Three-dimensional Path Planning for Unmanned ...

Three-dimensional Path Planning for Unmanned Aerial Vehicles Based on Fluid Flow Xiao Liang Science and Technology on Aircraft Control Laboratory Department of Automation Science and Electrical Engineering Beihang University Beijing, People’s Republic of China 100191 [email protected]

Honglun Wang Science and Technology on Aircraft Control Laboratory Department of Automation Science and Electrical Engineering Beihang University Beijing, People’s Republic of China 100191 [email protected]

Dawei Li Science and Technology on Aircraft Control Laboratory Department of Automation Science and Electrical Engineering Beihang University Beijing, People’s Republic of China 100191 [email protected]

Abstract—Using the principles of fluid mechanics for flow around objects, a three dimensional (3D) path planning method for unmanned aerial vehicles (UAVs) in complex environments is studied. As a potential field method, it theoretically guarantees to avoid local minima with smooth paths and the modeling of environment is simple. First, an analytical solution is derived to determine the steady 3D fluid flow acting on a single spherical obstacle. Subsequently, an interpolation function is introduced to multiple obstacles avoidance. Finally, the maneuverability constraints of the UAV are imposed and flight paths are obtained. Added the effect of human factors, a Generalized Fuzzy Competitive Neural Network (G-FCNN) is proposed to evaluate the flight paths. In simulation, the path is smoother and more reasonable. In terms of evaluation, G-FCNN could considerate multiple factors and the result is satisfied.

Chang Liu Science and Technology on Aircraft Control Laboratory Department of Automation Science and Electrical Engineering Beihang University Beijing, People’s Republic of China 100191 [email protected]

the three main issues in this area are path planning in a dynamic environment, cooperative path planning and 3D path planning. In particular, 3D path planning can make full use of the maneuverability of a UAV, and a substantial amount of research has been carried out in this field. Many important path planning methods have been proposed in or applied previously to robots, including PRM (Probabilistic Roadmap), grid-based methods, artificial potential fields, A* search, and Genetic Algorithm (GA). In recent years, these methods have been improved and applied to 3D path planning for UAVs. Initially, Voronoi diagram [1], and A* search [2] is used to make 2D path planning. Now, these methods have been gradually developed to 3D. The common problem of this extension is the additional third dimension will result in geometric growth of the search space. Moreover, A* search use the expanding nodes and the tree structure to express terrain information, respectively, so when facing more complex environments or those involving high-dimensional calculations, these methods will require more storage space and time to solve because of the combinatorial explosion problem.

TABLE OF CONTENTS 1. INTRODUCTION .................................................1  2. PATH PLANNING BASED ON PRINCIPLES OF FLUID MECHANICS FOR FLOW AROUND OBSTACLE.............................................................2  3. ANALYTICAL SOLUTION FOR A STREAM TO AVOID OBSTACLES ...............................................3  4. COMPREHENSIVE EVALUATION AND OPTIMAL CHOICE OF THE PATH ..........................................5  4. SIMULATIONS AND ANALYSIS ...........................8  7. SUMMARY .......................................................11  ACKNOWLEDGMENTS .........................................11  REFERENCES.......................................................11  BIOGRAPHY ........................................................13 

PRM (Probabilistic Roadmap) [3] and RRT (RapidlyExploring Random Tree) [4] are both suitable to solve the high-dimensional problem and does not need complex environment modeling. These incremental algorithms allow termination as soon as a solution is found. During the last decade, more researches focus on the probabilistic completeness and optimum [5, 6]. When the search space is discrete for path planning, the paths are composed of some roadmaps or waypoints with distant intervals, which requires these paths to be smoothed when applied to UAV path planning. Nikolos et al. [7] used B-spline curves to simulate the 3D flight trajectory of aircrafts, and then used an evolutionary algorithm to optimize the B-spline curve control points. This method can generate a smooth 3D path, but it will become complex to

1. INTRODUCTION Efficient path planning is the fundamental requirement for a UAV to accomplish its mission autonomously. Currently, 978-1-4799-1622-1/14/$31.00 ©2014 IEEE

1

avoid passing through obstacles. GA can handle a variety of constraints such as the environment and maneuverability [8], and has been applied to multi-UAV cooperative path planning [9]. Compared with the complexity of the GA, the theory and application of PSO (Particle Swarm Optimization) to the multi-objective optimization problem is relatively simple. Swartzentruber et al. [10-12] took threats, fuel and deviation from the original path as constraints to make use of PSO. Alternate paths were generated using Bspline curves, and the method was simulated in a virtual battle space environment. But the path was a polyline composed of waypoints and not smooth enough.

based on the principles of a fluid flowing around objects and the phenomenon of stream flow from start to end. Through the combination of streamlines with constraints on UAV maneuverability, the resulting flight paths are smooth like the streamlines and can avoid local minima of potential field. Experimental evidence demonstrates the effectiveness of the method proposed and supports the theoretical claims.

Artificial potential fields perform well in real time. However, the method easily converges to local minima when path planning, especially in complex environments. Thus, many researchers have proposed improvements to solve the problem, including Khatib [13], the originator of the method. Frew et al. [14] constructed a vector field parameterized by a function of time, and vehicle motion can be specified to equal desired vector field. The method was 3D, real-time and robust to control. Then it succeeded in coordinate flight and tracking moving targets [15]. Recently, the Stream Function [16] has attracted attention with its smooth path and fast computation property, and has good performance in the flight test [17]. Then the method was extended to complex shaped obstacles [18]. But because of the restrictions of fluid concepts and calculations, this method is only suitable for 2D path planning.

The relation between the characteristic of ideal flow and path is studied first. By analyzing the principle of fluid mechanics, streamline is taken as the path in potential field navigation. Then it has been found that the path planning method has some similarities with potential field, and also some differences. Both two methods generate smooth paths with short period of time. On the other hand, the differences indicate that the method based on fluid mechanics can avoid local minima problems of potential field. Moreover, the method can guarantee the shortest distance.

2. PATH PLANNING BASED ON PRINCIPLES OF FLUID MECHANICS FOR FLOW AROUND OBSTACLE

Path Planning Method Based on Principles of Fluid Mechanics Figure 1 shows the streamlines in the ideal flow field. The streamlines are smooth and behave in the way of obstacle avoidance.

