Prediction and Compensation Model of Longitudinal and Lateral Deck Motion for Automatic Landing Guidance System

: This paper mainly studies the longitudinal and lateral deck motion compensation tech ‐ nology. In order to ensure the safe landing of the carrier ‐ based aircrafts on the flight decks of carriers during the landing process, it is necessary to introduce deck motion information into the guidance law information of the automatic landing guidance system when the aircraft is about to land so that the aircraft can track the deck motion. To compensate the influence of the height change in the ideal landing point on the landing process, the compensation effects of the deck motion compensators with different design parameters are verified by simulation. For further phase ‐ lead compensation for the longitudinal automatic landing guidance system, a deck motion predictor is designed based on the particle filter optimal prediction theory and the AR model time series analysis method. Be ‐ cause the influence of up and down motions on the vertical motion of the ideal landing point is the largest, the compensation effects of the designed predictor and compensator are simulated and ver ‐ ified based on the up and down motion of the power spectrum. For the compensation for the lateral motion, a tracking strategy of the horizontal measurement axis of the inertial stability coordinate system to the horizontal axis of the hull coordinate system (center line of the deck) is proposed. The tracking effects of the horizontal measurement axis of the designed integral and inertial tracking strategies are simulated and compared. Secondly, the lateral deck motion compensation commands are designed, and the compensation effects of different forms of compensation commands are ver ‐ ified by simulations. Finally, the compensation effects for the lateral deck motion under integral and inertial tracking strategies are simulated and analyzed.


Introduction
The influence of deck movement on ship landing can be divided into longitudinal and lateral dimensions. The longitudinal deck motion causes the height change in the ideal landing point on deck, which increases the difficulty of carrier landing and seriously affects the safety of the landing process [1,2]. The lateral deck motion includes three components, namely swaying, roll and yaw, among which the lateral deviation caused by the yaw has the greatest influence on the landing [3]. When there is a lateral deviation, it is easy to generate decentered hooking between the aircraft and the arrester wire, which severely affects the landing performance of carrier-based aircrafts. Therefore, it is necessary to design a deck motion predictor and compensator to improve the carrier landing performance and enhance the safety and accuracy during landing. There are many different aspects and subjects regarding the deck motion. A prediction method of deck lateraldirectional motion for the control of landing trajectory of aircraft is proposed by Xu L. H. et al. [4]. Apart from that, an innovative autonomous carrier landing system (ACLS) using the novel robust adaptive preview control (RAPC) method, which can assure safe and successful autonomous carrier landing under the influence of airwake disturbance and irregular deck motion, is proposed by Ajeet Kumar Bhatia et al. [5]. The autonomous carrier landing problem of an aircraft is addressed through the proposed autonomous carrier landing system (ACLS) composed of pre-viewable guidance and control systems by Bhatia Ajeet Kumar et al. [6]. What is more, an adaptive fault-tolerant control (AFTC)1 method for an automatic carrier landing system (ACLS)2 with actuator failures is proposed by Xue Y. et al. [7]. Zhen Z Y et al. [8] design an innovative ACLS system for carrierbased UAVs, which is composed of the flight deck motion prediction, reference glide slope generation and integrated guidance and control (IGC) modules. The particle filtering method is used to online predict the magnitudes and frequencies of the deck motion. An autonomous system for unmanned aerial vehicles (UAVs) to land on moving platforms, such as an automobile or a marine vessel, is developed by Chang C. W. et al. [9]. An UAV equipped with an automatic control system for educational purposes, such as navigation flight or autonomous flight, is proposed by Park, Myeong-Chul, Hur, Hwa-ra, and the proposed UAV is capable of automatic navigation flight and it is possible to control more precisely and delicately than existing UAV which is directly controlled [10].
This paper mainly studies the prediction and compensation for deck motion along longitudinal and lateral directions. In Section 3, a deck motion predictor is designed and a compensator for longitudinal deck motion is designed. The compensation effects are then simulated and verified under different kinds of conditions. In Section 3, two tracking strategies and three compensation commands are proposed and their compensation effects are simulated and verified, respectively, under different situations.
The focus of Section 2 is on the overall framework of the prediction and compensation of longitudinal deck motion. Based on particle filter optimal estimation theory and the time series AR model, respectively, a deck motion predictor is designed and a deck motion compensator based on phase-lead network is proposed. The compensation effects of predictors and compensators with different design parameters are simulated and compared by using the deck motion model based on power spectrum. In order to fully verify the compensation effects of the proposed deck motion prediction and compensation strategy, the vertical motion prediction and compensation effects at an ideal landing point under different conditions (different ship sizes and navigation states) are simulated.
The influence of lateral deck motion on the lateral automatic landing guidance system is mainly reflected in the lateral deviation of aircraft [11][12][13][14]. The lateral deck motion includes the swaying, roll and yaw of the deck, among which the lateral deviation caused by the yaw has the greatest influence on the landing. When there is a lateral deviation, it is easy to generate decentered hooking between aircraft and the arrester wire, which affects the landing performance of aircraft. When the lateral deviation is too large, it will easily lead to the aircrafts' failure to be hooked [15,16]. Therefore, the lateral deck motion compensation is studied in Section 4 in order to ensure that the aircraft can safely land at the predetermined ideal landing point. Two different tracking strategies for lateral deck motion, namely inertial tracking strategy and integral tracking strategy, are proposed in order to track the lateral deck motion to realize a better compensation to lateral deviation. Three compensation commands, roll, yaw and the parallel one of yaw and roll, are designed. Through respective simulations and verifications of the effects of the tracking strategies and compensation commands mentioned above, a conclusion is drawn: on one hand, lateral error can be reduced by both rolling compensation command and the parallel compensation command of yaw and roll, while the introduction of the parallel compensation command of roll and yaw does not significantly improve the compensation ef-fect for deck motion, but increases the system complexity; on the other hand, inertial tracking strategy can ensure a small tracking error of the horizontal measurement axis, and meanwhile, it can reduce the lateral deviation and motion parameters.
In this article, Section 2 presents the research on the prediction and compensation strategy of the longitudinal deck motion; Section 3 introduces the research on the compensation strategy, its detailed design and its simulation results; Section 4 describes the counterpart content of lateral deck motion. In Section 5, a conclusion is drawn according to the simulation results under some certain conditions.

