cooperative control of multiple ucavs for suppression of ...은연주... · objectives.1 first,...

Cooperative Control of Multiple UCAVs for Suppression of Enemy Air Defense

Yeonju Eun* and Hyochoong Bang† Division of Aerospace Engineering, Korea Advanced Institute of Science and Technology,

Daejon, 305-701, Republic of Korea

In recent years, there is a growing interest in employing Unmanned Combat Air Vehicles (UCAVs) for various military missions. One of typical examples is the suppression of Enemy Air Defense (SEAD) mission, in which the objective is to fly to the enemy territory to destroy, or suppress surface-based air defenses such as radars and surface-to-air missile sites. This paper deals with a hierarchical and fully automated control scheme which is suitable to typical SEAD mission using multiple UCAVs. For the path finding and planning, most of the existing control scheme use the Voronoi-diagram concept, but a new path planning method which is based on the potential field concept is proposed. The advantages and disadvantages of existing method and a new method are presented with virtual battle field simulation results.

I. Introduction

AS a typical battle field scenario of the SEAD mission, let us consider a group of N UCAVs required to drop munitions on M known targets locations. In addition, there are already known threats and unknown pop-up

threats that may appear during operations. Figure 1 is a typical example of this situation. There are some requirements for the UCAV system to be adopted in a SEAD mission. The UCAVs have to be true air vehicles, more like manned aircrafts than intelligent cruise missiles for the variety of submissions, reconnaissance or return to the base operations. It has to be possible to fly the UCAVs into the battle field and the cost has to be lower than manned air vehicles. Moreover, the most important requirement is that the entire progress of the mission, such as path finding and planning, detection of threats (for example, radars), assignment of each UAVs to the target, determination of feasibility of attack, and return to the base safely in an infeasible attack condition, has to be fully automated, and the mission can be achieved without intercept of human’s operations during the planning and operation. In Fig. 1, each target is assumed to be equipped with anti-aircraft facilities with radars that detect the location of attacking aircraft, but these facilities are only capable of firing on one aircraft at a time. To maximize the probability of mission success in the situation like these, there are some requirements for control scheme. First, efficient assignment of UCAVs to each target location is required. Second is the optimal path planning is required to avoid the dangerous threat regions and to save fuel and time. Third, when all of the UCAVs which left different locations access to one target, each UCAV is required to arrive on the boundary of target radar detection region simultaneously and enter the attack region through different boundary points. And all of these tactics are

Pop-up

threats

Known

threats

Targets

Radius of

dangerous area

UCAVs

Figure 1. Typical battle field of SEAD mission. * Graduate Student, Division of Aerospace Engineering, Department of Mechanical Engineering, KAIST, Korea † Associate Professor, Division of Aerospace Engineering, Department of Mechanical Engineering, KAIST, Korea

American Institute of Aeronautics and Astronautics

1

AIAA 3rd "Unmanned Unlimited" Technical Conference, Workshop and Exhibit20 - 23 September 2004, Chicago, Illinois

AIAA 2004-6529

Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

required to be achieved in real-time, to cope with accidental situations such as appearance of unknown threats. For the first requirement, efficient assignment in the previous section, Beard and McLain suggested target management which use acceptability and rejectability functions to each target1. For the second requirement of optimal path, Voronoi-diagram and so-called k-shortest paths search algorithm have been used in some papers2-4. But the general case of k shortest paths search problem is so complicated and many researchers have been working on this issue with vigor. For the third requirement, the decomposition strategy is suggested in Ref. 4. All of these introduced methods are using the candidate paths generated by using the Voronoi-diagram. Robot path planning using the Voronoi-diagram has been studied since mid-1980s5, and cooperative path planning study for multiple UCAVs started in late 1990’s. The Voronoi-diagram method is considered as a most suitable method for SEAD. It provides many strong points but at same time has many disadvantages. For example, it is rather difficult to extend the path planning to 3-D space with the Voronoi-diagram, and it is very inefficient in case the real obstacle avoidance is required. And as the path consists of the discrete edges of the Voronoi-diagram, it is very unsuitable for the trajectory generation task. For that reason, in this study, a new path planning method for SEAD is proposed. It’s based on the robot path planning using potential filed methods. The potential field is modified for the SEAD mission. This method needs a relatively heavy numerical computation in the first stage of planning, but it is just only once before the UCAVs begin their missions. All of these strategies introduced are required for target assignment and path planning, and thereby not iterative in nature. So, against the accidental situations such as appearance of unknown threats or a certain UCAV dropping out of the assigned group, re-assignment and re-planning should be achieved in real-time. The main contents of this paper consist of four sections. In section II, the existing control scheme based on 1 for SEAD is introduced. In section III, a new cooperative control scheme based on the potential field path planning method is proposed. This new cooperative control scheme is compared with the conventional control approaches by simulation results in section IV. Finally, the conclusion is presented in section V.

