MIT16_410F10_lec15

MIT16_410F10_lec15 - 16.410/413 Principles of Autonomy and...

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Unformatted text preview: 16.410/413 Principles of Autonomy and Decision Making Lecture 15: Sampling-Based Algorithms for Motion Planning Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology November 3, 2010 Reading: LaValle, Ch. 5 S. Karaman and E. Frazzoli, 2011 E. Frazzoli (MIT) L15: Sampling-Based Motion Planning November 3, 2010 1 / 30 The Motion Planning problem Get from point A to point B avoiding obstacles E. Frazzoli (MIT) L15: Sampling-Based Motion Planning November 3, 2010 2 / 30 The Motion Planning problem Consider a dynamical control system defined by an ODE of the form dx / dt = f ( x , u ) , x (0) = x init , (1) where x is the state, u is the control. Given an obstacle set X obs R d , and a goal set X goal R d , the objective of the motion planning problem is to find, if it exists , a control signal u such that the solution of (1) satisfies x ( t ) / X obs for all t R + , and x ( t ) X goal for all t > T , for some finite T 0. Return failure if no such control signal exists. Basic problem in robotics (and intelligent life in general). Provably very hard: a basic version (the Generalized Piano Movers problem) is known to be PSPACE-hard [Reif, 79]. E. Frazzoli (MIT) L15: Sampling-Based Motion Planning November 3, 2010 2 / 30 Mobility, Brains, and the lack thereof The Sea Squirt , or Tunicate , is an organism capable of mobility until it finds a suitable rock to cement itself in place. Once it becomes stationary, it digests its own cerebral ganglion, or eats its own brain and develops a thick covering, a tunic for self defense. [S. Soatto, 2010, R. Bajcsy, 1988] E. Frazzoli (MIT) L15: Sampling-Based Motion Planning November 3, 2010 3 / 30 Image of sea squirts removed due to copyright restrictions. Please see: http://en.wikipedia.org/wiki/File:Sea-tulip.jpg. Motion planning in practice Many techniques have been proposed to solve such problems in practical applications, e.g., Algebraic planners : Explicit representation of obstacles. Use complicated algebra (visibility computations/projections) to find the path. Complete, but impractical. Discretization + graph search : Analytic/grid-based methods do not scale well to high dimensions. Graph search methods (A * , D * , etc.) can be sensitive to graph size. Resolution complete. Potential fields/navigation functions : Virtual attractive forces towards the goal, repulsive forces away from the obstacles. No completeness guarantees, unless navigation functions are availablevery hard to compute in general. These algorithms achieve tractability by foregoing completeness altogether, or achieving weaker forms of it, e.g., resolution completeness. E. Frazzoli (MIT) L15: Sampling-Based Motion Planning November 3, 2010 4 / 30 Source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse . Sampling-based algorithms A recently proposed class of motion planning algorithms that has been very...
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MIT16_410F10_lec15 - 16.410/413 Principles of Autonomy and...

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