The lateral control during aircraft-on-ground deceleration phases

Aerospace Science and Technology 95 (2019) 105482

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The lateral control during aircraft-on-ground deceleration phases Yaqi Dai, Jian Song, Liangyao Yu ∗ , Zhenghong Lu, Sheng Zheng, Fei Li State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Shuangqing Road, Beijing 100084, China

a r t i c l e

i n f o

Article history: Received 21 June 2018 Received in revised form 13 September 2019 Accepted 12 October 2019 Available online 17 October 2019 Keywords: Aircraft-on-ground Lateral dynamics control Parameter variation Nonlinear system Linear quadratic control Predictive control

a b s t r a c t Lateral control during aircraft-on-ground deceleration, which is one of the most important issues in aircraft-on-ground maneuvers, is discussed in this paper. Based on linear quadratic and predictive control theories, two controllers are developed to replace pilot control, which is effective only to a limited level under severe conditions. Particularly, because the aircraft forward speed decreases during deceleration, we propose a novel weighted model predictive control (MPC) method to control this parameter-varying system. Under the same severe conditions, simulation results show different efficiencies for these two controllers and both exhibited better performance than pilot control. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction Over the past few years, aircraft-on-ground (AOG) control issues have received much attention from both the industry and academia. Among these issues, lateral control during the deceleration phase is extremely important because of the high initial longitudinal velocity and the uncertain effects of tire–ground friction or gust. Therefore, although control laws are quite simple under normal conditions, explicit pilot control plays only a limited role in severe conditions, such as asymmetric braking after touchdown. Thus, a pilot might not be able to handle all kinds of conditions during aircraft-on-ground deceleration, especially the lateral control for lane keeping. Fly-by-wire (FBW) and autopilot systems, however, may be used for AOG control in conditions that humans might not handle well enough, which allows the use of more complicated control strategies. The main purpose of this investigation is to solve this kind of problem and propose relevant control strategies. The characteristics of aircraft-on-ground maneuvers have been studied in the past by several researchers. Several mathematical aircraft ground models were proposed in [1–3]. These models were utilized to study not only ground-handling characteristics but also the relationships among aircraft speed, steering radius, lateral control, and other parameters. As for different speed conditions, the characteristics of low-speed ground maneuvers were discussed in

*

Corresponding author. E-mail address: [email protected] (L. Yu).

https://doi.org/10.1016/j.ast.2019.105482 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

[4] and [5]. Furthermore, the authors proposed a yaw control law using the feedback linearizing theory, which could be applied in the high-speed range as well. The asymmetric aircraft-on-ground condition is also important and should be discussed. The authors of [6] worked on asymmetric landing and aircraft-on-ground maneuvers and simulated different cases of cornering, braking, and asymmetric landing using commercial software. Asymmetric landing under unsteady aerodynamic effects was also studied in [7], where the proposed control strategy was mainly based on ailerons and rudders. As discussed above, advanced fly-by-wire and autopilot systems enable precise and integrated control of an aircraft. Based on this fact, various strategies for aircraft-on-ground automatic control have been discussed in the past. The studies in [8] and [9] were mainly concerned with the AOG path following control problem and yaw rate control of taxiway maneuvers, respectively. In these two studies, a nonlinear adaptive controller with a dynamically adaptive backstepping design and a sliding mode controller (SMC) were employed to find the control law of nose-wheel deflection. Meanwhile, integrated control has received much attention recently; it refers to simultaneous control over the nose wheel steering, differential brake, rudder, and even engine thrust ([10] and [14]). Meanwhile, during braking or takeoff processes, aircraft longitudinal velocity varies over a wide range. Such characteristics create parameter-varying problems, but few studies focused on these issues. [11] and [12] developed integrated controllers for automatic takeoff and braking processes, in which the control law varied with an increase or decrease in the forward speed. Moreover, the study in [11] also assigned priorities to nose wheel steer-

2

Y. Dai et al. / Aerospace Science and Technology 95 (2019) 105482

Fig. 1. Structure of the aircraft-on-ground simulation system.

ing and differential brake and rudder at different forward speeds. Some researchers concentrated on parameter-varying maneuvers at low speeds [13], [14]. To achieve trajectory tracking of the varying longitudinal velocity and yaw rate, the authors of [13] proposed an integrated control system based on sliding model control (SMC). Compared with flight phases, AOG deceleration phases are more complex. This is because when braking on the ground, an aircraft might not only be affected by aerodynamics but also by nonlinear tire-ground friction forces, which make the problem closer to the ground vehicle control. In addition, lateral-control research on ground vehicles is more comprehensive than that on aircraft-onground problems. Therefore, literature on road vehicles may provide a good reference. There are several studies ([15], [17]) on vehicle steering control, which is one of the most important lateralcontrol issues. The two degree-of-freedom model described in [16] and [17] has been widely utilized in vehicle lateral-control design; further, the μ-split road simulation described in [15] and [16] is generally adopted to test disturbance rejection properties. In [18], an integrated multiple-in-multiple-out (MIMO) controller involving steering and braking actuators was proposed; this controller was based on H ∞ control and could collaborate with the anti-brake system (ABS). The path-following issue had been studied comprehensively by several researchers, who used linear quadratic and predictive control theories to solve this problem. In [19], optimal linear preview controllers based on these two theories were compared and identical results were observed for long preview and control horizons. Based on the results in [19], the authors of [20] focused on interactions between the driver and vehicle active front steering (AFS) controller and developed a driver-AFS interactive steering control model using the linear quadratic (LQ) and MPC method. However, these path-following issues were mainly concerned with conditions with a constant forward speed. In some studies on flight-phase control, time-varying problems of output constraints as well as uncertainties were considered [21], [22]. Therefore, similar LQ and MPC methods for ground vehicles would show different effects in the AOG braking process, where the predictive controllers would face parameter-varying problems. The main contribution of this study is that we solve the automatic lateral-control problem during aircraft-on-ground deceleration to overcome the limitations faced by human pilots under severe landing conditions, such as asymmetric braking processes. To achieve this objective, two approaches based on the LQR and MPC theories were proposed and improved successively to replace pilot control. The main control objective of the proposed controllers is to keep the aircraft near the center line with limited deviation in the directional angle as well as to ensure stability during the braking process. Fig. 1 shows the structure of the simulation system for