Research on Prediction Technology of Deck Motion
In the process of landing on ships, in order to ensure a safe landing on the flight deck of aircraft carriers, the motion information of the deck must be introduced into the guidance law information of the longitudinal automatic landing guidance system Long ACLS when the aircraft is approaching the ship so that the aircraft can track the deck. There are inevitably some phase delays in longitudinal automatic landing guidance systems within effective frequency range [17][18][19], and these phase delays will possibly lead to the pose of aircrafts failing to change coordinately with the motion of the deck, as well as the exact position of the ideal landing point. As a result, in order to reduce the tracking error, longitudinal deck motion prediction and compensation technology should be studied to ensure that the aircraft can safely land at the predetermined ideal landing point. The phaselead network of deck motion can provide some compensations to the phase delay of Long ACLS within effective frequency range, improving the aircrafts' capabilities of tracking the motion of the deck. However, due to the limitation of the deck motion compensation network structure, the phase delay of Long ACLS , which cannot be completely compensated by the deck motion compensation network alone, will still lead to a large landing error, which fails to meet the military standards on landing guidance. In such cases, the phase-lead compensation to Long ACLS can be achieved by deck motion prediction to reduce the tracking error of the aircraft. The structure of the longitudinal automatic landing guidance system to which the deck motion predictor and compensator are introduced is shown in Figure 1. Under a large frequency of encounter, the tracking abilities of carrierbased aircrafts to the deck motion will deteriorate and, therefore, the velocity signal of the deck motion as compensation to improve aircrafts' tracking capabilities to the deck motion. The structure of the longitudinal automatic landing and guidance system   Introducing the deck motion information too early can cause unnecessary maneuvering flight. The carrier-based aircrafts fly on an ideal path regardless of the deck motion in the most time of the landing process. Only the deck motion information is introduced into the guidance law information of the longitudinal automatic landing guidance system, making aircrafts track the motion of the deck. The working sequence of the longitudinal automatic landing guidance system for carrier-based aircraft in the process of landing is shown in Figure 3.

Research on Deck Motion Prediction
In the landing process, the real-time and accuracy of deck motion prediction information are very important. Inaccurate deck motion prediction information will pose a great threat to the safe landing of carrier-based aircrafts. Therefore, it is exceedingly necessary to study the online deck motion predictor with accurate prediction and good real-time performance. Due to the inevitable phase lag phenomenon in the characteristic frequency of deck motion in the longitudinal automatic landing guidance system Long ACLS , there will be a delay in the tracking of deck motion during the landing process, resulting in the landing error. Therefore, it is necessary to introduce the future information of deck motion into the guidance law of the automatic landing flight control system to eliminate the landing tracking error caused by the phase delay in Long ACLS .

