Why is bundled linearization better than exact linearization for planning?

[hjsuh94]

I feel that this is one of the core stories of the paper that we should strengthen.

There are several planners out there that attempt to work on local linearizations (exact) of the dynamics to compute reachable sets, and they've been somewhat costly to implement. We should prove our edge over them. Examples: LQRTrees, R3T.

Theorem: fix step e. Then, the error between true dynamics and exact linearized dynamics under that step is uniformly less than error between bundled dynamics and bundled linearized dynamics.
Proof: simple Lipschitz argument. bundled dynamics has less Lipschitz due to convolution.

This is good, but somewhat unsatisfying since what we really want is error between true dynamics and bundled linearized dynamics. Maybe there is some kind of mean value theorem-like argument we can use here under locally convex assumptions about the dynamics?

Do we have a pathological example where you cannot take large steps under linearization but can under bundled dynamics?

Separate question is #2 , which can be divided into: "what is it about quasidynamic systems that further allows us to take large steps?" But to some degree, I believe this has been answered in Pang's simulation paper. The second part is a question specific to the hand system, but not sure if this is something we want in the paper.

[hjsuh94]

Per conversaion w/ Russ, a theoretical justification of this does not seem "as interesting".