controller property testing. The aircraft-on-ground model suited to the problem description and simulation tests is described in Section 2, including aircraft dynamics, tire model, and aerodynamics model. As shown in Fig. 1, a control command could be produced by the pilot model or automated controllers independently to evaluate properties of different control methods. To test the properties of disturbance rejection and stability control, the control process was simulated under the asymmetric braking conditions of μ-split roads. The rest of this article is organized as follows. The aircraft-onground model used for simulations is described in Section 2. In Section 3, the pilot model is discussed and the control process is simulated under different road conditions. In Section 4, to replace pilot control, a receding horizon LQR controller is proposed and compared with the pilot model. A controller reference model for the controlled aircraft is proposed in Section 4.2. In Section 5, a novel weighted MPC controller is designed to improve automatic control using the controller reference model in Section 4. Finally, the simulation results obtained with different control commands are compared in Section 6. Concluding remarks and potential future research avenues are presented in Section 7. 2. Aircraft-on-ground model 2.1. Aircraft-on-ground dynamics model When considering lateral control, rolling and pitch dynamics have been neglected in many studies [8], [14]. In relevant studies such as [3], [8], and [23], because there is a similarity between the deceleration phase of ground vehicles and on-ground aircraft, a typical three-wheel AOG model was used, which represents a transformed 6-DOF vehicle model. Therefore, we adopted a typical AOG model to describe aircraft ground in this study (Fig. 2). In Fig. 2, F x_ f , F x_rl , F x_rr , F y_ f , F y_rl , and F y_rr represent the total longitudinal and lateral tire forces in different wheels; F xair is the equivalent air friction on C.G, F δr is the lateral aerodynamic force on the rudder, a is the distance between the nose wheel and C.G, b is the distance between the main wheel and C.G, br is the distance between the equivalent center of the rudder and C.G, h is the height of the C.G, c is the half wheel track, δ f is the steering angle, δr is the rudder angle, v x is the longitudinal velocity, v y is the lateral velocity, and ωz is the yaw velocity. Therefore, longitudinal acceleration ax and lateral acceleration a y are given by:

ax = v˙ x − v y ωz

and a y = v˙ y − v x ωz

(1)

Thus, the equation set corresponding to the aircraft ground braking model can be described as follows:

Y. Dai et al. / Aerospace Science and Technology 95 (2019) 105482

3

where s f , srl , and srr represent slip ratio at the nose wheel and left and right main wheels, respectively; ω w f , ω wrl , and ω wrr represent their respective wheel speeds while α f , αrl , and αrr represent their respective sideslip angles. In some studies conducted in normal conditions [24], [25], tire force was given little importance and was described as a linear function of the sideslip angle. However, considering the possibility of landing on the runway in severe conditions, such as wet or icy conditions in [8], several researchers built precise models based on the Magic Formula (MF), which is generally utilized for vehicle control [26]. Therefore, in this investigation, the Magic Formula is used to describe tire force. The longitudinal and lateral forces can be represented individually as shown below,









F (x, F z ) = Dsin Carctan Bx − E Bx − arctan( Bx)

(8)

where F represents the longitudinal or lateral tire force. Here, x represents the slip ratio s or sideslip angle α . Meanwhile, the tire force factors D, C , B, and E are decided by F z , which is related to longitudinal and lateral acceleration. Longitudinal force factors are given by:

D x = b1 F z2 + b2 F z ,

C x = b0 , Bx =

b3 F z2 + b4 F z C x D x eb5 F z

E x = b6 F z2 + b7 F z + b8

and

,

(9)

Lateral force factors are given by:

D y = a1 F z2 + a2 F z ,

C y = a0 , By = Fig. 2. The 6-DOF aircraft ground braking model. (a) Side view of the model. (b) Top view of the model. C.G: Center of gravity.

max = F x_ f cosδ f − F y_ f sinδ f + F x_rl + F x_rr + F xair

(2)

ma y = F x_ f sinδ f + F y_ f cosδ f + F y_rl + F y_rr + F δr ˙ z =c (− F x_rl + F x_rr ) + a( F x_ f sinδ f + F y_ f cosδ f ) Izω

(3)

− b( F y_rl + F y_rr ) + M δ

(4)

where m is the total mass of the aircraft, I z is the rotational inertia of the aircraft about the C.G, and M δ is the additional yaw moment provided by F δr . In the rest of this section, we will describe the different forces and yaw moments observed.

