Thirteenth USA/Europe Air Traffic Management Research and Development Seminar (ATM2019)

Extraction and Interpretation of Geometrical and

Topological Properties of Urban Airspace for UAS

Operations

Jungwoo Cho, Yoonjin Yoon

Department of Civil and Environmental Engineering

KAIST (Korea Advanced Institute of Science and Technology)

Daejeon, South Korea

jjw9171@kaist.ac.kr

Abstract— With the rapid adoption of operational concepts of

Unmanned Aerial Systems (UAS), a large amount of traffic is

expected to flow into low-level airspace. However, this low-level

airspace contains the existing environment of people and

surrounding structures that are vulnerable to the risks posed by

UAS operations. In order to provide necessary separation in flight

planning services, UAS traffic management (UTM) system needs

to first identify airspace in which vehicles can operate within an

acceptable level of safety. This study highlights that what is

perceived as an empty airspace may not all be operationally

available and that airspace in highly urbanized areas has a

complex geometric form. For the maximum utilization of the finite

UTM airspace, such complex forms of airspace needs to be

examined geometrically and topologically. In this paper, we

present topography map and skeletal graph to interpret underlying

geometrical and topological features of urban airspace.

Specifically, we present topographic map to provide a quantitative

representation of vertical and horizontal dimension of available

airspace. A skeletal graph is extracted from airspace to uncover

underlying connectivity of its component parts. Both methods not

only provide a compact and informative abstraction of airspace

but also can be used to partition the entire airspace into different

levels based on geometrical and topological properties.

Keywords-UAS traffic management (UTM); airspace

availability; airspace topography map; airspace skeletonization;

airspace partitioning;

I. INTRODUCTION

With the rapid adoption of the operational concept of unmanned aerial systems (UAS), a large amount of traffic is expected to flow into low-level airspace [1-3]. Several air navigation service providers (ANSPs) have initiated the development of UAS traffic management (UTM) system to accommodate potential UAS traffic [4-9], but there still remain challenges in identifying available airspace that unmanned aircraft (UA) can fly within an acceptable level of risk.

Low-level airspace contains existing environment of people and surrounding structures that are sensitive to the risks of UAS operations. In such environment, not all free airspace is available for operational use. In many literatures, however, UTM airspace

is often regarded nearly free of obstacles, and the geospatial complexity arising from geometric distribution of surface obstacles is not fully addressed [10-19]. Our previous study was the first attempt highlighting the importance of incorporating geospatial elements in assessing airspace availability [20]. By applying two types of geofencing to account for safety clearances, a notable difference was found between free and available airspace. One of the major findings was that urban airspace has a complex geometric shape, which necessitates further analysis of underlying spatial characteristics such as horizontal and vertical connectivity.

In this study, we propose a topological airspace assessment method for extracting and interpreting the geometrical and topological properties of low-level airspace. We first introduce topographic map to provide a quantitative representation of vertical and horizontal dimension of available airspace. Second, we extract the topological skeleton of airspace to uncover underlying connectivity of airspace component parts. Furthermore, we partition the whole airspace into different levels using the features discovered by skeletal abstraction of airspace.

The rest of the paper is organized as follows. Section 2 explains the airspace availability assessment framework. Geometrical and topological assessment methodology is presented and discussed in Section 3, and the case study results for the actual urban environment are presented in Section 4. Section 5 provides conclusions and future study ideas on capacity management.

II. AIRSPACE AVAILABILITY ASSESSMENT FRAMEWORK

An important first step to extract geometric and topological properties of airspace is to identify available airspace. In this study, we adopt and improve the airspace availability assessment framework introduced in our previous work [20]. We apply two types of geofencing to incorporate vehicle operational requirements as well as provide a protection boundary to surrounding environments. Notable modifications to data processing have been made to handle large-scale obstacle data. This section describes terminologies associated with

This research is supported by Ministry of Land, Infrastructure and

Transport of Korean government under Grant 18USTR-B127901-02.

airspace availability, and details the modified data processing steps.

