E-motion-planning

E-motion-planning - (Its all in the discretization R&N Chap...

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1 1 Motion Planning (It’s all in the discretization) R&N: Chap. 25 gives some background 2 Motion planning is the ability for an agent to compute its own motions in order to achieve certain goals. All autonomous robots and digital actors should eventually have this ability 3 Digital Actors ± video 1 ± video 2 4 Basic problem ± Point robot in a 2-dimensional workspace with obstacles of known shape and position ± Find a collision-free path between a start and a goal position of the robot 5 Basic problem ± Each robot position (x,y) can be seen as a state ± Continuous state space ± Then each state has an infinity of successors ± We need to discretize the state space (x,y) 6 Two Possible Discretizations Grid-based Criticality-based
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2 7 Two Possible Discretizations Grid-based Criticality-based But this problem is very simple How do these discretizations scale up? 8 Intruder Finding Problem ± A moving intruder is hiding in a 2-D workspace ± The robot must “sweep” the workspace to find the intruder ± Both the robot and the intruder are points robot’s visibility region hiding region 1 cleared region 23 4 5 6 robot 9 Does a solution always exist? Easy to test: “Hole” in the workspace Hard to test: No “hole” in the workspace No ! Information State ± Example of an information state = (x,y,a=1,b=1,c=0) ± An initial state is of the form (x,y,1, 1, . .., 1) ± A goal state is any state of the form (x,y,0,0, . .., 0) (x,y) a = 0 or 1 c = 0 or 1 b = 0 or 1 0 Æ cleared region 1 Æ hidding region 11 Critical Line a=0 b=1 a=0 b=1 Information state is unchanged a=0 b=0 Critical line 12 A B C D E Criticality-Based Discretization Each of the regions A, B, C, D, and E consists of “equivalent” positions of the robot, so it’s sufficient to consider a single position per region
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3 13 Criticality-Based Discretization A BC D E (C, 1, 1) (D, 1) (B, 1) 14 Criticality-Based Discretization A D E (C, 1, 1) (D, 1) (B, 1) (E, 1) (C, 1, 0) 15 Criticality-Based Discretization A D E (C, 1, 1) (D, 1) (B, 1) (E, 1) (C, 1, 0) (B, 0) (D, 1) 16 Criticality-Based Discretization A CD E (C, 1, 1) (D, 1) (B, 1) (E, 1) (C, 1, 0) (B, 0) (D, 1) Much smaller search tree than with grid-based discretization ! B 17 Grid-Based Discretization ± Ignores critical lines Æ Visits many “equivalent” states ± Many information states per grid point ± Potentially very inefficient 18 Example of Solution
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4 19 But . .. Criticality-based discretization does not scale well in practice when the dimensionality of the continuous space increases (It becomes prohibitively complex to define and compute) 20 Motion Planning for an Articulated Robot Find a path to a goal configuration that satisfies various constraints: collision avoidance, equilibrium, etc. .. 21 Configuration Space of an Articulated Robot ± A configuration of a robot is a list of non-redundant parameters that fully specify the position and orientation of each of its bodies ± In this robot, one possible choice is: (q 1 , q 2 ) The configuration space ( C-space ) has 2 dimensions 22 How many dimensions has the C-space of these 3 rings?
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E-motion-planning - (Its all in the discretization R&N Chap...

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