Presented by David Stavens

Autonomous Inspection Compute a path such that every point on the boundary of the workspace can be inspected from some point on the path. Can think of this as a watchman’s route. Key Innovation: Handles sensors with limited range and visibility (incidence).

Incidence Constraint θ must be less than θ max.

Background on Watchman Problem Chin and Ntafos, 1988: – O(n 4 ) in simple polygons. – But NP-Hard if the polygons have holes. Hoffmann et. al. (1998) gives online, constant- factor approximation algorithm.

One Out-of-Date Assumption? They assume sensing “is a time consuming operation” thus “it makes sense to identify at which points on the inspection path it is actually necessary to perform the sensing.” This assumption seems out-of-date with modern robots. Sensing for this application is computationally cheap (eg: SICK Lidar).

Randomized Algorithm Expands Gonzalez-Banos and Latombe [1998]. Pick point P from the set of points on the border of the workspace that aren’t guarded. – (Efficient implementation might use trees.) Region R that can see P (equals what P sees) is constructed and clipped according to sensing limitations.

Randomized Algorithm (cont’d) Region R is sampled K times. Each sample is evaluated as a new guard. Sample that guards most NEW length is kept. – (Would backtracking be an improvement?) Repeat above until everything is guarded.

Algorithm Illustration Selected Point Already Covered

Algorithm Illustration Selected Point Already Covered Visibility Region (Note we must check orientation in this case – not discussed in algorithm. The sensor is NOT omnidirectional.)

Algorithm Illustration Selected Point Already Covered Visibility region Two Sensing Candidates

Algorithm Illustration Newly Covered Area Already Covered Visibility region Keep Candidate Seeing More New Boundary

Solving TSP Connecting the guards is the well-known Traveling Sales(wo)man Problem (TSP). – TSP is NP-Complete. Fortunately: Simple 2-approximation exists. – Use the pre-order MST walk. – Works provided graph obeys triangle inequality: A -> B + B -> C ≥ A -> C

Building the Graph Authors use the shortest-paths graph. Fully connected graph: – One edge for each pair of guards. – O(N 2 ) in total. Edge weight for (i,j) is length of the shortest path between (i,j). How do we compute this?

Computing Shortest Paths Start with a SECOND graph containing: – Nodes: All guard positions and workspace vertices. – Edges: Any free straight line between two nodes. – Weights: Euclidean distance Then search the graph. Optimization: Only connect nearby points. – Minimal effect since MST uses short edges.

2D Result 1

2D Result 2

2D Result 3

What About 3D? Algorithm works “largely unchanged.” But authors want an optimization to avoid “visibility volumes,” which are expensive to compute. – Recall these are used to pick and evaluate guard locations. Key: Optimize separately for “visible surface enumeration” and to determine where to sample.

Determining Where to Sample Range constraint is a sphere. Incidence constraint is an (infinite) cone. Thus: – Calculate sphere-cone intersection. – Ray-trace between workspace boundary point and points on the intersection to verify visibility.

Visible Surface Enumeration …allows us to determine what surfaces a guard location candidate can see. Uses very basic algorithm from graphics: – Iterate over surfaces, front-to-back order. – Obviously, surfaces occlude those behind them. Many ways to deal with this, paper doesn’t give great depth on this subject.

One Problem Iterating can have circular dependences!

3D Result 1

3D Result 2

Closing Remark Partially Observable Markov Decision Processes (POMDPs) are another way to generate policies that inspect or clear spaces. Can even make guarantees about intruder not being able to “slip by” the robot. – Doesn’t seem to be the case here.

POMDP Example – Will show file-not-available if streamed. – Must be saved. Work by Joelle Pineau, McGill University Formerly part of Thrun’s lab at CMU.