II. Conventional Cooperative Control Scheme for SEAD To avoid the dangerous threat regions and save fuel and time, it is

required to construct an optimal path to the target. For this problem, the Voronoi-diagram is adopted in Ref. 2-4, and one can get some suboptimal initial paths using this method. In some interested domain such as in Fig. 2, Let be a set of n distinct points in the plan. The Voronoi-diagram of , Vor is defined as the subdivision of the plane into n cells, one for each site in P with the property that any point

is for each

:P =

( iq V p

1 2, ,p p

) dist

, npP

( , )q p <

(P

i dist

)

j P∈ ( )q, p ip ∈ with i j≠ , where is defined as the distance between two points q and . In this

problem, the point is treated as threat locations. Then, the edges of the cells can be the best path to follow avoiding the site of two nearest threats. The algorithm to construct this diagram is described in Ref. 6, and MATLAB offers the function which can produce this diagram.

( , iq p )dist ip

ip

ip( )iV p

Figure 2. Voronoi-diagram

(a) (b)

Figure 3. Ground threat position and Voronoi-diagram


2

Figure 3 depicts initial known threat locations and Voronoi-diagram. The Voronoi-diagram consists of nodes and cells. The edges of the cells are available paths to the nearest node to the target positions. Now we have to search the shortest and safest path to go to the nearest node to the target positions. Before this work, the costs of each Voronoi edge should be decided. The total cost of each edge consists of both length and exposure cost.

From Fig. 4, are the Voronoi nodes and T denotes i-th threat. Then, the cost of the edge

1( ), ( )j jP t P t + i

1( ) ( )j jP t P t + is given by Eq. (1)~(3).

( )( )

j+1

( , 1) ( , 1)length 1 exposure 4i=1

ˆ ˆ( ), ( ) , =( ),j j j j

jj j t

i

M 1tJ dist P t P t J d

dist P Tτ

τ+ ++= ∑∫ (1)

iT

( )jP t

1( )jP t +

( )jP t τ+

( , )idist T P

Figure 4. Edge cost calculation

( )( ) ( ) ( ) ( )

( , 1) ( , 1) ( , 1) ( , 1)

( , 1) ( , 1)

( , 1) ( , 1) ( , 1) ( , 1)

length length exposure exposure

length exposure

length length exposure exposure

min min ,

ˆ ˆ ˆ ˆmax min max minj j j j j j j j

j j j j

j j j j j j j j

J J J JJ J

J J J J+ + + +

+ +

+ + + +

− −= =

− −

ˆ ˆ ˆ ˆ

(1 )J J Jλ λ+

= + −

(2) ( )

( , 1) ( , 1)( , 1) length exposurej j j jj j ++ (3)

where M it is the number of threats, and represents the distance between two points, T and . The total cost consists of two terms, first term is associated with length cost and second one is the exposure cost. And

is a weighting factor of the length cost. Exposure cost based on a UCAV degree of exposure to

enemy radars, is proportional to 1/ , since the strength of a radar signature is proportional to 1/ . Instead of numerical integration, discrete sum is capable of computing of exposure cost.

( , )dist P T P

4

(0 )1 λ λ≤ ≤4dist dist

Now, one can find the best route which minimizes the cost sum of the path. Now the best route with a minimum cost is named ‘shortest path’. Not only the shortest path but also k-shortest paths should be found for rendezvous. The k-shortest paths of each UCAV from start point to target may constitute some candidates for the simultaneous arrival. The searching algorithms by the k-shortest path has been investigated in many papers.7,8 By using the Voronoi-diagram and k-shortest paths search algorithm, each UCAV can secure own k-shortest paths to each target. Next task is to assign each vehicle to the appropriate target such that the overall group cost is mitigated, while maximizing the number of targets to be destroyed. In assigning the UCAVs to targets, there are four objectives.1 First, minimizing the group path length to the targets, second, minimization of the group threat exposure, third, maximize the number of vehicles attacking each target (to maximize survivability), and fourth, maximization of the number of targets attacked. Details about this assignment problem are presented in Ref. 1.

