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Unformatted text preview: 1 SA1 1 Probabilistic Robotics Planning and Control: Markov Decision Processes 2 Problem Classes • Deterministic vs. stochastic actions • Full vs. partial observability • Today how to make decisions under uncertainty Uncertainty and decisions • Previously how to do state estimation under uncertainty • Uncertainty can affect how the robot makes decisions • How to encode preferences, between different outcomes of the planes (e.g. going to the airport – lots of options, risks) • Utility theory – reasoning about preferences (utility – quality of being useful) • Every state has some utility • Decision theory = probability theory + utility theory • Principle of maximum expected utility – agent is rational if it chooses an action with the highest expected utility 3 Designing control systems • Often in addition to stability, observability, controllability, we want to have some optimality • Such that the goal it that the trajectory will maximize certain performance index (e.g. time travelled, fuel cost, quadratic cost for trajectory tracking …) • Using techniques from calculus of variations to solve for functions which maximize the performance index V • Special class of systems nstage decision processes • Find such V and choices of action such that the V is maximal • Blackboard example : Recursive computation of V in deterministic case (in case of grid world similar to waverfront planner) • Principle of dynamic programming – decompose the problem in nstages; at each stage relaxation • Next what if the outcomes of actions are uncertain ? 4 2 Making decisions under uncertainty Suppose I believe the following: P(A 25 gets me there on time  …) = 0.04 P(A 90 gets me there on time  …) = 0.70 P(A 120 gets me there on time  …) = 0.95 P(A 1440 gets me there on time  …) = 0.9999 • Which action to choose? Depends on my preferences for missing flight vs. time spent waiting, etc. • Utility theory is used to represent and infer preferences • Decision theory = probability theory + utility theory A simple knowledgebased agent • The agent must be able to: • Represent states, actions, etc. • Incorporate new percepts • Update internal representations of the world • Deduce hidden properties of the world • Deduce appropriate actions Markov decision processes • Framework for represention complex multistage decision problems in the presence of uncertainty • Efficient solutions • Models the dynamics of the environment under different actions • Markov assumptions : next state depends in the previous state, and action not the past CS 685 7 Markov Decision Process • Formal definition • 4tuple (X, U, T, R) • Set of states X  finite • Set of actions A  finite • Transition model Transition probability for each action, state • Reward model 8 T : X × U × X → [0,1] X × U × X → R 3 Example • Robot navigating on the grid • 4 actions – up, down, left, right • Effects of moves are stochastic, we may end up in...
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This note was uploaded on 04/07/2010 for the course CS 685 taught by Professor Luke,s during the Fall '08 term at George Mason.
 Fall '08
 Luke,S

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