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ICRAT 2020
1
Obstacle Clustering and Path Optimization for Drone
Routing
Ang Li, Mark Hansen
Department of Civil and Environmental Engineering
University of California, Berkeley
Berkeley, CA, USA
angli@berkeley.edu, mhansen@ce.berkeley.edu
Abstract—To enable safe and efficient Unmanned Aircraft
Systems (UAS) operations at low altitudes, it is necessary to
conduct airspace management and operations for UAS traffic.
This study focuses on deterministic clustering-based drone
routing, with specific emphasis on the trade-off between
horizontal and vertical travel costs. The routing problem is
simplified to a 2D problem that we solve at several altitude
candidates. Altitude candidates were generated based on clustered
static obstacles in low urban airspace. Fast-Marching algorithm is
performed to generate the shortest path at each altitude candidate.
The optimal altitude is determined by weighing the vertical cost
for ascent and descent over the horizontal cruising cost at certain
altitude. Experiments are conducted to choose proper number of
clusters and weight given to building height in the clustering
procedure, and different shortest path algorithms are compared.
Larger scale of Unmanned Aerial Vehicles (UAV) missions are
simulated, based on which we analyze the relationship between
optimal travel altitude and shortest cruise path, and estimate the
UAV cost function.
Keywords-UAV path planning; UAV cost function; Fast-
Marching; A star (JPS);
I. INTRODUCTION
UAV have received increasing attention over the last decade,
because of their immense potential to benefit commercial and
industrial activities [1]. With large potential demand, it
becomes necessary to manage UAV traffic in urban airspace.
Organizations including National Aeronautics and Space
Administration (NASA) [2] and Netherlands Aerospace Centre
(NLR) [3], have undertaken the task of developing traffic
management methods for UAV operations. There are also
emerging UAS Traffic Management projects from Europe,
Singapore and Korea.
The aim of UAV path planning is to identify efficient, safe
flight trajectories in a timely manner, so that UAV can
accomplish their missions and avoid threats. Performance of
multiple UAV path planning algorithms were compared in
various environments. These include Dijkstra’s algorithm,
Bellman Ford’s algorithm, Floyd-Warshall’s algorithm and the
A Star algorithm in [4], and A star is found to perform better
than others. A Star (JPS) is an improved path planning
algorithm based on A Star [5,6,7]. Reference [8] finds that A
Star (JPS) has better performance than Rectangular Symmetry
Reduction (RSR). Fast-Marching methods have been found to
yield consistent, accurate and highly efficient algorithms in
optimal path planning around obstacles [9]. Reference [10] uses
Fast-Marching algorithms to navigate a small quadrotor on an
optimal collision avoidance path with a helicopter. Reference
[11] presents both an offline optimal path planning algorithm
based on A Star without considering the computation cost, and
real-time suboptimal path planning algorithm based on genetic
algorithm and potential fields technology. A spline-based path
planning scheme that generates feasible flight routes for an
UAV is developed in [12], which allows quick computation
using a decomposition strategy.
An understanding of the path-related costs of UAV
operation is needed for path planning. Horizontal and vertical
cost ratio was used in [13] for drone routing. Horizontal path
length cost and height cost were specified in [14] to perform 3D
UAV trajectory planning. Reference [15] used required number
of turns to measure cost function. Dublin path length was used
in [16] as travel cost.
The primary purpose of this paper is to investigate the
combined cost of horizontal and vertical distance when routing
drones in low-altitude airspace in the presence of tall buildings.
Using San Francisco as a case study, we investigate the trade-
offs between routing a single drone at lower altitude with the
resulting need to avoid many obstacles and using a higher
altitude, which allows more direct horizontal paths but entails
more vertical flight. The “sweet spot” in this trade-off depends
on the relative cost of horizontal and vertical flight, and we
study this relationship parametrically.
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As part of this investigation, we consider different
approaches to represent tall building obstacles for purposes of
path planning, and also compare the performance of Fast-
Marching and A Star routing algorithms. We assume that
airspace structures will incorporate “no fly zones” that keep
UAVs from tall buildings, but that to avoid undue complexity,
these zones will be defined based on a relatively small number
of building groups rather than a multiplicity of zones, each
corresponding to an individual building. We therefore propose
a methodology for identifying these building groups based on
clustering, and consider how the number of clusters and the
weight attached to building altitude in the clustering algorithm
affects route efficiency and the computation time. The choice
of routing algorithm is also critical in our analysis. In this
regard, we show that Fast-Marching dominates A Star, even
when a faster variant of the latter is used.
