Design and performance of directional rectification control system in an aircraft with a novel type of wheel-ski landing gear

Abstract

For a hypersonic-speed aircraft with a flat fuselage structure that has narrow space for a traditional wheel-type landing gear retraction, a novel type of wheel-ski landing gear is designed, which is different from traditional landing gears in force distribution and actuation methods. In order to capture the direction control performance of an aircraft with the wheel-ski landing gear, the aircraft ground taxiing nonlinear dynamic mathematical model is built based on a certain type of aircraft data. The experiment of the wheel-ski landing gear actuator and the differential brake control system is carried out to verify that the electric wheel-ski actuator model with the pressure sensor is in good agreement with the test results, indicating the model validity and the speediness of the differential brake response. Then a new fuzzy combined direction rectifying control law is designed based on the optimisation method and the fuzzy control theory. Comparing with the PD wheel-ski differential brake control, the direction rectifying efficiencies increase higher than 140% during the whole taxiing process. In addition, the combined control law can also decrease the overshoots of the yaw angle responses effectively. Finally, the stability and robustness of the designed combined direction control law are verified under various working conditions.

Nomenclature

Symbols

${a_m},{a_n}$

distances from the aircraft gravity centre to the main and nose landing gears

${b_w}$

main wheel track width

e

weighted summation of the lateral displacement and the yaw angle $\psi $

ec

rate of e

${F_*},\;{F_{s\& }}$

lateral forces on the wheel and the ski

$G$

gravity

${K_{\beta *}}$

tire cornering stiffness

${K_P},{K_d}$

proportional coefficient and differential coefficient in the fuzzy-PD controller in real time

${K_{P0}},{K_{d0}}$

initial proportional coefficient and differential coefficient in the traditional PD controller

$\Delta {K_P},\Delta {K_d}$

the increments of ${K_{P0}}$ and ${K_{d0}}$

$K_p^\psi ,K_p^y$

proportional coefficients of the yaw angle and the lateral displacement

$K_d^\psi $ , $K_d^y$

differential coefficients of the yaw angle and the lateral displacement

$K_{sp}^\psi ,K_{sp}^y$

proportional coefficients of the nose wheel steering control system

$K_{sd}^\psi ,K_{sd}^y$

differential coefficients of the nose wheel steering control system

$K_{rp}^\psi ,K_{rp}^y$

proportional coefficients of the rudder control system

$K_{rd}^\psi ,K_{rd}^y$

differential coefficients of the rudder control system

${l_r}$

vertical distance from the rudder to the aircraft gravity centre

$L,\;D,\;Y$

aerodynamic lift, drag and side force

$m$

aircraft mass

n

number of the fuzzy subsets

$NR$

aerodynamic rolling moment

${P_r}$

rudder force

${P_*},{P_{s\& }}$

ground reaction forces on the wheel and the ski

${P_x},\;{P_y}$

aircraft taxiing distance and lateral offset along x- and y-direction in the ground coordinate system ${S_g} - {O_g}{X_g}{Y_g}{Z_g}$

$q$

pitch angular velocity

${Q_*},\;{Q_{s\& }}$

friction forces on the wheel and the ski

r

yaw rate

R ²

value of the goodness of fit

$T$

engine thrust

$u,\;v$

velocity components of the aircraft gravity centre along x- and y-direction in the body coordinate system ${S_b} - {O_b}{X_b}{Y_b}{Z_b}$

${u_s},{v_s}$

velocity components of the aircraft gravity centre along x- and y-direction in the stable coordinate system ${S_s}{ - _s}{X_s}{Y_s}{Z_s}$

$V$

speed

${x_a}$

distance from the aerodynamic centre to ${O_b}{Y_b}{Z_b}$ plane

$y$

lateral offset

${y_{max}}$

maximum lateral displacement

${z_0}$

actual output of the fuzzy controller

${z_i}$

fuzzy subset

$\alpha $

angle-of-attack

$\beta $

sideslip angle of the fuselage

${\beta _*},{\beta _{s\& }}$

sideslip angles of the tire and the ski

${\delta _b}$

direction rectification command

${\delta _{bl}},{\delta _{br}}$

direction rectification commands distributed on the left and right braking mechanisms

${\delta _s}$

control signal

${\eta _{b0}}$

initial value of ${\eta _b}$

${\eta _r},{\eta _b},{\eta _s}$

weighting coefficients of the rudder subsystem, the wheel-ski differential brake subsystem and the nose-wheel steering subsystem

$\theta $

pitch angle

${\theta _l}$

nose wheel steering angle

${\mu _c}\left( {{z_i}} \right)$

membership value of the fuzzy subset ${z_i}$

${\mu _*},{\mu _{s\& }}$

frictional coefficient of the tire and the ski

${\sigma _c}$

direction rectification performance criteria

${\varphi _p}$

angle of the engine thrust and the x-axis ${X_b}$ in the body axis system

$\phi $

roll angle

$\dot \phi ,\;\dot q$

derivatives of $\phi ,\;q$ with respect to time

$\psi $

yaw angle

${\psi _f}$

yaw angle at the first crest

1.0 Introduction

Common aircraft landing gears are wheel-type and ski-type [1, 2]. It is difficult for wheel-type landing gears to roll out on flexible pavements covered with snow and sand. In addition, since the kinetic energy of a wheel-type aircraft during the taxiing process is usually absorbed by the brake pads on the braking wheels [3], sometimes the volume of the wheel-type landing gears is too big to retract into the limited space of the fuselage of a hypersonic-speed aircraft with a flat fuselage structure. However, a ski-type landing gear is simple in structure, light in weight, small in size and reliable in performance. It can land and taxi on a flexible runway safely, but lack of the capability to correct the taxiing direction due to the fixed structure of the ski leads to a poor direction stability during the aircraft rollout process. For traditional ski-type fixed-wing aircrafts, since there are no differential braking direction control systems on this kind of aircrafts, they usually land on a broad dry lake bed which can provide enough lateral space for rolling out safely [4]. Therefore, to improve the direction control ability of this kind of aircraft, a novel type of wheel-ski landing gear is proposed in this study, which can decelerate the aircraft through the friction between the skis and the ground rather than traditional heavy brake pads. Also, the up-down actuation of the skis provide the differential brake ability for the wheel-ski main landing gear to correct the aircraft taxiing direction effectively.

Several asymmetric factors [5, 6] such as airframe structure, rough runway, crosswind, asymmetric braking forces and initial yaw angles, are all likely to lead an aircraft to veer off the runway [7]. Thus the aircraft taxiing direction need to be rectified to keep the aircraft running along the runway centreline all the time through the direction control systems. This brings about the direction rectification problem, where the design and study of the landing gear structure and the corresponding direction control system need to be carried out. Moreover, as high-speed take-off and landing aircrafts developing [8, 9], the lateral velocity increases when a side disturbance exists so that it is more difficult to correct the direction at a higher taxiing speed, which brings a higher demand for the direction control system. Comparing with a ski-type landing gear, the structure of the designed wheel-ski landing gear in this study is more complicated. In order to improve the wheel-ski aircraft direction stability and environmental suitability during the rollout process, the study on how to properly distribute the ground reaction force on the wheel and the ski under differential brake control, and also how to efficiently introduce other auxiliary direction rectifying devices into the direction control system to optimise the control law need to be conducted.

