Increasing Operational and Fuel Efficiency for Multi

AIAA Infotech@Aerospace (I@A) Conference August 19-22, 2013, Boston, MA

AIAA 2013-4732

Increasing Operational and Fuel Efficiency for Multi-UAV Missions Sunghun Jung ∗ and Kartik B. Ariyur

†

Downloaded by PURDUE UNIVERSITY on August 23, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4732

School of Mechanical Engineering, Purdue University, West Lafayette, IN, 47907, USA Multiple Traveling Salesman Problem (mTSP) with n number of nodes to be visited by m identical UAVs can be solved by transforming into m Traveling Salesman Problems (TSP) and then solved with improved Genetic Algorithm (GA). Within our framework of spatiotemporal discretization and offline calculation, this is done with the objective of maximal fuel efficiency or minimal fuel consumption. While m TSPs correspond to m regions are running, each UAV under this algorithm can exchange nodes to equalize total traveling distances of each UAV. Exchanged nodes are prioritized in the shortest distance between two adjacent regions among m regions, that is, region which has longer total traveling distance gives a node located most closely to the neighboring region which has shorter total traveling distance. This give and take process will continue until differences among total traveling distances of each region become less than a certain threshold. Equalized traveling distances of each UAV keeps all UAVs in the air for the about the same time, increasing operational efficiency. Extensions to heterogeneous UAV types are discussed toward the end of this work.

Nomenclature GA KVRD mTSP NEA PSA TSP UAV URD

Genetic Algorithm K-means Clustering with Voronoi Region Division multiple Traveling Salesman Problem Node Exchange Algorithm Path Sliding Algorithm Traveling Salesman Problem Unmanned Aerial Vehicle Uniform Region Division

I.

Introduction

ultiple Traveling Salesman Problem (mTSP) can be solved using various methods.,1–4 but none of previous works considered fuel efficiency of Unmanned Aerial Vehicles (UAVs) as another constraint for optimization methods. In real life, military uses mostly homogeneous UAVs for surveillance and so those UAVs can carry approximately equal amount of fuel. In order to maximize surveillance coverage, UAVs should be in operation as long as possible until the amount of leftover fuel is sufficient to fly back to the base station. If total distance of each trajectory resulted from mTSP method is biased to specific trajectories, UAVs flying these trajectories will consume fuel quickly and return to the base station. Rather, having equalized trajectories guarantees better operational and fuel efficiency by equally distributing and consuming fuel. In addition, fuel efficiency can be maximized by letting UAVs fly with the minimum velocity since high velocity introduces bigger value of drag force5 as, Fd = 12 ρv 2 cd A, where ρ is the mass density of the fluid, v is the speed of the object relative to the fluid, cd is the drag coefficient, and A is the reference area. The drag coefficient and the velocity is proportional each other.

M

∗ PhD student, School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette IN 47907, Email: [email protected]. † Assistant Professor, School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette IN 47907, AIAA professional membership, Email: [email protected].

1 of 10 American Institute of Aeronautics and Astronautics Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