The contribution of this paper is that we apply obstacle
clustering to efficiently reduce the obstacle complexity for
routing. We applied Fast-Marching to horizontal drone routing
and combined with obstacle clustering, which quickly returns
better routes than many other shortest path algorithms. Finally,
this paper proposes a UAV path cost function that predicts the
cost of the least-cost path as a function of direct-line distance
and the relative cost of horizontal and vertical travel.
II. DATA
This paper uses the financial district in San Francisco (SF)
as study area. Only buildings are considered as static obstacles
in urban airspace. San Francisco building footprints data from
DataSF was used. The data contains San Francisco building
footprint features, including roof boundary and building height.
The research was performed using the projected coordinate
system of EPSG 32610, WGS 84 / UTM zone 10N.
III. ASSUMPTIONS
The deterministic clustering-based single drone routing
focuses on the trade-off between horizonal and vertical costs.
For the purposes of our analysis make several simplifying
assumptions. Since wind or any other features that cause
uncertainties are not considered in this stage, cost is insensitive
to where along its path the drone ascends and descends. We
assume the drone flies at a single altitude, after vertically
ascending at the origin and prior to descending at the
destination, and correspondingly that the cost of the route
depends on the vertical distance and the horizontal distance of
the route. Cost of turning is not considered in this research.
Though we didn’t explicitly consider random deviations
between the actual path of the drone and its nominal path, in
order to ensure safety, a keep-out geofence, the safety distance
that drones are required to keep away from buildings, is
considered when we generate aggregated obstacles by
clustering. We simply add the keep-out geofence distance by
expanding the actual building boundary outward a certain
distance. All the following research is using 10-meter keep-out
geofence distance. In addition, we don’t consider any
geographical ground level in the current stage. Above Ground
Level (AGL) or Median Sea Level (MSL) can be added by
performing this research in corresponding projected coordinate
system.
IV. METHODOLOGY
The routing approach can be simplified from 3D path
planning to 2D by routing at several attitudes with assumptions
above. The optimal travel altitude will be determined by
weighing the horizontal travel cost over vertical cost for ascent
and descent. A set of altitude candidates is needed to compare
the vertical and horizontal cost since exhaustive search over all
altitudes is computational expensive. Obstacle complexity will
influence the computation time of shortest path algorithm. In
order to generate the most appropriate altitude candidates and
reduce complexity of the obstacle field, employ a clustering
approach to summarize the height and location of the numerous
static obstacles. Based on the generated altitude candidates set,
horizontal shortest paths that avoid obstacles will then be
generated for each altitude candidate. We compare the vertical
and horizontal costs to decide the optimal travel altitude and 2D
cruise path at optimal altitude.
A. Static Obstacles Clustering for Altitude Candidates
A set of altitude candidates was generated by clustering
obstacles. The K-means clustering algorithm is applied to
perform clustering over all buildings in SF financial district.
We first generate the minimum bounding rectangle containing
the footprint of each building, since the K-means algorithm
requires the same feature dimensions for every building
observation. Each building is summarized by nine features: X
and Y coordinates of four minimum bounding rectangle
building vertices plus building height. While all variables are in
units of meters, height is unique because it varies far less than
the X and Y coordinates. For this reason, building heights are
rescaled by different factors. We presented the results of 20, 30,
and 40 clusters, with scale of 10, 20, 30, 40, and 50 times for
building height in this paper.
After clustering, the convex hull of all minimum bounding
rectangle vertices of buildings in the same cluster forms an
aggregated obstacle whose altitude is the maximum height of
buildings in the cluster. The path planning method in later
section is performed with the aggregated obstacles. Two trade-
offs of obstacle clustering, number of clusters and building
height rescaling factor, are shown in Fig. 1. On the one hand,
when the number of clusters is small, more airspace will be
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made unavailable as a result of being included in the polygons
of aggregated obstacles. On the other hand, when the number
of clusters increases the complexity of the obstacle field
increases and it will be more computationally expensive to
generate shortest paths. With larger rescaling factor, buildings
with similar height will be more likely to be clustered in the
same aggregated obstacle, instead of buildings that are located
closer. Aggregated obstacles are most dispersed but of more
uniform height in the case with more scaling. We perform
sensitivity analysis for these two factors in Section V.
Fig. 2 shows an example of aggregated obstacles at altitude
146.07m in 20-cluster case. Red dots are the boundary points
of aggregated obstacles after clustering, and the aggregated
obstacles area is filled with green. Dots with the same color
within the boundary, as well as the red boundary points, are the
minimum bounding rectangles vertices of buildings belong to
the same cluster. Grey points represent all the other buildings
lower than the current altitude in SF financial district.