Establishing an aircraft ground taxiing dynamic model is the basis of studying its taxiing direction control system performance. In terms of aircraft ground dynamic models, researchers mainly concentrated on wheel-type aircrafts, where the force characteristics and the taxiing direction stability were studied during the asymmetric landing and the steering rollout processes. In the 1980s, Barnes [10] built a six-degree-of-freedom (6-DOF) wheel-type aircraft taxiing dynamic model, and the longitudinal and lateral stability was studied. Pi [11] developed an aircraft ground taxiing dynamic procedure to obtain the aircraft asymmetric landing dynamic response. Khapane [12] established an aircraft ground model considering the nonlinear tire forces through SIMPACK and the force characteristics of the landing gear were captured. Gu [13] built the aircraft 6-DOF roll-out dynamic model under nonrectilinear motion and the research of the directional stability was carried out. Zhang [14] took the landing gear flexibility into account when building an aircraft ground taxiing virtual prototype model. Hou [15] and Qiu [16] both considered the relationship between the tire lateral force and the sideslip angle as an unmanned aerial vehicle (UAV) ground taxiing model was established. Zhao [17] built the ground dynamic model of an aircraft with the four-wheeled bogie landing gear at a low taxiing speed and the nose wheel steering control law was designed to improve the directional stability. Knowles [18] used the numerical-continuation method to build a wheel-type landing gear and the influence of the elastic joints on the landing gear loading status was studied. Yin [19] established a nonlinear UAV ground wide-angle-steering model considering aerodynamic force, tire force and landing gear absorber force. Thus it can be seen researchers have built accurate 6-DOF aircraft ground taxiing dynamic models, laying the foundation for the wheel-ski type aircraft dynamic model establishment in this study. However, the tire force models rather than the ski-type landing gear were analysed in the previous studies above. In the aspect of ski-type aircraft dynamic modeling, the researchers mainly put emphasis on material properties, structure optimisation [20] and drop test performance [21] of a ski landing gear. For the study of the ski-type aircraft taxiing direction stability, Liang [22] established the equipped-ski aircraft planar taxiing dynamic model where the accurate frictional force on the ski is considered, while the vertical, roll and pitch motions of the aircraft are ignored to study the effects of the landing gear layout on the safe rollout envelope. Existing research of wheel-type and ski-type aircraft dynamic models provided the reference for analysing the tire and the ski ground forces in this study. But different from traditional wheel-type and ski-type landing gears, there is no brake mechanism on the wheel of the landing gear in this study so that the longitudinal rolling frictional force on the tire is very small and the frictional force between the ski and the ground plays an important role in the differential brake control system. Also, the lateral forces of both the ski and the tire need to be considered during the rollout process. Therefore, the complete wheel-ski type aircraft ground taxiing nonlinear dynamic model is built considering the precise model of the tire and ski forces in this study, which lays the foundation for the direction control system design and the performance study of the proposed wheel-ski type main landing gear.

In order to improve the aircraft taxiing direction stability and rollover prevention performance, the active direction rectifying control law need to be designed. A direction rectifying control system usually includes differential brake, nose wheel steering and rudder control subsystems. In terms of the study of the direction control law, Abzug [23] and Deng [24] compared the effects of the three control subsystems on the aircraft direction performance. Li [25] adopted a hydraulic differential brake system to correct the aircraft taxiing direction under side wind. Chen [26] put forward that the nose wheel steering control could be better used under low speed while the differential brake control could be used under higher speed. Wang [27] combined the differential brake and rudder control to correct the UAV taxiing direction. As the direction rectifying control law developing, researchers introduced fuzzy control into aircraft differential brake subsystem or a combined direction control system [28, 29] to improve the system adaptability and stability. To sum up, the study of the aircraft direction control system developed from the single subsystem to the combined control system. A novel differential brake control method which is applicable to the wheel-ski landing gear is proposed in this study. Moreover, to improve the direction control efficiency, a combined direction control law is designed and the allocation method of the three direction control subsystems is carried out according to the features of each subsystem.

Rudder control has a high efficiency under a high taxiing speed, while a nose wheel steering control system is more stable during the low-speed process. In addition, only after the brake system starting to work, the differential brake can play a part. Thus, according to the application scope of these subsystems, researchers have put forward some design methods of the allocation coefficients in the aircraft taxiing combined direction control system. Zhang [30] and Wang [31] introduced an allocation coefficient into the combined control system and the coefficient varied along with the aircraft speed. The rudder control occupied a significant proportion under a higher taxiing speed while the differential brake played a more important role under a lower speed. Both Hu [32] and Cassaro [33] designed the control allocation methodology considering rudder and nose wheel steering actions to guarantee robustness towards lateral disturbances during the rollout process. Hao [34] and Wang [35] synthesised a guidance control loop under the combination of all the three subsystems and the pseudo-inverse technique was used to allocate the coefficient dynamically. From the previous studies, it can be seen that though the design approaches of the allocation coefficients are different, it is always based on the aircraft taxiing speed. In this study, according to the features of the wheel-ski landing gear, the optimisation method is put forward to design the allocation coefficients of the combined direction control law and the optimum curves of the allocation coefficients are given at various taxiing speeds, aiming at improving the aircraft taxiing direction control efficiency. Also, the direction control performances are compared under different control systems in various working conditions.

The purpose of this study is to design a combined ground direction control system for a wheel-ski type aircraft to improve the directional stability and direction correction efficiency. In Section 2, the nonlinear dynamic aircraft taxiing model is built and also, the tests of the wheel-ski actuator and the direction control law are carried out to verify the accuracy of the wheels-ski differential braking control system. Then in Section 3, the direction rectifying performance of the differential brake on the wheel-ski landing gear, the nose wheel steering and rudder control subsystems are studied and compared. In Section 4, a new piecewise combined direction control system is designed based on the optimisation method and fuzzy control is also brought in to improve the adaptability of the control system. In Section 5, the designed wheel-ski landing gear system and the self-adaptive fuzzy-PD combined direction control system are analysed. The comparison of several control systems helps to verify the good performance of the designed operating control system. Also, the stability and robustness are validated. Conclusions are drawn in Section 6.

2.0 Wheel-Ski-Type Aircraft Ground Taxiing Dynamic Modeling and Verification

2.1 Wheel-ski force

During the wheel-ski-type aircraft ground taxiing process, the forces acting on the three tires include the ground reaction forces ${P_n},\;{P_{ml}},\;{P_{mr}}$ , the frictional forces ${Q_n},\;{Q_{ml}},\;{Q_{mr}}$ and the lateral forces ${F_n},\;{F_{ml}},\;{F_{mr}}$ on the nose landing gear, the left and right main landing gears, respectively. In addition, the forces acting on the two main skis also include the ground reaction forces ${P_{sl}},\;{P_{sr}}$ , the frictional forces $\;{Q_{sl}},\;{Q_{sr}}$ and the lateral forces ${F_{sl}},\;{F_{sr}}$ on the left and right main landing gears, respectively. The wheel-ski type aircraft ground force analysis is shown in Fig. 1.

Figure 1. Wheel-ski-type aircraft ground force analysis diagram.