Design of the Predictor Based on Particle Filter Theory
Particle filter is a statistical filtering method based on the non-parametric Monte Carlo method and recursive Bayesian estimation. It is suitable for any stochastic systems which can be expressed by state-space models and for nonlinear systems which cannot be realized by traditional Kalman filter, and its accuracy is approximately the same with that of the optimal estimation. The essence of particle filtering is to approximate the probability distribution related to the discrete random measure composed of particles and their weights and update the discrete random measure according to the recursive algorithm. Its basic idea is expressed as follows: first, randomly draw samples (particles) set in the state space according to the prior probability distribution of system, and then adjust the weight of the particles constantly based on the measurement value of system. The minimum variance distribution of system state quantity is obtained by recursive algorithm.
When the sample number N   , it can approximate the posterior probability density functions of state quantities of any kinds.
In the particle filter algorithm, the selection of the importance function is a key problem, which not only affects the calculation efficiency of the algorithm, but also is related to the degradation of the weight coefficient (the particle weights are concentrated on a few particles) speed. Therefore, many scholars put forward various improved particle filtering algorithms for the purpose of obtaining appropriate importance functions to generate high-quality particles based on the standard particle filter framework. For example, EKFPF (Extended Kalman particle Filter), APF (Auxiliary particle filter), UPF (Unscented Kalman particle filter), IMMPF (Interactive Multi-model particle filter) and so on.
Because the particle filter algorithm is applicable to any stochastic systems that can be expressed by state-space models, when the particle filter algorithm is utilized to design the deck motion predictor, the deck motion state-space equation should be firstly built. In this section, the up and down motion model is introduced to study the deck motion prediction technology; the power spectrum density curve of up and down motion of a carrier under moderate sea conditions is shown in Figure 4. According to the power spectrum density curve in Figure 4, the simplified form of shaping filter of the up and down motion can be obtained through calculation, shown as follows: Using the minimum realization method of the transfer function of the linear timeinvariant system, the state-space expression of the up and down motion can be obtained from Equation (1) as follows: x is the displacement of the up and down motion, 2 x is constantly the speed of up and down motion, z is the observation signal, w is the dynamic noise of the system, v is the observation noise and After the discretization of Equation (2), the discrete model of up and down motion can be obtained: The calculation process of deck motion predictor based on standard particle filter algorithm and optimal estimation theory is shown as follows: (1) Discrete system equation sets are used to calculate the optimal filtering value of the system ˆk k x (including deck motion displacement 1( ) k k x and deck motion speed 2( ) k k x ) at k moment. The specific calculation process is shown as follows: (a) The particle swarm   (2) The expression of the optimal estimation value ˆk mk x  of the displacement information of the up and down motion k x at the future moment  : The up and down movement information in the time domain can be obtained through the transfer matrix Formula (3) with the unit white noise. The dynamic noise power of the system k w is set as 1, and the observation noise power k v is set as 0.225.
The time domain information of the up and down motion and predicted up and down motion is shown in Figure 5, where curve 1 is the up and down motion curve, curve 2 is the up and down motion curve after particle filtering and curve 3 is the up and down motion prediction curve after particle filtering and its predicted time is 2 s. The time range starts from 44 s because the deck motion begins at 44 s. It can be seen from Figure 5 that after particle filtering, the sink and float motion curve become relatively smooth, indicating that the particle filter designed in this paper has an excellent noise-filtering function. It can be seen from curve 2 and curve 3 that after optimal estimation, the phase of sink and float motion is advanced by 2 s, and the prediction is relatively accurate.

