1. Introduction
Locating the source of chemical releases is an important challenge in the field of robotics. Traditionally, human searchers and animals with better olfaction capabilities have been used to perform localization tasks for search and rescue, chemical leaks, and finding concealed narcotics and explosive devices such as land mines. However, there are many challenges inherent in this approach. Training humans and animals requires a large amount of time and effort; they have limited endurance, there is a limit to the set of chemicals they can sense [
1], and there are situations in which the safety of human and animal searchers precludes their application. As a result, the application of robots with advanced olfactory and navigation capabilities has emerged as a machine-based alternative. One of the major research challenges with robots in olfactory applications is the efficient and effective localization of multiple unknown sources in outdoor environments under turbulent conditions. This type of situation commonly arises with fugitive chemical emissions in industrial settings.
Chemical or odor dispersion occurs through two primary mechanisms: diffusion and advection. Diffusion occurs due to a difference in chemical concentration. When diffusion is the dominant dispersion mechanism, chemical gradients form, with concentration decreasing with distance from the source. Initial research took advantage of this fact and developed algorithms to follow the chemical gradient to its source. These algorithms are thus classified as ‘Chemotaxis’ algorithms. Examples include the E. Coli, sometimes called the Biased Random Walking (BRW) algorithm [
2], and gradient-based algorithms [
1,
3].
In outdoor environments, advection, not diffusion, is the dominant form of dispersion [
4]. Chemotaxis algorithms typically rely on smooth chemical gradients to find the odor source. Under turbulent airflow conditions, research has shown that Chemotaxis algorithms have a much lower success rate than Anemotaxis algorithms, which use fluid flow measurements along with chemical concentration measurements [
1,
5]. The odor plume under turbulent conditions typically exhibits spatio-temporal variability and does not follow a smooth chemical gradient. To overcome fluctuations in the concentration gradient, Anemotaxis algorithms such as Silkworm moth [
3], Zigzag [
6], and Dung Beetle use the wind flow direction to follow the plume upwind to the source.
The aforementioned algorithms are forms of reactive algorithms. One of the shortcomings of reactive algorithms is that under dilute conditions, detectable events are separated by gaps with no detectable anomaly. As the delay between detections increases, the performance of reactive algorithms decreases [
7]. Cognitive algorithms differ from reactive algorithms in that they use information from current and past measurements to guide the searcher’s movements. Even when a plume is dilute and very few odor detections occur, cognitive algorithms are still able to find the odor source, as each odor detection is used to improve the belief state of the possible leak source location. Several different methods have been developed to retain knowledge of the current state, including concentration grid maps [
8,
9] and the ‘Source Likelihood Map’ [
9].
The above algorithms only consider one odor source; however, multiple odor sources are common in many applications such as fugitive chemical emissions from industrial facilities. Typically, the plume is detected far downwind of the facility, and it is difficult to ascertain if the plume originates from a single source or multiple sources. In this case, multiple odor source localization (MOSL) algorithms need to be used. MOSL algorithms for mobile platforms currently fall into three categories: occupancy grid mapping algorithms [
10,
11,
12,
13], particle swarm optimization (PSO) algorithms [
14,
15,
16,
17], and learning algorithms [
18,
19]. In [
19], a sampling strategy for a multi-robot system using sparse Bayesian learning was presented and tested. When testing was performed with multiple sources, the same model was used for both the simulation environment and the model-based exploration strategy. The sampling strategy was also tested using a more complex simulation environment for the case of a single source to determine the effects of model mismatch, and a drop in source localization was observed. The localization performance comparison used the earth mover’s distance (EMD) as the metric. In [
19], reinforcement learning along with existing domain knowledge was used to develop a policy for the movement of a searcher for the problem of localizing multiple sources in a steady-state environment. Domain knowledge was developed using the methods from [
18]. The Asynchronous Actor Critic (A3C) algorithm was used, various reward functions and observation spaces were tested, and the localization performance was compared using the EMD.
In [
14], a modified particle swarm optimization (MPSO) technique for multiple sources was used that relied on turning a plume off after finding the source so that the next source could then be searched for. The ability to turn off plumes is a major assumption and is unrealistic in many real scenarios. A niching strategy was used in [
15] to split and merge the particles into niches to ensure each niche of particles followed different plumes to the sources, which parallelizes the search even in situations with multiple peaks. A Diverse-PSO that uses group dynamics was presented in [
16] and demonstrated better performance than MSPO and Niching-PSO in a turbulent flow environment with multiple sources. In [
17], a hybrid teaching learning particle swarm optimization (HTLPSO) was presented, and its plume localization performance was compared to a PSO algorithm and a teaching–learning-based optimization (TLBO) algorithm.
