2.2 Calculation model of arresting device

To calculate the arresting force, a dynamic model of the arresting device should be established. The damping force of the arresting device is provided by the energy absorber, and the damping force output by the energy absorber is related to the movement displacement of the discharge flow control valve, the area of the oil hole on the valve, and the internal pressure of the accumulator [Reference Peng, Xie, Wei and Nie11]. The working principle of the energy absorber of the arresting device is shown in Fig. 3.

Figure 3. Principle diagram of the arresting gear energy absorber.

For a certain arresting system, the output of the force is related to the damper stroke, and the same elongation of the cable corresponds to the same cable tension [Reference Liu and Nie19]. The main differences in the arresting process in the three cases in Fig. 2 are: There is a difference in the length of the initial cable on the left and right of the arresting hook, which results in the different extension speeds of the cables from the left and right pulleys, and finally causes the displacement and speed of the pistons on both sides to be different. So it can be concluded:

For the case of left off-centre arrest for aircraft, the piston displacement and movement speed are:

(4) \begin{align}\left\{ \begin{array}{l}{S_{rL}} = \dfrac{{\sqrt {{S^2} + {{(\textrm{D} - \Delta {\rm{D}})}^2}} - (\textrm{D} - \Delta {\rm{D}})}}{N}\\[13pt]{S_{rR}} = \dfrac{{\sqrt {{S^2} + {{(\textrm{D} + \Delta {\rm{D}})}^2}} - (\textrm{D} + \Delta {\rm{D}})}}{N}\\[10pt]{v_{rL}} = \dfrac{S}{{N\sqrt {{S^2} + {{(\textrm{D} - \Delta {\rm{D}})}^2}} }}v\\[13pt]{v_{rR}} = \dfrac{S}{{N\sqrt {{S^2} + {{(\textrm{D} + \Delta \textrm{D})}^2}} }}v\end{array} \right.\end{align}

where ${S_{rL}}$ is the displacement of the left piston, ${S_{rR}}$ is the displacement of the right piston, ${v_{rL}}$ is the speed of the left piston, and ${v_{rR}}$ is the speed of the left piston.

For the case of symmetrical arrest, the piston displacement and movement speed are:

(5) \begin{align}\left\{ \begin{array}{c}{S_rd} = \dfrac{{\sqrt {{S^2} + {{\rm{D}}^2}} - {\rm{D}}}}{N}\\[10pt]{v_r} = \dfrac{S}{{N\sqrt {{S^2} + {{\rm{D}}^2}} }}v\end{array} \right.\end{align}

where $N$ is the number of movable pulleys in the movable pulley group, ${S_rd}$ is the displacement of the piston, and $v$ is the speed of the aircraft.

For the case of right yaw of the aircraft, the piston displacement and movement speed are:

(6) \begin{align}\left\{ \begin{array}{l}{S_{rL}} = \dfrac{{\sqrt {{{(\textrm{D} + S\sin \varphi )}^2} + {{(S\cos \varphi )}^2}} - \textrm{D}}}{N}\\[13pt]{S_{rR}} = \dfrac{{\sqrt {{{(\textrm{D} - S\sin \varphi )}^2} + {{(S\cos \varphi )}^2}} - \textrm{D}}}{N}\\[13pt]{v_{rL}} = \dfrac{{v\cos {\varphi _L}}}{N}\\[13pt]{v_{rR}} = \dfrac{{v\cos {\varphi _R}}}{N}\end{array} \right.\end{align}

3.1 Aerodynamic force

Due to the vertical movement of the aircraft is ignored, the aerodynamic force consist of the drag ${F_D}$ , side force ${F_C}$ and yawing moment ${M_y}$ . The forces and moments are described respectively in the air axis system ( ${O_a}{x_a}{y_a}$ ) and the body axis system ( ${O_b}{x_b}{y_b}$ ). The expressions of force and moment are as follows:

