Papers on Thruster Mixing

I was wondering if someone had a link for some papers or ressources describing the theory of a thruster mixer from scratch, given a frame. What are the usual tricks and algorithms? I suppose that some ODEs are involved?

In particular I would like the general case, where planarity and symmetry are not assumed (so thrusters are not symmetric around the center of mass and the thrusters are not pointing along the same vector).

I would really appreciate if someone would have some information!
Have a nice day!

Hi, we have not done anything so complex with our vehicle software ArduSub. We’ve made assumptions/approximations and it works out generally well. You can see about it here.

I realize this could be improved, let me know if you find any literature you would like to share!


In my point of view, this is usually done when modelling the vehicle. After building the mathematical model, you can find try different approaches and arrangements. But you need to take in mind that this also is projected around some requirements and what you are aiming for.

I can point this one here, where some variables can be specified by the project and others from the control perspective. There is also an interesting point around β, that was created to help the roll stabilization.

“The inclusion of this inclination decreases the maximum vertical thrust the aircraft can perform, but the gain in controllability in the Y axis is worth the loss for a small β angle”

Dear Jacob, dear Patrick,
Thanks for replying and sorry for not seeing the replies before. I have actually found the following ressource:

And I feel that the math here is correct (,aybe the coefs need to be changed).

One thing stuck out though: the thrusters need to spin in a constant direction, if so then the computation is good. Otherwise the matrix to pseudo-invert changes as the spin of the thruster changes.

In the case where the thrusters CAN switch directions, then approximating the effect of the motor torque w.r.t. the robot to be negligeable also works.

The general case, when the thrusters can switch directions and the motor torque is not negligable, seems more difficult.
Cheers, let me know your thoughts about it

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I also found this, but this one is more for drones and I don’t really understand what is going on.