In this article we will look at the basic principles of landing on large jet aircraft as they apply to our environment. Although the Tu-154 was chosen as the basis for consideration, it should be taken into account that other types of aircraft generally use similar piloting principles. The information was taken based on real equipment, and we will tempt fate for now in MSFS98-2002, Microsoft has such a computer simulator, you may have even heard...

Aircraft landing configuration

Aircraft configuration- a combination of provisions for the mechanization of the wing, landing gear, parts and assemblies of the aircraft, which determine its aerodynamic qualities.

On a transport aircraft, even before entering the glide path, the wing mechanization and landing gear must be extended and the stabilizer must be repositioned. In addition, by decision of the aircraft commander, the crew can turn on the autopilot and/or autothrottle for an automatic approach.

Wing mechanization

Wing mechanization- a set of devices on the wing designed to regulate its load-bearing capacity and improve stability and controllability characteristics. The wing mechanization includes flaps, slats, flaps (interceptors), active boundary layer control systems (for example, its blowing off with air taken from the engines), etc.

Flaps

In general, flaps and slats are designed to increase the load-bearing capacity of the wing during takeoff and landing conditions.

Aerodynamically, this is expressed as follows:

1. flaps increase the wing area, which results in increased lift.
2. flaps increase the curvature of the wing profile, which leads to a more intense downward deflection of the air flow, which also increases lift.
3. flaps increase the aerodynamic drag of the aircraft, and therefore cause a decrease in speed.

Increasing the wing's lift allows the speed to be reduced to a lower limit. For example, if with a mass of 80 tons stall speed Tu-154B without flaps is 270 km/h, then after the flaps are extended completely (by 48 degrees) it decreases to 210 km/h. If you reduce the speed below this limit, the aircraft will reach dangerous angles of attack, causing stall shaking (buffeting)(especially with the flaps retracted) and, in the end, it will happen spinning.

A wing equipped with flaps and slats that form profiled slots in it is called slotted. Flaps can also consist of several panels and have slots. For example, on the Tu-154M they use double-slit, and on the Tu-154B three-slit flaps (pictured Tu-154B-2). On a slotted wing, air from the area of ​​high pressure under the wing flows at high speed through the slots onto the upper surface of the wing, which leads to a decrease in pressure on the upper surface. With a smaller pressure difference, the flow around the wing is smoother and the tendency to form a stall is reduced.

Angle of Attack (AoA)

Basic concept of aerodynamics. The angle of attack of the wing profile is the angle at which the profile is blown by the incoming air flow. In a normal situation, UA should not exceed 12-15 degrees, otherwise flow breakdown, i.e. the formation of turbulent “breakers” behind the wing, as in a fast stream, if you place your palm not along, but across the flow of water. A stall results in loss of lift on the wing and stalling airplane.

On "small" aircraft (including the Yak-40, Tu-134), releasing the flaps usually leads to "swelling"- the plane slightly increases its vertical speed and lifts its nose. On "big" planes there are systems for improving stability and controllability, which automatically counter the emerging moment by lowering the nose. Such a system is available on the Tu-154, so the “swelling” is small (in addition, there the moment of flap release is combined with the moment of repositioning the stabilizer, which creates the opposite moment). On the Tu-134, the pilot has to dampen the increase in lift by manually deflecting the control column away from himself. In any case, to reduce “swelling”, it is customary to release the flaps in two or three steps - usually first by 20-25, then by 30-45 degrees.

Slats

In addition to flaps, almost all transport aircraft also have slats, which are installed in the front part of the wing, and automatically deflect downward simultaneously with the flaps (the pilot hardly thinks about them). In principle, they perform the same function as flaps. The difference is as follows:

1. At high angles of attack, the downward slats cling like a hook to the incoming air flow, deflecting it down along the profile. As a result, the slats reduce the angle of attack of the rest of the wing and delay the stall moment at higher angles of attack.
2. Slats are usually smaller in size, which means less drag.

In general, the extension of both flaps and slats comes down to an increase in the curvature of the wing profile, which allows the incoming air flow to be more deflected downwards, and therefore increases the lift force.

As far as is known so far, the slats are not highlighted separately in the air file.

To understand why such complex mechanization is used on airplanes, watch the birds land. You can often notice how pigeons and similar crows land with their wings fluffed out, tucking their tail and stabilizer under themselves, trying to get a wing profile of great curvature and create a good air cushion. This is the release of flaps and slats.

Mechanization of a B-747 on landing

Interceptors (spoilers)

Interceptors, they are spoilers are deflectable brake flaps on the upper surface of the wing, which increase aerodynamic drag and reduce lift (unlike flaps and slats). Therefore, interceptors (especially on “silts”) are also called lift dampers.

Interceptors are a very broad concept, which includes many different types of dampers, and on different types they can be called differently and located in different places.

As an example, consider the wing of a Tu-154 aircraft, which uses three types of spoilers:

1) external aileron spoilers (spoilers, roll spoilers)

Aileron spoilers are an addition to ailerons. They deviate asymmetrically. For example, on the Tu-154, when the left aileron is deflected upward by an angle of up to 20 degrees, the left aileron-interceptor automatically deflects upward by an angle of up to 45 degrees. As a result, the lift on the left wing decreases and the plane rolls to the left. The same for the right half-wing.

Why can't we just use ailerons?

The fact is that in order to create a roll moment on a large aircraft, a large area of ​​deflected ailerons is needed. But because jets fly at speeds close to the speed of sound, they need to have a thin wing profile that doesn't create too much drag. The use of large ailerons would lead to its twisting and all sorts of bad phenomena such as aileron reverse (this, for example, can happen on the Tu-134). Therefore, we need a way to distribute the load on the wing more evenly. For this purpose, aileron interceptors are used - flaps installed on the upper surface, which, when deflected upward, reduce the lift force on a given half-wing and “sink” it down. The rotation speed along the roll increases significantly.

The pilot does not think about the aileron interceptors; from his point of view, everything happens automatically.

In principle, aileron interceptors are provided in the air file.

2) middle spoilers (spoilers, speed brakes)

Medium spoilers are what are usually understood as simply “interceptors” or “spoilers” - i.e. "air brakes". The symmetrical activation of spoilers on both halves of the wing leads to a sharp decrease in lift and braking of the aircraft. After the “air brakes” are released, the aircraft will balance at a higher angle of attack, begin to slow down due to increased drag, and descend smoothly.

On the Tu-154, the middle spoilers are deflected at an arbitrary angle of up to 45 degrees using a lever on the middle pilot console. This is about the question of where the stop valve is on the plane.

On the Tu-154, the outer and middle spoilers are structurally different elements, but on other aircraft the “air brakes” can be structurally combined with aileron spoilers. For example, on the IL-76, spoilers usually operate in aileron mode (with a deflection of up to 20 degrees), and, if necessary, in braking mode (with a deflection of up to 40 degrees).

There is no need to deploy the middle spoilers during landing. In fact, releasing spoilers after releasing the landing gear is usually prohibited. In a normal situation, spoilers are released for a faster descent from flight level with a vertical speed of up to 15 m/s and after the aircraft has landed. In addition, they can be used during aborted takeoff and emergency descent.

It happens that “virtual pilots” forget to release the throttle during landing and keep the mode almost on takeoff, trying to fit into the landing pattern at a very high speed, causing angry screams from the dispatcher in the style of “Maximum speed below ten thousand feet is 200 knots!” ” In such cases, you can briefly release the middle interceptors, but in reality, this is unlikely to lead to anything good. It is better to use this crude method of reducing speed in advance - only when descending, and it is not always necessary to extend the spoilers to the full angle.

3) internal spoilers (ground spoilers)

Also "brake flaps"

Located on the upper surface in the inner (root) part of the wing between the fuselage and landing gear nacelles. The Tu-154 automatically deviates to an angle of 50 degrees after landing when the main landing gear struts are compressed, the speed is more than 100 km/h and the throttle is in the “idle” or “reverse” position. At the same time, the middle interceptors also deflect.

Internal spoilers are designed to dampen lift after landing or during an aborted takeoff. Like other types of spoilers, they do not so much dampen the speed as they dampen the lifting force of the wing, which leads to an increase in the load on the wheels and improved traction of the wheels with the surface. Thanks to this, after releasing the internal spoilers, you can proceed to braking using the wheels.

On the Tu-134, brake flaps are the only type of spoilers.

In the simulator, internal interceptors are either absent or recreated rather conditionally.

Pitch trim

Large aircraft have a number of pitch control features that cannot be ignored. Trimming, centering, balancing, stabilizer repositioning, steering column consumption. Let's look at these questions in more detail.

Pitch

Pitch- the angular movement of the aircraft relative to the transverse axis of inertia, or, more simply, “bully”. Sailors call this bullshit "trim". Pitch opposed bank And yaw, which respectively characterize the position of the aircraft during its rotation around the longitudinal and vertical axis. Accordingly, pitch, roll and yaw angles are distinguished (sometimes called Euler angles). The term "yaw" can be replaced with the word "course", for example they say "in the course channel".

I hope there is no need to explain the difference between the pitch angle and the angle of attack... When the plane falls completely flat, like an iron, its angle of attack will be 90 degrees, and the pitch angle will be close to zero. On the contrary, when a fighter is climbing, in afterburner, at a good speed, its pitch angle can be 20 degrees, but the angle of attack, say, is only 5 degrees.

Trimming

To ensure normal piloting, the force on the control wheel must be noticeable, otherwise, any random deviation could send the plane into some kind of bad tailspin. As a matter of fact, this is why on heavy aircraft that are not designed to perform sharp maneuvers, yokes are usually used rather than sticks - they are not so easy to accidentally roll. (The exception is Airbus, which prefers joysticks.)

It is clear that with heavy control, the pilot’s biceps will gradually develop quite decent ones, moreover, if the aircraft unbalanced in effort it is difficult to pilot because any weakening of the force will push steering column (SHK) not where it should be. Therefore, so that during the flight, pilots can sometimes slap flight attendant Katya on the ass, trimmers are installed on airplanes.

