There have been many investigations into the development of pollution source seeking controllers. For example Baronov and Baillieul [23], used a radial function as a potential function to develop an ascending controller. Their algorithm is not fully dependent on gradient but the function to be mapped or transverse must be known a priori for tuning the controller. Mayhew et al. [24] used a hybrid controller that combines a line minimization-based algorithm and a vehicle path planning algorithm to find the extremum of a function. Their approach does not need GPS point measurements and does not need any knowledge of the function to be mapped a priori. Dhariwal [25] used the Keller and Segel bacteria model as a basis for developing their controller whilst Marques et al. [26] investigated the use of rule based approach to using bacteria chemotaxis behaviour on a robot. In their experiments, they also investigated the use of the silkworm moth algorithm and a direct gradient following method. Lilienthal et al. [27] used a Braitenberg vehicle approach to find an ethanol source in their own experiments.

However this work uses a controller based on the Berg and Brown model in [28]. Models such as the one in [29], and other modern complex models, were not considered because a random biased walk model is sufficient for environmental exploration and source declaration in an environment. A bacterium motion is composed of a combination of tumble and run phases. The frequency of these phases depends on the measured concentration gradient in the surrounding environment. The run phase is generally a straight line while the tumble phase is a random change in direction with a mean of about 68 degrees in the E. Coli bacterium. If the bacterium is moving up a favourable gradient, it tumbles less thereby increasing the length of the run phase and vice versa if going down an unfavourable gradient. This behaviour was modelled by Berg and Brown by fitting the results of their experimental observations in [30] with a best fit equation in [28]. This model is shown below: (1) (2) (3) where τ is the mean run time and τ₀ is the mean run time in the absence of concentration gradients, α is a constant of the system based on the chemotaxis sensitivity factor of the bacteria, P_b is the fraction of the receptor bound at concentration C. In our work, C was the present reading taken by our Robotic agent. k_D is the dissociation constant of the bacterial chemoreceptor. is the rate of change of P_b. is the weighted rate of change of P_b, while τ_m is the time constant of the bacterial system.

The above equations determine the time between tumbles and hence the length of runs between tumbles. During the tumble phase, the robot agent can randomly choose a range of angles in the set σ ∈ {0…… 360} by randomly choosing angles. This made it possible for the robot agents to backtrack if there is a favourable gradient behind it. The ability of the implemented bacteria chemotaxis controller to work over long distances has been presented in [31,32] and works fine even in the presence of small disturbances.

The above discussion about the bacterium behaviour can be depicted graphically, as shown in Figure 1.

A counter that increments every time tick is implemented. Once the counter is greater than τ a tumble phase is initiated otherwise the bacterium stays in the run phase. Once a tumble phase is completed through the assigning of a new σ, the run phase is resumed. A velocity controller which will be discussed in Section 2.3 was embedded into the bacterium controller.

It could be argued that the bacterium behaviour is a gradient based algorithm and, as such, an ordinary gradient based algorithm could be used in place of it. However, the Equations (1) to (3) have more to them than meets the eye. These equations take noise in the bacterium sensors, dynamics or environment into consideration. Equation (2) is an exponentially weighted low pass filter that is capable of filtering out noise. As a result of the filter, the Berg and Brown derived bacterium controller is not easily affected by noise and hence would not be readily caught in local maximums when they are encountered [33]. This is further aided by the random nature of the algorithm as a result of the tumble phase of the controller.

It could be argued as well that the random nature of the algorithm is not beneficial to bacteria. However, as was proven in [34], it is this very random nature that enables them to form a distribution corresponding to the profile of the pollutant in the environment. This is further supported by the works of Mach and Schweitzer, Hamann and Heinz [35,36] in which they discussed how a swarm of stochastic agents having a random component (bacterium tumble phase in this case) and a direct component (bacterium run phase in this case) can be represented by a Fokker Planck equation. The Fokker Planck equation would always converge to its stationary distribution that corresponds to the spatial function of interest.

It is the above properties that make the Berg and Brown derived bacterium controller different from an ordinary gradient ascent algorithm, and make it suited for use on noisy functions as used in the simulations in this work. In this work, the filter as represented by Equation (2) was implemented in discrete form using an ant limited memory resource length k = 4 to remember a history of past readings.

There has been a lot of work done on flocking controllers. The first flocking controller was developed by C.W. Reynolds in [37]. In order to achieve flocking, he defined that three rules need to followed. These rules were: Keep as close as possible to your neighbour (Aggregation), do not collide with your neighbour (Collision Avoidance), and move in a similar general heading as your neighbours.

