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Robust roll motion control of a vehicle using integrated control strategy
Control Engineering Practice 19 (2011) 820–827
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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Robust roll motion control of a vehicle using integrated control strategy Hyo-Jun Kim n Department of Mechanical Engineering, Kangwon National University, KyoDong, Samcheok, KangwonDo 245-711, Republic of Korea
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 March 2010 Accepted 4 April 2011 Available online 7 May 2011
This paper presents an electrically actuated roll motion control of a vehicle using simulation and experimental analysis. The controller is designed with an HN control scheme based on the 3 DOF vehicle model considering parameter variations, which affect the roll dynamics. To investigate the feasibility of the active roll control system, its performance is evaluated by simulation in a full vehicle model under various conditions. The Hil setup with the electrically actuated roll control system was devised and its performance was investigated through experimental works. Finally, to enhance the performance in a transient region, an integrated control strategy is presented. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Roll motion Robust control Hil setup Integrated control
1. Introduction In general, vehicle design presents a trade-off between performance and safety. Design parameters affecting lateral dynamics can influence maneuvering ability, but also have additional influence on dynamic stability including spinout and rollover. In a steering maneuver, vertical loads on the outer track of tires increase while those on the inner track decrease. This is referred to as lateral load transfer resulting from body roll motion. Roll stability is one of the dynamic factors that needs to be considered in a ground vehicle. Geometric dimensions, suspension characteristics as well as maneuvering conditions influence the dynamic roll behavior of a car. This is also an issue for railroad vehicles with a high center of gravity. To improve the roll characteristics of a car, the customary approach is to increase the roll stiffness using a stabilizer bar. However, this method has limited effectiveness due to additional stiffness that affects ride comfort. (For example, high roll disturbance from a single tire rolling over a bump.) In order to enhance vehicle performance in an active manner, advanced suspension systems, such as active suspension system, have been widely analyzed in the literature over the years as typical works of Hrovat (1993), Yoshimura, Nakaminami, Kurimoto, and Hino (1999) and Li and Goodall (1999). Despite its performance, commercialized systems are mostly semi-active type system due to practical requirements that include simplicity, cost, etc. However, a weakness of the semi-active system is that it cannot generate an active force for improvement of the diverse behaviors of a car.
n
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0967-0661/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2011.04.009
Currently, one of the most efficient methods that are being further developed is the roll control option in conjunction with a semi-active suspension system. The first function of this system is the use of a variable damper system to isolate the driver from uneven roadway noise and to improve road grip on irregular road surfaces. The second function is to provide safer steering using the active roll control system. This provides many of the benefits of a fully active system at a much reduced cost and less power consumption. Unfortunately, most studies have focused on the active or semi-active suspension system such as Yagiz and Hacioglu (2008) and Poussot-Vassal et al. (2008), and there have been few studies of the ARC (active roll control) system or its combined system. Lin, Cebon, and Cole (1996) performed a theoretical study of active roll reduction in heavy vehicles. Their system used an antiroll bar equipped with a hydraulic linear actuator, which provided the necessary torque to counteract the roll moment of the car body using a lateral acceleration feedback and an LQR control scheme. Darling and Ross-Martin (1997) performed a simulation study of the ARC system for passenger cars with a hydraulic rotary actuator and lateral acceleration feedback control. On a practical basis, the ARC system based on a hydraulically actuated system is too complicated to implement due to the many hydraulic components including pipe laying. This also leads to the possibility of contamination caused by oil leakage. In addition, as is the case with any vehicle system, an actual car is expected to operate in a variable environment. For instance, parameter variations resulting from loading pattern and driving conditions will influence vehicle dynamics. This raises questions about the robustness of the control system, which means that the controller must cope with these uncertainties successfully. In this paper, the active roll motion controller for passenger cars is designed in the framework of the output feedback HN
H.-J. Kim / Control Engineering Practice 19 (2011) 820–827
control scheme, taking into account the roll dynamic responses caused by parameter variations. In order to investigate its performance, simulation work is performed based on the full vehicle model with nonlinear tire characteristics considering loading condition, driving speed and maneuvering disturbance. To implement the control system in a car, an electrically actuated roll control system is devised. Control performance is evaluated using the Hil (hardware-in-the-loop) setup. Considering the characteristics of the actuating system related to dynamic and structural configuration, an integrated control method with a variable damping system is presented for further enhancement of control performance.
2. Dynamic model of a vehicle In general, for developing the active controller, it is not desirable to use a complex vehicle model due to difficulties in the implementation of the control system. In this paper, the linear three degrees of freedom vehicle model (Kim and Park, 2002) is used for the design of the controller. Fig. 1 shows a vehicle model, including the yaw and roll dynamics of a car, related to driver steering maneuvers while traveling on a road surface at a constant speed V with the tire steering angle di (i¼1y4). The coordinates xyz are a vehicle-fixed frame relative to XYZ of an earth fixed coordinates system. The vehicle is assumed to be symmetrical in the x–z plane, the tire characteristics and road conditions for the left and right tires are the same. In a conventional front steer vehicle, the tire has a steering angle df (df ¼ d1 ¼ d2) at the front only. In Fig. 1(a) of a yaw plane representation, the side slip angle b, at the center of gravity in a car body, is expressed in Eq. (1) when a vehicle is rotating at an angular velocity g relative to the inertial frame v b ¼ tan1 y ð1Þ vx where vx and vy are velocity components of V in x and y directions, respectively. Where the slip angle b is assumed to be small and if the magnitudes of 9tfg/29, 9trg/29 are small, tire slip angles at front and rear, expressed as bf ¼ b1 ¼ b2 and br ¼ b3 ¼ b4, have the following
matrix form if roll angle effects are considered: "
2
3
l
f # " # " # bf af 61 V 7 b df 6 7 f ¼4 br lr 5 g ar 0 1
ð2Þ
V
where af and ar are the fixed coefficients of roll effects. In practice, the lateral tire forces Yf and Yr are assumed to be linear functions with slip angles bf and br in general operating range, Eq. (3) is used Yf ¼ 2Kf bf ,
Yr ¼ 2Kr br
ð3Þ
where Kf and Kr are the cornering stiffness for the front and rear tires, respectively. In Fig. 1(b) of the roll plane representation, the car body has a roll motion with roll angle f and roll angular velocity p relative to the roll center (RC). Taking into consideration the previous forces and moments, equations of motion including lateral, yaw and roll motion are written as lf g lr g MVðb_ þ gÞ þ Ms hs p_ ¼ 2Kf df þ af fb þ 2Kr ar fb þ V V ð4Þ lf g lr g lf 2Kr ar fb þ lr Iz g_ ¼ 2Kf df þ af fb V V
ð5Þ
Ix p_ þMs hs Vðb_ þ gÞ ¼ Kf fCf p
ð6Þ
where Kf is roll stiffness and Cf is roll damping. From Eqs. (4)–(6), the state-space representation can be expressed as Eq. (7) Ex_ ¼ Ao x þBo u þLo df
ð7Þ
where u is an active roll moment to reduce the roll response resulting from steering disturbance df. The measured output equation (if the measured output variable is the car body roll angle) is written as Eq. (8) y ¼ Cp x þ Dp u
ð8Þ
where h x¼ v
g f p
iT
2
ðKf þKr Þ 6 V 6 6 6 ðKf lf Kr lr Þ Ao ¼ 6 6 V 6 6 0 4 0
Kf lf Kr lr MV þ V V
ðKf af þKr ar Þ
ðKf l2f Kr l2r Þ V 0
ðKf af lf Kr ar lr Þ
Ms hs V
Kf
h L o ¼ Kf
Bo ¼ 0
0
0
1
T
,
Cp ¼ 0
0
1
0
T
,Dp ¼ ½0
0 Iz
0 0
0
1
0
0
2
Fig. 1. Handling characteristics model. (a) Top view and (b) front view.
