UNC at CHAPEL HILL & Ewha Womans University Generalized Penetration Depth Computation Liang-Jun Zhang Gokul Varadhan Dinesh Manocha Dept of Computer Sci. University of North Carolina Chapel Hill, USA Young J. Kim Dept of Computer Sci. Ewha Womans University Seoul, Korea ACM Solid and Physical Modeling Conference, 2006

UNC at CHAPEL HILL & Ewha Womans University 2 Distance Measure Separation → Euclidean distance Interpenetration → Penetration depth d d

UNC at CHAPEL HILL & Ewha Womans University 3 Translational Penetration Depth (PD t ) Minimum translational distance needed to separate objects

UNC at CHAPEL HILL & Ewha Womans University 4 Translational Penetration Depth (PD t ) d Minimum translational distance needed to separate objects

UNC at CHAPEL HILL & Ewha Womans University 5 Applications of PD t Physically-based animation 6DOF haptic rendering Robot motion planning Tolerance verification Time of Contact Force

UNC at CHAPEL HILL & Ewha Womans University 6 PD g No Rotational Motion in PD t PD t

UNC at CHAPEL HILL & Ewha Womans University 7 Generalized Penetration Depth PD g ♦ Taking into account both translational and rotational motion

UNC at CHAPEL HILL & Ewha Womans University 8 Main Contributions Definition of PD g ♦ New distance metric, D g ♦ Properties Practical algorithms for PD g ♦ Convex/convex ♦ Convex/convex complement ♦ Non-convex/non-convex

UNC at CHAPEL HILL & Ewha Womans University 9 Outline Previous Work PD g definition PD g algorithms ♦ Convex/Convex ♦ Convex/Convex complement ♦ Non-convex/Non-convex Application to motion planning

UNC at CHAPEL HILL & Ewha Womans University 10 Previous Work Translational Penetration Depth Intersection depth ♦ [Dobkin et al. 93] Convex polytope ♦ [Agarwal et al. 00] ♦ Lower bound [van den Bergen01] ♦ Upper bound [Kim et al. 02b] Non-convex polyhedra ♦ Higher complexity: O(n 6 ) ♦ [Kim et al. 02a], [Redon et al. 05]

UNC at CHAPEL HILL & Ewha Womans University 11 Previous Work Generalized Penetration Depth No directly related work ♦ 6DOF configuration space: O(n 12 ) complexity Object Containment problem ♦ [Chazella 83], [Milenkovic 99], [Grinde and Cavalier 96], [Avniam and Boissonnat 89], [Agarwal et al 98] Rotational overlapping minimization ♦ [Milenkovic 98], [Milenkovic and Schmidl 01] ♦ Quadratic metric

UNC at CHAPEL HILL & Ewha Womans University 12 Outline Previous Work PD g definition PD g algorithms ♦ Convex/Convex ♦ Convex/Convex complement ♦ Non-convex/Non-convex Application to motion planning

UNC at CHAPEL HILL & Ewha Womans University 13 Configuration Space (C-space) Workspace C-space X Y X Y θ q 0 = q 1 = A

UNC at CHAPEL HILL & Ewha Womans University 14 To measure the distance for one object at two different configurations [LaValle06,Amato00,Kuffner04] ♦ L p metrics (L 2, L 1, L ∞ ) ♦ Displacement metric Maximum displacement ♦ Our D g distance metric Distance Metric in C-space X Y q0q0 q1q1 A(q 0 ) X Y θ A(q 1 ) d

UNC at CHAPEL HILL & Ewha Womans University 15 Min over every path connecting q 0 and q 1 Max trajectory length for distinct points D g (q 0, q 1 ) = D g Metric in C-space X Y θ q1q1 q0q0 l1l1 l2l2 Motion Paths in C-Space Trajectory length A(q 0 ) A(q 1 ) D g (q 0,q 1 )

UNC at CHAPEL HILL & Ewha Womans University 16 Properties of D g metric Non-negativityD g (q 0,q 1 )≥0 Reflexivity D g (q 0,q 1 )=0 ⇔ q 0 =q 1 SymmetryD g (q 0,q 1 )=D g (q 1,q 0 ) Triangle Inequality D g (q 0,q 1 )+D g (q 1,q 2 )≥D g (q 0,q 2 )