Unlike PSO methods and the learning-based methods, occupancy grid mapping algorithms are not reliant on a specific path for the searcher(s) for localization, nor do they require multiple searchers like the PSO methods and [
18] do. However, the relative performance of the IP and DS algorithms are not well understood, and particularly so in conditions with natural time-varying flow. Existing testing has been limited. For example, the IP algorithm has been tested in two situations in a weak flow environment, i.e., in a hallway and in an underwater environment. The DS algorithm has been tested in a single simulation setup and outdoor environment. Before these algorithms can be applied to localize fugitive emissions outdoors, further testing is needed. The IP algorithm has not yet been tested in time-varying environments [
10,
11], and the DS theory algorithm has been tested three times outdoors, but only the results from one of those experiments were presented [
12,
13].
To improve understanding of the performance of these algorithms in outdoor environments, we tested and compared the IP and DS algorithms under time-varying flows and a variety of conditions and evaluated their source-localization performance. We used a simulation approach with a filament dispersion model based on Farrell et al. [
20]. This model was chosen due to its ability to simulate dispersion with time-varying wind flow and its fast computation speed. The localization performance of the two algorithms was measured using the earth mover’s distance (EMD). Using the EMD, we evaluated how localization performance responded to changes in environmental parameters, which included search area size, source position, mean wind speed, puff random walk, bandwidth, and release rate, as well as the searcher parameters—searcher velocity and sample frequency—which were also varied. In this work, we refer to searchers as those entities tasked with discovering the source of a chemical release or emission. Each search was performed using a raster search, and both algorithms used the same wind, concentration, and non-detection/detection concentration threshold. In each set of simulations, a single parameter was varied, with the parameter being either at its high or low level. This was conducted to determine the sensitivity of the algorithms to each parameter.
2. Algorithm Description
While both MOSL algorithms are similar in that they use occupancy grid (OG) mapping, there are two primary differences. First, the algorithms differ in how they represent the belief state. The IP algorithm uses probability theory as its belief representation [
10,
11] for each cell in the occupancy map, while the DS algorithm uses DS theory [
21] instead of probability theory. Second, they differ in how the belief update is performed when there is a detection event. The DS algorithm assumes the conditional independence of detections, which means that each cell being updated is independent, so information from other cells is not considered during the probability update. By contrast, the IP algorithm assumes each cell is dependent for detection events, in which case, it considers information from other cells during the probability update.
When a detection occurs with multiple potential sources present, no clear information on possible upwind sources can be obtained. To address this, the IP algorithm uses an update rule for detection events where the cells are dependent. For the case where it is known that there is only a single source, a detection event provides information about the entire map [
10,
11]. In the case of multiple sources, a detection event can occur in response to one or more sources. The number of sources contributing to that detection event cannot be known without further information. Dependencies arise between the occupancy grid map cells associated with the possible source locations that could have contributed to the detection event. At each time step, the information from the measurement needs to be used to update the current occupancy grid map. To perform this grid map update, all dependencies need to be accounted for. The number of calculations needed to account for all the dependencies scales at
, where
is the number of cells in the occupancy grid map. This formulation quickly becomes intractable as the search area grows or the grid resolution increases.
Addressing the problem of exponential scaling with search area, the IP algorithm uses an update rule (Algorithm 1) that can handle multiple odor sources and scales linearly with number of grid cells. The IP algorithm calculates an approximation of the marginal posteriors of the cells within the occupancy grid map. The IP algorithm assumes the independence of the posterior probability and interprets the current measurement considering past measurements but does not reinterpret past data based on new information. Each update interprets the new marginal probabilities using the dependencies from the current measurement but does not maintain the dependencies after the measurements have been incorporated and become the new prior. In its updates, the IP algorithm does not reinterpret past data considering new data. While conducting this reduces computational complexity, this algorithm is path-dependent [
11]; therefore, the resulting map depends on the order in which the measurements are taken, and early interpretations can disproportionately affect the result.