The lateral force on the landing gear is produced by the aircraft lateral movement resulting from side disturbance. An angle between the aircraft velocity and the wheel/ski will appear during this process, which is called sideslip angle $\beta $ :

(1) \begin{align}\left\{ {\begin{array}{*{20}{c}}{{\beta _n} = arctan\dfrac{{\left( {{v_s} + r\cdot{a_n}} \right)\cos {\theta _l} - {u_s}\sin {\theta _l}}}{{{u_s}\cos {\theta _l} + \left( {{v_s} + r\cdot{a_n}} \right)\sin {\theta _l}}}}\\[7pt]{{\beta _{ml}} = arctan\dfrac{{{v_s} - r\cdot{a_m}}}{{{u_s} + r\cdot{b_w}/2}}}\\[7pt]{{\beta _{mr}} = arctan\dfrac{{{v_s} - r\cdot{a_m}}}{{{u_s} - r\cdot{b_w}/2}}}\end{array}} \right.\end{align}

where, ${u_s},{v_s}$ are the velocity components of the aircraft gravity centre along x- and y-direction in the stable coordinate system ${S_s} - {O_s}{X_s}{Y_s}{Z_s}$ [36].

The direction of the lateral force is vertical to the tire symmetry plane. The sideslip angles of the ski and the tire decrease gradually under the lateral force, leading the aircraft taxiing velocity direction to return to the tire symmetry plane. The tire lateral force ${F_*}$ and the ski lateral force $\;{F_{s\& }}$ can be expressed by:

(2) \begin{align}{F_*} = {K_{\beta *}}\cdot{\beta _*}\end{align}

(3) \begin{align}{F_{{\rm{s}}\& }} = {\mu _{s\& }}\cdot{P_{s\& }}\cdot\sin {\beta _{s\& }}\end{align}

The friction forces on the tire and the ski are proportional to the frictional coefficients:

(4) \begin{align}{Q_*} = {\mu _*}\cdot{P_*}\end{align}

(5) \begin{align}{Q_{{\rm{s}}\& }} = {\mu _{s\& }}\cdot{P_{s\& }}\cdot\cos {\beta _{s\& }}\end{align}

2.2 Aircraft ground taxiing dynamic model

In addition to the ground forces on the wheel-ski landing gear, some other forces include the gravity $G$ , the engine thrust $T$ , the rudder force ${P_r}$ , the aerodynamic force and moments are all acting on the aircraft, which are shown in Fig. 2. The aircraft pitching and rolling motions can be neglected during the taxiing process, thus:

(6) \begin{align}\left\{ {\begin{array}{*{20}{c}}{\phi = \dot \phi = 0}\\{q = \dot q = 0}\\{\theta = \alpha = const}\end{array}} \right.\end{align}

Based on the momentum theorem, the gravity centre dynamic equation set of the wheel-ski-type aircraft in the body axis system ${S_b} - {O_{}}{X_b}{Y_b}{Z_b}$ [36] is obtained:

Figure 2. Force analysis of aircraft movement on the ground.

(7) \begin{align}\left\{ \begin{array}{l}\dot u = rv + \left( {\begin{array}{*{20}{c}}{ - mg\sin \theta + T\cos {\varphi _p} - D\cos \alpha \cos \beta - Y\cos \alpha \sin \beta + L\sin \alpha }\\{ + \left( {{P_n} + {P_{ml}} + {P_{mr}} + {P_{sl}} + {P_{sr}}} \right)\sin \theta }\\{ + \left( { - {F_n}\sin {\theta _l} - {Q_n}\cos {\theta _l} - {Q_{ml}} - {Q_{mr}} - {Q_{sl}} - {Q_{sr}}} \right)\cos \alpha }\end{array}} \right)/m\\[10pt]\dot v = - ru + \left( {\begin{array}{*{20}{c}}{mg\cos \theta - D\sin \beta + Y\cos \beta }\\{ + \left( {{F_n}\cos {\theta _l} - {Q_n}\sin {\theta _l} + {F_{ml}} + {F_{mr}} + {F_{sl}} + {F_{sr}}} \right) + {P_r}}\end{array}} \right)/m\end{array} \right.\end{align}

Then according to the moment of momentum theorem, the dynamic function describing the rotational motion of the wheel-ski aircraft to the z-axis ${Z_b}$ in ${S_b} - {O_b}{X_b}{Y_b}{Z_b}$ is expressed as:

(8) \begin{align}\dot r = \left( {\begin{array}{*{20}{c}}{ - \left( {{Q_{ml}} + {Q_{mr}} + {Q_{sl}} + {Q_{sr}}} \right){a_m} - \left( {{F_n}\sin {\theta _l} - {Q_n}\cos {\theta _l}} \right){a_n}}\\{ + \dfrac{{\left( {{Q_{mr}} + {Q_{sr}} - {Q_{ml}} - {Q_{sl}}} \right){b_w}}}{2} + Y{x_a} - {P_r}{l_r} + NR}\end{array}} \right)/{I_z}\end{align}

In addition, the kinematical equation set describing the relationship between the aircraft velocity and displacement based on the first Euler law is given by:

(9) \begin{align}\begin{cases} {{\dot P}_x} = u\cos \psi \cos \theta - v\sin \psi \\ {{\dot P}_y} = u\sin \psi \cos \theta - v\cos \psi \end{cases} \end{align}

where, ${P_x},\;{P_y}$ are the aircraft taxiing distance and lateral offset along x- and y-direction in the ground coordinate system ${S_g} - {O_g}{X_g}{Y_g}{Z_g}$ [36].

According to the second Euler law, the angular momentum kinematical equation can be obtained:

(10) \begin{align}\dot \psi = r/\cos \theta \end{align}

Eqs. (7–10) form the mathematical dynamic ground taxiing model of the wheel-ski-type aircraft.

2.3 Design and experimental verification of wheel-ski-type landing gear

The structure of the designed wheel-ski-type landing gear is illustrated in Fig. 3(a). It consists of landing gear torque links, a rocker arm, a wheel, a ski and a main strut including an outer cylinder and a piston rod with the integration of a motor, a pressure sensor and a shock absorber in it. The motor is used for lifting and dropping the ski, and the pressure sensor is used for collecting the pressure between the ski and the ground. The scaling experimental prototype and the direction control system are demonstrated in Fig. 3(b)(c). The control system is composed of a pressure sensor transmitter, an attitude sensor, a differential Global Positioning System (GPS), two motor drivers and a digital signal processor (DSP) used as a direction controller in the experiment.

Figure 3. Wheel-ski landing gear design and direction control system test. (a) Structure of wheel-ski landing gear. (b) Test of wheel-ski landing gear. (c) Direction control system.

The specific direction rectifying control law is written in the DSP. When the direction controller is working, the voltages of the two motors and the expected ground reaction forces on the left and right skis of the main landing gears can be calculated based on the actual values of the aircraft lateral offset measured by a differential GPS and the yaw angle measured by an attitude sensor. The motor will press the ski onto the ground and the pressure sensor will return the actual value of the ski pressure to the controller, which ensures that the actual pressure can reach the expected value.