(7) \begin{align}{F_D} & = \frac{1}{2}{\rho _a}V_a^2{S_w}{C_D}\nonumber\\[3pt]{F_C} & = \frac{1}{2}{\rho _a}V_a^2{S_w}{C_C}\\[3pt]{M_y} & = \frac{1}{2}{\rho _a}V_a^2{S_w}{C_Y}b\nonumber\end{align}

where ${\rho _a}$ is the air density and ${V_a}$ is the aerodynamic velocity. ${S_w}$ is the wing area, $b$ is the wingspan, and $c$ is the mean aerodynamic chord. ${C_D}$ , ${C_C}$ and ${C_Y}$ are the aerodynamic coefficients. These coefficients are corresponding to the angle-of-attack $\alpha $ and the sideslip angle $\beta $ . The aerodynamic angles can be calculated as:

(8) \begin{align}\alpha & = \arctan \frac{{{w_a}}}{{{u_a}}}\nonumber\\[3pt]\beta & = \arctan \frac{{{v_a}}}{{\sqrt {u_a^2 + w_a^2} }}\end{align}

where ${u_a}$ , ${v_a}$ and ${w_a}$ are the components of aerodynamic velocity in the body axis system ( ${O_b}{x_b}{y_b}$ ).

The force generated by the rudder is calculated in ${O_a}{x_a}{y_a}$

(9) \begin{align}{C_{\delta r}} = \frac{1}{2}{\rho _a}V_a^2{S_r}{C_r}{\eta _r}{\delta _r}n\end{align}

The corresponding moment can be expressed in ${O_b}{x_b}{y_b}$ :

(10) \begin{align}{Y_{\delta r}} = - {C_{\delta r}}{L_{\delta r}}\end{align}

where ${S_ra}$ is the rudder area and ${\eta _r}$ is the efficiency of the rudder. ${\delta _r}$ is the control angle of deflection and ${C_r}$ is the side force coefficient of the rudder. ${L_{\delta r}}$ is the perpendicular distance between the force point on the rudder and the centre of gravity of the aircraft.

3.2 Dynamic equations of the aircraft

Before establishing the dynamic model, aerodynamic forces are converted to the body axis system. The direction of the force ${F_x}$ is positive along ${O_b}{x_b}$ and ${F_y}$ is positive along ${O_b}{y_b}$ . The direction of the ${M_z}$ is positive around ${O_b}{x_b}$ . The expressions are as follows:

(11) \begin{align}\left\{ {\begin{array}{l}{{F_x} = - {F_D}\cos \beta - ({F_C} + {C_{\delta r}})\sin \beta }\\[4pt]{{F_y} = - {F_D}\sin \beta + ({F_C} + {C_{\delta r}})\cos \beta }\\[4pt]{{M_z} = {M_y} + {Y_{\delta r}}}\end{array}} \right.\end{align}

This study focuses on the lateral movement of carrier-based aircraft during landing process, which is considered as planar. The rolling, pitching and vertical dynamics are neglected. And the displacement along the ${O_g}{x_g}$ is also not account into the analysis of the landing safety boundary. Therefore, a3-DOF (five-dimensional) dynamic model is simplified to the following format:

(12) \begin{align}\left\{ {\begin{array}{l}{\dot \psi = r}\\[3pt]{{{\dot P}_y} = u\sin \psi + v\sin \psi }\\[3pt]{\dot u = rv + {F_x}/m}\\[3pt]{\dot v = - ru + {F_y}/m}\\[3pt]{\dot r = {M_z}/{I_z}}\end{array}} \right.\end{align}

where $\psi $ represent the yaw angle, ${P_y}$ represent the lateral displacement which is defined as along the ${O_g}{y_g}$ , $m$ is the quality of the aircraft, ${I_z}$ is the moment of inertia around the ${O_b}{z_b}$ axis, $u$ is the longitudinal velocity, $v$ is the lateral velocity and $r$ is the angular velocity in yaw.

3.3 The particular H-J PDE for the viability

In order to obtain the safe boundary of the dynamic system, reachability theory is an effective method. According to Lygero‘s description [Reference Lygeros14], a continuous time control system is considered:

(13) \begin{align}\dot x = f(x,u)\end{align}

where $x \in C \in {R^n}$ , $u \in U \in {R^m}$ , $f( \cdot , \cdot ):{R^n} \times U \to {R^n}$ and $C$ is the permissible state set determined by the system.