UNC at CHAPEL HILL & Ewha Womans University 17 PD g definition The minimum D g distance over all possible collision- free configurations A B PD g

UNC at CHAPEL HILL & Ewha Womans University 18 PD t = Special Case of PD g PD t : only by translation d B A A(q)

UNC at CHAPEL HILL & Ewha Womans University 19 PD g (A,B) ≤ PD t (A,B) PD g PD t

UNC at CHAPEL HILL & Ewha Womans University 20 Outline Previous Work PD g definition PD g algorithms ♦ Convex/Convex ♦ Convex/Convex complement ♦ Non-convex/Non-convex Application to motion planning

UNC at CHAPEL HILL & Ewha Womans University 21 PD g for Convex Objects Theorem PD g (A,B) = PD t (A,B) Pf) 1.In general, PD g (A,B) ≤ PD t (A,B) 2.Show that PD g (A,B) < PD t (A,B) is impossible for convex objects

UNC at CHAPEL HILL & Ewha Womans University 22 Corollary Known PD t algorithms directly applicable to computing PD g PD g (A,B) = PD g (B,A) for convex objects In general, PD g (A,B)≠PD g (B,A) for non- convex objects

UNC at CHAPEL HILL & Ewha Womans University 23 Outline Previous Work PD g definition PD g algorithms ♦ Convex/Convex ♦ Convex/Convex complement ♦ Non-convex/Non-convex Application to motion planning

UNC at CHAPEL HILL & Ewha Womans University 24 Object containment ♦ Can Q contain P, when P is allowed to translate and rotate? PD g ♦ Consider as the container Q ♦ Harder because we need to find an optimal PD g vs Containment

UNC at CHAPEL HILL & Ewha Womans University 25 Object Containment Problem Complexities ♦ O(m 3 n 3 log(mn)) for 2D polygons ♦ O(mn 2 ) for 2D convex polygons Motivates us to consider PD g between convex and convex complement

UNC at CHAPEL HILL & Ewha Womans University 26 Algorithm Overview 1.Find a containment 2.Find a locally optimal containment A

UNC at CHAPEL HILL & Ewha Womans University 27 Step 1: Find a Containment Formulate as a linear programming problem Containment constraint: ♦ Each point on A is contained in a half space defined by each face in B

UNC at CHAPEL HILL & Ewha Womans University 28 Step 2: Find a locally optimal containment Another Linear programming Optimization objective ♦ D g (q 0, q 2 ) ≤D g (q 0, q 1 ) D g (q 0,q 1 ) D g (q 0,q 2 )

UNC at CHAPEL HILL & Ewha Womans University 29 Outline Previous Work PD g definition PD g algorithms ♦ Convex/Convex ♦ Convex/Convex complement ♦ Non-convex/Non-convex Application to motion planning

UNC at CHAPEL HILL & Ewha Womans University 30 PD g for Convex/Non-convex Difficult to compute the exact PD g ♦ May need to compute 6DOF C-space (O(n 12 )) Lower bound algorithm Upper bound algorithm

UNC at CHAPEL HILL & Ewha Womans University 31 Lower Bound on PD g 1.Convex decomposition 2.Eliminate non-overlapping pairs 3.PD t for overlapping pairs 4.LB(PD g ) = Max over all PD t s PD t

UNC at CHAPEL HILL & Ewha Womans University 32 Upper Bounds on PD g 1.PD g (A,B) ≤ PD t (CHull(A), CHull(B)) 2.PD g (A,B) ≤ PD t (A, B) More difficult 3. Better bounds

UNC at CHAPEL HILL & Ewha Womans University 33 Separating planes (b) (c) Convex separator Non-convex separator Separating Planes and Separators

UNC at CHAPEL HILL & Ewha Womans University 34 Separating Planes PD t (Chull(A), Chull(B)) yields a upper bound

UNC at CHAPEL HILL & Ewha Womans University 35 Convex Separators 1.Enumerating convex separators S 2.For each convex separator S, use convex/convex complement for an upper bound 3.Min over all these upper bounds

