**3.2 Heading angle**

*Use of Gamma Radiation Techniques in Peaceful Applications*

bution (e.g., uniform or based on a 1/*R*<sup>2</sup>

measurements.

inherent error in a 1/*<sup>R</sup>* <sup>2</sup>

improves with more UAS in the swarm.

rule, the gradient is estimated [20, 21] as:

∇*cTavg* ≈ \_\_\_2

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 distri-

> *πr*<sup>2</sup> <sup>∑</sup> *i*=1 *N*

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

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

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

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

model). Using the composite trapezoidal

scalar field. A relatively small change in distance can have a

*Tn*(*pi*)*pi*∆*s* (2)

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**Figure 7.**

*of UAS on the gradient estimation error.*

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 difference *es* with respect to the reference contour value *Tm*:

$$R\_c = K\_p e\_s + K\_d \frac{d e\_i}{dt} + K\_i f e\_s dt,\\ \phi = \tan^{-1} \frac{R\_c}{R} \tag{3}$$

where *Kp*, *Ki* , and *Kd* are the proportional, integral, and derivative gains, respectively. As shown in Eq. (3), a heading angle *ψ* becomes 90 degrees for a large value of *Rc* when the swarm is far away from the reference contour.

For the swarm that is near the reference contour, *Rc* becomes small and *ψ* is close to zero. This leads to the equation for determining the heading angle:

$$
\Psi = \Theta\_H - \frac{\pi}{2} + \Phi \tag{4}
$$

#### **Figure 8.**

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

Based on Eq. (4), the swarm will move in a tangential direction near the reference contour, and demonstrate a source-seeking behavior when it is far away from the source or the reference contour.

### **4. Computer simulation of radiation contour mapping**

#### **4.1 UAS swarm simulation in 1/***R***<sup>2</sup> field**

To validate the contour mapping algorithm, two types of radiation fields were used. The first is based on a 1/*R*<sup>2</sup> model for a radioactive source located at the distance *R* from a sensor. This simplified model was employed to properly tune and improve contour mapping and source-seeking behaviors of the swarm. The algorithm was improved by including the UAS dynamics and adaptive spinning of the swarm (for the reduced flight trajectory of each UAS). The improved algorithm was validated using a realistic gamma field formed by multiple sources placed in the area with physical obstacles that was computed using the Monte Carlo Neutron and Particles code (MCNP) [22]. MCNP is widely used for simulation of coupled photon, neutron, electron and particle transport in complex geometries [23, 24].

A stochastic nature of radiation sensing was taken into account in the sensor's data. A random noise was introduced to the radius value of the 1/*R*<sup>2</sup> model in the simulation: a noise of ±2.5 m was added, causing erratic gradient estimation and heading generation, as expected. To reduce this effect, a moving average filter was applied to both the gradient estimation and the heading angle generation. The simulation scheme is shown in **Figure 9**, including the UAS spinning formation used to estimate the steepest gradient. The UAS dynamics was incorporated for the realistic simulation of flight trajectories of aerial platforms during contour mapping and source seeking. Instead of using just kinematic motions of each UAS for this simulation, their dynamics was used to calculate their flight trajectories. The effect of inclusion of dynamics of UAS platforms on their trajectories that the swarm is able to follow is shown in **Figure 10**. A double integrator was incorporated in MATLAB to approximate the UAS dynamics. This provides more realistic simulation comparing to a case when only kinematics is considered.

A spinning formation of the swarm improves the estimation of radiation gradient direction [21, 25]. Spinning occurs around a virtual center of the UAS formation while the center of the formation moves along the desired vector to accomplish the contour mapping or source seeking. **Figure 11a** illustrates the swarm of three UAS platforms spinning around a virtual center to counteract error from the gradient estimation algorithm. A bold line shows a path traveled by the swarm's center in mapping a contour without spinning (**Figure 11b**) and with spinning (**Figure 11c**). The non-spinning formation exhibits poor mapping performance primarily due to large gradient estimation errors which depend on the relative direction to the source with respect to the swarm.

**95**

**Figure 11.**

**Figure 10.**

Having the swarm circling around a virtual center, while this center travels in the direction dictated by the swarm heading algorithm, increases the total flight paths significantly. This led to development of an adaptive spin rate adjustment scheme to avoid unnecessary spinning formation when it is not required.

