mathematics

Article Path Planning for Autonomous Landing of Helicopter on the Aircraft Carrier

Hanjie Hu 1,2, Yu Wu 3,* , Jinfa Xu 1 and Qingyun Sun 2

1 College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China; [email protected] (H.H.); [email protected] (J.X.) 2 Chongqing General Aviation Industry Group Co., Ltd., Chongqing 401135, China; [email protected] 3 College of Aerospace Engineering, Chongqing University, Chongqing 400044, China * Correspondence: [email protected]; Tel.: +86-023-6510-2510



Received: 23 August 2018; Accepted: 26 September 2018; Published: 27 September 2018


Abstract: Helicopters are introduced on the aircraft carrier to perform the tasks which are beyond the capability of fixed-wing aircraft. Unlike fixed-wing aircraft, the landing path of helicopters is not regulated and can be determined autonomously, and the path planning problem for autonomous landing of helicopters on the carrier is studied in this paper. To solve the problem, the returning flight is divided into two phases, that is, approaching the carrier and landing on the flight deck. The feature of each phase is depicted, and the conceptual model is built on this basis to provide a general frame and idea of solving the problem. In the established mathematical model, the path planning problem is formulated into an optimization problem, and the constraints are classified by the characteristics of the helicopter and the task requirements. The goal is to reduce the terminal position error and the impact between the helicopter and the flight deck. To obtain a reasonable landing path, a multiphase path planning algorithm with the pigeon inspired optimization (MPPIO) algorithm is proposed to adapt to the changing environment. Three experiments under different situations, that is, static carrier, only horizontal motion of carrier considered, and 3D motion of carrier considered, are conducted. The results demonstrate that the helicopters can all reach the ideal landing point with the reasonable path in different situations. The small terminal error and relatively vertical motion between the helicopter and the carrier ensure a precise and safe landing.

Keywords: helicopter; aircraft carrier; landing; path planning; pigeon inspired optimization

1. Introduction Landing on the flight deck of a carrier is one of the most dangerous tasks for fixed-wing aircraft [1]. Only 70% of aircraft can finish the landing task successfully for one time, and the percentage is even lower at night [2]. When fixed-wing aircraft land on the flight deck, the landing path is regulated. The fixed-wing aircraft just need to follow the path and reduce error as much as possible [3]. In addition to fixed-wing aircraft, helicopters are also introduced on the carrier to perform the tasks which are beyond the capability of fixed-wing aircraft, such as sea rescue, transportation, and antisubmarine warfare. One of the biggest advantages is that helicopters do not occupy the resource of runways, which reduces the waiting time of fixed-wing aircraft in the air [4]. Only an empty area of the flight deck is needed for a helicopter to land, and the mode of vertical landing is much safer than stopping by the arresting cable on the flight deck of a carrier [5]. In a typical combat mission, helicopters usually take off first to execute reconnaissance tasks, and then fixed-wing aircraft are launched in the air to attack enemy planes. After the combat mission is finished, fixed-wing aircraft return to the carrier and are maintained on the flight deck, and helicopters land on the specified empty area of flight deck. Unlike fixed-wing aircraft, the landing path of

Mathematics 2018, 6, 178; doi:10.3390/math6100178 www.mdpi.com/journal/mathematics Mathematics 2018, 6, 178 2 of 20 helicopters is not regulated and can be determined autonomously, satisfying certain constraints [6]. At present, the path of helicopter landing on the carrier is determined by the pilot in advance. In this mode, the pilot watches the moving ideal landing point all the time and manipulates the helicopter. When the weather is good and the near space of the carrier is free, manual operation is enough to finish the landing task. However, if bad weather or a busy situation in the near space of the carrier occurs, this working mode is not so reliable, due to the poor visibility and the dynamical environment. To be specific, the ideal landing point is changing with the movement of carrier, and a no-fly zone is set on the space near the carrier. Additionally, specific angles of reaching the ideal landing point are regulated for the helicopter to avoid a collision with other returning fixed-wing aircraft and to reduce the influence of wind field. Those constraints increase the difficulties of searching a feasible landing path by men, and an automatic path planner for helicopters is imperative to ensure the efficiency and safety of the landing task on the carrier [7].

2. Literature Review and the Work of this Paper The study on helicopter landing has been a hotspot in recent years. The estimation of position and altitude [8,9], landing in unusual conditions [10], and visualization [11,12] are the main concerns. In Reference [13], the movement state estimation method based on the vision image process for unmanned helicopters is proposed. In this method, a relative position to the landing pad and attitude estimation algorithm and a velocity and angular rate estimation algorithm are included to estimate the position, attitude, velocity, and angular rate with the vision image process data. Considering that it is difficult for a pilot to see obstacles or ground through snow or dust, a sensor detecting obstacles or ground inside aerosols has been designed in Reference [10]. The sensor can suppress the return signals from nearby aerosol scattering and has a sensitivity and dynamic range to detect obstacles or ground inside aerosol at the same time. To make an accurate landing, a vision-based navigation algorithm is developed to deal with the landing task in complex environments. The vision navigation system is integrated with algorithms of vision detection, target recognition, and navigation instruction calculation [11]. In the existing literature concerning path planning of helicopters, different situations are taken into account. In Reference [14], the helicopter is used for both the surveying and spraying of weeds in forest. This task is divided into two phases, and 2D and 3D path plans are both operated respectively with the rapidly exploring random tree (RRT) algorithm. The helicopter can also be applied in aerial photography. In this task, multiple points are set that the helicopter must pass through to take photos and shoot videos, and the optimal path is obtained by developing an improved probabilistic roadmap method (PRM) [15]. When a helicopter is used in the military, it must be guaranteed that the helicopter can avoid threats in low altitude areas [16] and can deal with emergency when one engine fails. In Reference [17], the landing trajectory of a multiengine helicopter in the event of a single engine failure is generated using the optimal control method. Time derivatives of thrust coefficients are treated as control variables, and constraints on rotor speed and thrust vector are imposed. The goal is to minimize the distance between the touchdown point and the original takeoff point, subject to impact speed limits at touchdown. Another important issue during helicopter landing is to reduce ground noise. In Reference [18], this problem is formulated into an optimal control problem that minimizes noise levels measured at points on the ground surface using equations of motion of three degrees of freedom. In summary, the studies of path planning for helicopters concentrate on the establishment of mathematical models and search algorithm designs in different tasks. Additionally, dividing the whole flight into several phases according to a specific mission also makes the problem easier to solve [19]. On the platform of flight deck operation on the carrier, path planning for aircraft taxiing and decision making for aircraft landing are concentrated on. The static obstacles (i.e., unemployed aircraft and island on the flight deck [20]), different states of wing (i.e., folded wings and unfolded wings [21]) and terminal constraints are taken into account in describing the path planning problem [22]. In the algorithm design, geometrical spatial search algorithms and swarm optimization algorithms are used Mathematics 2018, 6, 178 3 of 20 to obtain the optimal taxiing path. As to the landing task of carrier aircraft, the optimization of a landing sequence for a team of fixed-wing aircraft [23], the landing path determination for fixed-wing aircraft [24], and comparison between expert user heuristics and automatic planning tools [25] are given more attention. In Reference [23], a dynamic sequencing algorithm based on an ant colony optimization (ACO) algorithm was developed to optimize the landing sequence of aircraft, considering the landing failure of aircraft, which provides an intelligent tool for overall air traffic management on aircraft carriers. Due to the fuzziness of environmental information, a TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution)-based group decision-making method was proposed to assist the pilot determining the most reasonable landing path under various environments [24]. To the best of the authors’ knowledge, there are no reported studies focused on path planning for helicopter landing on a carrier. Similar studies focus on path planning for helicopter landing on a moving platform [26,27]. In Reference [26], the trajectory planner is based on the Variational Hamiltonian and Euler–Lagrange equations, and a kinematic model of the helicopter is used to derive an optimal controller that is able to track an arbitrarily moving target and then land on it. In Reference [27], the landing procedure is divided into four stages, and the goal of each flight stage is different. Finally, the landing path is obtained by linear programming piecewise. In this paper, the landing task of helicopters is divided into two phases, according to the distance between the helicopter and the ideal landing point, that is, approaching the carrier and landing on the flight deck. Then, the conceptual model is abstracted, which gives a general idea of solving the problem. The constraints by category and the cost functions in different flight phases are all included in the mathematical model. When designing the path planning algorithm, different search strategies are developed using the pigeon inspired optimization (PIO) algorithm, based on the feature of each flight phase. The PIO algorithm is a bioinspired hybrid metaheuristic [28] and was firstly proposed by Duan in 2014 [29], and the feasibility and superiority of the algorithm have been proved in image restoration problems [30] and model prediction control [31]. However, the PIO algorithm has not been applied in a multiphase path planning problem, and its performance needs to be further investigated. The rest of the paper proceeds as follows. Section2 describes the landing task of helicopters on a carrier. The mathematical model of the landing path planning problem is established in Section3. The details of the path planning algorithm are depicted in Section4. Experimental studies are reported in Section5 and the concluding remarks are contained in the last section.

