- (304) Since paths as discussed are simply a sequence of configurations, some method must be used to ensure actions can be taken to cause that sequence of configurations
- It may not be possible to actually achieve that sequence
- What happens why noise disrupts expectations?

- (305) Can try and use ctrl thry to deal w/ uncertainty
- (306) Discrete optimal feedback planning is under discrete state/action sets (state can be continuous though)
- (307) A
**feasible**plan (policy) is one that directs planning to goal state for all configurations - (309) Basically rehashing RL, but w/o noise
- (312) For maximum clearance, propogate values in waves out from obstacles, and follow skeleton
- (318) Given a vector fielad & starting pt, traj represents an integral curve of a differential eq
- Can have discontinuities, called
**hybrid systems**in ctrl thry (I think this is also what mixed discrete/continuous is called)

- Can have discontinuities, called
- (325) A
**tangent space**on an*n*-dim manifold is a hyperplane*R^m*that best approximates the the manifold at a point, related directly to the derivative - (328) A vector field defines tangent spaces all over the surface of a manifold, defines
*n*differential eqs, defines a solution traj from any pt on the manifold - In feedback motion planning, a path must be found, if one exists or must report failure if one doesn’t
- No initial condition, solution vector field instead of solution path

- (329) Vector fields that have a solution pt only converge asymptotically to the pt
- Can instead use unit speed everywhere except at the goal to have finitie-time convergence, but this may not always be possible to execute in a real system.

- (330)
**Feasible**feedback motion planning can be made optimal w/addition of cost function- Vector field can be defined as the negative gradient of the cost function
- Related to LQR

- (332) Planning over a cell-decomposed space works by
- Doing complete planning across cells
- Defining a vec field on each
*n*-cell, vector field should cause flow according to discrete plan

- (340) Using funnels. Should have superstates and superactions, because they overlap
- (342) Once funnels have been laid out, chosing the sequence of funnels to take is another discrete planning problem
- (343) Can also do sample based methods, seems efficient
- (346) VI for continuous state, discrete action
- (348) VI for continuous state, action
- (350) Using a timestep in VI is counter to most motion planning algs that work over continuous time
- Approximating with discrete time isn’t so bad for regular motion planning, but causes trouble in dynamic motion planning

- Stochasticity
- (352) VI and tracking states that can have values that possibly change
- (353) Dijkstra-like continuous state algo
- Intermediary values in Dijkstra and VI are different but end result is same
- (354) Wave-front propagation

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