1 Motion and Manipulation Configuration Space

Outline Motion Planning Configuration Space and Free Space Free Space Structure and Complexity

Motion Planning Given a (2- or 3-dimensional) workspace W, a robot A of fixed and known shape moving freely in W, a collection of obstacles B={B 1,…, B n } of fixed and known shape and location, 3

Motion Planning Problem and an initial and a final placement for A, find a path for A connecting these two placements along which it avoids collision with the obstacles from B, or report that no such path exists. 4

5 Modeling Solution of the motion planning problem in configuration space C: the space of parametric representations of all robot placements. Configuration q: unique characterization of robot placement by (minimum) number of parameters. Degrees of freedom (DOF) Subset A(q) of W covered by A in q should be uniquely determined.

6 Frames and Transformations Fixed world frame, moving frame attached to robot A, initially coinciding. Maintain transformation that maps moving frame back to fixed frame.

7 Displacements Any displacement in R 2 or R 3 is the composition of a rotation and a translation with respect to a selected point O (origin).

Configuration Space Configuration space C=R 2 8

Robot translates: C=R 2 Robot translates and rotates: C=R 2 x [0,2 π) Configuration Space 9

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Configuration Space 11

Robot translates: C=R 3 Robot translates and rotates: C=R 3 x [0,2 π ) x [0, π ] x [0,2 π ) Configuration Space 12

Configuration Space Robot translates and rotates: C=R 3 x [0,2 π ) x [0, π ] 13

14 Configuration Space Manifold: space locally behaves like R n for some fixed n Wrap around for rotational dimensions Complex numbers to encode rotational part of SO(2) Quaternions to encode rotational part of SO(3)

Configuration Types q is freeq is forbidden 15

Obstacles in Configuration Space Configuration Space Obstacle for B i : Forbidden Space Free Space BiBi 16

Free Space and Motion Planning Motion Planning problem: q init q goal 17

Configuration Space Obstacles Explicit computation often base step in exact methods. Motion planning problem not yet solved when CB is known; additional processing of C free =C\CB is necessary for finding path. q init q goal 18

1D Configuration Space Obstacle W=R, translating robot A=[-1,2], obstacle B i =[0,4] Placement of A specified by position of reference point, so A(q) = A+q = { a+q | a Є A } BiBi A 19

Minkowski Sum Minkowski sum of sets P and Q 20

1D Configuration Space Obstacle Boundary of CB i obtained by tracing reference point while sliding A along B i. CB i A -A BiBi 21

2D Configuration Space Obstacle W=R 2, translating convex polygonal robot A, convex polygonal obstacle B i Boundary of CB i obtained by tracing reference point while sliding A along B i. A BiBi 22

2D Configuration Space Obstacle Edges of CB i correspond to an edge of A touching a vertex of B i, or a vertex of A touching an edge of B i. 23

2D Configuration Space Obstacle CB i is convex. Edges of CB i share orientations of edges of A and B i. 24

2D Configuration Space Obstacle If convex A has m edges, and convex B i has n edges, then CB i has complexity O(m+n). Computable in O(m+n) time if edges are ordered. A BiBi 25

Exercises on Minkowski Sums in 2D 1.Find a convex translating polygonal robot A with four vertices and a convex polygonal obstacle B i with four vertices such that a.the configuration space obstacle CB i has eight vertices, b.the configuration space obstacle CB i has seven vertices, c.the configuration space obstacle CB i has as few vertices as possible. 2.Find a convex translating polygonal robot A and a non-convex polygonal obstacle B i without a hole such that the configuration space obstacle CB i has a hole. 3.What is the Minkowski sum of two disks with radius 1? 4.Consider two line segments of length 1. What is the a.largest area of their Minkowski sum? b.smallest area of their Minkowski sum? 26

Minkowski Sums Minkowski sum of a convex and a non-convex polygon can have complexity Ω(mn)! Minkowski sum of two non- convex polygons can have complexity Ω(m 2 n 2 )! 27

Translational Motions Translating polygonal amidst polygonal obstacles in W = point amidst polygonal Minkowski sums in C. 28

Mobility in Configuration Space Translations only Isolated point in free space: no translations possible 29

Mobility in Configuration Space Translations only Small translations possible, but escape impossible 30

Contact ‘Surfaces’ in a 2D C-Space An edge of CB i corresponds to an edge-vertex or vertex- edge contact. A vertex of CB i corresponds to a vertex-vertex contact = a double edge-vertex contact. A BiBi BiBi A 31

Contact Surfaces in 3D C-Space Translational motion in 3D: a surface patch corresponds to a vertex-facet contact an edge-edge contact a facet-vertex contact Arbitrary motion in 2D: a surface patch corresponds to a vertex-edge contact an edge-vertex contact 32

Contact Hypersurfaces f-Dimensional C: single contact: (f-1)-dim. boundary feature in CB/C free, double contact: (f-2)-dim. boundary feature in CB/C free, … f-fold contact: 0-dim. boundary feature in CB/C free. Number of multiple contacts determines complexity of C free, and complexity of (exact) motion planning algorithms. O(n f ) bound for constant-complexity robot. 33

Free Space Complexity Lower bound constructions: 2D translational motion Convex robot: roughly one order of magnitude smaller Single cell: roughly one order of magntude smaller Ω(n 2 ) double contacts 34

Free Space Complexity Lower bound constructions: 2D translation and rotation Convex robot: roughly one order of magnitude smaller Single cell: roughly one order of magntude smaller Ω(n 3 ) triple contacts 35

Free Space Complexity Similar construction in 3D Worst-case constructions cage the robot in most placements Better bounds on free space complexity under realistic assumptions, such as fatness, low density, unclutteredness, small simple cover complexity 36

Collision Detection Recall that the Minkowski sum of sets P and Q is Recall that the configuration space obstacle CB of B with respect to translating object A is which consists of all translations of A in which it intersects B 37

Collision Detection: Intersection Objects A and B intersect if and only if If A and B intersect then the point v closest to O on the boundary of corresponds to the shortest translation taking A and B apart. The length of v is referred to as the penetration depth. (Similar for the point w closest to O on the boundary of.) 38

Collision Detection: Non-Intersection Clearly, objects A and B do not intersect if and only if If A and B do not intersect then the point v closest to O on the boundary of corresponds to the shortest translation that makes A and B intersect. The length of v is the distance between A and B. (Similar for the point w closest to O on the boundary of.) 39