821
M 6 6 0 E¼6 6 0 4 Ms hs
3 Ms hs 7 0 7 7 0 7 5 Ix
Kf lf
0
0
0
iT
,
3 0 7 7 7 7 0 7 7 7 1 7 5 Cf
822
H.-J. Kim / Control Engineering Practice 19 (2011) 820–827
3. Design of active roll controller 3.1. ARC controller This section presents the design of an active roll motion controller in the framework of a general output feedback HN control scheme with configuration of Fig. 2 taking into account the additive modeling error resulting from parameter variations of a vehicle. The additional actuator dynamics and saturation are not considered in this section. As shown in Fig. 2 of control system configuration, the plant transfer function Pu(s) and disturbance transfer function Pw(s) based on the vehicle model in Section 2 are expressed as Pu ðsÞ ¼ Cp ðsIAp Þ1 Bp
ð9Þ
Pw ðsÞ ¼ Cp ðsIAp Þ1 Lp
ð10Þ
where, Ap ¼ E1 A0 , Bp ¼ E1 B0 and Lp ¼ E1 L0 Considering the input–output relationships with the steering disturbance at tire wd, noise to sensor n, controlled output z1, z2, control input u, the controller K(s) and the weighting functions Wa(s), Ws(s), Wn(s) yield 2 3 " # " # 0 0 Wa ðsÞ zðsÞ wðsÞ 6 W ðsÞP ðsÞ 0 Ws ðsÞPu ðsÞ 7 ¼4 s ð11Þ w 5 yðsÞ uðsÞ Wn ðsÞ Pu ðsÞ Pw ðsÞ where, w ¼ wd T z ¼ z1 z2
n
defined as 2 3 " # Aa 0 0 02x2 6 0 A BC 7 s s p 5, A¼4 B1 ¼ L p 0 , 0 0 Ap 2 3 " # Ba Ca 0 0 6B D 7 B2 ¼ 4 s p 5, C1 ¼ 0 C D C s s p Bp " D11 ¼ ½02x2 ,
D12 ¼
D22 ¼ Dp ,
h C2 ¼ 0
#
Da Ds Dp 0
D21 ¼ 0
,
Cp
i
In order to solve the mixed-sensitivity problem, the following procedures are carried out. The transfer function M(s) from disturbance to output is written as MðsÞ ¼ ðI þ Pu ðsÞKðsÞÞ1 Pw ðsÞ
Assuming that Wn(s) is the measurement noise with scalar Nw, the weighting functions Wa(s), Ws(s) can be expressed as statespace forms, respectively
ð17Þ
In order to suppress the roll motion caused by steering disturbance, the weighting function Ws(s) is properly chosen to satisfy Eq. (18) :Ws ðsÞMðsÞ:1 o1
ð18Þ
The transfer function N(s) from disturbance to control input is written as NðsÞ ¼ KðsÞðI þPu ðsÞKðsÞÞ1 Pw ðsÞ
T
Nw ,
ð19Þ
To guarantee robust stability relative to the additive modeling error resulting from parameter variations, the condition of Eq. (20) is to be satisfied. :Wa ðsÞNðsÞ:1 o 1
ð20Þ
Wa ðsÞ ¼ Da þ Ca ðsIAa Þ1 Ba
ð12Þ
where the weighting function Wa(s) should be properly chosen to match the following condition:
Ws ðsÞ ¼ Ds þ Cs ðsIAs Þ1 Bs
ð13Þ
smax fDa ðjwÞg r9Wa ðjwÞ9, 8w
As is well known, the first step in designing an HN controller is the formulation of the generalized plant. By rearranging Eq. (11), the state-space representations describing the generalized plant are _ ¼ AX þ B1 w þ B2 u X
ð14Þ
z ¼ C1 X þ D11 w þ D12 u
ð15Þ
y ¼ C2 X þD21 wþ D22 u
ð16Þ
where, state vector X includes xa, xs and xp. xp is the plant state vector, xs and xa are state vectors of weighting function Ws(s) and Wa(s), respectively. The matrices in Eqs. (14)–(16) are
Fig. 2. Configuration of control system.
ð21Þ
where Da(s) is additive modeling error. The magnitude and shape of the weighting functions Ws(s) and Wa(s) in Eqs. (18) and (20) have an influence on the characteristics of the controller K(s). Therefore, these weighting functions are design factors that need to be selected properly. 3.2. Numerical simulation In order to evaluate the performance of the ARC system in a car, simulation work is performed using the full vehicle model with nonlinear tire characteristics. The vehicle model, based on Kim and Park (2002), includes bounce, pitch and roll dynamics of sprungmass and longitudinal, lateral and yaw motions. This vehicle model has parameter values with sprungmass 1011 kg, roll moment of inertia 440 kg m2 and yaw moment of inertia 2400 kg m2. The wheel–axle (unsprungmass) dynamics were described in the vertical plane while the suspension characteristics were modeled as linear components with stiffness kf ¼ 10,947 N/m, kr ¼14,559 N/m and damping cf ¼ 526 N s/m, cr ¼925 N s/m. This vehicle model has ten degrees of freedom having dimensions with tf ¼ 1.51 m, tr ¼1.48 m, lf ¼1.13 m, lr ¼1.44 m, roll steer af ¼.2, ar ¼ .2 where subscript f means the front and r means the rear. Tire forces were calculated using a nonlinear tire model presented by Bakker, Pacejka, and Linder (1989). Avoiding the complexity of tire characteristics combined with braking and cornering, in this study, the side force caused by steering maneuver is considered without any brake pedal operations. In this case, the side force is proportional to slip angle if the slip angle is small, thus the cornering stiffness coefficient in Section 2
H.-J. Kim / Control Engineering Practice 19 (2011) 820–827
6000
823
-50
4000
f d a Magnitude (dB)
Tire force (N)
2000 0 -2000
b
-100
e c
-150
-4000 -6000
-5
-4
-3
-2
-1
0
1
2
3
4
-200
5
100
Slip angle (deg)
0:00031ðs2 þ 2s þ1Þ s2 þ 1:3s þ 36
ð22Þ
745000ðsþ 1Þ s4 þ 38:4s3 þ 1828:5s2 þ19814s þ 331776
ð23Þ
The controller K(s) is determined using a mixed-sensitivity approach, which satisfies Eqs. (18) and (20) in Section 3.1 with the chosen weighting functions. Thus the active roll moment was calculated by the aforementioned controller, based on the 3DOF model, using the feedback roll angle signal induced by a driver steering disturbance. The simulation is performed under general operating conditions taking into account loading conditions, handling of driver steering wheel and legal driving speed limits. Fig. 5 shows simulation results for the roll angle and rollrate under the steering maneuver of single period sine input with magnitude of 3.51 and the period of 0.8 s at a constant vehicle speed of 50 km/h. This maneuver seems like an obstacle avoidance operation. The perturbed vehicle parameters of sprungmass, roll moment of inertia and yaw moment of inertia are adopted to add 20% of nominal value considering loading conditions.
