# An Approach for Indoor Path Computation among Obstacles that Considers User Dimension

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## Abstract

**:**

## 1. Introduction

**Figure 1.**Minkowski sums of obstacles to a user and the minimum distance between obstacles. (

**a**) Minkowski sum of obstacles for a user approximated as a circle; (

**b**) union of the Minkowski sum of obstacles and the minimum distance.

**Figure 2.**The union of Minkowski sums contains inner rings and self-intersections. (

**a**) Union of Minkowski sums with one inner ring (the circle denotes a user); (

**b**) self-intersection and inner rings of the Minkowski sums.

**Figure 3.**Creating polygonal boundaries of objects according to inaccessible gaps between the objects (the circle denotes a user with tools, and red lines represent inaccessible gaps). (

**a**) The polygonal boundary for the six objects without inner rings; (

**b**) the simple boundary for the nine objects.

**Figure 4.**Computing a path with simple boundaries of obstacle groups for a user. (

**a**) Selecting groups of obstacles between two locations (the circle denotes the user; boundaries of obstacles are yellow; and the blue line is the direct path); (

**b**) the navigation network considering the inaccessible gaps between obstacles and walls (blue lines denote the buffer of walls; red lines denote inaccessible gaps; and black lines form the network); (

**c**) a schematic representation of the computed shortest path on the network (the path is black); (

**d**) a realistic path by taking into account the size of the user (circles denote the user; and black lines are the path).

## 2. Related Work

## 3. The Proposed Approach

#### 3.1. Compute MD between Obstacles and Find Inaccessible Gaps

#### 3.2. Group Obstacles

- Step 1. Pick an unchecked obstacle as the current obstacle obs. If there is no unchecked obstacle, then go to Step 5. Otherwise, create an empty group gop; add obs to gop; and go to Step 2.
- Step 2. In the linked list of obs, add all of the unchecked members to gop.
- Step 3. For the previously-added members, add all of the unchecked members in their linked lists to the gop.
- Step 4. Repeat Step 3 until no unchecked members are found. Then, the obstacles in the gop form a group. Go to Step 1.
- Step 5. All of the groups have been identified. Count the number of groups and assign each group an ID.

#### 3.3. Select Obstacle Groups

- Step 1. Find the obstacles intersecting the direct path.
- Step 2. Select all of the obstacles from the groups that have an obstacle intersecting the direct path.
- Step 3. Compute a CH with the nodes of all of the selected obstacles and the direct path.
- Step 4. If the current CH intersects or contains new obstacles, look up the groups of the new obstacles, and select all of the obstacles from the new groups. Re-compute a CH.
- Step 5. Iterate Step 4 until there are no other obstacles included by the current CH.

**Figure 8.**Selecting obstacle groups with respect to the start and target location of a user. (

**a**) The direct path intersects two obstacles; (

**b**) selecting the groups of the intersected obstacles; (

**c**) computing a convex hull (CH), and the CH intersects other obstacles; (

**d**) selecting the group of new obstacles in and re-computing the CH; (

**e**) no more obstacles intersect the CH, and then, three groups are selected.

#### 3.4. Create Boundaries for Selected Groups

- Step 1. For a selected obstacle group, check the number of obstacles. If the group includes only one obstacle, then the obstacle’s polygon is the boundary. Otherwise, go to Step 2.
- Step 2. Create a DT with the vertices of all obstacles in the group, and assign the given dimension of the user to the alpha value.
- Step 3. Compute all of the lengths of edges of each triangle in the DT. Preserve a triangle if its edges’ lengths are all less than the alpha value.
- Step 4. In all of the preserved triangles, find the edges only used for one triangle. Form the edges into a boundary.

#### 3.5. Create a Network Considering Inaccessible Gaps with Walls

**Figure 11.**Detecting bottlenecks between a wall and the boundary of an obstacle group and identifying inaccessible edges crossing the bottlenecks.