The viability is the set of states [Reference Liang, Yin and Fang20], which can remain in set $C$ for time $t - T$ under permissible control, as shown in Fig. 5.

Figure 5. Controllable set diagram.

In this study, if the initial landing state is inside the unsafe set $Q$ , no control law can be adopted to make the aircraft remain in the target set. The description of the viability by set is as follows:

(14) \begin{align}Viab(t,C) = \left\{ {x \in {R^n}\left| {\exists U,\forall \tau \in [t,T],x = \varphi (\tau ,t,x,u( \cdot )) \in S} \right.} \right\}\end{align}

where $x = \varphi (\tau ,t,x,u( \cdot ))$ are the trajectories of the system and $U$ is the permissible input.

The calculation of the controllable sets has been extensively studied in the last 20 years. A compelling method that involves a time-dependent H–J PDE is proposed. The method builds up a connection between viability and the SUPMIN optimal control problem.

Assume that the set $C$ can be represented by a zero-level set of a continuous directional distance function: $l:{R^n} \to R$ by $C = \left\{ {x \in {R^n}\left| {l(x) \gt 0} \right.} \right\}$ , the controllable set can be obtained by solving the following partial differential equations:

(15) \begin{align}\frac{{\partial V}}{{\partial t}}(x,t) + \min \left\{ {0,{H^*}} \right\} = 0\end{align}

where $V(x,t) = l(x)$ , $l(x)$ is the direction distance function of set $C$ . The expression of the controllable set is as follows:

(16) \begin{align}S = Viab(0,C) = \left\{ {x \in {R^n}\left| {V(x,0) \gt 0} \right.} \right\}\end{align}

In this paper, direction distance function $l(x)$ is defined as

(17) \begin{align}l(x) = \min \left\{ {{\psi _{\max }} - \psi ,\psi - {\psi _{\min }},{D_{y\max }} - {D_y},{D_y} - {D_{y\min }}} \right\}\nonumber\\[4pt]\left\{ {{u_{\max }} - u,u - {u_{\min }},{v_{\max }} - v,v - {v_{\min }},{r_{\max }} - r,r - {r_{\min }}} \right\}\end{align}

Hamiltonian is defined as:

(18) \begin{align}{H^*} = \mathop {\sup }\limits_{u \in U} {p^T} \cdot f(x,u) = {p_1}\dot \psi + {p_2}{\dot D_y} + {p_3}\dot u + {p_4}\dot v + {p_5}\dot r\end{align}

where ${p_i}(i = 1,2,3,4,5)$ is the spatial derivative of the state value for each dimension.

(19) \begin{align}p = \nabla V(x)\end{align}

Obviously, the analytic solution to the PDE of high dimensions is difficult to obtain. So, a numerical level set tools developed by Mitchell [Reference Mitchell12] are employed to obtain the viability of five-dimensional model. The tools are based on algorithms proposed by Osher and Fedkiw [Reference Osher and Fedkiw21]. The optimal control corresponding to the maximum controllably set can be solved by equation (20). The value of ${\mu ^ * }$ maximise the Hamiltonian on the certain state:

(20) \begin{align}{\mu ^ * } = \arg \mathop {\sup }\limits_{u \in U} {H^ * }\end{align}

4.1 Off-centre and yaw arrest matching envelope

After the aircraft touches down the carrier, the hook engages with the arresting cable and starts to slow down under the action of the arresting device, and the arresting trajectory should be along a part of the inclined deck on the carrier [Reference Huafeng22]. Since the direction of movement of the carrier-based aircraft is uncontrollable during the arresting, if the yaw angle or off-centre distance of the aircraft is too large before the arresting, the aircraft may impact the both sides of the blocking runway or parked aircraft on the deck. Figure 6 is the schematic diagram of the arresting and rolling process of carrier-based aircraft.

Figure 6. The compositions of angled deck.