Trimmer is a device that in one way or another fixes the steering wheel (control stick) in a given position so that the papelats can descend, gain altitude and fly in horizontal flight, etc. without applying any force to the steering column.

As a result of trimming, the point to which the steering wheel (handle) is pulled will not coincide with the neutral position for a given steering wheel. How further from the trim position, the big effort has to be made to hold the steering wheel (handle) in a given position.

Most often, by trimmer they mean a trimmer in the pitch channel - i.e. Elevator trimmer (ER). However, on large aircraft, just in case, trim tabs are installed in all three channels - there they usually perform an auxiliary role. For example, in the roll channel, trimming can be used when the aircraft is longitudinally unbalanced due to asymmetrical fuel production from the wing tanks, i.e. when one wing pulls the other. In the heading channel - in case of engine failure, so that the plane does not yaw to the side when one engine is not working. Etc.

Trimming can be technically implemented in the following ways:

1) using a separate aerodynamic trimmer, as on the Tu-134 - i.e. a small “knob” on the elevator, which holds the main rudder in a given position using aerodynamic compensation, i.e. using the force of the oncoming flow. On the Tu-134, such a trimmer is used to control trimmer wheel, on which the cable going to the RV is wound.

2) by using MET (trimming effect mechanism), as on the Tu-154 - i.e. simply by adjusting the tension in the spring system (it would be more correct to say spring loaders), which purely mechanically holds the steering column in a given position. When the MET rod moves back and forth, the loaders are either loosened or tightened. To control the MET, small push switches are used on the steering wheel handles, when turned on, the MET rod, and behind it the steering column, slowly move to a given position. There are no aerodynamic trim tabs like on the Tu-134 or on the Tu-154.

3) using adjustable stabilizer, as on most Western types (see below)

In the simulator it is difficult to recreate a real elevator trimmer; for this you will have to use a fancy joystick with a trimming effect, because what is called a trimmer in MSFS, in fact, should not be perceived as such - it would be more correct to cover the joystick with plasticine or chewing gum or simply put mouse on the table (in FS98) - here you have a trimmer. I must say that control is generally a sore point of all simulators. Even if you buy the most sophisticated steering wheel and pedal system, it will still most likely be far from the real thing. An imitation is just that, an imitation, because to get an absolutely exact copy of a real plane you need to spend the same amount of effort and process the same amount of information as to build a real plane...

Centering (CG)

Center of Gravity (CG) position- the position of the center of gravity, measured as a percentage of the length of the so-called mean aerodynamic chord (MAC)- i.e. chords of a conventional rectangular wing, equivalent to a given wing, and having the same area as it.

Chord is a straight segment connecting the leading and trailing edges of the wing profile.

center of gravity position 25% MAR

The length of the average aerodynamic chord is found by integrating over the lengths of the chords along all half-wing profiles. Roughly speaking, the MAR characterizes the most common, most probable wing profile. those. it is assumed that the entire wing with all its diversity of profiles can be replaced by one single averaged profile with one single averaged chord - MAR.

To find the position of the MAR, knowing its length, you need to intersect the MAR with the contour of the real wing and see where the beginning of the resulting segment is located. This point (0% MAR) will serve as a reference point for determining alignment.

Of course, a transport aircraft cannot have a constant alignment. It will change from departure to departure due to cargo movements, changes in the number of passengers, and also during the flight as fuel is used up. For each aircraft, an acceptable range of alignments is determined, which ensures its good stability and controllability. Usually distinguish front(for Tu-154B - 21-28%), average(28-35%) and rear(35-50%) alignment - for other types the numbers will be slightly different.

The alignment of an empty aircraft is very different from the alignment of a fueled aircraft with all cargo and passengers, and to calculate it before departure, a special centering chart.

An empty Tu-154B has an alignment of about 49-50% of the MAC, despite the fact that at 52.5% it already tips over onto its tail (the engines on the tail are pulled). Therefore, in some cases it is necessary to install a safety rod under the rear fuselage.

Balancing in flight

An airplane with a swept wing wing lift center located at a point of approximately 50-60% of the MAR, i.e. behind the center of gravity, which in flight is usually located in the region of 20-30% of the MAR.

As a result, in horizontal flight a lift lever who wants to tip the plane over on its nose, i.e. in a normal situation the aircraft is under the influence diving moment.

To avoid all this, you will have to counter the resulting diving moment throughout the flight. balancing deviation РВ, i.e. The elevator deflection will not be zero even in level flight.

Basically, in order to keep the plane from “pecking” you will need to create pitching moment, i.e. The RV will need to deflect upward.

To trim - from fr. cabrer, "to rear."

Always up? No not always.

As the speed increases, velocity head will increase, which means the total lift force on the wing, stabilizer and elevator will increase proportionally

F under = F under1 – F under2 – F under3

But the gravity will remain the same, which means the plane will go into climb. To rebalance the papelats in horizontal flight, you will have to lower the elevator lower (move the steering wheel away from you), i.e. reduce the term F sub3. Then the nose will drop, and the plane will again balance in level flight, but at a lower angle of attack.

Thus, for each speed we will have our own balancing deviation of the RT - we will get quite a whole balancing curve(dependence of the deviation of the aircraft on the flight speed). At high speeds, you will have to push the steering column away from you (RV down) to keep the Samik from pitching up; at low speeds you will have to take the steering column toward you (RV up) to keep the Samik from diving. The helm and elevator will be in a neutral position only at one specific indicated speed (about 490 km/h for the Tu-154B).

Stabilizer (Horizontal Stabilizer)

In addition, as can be seen from the diagram above, the aircraft can be balanced not only by the elevator, but also by an adjustable stabilizer (component Fpod2). Such a stabilizer can be completely installed at a new angle using a special mechanism. The efficiency of such a transfer will be approximately 3 times higher - i.e. 3 degrees of deflection of the radio will correspond to 1 degree of deflection of the stabilizer, because its area of ​​the horizontal stabilizer at the “carcass” is approximately 3 times larger than the area of ​​the RV.

What is the advantage of using an adjustable stabilizer? First of all, in this case Elevator consumption is reduced. The fact is that sometimes, due to too forward alignment, in order to keep the plane at a certain angle of attack, you have to use the entire stroke of the control column - the pilot chose control completely over himself, and the plane can no longer be lured upward by any carrot. This can especially occur on landings with extremely forward centering, when when attempting a go-around, the elevator may not be sufficient. As a matter of fact, the value of the maximum forward alignment is set on the basis that the available deflection of the elevator is sufficient in all flight modes.

Since the RV deviates relative to the stabilizer, it is easy to see that the use of an adjustable stabilizer will reduce steering wheel consumption and increase the available range of alignments and available speeds. This means it will be possible to take more cargo and arrange it in a more convenient way.

In horizontal flight at flight level, the Tu-154 stabilizer is at a pitch-up angle of -1.5 degrees relative to the fuselage, i.e. almost horizontal. On takeoff and landing, it is further shifted to pitching up at an angle of up to -7 degrees relative to the fuselage in order to create a sufficient angle of attack to maintain the aircraft in level flight at low speed.

A special feature of the Tu-154 is that the stabilizer is rearranged only on takeoff and landing, and in flight it is retracted to position -1.5 (which is considered zero), and the plane is then balanced with one elevator.

At the same time, for the convenience of the crew and for a number of other reasons, the relocation combined with the release of flaps and slats, i.e. when moving the flap handle from position 0 to the release position, automatically The slats are extended and the stabilizer is moved to the agreed position. When retracting the flaps after takeoff, do the same in reverse order.

Let's give a table that hangs in the cockpit to constantly remind him that they don't produce a damn thing...

Thus, everything happens by itself. On the circle before landing at a speed of 400 km/h, the crew only needs to check whether the balancing deviation of the aircraft corresponds to the position of the stabilizer adjuster and, if not, then set the adjuster to the desired position. Let's say the arrow of the position indicator of the PV is in the green sector, which means we set the set pointer to the green “P” - everything is quite simple and does not require significant mental effort...

In case of automation failures, all releases and relocations of mechanization can be done manually. For example, if we are talking about a stabilizer, you need to fold back the cap on the left in the photo and move the stabilizer to the agreed position.

On other types of aircraft, this system works differently. For example, on the Yak-42, MD-83, B-747 (I find it difficult to say for the whole of Odessa, but this should be the case on most Western aircraft) the stabilizer deflects throughout the flight and completely replaces the trimmer. This system is more advanced because it allows you to reduce drag in flight, since the stabilizer, due to its large area, deflects at smaller angles than the flywheel.

On the Yak-40, Tu-134, the stabilizer is also usually adjusted independently of the wing mechanization.

Now about MSFS. In the simulator we have the situation of a “trimming stabilizer”, as on Western types. There is no separate virtual trimmer in MSFS. That rectangular thing (like on a Cessna), which Microsoft calls a “trimmer,” is actually a stabilizer, which is noticeable by its independence of operation from the radio.

Why is that? Probably the whole point is that initially (in the late 80s) FS was used as a software base for full-featured simulators on which there were real steering columns and real METs. When MS bought (stole?) FS, it did not delve deeply into the features of its operation (and perhaps did not even have a complete description for it), so the stabilizer began to be called a trimmer. At least, this is the assumption I would like to make when studying MS+FS, because the description for the air file has never been published, and judging by the quality of the default models and a number of other signs, we can conclude that Microsoft itself is not particularly versed in it.

In the case of the Tu-154, you should probably set the microsoft trim once before landing in level flight, so that the elevator indicator is approximately in the neutral position, and not return to it again, but work only with the joystick trim, which no one has.. Or work with the “rectangular thing”, close your eyes and repeat to yourself: “This is not a stabilizer, this is not a stabilizer...”

Auto Throttle

In helm mode, KVS or 2P controls the engines using Thrusters (motor control levers) on the middle console or by giving commands to the flight engineer: “Mode such and such”

Sometimes it is convenient to control engines not manually, but using automatic traction (auto throttle, AT), which tries to keep the speed within acceptable limits by automatically adjusting the engine mode.