Algorithm 1 IP algorithm () [11] |
1: Time starts at instant |
2: For all cell do |
3: if is in the perceptual field of then |
4: if ( results in a detection) |
5: |
6: |
7: else (, results in a non-detection) |
8: |
9: end if |
10: else |
11: |
12: end if |
13: end for |
14: return |
where . It is trivial from to recover . In fact |
To address the problems associated with the search path dependence as described in [
11], DS theory was used in [
12] instead of probability theory. For the task of localization, each cell has two states:
(source) and
(not a source), which compose the frame of discernment
. In probability theory, the entire belief is split between the elements making up the frame of discernment. DS theory, on the other hand, assigns belief to the power set of the frame of discernment
. This power set for the given frame of discernment results in the following power set:
. The basic probability assignment (BPA), or
, is the measure of available evidence for a given state in the subset
. The BPA assigned to
represents the amount of ignorance about the state of the cell. Following the convention of [
21], the state
will be replaced with
for the unknown state. In its update rule, the DS algorithm ignores any potential dependencies between cells and assumes they are independent. The belief mass functions for detection and non-detection events are given as:
where
is the mass belief,
is the evidence of a given state,
is the single source detection probability for cell
i,
is the reliability of the gas transport model,
is a detection event,
is a non-detection event,
is the evidence that
is a source,
is the evidence that
is not a source,
is the evidence that
is the unknown state and is the set of both
and
(
), and
is the false-negative probability of a non-detection event. The reliability of the gas transport model
is a term that is unique to [
12,
13]. A conservative estimate for
is used in [
12], whereas in [
13], the value of
is thought to be a function of distance and the relationship was empirically derived. In this work,
was selected as 0.8 and the odor patchiness was chosen as 0.75. The OG map for the DS theory algorithm updates each cell independently. The three possible states
,
, and
result in three different mass beliefs for each cell in the occupancy grid map. The mass belief update is given as:
where
represents the amount of conflict between the two pieces of evidence and is given by:
The DS theory algorithm in [
12,
13] uses the air mass path (AMP) [
22] to determine the range of possible locations where the air passing over a searcher could have come from. The AMP was also used to calculate the probability of the various paths that could have been taken by the air mass to reach the searcher’s position (
Figure 1). The path is created using the history of wind velocities. The AMP probability calculation is restricted to only the locations within the path to reduce computation time. The edges of the path are found using a threshold-detection probability. The width of the path is a function of the variability in wind velocity and time expired prior to the present measurement. The probabilities obtained by the AMP are then used as an input to the algorithms as the single-source-detection probability. The full AMP formulation used here is described in [
12,
13]. The IP algorithm in [
11] was tested in an environment with no strong airflow, and the update rule used did not take advection into account. For situations with strong airflow, the IP algorithm uses the AMP as its single-source-probability model.
5. Discussion
Overall, the IP algorithm outperformed the DS theory algorithm in terms of source localization. The IP algorithm’s predicted sources were closer to the true source locations than the DS algorithm’s for all parameter levels, with a few exceptions. The OG maps had bands of predicted sources that extended from the true source location to the downwind edge of the search area. The width of the bands of predicted sources were narrower for the IP algorithm than the DS algorithm. The density of the sources in the bands were greater when they were closer to the true source locations. Both algorithms overpredict the total number of sources, with the DS algorithm overpredicting the number of sources to a greater extent than the IP algorithm. All the predicted sources of the IP algorithm were subsets of the predicted sources of the DS algorithm. Additionally, the DS algorithm had predicted sources that were observed on the upwind side of the true source locations in the DS algorithm’s OG maps and were not observed with the OG maps produced by the IP algorithm. The IP algorithm also outperformed the DS algorithm in terms of EMD scoring, with a few exceptions, demonstrating consistency between the qualitative OG map observations and the quantitative EMD scores. The exceptions include the low source position parameter level in the staggered configuration, where there was not a statistical difference between the EMD scores, and the low release rate parameter level, where the DS algorithm had a lower EMD score than the IP algorithm. With the low release rates, the predicted sources often failed to include the true source location at low release rates with both algorithms. While the IP algorithm overall had lower median EMD scores, the variance in EMD scores was much higher than the DS algorithm. The difference between the 75th percentile and the 25th percentile EMD scores was greater for the IP algorithm than the DS algorithm. The range of observed scores for each parameter level was also greater. The IP algorithm was more sensitive to changes in parameter level than the DS algorithm and had greater changes in median EMD score based on changes in parameter level. The IP algorithm had the lowest median EMD score, which was with the high sample frequency with inline configuration. The parameter levels across all configurations with the lowest EMD scores were high mean wind speed and high sample frequency with the IP algorithm. The median EMD scores for the DS algorithm were relatively similar for most parameter levels, with the exception of the low release rate and low and high source positions parameter levels. Generally, the results were the same across all configurations. However, the median EMD scores were higher for both algorithms in the staggered source configuration compared to the other configurations.
