Randomized Planning for Short Inspection Paths Tim Danner Lydia E. Kavraki Department of Computer Science Rice University

Outline Introduction Art Gallery and Watchman Route Problems Adding Realism 2-D Algorithm –Selecting Guards –Connecting Guards –Results 3-D Algorithm –Differences –Preliminary Results Future Work

Introduction Problem Definition Robot with vision capabilities Workspace W Path p Boundary dW W dW p

Introduction Problem Definition Find a short p from which every point on dW is visible W dW p

Introduction Uses Autonomous inspection of spacecraft exterior Flying camera building inspection Virtual reality architecture walkthrough

Art Gallery Problem Find minimal number of positions for guards to stand so that every point in a gallery is “visible” to at least one guard “Visible” is defined as the line of sight between the guard and the point lies entirely in W

Watchman Route Problem For workspace W, find the shortest path p in W such that every point on the boundary dW can be seen by a point on path p

Adding Realism “Straight-line” visibility is not very realistic for real sensors Length of line of sight must have a maximum Angle of incidence of line of sight must have a maximum – 60 degrees is a typical value System is adaptable for different sensors 2-D vs 3-D

Angle of Incidence

2-D Algorithm Sensing with real sensors is time consuming Two parts to the algorithm: –Solve “art gallery problem” to find locations for sensing locations - “guards” –Connect the guards with a short path, the “watchman route”

2-D Algorithm Selecting the Guards True minimal set of guards is an NP-hard problem Randomized planner is used in this case Uses Gonzalez-Banos and Latombe’s randomized, incremental algorithm

2-D Algorithm Selecting the Guards Main structure is a loop At each iteration, a point x on the border dW of W that is not yet guarded is chosen randomly Construct region which can see x (same as region which x can see) Apply two range constraints: limited length line of sight and angle of incidence

Choose a point x

Construct its visibility region

Apply maximum line of sight constraint

Apply angle of incidence constraint

Select new guards from blue outlined region

2-D Algorithm Selecting the Guards Sample region k times, evaluating each point as a possible new guard Sample which can cover the largest portion of the new length of border is chosen as the new guard and guarded border is updated Loop repeats until the entire border is guarded

2-D Algorithm Selecting the Guards One problematic case is sharp interior angles A “disproportionately large” number of guards may be needed and hard to place Incremental loop can be terminated

2-D Algorithm Connecting the Guards Basically, find an order to connect guards out of a possible n! orders Connect guards using a graph algorithm In this manner, the problem becomes a “traveling salesman problem”

2-D Algorithm Connecting the Guards Actually use an approximation to the TSP – pre- order walk of a Minimum Spanning Tree Applies in cases where the triangle inequality holds, which is the case for graphs in R 2 and R 3 (which our graph of guards is) Path length is bounded by 2X actual TSP for complete graphs If workspace is connected, then the graph is complete (for an inspection path to exist, the space must be connected)

2-D Algorithm Connecting the Guards Shortest Paths Graph (SPG) –One node for each guard –One edge for each pair of guards –Weight is assigned to each edge (i,j) that is equal to the shortest collision-free path from point i to j –May be straight line or more complex

SPG - Guard locations are nodes

2-D Algorithm Connecting the Guards Shortest path between two points is done by constructing and searching another graph, the workspace-guard roadmap Workspace-guard roadmap has one node for each vertex on dW and one node for each guard Has an edge between a pair of nodes i and j if and only if it is collision-free

SPG - Add nodes at vertices

SPG - Add edges

Use MST for short path

2-D Algorithm Connecting the Guards Complete graph has n 2 edges, but we can use a shortcut Only connect close points, by dividing workspace into rectangular grid About 10 nodes per cell Connection is made with a moving 3X3 window

2-D Algorithm Connecting the Guards

2-D Algorithm Note The default is to inspect the interior To inspect an exterior, surround entire workspace with a rectangle and mark it guarded W W

2-D Algorithm Experimental Results Most computing time spent creating visibility polygons Figure 1 Figure 2 Figure 3

3-D Algorithm Necessary for real workspaces Algorithm is very similar Difficulty – computing visibility polyhedrons in 3-D instead of visibility polygons in 2-D

3-D Algorithm Selecting the Guards Visual constraints remain simple Two steps will require visibility volumes –Determining a sampling region –Determining what surfaces a sampled point can see However, explicitly computing visibility polyhedron can be avoided

3-D Algorithm Selecting the Guards Determining sampling region utilizes constraint sphere and cone –Compute the intersection of these –Randomly sample this region and test if point is valid Both of these are much easier than computing a visibility polyhedron

3-D Algorithm Selecting the Guards Once points are sampled, need to: –Determine what surfaces can be seen by them –Subtract already guarded surfaces Use a front to back checking method, clipping each additional surface with the previous ones

3-D Algorithm Selecting the Guards Front Back

3-D Algorithm Selecting the Guards Complications –Need a way of defining order –Resolving circular problems –Selecting adequate data structure Binary Space Partitioning Tree for defining front to back order

3-D Algorithm Connecting the Guards No analog to creating the optimal shortest paths as in 2-D Shortest path is most likely not around vertices Instead of augmenting guard roadmap with workspace vertices, random planner is used

3-D Algorithm Preliminary Results Two Cubes 20 Seconds Four Cubes & Three Tetrahedra 143 Seconds

Future Work Considering dynamics in “path goodness” criteria Visiting areas rather than points Considering non-omnidirectional cameras