Turn on AT (Shift R key), set the desired speed to US-I(speed indicator), and the automation will try to maintain it without pilot intervention. On the Tu-154 speed when turned on AT-6-2 can be adjusted in two ways: 1) by rotating the ratchet on the left or on the right US-I 2) by rotating the regulator on PN-6 (= remote control for STU and autothrottle).

Types of landing systems

Distinguish visual approach And instrument approach.

Purely visual approaches are rarely used on large aircraft and can cause difficulties even for an experienced crew. Therefore, entry is usually carried out by instruments, i.e. using radio systems under the control and control of an air traffic controller.

Air Traffic Control (ATC)- control of the movement of aircraft in flight and on the maneuvering area of ​​the airfield.

Radio-technical landing systems

Let's consider approaches using radio-technical landing systems. They can be divided into the following types:

“according to OSB”, i.e. using DPRM and BPRM

“according to RMS”, i.e. using ILS

“according to RSP”, i.e. by locator.

Entry using OSB

Also known as "approach by drives".

OSB (landing system equipment)- a complex of ground-based equipment, including two drive radio stations with marker radio beacons, as well as lighting equipment (STO), installed at the airfield according to the approved standard layout.

Specifically, NSP includes

"distant" (locator beacon) (DPRM, Outer Marker, OM)- a long-range radio station with its own marker, which is located 4000 (+/- 200) m from the runway end. When a marker passes, a light and sound alarm is triggered in the cockpit. The Morse code of the signal in the ILS system looks like “dash-dash-dash...”.

"near" (locator beacon) (BPRM, Middle Marker, MM)- a near-range radio station, also with its own marker, which is located 1050 (+/- 150) m from the runway end. Morse code in the ILS system looks like “dash-dot-...“

Drive radios operate in the range of 150-1300 kHz.

When flying in a circle, the first and second sets automatic radio compass (ARK, Automatic Direction Finder, ADF) are tuned to the frequencies of DPRM and BPRM - in this case, one arrow on the ARC indicator will point to DPRM, the second to BPRM.

Let us recall that the arrow of the ARC indicator always points to the radio station, just as the arrow of a magnetic compass always points to the north. Therefore, when flying according to the pattern, the moment of the beginning of the fourth turn can be determined according to the heading angle of the radio station (KUR). Let's say, if the DPRM radio station is exactly on the left, then CUR = 270 degrees. If we want to turn towards it, then the turn needs to start 10-15 degrees earlier (i.e. at CUR = 280...285 degrees). Flying over the radio station will be accompanied by a 180 degree turn of the needle.

Thus, when flying in a circle, the heading angle of the DPRM helps to determine the moments when turning turns on the circle begin. In this regard, the DPRM represents something like a reference point, relative to which many actions during the landing are calculated.

Also attached to the radio station marker, or marker beacon- a transmitter that sends upward a narrowly directed signal, which, when flying over it, is perceived by aircraft receivers and causes the indicator light and electric bell to go off. Thanks to this, knowing at what height the DPRM and BPRM should be passed (usually this is 200 And 60 m respectively) you can get two points from which you can build a pre-landing straight line.

In the west, at airfields of categories II and III with difficult terrain, at a distance of 75..100 m from the end of the runway, they also install internal radio marker (Inner Marker, IM)(with Morse code “dot-dot-dot...”), which is used as an additional reminder to the crew that they are approaching the moment when visual guidance begins and the need to make a landing decision.

The OSP complex is a simplified landing system; it must provide the aircraft crew with a drive to the airfield area and a descent maneuver to the visual detection altitude of the runway. In practice, it plays an auxiliary role and usually does not replace the need to use an ILS or landing radar system. They enter purely using OSB only in the absence of more advanced landing systems.

When approaching only using the OSP, horizontal visibility must be at least 1800 m, vertical visibility at least 120 m. If this meteorological minimum is not observed, it is necessary to go to dispersal field.

Please note that the DPRM and BPRM at different ends of the band have the same frequency. In a normal situation, the radio stations at the other end should be turned off, but in the sim this is not the case, so when flying in a circle, the ARC often starts to glitch, picking up one radio station, then another.

Call by RMS

They also say "login". In general, this is the same as an ILS approach. (see also Dmitry Prosko’s article on this site)

In Russian terminology radio beacon landing system (RMS) is used as an umbrella term that includes various types of planting systems - in particular, ILS (Instrument Landing System)(as Western standard) and SP-70, SP-75, SP-80 (as domestic standards).

The principles of approaching the RMS are quite simple.

The ground part of the RMS consists of two radio beacons - localizer (LOB) And glide slope radio beacon (GRM), which emit two oblique beams (equal-signal zones) in the vertical and horizontal planes. The intersection of these zones forms the approach path. Aircraft receiving devices determine the position of the aircraft relative to this trajectory and issue control signals to PKP-1 flight control device(in other words, on the artificial horizon) and planning and navigation device PNP-1(in other words, to the course indicator).

If the frequency is set correctly, then when approaching the runway the pilot will see two moving lines on the large attitude indicator - a vertical course command arrow And horizontal glide slope command arrow, as well as two triangular indices indicating the position of the aircraft relative to the calculated trajectory.

pitch- pitching) - angular movement of an aircraft or vessel relative to the main (horizontal) transverse axis of inertia. Pitch angle - the angle between the longitudinal axis of the aircraft or ship and the horizontal plane. The pitch angle is symbolized by θ (theta). In aviation there are:

• positive pitch, with increasing angle (nose lift) - pitching up , steering wheel to yourself;
• negative, with a decrease in angle (dropping of the nose) - dive , steering wheel away from you.

This is one of the three angles (roll, pitch and yaw), which specify the inclination of the aircraft relative to its center of inertia along three axes. In relation to seagoing vessels, the term “ trim” is used with the same meaning. It is noteworthy that the trim has opposite ideas about positivity/negativity.

see also

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Notes

Links

• Aresti FAI Aerobatic Catalog = FAI Aresti Aerobatic Catalog. - Federation Aeronautique Internationale, 2002.

Excerpt characterizing Pitch

“Oh my God, the people are like beasts, where can a living person be!” - was heard in the crowd. “And the guy is young... he must be from the merchants, then the people!.. they say, he’s not the one... how could he not be the one... Oh my God... They beat another, they say, he’s barely alive... Eh, people... Who is not afraid of sin...” they were saying now the same people, with a painfully pitiful expression, looking at the dead body with a blue face, smeared with blood and dust and with a long thin neck severed.
The diligent police officer, finding it indecent the presence of a corpse in his lordship's courtyard, ordered the dragoons to drag the body out into the street. Two dragoons took hold of the mangled legs and dragged the body. A bloody, dusty, dead shaved head on a long neck, tucked under, dragged along the ground. The people huddled away from the corpse.
While Vereshchagin fell and the crowd, with a wild roar, was embarrassed and swayed over him, Rostopchin suddenly turned pale, and instead of going to the back porch, where his horses were waiting for him, he, without knowing where or why, lowered his head, with quick steps I walked along the corridor leading to the rooms on the lower floor. The count's face was pale, and he could not stop his lower jaw from shaking, as if in a fever.
“Your Excellency, here... where do you want?... here, please,” said his trembling, frightened voice from behind. Count Rastopchin was unable to answer anything and, obediently turning around, went where he was shown. There was a stroller on the back porch. The distant roar of the roaring crowd was heard here too. Count Rastopchin hastily got into the carriage and ordered to go to his country house in Sokolniki. Having left for Myasnitskaya and no longer hearing the screams of the crowd, the count began to repent. He now remembered with displeasure the excitement and fear that he had shown in front of his subordinates. “La populace est terrible, elle est hideuse,” he thought in French. – Ils sont sosche les loups qu"on ne peut apaiser qu"avec de la chair. [The crowd is scary, it is disgusting. They are like wolves: you can’t satisfy them with anything except meat.] “Count!” one god is above us!“ - Vereshchagin’s words suddenly came to his mind, and an unpleasant feeling of cold ran down Count Rastopchin’s back. But this feeling was instantaneous, and Count Rastopchin smiled contemptuously at himself. “J"avais d"autres devoirs,” he thought. – Il fallait apaiser le peuple. Bien d "autres victimes ont peri et perissent pour le bien publique", [I had other responsibilities. The people had to be satisfied. Many other victims died and are dying for the public good.] - and he began to think about the general responsibilities that he had in relation to his family, his (entrusted to him) capital and about himself - not as about Fyodor Vasilyevich Rostopchin (he believed that Fyodor Vasilyevich Rostopchin sacrifices himself for the bien publique [public good]), but about himself as the commander-in-chief, about representative of the authorities and the tsar’s authorized representative: “If I were only Fyodor Vasilyevich, ma ligne de conduite aurait ete tout autrement tracee, [my path would have been charted completely differently,] but I had to preserve both the life and dignity of the commander-in-chief.”

CONSTRUCTION OF A VERTICAL USING A PHYSICAL PENDULUM ON AN AIRPLANE

When piloting an airplane, you need to know its position relative to the plane of the earth's horizon. The position of the aircraft relative to the horizontal plane is determined by two angles: the pitch angle and the roll angle. Pitch angle is the angle between the longitudinal axis of the aircraft and the horizontal plane, measured in the vertical plane. Roll angle - the angle of rotation of the aircraft around its longitudinal axis, measured from the vertical plane passing through the longitudinal axis of the aircraft

Fig. 4.1 physical pendulum - vertical determinant on an airplane.

Thus, the position of the aircraft relative to the horizon plane can be determined if the direction of the true vertical is known on the aircraft, that is, the direction of the line passing through the center of the Earth and the aircraft, and the deviation of the aircraft from this direction is measured.

Deviation from the vertical on the ground is determined by an ordinary plumb line, i.e., a physical pendulum.

