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Probabilistic Robotics

Probabilistic Robotics. Planning and Control: Markov Decision Processes. Problem Classes. Deterministic vs. stochastic actions Full vs. partial observability. Deterministic, fully observable. Stochastic, Fully Observable. Stochastic, Partially Observable. Markov Decision Process (MDP).

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Probabilistic Robotics

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  1. Probabilistic Robotics Planning and Control: Markov Decision Processes

  2. Problem Classes • Deterministic vs. stochastic actions • Full vs. partial observability

  3. Deterministic, fully observable

  4. Stochastic, Fully Observable

  5. Stochastic, Partially Observable

  6. Markov Decision Process (MDP) r=1 s2 0.01 0.9 0.7 0.1 0.3 0.99 r=0 s3 0.3 s1 r=20 0.3 0.4 0.2 s5 s4 r=0 r=-10 0.8

  7. Markov Decision Process (MDP) • Given: • States x • Actions u • Transition probabilities p(x‘|u,x) • Reward / payoff functionr(x,u) • Wanted: • Policy p(x) that maximizes the future expected reward

  8. Rewards and Policies • Policy (general case): • Policy (fully observable case): • Expected cumulative payoff: • T=1: greedy policy • T>1: finite horizon case, typically no discount • T=infty: infinite-horizon case, finite reward if discount < 1

  9. Policies contd. • Expected cumulative payoff of policy: • Optimal policy: • 1-step optimal policy: • Value function of 1-step optimal policy:

  10. 2-step Policies • Optimal policy: • Value function:

  11. T-step Policies • Optimal policy: • Value function:

  12. Infinite Horizon • Optimal policy: • Bellman equation • Fix point is optimal policy • Necessary and sufficient condition

  13. Value Iteration • for all x do • endfor • repeat until convergence • for all x do • endfor • endrepeat

  14. Value Iteration for Motion Planning

  15. Value Function and Policy Iteration • Often the optimal policy has been reached long before the value function has converged. • Policy iteration calculates a new policy based on the current value function and then calculates a new value function based on this policy. • This process often converges faster to the optimal policy.

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