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a variant of flower pollination algorithm is proposed to solve the UUV path planning ... Osama Abdel-Raouf has used an improved FPA to solve Sudoku Puzzles in 2014. 7. FPA has been used to solve Large Integer Programming Problems by Ibrahim ..... based incremental learning)25 is a type of genetic algorithm where the ...
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International Journal of Pattern Recognition and Artificial Intelligence c World Scientific Publishing Company ⃝

An Improved Flower Pollination Algorithm for Optimal Unmanned Undersea Vehicle Path Planning Problem

yongquan Zhou∗ Key Laboratory of Guangxi Higher Schools for Complex System and Intelligent Computing Nanning,530006, China

E-mail: [email protected] Rui Wang College of Information Science and Engineering, Guangxi University for Nationalities, Nanning,530006,China

Path planning of Unmanned Undersea Vehicle is a rather complicated global optimum problem which is about seeking a superior sailing route considering the different kinds of constrains under complex combat field environment. Flower pollination algorithm is a new optimization method motivated by flower pollination behavior. In this paper, a variant of flower pollination algorithm is proposed to solve the UUV path planning problem in 2D and 3D space. Optimization strategies of particle swarm optimization are applied to the local search process of IFPA to enhance its search ability. And in the progress of iteration of this improved algorithm, a dimension by dimension based update and evaluation strategy on solutions is used. This new approach can accelerate the global convergence speed while preserving the strong robustness of standard flower pollination algorithm. The realization procedure for this improved flower pollination algorithm is also presented. To prove the performance of this proposed method, it is compared with nine population based algorithms. The experiment result shows that the proposed approach is more effective and feasible in UUV path planning in 2D and 3D space. Keywords: UUV path planning problem; flower pollination algorithm; dimension by dimension improvement; engineering optimization.

1. Introduction Since 1996 the US Navys Naval Undersea Warfare Center (NUWC) has been developing a concept for undersea warfare in the new century. Playing a major role in this futuristic concept is unmanned undersea vehicle (UUV) that will be capable of carrying out surveillance, tactical oceanography, mine warfare, and anti-submarine warfare missions. The Manta Test Vehicle (MTV) 1 is the first prototype. In modern society, Unmanned Undersea Vehicle (UUV) is one of inevitable trend of the modern nautical weapon equipment which develops in the direction ∗ Typeset

names in 8 pt roman, uppercase. Use the footnote to indicate the present or permanent address of the author. 1

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of unmanned attendance and intelligence. Research on UUV directly affects battle effectiveness of the navy force and is fatal and fundamental research related to safeness of a nation. Path planning is to generate a space path between an initial location and the desired destination that has an optimal or near-optimal performance under specific constraint conditions, and it is an imperative task required in the design of UUV 2 . That mission requires UUV to operate with contacts present for extended periods of time in complex, partially known environments. The ability to dynamically plan vehicle paths to avoid dynamic contacts and create safe vehicle trajectories in complex shallow water environments is extremely important for mission success and overall system survivability. Because of the complexity and the large searching space of the problem, few methods can solve the contradiction between the global optimization and the excessive information. Furthermore, the current work mainly focuses on UUV path planning in 2 dimensions. Many scholars have researched deeply in this hot topic 28 29 30 lately. In this paper, a variant of the flower pollination algorithm is proposed to solve the UUV path planning problem3 in 2D and 3D space. Flower pollination algorithm (proposed by Yang in 2012) 4 is a new population-based intelligent optimization algorithm by simulating flower pollination behavior. And it has been extensively researched in last two years by scholars. Yang and Xingshi He have used FPA to solve multi-objective optimization problem in 2013 5 . Marwa Sharawi has applied FPA for solving Wireless Sensor Network Lifetime global optimization in 2014 6 . Osama Abdel-Raouf has used an improved FPA to solve Sudoku Puzzles in 2014 7 . FPA has been used to solve Large Integer Programming Problems by Ibrahim El-henawy in 2014 8 . The remainder of this paper is organized as follows. Section 2 introduces the environmental modeling for UUV path planning; Section 3 describes the principle of basic flower pollination algorithm; while section 4 specified implementation procedure of our proposed improved flower pollination algorithm (IFPA); In section 5, series of comparison experiments are conducted; Our concluding remark and future works are described in section 6; And acknowledgements and references are shown in section 7 and section 8.

2. Environmental modeling for UUV path planning As a key component of mission planning system 9 , path planning for UUV is the design of optimal sailing route to meet certain performance requirements according to the special mission objective and is modeled by the constraints of terrain, data, threat information and fuel. The goals of path planning for both 2D and 3D cases are calculating the optimal or near-optimal sailing route for UUV. The mathematical model in UUV path planning we used in this paper can be described as follows.