The direction control system response is tested in this study. An input signal of the lateral displacement 5m from 1s to 10s is given both in the simulation model and also through the differential GPS in the test, as shown in Fig. 4(a). The data is then transmitted to the direction control system. Since the input signal is a positive value in this test, indicating that the aircraft is yawing rightwards, the left ski is pressed on the ground and the right ski keeps still. Then the frictional forces on the left and right skis can generate a left yawing moment to the aircraft, which can correct the aircraft taxiing direction. The blue curve in Fig. 4(c) illustrates the expected value of the ground reaction force on the left ski, which is calculated by the direction control law. Thus the correctness of the direction rectification control logic is verified.

Figure 4. Direction control system response under a ski pressure input signal. (a) Input signal of lateral displacement. (b) Motor rotational speed. (c) Ground reaction force on left ski.

After receiving the calculated expected value of the ground reaction force on the left ski, it can be seen in Fig. 4(b) that the left motor starts to work and its rotational speed rises to the expected value in less than 0.1s, indicating that the motor has a fast response. The motor rotor then converts the rotational movement into a rectilinear motion and presses the ski to the ground. The black curve in Fig. 4(c) indicates the simulation value of the ground force on the left ski, while the red curve shows the test value collected by the pressure sensor during the test. Figure 4(c) demonstrates that this ski actuation process takes approximately 0.5s, which leads to the error between the expected value and the actual actuation process. Once the ground reaction force on the ski reaching the expected value, the motor stops working also in less than 0.1s and the pressure of the ski can keep constant under the action of the motor self-lock. Figure 4(c) also shows that the pressure on the ski is smoother in the simulation, while the response vibrates at 1-2s in the test. This is owing to the fact that not only the pressure sensor noise exists in the test, but the ski stiffness may also bring about tiny vibration during the pressing process. However, the error of the overshoot is approximately 5% and can return to the steady state rapidly, which has little effect on the direction control performance. The results show that the wheel-ski landing gear and the direction control system model fits well with the actual mechanism. This validated model is then used in the following sections in this study.

3.0 Traditional Direction Rectifying Control Systems Applied in Wheel-Ski Aircraft

3.1 Differential brake in wheel-ski landing gear

Since there is no brake mechanism on the wheel of the wheel-ski landing gear, the wheel is always rolling without slipping, so that only the ski is used for stopping the aircraft through the frictional force between the ski and the ground. Thus, the ski can also be adopted to correct the aircraft direction by adjusting the ground reaction forces on the left and right skis. The friction forces on the skis will vary proportionally to the ground reaction forces and then, the differential frictional moments to ${O_b}{Z_b}$ can produce a yawing moment to correct the aircraft rollout direction. Moreover, the wheel on the landing gear not only improves the aircraft rollover stability, but can also generate the lateral force when a sideslip angle exists, which provides with an aligning torque for the aircraft.

The control schematic diagram of the wheel-ski differential brake is demonstrated in Fig. 5. The actual yaw angle $\psi $ and lateral offset $y$ are the input control signals of the proportion-differentiation (PD) differential brake controller, and the direction rectification command ${\delta _b}$ is obtained:

(11) \begin{align}{\delta _b} = K_p^\psi \left( {\psi - {\psi _0}} \right) + K_d^\psi \dot \psi + K_p^y\left( {y - {y_0}} \right) + K_d^y\dot y\end{align}

Figure 5. Control schematic diagram of wheel-ski differential brake.

The direction rectification command ${\delta _b}$ is processed in the allocation unit to output the specific braking commands to the left and right main landing gears. If the aircraft is yawing left on the runway, the braking signal ${\delta _b}$ is positive after tuning by the PD controller. Thus the left ski is lifted and the right ski is pressed on the ground to generate the frictional force. On the contrary, if the aircraft is taxiing rightwards, the right ski is raised while the left ski is put down to the ground. The specific control law is:

\begin{align*}\left\{ {\begin{array}{*{20}{c}}{{\delta _{bl}} = - {\delta _b}}\\{{\delta _{br}} = 0}\end{array}\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \right.{\delta _b} \le 0\end{align*}

(12) \begin{align}\left\{ {\begin{array}{*{20}{c}}{{\delta _{bl}} = 0}\\{{\delta _{br}} = {\delta _b}}\end{array}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \right.{\delta _b} \gt 0\end{align}

The yaw angle and the lateral displacement are captured by the displacement and the attitude sensors and controlled by the closed-loop differential brake control system to correct the aircraft taxiing direction during the whole rollout process.

Figure 6 shows the simulation results at various taxiing speeds of 30m/s, 50m/s and 70m/s under 1° initial yaw angle. Results indicate that the aircraft taxiing direction can be corrected effectively under the wheel-ski differential brake control in the whole speed range. As the taxiing velocity decreasing, the aerodynamic lift drops, leading to the increase of the ground reaction force and the friction force on the skis. Thus, the skis can provide with greater differential rectifying moments so that the direction rectification performance is better. Figure 6(d) illustrates that the left and right skis press towards the ground alternately to adjust the differential ground reaction forces.

Figure 6. Direction rectifying performance under 1° initial yaw angle using wheel-ski landing gear differential control at different initial taxiing velocities. (a) Lateral displacement. (b) Yaw angle. (c) Aircraft sideslip angle. (d) Left and right ground reaction forces on the skis.

However, from Fig. 6(a)(b), it can be seen that when the aircraft is running at the speed of 70m/s, the maximum lateral offset reaches 8m and it takes approximately 20s to return to the runway centreline, indicating that the direction rectification efficiency is relatively low. Moreover, the PD control parameters which satisfy the low-speed taxiing conditions may not be appropriate for the high-speed rollout conditions, showing that the robustness of PD control law with a set of fixed control parameters does not perform well in all conditions. As a result, if the control parameters can be tuned in real time, the direction rectifying performance will achieve better effects under various conditions.

3.2 Nose wheel steering control

Figure 7 demonstrates the nose wheel steering control schematic diagram. During the nose wheel steering control process, the aircraft taxiing direction is corrected by the forces between the ground and the nose wheel. If the aircraft deviates from the runway centreline, the nose wheel will steer under the control of the nose wheel steering control system. Thus the lateral force of the nose tire and the corresponding torque to the aircraft are then produced so that the aircraft can be adjusted to the right rollout direction.

Figure 7. Nose wheel steering control schematic diagram.

The inputs of the nose wheel steering control system are also the actual values of the yaw angle $\psi $ and the lateral displacement $y$ . Then the steering gear will turn the nose wheel to a certain angle ${\theta _l}$ according to the control signal ${\delta _s}$ calculated by the PD control law. If the aircraft runs rightwards, the deviation signal ${\delta _s}$ is negative, resulting in a left turn of the nose wheel and a leftward torque generated by the nose tire to correct the aircraft direction. The specific control law can be expressed by:

(13) \begin{align}{\delta _s} = K_{sp}^\psi \left( {\psi - {\psi _0}} \right) + K_{sd}^\psi \dot \psi + K_{sp}^y\left( {y - {y_0}} \right) + K_{sd}^y\dot y\end{align}

The nose wheel steering system is flexible to be operated so that it is one of the most important parts of the aircraft direction rectifying system. However, the aerodynamic lift will lead to a decrease of the tire force on the nose wheel and thus, the control efficiency reduces remarkably at a high speed. Therefore, the nose wheel steering control system is only applicable in low-speed taxiing conditions.