50
Ws (s)
0 Magnitude (dB)
can be defined as shown in Fig. 3. As well known, this value varies with operating conditions, therefore these parameter variations are also considered when selecting weighting functions. Fig. 4 shows the determined weighting functions. These weighting functions are selected on the basis of parameter sensitivity analysis and simulation work considering performance and robustness. The deliberation of Wa(s) includes not only definite parameter variations of the vehicle, but also variable parameters (tire characteristics, driving forward speed, etc.) through operation. The modeling error in Fig. 4(a) reflects each perturbed parameter in Table 1 with respect to the nominal vehicle values. As shown in Fig. 4(b), Wa(s) is selected as Eq. (22) that has robustness margin to modeling uncertainties and covers the entire modeling error induced by parameter variations. It also has effective magnitude in high frequency ranges considering the unmodeled higher frequency dynamics such as wheel hop mode. Performance degradation resulting from too large a weighting for robust stability is also avoided through simulation works. When the bandwidth of the weighting function Ws(s) is too wide, the control performance declines. Thus Ws(s) is chosen in Eq. (23), to have sufficient magnitude in the low frequency range including the roll mode frequency of the vehicle
Ws ðsÞ ¼
102
Frequency (rad/sec)
Fig. 3. Tire characteristics (o: non-linear tire model and line: fitted linear model).
Wa ðsÞ ¼
101
-50
Wa (s)
-100
Wn (s)
-150 100
101
102
Frequency (rad/sec) Fig. 4. Weighting functions and modeling error. (a) Modeling error and (b) weighting functions.
Table 1 Parameter range. Index
Parameter
Range
a b c d e f
Mass Roll moment of inertia Yaw moment of inertia Roll damping Cornering stiffness Driving speed
20% 20% 20% 30% 50% 110 km/h
In Fig. 5(a) and (b), the controlled responses of the roll angle and the rollrate can be reduced both in nominal and perturbed parameter cases although the responses of roll motion are affected by parameter perturbation. Fig. 5(c) shows the roll mode phase portraits, which easily display dynamic roll behaviors. In the figure, the trajectories of the uncontrolled and the controlled case converge toward the equilibrium position after the steering disturbance. Control performance can be displayed clearly despite the roll mode phase trajectory with perturbed parameters being more widely spread. In order to investigate control effectiveness with respect to variable operating conditions including forward speed and
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H.-J. Kim / Control Engineering Practice 19 (2011) 820–827
0.4
0.06 0.05 0.03 0.02
Rollrate (rad/sec)
Roll Angle (rad)
0.3
c
a
0.04
d
0.01
b
0 -0.01 -0.02
c
0.2 d
0.1 b
0 -0.1 -0.2
-0.03 -0.04
a
-0.3 0
0.5
1
1.5 2 Time (sec)
2.5
0
3
0.5
1
1.5 2 Time (sec)
2.5
3
0.4
Rollrate (rad/sec)
0.3
c
0.2 d 0.1 a 0 -0.1
b
-0.2 -0.3 -0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Roll Angle (rad) Fig. 5. Comparison of roll characteristics for passive and active vehicle (a: uncontrolled-nominal, b: controlled-nominal, c: uncontrolled-perturbed, d: controlledperturbed). (a) Roll angle responses; (b) rollrate responses and (c) roll mode phase portraits.
0.1
50–100 km/h and steering input range 3.5–7.01. Both in uncontrolled and controlled case, the roll angle value has an upward tendency with higher speeds and steering magnitudes, as a result of an increase in lateral acceleration. The figure shows that roll motion can be suppressed by roll control options relative to the passive type vehicle. These results indicate that the roll response is improved and the desired control performance is achieved under variable parameter conditions.
0.09 Max. Roll Angle [rad]
0.08
a b
0.07
c
0.06
Uncontrolled
d
0.05
4. Experimental work
0.04 Controlled
0.03
a
0.02
b
c
d
0.01 50
55
60
65
70
75
80
85
90
95
100
Forward Speed [km/h] Fig. 6. Comparison of roll angle responses in variations of forward speed and steering input. (Steering input a: 3.51, b: 4.81, c: 6.21 and d: 71).
steering magnitude variations, a series of simulations are performed under the previous obstacle avoidance operations. Fig. 6 shows the comparison of maximum roll angle in the speed range
In order to evaluate the control performance of the designed active roll control system in a car, a series of experiments were performed using a Hil (hardware-in-the-loop) setup. Before the actual vehicle test with a heavy burden (time, cost, etc.), the Hil technique is an efficient way to realistically test the effectiveness of the control system in a laboratory. A schematic diagram of the experiment using the Hil setup is shown in Fig. 7 and a configuration of the ARC system module is represented in Fig. 8. The test work part is divided into two groups; in one group, the computational work parts including 10 DOF vehicle dynamics with tire characteristics as well as control logic. In the other group, hardware parts with sensors, data acquisition-interface and the ARC system module. The overall experimental processing loop, with a sampling time of 2 ms, is constructed as follows; as the driver
H.-J. Kim / Control Engineering Practice 19 (2011) 820–827
performs a steering operation, the dynamic behavior and tire force in a vehicle are simulated in the computer. After the active roll moment is calculated by the control logic using the feedback signal of the car body roll angle, the control signal corresponding to the desired active moment is transmitted to the motor driver through an interfacing board with a D/A converter. The actual moment, generated by the prototype ARC system, is measured by a force transducer and delivered to the computer through an interfacing board with an A/D converter. The actual active roll moment is exerted on the vehicle dynamic model. From previous work (Lin et al., 1996), the active roll control mechanism was devised with a hydraulic linear actuator connecting the stabilizer bar ends to the suspension tie rods. In this mechanism, to make the ARC system module respond faster, a stabilizer bar with higher stiffness is needed. However, if there is too much additional roll stiffness, it will spoil the ride comfort due to high roll excitation, which is caused when a bump is met by the wheels on one side of the car. To address this problem,
Darling and Hickson (1998) devised a hydraulic actuating system with a P-port closed proportional valve, which allows the actuator to free-wheel during straight line driving. On the basis of considerations of above issues and complexity of hydraulic system, in this study, an electrically actuated roll control system is devised, which would effectively remove the additional roll stiffness introduced by the ARC system module and thus improve the high frequency isolation of the car body during periods when the system is inoperative. The prototype ARC system module, as shown in Figs. 7 and 8, is implemented as a hardware component in the Hil setup (Kim, Yang, & Park, 2002). This system module includes the electrically actuated system and stabilizer bar fixed with bush on the frame. The ARC system module is constructed as a single unit because real-time roll moment distribution control problem between front and rear axle is not focused on in this study. The actuator part has a maximum force of 7200 N, a maximum velocity of 320 mm/s and the stiffness of the stabilizer bar is 5100 N m/rad. Fig. 9 shows the experimental results of frequency responses of the devised ARC system module. In this figure, the bandwidth of actuator under stalled condition is wider than ARC system module when moved condition, which is tested with 1 V sinusoidal command signal. This shows the practical characteristics of actuating module. Fig. 10 shows the test results of the roll responses through the previous Hil setup. The results in these figures were obtained at constant forward speed of 50 km/h with nominal parameters and a driver steering input as a J-turn maneuver having parameters with ho ¼0, h1 ¼3.51, to ¼0 s and t1 ¼ 0.2 s in the following equation: 2
t oto 6 6 t o t ot 6 o 1 4 t 4t1
Fig. 7. Schematic diagram of experiment using hardware-in-the-loop setup.