B. Optimal Horizontal Travel Route
The height of each aggregated obstacle, which is the
maximum actual building height within each cluster, forms the
altitude candidates set. The optimal 2D cruise path is generated
at each altitude candidate in this subsection.
The Fast-Marching (FM) algorithm is used to generate the
shortest cruise path. Compared to the traditional Dijkstra
algorithm or A star algorithm, FM replaces the graph update by
a local resolution of gradient descent, instead of only
considering standard 8 directions of neighbors, which
significantly reduces the grid bias. The computation complexity
of FM is , where N is the total number of grid points,
which is the number of visited points during the computation in
practice. FM method has less computation complexity
compared to A star whose complexity depends on heuristic, and
FM yields a better approximation of the true shortest path.
In our research, the grid size is set to be 1m and the step size
is set to be 5m. Fig. 3 shows an example of shortest path results
for the case with 20 clusters and 50x building height rescaling.
The dark blue areas in the plots represent all aggregated
obstacles at given altitude. The red and green dots represent the
origin and destination. As the altitude increases, some obstacles
disappear, leaving more available airspace for drone to travel.
The shortest cruise path decreases accordingly.
C. Determine Optimal Travel Altitude
Shortest travel paths at different candidate altitudes are
generated as described above. To decide the optimal travel
altitude, we are interested in how the length of the shortest path
changes at different altitudes. Fig. 4 plots the pattern of shortest
path length at different altitude candidates of the same OD as
in Fig. 3 with 20 clusters and 50x rescaling of building height.
The red dots represent the data at each altitude candidate, and
Figure 1. Trade-offs of different number of clusters and building height rescaling factors
Figure 2. Sky view of aggregated obstacles in SF financial district
(20 clusters)
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we superimpose step lines. Based on the shortest path length
profile in this case, we can determine the optimal route altitude
for a given ratio of vertical cost to horizontal cost. (In this paper
we represent vertical unit cost as the average of climbing and
descending unit cost.) This ratio determines the slope of the
black iso-cost line in Fig. 4. The point where the lowest iso-cost
line touches the red plot will give the optimal travel altitude. A
given altitude will be optimal for a range of cost ratios.
V. EXPERIMENTS
This section performs sensitivity analysis for the number of
clusters and the building height rescaling factor, and compares
different shortest path algorithms. 10 random OD pairs were
generated for analysis in this section.
Figure 4. Shortest horizontal path lengths at different altitudes
Figure 3. An example of path planning at different altitudes for the same OD
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A. OD Sampling
A random sample of OD’s is generated within the red circle
containing all obstacles in San Francisco area shown in Fig. 5.
The center of the red circle is the middle point of maximum and
minimum X and Y coordinates of all obstacles. The radius is
1100 meters, which just contains all obstacles in study area.
Points are randomly sampled within the red circle. We assume
that only the points not located in the minimum bounding
rectangle of buildings can be used as O’s and D’s. Red and
green dots are the sampled origins and destinations. The
obstacles area (filled with blue) here uses the minimum
bounding rectangle of buildings without clustering. As
mentioned in the last section, there exists O’s or D’s located in
the clustered aggregated obstacle area but not in the actual
obstacles, because of wasted airspace by clustering. These
points located in the wasted airspace can be used as O’s and
D’s, but the shortest path is only feasible at altitude higher than
both altitudes of aggregated obstacles at origin and destination.
We only consider OD’s with Euclidean Distance longer than
1000 meters to reduce the possibility that the travel paths of OD
samples are obstacle free.
B. Sensitivity Analysis of the Number of Clusters
Sensitivity analysis of 20, 30 and 40 clusters, assuming 30x
height rescaling, for 10 OD’s is performed in this subsection in
order to determine the proper number of clusters. An example
of altitude vs. horizontal path length plot is shown in Fig. 6. The
shortest cruise path length is always shorter with more clusters,
since less available airspace is wasted. More clusters require
more computation time (see Table Ⅰ). The largest distance gap
(refer to Fig. 6) between 20 and 40 clusters cases is calculated
for all 10 OD samples. The maximum percentage savings of
shortest cruise path length are calculated as the largest distance
gap divided by corresponding 20-cluster shortest path length.
The average maximum path length savings is only about 10%
using 40 clusters compared to 20 clusters. Therefore, we pick
20 clusters for later study considering both path length savings
and computational convenience.
