Localizing Vapor-Emitting Sources Using
a Distributed
Chief Investigator:
Panos Tzanos
Problem Statement
With the ever increasing technological
advances in networking and the miniaturization of electromechanical systems,
the ability to deploy large groups of autonomous vehicles to perform various
coordinated tasks is increasing. Some of
the foreseeable tasks include search and recovery missions, hazardous waste
clean up, exploration, and
surveillance. Many of these tasks
cannot be performed by a single agent.
Furthermore, having many vehicles provides robustness to failures of a
single agent, or communication links.
In particular, we would like
to devise a method to estimate the location of a vapor-emitting source using
sensor mounted mobile robots that communicate through a distributed network. Such a
method would be useful in applications such as explosives detection, locating
hazardous chemical leaks, and pollution sensing.
Figure 1: Depiction of sensor mounted
robots measuring vapor concentration
Model
To model the concentration of
a diffusing substance at a point 
, we use the solution to the classical diffusion equation for
a source free volume and constant diffusivity:
where 
is the complimentary
error function. The concentration
depends on the location of the source 
, the diffusivity 
in m2/s,
the diffusion rate 
in Kg/s, and the
initial time of diffusion 
. These are all
unknown parameters.
Sensor Model
The concentration
measurements will invariably be affected by the presence for foreign materials,
and sensor noise. Therefore we may model
the sensor measurements in the following fashion:
where 
is the bias term
representing the sensor’s response to foreign substances, and 
is the sensor’s
noise. We may assume the noise is
Gaussian distributed with zero mean and variance 
.
Taking measurements at 
points at 
different times, we may collect the measurements into a
vector form
where 
, and 
. The vector 
is an 
-dimensional vector whose 
-th component is 
, 
, 
. The same can be said
for the vector 
. Finally, 
is an 
matrix whose 
-th row is
Parameter estimation
To obtain an estimate of the
unknown parameters, maximum likelihood estimation is used. To obtain a measure of the variability of
these estimates the Cramer-Rao bound (CRB) in computed.
The Network
Communication between the
mobile agents is conducted in a distributed fashion. Each agent has the
capability to communicate only with other agents in its communication range. This
is due in part to the spatially-distributed nature and limited communication
capabilities of a mobile network. Also,
we would like to reduce the computational complexity of our sensing
algorithm. Using a distributed network
the computational complexity grows linearly as opposed to quadratically in a
centralized communication scheme.
The Moving-Sensor Algorithm
Our tactic for moving the
mobile agents is as follows: Suppose we
have already taken 
measurements with a
time interval 
between
measurements. We want to place the
agents in locations where the CRB is a minimum.
To accomplish this, each agent computes the gradient of the CRB given
its measurements up to and including 
and then moves in the direction opposite to the gradient to a
point that it can reach at time 
. We continue this
tactic, and estimating the unknown parameters at each new location, until the
CRB computed by one of the agents goes below some pre-defined threshold.
Accomplishments
At this current time, we have
successfully simulated the moving-sensor algorithm for one mobile agent, and
for a distributed network of mobile agents.
We have also begun analyzing a linearized version of the sensor model. Here is a sample of our results for the case
of one mobile agent and multiple mobile agents.
One Sensor Algorithm
In this simulation, we have placed
the vapor source (denoted by the “x”) at the coordinates (-50, -50). Sensing begins at 100 s after the beginning
of emission. The sensor first collects
concentration data by traversing a circle centered at (0, 0) and having a
radius of 250 m. Data is collected every 10 s. At time 
= 400 s the moving-sensor algorithm begins using the data
accumulated during the first 400 s.
After 22 more measurements (220 s) the CRB of both the x and y
coordinates of the vapor source go below 0.1 and the algorithm stops. As can be seen from the plot of the estimates
of x and y that they converge to around (-50, -50).
Multiple Sensor Algorithm
In this multiple sensor
simulation, the vapor source (denote by the red circle) is again placed at the
coordinates (-50, -50). Again sensing
begins 100 s after the beginning of emission. The sensors (denoted by the
“x”’s) collect concentration data for 300 s and share this data with their
neighbors. Each agent has a
communication radius of 300 m. At time = 400 s the algorithm begins, after 6 more measurements one
of the agents measures the CRB of the x and y coordinates of the vapor source
to go below 0.1 and the algorithm stops.
The CRB of the two coordinates as measured by the algorithm terminating
agent are shown in the upper right hand corner.
The plot on lower left shows the same agent’s estimates of the x and y
coordinates of the vapor source.
Future Work
We hope to have a full
understanding of the convergence properties of the moving-sensor algorithm, and
have designed an optimum distributed communication network.
References
B. Porat and A. Nehorai.
“Localizing vapor-emitting sources by moving sensors,” IEEE Trans. Signal Processing, vol. 44, no. 2, pp.1018-1021, Apr. 1996.
J. Cortes,