How to find the Koopman eigenbasis for a given dynamical system (robotics)

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To implement Model Predictive Control (MPC) for a swerve drive in robotics, a linear state space representation is necessary, which requires identifying a finite-dimensional Koopman invariant subspace. The discussion seeks methods for finding this Koopman eigenbasis and explores its similarity to traditional eigenvalue problems in linear transformations. Swerve drive systems are characterized by their holonomic nature, allowing movement in all degrees of freedom through independent control of each wheel's speed and orientation. This complexity necessitates multiple motors and controllers to achieve precise maneuverability. Understanding the Koopman framework is crucial for effective control design in such dynamic systems.
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I am looking to do control for a swerve drive for FRC, and I would really like a linear state space representation so that I can formulate my MPC as a Quadratic Program. I understand that in order to do this I need to find an appropriate Koopman invariant subspace, ideally finite dimensional. What are the general means of finding this basis, and is it at all analogous to finding eigen things for a linear transformation?
 
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Welcome to the PF. :smile:

This is a pretty specialized question, but hopefully you will get some good replies soon. In the mean time, here is some background information for those interested:

https://www.mit.edu/%7Earbabi/research/KoopmanIntro.pdf

http://team484.org/programming/notes/swerve-drive/

Swerve drive is a holonomic drive system. Holonomic means that the drive train allows the robot to move in all degrees of freedom (It can rotate, move forward/backward, and slide left/right). In the case of swerve drive, this is achieved by independently pivoting and controlling the speed of each wheel on the drive train. Due to the nature of this control system, 8 motors and speed controllers are required but the result is a high traction drive system that can predictably move according to all three degrees of freedom at the same time.
 

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