Figure 8 shows the direction rectifying performance under 1° initial rightward yaw angle under the nose wheel steering control at low initial taxiing velocities. The nose wheel steers to the left side when the rightward yaw angle is detected. Then the yaw angle decreases rapidly to a negative value while the lateral displacement is still positive. So the nose wheel remains on the left side, but the steering angle reduces gradually as the aircraft is running towards the runway centreline. The direction rectifying performance improves as the aircraft taxiing speed decreases. When the taxiing speed increases from 20m/s to 30m/s, the maximum lateral offset rises slightly from 1.2m to 1.6m. And the aircraft can return to a steady state at approximately 11s when the taxiing speed is lower than 30m/s. However, if the speed is higher than 40m/s, the lateral displacement goes up to 2.4m. Also, a big overshoot appears at 10s and it takes 20s to become stable, indicating that only adopting the nose wheel steering system to correct the direction is inefficiency in the whole speed region. As a result, to improve the direction rectification system performance, a combined direction control system is introduced in this study and the nose wheel steering system is added as the taxiing speed is lower than 30m/s.

Figure 8. Direction rectifying performance under 1° initial yaw angle using nose wheel steering control at low initial taxiing velocities. (a) Lateral displacement. (b) Yaw angle. (c) Aircraft sideslip angle. (d) Nose wheel steering angle.

3.3 Rudder control

The rudder control efficiency is relatively high so that it often plays an important role in direction rectification during the high-speed rollout process. The aerodynamic lateral force and the corresponding moment generated by the rudder can correct the taxiing direction effectively without veering off the runway. The control principle is shown in Fig. 9.

Figure 9. Rudder control schematic diagram.

The input signals of the rudder control system also include the yaw angle $\psi $ and the lateral displacement $y$ . The steering gear turns the rudder to a certain angle according to the signal output from the PD controller. Then the lateral force produced by the rudder can turn the aircraft taxiing direction. The specific control law is given by:

(14) \begin{align}{\delta _r} = K_{rp}^\psi \left( {\psi - {\psi _0}} \right) + K_{rd}^\psi \dot \psi + K_{rp}^y\left( {y - {y_0}} \right) + K_{rd}^y\dot y\end{align}

where, $K_{rp}^\psi $ , $K_{rp}^y$ , $K_{rd}^\psi $ , $K_{rd}^y$ are the proportional and differential coefficients of the rudder control system.

Figure 10 demonstrates the direction rectifying performance under 1° initial yaw angle using the rudder control at different initial taxiing velocities. As the taxiing velocity decreases, the rudder works at the saturation point for a longer period and the direction correction efficiency reduces. When the aircraft is rolling out rightwards at 70m/s on the runway, the rudder deflects to the left, and a rightward aerodynamic lateral force is produced which gives the aircraft an anticlockwise yawing moment to correct the direction. Nevertheless, the lateral aerodynamic force will pull the aircraft away from the runway centreline, leading to a broader lateral offset. Then at 1.5s, to decrease the lateral displacement, the rudder defects to the right and a clockwise yawing moment to the aircraft is generated. Thus the fuselage slip angle reduces and the aircraft returns to the runway centreline gradually. The maximum lateral offset is 3.2m and it only takes 12s to rectify the taxiing direction, indicating that it is applicable to employ only the rudder control system during the high-speed rollout process to control the direction.

Figure 10. Direction rectifying performance under 1° initial yaw angle using rudder control at different initial taxiing velocities. (a) Lateral displacement. (b) Yaw angle. (c) Aircraft sideslip angle. (d) Rudder deflection angle.

However, when the taxiing speed drops to 50m/s, the aerodynamic efficiency decreases significantly. The red curve in Fig. 10(d) shows that the rudder works at the saturation points of -30° and 30° from 0.5s to 3s to try to correct the taxiing direction effectively. Moreover, if the rudder is still the only mechanism used for rectifying the direction when the aircraft velocity decreases to 45m/s, though the rudder deflection angle is always working at the saturation points (see black dashed line in Fig. 10(d)), the aircraft taxiing direction is still out of control. As a result, the rudder is designed to be applied when the taxiing speed is higher than 50m/s in the combined rectifying control system in Section 4.

4.0 New Combined Fuzzy Direction Rectification Control System Design

4.1 Combined direction rectification control system based on optimisation design method

4.1.1 Performance criteria

Based on the analysis results in Section 3, the nose wheel steering subsystem and the rudder subsystem are added in the taxiing direction control system to improve the direction rectifying performance. The allocation basis is the aircraft taxiing speed, which plays an important role in all the three subsystems, the weighting coefficient distribution of which is illustrated in Fig. 11. The new combined direction rectification control system can be divided into three phases. (1) If the aircraft is rolling out at a high speed ( $V \geqslant 50$ m/s), the rudder and the differential brake are adopted in combination since the aerodynamic efficiency is high. (2) If the taxiing speed is located in $30 \lt V \lt 50$ m/s at a middle speed, only the differential brake in wheel-ski landing gears is used to correct the direction. (3) Otherwise, if the taxiing speed is $0 \le V \le 30$ m/s, the ground forces on the wheels and skis increase so that the nose wheel steering unit is added in the direction control system.

Figure 11. Schematic diagram of weighting coefficient distribution of three direction rectifying subsystems.

From Fig. 11, it can be seen that the weighting coefficients of the three subsystems are ${\eta _r},{\eta _b},{\eta _s}$ . Under different speeds, the three coefficients satisfy the relationships below:

(15) \begin{align}\left\{ {\begin{array}{*{20}{c}}{{\eta _b} + {\eta _r} = 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;V \geqslant 50}\\{{\eta _b} = 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;30 \lt V \lt 50}\\{{\eta _b} + {\eta _r} = 1\;\;\;\;\;\;\;\;\;\;\;\;\;0 \le V \le 30}\end{array}} \right.\end{align}

(1) Objective Function

The three optimised weighting coefficients at various taxiing speeds are obtained through the Levenberg–Marquardt (LM) optimisation design algorithm [37] based on the simulated annealing. The direction rectification performance criteria ${\sigma _c}$ in this study is defined as the integral of the absolute values of the lateral offset $y$ and the yaw angle $\psi $ :

(16) \begin{align}{\sigma _c} = \mathop \smallint \limits_0^t \mathop \sum \limits_{i = 1}^n \left| {{x_i}} \right|\left( t \right)dt{\rm{\;\;\;\;\;}}\left( {n = 2,{x_1}\left( t \right) = y\left( t \right) - {y_0},{x_2}\left( t \right) = \psi \left( t \right) - {\psi _0}} \right)\end{align}

Therefore, the minimum value of ${\sigma _c}$ is taken as the objective function ${\rm{min}}\left( {{\sigma _c}} \right)$ to find out the optimal solutions of the direction rectifying coefficients.