3. Description of the Landing Path Planning Problem for Helicopters In this section, the background of the landing task for helicopters will be introduced, and the whole flight is divided into two phases. Then, the conceptual model solving the problem is proposed to present the idea and organize the elements that are taken into account in this problem.

3.1. Division of the Flight of the Landing Task The landing task of helicopter on the carrier specified in this paper can be described as follows: After the helicopter has finished the task in the air, it informs the air traffic control center on the carrier of the return message. Then, the carrier sails towards a certain direction with a constant velocity [32]. The helicopter keeps following the carrier at a safe altitude and then decreases its altitude to land on the ideal landing point on the flight deck. The whole flight can be divided into the following two phases: 1. Approaching the carrier (“approaching phase” for short for convenience in the rest of this paper) The helicopter flies with a constant altitude in this phase. The goal is to minimize the horizontal distance between the helicopter and the carrier as soon as possible. In this phase, the distance between the helicopter and the carrier is relatively far, so the constraint of a no-fly zone is not imposed. After the distance is reduced to a certain value, the helicopter moves to the next flight phase. Mathematics 2018, 6, x FOR PEER REVIEW 4 of 19

Mathematics 2018, 6, 178 4 of 20 2. Landing on the flight deck (“landing phase” for short for convenience in the rest of this paper) TheMathematics ideal 2018 landing, 6, x FOR point PEER REVIEW is located in the empty area of the center flight deck. The 4helicopter of 19 2. Landing on the flight deck (“landing phase” for short for convenience in the rest of this paper) reduces its altitude and reaches the flight deck with the minimum position error. Note that the heave 2. Landing on the flight deck (“landing phase” for short for convenience in the rest of this paper) motionThe of ideal the carrier landing is pointconsidered is located in this in phase the empty to reflect area the of thechange center of altitude flight deck. of the The ideal helicopter landing reducespoint. itsThe altitude ideal landing and reaches point is thelocated flight in the deck empty with area the of minimum the center positionflight deck. error. The helicopter Note that the heaveToreduces motion get itsa ofbetteraltitude the carrierunderstanding and reaches is considered the flightof the deck in above this with phase thelanding minimum to reflect phases, position the the change error. diagram ofNote altitude thatof the the of heavehelicopter the ideal motion of the carrier is considered in this phase to reflect the change of altitude of the ideal landing landingreturning point. is shown in Figure 1. point. To get a better understanding of the above landing phases, the diagram of the helicopter returning To get a better understanding of the above landing phases, the diagram of the helicopter is shownreturningApproaching in Figure is shown the1. carrier in Figure 1. Landing on the flight deck

Approaching the carrier Landing on the flight deck

Fly with constant altitude

Fly with constant altitude

Reduce the altitude and reach the ideal landing point Reduce the altitude and reach the ideal landing point

Aircraft carrier No-fly zone Aircraft carrier No-fly zone

Figure 1. Description of the landing task for the helicopter. FigureFigure 1. 1.Description Descriptionof of thethe landing task task for for the the helicopter. helicopter.

InIn FigureFigureIn Figure1 ,1, the the 1, content thecontent content of of each ofeach each flight flight flight phase phasephase is shown. is shown.shown. The The The no-fly no-fly no-fly zone zone zone is is set setis at setat the the at end endthe of endof the the flightof the deckflightflight to deck avoid deck to theavoid to helicopteravoid the the helicopter helicopter colliding colliding colliding with the withwith returning the returningreturning fixed-wing fixed-wing fixed-wing aircraft aircraft and aircraft to and reduce and to reduce to the reduce influence the the ofinfluence windinfluence field. of wind Theof wind taskfield. field. of The path The taskplanning task of of path path is to planningplanning generate isis a toto feasible generate generate landing a feasiblea feasible path landing forlanding the path helicopter path for the for andthe guaranteehelicopterhelicopter landingand andguarantee guarantee safety. landing landing safety. safety.

3.2. Conceptual3.2. Conceptual Model Model of the of the Path Path Planning Planning Problem Problem In Section 2.1, the whole flight was divided into two phases, which provides a frame for InIn SectionSection 3.1 2.1,, the the whole whole flight flight was was divided divided into into two phases,two phases, which which provides provides a frame a forframe solving for solving the problem. On this basis, the conceptual model is proposed to organize the idea and thesolving problem.elements the problem. Onof the this problem, basis, On this the as shown conceptualbasis, in the Figure modelconceptual 2. is proposed model tois organizeproposed the to idea organize and elements the idea of and the problem,elements asof shownthe problem, in Figure as shown2. in Figure 2.

Figure 2. Conceptual model of the problem.

Figure 2. Conceptual model of the problem.

Mathematics 2018, 6, 178 5 of 20

In Figure2, the problem is composed of three elements. The constraints are imposed on the helicopter and the task, respectively. The goal of the landing task is described by the cost function. In the path searching process, different strategies need to be developed to adapt to the situations. The above description summarizes the idea of solving the problem. Next, each element in Figure2 will be explained in detail.

4. Establishment of the Mathematical Model According to the conceptual model, path planning for helicopter landing on the carrier can be formulated into an optimization problem. In this problem, under a given model of helicopter, the control variables are the velocities in each direction which need to be optimized. The constraints can be grouped into two categories, that is, constraints imposed on the control variables and the state of the helicopter. In the constraints of control variables, the performance of the helicopter is limited by the velocity and acceleration. As for the constraints of state of helicopter, they limit the position and flight direction of the helicopter, considering the characteristics of the environment and the requirements of the landing task. As the landing task has been divided into two phases, the optimization goal of each phase is not the same, and they are proposed based on the characteristics of each phase. Next, each element of the optimization problem will be formulated into mathematical forms.

