**3. Multirobot contour mapping of radiation fields**

One of the approaches of cooperative multirobot sensing is the method of contour mapping of radiation areas. It is based on the use of several UAS platforms (the 'swarm') equipped with radiation sensors. This approach allows for the automatic determination of the contour in space that corresponds to a preset radiation dose. Thus, the multirobot system could locate and follow a boundary of the hazardous zone.

In this section, the contour mapping algorithm is presented along with a gradient direction estimation and heading angle calculation scheme for the swarm consisting of three UAS that are positioned in a circular formation in the two dimensional space. It is assumed that a gamma-ray sensor, CZT or CLYC, is mounted on each UAS platform. The gamma-ray data are time stamped and merged with the position data as discussed in the previous section.

## **3.1 Gradient direction estimation**

The contour mapping is based on two components: the gradient direction estimation and the average radiation level calculated using the radiation measurement data from sensors mounted on the UAS platforms of the swarm. The average of a scalar field is estimated over a circular area of radius *r* centered at a point *c* as shown in **Figure 6**. *Tavg* is the average radiation level calculated using the data from sensors of UASs flying in a circular formation:

$$T\_{\text{avg}}(\mathcal{X}, \mathcal{Y}) = \frac{\sum\_{i=1}^{N} T\_i(\mathcal{X}, \mathcal{Y})}{N} \tag{1}$$

Here, *N* is the number of UAS platforms, *Ti* is the intensity of the measured gamma peak of interest at a point(*x*, *<sup>y</sup>*) by *i*th UAS. The formation center moves toward the increasing (a source-seeking method) or the constant (a contour mapping method) value of the average of sensor readings. To find the required direction

#### **Figure 6.**

*Formation of three UAS platforms around a circle of radius r. Radiation measurements Tn by three UAS are based on 1/R<sup>2</sup> model.*

of motion of the swarm's center, the gradient of *Tavg* should be determined using multiple readings *Tn* from the UAS's sensors. It is assumed that *N* measurements are taken out of the readings distributed inside the circle according to a known distribution (e.g., uniform or based on a 1/*R*<sup>2</sup> model). Using the composite trapezoidal rule, the gradient is estimated [20, 21] as:

$$\nabla\_c T\_{\text{avg}} \simeq \frac{2}{\pi r^2} \sum\_{i=1}^{N} T\_n(p\_i) \, p\_i \, \Delta s \tag{2}$$

Here, *pi* <sup>=</sup> *xi* <sup>−</sup> *<sup>c</sup>* and ∆*<sup>s</sup>* <sup>=</sup> <sup>2</sup>*r*/*N*. It should be noted that the origin is moved to the center of the circle, and the integral is approximated by a finite number of measurements.

Three UAS platforms of the swarm (*<sup>n</sup>* <sup>=</sup> 3) are equally distributed around a circle circumference. Horizontal and vertical components of the gradient can be calculated using Eq. (2). Then the formation center *c* can be moved relative to its current position using the estimated direction of gradient for the source-seeking behavior for the swarm. In this technique, any number of UAS platforms can be used in the swarm. For accurate estimation of the gradient, these drones should be evenly distributed around a circle. In the gradient estimation algorithm, there is an inherent error in a 1/*<sup>R</sup>* <sup>2</sup> scalar field. A relatively small change in distance can have a large effect on gamma measurement for each UAS depending on relative orientation of the swarm to the source as well as its distance from the source. In the contour mapping technique, three UAS platforms rotate about the swarm's center to improve the gradient estimation by changing the relative direction of the radiation source with respect to the UAS. **Figure 7a** illustrates how a gradient estimation error ε is defined. It should be noted that ε increases as a distance to the source relative to the swarm's center decreases. Therefore, the estimation error decreases when the source is located far away from the swarm (**Figure 7b**). The number of UAS in the swarm affects the gradient error as well. **Figure 7c** shows that the gradient estimation improves with more UAS in the swarm.

#### **Figure 7.**

*(a) Gradient estimation, (b) effect of source distance on the gradient estimation error, and (c) effect of number of UAS on the gradient estimation error.*

**93**

**Figure 8.**

*Gamma Ray Measurements Using Unmanned Aerial Systems*

The bulk heading angle of the UAS swarm *ψ* depends on how far this swarm is from the desired radiation contour to be mapped. As the swarm approaches the desired contour, the heading angle must be directed to a tangential direction for this contour. When the swarm is far away from the contour, the heading angle will be directed toward the source, which is the source-seeking behavior of the swarm as shown in **Figure 8**. Here, <sup>θ</sup>*H* is an estimated steepest gradient direction, and *ϕ* is a control angle determining a bulk heading angle *Ψ* [21]. The angles are measured with respect to a positive *x*-axis. The control angle *ϕ* is determined by how far the desired contour is located based on the average radiation measurement: *es* <sup>=</sup> *Tr* <sup>−</sup> *Tm*, where *Tr* is the desired radiation intensity of the contour to be mapped and *Tm* is an average radiation intensity measured using three UAS platforms. Here, an arbitrary constant of small magnitude *R* is used along with *Rc* to calculate a heading angle. Value of *Rc* is determined based on the PID control action from the measurement

with respect to the reference contour value *Tm*:

to zero. This leads to the equation for determining the heading angle:

of *Rc* when the swarm is far away from the reference contour.

*des*

tively. As shown in Eq. (3), a heading angle *ψ* becomes 90 degrees for a large value

For the swarm that is near the reference contour, *Rc* becomes small and *ψ* is close

*π*

*dt* <sup>+</sup> *Ki*∫*esdt*, <sup>ϕ</sup> <sup>=</sup> tan−1 \_\_

, and *Kd* are the proportional, integral, and derivative gains, respec-

*Rc*

<sup>2</sup> <sup>+</sup> <sup>ϕ</sup> (4)

*<sup>R</sup>* (3)

*DOI: http://dx.doi.org/10.5772/intechopen.82798*

*Rc* = *Kp es* + *Kd* \_\_\_

ψ = θ*<sup>H</sup>* − \_\_

*Bulk heading angle of the swarm Ψ and the gradient angle* θ*H.*

**3.2 Heading angle**

difference *es*

where *Kp*, *Ki*
