1 / 15

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).

davis-bishop
Télécharger la présentation

Probabilistic Robotics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related