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Robotic Active Information Gathering for Spatial Field Reconstruction with Rapidly-Exploring Random Trees and Online Learning of Gaussian Processes^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- We provide a more detailed description of the algorithms and the underlying methods. This helps the reader to better understand the algorithm implementation and simulation results.
- This paper also performs an analysis of the algorithm’s computational complexities.
- Also, additional simulations are presented and analyzed. Specifically, we carry out a detailed analysis of the proposed RRT*-based informative path planner. Moreover, we include an additional scenario to test the whole exploration strategy described in the paper. Moreover, a metric that benchmarks state-of-the-art algorithms according to their solution quality is also introduced.
- In this paper we also include an evaluation of the online learning of the GP hyperparameters and discuss the effect of online hyperparameter learning on the algorithms performance.
- Finally, we include an experiment with a real robot performing on exploration and reconstruction of a magnetic field using a sensor. We describe in detail the experimental setup and discuss the obtained results.

## 2. Related Work

## 3. Problem Statement

- The physical process takes place in an environment populated with obstacles. The borders and obstacles that define the environment are a priori known. This assumption allows us to abstract the exploration of the physical process from the perception and mapping of the environment.
- The physical process is time-invariant during the information gathering task.
- The robot’s position is known exactly and is noise-free. We assume that there exists an external positioning system that provides us with a highly accurate localization, e.g., a Real Time Kinematic Global Positioning System (GPS-RTK) for outdoor scenarios, or a motion tracking system for indoor cases. Uncertainty in positioning can also be accounted for using e.g., GPs [35].

## 4. Gaussian Processes for Spatial Data

## 5. Efficient Information Gathering Using RRT-Based Planners and GPs

#### 5.1. Algorithm Overview

Algorithm 1.$\mathtt{SBSRE}\phantom{\rule{3.33333pt}{0ex}}\mathtt{Algorithm}({\mathbf{x}}_{r},{\mathcal{X}}_{free},b,StopAlgorithm)$ |

Algorithm 2.$\mathtt{SearchStation}({\mathbf{x}}_{r},{\mathit{\theta}}_{\ast},\mathbf{X},b,{\mathcal{X}}_{free})$ |

#### 5.2. Search for Highly Informative Stations

#### 5.3. Informative Path Planner Using RRT*

#### Non-Monotonicity of the Utility Function

#### 5.4. Information Metric

#### 5.4.1. Mutual Information

Algorithm 3.$\mathtt{InformativePlanner}({\mathbf{x}}_{r},{\mathbf{s}}_{\ast},{\mathit{\theta}}_{\ast},\mathbf{X},b,{\mathcal{X}}_{free})$ |

#### 5.4.2. Mean Entropy

#### 5.5. Computational Complexity

## 6. Simulations and Discussion of Results

#### 6.1. Simulations Setup

#### 6.2. Analysis of the Informative Path Planner

#### 6.2.1. Setup

- Scenario 1 recreates a physical process with low spatial correlation in which a robot has already gathered two patches of measurements. The blue areas correspond to the measured areas and the red areas to the non-measured positions. We employ the following $\mathit{\theta}$: $l=0.02$, ${\sigma}_{f}=0.084$, ${\sigma}_{n}=0.02$.
- Scenario 2 recreates the same scenario, but now we consider a process with higher spatial correlation. Here we set $l=0.13$, ${\sigma}_{f}=0.084$, ${\sigma}_{n}=0.09$.
- Scenario 3 corresponds to three measurements that are taken randomly for each of the simulation runs. For this case we consider the same hyperparameters as for scenario 2.

#### 6.2.2. Choice of the Information Function

#### 6.2.3. Performance Analysis

- (i) the technique of [31], where multiple paths are obtained by running the RRT planner several times, and the paths are then evaluated according to the information metric. This algorithm we will term $\mathtt{Multiples}\phantom{\rule{3.33333pt}{0ex}}\mathtt{RRT}$;
- (ii) the RIG-tree planner [25], to which we will refer as $\mathtt{RIG}\phantom{\rule{3.33333pt}{0ex}}\mathtt{Algorithm}$.

**Utility analysis.**The difference in terms of utility (see first row of Figure 4) with respect to the other algorithms ranges between $0.05$ and $0.15$ bits per second. We notice as well that the RIG algorithm only presents a minor improvement of the utility as the planning time increases. We believe this is due to the inclusion of the goal constraint, which the RIG algorithm is not able to handle.