If each UCAV is assigned to appropriate target, the members of a group attacking one target have similar costs. But it doesn’t guarantee the simultaneous arrival. Thus the detailed trajectory planning for each UCAV is required. For the coordinated control problem, the purpose of decomposition is to break up a single, very large optimization problem into a set of smaller, more tractable optimization problems that allow computations to be decentralized among the UCAVs, while taking into account the threat and fuel of each UCAV. This decomposition strategy was developed in Ref. 3. Using this decomposition strategy, one can determine TOT (time over target 3) of each team and each UCAV which assigned to same target can get the appropriate path for simultaneous arrival.

III. A New Cooperative Control Scheme for SEAD The new path planning for SEAD mission in this research is conceptually based on moving robot path planning

with obstacle avoidance. Voronoi-diagram also has been used in robot path planning already since mid-1980s. Path planning with obstacle avoidance or motion planning of manipulator with collision avoidance has been one of the principal issues in robotics in recent years and is very well-known problem.

The main approaches for solving the path-planning problem can be classified into global and local methods. The global method need complete information of the environment and they are based on a global world representation. Examples of this approach are the grid of visits, the numeric potential fields, the graphs using the A* algorithm


3

(graphs of visibility, of tangents, Voronoi) the behavior-based models and the genetic algorithms, and others.9 These approaches guarantee that either the goal position will be reached with the path planning problem solution, or the methods will entirely analyze the free space and conclude that the goal position is unreachable. In this latter case, the path planning problem has no solution. However, the construction and maintenance of a global model require heavy computational work as well as a large amount of memory resource.

On the other hand, the local path planners use only the local information in a purely reactive manner. At every control cycle, the robot performs the action corresponding to the configuration of the obstacles located in the neighborhood. The local path planners are typically simpler than the global ones, since they can directly map the sensor readings to actions. Different methods are used to choose the next action according to the local environment: the potential fields, the Bug algorithm, the behavior-based models, the probabilistic models (Virtual Field Histogram, occupancy grids) the fuzzy logic, and etc.9 Although the local planners are simpler to implement, they do not guarantee global convergence to the goal position. The mobile robot may get trapped in a local minimum, and the method will follow a divergent path or a loop while attempting to escape from the deadlock condition.

For this reason, we can see that global path planning methods are more appropriate than local one for SEAD mission. And in this study, the potential field method is used to find initial paths. The path planning using artificial potential field or physical potential field has been studied since mid-1980s. An elegant approach for obstacle avoidance in using artificial potential field method was pioneered by KhatibnR 10 as it was studied by many others.

A. Modified potential Field method for Path Planning of SEAD The main idea of path planning with global potential field is to construct a suitable potential field with an

attractive global minimum at the goal point and repulsive local maxima at the obstacles. The path is then generated by following the gradient of a weighted sum of potentials. Numerous artificial potential functions have been proposed in the past decade. In this study, we use the mass diffusion equation to construct the potential field. Goal point (target point of SEAD), G, is considered the location of a virtual source which emits some gaseous substance, e.g. a scent or a perfume. While concentration at point G is kept constant, substance diffuses steadily into the surrounding space. If complete and instantaneous absorption of any substance reaching obstacle points or natural and artificial space boundaries is assumed, concentration values will always remain equal to zero at the corresponding point sets. As a result of the diffusion process, a concentration distribution develops over time and space. In its equilibrium state, the distribution shows a monotonously decreasing concentration along a path between the goal point, the point of maximum scent concentration, and an arbitrary robot starting point S within the substance filled space. The path between S and G can be easily found by following a steepest ascent strategy, i.e., streamline.