Let us assume that a physical pendulum is installed on an airplane that is flying horizontally with acceleration A(Fig. 4.1). To the mass of the pendulum T forces will act from the acceleration of gravity g and inertial force from acceleration a. The sum of the moments from these forces relative to the pendulum’s suspension point is zero and is expressed by the equation

Where l- length of the pendulum;

α - pendulum deflection angle

From equation (4.1) we have

(4.2)

Consequently, a pendulum mounted on an object moving with acceleration is deflected in the direction opposite to the action of acceleration and shows the so-called “apparent vertical”. Modern transport aircraft can have accelerations commensurate in magnitude with the acceleration of gravity, so the angle α of the pendulum's deviation from the vertical can reach significant values. Thus, a physical pendulum is not suitable for determining the direction of the vertical position, i.e., for measuring roll and pitch angles if the aircraft is flying with acceleration.

AIRLINE HORIZONS

It was previously noted that a pendulum can be used to determine the vertical only during flight without acceleration, and a free three-degree gyroscope can maintain a given spatial position, regardless of the current accelerations, only for a short time.

Therefore, these two devices are connected together, using the positive properties of each. In the absence of acceleration using a pendulum, the main axis of the gyroscope is set vertically. At those moments when accelerations act on the pendulum, it is turned off and the gyroscope operates in “memory” mode.

The device by which the pendulum acts on the gyroscope is called a pendulum correction system. A gyroscope with such correction is called gyrovertical. The gyro vertical, visually showing the position of the aircraft relative to the earth's horizon, is called the attitude indicator.

Attitude indicators use an electrolytic pendulum (Fig. 4.2), which is a flat copper bowl 3, filled with conductive liquid 1 with high electrical resistivity. There is so much liquid in the bowl that there is room for an air bubble 2 . The bowl is closed with a lid made of insulating material, into which four contacts are mounted 4, the fifth contact is the bowl itself. If the pendulum is positioned horizontally, then all four contacts are evenly covered by liquid and the electrical resistance of the areas between them and the bowl is the same. If the bowl tilts, then the air bubble, occupying the upper position in the bowl, will expose one of the contacts and thereby change the electrical resistance of the area, which at small angles (up to 30") is proportional to the angle of inclination of the bowl.

The pendulum contacts are included in the electrical circuit, as shown in Fig. 4.3. When the pendulum tilts, the resistance between pins 0 and 1 will be greater than the resistance between pins 0 and 3. Then the current i 1 which passes through the control winding OY 1, there will be less current i 2 windings OY 2 correction motor. Windings OY 1 and OY 2 are wound counter-winding, so the difference current Δ i=i 2 -i 1 creates a magnetic flux, which, interacting with the magnetic flux of the field winding, causes torque. The engine rotor is fixed to the axis of the gimbal, therefore, a moment is applied to the axis of the gimbal, under the influence of which the gyroscope precesses. The precession of the gyroscope continues as long as there is a moment along the axis of the gimbal suspension, and this moment acts until the pendulum is installed in a horizontal position, at which the current i 1 =i 2. By connecting the pendulum with the inner , frame of a cardan suspension and placing correction motors along the axes of the suspension, we obtain a gyrovertical with electromechanical pendulum correction (Fig. 4.4). Thus, the electrolytic pendulum 1 , acting on the gyroscope through correction motors 2 And 3 , will always bring the main axis of the gyroscope to the vertical position. When correction is turned off, the gyroscope will maintain its previous position in space with an accuracy determined by its own errors, for example, due to precession caused by moments of friction along the axes of the gimbal.

Correction systems differ in types of characteristics. The correction characteristic is the law of change in the torque developed by the correction motor, depending on the deviation of the main axis of the gyroscope from the vertical position.

In aviation instruments, the mixed correction characteristic is most widespread (Fig. 4.5). Area ±Δ α defines the dead zone of the system. Up to certain extreme angles α etc,

β at the moment of correction M k varies proportionally to the angles α And β , and then becomes constant.

ERRORS OF GYROVERTICALS

Error from friction moments in the axes of the frame and the frame. There are inevitably friction moments in the axes of the gimbal, so the precession of the gyroscope under the influence of correction moments continues as long as the correction moment is greater than the friction moment. The movement of the gyroscope stops when these moments are equal:

It follows that the main axis of the gyroscope will not reach the vertical position at the angles α * And β *:

Thus, due to friction in the axes of the gimbal, the gyrovertical has a stagnation zone, which depends on the magnitude of the friction moment in the axes of the gimbal and, naturally, on the dead zone of the pendulum correction (see Fig. 4.5). The greater the specific torque developed by the correction motors, the smaller the stagnation zone. Too large a specific moment leads to significant errors in turns. For attitude indicators, the stagnation zone is usually 0.5-1°.

Turning error. When the plane makes a turn with angular velocity ω, then on the pendulum, in addition to the force of gravity mg, centrifugal force is still active mω 2 R, and the pendulum is not installed along the true vertical, but along the resultant of these forces (Fig. 4.7). Signals are sent to the correction motors, and the main axis of the gyroscope is set to an apparent vertical position. This process occurs the faster, the greater the specific moments k x , k y correction systems. As can be seen from Fig. 3.10, on a bend the lateral correction system generally does not work correctly. Therefore, in modern gyro verticals and artificial horizons, lateral correction on turns is disabled by a special device.

Naturally, linear accelerations of the aircraft, for example, with increasing speed, also lead to similar errors. Therefore, in such attitude indicators as AGD-1, longitudinal correction is also disabled. When correction is turned off, the gyrovertical operates in “memory” mode. After the aircraft completes the evolution associated with accelerations, the correction system turns on and brings the main axis of the gyroscope to a vertical position if it has deviated during operation in the “memory” mode.

An error appears in the gyrometers both due to the daily rotation of the Earth and due to the aircraft’s own flight speed, but for transport aircraft this error does not exceed several minutes of arc.

a red flag will appear 12. This switch connects the control windings of the transverse correction motor 4 with phase C, bypassing resistance R2, and thereby increases

current in the motor, and therefore the correction torque it develops.

After the device reaches the nominal operating mode, the switch 10 should be returned to its original position (the flag will disappear from view). In the nominal operating mode, the control windings of the correction motor 4 connected to phase C through the contacts of the correction switch VK-53RB.. When the aircraft makes turns, the correction switch turns off the transverse correction motor, otherwise a large turning error occurs.

AIR HORIZONT AGI-1s

The attitude indicator is designed to determine the position of the aircraft in space relative to the true horizon line; it has a built-in slip indicator device. An attitude indicator is installed on civil aviation transport aircraft.

The kinematic diagram of the device is shown in Fig. 4.8, simplified electrical - in Fig. 4.9, and the view of the scale is in Fig. 4.10.

Let's consider the operation of the device. Own axis of rotation of the gyroscope (see Fig. 4.8) according to signals from the electrolytic pendulum 8 using correction motors 3 And 10 installed and held in a vertical position.

A special feature of the AGI-lc attitude indicator is its ability to operate in an unlimited range of roll and pitch angles. This is possible thanks to the use of an additional tracking frame in the device. 4, the axis of which coincides with the longitudinal axis of the aircraft, and the frame itself can be rotated relative to the aircraft by the engine 11 . The purpose of the additional tracking frame is to ensure perpendicularity to the axis of the gyroscope’s own rotation and the axis of the external frame of the gimbal. When the aircraft rolls, the outer frame 5 The cardan suspension rotates around the axis of the internal frame. This rotation is fixed by a switch 9 (see Fig. 4.8 and 4.9), with which the engine is turned on 11 , turning the follower frame 4 , and with it the frame 5 in the opposite direction. Therefore, the perpendicularity of the gyroscope’s own axis 6 and the axes of the outer frame are not violated. When the aircraft performs pitch evolutions at angles greater than 90˚, using the switch 12 the direction of rotation of the engine changes 11. For example, if an airplane makes a “Nesterov loop” figure, then at the moment when it finds itself in an inverted state, i.e., changes its position relative to the main axis of the gyroscope by 180°, the direction of rotation of the engine 11 To rotate the follower frame, it should be reversed.

When an airplane performs a pitch evolution, the airplane rolls around the axis of the external gimbal frame and therefore has a 360° operating range.

Indication of the aircraft's position relative to the horizon plane in AGI-1s is carried out using the silhouette of the aircraft (see Fig. 4.8 and 4.10), mounted on the instrument body, and a spherical scale 2, connected to the axis of the internal frame 7 of the gyroscope gimbal suspension. Spherical scale 2 colored brown above the horizon line and blue below the horizon line. On the brown field there is the inscription “Descent”, on the blue field there is the inscription “Rise”. Thus, when climbing, the silhouette of the aircraft, along with the aircraft itself, will move to the blue field, as shown in Fig. 3.18, V, since the scale 2, associated with the gyroscope, will remain motionless in space. It should be noted that the AGI-lc attitude indicator's pitch readings are opposite to those of the AGB-2. This is extremely important since both instruments are sometimes installed on the same aircraft.

Fig. 4.9 electrical diagram of the attitude indicator AGI-1.

Reducing the time for the initial alignment of the axis of self-rotation of the gyroscope to a vertical position is achieved by sequentially switching on the excitation windings of the correction motors 3 And 10 with stator windings of the gyromotor. In addition, on the inner frame 7 there is a mechanical pendulum, which, when the device is not turned on, holds the frame system at approximately zero

position For the same purpose, a mechanical lock is used, when you press a button 15 which (see Fig. 4.10) the additional follower frame is installed in the zero position. The button says “Press before starting”. In order to reduce the turning error of the attitude indicator, a transverse correction engine 3 on a turn it is turned off by the correction switch VK-53RB. On the front side of the device, at the bottom, there is a slip indicator 13 and on the left - the handle 14 to change the position of the airplane silhouette.

AIR HORIZON AGD-1

The AGD-1 remote attitude indicator provides the crew with an easily perceptible large-scale indication of the aircraft’s position relative to the plane of the true horizon and

provides consumers (autopilot, heading system, radar stations) with electrical signals proportional to the aircraft's roll and pitch deviations.

