# CAGING OF RIGID POLYTOPES VIA DISPERSION CONTROL OF POINT FINGERS Peam Pipattanasomporn Advisor: Attawith Sudsang 1.

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CAGING OF RIGID POLYTOPES VIA DISPERSION CONTROL OF POINT FINGERS Peam Pipattanasomporn Advisor: Attawith Sudsang 1

Motivation? ! ! ! ! ! ! 2

Better Approach? 3

Overview Master Thesis Proposed Ph.d. Thesis Additional Chapters Fix Cage (2011) Imperfect Shape (2010) Robust Cage (2012) n-Squeeze (2008) n-Stretch (2008) 2-Squeeze (2006) 2-Stretch (2006) 4

Fix Cage (2011) n-Stretch (2008) 2-Stretch (2006) Overview Master Thesis Proposed Ph.d. Thesis Additional Chapters Imperfect Shape (2010) Robust Cage (2012) n-Squeeze (2008) 2-Squeeze (2006) 5

2-Squeeze, How? Keep distance below a value Given object shape, solve: – Where to place the fingers? – The upperbound distance? “ Distance ” 6

2-Squeeze Possible escape path (object frame) Along the path Distance 7

2-Squeeze “Better” escape path Distance “Better” Upperbound Initial 8 Along the path

2-Squeeze Find an Optimal Escape Path in C-Free Workspace (2D) b b a a Configuration Space (4D) (abstracted) C-Obstacle (a,b) 9 Abstracted set of escape configurations

2-Squeeze Find an Optimal Escape Path in C-Free Configuration Space (4D) (abstract) C-Obstacle 10 Abstract set of escape configurations (a,b)

C-Free Decomposition C-Obstacle 11

Paths connecting Terminals C-Obstacle 12

Finite Categorization of Paths C-Obstacle 13

Straight Path Distance : |a-b| 2 Along the path (linear interpolation) b b a a (a,b) 14

Moving Across Convex Subsets C-Obstacle 15

Through Convex Intersections C-Obstacle 16

Requirements For The Algorithm Distance(x) Rigid Transformation InvariantConvex x 17

Convex & RTI Examples d 1 + d 2 + d 3 d 1 2 + d 2 2 + d 3 2 max(d 1, d 2, d 3 ) d3d3 d1d1 d2d2 x1x1 x2x2 x3x3 18 Larger  Loose cage Fingers at a point  Smallest “Formation Size”

Results (n-Squeeze) Size: d 1 2 +d 2 2 +d 3 2 +d 4 2 19

1 23 1 23 1 23 Squeezing? 1 23 1-DOF Scaling ONLY 20

“Size” & “Deformation” 1 23 Reference Formation 1 23 Same size No deformation Larger size Deformed 1 2 3 Smaller size Slightly Deformed 1 2 3 21

Smaller size Slightly Deformed 1 2 3 1 23 Same size No deformation Same Formation Larger size Deformed 1 2 3 “Size” & “Deformation” Reference Formation 1 23 22

Smaller size Slightly Deformed 1 2 3 1 23 Same size No deformation Same Formation Larger size Deformed 1 2 3 “Size” & “Deformation” Reference Formation 1 23 23

Smaller size Slightly Deformed 1 2 3 1 23 Same size No deformation Same Formation Larger size Deformed 1 2 3 “Size” & “Deformation” Reference Formation 1 23 24

“Size” & “Deformation” |r| 2 2 (x) = |A † x| 2 2 – “Scale” or “Size” (w.r.t. reference) D(x) = |A(r; t) – x| 2 2 – “Deformation upto Scale” (w.r.t. reference) 1 2 3 1 23 A stores information of the reference. 25

Squeezing ? 1 23 1 23 1 23 1-DOF Scaling ONLY Size = |r| 2 2 < ??? D ≤ 0 Convex & RTI 26

Squeezing 1 23 1 23 1 23 1-DOF Scaling ONLY Size = |r| 2 2 < ??? D ≤ 0 x D > 0  D ≤ 0 |r| 2 2 ; D ≤ 0  ; D > 0 Size* = 27

Fix Formation Cage 1 23 Size* = 1 Convex Constraint Convex & RTI Size*  1 Size* ≤ 1 “Stretch” “Squeeze” 28

Robust Caging Independent Capture Regions 29 Keep error (deformation) below a value Given object shape, find: – Where to place the fingers – The upperbound error

n-Squeeze vs Fix Formation KEEP SIZEERROR (DEFORMATION) BELOW UPPERBOUNDBELOW UPPERBOUND OPTIMAL ESCAPE PATHSIZEMINIMIZE UPPERBOUND DISTANCEERROR (DEFORMATION) 30

