A novel hybrid grey wolf optimizer algorithm for unmanned aerial vehicle (UAV) path planning

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A novel hybrid grey wolf optimizer algorithm for unmanned aerial vehicle (UAV) path planning✩ , ✩✩ ∗

Chengzhi Qu, Wendong Gai , Jing Zhang, Maiying Zhong Shandong University of Science and Technology, Qingdao 266590, China

article

info

Article history: Received 16 May 2019 Received in revised form 13 December 2019 Accepted 14 January 2020 Available online xxxx Keywords: Hybrid meta-heuristic algorithm Unmanned aerial vehicle (UAV) Path planning

a b s t r a c t Unmanned aerial vehicle (UAV) path planning problem is an important component of UAV mission planning system, which needs to obtain optimal route in the complicated field. To solve this problem, a novel hybrid algorithm called HSGWO-MSOS is proposed by combining simplified grey wolf optimizer (SGWO) and modified symbiotic organisms search (MSOS). In the proposed algorithm, the exploration and exploitation abilities are combined efficiently. The phase of the GWO algorithm is simplified to accelerate the convergence rate and retain the exploration ability of the population. The commensalism phase of the SOS algorithm is modified and synthesized with the GWO to improve the exploitation ability. In addition, the convergence analysis of the proposed HSGWO-MSOS algorithm is presented based on the method of linear difference equation. The cubic B-spline curve is used to smooth the generated flight route and make the planning path be suitable for the UAV. The simulation experimental results show that the HSGWO-MSOS algorithm can acquire a feasible and effective route successfully, and its performance is superior to the GWO, SOS and SA algorithm. © 2020 Elsevier B.V. All rights reserved.

1. Introduction The development of unmanned flight technologies is inevitable trends in many countries [1,2]. As a kind of modern aerial weapon equipment, unmanned aerial vehicle (UAV) has attracted more attention due to its potential to work in complicated and hazardous environments [3]. The path planning design is a crucial mission required in the mission planning system of UAV, which needs acquire a safe and efficient route from the initial location to the desired destination based on the specific constraint conditions [4]. Therefore, the flight path planning problem could be considered as a complex optimization problem that requires efficient algorithms to solve. As effective methods to solve optimization problems, intelligent algorithms have been applied to various field [5–9], such as sustainable mobility project [10–12], sustainable manufacturing [13,14], modeling and optimal lotsizing of the replenishments [15,16] and design of supply-chain ✩ This work is supported by National Nature Science Foundation, China under Grant 61603220, 61873149, 61733009; the Research Fund for the Taishan Scholar Project of Shandong Province of China; SDUST Young Teachers Teaching Talent Training Plan under Grant BJRC20180503, BJRC20190504. ✩✩ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.knosys. 2020.105530. ∗ Corresponding author. E-mail address: [email protected] (W. Gai).

network [17–24]. Meta-heuristic algorithms are nature-inspired intelligent algorithms [25,26], which means that they are originated from mimicking physical phenomena or the interactive behaviors of the organisms. Meta-heuristic optimization algorithms can solve the complex optimization problems and search a set of relevant parameter values by minimizing or maximizing the objective functions [27]. Series of meta-heuristic algorithms have been used to solve the path planning problem. For instance, Ref. [28] presented a novel multi-frequency vibrational genetic algorithm (mVGA) to solve the unmanned aerial vehicles path planning problem. Ref. [29] formulated the UAV path planning problem as an extended traveling salesman problem (TSP), and an enhanced discrete particle swarm optimization (DPSO) algorithm was developed to solve this problem. A chaotic artificial bee colony algorithm was developed to solve the unmanned combat aerial vehicle (UCAV) path planning problem by Ref. [30]. Ref. [31] proposed a modified differential evolution (DE) method to acquire a feasible route to solve the global route planning problem for UAV. An social class pigeon inspired algorithm was presented for solving multi-UAV path planning problem by Ref. [32], and a time stamp segmentation path planning model was developed to simplify the handling of multi-UAV coordination cost. For performing the UAV swarm’s reconnaissance mission, Ref. [33] proposed various distributed particle swarm optimization path planning algorithms. Grey wolf optimizer (GWO) algorithm was a new metaheuristic algorithm and proposed by Mirjalili et al. in 2014. This

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Please cite this article as: C. Qu, W. Gai, J. Zhang et al., A novel hybrid grey wolf optimizer algorithm for unmanned aerial vehicle (UAV) path planning, Knowledge-Based Systems (2020) 105530, https://doi.org/10.1016/j.knosys.2020.105530.

