 14:27
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The attitude angles are explained in this wikipedia page, but it seems your confusion stems from the fact that they can present a singularity.

Each set of unique Euler angles, outside of the gimbal lock state, represent a unique attitude. In the singularity state, however:

The angles α, β and γ are uniquely determined except for the singular case that the xy and the XY planes are identical, i.e. when the z axis and the Z axis have the same or opposite directions. Indeed, if the z axis and the Z axis are the same, β = 0 and only (α + γ) is uniquely defined (not the individual values), and, similarly, if the z axis and the Z axis are opposite, β = π and only (α − γ) is uniquely defined (not the individual values). These ambiguities are known as gimbal lock in applications.

To avoid this singularity, you should use alternative representation methods, like quaternions. These are 4-dimensional vectors that nonetheless have only 3 degrees of freedom (because their modulus is normalized).

One of their main advantages is that they do not have discontinuities in their representation of the 3 dimensional attitude space, unlike the 3 dimensional representations.

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One very important aspect is that roll, pitch, and yaw, relative to the aircraft (and the pilot that controls it) never change. As you progress as a pilot, using sky/ground as your reference is usually where we all start, indeed, as land walking folks, this is how we see our world.

Practice flying model aircraft can help eliminate this mode of thought and replace it with simply using the controls to fly the plane where you want to.

"Inverted" flight becomes stick forward to the sky to keep flying. Rolling through 360 degrees, again, roll to the sky. Yaw (turn) to your desired compass heading or visual reference.

This is truly flying in 3 dimensions. Roll, Pitch, and Yaw are With Respect To the Aircraft.

Gyroscopes are used to keep track of the original sky/ground reference. Newer models have a safety switch that will "right" the plane in less than a second. These may be worth looking into.

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    $\begingroup$ hey, thanks for the answer , but your language came little bit confusing to me(probably because of me not you, I am not a native speaker). Does your 2nd paragrapgh implie "I should buy a model plane and control it to get used to these values" ? $\endgroup$ – muyustan Jul 25 at 18:18
  • $\begingroup$ Yes, absolutely. For a pilot. Mathematically, it may be more breaking the position into R, P, Y vectors WRT earth/sky. Interesting logic as a gyro displays position relative to where its original position was (starting out level). Visual takes real time info that (hopefully) the pilot interprets (correctly). $\endgroup$ – Robert DiGiovanni Jul 25 at 20:04
  • $\begingroup$ This is a truly interesting point, it relates to the mental picture of yourself vs world. A lot of research has been done in this area. However, I'm afraid it is not directly related to the original question and will only confuse the OP. $\endgroup$ – Zeus Jul 26 at 2:02
  • $\begingroup$ The statement that pitch/roll/yaw is related to aircraft axes is correct. $\endgroup$ – Koyovis Jul 26 at 2:14
  • $\begingroup$ @Zeus. Training with models gave me enormous confidence when steep bank angles were done in full scale training. I remember instructor sharply rocking the plane back and forth and asking me "you mean, you aren't afraid of this?". $\endgroup$ – Robert DiGiovanni Jul 26 at 7:10
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Euler angles are relative to earth axes. Pitch, roll and yaw are relative to aircraft axes. Pitch means: nose up/down. Roll means: wingtip up/down. Yaw means: nose left/right. All from the pilots viewpoint. It is that simple.

As confirmed by the Wikipedia article on Euler angles:, which first mentions proper Euler angles, then Tait-Brian angles pitch/roll/yaw in the moving aircraft reference frame.

In Full Flight Simulators, the code computes:

  • forces and moments on the airframe in response to flight control deflection.
  • Linear thrust, drag, speed, vertical speed - in aircraft axes.
  • Angular inertia response to moments. In aircraft axes, pitch/roll/yaw, no matter what the position of the nose is.

Only then is the world position and orientation of the aircraft integrated and computed, in 6 Degrees of Freedom. This cannot be done with Euler angles, due to the nose-up singularity, and has to be done with quaternions. More on these here and here.

The term "roll" is normally used for an intrinsic motion: axes frame attached to the moving body. Roll angle starts at any convenient zero angle. The choice is pretty clear for fixed wing craft and helicopters, not so clear for a rocket. Whichever zero was chosen, after rolling 360 degrees it comes back to zero again, regardless of where the nose is pointing now.

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  • $\begingroup$ Then pitch roll and yaw are meaningful only during a short amount of time while doing the motion. Because after the motion is ended, then according to your saying, R,P,Y are aleays 0 relative to aircraft axes. Did I get your point right? Thanks anyway $\endgroup$ – muyustan Jul 26 at 16:22
  • $\begingroup$ No, pitch acceleration is integrated into pitch velocity is integrated into pitch position. Zero position in pitch/roll/yaw can be arbitrarily chosen. For zero pitch and roll the earth gravity plane is chosen. Zero yaw does not have a very convenient consistent zero, north can be chosen, or position of the nose at the gate upon departure. $\endgroup$ – Koyovis Jul 26 at 16:27
  • $\begingroup$ then by "relative to aircraft axes" you meant a fixed coordinate system not changing always during the flight. Then how does it differ from the earth axes? $\endgroup$ – muyustan Jul 26 at 17:14
  • $\begingroup$ "Euler angles are relative to earth axes."-- is this really always true? For example, at en.m.wikipedia.org/wiki/Euler_angles, we read "Intrinsic rotations are elemental rotations that occur around the axes of a coordinate system XYZ, Therefore, they change their orientation after each elemental rotation." We also read "Euler angles can be defined by intrinsic rotations. The rotated frame XYZ may be imagined to be initially aligned with xyz, before under going the three elemental rotations represented by Euler angles. $\endgroup$ – quiet flyer Jul 26 at 19:02
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    $\begingroup$ @muyustan That is because you are using the term roll for an extrinsic motion, while it is meant for an intrinsic motion. Roll angle starts at any convenient zero angle - the choice of which zero position is not so clear for a rocket, but it still rolls. Whichever zero was chosen, after rolling 360 degrees it comes back to this same zero. $\endgroup$ – Koyovis Jul 26 at 22:43

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