Adaptive source localization for mobile robots based on Kriging prediction

This paper addresses the source localization for mobile robots under the assumption of spatio-temporal invariance. Two kriging-based source localization methods, namely global traversal and adaptive gradient extremum, are proposed and implemented using limited sampling data. The global traversal method controls the robot to traverse the whole region according to a fixed trajectory, and finally fits the information distribution of the sources in the whole target region, so as to obtain the location and number of sources. The adaptive gradient extremum method initially controls the robot to collect data by traversing the target search area. Simultaneously, it uses the collected data to fit the source distribution across the entire region. Then, it utilizes gradient and extremum principles to determine the next target point iteratively, eventually reaching the positions of the sources. The simulation results show that the global traversal can obtain more field distribution information, and the adaptive gradient extremum method is more efficient.


Introduction
The leakage of hazard will have an unpredictable impact on the surrounding environment, animals and plants.For example, the Fukushima nuclear accident in Japan in March 2011.When such accidents occur, it is usually to assign professional search and rescue personnel or animals to enter the dangerous area to find the location of the leakage source and handle it.This is not only harmful to the health of humans and animals but also inefficient.Therefore, the use of mobile robots to search and locate hazard sources has become a hot research topic.
According to the nature of the hazard sources can be divided into time-varying sources and timeinvariant sources.Time-varying sources are sources whose signal distribution as well as intensity of the target source in the field changes with time [1], such as chemical sources, pollution sources, odor sources [2], etc. Time-invariant sources are sources whose distribution in the field does not change with time [1], such as light sources [3], radiation sources [4], WiFi sources [5], etc.The scenarios explored by the mobile robots in this paper are limited to time-invariant sources.
In the beginning of the source localization phase, the robot moves according to a set traversal strategy, and collects information on the intensity and concentration of substances in the target field with the mounted sensors.Subsequently, the information collected by the robot is used to guide the robot's motion to the target location, and this type of source finding method can be roughly divided into three categories.The gradient method [6][7] [8] which estimates the gradient of the robot's current position based on the collected source information, guides the robot to move in the direction of the rising gradient until the robot reaches the position of the source.The extreme value method [9] does not rely on the method of solving the maximum value in the mathematical model [10].Get as much information about the source as possible by partial traversal search, and get the source location by iterating to find the maximum value.Global traversal type and probabilistic graph search [11][12], in which the robot performs a global traversal over the target search area, collecting and fitting data over the entire area to obtain an approximate distribution and thus determine the location of the source.
Therefore, in order to improve the utilization of data collected by robots, this article introduces the Kriging interpolation method [13], which uses the data as samples to predict the distribution of hazardous areas.An adaptive gradient polar source localization method based on Kriging interpolation prediction is proposed for robot search for source location.The feasibility as well as the effectiveness of the proposed method is verified by designing simulation experiments, and the time cost, path cost and positioning accuracy of the robot source localization are experimentally analyzed.

Kriging interpolation algorithm
Kriging interpolation algorithm [14]is a widely used prediction algorithm for geographic observations, which is used to compensate for the problem of insufficient sampling points [15].This algorithm assigns corresponding weight coefficients to each observation value after fully considering the relationship between them, and then performs weighted averaging to obtain the corresponding estimated value.It can predict various grid units in the target area by using continuous variation function for weighted summation, and has high statistical stability.
The Kriging model assumes that the true relationship between the response of the system and the independent variables can be expressed in equation (1).
Where ( ) yx is an unknown Kriging model, and ( ) fxis a known function about x .( ) Zx is a stochastic process, and its covariance can be expressed by equation (2), R is the correlation matrix, ( ) x is the correlation function of any two sampling points i x and j x .
( ) ( ) ( ) The expression of the Gaussian function as the correlation function is shown in equation ( 3), where ( ) is the unknown parameter, and i k x and j k x are the k th parameters of i x and j x .
( ) As shown in Figure 1, a 500*500 random field is designed, and random sampling is performed in the random field, where the black dots are the sampling locations, the blue area is the area with lower signal intensity value, and the yellow area is the area with higher signal intensity value.The Kriging interpolation prediction process is performed on all the sampled data, and the final obtained prediction results are shown in Figure 2. The intensity values close to the sampling point location are closer to the actual field intensity values, the predicted values far from the sampling point location are more different from the actual field intensity values.

Source localization method based on Kriging interpolation prediction
The source localization method based on Kriging interpolation algorithm utilizes partial data collected by robots in hazardous areas for interpolation prediction [8], thereby obtaining more data to approximate the information distribution of the source in the hazardous area.This article adopts two source localization methods based on Kriging interpolation, including global traversal source localization method and adaptive gradient extremum method.

Global traversal source location method
First set a specific traversal strategy for the robot, then control the robot to follow a predefined traversal strategy, such as square traversal, and collect the signal strength on the path in real time.Then, the sampled data is used to predict and reconstruct the unknown field using the Kriging method, in order to obtain the predicted field.Therefore, by sampling some data from hazardous areas to fit, the distribution of the entire signal strength can be obtained.The pseudocode of this algorithm is shown in Table 1.
Table 1 Global traversal source location algorithm -pseudo code.The (X, Y) is the source position.