825
df ¼ ho df ¼ ho þ ðh1 ho Þ
2 oÞ 32 ðtðtt 1 to Þ
ðtto Þ ðt1 to Þ
df ¼ h1
As shown in Fig. 10(a), the resulting roll angle is reduced from 2.61 to 0.461 (max. value), which is about 17% of the level relative to the uncontrolled passive vehicle condition. Also in Fig. 10(b),
Fig. 8. Configuration of ARC system module.
40 Magnitude (dB)
ð24Þ
20 0 -20 -40 -60 -80 101
102 Frequency (rad/s)
Fig. 9. Frequency response of ARC system module. (Dashed: actuator with stalled condition and solid: actuating module with moved condition.)
826
H.-J. Kim / Control Engineering Practice 19 (2011) 820–827
construct a multi-loop control structure for force tracking such as the slow active suspension system case (Priyandoko, Mailah, & Jamaluddin, 2009). In this section, as an alternative, an integrated control strategy is presented in order to further enhance the roll response. In general, a variable damper is used for vibration control induced by uneven road input. The damping force of the variable damper is determined by a controllable damping coefficient as in the following rule: " f d ½fd ðx_ s x_ u Þ 4 0 _ _ Cd ¼ ðx s x u Þ ð25Þ ½fd ðx_ s x_ u Þ r0 Cmin
0.01
Roll Angle (rad)
0 -0.01 -0.02 -0.03 -0.04 -0.05 0
0.5
1
1.5 Time (sec)
2
2.5
3
0.15
Rollrate (rad/sec)
0.1
Md ¼
2 X
Mj
ð26Þ
j¼1
0.05
Mi ¼ Fli
0 -0.05 -0.1 -0.15 -0.2
If Cd 4Cmax, then Cd ¼Cmax. Where Cd is desired damping coefficient, Cmax and Cmin are the maximum, minimum damping coefficients of the variable damper. fd is the desired force for _ xuÞ _ is relative velocity between car suppression of vibrations, ðxs body and wheel at each corner. From the previous work (Kim and Park, 2002), considering the roll motion control in a steering maneuver, the desired damping force at each damper on roll control can be calculated by the following relationships:
0
0.5
1
1.5 Time (sec)
2
2.5
3
Fig. 10. Dynamic roll responses. (a) Roll angle responses (dotted: uncontrolled and solid: controlled); (b) rollrate responses (dotted: uncontrolled and solid: controlled).
the rollrate can be reduced. In these figures, the responses of roll angle and rollrate are affected by the characteristics of the ARC system module in Fig. 8, caused by the dynamic and structural configuration. Including the global roll motion related to the dynamic response of the actuating module in Fig. 9, a dead-zone like behavior induced by the supporting fixture (friction, deflection of rubber insert) of the stabilizer bar in the module affects a minute control region related to oscillations near steady-state body roll position. These results imply that further improvement is needed, even though the roll behavior can be improved by the devised ARC system with respect to uncontrolled case
ti t Fri i 2 2
ðFli þ Fri ¼ 0Þ
ði ¼ 1,2Þ
where Md is the total desired roll moment, Mj is the moment component of subscript j with the front and rear axles, Fli and Fri are the desired forces of the left and right sides of the axle, ti is the track width with subscript i of left and right. The desired damping coefficient can then be determined by Eq. (25). The integrated control strategy is based on the ARC system as an option of the vehicle equipped with the semi-active type suspension system. The variable damper system is controlled with the devised ARC system simultaneously based on the structure of Fig. 11. This control structure was proposed in a previous study (Kim and Park, 2002) in the framework of lateral acceleration feedback control scheme. The results showed that this method could improve roll responses. In this section, on the basis of these studies, the integrated control is adopted in order to improve roll behavior without any extra equipment and any additional multi-loop control logic to the devised ARC system module. Fig. 12 illustrates the resulting roll angle responses, related to Fig. 10, with respect to passive, active roll control and integrated control cases, respectively. In the figure, the integrated control can give a smoother response even though there is no further improvement in the steady-state region. It results from the operating characteristics of a semi-active type actuator which cannot generate an active force. Fig. 13 shows the roll mode phase portrait with respect to passive, active roll control and
5. Integrated control strategy In general, the controlled responses are affected by the characteristics of an actuating system. Typically, the bandwidth of the actuating system causes some degradations of control performance (Darling, Dorey, & Ross-Martin, 1992). Previous experimental results in Section 4 are also affected by ARC system module characteristics. In order to improve on this problem, a high response actuating system or mechanism can be used. From a practical point of view, this is more costly. Another method is to
ð27Þ
Fig. 11. Schematic diagram of integrated control strategy.
H.-J. Kim / Control Engineering Practice 19 (2011) 820–827
0.01
6. Conclusion
Roll Angle (rad)
0 -0.01 -0.02 -0.03 -0.04 -0.05
0
0.5
1
1.5 Time (sec)
2
2.5
3
Fig. 12. Comparison of roll responses in J-turn maneuver. (Dashed: uncontrolled, dotted: ARC only and solid: integrated control.)
0.15 0.1 Roll Angular Vel. (rad/sec)
827
0.05 0 -0.05 -0.1 -0.15 -0.2 -0.045 -0.04 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 Roll Angle (rad)
0
Fig. 13. Comparison of roll mode phase portraits. (Dashed: uncontrolled, dotted: ARC only and solid: integrated control.)
integrated control case, respectively. This figure shows that control performance can be achieved both in a steady-state and in a transient region with a smoother response with respect to uncontrolled cases.
