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Dynamical Systems, Stochastic Processes, and Probabilistic Robotics

Dynamical Systems, Stochastic Processes, and Probabilistic Robotics. David Rosen. Goals. Overview of some of the big ideas in autonomous systems Theme: Dynamical and stochastic systems lie at the intersection of mathematics and engineering ZOMG ROBOTS!!!. What is a robot?.

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Dynamical Systems, Stochastic Processes, and Probabilistic Robotics

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  1. Dynamical Systems, Stochastic Processes, and Probabilistic Robotics David Rosen

  2. Goals Overview of some of the big ideas in autonomous systems Theme: Dynamical and stochastic systems lie at the intersection of mathematics and engineering ZOMG ROBOTS!!!

  3. What is a robot? • Actually, no universally accepted definition • For this talk: • Sensing (what’s going on?) • Decision (what to do?) • Planning (how to do it?) • Actuation & control (follow the plan)

  4. Dynamical Systems How do we begin to think about this problem?

  5. General Mathematical Framework: Dynamical Systems

  6. General Mathematical Framework: Dynamical Systems

  7. Sensing and Estimation What’s going on?

  8. Sensing and Estimation • How do we know what the state is at a given time? • Generally, we have some sensors: • Laser rangefinders • GPS • Vision systems • etc…

  9. Sensing and Estimation • Great! • Well, not quite… • In general, can’t measure all state variables directly. Instead, an observation function H : M → O maps the current state x to some manifold O of outputs that can be directly measured • Usually, dim O < dim M • Given some observation z = H(x), can’t determine x !

  10. Sensing and Estimation • Maybe we can use the system dynamics (f ) together with multiple observations? • Observability: Is it possible to determine the state of the system given a finite-time sequence of observations? • “Virtual” sensors! • Detectability(weaker): Are all of the unobservable modes of the system stable?

  11. Sensing and Estimation • What about noise? • In general, uncorrected/unmodeled error accumulates over time. • Stochastic processes: nondeterministic dynamical systems that evolve according to probability distributions.

  12. Sensing and Estimation • New model: for randomly distributed variables wt and vt. • We assume that xtconditionally depends only upon xt-1 and the control ut (completeness): Stochastic processes that satisfy this condition are called Markovchains.

  13. Sensing and Estimation • Similarly, we assume that the measurement ztconditionally depends only upon the current state xt :

  14. Sensing and Estimation Thus, we get a sequence of states and observations like this: This is called the hidden Markov model (HMM).

  15. Sensing and Estimation How can we estimate the state of a HMM at a given time? Any ideas?

  16. Sensing and Estimation Hint: How might we obtainfrom ?

  17. Bayes’ Rule

  18. Bayes’ Rule

  19. Bayes’ Rule Punchline: If we regard probabilities in the Bayesian sense, then Bayes’ Rule provides a way to optimally update beliefs in response to new data. This is called Bayesian inference. It also leads to recursive Bayesian estimation.

  20. Recursive Bayesian Estimation: The Bayes Filter • Define • Then by conditional independence in the Markov chain: and by Bayes’ rule:

  21. Recursive Bayesian Estimation: The Bayes Filter This shows how to compute given only and the control input . Recursive filter!

  22. Recursive Bayesian Estimation: The Bayes Filter • Initialize the filter with initial belief • Recursion step: • Propagate: • Update:

  23. Recursive Bayesian Estimation: The Bayes Filter • Benefits of recursion: • Don’t need to remember observations • Online implementation • Efficient! • Applications: • Guidance • Aerospace tracking • Autonomous mapping (e.g., SLAM) • System identification • etc…

  24. Example: Missile guidance • This clip was reportedly sampled from an Air Force training video on missile guidance, circa 1955. • It is factually correct. • See also: • Turboencabulator • Unobtainium Rudolf Kalman

  25. Trajectory Generation How do we identify trajectories of the system with desirable properties?

  26. Recap: Control Systems

  27. Key questions for trajectory generation • Controllability: given two arbitrary specified states p and q, does there exist a finite-time admissible control u that can drive the system from p to q ? • Reachability: Given an initial state p, what other states can be reached from p along system trajectories in a given length of time? • Stabilizability: Given an arbitrary state p, does there exist an admissible control u that can stabilize the system at p ?

  28. Trajectory Generation • Several methods for generating trajectories • Rote playback • Online synthesis from libraries of moves • etc… • Optimal control: Minimize a cost functional amongst all controls whose trajectories have prescribed initial and final states x0 and x1.

  29. The Pontryagin Maximum Principle • Provides a set of necessary conditions satisfied by any optimal trajectory. • Can often be used to identify optimal controls of a system. Lev Pontryagin

  30. The Pontryagin Maximum Principle

  31. The Pontryagin Maximum Principle

  32. The Pontryagin Maximum Principle Can also derive versions of the PMP for: • State-constrained control • Non-autonomous (i.e., time-dependent) dynamics. • etc…

  33. Nothing Could Possibly Go Wrong…

  34. Trajectory Following How can we regulate autonomous systems?

  35. The problem Real-world systems suffer from noise, perturbations If the underlying system is unstable, even small perturbations can drive the system off of the desired trajectory.

  36. Example: Pendulum on a Cart

  37. What to do? • We have a desired trajectory that we would like to follow, called the reference. • At each time t, we can estimate the actual state of the system . • In general there is some nonzero errorat each time t.

  38. What to do? Maybe we can find some rule for setting the control input u (t ) at each time t as a function of the error e (t ) such that the system is stabilized? In that case, we have a feedbackcontrol law:

  39. Trajectory Following: Feedback Control Many varieties of feedback controllers: • Proportional-integral-derivative (PID) control • Fuzzy logic control • Machine learning • Model adaptive control • Robust control • H∞ control • etc…

  40. The Moral of the Story We started with what (at least conceptually) were very basic problems from engineering e.g., make do this this

  41. The Moral of the Story and ended up investigating all of this: • Dynamical systems • Stochastic processes • Markov chains • The hidden Markov model • Bayesian inference • Recursive Bayesian estimation • The Pontryagin Maximum Principle • Feedback stabilization and this is just the introduction!

  42. Questions?

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