**Algorithm complexity analysis.**Another important figure that characterizes the algorithm is the number of nodes spanned by the path planner. We observe in the second row of Figure 4 that the $\mathtt{Multiples}\phantom{\rule{3.33333pt}{0ex}}\mathtt{RRT}$ variant has a limited number of nodes since we reset the algorithm each time we find a new path. Furthermore, Algorithm 3 requires a larger number of nodes than the $\mathtt{RIG}\phantom{\rule{3.33333pt}{0ex}}\mathtt{Algorithm}$. The latter employs a smaller number of nodes because of the pruning strategy that removes those co-located nodes that have a smaller utility than the new added node. However, this does not lead to a higher complexity per iteration, as we can observe in the third row of Figure 4, which shows the number of iterations of the algorithm vs. the planning time. Here, the $\mathtt{Multiples}\phantom{\rule{3.33333pt}{0ex}}\mathtt{RRT}$ alternative offers the lowest complexity.

**Paths output by Algorithm 3.**In the last row of Figure 4 we depict the paths output by Algorithm 3. We observe that, for scenario 1, the robot takes the path that has the most information and takes the least time, which results in a straight line. However, in scenario 2 the straight line corresponds to a path that has little information, and therefore the robot takes a path that is longer but allows it to gather more information as it visits not yet measured locations. These results illustrate the need for defining a utility function that trades off the information gathering and the path’s cost (6).

**Posterior entropy analysis.**We showed in Figure 4 that Algorithm 3 outperforms the considered state-of-the-art approaches in terms of our information function. However, this does not necessarily imply that our algorithm can find a more informative path. To make a fair comparison between the three considered algorithms we evaluate them in terms of the posterior entropy after measuring along the calculated path. In addition, we compare the cost of the resulting paths. Table 2 shows the results for scenario 2, for 180 s of planning time. We can conclude that Algorithm 3 offers the best ratio entropy-cost for all scenarios. Additionally Algorithm 3 results in a twofold and sevenfold increase respect to $\mathtt{RIG}\phantom{\rule{3.33333pt}{0ex}}\mathtt{Algorithm}$ and $\mathtt{Multiples}\phantom{\rule{3.33333pt}{0ex}}\mathtt{RRT}$, respectively, while offering a similar path cost.

#### 6.3. Analysis of the Exploration Strategy

#### 6.3.1. Setup

#### 6.3.2. Performance Analysis

- Random approach: an RRT is grown from ${\mathbf{x}}_{r}$ for the same planning time and budget b as in $\mathtt{SearchStation}$ algorithm. The next station is selected randomly as one of the leaves of the RRT. The path that links ${\mathbf{x}}_{r}$ to the selected leaf is then followed by the robot.

**RMSE analysis.**Figure 6 shows the mean and variance of the RMSE for all executions. This is done for the different strategies and for both scenarios. We also test the methods under assumption that the optimal $\mathit{\theta}$ are known and fixed (listed with an asterisk sign). Our goal is to shift the Mean(RMSE) curve to the left bottom corner. This implies a small RMSE that is achieved efficiently in terms of time resources. First fact that we can observe is that Algorithm 1 is able to obtain a performance comparable to Algorithm 1*, which uses pre-learned hyperparameters. This result indicates that Algorithm 1 performs a correct exploration-exploitation trade-off. That is, in the beginning of the IG task, we would expect a robot to perform exploratory actions to learn about the environment and about process of interest. Once this is done, the robot can use this knowledge to update and exploit the GPs model. In contrast, note that myopic and random strategies cannot perform a correct exploration-exploitation trade-off, which is reflected in their inferior performance when hyperparameters are unknown.

**Solution quality.**We analyze in Figure 7 the quality of the solution respect to the best possible performance that we could obtain by systematic sampling. We consider the best possible solution as the estimation over the complete environment that results after measuring at all $\mathbf{x}\in {\mathcal{X}}_{free}$. This we term it $RMS{E}_{best}$. Let us remark that this solution considers optimal $\mathit{\theta}$. More formally we define solution quality as: $Solution\phantom{\rule{3.33333pt}{0ex}}quality=\frac{1}{{n}_{sim}}{\sum}_{i=1}^{{n}_{sim}}100\frac{RMS{E}_{best}}{RMS{E}_{i}}$. We show in Figure 7 the solution quality for a myopic, random, and Algorithm 1. A percentage of $100\%$ indicates that Algorithm 1 is able to achieve an RMSE that is equal to the best possible RMSE that we could obtain. According to Figure 7, after 900 s Algorithm 1 is able to obtain a RMSE that is the $90\%$, while the myopic and random approach achieve only half of it.