(2) Constraints

The width of a standard 4C-4F airport runway is within the range of [45-60]m. Thus one of the constraints is that the aircraft maximum lateral offset is smaller than the half-width of the runway:

(17) \begin{align}{\left| y \right|_{{\rm{max}}}} \lt 22.5{\rm{m}}\end{align}

According to the airworthiness standard [38], the spin tendency of an aircraft is not allowed to appear on the ground beyond control. As a result, the yaw angle is restrained within 90° to prevent the aircraft from spinning around:

(18) \begin{align}{\left| \psi \right|_{{\rm{max}}}} \lt \frac{\pi }{2}{\rm{rad}}\end{align}

In addition, the weighting coefficient of the wheel-ski differential brake ${\eta _b}$ should meet the condition:

(19) \begin{align}0 \le {\eta _b} \le 1\end{align}

4.1.2 Low-speed phase $(0 \le V \le 30$ m/s)

During the low-speed taxiing phase, the initial value of ${\eta _b}$ is set as ${\eta _{b0}} = 0.5$ . Then several working conditions including 13 taxiing velocities are selected from 5m/s and the interval is 2m/s in $0 \le V \le 30$ m/s to carry out the single-objective optimisations:

(20) \begin{align}V = 5 + 2{n_l}\left( {{n_l} \in \left[ {0,12} \right],{n_l} \in N} \right)\end{align}

Then the 13 optimised points are obtained from the 13 optimisation procedures of the designed combined direction rectifying control system. And the corresponding polynomial fitting curve of the 13 points is shown in Fig. 12. The red points in Fig. 12 are the optimised values and they are distributed uniformly around the cubic polynomial fitting curve. The value of the goodness of fit R² is 0.9992, which is close to 1, indicating that it is reasonable to use this fitting curve to represent the coefficient ${\eta _b}$ of the combined direction control system at this low-speed phase. Thus the three coefficients can be expressed by:

(21) \begin{align}\left\{ {\begin{array}{*{20}{c}}{{\eta _s} = 1 - {\eta _b}}\\{{\eta _b} = 1.624e6{V^3} - 3.868e4{V^2} + 1.858e2V + 0.515}\\{{\eta _r} = 0}\end{array}} \right.\end{align}

Figure 12. Optimum values and the corresponding polynomial fitting curve of ${\eta _b}$ during low-speed phase.

4.1.3 High-speed phase ( $50 \le V \le 70$ m/s)

Both the coefficients of the wheel-ski differential brake ${\eta _b}$ and the rudder subsystem ${\eta _r}$ are optimised when designing the combined direction control system at the high-speed phase. The working conditions are chosen with the interval of 2m/s in $50 \le V \le 70$ m/s:

(22) \begin{align}V = 50 + 2{n_h}\left( {{n_h} \in \left[ {0,10} \right],{n_h} \in N} \right)\end{align}

Under 11 different taxiing velocities obtained by Eq. (22), 11 single-objective optimisations of the combined direction control system are carried out to find out the optimised weighting coefficients ${\eta _r}$ and ${\eta _b}$ . The 11 optimum points and the corresponding polynomial fitting curves are demonstrated in Fig. 13. The fitting curves are divided into two segments ( $50 \le V \lt 60$ m/s and $60 \le V \le 70$ m/s) based on the variation tendency of the obtained red and purple points. The values of the goodness of fit R² are 0.9563 (the black curve in Fig. 13) and 0.9989 (the blue curve in Fig. 13), respectively. The three weighting coefficients can be given by:

(23) \begin{align}\left\{ {\begin{array}{*{20}{c}}{{\eta _r} = 1 - {\eta _b}}\\{{\eta _b} = \left\{ {\begin{array}{*{20}{c}}{ - 1.011e5{V^3} + 1.632e3{V^2} - 8.820e2V + 2.190\;\;\;\;\;\;\;\;\;\;\;\;\;50 \le V \lt 60{\rm{m}}/{\rm{s}}}\\{ - 2.564e6{V^3} + 7.553e4{V^2} - 6.919e2V + 2.577\;\;\;\;\;\;\;\;\;\;\;\;\;60 \le V \le 70{\rm{m}}/{\rm{s}}}\end{array}} \right.}\\{{\eta _s} = 0}\end{array}} \right.\end{align}

Figure 13. Optimum values and the corresponding polynomial fitting curve of ${\eta _b}$ during high-speed phase.

In conclusion, the weighting coefficients of the three direction control subsystems are obtained in the whole speed region, as illustrated in Fig. 14. The blue curve demonstrating the variation trend of the nose steering system coefficient ${\eta _s}$ indicates that the nose wheel steering system occupies above 40% when the taxiing speed is lower than 5m/s and this percentage reduces gradually as the aircraft velocity increases, which shows that the nose steering wheel subsystem is more effective during a low-speed rollout process. In addition, as the taxiing velocity rising, the coefficient ${\eta _b}$ of the wheel-ski differential brake subsystem accounts for increasing percentage till 50m/s (see the red curve in Fig. 14). When the aircraft is taxiing in a high-speed region ( $50 \le V \le 70$ m/s), the rudder efficiency rise leads to an increase in the weighting coefficient ${\eta _r}$ of the rudder control subsystem.

Figure 14. Weighting coefficients of three direction control subsystems in whole speed region.

4.2 Direction rectification control system based on self-tuning fuzzy-PD control

4.2.1 Fuzzy-PD control principle

The simulation results of the wheel-ski differential brake PD control law in Section 3.1 indicate that using only one set of PD control parameters to correct the aircraft direction is inappropriate in the whole taxiing speed region. However, the wheel-ski differential brake subsystem plays an important role during the whole rollout process in every speed phase, thus to improve the direction rectifying control efficiency in the whole speed region, the fuzzy control method is introduced to identify the aircraft taxiing characteristic parameters online and to tune the PD control parameters in real time. The self-tuning fuzzy-PD differential brake control is designed in this Section in lieu of the classic PD control law in Section 3.1, where the interference immunity and the robustness can be both improved while the PD control merits are also retained. Then this fuzzy-PD wheel-ski differential brake subsystem is used in the combined optimised direction control system designed in Section 4.1 to ensure great performance in the whole speed region.

The specific self-adaptive fuzzy-PD control diagram of the wheel-ski differential brake subsystem is demonstrated in Fig. 15. The lateral displacement $y$ and the yaw angle $\psi $ detected by the sensors are transmitted to the control system. Then the weighted summation e of these two values and the corresponding rate ec are taken as the inputs of the fuzzy controller. They are fuzzified and then computed and reasoned by the fuzzy rules so that two outputs $\Delta {K_P}$ and $\Delta {K_d}$ can be obtained after defuzzification. $\Delta {K_P}$ and $\Delta {K_d}$ are the increments of the initial proportional coefficient ${K_{P0}}$ and the differential coefficient ${K_{d0}}$ in the traditional PD controller. As a result, the current control parameters used in the fuzzy-PD controller in real time are:

(24) \begin{align}\left\{ {\begin{array}{*{20}{c}}{{K_P} = {K_{P0}} + \Delta {K_P}}\\{{K_d} = {K_{d0}} + \Delta {K_d}}\end{array}} \right.\end{align}

Also, e and ec are the inputs of the wheel-ski differential brake to generate the direction rectification command ${\delta _b}$ to the two main wheel -ski type landing gears (see Eq. (12)). The differential forces and torques on the two landing gears can turn the aircraft rollout direction effectively.

4.2.2 Fuzzy-PD controller design

The key part of designing the differential brake control system of the aircraft with the wheel-ski type landing gears is the fuzzification of the inputs and outputs, as well as the fuzzy rules between them.