Another important part in this area is the optimization of paths, and many mathematical optimization algorithms have been used in these researches such as GA and PSO. Additionally, Mixed Integer Linear Programming (MILP) was introduced to path planning. How et al. [19] used MILP to compute optimal trajectories for agile autonomous vehicles in cluttered environments, and the method made full use of the mobility. Then the method was extended to 3D with new cost map and detailed paths [20]. Initial guess for MILP was constructed from the previous solution and to reduce the solution time. The quad rotor test flight demonstrated the effectiveness of the method [21]. The hot issue in optimization is how to reduce the error and improve the efficiency.

B

A Figure 1 – Ideal flow around circular obstacle The method is suitable for spherical obstacle or the obstacle can be replaced by sphere. First calculate the streamline by fluid mechanics and a starting point only has one streamline. Streamlines will not cross each other, so by changing starting point in a small degree, they will distribute in the ribbon from starting to finishing area. After treatment by UAV constraints, some of these streamlines are left as the alternative paths. Finally the paths are optimized and evaluated.

In summary, the researches of 3D path planning focus on how to solve the following issues: 1) The environment is modeled by complex grid or data structures, which consumes much of time. 2) The path is not sufficiently smooth or does not consider the constraints of UAV maneuverability.

Analysis of the Characteristic of Ideal Flow In this research, the analytical model is steady, incompressible, irrotational and nonviscous ideal flow. According to the definition of ideal flow [22], the schematic diagram of ideal flow around circular obstacle is shown in Figure 2.

3) Too many factors influence the evaluation of the path, and how to find an optimal or suboptimal path. In view of these issues, the research presented in this paper proposes a new theory and method that is suitable for UAV 3D path planning in complex environments. The method is 2

V∞

Vθ

1) In the area far from obstacles, path maintains the initial direction along straight line.

B

2) When close to obstacles, path will change direction along the approximate tangent of obstacles, which has the minimum deviation from original direction and ensures the shortest path.

P A

O

C

x

3) Path can avoid local minimum. Except stagnation points, the method must be able to find a path from starting point to finishing point.

Figure 2 – Ideal flow around circular obstacle

4) After through the area interfered by obstacles, path will return to the initial direction and maintain straight line.

A and C are stagnation points with minimum speed and maximum pressure. B is the point with maximum speed and minimum pressure. This issue will be proved in next section. Fluid will separate and converge on these points. Streamlines will not cross each other, so ideal flow in Figure 2 will flow along upper surface of the obstacle, rather than under surface where it will flow through stagnation point (chain line area in Figure 2). In addition, velocity decomposition can explain this phenomenon. Fluid flows straight forward to point P of the obstacle. Find the tangent of the obstacle at point P and the tangential

Based on the above, in the view of path planning, the method can ensure the shortest path, avoid local minima of potential field, and have outstanding features of obstacle avoidance. It is similar to potential field but the meanings of paths in these two methods are completely different. This distinction will result in potential field may encounter a balance of repulsive field and attractive field, and object does not reach the finishing point (local minima). However no force exists in ideal flow field, so the method has more advantages in the problem of local minima.

projection Vθ of V∞ . According to the direction of velocity projection, fluid will flow along upper surface. Ideal flow is always along the original direction of velocity, or the direction of velocity on tangential projection of obstacle until to the end. Therefore, the problem of local minima in potential field [23] does not exist in this path planning method. Specially, the method will find the path with the minimum deviation from the original direction, which means shortest path.

3. ANALYTICAL SOLUTION FOR A STREAM TO AVOID OBSTACLES The combination of fluid flow calculations with path planning originates from the Stream Function. The method has been successfully applied, for it can generate a smooth path and avoid local minima [24]. However, because of the restrictions of fluid flow concepts and calculations, this method is difficult to extend to three dimensions. This section introduces the analytical solution for steady 3D fluid flow acting on a single spherical obstacle, which is detailed in the work [25].

To explain another very important characteristic of this path planning method, potential field is employed as an analogy. The two methods both have the concept of field and their paths are similar to each other. In potential field, the value of repulsive field is determined only by the relative distance between object and obstacle and it has no relation with their positions. So in Figure 2, the value of repulsive force in A , B , and C is same. This situation is similar to the constant centripetal force in circular motion.

Problem Formulation Figure 2 is redrawn as Figure 3 in details. Suppose that the sphere is located at [0, 0] and that its radius is a . Consider the flow past the sphere in a uniform stream with velocity V∞ at infinity in the positive x direction. The calculation

In the method based on ideal flow, A and C have minimum speed and B has maximum speed in Figure 2. Suppose small fluid micelle has quality, then the force in A , B , and C is different. It means besides the relative distance, such kind of force also has relation with relative position. This is one of the most important differences in these two methods. It should be noted that there is no force exists in the ideal flow. The above analogy is to explain the difference and connection between the two methods.

θ

starts from axis Ox , and the distance between O and point P is OP = r . When θ = 90 , the point on the surface of sphere is B . Thus, the movement of the flow in plane I , which is through the axis Ox , is axisymmetric. According to the characteristic of axisymmetric movement, the flow in any plane that contains the axis Ox is the same as that in plane I . of

Properties of the method According to the characteristic of ideal flow, the path planning method based on principles of fluid mechanics for flow around obstacle will have following properties: 3

I

along the x , y , and z axis in three dimensions, respectively, which can be written as:

P

B

r

V∞

θ

O A

C

⎧ ⎛ ⎞ ∂ϕ 3a 3 x 2 a3 = V∞ ⎜ 1 + − V∞ ⎪u = 2 2 2 32 ⎟ 2 ∂x 2( x + y 2 + z 2 )5 2 ⎝ 2( x + y + z ) ⎠ ⎪ ⎪⎪ ∂ϕ 3a 3 xy = −V∞ (4) ⎨v = 2 ∂y 2( x + y 2 + z 2 )5 2 ⎪ ⎪ 3a 3 xz ∂ϕ = −V∞ ⎪w = ∂z 2( x 2 + y 2 + z 2 )5 2 ⎪⎩

x

a

Figure 3 – Flow past the sphere in a uniform stream at infinity The reason for which the 2D stream function cannot be extended to 3D will be given briefly. Consider the ideal expressions of a 3D stream function ψ and velocity potential ϕ in the cylindrical coordinates system [26].