A. Available airspace and geofencing

1) Terminologies: Available airspace can be categorized into free and usable airspace, as defined in [20]. Free airspace is space with no surface obstacle. It is equivalent to the empty space seen by the human eye. Usable airspace is a space that a spherical containment limit of UA can fit in. It is the space after removing geofenced space from free airspace (Fig. 1). Geofence is categorized into two types - keep-in and keep-out [21-31], and usable airspace is identified by applying the two geofencing methods. Keep-out geofence is used to put an artificial fence around designated areas, so that vehicles stay out from the fenced boundary, while keep-in geofence concerns vehicle’s safety margin in relation to potential flight deviations caused by mechanical instability or navigational inaccuracy. Note that UA is not treated as a point but as a sphere containing the aircraft, which is in its essence keep-in geofence. In Fig. 1, black cells are occupied by static obstacles, gray cells are closed by keep-out geofence, and orange cells are closed as the passage is not wide enough for a UA with a specific keep-in radius.

2) Geofencing implementation: Keep-out geofence is modeled as a buffer space around static obstacles such as buildings, whereas keep-in geofence is modeled using alpha-shape method (see Section II.B). Boundary points of obstruct polygons are used as input data for geofence implementation. Since discretizing the entire airspace into uniform grids creates an unnecessary computational load, we chose to use the boundary points use to handle large-scale (i.e. citywide) obstacle data. In preprocessing, a set of vertices that define an obstacle polygon is extracted, and a buffer of size δ is generated on each vertex of the obstacle polygon to apply keep-out geofencing. For keep-in geofencing, we constructed alpha shapes of radius r on sampled boundary points. The output is a

triangulated mesh representing airspace not usable by sUAV of keep-in radius r.

B. Alpha shape method

Alpha shape is a computational geometric technique used to construct a shape of a finite point set using a spherical ball or 𝛼-ball [32-35]. In this subsection, we provide an intuitive description of alpha shape and explain how it is applied in the airspace availability assessment.

1) Intuitive description of alpha shape: Consider there is a point set 𝑆 in ℝ3 and a real number 𝛼 such that 0 ≤ 𝛼 ≤ ∞. Imagine ℝ3 as a 3D space and 𝑆 as a set of obstacle points. Suppose there is a spherical eraser of radius 𝛼 (i.e. α-ball) that removes the 3D space with condition not to touch any obstacle points in 𝑆. The remaining space after removal will consist of curved boundaries enclosing 𝑆. The alpha shape of 𝑆 is then obtained by replacing the curves with straight lines or planes. For mathematical definition of alpha shape, please refer to [32-35].

2) Application of alpha shape to airspace availability assessment: The key idea of applying the alpha shape method to the airspace availability assessment is to relate α-ball with the vehicle’s spherical containment limit [20]. Note once again that a vehicle is not treated as a point but assumed as a sphere containing it, as shown in Fig. 2a. Eliminating where α-ball cannot fit in will result in airspace usable by vehicle with spherical containment limit of a specific size. By controlling the size of 𝛼-ball , one can observe that some portion of free airspace cannot accommodate vehicles with a specific keep-in radius, as depicted in Fig. 2b.

III. GEOMETRICAL AND TOPOLOGICAL ASSESSMENT

METHODOLOGY

To enhance the understanding of low-level airspace composed of various horizontally- and vertically- connected components, we introduce methodologies that generate a topography map and extract a skeletal graph from airspace availability information.

Figure 1. Illustration of keep-out and keep-in geofence effect to airspace

availability

Figure 3. Sample output of airspace availability assessment: (a) usable airspace layout and (b) percentage volume of airspace at altitude h (black

cells are those occupied by static obstacle; gray cells are closed by keep-

out geofence; red cells are closed by alpha shape application)

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Figure 2. Illustration of alpha shape method (a) α-ball (red) and obstacle

points (black); (b) change in available airspace depending on the radius

of α-ball (Fig.2a and 2b are adopted from [27] and [15], respectively)

A. Airspace topography map

Airspace topography map shows a 2D projection of the lowest navigable altitude at specified latitude and longitude. A sample airspace topography map of a built-up area in San Francisco in Fig. 4, where the lowest navigable altitude per unit cell is visualized in grayscale at resolution of 3m by 3m. The darker the cell is, the higher the airspace is occupied by geographic obstacles and geofences. A contour line connecting spaces of equal minimum navigable altitude is also be overlaid in the map, as shown in Fig. 4.