Predictor Design Based on Time Series AR Model
The state-space model of ship motion is necessary when particle filter method is used to predict ship motion. Because ship motion is a random signal, the state-space model will also change when the sea states and navigation states change, which causes great difficulties for the ship motion prediction process. This paper uses the historical data of ship motion to estimate the future value of ship motion by establishing time series AR model, avoiding the trouble of obtaining the state-space equation of ship motion.
Ship motion can be regarded as a stationary random process and it meets the statistical characteristics of normal distribution. Therefore, satisfactory prediction effects can be achieved by introducing AR (auto-regressive) model into the prediction of ship motion.
(1) General form of AR model sequence: (3) Determination of sequence order of AR model: FPE, AIC, BIC, corner pricing and other criteria are commonly used for model determination. In this paper, AIC criteria are used for model determination. Established and developed by the Japanese statisticians chek ChiHong (Akaike), AIC criterion (Akaike information criterion) is a measurement standard of statistical model fitting. It is an optimal criterion function method, that is, to determine a criterion function, which is responsible for not only the fitting proximity of some of the model to the original data, but also the influence of the number of undetermined parameters contained in the model on the calculation complexity. The specific algorithm is as follows:   (12) p is the order of the predicted model AR.
(4) Parameters prediction of AR model: The parameter estimation methods of AR model include recursive least square method (RLS), least mean square method (LMS), Yule-Walker equation solution method, Kalman filter algorithm (Kalman filter algorithm), particle filter algorithm, etc. This paper uses Kalman filter algorithm to estimate AR model parameters.
as the input vector of the system, then the corresponding expected output of the system is ( ) The state-space equation of parameter vector estimation is: In the equation above, ( ) v t and ( ) t  are white noise sequences which are mutually independent from each other and their mean value is zero, which meets the application conditions of Kalman filter theory. Assume that the variance of ( The calculation process of parameter predictor which is based on Kalman filter theory is shown as follows; set Optimal prediction error covariance matrix: Optimal gain matrix Kalman optimal filtering value ˆ( ) A t : Optimal filtering error covariance matrix:  Figure 6. The height of ideal landing point and its estimated error are shown in Figure 7. The estimated mean square error is 0.098 m, and the relative mean square error is 2.4%. When the encounter angle is 2 / 3  , the ideal landing point height and its estimated value are shown in Figure 8, and the ideal landing point height and its estimated error are shown in Figure 9. The estimated mean square error is 0.14 m, and the relative mean square error is 6.5%. According to the prediction simulation curves and the analyses of numerical calculation, the AR model can precisely and effectively predict the height change in ideal landing point. From Figure 6 to

Research on Vertical Motion Compensation
The influence of deck motion on longitudinal automatic landing guidance system Long ACLS is mainly reflected in the height change in the ideal landing point. Deck motion can be divided into longitudinal motions (surge, ups and downs, pitch) and lateral motions (sway, roll and yaw), among which the longitudinal deck motion is the main factor of the height change in the ideal landing point. The altitudinal change in the ideal landing point will cause the height error during landing, which poses a serious threat to aircrafts' safety when landing.

Design of Vertical Motion Compensator at Ideal Landing Point
In order to reduce the height error of aircrafts landing, the vertical motion compensation at the ideal landing point is studied in this section. The structure diagram of the longitudinal automatic landing guidance system under deck motion compensation is shown in Figure 10, also in which, the longitudinal automatic landing guidance system Long ACLS is inside the dashed box; h is the landing point height; H is the aircraft height; 0 H is the initial aircraft height; com H is the ideal descending trajectory command signal.
where 1  is a lead filter network, whose main function is phase compensation; is the compensation filter network, and its main function is to suppress the high-frequency noise and enable the deck motion compensation network to meet certain bandwidth requirements. ) of the system output signal and the requirement of suppressing the high-frequency noise. In this paper, two kinds of deck motion compensators with the same structure but different parameters are designed for the longitudinal automatic landing guidance system of a certain carrierborne aircraft. The transfer functions of these two compensators are shown as follows: The amplitude-frequency characteristic curves of longitudinal automatic landing  (21). As can be seen from Figure   11, the phase delay of high-frequency gain is larger. According to the control theory, the compensation network gives aircrafts a weaker tracking ability of deck motion but a stronger noise-suppression capability in comparison to 2 DMC G . Figure 11. The curve of amplitude vs frequency of ( )