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algorithm mimics the hunting behavior and the social leadership of grey wolves, which gives it great exploration ability [34]. Superior to other meta-heuristic algorithms, the GWO has the advantages of flexibility, simplicity and implementation. As an efficient and competitive optimization algorithm, the GWO has been applied to solve many engineering applications and control problems, such as economic load dispatch problems [35], PID controller design [36], multi-tracking target [37], wide-area power system stabilizer design [38]. Symbiotic organisms search (SOS) algorithm was originally developed by Cheng and Prayogo [39] in 2014, based on the symbiotic interaction between organisms in the ecosystem. In SOS, the offspring can be created by the symbiotic behaviors between the current organism and the other randomly organism or the best organism from the current ecosystem, and this approach owns great exploitation ability [40]. The SOS algorithm is widely adapted because of the characteristic that it is free from tuning parameters, and easy to implement [41]. In recent years, researchers turn towards to a new kind of technique called hybridization [42,43], which is a combination of multiple metaheuristic algorithms for optimization. In this technique, the skilled combination can display a more efficient optimization when dealing with real problems [44]. To solve the multi-robot path planning problem, Ref. [45] improved the particle swarm optimization (PSO) and gravitational search algorithm (GSA), and proposed a novel hybrid algorithm called IPSO–IGSA. Ref. [46] presented two hybridization methods based on whale optimization algorithm (WOA) to design different feature selection problem. A hybrid teaching learning-based PSO algorithm was proposed to acquire a great convergence in Ref. [47]. Ref. [48] developed a novel hybrid PSO–GWO method to solve unit commitment problem. The effectiveness of the proposed method is compared with various classical algorithms. Moreover, the convergence analysis is significant for metaheuristic algorithms to prove its stability. Ref. [49] proposed a novel hybrid algorithm by combining three classical algorithms such as particle swarm optimization (PSO), genetic algorithm (GA) and SOS. The modified mutualism phase of the proposed algorithm was proved to create a better convergence to the minimum by mathematical induction. Ref. [50] developed an improved gravitational search algorithm (IGSA) for solving optimization problems. They used discrete time linear system theory to prove the stability of IGSA. Ref. [51] presented a modified central force optimization (MCFO) algorithm and introduced the method of linear difference equation to finish the convergence analysis of the proposed algorithm. However, the convergence analysis of the GWO and its hybrid algorithm is scarce in current research. Meta-heuristic algorithms need to have an excellent balance between the exploration operation and the exploitation operation for achieving the global and local searches efficiently. The optimization pattern of the GWO algorithm owns great exploitation capacity. In contrast, the SOS algorithm has an aptitude for local exploitation capability. Meanwhile, both the GWO and SOS algorithms show the possibility of boosting performance when hybridized with other meta-heuristic algorithms. Therefore, we have combined the exploration and exploitation abilities of the wolves and the organisms and proposed a novel hybrid optimization algorithm called HSGWO-MSOS algorithm. The UAV optimal path is computed by the proposed algorithm off-line, and it is smoothed by the cubic B-spline curve. The main innovations of this paper are summarized as follows:

• A novel hybrid algorithm called HSGWO-MSOS is proposed, which consists of the simplified GWO phase and modified SOS phase. • The convergence analysis inspired by the linear difference equation has been finished for the proposed algorithm.

Fig. 1. Schematic diagram of UAV path planning.

• The HSGWO-MSOS algorithm has been applied for UAV path planning problem and the better experiment results are achieved with the proposed algorithm compared to other classical algorithms. The structure of this paper is organized as follows. The mathematical model in UAV path planning is described in Section 2. Section 3 explains the principles of the basic GWO and SOS algorithms. The detailed implementation of the proposed HSGWOMSOS algorithm and its convergence analysis are described in Section 4. Section 5 describes the route smoothing method by the cubic B-spline curve. In Section 6, the comparison experiments are finished. Finally, Section 7 summarizes the conclusions. 2. Mathematical model in UAV path planning Path planning is a critical component of UAV mission planning, and it is used to find the optimal flight route under the constraints such as terrain threat, radars threat, missiles threat, fuel and time. In this paper, we assume that the UAV maintains constant speed during its mission, and the mathematical model is described as follows. 2.1. Threat resource model in UAV path planning In the model, the starting point has been defined as S : (xS , yS , zS )T and the target point has been defined as T : (xT , yT , zT )T , which is shown in Fig. 1. The UAV path planning is to acquire an optimal route from S to T considering all the constraint condition and mission requirements. The UAV will be attacked by the ground defense installation if the path falls in the threat areas. Fig. 1. Schematic diagram of UAV path planning First we connect the starting point S to the target point T , and segment ST is divided into D + 1 equal portions by lines {Lk , k = 1, 2, . . . , D}. Take a point at each line {(x(k), y(k), z(k)), k = 1, 2, . . . , D} and connect these points to form a feasible path from S to T . Therefore, the path planning problem is considered as coordinate optimization problem to acquire a superior flight path. To accelerate the optimization process, A new coordinate frame is established and the segment ST has been considered as the new x axis and each path point (x(k), y(k), z(k)) needs to be taken the coordinate transformation according to Eq. (1). x∗ (k) y∗ (k) z ∗ (k)

[

]

[ =

cos θ − sin θ 0

sin θ cos θ 0

][

x(k) − xs y(k) − ys z(k) − zs

] (1)

where θ denotes the angle between the segment ST and the original X axis, (x∗ (k), y∗ (k), z ∗ (k)) is the path point after transformation of coordinates.