AGD-1 consists of two devices: 1) a three-degree gyroscope with pendulum correction, called a gyro sensor, which is installed as close as possible to the center of gravity of the aircraft; 2) indicators placed on the crew instrument panels. Up to three indicators can be connected to one gyro sensor.

The schematic electromechanical diagram of AGD-1 is shown in Fig. 4.12, a view of the pointer scale is shown in Fig. 4.13

Fig. 4.13 front side of the AGD-1 attitude indicator.

36-lock button, 37-lamp, other designations are the same as on 4.12.

The gyro sensor is a three-degree gyroscope, the axis of the external gimbal frame of which is mounted in the tracking frame 7. The purpose of the tracking frame is to ensure the roll operation of the device in an unlimited range of angles. Follower frame 7 ensures that the axis of the gyroscope's own rotation is perpendicular to the axis of the external frame of the suspension using an induction data

chica 3 and engine-generator 2, amplifier controlled 1 . Anchor 5 sensor is fixed on the axis of the inner frame, and the stator 3 rigidly connected to the outer frame 8 gimbal suspension.

Switch 4 changes the direction of rotation of the engine 2, when the aircraft performs pitch evolutions at angles greater than 90°. Thus, the tracking frame 7 performs the same functions as in the AGI-1s attitude indicator.

A special feature of the roll tracking system for frame 7 in the AGD-1 attitude indicator is the use of an amplifier based on semiconductor elements and an engine-generator. Pendulum correction AGD-1 is similar to the correction of AGI-lc and AGB-2, but differs in that the transverse correction engine 6 can be turned off not only by the switch 17, which is controlled by the correction switch VK-53RB, but also by a special lamella device (not shown in the diagram) at rolls of 8-10°. In addition, the longitudinal correction motor 10 controlled by an electrolytic pendulum 13 via liquid accelerometer 16. It is a device similar to a liquid pendulum. During longitudinal acceleration of the aircraft, the conductive liquid, under the influence of inertial forces, is shifted to one of the contacts and due to an increase in the electrical resistance of the circuit, the correction is weakened by 50%.

The aircraft's roll and pitch deviations are measured by a gyro sensor and transmitted to the pointer by two identical tracking systems:

1) roll tracking system, which consists of a synchro sensor 9, synchronizer-receiver 20, amplifier 18 and engine-generator 19;

2) pitch tracking system, which includes: synchro sensor 14, selsyn-receiver 23, amplifier 24, motor-generator 25.

Switch 15 included in the pitch tracking system for its proper operation at an angle of more than 90°. A feature of the tracking systems in AGD-1 is the use of motor-generators as actuators. A motor-generator is an electrical machine consisting of a motor and a generator mounted on the same shaft. The voltage produced in the generator is proportional to the engine speed. In the servo system, it serves as a high-speed feedback signal to dampen system oscillations. Engine generator 19 turns the gear 21 with airplane silhouette 22 relative to the device body, and the engine-generator 25 rotates the pitch dial 26,

having a two-color color: above the horizon line - blue, below - brown. Thus, the indications are indicated by the moving silhouette of the aircraft and the moving pitch scale.

The indication of the aircraft's position relative to the horizon in the AGD-1 is natural, i.e., it corresponds to the image that the crew imagines about the aircraft's position relative to the ground. A rough roll reading is possible using a digitized fixed scale on the instrument body and the silhouette of the aircraft; on a scale 26 and the silhouette of the aircraft are approximately determined by the pitch angles. The AGD-1 indicator indication for roll and pitch is shown in Fig. 4.11. In our opinion, determining the aircraft position in AGD-1 is more convenient than in AGB-2 and AGI-1s.

The AGD-1 attitude indicator uses a special device called a arrester, which allows you to quickly bring the frame of the device and the gyromotor into a strictly defined position relative to the body of the device and, consequently, the aircraft. The kinematic diagram of the electromechanical remote locking device AGD-1 is shown in Fig. 4.14.

The device works as follows. When you press the red button 36 (see Fig. 4.13), located on the front side of the indicator, supplies voltage to the motor 34 (see Fig. 4.14. which, rotating, causes the rod to move forward 33 using a finger moving along the screw slot, i.e. the rotating nut is stationary, and the screw moves. Stock 33 via video 32 rests against an additional follower frame 7, which has a wedge-shaped ring 35.

Due to this profile of the ring, when there is pressure on the frame from the side of the rod, the ring 35 together with the gyro unit, rotates around the axis of frame 7 until the roller 32 will not be in the lower position of the ring. In this case, the plane of the frame 7 is parallel to the plane of the aircraft wings. Next stock 33 moves the profile bar 31, which rests on the fist 30 and creates a moment around the axis of the outer frame 8. Under the influence of this moment, the gyroscope precesses around the axis of the inner frame and reaches the stop, after which the precession stops and the gyroscope begins to rotate around the axis of the outer frame until the protrusion of the bar 31 will not fit into the cam cutout 30, thus fixing the frame 8 in a position in which the axis of the internal frame is parallel to the longitudinal axis of the aircraft.

At the same time, the finger 28, resting on cam 27, installs the inner frame 12 to a position in which the axis of the gyroscope’s own rotation is perpendicular to the axes of the external and internal frames of the gimbal. Then the rod 33 under the action of the return spring contained in it, it reclines to its original position and allows the bar 31 release the cams 27 And 30.

Thus, the arrester, having installed the frames of the gyro unit in a certain position, immediately releases them. If arresting is performed on the ground when the aircraft is horizontal, or in horizontal flight, then the gyro's own axis of rotation is set in the direction of the vertical position. Locking should be carried out only in horizontal flight, as the crew is reminded of by the inscription on the button 36 "Catch in level flight."

If you perform arresting, for example during a roll, then when transitioning to level flight the attitude indicator will show a false roll. True, under the influence of the pendulum correction, the gyroscope’s own axis will be set to a vertical position, and, naturally, false readings will disappear, but this will take time sufficient for the crew to make mistakes in piloting. It should be noted that the electrical locking circuit is designed in such a way that when the AGD-1 is turned on under voltage, the locking occurs automatically, without pressing a button. When re-arresting, for example during a temporary power failure of AGD-1, pressing the button 36 mandatory, but only during horizontal flight.

There is a warning light on the front side of the indicator 37 (see Fig. 4.13), which lights up, firstly, if the arresting process occurs and, secondly, if there is a malfunction in the power supply circuits of the gyromotor and DC ±27 V.

AIR HORIZONT AGB-3 (AGB-Zk)

The main purpose of the AGB-3 attitude indicator is to provide the crew with an easily perceived large-scale indication of the position of an airplane or helicopter in roll and pitch angles relative to the plane of the true horizon. In addition, the attitude indicator allows you to issue electrical signals proportional to the roll and pitch angles to external consumers on the airplane and helicopter (autopilot, heading system, etc.).

The attitude indicator AGB-Zk is a modification of the attitude indicator AGB-3. It differs only in the presence of built-in red light fixtures to illuminate the front part of the device and the coloring of the elements: indication.

The electromechanical diagram of the AGB-3 attitude indicator is shown in Fig. 4.15, electrical diagram - in Fig. 4.16, and a view of its scale is in Fig. 4.17. The gyroscope's own axis is brought to a vertical position by a pendulum correction system, which includes two electrolytic pendulums 20 And 21, controlling correction motors 7 and 9. AGB-3 uses single-coordinate: electrolytic pendulums, operating on the same principle as two-coordinate ones, which are used in AGB-2, AGI-lc and AGD-1. A single-axis pendulum has three contacts and responds to tilts in only one direction. There is a contact in the lateral correction circuit 16 correction switch VK-53RB, which breaks the circuit when the aircraft makes turns, reducing the turning error.

The readiness time of the device for operation in the attitude indicator is reduced by a mechanical arrester (it is not shown in Fig. 4.15). If the aircraft is in a horizontal position, then the arrester sets the frames of the gyroscope to its initial state, in which the main axis of the gyroscope coincides with the vertical position. The arrester is used before starting the device, when for one reason or another it is necessary to quickly bring the frame of the device to its original position. The lock in AGB-3 is of a push type, i.e. for it to work you need to press a button 26 (see Fig. 4.17) to failure. The frames are automatically released from the lock when the button is released.

The operation of the arresting device is similar to the operation of the arrester in the AGD-1 attitude indicator. The AGB-3 attitude indicator has a mechanical arrester.

To provide consumers with signals for aircraft deflection in roll and pitch, a synthetic sensor is installed on the axis of the external frame of the gimbal. 14 (see Fig. 4.15, 4.16), and on the axis of the inner frame there is a synthetic sensor 15.

On an airplane, the attitude indicator is installed in such a way that the axis
outer frame 8 (see Fig. 4.15) is directed parallel to the longitudinal axis of the aircraft. This ensures the device operates in a roll range of 360°.

The axis of the internal frame of the gimbal is parallel to the transverse axis of the aircraft at the initial moment. Since additional

Since the AGB-3 does not have a tracking frame, like the AGI-lc and AGD-1, the operating pitch range in this attitude indicator is limited to angles of ±80°. Indeed, if the plane has a pitch angle of 90°, then the axis of the external frame will align with the axis of the gyroscope’s own rotation. The gyroscope, having lost one degree of freedom, becomes unstable. However, to provide the crew with a correct indication of the position of the aircraft relative to the horizon plane in an inverted state (for example, when performing the “Nesterov loop” figure), stops are used in the device 10 And 11 (see Figure 4.15). When performing complex evolutions in an aircraft with a pitch angle of more than 80°, the stop 10, located on the outer frame, will begin to press against the stop 11, fixed on the axis of the internal frame. This creates a moment around the axis of the inner frame. According to the law of precession, the gyroscope, under the influence of this moment, precesses, i.e., rotates around the axis of the external frame, trying to align the axis of its own rotation with the axis of application of the moment over the shortest distance. Thus, the external cardan frame is under. The weight rotates 180°. When the pitch angle is more than 90°, stop 11 will move away from the stop 10, precession will stop, and the silhouette of the airplane 4 will be flipped 180° relative to the pitch scale 3, which will indicate the inverted position of the aircraft by 180 relative to the horizontal plane.

