A few months ago, I published a video of a completely automatic launch to orbit in KSP. I’ve gotten a few questions about it in the comments, so I’ll try to answer them here in bulk.
Basic Tech
The software runs on kOS, a KSP plugin and scripting language that gives you access to the spaceship and most of the KSP world. My KSP save runs at twice the scale of the stock game, making it a bit more expensive to get to orbit, but makes singleburn to orbit launches closer to reallife launches.
A launch is made of five main phases.

Liftoff and vertical ascent: the rocket holds a purely vertical altitude, to gain some vertical speed and clear the launch pad.

Pitch & Roll programme: the rocket rolls towards the launch azimuth that matches the target orbit’s inclination, and pitches down to a set angle over a set duration. The angle and duration are given as parameters, and depend a lot on the rocket’s Thrust/Weight ratio and the orbit insertion altitude.

Zeroalpha boost phase: the rocket follows its velocity vector to keep the angle of attack close to 0° and avoid instabilities. This lasts until the first stages depletes its fuel. Depending on the profile, the acceleration can be limited by throttling towards the burn’s end.

Staging: stage separation is commanded, followed by a few seconds of RCS firing to push the second stage away. The second stage then ignites its engine and waits to be out of the tick layers of the atmosphere to get rid of the payload fairing if there is one.

Closedloop guidance kicks in. I implemented a version of the Powered Explicit Guidance algorithm, which was developed for the Space Shuttle. The algorithm tries to solve for a pitch, a pitch rate and a burn time based on the current state vector (velocity and location) to achieve the target orbit (defined by a target velocity and location).
FAQ
How do you figure out what time each event occurs, and especially, how do you calculate the pitch timetable?
The timing of the launch is a mix of controlled and automatic events. The pitch programme’s start time, angle and duration are variables set by me before the launch. Other events (throttle reduction around Mach 1, staging, SECO) are triggered automatically by the autopilot on certain conditions like fuel depletion, velocity or on reaching the target orbit.
The pitch programme is very simple: I give the programme the start and end time as well as the pitch angle, and the autopilot commands the pitch to move from 0° to the target angle over the duration of the programme. Finding the angle, start time and duration is trickier, and so far I’ve always started with a programme pitching by 12° from ~T+20s to ~T+50s. From there, the only method I have is trial and error. As a rule of thumb, a vehicle with more power or going to a lower orbit will start the pitch kick earlier, and pitch harder (my Atlas V lookalike with four solid boosters pitches by 28° from T+4s to T+30s on launches to 130km*130km orbits).
A good sign that I’ve found a good pitch programme is when the second stage corrects very little its attitude when closedloop guidance kicks in. In the future, I want to write a simulation tool that runs multiple iterations to try and find the optimum pitch programme (probably by minimising propellant use in the upper stage).
Can this take a rocket to an arbitrary orbit at any inclination?
It can, with a few constraints. The obvious one is that you can’t reach an inclination lower than the launch site’s latitude, but it shouldn’t be an issue in stock KSP since the space centre is located spot on the equator.
Inclination targeting is pretty tricky. The first version of the code would just compute a launch azimuth from the target inclination and just go with the velocity vector after the pitch and roll programme. The current version uses a modified version of BriarAndRye’s instantaneous azimuth calculations: the function returns the heading to burn at to reach the exact inclination by the time the upper stage reaches orbit insertion.
As far as orbit altitude is concerned, it depends a lot on the rocket, how much fuel the upper stage carries, and the upper stage’s thrust/weight ratio. as in real life, most launches target a low periapsis and an apoapsis close to the target orbit’s. If needed, the upper stage can finalise the orbit with a second burn, or let the spacecraft handle it.
Could you share your code?
You can download the three files (PEG & Launch library, inclination control and an example rocket script). It’s ugly and full of global variables (I should really do something about that) but it works. And seeing your upper stage guide itself pulling the same moves at Atlas V’s Centaur is pretty cool.