**5. Numerical results**

The proposed search algorithm has been applied to an experimental dataset, collected by COANDA Research & Development Corporation using their large recirculating water channel. The emitting source was releasing fluorescent dye at a constant rate from a narrow tube. The dataset comprises a sequence of 340 frames of instantaneous concentration field measurements in the vertical plane and is sampled at every 10/23 s. The size of a frame is 49 � 49 pixels, where a pixel corresponds to a square area of 2*:*<sup>935</sup> � <sup>2</sup>*:*935mm2. As the size of the data is relatively small, we follow the approach used in [24]: upscale each frame by a factor of 3 using bicubic interpolation and place the result in the top left corner of a 500 � 500 search area. A measurement obtained by a platform is, thus, the integer value of the concentration of the dye taken from the closest spatial and temporal sample from the experimental data.

An example of the search algorithm running on the experimental data is shown in **Figure 2**. All physical quantities are in arbitrary units (a.u.). The following environmental/sensing parameters were used: *D* ¼ 1, *τ* ¼ 250, *U* ¼ 0, *a* ¼ 1 and *t*<sup>0</sup> ¼ 1. Algorithm parameters are selected as follows: *κ*<sup>0</sup> ¼ 3, *ϑ*<sup>0</sup> ¼ 5*:*2, number of particles *<sup>M</sup>* <sup>¼</sup> 252 , V ¼ f g1 , O ¼ �f g 3*;* �2*;* �1*;* 0*;* 1*;* 2*;* 3 degrees per unit of time and T ¼ f g 0*:*5*;* 1*;* 2*;* 4*;* 8*;* 16*;* 32*;* 64 . The number of iterations, both for the exchange of

*Decentralised Scalable Search for a Hazardous Source in Turbulent Conditions DOI: http://dx.doi.org/10.5772/intechopen.86540*

measurement triples and in the consensus algorithm, was fixed to 30. The local search stopping threshold was *ϖ* ¼ 3.

**Figure 2** displays the top-down view of the search progress at step indices *k* ¼ 0*;* 12*;* 22*;* 32. The formation consists of *N* ¼ 7 platforms, whose trajectories are shown in different colours. The search algorithm terminated at *K* ¼ 33. Note that the plume size is much smaller than the search area. Panels **(a)–(c)** of **Figure 2** show the particles before resampling: the particles are placed on a regular grid, thus mimicking a grid-based approach, with the value of particle weights indicated by the grey-scale intensity plot (white means a zero weight). This provides a good visual representation of the posterior *p*ð Þ **r**0j*d*1:*<sup>k</sup>* . Panel **(d)** shows the situation after a non-zero concentration measurement was collected by the search team. The positional particles have been resampled at this point of time and moved closer to the true source location.

Using 200 Monte Carlo simulations, the mean search time for the algorithm was 2525 a.u., with a 5th and 95th quantile of 1840 and 3445 a.u., respectively. Note that in all simulations the formation started from the bottom right hand corner indicated in **Figure 2(a)**.

#### **Figure 2.**

and is given a value of one, otherwise it is zero. This local stopping criterion value (zero or one) becomes the initial state of the global stopping criterion on platform *i*,

*<sup>σ</sup>i*ð Þ¼ <sup>0</sup> 1 if ffiffiffiffiffiffiffiffiffiffiffiffiffi

(

*tr*½ � **C***<sup>k</sup>* p < *ϖ*

(22)

0 otherwise

The global stopping criterion is computed on each platform using the average consensus algorithm, using (21), but with *bi* replaced by *σi*. After a sufficient number of iterations, *S*, platform *i* decides to stop the search if at least one of the platforms in the formation has reached the local stopping criterion, that is, if

We point out that both estimation and control are based on the consensus algorithm. While the cooperative control is using the *average* consensus (21), the decentralised measurement dissemination of Section 3 achieves the consensus on the set of measurements at time *k*. The consensus algorithm is iterative, and hence its convergence properties are very important. First note that, although the network topology changes with time (as the robots move while searching for the source), during the short interval of time when the exchange of information takes place, the topology can be considered as *time-invariant*. Furthermore, assuming bidirectional communication between the robots in formation, the network topology can be represented by an undirected graph. The convergence of the consensus algorithm for a time-invariant undirected communication topology is guaranteed if the graph is connected [31–33]. Note that this theoretical result is valid for an infinite number of iterations. In practice, if the communication graph at some point of time is not connected, or if an insufficient number of consensus iterations are performed, it may happen that one or more robots are lost (they could re-join the formation only by coincidence). This event, however, does not mean that the search mission has failed: the emitting source will be found eventually, albeit by a smaller formation in

The proposed search algorithm has been applied to an experimental dataset, collected by COANDA Research & Development Corporation using their large recirculating water channel. The emitting source was releasing fluorescent dye at a constant rate from a narrow tube. The dataset comprises a sequence of 340 frames of instantaneous concentration field measurements in the vertical plane and is sampled at every 10/23 s. The size of a frame is 49 � 49 pixels, where a pixel corresponds to a square area of 2*:*<sup>935</sup> � <sup>2</sup>*:*935mm2. As the size of the data is

relatively small, we follow the approach used in [24]: upscale each frame by a factor of 3 using bicubic interpolation and place the result in the top left corner of a 500 � 500 search area. A measurement obtained by a platform is, thus, the integer value of the concentration of the dye taken from the closest spatial and temporal

An example of the search algorithm running on the experimental data is shown

, V ¼ f g1 , O ¼ �f g 3*;* �2*;* �1*;* 0*;* 1*;* 2*;* 3 degrees per unit of time and

in **Figure 2**. All physical quantities are in arbitrary units (a.u.). The following environmental/sensing parameters were used: *D* ¼ 1, *τ* ¼ 250, *U* ¼ 0, *a* ¼ 1 and *t*<sup>0</sup> ¼ 1. Algorithm parameters are selected as follows: *κ*<sup>0</sup> ¼ 3, *ϑ*<sup>0</sup> ¼ 5*:*2, number of

T ¼ f g 0*:*5*;* 1*;* 2*;* 4*;* 8*;* 16*;* 32*;* 64 . The number of iterations, both for the exchange of

denoted *σi*ð Þ 0 :

*Unmanned Robotic Systems and Applications*

*σi*ð Þ*S* >0.

possibly longer interval of time.

sample from the experimental data.

particles *<sup>M</sup>* <sup>¼</sup> 252

**24**

**5. Numerical results**

*Experimental dataset: an illustrative run of the decentralised multi-robot search using N* ¼ *7 platforms. Graphs (a)–(d) show the positions and trajectories of the platforms at step indices k* ¼ *0,12,22 and 32, respectively. The concentration of the plume is represented in grey-scale (darker colours represent higher concentration).*