*(a) UAS swarm spins around a virtual center to counteract error from the gradient estimation algorithm* 

*(b) without spinning and (c) with spinning in mapping a contour.*

*Gamma Ray Measurements Using Unmanned Aerial Systems*

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

*Contour mapping with UAS dynamics and without it.*

**Figure 9.** *Simulation scheme.*

#### *Gamma Ray Measurements Using Unmanned Aerial Systems DOI: http://dx.doi.org/10.5772/intechopen.82798*

*Use of Gamma Radiation Techniques in Peaceful Applications*

**4. Computer simulation of radiation contour mapping**

data. A random noise was introduced to the radius value of the 1/*R*<sup>2</sup>

tion comparing to a case when only kinematics is considered.

 **field**

To validate the contour mapping algorithm, two types of radiation fields

distance *R* from a sensor. This simplified model was employed to properly tune and improve contour mapping and source-seeking behaviors of the swarm. The algorithm was improved by including the UAS dynamics and adaptive spinning of the swarm (for the reduced flight trajectory of each UAS). The improved algorithm was validated using a realistic gamma field formed by multiple sources placed in the area with physical obstacles that was computed using the Monte Carlo Neutron and Particles code (MCNP) [22]. MCNP is widely used for simulation of coupled photon, neutron, electron and particle transport in complex geometries [23, 24]. A stochastic nature of radiation sensing was taken into account in the sensor's

simulation: a noise of ±2.5 m was added, causing erratic gradient estimation and heading generation, as expected. To reduce this effect, a moving average filter was applied to both the gradient estimation and the heading angle generation. The simulation scheme is shown in **Figure 9**, including the UAS spinning formation used to estimate the steepest gradient. The UAS dynamics was incorporated for the realistic simulation of flight trajectories of aerial platforms during contour mapping and source seeking. Instead of using just kinematic motions of each UAS for this simulation, their dynamics was used to calculate their flight trajectories. The effect of inclusion of dynamics of UAS platforms on their trajectories that the swarm is able to follow is shown in **Figure 10**. A double integrator was incorporated in MATLAB to approximate the UAS dynamics. This provides more realistic simula-

A spinning formation of the swarm improves the estimation of radiation gradient direction [21, 25]. Spinning occurs around a virtual center of the UAS formation while the center of the formation moves along the desired vector to accomplish the contour mapping or source seeking. **Figure 11a** illustrates the swarm of three UAS platforms spinning around a virtual center to counteract error from the gradient estimation algorithm. A bold line shows a path traveled by the swarm's center in mapping a contour without spinning (**Figure 11b**) and with spinning (**Figure 11c**). The non-spinning formation exhibits poor mapping performance primarily due to large gradient estimation errors which depend on the relative direction to the source with respect to the swarm.

model for a radioactive source located at the

model in the

the source or the reference contour.

**4.1 UAS swarm simulation in 1/***R***<sup>2</sup>**

were used. The first is based on a 1/*R*<sup>2</sup>

Based on Eq. (4), the swarm will move in a tangential direction near the reference contour, and demonstrate a source-seeking behavior when it is far away from

**94**

**Figure 9.** *Simulation scheme.*

**Figure 10.** *Contour mapping with UAS dynamics and without it.*

#### **Figure 11.**

*(a) UAS swarm spins around a virtual center to counteract error from the gradient estimation algorithm (b) without spinning and (c) with spinning in mapping a contour.*

Having the swarm circling around a virtual center, while this center travels in the direction dictated by the swarm heading algorithm, increases the total flight paths significantly. This led to development of an adaptive spin rate adjustment scheme to avoid unnecessary spinning formation when it is not required.

When the UAS swarm is far away from the source or the contour to be mapped, the direction to them is nearly fixed, thus it is not necessary to spin the formation. When the swarm is near the contour, it is necessary to spin the formation since most contours have a curvilinear shape. The criterion for spinning is chosen as a radius of curvature of the formation center's path.