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2.2. Cost function and performance constraints The performance evaluation index of the UAV flight path is composed of the fuel cost Jfuel and the threat cost Jthreat [30] as follows: Jcost = µ · Jfuel + (1 − µ) · Jthreat

= µ·

length

∫

Wfuel dl + (1 − µ) · 0

length

∫

Wthreat dl

(2)

0

where µ is a weighting parameter between 0 and 1, l represents the line segment of the whole route {lk k = 1, 2, . . . , D, D + 1}, Wfuel and Wthreat represent the fuel cost and threat cost on each path segment, and length denotes the length of the created flight path. The threat cost of the segment lk is calculated at five points (include the start point and target point of lk ). If the path segment falls into a threat, the Wthreat is calculated as follows: Wthreat ,lk =

lk 5

·

(

m ∑ ri i=1

+

·

10

1 dk0.25,i

+

1 dk0,i 1

dk0.5,i

+

1 dk0.75,i

+

1

)

(3)

dk1,i

where m denotes the number of threatening circles, dk0.25,i denotes the distance between the ith threat center and the 0.25 point on the segment, ri refers to the radius of the ith threat. It is assumed that the speed of the UAV is a constant, therefore, the Jfuel can be considered as the length of the path. Considering generating a suitable route for the UAV, the yawing angle and the pitch angle constraints are introduced as follows:

⏐ ( )⏐ ⏐ yk+1 − yk ⏐ ⏐ ≤ ϕ max , k = 1, 2, 3, . . . , n − 1 ϕk = ⏐⏐arctan xk+1 − xk ⏐ 2

(4)

⏐ )⏐ ( ⏐ zk+1 − zk ⏐ ⏐ ≤ θ max , k = 1, 2, 3, . . . , n − 1 ⏐ θk = ⏐arctan xk+1 − xk ⏐ 2

(5)

where ϕ max is the maximum yawing angle, θ max is the maximum pitch angle, ϕk and θk are the yawing angle and the pitch angle of the path point (xk , yk , zk ). 3. Preliminary knowledge and basic idea 3.1. The GWO algorithm The GWO algorithm is a novel meta-heuristic algorithm that can imitate the social hierarchy and hunting pattern of grey wolves in nature. In the GWO algorithm, the individual with the best fitness is considered as α wolf. The second and third individuals with better fitness are considered as β wolf and δ wolf. The rest of the individuals are considered as ω. To imitate the encircling behavior that the grey wolves hunt the prey, the individuals update their positions as follows:

⏐

⏐

Di = ⏐Ci · Xp (t) − Xi (t)⏐

(6)

Xi (t + 1) = Xp (t) − Ai · Di

(7)

where t is the current iteration, Xi indicates the position of a grey wolf in the search space. Xp is the position of the prey. The coefficient vectors Ai and Ci are shown as: Ai = 2a · r1 − a Ci = 2 · r2 a = 2 − 2t /tmax

{

(8)

where r1 and r2 are random parameters in [0,1], a is linearly decreased from 2 to 0.

3

It is assumed that the leader group (α , β and δ ) has the better information about the location of the prey. Therefore, each ω wolf can update its position according to the best search agents as follows: Dα = |C1 · Xα (t) − Xi (t)|

(9)

⏐ ⏐ Dβ = ⏐C2 · Xβ (t) − Xi (t)⏐

(10)

Dδ = |C3 · Xδ (t) − Xi (t)|

(11)

X1 = Xα (t) − A1 · Dα

(12)

X2 = Xβ (t) − A2 · Dβ

(13)

X3 = Xδ (t) − A3 · Dβ

(14)

Xi (t + 1) = (X1 + X2 + X3 )/3

(15)

where Xα , Xβ , and Xδ denote the position of the leader group, Xi is the current individual’s position. The GWO algorithm has been used for solving the UCAV path planning problem [52]. They have used three simulation cases to test GWO, and compared this algorithm with other classical meta-heuristic algorithms. There are comparative convergence curves of GWO for different cases and D values. However, the theoretical convergence analysis of the GWO is missing in this work. In addition, the simulation environment used is relatively simple. 3.2. The SOS algorithm The symbiotic organisms search (SOS) algorithm is a novel metaheuristic method. This method is inspired from the symbiotic behavior between organisms. In SOS, new organisms are updated by using the mutualism phase, commensalism phase and parasitism phase, which simulates the biological interaction model. 3.2.1. Mutualism phase The mutualism phase is an important part of SOS algorithm, which benefits two organisms synergistically. Xi represents the ith organism in the population, and Xj is selected randomly to communicate with Xi . Finally, new organisms coming from Xi and Xj are generated by these equations: Xinew = Xi + rand0,1 · (Xbest − Mutual_Vector · BF 1)

(16)

Xjnew = Xj + rand0,1 · (Xbest − Mutual_Vector · BF 2)

(17)

Mutual_Vector =

Xi + Xj 2

(18)

where Mutual_Vector represents the relationship between Xi and Xj . BF 1 and BF 2 are the benefit factors, which are randomly chosen either 1 or 2, rand0,1 is a random parameter in [0,1]. 3.2.2. Commensalism phase The commensalism phase represents a relationship between two organisms that only benefits one and the other is unaffected. The organism Xj is selected randomly to interact with Xi . Therefore, Xi benefits from the relationship while Xj is not, the new organism is generated as follows: Xinew = Xi + rand−1,1 · (Xbest − Xj )

(19)

where rand−1,1 is a random parameter in [−1,1]. 3.2.3. Parasitism phase The parasitism phase can benefit the organism by harming the other one. The parasite Parasite-Vector is generated by duplicating organism Xi . Then the Parasite-Vector is modified randomly