Indication of the aircraft's position relative to the horizon plane in AGB-3 is carried out as follows. During rolls, the body of the device, together with the aircraft, rotates around the axis of the outer frame by a roll angle, since the gyroscope’s own axis of rotation maintains a vertical direction. Airplane silhouette 4 At the same time, it participates in two movements: 1) portable - together with the device body to the roll angle at(Fig. 4.18) and 2) rotational (tribe 6 rolls the trib 5) motionless in roll to the same angle Y. As a result of these two movements, the silhouette of the aircraft in space rotates through double the roll angle of the aircraft. The crew observes the bank angle based on the movement of the airplane silhouette 4 relative to scale 3. In this case, the silhouette turns to a natural bank angle in the same direction as the aircraft.

Roll angles can be roughly measured using a scale 27 on the instrument body, and pitch angles - on the scale 3 and the silhouette of an airplane 4. The pitch scale follows the aircraft's pitch angles thanks to a tracking system that includes a synchronizer sensor 15, located on the internal axis of the cardan suspension, synchronizer receiver 19, amplifier 17 and motor-generator 18. In the slot of the scale.3 there is an axis on which the silhouette of the aircraft is attached.

Thus, the readings in AGB-3 for roll and pitch are natural and identical to the readings of AGD-1 (see Fig. 4.11).

AGB-3 has a failure signaling circuit in the device power supply circuits, containing the following elements: power failure motor 1 with checkbox 2 (see Fig. 4.15 and 4.16) and two relays 22 And 23. Motor windings 1 connected in series with the stator windings of the gyromotor 13. When the 36 V AC circuits are in good working order, the currents of the gyromotor and synchronous sensors flow through the motor windings 14 And 15.

As a result, torque occurs on the motor shaft 1, under the influence of which the checkbox 2 The signaling device mounted on the motor shaft is removed from the visible area of ​​the front part of the device.

If there is no AC voltage in the power supply circuit of the gyromotor or a phase loss occurs, then the motor torque drops sharply and, under the influence of a spring, the flag is thrown into the visible area of ​​the front part of the device.

Relay 22 And 23 are connected in parallel to the power supply circuit of the pitch tracking system amplifier. In the absence of 27 V DC voltage, the contacts 24 And 25 these relays close, shunting the two phases of the windings of motor 1, therefore, its torque decreases, and the spring throws a flag 2, which signals a power failure.

Thus, an open circuit in a circuit with a voltage of 36 V, a frequency of 400 Hz or in a circuit with a voltage of 27 V, as well as the absence of one of these types of power supply, can be determined by the presence of an indicator flag in the field of view of the instrument scale.

AVIAHORIZONT AGK-47B

The attitude indicator is combined, since three instruments are mounted in one housing: an attitude indicator, a turn indicator and a slip indicator.

The purpose of the attitude indicator is to provide the crew with information about the position of the aircraft relative to the horizon plane. The turn indicator is used to determine the direction in which the aircraft is turning, and the slip indicator measures slip. The direction indicator is discussed in section. 4.2, and the slip indicator - in section. 3.11. Simplified kinematic, electrical diagrams and the front side of the attitude indicator are presented in Fig. 4.19, 4.20, 4.21; All symbols in the figures are the same.

The own axis of rotation of the gyroscope 7 (see Fig. 4.19, 4.20) is brought to a vertical position using a pendulum correction system, which includes an electrolytic pendulum, /6 and two solenoids 13 And 14, Solenoid 13 located perpendicular to the external axis at gimbal suspension, and the solenoid 14 - perpendicular to the internal axis X cardan suspension on the internal frame 6, made in the form of a casing. Each of the solenoids has two windings, which create magnetic fields in the opposite direction when currents pass through them. Solenoids have metal cores that are able to move within the solenoids. If the gyroscope’s own axis of rotation coincides with the direction of the local vertical, then the same signals are received from the electrolytic pendulum to the solenoid windings and the cores, being in the middle position, do not create moments around the gimbal axes. When the main axis of the gyroscope deviates from the vertical direction, the currents flowing through the windings of the solenoids will not be equal due to unequal resistances between the contacts of the electrolytic pendulum. This will lead to movement of the cores in the solenoids, and due to their weight around the axes of the gimbal, moments will arise that will return the axis of the gyroscope’s own rotation to a vertical position. So solenoid 14 participates in creating torque around the internal axis of the gimbal, and the solenoid 13 - around the external axis of the suspension.

The external axis of the attitude indicator's gimbal is parallel to the transverse axis of the aircraft, so the pitch is indicated on a circular scale 4, associated with the external frame of the gimbal 5, and the horizon line associated with the body of the device. When diving or pitching up, the horizon line moves relative to a fixed scale - the pilot sees the opposite picture: the silhouette of an aircraft 1 along with the scale 4 falls or rises relative to the horizon line. The roll indication is carried out by the relative position of the silhouette of the aircraft / associated with the internal frame of the gimbal, and the scale 3, mounted on the external gimbal frame. In order for the roll indication to be natural, that is, the silhouette of the aircraft simulates a roll relative to the horizon plane, just like in the AGB-3, the AGK.-47B uses a pair of gears with a gear ratio of 1:1. The pitch scale is marked at 20° intervals, and the roll scale is marked at 15° intervals. The roll and pitch indication of the AGK-47B during aircraft evolutions is shown in Fig. 4.11.

The attitude indicator has a mechanical lock of a fixed type, i.e. if in the AGB-3 and AGD-1 the lock works only when the button is pressed, then in the AGK-47B it is possible by extending the lock rod 20 (Fig. 4.21) towards yourself, fix it in this position. When the device is locked, a red flag with the inscription “Locked” appears on the front side of the device. When the device is locked, the axis of the gyroscope’s own rotation coincides with the vertical axis of the aircraft, and the axes at and x coincide, respectively, with the longitudinal and transverse axes of the aircraft. On the lock control handle it is written “Pull the lock”.

Using a ratchet 22 It is possible, within certain limits, to change the position of the artificial horizon line relative to the instrument body, which is sometimes advisable to do for the convenience of maintaining the pitch flight path during a long non-horizontal flight.

Like any attitude indicator, the AGK-47B is subject to a turn error, but due to the fact that it is intended for installation on light-engine aircraft, where there may not be a correction switch, the correction cannot be turned off in it. At the same time, to reduce errors during a left turn, the device is designed in such a way that the normal position of the axis of its own rotation is its inclined position forward, along the flight, by 2°. The decrease in error specifically for the left turn can probably be explained by the fact that aircraft more often make left turns, since the pilot sits in the cockpit on the left seat. Indeed, during a left turn, the electrolytic pendulum will show an apparent vertical, which deviates into the turn at an angle

where ω is the angular velocity of the turn; V- aircraft flight speed; g- acceleration of gravity.

Under the influence of the lateral correction system using a solenoid 13 the gyroscope will begin to precess towards the apparent vertical at a speed

At the same time, when turning, the end of the gyroscope’s own rotation axis will rotate around the position of the true vertical at a speed

(4.5)

where α 0 is the initial angle of inclination of the axis of its own rotation of the gyroscope forward (Fig. 4.22), directed in the opposite direction, since the gyroscope strives to maintain the position of the axis of its own rotation in space unchanged. The direction of the speed ω γ is opposite to the direction of the gyroscope precession speed β.

Obviously, in order for there to be no error during a left turn, the condition must be met

or for small angles β 0 (4.6) can be written

(4.7)

(4.8)

Knowing K y attitude indicator and the most common speeds at which a turn occurs, you can determine the required angle α 0 of inclination of the gyroscope axis.

AIR HORIZONT AGR-144

The AGR-144 attitude indicator is a combined instrument; It contains three instruments: an attitude indicator, a turn indicator and a slip indicator.

The purpose of the attitude indicator is to provide the crew with information about the position of the aircraft relative to the horizon plane. The direction indicator is used to determine the presence and direction of the aircraft's turn around its vertical axis. The slip indicator measures the aircraft's slip. In addition, when coordinated

The section is very easy to use. In the field provided, just enter the right word, and we will give you a list of its values. I would like to note that our site provides data from various sources - encyclopedic, explanatory, word-formation dictionaries. Here you can also see examples of the use of the word you entered.

Meaning of the word pitch

pitch in the crossword dictionary

Encyclopedic Dictionary, 1998

pitch

PITCH (French tangage - pitching) is the angular movement of an aircraft or vessel relative to the transverse (horizontal) axis.

Pitch

(French tangage ≈ pitching), the angular movement of an aircraft or ship relative to the main transverse axis of inertia. Angle T. ≈ the angle between the longitudinal axis of an aircraft or ship and the horizontal plane. In aviation, a distinction is made between pitches with an increasing angle (pitch up) and with a decreasing angle (dive); caused by deflection of the elevator.

Wikipedia

Pitch

Pitch- angular motion of an aircraft or ship relative to the main transverse axis of inertia. Pitch angle - the angle between the longitudinal axis of the aircraft or ship and the horizontal plane. The pitch angle is denoted by θ. In aviation there are:

• positive pitch, with increasing angle - pitching up , steering wheel to yourself;
• negative, with decreasing angle - dive , steering wheel away from you.

Caused by elevator deflection.

This is one of the three angles (roll, pitch and yaw), which specify the inclination of the aircraft relative to its center of inertia along three axes. In relation to seagoing vessels, the term “trim” is used with the same meaning. It is noteworthy that the trim has opposite ideas about positivity/negativity.

Examples of the use of the word pitch in literature.

Moreover, if maintaining the course is carried out practically without much difficulty, then maintaining the glide path is associated with solving the complex problem of longitudinal balancing of the aircraft in terms of speed, engine operating mode and pitch, however, due to less distraction in selecting and maintaining a course, this task is easier to solve.