In this paper, an investigation of an active roll motion control system has been presented. A designed controller, which has taken into account the influence of parameter perturbation, was applied to a full vehicle model with nonlinear tire characteristics. An enhancement in roll behavior under loading patterns, speed and maneuvering disturbance conditions was achieved in the simulation work. In order to evaluate the feasibility as an actual system in a car, the electrically actuated roll control system (having a high compliance to external road excitation during inoperative periods) was devised. Through experimental works using the hardware-in-the-loop technique, the control performance was investigated. Finally, as an option of a car equipped with a semi-active suspension system, the integrated control method has been presented and it has been shown that this control strategy is feasible for providing improved steadystate and transient behaviors with respect to passive vehicle conditions. References Bakker, E., Pacejka, H. B., & Linder, L. (1989). A new tire model with an application in vehicle dynamics studies. SAE paper 890087 (pp. 83–95). Darling, J., Dorey, R. E., & Ross-Martin, T. J. (1992). A low cost active anti-roll suspension for passenger cars. Transactions of the ASME, Journal of Dynamic Systems, Measurement and Control, 114, 599–605. Darling, J., & Hickson, L. R. (1998). An Investigation of a roll control suspension hydraulic actuation system. Transactions of the ASME, Fluid Power Systems and Technology, 5, 49–54. Darling, J., & Ross-Martin, T. J. (1997). A theoretical investigation of a prototype active roll control system. Proceedings of Institute of Mechanical Engineers, Part D, Journal of Automobile Engineering, 211, 3–12. Hrovat, D. (1993). Applications of optimal control to advanced automotive suspension design. Transactions of the ASME, Journal of Dynamic Systems, Measurement and Control, 328–342. Kim, H. J., Yang, H. S., & Park, Y. P. (2002). Robust roll control of a vehicle; experimental study using hardware-in-the-loop setup. Proceedings of Institute of Mechanical Engineers, Part D, Journal of Automobile Engineering, 216, 1–9. Kim, H. J., & Park, Y. P. (2002). Hybrid attitude control in steering maneuver using ARC Hil setup. Control Engineering Practice, 10(12), 1339–1345. Li, H., & Goodall, R. M. (1999). Linear and non-linear skyhook damping control laws for active railway suspensions. Control Engineering Practice, 7(7), 843–850. Lin, R. C., Cebon, D., & Cole, D. J. (1996). Optimal roll control of a single-unit lorry. Proceedings of Institute of Mechanical Engineers, Part D, Journal of Automobile Engineering, 210, 45–55. Poussot-Vassal, C., Sename, O., Dugard, L., Gaspar, P., Szabo, Z., & Bokor, J. (2008). A new semi-active suspension control strategy through LPV technique. Control Engineering Practice, 16(12), 1519–1534. Priyandoko, G., Mailah, M., & Jamaluddin, H. (2009). Vehicle active suspension system using skyhook adaptive neuro active force control. Mechanical Systems and Signal Processing, 23, 855–868. Yagiz, N., & Hacioglu, Y. (2008). Backstepping control of a vehicle with active suspensions. Control Engineering Practice, 16(12), 1457–1467. Yoshimura, T., Nakaminami, K., Kurimoto, M., & Hino, J. (1999). Active suspension of passenger cars using linear and fuzzy-logic controls. Control Engineering Practice, 7(1), 41–47.
Contents lists available at ScienceDirect
Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Robust roll motion control of a vehicle using integrated control strategy Hyo-Jun Kim n Department of Mechanical Engineering, Kangwon National University, KyoDong, Samcheok, KangwonDo 245-711, Republic of Korea
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 March 2010 Accepted 4 April 2011 Available online 7 May 2011
This paper presents an electrically actuated roll motion control of a vehicle using simulation and experimental analysis. The controller is designed with an HN control scheme based on the 3 DOF vehicle model considering parameter variations, which affect the roll dynamics. To investigate the feasibility of the active roll control system, its performance is evaluated by simulation in a full vehicle model under various conditions. The Hil setup with the electrically actuated roll control system was devised and its performance was investigated through experimental works. Finally, to enhance the performance in a transient region, an integrated control strategy is presented. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Roll motion Robust control Hil setup Integrated control
1. Introduction In general, vehicle design presents a trade-off between performance and safety. Design parameters affecting lateral dynamics can influence maneuvering ability, but also have additional influence on dynamic stability including spinout and rollover. In a steering maneuver, vertical loads on the outer track of tires increase while those on the inner track decrease. This is referred to as lateral load transfer resulting from body roll motion. Roll stability is one of the dynamic factors that needs to be considered in a ground vehicle. Geometric dimensions, suspension characteristics as well as maneuvering conditions influence the dynamic roll behavior of a car. This is also an issue for railroad vehicles with a high center of gravity. To improve the roll characteristics of a car, the customary approach is to increase the roll stiffness using a stabilizer bar. However, this method has limited effectiveness due to additional stiffness that affects ride comfort. (For example, high roll disturbance from a single tire rolling over a bump.) In order to enhance vehicle performance in an active manner, advanced suspension systems, such as active suspension system, have been widely analyzed in the literature over the years as typical works of Hrovat (1993), Yoshimura, Nakaminami, Kurimoto, and Hino (1999) and Li and Goodall (1999). Despite its performance, commercialized systems are mostly semi-active type system due to practical requirements that include simplicity, cost, etc. However, a weakness of the semi-active system is that it cannot generate an active force for improvement of the diverse behaviors of a car.
n
Tel.: þ82 33 570 6322; fax: þ82 33 574 2993. E-mail address: [email protected]
0967-0661/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2011.04.009
Currently, one of the most efficient methods that are being further developed is the roll control option in conjunction with a semi-active suspension system. The first function of this system is the use of a variable damper system to isolate the driver from uneven roadway noise and to improve road grip on irregular road surfaces. The second function is to provide safer steering using the active roll control system. This provides many of the benefits of a fully active system at a much reduced cost and less power consumption. Unfortunately, most studies have focused on the active or semi-active suspension system such as Yagiz and Hacioglu (2008) and Poussot-Vassal et al. (2008), and there have been few studies of the ARC (active roll control) system or its combined system. Lin, Cebon, and Cole (1996) performed a theoretical study of active roll reduction in heavy vehicles. Their system used an antiroll bar equipped with a hydraulic linear actuator, which provided the necessary torque to counteract the roll moment of the car body using a lateral acceleration feedback and an LQR control scheme. Darling and Ross-Martin (1997) performed a simulation study of the ARC system for passenger cars with a hydraulic rotary actuator and lateral acceleration feedback control. On a practical basis, the ARC system based on a hydraulically actuated system is too complicated to implement due to the many hydraulic components including pipe laying. This also leads to the possibility of contamination caused by oil leakage. In addition, as is the case with any vehicle system, an actual car is expected to operate in a variable environment. For instance, parameter variations resulting from loading pattern and driving conditions will influence vehicle dynamics. This raises questions about the robustness of the control system, which means that the controller must cope with these uncertainties successfully. In this paper, the active roll motion controller for passenger cars is designed in the framework of the output feedback HN
H.-J. Kim / Control Engineering Practice 19 (2011) 820–827
control scheme, taking into account the roll dynamic responses caused by parameter variations. In order to investigate its performance, simulation work is performed based on the full vehicle model with nonlinear tire characteristics considering loading condition, driving speed and maneuvering disturbance. To implement the control system in a car, an electrically actuated roll control system is devised. Control performance is evaluated using the Hil (hardware-in-the-loop) setup. Considering the characteristics of the actuating system related to dynamic and structural configuration, an integrated control method with a variable damping system is presented for further enhancement of control performance.