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Instructions for Typing Manuscripts (An Improved FPA for Optimal UUV Path Planning Problem)

2.1. Mathematical Model for UUV Path Planning in 2D space The objective of UUV path planning is to design an optimal or sub-optimal route from start point to target point. And a good route has the ability to avoid the threats in the battle field with a short length. Firstly, the route planning problem should be transformed into a D-dimensional functional optimization problem(Fig. 1).

Fig. 1. Coordinates transformation relation in 2D space.



where original coordinate system Oxy is transformed into new coordinate OX ′ Y ′ in Fig. 1. And the horizontal axis is the connection line from start point to target point. Coordinate in two coordinate systems could be transformed by Eq. (1) 14 ,

γ = arcsin

[ ] [ ] [ ′] [ ] y2 − y1 x cos γ sin γ x x , =  ′ + 1 ⃗ y − sin γ cos γ y y1 |AB|

(1)

where the point (x, y) is the coordinate in OXY , and (x′ , y ′ ) is the coordinate in ′ the new rotating coordinate system OX ′ Y ′ . γ is the rotation angle of the coordinate system. From Fig. 1, horizontal axis X ′ is divided into D equal parts. And then we optimize the vertical coordinate Y ′ on vertical line for each node. It is also easy to get the horizontal coordinate of each node. At last, we can get a series of nodes with horizontal coordinate and vertical coordinate. Thus a path from start point to target point is designed through connecting these points together. And the route planning problem is transformed into a D-dimensional functional optimization problem in 2D space.

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2.2. Mathematical Model for UUV Path Planning in 3D space The mathematical model for UUV path planning in 3D space is more difficult than that of 2D case. In 3D case, we also transform the path planning problem into a D-dimensional functional optimization problem (Fig. 2). And the transformation of coordinate is the key component of the model. Thus it should be introduced firstly in this sub-section. In Fig. 2, OXY Z is the original coordinate system. And there

Fig. 2. Coordinates transformation relation in 3D space.

exist two specific points (start(x0 , y0 , z0 ) and target) in OXY Z . We should trans′′′ form the original coordinate system OXY Z into new coordinate system OX ′′ Y ′′ Z ′′ whose horizontal axis is the connection line from start point to target point. The transformation is complicated. We can divide it into three steps as follows: ′ Step1. we can transform OXY Z (colored black) to OX ′ Y ′ Z ′ (colored blue) with a spatial translation. ′ ′′ Step2. In order to transform OX ′ Y ′ Z ′ (colored blue) to OX ′′ Y ′′ Z ′′ (colored red), ′ we should rotate OX ′ Y ′ Z ′ an angle a (colored red in Fig. 2) around axis Z ′ . And this rotation must obey the right-hand rule. ′′ ′′′ Step3. After two steps above. We should rotate OX ′′ Y ′′ Z ′′ into OX ′′′ Y ′′′ Z ′′′ by ′′ rotate OX ′′ Y ′′ Z ′′ an angle b (colored green in Fig. 2) around axis Y ′′ . This rotation ′′′ should also obey the right-hand rule. Coordinates in OXY Z and OX ′′′ Y ′′′ Z ′′′ could

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Instructions for Typing Manuscripts (An Improved FPA for Optimal UUV Path Planning Problem)

be transformed by Eq. (2):        ′   x0 cos a − sin a 0 cos b 0 sin b x x I1 =  sin a cos a 0  , I2 =  0 1 0  ,  y  = I1 I2   y ′  +  y0  (2) z0 0 0 1 − sin b 0 cos b z z′ where a is the rotation angle in step 2, and b is the rotation angle in step3.In √ this paper, in order to simplify the calculating process, we set a = π4 ,b = arctan 22 . I1 and I2 are transformational matrix of axis rotation. (x, y, z) is the coordinate in ′′′ OXY Z , (x′ , y ′ , z ′ ) is the coordinate in OX ′′′ Y ′′′ Z ′′′ . And (x0 , y0 , z0 ) is the coordinate of start point in OXY Z . Then, we divide the horizontal axis X ′′′ into D equal partitions and then optimize the coordinate Y ′′′ and coordinate Z ′′′ on the plane perpendicular to the axis X ′′′ for each node to get a group of points composed by coordinates of D points. Obviously, it is easy to get horizontal abscissas of these points. We can get a path from start point to end point through connecting these points together, so that the route planning problem is transformed into a D-dimensional functional optimization problem. 2.3. The performance evaluation function of route optimization The performance indicators of the UUV route mainly include the threat cost Jt and Jf the fuel cost, which can be evaluated as follow 10 11 : ∫ L Jt = wt dl (3) o

∫ Jf =

L

wf dl

(4)