If the vertical speed is not taken into account, as well as the swings that usually accompany its jumps pitch, then, if the course and glide path are formally maintained, and the indicated speed is constant, it is still quite possible in front of the end to have an undesigned high vertical speed, the correction of which makes an adjustment to the glide path maintenance, and the correction of an error in glide path maintenance can result in an already undesigned vertical speed.

As I gained experience, I realized that the basis of a soft landing is strict adherence to the course, which means freeing up my mental abilities to analyze the behavior of the machine along the longitudinal channel: pitch, glide path, thrust, vertical speed.

Sensitive gyroscopic sensors detect vibrations of the aircraft around three conventional axes and send signals to deflect certain rudders to correct the roll, pitch or course.

While all these manipulations are going on, I fix the angle using the artificial horizon pitch, I monitor the speed and variometer and out of the corner of my eye I notice the red landing gear warning lights going out.

In this case, it will be very problematic to accelerate the car to such a speed at which it is possible to remove the engine mode from the nominal one, and the aircraft will reduce pitch to acceptable drag.

Very low and very clear alignment, with a clear fixation of the landing pitch, grinds against the concrete inaudibly.

Sudden disconnection of the autopilot with an accumulated error of unbalanced roll forces and pitch can lead to an energetic throw of the aircraft towards the direction of the released rudders.

If the increase in vertical speed is associated with suction under the glide path, then the director needle will vigorously go up at the same pitch and at the same speed.

This confidence is that a heavy machine approaches the concrete with a small vertical speed, ensuring a soft landing, and that the reduction of this vertical speed on leveling is ensured by sufficient controllability pitch.

Upon reaching a speed of 550, a constant rate of climb is established, the aircraft is trimmed according to pitch, and then the indicated speed is maintained by lightly pressing the trimmer.

So, in addition, hammer home to the student that it is better to hang himself and swing in a noose than to swing pitch in front of the ground.

As soon as the slats were retracted, the speed jumped over 500, and the further climb, with a hundred passengers in the cabin, was carried out lying on the back: pitch 20 degrees, the variometer, having scrolled the circle with the arrow, froze at 33.

I removed the spoilers and started balancing again with trimmers: pitch, roll.

Exactly takeoff pitch and - out of the corner of the eye - the variometer determines the termination of taking the helm.

Basic dynamic forces

A jump is a complex concept: the result of the interaction of two or more variables, the action of the laws of physics and man. To understand how this interaction occurs, we need to consider each quantity separately.

"Magnet under the table"

If I scattered metal filings on the table, you would probably look at me in surprise. But if I placed a magnet under the surface of the table and moved it, you would think I was a magician. Of course, there are no miracles here. This is a simple operation of the laws of physics. An obvious reality is the movement of metal filings across the surface of the table for no apparent reason. In fact, the magnet acts on the sawdust as it should act without any interference from otherworldly forces. Approximately the same thing happens with flight. Until we understand the underlying dynamic forces, we will assume that some kind of miracle is happening. To learn to fly, you must understand how these forces work.

It is necessary to learn to understand the situation as a whole. Let's take birds, for example. They are considered not the smartest in the world. They haven't even been to kindergarten, yet they have a comprehensive understanding of the basic principles of flight, allowing them to fly safely and more gracefully than a human can. Maybe we are thinking too much? However, man can fly. We can learn to understand situations and relationships. It is our rational understanding of the principles of flight that makes it possible. We will never get to where our thoughts have not already been. When you have thought about it and analyzed it, you realize that there are a huge number of parts that control the flying body. We must study each component part of the jump, look at it under a microscope in order to understand how the whole is formed from the individual parts. I suggest starting with learning the language of flight.

Spatial orientation language

Various variables related to flight require clarification (definition), which can be done using language. This language is very specific to aviation, when ordinary and familiar words take on a different meaning depending on the specific situation.

Roll, pitch and yaw

Orientation or location should only be understood in relation to something. This “something” is the celestial body closest to us, i.e. the Earth. When we start skydiving on others celestial bodies With less gravity than that of the earth, we will determine our location in relation to the nearest planets. The system we use to determine our location requires the construction of three orientation axes. Let's simplify our task by taking the human body to be a flying body. If you spread your arms out to the sides, your arms will represent the “Pitch Axis.” Off-axis deviation can be demonstrated by tilting the body forward and backward. The "Roll Axis" is the pole that runs through your chest. Deviation from this axis will be tilts to the sides. The third axis is the “Yaw Axis” (the axis of rotation in the horizontal plane around the vertical axis). It can be thought of as a pole running through your body from the top of your head to your feet. A deviation from this axis will be a pirouette turn to the right or left.

Let's check your understanding of these terms using specific examples. Imagine that you are an airplane flying at a certain altitude. If you are asked to pitch downward, you will force the airplane's nose down. Increasing the axle will cause you to lift the nose up in relation to the tail. If you need to roll to the right, you lower the right wing and raise the left. "Yaw" to the right would be a simple turn to the right in the horizontal plane.

Attention! This site is not updated. New version: shatalov.su

Transformations: The Last Stand

Creation date: 2009-10-20 03:43:37
Last edited: 2012-02-08 09:36:52

Preliminary lessons:
1. Trigonometry. Go.
2. Vectors. Go.
3. Matrices. Go.
4. Coordinate spaces. Go.
5. Transformations of coordinate spaces. Go.
6. Perspective projection. Go.

We haven’t thought about transformations for a long time! Perhaps, my dear reader, you already miss them? As practice shows, transformations are the most favorite topic among students of 3D programming.

By now you should have a good understanding of conversions.

45. The principle of operation of the roll, pitch and yaw channels of the autopilot.

If not, then watch the preliminary lessons.

When we first started studying transformations, I wrote that with the help of matrices you can manipulate objects in space: move, rotate, enlarge. If you have studied all the previous lessons and tried to apply the acquired knowledge in practice, then most likely you had to face certain difficulties: how to move objects in an arbitrary direction, how to create a matrix for transformation into camera space, how to rotate objects in an arbitrary direction direction?

We will consider these issues today.

Moving in space

A small note: We will denote the world coordinate space by the x, y, z axes. We will denote the basis vectors forming the local (object, camera) space as i=(1,0,0), j=(0,1,0), k=(0,0,1) (vector names read as: And, live, ka). Vector i— parallel to the x-axis, vector j— y-axis, vector k— z axis.

Let me remind you that using a linear combination (sum) of basis vectors you can express any vector in space. Also, do not forget that the length of the basis vectors is equal to one.

Now look at the picture:

For simplicity, we discarded one dimension - vertical. Accordingly, the pictures show a top view.

Let's say we are at some point in world space. In this case, the pronoun “we” can mean anything: an object in the game world, a character, a camera. In this case ( Fig.a) we look towards the point A. How do we know that the “gaze” is directed towards the point A? Well, when we discussed cameras, we agreed that the vector k indicates the direction of view.

We are separated from the center of the world (world coordinate space) by a vector v. And suddenly! We really wanted to get to the point A. First thought: remove the value (dz) from the forward arrow and add it to the third component of the vector v. The result of this misunderstanding can be seen in Fig.b. It would seem that everything is lost - goodbye to dreams of your own Quake. Stop panicking! You just need to think carefully about the current situation.

Let's imagine that we are already at the point A- let's look at Fig.c. As can be seen from the figure, after moving the vectors k And i not changed. Accordingly, we will not touch them.

Let's look at the rest of the picture: vector v after moving, this is the sum of two vectors: vector v before moving and a vector unknown to us, coinciding in direction with the vector k... But now we can easily find the unknown vector!

If you carefully studied the lesson about vectors, then you remember that multiplying a scalar by a vector increases (if the scalar is greater than one) the vector. Therefore the unknown vector is equal to k*dz. Accordingly, the vector v after moving can be found using the formula:

Well, isn't it simple?

Rotation around axes

We already know the formulas for rotation around axes. In this section I will simply explain them more clearly. Let's consider the rotation of two vectors around the center of coordinates in two-dimensional space.

Since we know the rotation angle (angle alpha), then the coordinates of the basis vectors of space can be easily calculated using trigonometric functions:

i.x = cos(a); i.z = sin(a); k.x = -sin(a); k.y = cos(a);

Now let's look at the rotation matrices around axes in three-dimensional space and the corresponding illustrations.

Rotation around x axis:

Rotation around y axis:

Rotation around the z axis:

The figures show which vectors change their coordinates.

A small note: It is incorrect to talk about rotation around axes. Rotation occurs around vectors. We do not know how to represent straight lines (axes) in computer memory. But vectors are easy.

And one more thing: how is positive and negative rotation angle determined? This is easy: you need to “stand” at the center of coordinates and look towards the positive direction of the axis (straight line). Counterclockwise rotation is positive, clockwise rotation is negative. Accordingly, in the figures above, the rotation angles around x and y are negative, and the rotation angle around the z axis is positive.

Rotation around an arbitrary line

Imagine this situation: you rotate the camera using the matrix around the x-axis (tilt the camera) by twenty degrees. Now you need to rotate the camera twenty degrees around the y axis. Yes, no problem, you say... Stop! Now what do you need to rotate the object around? Around the y-axis, which was before the previous rotation or after? After all, these are two completely different axes. If you simply create two rotation matrices (about the x-axis and about the y-axis) and multiply them, the second rotation will be about the original y-axis. What if we need a second option? In this case, we will need to learn how to rotate objects around an arbitrary line. But first, a little test:

How many vectors are there in the following picture?

The correct answer is three vectors. Remember: vectors are length and direction. If two vectors in space have the same length and direction, but are in different places, then we can assume that they are the same vector. In addition, in the figure I depicted the sum of vectors. Vector v = v 1 + v 2 .

In the vectors lesson, we briefly looked at the dot product and cross product of vectors. Unfortunately, we did not study this topic in more detail. The formula below will use both the dot product and the cross product. Therefore, just a few words: the value of the scalar product is the projection of the first vector onto the second. With the cross product of two vectors: a x b = c, vector c perpendicular to the vectors a And b.

