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Abstract

The rolling control of a car that focuses on reducing the roll angle passively has limited performance of increasing handling stability, passing speed, ride comfort, and rollover prevention while turning. This project presents a method for controlling an automobile to tilt toward the turning direction using active suspension. A 6-degree-of-freedom vehicle model with a 2-degree-of-freedom steering model and a 4-degree-of-freedom tilting model is established. The active tilt sliding mode controller, which causes zero steady-state tilt angle error, is established after the desired tilt angle is determined by dynamic analysis. Simulation results confirm the effectiveness of the control method. The proposed controller reduces the perceived lateral acceleration and the lateral load transfer rate, thereby effectively improving handling stability, ride comfort, and vehicle speed, meanwhile decreasing the possibility of rollover while turning.

Keywords

Automobile active tilt active suspension sliding mode control perceived lateral acceleration lateral load transfer rate

Introduction

Traditional passive suspension (PS) prevents vehicle roll by enhancing suspension stiffness and installing a stabilizer bar but decreases vehicle ride comfort and is unable to prevent rolling and improve ride comfort simultaneously. Active or semi-active roll control can solve this contradiction. A control method performs active or semi-active control by installing an actuator in series with a stabilizer bar,^1,2 and another method conducts attitude control on the vehicle body to control rolling through active or semi-active suspensions.^3–6 Active or semi-active suspensions can also be used to control rollover; however, this control strategy is not the same as the anti-roll control when the vehicle travels normally.⁷ In addition, hydraulically interconnected suspensions also have a good effect on anti-roll.⁸ Vehicle roll deteriorates ride comfort and handling stability and even leads to rollover when turning. However, these roll controls mostly focus on reducing the roll angle or causing zero roll angle at most.^9,10 In the case of quasi-static rollover, the rollover threshold of a rigid car without roll can be improved at most by approximately 10% compared with a normal vehicle considering suspension and tire deformation. Thus, the effectiveness of these roll controls mentioned above is limited in terms of improving vehicle performance when steering.

A vehicle will roll outwards due to the centrifugal force while turning. However, if the vehicle can tilt inwards similar to a bicycle, this reverse tilt can cause the moment generated by gravity to offset that generated by the centrifugal force, and therefore reduce the lateral load transfer rate (LTR) and the lateral acceleration experienced by passengers even to zero. Reverse tilt can extremely improve handling stability, ride comfort, vehicle speed, and rollover prevention when turning, and improve maneuverability and agility of manipulation. Meanwhile, reverse tilt can not only improve performance while turning but also control the attitude of the vehicle and reduce the transversal load transfer. Moreover, the transversal load transfer on the front or rear axles can be changed when actively tilting the front or rear suspension alone. Active tilting technology¹¹ was applied on railway vehicles in the early 1980s and started to be widely used in high-speed trains after decades of research and development. The speed of the train is increased without changing existing track lines and infrastructure.¹² Another application of active tilt control is the narrow commuter vehicle, the width of which is only half that of the common car. This vehicle, which has aroused great interest among researchers in recent decades, is suitable for driving in urban areas. The vehicle needs the active tilt technique to ensure ride stability because it is narrow and has a high center of gravity (c.g.). Some researchers have developed some prototype vehicles.^13–15

Research on automobile active tilt control started fairly late, and only few studies can be found. Wang and Shen¹⁶ explained the feasibility of actively tilting the vehicle inwards via active suspension. Considering vehicle ride comfort, a robust H-infinity active tilt integrated controller was designed. The simulation results showed that lateral acceleration can be reduced via effective active tilt control. Taking zero acceleration as the control target, Youn et al.¹⁷ designed an attitude-tracking controller to control vehicle attitude by active tilt control that is based on active suspension. The simulation results verified the feasibility of the controller. The automobile active tilt control needs further research to find some more efficient control strategies and evaluation methods, as well as further explanation for the significance of active tilt control.

This article presents a sliding mode controller (SMC) for controlling automobile to tilt toward the turning direction via active suspension. The perceived lateral acceleration and the LTR are proposed for evaluating and explaining the significance of this active tilt control.

