# An Integrated Approach to Goal Selection in Mobile Robot Exploration

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## Abstract

**:**

## 1. Introduction

- We formulate the objective of the integrated approach to candidates generation and goal selection as the d-Watchman Route Problem, which enables a theoretically sound interpretation of the integrated approach to the goal determination problem. Our solution to the problem then leads to a definition of the objective of a goal selection itself as a variant of the Generalized Traveling Salesman Problem (GTSP).
- The introduced GTSP variant involves additional constraints to the original GTSP, which renders standard GTSP solvers inapplicable here. A novel evolutionary algorithm taking into account the added constraints is introduced. It uses an indirect representation and an extended nearest neighbor constructive procedure which circumvent the candidate solutions feasibility issue encountered when using traditional evolutionary algorithms with direct encodings.
- A novel approach for generating nodes/vertices for GTSP is presented as a mixture of the techniques for goal candidates generation mentioned above.
- The whole exploration framework is evaluated in a simulated environment and compared to the state-of-the-art methods. Moreover, we implemented the framework on a real robot to show that the proposed strategy is applicable in real conditions.

## 2. Problem Definition

**next goal assignment problem**(step 5), formally defined as follows:

**exploration strategy**. More formally, exploration strategy aims to find a policy $\pi :\mathit{M}\times \mathcal{M}\to \mathit{M}$, where $\mathcal{M}$ is a set of all possible maps and $\mathit{M}\in \mathcal{M}$ is a map. Given the robot position $p\in \mathit{M}$ and the map $\mathit{M}\in \mathcal{M}$, the policy determines the next goal ${g}^{*}\in \mathit{M}$ so that the following cost is minimized:

## 3. Exploration Framework

#### 3.1. Proposed Exploration Strategy

**Generation**: of a set of goal candidates so that union of their visibility regions (i.e., areas visible from the particular goal candidates) contains all frontier cells, and**Construction**of a route connecting a subset of the goal candidates, whose visibility regions cover all frontier cells and a path traversing all of them from a current robot position is minimal.

#### 3.1.1. Goal Candidates Generation

Algorithm 1: Proposed exploration strategy |

#### 3.1.2. Route Construction

- a set of m frontiers $\mathbf{F}={\left\{{F}^{i}\right\}}_{i=1}^{m}$, where each frontier ${F}^{i}$ is a set of frontier cells ${F}^{i}=\left\{{f}_{j}^{i}\right\}$,
- a set of n goal candidates $\mathbf{G}$ clustered into m clusters. Each cluster ${G}^{i}={\left\{{g}_{j}^{i}\right\}}_{j=1}^{{n}_{i}}$ associated to a frontier ${F}^{i}$ is represented by ${n}_{i}$ candidate goals,
- coverage $\mathbf{R}\left({g}_{j}^{i}\right)\subset {F}^{i}$ for each ${g}_{j}^{i}$,
- a current position r of the robot,
- a matrix D of mutual distances between all goal candidates.

## 4. Proposed Approach to Solve the GTSPC

#### 4.1. Original IREANN Algorithm

**extended nearest neighbor**constructive procedure (ENN) determines the tour generated. When the cities are presented to (ENN) procedure in the right order, an optimal solution can be produced. This is illustrated in Figure 6 where the optimal solution is produced by processing cities in the order

`AJCEIFBHGD`.

**IREANN algorithm**. The pseudo-code of IREANN is shown in Algorithm 2. It starts with a random initialization of the population of candidate priority lists (line 1). Then, each individual is evaluated (line 2), i.e., a tour is constructed using the ENN procedure applied to its priority list (described in Algorithm 3) and its length is used as the individual’s fitness value. After the population has been evaluated, the counter of fitness function evaluations is set to the population size (line 3). Then, the algorithm runs until the number of calculated fitness evaluations reaches the maximum number of fitness evaluations (lines 4–16). The first parental individual is selected in each iteration using a tournament selection (line 5) and a new individual is created using either crossover (lines 7–10) or mutation (lines 12–13) operator. When the crossover operator is chosen with the crossover rate ${P}_{C}$, the second parental individual is selected (line 3) and the offspring produced undergoes the mutation operation (lines 9–10). The newly created individual is evaluated and the worst individual in the population is replaced by it (line 16).