4.1. Kinematic Model and Performance Constraints of Helicopter In this study, a simplified kinematic model of helicopter presented in Reference [27] is used, and a more complicated form of equations can be further considered in the design of a landing controller. Firstly, a coordinate system must be defined to describe the motion of the helicopter and the carrier. The coordinate origin O is fixed and is defined as the initial position of the ideal landing point, and the axis Ox points to the direction where the carrier sails. The axis Oz is vertically downward and the axis Oy can be determined by the right hand rule. Under the above coordinate system, (x, y, z) and (Vx, Vy, Vz) are the position and the velocity of helicopter in each direction, and ϕ is the yaw angle. In this path planning problem, x, y, z, ϕ are the state variables and can be denoted as the vector h i x = x y z ϕ . Vx, Vy and Vz are the control variables to be optimized and can be expressed as h i u = Vx Vy Vz . Note that in different flight phases, the control variables are not always the same. For example, when the helicopter is approaching the carrier, the flight altitude remains unchanged. In this phase, Vz = 0 and only Vx and Vy are the control variables. While in the landing phase, all of the three control variables need to be optimized to make a precise landing. In the landing task, the helicopter must fly within its maneuverability. The performance constraints imposed on the helicopter can be denoted as follow:

max Vx ≤ Vx (1)

max Vy ≤ Vy (2) max Vz ≤ Vz (3) max axy ≤ axy (4) max az ≤ az (5) In the above five equations, the flight velocity in each direction is constrained by Equations (1)–(3). Equations (4) and (5) are added to limit the acceleration in the plane xOy and axis Oz, r . 2 . 2 and axy = Vx + Vy. Mathematics 2018, 6, 178 6 of 20 Mathematics 2018, 6, x FOR PEER REVIEW 6 of 19

4.2. Constraints Constraints Regarding Regarding the Landing Task InIn the landinglanding task,task, certaincertain constraints constraints must must be be imposed imposed on on the the state state of helicopterof helicopter to meet to meet the taskthe taskrequirements. requirements. These These constraints constraints are described are described as follow: as follow: 1. Determination of the feasible flying flying space To ensure thethe safetysafety ofof the the landing landing task, task, the the no-fly no-fly zone zone is setis set to to avoid avoid the the helicopter helicopter disturbing disturbing the thenormal normal landing landing of the of fixed-wing the fixed- aircraftwing oraircraft even collidingor even withcollidin theg fixed-wing with the aircraft.fixed-wing Additionally, aircraft. Additionally,the wind field the in thewind near field space in the of near the carrier space of also th affectse carrier the also landing affects of the the landing helicopter. of the The helicopter. position Theof the position helicopter of the cannot helicopter be ahead cannot of be the ahead carrier of the in thecarrierOx direction,in the Ox direction, or the landing or the tasklanding will task fail. willThose fail. constraints Those constraints limit the limit position the ofposition the helicopter of the helicopter and are described and are described in Figure in3. Figure 3.

Position of the helicopter Feasible flying space

θ = 45 Boundary of the influenced wind field f

Helicopter can not be ahead of carrier

O x

Ideal landing point y

Space used for fixed-wing landing

No fly zone FigureFigure 3. DiagramDiagram of of feasible feasible flying flying space in the landing task.

InIn Figure 33,, thethe feasiblefeasible flyingflying spacespace onlyonly occupiesoccupies aa relativelyrelatively smallsmall partpart ofof thethe spacespace nearnear thethe carrier, which adds the difficultydifficulty of searching forfor the optimaloptimal landinglanding path.path. The The constraint of of the direction of reaching the ideal landing point will be described later on. 2. Direction Direction of of reaching the ideal landing point When the helicopter is near the flight flight deck, the helicopter is required to reach the ideal landing ◦ point with the constraint of θf == 45 to conform to the landing rules [33], as shown in Figure3. point with the constraint of f 45 to conform to the landing rules [33], as shown in Figure 3. Note that this constraint is proposed based on the assumption that the carrier sails in the opposite Note that this constraint is proposed based on the assumption that the carrier sails in the opposite direction of the wind field. Under this condition, the wind in the two sides of the carrier is symmetrical, direction of the wind field. Under this condition, the wind in the two sides of the carrier is and the ideal direction for the helicopter reaching the landing point is set as θ = 45◦. In other cases of f θ =  symmetrical,asymmetric wind, and the the ideal ideal direction direction for must the behelicopter decided reaching according the to landing the wind point graph. is set The as valuef 45 of θ. Induring other landing cases of can asymmetric be calculated wind by, the the ideal following direction equation: must be decided according to the wind graph. The value of θ during landing can be calculated by the following equation: y − y θ = tan−1( t ) (6) − yy− θ = tan1 (xtt − x ) − (6) xt x where (x, y) is the current position of helicopter, and (xt, yt) is the position of the ideal landing point.

The above constraint can be denoted as the following form, where eθ max is the maximum θ permitted error of f :

Mathematics 2018, 6, 178 7 of 20

where (x, y) is the current position of helicopter, and (xt, yt) is the position of the ideal landing point. The above constraint can be denoted as the following form, where eθmax is the maximum permitted error of θ f :

eθ = θ(t f ) − θ f ≤ eθmax (7)

4.3. Determination of the Cost Function The helicopter should finish the landing task as soon as possible after it performs the specific task in the air. In each flight phase, the goal is not exactly the same. To be specific, when the helicopter is approaching the carrier, the prime goal is to reduce the distance between the helicopter and the carrier. To satisfy the constraint in Equation (7) more easily, the value of θ becomes important, as the distance is smaller. The cost function in the approaching phase can be written as:

J1 = dis_hor + t1 · n · eθ (8)

J1 is the cost function of the approaching phase, and dis_hor is the horizontal distance between the helicopter and the ideal landing point. n is the serial number of path point, and t1 makes the two items in Equation (8) have the same order of magnitude. Note that Equation (8) is a form of dynamic weight function. With the increasing of flight time, the value of θ will be given more attention. In the landing phase, the terminal position error between the helicopter and the ideal landing point is important. Moreover, to reduce the heavy impact at the landing moment, the difference of vertical velocity between the helicopter and the ideal landing point is expected to be smaller. The cost function in this phase can be denoted as:

J2 = dis + t2· Vz(t f ) − Vzc(t f ) (9)

J2 is the cost function in the landing phase, and dis is the distance between the helicopter and the carrier in 3D space. Vzc is the vertical velocity of the ideal landing point, and t2 makes the two items in Equation (9) comparable. Equation (9) gives a comprehensive consideration of both the landing accuracy and safety.

5. Path Planning Algorithm Design for Helicopter Landing As the whole flight has been divided into two phases, a multiphase path planning strategy will be developed in this section. Then, the general principle of the PIO algorithm is introduced, based on which the path planning algorithm is proposed. The flow of the algorithm is also presented.

