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Trajectory Design and Coverage Control for Solar-Powered UAVs

Trajectory Design and Coverage Control
for Solar-Powered UAVs
Soumya Vasishtand Mehran Mesbahi
William E. Boeing Department of Aeronautics and Astronautics,
University of Washington, Seattle, WA 98195
This paper explores the benefits and challenges of using solar energy to power unmanned aerial vehicles (UAVs)
for surveillance purposes. The task of persistent surveillance requires constant supply of input energy and is of partic-
ular significance in a number of applications such as weather monitoring, wildfire control, pollution or contamination
detection, target search and other long endurance missions. Here, we consider the task of monitoring a geographical
area for events of varying priorities and design optimal trajectories for sufficient continuous coverage of the region
and eliminate the need to devise a refueling policy. Trajectories are designed based on how the UAVs collect the solar
energy to carry out the mission under a variety of dynamic and endurance constraints. An algorithm is proposed to
handle the energy collection and the coverage of the target area. Different goals are set and stimulation results are
presented to demonstrate the ability of the UAVs to meet these goals.
1 Introduction
The idea of a futuristic, regenerative power system has been of interest for several years. Hybrid power systems have
been envisioned and implemented using naturally occurring elements such as wind, water and the sun. On a bright,
sunny day, the sun’s rays give off approximately 1,000 watts of energy per square meter of the planet’s surface. If we
could harvest all of that energy, we could easily power our homes and offices for free. Therefore, powering a group of
small aircraft during atmospheric flight should be quite straightforward. However, designing trajectories to achieve a
certain goal increases the complexity of the problem. More so, when we increase the number of vehicles in the UAV
team to form a coordinated workforce.
Trajectory optimization of more conventional UAVs has been extensively studied over the past few years [1, 2, 3].
Coverage control has also piqued the UAV community’s interest over the years. There has been a lot of research on
designing sensor networks to effectively cover an area for a variety of applications [4, 5]. Research soon moved to
larger domains requiring mobile sensors [6]. Problems arise when the surveillance region is very large with too few
sensors to monitor the region. One way to remedy this is to have the sensors or in our case the UAVs trace the area
back and forth to essentially map the space and achieve an effective total coverage. There have been several attempts
to achieve effective coverage of an area under a variety of dynamic, energy and endurance constraints. The benefits
of formal derivation based and heuristic methods have been explored in [7] to address the persistence problem. They
study the coupling between the control policy and the constraints and demonstrate their claims of maintaining good
performance under limitations and failures with flight test results. They also consider the refueling problem in order to
extend the life of the mission. More recently, [8] studied the cooperative search area to track stationary targets scattered
over an area which translates to a short-term coverage problem. Like in [7], in [8], they partition the region in to cells
over which they move the UAVs in order to minimize a cost function. However, they use a probabilistic approach in
order to determine whether a target exists within a cell and then perform individual probability map fusions until there
is convergence. This is highly effective, intuitive and relatively easy to implement for short-term missions without
endurance constraints.
A novel dynamic awareness model is proposed in [9]. Each vehicle has a state of awareness which is updated at
different time intervals when more information becomes available. They focus on coverage control of mobile sensor
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networks and guarantee a satisfactory state of awareness under an arbitrary dynamic communication structure and/or
faulty sensors. Again, they consider only short term missions with conventional UAVs for one-time coverage.
An interesting coverage problem is the rendering of 3D models or remote sensing of geographical landscapes.
Three UAVs or drones were used for mapping the Matterhorn, one of the highest peaks in the Swiss Alps, in a recent
experiment by a UAV company [10] in under 6 hours. This is again highly useful in extra-terrestrial exploration or
in unknown environments. We can motivate the use of solar powered aircraft for similar applications and map larger
regions of earth or other planets.
In [11] and [12], the authors discuss the energy-optimal path planning and perpetual endurance for unmanned
aerial vehicles using solar cells alone. They consider point to point travel as well as perpetual loiter from a given
point and establish conditions for perpetuity. In [13], a reduced model is proposed incorporating both total energy
optimization and battery dynamics to achieve a 24 hour long mission which can be extended to achieve perpetual
surveillance. They use this reduced model in [14] to solve a version of the coverage problem in which they maximize
the total area covered in a day’s time.