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to generate a new organism. The organism Xj is selected randomly and regarded as a host of the Parasite-Vector. If ParasiteVector is superior to Xj , then Xj will be killed and replaced by Parasite-Vector, otherwise Xj will survive and Parasite-Vector will be killed. The SOS algorithm could be operated with simple mathematical operations, and it requires no tuning parameters. However, the application of SOS algorithm is scarce in the field of UAV path planning. 3.3. Basic idea of the proposed algorithm To solve complex optimization problems, all the metaheuristic algorithms like GWO and SOS have been designed to find a proper balance between global exploration ability and local exploitation ability. In GWO, the positions of individuals are updated according to the α wolf, β wolf and δ wolf. Meanwhile, the exploration ability and the exploitation ability mainly depend on the parameterA. When the random values of A are in [−1,1], the wolves move with smaller distance, which means a process of local search. When |A| > 1, the wolves are forced to make a global search. In SOS, the update of the organisms’ positions is not only related to the current optimal organism, but also affected by another random one in the population. In this way, the exploration and the exploitation can be guaranteed by the behavior of mutualism and commensalism phase, and parasitism phase. Consequently, the optimization pattern of the GWO algorithm gives it a great exploration ability. In contrast, the SOS algorithm has an aptitude for local exploitation capability. Therefore, to better solve the UAV path planning problem under complex environment, we have simplified the GWO algorithm to retain the exploration ability and accelerate the convergence rate. Then the commensalism phase of the SOS algorithm is modified to enhance the exploitation capability. Finally, to combine the advantages of simplified GWO (SGWO) and modified SOS (MSOS), we present a novel hybrid algorithm called the HSGWO-MSOS algorithm, and the convergence analysis of the proposed method is given. 4. The development of the HSGWO-MSOS algorithm

Fig. 2. Upper and low bounds of the population initialization range.

The size of population N, the maximum number of iterations tmax are set in this step. We constrain the population initialization range to make the search process more efficient. As shown in Fig. 2, two dash lines are set to limit the lower bound and upper bound. Those bounds are determined by the threats that have the maximum distance from the line ST . Therefore, the positions of the initial population are restricted in [Pmin , Pmax ]. And the fitness of each individual in the initial population is calculated and the best individual is denoted as a wolf (Xα ). [Pmin , Pmax ] is calculated as follows: Pmin = min{min{y∗ threat ,i − Ri }, 0} − ∆d

(20)

Pmax = max{max{y∗ threat ,i + Ri }, 0} + ∆d

(21)

i

i

where Ri is the radius of the ith threat, y∗ threat ,i is the vertical coordinate of ith threat after transformation of coordinates, ∆d is a definable distance. Step 2: Updating individual position with SGWO To accelerate the convergence rate, instead of exploring the solutions influenced by the best three wolves in Eq. (15), individuals update their positions only affected by α wolf shown in Eqs. (22), (23) and (24). By this alteration, the advantage of the GWO algorithm in exploration has been retained.

⏐

In the proposed HSGWO-MSOS algorithm, the advantages of both GWO and SOS are combined to solve the optimization problems. The GWO algorithm is simplified to explore the possible solutions available in the search space and generate offspring. Then the modified commensalism phase of the SOS algorithm is performed to exploit local solutions. 4.1. The HSGWO-MSOS algorithm In each iteration cycle, the current population is updated by the GWO algorithm firstly. In this phase, the current best individual is denoted as α wolf. Instead of exploring the possible solutions influenced by the best three wolves, individuals update their positions only affected by α wolf to accelerate the convergence rate. Next step is to gain better results by the modified commensalism phase. In this phase, the positions of individuals are influenced by both current optimum and a random one. If the specific phase causes better fitness value, the individual’s position is updated, otherwise the position does not change. The algorithm will be stopped if the termination criterion is reached, otherwise the next iteration will start and pass new cycle. Fig. 3 shows the flowchart of the proposed HSGWO-MSOS algorithm. To implement the proposed algorithm, the steps to be followed are as follows: Step 1: Initialization

⏐

D∗ = ⏐C ∗ · Xα (t) − Xi (t)⏐ ∗

(22) ∗

Xi (t + 1) = Xα (t) − A · D A∗ = (2r0,1 − 1) · a C ∗ = 2 · r0,1 a = 2 − 2t /tmax

(23)

{

(24)

Step 3: Updating individual position with MSOS The commensalism phase in the SOS algorithm plays a significant role in exploiting local solutions of the optimization process. To improve the exploitation ability, a modified commensalism phase of SOS has been proposed for ensuring the efficiency of the proposed hybrid algorithm. The idea of mutualism has been introduced. In this phase, an individual Xj is selected randomly from the population to exchange information with the ith individual Xi . The new offspring of Xi is affected by the distance between Xα and Xj . The same operation is taken for Xj simultaneously. The equations of modified commensalism phase are as follows: Xinew = Xi + rand−1,1 · (Xα − Xj )

(25)

Xjnew = Xj + rand−1,1 · (Xα − Xi )

(26)

Step 4: Go to step2 until the termination condition is met If the number of iterations reaches maximum number, or the optimum solution has been found, stop the algorithm, otherwise, continue the algorithm to search for optimum result. The pseudocode is defined as follows:

Please cite this article as: C. Qu, W. Gai, J. Zhang et al., A novel hybrid grey wolf optimizer algorithm for unmanned aerial vehicle (UAV) path planning, Knowledge-Based Systems (2020) 105530, https://doi.org/10.1016/j.knosys.2020.105530.

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5

Fig. 4. Representation of the Rosenbrock function.