∂ 2ψ ∂ 2ψ 1 ∂ψ ∂ 2ϕ ∂ 2ϕ 1 ∂ϕ , + − = 0 + + =0 ∂x 2 ∂r 2 r ∂r ∂x 2 ∂r 2 r ∂r

Assume that the coordinates of a UAV are

[u, v, w] describes the movement of the UAV along the streamline. By imitating the motion of fluid flow, the method achieves obstacle avoidance.

(1)

Multiple Obstacles Avoidance for Arbitrary Position and Arbitrary Flow Direction

From Equation 1, the equation for the 3D stream function ψ is not Laplace’s equation, and its solution is not a harmonic function, so ψ and ϕ do not satisfy the CauchyRiemann condition. It means that in the 3D case, the complex potential does not exist and cannot be used to solve the problem (unlike in 2D, complex potential is w( z ) = ϕ + iψ ). However, the equation for the velocity potential ϕ is Laplace’s equation, and it can be solved by the separation of variables or the source-sink method. Therefore, the solution can start from the equation of the velocity potential ϕ .

In work [25], the multiple obstacles avoidance is realized by data processing and curve fitting. Here, an interpolation function is introduced to multiple obstacles avoidance. Abbreviating [24], below is a method of weighting velocity when there are multiple obstacles. Let the configuration space contain m obstacles, with the i th obstacle

(

Let ui , vi and wi denote the x , y and z velocity components that would exist for a sink flow about obstacle i on its own. Define m distance functions bi : 3 → ,

Obstacle

bi =

According to work [25], the velocity potential in Figure 3 is:

a3 ) 2r 3

(x

2

− bx2i

) +(y 2

2

− by2i

) +(z 2

2

− bz2i

Which is the distance from the point

(2)

)

2

− ai

( x, y )

(5)

to the

boundary of the i th obstacle. The interpolation function is defined as below,

On the surface of sphere ( r = a ), here is vr = 0 ,

αi = ∏

3 vθ = − V∞ sin θ , vλ = 0 . Using Bernoulli integral, the 2 pressure coefficient on the surface is:

j ≠i

bj

w = ∑ α i wi .

(3)

(6)

bi + b j

Then, the final velocity field is u =

2

⎛V ⎞ p − p∞ 9 p= = 1 − ⎜ ⎟ = 1 − sin 2 θ 1 4 ⎝ V∞ ⎠ ρV∞2 2

)

( i = 1,..., m ), having radius ai and location bxi , byi , bzi .

Analytical Solution for a Stream to Avoid a Spherical

ϕ = V∞ r cos θ (1 +

( x, y, z ) . Then,

∑α u , v = ∑α v , i i

i

i i

i

i

Terrain Pretreatment and Virtual Obstacle

From Equation 3, it is obvious that ideal flow has minimum speed and maximum pressure in A and C , but maximum speed and minimum pressure in B . Notice that in Cartesian coordinate system (the x axis of the Cartesian coordinate x system is axis Ox ), cos θ = and r = x 2 + y 2 + z 2 . Let r u , v , and w represent the velocity component of flow

Terrain needs pretreatment so that the planed paths are suitable for UAV to flight. In addition, artillery position and radar are also equivalent to spherical obstacle. Suppose UAV flies with constant velocity. The movement of UAV can be separated into both longitudinal and latitudinal direction. In latitudinal direction, assume that the velocity of 4

Assume that the minimum turning radius of UAV is R0 . Make a vertical line of x axis and it passes through the point O1 . Define the crossover point of the vertical line and

UAV is v0 and the latitudinal maximum available acceleration is alat . Then the turning process of UAV is a circular arc and its radius is

Rlat = v02 alat . Similarly, the

obstacle O1 is A . Then a point O2 can be found on the

radius of circular arc in longitudinal direction is

extension line of AO1 , and AO2 = R0 . Thus the virtual

Rlon = v02 alon . The initial radius of obstacle Oi is Ri and

obstacle is the sphere whose radius is R0 and center is O2 .

new radius is Rnew = max{Ri , Rlat , Rlon } after pretreatment. So the streamlines which are generated in the terrain after pretreatment can satisfy the constraint of turning radius.

x

When obstacles overlap each other, the design of virtual obstacle is shown in Figure 4. Oi and Oi +1 represent the

A O1

i th and i + 1 th obstacles respectively, and their respective radius are Ri and Ri +1 . Thus the virtual obstacle is the circumscribed sphere of obstacle Oi and

centers of the

O2 r0

Oi +1 , and its radius is ( Ri + Oi Oi +1 + Ri +1 ) / 2 .

Figure 6 –Virtual obstacle (when real obstacle radius is larger than minimum)

Ri

Oi O new

When the real obstacle radius is larger than the minimum turning radius, the problem is simpler as Figure 6 where the variables have the same meaning with Figure 5. By defining a small parameter r0 > 0 , a point O2 can be found on the

Ri +1

x

Oi +1

Si

extension line of AO1 and O1O2 = r0 . The design can Figure 4 – Virtual obstacle (when obstacles overlap each other)

ensure the radius of obstacle O2 is larger than that of obstacle O1 , and the radius of obstacle O1 is larger than turning radius.

When the line determined by current position of UAV and the target comes across the center of circular obstacle, the method will fail because of the stagnation point. A stagnation point is the place where velocity becomes zero. The random walk method has been proposed to solve this problem, but it also brings some disadvantages on smoothness and feasibility for UAV. Here a virtual obstacle is designed for stagnation point according to the constraint of UAV as Figure 5 and Figure 6.

Because virtual obstacle can satisfy the constraint of UAV and avoid the stagnation point, it will replace the real obstacle and be taken into account when making path planning.