**Comparison with RIG algorithm.**To get a better understanding of Algorithm 1 capabilities we include a comparison with state-of-the-art $\mathtt{RIG}\phantom{\rule{3.33333pt}{0ex}}\mathtt{Algorithm}$. Specifically, we consider the following for the $\mathtt{RIG}\phantom{\rule{3.33333pt}{0ex}}\mathtt{Algorithm}$: (i) the model is a priori known; i.e., we know the GPs hyperparameters and they do not need no be estimated, (ii) the utility function corresponds to the MI, as suggested by the authors in Ref. [25], and (iii) the planning time is 600 s and then we let the robot follow and measure along the planned path. Let us remark that these are favorable conditions for the $\mathtt{RIG}\phantom{\rule{3.33333pt}{0ex}}\mathtt{Algorithm}$ as our algorithm assumes an a priori unknown model that needs to be estimated online. Please also note that complexity of $\mathtt{RIG}\phantom{\rule{3.33333pt}{0ex}}\mathtt{Algorithm}$ is $\mathcal{O}\left({N}_{p}^{3}\right)$. Algorithm 1 complexity is $\mathcal{O}({N}_{p}log{N}_{p}+{n}_{b}^{3})$, with ${n}_{b}<<n,{N}_{p}$. We run the simulation 40 times starting from different positions in the environment. Then we calculated the RMSE after measuring along the calculated path. The average RMSE that we obtained for the $\mathtt{RIG}\phantom{\rule{3.33333pt}{0ex}}\mathtt{Algorithm}$ is $0.27$, which is much higher than the one obtained by Algorithm 1 that is $0.05$ (see Table 3). We believe that the lower performance of the $\mathtt{RIG}\phantom{\rule{3.33333pt}{0ex}}\mathtt{Algorithm}$ is due to the fact that the algorithm grows a single tree to explore the complete environment. Notice that the complexity of adding a new sample grows exponentially as the tree grows, which complicates the exploration of the complete environment. In contrast, Algorithm 1 runs multiple consecutive trees using our devised two-step approach that permits an efficient online exploration.

#### 6.3.3. Hyperparameters Analysis

## 7. Experiments and Discussion of Results

#### 7.1. Experimental Setup

#### 7.2. Experimental Results

## 8. Conclusions and Future Work

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Search for highly informative stations. The color scale represents the informativeness, as measured by a predefined information metric, at a particular location. In particular, dark blue corresponds to low informativeness and red represents high informativeness. Algorithm 2 selects ${\mathbf{s}}_{\ast}$ as the location with the highest informativeness among all $\mathbf{x}\in \mathcal{V}$.

**Figure 3.**Simulation scenarios used to test the performance of Algorithm 3. We represent the entropy of the process after measuring at some spots. Scenario 1 and 2 correspond to two patches of measurements of processes that have a low and high spatial correlation, respectively. Scenario 3 corresponds to three measurements taken at random positions for each simulation run. In this latter case, we show just one example.

**Figure 4.**Performance analysis of Algorithm 3 as we increase the planning time. Here, from the first to the last row, we evaluate the utility of the best path, the number of nodes spanned by the tree, and the algorithm’s complexity that is represented as the curve number of iterations vs. planning time. In addition, we plot the 40 paths output by Algorithm 3 given 180 s of planning time.

**Figure 5.**Scenarios employed to test Algorithm 1. Black polygons correspond to the obstacles and the underlying picture is the magnetic field intensity we aim to explore.

**Figure 6.**RMSE between the estimation of the process and the ground truth. Top: scenario A. Bottom: scenario B. We represent the mean and variance of the RMSE over the 40 simulations we carried out. Here we test three different trajectories: (i) Algorithm 1, (ii) a myopic approach, and (iii) random trajectories. For all of them we compare their performance assuming: (i) no prior knowledge about the process, which implies an online learning of $\mathit{\theta}$, and (ii) assuming they know the optimal $\mathit{\theta}$ a priori (marked with an asterisk).