2. Dynamic model of a vehicle In general, for developing the active controller, it is not desirable to use a complex vehicle model due to difficulties in the implementation of the control system. In this paper, the linear three degrees of freedom vehicle model (Kim and Park, 2002) is used for the design of the controller. Fig. 1 shows a vehicle model, including the yaw and roll dynamics of a car, related to driver steering maneuvers while traveling on a road surface at a constant speed V with the tire steering angle di (i¼1y4). The coordinates xyz are a vehicle-fixed frame relative to XYZ of an earth fixed coordinates system. The vehicle is assumed to be symmetrical in the x–z plane, the tire characteristics and road conditions for the left and right tires are the same. In a conventional front steer vehicle, the tire has a steering angle df (df ¼ d1 ¼ d2) at the front only. In Fig. 1(a) of a yaw plane representation, the side slip angle b, at the center of gravity in a car body, is expressed in Eq. (1) when a vehicle is rotating at an angular velocity g relative to the inertial frame v b ¼ tan1 y ð1Þ vx where vx and vy are velocity components of V in x and y directions, respectively. Where the slip angle b is assumed to be small and if the magnitudes of 9tfg/29, 9trg/29 are small, tire slip angles at front and rear, expressed as bf ¼ b1 ¼ b2 and br ¼ b3 ¼ b4, have the following
matrix form if roll angle effects are considered: "
2
3
l
f # " # " # bf af 61 V 7 b df 6 7 f ¼4 br lr 5 g ar 0 1
ð2Þ
V
where af and ar are the fixed coefficients of roll effects. In practice, the lateral tire forces Yf and Yr are assumed to be linear functions with slip angles bf and br in general operating range, Eq. (3) is used Yf ¼ 2Kf bf ,
Yr ¼ 2Kr br
ð3Þ
where Kf and Kr are the cornering stiffness for the front and rear tires, respectively. In Fig. 1(b) of the roll plane representation, the car body has a roll motion with roll angle f and roll angular velocity p relative to the roll center (RC). Taking into consideration the previous forces and moments, equations of motion including lateral, yaw and roll motion are written as lf g lr g MVðb_ þ gÞ þ Ms hs p_ ¼ 2Kf df þ af fb þ 2Kr ar fb þ V V ð4Þ lf g lr g lf 2Kr ar fb þ lr Iz g_ ¼ 2Kf df þ af fb V V
ð5Þ
Ix p_ þMs hs Vðb_ þ gÞ ¼ Kf fCf p
ð6Þ
where Kf is roll stiffness and Cf is roll damping. From Eqs. (4)–(6), the state-space representation can be expressed as Eq. (7) Ex_ ¼ Ao x þBo u þLo df
ð7Þ
where u is an active roll moment to reduce the roll response resulting from steering disturbance df. The measured output equation (if the measured output variable is the car body roll angle) is written as Eq. (8) y ¼ Cp x þ Dp u
ð8Þ
where h x¼ v
g f p
iT
2
ðKf þKr Þ 6 V 6 6 6 ðKf lf Kr lr Þ Ao ¼ 6 6 V 6 6 0 4 0
Kf lf Kr lr MV þ V V
ðKf af þKr ar Þ
ðKf l2f Kr l2r Þ V 0
ðKf af lf Kr ar lr Þ
Ms hs V
Kf
h L o ¼ Kf
Bo ¼ 0
0
0
1
T
,
Cp ¼ 0
0
1
0
T
,Dp ¼ ½0
0 Iz
0 0
0
1
0
0
2
Fig. 1. Handling characteristics model. (a) Top view and (b) front view.
821
M 6 6 0 E¼6 6 0 4 Ms hs
3 Ms hs 7 0 7 7 0 7 5 Ix
Kf lf
0
0
0
iT
,
3 0 7 7 7 7 0 7 7 7 1 7 5 Cf
822
H.-J. Kim / Control Engineering Practice 19 (2011) 820–827
3. Design of active roll controller 3.1. ARC controller This section presents the design of an active roll motion controller in the framework of a general output feedback HN control scheme with configuration of Fig. 2 taking into account the additive modeling error resulting from parameter variations of a vehicle. The additional actuator dynamics and saturation are not considered in this section. As shown in Fig. 2 of control system configuration, the plant transfer function Pu(s) and disturbance transfer function Pw(s) based on the vehicle model in Section 2 are expressed as Pu ðsÞ ¼ Cp ðsIAp Þ1 Bp
ð9Þ
Pw ðsÞ ¼ Cp ðsIAp Þ1 Lp
ð10Þ
where, Ap ¼ E1 A0 , Bp ¼ E1 B0 and Lp ¼ E1 L0 Considering the input–output relationships with the steering disturbance at tire wd, noise to sensor n, controlled output z1, z2, control input u, the controller K(s) and the weighting functions Wa(s), Ws(s), Wn(s) yield 2 3 " # " # 0 0 Wa ðsÞ zðsÞ wðsÞ 6 W ðsÞP ðsÞ 0 Ws ðsÞPu ðsÞ 7 ¼4 s ð11Þ w 5 yðsÞ uðsÞ Wn ðsÞ Pu ðsÞ Pw ðsÞ where, w ¼ wd T z ¼ z1 z2
n
defined as 2 3 " # Aa 0 0 02x2 6 0 A BC 7 s s p 5, A¼4 B1 ¼ L p 0 , 0 0 Ap 2 3 " # Ba Ca 0 0 6B D 7 B2 ¼ 4 s p 5, C1 ¼ 0 C D C s s p Bp " D11 ¼ ½02x2 ,
D12 ¼
D22 ¼ Dp ,
h C2 ¼ 0
#
Da Ds Dp 0
D21 ¼ 0
,
Cp
i
In order to solve the mixed-sensitivity problem, the following procedures are carried out. The transfer function M(s) from disturbance to output is written as MðsÞ ¼ ðI þ Pu ðsÞKðsÞÞ1 Pw ðsÞ
Assuming that Wn(s) is the measurement noise with scalar Nw, the weighting functions Wa(s), Ws(s) can be expressed as statespace forms, respectively
ð17Þ
In order to suppress the roll motion caused by steering disturbance, the weighting function Ws(s) is properly chosen to satisfy Eq. (18) :Ws ðsÞMðsÞ:1 o1
ð18Þ
The transfer function N(s) from disturbance to control input is written as NðsÞ ¼ KðsÞðI þPu ðsÞKðsÞÞ1 Pw ðsÞ
T
Nw ,
ð19Þ
To guarantee robust stability relative to the additive modeling error resulting from parameter variations, the condition of Eq. (20) is to be satisfied. :Wa ðsÞNðsÞ:1 o 1
ð20Þ
Wa ðsÞ ¼ Da þ Ca ðsIAa Þ1 Ba
ð12Þ
where the weighting function Wa(s) should be properly chosen to match the following condition:
Ws ðsÞ ¼ Ds þ Cs ðsIAs Þ1 Bs
ð13Þ
smax fDa ðjwÞg r9Wa ðjwÞ9, 8w
As is well known, the first step in designing an HN controller is the formulation of the generalized plant. By rearranging Eq. (11), the state-space representations describing the generalized plant are _ ¼ AX þ B1 w þ B2 u X
ð14Þ
z ¼ C1 X þ D11 w þ D12 u
ð15Þ
y ¼ C2 X þD21 wþ D22 u
ð16Þ
where, state vector X includes xa, xs and xp. xp is the plant state vector, xs and xa are state vectors of weighting function Ws(s) and Wa(s), respectively. The matrices in Eqs. (14)–(16) are
Fig. 2. Configuration of control system.
