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Unformatted text preview: 16.410/413 Principles of Autonomy and Decision Making Lecture 15: SamplingBased 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: SamplingBased Motion Planning November 3, 2010 1 / 30 The Motion Planning problem Get from point A to point B avoiding obstacles E. Frazzoli (MIT) L15: SamplingBased 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 Mover’s problem) is known to be PSPACEhard [Reif, ’79]. E. Frazzoli (MIT) L15: SamplingBased 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: SamplingBased Motion Planning November 3, 2010 3 / 30 Image of sea squirts removed due to copyright restrictions. Please see: http://en.wikipedia.org/wiki/File:Seatulip.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/gridbased 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 available—very 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: SamplingBased 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 . Samplingbased algorithms A recently proposed class of motion planning algorithms that has been very...
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 Fall '10
 Prof.BrianWilliams
 Mass, Analysis of algorithms, Computational complexity theory, samplingbased motion

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