4. COMPREHENSIVE EVALUATION AND OPTIMAL CHOICE OF THE PATH By changing starting point in a small area, streamlines will distribute in the ribbon from starting to finishing area. In this way, streamlines constitute a set. The streamlines in set are discretely distributed in configuration space and will not cross. In addition, the flight paths must be evaluated according to certain criteria so that the UAV can choose the optimum one to fly. To perform a comprehensive evaluation and make the optimal choice, a Generalized Fuzzy Competitive Neural Network (G-FCNN) is proposed. GFCNN can find the optimum path which accord with the characteristic of flow around obstacles.

x

A O1

R0

O2

Calculation of Sub-objective Functions

Figure 5 – Virtual obstacle (when real obstacle radius is smaller than minimum turning radius)

As a result of the 3D fluid flow calculations, there are many streamlines and they comprise a set of acceptable flight paths. To evaluate the flight paths, the set P should be 5

classified. Here, four sub-objective functions are designed to represent the properties of the paths: length of the path, safe distance for obstacle avoidance, and times of motion in the longitudinal and latitudinal directions. Note that the i th path of set P is P i (i = 1, 2...n) . P i consists of m discrete

points,

and

f1 ( Pji+1 )

P d

j th discrete point is

the

P ( j = 1, 2...m) .

a) f1 ( P ) and f 2 ( P )

i

1) Sub-objective function f1 ( P ) — length of the path

flon ( Pji )

The fuel constraint can be transformed into this subobjective function. To reduce the calculations, the length of the path will not be calculated precisely, instead, the 3D Manhattan distance between discrete points will be used. Sub-objective function

Pji+3

i j

Pji

i j

2)

d ij +1

i j +1

f1 ( Pji )

Pji+ 2

θ ij i j

P

Pji+1

θ

f lon ( Pji+1 )

i j +1

i

f 2 ( P ) — safe distance for

Pji+ 2

obstacle avoidance

vertical direction

Pji+3

i

Define a threshold Ds . d j is the minimum distance between the j th route segment of the

b) f 3 ( P )

i th path and all

i

obstacles. Thus, f 2 ( P ) can be calculated as

Figure 7 – Calculation of the sub-objective functions Comprehensive Evaluation and the G-FCNN

⎧ d ij > Ds f 2 ( P i ) = max f D ( Pji ) , f D ( Pji ) = ⎨1 / Dis , (7) i j =1...m −1 ⎩1 / d j , 0 < d j ≤ Ds

Because the dimension of each sub-objective function is not uniform, when choosing the weighting expressions of

f1 ( P i ) ~

i

3) Sub-objective function f 3 ( P ) — times of motion in the

evaluation, it is difficult to set the weights. On the other hand, many factors affect the optimality of path. Especially, an optimal path will prefer some certain factors in different task, such as shortest path or most concentrated fire. In fact, it is hard to find the optimal path finally. Because these factors are all artificially predetermined as threat model or cost function. It means human factors should engage in the evaluation. Therefore, the G-FCNN is proposed to make the optimal choice. The G-FCNN uses fuzzy operations and competitive learning, so it will tolerate some uncertain factors and contain human factors. To training samples, first generate the paths in a typical terrain. Second invite path planning experts to evaluate these paths. Finally the evaluated results are taken as training samples. Therefore, the G-FCNN after training will have the expertise in this area. When the G-FCNN evaluates the paths in a different environment, it can directly make the optimal choice similar to those of experts, achieving generalization.

longitudinal direction As shown in Figure 7(b), calculate the angle

θ ij

between

the j th and j + 1 th route segments in the longitudinal direction (the absolute value of the difference between the angles of climb). These angles are then compared with the standard threshold value α s . When θ > α s , the value of

f 3 ( P ) adds 1. m−2 ⎧1, θ ij > α s f 3 ( P i ) = ∑ flon ( Pji ) , flon ( Pji ) = ⎨ (8) i j =1 ⎩0, 0 ≤ θ j ≤ α s

4) Sub-objective function

f 4 ( P i ) — times of motion in

the latitudinal direction The calculation of

f 4 ( P i ) to perform the comprehensive

f 4 ( Pi ) is similar to that of f3 ( Pi ) .

Structure of G-FCNN—The network structure of G-FCNN is shown in Figure 8.

6

Here, i1 ∈ {1, 2...m1} , i2 ∈ {1, 2...m2 } , ... , ik ∈ {1, 2...mk } , j = 1, 2...m . The neurons in the third layer consist of one of

α1 w11

μ11

y1

f1 ( P i )

α2

f2 (Pi )

the fuzzy segmentations of each f k . The number of nodes

μ1m

1

in third layer is N 3 = m , m =

Pi f3 ( Pi )

k

∏m . i =1

i

a = [0,1, 0,0, 0]

μ41

The fourth layer is the competitive layer and consists of two parts: the input of the previous layers and the mutual inhibitory value within the competitive layer. The output of the neurons in this layer is the result of competition. Because the G-FCNN has more latent layers, the competitive layer is useful for improving the training speed. The inputs of the neurons in this layer are

α m−1

i

f4 (P )

yr

wrm

μ4m

αm

4

Figure 8 – Generalized Fuzzy Competitive Neural Network (G-FCNN)

m

yi = ∑ wijα j

calculates the value of the sub-objective functions

f k ( Pi )

j = 1...m , and r represents the number of levels. If the value of yi is greater than the value of the other neurons in this layer, the outputs of yi and the other neurons are ai = 1 and ak = 0 (k = 1...r , k ≠ i ) , respectively. Finally, the output of the G-FCNN is an r dimensional vector a = [ a1 ,..., ar ] . When ai = 1 , the level of the flight path is i .

Here, i = 1...r ,

of each path. Because the numerical range of each subobjective function in different environments will not be the

f k ( Pi ) should be

same, before using the G-FCNN, i

normalized to f k ( P ) first using Equation 9. The number of nodes in this layer is N1 = k . In the following, and

f k ( Pi )

f k ( Pi ) will be denoted as f k and f k for short,

respectively.