Vehicle active tilt model based on active suspension

A 6-degree-of-freedom (DOF) vehicle model with a 2-DOF steering model and 4-DOF tilt model aims to study the dynamics of active tilt during steering. The established vehicle model is shown in Figure 1. From D’Alembert’s principle and referring to the work of Jin et al.,¹⁸ the equations of the vehicle model are as follows:

Vehicle steering dynamics equation

m \overset{\cdot\cdot}{y} + m \overset{\cdot}{ψ} v + m_{s} h \overset{\cdot\cdot}{θ} = F_{f} + F_{r}

(1)

I_{z} \overset{\cdot\cdot}{ψ} = a F_{f} - b F_{r}

(2)

Dynamics equation of sprung mass vertical movement

m_{s} \overset{\cdot\cdot}{z} = F_{s 1} + F_{s 2} + f_{1} + f_{2}

(3)

Dynamics equation of vehicle tilt movement

I_{x} \overset{\cdot\cdot}{θ} = (F_{s 1} - F_{s 2}) d - m_{s} (\overset{\cdot\cdot}{y} + \overset{\cdot}{ψ} v) h + m_{s} g h θ + M_{t}

(4)

Dynamics equation of unsprung mass vertical movement

m_{u 1} {\overset{\cdot\cdot}{z}}_{u 1} = - F_{s 1} - k_{t 1} (z_{u 1} - z_{r 1}) - f_{1}

(5)

m_{u 2} {\overset{\cdot\cdot}{z}}_{u 2} = - F_{s 2} - k_{t 2} (z_{u 1} - z_{r 2}) - f_{2}

(6)

where

\begin{matrix} F_{f} = 2 c_{f} (δ - \frac{\overset{\cdot}{y} + a \overset{\cdot}{ψ}}{v}) \\ F_{r} = 2 c_{r} (- \frac{\overset{\cdot}{y} - b \overset{\cdot}{ψ}}{v}) \\ F_{s 1} = - k_{s 1} (z_{s 1} - z_{u 1}) - c_{s 1} ({\overset{\cdot}{z}}_{s 1} - {\overset{\cdot}{z}}_{u_{1}}) \\ F_{s 2} = - k_{s 2} (z_{s 2} - z_{u 2}) - c_{s 2} ({\overset{\cdot}{z}}_{s 2} - {\overset{\cdot}{z}}_{u_{2}}) \\ z_{s 1} = z + d θ, z_{s 2} = z - d θ \\ {\overset{\cdot}{z}}_{s 1} = \overset{\cdot}{z} + d \overset{\cdot}{θ}, {\overset{\cdot}{z}}_{s 2} = \overset{\cdot}{z} - d \overset{\cdot}{θ} \\ M_{t} = (f_{1} - f_{2}) d \end{matrix}

where $m$ is the vehicle total mass, $m_{s}$ is the vehicle sprung mass, $m_{u 1}$ and $m_{u 2}$ are the vehicle unsprung mass of the left and right suspensions, respectively, $z$ is the vertical displacement of the c.g. of the sprung mass, $I_{x}$ is the roll moment of inertia about the roll axis, $I_{z}$ is the yaw moment of inertia about the yaw axis, $a$ and $b$ represent the distance from the c.g. to the front and rear axles, respectively, $k_{s 1}$ and $k_{s 2}$ represent the spring stiffness coefficients of the left and right suspensions, respectively, $c_{s 1}$ and $c_{s 2}$ represent the constant damping coefficient of the left and right suspensions, respectively, $k_{t 1}$ and $k_{t 2}$ represent the vertical stiffness of the left and right tires, respectively, $θ$ is the tilt angle of the sprung mass, $F_{f}$ and $F_{r}$ represent the lateral force in the front and rear tires, respectively, $v$ is the longitudinal speed of the vehicle, $z_{u 1}$ and $z_{u 2}$ are the vertical displacements of the unsprung mass of the left and right suspensions, respectively, $z_{r 1}$ and $z_{r 2}$ represent the road input on the left and right wheels, respectively, $f_{1}$ and $f_{2}$ represent the controllable damping forces of the left and right suspensions, respectively, $M_{t}$ is the tilt control moment of active suspension, $d$ is the width from the c.g. of the vehicle to the left or right wheels, $h$ is the distance from the c.g. to the roll center, $ψ$ is the yaw angle, $δ$ is the steering angle of the front wheel, and $c_{f}$ and $c_{r}$ are the cornering stiffness of the front and rear tires, respectively.

