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Arguably the darling and most glamorous branch in this I am not a person you can put on speakerphone shirt. The skills you would require would be: A good understanding of Newtonian mechanics, maybe classical but not quantum mechanics. Very good understanding of Kinematics and dynamics of machines including vibrations, differential equations, linear algebra, integral and differential calculus. Knowing electrical stuff like control theory, electrical actuation etc are a bonus. Till now all the listed skills need only an understanding of maths and physics so you are good.
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Other than these you should be able to code in I am not a person you can put on speakerphone shirt, and depending on whether you want to go into research or not a practical knowledge of open source hardware like Arduino or Rasberry Pi( avoid if research is an aim). Other than a good undestanding of kinematics, waves and vibrations you might need a good understanding of how internal combustion engines work, this needs a decent amount of practical knowledge but i guess you can skip that too if research is a criteria. But that too may turn out redundant as the industry tunrs to electrical actuation.