The IP algorithm typically had a lower median EMD score compared to the DS algorithm because the IP algorithm’s OG maps had a lower number of predicted sources, and the predicted sources were typically closer to the true source locations. The primary reason for the lower number of predicted sources was that the DS algorithm assumed that all detection events were independent, whereas the IP algorithm assumed detection events were dependent. Both algorithms assumed that non-detection events were independent. When the DS algorithm updated cells independently during a detection event, the algorithm did not consider that the other potential source locations calculated using the AMP could also have been the source, resulting in the increased number of predicted sources. To minimize the overidentification of sources during detection events, the IP algorithm conditioned its current probability update received from the AMP on the previous OG map, implicitly considering previous measurements in its update. Interpreting current measurements in the context of past measurements which were implicitly held in the OG map resulted in the updates for detection events being context-dependent, which resulted in the IP algorithm being path-dependent [
11]. Path dependence means that the order in which the measurements are received will have an impact on the final OG map. Note that past measurements were also not reinterpreted in the context of the new measurement.
Contextualizing detection events on the current map led to one of the primary drawbacks of the IP algorithm: early measurements influence the interpretation of later measurements. Early in the search, high-variance events, such as large amplitude plume meander or higher-variance random walk, have a greater influence on the IP algorithm’s EMD score compared to the DS algorithm’s EMD score. This was because high-variance events that occurred early in the search developed an OG map with predicted source locations with a skewed position, and all future measurements were conditioned on the skewed OG map. Furthermore, the IP algorithm had fewer total sources in its OG maps, which resulted in greater weighting to movement of potential source positions on the EMD score. The effect of having a lower number of predicted sources was highlighted by both algorithms at the low release rate parameter level, where the OG maps were very sparse and the greatest range in EMD scores were observed.
Potential source locations erroneously identified early in the search often do not end up in the IP algorithm’s final OG map. This was because subsequent non-detection events lowered the probability of the potential source locations. The IP algorithm required fewer non-detections than the DS algorithm to lower the probability of potential source locations. This was evident from the number of sources upwind of the true source locations in the DS algorithm’s OG maps and the lack thereof with the IP. Once the searcher was upwind of the true source locations, only non-detection events occurred. Each non-detection lowered the probability of any potential source locations upwind of the searcher that were identified as potential sources earlier in the search. The number of non-detections upwind of the true source locations was sufficient for the IP algorithm to eliminate potential sources upwind of the true source locations, but not for the DS algorithm.
Comparing the DS algorithm’s performance to the literature is difficult because of the limited number of experiments presented and missing environmental parameters for those experiments, making results difficult to replicate. The pattern of taking many non-detection events to lower the belief of potential sources was observed in the original papers [
12,
13], which detail the DS algorithm. Bands of cells identified as occupied by a source
were observed extending upwind from the true source locations early in the search in both [
12,
13]. Both of these papers also used the raster search pattern, but unlike our work, [
12] completed multiple ‘rounds’, repeating the raster search pattern multiple times per search. In [
12], the bands begin to shrink in size after five rounds of searching the entire search area. In [
13], after the second round of searching the search area, the bands had grown in thickness and formed a large patch around each of the source locations. Results on further rounds were not shown. The results after multiple rounds of searches showed that the DS algorithm was capable of lowering the probability of upwind sources but required more non-detection events than are measured in several rounds of searching. This could be a result of using the Dempster rule of combination as the combination rule, which lacks robustness when handling conflicting evidence [
23]. The use of a different combination rule that handles conflicting evidence better may result in better performance with the DS algorithm.
The IP and DS theory algorithms were most sensitive to parameter-level changes in release rate. This sensitivity was asymmetric, with the largest observed difference in median EMD score occurring between the low release rate and normal parameter level, and minimal difference in median EMD score between the high release and the normal parameter level. The asymmetry is a result of using binary detection events for source localization, as the searcher only cares whether the concentration is above the threshold concentration for detection. At the normal release rate and high release rate, the plume is continuous for both cases and the searcher would observe a similar number of detection events. The asymmetry arises when the release rate is low enough that a discontinuous plume forms with a concentration below the detection limit dispersed along the instantaneous plume centerline. Above this release rate, all release rates appear the same to the searcher, and below this release rate, non-detection events occur even when the searcher is directly where the instantaneous plume would normally be at a higher release rate, resulting in a sparser OG map for both algorithms. With the low release rate parameter level, the release rate was very low, and very few detections were observed by the searcher, which resulted in a very sparse OG map, which contributed to the greater variance in EMD score.