Algorithm 2: Evolutionary algorithm for the optimal tour generation problem. |

**IREANN evolutionary operators**. The evolutionary algorithm uses the order-based crossover defined in [65] and a simple point mutation. The crossover works so that a set of goals randomly chosen from the priority list of the first parent are copied to the offspring into the same positions as they appear in the first parent. Remaining positions are filled in with goals taken in the same order as they appear in the priority list of the second parent.

#### 4.2. IREANN Adaptation for GTSPC

**Representation**. A candidate solution is a tour through a subset of candidate goals ${\mathbf{G}}^{\prime}\subseteq \mathbf{G}$ such that the union of visibility regions of goals $g\in {\mathbf{G}}^{\prime}$ covers all frontier cells. Generally, there might be multiple subsets of goals that produce a feasible solution. Thus, the representation should cover all the possibilities. Therefore, the representation used in this work is a priority list, i.e., the permutation, overall candidate goals in the original set $\mathbf{G}$. Note that, even though the priority list contains all candidate goals, the resulting tour can be composed of just a subset of these goals, as described in the following paragraphs.

**Tour construction procedure**. For the sake of efficiency, the set of frontiers $\mathcal{F}$ is split into two sets—a set ${\mathcal{F}}_{\mathcal{N}}$ of K frontiers nearest to the current position of the robot and a set ${\mathcal{F}}_{\mathcal{D}}$ of remaining distant frontiers—which are treated differently. For each frontier in the set ${\mathcal{F}}_{\mathcal{D}}$, a tour component C is constructed using the standard nearest neighbor algorithm such that C consists of a minimal set of goals completely covering the given frontier. The set of disjoint tour components, $\mathcal{C}$, constitutes a so-called embryo, which is created once before the evolutionary optimization starts. The tour components are immutable, only their connection within the whole tour is subject to further optimization.

**ENN procedure**. The ENN procedure takes a priority list $\mathrm{P}$ and the initial set of tour components $\mathcal{C}$ as input and produces a complete tour through the goals so that all frontier cells are visible from (or covered by) at least one of the used goals. The procedure starts with the initialization of the set of uncovered frontier cells with the set of all frontier cells of uncovered frontiers (line 1). The number of available connections of each goal is set to 2 (line 2). The value of $availConn\left[{g}_{i}\right]$ indicates whether the goal ${g}_{i}$ is already fully connected in the generated tour (i.e., $availConn\left[{g}_{i}\right]=0$) or it is partly connected with one link and the other link is still available (i.e., $availConn\left[{g}_{i}\right]=1$) or it is not used in the generated tour yet (i.e., $availConn\left[{g}_{i}\right]=2$). Then, the algorithm iterates through lines 4–32 until a single tour has been constructed that covers all frontiers’ cells. In each iteration i, the goal to be processed g is taken from the priority list $\mathrm{P}$ (line 5). If g is not yet used in the partial solution and the set of frontier cells visible from g does not contain any of the uncovered frontier cells, then it is skipped as it does not contribute to the overall coverage of the solution (lines 7–8). Otherwise, g is used to extend the current partial solution in three possible ways:

- The goal g is not connected in the solution yet. Its nearest available neighbor goal ${g}_{nn}$ is found, the two goals g and ${g}_{nn}$ are connected, the set of uncovered frontier cells is updated accordingly and the available connections of g and ${g}_{nn}$ are decremented (lines 9–14).
- The goal g is already linked to one other goal. Its nearest available neighbor goal ${g}_{nn}$ is found, the two goals g and ${g}_{nn}$ are connected and the available connections of g and ${g}_{nn}$ are updated (lines 15–19). The set of uncovered frontier cells remains unchanged.
- The goal g is already fully connected, i.e., it is linked to two other goals in one tour component ${C}^{g}$. First, the nearest available neighbors $star{t}_{nn}$ and $en{d}_{nn}$ of the component’s boundary goals are found. Then, the shorter link of the two possible links ($start$, $star{t}_{nn}$) and ($end$, $en{d}_{nn}$) is added to the solution and the available connections of newly connected goals are updated (lines 20–32).

