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Modeling & controlling hybrid systems like walking robots, whose combination of continuous dynamics (e.g. pendulum swing) and discrete dynamics (e.g. heel strike) are devilishly hard to deal with. Boston Dynamics takes a trial-and-error approach. Others are more mathematical.
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Is there a specific part of mechanical engineering? It's barely a single field: Thermo, fluid dynamics, structural engineering, dynamics and controls, robotics - basically "engineering while atoms that don't involve chemical or biological reactions."
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I'm sort of asking the original question to understand how to answer this question. I realized I don't understand the field's shape, and I imagined that by reading about the foundational questions, I'd be able to see some of the decomposition.
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