Learning algorithm for parameter adjustment—Assume that the level of the sample from the experts is

f k ( Pi ) − min( f k ( P )) fk (P ) = , k = 1...4, i = 1...n (9) max( f k ( P )) − min( f k ( P )) i

r aact = [a1act ,..., aact ] (consists of 0 and 1). The error

function of the G-FCNN is

The second layer is the fuzzy layer, and each node represents a fuzzy variable. This layer makes a fuzzy segmentation of each sub-objective function and calculates the membership function

μkl

⎧( ymax + α − yi ) 2

1 r ⎪ E = ∑ Ei , Ei = ⎨ 2 i =1 ⎪

of input components in the

⎛ ( f k − ckl ) 2 ⎞ ⎟ , k = 1...4, l = 1...mk σ kl2 ⎝ ⎠

4

k =1

The third layer is the fuzzy inference layer, which is used to match the antecedent of the fuzzy rules and calculate the relevance grade of each rule. 2

k

(13)

i ai > aact

α and β are adjustment coefficients that are greater than zero, and the greater the values of and are, the greater the training times of the samples may be. This kind of dynamic error can greatly reduce the training times compared with constant error. If the level is different from that of the experts, the competitive neurons with the incorrect output only need to learn the one with the maximum or minimum output instead of training all of the competitive neurons. For the competitive neurons that need training, as long as their outputs are greater or less than the outputs of some neurons according to the requirement of grading, the G-FCNN can provide correct results. The reason for designing this form of error function is not the pursuit of minimum error, which is the traditional goal, but rather the highest correct rate of comprehensive evaluation.

N 2 = ∑ mk .

1

i ai < aact i ai = aact

Here, ymax = max{ y1 ,..., yr } , ymin = min{ y1 ,..., yr } .

(10)

Here, mk is the fuzzy segmentation of the k th subobjective function, and the number of nodes in this layer is

α j = min {μ1i , μ2i ...μki }

0

2 ⎩ ( ymin − β − yi )

fuzzy variable set.

μkl = exp ⎜ −

(12)

j =1

The first layer in Figure 8 is the input layer, which

(11)

7

Differential equations 4 can be solved by functions Ode45 or Ode23 in MATLAB. Using both the fixed-step and variable-step methods, all of the streamlines can be generated within 10−2 s after determining the calculation interval. The amount of computation in this ideal situation is small, and the streamlines can satisfy the requirements of obstacle avoidance in path planning. From the results shown in Figure 9, the streamlines are smooth and envelope the spherical obstacle well.

The fuzzy partitions of the input components are predetermined, so the parameters that need to learn are the connection weights between the third and fourth layers, as well as the fuzzy center and fuzzy width of the membership function in the second layer. The learning algorithm of the G-FCNN is based on the error backpropagation algorithm; the algorithm for adjusting the parameters is shown in Equation 14. Assume that the learning rate is η . Then, i ∂E / ∂yi = −( ymax + α − yi ) when ai < aact , and

r ⎧ ∂E ∂E ∂E ∂E , , δ (2) , i = 1, 2...r , j = 1, 2...m =αj = −∑ wij ⎪ wij (k + 1) = wij (k ) − η j ∂ w ∂ w ∂ y ∂ yi i =1 ij ij i ⎪ ⎪ 2( fi − cij ) ∂E ∂E ⎪ , , i = 1...4, j = 1, 2...mi = −δ ij(1) ⎨cij (k + 1) = cij (k ) − η σ ij2 ∂cij ∂cij ⎪ ⎪ 2 ⎪σ (k + 1) = σ (k ) − η ∂E , ∂E = −δ (1) 2( fi − cij ) , i = 1...4, j = 1, 2...m ij ij i ⎪⎩ ij σ ij3 ∂σ ij ∂σ ij

Here,

Multiple Obstacle Avoidance of the Flight Path—The range for path planning is 40 × 40 × 20 , the starting point is [ 0, 0] ,

fi is the value of the i th sub-objective function after

normalization,

δ

m

(1) ij

= ∑δ k =1

(2) k

⎛ ( f − c )2 ⎞ i ij ⎟ and Sij exp ⎜ − 2 ⎜ ⎟ σ ij ⎝ ⎠

the finishing point is [33,33] , and the information about the obstacles is listed in Table 1. Table 1 shows that, the complex environment consists of several spherical obstacles with different radii and that their centers not all in plane xy .

m = ∏ mi . According to Equation 11, in the learning algorithm, if the result of the minimum is

(14)

μi j , the value of

Table 1. Information of obstacles

i Sij is Sij = 1 ; otherwise Sij = 0 . The case of ai > aact is

No.

similar to the above.

4. SIMULATIONS AND ANALYSIS Three-dimensional path based on flow around obstacles

1

2

3

4

5

6

7

8

9

10

2

4

7

8

9

12

12.5

13

13

15

10

5

14

8

19

17.5

11.5

21

6

8

0.3

0.3

0

0.5

0

-0.1

-0.2

-0.2

0.2

0

radius

1.5

0.7

0.7

1

0.5

1

1.7

1.3

1.5

0.5

No.

11

12

13

14

15

16

17

18

19

20

16.5

17

20

21

23.5

25

25

26

27

28

17.5

25

11.5

27

23

27

17.5

15

21

26

0.5

0.2

-0.2

0

0.4

0

0.3

0

-0.1

0

2

1.5

1.5

1

1

1.3

2

0.5

1.3

1.3

x y z

x y z

Streamlines past Spherical Obstacle—The application of the method described in Section 3 is demonstrated in Figure 9. The direction of the flow is in the positive direction of the x axis, and the center of the spherical obstacle is located at [0, 0] . The fluid model is steady, incompressible, irrotational and nonviscous ideal flow, so the streamlines will not separate from the sphere after passing by it.

radius

Stream Function [16] and the path planning method based on principles of stream avoiding obstacles [25] are employed to verify the effectiveness of the proposed method. Because Stream Function is a two-dimensional path planning method based on fluid mechanics, the simulation will be helpful to find the characteristics of this method. Figure 10 shows the paths around multiple obstacles. Here, only one path of the proposed method is displayed, which is compared with the ones calculated by Stream Function and the method in Ref. [25]. Figure 10(a) and 10(b) are two different viewing angles of the paths, which have the same starting point, finishing point and obstacles.