To make this approach suitable to SEAD, we define the dangerous area of each threat. Dangerous area is a circle with its center at the threat location. It can be obtained from initially known information, the power of radar, and etc. In real battle field, the radar powers cannot be all the same. And if we can get the information of them, it is better to use for improved results. The dangerous area consists of two areas, a critical area (very dangerous area), and a less dangerous area. The boundaries of critical areas are treated as boundaries of obstacles in diffusion field, and the concentration in this area are given as a constant value, zero. Then the path, streamline, cannot enter this critical areas, and the critical areas should be determined as the most dangerous areas being very close to threat and too dangerous to pass. The less dangerous area is less dangerous than critical areas. The critical areas or less dangerous area could be overlapped as shown in Fig. 5.

threats

(less)

dangerous

areas

critical

areas

Figure 5. Dangerous areas due to threats

Now, solve the unsteady diffusion equation given by Eq. (4) with boundary conditions and source at goal position, G. Then we can construct the potential field. The boundary condition is constant concentration, being the same with initially given concentration on whole field, . An unsteady diffusion equation is (0, )u x

2 2u a u gt

∂ u= ⋅∇ − ⋅∂

(4)


4

with the boundary conditions

2

20 0

( , ) 1000,

( , ) 0, G Gu t x x R

u t x x Rδ

= ∈

= ∈

Ω ⊂

Ω ⊂ (5)

and initial value (0, ) 0=u x , for 2x R∈Ω ⊂ . ( , )u t x denotes the concentration function over time and Cartesian space with ( ) 2[ , ]Tx x y Rδ= ∈ Ω∪ Ω ⊂ . Ω

defines a closed connected region, the battle field, including goal point, Gx . The boundry set δΩ is formed by obstacles and free space boundaries. a and are the diffusion constant and the substance disintegration rate, respectively. The solution of diffusion equation can be obtained by analytical method or numerical method. In this case, the boundaries are too complicated to solve analytically and in this study the numerical method, FDM(Finite Difference Method) is applied. By application of the standard finite difference methods, the following time and space discretized analogy of Eq. (6) can be obtained for a grid point or node

2 0g ≥

( )( , )r x y δ= ∈ Ω∪ Ω .11

( )2

1; ;; ;2

1( )

Mi r i r

i m i r i rmm N r

u u a u u g uhτ

+

=∈

−;= ⋅ − −∑ ⋅ (6)

where, τ denotes the time step size and h is grid width. and M represent the set and number of neighboring nodes of

( )N rτ considered for an approximation of the Laplace operator, 2 ( )∇ ⋅ . For g 0= fastest convergence of the

difference scheme is achieved by selecting 2 2 τa h M= ⋅ . In the simulation in section IV, the battle field, assumed as a 25km by 25km square, is divided into 250x250 elements (resolution:100m) and all of the elements in FDM are squares of the same sizes. The smaller elements the field is divided into, the more detailed results can be obtained. But, the excessive computational load is required also. This work is not for the real mass diffusion analysis and it just to derive the potential field which has the similar character as that of the mass diffusion in the real world. Then, it is not needed to divide it into very small elements. In addition, this work should be finished before the UCAVs take off, and it may be possible to make use of the parallel computer.

If the potential field is obtained by numerical method described as above, a steepest ascent path, following the gradient of ( , )x∞u t from arbitrary starting points (except the critical are) to the goal point, can be computed and each path never comes into the critical area, and it is continuous and smooth. Especially, as critical areas were treated as obstacles which obstruct the mass diffusion, paths avoid the critical areas with some distance. But if you want safer path than it, it is possible to modify it to avoid the dangerous areas. When the path is constructed by steepest ascent rule, if it enters the dangerous area, the repulsive force from threat is applied to it. This repulsive force is determined by Eq. (7)

22 2Repulsive force: ( ) ( ) ( , )r rF k x x y y u t x∞= − + − ∇

(7)

where, [ ], Tx x y= is current position and ( , )r rx y

denotes the position of threat which has the dangerous area concerned now. Also, is the fourth power of

distance to the threat.

22( ) ( )r y y− + − 2rx x

( ,u t x∞∇

)) denotes the norm of

gradient at point ( ,x y . And it implies the magnitude of tendency to go through original paths. And gain (0,1)k ∈ has to be determined by considering these factors,

and ( )x x− + 22 2( )r ry y− ( , )u t x∞∇ . Empirically, when

determined as in the range k , the better results can be obtained. In Fig. 6, a dashed line denotes the (0,0.5)∈

Figure 6. Path modification using repulsive force


5

original path found without repulsive force and a solid line denotes the modified path found with repulsive force. It is shown that the modified path avoids the dangerous area successfully by virtue of the repulsive force.