Error Tolerance inf r,tϵ R 2 2 2 2 2 3 3 3 3 1 1 1 1 2 2 2 3 3 1 1    D 2 = 2 2 3 3 1 1 inf |r| 2 =1 tϵ R 2 2 2 2 2 3 3 3 3 1 1 1 1 p 2 2 3 3 1 1    E p = 2 2 3 3 1 1 NOT CONVEX! r + t - “Placement Error” “Placement Error upto Scale” 31

 Approximation inf g(r) |r| 2 =1  inf g(r) r ϵ R i min i ϵ{1,…, m} R1R1 R2R2 R3R3 R4R4 32

 Approximation inf g(r) |r| 2 =1  inf g(r) r ϵ R i min i ϵ{1,…, m} R1R1 R2R2 R3R3 R4R4 Min of Convex Functions (not convex) 33

Optimal Path Min of a Convex Function is Convex f = f 1 = min(f 1 )

Optimal Path 35 Min of Two Convex Functions f = min(f 1, f 2 ) f 1 = ff = f 2 f 1 = f = f 2 35

Optimal Path 36 Min of Two Convex Functions f = min(f 1, f 2 ) f 1 = ff = f 2 f 1 = f = f 2 36 ???

Optimal Path 2 1 37 x f(x)f(x) f 1 =f=f 2 f=f 2 f=f 1 What is the optimal path, starting from the minimal points?

Critical Point 2 1 38 x f 1 =f=f 2 f=f 2 f=f 1 Consider… 1,2 Only the points under the water level are reachable when the maximum deformation is limited to below the water level. f(x)f(x)

Optimal Path 1,2 2 1 39 x f 1 =f=f 2 Critical Value f=f 2 f=f 1 : minimizer for a CONVEX optimization problem: minimize L s.t. f 1 (x) < L f 2 (x) < L f(x)f(x)

Critical Point 1,2 2 1 40 x f 1 =f=f 2 Critical Value f=f 2 f=f 1 f(x)f(x)

Min of Multiple Convex Functions 41 Min of Multiple Convex Functions f = min(f 1, f 2, f 3 ) f= f 1 f= f 2 f= f 3

Min of Multiple Convex Functions 42 Min of Multiple Convex Functions f = min(f 1, f 2, f 3 ) f= f 1 f= f 2 f= f 3 1,2 2,3 1,3 1 1 3 3 2 2

Search Space Min of Multiple Convex Functions f = min(f 1, f 2, f 3 ) 1,2 2 1 3 1,3 2,3 43 Include all possible between any two regions: f=f i, f=f j

Optimal Path 44

Results 45

Results 46

Shape Uncertainty Exact Object (Unknown) Scanned Object sensor 47

Idea Cage subobject  Cage object ? Fingers must not penetrate the object. 48

Exact Object (Unknown) Idea Exact boundary (unknown) but inbetween the bounds. Find placements that cage subobject, outside superobject. 49

Applications Simplification Curved Surface, Spherical Fingers Shape Uncertainty Slightly Deformable Object Partial Observation 50

Results 51

Results 52

Conclusion O(c log c) exact algorithms – Squeeze, Stretch, Squeeze & Stretch – c : # decomposed convex features O(cm 2 log( cm 2 ) ) approximate algorithm – m : # approximation facets Extension to three dimension. Trade error tolerance with uncertainty. 53

Q&A :V :o 54

Publications Journal Papers – Peam Pipattanasomporn, Attawith Sudsang: Two-Finger Caging of Nonconvex Polytopes. IEEE Transactions on Robotics 27 (2011)IEEE Transactions on Robotics 27 (2011) – Thanathorn Phoka, Pawin Vongmasa, Chaichana Nilwatchararang, Peam Pipattanasomporn and Attawith Sudsang: Optimal independent contact regions for two-fingered grasping of polygon. Robotica (2011) Conference Papers – Peam Pipattanasomporn, Attawith Sudsang: Object caging under imperfect shape knowledge. ICRA 2010 ICRA 2010 – Thanathorn Phoka, Pawin Vongmasa, Chichana Nilwatchararang, Peam Pipattanasomporn, Attawith Sudsang: Planning optimal independent contact regions for two-fingered force- closure grasp of a polygon. ICRA 2008ICRA 2008 – Peam Pipattanasomporn, Pawin Vongmasa, Attawith Sudsang: Caging rigid polytopes via finger dispersion control. ICRA 2008 ICRA 2008 – Peam Pipattanasomporn, Pawin Vongmasa, Attawith Sudsang: Two-Finger Squeezing Caging of Polygonal and Polyhedral Object. ICRA 2007 ICRA 2007 – Peam Pipattanasomporn, Attawith Sudsang: Two-finger Caging of Concave Polygon. ICRA 2006 ICRA 2006 – Thanathorn Phoka, Peam Pipattanasomporn, Nattee Niparnan, Attawith Sudsang: Regrasp Planning of Four-Fingered Hand for Parallel Grasp of a Polygonal Object. ICRA 2005ICRA 2005 55

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