Figure 1.

Steer–tilt dynamic model of the full vehicle: (a) lateral and yaw movement and (b) tilt and vertical movement.

Design of sliding mode variable structure controller

Desired tilt angle for active tilt control

When a vehicle is turning, it is tilted inward actively to cause the vehicle body gravity generating roll moments which can offset the moments generated by the centrifugal force. Thus, the value of the lateral acceleration experienced by passengers is equal or close to zero. The roll moment generated by centrifugal force is

M_{steer} = m_{s} (\overset{\cdot\cdot}{y} + v \overset{\cdot}{ψ}) h \cos θ

(7)

The roll moment generated by gravity is

M_{G} = m_{s} gh \sin θ

(8)

When the vehicle is in the steady state, $\overset{\cdot\cdot}{y} = 0$ , $\overset{\cdot}{ψ} = {\overset{\cdot}{ψ}}_{des} = δ (v / (a + b))$ . ${\overset{\cdot}{ψ}}_{des}$ is the yaw rate of the 2-DOF ideal steering model (in neutral steer). Then, the desired tilt angle for the active tilt can be obtained via $M_{steer} = M_{G}$

θ_{des} = \arctan (\frac{v {\overset{\cdot}{ψ}}_{des}}{g})

(9)

The lateral acceleration experienced by passengers is mainly composed of three parts, namely, lateral acceleration $a_{y}$ , component of gravitational acceleration, and tilting acceleration of the vehicle body, and can be represented as

a_{per} = a_{y} \cos θ + h \overset{\cdot\cdot}{θ} - g \sin θ

(10)

where $a_{y} = \overset{\cdot\cdot}{y} + v \overset{\cdot}{ψ}$ .

LTR is a commonly used index for predicting non-tripping rollover of vehicles and can be represented as¹⁸

LTR = \frac{m_{s} a_{y} h + I_{x} \overset{\cdot\cdot}{θ} - m_{s} gh θ}{m_{s} gd}

(11)

Controller design

The SMC technique is adopted to control the active suspension to generate actuating torque that will allow the actual active tilt angle to track the desired tilt angle.

This control system is called the single-input single-output (SISO) system. The system input is $u = M_{t}$ , the output is $w = θ$ , the state vector is $x = [y, \overset{\cdot}{y}, ψ, \overset{\cdot}{ψ}, θ, \overset{\cdot}{θ}, z, \overset{\cdot}{z}, z_{u 1}, {\overset{\cdot}{z}}_{u 1}, z_{u 2}, {\overset{\cdot}{z}}_{u 2}]^{T}$ , and the system disturbance is $u_{r} = {[δ, z_{r 1}, z_{r 2}]}^{T}$ .

The sliding surface $s$ is defined as

s = C e

where $e = {[\overset{\cdot}{w} - {\overset{\cdot}{w}}_{d e s}, w - w_{d e s}, \int (w - w_{d e s})]}^{T}$ , $C = [1, 2 λ, λ^{2}]$ , $λ$ is a positive real number, and $w_{des}$ is the desired system output. Thus

s = (\overset{\cdot}{w} - {\overset{\cdot}{w}}_{des}) + 2 λ (w - w_{des}) + λ^{2} \int (w - w_{des}) dt

(12)

In order to let the sliding motion become stable asymptotically and have a good dynamic quality, the pole placement method is used to obtain the sliding surface. All the roots of the characteristic equation $C (s) = s^{2} + 2 λ s + λ^{2}$ should be on the left-half $s$ plane.

By taking the derivative of the sliding surface $s$ , one can obtain

\begin{matrix} \overset{\cdot}{s} = \overset{\cdot\cdot}{w} + 2 λ (\overset{\cdot}{w} - {\overset{\cdot}{w}}_{des}) + λ^{2} (w - w_{des}) = \overset{\cdot\cdot}{θ} + 2 λ (\overset{\cdot}{θ} - {\overset{\cdot}{θ}}_{des}) \\ + λ^{2} (θ - θ_{des}) \\ = \frac{f}{I_{x}} + \frac{u}{I_{x}} + 2 λ (\overset{\cdot}{θ} - {\overset{\cdot}{θ}}_{des}) + λ^{2} (θ - θ_{des}) \end{matrix}

(13)

where $f = m_{s} gh θ + (F_{s 1} - F_{s 2}) d - m_{s} (\overset{\cdot\cdot}{y} + \overset{\cdot}{ψ} v) h$ and $θ_{des}$ is obtained via equation (9).