Compensation Effects Verification of Vertical Motion of the Ideal Landing Point
Since the up and down motion has the largest influence on the vertical movement of aircraft landing point [20][21][22], therefore, the up and down motion model is used to verify the designed deck motion predictor and compensator. The up and down movement information in the time domain can be obtained by putting the unit white noise through the transfer matrix Formula (3), among which, the dynamic noise power of the system k w is set as 1, and the power of observation noise k v is set as 0.225. The structure diagram of longitudinal automatic landing guidance system with deck motion predictor and compensator is shown in Figure 1, where the deck motion predictor based on particle filter optimal estimation theory is selected as deck motion predictor, and deck motion compensators are expressed as Formulas (20) and (21), respectively. When 3.7~7.4 km far away from the tail of the ship, the aircraft enters the ACLS radar intercept window. In landing process, assume that the carrier is traveling at a speed of 30 knots, the glide angle of the base glide line of the aircraft guidance system is −3.5 degrees, the aircraft speed is 66.7 m/s and the aircraft lands at 56.3 s since the beginning of landing guidance. In order to avoid unnecessary maneuvering flight caused by introducing the deck motion too early, at 12.5 s before aircraft landing, the information of up and down motion is introduced into the guidance law of longitudinal automatic landing guidance system through deck motion predictor and compensator to verify the effects of deck motion compensation. The response curve of the aircraft tracking the up and down motion after the introduction of deck motion compensator is shown in Figure 12, and the response curve after the introduction of both predictor and compensator is shown in Figure 13. In simulation results from Figures 12 and 13, Z is the displacement of the up and down motion, and H is the height of the aircraft, with unit of meter.  It can be concluded from curves 3 and 4 in Figure 13 that the tracking error of aircraft's tracking deck motion with deck motion compensator 2 DMC is smaller than that with deck motion compensator 1 DMC because the phase-lead of 2 DMC is larger and the aircraft can track deck motion faster, which is consistent with the results of amplitudefrequency characteristics of the transfer function in Figure 11. The aircraft motion curve is smoother with 1 DMC compared with that of 2 DMC . This is because the real deck motion signal is a random signal. Although filtered, there is still some noise in the deck motion signal, and the high-frequency gain of 2 DMC is exactly large, which will amplify the noise and cause frequent operation of the aircraft.
In Figure 13, curve 1 is the up and down motion curve, curve 2 is the aircraft motion curve after introducing DMP (with a prediction time of 2 s) and  Figures 12 and 13 that the error of planes tracking the deck motion decreases significantly after the introduction of deck motion predictor; this is because the phase delay of longitudinal automatic carrier landing guidance system is further compensated after introducing deck motion predictor, thus further enhancing the capabilities to track the up and down motion, improving the landing accuracy of carrier-borne aircrafts. According to curves 3 and 4 in Figure 12 and curves 2 and 3 in Figure 13, it can cause frequent operations of aircrafts, leading to unnecessary maneuvering flight when 2 DMC is adopted, so the choice of 1 DMC as a compensation to the longitudinal automatic landing guidance system better meets the actual requirements; further phase-lead compensation can be realized through deck motion predictor.

Simulation on the Compensation for the Vertical Motion at Ideal Landing Point
In this section, the prediction and compensation effects of the vertical motion of ideal landing point under different conditions (different ship sizes and navigation states) is verified by simulation. The landing environment of carrier-borne aircraft is the complex and changeable ocean [23,24]. Due to the influence of uncertain factors, such as waves and wind, the deck motion changes constantly [25][26][27][28][29]. Therefore, the designed deck motion predictor and compensator should be able to achieve good compensation effects under different deck motion conditions to ensure a safe landing in complex landing environments. So, it is necessary to verify the compensation effects of deck motion predictor and compensator for vertical motions at landing points under different situations. Some characteristics of sea waves under different sea conditions used in the simulations are shown in Table 1, frequency band and frequency increment selected for the simulations under different sea conditions are shown in Table 2 and some hull parameters of the carrier used in simulations are shown in Table 3. In the following sections, compensation results and its analysis of the vertical motions at ideal landing points under different ship sizes, sea conditions, navigational speeds and encounter angles are described, respectively.  The influence of deck motion on the landing process is mainly reflected in the height change in the ideal landing point, which is mainly related to the roll, pitch and up and down motion of the ship. The height of landing point h can be expressed by the following expression: In the above formula, z is the displacement of up and down motion,  is the pitching motion angle,  is the roll motion angle, TD L is the horizontal distance be- Based on the time series analysis method, this section introduces the AR model to estimate the vertical motion at the landing point for a certain longitudinal automatic landing guidance system, and the estimation time is 2 s. The deck motion compensator is shown in Equation (20). The structure diagram of longitudinal automatic landing guidance system after introducing deck motion predictor and compensator is shown in Figure  1. When the aircraft was 3.7~7.4 km away from the tail of the ship, the aircraft entered intercept window of the radar ACLS . In the landing process, under the assumption that the glide angle of the base glide line of the aircraft guidance system is −3.5 degrees and the velocity of aircraft is 66.7 m/s, the aircraft lands 56.3 s after the landing guidance begins. In order to avoid unnecessary maneuvering flight caused by introducing the deck motion too early, the information of height change in the landing point can be added into the guidance law information of longitudinal automatic landing guidance system through deck motion predictor and compensator 12.5 s before landing, so that the aircraft can accurately track deck motion.   Figure 15. In the figure, h is the height of ship point (curve 1), and H is the height of aircraft (curve 2). As can be seen from Figure 15, under the same navigational speed and encounter angle, the higher the sea state level is, the more intense the vertical motion of the ideal ship landing point is. Carrier-borne aircraft can accurately track the vertical motion of ideal landing point under different sea state levels, which indicates that the deck motion predictor and compensator designed in this paper have a certain adapting ability to the changes in state of sea scale.

