# A Behavioral Robotics Approach to Radiation Mapping Using Adaptive Sampling
## 2 Methodology
### Bayesian Optimization  
Exhaustive sampling, which involves measuring radiation at every reachable point on a grid within an environment, is a time-consuming process, especially when dealing with the long-term surveillance of large areas. Our research addresses this challenge by employing an adaptive sampling approach, which leverages a Gaussian Process (GP) to model the radiation map. This GP-based approach is particularly effective for scenarios involving uncertainty, as it provides a flexible and continuously improving estimate of the true radiation distribution. Initially, our Gaussian Process model offers an inaccurate guess of the true radiation map. However, it becomes more accurate as additional radiation measurements are incorporated. We use Bayesian optimization to iteratively determine the most informative locations to measure next. In essence, this is an optimization problem through which we seek to approximate the radiation map by balancing two key objectives: reducing uncertainty across the entire area and identifying regions with high radiation levels. Specifically, our approach aims to exploit areas with potentially high radiation for further measurements while also exploring the broader area to discover other regions of elevated readings. In Equation (1), \(\hat{f}(x)\) represents the radiation map as a function whose input, \(x\), refers to a coordinate pair, \(i\) (\(x_{i},y_{i}\)), and the output is the radiation reading at  
Figure 1: Spot quadruped platform equipped with a 6-DOF manipulator, six depth cameras (with five mounted on the body and one on the end effector), an onboard Spot Core computer, a 564 Wh battery providing a typical runtime of 90 min, and a Kromek GR1 gamma radiation sensor mounted on the jaw. that point in space. This function is represented as a Gaussian process defined by a mean function, \(m(x)\), and a kernel, \(k(x,x^{\prime})\). \[\hat{f}(x)\sim GP(m(x),k(x,x^{\prime})) \tag{1}\]  
The mean function outputs the expected value at any coordinate and is visualized as the radiation map when uncertainty information is not shown. The kernel, also called the covariance function, outputs the relatedness between any coordinate, \(x\), and any other coordinate, \(x^{\prime}\). This kernel can be thought of as the fabric of the estimated surface, influencing the shape and smoothness of the map in regions that have not been directly measured. Equation (1) provides the necessary information to evaluate the normal distribution given in Equation (2), which represents the estimated range of possible radiation values at any specific coordinate, \(x\). Here, \(\mu(x)\) is the predicted mean radiation level at that coordinate, and \(\sigma(x)^{2}\) is the variance that dictates the width of the distribution curve. \[f(x)\sim\mathcal{N}(\mu(x),\sigma(x)^{2}) \tag{2}\]  
Acquisition functions allow us to calculate the value of measuring at any coordinate, \(x\). We selected the probability of improvement as the acquisition function due to its simplicity, intuitive appeal, and ease of adjusting the balance between exploiting regions of high radiation and exploring areas with greater uncertainty. Equation (3) shows that the probability of improvement at coordinate \(x\) is the probability that the value from Equation (2) is greater than the highest currently measured value, \(y^{*}\). \[\text{PI}(x)=\text{Pr}(f(x)>y^{*}) \tag{3}\]  
This probability value can be found by inputting the Z-score \(\frac{\mu(x)-y^{*}}{\sigma(x)}\) into a cumulative distribution function (CDF), \(\phi\). The Z-score in this form represents how many standard deviations the highest value we have measured, \(y^{*}\), is from the mean of that distribution. The area under the distribution curve that lies above this highest value corresponds to the probability that the point will have a higher value. The CDF, being the integral of the normal distribution, provides this probability.