Considering the rolling resistance and rotational inertia of wheels, the wheel model can be written as

˙ w = R w (F x − f F z) − Tb Jwω

(5)

where rotational inertia is represented by J w , R w is the wheel radius, f is the rolling resistance coefficient, F x and F z represent the longitudinal tire force and vertical load, respectively, and T b is the braking torque provided by the braking actuator. Tire force is determined from the slip ratio and sideslip angle using the following equations,

sf =

v x cosδ + ( v y + aωz )sinδ

,

v x + c ωz − R w ω wrr , srr = (6) v y − c ωz v y + c ωz     v y + aω z v y − bωz − δf , , = arctan αrl = arctan vx v x − c ωz

srl =

αf

v x cosδ + ( v y + aωz )sinδ − R w ω w f v x − c ωz − R w ω wrl



αrr = arctan

v y − bωz v x + c ωz



(7)

C y D y e a5 F z

E y = a6 F z2 + a7 F z + a8

and

,

(10)

When both longitudinal and lateral forces are exerted on a tire, the total force can be modified as follows:

σx =

s 1+s

σy =

,



Fx =

|σ x |

σ∗



∗

Fx λ , Fz

tanα 1+s

λ∗ =

σ = σx2 + σ y2 , 

and

,

  σ∗ , and α ∗ = arctan σ ∗ ∗ 1+σ Fy =

|σ y |

σ∗

Fy



∗

α , Fz



(11) (12)

On different kind of roads, tire force can be modified as follows,

F x∗ =

2.2. Wheel & tire model

a3 F z2 + a4 F z



μ μ0 ∗ Fx λ , Fz μ0 μ



and

F ∗y =



μ μ0 ∗ α , Fz Fy μ0 μ

where μ0 is the MF initial adhesion coefficient and adhesion coefficient.



(13)

μ is the actual

2.3. Aerodynamics model During aircraft-on-ground deceleration, yaw velocity and the sideslip angle of the C.G are supposed to be so small that they can be neglected in the aerodynamics model. Therefore, the aerodynamics effects can be given as [27]

1 F xair = − C d S w ρ v 2x , 2 1 M δ = C δr S r δr ρ v 2x br 2

1 F δr = − C δr S r ρ δr v 2x , 2

and (14)

where F xair is the longitudinal aerodynamic resistance, F δr is the lateral aerodynamic force created by the rudder, M δ is the additional yaw moment created by the rudder, C d and C δr are aerodynamic coefficients of the aircraft body (including wings) and rudder, S w and S r are the reference areas of the aircraft body (including wings) and rudder, respectively, br represents the distance between C.G and the rudder, and ρ is air density.

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Y. Dai et al. / Aerospace Science and Technology 95 (2019) 105482

By applying the infinite time LQR solution, we can obtain the full-state feedback,

u˙ p = − g p · xˆ o = −[ g 1 , . . . , gn , gn+1 ] · xˆ o ,

g p = f −1 ( B o ) T K (18)

where K is the solution of the Ricatti equation and B o is an aug T mented input coefficient matrix. xˆ o = xˆ xˆ k uˆ p is the augmented state vector, which was estimated using a Kalman filter (the Estimator in Fig. 3). Finally, the input of the Kalman filter u c is given by the Neuro-motor lag as:

1

τη u˙ p + u p = u c and τη =

Fig. 3. Structure of a typical MOCM.

(19)

g n +1

3. Modeling and simulation

3.2. Actuator system model

3.1. Modified optimal control model (MOCM)

In this study, the actuator system model can be divided into two parts, the rudder system and differential braking system. Because the structure of the actuator system is not the main influencing factor, in this study, a PID control was adopted in the model, as shown in Fig. 4. As shown in Fig. 4(a), T r is the input torque of the rudder, which is decided by the PID controller and target rudder angle  T δr∗ and as illustrated in Fig. 4(b), T b = T brl T brr is the input braking torque, which is decided by the PID controller and target

Ever since its introduction in the 1960s, the pilot model based on optimal control has been used widely to replace earlier models based on the transfer function. For example, Kleinman et al. proposed the optimal control model (OCM), which was the first model to describe pilot behavior in an optimal control framework with respect to the time domain [28]. Over the years, a number of pilot models based on the OCM method have been proposed. In 1992, Davidson et al. proposed a modified optimal control pilot model (MOCM) based on earlier optimal control models, which is currently one of the most commonly used pilot models [29]. In this study, the MOCM was utilized to simulate pilot control results of aircraft ground braking condition after landing. A typical MOCM is shown in Fig. 3. As shown in the figure, u p and uk represent a pilot’s output



T

and delayed output, respectively, and uk = δr∗ M t∗ . δr∗ and M t∗ represent the target rudder angle and additional yaw moment, respectively, u c is the pilot input before the neuro-motor lag, and x and y represent the state vector and observational vector, respectively. In this case, the Pade approximation was used to model the pilot’s time delay, which is given as:

uk up

=

(15)

1 + 12 (τ s) + 18 (τ s)2

where τ is the delay interval. Thus, in the state space form, the delay system is:

and

uk = C k xk + D k uk

(16)

where xk is the state vector of Pade delay. As the cost function of MOCM takes u˙ p into account, the quadratic performance index J P ilot is given by:

J P ilot =

∞  0

F rl∗ =

λrl∗ =

1 − 12 (τ s) + 18 (τ s)2

x˙ k = A k xk + B k u p





y T Q y y + u Tp Ru p + u˙ Tp f u˙ p dt

(17)

T

∗ λ∗ slip ratio λ∗ = λrl . The target slip ratio of the differential rr braking system is decided by the target longitudinal acceleration a∗x and additional yaw moment M t∗ in the braking torque allocation part. Considering that the computing power of the actuator controller is limited, when additional yaw moment is needed, the controller often selects one gear to generate additional torque based on the slip ratio control. For example, to achieve M t∗ and a∗x , the braking torque allocation part decides the desired tire force as follows,

a∗x m 2 F rl∗ F rlmax

−

M t∗ c

· s∗ ,

∗ F rr =

, and

a∗x m

∗ λrr =

,

2 F rl∗

max F rr

· s∗

(20)

∗ represent the desired tire force of the left and where F rl∗ and F rr right main wheels, respectively. As the main goal in this section is to simulate the pilot handling process, the road adhesion characmax ter is assumed to be already estimated. Therefore, F rlmax and F rr represent the maximum tire force of left and right main wheels, respectively, which can be easily obtained depending on vehicle acceleration and road character. s∗ is the slip rate at which the longitudinal tire force would reach the maximum value.

3.3. Simulation of pilot control on a split road As discussed earlier, explicit pilot control plays only a limited role in severe conditions. To test the properties of the pilot con-

Fig. 4. The structure of actuator system model. (a) Rudder system. (b) Differential braking system.

Y. Dai et al. / Aerospace Science and Technology 95 (2019) 105482

Fig. 5. Pilot-controlled and uncontrolled results on a split road of of the C.G, (e) rudder angle, and (f) additional yaw moment.

5

μl = 0.9 and μr = 0.7. (a) Aircraft movement, (b) longitudinal velocity, (c) direction angle, (d) sideslip angle

troller, we simulated the braking process after touchdown on a μ-split road. The runway was 24 m wide (with reference to the aircraft’s wheel track). The road surface was assumed to be flat with an adhesion coefficient of 0.9 on the left main wheel (μl = 0.9) and 0.7 on the right main wheel (μr = 0.7). The initial aircraft longitudinal velocity v x0 was assumed to be 72 m/s and the target longitudinal acceleration a∗x was about 2 m/s2 . And we also



T

simulated the braking process without the input of δr∗ M t∗ to compare with the pilot control. The simulation results of the above-described braking process are shown in Fig. 5. Fig. 5(a)–(d) show the effect of pilot control, including aircraft movement, longitudinal velocity v x , direction angle θ , and slip angle of the C.G β . Fig. 5(e)–(f) show the control input, including the rudder angle δr and additional yaw moment M t . As shown in Fig. 5(a) and (b), the longitudinal velocity decreased as expected, regardless of the existence of pilot control. However, the aircraft veered off the runway without control input (Fig. 5(a)), while the pilot could always keep it on the runway during the braking process. It can be seen in Fig. 5(c) and (d) that the direction angle and sideslip angle of the C.G were always within a narrow range. Therefore, a sideslip never occurred and the pilot felt that the aircraft could be stably handled. However, as shown in Fig. 5(e) and (f), at the end of the braking process, the rudder angle output reached the maximum value and additional yaw moment increased. When aircraft braking occurred on a split road with a high adhesion coefficient, the observed results were quite dangerous. Fig. 6(a)–(d) show the simulation results with the same pilot control; in this case, the split road conditions were μl = 0.9 and μr = 0.3. As shown in Fig. 6(a) and (b), an aircraft without control veered off the runway after only 500 m of taxiing. Sideslip occurred in ∼10 s when the longitudinal velocity reduced rapidly. In

contrast, pilot control kept the aircraft on the runway for a longer distance and sideslip did not occur for a longer period of time. Although pilot control exhibited good performance at first, it still failed to keep the aircraft stable in the second half of the braking process. As shown in Fig. 6(a), the aircraft veered off the runway after about 1100 m of taxiing. Fig. 6(c) and (d) show that both the direction angle and sideslip angle of the pilot-controlled aircraft increased quickly after about 23 s. Finally, the sideslip angle of the C.G reduced to a value lesser than −5◦ , which indicates a loss of stability.

3.4. Analysis of pilot control based on the MOCM model

The results in Fig. 5(a)–(f) show the effectiveness of pilot control on aircraft braking after landing when the difference between the two sides of a split road is low. However, as shown in Fig. 6(a)–(d), the effectiveness of pilot control is limited. It could not maintain the stability of the aircraft for the overall braking process when the difference between the two sides of the split road was high. This limited effectiveness of pilot control can be mainly attributed to its inherent characteristics. Because longitudinal velocity reduces continuously during braking, it is difficult for a pilot to predict the lateral dynamics of the aircraft. The MOCM pilot model takes this factor into account and optimization is carried out according to the infinite LQR process. According to the simulation results, the pilot model couldn’t deal with the situation, when velocity is decreasing. In the following sections, a more effective controller is described to solve this problem.