Fuel consumption management for a single UAV by using the glider type UAV (described as soaring flights through linear wind gradients) was proposed and formulated.6 Although it was only applied to a single UAV, it would be a good approach to extend multiple glider type UAVs to minimize fuel consumption for surveillance mission. Fuel consumption management for the group of UAVs is formulated for the persistent surveillance coverage by considering the vehicle failures and degradations,7 but this work focused only on the fuel consumption level of UAVs by excluding the consideration on the surveillance coverage. In contrast, surveillance coverage was dealt with a cost function to achieve better fuel efficiency by accommodating the threat avoidance and path length by other researchers,8 but still this work lacked the application of the proposed method to the multiple UAVs surveillance problem. In another work, genetic algorithm (GA) based Traveling Salesman Problem (TSP) was used for the path planning of the multiple vehicles, but the optimization formulation setup in this work did not consider the method of finding the optimum number of vehicles to minimize the fuel consumption for surveillance missions and also did not consider the fuel carriage limit per each vehicle.9 The work done by Naval Research Laboratory10 was in succeed to integrate various variables including fuel, obstacles, sample points to visit, and mission complete time, but it lacked descriptions on the intelligent methods of UAV distributions to each cite. To compensate the listed problems, we develop the hierarchy of mission planning (Figure 1) which results the optimized number of vehicles to obtain the minimum fuel consumption and the minimum mission complete time when we perform surveillance missions using multiple UAVs. Vehicles have to follow definite paths while in motion and so the energy consumption of which can be calculated through integration of the product of thrust and vehicle velocity over the path. First of all, total region is divided into m number of regions with two different region division algorithms; Uniform Region Division (URD) and K-means Clustering with Voronoi Region Division (KVRD). Then, independent TSPs are solved for each of the m regions and all TSPs are solved in parallel using Genetic Algorithm (GA)(step (1) - step (6)).11 In step (7), we find the optimum number of UAVs, m, to have both the minimum fuel consumption and mission complete time. In step (8), we apply the Node Exchange Algorithm (NEA) to minimize the difference of the trajectory distances among m number of UAVs. In step (9), we finally apply UAV dynamic model to simulate real-time trajectories.

Figure 1

Hierarchy of mission planning.

In this paper, we give two types of region division algorithms in Section 2, formulation of the optimization problem in Section 3, detailed description on the NEA in Section 4, computational results in Section 5, and concluding remarks in Section 6.

II.

Region Division Algorithm

To utilize multiple UAVs for surveillance missions, we need to efficiently deploy UAVs over the interested area. In here, we assume that UAVs have homogeneous system properties. To maximize the fuel efficiency, each UAV should fly over the shortest distance without any overlapped searching area and it can be solved via mTSP. To achieve the goal, we apply region division algorithms; URD (Figure 2(a)) and KVRD (Figure 2(b)).12, 13


(a) URD method.

Figure 2

Region division with URD and KVRD algorithms using six UAVs (unit: m).

III. Downloaded by PURDUE UNIVERSITY on August 23, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4732

(b) KVRD method.

Optimization Problem Formulation

To equalize trajectories on the m number of regions, optimization cost function is introduced with some constraints. Using an assignment-based double-index integer programming, mTSP formulation can be written as,     m n X n X X  min E tmission = cijk xijk  + wk + ok  (1) k=1

s.t.

i=1 j=1

m m X X tp − tq < Tt , p=1 q=1,p6=q n X j=2 n X

xijk = 1

for any k,

xjik = 1

for any k,

xijk = 1

for any k,

xijk = 0

for any k,

j=2

X i6=j

X i=j

cijk = Gt (i, j) for i 6= j, wk , ok ≥ 0, where, cijk is the time taken by kth UAV between ith and jth nodes, xijk = {0, 1}, n is the number of vertices, m is the number of UAVs, Gt is the time array among nodes (n × n), wk is the time increase due toPwind, ok is the time increase due to unexpected obstacles, tp is the time taken by pth UAV which is n Pn ( i=1 j=1 cij xij + w + o)p , and Tt is the threshold. The first constraint equation is used to result similar total distance of trajectories among m regions. Here, an appropriate threshold, Tt , is desired since the simulation time is getting longer as Tt is getting smaller. Since the mission requires high fuel efficiency, we need to consider how to choose appropriate number of UAVs. To achieve this, we need to acknowledge that the fixed wing vehicle consumes more fuel when it takes off and climbs to certain altitude than flies at certain altitude and lands on the ground. So, to save more fuel, less takeoff and altitude change are required. The total fuel consumption of a fixed wing vehicle can be expressed as, nf = Ft + Fc + Fn + Fl , (2) where nf is the total number of fuel units burned, Ft is the fuel consumption during take off, Fc is the fuel consumption during increasing altitude, Fn is the fuel consumption when the fixed wing vehicle keeps at constant altitude, Fl is the fuel consumption during landing. With an assumption that the Ft , Fc , Fn , and