In this paper, we attempt to solve the coverage problem, which is to survey a given fixed area completely and
continuously with multiple UAVs powered by the collected solar energy. The path planning problem is posed as an
optimal control problem that is solved at each time step to maximize the total energy of the UAVs and hence increase
their endurance. Section II introduces the target space and the terminologies associated with it. The solar energy
collection model is also introduced along with the vehicle kinematic model. The optimization problem is formally
defined and a solution strategy is discussed in Section III. Simulation results are presented for different cases along
with discussions in Section IV.
2 Background and Problem Formulation
Remote sensing and monitoring applications would require the UAVs to fly at a certain minimum altitude and at near
constant speed to record or capture the scene below with a reasonable resolution. The position of the UAV in 3-D
space can be projected on to a plane to observe the region it is flying over. The area on the plane “seen” by the UAV
is called the Field of View (FOV) or the footprint, which is the area over which the UAV takes its measurements at
a given instant of time. This area will vary according to the instruments (visual cameras, thermal sensors, etc.) on
board the UAV, the altitude at which the aircraft is flying and the application. It is, hence, reasonable to assume that
the physical area to be surveyed is 2-D and that the UAV is moving on a fixed plane over this surveillance region
performing level flight.
2.1 Surveillance Region
Consider a simple closed, bounded region WR2which is divided in to cells of equal size. Since we assume level
flight, the FOV remains a constant for all time. This lets us assume that the cell area may be adjusted according to the
footprint or FOV of the aircraft such that when the vehicle is above that cell, the entire cell is covered.
Figure 1: The target space to be surveyed. It is divided in to cells with the centroids represented by the ’o’s
A similar formulation is used in [7] for conventional UAVs. We will study two cases using this setup, one of which
uses a probability distribution function to specify “hotspots” or subareas that require more vigilant monitoring than
others. These points in the region need to be visited more frequently to catch any event that might occur. This setup is
different from the one in [8] where conventional UAVs attempt to produce their version of the probability map. Here
we assume a distribution and drive the vehicles towards these points.
2.2 Vehicle model
Consider a team of UAVs, where each aircraft is denoted by Aiwhere i=1,2,...,N, and Nis the number of vehicles.
Since we assume level flight for the problem at hand, the kinematic model is assumed to be the following
where xi,yi,yiand fiare respectively the xand ycoordinates, heading angle and bank angle for agent Ai.We
assume the UAVs are flying at a constant speed Vand at a fixed altitude with initial conditions (x0,y0,y0)and with
a minimum turning radius r.
2.3 The Energy Collection Model
Since we want to explore the benefits of solar energy for flight, let us formally introduce the model for the solar cells.
The photovoltaic cells are expected to be mounted on the wings and/or other surfaces of the aircraft. These solar cells
collect solar energy which is converted to useful power for actuating the aircraft. The power gained from these solar
cells is a function of the incidence angle iof the sun’s rays with respect to the z-axis in the body frame.
Figure 2: (X,Y,Z) represents the Earth-fixed frame and (x,y,z) represents the body frame. The azimuth and elevation angles are
defined in the Earth-fixed frame and the Euler angles g,fand yare represented in the body frame. gis ignored in the level-flight
The input power obtained is given by
in =hsol P
sd Scosi(2)
where hsol is the efficiency of the solar cells, S is the overall cell area and P
sd is the solar spectral density. The position
of the sun plays an important role in the amount of energy collected. This model is more formally presented in [15].
If aand eare the azimuth and elevation angles of the sun respectively, then the input fwhich is the bank angle of the
aircraft is related to the incidence angle ias
The power lost by the vehicle to drive the propeller is given by
out =TV
where Tis the thrust of the aircraft, hprop is the efficiency of the propeller, Vis the velocity of the aircraft. The thrust
of the aircraft is related to the drag and the lift coefficients, CDand CL.