6

Y. Dai et al. / Aerospace Science and Technology 95 (2019) 105482

Fig. 6. Pilot-controlled and uncontrolled results on a split road of angle of the C.G.

μl = 0.9 and μr = 0.3. (a) Aircraft movement, (b) longitudinal velocity, (c) direction angle, and (d) sideslip

x(t ) =

⎡ A=⎣

 Fig. 7. Structure of the aircraft ground braking system with the designed optimization controller.





K f +Kr mV x aK f −bK r Iz V x

aK f −bK r − Vx mV x a2 K f +b2 K r Iz V x

r Sr ρ − C δ2m V x2

C r δ S r br ρ 2I z

u (t ) =

and

0

δr (t ) M t (t ) ⎤ ⎦

 (22)

and



(23)

1 Iz

V x2

where K f and K r represent the cornering stiffness of the nose wheel and main wheels respectively. V x is the reference longitudinal velocity in the present control period. As nose wheels usually do not provide braking force, V x could be understood to refer to the rotational speed of the nose wheel.

4. Receding horizon LQR controller design 4.1. General controller design concept From our analysis of pilot control, it can be inferred that due to variations in longitudinal velocity, the form of the state space is not constant. Therefore, the optimization process of the controller we propose was not set in an infinite time domain, but in a finite time domain. Finite-time control is always considered in control problems corresponding to individual or multiple aircrafts and flight vehicles [30–32]. In the area of vehicle dynamic control, the LQ theory has been used to eliminate lateral offset [33]. To overcome the shortages of pilot control, we designed a receding horizon LQR controller in the finite time domain. As shown in Fig. 7, with the control objective T (k) at time step k, the optimization controller would output a target rudder angle δr∗ and an additional yaw moment M t∗ . In this case, we used the finite LQR method to design the optimization controller. As described in the introduction section, the main objective of the proposed controller is to keep the aircraft near the center line with limited deviation in the direction angle and ensure stability during the braking process. The detailed description of T (k) can be found in Section 5.3. 4.2. Reference model for controller design Based on the aircraft-on-ground model in Section 2 and the similarity between road vehicle and aircraft-on-ground problems, we adopted the linear 2-DOF vehicle model as the reference model for our controller design. This model can be described as follows,

x˙ (t ) = Ax(t ) + Bu (t )

B=

v y (t ) ωz (t )

(21)

4.3. State space equation The augmented reference model is given by

x˙ e (t ) = A e xe (t ) + B e u e (t )



xe (t ) = v y

⎡ ⎢ ⎢ ⎢ Ae = ⎢ ⎢ ⎣ ⎡ ⎢ ⎢

Be = ⎢ ⎢

⎣

ωz y θ y int

K f +Kr mV x aK f −bK r Iz V x

aK f −bK r − Vx mV x a2 K f +b2 K r Iz V x

1 0 0

0 1 0

r Sr ρ V x2 − C δ2m

C r δ S r br ρ 2I z

0 0 0

T

V x2

0

⎤

⎥ ⎥ ⎥ 0 ⎥ 0 ⎦

1 Iz

and

u e (t ) =

⎤

0

0

0

0 0 0 1

0 Vx 0 0

0⎥ ⎥ ⎥ 0⎥ ⎦ 0 0

δr (t ) M t (t )



(24) (25)

⎥ and

(26)

0

where y indicates lateral movement (integral of v y ) and θ is the yaw angle, which is the integral of the yaw moment ωz . Particularly, a new state variable y int is introduced and it is an integral of y. Thus, with the feedback y int , we expect a small steady-state error. Therefore, state space in discrete time can be denoted as follows,

Y. Dai et al. / Aerospace Science and Technology 95 (2019) 105482

xd (k + 1) = A d xd (k) + B d ud (k)

(27)

where k is the time series index. xd and ud represent the discrete time state vector and input, respectively.



xd (k) = v y (k)

ωz (k) y (k) θ(k) y int (k)

 δr∗ (k) , ∗ M t (k)



ud (k) =

Ad = I + T s A e ,

T

, Bd = T s Be

and

(28)

Besides, the observation vector (output) is given by:

zd (k) =

⎡



ωz (k) y (k) θ(k) y int (k)

0

1

0

0

0

⎤

T

= C d xd (k) and

⎢0 0 1 0 0⎥ ⎥ Cd = ⎢ ⎣0 0 0 1 0⎦ 0

0

0

0

(29)

1

This result is obtained after including variables that can be estimated by aircraft sensors.

7

4.6. Analysis of pilot control based on the MOCM model The results in Fig. 8(a)–(g) show that the LQR control we proposed performs better than the pilot control under similar conditions. However, as shown in Fig. 8(a), even if the aircraft is always within the runway limits under LQR control, the maximum lateral movement y is still as large as 5 m. This observation showed that it is not enough to consider only receding horizon optimization in the finite time domain. The main reason for this observation is that as longitudinal velocity reduces continuously, the further a predictive time step is, greater is the difference between the reference model and the actual situation. However, as shown in (31) and (32), the finite LQR control method provides a contrary recursion procedure to obtain the control law at present, which is based on the terminal constraint condition. Thus, the error of this method might increase when the state space varies continuously. To solve these problems, we proposed another controller as described in the next section. 5. Weighted MPC controller design

4.4. LQR controller 5.1. General design concept of the weighted MPC controller Therefore, in the finite time domain, we set the performance index as follows,