The fuzzy subsets of the inputs e, ec and the outputs $\Delta {K_P}$ , $\Delta {K_d}$ are all defined as {NB, NM, NS, ZO, PS, PM, PB}. The membership functions of these variables are chosen as the triangular type. Then, designing the fuzzy rules is a process that deducing empirical regulations according to the typical step responses. Firstly, tuning ${K_P}$ is for reducing the systematic deviation and improving the response rate. Secondly, the adjustment of ${K_d}$ is for decreasing the overshoots. If the inputs are small, ${K_P}$ should rise to reduce the steady-state error and to improve the control precision. Moreover, if ${K_d}$ results in a time expansion during the tuning process, the value should be decreased. Also, if the input error is a medium value, ${K_P}$ should be reduced, leading to a smaller overshoot and a faster response. Then according to the adjustment rules above, the fuzzy rules are designed as shown in Tables 1 and 2.

Table 1. Fuzzy control rules of $\Delta {K_P}$

Table 2. Fuzzy control rules of $\Delta {K_d}$

Figure 15. Self-tuning fuzzy-PD control diagram of wheel-ski differential brake system.

Finally, the fuzzy subsets are obtained after the self-adaptive fuzzy-PD control so that the defuzzification step is needed to acquire the real values of ${K_P}$ and ${K_d}$ . The centroid method is adopted in this study to defuzzify the output fuzzy subsets:

(25) \begin{align}{z_0} = \frac{{\mathop \sum \nolimits_{i = 0}^n {\mu _c}\left( {{z_i}} \right) \cdot {z_i}}}{{\mathop \sum \nolimits_{i = 0}^n {\mu _c}\left( {{z_i}} \right)}}\end{align}

5.0 Simulation and Analysis of New Combined Self-Tuning Fuzzy-PD Direction Control System

Based on the optimisation of the weighting coefficients in Section 4.1 and the self-tuning fuzzy-PD control law design in Section 4.2, a new combined self- tuning fuzzy-PD direction control system which applies to the aircraft with wheel-ski type main landing gears is developed in this study. The direction rectification properties of the three subsystems and the proposed combined control system are compared under various working conditions in this section.

5.1 Comparison of different directional rectifying control laws

In this subsection, the maximum lateral displacement ${y_{max}}$ , the first crest of the yaw angle ${\psi _f}$ , the reciprocal of the direction rectification performance criteria $1/{\sigma _c}$ are all taken into consideration to demonstrate the system performance during the whole taxiing process under 1° initial yaw angle.

5.1.1 At a Low Taxiing Speed (30m/s) under 1° Initial Yaw Angle

Table 3 shows the performance of three directional control laws at 30m/s taxiing velocity. The second and fourth columns in Table 3 indicate that the maximum lateral displacement ${y_{max}}$ and the integral of the lateral displacement $\mathop \smallint \limits_0^t \left| {y\left( t \right)} \right|dt$ in the whole time domain both stay at the minimum values under the control of the combined direction control system. Comparing with the PD wheel-ski differential brake and the nose wheel steering control, the integral of the lateral displacement reduces 72.65% and 4.33%, respectively. The third and fifth columns demonstrate the first crest of the yaw angle ${\psi _f}$ and the integral of the yaw angle $\mathop \smallint \limits_0^t \left| {\psi \left( t \right)} \right|dt$ . The ${\psi _f}$ under the nose wheel steering control is the maximum among three control systems. In addition, the $\mathop \smallint \limits_0^t \left| {\psi \left( t \right)} \right|dt$ under the combined control system in the whole-time domain decreases 18.33% and 16.99% than that under the differential brake and the nose wheel control. The new combined control system at a low taxiing velocity consists of the nose wheel control system and the fuzzy differential brake control system. Comparing to a single nose wheel steering system, the frictional force difference between the two skis can provide with an additional yawing moment for the aircraft after adding the differential brake in the direction control law, so that the maximum lateral offset and yaw angle both can be reduced. Besides, the control parameters can be tuned online through the fuzzy controller in the combined control law so that the accumulative errors of the displacement and the yaw angle responses generated by the original differential brake system can be eliminated. The sixth column illustrates the direction control performance through the criteria $1/{\sigma _c}$ , the greater value of which represents a better system performance. From the fifth row of this column, it can be seen that comparing with the differential brake and the nose wheel steering control, the direction rectifying efficiency improves 212.96% and 7.92% respectively, indicating that at a low taxiing velocity, the directional safety can be improved during the whole process under the combined control system.

Table 3. Direction performance comparison at a low taxiing speed under 1° initial yaw angle

^*: The percentage in the first row is the improvement of the new combined fuzzy-PD control comparing to PD wheel-ski differential brake control; the percentage in the second row is the improvement of the new combined fuzzy-PD control comparing to nose wheel steering control.

5.1.2 At a medium taxiing speed (50m/s) under 1° initial yaw angle

Table 4 shows the comparison of the PD differential brake control and the new combined control laws at 50m/s taxiing velocity. The wheel-ski differential brake system plays a leading role during the direction control process at the medium taxiing speed. However, the participation of the rudder control system can increase the yawing control moment so that the lateral offset decreases above 50% under the control of the combined direction system, as can be seen in the second and fourth columns in Table 4. In addition, the accumulative errors of the responses also can be removed under the control of the combined direction system due to the control of the fuzzy control law. Comparing with the Fig. 10(b)(d), the rudder will no longer work at the saturation points and the maximum yaw angle decreases remarkably. Comparing to the PD wheel-ski differential brake control, though the maximum yaw angle rises slightly, the direction rectifying efficiency increases 140%. The results show that the novel designed directional control law is appropriate for the medium-speed taxiing phase, where the lateral offset can be reduced markedly so that the danger of veering off the runway decreases significantly.

Table 4. Direction performance comparison at a medium taxiing speed under 1° initial yaw angle

^#: The percentage is the improvement of the new combined fuzzy-PD control comparing to PD wheel-ski differential brake control.

5.1.3 At a high taxiing speed (70m/s) under 1° initial yaw angle

Table 5 presents the rectification performance of the three control laws under 70m/s taxiing velocity. From the second and fourth columns, we can see that in comparison with the PD differential brake control, the lateral displacement decreases greatly during the whole taxiing process, especially ${y_{max}}$ drops from 8m to 3m under the combined control system. Besides, the third and fourth rows in Table 5 demonstrate that the lateral displacements are both small under the control of the rudder and the designed control systems, while the maximum yaw angle reduces greatly under the combined fuzzy-PD control system. The right column shows that comparing with the differential brake control, the direction rectifying efficiency increases approximately 300%, which is owing to the fact that the aerodynamic lift is large during a high-speed taxiing process so that the ground reaction force and the frictional force between the ski and the ground are relatively small. Thus a rudder control system need to be introduced into the combined direction control system, which is of high efficiency and can generate a larger yawing moment to the aircraft in high-speed rollout working conditions to improve the direction control performance.

Table 5. Direction performance comparison at a high taxiing speed under 1° initial yaw angle

^&: The percentage in the first row is the improvement of the new combined fuzzy-PD control comparing to PD wheel-ski differential brake control; the percentage in the second row is the improvement of the new combined fuzzy-PD control comparing to rudder control.