The energy gained during the time interval t0,tfis Ein =´tf
in (f)dt and the energy lost during the same time
period is given by Eout =´tf
out (f)dt. The total effective energy of the UAV is given by
Eout =(Ein Eout )
In order to enable the UAV to function solely on solar power, we need to maximize the total energy of the system. If
we assume small bank angles and a constant velocity, we will need to maximize only the input energy which simplifies
the cost function.
2.4 Navigating the cells
Now that we’ve established the vehicle dynamics and the solar cell model, let us examine how we can move the aircraft
over the target region and track the coverage achieved. Since the energy collection process dictates the heading of the
UAVs, we need to devise a strategy to visit every cell in the target area without compromising the power to the aircraft.
Let us start by assigning a specific value to each cell based on certain factors. We want to ensure that the region is
continuously surveyed to catch anything of interest. In other words, we do not want to leave any region unnoticed for
too long. The priority here is to collect maximum solar energy to stay in flight while achieving sufficient coverage.
At every time step, the UAV needs to maximize the input solar energy and pick cells which have not been visited in a
while. To ensure this, for each cell, we track the time elapsed since the last visit. Intuitively, we want to reduce this
time or equivalently, we want to minimize the maximum time elapsed since the last visit for all cells. Another factor
to consider is the time that it takes for the UAV to travel to a given cell from its current position. Suppose we assign
a value, Vjto the cell j. From [7], we realize that for a single UAV, the value Vjthat is associated with a cell can be
defined as
Vj=((Aj+w0dj)when (Aj+w0dj)>0
0 when (Aj+w0dj)0(5)
where Ajis the time elapsed since the UAV’s last visit to the cell jand djis the distance of the cell jfrom the UAV.
At each time-step, or at regular intervals, we calculate the value associated with each cell and move the UAV to the
cell with the maximum value. This way no region is left unsurveyed for long periods of time. If multiple cells get
assigned the same maximum value due to a combination of Aand d, the cell that enables maximum energy collection
is chosen. If multiple cells are visited en route to a target cell, all these cells must be deemed visited to ensure that the
UAV chooses the cell with the right value.
When the region is surveyed by a team of UAVs, each UAV needs to be aware of the distance of every other UAV
to a given cell. In other words, a UAV would not go to a cell which is closer to another UAV. The value of each cell
calculated by UAV iis
Vij =((Aj+w1minidij)when (Aj+w1minidij)>0
0 when (Aj+w1minidij)0(6)
The weights w0and w1may be optimally chosen based on the mission specifications. The most common weight to
use would be 1/V, where Vis the aircraft velocity. But this of course may not be optimal for most missions. This
drives the regular information exchange between the UAVs to ensure the region is sufficiently surveyed and the mission
requirements are met. It also curbs any redundant surveillance of visited cells while there are still cells not surveyed.
The major takeaway from this is that the region would be covered in much less time as compared to a single UAV.
3 The Optimal Control Problem
The mission is to completely cover and continuously monitor a region over long periods of time. In order to enable
this, we look to maximize the incoming solar energy and design our trajectory such that most of the cells are visited.
This optimal path planning problem is broken down in to smaller problems, when we consider travel from one cell to
the next. We will need to place a restriction on the rate of change of heading in order to obtain a smoother trajectory,
which in turn places bounds on the input. We assume that the bank angle should be limited as |f|p
2. The path
planning problem between the cells is then defined as following:
in (f)dt
[x(t0),y(t0)] = [x0,y0]
xtf,ytf⇤ =xf,yf
Assuming that the UAVs fly at a constant altitude and at a constant speed has reduced the complexity of the problem
to controlling only one input, which is the bank angle, f. Due to this assumption, we find that the energy lost is a
constant over time. This allows us to calculate the minimum velocity required to drive each UAV, which is obtained as
out =TV
which leads to
Vmin =4KW2
where ris the atmospheric density, S is the wing area, CL,CDand CD0are the lift and drag coefficients. The aerody-
namic coefficient K=1
epAR represents the induced drag. AR =b2/Sis the aspect ratio, bis the wing span, eis the
Oswald efficiency factor. In order to calculate the value of each cell, we need to calculate the actual distance traveled
by the UAV from its current position to each cell. This means we will have to solve for the optimal trajectory for travel
to each cell by a single UAV at every time step. For conventional UAVs, this is merely including the dynamic con-
straints. But, for the solar-powered UAV, the dynamic and endurance constraints are so closely coupled that solving for
the optimal trajectory increases the computational complexity to an extent that is undesirable. It is safe to approximate
this distance in our calculations by either the euclidean distance or the actual distance computed for the conventional
UAV. At every time step, each UAV assigns values for each cell and optimizes its path to collect enough solar energy
to reach that cell. The algorithm that a single UAV follows is given below.
