multiple 3D rotations on one object

Question

I'm working on a 3D space game with no gravity but multiple engines at the spaceship. My problem is collecting the different rotations and put them together.

For example, this is the prototype of my spaceship:

If I put force to the upper and lower part (A) the ship will rotate like a screw as shown in the picture. This is working very well... but if I add a second rotation onto the spaceship, like in picture number two:

The ship rotates weird. I want to have it like in this video:

https://www.youtube.com/watch?time_continue=39&v=-lX0O65ieHM

I know, that the problem is, that I multiply the rotation-matrices in the wrong order... but I tried every order and then I figured out, that with the euler-angles I always get the problem with the gimbal lock, because I split the rotation into x/y/z rotations. I don't know an other way, because in space if you make a rotation, it'll rotate infinite long... and I somehow have to save the different rotations and add them together. I also don't need a 100% physical correct rotation, and it also has to be calculated in real-time so 60 frames per second. Does someone have an idea of implementing it? BTW: At the end the ship will have a lot of engines and they are separate controllable.

More details:

I usually have some matrices:

1: model-matrix: this matrix is a 4x4 matrix which describes the position rotation and scale of the spaceship. the upper-left 3x3 matrix inside the 4x4 matrix describes the rotation and scale of the ship.

2: Rotation-matrix: this is a 3x3 matrix which describes an axis and an angle for rotating the model-matrix.

to make a rotation, I multiply the rotation-matrix with the model-matrix (upper-left 3x3 matrix). every engine(jet) makes a torque. And I split this rotation into x/y/z rotations. and add them to the rotation-matrix. That means, to stop a rotation (for example x rotation), I add the -x rotation onto the rotation-matrix.

I hope this description helps ;)

also: I don't need a 100% correct rotation. the spaceship should rotate like a sphere... maybe different models would rotate different.

Answer 1

Judging by the way you've worded your question, the problem could be in how you are "adding" rotation matrices. You can't just add rotation matrices. Let's introduce the notation $\textbf{R}\left(\hat{a}, \theta \right)$ to be a rotation matrix around the axis $\hat{a}$, by an amount $\theta$. Say we want to rotate a vector $\vec{v}$ around $\hat{x}$ by $\varphi$. That is $$\vec{w}=\textbf{R}\left( \hat{x}, \varphi \right) \vec{v}.$$ This is a new vector, and if we want to rotate it around say $\hat{y}$, we just treat it as rotating $\vec{w}$. So we get: $$\vec{u}= \textbf{R}\left(\hat{y}, \theta \right) \vec{w} = \textbf{R}\left(\hat{y}, \theta \right)\textbf{R}\left( \hat{x}, \varphi \right) \vec{v}.$$ Hence to do multiple rotations, we have to chain these rotations together, by multiplying the matrices together. Note that $$\textbf{R}\left(\hat{y}, \theta \right)\textbf{R}\left( \hat{x}, \varphi \right) \neq \textbf{R}\left(\hat{y}, \theta \right)+\textbf{R}\left( \hat{x}, \varphi \right).$$ You also speak about undoing rotations. The inverse of a rotation matrix is handily the same rotation matrix, but with $\theta \rightarrow -\theta$. However matrices don't commute, so you need to be careful about the order you use them in. If we want to get $\vec{v}$ from $\vec{u}$, we first need to undo the rotation about $\hat{y}$, and then undo the rotation about $\hat{x}$. That is: $$\vec{v} = \textbf{R}\left(\hat{x}, -\varphi\right)\textbf{R}\left(\hat{y}, -\theta\right)\vec{u} =\textbf{R}\left(\hat{x}, -\varphi\right) \textbf{R}\left(\hat{y}, -\theta\right)\textbf{R}\left(\hat{y}, \theta \right)\textbf{R}\left( \hat{x}, \varphi \right) \vec{v} =\textbf{R}\left(\hat{x}, -\varphi\right) \textbf{I}~\textbf{R}\left( \hat{x}, \varphi \right) \vec{v} = \textbf{R}\left(\hat{x}, -\varphi\right) \textbf{R}\left( \hat{x}, \varphi \right) \vec{v} = \textbf{I} ~\vec{v} =\vec{v},$$ where I've used $\textbf{I}$ as the identity matrix. You can see how the order I've applied the un-rotations on the left means they're able to cancel out the original rotations.

Not sure if this will have answered your questions fully, feel free to comment questions.

Answer 2

The short answer is, use quaternions. You start with an orientation $M = (0,x,y,z)$.

You do a rotation like this: choose your axis of rotation $R = (0,a,b,c)$.

Choose how much to rotate, probably a small angle $\alpha$, the amount it rotates in 1/60 second.

Calculate $A = (\cos(\alpha),0,0,0) + sin(\alpha)R$.

To get the rotated version, do 2 quaternion multiplications, $M_1 = A M A'$.

Next time do $M_2 = A M_1 A'$. Simple.

I won't try to explain that here, you'll look it up anyway unless you just find the code and use it.

To do a second rotation at the same time, just calculate $C = BA$ and $C' = A'B'$. Once you've calculated those, you can multiply $C M_2 C'$, $C M_3 C'$ $C M_4 C'$ etc. Simple.

Rounding error will eventually build up, and you can interpolate every now and then to reduce that.

This approach won't do gimbal lock. It just works.