B. Airspace skeletonization

To capture how airspace segments are connected to each other to form the whole, we extract a skeletal graph of usable airspace based on a topological concept called Reeb graph. Reeb graph is a shape descriptor widely used in computational geometry and computer graphics applications such as 3D shape encoding, segmentation, and comparison [36-41]. In this study, we derive an approximated Reeb graph 𝐺 ∈ ℝ1 from usable airspace 𝐴 ∈ ℝ3 using Mapper algorithm [42,43]. Topological features of airspace is described by connectivity of the graph, providing a structural overview of an entire airspace. The following subsections introduce necessary topological notions of Reeb graph and implementation details of Mapper algorithm.

1) Formal definition of Reeb graph: Let 𝑓: 𝑋 → ℝ be a continuous function defined on a domain 𝑋. For each scalar value 𝑎 ∈ ℝ , the level set 𝑓−1(𝑎) = {𝑥 ∈ 𝑋|𝑓(𝑥) = 𝑎} may have multiple connected components. Reeb graph of 𝑓, denoted by ℛ𝑓(𝑋), is obtained by reducing each connected component

in a level set to a single point. ℛ𝑓(𝑋) is the image of a mapping

function Φ:𝑋 → ℛ𝑓(𝑋) where Φ(𝑥) = Φ(𝑦) if and only if 𝑥

and 𝑦 belong to the same connected component in a level set of 𝑓 (Fig. 5). As the value 𝑎 increases, connected components in the level set 𝑓−1(𝑎) appear, disappear, split and merge, and Reeb graph of 𝑓 tracks such changes.

2) Approximated Reeb graph computation using Mapper algorithm: The best known algorithms of computing exact Reeb graph from a 3D mesh structure require expensive temporary data structures [41]. In this study, we compute approximated Reeb graph from a point set [42-44]. Specifically, we adopt a clustering-based graph construction algorithm called Mapper to compute an approximation of Reeb graph. Mapper algorithm is partial clustering of overlapping data subsets [43]. The basic idea is to divide the original data into overlapping subsets, apply a clustering algorithm to each subset, and interpret the connectivity among those partial clusters. Any partial clusters are ‘connected’ if there exist any non-empty intersections. By representing each cluster as a node and each connection as an edge, a graph-like structure is derived as shown in Fig.6.

To apply Mapper algorithm to extract a skeletal graph, we first sampled points of usable airspace and divide the sampled points into overlapping subsets based on altitude. Suppose an altitude function ℎ: 𝑋 → 𝑍 for a point set 𝑋 ∈ ℝ3 and 𝑧 coordinate 𝑍 ∈ ℝ . 𝑍 is divided into a set of overlapping intervals 𝒰 = {𝑈𝑘} , so that 𝑋 is decomposed into subsets ℎ−1(𝒰) = {ℎ−1(𝑈𝑘)}. For each subset ℎ−1(𝑈𝑘), we apply the density-based clustering algorithm, DBSCAN, to divide ℎ−1(𝒰) into a set of disjoint partial clusters. Note that each partial cluster is a path-connected space, meaning that for any two points in the partial cluster there exists a path between them. Each partial cluster is represented as a node, and the nodes are connected by an edge if any clusters have points in common. In this study, the length of interval is 10 meters, and the overlapping portion between successive intervals is 50%.

C. Airspace partitioning

Abstraction of airspace into a compressed graph-like structure allows us to infer, analyze, and interpret spatial characteristics such as connectivity. This low-dimensional skeletal representation, instead of bulks of grids, point clouds or triangulated surfaces, is simple and practical not only in visualizing the connectivity and but also in partitioning airspace into multiple segments based on geometrical and topological properties.

