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Understanding Markov Decision Processes in Probabilistic Robotics: Concepts and Applications

This overview covers key concepts in probabilistic robotics, focusing on Markov Decision Processes (MDPs). We explore planning and control mechanisms, differentiating between deterministic and stochastic actions and the traits of full versus partial observability. The analysis includes critical elements such as states, actions, transition probabilities, and reward functions. We also discuss optimal policies, value functions, and various time horizons for decision-making. Furthermore, techniques like value iteration and policy iteration are examined for their efficacy in reaching optimal solutions in robotics systems.

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Understanding Markov Decision Processes in Probabilistic Robotics: Concepts and Applications

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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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