Influence of Ship Speed on Compensation Effect
In the level-5 sea state, the response curve of carrier-borne aircraft tracking the vertical motion of the ideal landing point under the condition with a displacement mass of 65,000 tons, an encounter angle / 4    , a navigational speed of 10 and 30 knots is shown in Figure 16. In the figure, h is the height of ship point (curve 1), and H is the height of aircraft (curve 2). As can be seen from Figure 16, under the same sea condition and encounter angle, and when the ship is outside the synchronous rolling zone, the lower the speed is, the more intense the vertical motion of the ideal landing point is. Carrier aircraft can accurately track the vertical motion of ideal landing point under different navigational speeds, which shows that the deck motion predictor and compensator designed in this paper have a certain adapting ability to the changes of navigational speed.

Overall Structure of Lateral Deck Motion Compensation Strategy
The deck motion compensation of the lateral automatic landing guidance system is mainly the elimination of the lateral landing deviation caused by yaw motion, that is, the deviation error of the deck central line to the horizontal measurement axis in the inertial stable coordinate system ACLS . The lateral geometry of the landing process of the carrier-based planes is shown in Figure 22. In order to avoid the unnecessary maneuvering flight caused by the measurement horizontal axis ox of inertial stability system ACLS tracking the horizontal axis s ox of hull coordinate system too early, usually the lateral deck motion information is introduced into the lateral automatic carrier landing guidance system at the end of the landing process, namely the moment when the horizontal distance between the plane and ideal landing point is According to the lateral geometry of carrier-based aircrafts' landing process and the generation mechanism of lateral deviation, the structural diagram of the lateral automatic landing guidance system is established as shown in Figure 23.

Strategy Design of Measurement Axis Tracking Deck Center Line
The tracking strategy of measuring horizontal axis ox of inertial stability coordinate system ACLS to horizontal axis s ox of ship coordinate system (i.e., deck center line) is as follows: the axis ox is required to be able to quickly track the axis s ox , and the closer the aircraft is to the landing, the higher the tracking rate of axis ox to axis s ox is, so as to eliminate the tracking error when the aircraft is landing.

Integral Tracking Strategy Design
The integral tracking strategy structure diagram of measuring horizontal axis ox tracking deck center line axis s ox is shown in Figure 24, and the open-loop transfer function of the system is in the integral form. Set PID parameters for the measurement axis integral tracking strategy, which is shown in Figure 24, by the experimental trial method. Assume that the speed of the carrier is 30 knots, the yaw motion of deck is  , if k is too large, the tracking rate of the measuring axis will be also too large, leading to the aircrafts' failure to respond quickly and then, an exceedingly large lateral deviation.

Inertial Tracking Strategy Design
In order to solve the contradiction between the tracking rate of measurement horizontal axis and the maneuverability of aircrafts, the inertial tracking strategy, as shown in Figure 26, is adopted. The open-loop transfer function of the system is the first-order damp elements. Adopting the tracking strategy in Figure 26 can not only generate a small time constant 0 T  , but also ensure the tracking rate of the measurement horizontal axis ox tracking deck center line axis s ox , and it will not result in a too large tracking rate cl  even under a large k , so that the tracking maneuverability of the aircrafts is also good.  It can be concluded from Figure 28 that when the value of T is fixed, the larger k is, the smaller the time constant of the measuring axis tracking strategy is, the better the effect of ox tracking s ox is and the smaller the tracking error during landing is. When the proportional coefficient k is set to increase exponentially with the decrease in x , the rate of ox tracking s ox also increases, so that the measurement horizontal axis gradually catches up with the center line of deck, which can ensure a relatively satisfactory maneuverability during landing. It can be concluded from Figure 28 that when k is constant, the larger T is and the worse the tracking effect of axis ox to axis s ox is. This is because the larger T is, the larger the time constant of the system is and the tracking rate of measurement axis decreases, resulting in a larger tracking error. Therefore, in the PID parameters design, the time coefficient T and scale coefficient k should be adjusted coordinately to obtain the optimal tracking strategy.