4.2. Visualization of convergence process To show the convergence process of the proposed algorithm vividly, we have finished a simple numerical experiment. Rosenbrock function Eq. (27) has been used to illustrate the iteration process of the proposed algorithm in 2D (x1 , x2 ) searching space.

F (x) =

D−1 ∑

[100(xi+1 · x2i )2 + (xi − 1)2 ]

(27)

i=1

Fig. 3. Flowchart of the HSGWO-MSOS algorithm.

Rosenbrock function is a multimodal and non-separable function with a global minimum fmin = 0 at (1,1,. . . ,1). Initialization range for this function are denoted as [−32,32]. Fig. 4 shows the representation of the Rosenbrock function. The visualization of the HSGWO-MSOS algorithm convergence process is shown in Fig. 5. The Rosenbrock function is designed to imitate the characteristic ‘‘banana function’’. This function is more difficult to find the global minimum because of which lies inside a narrow, curved valley [53]. Surprisingly, in Fig. 5, HSGWO-MSOS finds the global optimum before the maximum iteration is achieved. The best solution with a fitness value of 1.018E−6 is found after 20 iterations. 4.3. Computing complexity As is shown in Fig. 3, the HSGWO-MSOS algorithm can be divided into three phases. As the first phase of the program, the initialization phase is executed one time at the start, and the other phases are executed in each cycle. The computation is applied to the population with size of N. The individual’s position in population is a vector with size of D. The computational complexity is mostly affected by the phase of the algorithm. The phases that affect the computing complexity are: simplified GWO position update operator and modified SOS commensalism operator. The computing complexity of each phase is shown as follows: Phase 1: Initialization The population are initialized for the next work and the computing complexity of this phase is O (N .D). Because the complexity of deciding on stopping criteria ended is O (1), hence this phase runs in O (N .D) complexity. Phase 2: Simplified GWO position update operator SGWO creates a new position for each individuals only affected by α wolf. All the original positions will be replaced by the new positions. The computing complexity of this phase is O (N .D). Phase 3: Modified SOS commensalism operator MSOS calculates the new position affected by another random selected individual’s position. The original position will be replaced if the new position offers fitter value. The same operation

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Fig. 5. Visualization of convergence process for Rosenbrock function.

is applied to the random selected individual. The computing complexity of this phase is O (N .D). Therefore, the computing complexity of the proposed algorithm in each iteration is O (N .D), which proves that this algorithm owns fastest execution speed.

where Xα (t) means the best individual in the current iteration. Define: A(t) = A∗

(31) ∗

∗

fi (t) = (1 − A · C + r−1,1 ) · Xα (t) − r−1,1 · Xj (t)

(32)

Thus Eq. (30) can be written as: 4.4. Convergence analysis

Xi (t + 1) = A(t) · X (t) + fi (t)

After the description of the HSGWO-MSOS algorithm, we have introduced the convergence analysis of the proposed method in this section. Definition 4.1 ([51] Linear Difference Equations). Define A(·) : J + → Rn×n is a square nonsingular matrix, J + is a set of the nonnegative integer, f ∗ (·) : J + → Rn , x(t) ∈ Rn , for each t ∈ J + , the linear difference equation and corresponding homogeneous linear equation with variable coefficients are as follows: x(t + 1) = A(t) · x(t) + f (t)

(28)

x(t + 1) = A(t) · x(t)

(29)

Theorem 4.1. The position update equation of the HSGWO-MSOS algorithm is a first-order linear time-varying difference equation. Proof. Assume that D∗ ≥ 0. According to the Eqs. (22), (23), (24), (25) and (26), the whole position update equation of the HSGWO-MSOS algorithm is shown as: Xi (t + 1) = Xα (t) − A∗ · C ∗ · Xα (t) + A∗ · Xi (t) +r−1,1 · (Xα (t) − Xj (t)) = A∗ · Xi (t) + (1 − A∗ · C ∗ + r−1,1 ) · Xα (t) −r−1,1 · Xj (t)

(33)

So the position update equation of the HSGWO-MSOS algorithm can be written as the form of Eq. (28), and Theorem 4.1 is proved. Lemma 4.1 ([51]). ∀f (t) ∈ [J + , Rn ], the stability degree of the linear difference equation is equivalent to the stability degree of the general solution of the corresponding homogeneous linear equation. Lemma 4.2 ([51]). Eq. (28) can be guaranteed the uniformly asymptotically stability if there is a kind of matrix norm that satisfy Eq. (34):

∥A(t)∥X ≤

t +1 t +2

, t = t0 , t0 + 1, . . .

(34)

where ∥·∥X means a certain norm, ∀t0 ∈ R. Theorem 4.2. When the condition of t → tmax is met, individuals {Xi , i = 1, 2, . . . , N } will converge to the local or global optimum solutions. Proof. According to Eqs. (24) and (31), we have:

(30)

lim A(t) =

t →tmax

lim (2r0,1 − 1) · (2 − 2t /tmax )

t →tmax

= (2r0,1 − 1) · lim (2 − 2t /tmax ) = 0

(35)

t →tmax

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7

where di (i = 0, 1, . . . , n) are control points, Ni,k (u) are the k-order normalized B-spline basic functions defined by the following Cox–de Boor recursion formulas:

{ ⎧ ⎪ 1, if ui ≤ u ≤ ui+1 ⎪ ⎪ Ni,k (u) = ⎪ ⎪ ⎪ 0, other w ise ⎨ ui+k+1 − u u − ui Ni,k (u) = Ni,k−1 (u) + Ni+1,k−1 (u) ⎪ ⎪ u − u u ⎪ i + k i i +k+1 − ui+1 ⎪ ⎪ ⎪ ⎩define 0 = 0

(42)

0 The basic functions are determined by a non-decreasing sequence of parameters called parametric knots {u0 ≤ u1 ≤ · · · ≤ un+k }. Moreover, Ni,k (u) is a piecewise polynomial on each interval and satisfy: n ∑

Fig. 6. The cubic B-spline curve.