Adaptive Gradient Extreme Source Localization Method
The method is a fusion improvement based on the gradient and the extreme value method.Although the path cost of the gradient method is low, the early random walk stage consumes too much time and has randomness; The time cost of the extremum method is low, but it is prone to falling into local minima, leading to mis-localization.Therefore, the random walk method in the data acquisition phase is replaced by the specified traversal method.This method can ensure that the robot moves purposefully from beginning to end, improving the efficiency of robot search.The pseudo code of the algorithm is shown in Table 2.  Firstly, the pre-sampling strategy of the gradient-based source localisation method uses random wandering, which is too random and leads to uncertainty in the time and path length consumed by the source search, and therefore the success rate of the robot in locating the source is reduced.
Second, although the polar source localization method can adjust the robot's step size adaptively by determining the polar value during localization to achieve fast localization, the fitted contour distribution is also inaccurate if the robot does not collect data on the true source location, resulting in inaccurate localization of the source by the robot as well.
Finally, two adaptive gradient-polar source localization methods based on Kriging interpolation prediction are proposed by combining the advantages of gradient and polar methods, which belong to the source localization methods of partial traversal.And the traversal source localization method based on Kriging interpolation prediction belongs to the global traversal localization method.Therefore, data comparison was conducted between the two source localization methods.In the scenario where the source is in the bottom left corner, the square traversal strategy was used for 10 experiments, and the average performance data of each method is shown in Table 3.The localization time and the consumed path length of the adaptive gradient extremum source localization method are much smaller than those of the traversal source localization method, while the former is larger than the latter in terms of root-mean-square error.Therefore, in order to enable the robot to quickly and accurately locate the source location, and considering the robot's own conditions, the latter can be chosen for a large range but hazardous source location task.While the former can be chosen for small range and high localization requirements.

Experimental verification of light source positioning
Through the simulation experiment results in Section 3, it can be concluded that the square traversal method helps to reduce the range of hazardous areas and achieve more uniform data collection.Therefore, the square traversal approach is used to experiment with the source location algorithm proposed in this paper.

Experiment platform introduction and scene construction
The experiments simulate the distribution of a time-invariant source by means of a light source, which is easily accessible in a laboratory scenario and free of hazards, while the distribution of the light source conforms to the form of a Gaussian distribution.The light sensor module is composed by A/D conversion of the Analog signal collected by the photosensitive sensor via STM32 microcontroller and thus transmitted to the robot platform.The mobile robot platform used in the experiments is the Turtlebot3 robot equipped with a light source sensor module.The light source sensor module is shown in Figure 5 and the mobile robot platform is shown in Figure 6.The range of the experimental scene is set to 3m×3m, to prevent the influence of other light sources on the experiment, the experiment was chosen to be conducted at night while turning off all extraneous light sources in the laboratory.

Global traversal source location
The global traversal source localization method is to set multiple path navigation points for the robot, complete data acquisition by controlling the robot according to the square traversal search strategy.The collected data and the robot's position data are transmitted to the Kriging interpolation prediction model to predict the intensity distribution of the source in the experimental area, and finally the location of the source is determined by the distribution of the obtained source intensity in the experimental area.A light source is placed in the lower left corner of the experimental area to simulate a single-source experimental scenario.The traversal path of the robot is shown in Figure 7.The blue arrow represents the robot's movement trajectory, and the red circle represents temporary obstacles.The predicted field intensity distribution of the source after the robot traversal is shown in Figure 8, and the location of the source in the lower left corner of the region can be obtained by the fitted source field intensity distribution.In summary, from the experimental results of single-source traversal, it can be concluded that the square global traversal method can be used to effectively localize a single source.Since the robot is globally traversing, it can collect a large amount of effective data evenly, so the localization accuracy is high and it is suitable for localization in small areas.

Adaptive gradient extremum source localization
The adaptive gradient extremum source localization method is validated for single source localization.Two obstacles are added to the experimental area.The robot is partially traversed according to the square traversal, and collect part of the source data simultaneously.By using the collected data, the gradient of the source is estimated by the Kriging interpolation prediction, the robots will select the maximum of this direction as its next target, until the source is localized by robots.
The motion trajectory of the robot is shown in Figure 9, and the light intensity distribution predicted by Kriging interpolation is shown in Figure 10.The location of the source is in the lower left corner of the experimental area, which is the same as the source location in the actual scene.In summary, the analysis from Figure 10 shows that for single source localization, the algorithm can successfully localize the source position and determine the intensity magnitude of the source at the same time.

Conclusion
This article focuses on the research of robot source localization methods.In order to achieve the contour distribution of the source in the region using limited data, the Kriging interpolation prediction algorithm is used to interpolate and predict the data collected by the robot, thus improving the data utilization.The source is localized by the method of global traversal and adaptive gradient extremum algorithm, and through experimental verification, the method in this paper can successfully localize the location of the source.Among them, square traversal can collect data evenly, and can effectively determine the safety critical area around the area.The adaptive gradient extremum method has better localization efficiency for sources.

Figure 1 .
Figure 1.Sampling points in the random field.Figure 2. Prediction results.

Figure 2 .
Figure 1.Sampling points in the random field.Figure 2. Prediction results.

Figure 3 .
Figure 3. Single source square traversal source location.

Figure 9 .
Figure 9. Robot traversing path.Figure 10.Source in the lower left corner of the experimental result.

Figure 10 .
Figure 9. Robot traversing path.Figure 10.Source in the lower left corner of the experimental result.

Table 3 .
Comparison of two source location methods.