As shown in Fig. 6, it can be obtained:

(21) \begin{align}\left\{ \begin{array}{l}{B_{deck}} = {B_a} + 2{B_T}\\[4pt]{S_{deck}} = d + \Delta {x_1} + \Delta {x_2} + \Delta {x_3} + {S_l} + {S_t} + {S_S}\end{array} \right.\end{align}

where ${B_{deck}}$ is the width of the landing runway, ${B_a}$ is the wingspan of the aircraft, ${B_T}$ is the distance between the aircraft wingtip and the safety boundary, ${S_{deck}}$ is the length of runway, $d$ is the length of landing zone, $\Delta {x_1} + \Delta {x_2} + \Delta {x_3}$ is the length of the arresting zone, ${S_l}$ is the length of the deceleration zone, ${S_r}$ is the length of the revolving zone and ${S_S}$ is the length of the reserve zone.

In this section, it is assuming that ${B_{deck}} = 34m$ , ${B_a} = 14m$ and ${B_T} = 10m$ . At the end of the arresting, if the ${B_T}$ ≤0 when the aircraft is stop, the aircraft will rush out of the safe landing zone, and the accident may occur.

In this section, taking the lateral displacement of the carrier-based aircraft after being stopped by the arresting device as the research objective, the dynamic model of the aircraft arresting and rolling was established. Then taking the different landing speed ${v_h}$ , yaw angle $\varphi $ and off-centre distance ${D_y}$ of the carrier aircraft as input parameters to perform batch simulation [Reference Zhang, Peng, Wei, Nie, Chen and Li23]. Monitor the value of ${B_T}$ in real time during the process, and judge the critical operating conditions that exceed the safety cordon after the carrier-based aircraft stop. When ${B_T}$ is greater than 0m, it means that the aircraft is stopped in the safe area, the input parameter constraint range as follows

(22) \begin{align}\left\{ \begin{array}{l}{v_h} = [40,65], \quad{\Delta _{vh}} = 1m/s\\[5pt]{D_y} = [ - 12,12] \quad{\Delta _{Dy}} = 1m \\[5pt]\varphi = [ - 10,10] \quad{\Delta _\varphi } = {1^ \circ }\end{array}\right.\end{align}

In the simulation results, the parameters satisfying ${B_T}$ greater than 0m are extracted, and the three-dimensional representation diagram is drawn as shown in Fig. 7. In order to show the relationship between off-centre distance and yaw angle clearly, the 3D representation map is sliced at the landing speed of 55m/s, and the 2D projection map is obtained as shown in Fig. 8.

Figure 7. Arresting horizontal security boundaries.

Figure 8. Projection of arresting horizontal security boundaries.

As shown in Fig. 7, the yaw angle and off-centre distance have a great impact on the arresting safety boundary of aircraft, while the landing speed has relatively little influence. As the landing speed increases, the range of the off-centre and yaw angle matching envelope gradually decreases. It means that the greater the landing speed of the carrier-based aircraft, the smaller the off-centre and yaw range that allow safe arresting during the interception process.

As shown in Fig. 8, the lateral dynamics safety boundary matching envelope of the aircraft arresting is centrally symmetrical. When the landing speed is 55m/s, the maximum safe yaw angle range is (−6.2°∼ + 6.2°), and the maximum safe eccentric distance range is (−9.8m∼ + 9.8m). It can be seen that if the carrier-based aircraft’s yaw angle and off-centre distance are within the above range (Fig. 7), the aircraft can remain within the safe boundary after stopping and will not collide with the island or aircraft on both sides of the deck. This conclusion is similar to literature [Reference Peng, Xie, Wei and Nie11], which proves that the conclusion is accurate to some extent.

In order to verify the accuracy of the safety envelope obtained above, three representative key points are extracted from the obtained lateral dynamic safety boundary of arresting as parameters for the arresting dynamic simulation of carrier-based aircraft. The result as shown in Fig. 6, selected points are $Po{\mathop{\rm int}}\ A({0^ \circ },0m)$ , $Po{\mathop{\rm int}}\ B({6.2^ \circ },0m)$ and $Po{\mathop{\rm int}}\ C({0^ \circ },9.8m)$ . The selected three key points are taken as parameters and brought into the carrier-based aircraft arresting dynamics model. The arresting force, acceleration, lateral displacement and velocity curve of the carrier-based aircraft during the arresting process are obtained, respectively, as shown in Figs. 9-12.