Modeling and Simulation of Lateral Deck Motion Compensation Commands
The structural diagram of the lateral automatic landing guidance system is shown in Figure 23. It ensures the rate of lateral deviation a y  caused by the compensation commands is equal to the lateral deviation rate The lateral deviation rate of the aircraft a y  can be expressed as: According to Formulas (23) and (24), the value of a y  can be obtained: Because the lateral deviation rate cl y  is generated when the horizontal axis ox of the inertial stabilization coordinate system ACLS tracks, the horizontal axis s ox of the hull coordinate system is: The expression of ac   can be obtained from Equations (29) and (30): The simulation model of yaw compensation instruction for lateral deck motion established under the MATLAB/Simulink environment is shown in Figure 29.  In Figure 30, curve 1 represents the response curve without compensation commands, and curve 2 is the response curve with compensation commands. According to the simulation curve, the lateral deviation er y performs a decreasing trend only in initial phase of compensation under the compensation commands. It does not reduce during landing, and the main motion parameters of the plane become large under the compensation commands, proving that using the yaw command as compensations to the lateral deck motion cannot effectively eliminate the lateral deviation during landing.

Roll Compensation Instruction Design and Simulation Analysis
According to the design idea of lateral deck motion compensation command, namely The lateral deviation rate a y  of the aircraft can be expressed as: The expressions of a y  can be obtained from formulas (29) and (30) as follows: The lateral deviation rate cl y  generated when the horizontal axis ox of the inertial stabilization coordinate system ACLS tracks the horizontal axis s ox of the ship coordinate system is: The expression of ac   can be obtained from Equations (35) and (36):

[ 2 ]
ac cl cl cl Since the landing process is short, the speed of the aircraft carrier can be assumed to be constant, and because the speed of the aircraft remains constant during landing process; thus 0 x   , then the expression of ac   can be simplified as: The simulation model of the compensation commands for lateral deck rolling is established under the MATLAB/Simulink environment, as shown in Figure 31.  In Figure 32, curve 1 is the response curve of the aircraft without compensation commands, and curve 2 is the response curve of the aircraft with compensation commands. It can be seen from the simulation curve that the lateral deviation of aircraft er y decreases obviously under the compensation command, which proves that it is correct and valid to utilize the rolling compensation command as compensations to lateral deck motion.  In Figure 33, curve 1 represents the response curve of the aircraft without compensation commands, and curve 2 is the response curve with compensation commands. It can be seen from the simulation curve that the lateral deviation of aircraft er y becomes smaller obviously under the compensation commands, which proves the validity and correctness of introducing the parallel yaw and roll commands as compensations to the lateral deck motion. As can be seen from Figures 32 and 33, compared with roll compensation command, after the introduction of parallel compensation instruction of roll and yaw, the complexity of the system increases while there is not apparent improvement in the compensation effects for deck motion.

Simulation on the Compensation Effect for Lateral Deck Motion under Different Tracking Strategies
As shown in Figure 23, it is assumed that the carrier's speed is 30 knots, the deck yaw motion is s before landing, and lateral deck motion compensation command is also added. The command model is the roll compensation command model, which is shown in Figure 31. In order to clearly observe and compare the compensation effect for deck motion, the time range shown in the simulation figures in this paper is from 38.3 s to 56.3 s, that is, from the introduction of deck yaw from Lat ACLS to the landing moment.

Compensation Effect under Integral Tracking Strategy
The integral tracking strategy of measurement axis is shown in Figure 24.