ð21Þ
where Da(s) is additive modeling error. The magnitude and shape of the weighting functions Ws(s) and Wa(s) in Eqs. (18) and (20) have an influence on the characteristics of the controller K(s). Therefore, these weighting functions are design factors that need to be selected properly. 3.2. Numerical simulation In order to evaluate the performance of the ARC system in a car, simulation work is performed using the full vehicle model with nonlinear tire characteristics. The vehicle model, based on Kim and Park (2002), includes bounce, pitch and roll dynamics of sprungmass and longitudinal, lateral and yaw motions. This vehicle model has parameter values with sprungmass 1011 kg, roll moment of inertia 440 kg m2 and yaw moment of inertia 2400 kg m2. The wheel–axle (unsprungmass) dynamics were described in the vertical plane while the suspension characteristics were modeled as linear components with stiffness kf ¼ 10,947 N/m, kr ¼14,559 N/m and damping cf ¼ 526 N s/m, cr ¼925 N s/m. This vehicle model has ten degrees of freedom having dimensions with tf ¼ 1.51 m, tr ¼1.48 m, lf ¼1.13 m, lr ¼1.44 m, roll steer af ¼.2, ar ¼ .2 where subscript f means the front and r means the rear. Tire forces were calculated using a nonlinear tire model presented by Bakker, Pacejka, and Linder (1989). Avoiding the complexity of tire characteristics combined with braking and cornering, in this study, the side force caused by steering maneuver is considered without any brake pedal operations. In this case, the side force is proportional to slip angle if the slip angle is small, thus the cornering stiffness coefficient in Section 2
H.-J. Kim / Control Engineering Practice 19 (2011) 820–827
6000
823
-50
4000
f d a Magnitude (dB)
Tire force (N)
2000 0 -2000
b
-100
e c
-150
-4000 -6000
-5
-4
-3
-2
-1
0
1
2
3
4
-200
5
100
Slip angle (deg)
0:00031ðs2 þ 2s þ1Þ s2 þ 1:3s þ 36
ð22Þ
745000ðsþ 1Þ s4 þ 38:4s3 þ 1828:5s2 þ19814s þ 331776
ð23Þ
The controller K(s) is determined using a mixed-sensitivity approach, which satisfies Eqs. (18) and (20) in Section 3.1 with the chosen weighting functions. Thus the active roll moment was calculated by the aforementioned controller, based on the 3DOF model, using the feedback roll angle signal induced by a driver steering disturbance. The simulation is performed under general operating conditions taking into account loading conditions, handling of driver steering wheel and legal driving speed limits. Fig. 5 shows simulation results for the roll angle and rollrate under the steering maneuver of single period sine input with magnitude of 3.51 and the period of 0.8 s at a constant vehicle speed of 50 km/h. This maneuver seems like an obstacle avoidance operation. The perturbed vehicle parameters of sprungmass, roll moment of inertia and yaw moment of inertia are adopted to add 20% of nominal value considering loading conditions.
50
Ws (s)
0 Magnitude (dB)
can be defined as shown in Fig. 3. As well known, this value varies with operating conditions, therefore these parameter variations are also considered when selecting weighting functions. Fig. 4 shows the determined weighting functions. These weighting functions are selected on the basis of parameter sensitivity analysis and simulation work considering performance and robustness. The deliberation of Wa(s) includes not only definite parameter variations of the vehicle, but also variable parameters (tire characteristics, driving forward speed, etc.) through operation. The modeling error in Fig. 4(a) reflects each perturbed parameter in Table 1 with respect to the nominal vehicle values. As shown in Fig. 4(b), Wa(s) is selected as Eq. (22) that has robustness margin to modeling uncertainties and covers the entire modeling error induced by parameter variations. It also has effective magnitude in high frequency ranges considering the unmodeled higher frequency dynamics such as wheel hop mode. Performance degradation resulting from too large a weighting for robust stability is also avoided through simulation works. When the bandwidth of the weighting function Ws(s) is too wide, the control performance declines. Thus Ws(s) is chosen in Eq. (23), to have sufficient magnitude in the low frequency range including the roll mode frequency of the vehicle
Ws ðsÞ ¼
102
Frequency (rad/sec)
Fig. 3. Tire characteristics (o: non-linear tire model and line: fitted linear model).
Wa ðsÞ ¼
101
-50
Wa (s)
-100
Wn (s)
-150 100
101
102
Frequency (rad/sec) Fig. 4. Weighting functions and modeling error. (a) Modeling error and (b) weighting functions.
Table 1 Parameter range. Index
Parameter
Range
a b c d e f
Mass Roll moment of inertia Yaw moment of inertia Roll damping Cornering stiffness Driving speed
20% 20% 20% 30% 50% 110 km/h
In Fig. 5(a) and (b), the controlled responses of the roll angle and the rollrate can be reduced both in nominal and perturbed parameter cases although the responses of roll motion are affected by parameter perturbation. Fig. 5(c) shows the roll mode phase portraits, which easily display dynamic roll behaviors. In the figure, the trajectories of the uncontrolled and the controlled case converge toward the equilibrium position after the steering disturbance. Control performance can be displayed clearly despite the roll mode phase trajectory with perturbed parameters being more widely spread. In order to investigate control effectiveness with respect to variable operating conditions including forward speed and
824
H.-J. Kim / Control Engineering Practice 19 (2011) 820–827
0.4
0.06 0.05 0.03 0.02
Rollrate (rad/sec)
Roll Angle (rad)
0.3
c
a
0.04
d
0.01
b
0 -0.01 -0.02
c
0.2 d
0.1 b
0 -0.1 -0.2
-0.03 -0.04
a
-0.3 0
0.5
1
1.5 2 Time (sec)
2.5
0
3
0.5
1
1.5 2 Time (sec)
2.5
3
0.4
Rollrate (rad/sec)
0.3
c
0.2 d 0.1 a 0 -0.1
b
-0.2 -0.3 -0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Roll Angle (rad) Fig. 5. Comparison of roll characteristics for passive and active vehicle (a: uncontrolled-nominal, b: controlled-nominal, c: uncontrolled-perturbed, d: controlledperturbed). (a) Roll angle responses; (b) rollrate responses and (c) roll mode phase portraits.
0.1
50–100 km/h and steering input range 3.5–7.01. Both in uncontrolled and controlled case, the roll angle value has an upward tendency with higher speeds and steering magnitudes, as a result of an increase in lateral acceleration. The figure shows that roll motion can be suppressed by roll control options relative to the passive type vehicle. These results indicate that the roll response is improved and the desired control performance is achieved under variable parameter conditions.