5.1. The Multiphase Path Planning Strategy The path planning strategy is developed according to different flight phases, and the details are presented below: 1. Strategy in the approaching phase Considering that the helicopter is far from the carrier and the environment is changing, only the motion of helicopter in a short time window of future (denoted as Tw) is planned in one step of search, assuming that the discrete time interval is ∆t, and the number of path point in one step of search (denoted as Nw) can be determined (Nw = Tw/∆t). Based on the ideas of model predictive control [34], this “short-term” planning strategy will last before the horizontal distance between the helicopter and the carrier is reduced to a certain value. Note that the heave motion of the carrier is not considered in this phase because it will not have an influence on the result. 2. Strategy in the landing phase Mathematics 2018, 6, 178 8 of 20

In this phase, the helicopter is near the carrier. As the motion of the carrier is known, the remaining unplanned path can be obtained all at once. To reduce landing time, the helicopter must decrease max its altitude as soon as possible. This makes the helicopter fly with Vz in most of the flight time, and only the values of Vz during a certain period of time, which is before the helicopter touches the flight deck, need to be optimized. This operation reduces the computational load of online planning and makes the landing task more efficient at the same time. To reduce the terminal position error, the Chebyshev pseudo spectral method is used to generate the collocation points [35]. The collocation points are distributed according to the value of Tk, as shown in Equation (10): (Nc − k) · π Tk = cos( )(k = 0, 1, 2, . . . , Nc) (10) Nc

The number of collocation points is Nc + 1. The Chebyshev points are dense at both ends and sparse in the middle. This feature just satisfies the demand that there should be more collocation points at the end of landing path to reduce the position error. Note that Tk ∈ [−1, 1], and it should be converted to the time interval [t0, tf] using the following equation:

t f − t0 t f + t0 t = · T + (11) k 2 k 2

Equation (11) demonstrates that only the control variable at the moments tk (k = 0, 1, ... , Nc) is optimized, and the values of control variables at other moments (denoted as ti) can be obtained by linear interpolation, as expressed in Equation (12) (take Vxti (tk < ti < tk+1) as an example):

Vxtk+1 − Vxtk Vxti = · (ti − tk) + Vxtk (12) tk+1 − tk

5.2. A Brief Introduction of the Pigeon Inspired Optimization (PIO) Algorithm The pigeon inspired optimization algorithm is a kind of bioinspired hybrid metaheuristic. In recent years, the bioinspired hybrid metaheuristic has attracted a lot of attention as an effective method to solve difficult real-world optimization problems because, in those algorithms, the solution quality and the computation load can be made a good balance, and an optimal or near optimal solution can be obtained in short time. The studies on the bioinspired hybrid metaheuristic have turned to problem-oriented, that is, an algorithm is applied or improved, based on the characteristics of the specific problem [36]. For example, to solve the capacitated network design problem (CNDP), a robust optimization model based on multiband robustness was proposed, and a hybrid primal heuristic was developed [37]. In this algorithm, a randomized fixing strategy inspired by ant colony optimization and an exact large neighborhood search were combined, and solutions of extremely high quality associated with low optimality gaps can be obtained by this new algorithm. Another case is to minimize the electricity cost in electric grids. To achieve this goal, hybridization of the genetic algorithm (GA) and PIO algorithm (HGP) in demand side management were proposed to apply for residential load management, and the performance of HGP is better than GA and PIO in optimizing the fitness function [38]. In the path planning problem, a predator–prey pigeon inspired optimization (PPPIO) algorithm was proposed to generate the three-dimensional path for uninhabited combat aerial vehicles (UCAV) in a dynamic environment [39]. In the PPPIO algorithm, the prey–predator concept is adopted to improve global best properties and to enhance the convergence speed of the standard PIO algorithm, and comparative simulation results verified the efficiency of the proposed PPPIO algorithm in solving the UCAV three-dimensional path planning problem. As for the PIO algorithm, it is motivated by the self-organized homing behavior of pigeons. To be specific, pigeons can always find their homes easily by using their unique homing tools, that is, magnetic fields, the sun, and landmarks. In the PIO algorithm, there are two operators to change Mathematics 2018, 6, 178 9 of 20 the position of pigeons. Next, the details of the PIO algorithm will be explained, combining the characteristics of the path planning problem for helicopters landing on the carrier. 1. The map and compass operator

In the PIO algorithm, each pigeon i has its position Xi and velocity Vi, and Xi and Vi are both D-dimension vectors according to the scale of problem. In each iteration, Xi and Vi are updated with the following equations: t+1 t −R(t+1) t Vi = Vi · e + rand · (Xg − Xi ) (13) t+1 t t+1 Xi = Xi + Vi (14) where t is the times of iteration, R is the operator factor, rand is a uniform number in the interval [0, 1], and Xg is the current global best position. In the path planning problem studied in this paper, the position Xi denotes a path consisting of a certain number of points, and the number of points in a path is the dimension of Xi and is determined by the phase that the path belongs to. In the approaching phase D = Nw, and in the landing phase D = Nc + 1. Vi is the disturbing operator that changes the value of the path point, and Xg represents the control variable vector corresponding to the minimal value calculated by the cost function in Equations (8) or (9), according to the different phases of the landing task. 2. Landmark operator When the pigeons are closer to their home (after certain times of iterations), the landmark operator is used in the rest of the journey. With the landmark operator, the number of pigeons is decreased by half after every iteration. Additionally, the center of the pigeons’ positions is referenced in the landmark operator to make the pigeons reach their home quickly. Assuming the center of the pigeons’ t positions at the tth iteration is Xc, the rule of updating the position of pigeon i can be written as the following equations: Nt−1 Nt = P (15) P 2 t t t ∑ Xi · f itness(Xi ) Xc = t t (16) NP∑ f itness(Xi ) t+1 t t t Xi = Xi + rand · (Xc − Xi ) (17) t t where NP is the number of pigeons at the tth iteration, and Xc is the geometrical center of the pigeons. t fitness(Xi ) is the fitness value of pigeon i, which can be calculated by Equation (8) or Equation (9), according to the different landing phase, and rand is a uniform number in the interval [0, 1]. In the landmark operator, the center of all pigeons is regarded as the destination in each iteration. Half of all the pigeons with the worst fitness values will be abandoned in the next iteration, and the rest of the pigeons that are close to the destination will fly home quickly. In the landmark operator, the number t of spare paths is reduced by half after an iteration, and Xc is the control variable vector corresponding to the mean fitness value of all the spare paths. Similar to the map and compass operator, the value of the control variable is updated by adding a disturbing operator on the old value.

