1.2 Contributions
1.2.1 Sensor Placement for Visibility-Based Coverage with Orientation
Cameras are one of the most commonly used sensors for coverage tasks. The capability of seeing every point in the environment is useful in a large number of applications. The problem of placing cameras for environment coverage is a classical one, broadly known as the art gallery problem. In the standard formulation, the environment is represented by an n sided 2D polygon and the sensors, also called guards, are modeled as points with omnidirectional vision. A guard is said to cover a point in the environment if
Figure 1.3: (Left) Autonomous boat developed for monitoring radio-tagged invasive fish during field experiments in lake Keller, Maplewood, MN. (Middle) During winters, we use a mobile robot on frozen lake. (Right) UAV obtaining multispectral images to form a nitrogen map of a corn plot in Janesville, MN. We study mobile sensing problems motivated by these applications.
the line segment joining them lies completely within the environment. The art gallery problem asks for the minimum number of guards sufficient to cover all points in the environment [1]. A summary of related research on art gallery problems is presented in the next chapter, in Section 2.2.
The standard formulation of the art gallery problem does not consider self-occlusions. Even when each point in the environment is covered, if some person starts moving in the environment, his/her back may occlude the front view. Obtaining a good view from all orientations is seen as an important requirement for many applications such as surveillance and video-conferencing.
Motivated by such applications, we study the coverage problem by imposing a new constraint termed △-guarding. The △-guarding constraint, introduced by Smith and Evans [13], states a point is covered if it is visible from two or more guards and it lies in the convex hull of the visible guards. If all points in the environment satisfy the △-guarding constraint, then even if any convex object is introduced anywhere in the environment all points on its perimeter will always be visible, in spite of self-occlusion. The △-guarding problem is to place the fewest number of guards such all points in a given input polygon satisfy the △-guarding constraint. Smith and Evans [13] proved the problem is NP-Hard. Efrat et al. [14] presented a randomized algorithm achieving
O(log copt)–approximation for polygons without holes,1where coptis the optimal number
of guards.
Our contributions are as follows: First, we prove a lower bound on the number of guards required for △-guarding any input polygon. We show that any △-guarding set uses at least Ω(√n) guards for any n-sided simple polygon. Second, we use this lower bound to present an O(log copt) approximation algorithm for polygon with and without
holes, when the guards are restricted to vertices of the polygon. Since copt itself can be
very large in practice, we restrict the input to a set of chords in the polygon. These chords can represent, for example, paths a target is likely to take in the environment. Our goal is to △-guard at least one point per chord. We present an approximation algorithm that is guaranteed to use at most 12 times the optimal number of guards. This result is one of the few constant-factor approximations for visibility-based coverage problems.
The results on this problem were first presented at ICRA 2014 [15] and a journal version is currently under review.
1.2.2 Bearing Sensor Placement for Target Localization
While a single camera can detect a target, information from multiple cameras might be necessary to precisely localize it. Cameras give bearing measurements towards targets in their field-of-view. Measurements from multiple sensors can be combined to estimate the target’s position. The uncertainty in estimation decreases as more sensors are used. Furthermore, the uncertainty is a function of the relative position of the target and the sensors.
In such a scenario, it is no longer sufficient to say a point is covered if it is sensed by one or more sensors. Instead, a richer notion of coverage is required. We will require each point to be sensed with multiple sensors placed in such a way so as to guarantee some upper bound on the estimation uncertainty. Thus, the coverage requirement is to guarantee that if the target were at any point in the environment, measurements from all sensors can be combined to yield an estimate with sufficiently good quality.
We model the sensors as measuring a bearing towards the target corrupted by an
1A polygon with holes is a polygon which contains one or more non-overlapping polygons within it. See Section 2.2 for more related terminology.
unknown but bounded amount of noise. We seek worst-case guarantees for our place- ment: Given the true location of the target, imagine an adversary choosing the noise values for sensors. The true target location can be anywhere in the intersection and we would like this set to be “small” no matter where the target is. One way of solving this problem would be to place sensors everywhere in the environment. There is a trade-off between the number of sensors and the guarantee on resulting uncertainty. We study the bi-criteria optimization problem of minimizing the number of sensors used and the resulting uncertainty achieved. We consider a simple square environment. Even in this basic setting, devising a sensor placement scheme and analyzing its performance turns out to be challenging, as we will see in Chapter 4.
Our first contribution is to present a lower bound on the number of sensors required for any placement algorithm as a function of the desired uncertainty. Our main result shows that by placing sensors on a triangular grid-like placement, 9 times as many sen- sors as an optimal algorithm are sufficient to guarantee 6 times the desired uncertainty when the maximum sensing noise is less than π4. We also show that in the triangu- lar grid placement, only a constant number of sensors need to be activated to achieve the desired uncertainty, a property that can be used for designing energy/bandwidth efficient sensor selection schemes.
The results on this problem were first presented at ICRA 2013 [16] and a journal version is currently under review.
1.2.3 Multi-Target Visual Tracking with Teams of Aerial Robots