Figure 9 – Streamlines around the sphere obstacle 8

the reason makes the proposed method have a good performance on terrain following and terrain avoiding. Therefore, the paper improves the method in [25] on the amount of calculation and the rationality. Comprehensive Evaluation of the Path Training of G-FCNN—According to the structure of the GFCNN in Figure 8, each sub-objective function is divided into five fuzzy sets, so mk =5 ( k = 1...4 ). Then, the 4

number of nodes in the third layer is m = ∏ mk = 625 . A

a)

k =1

larger value of will lead to excessive latent layers, which will raise more stringent requirements on both the samples

b)

and training time. Because the value of

f k is normalized in

[0,1] , the fuzzy width of the membership function is initially set to

σ kl = 0.2 ( k = 1...4, l = 1...mk ),

and the

fuzzy center ckl ( l = 1...mk ) of the membership function is initially set to 0.1, 0.3, 0.5, 0.7, and 0.9, respectively (in accordance with the principle of the average division). The connection weights between the third and fourth layers, as well as the parameters α and β of the error function, are also initially set to 1. When training, the values of α and β are gradually reduced along with the increase in the accuracy of the G-FCNN.

Figure 10 – Paths around multiple obstacles

In the fourth layer, the number of levels is set to five, so r = 5 . All of the computations are performed using the programming environment MATLAB (version 2010b) on a computer with 8GB of RAM and an Intel I7 64-bit processor (3.2 GHz) running the operating system Windows Vista. Through the analysis of papers on path planning, several typical environments are designed which cover some special cases such as obstacles are closer to the starting and finishing points, some special large obstacles, and overlapped obstacles. Then, paths will be obtained by the method of this paper. After evaluation by experts, the results will be taken as the samples of the network.

The red path is the 3D path proposed in this paper, the black one is from the Stream Function and the blue one is Ref. [25]. According to the UAV constraints and the calculation of the sub-objective functions in Section 5, the simulation is specified with the following parameters: Φ =40 , Ds =0.05 , and α s =1 . Because a single path cannot be normalized, the values of f1 ~ f 4 are: 78.1408, 23.26, 89, and 41 in red path; 83.2757, 22.22, 0, and 152 in black path; 62.8158, 21.27, 68, and 57 in blue path. Here, the Stream Function is developed in 2D, so f 3 =0 in black path. The black path has the longest journey and most motions in the latitudinal direction. Although the red path is not very outstanding in safe distance and times of motion, it has better performance of adaption to terrain. In 3D, the obstacles located far from the path only have a few influences on the proposed method, which is similar in 2D with the Stream Function and is a property of the artificial potential fields. It is consistent with the analysis in Section 2. However, compared with the 2D Stream Function method, the 3D path planning method will make more reasonable use of environmental information and make full use of the UAV’s 3D maneuverability.

As the form of the error function in Equation 13, the GFCNN pursues the correct rate of grading. In a typical environment, the initial correct rate of 2,000 standard training samples is 6.7%. Using the backpropagation method [27] to train the network, the correct rate reaches 97.5% after 13,000 training times. The learning algorithm of Equation 14 is derived based on a first-order gradient algorithm, and the speed of convergence will reduce after the correct rate reaches 97%. The G-FCNN with this correct rate is considered to be acceptable for use, having the expertise for path planning, and being able to give the evaluation results that are similar to those of experts.

In Figure 10, the red path in this paper is similar to the blue one, but it is more acceptable. Because an entire 3D path is obtained by data processing in Ref. [25], it can be seen that there is a turning point existing in the blue path. Moreover,

Optimal Choice from Multiple Paths—Using the positions of the obstacles in Table 1, parts of alternative streamlines from the starting point to the finishing point are shown in Figure 11. Only 34 streamlines are shown here, and these 9

streamlines can satisfy the accessibility requirements while avoiding obstacles. After the consideration of UAV maneuverability and the optimal choice from the G-FCNN, the evaluated flight paths are shown in Figure 12. After comparison with Figure 11 and 12, 18 paths not suitable for the UAV are removed in Figure 11, and the remaining 16 paths that satisfy the constraints on UAV maneuverability will be sent to the G-FCNN to perform a comprehensive evaluation.

es =

des qS CT v0 1 =− , q = ρ v02 dl mg ηecη p 2

(15)

where v0 is airspeed;

S is wing area; CT is thrust coefficient; ρ is density of the air; ηec is the motor efficiency; η p is the propeller efficiency. The energy change is always negative, because UAV constantly consume energy. The integral of Equation 15 on li is:

es (li ) = −li i

qS CT v0 mg ηecη p

(16)

This is consistent with the reality that onboard energy of UAV relates to engine thrust, airspeed and flight distance. Take potential energy and kinetic energy into account, so total energy of UAV from i th to i + 1 th waypoint is:

E( i ,i +1) = mghi +1 +

m 2 v0 + mges (li ) 2

(17)

Here, the altitude from i th to i + 1 th waypoint is

Figure 11 – Set of streamlines

considered to be constant

hi +1 . Thus, the total energy of the

path for UAV is Etot =

M −1

∑E i =1

( i ,i +1)

. The terrain is

dimensionless, so the energy in Table 2 is also dimensionless and normalized to [−1, 0] . In simulation, the parameters of UAV are v0 = 80m / s , S = 30m 2 , CT = 140 ,

ρ = 1.2kg / m3 , ηec = 0.8 , η p = 0.8 , g = 9.8m / s 2 , and m = 6000kg . From Table 2, the results of evaluation in G-FCNN are similar to those in energy. The interval of each grade and values of energy can basically constitute bijection. To IIIc and IVc, the results of evaluation are different between GFCNN and energy. It is because safe distance for obstacle

Figure 12 – Evaluated flight paths Finally, the flight paths are divided into five grades. Red, yellow, green, blue and black represents the grades I to V, respectively, and the paths in grade I are the optimal ones. In Figure 12, there are two red paths with grade I, two yellow paths with grade II, four green paths with grade III, three blue paths with grade IV and five black paths with grade V. In Table 2, two flight paths of each grade are chosen and the values of their normalized sub-objective functions are listed in detail. Here the evaluation by energy is also employed to verify the effectiveness of G-FCNN. Suppose a path consists of N route segments and N + 1 waypoints. The route segment from i th to i + 1 th waypoint

avoidance

f 2 is one of sub-objective functions in G-FCNN.