TABLE I. COMPUTATION TIME WITH DIFFERENT CLUSTER NUMBERS
# clusters
20
30
40
Computation time for 10 OD’s /s
386.2
549.7
950.2
The influence of different numbers of clusters on path cost
is also analyzed. We assume constant unit costs for vertical and
horizontal travel. Given vertical and horizontal cost ratio (V/H),
the adjusted cost, in horizontal distance units can be calculated
by the following equations:
Where is the shortest horizontal path length at altitude ,
and is the optimal altitude that minimizes total cost. We
use simple enumeration among all the altitude candidates to find
. Cost ratios of 1, 2, 5, and 10 are used to compare cost
results since vertical cost is higher than horizontal travel cost in
most cases [17]. As shown in Fig. 7, total travel cost has lower
mean with more clusters for all four cost ratios. The optimal
travel altitude is lower with more clusters, since less airspace is
wasted and horizontal path is shorter. The cost difference with
different numbers of clusters increases as the V/H cost ratio
increases. When V/H cost ratio is small (e.g. V/H=1 in Fig. 7),
cost is quite insensitive to the number of clusters, since the
optimal paths are higher and therefore avoid most obstacles. The
sensitivity to the number of clusters becomes greater when high
Figure 5. OD Sampling
Figure 6. Horizontal shortest path lengths at different altitudes with
30X scaling of building height
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costs of vertical movement push optimal paths toward lower
altitudes, where obstacles matter more.
C. Sensitivity Analysis of Building Height Rescaling Factor
While all features in clustering are in units of meters, height
is unique because it varies far less than the X and Y coordinates.
The height of a cluster is the height of the tallest building in the
cluster. For this reason, building heights are rescaled by
different factors. The impact of changing the building height
rescaling factor from 10 to 50 times is analyzed in this
subsection. In Fig. 8, the total travel cost is larger with higher
V/H cost ratio. The cost is not monotonically increasing or
decreasing as building height rescaling factor changes when
V/H cost ratio is small (V/H=1 or 2). This can be explained by
the trade-off of rescaling factor described in Fig. 1. The more
we scale, the more likely available airspace between buildings
with similar height is regarded as obstacles. However, if we
don’t scale enough, more airspace is wasted because of building
height difference. Drones will be able to travel at lower altitude
with higher rescaling factor, since a higher rescaling factor
saves more airspace associated with the height difference of
buildings, which results in lower cost with higher rescaling
factor if the cost ratio is large (V/H=5 or 10).
In order to determine the proper rescaling factor, we plot the
total volume of obstacles in study airspace under different cases
in Fig. 9. The total volume of obstacles decreases with more
clusters, since less available airspace will be counted as
obstacles. The total volume of obstacles in the airspace
decreases at first as rescaling factor increases, then stays almost
stable, after the 30x rescaling factor. Fig. 8 suggests that a
higher rescaling factor (e.g. 50) yields a lower cost when V/H
is high, without any significant cost penalty when V/H is small.
The computation time does not change significantly with
different building height rescaling factors.
D. Comparison with A star (JPS)
The A Star Jump Point Search (JPS) algorithm makes
pathfinding on a rectangular grid more efficient, especially in
open spaces. It performs very well on quickly generating a path.
This algorithm is compared with Fast-Marching method.
The cumulative frequency diagram of cost difference
between A star (JPS) and Fast-Marching algorithms with
different cost ratios is presented in Fig. 10. Cost difference of
two algorithms is distributed in a larger range with higher cost
ratio. Given different vertical and horizontal cost ratios, cost
using A star (JPS) is always larger since Fast-Marching gives a
near-optimal shortest path and A star (JPS) does not necessarily
do so. The travel cost difference between two algorithms
amplifies with larger vertical and horizontal cost ratio. In
addition, by comparing computation time in Table Ⅲ, we see
Fast-Marching has better performance. For this reason, we
subsequently used Fast Marching.
Figure 7. Sensitivity analysis of different numbers of clusters
Figure 8. Building height rescaling factors sensitivity analysis
Figure 9. Total volume of obstacles in different scenarios
Figure 10. Cumulative frequency diagram of cost difference percentage
( (A star (JPS) cost - Fast-Marching cost)/ Fast-Marching cost)
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TABLE II. COMPUTATION TIME OF DIFFERENT ALGORITHMS
Shortest path algorithm
A star (JPS)
Fast-Marching
Computation time for 10 OD’s /s
585.07
386.2
VI. LARGER SAMPLE ANALYSIS
More OD pairs are simulated in this section to analyze the
relationship between shortest cruise path length and travel
altitude, and trade-off between horizontal and vertical cost. 200
OD pairs are generated using the same sampling strategy in
Section Ⅴ. Results in this section are based on 20 clusters and
30x rescaling factor case.