Figure 4. Airspace topography map of a built-up area in San Francisco.

(The darker the cell is, the higher the airspace is occupied by geographic

obstacles and geofences. The zoomed-in plot shows that four disjoint airspace segments, which are usable from 60m, are merged to a larger

segment at altitude of 80m.)

Figure 5. Illustration of Reeb graph 𝓡𝒇(𝑿) (adopted from [41])

Figure 6. Skeletal graph extraction using Mapper algorithm

Figure 6. Skeletal graph extraction using Mapper algorithm

1) Skeletal graph decomposition: Given a skeletal graph 𝐺, let 𝑣 represent a node of 𝐺 and ℎ(𝑣) indicate the altitude of 𝑣. Note that each node in a skeletal graph represents a path-connected space, where a path exists between any two points inside the space. We classify each node into leaf, middle, and

branch node, where leaf nodes 𝒗𝒍 is of degree 1, middle nodes

𝒗𝑴 of degree 2, and branch nodes 𝒗𝑩 of degree 3. As altitude increases, leaf nodes gradually develop and eventually merge into others at branch nodes. The evolution of leaf nodes is summarized by information at which altitude it is first generated (birth-altitude) and is finally merged to another (death-altitude), as shown in Fig. 7a.

Nodes of 𝐺 can then clustered into a group of partitions, where each stem is defined by either leaf nodes or branch nodes.

Suppose nodes 𝑣𝑖 , 𝑣𝑗 ∈ 𝒗𝒍 ∪ 𝒗𝑩 such that ℎ(𝑣𝑖) < ℎ(𝑣𝑗) . We

define that 𝑣𝑖 and 𝑣𝑗 form a partition 𝑆(𝑣𝑖 , 𝑣𝑗) if the two nodes

are connected via 𝑣𝑝1 , … , 𝑣𝑝𝑛 ∈ 𝒗𝑴 such that ℎ(𝑣𝑖) < ℎ(𝑣𝑝1) <

⋯ < ℎ(𝑣𝑝𝑛) < ℎ(𝑣𝑗) for 𝑛 ≥ 0. For 𝑆(𝑣𝑖 , 𝑣𝑗), we name ℎ(𝑣𝑖)

as birth-altitude and ℎ(𝑣𝑗) as death-altitude. Then, each

partition is summarized as a point in birth-death diagram. For example, a skeletal graph 𝐺 in Fig. 7a can be decomposed into a set of partitions: 𝑆(𝑣1, 𝑣4), 𝑆(𝑣3, 𝑣4), 𝑆(𝑣2, 𝑣5), 𝑆(𝑣4, 𝑣5), and 𝑆(𝑣5, 𝑣6). The birth- and death- altitude of 𝑆(𝑣1, 𝑣4) are ℎ(𝑣1) and ℎ(𝑣4), respectively and shown accordingly in birth-death diagram. Note that the notion and objective of birth-death diagram in this study differ from those of persistence diagram.

2) Airspace parititioning logic: In this paper, we categorize partitions into four levels based on two factors (see Fig. 7b). We first determine if a partition has a leaf node. Then, we compare airspace volume per altitude for each partition. Airspace partitions are color-coded as shown in Fig. 7b. Characteristics and potential use of the four-level airspace are described below.

a) Level 1: This level of airspace is characterized by

relatively low volume compared to Level 2 airspace and

proximity to ground level. Operational use may be limited to

ascent/descent operations for a small number of traffic.

b) Level 2: This level of airspace is characterized by

relatively high volume compared to Level 1 airspace and

proximity to ground level. It may also be utilized soely for

ascent/descent operations and may require demand/capacity

management at intrasegment level.

c) Level 3: This level of airspace is characterized by

relatively low volume compared to Level 4 airspace and by

intermediacy in terms of its situated altitude. It may be utilized

to transit traffic to Level 4 airspace from Level 1 or 2 airspace.