Algorithm 3: Extended nearest neighbor constructive procedure for the feasible tour generation problem. |

## 5. Experimental Evaluation

#### 5.1. Simulations

`greedy`approach (greedy) [10], probably the most popular and used strategy nowadays—see, e.g., [66,67,68]. The second one is our previous strategy [27], which minimizes the cost over several steps. As the number of steps corresponds to the number of candidates, we call the strategy Full Horizon Planning (FHP). To the best of our knowledge,

`FHP`is currently one of the best strategies which do not use background knowledge. For example, although the recently published approach [28] benefits from a priori known topological map of the environment, it produces similar results to

`FHP`in similar environments in comparison to

`greedy`. The third method is an information-based strategy [22] that determines the cost of a candidate as a weighted sum of the information gained from visiting the goal and the Euclidean distance to the candidate. A hysteresis gain is added to prefer goals in robot’s vicinity. This method is further referred to as

`UMARI`.

`greedy`selects the nearest cell among all frontiers cells,

`FHP`clusters frontier cells using k-means, which is also used in our implementation of

`UMARI`. The proposed approach (EA) employs a two-stage sampling process as described in Section 3.

`UMARI`strategy, the settings recommended by the authors is used: $\lambda =3$, ${h}_{gain}=2$, ${h}_{rad}=$ sensor range (see [22] and https://github.com/hasauino/rrt_exploration).

`greedy`,

`FHP`and

`EA`are statistically summarized in Table 1, where avg stands for the time needed to explore the whole environment (exploration time ${T}_{exp}$) averaged over all 50 runs, min and max are minimal and maximal ${T}_{exp}$ over these runs and stdev stands for the standard deviation of ${T}_{exp}$. ${R}_{gr}$ expresses a ratio of the average ${T}_{exp}$ obtained with the proposed method to the average ${T}_{exp}$ obtained with the

`greedy`one. Similarly, ${R}_{FHP}$ is a ratio of average ${T}_{exp}$ values of

`EA`and

`FHP`. A value of less than 100% indicates the proposed approach is better than the respective compared one, and vice versa. Differences between the average ${T}_{exp}$ values of the compared algorithms were evaluated using a two-sample t-test with the significance level $\alpha =0.01$. The null hypothesis being the two data vectors are from populations with equal means. Results of the statistical tests are presented in the last column of the table. A sign ‘+’ means the average value obtained with the proposed algorithm is significantly better than the one of the compared algorithm. A sign ‘-’ indicates the opposite case. A situation when the two compared means are statistically indifferent is indicated by the ‘=’ sign.

`FHP`and

`EA`significantly outperform the

`greedy`approach for the empty map as they benefit from longer planning horizon allowing them to explore the space systematically. Moreover,

`EA`provides better results than FHP, especially for $\rho =3.0$, where the difference is more than 22%. This is because

`EA`directs a robot to a distance $\rho $ from obstacles contrary to FHP, where k-means generates goals to be visited closer to obstacles—see Figure 9a.

`is outperformed by the sophisticated approaches, although the difference is not as big as for empty. In addition,`

`greedy``EA`gives better results than

`FHP`in all cases except one.

`EA`performs better by approx. 8–15% than

`for jari-huge. On the other hand, it is slightly (up to 5%) outperformed by`

`greedy``FHP`due to the same reason that it is better than

`FHP`for empty. Here, directing the robot far from obstacles when going between neighboring rooms is contra-productive (see in Figure 9b). Conversely,

`EA`is more effective in the empty area in the middle.

