The F-22 Raptor has vectored thrust. This means that the engines don’t just push towards the front of the aircraft. Instead, the thrust can be directed upward or downward (from the rear of the jet). With this vectored thrust, the Raptor can do some cool stuff like this power loop. I first saw this on thedrive.com.

F-22猛禽具有矢量推力。 这意味着引擎不仅会推向飞机的前部。 取而代之的是，推力可以指向上方或下方(从射流的后部)。 有了这个矢量推力，猛禽可以做一些很酷的事情，例如功率循环。 我首先在thedrive.com上看到了这一点。

There are countless F-22 power loop videos online (I mean, I didn’t actually count). However, I like this one since it has the following features.

在线上有无数F-22电源回路视频(我是说，我实际上没有数过)。 但是，我喜欢它，因为它具有以下功能。

• The motion of the aircraft is mostly perpendicular to the view of the camera. This makes it appear to move in a 2D motion of the video frame.

  飞机的运动大部分垂直于摄像机的视野。 这使其看起来像在视频帧的2D运动中移动。

• The Raptor is fairly far away from the camera such that you can see the whole motion of the power loop.
  猛禽离摄像机很远，因此您可以看到电源回路的整个运动。
• Finally, there are these things called “mountains” in the background. Who cares about mountains? I do. The mountains can serve as points to mark the background so that I can account for the panning of the camera.
  最后，在后台有称为“山”的东西。 谁在乎山脉？ 我做。 山脉可以作为标记背景的点，这样我就可以说明相机的摇摄。

Oh, well then let’s do it. Of course I’m going to use my favorite video analysis program, Tracker Video Analysis. In this case, I’m going to start off by scaling the video based on the length of the F-22 with a value of 18.9 meters. After that, I need to mark points on the mountain range to keep track of the origin using calibration point pairs (here is an older tutorial if you want to see how to do that).

哦，那就开始吧。 当然，我将使用我最喜欢的视频分析程序Tracker Video Analysis 。 在这种情况下，我将首先根据F-22的长度(值为18.9米)缩放视频。 之后，我需要在山脉上标记点，以使用校准点对来跟踪原点( 如果您想了解如何进行校准，可以使用旧的教程 )。

Once you adjust the video for the motion of the camera, you can make these awesome “frame corrected” views.

一旦根据摄像机的运动调整了视频，就可以制作出这些很棒的“帧校正”视图。

I love that.

我喜欢那个。

OK, now for some physics. I want to mark the location of both the front and the rear of the F-22 so that I can measure it’s position AND it’s orientation. It’s a little easier to do the calculations in python. Here is a plot of the front and back of the aircraft. Oh, this is the trajectory (x vs. y). Here is the python code with all the data.

好的，现在开始一些物理学。 我想标记F-22前后的位置，以便可以测量它的位置和方向。 在python中进行计算要容易一些。 这是飞机正面和背面的图解。 哦，这就是轨迹(x与y)。 这是包含所有数据的python代码。

But that doesn’t really show you the orientation of the aircraft since you can’t easily see with point on the blue curve goes with which point on the red curve. So, here’s how I can calculate the orientation angle of the aircraft. Let me start with a diagram.

但这并不能真正显示飞机的方向，因为您很难轻易看到蓝色曲线上的点与红色曲线上的点。 因此，这就是我如何计算飞机的方位角。 让我从图开始。

Just to be clear, I’m going to first find the vector value from the rear to the front of the F-22. From the x and y-components of this vector, I can calculate the angle.

为了清楚起见，我将首先找到F-22从后到前的向量值。 从这个向量的x和y分量，我可以计算出角度。

Just to test stuff, here is my plot of angle (in degrees) as a function of time. Also, the code.

仅出于测试目的，这是我随时间变化的角度图(以度为单位)。 另外，代码 。

Note: you have to do a little angle trick when the vector is in the 3rd quadrant (when both components are negative). But still, this looks fine.

注意：向量在第三象限时(两个分量均为负值)，您必须做一点角度技巧。 但是，这看起来还不错。

But what about the angle of attack? This is the angle between the velocity vector and the “orientation” vector for the aircraft. I already have the orientation vector (not sure if that’s the correct technical name), so I just need to get the velocity vector.

但是攻角呢？ 这是速度矢量和飞机的“方向”矢量之间的角度。 我已经有了方向矢量(不确定这是否是正确的技术名称)，所以我只需要获取速度矢量。

Here’s what I’m going to do. First, I will find a new point — the center of mass of the plane. OK, technically this will just be the center of the aircraft since I will just average the vector position of the front and rear.

这就是我要做的。 首先，我将找到一个新点-平面的质心。 好的，从技术上讲，这将只是飞机的中心，因为我将平均前后向量位置。

Now to find the velocity vector, I am going to take three points (labeled 1, 2, 3). The velocity at point 2 will be the change in position from 1 to 3 divided by the time interval.

现在找到速度矢量，我将取三个点(标记为1、2、3)。 点2处的速度将是位置从1到3的变化除以时间间隔。

Hopefully, by using 3 data points it will sort of “smooth” out the velocity. Here is my plot of the velocity of the center of the F-22. Code.

希望通过使用3个数据点，可以“平滑”速度。 这是我对F-22中心速度的曲线图。 码。

I lied. I decided to use the a wider range of points (2 times before to 2 times after) for the average velocity. The other was a bit too jagged. I suspect there is a problem with the video frame rate — you can see there are some repeated frames that mess up the velocity. But this should be good enough.

我撒了谎。 我决定对平均速度使用更大范围的点(前2次到后2次)。 另一个太参差不齐。 我怀疑视频帧速率存在问题-您可以看到有些重复的帧会扰乱速度。 但这应该足够了。

Now that I have a velocity vector and an orientation vector, I just need to find the angle between these two vectors. Here is the classic trick that uses the definition of the dot-product for vectors.

现在我有了速度矢量和方向矢量，我只需要找到这两个矢量之间的夹角即可。 这是将点积定义用于矢量的经典技巧。

Oh darn. I just realized that I am using θ for the angle of attack, but I already used to for the orientation angle. Oh well — we are all adults here right? Sorry about that.

真是的 我刚刚意识到我将θ用于迎角，但我已经习惯于将方位角用于迎角。 哦，对了-我们都是大人吗？ 对于那个很抱歉。

OK, so here is the angle of attack as a function of time. You know I’m going to give you the code — you don’t have to ask.

好的，这是迎角随时间变化的函数。 您知道我将为您提供代码-您不必问。

I’m not sure about that spike at the beginning, but I’m pretty happy with this. Oh, one last thing — the motion of the Raptor was NOT perpendicular to the camera. You can see this because the apparent size of the aircraft changes over time. But still, it’s a pretty sweet video for analysis.

我不确定一开始的峰值，但是对此我感到非常满意。 哦，最后一件事-猛禽的运动不垂直于相机。 您会看到这是因为飞机的外观尺寸会随时间变化。 但是，这仍然是一个很好的分析视频。