0

where wt and wf are variables closely related with the current path and changing along with l, respectively presenting the threat cost and fuel cost of each line segment on the path, while is the total length of the generated path. In order to simplify the calculation, more efficient approximation to the exact solution is adopted. In this work, each path segment is discretized into five subsegments and the threat cost is calculated on the end of each sub-segment. And the model is shown in Fig. 3. If the distance from the threat point to the end of each sub-segment is less than the threat radius of threat point, the threat cost associated with this threat is given by Eq. (5)13 . t 1 1 1 1 L5i ∑ 1 • + 4 + 4 + 4 + 4 ) tk • ( 4 5 d0.1,i,k d0.3,i,k d0.5,i,k d0.7,i,k d0.9,i,k

N

wt,Li =

(5)

k=1

where Nt is the number of threatening areas, Li is the ith sub-path length, d0.1,i,k is the distance from the 1/10 point on the ith edge to the kth threat, and

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Fig. 3. Modeling of the UUV threat cost 12

tk is the threat level of kth threat. Assuming that the speed of UUV is a constant, and then the fuel cost of the path Jf can be considered equal to L, the total length of path. The total cost of traveling along the trajectory comes from a weighted sum of the threat and fuel costs, as is given by Eq. (6), J = kJt + (1 − k)Jf

(6)

where k is a variable determining the relative emphasis of the various cost components with respect to the overall cost function, which gives the designer certain flexibility to dispose relations between the threat exposition degree and the fuel consumption. The value k of is normalized between zero to one (0.5 in this paper). When k is more approaching 0, a shorter path is need to be planned, and less attention is paid to the collision avoidance. When k is more approaching 1 it requires avoiding the threat as far as possible on the cost of sacrificing the trajectory length. 3. Flower Pollination Algorithm (FPA) Flower Pollination Algorithm (FPA) was founded by Yang in the year 2012. Inspired by the flower pollination process of flowering plants are the following rules 15 4 : Rule 1: Biotic and cross-pollination can be considered as a process of global pollination process, and pollen-carrying pollinators move in a way that obeys L´evy flights. Rule 2: For local pollination, a biotic and self-pollination are used. Rule 3: Pollinators such as insects can develop flower constancy, which is equivalent to a reproduction probability that is proportional to the similarity of two flowers involved. Rule 4: The interaction or switching of local pollination and global pollination

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Instructions for Typing Manuscripts (An Improved FPA for Optimal UUV Path Planning Problem)

can be controlled by a switch probability p ∈ [0, 1], with a slight bias toward local pollination. In order to formulate updating formulas, we have to convert the aforementioned rules into updating equations. For example, in the global pollination step, flower pollen gametes are carried by pollinators such as insects, and pollen can travel over a long distance because insects can often fly and move in a much longer range. Therefore, Rule 1 and flower constancy can be represented mathematically as: = xti + γL(λ)(xti − B) xt+1 i

(7)

where xti is pollen i or solution vector xi at iteration t, and B is the current best solution found among all solutions at the current generation/iteration. Here γ is a scaling factor to control the step size. In addition, L(λ) is the parameter that corresponds to the strength of the pollination, which essentially is also the step size. Since insects may move over a long distance with various distance steps, we can use a L´evy flight to imitate this characteristic efficiently. That is, we draw L > 0 from a L´evy distribution: L(λ) =

λΓ(λ)sin(πλ/2) 1 × 1+λ , (S ≫ S0 > 0) π S

(8)

here, Γ(λ) is the standard gamma function, and this distribution is valid for large steps s > 0 .Then, to model the local pollination, both Rule 2 and Rule 3 can be represented as xt+1 = xti + U (xtj − xtk ) i

(9)

where xti and xtk are pollen from different flowers of the same plant species. This essentially imitates the flower constancy in a limited neighborhood. Mathematically, if xti and xtk comes from the same species or selected from the same population, this equivalently becomes a local random walk if we draw from U a uniform distribution in [0,1]. Though flower pollination activities can occur at all scales, both local and global, adjacent flower patches or flowers in the not-so-far-away neighborhood are more likely to be pollinated by local flower pollen than those faraway. In order to imitate this, we can effectively use the switch probability like in Rule 4 or the proximity probability p to switch between common global pollination to intensive local pollination. To begin with, we can use a na¨ive value of p = 0.5 as an initially value. A preliminary parametric showed that p = 0.8 might work better for most applications 4 . 4. Improved Flower Pollination Algorithm (IFPA) In order to enhance the search ability of Flower Pollination Algorithm for solving UUV path planning problem, three optimization strategies are applied, those are PSO in local search (PSOLS), dimension by dimension evaluation and improvement strategy (DDEIS), and dynamic switching probability strategy (DSPS).

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