#### **Figure 12.**

*(a) Adaptive tuning of spin rate based on radius of curvature of a formation center path. In this example, as an estimated radius of curvature approaches the target contour located at 10 m, a spin rate converges to the desired 0.25 rad/s and (b) plot of average path lengths in terms of spin rates.*

**97**

**Figure 13.**

*Contour mapping for three radiation sources in the 1/R<sup>2</sup>*

 *model.*

*Gamma Ray Measurements Using Unmanned Aerial Systems*

As shown in **Figure 12a**, the swarm starts to spin when it approaches the contour to be followed. In this simulation, a spin rate control was bound with a lower value of 0.05 rad/s and an upper value of 0.3 rad/s. There is a big spike in a radius of curvature plot due to the nearly linear path the swarm has to follow. During the sourceseeking stage, a minimum spinning was maintained to save on the total flight paths. Total path length in terms of spin rates is shown in **Figure 12b** which demonstrates a nearly negligible difference in path length if the swarm spins clockwise or counterclockwise. Rotation rates of 0, ±0.075, ±0.1, ±0.2, and ± 0.3 rad/s were used to

Multiple radiation sources were also considered. **Figure 13** shows contour mapping simulation for three sources; the algorithm was capable to trace the desired

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

compare the average of the flight path lengths.

*Gamma Ray Measurements Using Unmanned Aerial Systems DOI: http://dx.doi.org/10.5772/intechopen.82798*

*Use of Gamma Radiation Techniques in Peaceful Applications*

curvature of the formation center's path.

When the UAS swarm is far away from the source or the contour to be mapped, the direction to them is nearly fixed, thus it is not necessary to spin the formation. When the swarm is near the contour, it is necessary to spin the formation since most contours have a curvilinear shape. The criterion for spinning is chosen as a radius of

*(a) Adaptive tuning of spin rate based on radius of curvature of a formation center path. In this example, as an estimated radius of curvature approaches the target contour located at 10 m, a spin rate converges to the* 

*desired 0.25 rad/s and (b) plot of average path lengths in terms of spin rates.*

**96**

**Figure 12.**

As shown in **Figure 12a**, the swarm starts to spin when it approaches the contour to be followed. In this simulation, a spin rate control was bound with a lower value of 0.05 rad/s and an upper value of 0.3 rad/s. There is a big spike in a radius of curvature plot due to the nearly linear path the swarm has to follow. During the sourceseeking stage, a minimum spinning was maintained to save on the total flight paths. Total path length in terms of spin rates is shown in **Figure 12b** which demonstrates a nearly negligible difference in path length if the swarm spins clockwise or counterclockwise. Rotation rates of 0, ±0.075, ±0.1, ±0.2, and ± 0.3 rad/s were used to compare the average of the flight path lengths.

Multiple radiation sources were also considered. **Figure 13** shows contour mapping simulation for three sources; the algorithm was capable to trace the desired

**Figure 13.** *Contour mapping for three radiation sources in the 1/R<sup>2</sup> model.*

contour reasonably well. Simulation was also done for a moving radiation source. The algorithm can accomplish tracing of the desired radiation contour well if the source moves reasonably slow. **Figure 14** shows a moving source traveling at 0.07 meters per second along the line from a point at (10, 40) m to a point (40, 10) m.

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**Figure 16.**

*Contour mapping in a radiation field computed using MCNP.*

**Figure 15.**

*radiative map.*

*Gamma Ray Measurements Using Unmanned Aerial Systems*

*Simulated radiation field: (a) model of the monitored volume; (b) radiation source locations; (c) simulated* 

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

**Figure 14.** *Contour mapping simulation for a moving source.*

*Gamma Ray Measurements Using Unmanned Aerial Systems DOI: http://dx.doi.org/10.5772/intechopen.82798*

#### **Figure 15.**

*Use of Gamma Radiation Techniques in Peaceful Applications*

contour reasonably well. Simulation was also done for a moving radiation source. The algorithm can accomplish tracing of the desired radiation contour well if the source moves reasonably slow. **Figure 14** shows a moving source traveling at 0.07 meters per second along the line from a point at (10, 40) m to a point (40, 10) m.

**98**

**Figure 14.**

*Contour mapping simulation for a moving source.*

*Simulated radiation field: (a) model of the monitored volume; (b) radiation source locations; (c) simulated radiative map.*

**Figure 16.** *Contour mapping in a radiation field computed using MCNP.*

Mapping this source was possible in this particular case because the speed of the swarm was roughly seven times the speed of the source.