`UMARI`are presented in Table 2. The meaning of the symbols is the same as in Table 1 except ${R}_{gr}$, which expresses a ratio of the average ${T}_{exp}$ obtained with

`UMARI`to the average ${T}_{exp}$ obtained with

`greedy`. Similarly, ${R}_{EA}$ is a ratio of average ${T}_{exp}$ values of

`UMARI`and

`EA`. It can be seen that the performance of

`UMARI`is the worst in all cases even in comparison to

`greedy`.

`approach. Chen et al. [26] show that the`

`greedy``approach outperforms imitation learning, reinforcement techniques as well as Curiosity-based Exploration [70] when localization error is below 3% (which is the case we are focused on). On the other hand, their strategy performs better than`

`greedy``when localization error increases. Zhu et al. [25] present the evaluation of their strategy based on Reinforcement Learning supervised Bayesian Optimization on ten office plans (similar to the jari-huge map but smaller). Their method is worse than`

`greedy``in four cases (by 41% in one case), while better in the other cases (by up to 32%). Table 1 shows that EA, on the other hand, performs better than`

`greedy``greedy`on jari-huge in all cases by 8–17%.

#### 5.2. Time Complexity

`EA`algorithm on a workstation with the Intel

^{®}Core™ i7-3770 CPU (Intel Corporation, Santa Clara, CA, USA) at 3.4 GHz running Sabayon Linux with the kernel 3.19.0. Fifteen trials of full exploration were run in the most complex setup: the jari-huge map and the visibility range $\rho =1.5$ m, while the other parameters were set to the same values as in the previous case. Figure 10a shows how the number of candidates (NoC) and computational time (T) of EA change during a typical trial. It can be seen that approximately 500 runs of EA were executed during the trial and that NoC grows as the robot explores new areas at the beginning, while it starts to decrease in the middle of the process. The curve of computational time follows the one of NoC but not so exactly as one would expect. This is caused by the fact that the complexity of the GTSPC problem does not only depend on NoC, but also on the number of clusters and distribution of candidates over them. The number of clusters varied up to 22 in this particular case.

`EA`are shown in Figure 10b in the form of dependency of T on NoC together with the averaged times for particular NoCs and the confidence interval computed for the means on a 95% confidence level. In fact, the averages and the confidence intervals are too noisy, so local polynomial fitting is applied to make the curves smooth. The maximal number of candidates was 321, while computational time does not exceed 1.3 s. This qualifies the method to be deployed in real time.

#### 5.3. Real Deployment

## 6. Conclusions and Future Work

`approach and up to $12.5\%$ lower than for`

`greedy``FHP`in some cases. On the other hand, the proposed method is worse than

`FHP`in narrow corridors by up to $4.5\%$, but still better than

`by more than $10\%$ on average.`

`greedy``UMARI`is even worse. In general, the method exhibits the best overall performance. Thus, our approach is a good choice when the type of the environment to be explored is not known in advance.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Sensor model. The robot (the red cell) measures an obstacle at the blue cell. The numbers in the cells represent sensor model values.

**Figure 4.**An exploration step: the robot (the green circle) selects one of the goal candidates (the blue dots; note that two different hues are used to visually distinguish overlapping dots.) as a next goal and plans a path to it (the pink curve). The green curve represents the already traversed trajectory.

**Figure 5.**Illustration of an ineffectiveness of the standard nearest neighbor constructive procedure on a simple instance of the TSP. (

**a**) shows an optimal solution; (

**b**) shows a solution constructed by the nearest neighbor when started from node B; (

**c**) shows a solution constructed when the procedure starts from node H.

**Figure 6.**Extended nearest neighbor constructive procedure applied to cities in the following order

`AJCEIFBHGD`. (

**a**,

**b**) show a partial solution after processing nodes {A, J} and {A, J, C, E, I, F}, respectively. (

**c**) shows the final solution. The last two nodes linked into the solution in the given stage of the construction process are shown in blue.