Fl are constants, then Equation 2 can be rewritten as, nf = F˙b (ct nt,t + cc nt,c + cn nt,n + cl nt,l ) ,

(3)

where ct , cc , cn , and cl are constants governing the fuel consumption rate which are dependent on the type of vehicle, F˙b is the rate of fuel burn, and nt,t , nt,c , nt,n , and nt,l are the time taken for each function. If total m number of homogeneous UAVs are used, only nt,n is changing with different total distance of given m trajectories. So, Equation 3 becomes, ! j X ˙ Nf = Fb j (ct nt,t + cc nt,c + cl nt,l ) + cn nt ,n < Tt , where, j = {1, ..., m}, (4) i

i=1


Nt

= nt,t + nt,c + nt,l + max(nti ,n ) < Tf ,

where, i = {1, ..., m},

where Nf is the total amount of fuel consumption of m UAVs and Nt is the total flight time of m UAVs, that is, Nf = Nf1 + Nf2 + · · · + Nfm and Nt = Nt1 + Nt2 + · · · + Ntm . Here, we introduce constraint equations, Nf < Tf and Nt < Tt , since the amount of fuel which the UAV can carry and the time for completing the mission are limited. The nti ,n can be calculated using the UAV velocity profile (Figure 3) since the distance between two nodes are managed to be equalized. That is, since the UAV increases its velocity when it leaves the current node, keeps the maximum velocity, and decreases its velocity before arriving at the next node, the flight time between two nodes can be expressed as,

(a) Acceleration stage.

(b) Constant velocity stage.

(c) Deceleration stage.

Figure 3 Velocity profile of the UAV during acceleration, constant velocity, and deceleration stages (1D Optimal Control (Step4 in Figure 1)). nf ≈ 2(nedge − 2)(nt,c + nt,n + nt,l ),

(5)

where nedge is the total number of edges of the given trajectory and nedge − 2 is used since it takes different amount of time at takeoff and landing stages. Then, Equation 4 becomes, ! j X Nf = F˙b j (ct nt,t + cc nt,c + cl nt,l ) + cn (nedge,i − 2)(nt,c + nt,n + nt,l ) < Tt , (6) i=1


where j = {1, ..., m}. Also, to compensate the unit difference between Nf and Nt , we normalize those as, Nf0

=

Nt0

=

Nf , max(Nf ) Nt . max(Nt )

(7)


To achieve the best operational efficiency, we need to consider both fuel consumption and mission complete time. So, the cost function g(m) is constructed to get the optimized number of UAVs, m, as m = arg min g(m) = Cf Nf0 (m) + Ct Nt0 (m) , (8) where Cf and Ct are relative costs of fuel consumption and mission complete time which are constrained by Cf + Ct = 1. Values of the Cf and Ct can be decided depends on the user preference on the either fuel consumption or mission complete time. That is, if we put more emphasis on the less fuel consumption, we choose higher value for the Cf and vice versa. Equation 8 is used to plot Figure 4 to Figure 6 by changing the coefficients, Cf and Ct with F˙b = 0.0067kg/s, ct = 1.5, cc = 1.3, cn = 1, cl = 1.2, nt,t = 5s, nt,c = 15s, and nt,l = 10s. When the coefficients are set as Cf = 0.1 and Ct = 0.9, we have the minimum cost, g, at four UAVs (Figure 4(a)). When the coefficients are set as Cf = 0.5 and Ct = 0.5, we have the minimum cost, g, at four and five UAVs (Figure 5(a)). When the coefficients are set as Cf = 0.9 and Ct = 0.1, we have the minimum cost, g, with five UAVs (Figure 6(a)). Choices of the coefficient values depend upon the mission purposes and we choose the usage of four UAVs through the rest of paper to achieve the minimum fuel consumption.