B. Strategies for a simultaneous attack Up until now, construction of the potential field and associated path finding is explained in the above. Each

UCAV should be assigned more than one candidate paths for simultaneous attack, as the main object of the cooperative control in SEAD. But, in the potential field constructed, the path from a certain start point to the goal is unique. Thus, strategies to find more candidate paths are required. There are three strategies; the first strategy is using the repulsive forces, the second strategy is utilizing the unsteady diffusion function, and the third strategy is new potential filed construction with a sink at the start point.

First, as mentioned previously, one can find more than one path; path with repulsive force and without repulsive force. Using repulsive force, we can find a safer path. The total cost of each path will be evaluated with the similar method in the section II. The threat exposure can be lowered, but the total cost (combination of threat exposure cost and length cost. The threat exposure cost is integrated by time. Thus it can be more dangerous if the total path length is elongated by modification) may not be lowered. It is impossible to decide which one is better for the team level cost so that these both paths could be candidates.

Second, using the fact that we use the unsteady diffusion equation, more paths could be found. As the unsteady diffusion equation is employed, one can get some different potential field from different time steps with just one iteration process. As a result of the diffusion process, there exists a minimum delay time before path computation from the start point to the goal point is initiated. If diffusion simulation is initiated at i 0= (in Eq. (6)) with initial function 0; 0xu = , a minimum of i time steps are required before diffusion reaches L= sr , i.e., u ; 0

zL r ≠ , with representing the number of grid nodes passed by the shortest path possible from the start point, S to goal point, G. For a fixed start and goal point, an average value for can be computed by

LL

, constantngL c n c= ⋅ = (8)

where and n gn denotes the dimension of space R (n 2n = , for planar case) and the total number of grid nodes, respectively. Finally, we can get some different paths from the potential field which has all different concentrations distributions due to different time steps, i . Figure 7 shows, two different paths from the same start point. The dotted line is the path obtained from the potential field of time, i 5000= , and solid line is the path of time step i 10000= .

critical area

(less) dangerous area

target location

Figure 7. Different paths of different time step

In Fig. 7, it is assumed that the radar powers are equal and the radiuses of dangerous areas are all the same in order to compare with the simulation results of the Voronoi-diagram method under the same condition in IV.

Third, some different paths from the new potential filed constructed with a sink at start point are generated. Considering the start point as a sink, ( , ) 10sxu t = − , the new potential field is constructed. The sink point is the singularity point and it is not possible to start exactly at a sink point. Despite the exact start point, new start points


6

can be selected as a point on the arc of a circle that is a sufficiently small distance away from the sink. However, there exist many candidate paths due to the choice of different initial directions around the circle, as one can see in Fig. 8.

critical area


target location

Figure 8. Various paths with sink at the start point

In Fig. 8, we can select the paths reaching the target location as candidates. Due to the local maxima at the boundaries the two paths cannot reach the target location. These local maxima exist only in the potential field with a source at the goal point (target location) and a sink at the start point.

After the candidate paths set of each UCAVs are determined, the length cost and threat exposure cost should be computed. And then we can select the best appropriate path of each UCAV for simultaneous attack using the same decomposition strategy mentioned in section II.

C. Strategies against the pop-up threat Each UCAV is assumed to be equipped with the sensors detecting the radar signal, and all information about the

pop-up threats are known to all UCAVs in the team when one of them detect the pop-up threat in almost real-time. Then, the counter-plan against unknown pop-up threats that may appear during the flight is necessary. The less dangerous area is determined as the circle with the radius as distance between current position and pop-up threat. And the critical area can be determined after the computation of hardware constant of the radar. The amount of power received by radar is the function of a distance12 and one can determine the radar power and critical area, as the UCAV moves on the initial path or some seconds. At this time, if each initial path of UCAV does not enter the dangerous area, replanning is not necessary, otherwise, all the paths should be replanned. Then, this replanning has two cases according to the current condition of each UCAV; the first one is initial path is affected by pop-up threat, and the other case is not affected case.