Correspondingly, evaluation in energy does not include the information of

f 2 . The evaluation of paths is influenced by

many factors and depends on different task, so subjective factor plays a significant role in the process of evaluation. In view of this consideration, G-FCNN is more suitable for expression of human experience and variety of factors.

is li . In constant-velocity flight with no wind, the rate of change of onboard energy lost is [28]:

10

Table 2. Values of normalized sub-objective functions and energy in each grade

Grade

Ia

I

IIb

II

III

IIIc

IVc

IVb

Vb

Va

f1

0.1837

0.2859

0.2714

0.3216

0.4313

0.6505

0.5281

0.8936

0.6546

0.7318

f2

0.1921

0.3012

0.3793

0.4159

0.6150

0.5738

0.7864

0.8721

0.4412

0.7923

f3

0.1314

0.3461

0.3123

0.4534

0.3907

0.4631

0.5613

0. 5781

0.7021

0.6735

f4

0.1537

0.1709

0.2371

0.1132

0.4211

0.5485

0.4702

0.5518

0.5983

0.6331

Energy -0.1998 -0.2573 -0.3107 -0.4321 -0.5294 -0.6501 -0.6212 -0.7913 -0.8121 -0.9089 Ia and IIb demonstrates that the paths with shorter lengths, REFERENCES fewer times of motion and higher levels of safety will be [1] Beard, R. W., McLain, T. W., Goodrich, M. A., and graded higher, which is consistent with the actual situation. Anderson, E. P., “Coordinated Target Assignment and Notice that in some cases, the length of the path, the times Intercept for Unmanned Air Vehicles,” IEEE Transactions of motion and the safe distance cannot be in the optimal b b on Robotics and Automation, IEEE, Vol. 18, No. 6, Dec. state simultaneously such as IV and V . When there are a 2002, pp. 911-922. little differences between the values of these four subobjective functions, the G-FCNN will consider times of doi: 10.1109/TRA.2002.805653 motion in the longitudinal direction f 3 first, the times of [2] Yang, H. I., and Zhao, Y. J., “Trajectory Planning for motion in the latitudinal direction f 4 second, then the safe Autonomous Aerospace Vehicles amid Known Obstacles and Conflicts,” Journal of Guidance, Control, and distance f 2 , and then the length of the path f1 . Dynamics, Vol. 27, No. 6, 2004, pp. 997-1008.

7. SUMMARY

doi: 10.2514/1.12514

In this paper, a three-dimensional path planning method for UAV is proposed. By imitating the motion of fluid flow, the method achieves obstacle avoidance. The method is similar to potential field but can avoid local minima and ensure shortest smooth path. Under the ideal conditions, the equation of the velocity potential can be written in analytical form. Then, an interpolation function is introduced to multiple obstacles avoidance. After eliminating the streamlines that are not suitable for UAV, four sub-objective functions composed of the length of the path, the safe distance for obstacle avoidance, and the times of motion in the longitudinal and latitudinal directions are designed to represent the properties of flight paths. The Generalized Fuzzy Competitive Neural Network (G-FCNN) with professional knowledge in path planning evaluates the flight paths so that the UAV can choose the suitable path in different tasks. The simulation result shows that the modeling of terrain and the computation of streamlines is relatively simple. Moreover, the path is smooth and the evaluation of G-FCNN is more convincible. Because the paths are parallel and will not cross, the method can be extended to cooperative path planning for multiple UAVs, which will be considered in future research.

[3] Kavaraki, L. E., Svestka, P., Latombe, J., and Overmars, M. H., “Probabilistic Roadmap for Path Planning in Highdimensional Configuration Spaces,” IEEE Transactions on Robotics and Automation, IEEE, Vol. 12, No. 4, Aug. 1996, pp. 566-580. doi: 10.1109/70.508439 [4] Prentice, S., and Roy, N., “The Belief Roadmap: Efficient Planning in Belief Space By Factoring the Covariance,” International Journal of Robotics Research, Vol. 28, No. 11-12, Nov./Dec. 2009, pp. 1448-1465. doi: 10.1177/0278364909341659 [5] Ladd, A. M., and Kavraki, L. E., “Measure theoretic analysis of probabilistic path planning,” IEEE Transactions on Robotics and Automation, IEEE, Vol. 20, Issue. 2, Apr. 2004, pp. 229-242. doi: 10.1109/TRA.2004.824649 [6] Karaman, S., and Frazzoli, E., “Sampling-based Algorithms for Optimal Motion Planning,” The International Journal of Robotics Research, Vol. 30, No. 7, Jun. 2011, pp. 846-894.

ACKNOWLEDGMENTS The authors are grateful to Professor Jianghao Wu for his insightful comments on fluid mechanics in this paper. This research has been funded by the National Natural Science Foundation of China under Grant 61175084/F030601.

doi: 10.1177/0278364911406761

11

[15] Frew, E. W., Lawrence, D. A., and Morris, S., “Coordinated Standoff Tracking of Moving Targets Using Lyapunov Guidance Vector Fields,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 2, 2008, pp. 290-306.

[7] Nikolos I. K., Valavanis K. P., Tsourveloudis N. C., and Kostaras A. N., “Evolutionary Algorithm Based Offline/Online Path Planner for UAV Navigation,” IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, IEEE, Vol. 33, No. 6, Dec. 2003, pp. 898912.

doi: 10.2514/1.30507 [16] Sullivan, J., Waydo, S., and Campbell, M., “Using Stream Functions for Complex Behavior and Path Generation,” AIAA Guidance, Navigation, and Control Conference and Exhibit, Austin, TX, AIAA Paper 20035800, Aug. 2003.

doi: 10.1109/TSMCB.2002.804370 [8] Anderson, M. B., Lopez, J. L., and Evers, J. H., “A Comparison of Trajectory Determination Approaches for Small UAVs,” AIAA Atmospheric Flight Mechanics Conference and Exhibit, Keystone, CO, AIAA Paper 2006-6644, Aug. 2006.

[17] Campbell, M. E., Lee, J. W., and Scholte, E., “Simulation and Flight Test of Autonomous Aircraft Estimation, Planning, and Control Algorithms,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 2, 2008, pp. 290-306.