1

J L Q R = xdT ( H p ) Sxd ( H p ) 2

+

H p −1 1 

2



xdT (k) Q x xd (k) + udT (k) Rud (k)

(30)

i =0

where H p is the predictive step length. Therefore, the optimal control sequence is given by

ud (k) = − K k · xˆ d (k) Kk = R

−1 T

B

Pk = Q + A



T

P k−+11



+ BR

P k−+11

 −1 T −1

+ BR

B

(31) A

 −1 T −1 B

A,

(32)

(k = N p − 1 . . . 1, 0) (33)

where xˆ d (k) is the estimated state vector. K k is the feedback gain at time step k and is calculated using the recursion Ricatti equation in a discrete time domain as shown in (45) and (46). P N p = S is the terminal constraint condition. 4.5. Simulation of LQR control To compare LQR and pilot controls, in this section, we simulated the braking process under the same road conditions as those in Fig. 6 and the results are presented in Fig. 8(a)–(d). Fig. 8(e) and (f) illustrate the control input in each of these systems. Unlike the performance of pilot control, as shown in Fig. 8(a) and (b), the LQR controller always maintained a lateral movement y (no more than 5 m) within the road width. Furthermore, longitudinal velocity declined steadily as expected during the braking process. Fig. 8(c) and (d) show that the LQR controller maintained the direction angle and sideslip angle of the C.G within a narrow range, which provided good handling stability for the aircraft. As shown in Fig. 8(e) and (f), the LQR control input is lesser than the input of pilot control. While the rudder angle reached both its upper (+20◦ ) and lower (−20◦ ) limits with pilot control, the LQR controller ensured that these limits were not reached. Furthermore, the range of additional yaw moment was lesser in LQR control. Fig. 8(g) shows the slip rate of different main wheels; the LQR controller could maintain this rate within a safe range. In contrast, the slip rate increased rapidly when sideslip occurred under pilot control.

As described in the previous section in which the pilot control and finite LQR control were compared, variations in longitudinal velocity increased the gap between the reference model and actual state space, owing to which the control results were not efficient. Thus, during the optimization process, the controller should include the prediction of prior time sequences. And the optimal control should not be based on the terminal condition alone. For vehicle lateral control, the MPC method has been studied by a few researchers to solve path-following issues without braking operation [19]. Meanwhile, the predictive control method has been widely used for spacecraft and aircraft control. The study in [34] adopted MPC for spacecraft proximity operations with a persistent disturbance and the study in [35] designed an LQGbased MPC method for flying aircrafts with dynamic gust loads. In this study, we propose a novel weighted MPC method, which provides a large penalty weight coefficient based on the prior time sequence of the state vector. A new optimization controller was developed based on this method. The reason for choosing the MPC theory to design this controller is that MPC provides a global optimization process unlike the contrary recursion procedure of LQR. Furthermore, MPC is a practical method to set different penalties on predictive time sequences. Because the varying velocity would increase the reference model error in following predictive time sequences, the previous prediction steps are more important than the following ones, and need larger weight coefficient for penalties. Therefore, a varying weight coefficient function is proposed as follows,

W (k) =

M + e −k t Np · M +

H p

i =1

e −i t

(34)

where W (k) is the weight coefficient in time step k and thus  W (k) = 1. M ≥ 0 is an adjustable parameter. Therefore, the smaller M is, the larger is the prior weight, as shown in Fig. 9. The weight coefficient function can have different forms and the function we proposed can easily change the distribution of W (k) to fix larger or smaller prior weights by changing the parameter M. Referring to the study in [36], time-to-line crossing (TLC) metrics are introduced to compare the influence of different M values, which represents the time available for a pilot until the moment an aircraft reaches the runway boundary. The lowest

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Y. Dai et al. / Aerospace Science and Technology 95 (2019) 105482

Fig. 8. LQR and pilot control results on a split road of μl = 0.9 and μr = 0.3. (a) Aircraft movement, (b) longitudinal velocity, (c) direction angle, (d) sideslip angle of the C.G, (e) rudder angle, (f) additional yaw moment, and (g) slip rate of main wheels. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

the boundary lane. For the MPC method with constant parameters, the TLC is much lower and even close to that of LQR method. Therefore, we choose the MPC parameters by varying function and set the weight parameter to be M = 0 for subsequent analysis. 5.2. Augmented state vector for MPC Based on the discrete time state space of LQR, the discrete increment of the state vector and control vector is given by:

xd (k + 1) = xd (k + 1) − xd (k) and ud (k + 1) = ud (k + 1) − ud (k)

(35)

Therefore, the recursion formula of zd (k) is Fig. 9. Weight coefficient allocation.

zd (k + 1) = zd (k) + C d A d xd (k) + C d B d ud (k) TLCs at different M values during each simulation test are shown in Fig. 10. As shown in Fig. 10, when M = 0, with the largest prior penalty weight, the lowest TLC can be obtained. The bigger the value of M, the lower the TLC, which means that an aircraft is inclined to cross

(36)

The augmented state space for MPC is given as follows:

xo (k + 1) = A o xo (k) + B o u o (k) where xo is given by:

(37)

Y. Dai et al. / Aerospace Science and Technology 95 (2019) 105482

9

Fig. 10. TLC comparison of different weight coefficients.





xd (k) , u o (k) = ud (k), zd (k)  



Ad 0 Bd , and B o = Ao =

The penalty matrices of state vector and control vector sequences are given by:

xo (k) =

C d Ad

1

(38)

Cd Bd



Q = S TQ SQ

⎡



Therefore, the observational vector is zd (k) = 0 1 xo (k).

