# An Integrated Strategy for Autonomous Exploration of Spatial Processes in Unknown Environments

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Related Work

#### 1.3. Contribution

- It selects intermediate goals in multi-step exploration for efficiently exploring the environment, while reducing the reconstruction error of the spatial process.
- It imposes visitation of intermediate goals as a routing problem for minimizing the traversed distance between two multi-exploration steps.
- It combines the strategy with efficient modelling using the GRBCM [29] to maintain online computational capabilities when exploring larger areas.

## 2. Gaussian Process Regression

#### 2.1. Gaussian Process

#### 2.2. Large-Scale Gaussian Process Regression

## 3. Integrated Exploration

#### 3.1. Sensors and Robot

#### 3.1.1. Sensors

#### 3.1.2. Robot

#### 3.2. Mapping and Localization, Navigation

#### 3.2.1. Mapping

#### 3.2.2. Localization

#### 3.2.3. Navigation

#### 3.3. GP Estimator

#### 3.4. Exploration Strategy

- Efficient spatial process exploration: Minimization of the process error in comparison to the ground truth.
- Efficient coverage strategy: Increasing coverage of the environment map to reduce unknown portions of the map.

#### 3.4.1. POI Detection

**Frontier.**First, to efficiently map the environment, we select classical frontiers, which maximize map coverage [6]. For efficient detection of frontiers, which separate known regions from unknown in $\mathcal{M}$, we apply WFD, a graph-based approach based on BFS [40]. This algorithm performs a search only on cells that are not yet traversed and represent free space, thus avoiding an expensive full map search. We weigh and sort frontiers by distance from the current location of the robot ${\mathbf{x}}^{\left[t\right]}$ to their centroids—the average position of all frontier points for a given frontier—and choosing the closest one. The distance is calculated with the A* path-planning algorithm.

**Candidates (POI).**Searching for all possible process measurement locations $\mathcal{X}$ on $\mathcal{M}$ is infeasible for large maps. Moreover, we need to keep in mind that map and process estimations are changing with robot traversing forward, meaning that planning over a too long horizon may produce an inefficient exploration strategy due to significant changes in the map.

Algorithm 1${\mathcal{X}}_{\mathrm{poi}}$ informative candidates sampling. |

Require: current location ${\mathbf{x}}^{\left[t\right]}$, map $\mathcal{M}$, radius r, sampling distance k, threshold $\u03f5$Ensure: POI ${\mathbf{X}}_{\mathrm{poi}}$1: Extract ${\mathcal{X}}_{\mathrm{free}}$ from the $\mathcal{M}$. 2: ${\mathcal{X}}_{\mathrm{poi}}=\left\{{\mathbf{x}}^{\left[t\right]}\right\}$ 3: BFS (${\mathcal{X}}_{\mathrm{free}},{\mathbf{x}}^{\left[t\right]}$): 4: $\widehat{\mathbf{x}}=\{{\mathbf{x}}_{*}\in {\mathcal{X}}_{\mathrm{free}}:H\left({\mathbf{x}}_{*}\right)>\u03f5,\left(\right)open="\parallel "\; close="\parallel ">{\mathbf{x}}_{*}-{\mathbf{x}}^{\left[p\right]}\ge k,\forall {\mathbf{x}}^{\left[p\right]}\in {\mathcal{X}}_{\mathrm{poi}},\}$ 5: ${\mathcal{X}}_{\mathrm{poi}}={\mathcal{X}}_{\mathrm{poi}}\cup \left\{\widehat{\mathbf{x}}\right\}$ |

#### 3.4.2. Goals Detection

**Distance matrix.**In order to find the optimal solution to traversing all ${\mathcal{X}}_{\mathrm{poi}}$ in terms of the distance travelled, we calculate the distance matrix between each pair of POIs ${\mathbf{x}}^{\left[i\right]},{\mathbf{x}}^{\left[j\right]}\in {\mathcal{X}}_{\mathrm{poi}}$, including the current pose ${\mathbf{x}}^{\left[t\right]}$. The distance between ${\mathbf{x}}^{\left[i\right]}$ and ${\mathbf{x}}^{\left[j\right]}$ is calculated with A* algorithm, where $i,j\in [1,\dots ,G]$.