[9] Darrah, M. A., Niland, W. M., and Stolarik, B. M., “Increasing UAV Task Assignment Performance through Parallelized Genetic Algorithms,” AIAA Infotech & Aerospace 2007 Conference and Exhibit, Rohnert Park, CA, AIAA Paper 2007-2815, May 2007.

doi: 10.2514/1.30507

[10] Swartzentruber, L., Foo, J. L., and Winer, E. H., “ThreeDimensional Multi-Objective UAV Path Planner Using Meta-Paths For Decision Making and Visualization,” 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Victoria, British Columbia Canada, AIAA Paper 2008-5830, Sep. 2008.

[18] Daily, R., and Bevly, D. M., “Harmonic Potential Field Path Planning for High Speed Vehicles,” American Control Conference, IEEE, Seattle, WA, Jun. 2008, pp. 4609-4614.

[11] Swartzentruber, L., Foo, J. L., and Winer, E. H., “ThreeDimensional Multi-Objective UAV Path Planner Using Terrain Information,” 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, CA, AIAA Paper 2009-2222, May 2009.

[19] Schouwenaars, T., Mettler, B., Feron, E., and How, J., “Hybrid Model for Trajectory Planning of Agile Autonomous Vehicles,” Journal of Aerospace Computing, Information, and Communication, Vol. 1, 2004, pp. 629651.

doi: 10.1109/ACC.2008.4587222

doi: 10.2514/1.12931

[12] Foo, J. L., Knutzon, J., Kalivarapu, V., Oliver, J., and Winer, E. H., “Path Planning of Unmanned Aerial Vehicles Using B-Splines and Particle Swarm Optimization,” Journal of Aerospace Computing, Information, and Communication, Vol. 6, No. 4, 2009, pp. 271-290.

[20] Kuwata, Y., and How, J., “Three Dimensional Receding Horizon Control for UAVs,” AIAA Guidance, Navigation, and Control Conference and Exhibit, Providence, Rhode Island, AIAA Paper 2004-5144, Aug. 2004. [21] Culligan, K., Valenti, M., Kuwata, Y., and How, J. P., “Three-Dimensional Flight Experiments Using On-Line Mixed-Integer Linear Programming Trajectory Optimization,” Proceedings of the 2007 American Control Conference, IEEE, Marriott Marquis Hotel at Times Square, NY, Jul. 2007. pp. 5322-5327.

doi:10.2514/1.36917 [13] Brock, O., and Khatib, O., “High-speed Navigation Using the Global Dynamic Window Approach,” Proceedings of the IEEE international conference on robotics and automation, IEEE, Detroit, Vol. 1, May 1999. pp. 341-346.

[22] Philip, J. P., “Fluid Mechanics: SI Version,” 8th International student edition. Washington: Wiley, 2011.

[14] Lawrence, D. A., Frew, E. W., and Pisano, W. J., “Lyapunov Vector Fields for Autonomous Unmanned Aircraft Flight Control,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 5, 2008, pp. 1220-1229.

[23] Eun, Y., and Bang, H., “Cooperative Control of Multiple Unmanned Aerial Vehicles Using the Potential Field Theory,” Journal of Aircraft, Vol. 43, No. 6, 2006, pp. 1805-1814.

doi: 10.2514/1.34896

doi: 10.2514/1.20345

12

[24] Waydo, S., and Murray, R. M., “Vehicle Motion Planning Using Stream Functions,” Proceedings of the 2003 IEEE International Conference on Robotics and Automation, IEEE, Taipei, Taiwan, Sep. 2003, pp. 24842491.

BIOGRAPHY Xiao Liang received the B.S. and M.S. in Northeastern University, and Ph.D. degrees in Navigation, guidance and control from Beihang University. He works in the school of automation, Shenyang Aerospace University. His research interests covers path planning, terrain following and terrain avoiding, situation awareness, and autonomous flight control for unmanned aerial vehicles.

[25] Liang, X., Wang, H. L., and Li D. W., “Threedimensional Path Planning for Unmanned Aerial Vehicles Based on Principles of Stream Avoiding Obstacles,” Acta Aeronautica et Astronautica Sinica, Vol. 34, No. 7, 2013, pp. 1670-1681. doi: 10.7527/S1000-6893.2013.0061

Honglun Wang received the B.S., M.S., and Ph.D. degrees in systems engineering from Northwestern Polytechnical University, China. Since 2004, he has been a professor affiliated with navigation, guidance and control. His research interests include path planning, autonomous flight control for unmanned aerial vehicles, sense and avoid technology for unmanned aerial vehicles.

[26] Liao, S. J., “An Analytic Approximation of the Drag Coefficient for the Viscous Flow past a Sphere,” International Journal of Non-Linear Mechanics, Vol. 37, Issue. 1, Jan. 2002, pp. 1-18. doi: 10.1016/S0020-7462(00)00092-5 [27] Horn, J. F., Schmidt, E. M., Geiger, B. R., and DeAngelo, M. P., “Neural Network-Based Trajectory Optimization for Unmanned Aerial Vehicles,” Journal of Guidance, Control, and Dynamics, Vol. 35, No. 2, 2012, pp. 548-562. doi: 10.2514/1.53889

Dawei Li Associate professor at Unmanned Aerial Vehicle Research Institute, Beihang University. He received a PH.D in Navigation, guidance and control from Beihang University in 2013. He has been with UAV design for more than 14 years. His research interest covers UAV design method, modern flight control and simulation, computational fluid dynamics. Chang Liu received the B.S. and M.S., from China Agriculture University. She is presently a Ph.D. candidate in aerospace engineering from Beijing University of Aeronautics and Astronautics. Her research interests are in the areas of guidance of unmanned aerial vehicle, data fusion, and dynamic control theory.

[28] Chakrabarty, A., and Langelaan, J. W., “Energy-Based Long-Range Path Planning for Soaring-Capable Unmanned Aerial Vehicles,” Journal of Guidance, Control, and Dynamics, Vol. 34, No. 4, 2011, pp. 10021015. doi: 10.2514/1.52738

13

14

15