**Figure 12.**Creating a visibility graph (VG) and a room buffer, removing the inaccessible edges and computing a path for users with a size of 0.8 m. (

**a**) Creating a VG; (

**b**) computing a room buffer; (

**c**) finding inaccessible edges in the bottlenecks; (

**d**) removing the inaccessible edges; (

**e**) the computed path.

## 4. Use Cases

**Figure 13.**The floor plans of two buildings. (

**a**) The floor of the conventional neonatal intensive care unit (CNICU) at the hospital; (

**b**) ground floor of the Architecture Faculty building.

**Figure 14.**Testing the complete approach between two locations for a user with a size of 0.6 m. (

**a**) Computing the bottlenecks between obstacles; (

**b**) grouping the obstacles; (

**c**) selecting obstacle groups; (

**d**) creating the boundaries of the selected groups; (

**e**) creating a VG and removing the inaccessible edges.

**Figure 15.**Path-finding for the sizes of 0.5 m, 0.6 m and 0.8 m on the hospital floor plan. (

**a**) The shortest path for the 0.5 m size; (

**b**) the shortest path for the 0.6 m size; (

**c**) no path for the 0.8 m size.

**Figure 16.**The shortest paths for the sizes 0.5 m, 0.8 m and 1.0 m on the floor plan of the campus building. (

**a**) Path for the 0.5 m size; (

**b**) path for the 0.8 m size; (

**c**) path for the 1.0 m size.

**Figure 17.**(

**a**) The VG regardless of a user’s dimension and (

**b**–

**d**) all of the shortest paths between all doors for the sizes of 0.5 m, 0.6 m and 0.8 m. (a) The complete VG of all obstacles without respect to a user’s dimension; (b) all of the shortest paths for the user with a size of 0.5 m; (c) all of the shortest paths for a user with a size of 0.6 m; (d) all of the shortest paths for the user with a size of 0.8 m.