**Route calculation.**Once the matrix is created, we formulate our problem as a routing problem. Since TSP does not consider distinguishable depots for both start and finish, we apply a simplified case of VRP [41]—a generalized version of TSP problem. More specifically, we apply a variant of the multi-depot VRP introducing additional constraints on start/end depots to minimize travelling distance from current location ${\mathbf{x}}^{\left[t\right]}$ to goal location ${\mathbf{x}}_{\mathrm{G}}$ while visiting all ${\mathcal{X}}_{\mathrm{poi}}$. We call it “vanilla” VRP with only one vehicle with given constraints to minimize the distance travelled between start and finishing depots.

- As a start depot, set the current location ${\mathbf{x}}^{\left[t\right]}$.
- If the frontier centroid ${\mathbf{x}}_{f}\in {\mathcal{X}}_{\mathrm{poi}}$, then ${\mathbf{x}}_{\mathrm{G}}={\mathbf{x}}_{f}$.
- Otherwise, we set ${\mathbf{x}}_{G}$ to be ${\mathbf{x}}^{\left[p\right]}\in {\mathcal{X}}_{\mathrm{poi}}$, that has the shortest distance to frontier centroid ${\mathbf{x}}_{f}$ (to preserve the direction favoring area coverage).

#### 3.5. All Components of Our Integrated Exploration Strategy

#### Algorithm Work-Flow

- Mapping and Localization: The robot continuously perceives the environment and accordingly updates the map $\mathcal{M}$ and its current location estimate ${\mathbf{x}}^{\left[t\right]}$.
- Navigation: Until any unvisited ${\mathbf{x}}^{\left[g\right]}\in {\mathbf{X}}_{\mathrm{goal}}$ exists, it continues following precomputed goal poses ${\mathbf{X}}_{\mathrm{goal}}$ (ordered representation of ${\mathcal{X}}_{\mathrm{poi}}$).
- At each ${\mathbf{x}}^{\left[g\right]}$ reached, collect the process measurement ${z}^{\left[g\right]}$.
- GP Estimator: Estimate GP process at probe locations ${\mathbf{X}}_{*}$ over the whole environment.
- Integrated Exploration If ${\mathbf{X}}_{\mathrm{goal}}$ is empty, detect the next frontier on $\mathcal{M}$ according to the procedure described in Section 3.4.1 and:
- −
- Sample locations within r as described in Section 3.4.1—producing unordered ${\mathcal{X}}_{\mathrm{poi}}$, a list of candidates where we want to obtain our next measurements to increase knowledge about the process.
- −
- From ${\mathcal{X}}_{\mathrm{poi}}$, create a distance matrix, representing computed distances between POI.
- −
- Order POI according to the procedure described in Section 3.4.2 so that all POI are visited and total travelled distance is minimized, resulting in ${\mathbf{X}}_{\mathrm{goal}}$.
- −
- If Algorithm 1 finds no suitable candidates within the limited horizon r, extend the horizon to cover all discovered cells ${\mathcal{X}}_{free}$ on the map. Select only the closest candidate location that satisfies $H\left(\mathbf{x}\right)>\u03f5$ as the next goal location ${\mathbf{x}}^{\left[g\right]}$. Otherwise terminate the mission.

## 4. System Evaluation

- What is the scalability of GRBCM for exploration of spatial processes?—Simulations (Section 5.1).
- What is the correlation between sampling distance k and error decrease in the process reconstruction for the IE strategy? How does it affect total exploration distance?—Simulations, experiment (Section 5.2 and Section 6.2).
- How does the IE perform against the benchmarks in various scenarios?—Simulations, experiment (Section 5.3.3 and Section 6.2).