UNC at CHAPEL HILL & Ewha Womans University 36 Implementation and Results Timing measured on 2.8 GHz P4, 2G main memory Lower bound on PD g ♦ Pairwise PD t Upper bound on PD g ♦ UB1=Containment Optimization ♦ UB2=PD t (CH(A), CH(B))

UNC at CHAPEL HILL & Ewha Womans University 37 Performance Tris#28/1,692 8,452/336304/304 Convex Pieces 28/215 94/2844/44 Convex Separator t: LB (ms) t: UB 1 (ms)

UNC at CHAPEL HILL & Ewha Womans University 38 Hammer in Notch Hammer ♦ 1692 triangles ♦ 215 convex pieces Notch ♦ 28 triangles ♦ 28 convex pieces ♦ 1 convex separator Timings ♦ LB: 4.3 msec ♦ UB: 108 msec

UNC at CHAPEL HILL & Ewha Womans University 39 Comparison

UNC at CHAPEL HILL & Ewha Womans University 40 Hammer in Narrow Notch

UNC at CHAPEL HILL & Ewha Womans University 41 Comparison

UNC at CHAPEL HILL & Ewha Womans University 42 Spoon in Cup Cup ♦ 8452 triangles ♦ 94 convex pieces ♦ 53 convex separators Spoon ♦ 336 triangles ♦ 28 convex pieces

UNC at CHAPEL HILL & Ewha Womans University 43 Comparison

UNC at CHAPEL HILL & Ewha Womans University 44 Outline Previous Work PD g definition PD g algorithms ♦ Convex/Convex ♦ Convex/Convex complement ♦ Non-convex/Non-convex Application to motion planning

UNC at CHAPEL HILL & Ewha Womans University 45 Application to Motion planning C-obstacle query ♦ L. Zhang, Y.J. Kim, G.Varadhan, D. Manocha, Fast C-obstacle Query Computation for Motion Planning, ICRA 2006 C-space When the robot’s configuration changes within the C-space cell, Obstacle Workspace Whether the robot ‘escape’ from the obstacle at some moment? cell

UNC at CHAPEL HILL & Ewha Womans University 46 PD g for C-obstacle Query Query criterion: ♦ If the motion of the robot is less than the overlap extent (PD g ), the robot can not escape from the obstacle Motion bound PD g Is Motion bound < PD g ?

UNC at CHAPEL HILL & Ewha Womans University 47 Time: 110s Results: Motion Planning

UNC at CHAPEL HILL & Ewha Womans University 48 Summary Definition of generalized PD Proved PD g = PD t for convex objects Pose PD g problem for convex/convex complement as convex containment optimization Algorithms for lower and upper bounds on PD g PD g for motion planning

UNC at CHAPEL HILL & Ewha Womans University 49 Limitations Computes a lower bound and a upper bound for non-convex polyhedra. Can not guarantee a global minimum. Convex decomposition and convex separator enumeration impacts on the performance.

UNC at CHAPEL HILL & Ewha Womans University 50 Ongoing and Future work A Simple Path Non-Existence Algorithm for low DOF robots ♦ [L. Zhang, Y.J. Kim, D. Manocha] WAFR2006 Theoretical side: ♦ Formulate D g metric in a computational tractable way for exact PD g computation. Practical side: ♦ Apply PD g for higher DOFs motion planning, dynamic simulation, and tolerance verification.

UNC at CHAPEL HILL & Ewha Womans University 51 Acknowledgements Army Research Office, DARPA/REDCOM NSF ONR Intel Corporation KRF, STAR program of MOST, Ewha SMBA consortium, the ITRC program (Korea)

UNC at CHAPEL HILL & Ewha Womans University Thank you!

UNC at CHAPEL HILL & Ewha Womans University 53 Discussion: why difficult for PD g ? 3D polyhedra PD t PD g Minkowski sumConfiguration space Dimension 3D6D DOFs 3T3T + 3R Arrangement Computation PlanesNon-linear hypersurfaces Non-convexO(n 6 )O(n 12 ) Distance metricEuclidean distance Easy to compute D g distance Difficult to compute

UNC at CHAPEL HILL & Ewha Womans University 54 PD g (A,B)≠ PD g (B,A) A