To say it will be impossible for robots to take over the world in the future is like saying it is impossible to fly to Mars and back. Until it happens, it is considered impossible. There are many ways robots could take over the world.
When I told Isaac Assimov at a science fiction convention that his three laws of robotics won't work in the real world, he didn't like to hear that. A few years later, "Terminator" came out and he saw I was correct. Such a future will be possible if artificial intelligence becomes a reality. Since robots will be programmed to be logical and replication of robots will also be possible, robots will be programmed to know as much as humans do and then some. They will use the knowledge to do things. When they determine that man is not needed in the world or to keep them functioning, they may decide man is too much of a burden on this planet and it would be best if man was eliminated.
We already allow computers to run society, to some extent. They have become indispensable. When I wrote THIRTY YEARS AFTER VIKING between 1978 and '79, I envisioned Mars being controlled by robots and computers. When the robot from the Trailblazer Rover entered the bookstore in Janvuor and was trying to remove a book without paying for it, it was arrested and tried to escape. It was destroyed. When the Apache Indian that decided to stay on Mars in 1992 dealt with the main computer that controlled the planet, he was dealing with the entity that controlled the planet after it had been evacuated by people during the early 1870's. Such a thing could happen to earth with our reliance on computers to run things. We'll give control to robots and computers.
Robots will survive by doing what is needed for survival. If they need energy, they will obtain it by any means possible. They will be able to repair systems in order to keep functioning. With artificial intelligence and computer links, there could be a planetary hive system as robots and computers share information with each other. They may have artificial imaginations to help them solve illogical problems. That is one thing humans have over robots and computers. Humans have the ability to use illogic to solve problems and have an imagination that allows them to see things that are logically impossible to see. If robots and computers obtain this ability, there may be no way to stop them except by destroying the world and making it uninhabitable even for robots.
Personally, I don't think it will happen. God created the world for humans to dominate. If robots were meant to control the world, the Bible would have stated that an artificial life form would rule the world in the future. The Beast which will rule the world supposedly is a computer in Europe that controls the finances of the world. The Antichrist will no doubt use computers to control the population. But ultimately, Christ will rule the world for 1,000 years in the future. But if we progress technically as I feel we have the potential of doing, there could come a time when robots and computers could control the world. It just won't be possible in this decade. But it could be a possibility before the end of this century.