First, the case of, the pop-up threat affecting the initial path, is considered. The potential field which was computed already cannot represent the presence of pop-up threat. Thus, the some strategies in the following is proposed to find some new candidate paths. 1. Modification of the current path

This approach is to modify only the section which enters the dangerous area of the pop-up threat in current path, as it is illustrated in Fig. 9. This modification is to employ chain of masses system which suffers the virtual forces from threats 2. The initial path is divided into n sections of all the same size, and each nodes is treated as a mass. These masses are connected in a single line. And between masses, spring and damper system exist. Then we call this system chained masses systems. Each mass is suffering not only forces by spring and damper but virtual repulsive forces by threats. These virtual forces arise from the threats in their own dangerous


7

dangerous area

of popup threat

Pop-up

threat known

threat

current path

modified path

Figure 9. Path modification against the pop-up threat

area where the masses exit. The direction of this virtual force is opposite direction of the vector from a mass to the threat. And the magnitude of this virtual force depends on the priority of threat which is determined by radar power. The dangerous priority of popup-threat could be higher than the original, in order to make its effect strong, since it didn’t affect the initial potential field and path. The start node and end node are assumed as boundary condition of constant position. Then, if we solve the dynamic equation of the chained masses system in the section, the modified path can be derived as the details described in Ref. 2. 2. Suitable path finding and modification

This strategy is used in order to find some paths avoiding the popup threat and approach the goal point. First of all, it is necessary to find point A, in Fig. 10, at a distance of determined value on the initial path. It is determined as a local goal point. The global goal point can be a local goal point in case the distance between current and global local point is shorter than value determined. Then one has to find some paths which approach the local goal point avoiding dangerous areas of pop-up threat. These paths can be found by examination through the scanning route. This scanning route is determined as follows; It turns to the left and right with minimum turn radius at current point and goes forward to escape from the dangerous area. After escaping from the dangerous area, it follows along the boundary of dangerous area. If this scanning route meets the critical area of different threat, it stops at that point. Now one can find some points where an appropriate path starts. For convenience, let the straight line that connects the arbitrary point to the local goal point be LOS(Line of Sight) of the point. For some discrete points on the scanning route, examining the LOSs, some points whose LOSs don’t overlap with critical areas such as point B~H in Fig. 3.8 could be selected.. Then, the section of each LOS overlapped with dangerous area is selected and modified by the same method in 2. The scanning route can connect current position to the modified path, and we can get the paths just escaping the dangerous area. Now the path to the global goal point should be determined. This path can be easily found by using the potential field which was constructed in the initial planning step.

Initial path

Minimum

turn radius

Current

position

Popup

threat

Scanning route

Selected sections to be modified

Figure 10. Suitable path finding and modification

Now new paths of the UCAV which would fly through the pop-up threat dangerous area could be constructed. Then one should find some candidate paths of other UCAVs for TOT* matching1. Using the unsteady diffusion equation, it is possible to construct the potential field again with sink point at the current position of each UCAV. The initial values are given as the potential field computed already. It physically makes sense that a sink suddenly appears during unsteady diffusion. And at this time the minimum time step for numerical computation, L in Eq. (8), is quite small due to the shortened distance to the goal point.

IV. Simulation Results In this section, simulation of simple case of SEAD is presented. As the conditions of mission for 3 UCAVs

departing from different sites, one target and 20 known threats and their positions are given initially. And there exist unknown pop-up threat.

A. Simulation Results in case of only initially known threats As the first step, Voronoi node and edges presented in Fig. 3 in section II. And using the strategies, the best path

set that satisfies the simultaneous arrival at the target location is determined as it is shown in Fig. 11. Figure 12 (a) presents the dangerous areas for potential field method. As it was mentioned in previous section, it

is assumed that the radar powers are equal and the radii of dangerous areas are all the same in order to compare with the simulation results of the Voronoi-diagram method under the same condition. After the computation of the potential field, some candidate paths for each UCAV could be generated, as shown in Fig. 12 (b). Using the strategy for TOT* matching and simultaneous attack, the best path set is determined as shown in Fig. 4.12 (c). This is the result of the initial path planning. In TOT* matching scheme, the weighting factorλ of Eq. (3) is identical to that in TOT* matching of the Voronoi-diagram method. Accordingly, the similar path set is selected with that in Voronoi diagram method. TOT* is determined as 379 sec, being shorter than TOT*(= 408sec) of the Voronoi method. And we can see also the path length is shorter than that of Voronoi-diagram.


