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I'm reading a book on robotics, and it has a different section on each term. The way both concepts are worded and phrased mathematically make them seem like they both constrain the configuration space of a closed loop mechanism in more or less the same way. I'm not sure if I'm missing something, but they seem like the same thing.

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Is a cell phone and a pocket calculator the same thing? It's not, but a cell phone is often used as one (using the app).

I think it depends on the context. In many multi body simulation solvers, holonomic constraints are the usual way to model joints. Think of it as using "loop closures" in all places, as a convenient way to not set up equations recursively. I am not sure about that, but there may also be other ways to close the (algebraic - this what you mean?) loop.

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