#### 4.1. General System Setup

#### 4.1.1. Robotic Platform

#### 4.1.2. Perception Sensor

#### 4.1.3. Process Sensor

## 5. Simulations

#### 5.1. Scalability of Gaussian Processes for Spatial Modelling

#### Simulation Results

#### 5.2. Sampling Distance

#### Simulation Results

#### 5.3. Evaluation of the Strategy in Simulation

#### 5.3.1. Applied Baselines

**GGE**. At each step, selecting one ${\mathbf{x}}_{g}$ from all possible reachable cells ${\mathcal{X}}_{free,r}$ within radius covering all discovered cells $r=full$:

**GLGE**. At each step, selecting one ${\mathbf{x}}_{g}$ from all possible reachable cells ${\mathcal{X}}_{free,r}$ within radius r covering all discovered cells that is the solution of:

**SS**. Goal pose is ${\mathbf{x}}_{g}={\mathbf{x}}_{f}$, the centroid of the frontier. If at least one reachable frontier exists, measurements are collected each k meters distance travelled towards ${\mathbf{x}}_{g}$, where parameter k is the sampling distance described in Section 3.4.1. Baseline for the map exploration frontier originates from commonly used frontier-based exploration, both in the map coverage tasks [6,17,20], as well as a part of active SLAM exploration component [5,15]. Once no more frontiers are presented, i.e., map is fully explored, the GLGE strategy is used to complete an entropy driven process exploration.

#### 5.3.2. System Simulation Setup

#### 5.3.3. System Simulation Results

**Obstacle-free small room-like environment**(S1 in Table 1). We assume that the process is fully explored when process NMSE reaches 0.18 with respect to the ground truth. We are not able to collect samples around borders due to physical limitations of the robotic platform.

**S1 simulation results.**In Table 5 we observe that our IE strategy outperforms benchmarks with respect to the total travelled distance. While GGE collects the least amount of process samples ${N}_{\mathrm{proc}}.$ while reaching the same NMSE (making it computationally less expensive), it produces significantly longer trajectories. As a reminder, with GRBCM we can compensate computation costs introduced due to a higher number of process samples, thus making the total distance travelled our main comparison tool between different strategies.

**Small room-like environment, obstacles introduced**(S2 in Table 1). In this evaluation, we assume that the process is explored when NMSE reaches 0.21 with respect to the ground truth. As before, we are not able to collect samples in close proximity of obstacles.

**S2 simulation results.**When we add obstacles, we observe that IE tends to produce lower process error for the same map coverage (Table 6). We also see that our strategy collects more samples and traverses larger distances before the map is fully explored. This is due to its focus on exploring the surrounding process before continuing further towards unexplored areas. The compared benchmarks favor moving faster towards regions with high process uncertainty while ignoring some less uncertain but still important areas around its current measurement location. As a consequence of exploring room by room before moving further, IE also produces shorter final trajectories (Table 6).

**Large room-like environment, obstacles introduced**(S3 in Table 1). In addition to the multiple tests performed with S1 and S2, we test our strategy against GLGE in a large scale environment covering a surface of 2154.24 ${\mathrm{m}}^{2}$ (Figure 11). We assume that the process is explored when process NMSE reaches 0.084 with respect to the ground truth.

**S3 simulation results.**We note a lower final NMSE error in S3, in comparison with S1/S2. This is due to higher process correlation in process 2, resulting in better estimation of unreachable areas $\{\mathcal{X}\backslash {\mathcal{X}}_{\mathrm{free}}\}$. The estimated process for $r=3$ m. is shown in Figure 12.

## 6. Experiments

#### 6.1. Experimental Setup

- Finite horizon, $r=[2,3]$ m.
- Sampling distance $k=[2l,3l]$.