Demand/capacity management may be required at both

intersegment and intrasegment level.

d) Level 4: This level of airspace is characterized by

relatively high volume compared to Level 3 airspace and

remoteness from ground level. It may be used mainly for

cruising between distant locations (e.g. air highway).

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b c

Figure 7. Skeletal graph decomposition and summarization: (a) birth-

death diagram; (b) partitioning logic; (c) conceptual diagram

If partition has a leaf node of G

Y N

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Low High

Volumeper altitude

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Level 1 Level 2 Level 3 Level 4

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Figure 8. San Francisco case study area: a) building distribution; b) sample airspace topography map

IV. CASE STUDY RESULTS AND DISCUSSIONS

We present our case study results of San Francisco airspace in this section. Note that we divided the city area into multiple parts and selected two areas of distinct geometries – D31 and D35 (see Fig. 8). Using the building distribution data, we generated airspace topography maps and skeletal graphs with respect to specific geofence parameter combinations. Altitude range is up to 150 meters above ground level.

A. Airspace topography map of case study areas

Fig. 9 shows airspace topography map of D31 and D35 with respect to geofence combinations of (𝛿, 𝑟) = (30,10) and (30,30). In the map, the lowest altitude that sUAV can fly is represented in grayscale at resolution of 3m by 3m unit cell. The darker the map is, the higher the flight altitude should be to avoid geographic obstacles and geofences. One can easily see that D35 airspace is filled with darker areas than D31 in both geofence

combinations. More importantly, there is a high variability of brightness (i.e. minimum navigable altitude) in D35. It implies that there exist peaks and ridges in D35 that vehicles need to either fly over or detour in order to pass such regions. Also, there are U-shaped valleys or conical-shaped holes of varying depths, where vehicles need to climb up or down. On the contrary, D31 airspace is barely influenced by static obstacle or geofencing, resulting in a low variability of brightness in the map. It is inferred from the map that D31 airspace is mostly open at above around 40 meters.

When airspace is more conservatively configured with a larger keep-in parameter, the overall topography changes as depicted in Fig. 9. By the effect of increased 𝛼-ball radius from 10 to 30 meters in D35 airspace, some portion of aerial valleys and holes that are not wide enough to fit in a vehicle with a larger spherical containment limit is blocked. Ridges are formed between adjacent peaks, blocking some airways that were originally usable when 𝑟=10. On the other hand, D31 airspace is only marginally affected by the increased keep-in parameter due to its relatively dispersed distribution of geographic obstacles.

B. Skeletal graphs and Birth-Death diagram

Now, we present skeletal graphs and birth-death diagrams of D31 and D35 airspace with respect to geofence combination (𝛿, 𝑟) = (30,30) in Fig.10 and Fig.11, respectively. The skeletal graphs visualize the vertical connectivity of disjoint airspace segments, while the birth-death diagrams summarize the structure of skeletal graphs. Each point in a birth-death diagram is positioned based on the birth- and death- altitude of a stem, where each point is color-coded according to airspace level. For example, (20,30) represents a path-connected airspace segment that has its minimum navigable altitude of 20 meters and merges to other segment at 30 meter altitude.

The birth-death diagram of D31 shows that its skeletal graph has fewer stems compared to D35, where all nodes are merged to a singular node at altitude of 35 meters. In contrast, that of

Figure 9. Airspace topography map of D31 and D35 with respect to two

geofence combinations (𝛿, 𝑟) = (30,10) and (30,30)

( , )=( , )

( , )=( , )

Figure 10. Illustration of D31 airspace: (a) skeletal graph, (b) birth-death

diagram, (c) color-coded airspace partitions, and (d) tree graph of

airspace partitions

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Altitude (

m)

(Birth-death diagram)(Skeletal graph)

(3D view of airspace)

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m)

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Figure 11. Illustration of D35 airspace(a) skeletal graph, (b) birth-death

diagram, (c) color-coded airspace partitions, and (d) tree graph of

airspace partitions

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D35 shows this airspace is characterized by complex merging of disjoint segments at various altitudes.