(a) Number of UAVs Vs. g in Equation 8.

Figure 4

(c) Number of UAVs Vs. complete time.

mission

Application to choose the total number of UAVs (when Cf = 0.1 and Ct = 0.9).


Figure 5

(b) Number of UAVs Vs. total fuel consumption.





mission



Figure 6



mission


IV.

Node Exchange Algorithm (NEA)

Even if UAVs carry equal amount of fuel, fuel efficiency might degrade if total distance of each trajectory is not equalized among UAVs since some UAVs fly longer and some UAVs fly less. Trajectory equalization using NEA can solve this problem and this process is done by giving and taking the adjacent nodes among m trajectories (Figure 7). The processes of NEA is explained below.

(a) Before NEA is applied.

(b) After NEA is applied.

(c) NEA operation.

Figure 7

Procedure of NEA.

1. Pick up the closest two nodes among nodes in the longest trajectory and nodes in its adjacent trajectories. We assume those two nodes are A and C. 2. Move node C in the longest trajectory to the adjacent trajectory which contains the closest node from the longest trajectory. Then, re-run TSP. 3. If step 1 and step 2 fails to find two nodes due to the local minimum, pick up the closest two nodes among nodes in the shortest trajectory and nodes in its adjacent trajectories. 4. Continue step 1 to step 3 until the difference between the maximum and minimum trajectory distances becomes less than a certain threshold Td . That is, every adjacent two trajectories keeps comparing and exchanging nodes until the difference between two total distances becomes less than Td . Often times, new trajectories after the node exchange 6 of 10 American Institute of Aeronautics and Astronautics Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


algorithm is applied cross neighbor blocks and so we develop the Path Sliding Algorithm (PSA) to avoid collisions by letting trajectories go along the block with an assumption that UAV can only detour blocks because of the height of the buildings (Figure 8). Due to the unexpected shape of the encountered block, there are additional distances in the amount of A, C, B − A, C1 , C, C3 , C2 , B (Figure 8(b)). The size of the buffer zone of the building is discussed in paper11 so we skip the description. When PSA determines a trajectory along the building, there appears two candidate trajectories (Figure 9) and we choose the shorter trajectory. However, we oftentimes encounter a shorter trajectory which the UAV already flied before when new trajectory crosses several buildings. In that case, we choose a longer trajectory to avoid overlaps of the trajectories.

(a) Before PSA is applied.

(b) After PSA is applied.

(c) PSA operation.

Figure 8

Procedure of PSA.

(a) First candidate with a longer trajectory (· · · − A − C1 − C2 − C3 − C − C3 − C2 − B − · · · ).

Figure 9

(b) Second candidate with a shorter trajectory (· · · − A − C1 − C − C3 − C2 − B − · · · ).

Two candidates appearing along the building when the PSA is applied.

This method can be applied to Figure 2(a). To search the given region, it is not necessary to visit all nodes as shown in Figure 2(a), so we minimize the number of nodes by selecting the coincident points of three vectors (Figure 10 and Figure 11). In Figure 11(a), red and green circles represent the nodes which are exchanged between two regions to equalize the total trajectory distances. Here, the number at each region represents the order of UAVs.


(a) Total node before NEA is applied.


Figure 10

(b) Final trajectories before NEA is applied.

Before NEA is applied using four UAVs with regions divided with URD (unit: m).

(a) Total node after NEA is applied.

Figure 11

(b) Final trajectories after NEA is applied.

After NEA is applied using four UAVs with regions divided with URD (unit: m).

V.