A. Analysis of Path Length and Altitude
The ratio of shortest cruise path length and horizontal
Euclidean Distance at different altitudes is presented in Fig. 11.
The median of shortest path length and Euclidean Distance ratio
decreases with altitude, since less obstacles must be avoided.
From altitude 258.49m to 23.26m, the median distance ratio
increases from 1 to around 1.25, and the 75th percentile
increases from 1.05 to around 1.35. At the highest altitude of
20 clusters, 258.49m, more than 75% of the paths are of
Euclidean Distance, while for the balance the path must be
adjusted to avoid the single obstacle cluster that has this
maximum altitude.
The relationship between additional shortest cruise path
length compared to Euclidean Distance and Euclidean Distance,
at altitudes 258.49m, 109.71m, and 23.26m are presented in Fig.
12. At very high altitude, 258.49m, most of shortest cruise paths
equal to Euclidean Distance, as drones fly direct Euclidean
Distance for most OD’s. The shortest cruise path length varies
much more at the lower bound altitude 23.26m. At the
intermediate altitude of 109.71m, the shortest cruise path length
varies more with longer distance between OD, because at
shorter distances it is more likely that the shortest cruise path is
obstacle free.
B. Cost Function Estimation
Based on the analysis of the relationship between cost and
other features, we generate the UAV path cost function. This
function predicts the cost of the least cost path in horizontal
distance units, taking into account both the vertical and
horizontal cost. We propose the following cost function
specification:
where ED is the Euclidean Distance between OD, and are
the coefficients to be estimated. The intuition for this functional
form is that if V/H=0, the optimal solution is to climb to an
obstacle-free altitude and fly the Euclidean Distance. However,
as this ratio increases, the optimal altitude will decrease,
resulting in more circuitous paths as well as a larger vertical cost
component.
Assigning cost ratios from 1 to 20 with increment 0.5, cost
function is estimated based on 200 OD samples using linear
regression. As before the minimum cost for a given OD is found
by simple enumeration of all the altitude candidates. The
estimated result is:
The R-square is 0.7, indicating the cost function is a good fit.
The beta coefficient in the cost function is 0.617 from estimation.
It is intuitive that this coefficient should be less than 1, since a
higher cost ratio reduces the optimal altitude.
Predicted cost and actual cost are compared in Fig. 13. The
blue scatter points show obvious quasi-linear patterns, which
correspond to the results for different OD pairs, and the cost
function captures the overall linear trend very well. Systematic
differences between the OD pairs are also evident. The different
curvatures of the OD-specific quasi-linear patterns show that
least-cost paths for different OD’s have different sensitivities to
the V/H value; the results in equation (4) thus reflect the average
of this sensitivity across the 200 OD’s. Further analysis is
expected to yield a cost model that is more sensitive to
differences between OD pairs.
VII. CONCLUSIONS
This paper finds that a clustering-based method can
efficiently summarize the trade-off between low altitude routes
that must avoid many buildings and high-altitude routes that
involve larger vertical cost. In the case of San Francisco, we
Figure 12. Horizontal shortest path length difference at three altitudes
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represented 931 buildings with 20-40 clusters. Each of these
clusters has an altitude defined by the tallest building it contains
and thus defines a candidate altitude for drone routing. Thus we
can capture the essential trade-off with a small number of
altitude candidates. For example, we find that the median ratio
of horizontal path length to Euclidean Distance decreases from
1.25 to 1.0 if the drone climbs from about 30 meters to 250
meters. For a given ratio of vertical to horizontal cost, one of
these candidates yields the lowest total path cost. The trade-off
can be succinctly summarized with a cost function that gives the
lowest total cost (vertical plus horizontal) for a route as a
function of the Euclidean Distance between the origin and
destination and the value V/H, which, despite having a very
simple form, has very good predictive performance.
Future work should move along several lines. First, the cost
function should be improved by considering other features of the
OD pair aside from Euclidean Distance. Second, the analysis
should be extended to other cities. Third, topography should be
taken into account by performing the routing in a projected AGL
coordinate system. Third, path costs should capture additional
factors as such turning, operator-drone connectivity, population
density, drone type, payload, and stochastic factors such as wind.
Finally, once the single-drone problem have been satisfactorily
solved, we must move on to the multiple drone routing problem,
which requires de-conflicting drone paths in space and time.
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Figure 11. Horizontal path length ratio distribution at different altitudes of 200 samples Figure 13. Comparison between actual and predicted cost