The IP algorithm had poorer localization performance compared to the DS algorithm at the low parameter level in terms of EMD score and qualitative observations of the OG maps. The IP algorithm’s potential source locations frequently failed to include the true source locations, while the DS algorithm’s potential sources occasionally did include the true source locations. At the low release rate, the effects of the IP algorithm’s path dependence were magnified, as the locations in which detection events occurred were very variable.
The only parameter where the median EMD score was not statistically significant was the low source position parameter level in the staggered configuration. However, the EMD score did not properly evaluate the localization ability of the algorithms at the low source position parameter level based on qualitative observations of the OG maps. From the OG maps, we found that the DS algorithm’s localization was worse than the IP algorithm’s. The DS algorithm had more predicted sources than the IP algorithm, with sources extending upwind of the true sources to the upwind edge of the search area, whereas all the IP algorithm’s predicted sources were concentrated between the true source locations and the downwind edge of the search area. Because of the upwind predicted sources, the maximum distance between a predicted source and the true source locations was far greater for the DS algorithm than the IP algorithm. As the DS algorithm’s downwind sources were very dense and, at the low source position, the downwind predicted sources were all close to the true source location, the lower number of upwind sources had a very small contribution to the overall EMD score, resulting in the DS algorithm having a non-statistically significant difference in median EMD score from the IP algorithm. This was a result of the EMD score not being able to directly compare OG maps with different mass and requiring the mass of the predicted sources to be normalized to equal that of the true source locations.
Further evidence of issues with the EMD score to evaluate localization performance at the low source position was demonstrated by the IP algorithm’s median EMD score in the low source position, for the staggered configuration was higher than its median EMD score at the normal parameter level, despite the fact that the predicted sources were fewer in number and closer to the true source location compared to the normal parameter level. At the low source position, the true source locations were close to the downwind edge, which resulted in the fewest number of sources of any source position, and for the IP algorithm, all of the predicted sources were near the true source location. At the normal parameter level, there were far more predicted sources and a larger distance between many of the predicted sources and the true sources than the maximum distance observed at the low source position parameter level. Further issues with using the EMD score were highlighted, with the IP algorithm’s OG maps at the high source position having a lower score than the low source position for the inline and staggered source configurations, despite having more sources, and with many sources being the furthest distance from the true source location than with any other parameter level.
The underlying mechanisms responsible for the IP algorithm’s sensitivity to changes in parameter level were the same as the mechanisms that caused the IP algorithm to have greater variance in EMD score compared to the DS algorithm, namely, the path dependence of the IP algorithm and the lower number of predicted sources. The sensitivity to changes in parameter level due to path dependence was demonstrated through the best localization performance occurring with the high sample frequency and high wind speed parameter levels. Both high wind speed and high sample frequency parameter levels minimized high-variance measurements. Higher sample frequency provides better resolution on the exact positioning of the plume. In the case of the IP algorithm, due to better information, early measurements were less likely to produce a skewed map and cause future results to be interpreted poorly. By contrast, the DS theory algorithm’s median EMD score did not improve to the same extent as the IP algorithms with higher frequency.
The high wind speed parameter level also produced low median EMD scores. This was due to the fact that higher wind speed produces narrower plumes with less variable concentration at the edges of the plume when compared to a lower wind speed, as there is less time for dispersion to occur for the same distance downwind. Due to the lower amount of meander and lateral dispersion at the high wind speed parameter level, the IP algorithm had fewer high-variance measurements and produced narrower bands of probability, resulting in a better EMD score compared to the normal or low wind speed parameter level.
In general, the difference in the DS theory algorithm’s median EMD score between the high and low parameter levels was less than the IP algorithms due to the consistently high occupancy in the DS algorithms OG map. With the high number of predicted sources, there was little variation in predicted sources from one simulation to the next. Furthermore, the high number of predicted sources reduced the relative weighting on the small changes in source position that did occur, resulting in minimal difference in EMD score between simulations.
With the staggered configuration, the total width of the band of probability is thicker than the other two source configurations, resulting in higher median EMD scores for all parameter levels. The staggered configuration was designed to produce partially overlapping plumes. The partially overlapping plumes produced the largest crosswind width of detection events, resulting in the greater thickness of the bands of probable source locations.