The Multi-UAV case follows similarly. Here, each UAV has access to the distance vectors to every cell of every
other vehicle in the region. This way each UAV can calculate the value of each cell by equation 6. There may be a
case where a cell gets assigned to more than one UAV and the cell is easily accessible to all of them, a heuristic policy
may be required to pick a UAV to go to the given cell.
Algorithm 1: Solar-Powered Coverage Control for a Single UAV
1Choose initial conditions [x0,y0]. Time elapsed since last visit, A=0 for all cells.
2for t=0to T do
3for each cell j do
5Vj=max (Aj+w0dj),0
8Maximize ´t1
indt over all fto reach cell j
3.1 Hotspots
There may be applications with subregions which need to be more frequently visited. This scenario can be easily
represented as a contributing factor for choosing the candidate cell. We may influence the trajectory of the UAV by
increasing the value associated with certain cells. If the target region is characterized by some probability distribution
representing the priorities of every cell, the value the UAV calculates at every time step is given by
where Pjis the priority associated with cell j. As before, if multiple cells get assigned the same maximum value, then
the cell which maximizes the energy collection is chosen.
4 Simulation Results
In this section, we will test the performance of the proposed plan to cover a region continuously by solar-powered
UAVs. We will first consider a single UAV whose parameters are given in table 1. With these parameters, we calculate
minimum velocity to be Vmin =9.22m/s. We survey a region that is 2km x 2km in area with cells which are 200m
x 200m wide. We consider two versions of the target space with and without hotspots. Intuitively, the case without
hotspots will record uniform coverage over the area whereas for the case with hotspots, we will notice some extra
flight time over these regions. For both cases, the UAV will start from one of the boundary points and fly with in the
region and cover as much area as possible. The extent of coverage required can again depend on the application and
the sensor capabilities. For remote sensing and geographical mapping of a landscape would require tighter coverage to
image every detail of the land below. For a more casual long term surveillance like weather and pollution monitoring,
a trade-off may be required to extend the life of the mission.
Table 1: UAV Parameters
sd W
m2hsol b(m)Sm2m(kg)hprop
380 0.22 0.711 0.1566 4.201 0.7
4.1 Case 1: Without hotspots
Single UAV
For this case, all points in the region are given equal priority and the value Vjfor cell jreflects the amount of coverage
it has received or whether it has been visited at all. At each time step, the value of each cell increases (or decreases if it
is visited very recently) drawing the UAV closer to it. Inversely, closer a cell is to the UAV along its heading direction,
higher is its value. Once the value for each cell is calculated, distance vectors to all the maximum valued cells are also
considered. The resulting cells may be scattered all over the target space. The cell that requires minimum effort and
that maximizes energy collection is chosen. There will be instances when the UAV may have to leave the boundaries
of the region entirely to collect more solar energy. For readability purposes and to demonstrate the amount of coverage
achieved by a single UAV, the simulation time was restricted to 90 minutes and the trajectory was recorded as in Figure
3. Nearly 100% of the region is surveyed at least once under a period of 90 minutes. The computation time of the
optimal trajectory at each time step was at an average of 2s.
Figure 3: Trajectory of a single solar-powered UAV over a uniform region.
Figure 4: Changing input power during flight (shown for 500s)
The fluctuating input power is shown in Figure 4. The output power required is a constant 18.15 W and the total
energy never falls below 4W. The average input energy collected is about 29W per time step and may be improved
with better cell selection.