**Figure 8.**An example with a single frontier, six candidate goals equally distant from the frontier and the priority list $\mathrm{P}=\left[{g}_{1},{g}_{6},{g}_{4},{g}_{2},{g}_{3},{g}_{5}\right]$. (

**a**) the scenario; (

**b**,

**c**) partial coverages of frontier f after applying first two and three steps of the ENN procedure to goals ${g}_{1}$, ${g}_{6}$ and ${g}_{4}$; (

**d**) the complete coverage of frontier f induced by processing of the goals in the order ${g}_{1}$, ${g}_{6}$, ${g}_{4}$ and ${g}_{2}$; (

**e**) an alternative coverage of frontier induced by processing of the goals in the order ${g}_{1}$, ${g}_{2}$, ${g}_{6}$ and ${g}_{5}$.

**Figure 9.**The best results found by

`EA`(red) and

`FHP`(green) on (

**a**) empty with $\rho =3.0$ m and (

**b**) jari-huge with $\rho =2.0$ m.

**Figure 10.**(

**a**) evolution of computational time of EA and the number of goal candidates during a typical run on jari-huge with $\rho =1.5$ m; (

**b**) dependency of computational time of EA on a number of candidates. The dots represent measurements from 15 runs on jari-huge with $\rho =1.5$ m. The red curve is a smoothed computational time averaged over the number of candidates, while the grey area represents the confidence interval computed for mean on a 95% confidence level.

**Figure 11.**Experiment with a real robot in the SyRoTek system. (

**a**) the robot during an experiment; (

**b**,

**c**) phases of the exploration process; (

**d**) the final map and the performed path.

Map | $\mathit{\rho}$ | Greedy | FHP-Based | EA-Based | ${\mathit{R}}_{\mathit{gr}}$ | ${\mathit{R}}_{\mathit{FHP}}$ | +/− | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Avg | Min | Max | Stdev | Avg | Min | Max | Stdev | Avg | Min | Max | Stdev | % | % | |||||||

1.5 | 2419.70 | 2171.00 | 2701.00 | 111.43 | 2241.90 | 2041.00 | 2436.00 | 89.45 | 2118.60 | 1911.00 | 2226.00 | 68.67 | 87.56 | 94.50 | +/+ | |||||

2.0 | 1924.90 | 1741.00 | 2186.00 | 82.77 | 1679.90 | 1441.00 | 1816.00 | 86.97 | 1524.70 | 1381.00 | 1621.00 | 45.78 | 79.21 | 90.76 | +/+ | |||||

empty | 3.0 | 1330.70 | 1101.00 | 1556.00 | 91.78 | 1050.40 | 966.00 | 1196.00 | 64.12 | 918.50 | 881.00 | 966.00 | 18.77 | 69.02 | 87.44 | +/+ | ||||

5.0 | 739.90 | 621.00 | 856.00 | 65.79 | 585.20 | 571.00 | 621.00 | 9.44 | 542.20 | 516.00 | 566.00 | 10.91 | 73.28 | 92.65 | +/+ | |||||

10.0 | 428.50 | 341.00 | 471.00 | 39.70 | 305.30 | 286.00 | 326.00 | 7.83 | 294.90 | 271.00 | 361.00 | 13.07 | 68.82 | 96.59 | +/+ | |||||

1.5 | 2604.80 | 2361.00 | 2886.00 | 125.88 | 2391.80 | 2171.00 | 2506.00 | 78.33 | 2172.50 | 2021.00 | 2311.00 | 69.24 | 83.40 | 90.83 | +/+ | |||||

2.0 | 2021.40 | 1846.00 | 2321.00 | 105.73 | 1816.40 | 1581.00 | 1931.00 | 73.48 | 1724.60 | 1606.00 | 1856.00 | 47.20 | 85.32 | 94.95 | +/+ | |||||

potholes | 3.0 | 1293.60 | 1271.00 | 1461.00 | 36.65 | 1321.00 | 1141.00 | 1451.00 | 75.07 | 1259.20 | 1176.00 | 1346.00 | 42.91 | 97.34 | 95.32 | +/+ | ||||

5.0 | 1032.90 | 1011.00 | 1141.00 | 27.86 | 957.40 | 876.00 | 1056.00 | 34.70 | 911.90 | 856.00 | 951.00 | 23.20 | 88.29 | 95.25 | +/+ | |||||