5.3. Process of the Multiphase Path Planning Algorithm The path planning strategy and PIO algorithm are integrated into the path planning algorithm design. The pseudocode is presented below (Algorithm 1), and the corresponding flow chart is summarized in Figure4. Mathematics 2018, 6, x FOR PEER REVIEW 10 of 19

AlgorithmMathematics 2018 1: PIO-based, 6, 178 algorithm for helicopter landing on the carrier 10 of 20 1: initialization: length of time window Tw, time interval ∆𝑡, times of iteration in map and compass operator Nc1, times of iteration in landmark operator Nc2, scale of problem Np, other Algorithm 1: PIO-based algorithm for helicopter landing on the carrier parameters in PIO algorithm, helicopter performance, initial path Xi and velocity Vi 2:1: calculate initialization: the fitness length value of time of window each pathTw, timeand intervaldetermine∆t, timesthe global of iteration best X ing. map and compass operator 3: Nmapc1, times and ofcompass iteration operator in landmark operator Nc2, scale of problem Np, other parameters in PIO algorithm, helicopter performance, initial path X and velocity V 4: for t1 = 1: Nc1 i i 2: calculate the fitness value of each path and determine the global best Xg. 5: for i = 1: Np 3: map and compass operator 6: calculate Xi and Vi 4: for t1 = 1 : Nc1 7: end 5: for i = 1: Np 8: for j = 1: D 6: calculate Xi and Vi 9:7: end if Xi is beyond its permitted range 10:8: for j =set1: X Di as the closest threshold 11:9: end if Xi is beyond its permitted range 12:10: end set Xi as the closest threshold 13:11: calculate end the fitness value of each updated path and determine the current global best Xg. 14:12: end end 15:13: landmark calculate operator the fitness value of each updated path and determine the current global best Xg. 14: end 16: for t2 = 1: Nc2 15: landmark operator 17: Np = Np/2 (keep half of the spare paths with better fitness value and abandon the other 16: for t2 = 1 : Nc2 half) 17: Np = Np/2 (keep half of the spare paths with better fitness value and abandon the other half) 18: for k = 1: Np 18: for k = 1: Np c 19:19: calculate calculate XXc ofof the eacheachremaining remaining paths paths 20:20: update update XXkk 21:21: end end 22:22: for for l = 1: D 23:23: if if Xl isis beyond its its permitted permitted range range 24:24: set XXll asas thethe closest closest threshold threshold 25:25: end end 26:26: end end 27: calculate the fitness value of each updated path. 27: calculate the fitness value of each updated path. 28:28: end end 29: output: Xg is returned as the global optimal control variables and the corresponding path is the landing 29: output: Xg is returned as the global optimal control variables and the corresponding path is path for helicopter. the landing path for helicopter.

FigureFigure 4. 4.TheThe flow flow of ofpath path planning planning algorithm. algorithm.

Mathematics 2018, 6, 178 11 of 20 Mathematics 2018, 6, x FOR PEER REVIEW 11 of 19

InIn Figure Figure4, path4, path planning planning strategy strategy No. 1No. is applied 1 is applied in the approachingin the approaching phase, and phase, path planningand path strategyplanning No. strategy 2 is usedNo. 2 in is theused landing in the landing phase to phase obtain to theobtain remaining the remaining unplanned unplanned path all path at once.all at PIOonce. algorithm PIO algorithm is adopted is adopted in both in of both the twoof the phases two phases to search to search the landing the landing path for path helicopter. for helicopter.

6. Results 6. Results To validate the effectiveness of the established mathematical model and the multiphase path To validate the effectiveness of the established mathematical model and the multiphase path planning algorithm in solving the problem of helicopter landing on the carrier, three experiments planning algorithm in solving the problem of helicopter landing on the carrier, three experiments under different situations are conducted. In the first case, the carrier is static during the whole landing under different situations are conducted. In the first case, the carrier is static during the whole process of the helicopter, and the landing path of the helicopter under this situation is obtained. In the landing process of the helicopter, and the landing path of the helicopter under this situation is second case, the carrier always moves with a velocity of Vc = 10 m/s. Based on the setting in case obtained. In the second case, the carrier always moves with a velocity of Vc = 10 m/s. Based on the 2,setting the heave in case motion 2, the ofheave the carriermotion is of taken the carrier into account is taken in into case account 3, which in case increases 3, which the increases difficulty the of generatingdifficulty of the generating landing path. the Inlanding all three path cases,. In theall three starting cases, point the of starting the helicopter point andof the the helicopter ideal landing and point are set as (−800, −600, −80) and (0, 0, 0), respectively. the ideal landing point are set as (−800, −600, −80) and (0, 0, 0), respectively. TheThe parametersparameters usedused inin thethe PIOPIO algorithmalgorithm areare setset accordingaccording toto thethe scalescale ofof thethe problemproblem andand areare basedbased onon ReferenceReference[ 30[30]] and and Reference Reference [ 31[31]] to to result result in in the the best best performance, performance, as as presented presented inin TableTable1 .1. TheThe constraints constraints on on helicopter helicopter performance performance and and landing landing task task are are set byset referringby referring to Reference to Reference [27] and[27] Referenceand Reference [33], and[33], they and arethey given are given in Table in 2Table. All the2. All experiments the experiments are conducted are conducted on a desktop on a desktop with Intelwith Core Intel i7-3370Core i7 3.40-3370 GHz 3.40 withGHz MATLAB with MATLAB R2017a. R2017a. Table 1. Parameters setting in pigeon inspired optimization (PIO) algorithm. Table 1. Parameters setting in pigeon inspired optimization (PIO) algorithm. Parameter Tw ∆t Np RNc1 Nc2 Parameter R Value 5 s 1 s 1000 0.2 20 30 Value 5 s 1 s 1000𝒑𝒑 0.2 20 30 𝑻𝑻𝒘𝒘 ∆𝒕𝒕 𝑵𝑵 𝑵𝑵𝒄𝒄𝒄𝒄 𝑵𝑵𝒄𝒄𝒄𝒄

TableTable 2. 2.Constraints Constraints of of helicopter helicopter performance performance and and landing landing task. task. max max max max max Item Item Vx V y Vz axy az eθ Value Value20 𝐦𝐦𝐦𝐦𝐦𝐦m/s 20 m/s10 𝐦𝐦𝐦𝐦𝐦𝐦m/s 10 m/s 3 3 m/s𝐦𝐦𝐦𝐦𝐦𝐦 m/s 3 m/s3 m/s𝐦𝐦2𝐚𝐚𝐚𝐚2 1 m/s12 m/s𝐦𝐦𝐦𝐦𝐦𝐦2 5 deg 5 deg 𝑽𝑽𝒙𝒙 𝑽𝑽𝒚𝒚 𝑽𝑽𝒛𝒛 𝒂𝒂𝒙𝒙𝒙𝒙 𝒂𝒂𝒛𝒛 𝒆𝒆𝜽𝜽

6.1.6.1. HelicopterHelicopterLanding Landing onon thethe StaticStaticFlight FlightDeck Deck InIn this this case, case, the the carrier carrier does does not not move move during during the landing the landing task of task the helicopter.of the helicopter. With the With settings the insettings Tables 1in and Table2, thes 1 landingand 2, the path landing of the path helicopter of the ishelicopter obtained is with obtained the proposed with the method, proposed as shownmethod, inas Figure shown5. in Figure 5.

Approaching phase

Landing phase

Starting point of the helicopter

The ideal landing point

-80

-60

-40

[m] -20 z

0

20

-600

-400 200

0 -200 -200

-400 0 -600

y [m] 200 -800 x [m]

Figure 5. The landing path of the helicopter when the carrier is static. Figure 5. The landing path of the helicopter when the carrier is static.