To improve the dynamic quality of the sliding mode, the exponential reaching law is chosen as

\overset{\cdot}{s} = - η s - k sgn (s)

(14)

where $η$ is the control gain, $k$ is the uncertainty of the control system, and both of them are positive constants. $k$ represents the reaching speed of approaching the sliding surface, which should be chosen properly. If the value of $k$ is big, then the reaching speed is fast but the chattering is large, and if the value of $k$ is small, then the reaching speed is slow. Thus, the value of $η$ should be reduced while increasing the value of $k$ to ensure fast reaching and reduce chattering simultaneously.

The equivalent moment $u_{eq}$ generated by active suspension on the vehicle active tilt situation can be obtained via equations (13) and (14)

\begin{matrix} \overset{\cdot}{s} = \frac{f}{I_{x}} + \frac{u_{eq}}{I_{x}} + 2 λ (\overset{\cdot}{θ} - {\overset{\cdot}{θ}}_{des}) + λ^{2} (θ - θ_{des}) \\ = - η [\overset{\cdot}{θ} + 2 λ (θ - θ_{des}) + λ^{2} \int (θ - θ_{des}) dt] - k sgn (s) \end{matrix}

The steady-state value for ${\overset{\cdot}{θ}}_{des}$ is zero, and thus

\begin{matrix} u_{eq} = - f - (2 λ I_{x} + η I_{x}) \overset{\cdot}{θ} - (λ^{2} I_{x} + 2 λ η I_{x}) (θ - θ_{des}) \\ - η λ^{2} I_{x} \int (θ - θ_{des}) dt - k I_{x} sgn (s) \end{matrix}

(15)

Figure 2 shows the block diagram of the active tilt SMC system.

Figure 2.

Block diagram of the active tilt control system.

Simulation results and analyses

A numerical simulation is performed using MATLAB/Simulink to verify the performance of the designed controller. The parameters of the vehicle model are listed in Table 1.

Table 1.

Parameters of the steer–tilt model.

Parameters	Values	Parameters	Values
$m_{s} (kg)$	1500	$m_{u 1} = m_{u 2} (kg)$	120
$I_{z} (kg m^{2})$	2500	$k_{s 1} = k_{s 2} (N / m)$	35,000
$I_{x} (kg m^{2})$	460	$c_{s 1} = c_{s 2} (N s / m)$	2000
$k_{t 1} = k_{t 2} (N / m)$	380,000	$g (m / s^{2})$	9.81
$a (m)$	1.4	$h (m)$	0.45
$b (m)$	1.7	$d (m)$	0.74
$c_{f} (N / rad)$	76,339	$c_{r} (N / rad)$	70,351

The vehicle is traveling in a straight line on a smooth road at a longitudinal speed of 80 km/h without considering the exogenous disturbances of vertical road vibration. The simulation time is 20 s, and the sampling time is 0.01 s. The simulation is performed while using 100% and 70% of the desired tilt angles as the control targets when the vehicle is given a step steer input. The vehicle is turned at t = 5 s, and the steering angle reaches its final value $δ = 1^{\circ}$ at t = 10 s. The front wheel steering angle is shown in Figure 3.

Figure 3.

Front wheel steering angle.

The response curves of the tilt or roll angle of the vehicle body are shown in Figure 4, where SMC, PS, and Desired represent the sliding mode active tilt control, PS, and desired tilt angle, respectively. The SMC can track the desired tilt angle well as its tilt angle overlaps the desired values. The tilt angle of the vehicle body with respect to active tilt control tilts inward to approximately 15°, which is opposite to the PS case. Figure 5 shows the response curve of the lateral acceleration experienced by passengers. The steady-state value of the lateral acceleration in PS is 3.3 m/s². Meanwhile, in the SMC case, the lateral acceleration initially has a small fluctuation and then approaches zero as the torque generated by gravity offsets that by the centrifugal force. The SMC significantly improves lateral stability and ride comfort. The response curve of the LTR of the vehicle is shown in Figure 6. The LTR value of the PS system is 0.21. Meanwhile, the LTR value in the SMC case is similar to that of the lateral acceleration experienced by passengers. A small fluctuation initially occurs, and then the LTR value approaches zero. The active tilt control reduces the LTR, decreases the possibility of rollover, and can increase speed while cornering.