Let's look at the following figure: a vector is defined in space v. And this vector needs to be rotated around the straight line l (el):

We don't know how to represent straight lines in programs. Therefore, we will represent the straight line as a unit vector n, which coincides in direction with straight line l (el). Let's look at a more detailed drawing:

What we have:
1. Line l represented by a vector of unit length n. As mentioned above, vector rotation v will be carried out around a vector, not a straight line.
2. Vector v, which needs to be rotated around the vector n. As a result of rotation, we should get a vector u(read as at).
3. The angle by which the vector needs to be rotated v.

Knowing these three quantities, we must express the vector u.

Vector v can be represented as the sum of two vectors: v = v ⊥ + v|| . In this case, the vector v || — parallel to the vector n(you can even say: v || - projection v on n), and the vector v⊥ perpendicular n. As you might guess, you only need to rotate the one perpendicular to the vector n vector part v. That is - v ⊥ .

There is one more vector in the figure - p. This vector is perpendicular to the plane formed by the vectors v|| And v ⊥ , |v ⊥ | = |p| (the lengths of these vectors are equal) and p = n x v.

u ⊥ = v⊥ cosa + p sina

If it's not clear why u⊥ is calculated exactly like this, remember what sine and cosine are and what is multiplication of a scalar value by a vector.

Now we need to remove from the last equation v⊥ and p. This is done using simple substitutions:

v || = n(v · n) v ⊥ = v — v || = v — n(v · n) p = n x vu || = v || u ⊥ = v⊥ cosa + p sina = ( v — n(v · n))cosa + ( n x v)sina u = u ⊥ + v || = (v — n(v · n))cosa + ( n x v)sina + n(v · n)

What a squiggle!

This is the vector rotation formula v by the angle a (alpha) around the vector n. Now with this formula we can calculate the basis vectors:

Exercises

1. Required: substitute the basis vectors into the formula for rotating a vector around an arbitrary line. Count (using a pencil and a piece of paper). After all the simplifications, you should end up with basis vectors like in the last picture. The exercise will take you about ten minutes.

That's all.

Roman Shatalov 2009-2012

Introduction.
Quaternion
Basic operations on quaternions.
Unit length quaternions
Interpolation
Conversion from two directions
Composition of rotations
Physics

Introduction.

Let's briefly define the terminology. Everyone has an idea of ​​what an object's orientation is. The term "orientation" implies that we are in some given frame of reference. For example, the phrase “he turned his head to the left” is meaningful only when we imagine where “left” is and where the head was before. This is an important point to understand, because if it were a monster with its head on its stomach with the top of its head down, then the phrase “he turned his head to the left” would no longer seem so unambiguous.

A transformation that rotates in a certain way from one orientation to another is called a rotation. Rotation can also be used to describe the orientation of an object if you enter a default orientation as a reference point. For example, any object described by a set of triangles already has a default orientation. The coordinates of its vertices are described in the local coordinate system of this object. The arbitrary orientation of this object can be described by a rotation matrix relative to its local coordinate system. You can also distinguish such a concept as “rotation”. By rotation we mean a change in the orientation of an object in a given way over time. To uniquely define rotation, it is necessary that at any moment in time we can determine the exact orientation of the rotated object. In other words, rotation specifies the "path" taken by an object when changing orientation. In this terminology, rotation does not specify an unambiguous rotation of the object. It is important to understand that, for example, the matrix does not specify a unique rotation of the body; the same rotation matrix can be obtained by rotating the object 180 degrees around a fixed axis and 180 + 360 or 180 - 360. I use these terms to demonstrate the differences in concepts , and in no way do I insist on using it. In the future I will reserve the right to say “rotation matrices”.

The word orientation often evokes an association with direction. You can often hear phrases like “he turned his head towards the approaching locomotive.” For example, the orientation of a car could be described by the direction in which its headlights are facing. However, direction is given by two parameters (for example, as in a spherical coordinate system), and objects in three-dimensional space have three degrees of freedom (rotation). In the case of a car, he can look in one direction while standing on wheels, or lying on his side or on the roof. Orientation can indeed be set by direction, but you will need two of them. Let's look at targeting simple example human head.

Let's agree on the starting position in which the head is oriented by default (without rotation). As the initial position, we will take the position in which the head looks with its face in the direction of the “z” axis, and looks upward (with the crown) in the direction of the “y” axis. Let's call the direction in which the face is turned "dir" (without rotation it coincides with "z"), and the direction in which the top of the head is looking "up" (without rotation it coincides with "y"). Now we have a reference point, there is a local coordinate system of the head “dir”, “up” and a global one with axes x, y, z. Let's randomly turn our head and note where the face is looking. Looking in the same direction, you can rotate your head around an axis that coincides with the direction of view "dir".

For example, by tilting our head to the side (with our cheek pressed to our shoulder), we will look in the same direction, but the orientation of the head will change. To fix the rotation around the direction of view, we also use the “up” direction (directed towards the top of the head). In this case, we have unambiguously described the orientation of the head and will not be able to rotate it without changing the direction of the “dir” and “up” axes.

We looked at a fairly natural and simple way to set orientation using two directions. How can we describe our directions in the program so that they are convenient to use? A simple and familiar way to store these directions as vectors. Let's describe the directions using vectors of length one (unit vectors) in our global xyz coordinate system. The first important question is how to convey our directions in an understandable form to the graphical API? Graphics APIs work primarily with matrices. We would like to obtain a rotation matrix from the available vectors. The two vectors describing the direction “dir” and “up” are the same rotation matrix, or rather two components of the 3x3 rotation matrix. We can obtain the third component of the matrix from the vector product of the vectors "dir" and "up" (let's call it "side"). In the head example, the "side" vector will point in the direction of one of the ears. The rotation matrix is ​​the coordinates of the three vectors "dir", "up" and "side" after the rotation. Before rotation, these vectors coincided with the axes of the global xyz coordinate system. It is in the form of a rotation matrix that the orientation of objects is very often stored (sometimes the matrix is ​​stored in the form of three vectors). The matrix can be used to specify orientation (if the default orientation is known) and rotation.

A similar way of representing orientation is called Euler Angles, with the only difference that the direction "dir" is specified in spherical coordinates, and "up" is described by a single angle of rotation around "dir". As a result, we get three angles of rotation around mutually perpendicular axes. In aerodynamics they are called Roll, Pitch, Yaw or Bank, Heading, Attitude. Roll is a tilt of the head to the right or left (towards the shoulders), a rotation around an axis passing through the nose and back of the head. Pitch is the tilt of the head up and down, around an axis passing through the ears. And Yaw is turning the head around the neck. We must remember that rotations in three-dimensional space are not commutative, which means that the order of rotations affects the result. If we rotate to R1 and then to R2, the orientation of the object is not necessarily the same as the orientation when rotated to R2 and then to R1. This is why the order of rotations around the axes is important when using Euler Angles. Please note that the mathematics of Euler angles depends on the chosen axes (we used only one of the possible options), on the order of rotation around them, and also on the coordinate system in which the rotations are made, in the global or local object. Euler angles can store both rotation and rotation.

A huge drawback of this representation is the lack of a rotation combination operation. Do not try to add Euler angles component by component. The resulting rotation will not be a combination of the original rotations. This is one of the most common mistakes novice developers make. To rotate an object by storing the rotation in Euler angles, we will have to translate the rotation into another form, such as a matrix. Then multiply the matrices of the two rotations and extract the Euler angles from the final matrix. The problem is further complicated by the fact that in special cases the direct addition of Euler angles works. In the case of a combination of rotations around the same axis, this method is mathematically correct. Rotating it 30 degrees around the X axis and then rotating it 40 degrees around X again gives us a 70 degree rotation around X. In the case of rotations along two axes, simple addition of angles can give some “expected” result.

Roll, pitch and yaw

But as soon as rotation along the third axis appears, the orientation begins to behave unpredictably. Many developers spend months of work to get the camera to work “correctly”. I recommend paying close attention to this shortcoming, especially if you have already decided to use Euler angles to represent rotations. It seems to novice programmers that using Euler angles is the easiest way. Let me express my personal opinion that the mathematics of Euler angles is much more complex and insidious than the mathematics of quaternions.

Euler angles are a combination (composition) of rotations around the base axes. There is another, simple way to specify rotation. This method can be called a "mixture" of rotations around the base coordinate axes, or simply rotation around an arbitrary fixed axis. The three components describing the rotation form a vector lying on the axis around which the object rotates. Typically, the rotation axis is stored as a unit vector and the rotation angle around this axis is stored in radians or degrees (Axis Angle). By selecting the appropriate axis and angle, you can set any orientation of the object. In some cases it is convenient to store the rotation angle and axis in one vector. The direction of the vector in this case coincides with the direction of the axis of rotation, and its length is equal to the angle of rotation. In physics, angular velocity is thus stored. A vector whose direction is the axis of rotation and whose length represents the speed in radians per second.

Quaternion

After brief overview About representations of orientation, we can move on to getting acquainted with the quaternion.

Quaternion- these are four numbers that were introduced into circulation (according to historians) by William Hamilton in the form of a hypercomplex number. In this article I propose to think of a quaternion as four real numbers, for example as a 4d vector or 3d vector and a scalar.

q = [x, y, z, w] = [v, w]

There are other representations of the quaternion that I will not consider.
How is rotation stored in a quaternion? Almost the same as in the "Axis Angle" representation, the first three components represent a vector lying on the axis of rotation, and the length of the vector depends on the angle of rotation. The fourth component depends only on the magnitude of the rotation angle. The dependence is quite simple - if you take a unit vector V for the axis of rotation and the angle alpha for the rotation around this axis, then the quaternion representing this rotation
can be written as:

q = [ V*sin(alpha/2), cos(alpha/2) ]

To understand how a quaternion stores rotation, let’s remember about two-dimensional rotations. Rotation in a plane can be specified by a 2×2 matrix, in which the cosines and sines of the rotation angle will be written. You can think of a quaternion as storing a combination of an axis of rotation and a matrix of half the rotation around that axis.

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#quaternions, #mathematics