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I'm new to MIT Drake, I was using CasADi (ipopt) for a mecanum wheeled model for MPC.

In CasADi, I can create a symbolic variable for the optimization parameters (i.e., initial state and target state).

Then use it to compute the error in state, how can I get an equivalent functionality in Drake?

ZINE Mahmoud
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I've definitely seen mechanum wheels work in drake, but doing trajectory optimization for them requires some thought. For most every wheeled robot trajectory optimization, you want to use a minimal coordinates model. I discuss that briefly in the ballbot example in the notebook associated with http://underactuated.csail.mit.edu/acrobot.html

And for formulating your MPC, I would recommend the notes in http://underactuated.csail.mit.edu/trajopt.html

Russ Tedrake
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