8

(a) (b) (c)

Figure 11. Path planning result without pop-up threats (Using Voronoi-diagram) (a)Trajectory generation result (b)Coordinated variable of each UCAV and TOT* matching (c) Time vs. distance to the target

(a) (b)


critical area


9

B. Simulation Results in case of unknown pop-up threat Figure 13 and 14 present the result of replanning case. It was assumed that all UCAVs engaging in this mission

can detect the pop-up threat in the range of 1,500m, and if one detects the pop-up threats, the information of this pop-up threat is shared with all other UCAVs. As shown in Fig. 13 (a), during the operations UCAV1 (displayed with solid line) detected the pop-up threats. In Figure 13 (b), the thin dashed line is Voronoi-diagram of initial condition and Each UCAV regenerates the Voronoi diagram and replans the path. Similarly, in Fig.14 (a) UCAV1 detect the pop-up threats and each UCAV computes their own candidate paths against the pop-up threat. And in the same manner as the initial path planning, the best appropriate path set is determined as shown in Fig. 14 (b).

(a) (b) Figure 13. Replanned path result in case of pop-up threat (Using Voronoi-diagram) (a)Result of Replanned Path in Real-time, (b)Final Results in case of pop-up threat

(c) (d) (e)

Figure 12. Path planning result without pop-up threats (Using Potential field method) (a)Dangerousareas of the virtual battle field, (b)Candidate paths, (c)Trajectory generation result (d)Coordinated variableof each UCAV and TOT* matching (e) Time vs. distance to the target


10

(a) (b)

Figure 14. Replanned path result in case of pop-up threat (Using Potential Field Method) (a)Candidate paths against the pop-up threat, (b)Final Results in case of pop-up threat

C. Analysis The advantages of new path planning in comparison with the Voronoi method are summarized as: 1) As we can see in the simulation results, the new method shortens the unnecessary elongated path lengths and

results in small TOT*. The threat exposure cost is integrated by time. This implies if UCAV is exposed to the weak threat for a long period of time, the cost can be larger than that of exposure to the strong threat for a relatively short moment. Consequently, maintaining a small TOT* can reduce the total cost almost always. And the new method can reduce the total cost as well.

2) The path made by the new method is continuous and smooth. Then, if real-time trajectory is generated, the computed control input does not violate the input magnitude constraints, and the off-tracking error is reduced. But the path made by the Voronoi-diagram method consists of straight lines, and input magnitude constraints are easily violated and the off-tracking error remains large.

3) Since the diffusion equation holds in R , generally, the new method using the diffusion equation can be applied to 3-D easily. The Voronoi-diagram also can be generated in , but the application to the path planning in

is not easy.

n

nR; ( 2)n n >R4) If a real obstacle such as a mountain and a building exists in battle field, the new method can easily find the

appropriate paths by including the boundaries of real obstacles in the boundary conditions of potential field computation. But for the Voronoi-diagram method, the Voronoid nodes or edges overlapped with real obstacles have to be excluded in path planning, or the edges should be modified as we can see in Fig. 15 (a). And in this figure, the shortest path is not available due to the real obstacle. But in the same situation, the new method can find the best path very easily. See Fig. 15 (b).


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Available path

Not available path

Real obstacle

(a) (b)

Figure 15. Shortest paths in case of existence of a real obstacle

5) Assume that more than one radar exist close to each other. By Voronoi diagram method, the path represented

by bold line in Fig. 16 (a), could be a candidate path although it may be too dangerous to pass the space between two radars. But, with the new method, such a path described in Fig. 16 (b) never can be a candidate path.

(a) (b)

Figure 16. Candidate paths in case of very close threats

On the order hand, there exist some disadvantages as follows: 1) The heavy computational time is required for numerical method. But, since this computation is performed

before the UCAVs take off, it may not degenerate significantly the original merit of the new method. 2) It may not generate various candidate paths. The phase of potential field used on this study is similar to the

harmonic function as Fig. 17 (a). And it is impossible to make a candidate path such as the dashed line in Fig. 17 (b). But, with the Voronoi-diagram it is possible to make a candidate path such as the bold solid line in Fig. 17 (b). The new method makes mostly similar paths and it helps TOT matching sometimes, but not always. These characteristics could be understood through the TOT vs. cost graphs.