Figure 5: Contour plots to show coverage over time. (a) Coverage at t=20min (b) Coverage at t=40min (c) Coverage at
t=60min (d) Coverage at t=70min
Figure 6: Multiple UAV case: Trajectories of two UAVs to cover the same region as before. The stars indicate the initial positions
of the UAVs.
Multiple UAVs
Let us consider two UAVs, each with the same parameters over the same target region. We assume that they are in
constant communication with each other or with a central authority and hence have access to each other’s distance
vectors. In order to avoid collisions, let us assume they fly at different altitudes, z1and z2, so that their flight paths
never intersect in the same plane. We notice in Figure 6 that both the UAVs follow more or less a lawn-mower pattern
while leaving the boundaries of the region a few times in order to maximize energy collection. Again, the simulation
here was stopped prematurely at about an hour to demonstrate the coverage achieved by the team. The coverage
achieved was 100% in about 40 minutes for this case and the computation time per time step was about 3s on an
4.2 Case 2: With hotspots
Certain applications may require extra flight time over certain sensitive sub-regions. The surveillance region may be
assumed to be characterized by a random distribution of high priority sub-regions or ’hotspots’ that the UAV must
frequent. To make these sub-regions more visible to the UAV, a slightly modified version of the value function, that
the UAV calculates at each time step, is used in the algorithm mentioned in the previous section.
where Pjspecifies the priority of cell j. This step ensures that the value of the cell is kept high enough that the UAV
revisits it regularly.
Single UAV
The implementation of this case is similar to the previous case, with the exception of how the values are calculated.
The underlying priority distribution is visible to the UAV throughout the duration of the flight. The trajectory obtained
with the UAV as described earlier is given in Figure 7.
Multiple UAVs
Similarly, for a group of two UAVs, the flight time over these hotspots is divided between them and they cover the
area cooperatively. The percentage area covered depends on the weight associated with the priority Pj. If we weight it
too high the coverage deteriorates as the trajectories generated would neglect the majority of the target space as seen
below. The total simulation time was fixed at about 75 minutes for most of the cases.
Figure 7: Trajectory of the solar-powered UAV over a region characterized by a distribution of hotspots.
Figure 8: Multi UAV case: Trajectories of two UAVs to cover the same region as before. The initial positions of the UAVs are
same as before.
5 Conclusion and Future Work
In this paper, we proposed the use of solar energy to power UAVs for persistent surveillance purposes with total
coverage control. The main objective was to collect maximum solar energy to stay aloft for long endurance missions.
The surveillance region was divided in to cells and a value was assigned to each cell based on some factors. The UAV
was then moved to the cells in the order of priority. In order to have truly long missions (that last for days or months),
we can look at incorporating battery dynamics in to the problem in future work. An implementation of the algorithm
on a hardware testbed with actual flight results would be favorable to assess the performance and effectiveness.
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... Besides, specific type of missions such as ground target tracking and multi-aircraft surveillance were investigated. [15][16][17][18][19] In these studies, the aircraft were assumed to fly at a constant altitude. ...
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This paper considers energy-optimal path planning in a loitering mission for solar-powered unmanned aerial vehicles (UAVs) which collect solar energy from the sun to power their flight. We consider ascending and descending flight maneuvers in a periodic mission constrained to the surface of a vertical cylinder. The coupling of the aircraft kinematic and energetic models is treated in a novel scheme that implements both the periodic and cylindrical constraints. Optimum trajectories are identified by specifying the heading angle and altitude by splines. Given the periodic splines, we are able to solve for the other aircraft parameters, including the aerodynamic, propulsive, and energetic properties of the aircraft. In an example problem trajectories are obtained that generate better energy properties than those given by constant altitude circular flight. Numerical simulation results are presented that help demonstrate the properties of the optimum trajectories. Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All right reserved.