Figure 4.

Roll or tilt angle of the vehicle body.

Figure 5.

Perceived lateral acceleration.

Figure 6.

Lateral load transfer rate (LTR).

Taking 100% of the desired tilt angle θ_des as the control target

Figures 7 and 8 show the control forces of the left and right active suspensions, respectively. The control forces are still within the allowable limits as the force of each actuator is half the control force on each side after the control forces are decomposed in four suspension actuators. Figure 9 shows the time response curve of the vehicle yaw rate, and the vehicle slip angle response curve is shown in Figure 10. In Figure 9, Ideal denotes the ideal yaw rate, ${\overset{\cdot}{ψ}}_{des}$ . As shown in the two figures, the yaw rate in the SMC case overlaps the one in the Ideal case, while the yaw rate in the PS is lower than that in Ideal. Although the direction of slip angle in SMC is opposite to that in PS, the slip angle in SMC is smaller than the one in PS. Thus, the SMC improves vehicle handling stability. Figures 11 and 12 show the time response curves of the suspension deflections. The deflections in the SMC are larger than those in the PS but are still below the bounds of available suspension deflections as the suspension travel of a typical passenger vehicle is approximately 0.25 m. The direction of suspension deflections in SMC is opposite to that in PS, thereby illustrating the actual situation of the active tilt control and the PS.

Figure 7.

Control force of the left suspension.

Figure 8.

Control force of the right suspension.

Figure 9.

Yaw rate.

Figure 10.

Vehicle slip angle.

Figure 11.

Left suspension deflection.

Figure 12.

Right suspension deflection.

Taking 70% of the desired tilt angle θ_des as the control target

The values of the desired tilt angle can be decreased properly within the acceptable range of LTR and perceived lateral acceleration experienced by passengers so as to reduce the actuator forces and energy consumption of the active suspension as well as reduce the suspension dynamic deflections. A total of 70% of the desired tilt angle is chosen as the active tilt control objective. Figure 13 shows the time response curve of the tilt or roll angle of the vehicle body. The tilt angle in SMC and the desired value overlap; thus, the tilt angle of the vehicle body in the active tilt control is consistent with the desired value. In total, 70% of the desired tilt angle, which is approximately 10°, is lower than that in the 100% of the desired tilt angle. Figure 14 shows the steady-state values of the lateral acceleration experienced by passengers. The lateral acceleration in the SMC is larger than the SMC’s acceleration in the 100% of the desired tilt angle case shown in Figure 5. However, the lateral acceleration in SMC, the amplitude of which is only 18% of the acceleration in PS, is still considerably less than that in PS. The steady-state values of the LTR are shown in Figure 15. The LTR in SMC is larger than that in the 100% of the desired tilt angle case shown in Figure 6. However, the LTR in SMC, the amplitude of which is only 25% of the LTR in PS, is still considerably less than that in PS.

Figure 13.

Roll or tilt angle of the vehicle body.

Figure 14.

Perceived lateral acceleration.

Figure 15.

LTR.

The control forces of the left and right vehicle active suspensions are shown in Figures 16 and 17, respectively. The forces are reduced to approximately 20% compared with that in the 100% of the desired tilt angle.

Figure 16.

Control force of the left suspension.

Figure 17.

Control force of the right suspension.

Figures 18 and 19 show the time response curve of the yaw rate and the vehicle slip angle response curve, respectively. Compared with the yaw rate in PS, the yaw rate in SMC is closer to the one in the Ideal case. And the amplitude of the slip angle in SMC is much smaller than the one in PS. Therefore, the SMC can achieve better turning performances. Figures 20 and 21 show the time response of the suspension deflections. The values of suspension deflections in SMC are reduced to approximately 30% and are significantly lower than those in the 100% of the desired tilt angle, as shown in Figures 11 and 12.

Figure 18.

Yaw rate.

Figure 19.

Vehicle slip angle.

Figure 20.

Left suspension deflection.

Figure 21.

Right suspension deflection.

