Sensing, Tracking, and Reasoning with Relations Leonidas Guibas, Feng Xie, and Feng Zhao Xerox PARC and Stanford University

Sensing for Reasoning and Acting Lead to high-level understanding of a situation Support efficient decision- making Allow for efficient updates as the world-state changes A system of collaborating sensors should provide data that can: Clustering of enemy forces Optimal route selection

Focus on Relations between Objects Data size: relational or aggregated information can be compactly encoded and efficiently transmitted Robustness: aggregated information can be statistically more robust than individual data Sensor control: Reasoning is naturally relation- based; well-established theories exist on how to go from observations to conclusions a c High-level reasoning is often based on relations between objects, or other aggregate information. Focusing on relations has many advantages:

Example: “Am_I_Surrounded?” We have a relatively flat terrain with friendly (white) and enemy (black) vehicles, stationary or moving. The field contains PIR and acoustic amplitude sensors. We want to know when a friendly vehicle is surrounded by enemy vehicles. Say “surrounded”  inside the convex hull

“Am_I_Surrounded” via CCW Relations The notion of being surrounded can always be captured by a a number of CCW predicates on the vehicle positions. CCW(a,b,c) asserts that the triangle defined by a, b, and c is counterclockwise oriented. CCW predicates can capture the notion of the convex hull of a set of points, and the notion of inclusion in a convex polygon or triangle.

Direct Sensing of CCW Relations s1, s2, and s3 are PIR sensors They directly sense the relations CCW(f,e1,e2), CCW(f,e2,e3), and CCW(f,e3,e1). From these relations, the containment of f in the triangle defined by e1, e2, and e3 follows.

Target Localization for CCW Relations In general, direct sensing is not possible. However, by sufficiently localizing the targets, CCW relations (or their negations) can be estimated with high confidence. CCW relations can be estimated using the current belief state of the system about the location of each target. Improved estimation requires a non-trivial sensor selection strategy. For CCW(t1,t2,t3), s2 is more valuable than s1.

An Example: CCW Using Distance Sensors We are given three targets t1, t2, and t3. Two acoustic amplitude measurements per target were used to obtain the distributions shown. Our goal is to have (say) 90% confidence that CCW(t1,t2,t3) is true. This is not yet the case in the current belief state – additional sensing is necessary to improve localization. We assume the sensor field is known. Darker color indicates higher likelihood. The x indicates actual target location, the x expected location. The + indicates location of the last sensor used. t1 t2 t3

Overview Initialization Compute Multi-res Belief States Compute Initial Relation Uncertainty If uncertain, else Select new sensor-target readingUpdate Relation Uncertainty Report Relation

Initialization Set up sensor model Generate sensor map Generate target locations with three different geometric relation, acute, round and skinny Select initial sensors whose readings would generate desired initial belief state shapes.

Data Structure Setup Estimate initial belief states Construct multi-res belief states Initialize the triplet bins, ccw bin and non cw bins are both empty, uncertain bin have one triplet of mass 1 formed by the 3 pyramid tops

Computing the CCW confidence Target location belief states are represented as 2-d arrays. A quad-tree-like multi-resolution structure is superimposed, to speed up computation. Undetermined tile triplets are refined until the mass in the “CCW’ or “not CCW” bins exceeds the desired threshold, or until a triplet budget is exceeded. Triplets of quad-tree tiles, one from each target, are classified as CCW, not CCW, or undetermined.

Sensor Selection Strategies Use Mahalanobis distance and cycle among the targets. Use a geometric heuristic – this selects a sensor close to the perpendicular from one target to the line through the other two. Use expected reduction in uncertainty, based on the current belief states and the known sensor locations. To have ground truth, simulate the effect of using each sensor, and choose the one that actually reduces uncertainty the most.

Geometric Heuristic

Round Robin using IDSQ

Expected Ideal Compute expected belief state using current belief state of a target and the new sensor location. Select the sensor-target pair whose expected belief state gives the best improvement in resolving relation uncertainty Because not all triplet are resolved, the reduction in relation uncertainty is only an estimate. This method is very expensive and not always effective.

Using the Target Cycling Heuristic Uncertain mass CCW mass 23 45

Using the Geometry Heuristic Reduction in uncertainty is less.

Result Analysis Geometric Heuristics effective for a variety of target configurations effective for a variety of sensor maps very cheap to compute Round Robin effective for a variety of target configurations sensitive to sensor map distribution (best for uniform sensor map) can be computed in real time

Tracking the “Am_I_Surrounded” Relation As vehicles move, we maintain three CCW relations establishing that the friendly vehicle is indeed in a triangle formed by three enemy vehicles. When the friendly vehicle escapes the triangle, the sensor net searches for a new enemy vehicle that can still establish containment. The triplet basis is maintained under motion.

Conclusion Sensing and tracking with relations provides a new approach to sensor exploitation. The approach allows the sensor net to focus its resources towards the task at hand and take advantage of discrete reasoning. It also raises many new and interesting problems on task decomposition, sensor selection, recovery from failure, etc.