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Yaw or pitch basically.
Anyways short answer: Yes, you generally need a tail stabilizer of some form. As far as pitch control, you can do that with front mounted canards if needed to move COL forward.
It's certainly possible to build a plane without Yaw control in KSP, but it is highly UN-recommended. (In real life the tail stabilizer would be just about the last thing you'd ever want to lose. It's pretty much a death sentence.)
Unless you go nuts on the Reaction wheels I guess. But that's just Yaw control of a different flavor. You could also do "airbrakes" on the left and right wings to affect a sort of Yaw control, similar to the IRL tailess stealth bombers.
Also, adding a tail fin shouldn't change the COL much if at all, it's lift neutral if you attached it correctly. (Straight up and down.)
Have you rotated yaw surfaces (2??) away from 90° so they form lifting surface as well?
nope, all flat. When I remove the "Elevon 1" it lowers the CoL a bit, then removing the "Small Delta Wing" (i.e., my tail fin) the CoL returns to normal. Idk why it's doing that
My case was rather specific with rotated craft and merging to a new nose so I quickly solved that issue just rebuilding some parts completely - not paying too much attention to the possible marker "lag" (old angular offset preserved?) issue. Not sure whether I have seen it earlier. May be an older issue of from one of the recent patches.
The situation was rather unique for me. Or so rare I have forgotten previous ones. So I didn't consider that could be a reoccuring problem or happen to you as well. Glad you solved it.
In case you can provide more details about how it happened to you, probably we can try to recreate the case with yours and mine experience and eventually localize a small bug. Or not. Anyway, let us know if you have any suspicions about how and why.
The closest aircraft that comes to mind would be the Ho 229, a WW2 german delta-winged jet that had no tail or vertical stabilizer, is that what you’re going for?
https://en.m.wikipedia.org/wiki/Horten_Ho_229
You want at least some kind of tail fin so it will automatically fight against flat spinning. But you can maybe get buy with a tail fin that is fixed, and do all your steering with roll control only. In a real world plane you'd probably not want that because for landing in the wind you need that rudder control, but KSP has no wind so you can always land straight-on.
* - There have been some real-world aircraft that tried to eliminate the need for a tail by having computer-controlled differential thrust on left and right engines, but that would be hard to accomplish in KSP where you don't have a separate throttle lever for each side and would instead have to do it by moving the thrust limiter sliders on the engine's context menus, with the mouse pointer. Also, the second problem is that to just have the goal of using differential thrust to hold it straight like a fixed tail would requires too fine-grain of a control for the human playing the game to be good at it. (i..e give it 0.01% more thrust on the left side now, for about 0.5 seconds, then bring it back to normal... That's not the sort of thing the KSP interface would let you manipulate fast enough.)
Can't RCS and gimbles provide my yaw?
Still some tail (even fixed small fins) will improve your directional stability considerably. So doable, but taking into account the low mass may not be worth the trouble to circumvent.