C. Airspace partitioning results

Fig. 10c and Fig. 11c show the results of airspace partitioning in D31 and D35 airspace. Each level of airspace is color-coded according to the partitioning logic described in Section III.C. Level 1 airspace (in yellow) indicates airspace segment of small volume that is located near ground level. The use of this airspace will be largely affected by operational environmental factors such as population density and surrounding structures. Its shape is usually a shallow hole, and its utility may be optimized for ascent and descent operations of a small number of traffic. On the contrary, Level 4 airspace (in blue) is by far the largest airspace segment that other segments finally merge into. It is located above 35 meters for D31 and 120 meters for D35. This airspace is mostly obstacle-free and is suitable for placing ‘air highway’ for long-distance trips.

Level 2 airspace (in orange) has relatively large volume compared to Level 1. Its shape is usually a deep and wide valley. It may best be utilized for ascent/descent operations and require demand/capacity management at intra-segment level. Level 3 partition (in green) is an intermediate airspace located in between Level 1 or 2 and Level 4, and thus traffic transition between Level 1 or 2 and Level 4 will frequently occur. Thus, capacity management might be needed at both intra- and inter- segment level. In D31 airspace, there are three disjoint Level 3 airspace segments at altitudes from 20 to 30 meters. In D35 airspace, there are sixteen Level 3 airspace clusters located at a wide range of altitudes from 35 to 115 meters, implying that D35 airspace has a highly complex geometric structure characterized by dynamic merging of airspace segments.

V. CONCLUSION AND FURTHER STUDIES

This paper introduced methodologies to uncover geometrical and topological properties of urban airspace. The proposed approaches of topography map and skeletal graph capture not only geometrical properties of urban airspace landscape but also topological properties of how airspace segments are connected to each other. Both approaches provide a graphical visualization of available airspace, can be readily modified when there is any change in operational environment, and are applicable to any region of any size. Moreover, skeletal representation, instead of bulks of grids, point clouds, or tessellated surfaces, yields a compact and informative abstraction of the entire airspace, and one can utilize this skeletal structure for dividing airspace into smaller partitions. If any airspace management strategies such as scheduling, slot allocation or altitude assignment are to be adopted in UTM environment, subdivision of airspace into smaller manageable partitions may add flexibility in managing airspace. Although only the surface obstruction is considered in this study, the resulting topographic map and skeletal structure can be used as a baseline and can always be updated in accordance with any new flight restrictions. This research is also in line with U-Space’s concept of dividing Class G airspace into three types, proposed by SESAR [9], and ICAO’s mention on potential need for a new airspace classification to accommodate UAS operations [3].

In future research, we will verify our conjecture on the expected traffic phases at each level of airspace mentioned in III. C. 2), by generating traffic for all possible OD pairs and analyzing traffic patterns at each partition. We also plan to evaluate the capacity of each partition, by applying related concepts introduced in pioneering researches [45-48]. Dynamic information related to weather and wind uncertainty is also a significant factor affecting the airspace availability [49-52] and will be incorporated by updating the tree structure and corresponding airspace information in future studies.

ACKNOWLEDGMENT

We would like to show our gratitude to Prof. Wonjong Rhee for sharing his insights and pearls of wisdom that greatly improved this study. We also thank Yuyol Shin for his insightful suggestions on application of Mapper algorithm.

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BIOGRAPHIES

Jungwoo Cho is a Ph.D. candidate in civil and environmental engineering at KAIST. His research concerns geometrical and topological approaches to data analysis with a current focus on airspace design and management for UTM in urban environments.

Dr. Yoonjin Yoon is an Associate Professor of Transportation Engineering in the Department of Civil and Environmental Engineering of KAIST, leading one of Korean UTM task groups. She received her B.S. in mathematics from Seoul National University, dual M.S. in computer science and management science and engineering from Stanford University, and Ph.D. in civil and environmental engineering from University of California, Berkeley.