Figure 9. Curve of arresting force–time.

Figure 10. Curve of arresting acceleration–time.

Figure 11. Curve of arresting lateral offset–time.

Figure 12. Curve of arresting speed–time.

As shown in Fig. 9, at the time of 1.6s, the arresting force of aircraft reaches the maximum value. The maximum arresting force of $Po{\mathop{\rm int}}\ B$ is 953.4KN, which is slightly greater than $Po{\mathop{\rm int}}\ A$ (840.8KN). The maximum arresting force of $Po{\mathop{\rm int}}\ C$ is 1018.6KN, which is greater than $Po{\mathop{\rm int}}\ A$ and $Po{\mathop{\rm int}}\ B$ . Due to the off-centre distance and the yaw angle of the aircraft are 0 under the condition $Po{\mathop{\rm int}}\ A$ , it can be concluded that the existence of off-centre distance and yaw angle will increase the arresting force, and the arresting force is the smallest when the carrier-based aircraft is in symmetrical arresting.

Figure 10 shows the acceleration variation curve of carrier-based aircraft during arresting under the selected key point conditions. As shown in Fig. 10, the change trend of the acceleration is consistent with the change trend of the arresting force, but in the opposite direction. At the time of 1.6s, the arresting acceleration of the three points reaches the maximum. The maximum acceleration of $Po{\mathop{\rm int}}\ A$ is −33 $m/{s^2}$ , $Po{\mathop{\rm int}}\ B$ is −34 $m/{s^2}$ and $Po{\mathop{\rm int}}\ C$ is −41 $m/{s^2}$ .

Figure 11 shows the curve of lateral displacement of carrier-based aircraft during arresting under the selected key point conditions. As shown in Fig. 11, the lateral displacement of carrier-based aircraft is equal to the off-centre distance in the conditions of $Po{\mathop{\rm int}}\ A$ and $Po{\mathop{\rm int}}\ C$ , the reason is that the yaw angle of aircraft is 0°. So the lateral displacement of $Po{\mathop{\rm int}}\ A$ is 0m, and the lateral displacement of $Po{\mathop{\rm int}}\ C$ is 9.8m. Under the condition of $Po{\mathop{\rm int}}\ B$ , due to the existence of yaw angle, the lateral displacement of aircraft increases with time, and the maximum lateral displacement is 9.95 m at the end of arrest.

Figure 12 shows the curve of speed change of carrier-based aircraft during arresting under the selected key point conditions. It can be seen from the simulation results that the change trend of the deceleration curve of the aircraft at the three boundary points is similar. Under working condition $Po{\mathop{\rm int}}\ C$ , due to the larger arresting force, its deceleration trend is slightly larger than $Po{\mathop{\rm int}}\ A$ and $Po{\mathop{\rm int}}\ B$ .

Through the above analysis, it can be seen that three key points are selected in the above off-centre and yaw matching envelope for simulation. Under the key point conditions in the safety boundary, the carrier-based aircraft successfully realises the arresting and stopping, and the lateral displacement of the arresting process does not exceed the safety boundary, which proves that the obtained lateral safety boundary of arresting is reliable to some extent.

4.2 Influence of landing parameters on safe boundary of arresting

When the carrier-based aircraft is landing arresting, once the arresting hook is engaged with the arresting cable upon landing on the carrier, the pilot cannot control the arresting process. Typically, during this process, the pilot is required to control the engine thrust of the carrier-based aircraft to at least 40% of the aircraft weight [Reference Zhang24] in order to be able to complete the re-flight operation after the arresting fails. According to the description in Reference [Reference Zhang24], the weight of the aircraft is the key parameter affecting the arresting. Therefore, in order to study the influence of engine thrust and weight of the aircraft on the safety boundary of arresting taxiing, in this section, the original quality of carrier aircraft $M = 22680kg$ , and $G = Mg = 226800N$ . Different engine thrust and weight are selected for batch simulation. The engine thrust is taken as $40{\rm{\% }}G$ , $80{\rm{\% }}G$ and $120{\rm{\% }}G$ respectively, where $G$ represents the weight of carrier-based aircraft. It is worth mentioning that the engine thrust depends on the weight of the carrier-based aircraft, when the weight of the aircraft is changed, the engine thrust also changes accordingly.