Moreover, the comparison between Tables 3–5 shows great adaptability of the combined control system in that the maximum lateral offset varies little at different velocities, indicating that working condition has a slight influence on the designed control system.

In conclusion, from Tables 3–5, we can see the new combined fuzzy-PD control system applies to the aircraft with wheel-ski type landing gears in the whole speed region, which can overcome the disadvantages of the traditional single direction control subsystem and the aircraft direction rectification efficiency is improved markedly.

5.2 Stability and robustness of the combined fuzzy-pd control system

In this subsection, six lateral interference conditions including various yaw angles and crosswind velocities are conducted to analyse the direction control performance of the three single subsystems and the combined fuzzy-PD system. The simulation results of the lateral displacement and the yaw angle are shown to verify the stability and robustness of the combined control system.

5.2.1 Influence of yaw angle

Figure 16 illustrates the direction rectifying performance under various initial yaw angles using different directional control laws. Figures 16(a)(b) give the comparison of the rectifying results under 1°, 2°, 3° initial yaw angles at 30m/s and also, the lateral offset and the yaw angle responses under the PD wheel-ski differential brake and the nose wheel steering control are shown under 1° initial yaw angle. As the initial yaw angle rises, we can see that though the maximum lateral offset and yaw angle both increase, the aircraft can eventually return to the runway centreline. In addition, the green simulation curves representing the differential brake control results show that though the yaw angle is the smallest under this kind of control system, the lateral offset is the largest and the aircraft does not return to the runway centreline. Comparing with Fig. 16(a)(b)(c), it can be seen that as the taxiing speed grows, the aircraft lateral displacement converges to 0 gradually, which indicates that only a fixed set of PD control parameters is difficult to satisfy all the working conditions. Moreover, the nose wheel steering control system accounts for a great proportion in the combined fuzzy-PD system in the low-speed region, so that the direction control performance under the combined system at 30m/s is close to but slightly better than that of a single nose wheel steering system.

Figure 16. Direction rectifying performance under various initial yaw angles using different directional control laws at different initial taxiing velocities. (a) Lateral displacement at 30m/s. (b) Yaw angle at 30m/s. (c) Lateral displacement at 50m/s. (d) Yaw angle at 50m/s. (e) Lateral displacement at 70m/s. (f) Yaw angle at 70m/s.

Figures 16(c)(d) give the direction control performance under the control of the combined fuzzy-PD control and the single PD wheel-ski differential brake systems at 50m/s. As the initial yaw angle goes up, the lateral velocity rises leading to the increase the maximum lateral displacement and yaw angle. However, the aircraft can return to the equilibrium position in 15s under the combined control system. Comparing Fig. 16(a)(c), it can be seen that as the taxiing velocity goes up from 30m/s to 50m/s, the maximum lateral displacement increases 50.3% from 1.55m to 2.33m under the new control system, while this value increases 116.7% from 2.40m to 5.20m under the differential brake control, which indicates that the new control law is of better robustness so that the working condition variation has less effect on the system performance.

The results under the combined control system at 70m/s under various initial yaw angles are shown in Fig. 16 (e)(f), and the comparison curves under the differential brake and the rudder control are also displayed. As the rollout speed increases, the aerodynamic force rises resulting in the decrease of the ground reaction force and the friction force on the ski. Thus, the rudder control is introduced into the new control law to correct the aircraft taxiing direction through the yawing moment produced by the rudder. The maximum lateral offset is reduced and the direction rectification duration decreases from 23s to 10s, indicating that the direction rectifying efficiency is improved remarkably. Figure 16(f) also demonstrates that the aircraft yaw angle can converge to the equilibrium state in 10s under the combined control law while it takes approximately 20s to become stable under the control of the single differential brake system.

As a result, the new combined braking control law is of great robustness and adaptability, which can correct the aircraft direction under various initial yaw angles effectively. In addition, the steady-state error can be eliminated under the new control law so that the control efficiency is then improved.

5.2.2 Influence of crosswind

Figure 17 shows the direction rectifying performance under continuous crosswind 1m/s, 2m/s and 3m/s from 1-5s using different directional control systems. Also, the results of the PD wheel-ski differential brake control and the nose wheel steering control under continuous crosswind 1m/s are given as a contrast.

Figure 17. Direction rectifying performance under crosswind using different directional control laws under different initial taxiing velocities. (a) Lateral displacement under 30m/s. (b) Yaw angle under 30m/s. (c) Lateral displacement under 50m/s. (d) Yaw angle under 50m/s. (e) Lateral displacement under 70m/s. (f) Yaw angle under 70m/s.

From the comparisons in Fig. 17(a)(c)(e), we can see that as the taxing speed drops, the crosswind has a greater influence on the aircraft yawing motion. This is because a lower taxiing speed will lead to a larger crosswind-to-taxiing-speed ratio and the aircraft slip angle will increase, resulting in a wider yaw angle and a larger lateral displacement. However, the aligning torque generated by the friction force on the ski becomes larger, and then the aircraft can return to a stable state faster.

Figures 17(c)(d)(e)(f) illustrate that the aircraft can return to the runway centreline in 25s under the designed combined control law. In Fig. 17(a)(b), it can be seen that in the severest working condition that the continuous crosswind speed is 3m/s, the aircraft will become stable after 12s without veering off the runway even the maximum lateral offset and yaw angle are 14.1m and 3.8°, respectively.

To sum up, the fuzzy self-tuning control is added in the direction control law of the wheel-ski landing gear to tune the proportional and differential control parameters in real time, which can improve the anti-interference performance and the robustness of the aircraft taxiing direction control system. Besides, the combination of the rudder control, the nose wheel steering system and the differential brake can improve the direction control performance in the whole taxiing speed region.

6.0 Conclusions

A new type of wheel-ski landing gear is designed, and its force analysis is conducted. The aircraft rollout dynamic model is built firstly. Then the landing gear actuator and the corresponding differential brake control system tests are carried out to verify that the wheel-ski actuation system has a fast response, and the wheel-ski landing gear differential brake system model is accurate. In addition, the traditional direction control subsystems including the differential brake, the nose-wheel steering mechanism and the rudder are all applied in the wheel-ski aircraft separately as baselines. Then a new combined fuzzy-PD direction control law is proposed, the weighting coefficients of which are optimised and the control parameters are tuned in real time. The simulation results in several working conditions are conducted and compared. The conclusions are drawn as follows:

1. (1) The new combined optimised control system can overcome the disadvantages of the traditional single direction control subsystem in the whole speed region so that the aircraft direction control efficiency is improved markedly. Especially, comparing with the differential brake control, the direction rectifying efficiencies increase by 212%, 140% and 299% in low, medium- and high-speed regions, respectively. In addition, comparing with the rudder control and nose wheel steering control subsystems, the overshoots of the yaw angle responses also can be reduced significantly.

2. (2) The combined direction control law is of great adaptability in that the working condition variation has less effect on the system performance.

3. (3) Only a fixed set of PD control parameters is difficult to satisfy all working conditions. Thus the fuzzy control is introduced for self-tuning the proportional and differential control parameters online to improve the system suitability. Results show that the steady-state errors can be eliminated under the new control law even there is no integral element in the system.