Compensation Effect under Inertial Tracking Strategy
The inertial tracking strategy of the measurement axis is shown in Figure 26 Figure 37 that the landing error of the aircraft is reduced significantly after the introduction of compensation command for lateral deck motion is added. Compared with the constant scale coefficient, the exponential scale coefficient can ensure that a small tracking error of the measurement axis when landing and meantime reduces the sideslip error of aircraft. It proves that it is more correct and effective to adopt the inertial tracking strategy with exponential scale coefficient in the design of the compensation strategy.
It can be concluded from Figure 38 that when the time coefficient T is constant and the scale coefficient k is in exponential form, the larger the exponential coefficient is, the larger the value of k will be, the larger the tracking rate of measurement horizontal axis ox to deck center line s ox will be and the smaller the tracking error of measurement horizontal axis ox to deck center line s ox will be during landing. In Figure 39 Figure 39 that the landing error of aircraft decreases significantly after the compensation instruction for lateral deck motion is introduced. When the exponential coefficient k is larger, the lateral deviation and the motion parameters of the aircraft also increase, and this is because the larger the exponential coefficient of scale coefficient k is, the larger the tracking rate is. Since the aircraft cannot respond too quickly, a large lateral deviation is caused.
It can be concluded from Figure 40 that when the time coefficient T is constant and the scale coefficient k is in exponential form, the larger the static coefficient is, the larger the value of k will be, the larger the tracking rate of the horizontal measurement axis ox to the deck center line s ox will be and the smaller the tracking error will be. In Figure with compensation command added. It can be seen from Figure 41 that the landing error is significantly reduced after the lateral deck motion compensation command is added. When the static coefficient of k is large, the aircraft's lateral deviation error and the values of aircraft motion parameters are also large. This is because the larger the static coefficient of the scale coefficient k is, the larger k is and the larger the tracking rate is. Since the aircraft cannot respond too quickly, a large lateral deviation error is generated.
It can be seen from Figure 42 that when the scale coefficient k is exponential and its value is constant, the larger the time coefficient T is, the smaller the tracking rate of the horizontal measurement axis ox to the center line axis  Figure 43 that the landing error of the aircraft is significantly reduced after the compensation command for lateral deck motion is added. When k is constant, the larger T is, the worse the tracking effect is and the greater the tracking error is. This is because the larger the value of T is, the larger the system time constant is and the tracking rate decreases. However, when the value of T is large, the value of aircraft lateral deviation error and motion parameter are both small because the larger the value of time coefficient T is, the smaller the tracking rate of the measurement axis is, leaving the aircraft enough response time to avoid lateral landing deviation caused by the slow response.

The Comparison and Analysis of the Compensation Effects under Two Strategies
The integral tracking strategy of the measurement axis is shown in Figure 24, PID parameters are set as   As shown in Figure 44, within 5 s before landing, the mean square error of the tracking under integral tracking strategy is 0.0018 radians, while that by inertial tracking strategy is 0.0023 radians. In Figure  T  ) is adopted. It can be seen from Figure 45 that under integral and inertial measurement axis tracking strategies, the lateral deviation of the aircraft er y reduced significantly, which proves that it is correct and valid to adopt the integral and inertial tracking strategies in the strategy design of lateral deck motion compensation. Compared with integral tracking strategy, inertial tracking strategy can ensure a small tracking error of horizontal measurement axis and it can reduce the lateral deviation and motion parameters. This proves that it is more effective and feasible to adopt inertial tracking strategy than integral tracking strategy in the strategy design of lateral deck motion compensation.

Conclusions
The landing environment of carrier-based aircrafts is on the complex and changeable ocean. Due to the influence of uncertain factors, such as waves and wind, the location of the ideal landing point changes constantly. Therefore, the designed deck motion predictor and compensator should achieve good compensation effects for landing points under different situations over the state-of-art compensator in order to ensure a safe landing of aircraft in complex landing environment. In this section, the longitudinal deck motion compensation is studied firstly, the focus of which is the compensation to the impact of the height change in the ideal landing point on the landing process. The deck motion compensator is designed based on the idea of lead-network, and the compensation effects of the deck motion compensators with different design parameters are verified and simulated. For further phase-lead compensation for the longitudinal automatic landing guidance system, the deck motion predictor is designed based on the particle filter optimal prediction theory and AR model time series analysis method. Because the influence of up and down motions on the vertical motion of the ideal landing point is the largest, the compensation effects of the designed predictor and compensator are simulated and verified based on the up and down motion of power spectrum. The simulation results show that the predictor and compensator proposed in this paper can effectively compensate the landing error caused by the phase lag in the longitudinal automatic landing guidance system so that the aircraft can accurately track the deck motion. Secondly, the lateral deck motion compensation is studied. A tracking strategy of the horizontal measurement axis of the inertial stability coordinate system to the horizontal axis of the hull coordinate system (center line of the deck) is designed. The tracking effects of the horizontal measurement axis of the designed integral and inertial tracking strategy are simulated and compared. The simulation results show that compared with the integral tracking strategy, the inertial tracking strategy can reduce the lateral deviation of the aircraft and the motion parameters with a small tracking error of the measurement axis, proving that the inertial tracking strategy is more effective and feasible than the integral tracking strategy when designing the lateral deck motion compensation strategy. The proposed compensation model further improves the computational efficiency of the algorithm, saves the computational resources and thus achieves a rapid convergence in the landing process, which prepares itself well for later embedding into the resource-limited flight control system and makes the real applications feasible.
Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable
Informed Consent Statement: Not applicable Data Availability Statement: Data available on request due to restrictions eg privacy or ethical. The data presented in this study are available on request from the corresponding author. The data are not publicly available due to commercial use.

Conflicts of Interest:
The authors declare no conflict of interest.