Conference Paper
In this paper, the energy optimal trajectories for Unmanned Aerial Vehicles (UAV) surveillance are explored. The main objective is to have maximum sensor coverage range while maintaining a perpetual flight. A solar powered UAV equipped with photovoltaic cells mounted on its wing and rechargeable batteries is considered. The photovoltaic cells generate solar energy based on the position of the sun, attitude of the UAV, and sky clarity. The vehicle aims to optimize the energy storage in the batteries and coverage during the day. However, the sensor resolution diminishes due to altitude gain. A model for optimal coverage, path planning, and power allocation in a solar powered UAV is proposed and the corresponding simulation results are presented. In addition, the effect of maximum altitude gain on the energy storage is studied based on a reduced hybrid model.
This paper addresses cooperative search for multiple stationary ground targets by a group of unmanned aerial vehicles with limited sensing and communication capabilities. The whole surveillance region is partitioned into cells where each cell is associated with a probability of target existence within the cell, which constitutes a probability map for the whole region. Each agent keeps an individual probability map and updates the map individually with measurements according to Bayesian rule. A nonlinear transformation of the probability map is introduced to simplify the computation by linearizing the Bayesian update. A consensus-like distributed fusion scheme is proposed for multiagent map fusion. We prove that all the individual probability maps converge to the same one that reflects the true existence or nonexistence of targets within each cell. Coverage and topology control algorithms are designed for the path planning of mobile agents. Moreover, the performance of the fusion scheme for asynchronous implementations of sampling and communication is analyzed. Finally, the effectiveness of the proposed algorithms is illustrated via simulations.
This paper provides a tutorial-style overview of sensor networks from a systems and control theory perspective. We identify key sensor network design and operational control problems and present solution approaches that have been proposed to date. These include deployment, routing, scheduling and power control. In the case of mobile nodes, a sensor network is called upon to perform a “mission”. We present solution approaches to two types of missions both involving stochastic mission spaces and cooperative control: reward maximization missions and coverage control missions. We conclude by outlining some fundamental research questions related to sensor networks and the convergence of communication, computing and control.
This paper considers energy-optimal path planning and perpetual endurance for unmanned aerial vehicles equipped with solar cells on the wings, which collect energy used to drive a propeller. Perpetual endurance is the ability to collect more energy than is lost during a day. This paper considers two unmanned aerial vehicle missions: 1) to travel between given positions within an allowed duration while maximizing the final value of energy and 2) to loiter perpetually from a given position, which requires perpetual endurance. For the first mission, the subsequent problem of energy-optimal path planning features the coupling of the aircraft kinematics and energetics models through the bank angle. The problem is then formulated as an optimal control problem, with the bank angle and speed as inputs. Necessary conditions for optimality are formulated and used to study the optimal paths. The power ratio, a nondimensional number, is shown to predict the qualitative features of the optimal paths. This ratio also quantifies a design requirement for the second mission. Specifically, perpetual endurance is possible if and only if the power ratio exceeds a certain threshold. Comparisons are made of this threshold between Earth and Mars. Implications of the power ratio for unmanned aerial vehicle design are also discussed. Several illustrations are given.
] Motivated by long-endurance requirements in surveillance, reconnaissance and exploration, this paper considers level flight path planning for unmanned aerial vehicles (UAVs) equipped with solar cells on the upper surface of the wings. These solar cells collect energy that is stored in a battery and used to drive a propeller. The mission of the UAV is to travel from a given initial position to a given final position, in less than an allowed duration, using energy from the battery. The subsequent problem of energy-optimal path planning features the interaction between the aircraft kinematics, the energy collection model, and the energy loss model. All three models are coupled by the bank angle of the UAV. Within this framework, the problem is formulated as an optimal path planning problem, with the aircraft bank angle and speed serving as control inputs. Necessary conditions for optimality are given, whose solution yields extremal paths. These necessary conditions are utilized to study analytically the properties of extremal paths. An efficient numerical procedure is also given. It is shown that optimal paths belong to one of two regimes: solar or drag. The Power Ratio, a non-dimensional number that can be computed before flight, is identified. It is shown that this ratio predicts the regime of the optimal path, which facilitates the solution. Implications of the power ratio for UAV design are discussed. Several illustrations are given.