Mathematics 2018, 6, 178 12 of 20 Mathematics 2018, 6, x FOR PEER REVIEW 12 of 19 Mathematics 2018, 6, x FOR PEER REVIEW 12 of 19 In Figure 5, the curves with different colors denote different flight phases of the helicopter. The InIn FigureFigure5 5,, thethe curves with with different different colors colors denote denote different different flight flight phases phases of the of thehelicopter. helicopter. The helicopter keeps its altitude when it is far from the carrier. As the distance decreases, the helicopter Thehelicopter helicopter keeps keeps its altitude its altitude when when it is it isfar far from from the the carrier. carrier. As As the the distance distance decreases, decreases, the helicopter reduces the flight altitude and finally lands on the flight deck. To get a better understanding of the reducesreduces thethe flightflight altitudealtitude andand finallyfinally landslands onon thethe flightflight deck.deck. ToTo getget aa betterbetter understandingunderstanding ofof thethe result, the state of the helicopter and the control variable during the landing task are presented in result,result, thethe statestate ofof thethe helicopterhelicopter andand thethe controlcontrol variablevariable duringduring thethe landinglanding tasktask areare presentedpresented inin Figures 6 and 7 respectively. FiguresFigures6 6 and and7 7respectively. respectively. ] g ] e g d [ e d [ ] m ] [

m

[ z

z ] m ] [

m

[ y

y

Figure 6. State of the helicopter during the landing task. FigureFigure 6.6. StateState ofof thethe helicopterhelicopter duringduring thethe landinglanding task.task.

20 20 Approaching phase Approaching phase 10 Landing phase 10 Landing phase 0 00 10 20 30 40 50 60 70 80 0 10 20 30 t 40[s] 50 60 70 80 10 t [s] 10 5 5 0 00 10 20 30 40 50 60 70 80 0 10 20 30 t 40[s] 50 60 70 80 40 t [s] 40 20 20 0 00 10 20 30 40 50 60 70 80 0 10 20 30 t 40[s] 50 60 70 80 4 t [s] 4 2 2 0 00 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 t [s] t [s] Figure 7.7. Velocity ofof the helicopterhelicopter duringduring thethe landinglanding task.task. Figure 7. Velocity of the helicopter during the landing task. In Figure 6, the motion of the helicopter in the xOy plane and Oz axis are given, respectively. In In Figure 6, the motion of the helicopter in the xOy plane and Oz axis are given, respectively. In the approaching phase, the value of z remains unchanged, which is consistent with the fourth the approaching phase, the value of z remains unchanged, which is consistent with the fourth

Mathematics 2018, 6, x FOR PEER REVIEW 13 of 19 subplot in Figure 7. In the landing phase, Vz keeps to the maximum most of the time to make the helicopter reach the ideal landing point as soon as possible. In general, the same conclusion can be reached from Figure 6 or Figure 7, thus making those results reasonable. The terminal state of the helicopter and the ideal state are listed in Table 3.

Table 3. The terminal state of the helicopter and the ideal state.

Item x (m) y (m) z (m) (deg) (m/s) (m/s) (m/s) (deg) ActualMathematics 2018−0.009, 6, 178 −0.010 −0.081 66.7 2.717 6.309 0.278 1350.0 of 20 𝝋𝝋 𝑽𝑽𝒙𝒙 𝑽𝑽𝒚𝒚 𝑽𝑽𝒛𝒛 𝜽𝜽 Ideal 0 0 0 - - - 0 45 ErrorIn Figure−0.0096, the motion−0.010 of the− helicopter0.081 in the- xOy plane- and Oz axis are- given, respectively.0.278 In5.0 the approaching phase, the value of z remains unchanged, which is consistent with the fourth subplot Thein symbol Figure7. In“-” the means landing that phase, thereVz keeps is no to theideal maximum value mostfor this of the item. timeto In make Table the helicopter3, the deviation reach the ideal landing point as soon as possible. In general, the same conclusion can be reached from betweenFigure the actual6 or Figure and7, thusideal making landing those point results is reasonable. small, and The the terminal relative state ofvertical the helicopter velocity and between the the helicopterideal and state the are carrier listed in is Table 0.2738. m/s, which can reduce the impact on the helicopter and the carrier. Additionally, the direction of reaching the ideal landing point also meets the constraint. The above Table 3. The terminal state of the helicopter and the ideal state. results demonstrate that the helicopter can land on the static ideal landing point, and the generated path satisfiesItem variousx (m)constraintsy (m) to ensurez (m) flightϕ (safety.deg) Vx (m/s) Vy (m/s) Vz (m/s) θ (deg) Actual −0.009 −0.010 −0.081 66.7 2.717 6.309 0.278 50.0 Ideal 0 0 0 - - - 0 45 6.2. Results Errorof Landing−0.009 on the M−oving0.010 Flight−0.081 Deck - - - 0.278 5.0

In this experiment, the carrier is set to move with the velocity of Vc = 10 m/s, and other settings are the sameThe as those symbol in “-” case means 1. The that landing there is nopath ideal of the value helicopter, for this item. the Instate Table, and3, the the deviation control variable between the actual and ideal landing point is small, and the relative vertical velocity between the during flight are presented from Figures 8–10. helicopter and the carrier is 0.278 m/s, which can reduce the impact on the helicopter and the carrier. ComparedAdditionally, to the the results direction in of case reaching 1, the the helicopter ideal landing spent point more also meetstime theto finish constraint. the landing The above task. The helicopterresults can demonstratetrack the movement that the helicopter of the can ideal land landing on the static point ideal and landing finally point, land and on the it. generated In the landing max max pathz satisfies various constraints to ensure flight safety. phase, V is accelerated to Vz with the shortest time at the beginning, keeps with Vz most of the time6.2., and Results decreases of Landing gradually on the Moving in Flightthe last Deck several seconds. This mechanism can play the flight performance and reduce the landing time of the helicopter. To check the position error and the In this experiment, the carrier is set to move with the velocity of Vc = 10 m/s, and other settings directionare of the reaching same as thosethe ideal in case landing 1. The landing point, path the ofterminal the helicopter, state theof state,the helicopter and the control and variable the ideal state are listedduring in Table flight 4. are presented from Figures8–10.

Approaching phase

Landing phase

Motion of ideal landing point

Starting point of the helicopter

Starting point of the ideal landing point

-80

-60

-40 [m] -20 z

0

20 1500

-600 1000

-400 500

-200 0

0 -500

200 -1000 x [m] y [m]

Figure 8. Landing path of the helicopter and motion of the ideal landing point. Figure 8. Landing path of the helicopter and motion of the ideal landing point.

