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by: Jada Daniel

RoboticsII MEM456

Jada Daniel
GPA 3.65


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Class Notes
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This 7 page Class Notes was uploaded by Jada Daniel on Wednesday September 23, 2015. The Class Notes belongs to MEM456 at Drexel University taught by Mong-YingHsieh in Fall. Since its upload, it has received 37 views. For similar materials see /class/212392/mem456-drexel-university in Mechanical Engineering at Drexel University.

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Date Created: 09/23/15
M EM456800 Potential Functions Week 3 Am Hsieh g Another Method of Thinking Think of the goal as the bottom ofa goal The robot is at the rim ofthe bowl What will happen Potential Functions E The basic idea 7 Suppose lna goal is a palm gs 211 7 Suppose the robot is a point r s 92 7 Thlrlk of a sprlng39 drawlng the robot toward 1 the goal and away from obstacles 7 Can also thlnk of llke and opposlle charges 39 mp vwrorKl m u emu mm molm The General Idea Both the bowl and the spring analogies are ways of storing potential energy The robot moves to a lower energy con guration A potentialfunction is a function U Wquot a 91 Energy is minimized by following the negative gradient ofthe potential energy function W lql DUlqlT 7 m l magnum T We can now think ofa vector eld overthe space of all q39s gt at every polnl ln time the robot looks atthe veljor at the polnt and goes ln that dll39eCthl l j AttractiveRepulsive Potential Field UQ UattQ UrepltQgt U3quot is the attractive potential move to the goal Urep is the repulsive potential avoid obstacles Attractive Potential Conical Potential I39m dJi dw n l lqitjwull VFW Quadratic Potential JK39I 1mm lmnWl 1 i leq Trivia vrmiqqgom l quotW qqg i A1 mm The Repulsive Potential PM l llum39 qrmliviil i5 Vrmlntl I U 11171239 lql 2 The Total Potential 439 W a 15 722 uquot 75 7qUantJUmpq 1215322 39 2 1241 quotmi F01 V Uq Potential Fields The Hessian For a 1D function howdo we know ifwe are at the unique max or unique min The Hessian is an nxn matrix ofsecond derivatives lfthe Hessian is nonsingular DetH neq 0 the critical point is a uni ue gt if H is positive deiiiite 7 mlHlWle gt if H is iiegative defl lle 7gt m imiim gt if H is iiideiiiiite rgt saddie p0 iit Gradient Descent A simple way to get to the bottom ofthe potential r i l liiiiii A critical point is a point x st q where Uq 0 q ation is Stationarv al a Critical point Max mll i saddle gt stabiiiiyzi iJV li l Gradient Descent Gradient Descent 2 0 415m 7 while v Uqi u do saw an in v Uql i Matlab Potential Function Demo In Practice Computing Distances Senmrlkxuiunrm fun 5 L Banmum Computing Distances Use a Grid Brushfi re Algorithm Use a discrete version of space and work from there Initially create a queue L of pixelscells on the boundary of all obstacles gt One way to do this Brush re Algorithm Need to de ne a grid on the workspace While L Not Empty Need to de ne connectivity 4 or 8 gt Pop the top element of L Obstacles start with a 1 in grid free space is zero gt If dt 0 Set dt 1 mintyE m 1 dt ndt Add all t E Nt with dt 0 to L at the end m H H l The result is a distance map d where each cell holds the minimum distance to an obstacle 4 E Gradient of the distance is found bytaking differences with neighboring cells Brushfire Algorithm Example gt 5 Potential Functions Question o How do we know that we have only a single global minimum We have two choices gt not guaranteed to be a global minimum do something other than gradient descent what gtgt make sure only one global minimum a navigation function which we ll see later The Wavefront Planner The Wavefront Planner Setup a Apply the brushfire algorithm starting from the goal Labe the goal pixe2and add all zero neighborstoL 7 39J D U D U 393 0 D D D 393 D 393 393 D 393 gtWhileLisNOTempty E o a o o a u a u o o o o o o o o popthetopelementofLJ 5 a El D 0 El D D D D D I El ti D D I Setd mintemmmmd 4 u u n u 1 391 1 1 139 i 1 1i 0 o o 0 AddalllEN MVllh dt0toLattheend 3 0 D U a 1 11 1 1 1 1 l U D U U 2 u u u u u o m n n n n n n u u n o The result is now a distance for every cell 1 o a o a n o a o 0 o o o a o n o gtgradientdescentisagainamatterofmovmgtotheneighborWIththe U D D D D D D D U U n U D D D D 2 lowest distance value 0123455739101112131415 The Wavefront in Action Part 1 Starting with the goal set all adjacent cells with 0quot to the current cell 1 gt 47Poinl Connectivily or SrPoinl Connectivim gt Your Choice 7 use Brpl in this example mwluw mmq 131415 E The Wavefront in Action Part 3 Repeat again The Wavefront in Action Part 2 Now repeat with the modi ed cells gt a 0 5 Wlll rernaln wnen regions are unreachable awnm me n U432 55739101112131415 The Wavefront in Action Part 4 And again 1 owwwamm 006543 3 9 1D 11 12 13 1415 The Wavefront in Action Part 5 And again until DEIDulUCIUD The Wavefront Now What To nd the shortest path according to your metric simply always move toward a cell with a lower number gt The numbers generated by tne waveront planner are rougnly oroportlonal to tnelrolstance to tne goal 7 5 Two Posslble 5 snortest Paths 4 Shown a 2 1 n 01234nslzg101112131415 The Wavefront in Action Done You re done gt Remembel 0 5 Should only rernaln n unreachable reglons exlsts umwsmmq Wavefront Overview Divide the space into a grid Number the squares starting at the start in either 4 or 8 point connectivity starting at the goal increasing until you reach rt Your path is de ned by any uninterrupted sequence of decreasing numbers that lead to the goal


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