Conclusion

The proposed SMC for active suspension can actively tilt a vehicle inward to decrease the lateral acceleration experienced by passengers and LTR when the vehicle is steering, thereby improving handling stability, ride comfort, and traveling speed and preventing rollover effectively. In addition, the desired tilt angle and actuator force of the active suspension can be decreased according to the requirements of different driving conditions, thereby potentially reducing energy consumption.

Footnotes

Handling Editor: Muhammad Akhtar

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by the “Summit of the Six Top Talents” Program of Jiangsu Province, China (grant no. JXQC-005), the Natural Science Foundation of Jiangsu Province (grant no. BK20181403), and the China Scholarship Council (CSC; file no. 201808320051).

ORCID iD

Jialing Yao

References

Gaspar

Szaszi

Bokor

Reconfigurable control structure to prevent the rollover of heavy vehicles. Control Eng Pract 2005; 13: 699–711.

Zhang

Han

Zhong

et al . Output feedback anti-roll robust H∞ control for heavy trucks subject to parameter uncertainties. Automot Eng 2005; 13: 699–711 (in Chinese).

Chen

Zong

Zhang

et al . Research on integrated control strategy of ride and roll attitude via active suspension. J Jilin Univ 2011; 41: 59–64 (in Chinese).

Dong

Liao

et al . Analysis of improvement on vehicle roll stability using magneto-rheological semi-active suspension system. J Shanghai Jiaotong Univ 2009; 43: 1541–1544 (in Chinese).

Yao

Ren

et al . Control of the lateral stability based on roll moment distribution via semi-active suspension. Automot Eng 2016; 38: 974–980+988 (in Chinese).

Xuan

Song

Yang

et al . Magneto-rheological damper model and roll stability control strategy. J Jilin Univ 2009; 39: 43–46 (in Chinese).

Yim

Park

Design of active suspension and electronic stability program for rollover prevention. Int J Automot Techn 2010; 11: 147–153.

Zhang

Zheng

MY.

Effects of HIS structure parameter variation on its stiffness and damping. Comput Simul 2014; 31: 151–155 (in Chinese).

Ikenaga

Lewis

Campos

et al . Active suspension control of ground vehicle based on a full-vehicle model. In: Proceedings of the American control conference, Chicago, IL, 28–30 June 2000, pp.4019–4024. New York: IEEE.

10.

Youn

Tomizuka

Level and attitude control of the active suspension system with integral and derivative action. Vehicle Syst Dyn 2006; 44: 659–674.

11.

Goodall

Tilting trains and beyond—the future for active railway suspensions. Part 1: improving passenger comfort. Comput Control Eng J 1999; 10: 153–160.

12.

Halikias

Goodall

Zolotas

AC.

Recent results in tilt control design and assessment of high-speed railway vehicles. Proc IMechE, Part F: J Rail and Rapid Transit 2007; 221: 291–312.

13.

Piyabongkarn

Keviczky

Rajamani

Active direct tilt control for stability enhancement of a narrow commuter vehicle. Int J Automot Techn 2004; 5: 77–88.

14.

Tan

JTC

Huang

Suda

et al . Cornering stability improvement by gyro moment for narrow tilting vehicle. J Mech Sci Technol 2015; 29: 2705–2711.

15.

Robertson

Darling

Plummer

AR.

Combined steering-direct tilt control for the enhancement of narrow tilting vehicle stability. Proc IMechE, Part D: J Automobile Engineering 2014; 228: 847–862.

16.

Wang

Shen

Integrated vehicle ride and roll control via active suspensions. Vehicle Syst Dyn 2008; 46: 495–508.

17.

Youn

et al . Attitude motion control of the active suspension system with tracking controller. Int J Automot Techn 2015; 16: 593–601.

18.

Jin

Zhang

et al . Stability and optimised H∞ control of tripped and untripped vehicle rollover. Vehicle Syst Dyn 2016; 54: 1405–1427.

Automobile active tilt control based on active suspension

Abstract

Keywords

Introduction

Vehicle active tilt model based on active suspension

Design of sliding mode variable structure controller

Desired tilt angle for active tilt control

Controller design

Simulation results and analyses

Taking 100% of the desired tilt angle θdes as the control target

Taking 70% of the desired tilt angle θdes as the control target

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Taking 100% of the desired tilt angle θ_des as the control target

Taking 70% of the desired tilt angle θ_des as the control target