Following this discovery, many researchers such as [38,39,40] have worked on these three basic rules in investigating various approaches of achieving flocking. Most of the approaches agree that Equation (4) should be satisfied in order to achieve flocking [38]. (4) where r is the distance between agents, G_R represents the repulsion function and G_A is the attractive function. Following this, we use exponential functions, as in Equation (5), to achieve flocking. This results in the Morse potential as in Equation (5) [41]. (5) (6) where i is the index of the agents. Gains of 1 for the repulsion term G_R and 0.99 for the attractant term G_A were used. It was discovered that it is possible to control how closely the agents get to each other whilst not colliding by reducing the G_G gain. r in Equation (5) is the distance between the two closest agents. This value was obtained as follows: Each agent had a communication radius l = 50 and can only buffer up to N = 5 readings obtained from agents within this range. From these readings, the closest agent r is obtained and this value is used in Equation (5) to avoid collisions with other agents. Despite using a limited number N of neighbours, the agents were still able to avoid collisions whilst maintaining cohesion as a flock. Using a buffer of five readings was used only for flocking. This buffer is not necessary if proximity sensors are used instead, making sure that the ant like feature is maintained.

The final output from the flocking controller and bacteria controller is calculated as shown in Equation (6) where G_F and G_B are gains applied to both the flocking controller and bacteria controller output respectively.

The agent velocity updates are such that by controlling them, it is possible for the agents form a distribution of the pollutant in the environment and in works of [32,42] it was obtained by an adaptable velocity that was determined by Equation (7): (7) where β is a dynamic velocity that depends on the present reading of the environmental quantity, β_o is the standard velocity without any reading, v_k is a constant for tuning the dynamic velocity and C is the pollutant reading obtained by the robot’s sensor. The separate controllers are combined by using the architecture shown in Figure 2.

The relationship between the bacterium controller and the velocity controller is shown in Equation (8), where dist_τ is the distance covered by the agent during a run phase and t_τ is the time duration from the last tumble phase. (8)

By using the velocity Equation (7), it was possible to enable agents achieve various visual distributions of pollution profiles as can be seen in Figure 2, Figure 3, Figure 4 and Figure 5. However, it was also discovered that the constant v_k, can be used to control the spread of the agents in the pollution profile with high v_k values resulting in agents behaving more like gas molecules, covering a smaller area and forming less detailed features of the pollution profile whilst lower v_k values result in agents behaving more like liquid-like molecules, covering a higher area and showing more detailed features of the pollutant profile as can be seen in Figure 6 and Figure 7.

By using the velocity Equation (7), it was possible to enable agents achieve various visual distributions of pollution profiles as can be seen in Figure 2, Figure 3, Figure 4 Figure 5. However, it was also discovered that the constant v_k, can be used to control the spread of the agents in the pollution profile with high v_k values resulting in agents behaving more like gas molecules, covering a smaller area and forming less detailed features of the pollution profile whilst lower v_k values result in agents behaving more like liquid-like molecules, covering a higher area and showing more detailed features of the pollutant profile as can be seen in Figure 6 and Figure 7.

However, as choosing the right v_k value is presently guess work, a way of using machine learning to do this can be investigated. The approach used in this paper relies on using a Genetic Algorithm based on Gaussian models. In order to do this, some modifications are made to Equation (7) as shown in Equation (9): (9) where T can be viewed as a system temperature constant and can be used for tuning the system using an adaptable scheme.

By adjusting this constant, the spread of the agents in the environment could be controlled to achieve different effects. βⁱ(t) is a dynamic velocity of agent i that depends on the present reading Cⁱ(t) of the environmental quantity, is the standard velocity without any reading. This velocity function was embedded in the bacteria controller. The present reading Cⁱ(t) of the agent adapts the velocity of the agent so that in an area of higher concentration, it moves slowly covering a smaller area and vice versa. How this achieves coverage can be explained using (10) where Aⁱ is the area covered by agent i, τ is the mean run time or mean run length of the bacteria controller and A = πτ² is the area covered by the agent when the bacteria controller is tuned so that the agents motion is circular. As the agents get closer to the source, the area covered by them decreases due to a lower velocity and higher concentration reading while they cover a larger area if they have a higher velocity due to a lower concentration. The area covered by the pollutant is covered if the individual areas are covered by the agents as follows: (11)

This could be used to develop a control law as in Equations (12) and (13) if an estimate of the pollutant profile F(S(x)) in the environment S(x) were known and used to tune the system temperature T so that the spread of the agents in the environment is controlled. As a result, Equation (9) becomes Equation (14). (12) (13) (14) where γ is an integral gain and is a proportional gain. A Gaussian model based Genetic algorithm explained in Section 3 was used to estimate parameters of the pollutant profile F(S(x)) locally. The standard deviation parameters of the locally estimated Gaussian are then used in Equation (12) so that it becomes (15) where A_T (x)^std is the standard deviation of the flock positions. From machine learning, combination of Gaussian curves could be used to form various functions [43] so that the agents using the present approach are able to form various complex pollutant shapes. By using machine learning, it is possible to correctly get the shape of the pollutant and increase the rate of convergence. The disadvantage however is that the kernel function used to estimate the pollutant profile must be capable of forming the distribution of the pollutant profile effectively. Investigating the use of machine learning in improving the proposed control system opens up the possibility of studying how various real world conditions could affect the agents’ distribution. This knowledge can then be incorporated into the control system so that the flock of agents are better prepared to deal with the environmental conditions as they arise.