## 5. Conclusions

**Figure 18.**The computed and genuine minimum distances (MDs) between a convex and a non-convex obstacle. (

**a**) The computed MD in our approach; (

**b**) the genuine MD.

**Figure 19.**The boundaries of three obstacles in our method and in a strict condition. (

**a**) Three obstacles; (

**b**) our boundary for the group of three obstacles; (

**c**) a strict boundary for the group of three obstacles.

**Figure 20.**(

**a**) A user can pass through a gap larger than the diameter of the user’s circle; (

**b**) the user cannot pass through the gap when the diameter is larger; and (

**c**) the user can pass through the gap after a 90-degree turn. (a) The user can pass through the gap when its length is longer than the diameter of the circle; (b) the user cannot pass the gap when its length is shorter than the diameter of the circle; (c) the user can pass through the gap because the user’s short side is smaller than the gap, though the diameter of the circle is larger than the gap.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Goetz, M.; Zipf, A. Formal definition of a user-adaptive and length-optimal routing graph for complex indoor environments. Geo-Spat. Inf. Sci.
**2011**, 14, 119–128. [Google Scholar] [CrossRef] - Lertlakkhanakul, J.; Li, Y.; Choi, J.; Bu, S. GongPath: Development of BIM based indoor pedestrian navigation system. In Proceeding of the Fifth International Joint Conference on INC, IMS and IDC, Seoul, Korea, 25–27 August 2009; pp. 382–388.
- Swobodzinski, M.; Raubal, M. An indoor routing algorithm for the blind: Development and comparison to a routing algorithm for the sighted. Int. J. Geogr. Inf. Sci.
**2009**, 23, 1315–1343. [Google Scholar] [CrossRef] - Höcker, M.; Berkhahn, V.; Kneidl, A.; Borrmann, A.; Klein, W. Graph-based approaches for simulating pedestrian dynamics in building models. In Proceeding of the 8th European Conference on Product and Process Modelling (ECPPM), Cork, Republic of Ireland, 14–16 September 2010.
- Kneidl, A.; Borrmann, A.; Hartmann, D. Generation and use of sparse navigation graphs for microscopic pedestrian simulation models. Adv. Eng. Inf.
**2012**, 26, 669–680. [Google Scholar] [CrossRef] - Liu, L.; Zlatanova, S. A Two-level Path-finding Strategy for Indoor Navigation. In Intelligent Systems for Crisis Management; Zlatanova, S., Peters, R., Dilo, A., Scholten, H., Eds.; Springer: Berlin, Germany, 2013; Volume 7418, pp. 31–42. [Google Scholar]
- Schougaard, K.; Grønbæk, K.; Scharling, T. Indoor pedestrian navigation based on hybrid route planning and location modeling. In Pervasive Computing; Kay, J., Lukowicz, P., Tokuda, H., Olivier, P., Krüger, A., Eds.; Springer: Berlin, Germany, 2012; pp. 289–306. [Google Scholar]
- Berg, M.D.; Cheong, O.; Kreveld, M.V.; Overmars, M. Computational Geometry: Algorithms and Applications, 3rd ed.; Springer: Santa Clara, CA, USA, 2008. [Google Scholar]
- Coenen, S.; Steinbuch, M.M.; Molengraft, V.D.M.; Lunenburg, J.; Naus, G. Motion Planning for Mobile Robots—A Guide. Master’s Thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 2012. [Google Scholar]
- Open Geospatial Consortium, Inc. OpenGIS Implementation Specification for Geographic information — Simple feature access — Part 1: Common architecture; Open Geospatial Consortium, Inc.: Wayland, MA, USA, 2011. [Google Scholar]
- Yuan, W.; Schneider, M. 3D indoor route planning for arbitrary-shape objects. In Database Systems for Adanced Applications; Xu, J., Yu, G., Zhou, S., Unland, R., Eds.; Springer: Berlin, Germany, 2011; pp. 120–131. [Google Scholar]
- Chen, L.C.; Wu, C.H.; Shen, T.S.; Chou, C.C. The application of geometric network models and building information models in geospatial environments for fire-fighting simulations. Comput. Environ. Urban Syst.
**2014**, 45, 1–12. [Google Scholar] [CrossRef] - Lee, J. A spatial access-oriented implementation of a 3-D GIS topological data model for urban entities. GeoInformatica
**2004**, 8, 237–264. [Google Scholar] [CrossRef] - Lorenz, B.; Ohlbach, H.; Stoffel, E.P. A hybrid spatial model for representing indoor environments. In Web and Wireless Geographical Information Systems; Carswell, J., Tezuka, T., Eds.; Springer: Berlin, Germany, 2006; pp. 102–112. [Google Scholar]
- Meijers, M.; Zlatanova, S.; Pfeifer, N. 3D geo-information indoors: Structuring for evacuation. In Proceedings of the First International Workshop on Next Generation 3D City Models, Bonn, Germany, 21–22 June 2005.
- Thill, J.C.; Dao, T.H.D.; Zhou, Y. Traveling in the three-dimensional city: Applications in route planning, accessibility assessment, location analysis and beyond. J. Transp. Geogr.