Ni,k (t) ≡ 1

(43)

i=0

Thus, when the condition of t → tmax is met, according to Lemmas 4.1, 4.2 and Eq. (35), it can be ensured that the position update equation based on the HSGWO-MSOS algorithm is stable. In other words, each individuals Xi will converge to a steady state Xe when the condition of t → tmax is met (limt →tmax Xi (t) = Xe ). So we have: lim Xi (t + 1) = lim Xi (t) = Xe

t →tmax

t →tmax

(36)

Substitute Eq. (30) into Eq. (36), we have: lim Xi (t + 1) =

t →tmax

lim [Xα (t) − A∗ · C ∗ · Xα (t) + A∗ · Xi (t)

t →tmax

+r−1,1 · (Xα (t) − Xj (t))] = Xe

(37)

Based on the Eqs. (31) and (35), Eq. (37) is formed into: lim Xi (t + 1) =

t →tmax

lim [Xα (t) − A∗ · C ∗ · Xα (t) + A∗ · Xi (t)

t →tmax

+r−1,1 · (Xα (t) − Xj (t))] = lim [Xα (t) + r−1,1 · (Xα (t) − Xj (t))]

(38)

t →tmax

= Xe Because of the randomness of r−1,1 , we can get that:

⎧ ⎨ lim Xj (t) = Xα t →tmax

(39)

⎩ lim Xi (t + 1) = Xα t →tmax

Therefore, Eq. (36) can be written as: lim Xi (t + 1) = lim Xi (t) = lim Xj (t) = Xe = Xα

t →tmax

t →tmax

t →tmax

(40)

So, when the condition of t → tmax is met, all individuals can converge to the optimum solutions, and Theorem 4.2 is proved. In summary, the convergence of the HSGWO-MSOS algorithm can be insured when the condition of t → tmax is met.

The B-spline curve method is particularly useful for smoothing path. Compared with the Bezier curves, the B-spline curve method overcomes the disadvantage that moving a control point will affect the entire curve, and the degree of the polynomial will not be increased no matter how many points are added. 6. Experiment results In this section, there are three simulation cases designed to evaluate the feasibility and effectiveness of the HSGWO-MSOS algorithm. To show the superiority of the proposed algorithm, the simulation experiments are performed in two-dimension (2D) field and three-dimension (3D) field respectively. In the twodimension field, the size of the flight planning space is 1000 m ∗ 1000 m. The start point is set to (0 m, 0 m). The target point is set to (1000 m, 1000 m). In the three-dimension field, the size of the flight planning space is 1000 m ∗ 1000 m ∗ 1000 m. The start point is set to (0 m, 0 m, 0 m). The target point is set to (1000 m, 1000 m, 1000 m). The weighting parameters µ are set as 0.4. The maximum iteration number tmax is set as 500, the dimension D is set as 10, and the population size N of the three algorithms are set as 50. The results are averaged 30 independent runs. In the twodimension planning field, the threats are denoted by eight oval areas. The comparative results with the simulated annealing algorithm (SA) [7], GWO and SOS algorithms are given. In the three-dimension planning field, the threats are denoted by eight cylinders. The comparative results with the GWO and SOS algorithms are also given. The related information of the threat areas is shown in Table 1. The two-dimension experiment comparison results are listed in Table 2. The three-dimension experiment comparison results are listed in Table 3. In those tables, the mean, std, worst and optimal represent the mean fitness value, the standard deviation, worst fitness value and the optimal fitness value, respectively.

5. Path smoothing

6.1. The two-dimension experiment comparison results

After the application of the HSGWO-MSOS algorithm, the solutions generate a route composed of line segments. To ensure the route flyable and smooth, the strategy of cubic B-spline is introduced in Fig. 6. The B-spline curve has evolved from Bezier curves and inherits all the advantages, which include geometrical invariability, convexity preserving and affine invariance [54]. The B-spline curve is defined by:

For the first case, Fig. 7 shows that the experimental results of the four algorithms have differences in the 2D planning field. We can see that the results planned by SOS and HSGWOMSOS can satisfy the path planning requirements. But the trajectory planned by SA and GWO are failure. The trajectory planned by the SOS is not the optimal route. In addition, the result of the HSGWO-MSOS algorithm owns significant performance, and proves the superiority of the proposed algorithm. The convergence curves of the four algorithms in Case 1 are illustrated in Fig. 8. It is obvious that the convergence effect of the proposed

P(u) =

n ∑ i=0

di Ni,k (u)

(41)

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Table 1 The information of threat areas. Case number

Threat center

Threat radius

Height(3D)

1

(300,150) (250,600) (600,100) (500,750) (850,550) (450,300) (750,350) (700,800)

