Could aircraft already do a safe auto-takeoff with todays equipment? If a plane can already autoland and stay aligned while roll-out by itself, on a CAT-3 ILS-Approach, I asked why there is no auto-takeoff-function on todays autopilots.

Because it should be technically possible and much safer than a hand flown takeoff.


For the technic:
When on the runway, the pilot presses an auto-take-off button. The plane stays aligned by following a normal CAT-3 Localizer. Takeoff-Thrust is made with autothrottle. At Vr, the plane rotates by itself, smoothly,and activates VNAV and LNAV and follows the SID. It is also possible to implement a live wind sensor to dynamically update V-Speeds.
For safety:
In case of an emergency during take-off, a computer take-off would be more safer. The plane knows the speed and runway length. So in case of an emergency, it knows if it's below V1 and also if the available runway length is sufficient for a safe rejected take-off. A computer always has a lower reaction time than humans and they can response faster. A computer can immediately decide wether to take off or reject based on lots of live data. A (rejected) take-off performed by a computer should be times safer.


So, why has an autotake-off-function not been developed yet? And what arguments would speak against an auto-take-off function?

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There is not an answer to my question here: On modern commercial airliners, how much of the flight could be fully taken care of by the auto pilot? as stated below. I want to know what reasons, if there are any, speak against auto-take-off.
With some refinement and programming, the technology is easily there, sure; it just needs a few million dollars of design and certification costs for very little benefit. 

The main reason we have autolands is because it saves airlines money, by allowing the plane to land safely in minimal visibility instead of diverting to another airport. Aircraft taking off don't have this problem as they can do it in just about zero visibility. If there is no commercial incentive, the reality is it is not going to happen. Only when pilots are considered surplus to requirements will this ability be developed in earnest. 

There could be an argument made that it should be done for safety's sake, but I don't buy it. Pilot errors made on the takeoff run, resulting in an accident are incredibly rare. Maybe an auto takeoff could prevent a few tailstrikes but I can't see much more benefit.

Whenever you introduce something new you have to think long and hard about the ramifications of it. There's a whole discussion to be had on how rejected takeoffs would work - generally speaking, you do not want automation doing something that the pilots aren't expecting. It is sometimes better to take an issue into the air than take the risk of a high-speed RTO even under V1, how will the computer judge that when there are few hard and fast rules for it? Do you trust the human programmers to account for every single scenario? Then there's pilot competency, how do they keep their skills up when they could go months without flying the aircraft? If they are in the habit of delegating their decision making to the computer, will they be able to handle the heat in that career-defining moment when the computer fails and it's all on them?