Orientation and angle displacement in 3D (2)

Another common method to describe the azimuth is ouarla. This technique is named by the famous mathematician Leonhard Euler (1707-1783). He proves that the angle displacement sequence is equivalent to a single angle displacement.

 

What is ouarla

The basic idea of the orah angle is to break down the angle displacement into a sequence composed of three rotations round three mutually vertical axes. This sounds complicated, but it is actually very intuitive (in fact, ease of use is one of its main advantages ). The reason why there is "Angular Displacement" is that the orah angle can be used to describe arbitrary rotation.

Which of the three axes is the rotation of the three vertical axes? In what order? In fact, any three axes and any order can be used, but the most significant thing is the rotation sequence consisting of Cartesian coordinate systems and in a certain order. The most common convention is the so-called "heading-pitch-bank" convention. In this system, an orientation is defined as a heading angle, a pitch angle, and a bank angle. The basic idea is to make the object begin with the "standard" orientation-that is, the coordinate axis of the object is aligned with the inertial coordinate axis. In terms of the Standard Orientation, let the object rotate as heading, pitch, bank, and finally reach the orientation we want to describe.

10.4. At this time, the object coordinate system and the inertial coordinate system coincide. heading is the amount of rotation around the Y axis, and the right rotation is positive (if from above, the positive direction of rotation is clockwise ).

After the heading rotation, pitch is the amount of rotation around the X axis. Note that the X axis of the object coordinate system is not the X axis of the original inertial coordinate system. Still follow the left-hand rule, and the downward rotation is positive, as shown in 10.5:

Finally, after heading and pitch, the bank is the amount of rotation around the Z axis. Again, it is the Z axis of the object coordinate system, not the Z axis of the original inertial coordinate system. According to the left-hand rule, the clockwise direction is positive from the origin point to + Z. 10.6:

Remember, when we say the order of rotation is heading-pitch-bank, it refers to from the inertial coordinate system to the object coordinate system. If you change from an object coordinate system to an inertial coordinate system, the order of rotation is the opposite. "Heading-pitch-bank" is also called "roll-pitch-Yaw". Roll is similar to bank, and yaw is similar to heading (in fact, yaw is not strictly equal to heading ). Note: In the roll-pitch-yaw system, the angle naming order is consistent with the rotation order from the object coordinate system to the inertial coordinate system.

 

Other agreements on oarla

The heading-pitch-bank system is not the only Oula Kok system. Any rotation around any three vertical axes of each other can define a azimuth. Therefore, multiple choices lead to the diversity of the orah angle conventions:

(1) The heading-pitch-bank system has multiple names. Of course, different names do not represent different conventions, which is not important. A group of commonly used terms is roll-pitch-yaw, where roll is equivalent to bank, and yaw is basically equivalent to heading. Note that the order is the opposite to that of heading-pitch-bank, which is only semantic. It defines the rotation sequence in which a vector is transformed from an object coordinate system to an inertial coordinate system. (In fact, there is still a technical difference between yaw and heading. Yaw rotates around the Y axis of the object coordinate system and heading rotates around the Y axis of the inertial coordinate system. Because the rotation here is performed when the Y axis of the object coordinate system and the Y axis of the inertial coordinate system are duplicated, the difference is not important .)

(2) Any three axes can be used as the rotation axis. Not necessarily Cartesian axes, but Cartesian axes are the most meaningful.

(3) When determining the positive direction of each rotation, the Left or Right Hand Rule may not be observed. For example, you can define that the positive direction of the pitch is upward, and this definition method is very common.

(4) It is also the most important. rotation can be performed in different order. Order is not important. Any system can be used to define a direction, but heading-pitch-bank order is the most practical. The main reason why heading measures rotate around the vertical axis is that our environment often has some form of "ground ". Generally, the rotation around the X or Z axis of the inertial coordinate system is meaningless. The other two angles in the heading-pitch-bank order mean that the pitch measures the horizontal direction of the angle, and the bank measures the amount of rotation around the Z axis.

 

Advantages

The orah corner uses only three numbers to express the azimuth, and these three numbers are all angles. These two features enable Orla Dorado to have advantages that are not available in other forms:

(1) It is easy to use for us. It is much simpler than matrix and Quaternary, probably because the numbers in the angle are all in line with the way people think about the orientation. If we select the most consistent agreement with the situation to be handled, we can directly describe the most important angle, for example, the heading-pitch-bank system can be used to directly describe the deviation angle. Ease of use is its biggest advantage. When you need to display the orientation or use the keyboard to enter the orientation, the ouarla corner is the only choice.