**Figure 7.**Quality of the solution achieved by Algorithm 1, a myopic, and a random trajectories after a 900 s mission. A $100\%$ corresponds to the best possible solution. These curves correspond to the mean value achieved over the simulations carried out for scenarios A and B.

**Figure 8.**GPs hyperparameters learned during the information gathering task for Algorithm 1, a myopic, and a random trajectory. We represent the mean and variance over the 40 simulation runs. We show the hyperparameters $\mathit{\theta}={[{\sigma}_{f}^{2},l,{\sigma}_{n}^{2}]}^{T}$.

**Figure 9.**Ground-based robot exploring the magnetic field intensity within an indoor environment populated with obstacles. The projection on the ground corresponds to the actual magnetic field intensity, which we measured prior to the experiment to use it as ground truth. The magnetic field intensity ranges between 5 and 84 $\mathsf{\mu}$T.

**Figure 10.**Screenshots of Algorithm 1 execution. The three rows correspond to three instants of times: 133, 502 and 946 s. From left to right, the columns are the estimation of the process (measured in $\mathsf{\mu}$T), the entropy, and the planned path using Algorithm 3.

**Figure 11.**Evolution of the RMSE during a 940 s exploration task that was carried out with a ground-based robot in our lab.

**Table 1.**Analysis of the information function. We compare Equation (7) with the MI, in terms of: time to find a first ${\mathcal{P}}_{{\mathbf{x}}_{r},{\mathbf{s}}_{\ast}}$ (${t}_{first}$), process differential entropy that results after measuring along ${\mathcal{P}}_{{\mathbf{x}}_{r},{\mathbf{s}}_{\ast}}$ output by Algorithm 3, and cost of the resulting path. Results correspond to the average $\pm 3\sigma $ variation, calculated over 40 simulations runs.

${\mathit{t}}_{\mathit{first}}$ [s] | Differential Entropy [bits] | Cost [s] | |
---|---|---|---|

Mean Entropy (6) | $6.31\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}2.1$ | $-6.93\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.18$ | $6.79\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.06$ |

Mutual Information | $46.71\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3.2$ | $-3.54\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.12$ | $6.86\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.08$ |

**Table 2.**Posterior differential entropy and path cost evaluated over the completeenvironment after measuring ${\mathcal{P}}_{{\mathbf{x}}_{r},{\mathbf{s}}_{\ast}}$, calculated for 180 s of planning time.

Differential Entropy [bits] | Path Cost [s] | |
---|---|---|

Algorithm 3 | $-6.93\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.18$ | $6.79\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.06$ |

$\mathtt{Multiples}\phantom{\rule{3.33333pt}{0ex}}\mathtt{RRT}$ | $-3.68\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.32$ | $6.23\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.09$ |

$\mathtt{RIG}\phantom{\rule{3.33333pt}{0ex}}\mathtt{Algorithm}$ | $0.03\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.29$ | $5.65\pm 0.12$ |

RMSE at t = 600 s | |
---|---|

Algorithm 1 | $0.05\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

$\mathtt{RIG}\phantom{\rule{3.33333pt}{0ex}}\mathtt{Algorithm}$ | $0.27\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

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## Share and Cite

**MDPI and ACS Style**

Viseras, A.; Shutin, D.; Merino, L.
Robotic Active Information Gathering for Spatial Field Reconstruction with Rapidly-Exploring Random Trees and Online Learning of Gaussian Processes. *Sensors* **2019**, *19*, 1016.
https://doi.org/10.3390/s19051016

**AMA Style**

Viseras A, Shutin D, Merino L.
Robotic Active Information Gathering for Spatial Field Reconstruction with Rapidly-Exploring Random Trees and Online Learning of Gaussian Processes. *Sensors*. 2019; 19(5):1016.
https://doi.org/10.3390/s19051016

**Chicago/Turabian Style**

Viseras, Alberto, Dmitriy Shutin, and Luis Merino.
2019. "Robotic Active Information Gathering for Spatial Field Reconstruction with Rapidly-Exploring Random Trees and Online Learning of Gaussian Processes" *Sensors* 19, no. 5: 1016.
https://doi.org/10.3390/s19051016