Mathematics 2018, 6, 178 14 of 20 MathematicsMathematics 2018 2018, ,6 6, ,x x FOR FOR PEER PEER REVIEW REVIEW 1414 ofof 1919 ] ] g g e e d [ d [ ] ] m [ m

[

z

z ] ] m [ m

[

y y

FigureFigure 9.9 9. .State State of of the the helicopter helicopter in in the the landing landing tasktask task consideringconsidering considering thethe the movingmoving moving ofof of thethe the carrier.carrier carrier. . ] s ] / s / m [ m

[

x x V V ] s ] / s / m [ m

[

y y V V ] s ] / s / m [ m

[

y y x x V V ] s ] / s / m [ m

[

z

z V V

Figure 10. Velocity of the helicopter in the landing task considering the moving of the carrier. FigureFigure 10 10. .Velocity Velocity of of the the helicopter helicopter in in the the landing landing task task considering considering the the moving moving of of the the carrier carrier. . Compared to the results in case 1, the helicopter spent more time to finish the landing task. TableTable 4 4. .Comparison Comparison between between the the actua actual lstate state of of the the helicopter helicopter and and the the ideal ideal state state. . The helicopter can track the movement of the ideal landing point and finally land on it. In the landing Item x (m) y (mmax) z (m) (deg) (m/s) (m/s) (m/s) max (deg) phase, ItemVz is acceleratedx (m) toyV (zm) withz (m the) shortest (deg) time at (m/s the) beginning, (m/s) keeps ( withm/s) Vz (degmost) of the time,ActualActual and decreases1009.9311009.931 gradually−−0.0770.077 −− in0.0780.078 the last29.129.1 several seconds.11.53711.537 This6.4166.416 mechanism 0.2440.244 can play48.248.2 the flight performanceIdealIdeal and1010 reduce1010 the landing00 time00 of the helicopter.- - To- - check the- -position error00 and the4545 direction of reachingErrorError the − ideal−0.0690.069 landing −−0.0770.077 point, −−0.078 the0.078 terminal - - state of the- - helicopter- - and the0.2440.244 ideal state3.23.2 are listed in Table4. InIn Table Table 4, 4, the the small small position position error error and and the the relative relative vertical vertical velocityvelocity make make the the helicopter helicopter land land precisely.precisely. A Additionally,dditionally, the the direction direction of of reaching reaching the the ideal ideal landing landing point point satisfies satisfies the the constraint. constraint. The The

Mathematics 2018, 6, x FOR PEER REVIEW 15 of 19 above results further explain the validity of the proposed path planning method for helicopter landing.

6.3. Formatting of Mathematical Components In a real situation, the sea water will move the carrier up and down, which makes the ideal landing point change in the Oz axis. In this experiment, the heave motion of the carrier is considered, based on the settings in case 2. The heave motion of the carrier can be represented by a sum of sine waves [40Mathematics], which2018 can, 6, 178 be denoted as: 15 of 20

ht( )=⋅ 0.2172 sin(0.4Table 4. Comparisont ) +⋅ 0.4714 between sin(0.5 the actualt ) +⋅ state 0.3592 of the helicopter sin(0.6 andt ) +⋅ the 0.2227 ideal state. sin(0.7t ) (18) where h is theItem positionx (m) of the yideal(m) landingz (m) pointϕ ( indeg the) OzVx ( axis,m/s) andVy ( tm/s is) theV currez (m/s)nt flightθ (deg )time. As wasActual explained 1009.931 in Section−0.077 4.1,−0.078 the heave 29.1 motion 11.537 of the 6.416carrier is 0.244 not considered 48.2 in the Ideal 1010 0 0 - - - 0 45 approachingError phase,− and0.069 the −path0.077 planning−0.078 results - of the approaching - - phase 0.244 in case 2 3.2 and 3 are the same. For this reason, only the path planning results in the landing phase are drawn when making a comparison.In The Table path4, the planning small position results error are and shown the relativefrom Figur verticales 11 velocity–13. make the helicopter Comparedland precisely. to Figure Additionally, 8, the carrier the has direction a 3D ofmotion reaching in Fig theure ideal 11, landing which pointmakes satisfies the ideal the landing point changeconstraint. in the The Oz above axis. results Under further this explain situation, the validity the helicopter of the proposed can still path reach planning the method ideal landing point andfor helicopterfinish the landing. landing task. Compared to case 2, the landing time is reduced by several seconds 6.3.in case Formatting 3. Furthermore of Mathematical, the Components curves representing the vertical motion in case 2 and 3 are close. This demonstratesIn a real that situation, the helicopter the sea water flies will for move as less the carriertime as up possible and down, to whichfinish makes the landing the ideal task. The vertical landingvelocity point of changethe carrier in the Oz isaxis. much In this smaller experiment, than the that heave of motionthe helicopter. of the carrier The is considered, terminal states correspondingbased on to the case settings 3 are in presented case 2. The heavein Table motion 5. of the carrier can be represented by a sum of sine In Tablewaves 5, [40 the], which position can be error denoted in the as: Oz axis increases, compared to the same item in Table 3 and

Table 4. This is becauseh(t) = 0.2172 the ·altitudesin(0.4t) +of0.4714 the ideal· sin(0.5 landingt) + 0.3592 point· sin (is0.6 changingt) + 0.2227 in· sin case(0.7t )3, which(18) makes it difficult to reach the ideal terminal state precisely. Note that the vertical motions of the helicopter and the whereideal hlandingis the position point of are the almost ideal landing in synchrony, point in the Ozandaxis, the and impactt is the can current be avoided flight time. to the greatest degree. OtherAs items was explained in Table in5 meet Section the 4.1 constraint, the heave and motion ensure of the a precise carrier is landing not considered path. in the approaching phase, and the path planning results of the approaching phase in case 2 and 3 are To demonstrate the variation of fitness value during iterations in the landing phase, the best the same. For this reason, only the path planning results in the landing phase are drawn when making fitness valuea comparison. of each Theiteration path planning in all three results cases are shown is drawn from Figuresrespectively, 11–13. as shown in Figure 14.

Approaching phase

Landing phase

Motion of the ideal landing point

Starting point of the helicopter

Starting point of the ideal landing point

-80

-60

-40

[m] -20 z

0

20

-600 1000

500 -400

0

-200 -500

y [m] x [m] 0 -1000

Figure 11. Landing path of the helicopter and 3D motion of the ideal landing point. Figure 11. Landing path of the helicopter and 3D motion of the ideal landing point.

Mathematics 2018, 6, 178 16 of 20 MathematicsMathematics 20182018,, 66,, xx FORFOR PEERPEER REVIEWREVIEW 1616 ofof 1919

11 CaseCase 2 2 ResultResult considering considering the the vertical vertical motion motion of of carrier carrier

0.50.5

00 7070 7575 8080 8585 9090 9595 100100 105105 tt [s][s] 5050

00

-50-50

-100-100 7070 7575 8080 8585 9090 9595 100100 105105 tt [s][s] 100100

00

-100-100

-200-200 550550 600600 650650 700700 750750 800800 850850 900900 950950 10001000 10501050 x [m] x [m] FigureFigure 12 12.12.. StateState of of the the helicopter helicopter in in the the landing landing task task,task,, considering considering the the 3D 3D motion motion of of the the carrier carrier.carrier..

