When Apollo 11 landed on the moon, was The Eagle piloted by Armstrong or remote controlled from Mission Control or by Computer Guidance system on board?
And or at what times during the landing was each system utilized?
Piloted by Armstrong, with Aldrin as his flight engineer. That's why they were there. They were also there to deploy the scientific experiments and take pictures around the landing site, but that was secondary on Apollo 11. Surface EVAs became far more elaborate and important on the later landings.
How
could the LM be remote-landed from Mission Control? The only TV camera on board was packed away in the side of the descent stage, and you know how crappy its pictures were. How would Mission Control have seen where they were going?
That was the whole point of having a crew on board. They had eyes to see where they were going, so they could find a good landing spot in the limited time available. And they could do it without the 3 second round trip delay to earth that will always be there no matter how good our technology gets.
One of the major tasks of today's LRO with 50cm (or better) resolution is to map prospective landing sites well enough that safe landing spots can be pre-identified. If you know where the safe spots are
and you can accurately navigate to them, then certainly with today's technology you can reliably land without local eyeballs and skilled judgment. And even when you do have a crew, you improve their safety by eliminating as many unknowns as possible.
But there was no LRO with 50cm resolution in those days. There
was Lunar Orbiter, whose primary job was to image the prospective Apollo sites at the highest resolution possible, but even it could not spot every boulder and crater large enough to cause a problem for a LM. Each landing was a trip into unknown (at high resolution) territory. The planners picked areas that, from the medium resolution LO images,
looked flat and clean, but they were often surprised (as on Apollo 11).
Navigating a lunar landing is a
big issue. The moon doesn't have GPS, and its detailed gravity field is only now being worked out. The incomplete lunar gravity model (especially for the farside) inherently limited the accuracy of the LM's onboard guidance system no matter how good the hardware and software were. These errors in the gravity model are the favored explanation for Apollo 11 landing so far downrange.
These errors were tolerable in orbital flight, and Mission Control did have a limited ability to remotely command simple LM maneuvers (e.g., the final deorbit of the ascent stage into impact after jettison, and the entire Apollo 5 mission). But landing is an entirely different problem. Because of the inherent accuracy of the unaided guidance system was nowhere near good enough, the Apollo LM carried a landing radar to tell how far away the surface
really was and how fast they were
really moving over it. They simply couldn't have landed without it, though Alan Shepard probably would have given it a really good try. And the LM simply wasn't designed to land without a crew, though there were proposals to create cargo versions that could.