The simulation results are shown in Figs. 13 and 14.

Figure 13. Influence of engine thrust on arresting lateral safety boundary.

Figure 14. Influence of aircraft quality on arresting lateral safety boundary.

As shown in Fig. 13, the change of engine thrust has an effect on the safety boundary of aircraft arresting, but the effect is small. The change of engine thrust will not change the maximum yaw angle of the safety boundary. The maximum range of off-centre distance decreases from 6.6 m to 6.4 m with the increase of the engine thrust from $40{\rm{\% }}G$ to $80{\rm{\% }}G$ , the area enclosed by the safety boundary increases by 1.1%, which means the safety boundary increases by 1.1%. When the engine thrust increases to $120{\rm{\% }}G$ , the safety boundary increases by 1.6%. According to the result in Fig. 14, the safe boundary has great influence under the change of aircraft weight. The reason is that with the change of aircraft weight, in order to ensure the well aerodynamic characteristics, engine thrust will change accordingly.

With the decrease of the weight of the carrier-based aircraft, the maximum yaw angle range of the safety boundary will not change, but the maximum safety off-centre distance decreases with the increase of the weight. When the quality of the aircraft is $0.5M$ , the maximum safety off-centre distance is 7.4m. The quality is $M$ , the maximum safety off-centre distance is 6.6m and the safety boundary range increases by 9.2%. When the quality of the carrier-based aircraft is $1.5M$ , the maximum safety off-centre distance is 6.4m, and the safety boundary range increases by 10.7%.

4.3 Influence of landing parameters on the safety of arresting

In this section, a method for defining the safety evaluation index of the carrier-based aircraft arresting is proposed based on the lateral dynamic safety matching envelope in the above study, which lays the foundation for in-depth analysis of the influence of relevant parameters on the safety of the arresting and obtaining the parameter combination with the highest safety in the arresting.

4.4 Safety evaluation index of arresting

Through the above analysis, it can be seen that the lateral dynamic safety boundary of the carrier-based aircraft during the arresting is affected by the engine thrust and weight of the carrier-based aircraft. Considering the actual situation of the aircraft arresting process, it is defined that the smaller the lateral displacement of the carrier-based aircraft at the end of the arresting taxiing, the higher the arresting safety.

According to GJB67.4A3.1.19 [25], the various variables of carrier-based aircraft landing arrest are determined in a certain form of distribution, in which the aircraft longitudinal speed, yaw angle and off-centre distance are determined by Gaussian distribution function. Therefore, the value evaluation index of the aircraft during the arresting phase is defined by:

(23) \begin{align}\left\{ {\begin{array}{l}{{Q_s} = \dfrac{1}{{\sqrt {{B_T}} }} \cdot {R_G}({X_{sys}})}\\[12pt]{{X_{sys}} = \left\{ {\left. {{v_h},{D_y},\varphi } \right\}} \right.}\end{array}} \right.\end{align}

where ${Q_s}$ is the value of each parameter combination, which is based on the three parameters of ${X_{sys}}$ . $\frac{1}{{\sqrt {{B_T}} }}$ is the reference value for the current state and indicates that the smaller the lateral displacement of the carrier-based aircraft, the higher the value. The longitudinal velocity, yaw angle and off-centre distance of the aircraft are taken into Gauss function, and the cumulative product is taken as the value weight of the current state. The sum of values of all states within the safety boundary is used as the evaluation index of aircraft arresting safety.

4.5 Change rule of safety of arresting under the influence of two parameters

In the parameter range of formula (22), the simulation is carried out according to formula (23) of the value function, and the state value obtained by the simulation is processed to obtain the two-dimensional density map that reflects the safety value of arresting, as shown in Fig. 15.

Figure 15. The variation trend of transverse safety of arresting under the influence of two-parameters.

