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Learning from Demonstrations: Kalman Filtering, Smoothing, and Dynamic Time Warping Techniques

This document presents the integration of Kalman filtering and smoothing techniques with the Expectation-Maximization (EM) algorithm for learning optimal trajectories from demonstrations. It discusses applications such as autonomous helicopter aerobatics and surgical tasks, analyzing how imperfect human demonstrations encode an ideal trajectory. The methodology employs the Kalman smoother to estimate states and introduces dynamic time warping to handle varying lengths of demonstrations, providing a robust framework for inferring ideal trajectories in complex dynamic systems.

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Learning from Demonstrations: Kalman Filtering, Smoothing, and Dynamic Time Warping Techniques

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  1. Learning from Demonstrations Jur van den Berg

  2. Kalman Filtering and Smoothing • Dynamics and Observation model • Kalman Filter: • Compute • Real-time, given data so far • Kalman Smoother: • Compute • Post-processing, given all data

  3. EM Algorithm • Kalman smoother: • Compute distributions X0, …, Xtgiven parameters A, C, Q, R, and data y0, …, yt. • EM Algorithm: • Simultaneously optimize X0, …, Xtand A, C, Q, Rgiven data y0, …, yt.

  4. Learning from Demonstrations • Application of EM-algorithm • Example: • Autonomous helicopter aerobatics • Autonomous surgical tasks (knot-tying)

  5. Motivation • Learning an ideal “trajectory” of system • Human provides demonstrations of ideal trajectory • Human demonstrations imperfect • Multiple demonstrations implicitly encode ideal trajectory • Task: infer ideal trajectory from demonstrations

  6. Acquiring Demonstrations • Known system dynamics (A, B, Q) • Observations with known sensors (C, R) • Inertial measurement unit • GPS • Cameras • Use Kalman smoother to optimally estimate states x along demonstration trajectory

  7. Multiple Demonstrations • D demonstration trajectories of duration Tj • Hidden ideal trajectory z of duration T*

  8. Model of Ideal Trajectory • Main idea: use demonstrations as noisy observations of hidden ideal trajectory • Dynamics of hidden trajectory • Observation of hidden trajectory

  9. Inferring Ideal Trajectory • Dynamics model: Parameter N controls smoothness; A, B, Qknown • Observation model: Parameters S encode relative quality of demonstrations • Use EM-algorithm with Kalman smoother to simultaneously optimize z and S (and N). • Initialize S with identity matrices

  10. Time Warping • But, this assumes demonstrations are of equal length and uniformly paced • Include Dynamic Time Warping into EM-algorithm • Such that demonstrations map temporally

  11. Time Warping • For each demonstration j, we have function tj(t) • Maps time t along zto time tj(t) along dj • Adapted observation model:

  12. Learning Time Warping • tj(t) is (initially) unknown • Assume (initially): • T* = (T1 + … + TD) / D • tj(t) = (Tj / T*) t • Adapted EM-algorithm: • Run Kalman smoother with current S and t • Optimize S by maximizing likelihood • Optimize t by maximizing likelihood (Dynamic Time Warping)

  13. Dynamic Time Warping • Match demonstration j with z • Assume that demonstration moves locally • twice as slow as z • same pace as z • twice as fast as z • Dynamic Programmingto find optimal “path” • Cost function: likelihood of

  14. Example: Helicopter Airshow • Thesis work of Pieter Abbeel • Unaligned demonstrations: • Movie • Time-aligned demonstrations: • Movie • Execution of learnt trajectory • Movie

  15. Example Surgical Knot-tie • ICRA 2010 Best Medical Robotics Paper Award • Video of knot-tie

  16. Conclusion • Learning from demonstrations • Includes Dynamic Time Warping into EM-algorithm

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