#### 6.2. Experimental Results

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GP | Gaussian process |

NBC | nuclear, biological, chemical |

SLAM | Simultaneous Localization and Mapping |

BCM | Bayesian Committee Machine |

RBCM | Robust Bayesian Committee Machine |

TSP | Traveling Salesman Problem |

FOV | field of view |

GRBCM | Generalized Robust Bayesian Committee Machine |

SE | squared exponential |

LIDAR | Light Detection and Ranging |

POI | Point of Interest |

ICP | Iterative Closest Point |

TEB | Time Elastic Band |

WFD | Wavefront Frontier Detector |

BFS | Breadth-first search |

VRP | Vehicle Routing Problem |

IE | Integrated exploration |

NMSE | Normalized Mean Square Error |

GGE | Greedy global entropy |

GLGE | Greedy local-global entropy |

SS | Sequential strategy |

ROS | Robot Operating System |

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**Figure 3.**Main components of our exploration strategy. (

**a**) Current position ${\mathbf{x}}^{\left[t\right]}$ is marked with a green circle, light red cross marks goal location ${\mathbf{x}}_{\mathrm{G}}$ within planning radius r (shaded green area), blue crosses represent POI ${\mathcal{X}}_{\mathrm{poi}}$, while orange circles are frontier with its centroid ${\mathbf{x}}_{\mathrm{f}}$ (red circle); (

**b**) optimized by distance travelled, ${\mathcal{X}}_{\mathrm{poi}}$ are represented as ordered points, i.e., goal poses ${\mathbf{X}}_{\mathrm{goal}}$, which describe the sequence that the robot follows, obtaining measurement ${z}^{\left[g\right]}$ on each ${\mathbf{x}}^{\left[g\right]}$.

**Figure 4.**One-robot solving a multi-depot “vanilla” VRP: ${\mathbf{x}}^{\left[t\right]}$—current location; ${\mathbf{x}}_{\mathrm{G}}$—goal location. Goals (${\mathbf{X}}_{\mathrm{goal}}$) are represented with blue asterisks and depots with black circles.

**Figure 5.**All components of a single multi-step of our integrated exploration strategy: Grey circles represent previously obtained measurements and the area around those circles is considered explored (light blue). Robot, at the current location (green circle), samples Point of Interest (POI, blue crosses)—in the obstacle-free region of the map, where the process is unexplored (red area)—within radius r (green line). The routing algorithm provides a multi-step exploration solution (black dotted line) from the current location to the goal location (red cross). The goal location is the POI closest to the frontier (brown circles) and the corresponding centroid (red circle). The frontier is defined by the LIDAR FOV (red dotted line). Obstacles are marked by a black line and unexplored area in grey.

**Figure 6.**Algorithm state diagram. Colored blocks represent the following: Red—mapping and localization; yellow—estimation of GP; green—core algorithm (our integrated exploration); and grey—navigation.

**Figure 8.**Example trajectory for scenario S1 and starting location (−3.3,−3.3) for the different strategies.

**Figure 10.**S2: IE $r=3m,\phantom{\rule{3.33333pt}{0ex}}k=3l$ with red diamonds in (c) representing measurements and white lines trajectory.

**Figure 16.**E1: IE, $r=3m,\phantom{\rule{3.33333pt}{0ex}}k=3l$ with red diamonds in (c) representing measurements and white lines trajectory.

Scenario | Process | Operating Environment | Dimensions $\left({\mathbf{m}}^{2}\right)$ |
---|---|---|---|

S1 | Process 1 | Obstacle-free | $8.0\times 8.0$ |

S2 | Process 1 | Small room-like environment, obstacles introduced | $8.0\times 8.0$ |

S3 | Process 2 | Large room-like environment, obstacles introduced | $56.1\times 38.4$ |

E1 | Magnetic field intensity | Obstacles introduced | $12.0\times 6.0$ |

Process | Dimensions (${\mathbf{m}}^{2}$) | ${\mathit{\sigma}}_{\mathit{f}}^{2}\left(\mathbf{m}\right)$ | l (m) | ${\mathit{\sigma}}_{\mathit{n}}^{2}$ (m) |
---|---|---|---|---|

Process 1 | $8.0\times 8.0$ | 0.03 | 0.2 | 0.0001 |

Process 2 | $56.1\times 38.4$ | 0.04 | 0.25 | 0.0001 |

Magnetic field | $4.3\times 9.6$ | $0.07$ | $0.2$ | 0.0001 |

**Table 3.**Performance evaluation for 2D field estimation with dataset with $N=7592$ measurements, R = 214,500 2D test points ($390\times 550$), grid resolution 0.1 m, hyperparameter $l=0.25$ m, sampling $k=3l$. We variate P and ${p}_{0}=N/P$ (number of local experts and their size, respectively).