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(a) (b)

Figure 17. Variety of candidate paths (a)Phase of potential field, (b)An impossible path in new method and a possible path in conventional method

In Fig.18 the solid line represents ideal TOT distribution for TOT* matching. It is continuous and spread into a wide range. The dashed line, TOT distribution by a new method is continuous, respectively, but large TOT does not exist. The dotted line, TOT distribution by a Voronoi diagram, is discontinuous, but large TOT exists. The new potential method based on the potential field method has advantages and disadvantages in terms of TOT distribution.

3) Sometimes, each path from different UCAVs can be overlapped in close position to the target. Then it is not possible to attack from all different directions and we need some control laws for final attack angle. A strategy for an attack from different directions has to produce a feasible trajectory in the attack area. The attack area indicates the detectable area of anti-aircraft facility at the target location. In this study, path planning is accomplished in 2-D space. Since UCAV is not a free flying vehicle, steering is required and corresponding suitable control schemes are necessary. Exponential control law for a mobile robot, proposed in 14, could be a solution about this problem.

Figure 18. An example of a TOT distribution for one UCAV

V. Conclusion As a typical example of the cooperative control scheme for multiple UAVs, a suitable control scheme for SEAD

mission as one of the military tactics is studied. A new control method, for multiple UAVs departing from all different positions to arrive at the target position simultaneously, was proposed and the performance of the new approach is compared with the existing method by simulation results under the same conditions. It is shown that the total cost is decreased by the new method in simulation results. Moreover, the new method is more useful when the path planning in 3-D is required or the real obstacles such as mountains and buildings exist in a battle field.

References 1Randal W. Beard, Timothy W. McLain, and Michael Goodrich, "Coordinated Target Assignment and Intercept for

Unmanned Air Vehicles”, IEEE International Conference on Robotics and Autonomous, IEEE, Washington DC, May 2002. 2Timothy W. McLain, and Randal W. Beard, “ Trajectory Planning for Coordinated Rendezvous of unmanned Air Vehicles”,

AIAA Guidance, Navigation, and Control Conference and Exhibit, AIAA, Denver, August 2000. 3P. R. Chandler, and S. Rasmussen, “UAV Cooperative Path planning”, AIAA Guidance, Navigation, and Control

Conference and Exhibit, AIAA, Denver, August 2000. 4Timothy W. McLain, and P. R. Chandler, S. Rasmussen, and M. Pachter, “Cooperative Control of UAV Rendezvous”,

American Control Conference, Ailington, June 2001.


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5S. S. Iyengar et al., “Robot navigation algorithm using learned spatial graphs”, Robotica, Vol. 4, No. 2, 1986, pp.93-100. 6T. H. Cormen, C. E. Leiserson, and R. L. Rivest, Introduction to algorithms, Cambridge, MA:The MIT Press, 1990. 7David Eppstein, “Finding the K Shortest Paths”, Society for Industrial and Applied Mathematics Journal of Computations,

Vol. 28, No.2, 652-673, 1998. 8B. L. Fox, “K Shortest Paths and Applications to the Probabilistic networks”, OSRA/TIMS Joint National Mtg., Vol. 23,

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Representation in the Velocity Space”, Robotica, Vol. 19, 2001, pp.543-555. 10O. Khatib, “Real-time Obstacle Avoidance for Manipulators and Mobile Robots”, International Journal of Robotics

Research, Vol. 5, No.1, 1986, pp.90-98. 11G. K. Schmidt and K. Azarm, “Mobile Robot Navigation in a Dynamic World Using an Unsteady Diffusion Equation

Strategy”, Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Raleigh, NC, July 1992. 12M. C. Novy, “Air Vehicle Optimal Trajectory Between Tow Radars”, American Control Conference, AK, May 2002. 13M. J. Van Nieuwstadt, and R. M. Murray, “Real-time Trajectory Generation for Differentially Flat Systems”, International

Journal of Robust and Nonlinear Control, Vol. 8, 1998, pp.995-1020. 14O. J. Sordalen, and C. Canudas de Wit, “Exponential Control Law for a Mobile Robot: Extension to Path Following”, IEEE

international Conference on Robotics and Automation, Nice, France, May. 1992.


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