⎢ 0 =⎢ ⎣ ...

5.3. Prediction & optimization

Y (k) = F xo (k) + E U (k)

(39)

⎡ ⎤ u o (k) ⎢ u o (k + 1) ⎥ ⎢ ⎥ and U (k) = ⎢ u o (k + 2) ⎥ ⎣. . . ⎦ u o (k + H c − 1)

⎤

zo (k + 1 | k) ⎢zo (k + 2 | k) ⎥ ⎢ ⎥ Y (k) = ⎢ zo (k + 3 | k) ⎥

⎣. . . ⎦ zo (k + H p | k)

⎡

⎤

⎣ ...

⎥ ⎥ ⎥ ⎥ ⎦

C o Ao ⎢ C o A o2 ⎢ 3 F =⎢ ⎢ C o Ao

(40)

λu1 ⎢ 0 R = S TR SR = ⎢ ⎣ ... 0

⎢ ⎢ E =⎢ ⎢ ⎣

C o Ao

Co Bo C o Ao B o C o A o2 B o

...

H p −1 C o Ao Bo

⎤T ⎡

0

λ y1

⎥ ⎢ 0 ⎥ ⎢ ... ⎦ ⎣ ... λyHp 0 ⎤T 0 ... 0 λu2 . . . 0 ⎥ ⎥ ... ... ... ⎦ 0 . . . λu H c

λ y2 ... 0

⎡

λu1 ⎢ 0 ⎢ ⎣ ... 0

... ... ... ...

0 0

⎤

⎥ ⎥ (44) ... ⎦ λyHp ⎤ 0 ... 0 λu2 . . . 0 ⎥ ⎥ ... ... ... ⎦ 0 . . . λu H c (45)

where λ yi is the penalty matrix multiplied by the weight coefficient at time step i, which is given by:

λTyi · λ yi = W (k) · Q y (k = 1, 2, . . . , H p )

(46)

Then the cost function can be written as



  S Q ( E U (k) − ε (k)) 2   J M P C (k) =   S R U (k)

(47)

where ε (k) = T (k) − F xˆ o (k) represents the error and xˆ 0 (k) is the state vector estimation. Thus, the target optimal control sequence, which is also the solution of this optimization problem is given by:

and

Hp

⎡

0

⎡

where Y (k) is the prediction sequence at time step k. U (k) is the control sequence of the next H c time steps.

0 0

... ... ... ...

λ y2 ...

0

From the above analysis, at time step k, the prediction equation is given by

⎡

0

λ y1

0 Co Bo C o Ao B o

...

H p −2 C o Ao Bo

... ... ... ... ...

0 0 0

...

H p −Hc C o Ao Bo

⎡

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(41)

The optimization problem can be described as follows:



2



2

min J M P C (k) = Y (k) − T (k) Q −  U (k) R

(42)

where T (k) is the control objective at time step k. Consider the controller design concept in Section 4.1; in this case, T (k) can be written as follows:

⎡

⎤

⎡

⎣ ...

⎦

⎣ ... ⎦

zd (k + 1 | k) ⎢ zd (k + 2 | k) ⎢ T (k) = ⎢ zd (k + 3 | k) zd (k + H p | k)

⎤

0 ⎥ ⎢0 ⎥ ⎥ ⎥ ⎢ ⎥=⎢0 ⎥ 0

U (k) =

SQ E SR

−1

SQ 0

⎤ u o (k)∗ ∗  ⎢ u o (k + 1) ⎥ ⎢ ⎥ ε(k) = ⎢ uo (k + 2)∗ ⎥ ⎣ ... ⎦ ∗ u o (k + H c − 1)

(48)

The control vector at time step k is u o (k)∗ = u o (k − 1) + ∗ ∗ u  can obtain the target control u o (k) =  o (k) ∗. Thus, we δr (k) M t (k)∗ . To show the difference between pilot, LQR, and MPC controls clearly, MPC simulation results will be compared the results of the first two methods in the next section. 6. Simulation results and comparison 6.1. Simulation comparison of MPC, LQR, and MOCM

(43)

Fig. 11(a)–(g) show simulation results under the same road conditions for MPC, LQR, and MOCM controls. As shown in Fig. 11(a) and (b), from the beginning to the end of the braking process, the

10

Y. Dai et al. / Aerospace Science and Technology 95 (2019) 105482

Fig. 11. Comparison of MPC, LQR, and pilot control results obtained on a split road with μl = 0.9 and μr = 0.3. (a) Aircraft movement, (b) longitudinal velocity, (c) direction angle, (d) sideslip angle of the C.G, (e) rudder angle, (f) additional yaw moment, and (g) slip rate of main wheels.

MPC controller maintained a lateral movement y much smaller than the pilot and LQR controls. Meanwhile, the braking effectiveness of MPC and LQR methods was nearly the same. Fig. 11(c) and (d) show that MPC resulted in a greater stability than the pilot and LQR control methods with an extremely small direction angle deviation and sideslip angle. Fig. 11(e) and (f) show that the MPCcontrolled rudder angle was in the same range as that obtained with LQR, while the additional yaw moment obtained with MPC control was much smaller. As shown in Fig. 11(g), the slip rate of the main wheels was always within the safety range in both LQR and MPC methods.