In this work, a very slow (near negligible) diffusion was assumed. The Genetic Algorithm was used to estimate the parameters (amplitude, source location, and standard deviation) of a Gaussian function that was used to simulate the pollutant. The estimated standard deviation F(S(x))^std was fed back to the robotic flock according to Equations (15), (13) and (14) in order to adjust the velocity of the agents to achieve optimal coverage of the environment.

The Genetic Algorithm was designed with 35 genes on each chromosome, with 7 genes representing each of the Gaussian’s amplitude, x and y standard deviations, and x and y means. Each gene could take a value of either 0 or 1. As a result, the estimation range for each of the parameter using the Genetic Algorithm was between 0 to 128. A gene pool size of 400 chromosomes, tournament size of 10, crossover, mutation, reproduction ratios of 10%, 75% and 15% were used respectively with 10 generations per run.

During simulation runs, agents explored the environment using both the bacteria and flocking algorithm whilst collecting spatial pollutant data. The spatial pollutant data was stored in a running memory having n = 20 elements. This data was then passed to the Genetic Algorithm in order to estimate the pollutant parameters.

A pollutant is known to break up into pieces due to the turbulent nature of the environment. Hence, the pollutant’s profile becomes one made up of various local maximums and one global maximum. This could also be viewed as noise. Nevertheless, the aim of the method proposed in this paper is to represent a pollutant’s profile as closely as possible. This would be a problem for the method discussed in Section 3.1 as it would attempt to estimate a single global estimate from the data collected by the agents.

In order to solve this problem, it is assumed that each agent i transmits its individual estimated Gaussian parameter values to its local neighbours j ∈ {1…N} present within an l = 5 units radius together with its position x_i. A reduced communication radius value ensures that only local agents to a local maximum contribute to each other’s estimation.

Based upon this information, each agent uses an averaging scheme depicted by Equation (18) to reach a consensus as to the parameter values of the local maximum.

The consensus scheme works by using a weighted averaging mechanism so that values closer to what an individual agent i is estimating are given more weight than the values farther from its estimation: (18) where k^ave_i is the k-average of the parameters estimate by agent i, G(||.||) is a Gaussian curve with agent i’s estimate as the centre value, N is the number of agents in the neighbourhood of agent i. N ≥ 5 as this is used to simulate the communication limitation of the agent i in that it can only buffer up to five readings.

By using the parallelism offered by swarm robotics and the consensus scheme as discussed above, it is possible to obtain an estimation of the pollutant’s profile in addition to recovering the function representing it.

The overall structure of the algorithm can be presented as in Algorithm 1. Lines four to six are necessary for a moving or changing pollutant profile. This has been tested in simulations with satisfactory results. The use of the model based genetic algorithm with the previous architecture is shown in Figure 9.

In order to test the approach, different pollutant profiles were generated with 50 agents deployed. A Gaussian having the following parameters of mean (x, y) = (400, 200), spread (x, y) = (38, 38) and amplitude of 100 was used. As can be seen in Figure 10, the agents were able to form the distribution even though their motion from Figure 10(b) seems erratic. This behaviour is as a result of the bacteria controller searching for a favourable path towards the optimal of the Gaussian distribution. Figure 10(c) shows how the error between the agents spread and the pollution spread reduces with time. The error was obtained by using Equation (14).

Another simulated pollutant profile generated by adding three Gaussian functions of randomly chosen parameters: means (x, y) = (80, 70), (170, 150), (200, 200), spreads (x, y) = (60, 40), (30, 80), (40, 80) and amplitudes = 100, 45 and 80 respectively were used as in Figure 11. It was observed that the approach enabled the agents to distribute according to the simulated pollutant’s profile.

Using the approach discussed in Section 3.2, tests on multimodal Gaussian functions generated by C = 160+ (x² – 150cos(2πxP₁) + y² – 150cos(2πyP₂)) were carried out with P₁ = P₂ = 0.065. During these experiments, 100 agents were deployed with sensor readings for the agents normalized between the ranges of 0 to 100 as that is the range that the Genetic Algorithm was designed to handle. A higher number of agents were deployed because of the nature of the pollutant profile.

As can be observed in Figure 12, 60 of the agents were able of form the shape of the pollutant to a limited extent with the dips in the multimodal function seen in the agents distribution at (x, y) = (2.5, 16) and (x, y) = (15, 16). The rest of the agents were exploring the environment.