As shown in Fig. 15, the safety of arresting decreases with the increase of engine thrust when the landing weight of the carrier-based aircraft is determined. And the safety of arresting decreases with the increase of engine thrust when the weight of the aircraft is determined.

It can be concluded that the engine thrust and landing quality should be minimised within the allowable range in order to ensure that the arresting and sliding process of shipborne aircraft does not exceed the lateral safety boundary. The smaller the thrust of the engine and the smaller the quality, the more conducive to improving the safety of the arresting.

5.1 Results of landing lateral safe set

In order to visualise the safety boundary, high-dimensional results need to be sliced. Figure 17 shows the safety boundary of the other three dimensions when the initial lateral velocity and yaw angular velocity are 0. Axis of $x,y,z$ represent yaw angle, lateral displacement and longitudinal velocity, respectively.

Figure 17. Lateral safety boundary during landing descent.

As shown in Fig. 17, under the optimal control input of rudder, the lateral safety boundary of carrier-based aircraft during landing is centrally symmetric, and its lateral displacement and yaw angle range decreases with the increase of longitudinal speed. When the longitudinal speed is 40 m/s, the maximum safe yaw angle is ±1.13°, and the safe lateral displacement is ±4.58 m. When the longitudinal speed is 64 m/s, the maximum safe yaw angle is ±0.77°, and the lateral displacement distance is ±4.58 m. When the longitudinal speed is 55m/s, the two-dimensional safety boundary is obtained by slicing Fig. 17, as shown in Fig. 18. The maximum safe yaw angle is ±0.9°, and the safe lateral displacement distance is ±4.6 m. When the carrier-based aircraft landing at a longitudinal speed of 55m/s, keep the yaw angle in the range of (−0.9°∼ + 0.9°), and the lateral displacement in the range of (−4.6m∼ + 4.6m), the carrier-based aircraft can successfully reach the target set under the optimal control of the rudder, and complete the arresting safely.

Figure 18. Safety boundary section when longitudinal speed is 55m/s.

5.2 Analysis of the influence of parameters on safe set

In this section, the engine thrust $T$ and quality of the carrier-based aircraft $M$ in the landing phase are selected as the parameters to analyse the influence of its change on the lateral safe set.

5.2.1 The influence of engine thrust

Figure 19 shows the calculation results of the safe set under the original engine thrust $T$ and $140\%\ T$ . Figure 20 is the two-dimensional envelope curve under the longitudinal speed is 55 m/s.

Figure 19. Lateral safety boundary of landing glide process under the influence of different engine thrust.

Figure 20. Safety boundary section when longitudinal speed is 55m/s.

It can be concluded that when the longitudinal speed is 55 m/s, the increase of engine thrust by 40% reduces the maximum safe yaw angle of aircraft from 0.9° to 0.87°, and reduces the safe boundary range by 4.2%. When the engine thrust increases to 80%, the maximum safety yaw angle of the aircraft decreases from 0.87° to 0.86°, and the safety boundary range decreases by 5.5%. It can be seen that the carrier-based aircraft cannot blindly increase the engine thrust during the landing glide process. Increasing the engine thrust will reduce the lateral safety boundary, thereby reducing the safety of the landing arrest.

5.2.2 Influence of aircraft quality

Figures 21 and 22 show the safe set of the original quality of aircraft, quality increase of 40% and quality increase of 80%, respectively.

Figure 21. Lateral safety boundary of landing glide process under the influence of different aircraft quality.

Figure 22. Safety boundary section when longitudinal speed is 55m/s.

It should be pointed out that when the quality of the carrier-based aircraft is changed, engine thrust should be guaranteed to be 40% of carrier aircraft quality. With the increase of the landing quality, the engine thrust will also increase accordingly.

As shown in the Figs. 21 and 22, the maximum safety yaw angle decreases from 0.9° to 0.86° with the quality of the carrier-based aircraft increases, and the lateral safety boundary range decreases by 2.8%. When the quality of the carrier-based aircraft increases by 80%, the maximum safety yaw angle decreases from 0.9° to 0.83°, and the lateral safety boundary range decreases by 3.7%, showing a decreasing trend.