A successful controller design depends on the fitness of the characteristics of the controlled objects. In this study, we observed that aircraft braking dynamics were affected not only by asymmetric conditions but also by the continuously declining velocity. Therefore, we initially took the declining longitudinal velocity into account and designed an infinite LQR controller. Later, after considering the large deviation between actual and predicted values, we designed a weighted MPC controller with different penalty weights as shown in Equation (34). The simulation results in Fig. 8(a)–(g) and Fig. 11(a)–(g) prove the validity of our proposed designs. 7. Conclusion

6.2. Analysis of controller design and results From Fig. 5(a)–(f) and Fig. 6(a)–(d), we could deduce that pilot control was effective only under slightly lateral asymmetric conditions. Meanwhile, as shown in Fig. 8(a)–(g) and Fig. 11(a)–(g), both LQR and weighted MPC controllers provided good lateral dynamic stability under aircraft braking conditions. However, the weighted MPC controller restrained shift in the lateral movement and the direction and sideslip angles to extremely small values and hence resulted in a better performance.

In this study, two different receding horizon optimization dynamic control strategies have been proposed to control the aircraft braking process under asymmetric conditions, which are not conducive for efficient pilot control. Unlike a conventional feedback control, the control strategies proposed in this study consider the declining longitudinal velocity and large deviations between the reference model and real aircraft conditions. To evaluate the effectiveness of the proposed control strategies, we conducted simulations on a 6-DOF three wheel aircraft

Y. Dai et al. / Aerospace Science and Technology 95 (2019) 105482

model with a MF tire model, which was previously proved to be highly accurate. The pilot control method employs the widely utilized MOCM model. As shown in the simulation results, pilot control was effective only when the difference between two sides of a split road was low (μl = 0.9 and μr = 0.7). When the adhesion coefficient of one side was 0.3 (μr = 0.3), the aircraft experienced sideslip. Compared with the pilot control model, both LQR and weighted MPC control strategies exhibited better performance; further, the weighted MPC controller resulted in the best lateral dynamic stability. Meanwhile, the control output of the rudder angle was always within the safety range with LQR and MPC controls; however, it reached both the upper and lower limits with pilot control. In our future work, we will employ models to recognize the pilot’s intentions to study the pilot-machine cooperation. Therefore, the controller is expected to correct the control output from the pilot based on the predicted aircraft dynamics and the pilot’s intentions. Furthermore, nose-wheel control should also be included, to enhance the mobility under braking and steering condition at medium and lower speed. Declaration of competing interest None declared. Acknowledgements The authors would like to thank for support from the National Science Foundation of China (Grant No. 51775293, No. U1664263) and the State Key Laboratory of Automotive Safety and Energy (Grant No. ZZ2019-042). References [1] T. Yong, A parameterized mathematical model of aircraft turning ground maneuvers, Bol. Téc. (ISSN 0376-723X) 55 (12) (2017). [2] M. Huang, H. Nie, M. Zhang, Analysis of ground handling characteristic of aircraft with electric taxi system, Proc. Inst. Mech. Eng., Part D, J. Automob. Eng. (2018) 0954407018764163. [3] J. Biannic, A. Marcos, M. Jeanneau, C. Roos, Nonlinear simplified LFT modelling of an aircraft on ground, in: Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control, 2006 IEEE, IEEE, 2006, pp. 2213–2218. [4] J. Duprez, F. Mora-Camino, F. Villaume, Aircraft-on-ground lateral control for low speed maneuvers, IFAC Proc. Vol. 37 (6) (2004) 475–480. [5] J. Duprez, F. Mora-Camino, F. Villaume, Control of the aircraft-on-ground lateral motion during low speed roll and manoeuvers, in: Aerospace Conference, 2004. Proceedings, vol. 4, 2004 IEEE, IEEE, 2004, pp. 2656–2666. [6] P.D. Khapane, Simulation of asymmetric landing and typical ground maneuvers for large transport aircraft, Aerosp. Sci. Technol. 7 (8) (2003) 611–619. [7] C.E. Lan, R.C. Chang, Unsteady aerodynamic effects in landing operation of transport aircraft and controllability with fuzzy-logic dynamic inversion, Aerosp. Sci. Technol. 78 (2018) 354–363. [8] B. Chen, Z. Jiao, S.S. Ge, Aircraft-on-ground path following control by dynamical adaptive backstepping, Chin. J. Aeronaut. 26 (3) (2013) 668–675. [9] J. Duprez, F. Mora-Camino, F. Villaume, Robust control of the aircraft on ground lateral motion, in: Proceedings of ICAS “24th international congress”, Yokohama, September 2004, 2004. [10] Z. Huang, M.C. Best, J.A. Knowles, Numerical investigation of aircraft high-speed runway exit using generalized optimal control, in: 2018 AIAA Guidance, Navigation, and Control Conference, 2018, p. 0879. [11] Y. Zhang, H. Duan, A directional control system for UCAV automatic takeoff roll, Aircr. Eng. Aerosp. Technol. 85 (1) (2013) 48–61.

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