Computational Results

We use tracking models of UAVs from prior work.14 These models are accurate approximations to the dynamics of attitude stabilized vehicles. This is because global or semi-global closed loop stabilization transforms UAV dynamics into target dynamics of point mass models with definite tracking time constants for position, τx , and velocity, τv . Path following aircraft dynamics which is dominated by the slowest or dominant poles is far more accurate than simplistic point mass or constant velocity models. The UAV tracking model is defined as, xc (k + 1)

=

vc (k + 1)

=

xc (k) + T vc (k), T T T ref − xc (k) + 1 − vc (k) + x (k), τx τv τv τx τv c

(9)

where T is the sampling time, xp is the planar position of prey [xpx , xpy ], vp is the velocity vectors of prey, xc is the position of chaser [xcx , xcy , xcz ], vc the velocity vectors of chaser, and xref is the tracking c set point. All the simulations are done with a fixed wing vehicle which has following system properties; vmin = 5m/s, vmax = 10m/s, vinitial = [0, 0, 0]m/s, amax = 5m/s2 , altitudemin = 25m, altitudemax = 50m, altitudenormal = 30m, M t = 0.01s, τx = 0.25s, and τv = 0.5s. All simulations are done using a desktop with Intel(R) Core(TM) 2 Duo CPU 3.00 GHz Processor, 64-bit Operating System, and 4.00 GB RAM. Flight trace of Figure 11(b) is simulated as in Figure 12. Red rectangles represent the starting location of UAVs and red circles represent the ending locations of UAVs. There are ripples along the trajectories largely due to the time constants, τx and τv , but UAVs generally fly well along the predetermined trajectories. Also, those flight ripples does not cause any collision on UAVs since we included buffer zones around buildings large enough by concerning the maximum velocity and turning angle of UAVs.11 Those ripples will be decreased if 8 of 10 American Institute of Aeronautics and Astronautics Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


we use hover capable UAVs instead which have the zero minimum velocity. From the simulated trajectories (Figure 12), we can predict the approximate traveling time and the amount of fuel consumption of each UAV.

Figure 12

Flight traces of four UAVs at Purdue Mall (lat: 40.428647◦ , lon: -86.913757◦ )(unit: m).

Now, NEA is applied to different number of UAVs to see the effects on the max, min, max-min, and total distances. In the aspect of the maximum and the minimum traveling distances, the case without NEA has larger maximum distance than the one with NEA (Figure 13(a)) and the case without NEA has lower minimum distance than the one with NEA (Figure 13(b)). Also, the case with NEA has much less M ax − M in value compare to the case when NEA is not applied (Figure 13(c)) and this result shows that the NEA processes indeed flatten the differences among m trajectories. In addition, we should acknowledge the fact that the total traveling distances without NEA and with NEA do not have major difference which indicates that NEA does not always bring increased trajectories (Figure 13(d)).

(a) The max traveling distance changes with increasing number of UAVs.

(b) The min traveling distance changes with increasing number of UAVs.

(c) The max − min traveling distance changes with increasing number of UAVs.

(d) The total traveling distance changes with increasing number of UAVs.

Figure 13

NEA application with various number of UAVs.


VI.

Conclusions

We have shown a method of achieving the maximum fuel efficiency when we perform surveillance missions by converting mTSP to m TSPs based on GA with the optimized number UAVs. The optimized value m will be different depends on the missions (minimum traveling time, minimum fuel consumption, etc) and it can be achieved by choosing proper weights, Cf and Ct . To compensate the achievement of the minimum fuel consumption, NEA is developed to equalize the total traveling distances among UAVs and PSA is developed to prevent any possible collision of UAVs. Once we have m number of equalized trajectories, we can send m homogeneous UAVs to each region. This method is generally a much safer surveillance mission planning than the normal swarming surveillance mission planning in the sense that each UAVs have the least number of coming across. In addition, our work opens up several possibilities for both system and algorithm developments:


1. Come up with more realistic relationships between fuel consumption and UAV vehicle dynamics, 2. Present a comparison table about fuel consumption rate between normal mTSP trajectories and mTSP trajectories with increased fuel efficiency.

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