$\mathit{P},{\mathit{p}}_{\mathit{o}}$ | Method | Prediction Time [s] | NMSE |
---|---|---|---|

37, 100 | GRBCM | 21.28 | 0.151 |

18, 200 | GRBCM | 20.74 | 0.093 |

12, 300 | GRBCM | 20.26 | 0.073 |

10, 350 | GRBCM | 19.26 | 0.070 |

9, 400 | GRBCM | 22.42 | 0.068 |

7, 500 | GRBCM | 24.31 | 0.062 |

1, 7592 | Full GP | 102.34 | 0.060 |

Strategy | Radius r (m) | Step Size k (m) | Multi-Step Planner |
---|---|---|---|

GGE | explored map | $[3l$, explored map] | No. |

GLGE | (i) fixed r, (ii) explored map | (i) $[3l,r]$, (ii) $[3l$, explored map] | No. |

SS | explored map | (i) fixed = $3l$, (ii) $[3l$, explored map] | No. |

our IE | (i) fixed r, (ii) explored map | (i) $[3l,r]$, (ii) $[3l$, explored map] | Yes. |

**Table 5.**Mean value and standard deviation over five runs for scenario S1. Process exploration: Total travelled distance and number of collected process measurements (${N}_{\mathrm{proc}}.$) required to achieve NMSE of 0.18 with respect to the ground truth.

Strategy | Distance (m) | ${\mathit{N}}_{\mathbf{proc}}.$ |
---|---|---|

GGE | $140.18\pm 13.36$ | $109\pm 2$ |

GLGE | $107.06\pm 2.12$ | $134\pm 3$ |

IE $r=2$ m | $102.82\pm 3.51$ | $134\pm 3$ |

IE $r=3$ m | $105.39\pm 1.84$ | $134\pm 2$ |

IE $r=\mathrm{full}$ | $104.49\pm 2.79$ | $138\pm 4$ |

**Table 6.**Mean value and standard deviation over five runs for scenario S2: (i) mapping: Total travelled distance needed to explore the map with accompanied number of obtained process measurements ${N}_{\mathrm{proc}.,\mathrm{map}}$ and process NMSE; (ii) process exploration: Total exploration distance needed to explore the process and number of collected measurements (${N}_{\mathrm{proc}}.$) required to achieve NMSE of 0.21 with respect to the ground truth.

Strategy | Distance (m) Map Explored | ${\mathit{N}}_{\mathbf{Proc}.\mathbf{Map}}$ | Proc. NMSE Map Explored | Distance (m) Proc. Explored | ${\mathit{N}}_{\mathbf{proc}}.$ |
---|---|---|---|---|---|

GGE | $15.84\pm 4.38$ | $13\pm 5$ | $0.6\pm 0.1$ | $126.31\pm 5.13$ | $84\pm 1$ |

GLGE | $49.77\pm 10.72$ | $64\pm 12$ | $0.48\pm 0.05$ | $94.54\pm 2.56$ | $114\pm 4$ |

SS | $25.72\pm 6.63$ | $35\pm 9$ | $0.62\pm 0.02$ | $108.17\pm 7.38$ | $118\pm 4$ |

IE $r=2$ m | $69.94\pm 12.58$ | $89\pm 15$ | $0.42\pm 0.07$ | $89.89\pm 4.97$ | $114\pm 2$ |

IE $r=3$ m | $64.89\pm 17.68$ | $85\pm 17$ | $0.47\pm 0.12$ | $90.97\pm 3.96$ | $116\pm 2$ |

**Table 7.**Scenario S3: (i) mapping: Total travelled distance needed to explore the map with accompanied number of obtained process measurements ${N}_{\mathrm{proc}.,\mathrm{map}}$ and process NMSE; (ii) process exploration: Total exploration distance to explore the process and number of collected measurements (${N}_{\mathrm{proc}}.$) required to achieve NMSE of 0.084 with respect to the ground truth.