75 100 100 100 75 75 50 75

1000 800 500 1000 500 750 1000 1000

2

(250,50) (380,420) (550,100) (250,750) (700,500) (850,250) (450,700) (750,800)

150 150 150 200 100 200 150 150

600 800 1000 1000 1000 600 600 800

3

(200,200) (380,500) (550,150) (250,750) (750,450) (800,250) (500,700) (800,800)

100 150 125 150 100 150 150 100

1000 800 1000 700 1000 800 600 1000

algorithm is better than the other algorithms. The HSGWO-MSOS attain the global optimal value in iteration 10. But GWO and SOS approach to their optimal value in iteration 80. The SA cannot find its optimal value in iteration 500. The statistical results are illustrated in Fig. 9 and Table 2. The optimal value of the GWO is 286.7, but the worst value is 1673.4, and the standard deviation is 502.4. These data indicate that GWO algorithm has lower success rate for multiple independent runs. For the SOS algorithm, the optimal value is 286.3, but the worst value and the standard deviation are 569.8 and 111.7. These denote that it is easy for SOS to fall into local optimum. The statistical results of SA are poor. Compared with the other algorithms, the simulation result of the proposed algorithm has smaller optimal, worst, and mean values. Meanwhile, the standard deviation of the HSGWO-MSOS is 0.1349, which proves that HSGWO-MSOS can search for the optimal path stably. For the second case, we have increased the complexity of the flight area to better demonstrate the performance of the algorithm. Fig. 10 shows the comparative experimental results. The path planned by SA and GWO cannot satisfy the path planning requirements. The results planned by SOS and HSGWO-MSOS can satisfy the path planning requirements. And the path planned by the HSGWO-MSOS is better than SOS. The convergence curves of the four algorithms in Case 2 are illustrated in Fig. 11. From these curves we can see that the best convergence rate belongs to the HSGWO-MSOS algorithm. The SA algorithm has a poor convergence rate, and the convergence effect of GWO and SOS algorithm are preferable but worse than the proposed algorithm. The statistical results of Case 2 are illustrated in Fig. 12 and Table 2. The HSGWO-MSOS algorithm owns the minimum mean and worst values compared with the other two algorithms. The SA algorithm cannot find the optimal route. The GWO and SOS can find the optimal route, but the standard deviations of the two algorithms are larger. The standard deviation of the proposed algorithm is 1.6418, which shows its superior performance. In the third case, a complex flight environment is established. Fig. 13 shows that the four algorithms can satisfy the path planning requirements. But the proposed algorithm completes the path planning mission perfectly. Fig. 14 shows the convergence curves of the four algorithms in Case 3. The HSGWO-MSOS has

Fig. 7. The comparative path planning results in Case 1, 2D.

Fig. 8. The convergence curves of four algorithms in Case 1, 2D.

Fig. 9. The statistical results of the four algorithms in Case 1, 2D.

the faster convergence rate. The SA has a poor performance. The convergence rates of other three algorithms have slight differences. As the statistical results shown in Fig. 15 and Table 2, the proposed algorithm still provides the best results in optimal, worst, mean and standard deviation values. These results prove that HSGWO-MSOS has excellent capability for UAV path planning in 2D environment. In summary, from the above experimental results we can see that the HSGWO-MSOS algorithm can search for a satisfactory path. Fig. 16 shows the average execution time for the three

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Fig. 10. The comparative path planning results in Case 2, 2D. Fig. 13.

The comparative path planning results in Case 3, 2D.

Fig. 11. The convergence curves of four algorithms in Case 2, 2D. Fig. 14.

Fig. 12.

The convergence curves of four algorithms in Case 3, 2D.

The statistical results of the four algorithms in Case 2, 2D.

cases. The SA has the shorter execution time but the worst search effect. The GWO has the shortest execution time but the poor search effect. The SOS has long execution time but the better search effect. And the proposed algorithm owns the favorable performance and the shorter execution time. 6.2. The three-dimension experiment comparison results For the first case, Fig. 17 shows that the experimental results of the four algorithms have remarkable differences. We can see that only the route planning by HSGWO-MSOS can satisfy the path planning requirements. The path planned by the GWO and

Fig. 15. The statistical results of the four algorithms in Case 3, 2D.

SOS are trapped in local optimization. The path planned by SA is oscillating. The convergence curves of the four algorithms in Case 1 are illustrated in Fig. 18. It is obvious that the convergence effect of the proposed algorithm is better than the other algorithms. The HSGWO-MSOS attain the global optimal value in iteration 20. The GWO approach to its optimal value in iteration 160. The SOS approach to its optimal value in iteration 210. The SA acquires its optimal value in iteration 170. The statistical results are illustrated in Fig. 19 and Table 3. The SA has the worst performance on

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Fig. 16.

The average execution time for the three cases, 2D.

Table 2 Two-dimension comparison results of the three cases. SA

GWO

SOS

HSGWO-MSOS

Case 1

Mean Std Worst Optimal

2804.7 1051.2 3922.3 1131.1

591.4 502.4 1673.4 286.7

344.1 111.7 569.8 286.3

286.3 0.1349 286.7 286.3

Case 2

Mean Std Worst Optimal

8092.2 2949.3 12 775.6 2363.7

1080.5 205.3 1187.1 499.1

574.5 163.4 912.4 495.4

499.7 1.6418 501.3 495.3

Case 3

Mean Std Worst Optimal

8339.9 3380.6 12 367.2 3103.8

778.5 476.9 1736.1 527.1

1221.1 632.7 1712.1 476.9

485.9 2.1246 488.9 482.1

all values. The GWO has the bigger mean and standard deviation values. The SOS has the smaller optimal value, but poor standard deviation value means its poor performance. Compared with the other three algorithms, the simulation result of the proposed algorithm has smaller worst, and mean values. Meanwhile, a lower standard deviation value proves that HSGWO-MSOS can search for the optimal path stably.