10.0 | 951.30 | 741.00 | 1026.00 | 94.23 | 684.40 | 631.00 | 771.00 | 32.27 | 696.90 | 641.00 | 796.00 | 29.18 | 73.26 | 101.83 | +/= | |||||

1.5 | 2703.70 | 2511.00 | 2916.00 | 69.50 | 2331.40 | 1831.00 | 2391.00 | 77.58 | 2392.80 | 2331.00 | 2461.00 | 26.66 | 88.50 | 102.63 | +/− | |||||

2.0 | 2022.90 | 1841.00 | 2281.00 | 135.65 | 1776.40 | 1711.00 | 1841.00 | 27.51 | 1855.70 | 1826.00 | 1896.00 | 21.34 | 91.73 | 104.46 | +/− | |||||

jari-huge | 3.0 | 1370.00 | 1256.00 | 1521.00 | 93.48 | 1194.60 | 1136.00 | 1236.00 | 36.20 | 1190.20 | 1146.00 | 1251.00 | 30.61 | 86.88 | 99.63 | +/= | ||||

5.0 | 1254.70 | 1166.00 | 1331.00 | 70.15 | 1055.10 | 986.00 | 1101.00 | 28.51 | 1079.60 | 1041.00 | 1101.00 | 15.05 | 86.04 | 102.32 | +/− | |||||

10.0 | 1168.50 | 1136.00 | 1201.00 | 14.85 | 952.90 | 896.00 | 1001.00 | 22.56 | 973.30 | 926.00 | 1086.00 | 39.80 | 83.29 | 102.14 | +/− |

Map | $\mathit{\rho}$ | Greedy | ${\mathit{R}}_{\mathit{gr}}$ | ${\mathit{R}}_{\mathit{EA}}$ | |||
---|---|---|---|---|---|---|---|

Avg | Min | Max | Stdev | % | % | ||

empty | 1.5 | 2766.21 | 2506.00 | 3066.00 | 132.46 | 114.32 | 130.57 |

2.0 | 2196.61 | 1971.00 | 2431.00 | 112.40 | 114.12 | 144.07 | |

3.0 | 1596.71 | 1406.00 | 1816.00 | 101.32 | 119.99 | 173.84 | |

5.0 | 1290.15 | 1136.00 | 1446.00 | 77.96 | 174.37 | 237.95 | |

potholes | 1.5 | 2711.93 | 1966.00 | 3026.00 | 268.21 | 104.11 | 124.83 |

2.0 | 2198.11 | 1451.00 | 2441.00 | 194.48 | 108.74 | 127.46 | |

3.0 | 1495.38 | 436.00 | 2026.00 | 402.12 | 115.60 | 118.76 | |

5.0 | 1429.27 | 471.00 | 1686.00 | 189.93 | 138.37 | 156.73 | |

jari-huge | 1.5 | 2845.89 | 2651.00 | 3061.00 | 91.46 | 105.26 | 118.94 |

2.0 | 2447.43 | 2191.00 | 2651.00 | 102.75 | 120.99 | 131.89 | |

3.0 | 1738.65 | 1571.00 | 1866.00 | 65.00 | 126.91 | 146.08 | |

5.0 | 1596.52 | 1511.00 | 1661.00 | 41.87 | 127.24 | 147.88 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kulich, M.; Kubalík, J.; Přeučil, L. An Integrated Approach to Goal Selection in Mobile Robot Exploration. *Sensors* **2019**, *19*, 1400.
https://doi.org/10.3390/s19061400

**AMA Style**

Kulich M, Kubalík J, Přeučil L. An Integrated Approach to Goal Selection in Mobile Robot Exploration. *Sensors*. 2019; 19(6):1400.
https://doi.org/10.3390/s19061400

**Chicago/Turabian Style**

Kulich, Miroslav, Jiří Kubalík, and Libor Přeučil. 2019. "An Integrated Approach to Goal Selection in Mobile Robot Exploration" *Sensors* 19, no. 6: 1400.
https://doi.org/10.3390/s19061400