Strategy | Distance (m) Map Explored | ${\mathit{N}}_{\mathbf{Proc}.\mathbf{Map}}$ | Proc. NMSE Map Explored | Distance (m) Proc. Explored | ${\mathit{N}}_{\mathbf{proc}}.$ |
---|---|---|---|---|---|

GLGE | 1398.45 | 2027 | 0.29 | 2977.52 | 3228 |

IE $r=3$ m | 1393.46 | 1852 | 0.33 | 2847.06 | 3423 |

IE $r=6$ m | 1888.78 | 2530 | 0.32 | 2872.52 | 3567 |

**Table 8.**Experiment E1—variation of sampling size: (i) mapping: Total travelled distance needed to explore the map with accompanied number of obtained process measurements up to that point ${N}_{\mathrm{proc}.,\mathrm{map}}$ and process NMSE; (ii) process exploration: Total exploration distance to explore the process and number of collected measurements (${N}_{\mathrm{proc}}.$) required to achieve NMSE of 0.34 with respect to ground truth.

Strategy | Distance (m) Proc. Explored | ${\mathit{N}}_{\mathbf{proc}}.$ | Distance (m) Proc. Explored | ${\mathit{N}}_{\mathbf{proc}}.$ |
---|---|---|---|---|

$k=2l$ | $k=3l$ | |||

IE $r=2$ m | 92.84 | 144 | 55.63 | 75 |

IE $r=3$ m | 93.56 | 146 | 58.23 | 74 |

**Table 9.**Experiment E1: (i) mapping: Total travelled distance needed to explore the map with accompanied number of obtained process measurements up to that point ${N}_{\mathrm{proc}.,\mathrm{map}}$ and process NMSE; (ii) process exploration: Total exploration distance to explore the process and number of collected measurements (${N}_{\mathrm{proc}}.$) required to achieve NMSE of 0.34 with respect to ground truth.

Strategy | Distance (m) Map Explored | ${\mathit{N}}_{\mathbf{proc}.\mathbf{map}}$ | Proc. NMSE Map Explored | Distance (m) Proc. Explored | ${\mathit{N}}_{\mathbf{proc}}.$ |
---|---|---|---|---|---|

GGE | 17.28 | 17 | 0.71 | 170.94 | 53 |

GLGE | 44.91 | 54 | 0.62 | 64.81 | 75 |

SS | 33.24 | 48 | 0.68 | 73.36 | 77 |

IE $r=2$ m | 47.45 | 62 | 0.54 | 55.62 | 75 |

IE $r=3$ m | 53.08 | 66 | 0.53 | 58.23 | 74 |

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## Share and Cite

**MDPI and ACS Style**

Karolj, V.; Viseras, A.; Merino, L.; Shutin, D.
An Integrated Strategy for Autonomous Exploration of Spatial Processes in Unknown Environments. *Sensors* **2020**, *20*, 3663.
https://doi.org/10.3390/s20133663

**AMA Style**

Karolj V, Viseras A, Merino L, Shutin D.
An Integrated Strategy for Autonomous Exploration of Spatial Processes in Unknown Environments. *Sensors*. 2020; 20(13):3663.
https://doi.org/10.3390/s20133663

**Chicago/Turabian Style**

Karolj, Valentina, Alberto Viseras, Luis Merino, and Dmitriy Shutin.
2020. "An Integrated Strategy for Autonomous Exploration of Spatial Processes in Unknown Environments" *Sensors* 20, no. 13: 3663.
https://doi.org/10.3390/s20133663