**2011**, 19, 405–421. [Google Scholar] [CrossRef] - Han, C.S.; Law, K.H.; Latombe, J.C.; Kunz, J.C. A performance-based approach to wheelchair accessible route analysis. Adv. Eng. Inf.
**2002**, 16, 53–71. [Google Scholar] [CrossRef] - Kostic, N.; Scheider, S. Automated generation of indoor accessibility information for mobility-impaired individuals. In AGILE 2015; Bacao, F., Santos, M.Y., Painho, M., Eds.; Springer: Berlin, Germany, 2015; pp. 235–252. [Google Scholar]
- Otmani, R.; Moussaoui, A.; Pruski, A. A new approach to indoor accessibility. Int. J. Smart Home
**2009**, 3, 1–14. [Google Scholar] - Pruski, A. A Unified Approach to Accessibility for a Person in a Wheelchair. Robot. Auton. Syst.
**2010**, 58, 1177–1184. [Google Scholar] [CrossRef] - Chin, F.; Wang, C.A. Minimum vertex distance between separable convex polygons. Inf. Process. Lett.
**1984**, 18, 41–45. [Google Scholar] [CrossRef] - Toussaint, G.T.; Bhattacharya, B.K. Optimal algorithms for computing the minimum distance between two finite planar sets. Pattern Recognit. Lett.
**1983**, 2, 79–82. [Google Scholar] [CrossRef] - Toussaint, G. Solving geometric problems with the rotating calipers. In Proceedings of 1983 Mediterranean Electrotechnical Conference, Athens, Greece, 24–26 May 1983.
- Yang, C.L.; Qi, M.; Meng, X.X.; Li, X.Q.; Wang, J.Y. A new fast algorithm for computing the distance between two disjoint convex polygons based on Voronoi diagram. J. Zhejiang Univ. Sci. A
**2006**, 7, 1522–1529. [Google Scholar] [CrossRef] - Li, Z.; Yan, H.; Ai, T.; Chen, J. Automated building generalization based on urban morphology and Gestalt theory. Int. J. Geogr. Inf. Sci.
**2004**, 18, 513–534. [Google Scholar] [CrossRef] - Akkiraju, N.; Edelsbrunner, H.; Facello, M.; Fu, P.; Mcke, E.P.; Varela, C. Alpha shapes: Definition and software. In Proceedings of the 1st International Computational Geometry Software Workshop, Minneapolis, MN, USA, 20 January 1995; pp. 63–66.
- Edelsbrunner, H.; Mücke, E.P. Three-dimensional alpha shapes. ACM Trans. Graph.
**1994**, 13, 43–72. [Google Scholar] [CrossRef] - Alt, H.; Welzl, E. Visibility graphs and obstacle-avoiding shortest paths. Z. Oper. Res.
**1988**, 32, 145–164. [Google Scholar] [CrossRef] - Asano, T.; Asano, T.; Guibas, L.; Hershberger, J.; Imai, H. Visibility of disjoint polygons. Algorithmica
**1986**, 1, 49–63. [Google Scholar] [CrossRef] - Geraerts, R. Planning short paths with clearance using explicit corridors. In Proceedings of the IEEE International Conference on Robotics and Automation, Anchorage, AK, USA, 3–7 May 2010; pp. 1997–2004.
- Ghosh, S.K.; Mount, D.M. An output-sensitive algorithm for computing visibility graphs. SIAM J. Comput.
**1991**, 20, 888–910. [Google Scholar] [CrossRef] - Hershberger, J.; Suri, S. An optimal algorithm for euclidean shortest paths in the plane. SIAM J. Comput.
**1999**, 28, 2215–2256. [Google Scholar] [CrossRef] - Kapoor, S.; Maheshwari, S.N.; Mitchell, J.S.B. An efficient algorithm for euclidean shortest paths among polygonal obstacles in the Plane. Discret. Comput. Geom.
**1997**, 18, 377–383. [Google Scholar] [CrossRef] - Overmars, M.H.; Welzl, E. New methods for computing visibility graphs. In Proceedings of the 4th Annual Symposium on Computational Geometry, Urbana-Champaign, IL, USA, 6–8 June 1988; pp. 164–171.
- Welzl, E. Constructing the visibility graph for n-line segments in O(n
^{2}) time. Inf. Process. Lett.**1985**, 20, 167–171. [Google Scholar] [CrossRef] - Smith, M.J.D.; Goodchild, M.F.; Longley, P.A. Geospatial Analysis—A Comprehensive Guide to Principles, Techniques and Software Tools, 2nd ed.; Troubador Publishing Ltd.: Leicester, UK, 2007. [Google Scholar]
- Dijkstra, E. A note on two problems in connexion with graphs. Numer. Math.
**1959**, 1, 269–271. [Google Scholar] [CrossRef] - Stevens, D.; Akram-Khan, M.; Munson, D.; Reid, E.J.; Helseth, C.; Buggy, J. The impact of architectural design upon the environmental sound and light exposure of neonates who require intensive care: An evaluation of the Boekelheide Neonatal Intensive Care Nursery. J. Perinatol.
**2007**, 27, 20–28. [Google Scholar] [CrossRef] [PubMed]

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**MDPI and ACS Style**

Liu, L.; Zlatanova, S.
An Approach for Indoor Path Computation among Obstacles that Considers User Dimension. *ISPRS Int. J. Geo-Inf.* **2015**, *4*, 2821-2841.
https://doi.org/10.3390/ijgi4042821

**AMA Style**

Liu L, Zlatanova S.
An Approach for Indoor Path Computation among Obstacles that Considers User Dimension. *ISPRS International Journal of Geo-Information*. 2015; 4(4):2821-2841.
https://doi.org/10.3390/ijgi4042821

**Chicago/Turabian Style**

Liu, Liu, and Sisi Zlatanova.
2015. "An Approach for Indoor Path Computation among Obstacles that Considers User Dimension" *ISPRS International Journal of Geo-Information* 4, no. 4: 2821-2841.
https://doi.org/10.3390/ijgi4042821