Fig. 18. The convergence curves of the four algorithms in Case 1, 3D.

In the second case, Fig. 20 shows that the HSGWO-MSOS can find a feasible path to satisfy the path planning requirements with the smallest cost. The route planned by the SOS is trapped in local optimization. The routes planned by the GWO and SA cannot satisfy the path planning requirements. The convergence curves of the four algorithms in Case 2 are illustrated in Fig. 21. From these curves we can see that the best convergence rate belongs to the HMGWO-SOS algorithm. The GWO and SA have poor convergence rates, and the convergence effect of the SOS algorithm is preferable but worse than the proposed algorithm. For the statistical results shown in Fig. 22 and Table 3, the SOS has the minimum optimal value, but has poor performance of mean and standard deviation values. The statistical values of SA algorithm are much worse than other algorithms. The GWO has poor search performance. The HSGWO-MSOS has the smallest standard deviation value and the best search results. In the third case, we have established a more complex flight environment to examine the performance of the four algorithms. Fig. 23 shows that only HSGWO-MSOS can complete the path planning mission perfectly. The trajectory planned by GWO, SOS and SA algorithms cannot satisfy the path planning requirements. Fig. 24 shows the convergence curves of the four algorithms in

Fig. 17. The comparative path planning result in Case 1, 3D.

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Fig. 19.

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The statistical results of the four algorithms in Case 1, 3D. Fig. 21. The convergence curves of the four algorithms in Case 2, 3D.

Table 3 Three-dimension comparison results of the three cases. SA

GWO

SOS

HSGWO-MSOS

Case 1

Mean Std Worst Optimal

5569.1 1965.5 8032.5 2815.1

1387.4 592.1 2245.2 638.2

933.3 317.2 1369.4 532.6

774.8 100.8 833.5 588.4

Case 2

Mean Std Worst Optimal

10 016.6 2859.6 14 847.6 6194.3

2448.5 730.1 3142.6 912.1

1046.2 338.1 1816.3 704.9

818.1 103.8 1107.8 765.4

Case 3

Mean Std Worst Optimal

11 158.2 2522.1 16 206.8 6504.9

1585.3 675.8 2666.8 754.4

1917.9 637.6 2361.8 702.1

777.8 84.7 939.4 732.8

Case 3. The proposed algorithm has the faster convergence rate. The convergence rate of SA shows its worst performance. The GWO and SOS have poor convergence rates. As the statistical results shown in Fig. 25 and Table 3, the SOS has the minimum optimal value, but the proposed algorithm still provides the best results in worst, mean and standard deviation values. These results prove that HSGWO-MSOS has excellent capability for UAV path planning in complex environment.

Fig. 22.

The statistical results of the four algorithms in Case 2, 3D.

In summary, from the above experimental results we can see that using GWO and SOS algorithms could not possibly lead to a satisfactory path, especially in the complex flight environment.

Fig. 20. The comparative path planning result in Case 2, 3D.

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Fig. 23.

The comparative path planning result in Case 3, 3D.

Fig. 26.

The average execution time for the three cases, 3D.

Fig. 24. The convergence curves of the four algorithms in Case 3, 3D.

has the shortest execution time but the poor search effect. The execution time of SA is similar to the HSGWO-MSOS, but SA owns the poorest search effect. And the SOS has the longest execution time but the better search effect. 7. Conclusions

Fig. 25.

The statistical results of the four algorithms in Case 3, 3D.

However, because of combining the advantages of the two algorithms, the HSGWO-MSOS algorithm has favorable performance and can obtain an optimal path. Fig. 26 shows the average execution time for the three cases. the proposed algorithm owns the favorable performance and the shorter execution time. The GWO

In this paper, a novel hybrid grey wolf optimizer method named HSGWO-MSOS algorithm is presented for solving path planning problem of UAV in complex and hazardous areas. At first, the exploration and exploitation abilities of the GWO and SOS algorithms are combined efficiently in the proposed algorithm. The phase of the GWO algorithm is simplified to accelerate the convergence speed and retain the exploration ability of the population. The commensalism phase of the SOS algorithm is modified to improve the exploitation ability. Second, the convergence analysis of the HSGWO-MSOS algorithm is finished by the method of linear difference equations. Meanwhile, the cubic Bspline curve is used to smooth the generated flight route and make the planning path suitable for the UAV. Eventually, the experimental results show that the HSGWO-MSOS algorithm can acquire an effective and safe route successfully, and the comparative results also show the superiority of the HSGWO-MSOS over the GWO, SOS and SA in solving the path planning problem of UAV.

Please cite this article as: C. Qu, W. Gai, J. Zhang et al., A novel hybrid grey wolf optimizer algorithm for unmanned aerial vehicle (UAV) path planning, Knowledge-Based Systems (2020) 105530, https://doi.org/10.1016/j.knosys.2020.105530.

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