(2) The most concise expression. The orah corner uses three numbers to express the azimuth. In 3D, the expression orientation cannot be less than three numbers. If you need to consider the memory factor, the Euclidean angle is the most suitable method to describe the azimuth.

(3) Any three numbers are valid. Take any three numbers, they can constitute a valid orah, and can be considered as a description of the azimuth. On the other hand, there is no "illegal" orah. Of course, the values may not be correct, but at least they are valid. This is not necessarily the case with the support matrix and the Quaternary element.

 

Disadvantages of ouarla

The following are the main disadvantages of expressing the orientation with the orah angle:

(1) The expression of the given orientation is not unique.

(2) interpolation between two angles is very difficult.

Let's discuss these issues carefully. The first problem is that for a given azimuth, there are multiple orazs to describe it. This is called an alias problem, which sometimes causes problems. For this reason, there are some basic problems (for example, "are the two groups of ouarla angles representing the same angle displacement? ") Are difficult to answer.

First, when we add an angle multiple of 360 degrees, we will encounter the simplest form alias problem. Obviously, adding 360 degrees does not change the orientation, although its value has changed.

Second, the more troublesome alias problem is caused by the fact that the three angles are not independent of each other. For example, pitch135 is equivalent to heading180 degrees, pitch45 degrees, and bank180 degrees. To ensure that all directions are unique, the angle range must be limited. A common technique is to limit heading and bank to + 180 degrees to-180 degrees, and pitch to + 90 degrees to-90 degrees. This sets up a "restriction scope" of the orah ". There is only one restriction on any location that can represent this location (in fact, there is another violation of uniqueness that needs to be addressed .)

The most famous alias problem in ouarla is the following: First heading45 degrees and then pitch90 degrees, which is equivalent to the first pitch90 degrees and then bank45 degrees. In fact, once + (-) 90 degrees is selected as the pitch angle, it is limited to only rotating around the vertical axis. This phenomenon, the angle is + (-) 90 degrees of the second rotation so that the first and third rotation of the rotating axis is the same, calledUniversal lock. In order to eliminate this alias that limits the ouarla corner, it is required that the heading completes all rotation around the vertical axis in case of a universal lock. In other words, if the pitch is + (-) 90 degrees, the bank is 0.

If you want to describe the location, especially when you use a restriction on the orah corner, the alias will not cause too many problems. Now let's look at the interpolation problem between two orientations A and B. That is to say, if the given parameter T, 0 ≤ T ≤ 1, calculate the temporary orientation C. When T changes from 0 to 1, C also smoothly changes from A to B.

The simple solution to this problem is to perform standard linear interpolation for three angles respectively. The formula is as follows:

But there are many problems.

First, if there is no limit on the use of the oaram, it will get a very large angle difference. For example, the heading of A is 720 degrees, the heading of B is 45 degrees, 720 = 360X2, that is, 0 degrees. Therefore, the heading value is only 45 degrees different, but the simple interpolation will go around for nearly two weeks in the wrong direction. 10.7:

The solution to the problem is to restrict the use of ouarla, however, even the restriction of ouarla can not completely solve the problem. The second problem of interpolation is caused by the periodicity of the rotation angle. Set heading of a to-170 degrees, and heading of B to 170 degrees. These values are between-180 degrees and 180 degrees within the range of heading. The two values are only 20 degrees different, but the interpolation operation has another error. The rotation is round 340 degrees along the "long arc" instead of 20 degrees shorter, as shown in 10.8:

The solution to this problem is to fold the interpolation's "difference" angle to-180 degrees to 180 degrees to find the shortest arc.

Even if these two angle restrictions are used, the Oula angle interpolation may still encounter the problem of the universal lock. In most cases, it may produce jitters, path errors, and other phenomena, the object will suddenly float like "hanging" somewhere. The fundamental problem is that the angular velocity is not constant during interpolation.

Although the first two problems of orah angle interpolation are annoying, they are not insurmountable. It provides a simple solution to limit the angle difference between the orah angles and within a certain range. Unfortunately, the universal lock is a very annoying underlying problem. You may consider re-planning rotation to invent a system that will not encounter these problems. Unfortunately, this is impossible. This is an inherent problem of expressing the 3D Orientation with three numbers. We can change the problem, but we cannot eliminate them. Any system that uses three numbers to express the 3D Orientation will encounter these problems if it can ensure the uniqueness of the space, such as the Universal lock.