2020

1515

1010 7070 7575 8080 8585 9090 9595 100100 tt [s][s] 1010

55

00 7070 7575 8080 8585 9090 9595 100100 tt [s][s] 3030 CaseCase 2 2 ResultResult considering considering the the vertical vertical motion motion of of carrier carrier Vertical velocity of the ideal landing point 2020 Vertical velocity of the ideal landing point

1010 7070 7575 8080 8585 9090 9595 100100 tt [s][s] 55

00

-5-5 7070 7575 8080 8585 9090 9595 100100 t [s] t [s] FigureFigure 13 13.13.. VelocityVelocity of of the the helicopter helicopter in in the the landing landing tasktask,task,, consideringconsidering thethe 3D3D motionmotion ofof thethe carriercarrier.carrier.. Compared to Figure8, the carrier has a 3D motion in Figure 11, which makes the ideal landing TableTable 55.. TerminalTerminal statesstates correspondingcorresponding toto casecase 33.. point change in the Oz axis. Under this situation, the helicopter can still reach the ideal landing point Item x (m) y (m) z (m) (deg) (m/s) (m/s) (m/s) (deg) and finishItem the landingx (m)task. y Compared (m) z (m to) case 2,(deg the) landing (m/s) time isreduced (m/s) by ( severalm/s) seconds (deg) in case 3. Furthermore,ActualActual the999.992999.992 curves −− representing0.0080.008 −−0.0620.062 the vertical36.436.4 motion12.93812.938 incase 29.5389.538 and 3 are close.0.0890.089 This demonstrates45.545.5 that theIdealIdeal helicopter 10001000 flies for as00 less time0.1010.101 as possible-- to finish-- the landing-- task.0.0890.089 The vertical 4545 velocity of the carrierErrorError is much −−0.0080.008 smaller −− than0.0080.008 that −−0.1630.163 of the helicopter.-- The-- terminal states-- corresponding00 0.50.5 to case 3 are presented in Table5.

Mathematics 2018, 6, 178 17 of 20

Table 5. Terminal states corresponding to case 3.

Item x (m) y (m) z (m) ϕ (deg) Vx (m/s) Vy (m/s) Vz (m/s) θ (deg) Actual 999.992 −0.008 −0.062 36.4 12.938 9.538 0.089 45.5 Ideal 1000 0 0.101 - - - 0.089 45 Error −0.008 −0.008 −0.163 - - - 0 0.5

In Table5, the position error in the Oz axis increases, compared to the same item in Tables3 and4. This is because the altitude of the ideal landing point is changing in case 3, which makes it difficult to reach the ideal terminal state precisely. Note that the vertical motions of the helicopter and the ideal landing point are almost in synchrony, and the impact can be avoided to the greatest degree. Other items in Table5 meet the constraint and ensure a precise landing path. To demonstrate the variation of fitness value during iterations in the landing phase, the best Mathematicsfitness value 2018 of, 6, eachx FOR iteration PEER REVIEW in all three cases is drawn respectively, as shown in Figure 14. 17 of 19

FigureFigure 14 14.. TheThe best best fitness fitness value value of of each each iteration iteration..

TheThe fitness fitness values values of of the the three three cases cases can can all all converge converge within within 20 20 times of of iteration, thus thus verifying verifying thethe fast fast convergence convergence of of the the PIO PIO algorithm. algorithm. Note Note that that Eq Equationuation (9) (9) is isregarded regarded as asthe the fitness fitness value value in thein the landing landing phase, phase, and and the thefitness fitness value value in case in case 3 is 3the is thesmallest smallest because because the vertical the vertical motions motions of the of helicopterthe helicopter and andthe theideal ideal landing landing point point are are almost almost in in synchrony, synchrony, which which makes makes the the second second item item of of EqEquationuation (9) (9) almost negligible.

77.. Conclusions Conclusions TheThe path path planning planning problem problem for for helicopter helicopter landing landing on on the the carrier carrier is is addressed addressed in in this this paper. paper. LiteratureLiterature investig investigationation showsshows thatthat thethe pathpath planningplanning problem problem for for helicopters helicopters is is important important and and there there is isno no reported reported study study concentrating concentrating on on the the landing landing path path planning planning problem problem on on the the carrier. carrier. Firstly,Firstly, the background of helicopter landing on on the carrier is introduced, and and the the wh wholeole flight flight isis divided divided into into two two phases phases according according to to different different goals goals of of each each phase, phase, that that is is,, the the approaching approaching phase phase andand the the landing landing phase. phase. The The feature feature of of each each phase phase is is described, described, and and the the conceptual conceptual model model of of the the path path planning problem problem is is abstracted abstracted on on this this basi basis,s, which which presents presents a general a general frame frame and and idea idea of solving of solving the problem.the problem. When When establishing establishing the the mathematical mathematical model, model, the the constraints constraints are are classified classified into into two categories,categories, that is, is, regardingregarding the the helicopter helicopter and and regarding regarding the the landing landing task. task. The The kinematic kinematic model model and andthe performance the performance of the ofhelicopter the helicopter are considered, are considered, and the no-fly and zonethe no and-fly the zone terminal and constraint the terminal are constraintdefined, taking are defined into account, taking the into influence account of the the influence fixed-wing of aircraft the fixed and-wing the wind aircraft field. and The the goal wind is field.to reduce The goal the terminal is to reduce position the terminal error and position the heavy error impact and the between heavy the impact helicopter between and the the helicopter carrier to andensure the acarrier precise to and ensure safe a landing. precise and safe landing. WhenWhen searching searching the the landing landing path, path, a a multiphase multiphase path path planning planning algorithm algorithm is is developed developed to to adapt toto the the environment environment in in different different flight flight phases, phases, and and the the PIO PIO algorithm algorithm is is used used to to obtain obtain the the path path in in both phases. In the experimental studies, three situations are considered to explain the validity of the proposed path planning method. In summary, the results demonstrate that the helicopters can all reach the ideal landing point with reasonable landing paths and a small position deviation. The landing safety is also guaranteed by meeting various constraints and reducing the relatively vertical motion between the helicopter and the carrier. The main contribution of this paper is to establish a conceptual model and the mathematical model of the path planning problem for helicopter landing on the carrier. A multiphase path planning algorithm (MPPIO) is also developed to solve this problem. The modeling idea proposed in this paper can also be applied to other similar tasks for helicopters. In the future, a more complicated model would be considered to describe the motion of the helicopter, and quantitative description of wind field is needed to calculate the influence on the helicopter. Moreover, a landing controller for the helicopter is also expected to track the landing paths generated in this paper under various situations.

Mathematics 2018, 6, 178 18 of 20 both phases. In the experimental studies, three situations are considered to explain the validity of the proposed path planning method. In summary, the results demonstrate that the helicopters can all reach the ideal landing point with reasonable landing paths and a small position deviation. The landing safety is also guaranteed by meeting various constraints and reducing the relatively vertical motion between the helicopter and the carrier. The main contribution of this paper is to establish a conceptual model and the mathematical model of the path planning problem for helicopter landing on the carrier. A multiphase path planning algorithm (MPPIO) is also developed to solve this problem. The modeling idea proposed in this paper can also be applied to other similar tasks for helicopters. In the future, a more complicated model would be considered to describe the motion of the helicopter, and quantitative description of wind field is needed to calculate the influence on the helicopter. Moreover, a landing controller for the helicopter is also expected to track the landing paths generated in this paper under various situations.

Author Contributions: Conceptualization, Y.W. and J.X.; methodology, H.H. and Y.W.; software, H.H. and Q.S.; validation, H.H., Y.W. and Q.S.; formal analysis, J.X.; investigation, Y.W.; resources, J.X.; data curation, Q.S.; writing—original draft preparation, H.H.; writing—review and editing, Y.W.; visualization, Q.S.; supervision, J.X.; project administration, Y.W.; funding acquisition, Y.W. Funding: This research was funded by Fundamental Research Funds for the Central Universities with the project reference number of 106112017 CDJXY320003. Conflicts of Interest: The authors declare no conflict of interest.

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