0.09 Max. Roll Angle [rad]
0.08
a b
0.07
c
0.06
Uncontrolled
d
0.05
4. Experimental work
0.04 Controlled
0.03
a
0.02
b
c
d
0.01 50
55
60
65
70
75
80
85
90
95
100
Forward Speed [km/h] Fig. 6. Comparison of roll angle responses in variations of forward speed and steering input. (Steering input a: 3.51, b: 4.81, c: 6.21 and d: 71).
steering magnitude variations, a series of simulations are performed under the previous obstacle avoidance operations. Fig. 6 shows the comparison of maximum roll angle in the speed range
In order to evaluate the control performance of the designed active roll control system in a car, a series of experiments were performed using a Hil (hardware-in-the-loop) setup. Before the actual vehicle test with a heavy burden (time, cost, etc.), the Hil technique is an efficient way to realistically test the effectiveness of the control system in a laboratory. A schematic diagram of the experiment using the Hil setup is shown in Fig. 7 and a configuration of the ARC system module is represented in Fig. 8. The test work part is divided into two groups; in one group, the computational work parts including 10 DOF vehicle dynamics with tire characteristics as well as control logic. In the other group, hardware parts with sensors, data acquisition-interface and the ARC system module. The overall experimental processing loop, with a sampling time of 2 ms, is constructed as follows; as the driver
H.-J. Kim / Control Engineering Practice 19 (2011) 820–827
performs a steering operation, the dynamic behavior and tire force in a vehicle are simulated in the computer. After the active roll moment is calculated by the control logic using the feedback signal of the car body roll angle, the control signal corresponding to the desired active moment is transmitted to the motor driver through an interfacing board with a D/A converter. The actual moment, generated by the prototype ARC system, is measured by a force transducer and delivered to the computer through an interfacing board with an A/D converter. The actual active roll moment is exerted on the vehicle dynamic model. From previous work (Lin et al., 1996), the active roll control mechanism was devised with a hydraulic linear actuator connecting the stabilizer bar ends to the suspension tie rods. In this mechanism, to make the ARC system module respond faster, a stabilizer bar with higher stiffness is needed. However, if there is too much additional roll stiffness, it will spoil the ride comfort due to high roll excitation, which is caused when a bump is met by the wheels on one side of the car. To address this problem,
Darling and Hickson (1998) devised a hydraulic actuating system with a P-port closed proportional valve, which allows the actuator to free-wheel during straight line driving. On the basis of considerations of above issues and complexity of hydraulic system, in this study, an electrically actuated roll control system is devised, which would effectively remove the additional roll stiffness introduced by the ARC system module and thus improve the high frequency isolation of the car body during periods when the system is inoperative. The prototype ARC system module, as shown in Figs. 7 and 8, is implemented as a hardware component in the Hil setup (Kim, Yang, & Park, 2002). This system module includes the electrically actuated system and stabilizer bar fixed with bush on the frame. The ARC system module is constructed as a single unit because real-time roll moment distribution control problem between front and rear axle is not focused on in this study. The actuator part has a maximum force of 7200 N, a maximum velocity of 320 mm/s and the stiffness of the stabilizer bar is 5100 N m/rad. Fig. 9 shows the experimental results of frequency responses of the devised ARC system module. In this figure, the bandwidth of actuator under stalled condition is wider than ARC system module when moved condition, which is tested with 1 V sinusoidal command signal. This shows the practical characteristics of actuating module. Fig. 10 shows the test results of the roll responses through the previous Hil setup. The results in these figures were obtained at constant forward speed of 50 km/h with nominal parameters and a driver steering input as a J-turn maneuver having parameters with ho ¼0, h1 ¼3.51, to ¼0 s and t1 ¼ 0.2 s in the following equation: 2
t oto 6 6 t o t ot 6 o 1 4 t 4t1
Fig. 7. Schematic diagram of experiment using hardware-in-the-loop setup.
825
df ¼ ho df ¼ ho þ ðh1 ho Þ
2 oÞ 32 ðtðtt 1 to Þ
ðtto Þ ðt1 to Þ
df ¼ h1
As shown in Fig. 10(a), the resulting roll angle is reduced from 2.61 to 0.461 (max. value), which is about 17% of the level relative to the uncontrolled passive vehicle condition. Also in Fig. 10(b),
Fig. 8. Configuration of ARC system module.
40 Magnitude (dB)
ð24Þ
20 0 -20 -40 -60 -80 101
102 Frequency (rad/s)
Fig. 9. Frequency response of ARC system module. (Dashed: actuator with stalled condition and solid: actuating module with moved condition.)
826
H.-J. Kim / Control Engineering Practice 19 (2011) 820–827
construct a multi-loop control structure for force tracking such as the slow active suspension system case (Priyandoko, Mailah, & Jamaluddin, 2009). In this section, as an alternative, an integrated control strategy is presented in order to further enhance the roll response. In general, a variable damper is used for vibration control induced by uneven road input. The damping force of the variable damper is determined by a controllable damping coefficient as in the following rule: " f d ½fd ðx_ s x_ u Þ 4 0 _ _ Cd ¼ ðx s x u Þ ð25Þ ½fd ðx_ s x_ u Þ r0 Cmin
0.01
Roll Angle (rad)
0 -0.01 -0.02 -0.03 -0.04 -0.05 0
0.5
1
1.5 Time (sec)
2
2.5
3
0.15
Rollrate (rad/sec)
0.1
Md ¼
2 X
Mj
ð26Þ
j¼1
0.05
Mi ¼ Fli
0 -0.05 -0.1 -0.15 -0.2
If Cd 4Cmax, then Cd ¼Cmax. Where Cd is desired damping coefficient, Cmax and Cmin are the maximum, minimum damping coefficients of the variable damper. fd is the desired force for _ xuÞ _ is relative velocity between car suppression of vibrations, ðxs body and wheel at each corner. From the previous work (Kim and Park, 2002), considering the roll motion control in a steering maneuver, the desired damping force at each damper on roll control can be calculated by the following relationships:
0
0.5
1
1.5 Time (sec)
2
2.5
3
Fig. 10. Dynamic roll responses. (a) Roll angle responses (dotted: uncontrolled and solid: controlled); (b) rollrate responses (dotted: uncontrolled and solid: controlled).
the rollrate can be reduced. In these figures, the responses of roll angle and rollrate are affected by the characteristics of the ARC system module in Fig. 8, caused by the dynamic and structural configuration. Including the global roll motion related to the dynamic response of the actuating module in Fig. 9, a dead-zone like behavior induced by the supporting fixture (friction, deflection of rubber insert) of the stabilizer bar in the module affects a minute control region related to oscillations near steady-state body roll position. These results imply that further improvement is needed, even though the roll behavior can be improved by the devised ARC system with respect to uncontrolled case
ti t Fri i 2 2
ðFli þ Fri ¼ 0Þ
ði ¼ 1,2Þ
where Md is the total desired roll moment, Mj is the moment component of subscript j with the front and rear axles, Fli and Fri are the desired forces of the left and right sides of the axle, ti is the track width with subscript i of left and right. The desired damping coefficient can then be determined by Eq. (25). The integrated control strategy is based on the ARC system as an option of the vehicle equipped with the semi-active type suspension system. The variable damper system is controlled with the devised ARC system simultaneously based on the structure of Fig. 11. This control structure was proposed in a previous study (Kim and Park, 2002) in the framework of lateral acceleration feedback control scheme. The results showed that this method could improve roll responses. In this section, on the basis of these studies, the integrated control is adopted in order to improve roll behavior without any extra equipment and any additional multi-loop control logic to the devised ARC system module. Fig. 12 illustrates the resulting roll angle responses, related to Fig. 10, with respect to passive, active roll control and integrated control cases, respectively. In the figure, the integrated control can give a smoother response even though there is no further improvement in the steady-state region. It results from the operating characteristics of a semi-active type actuator which cannot generate an active force. Fig. 13 shows the roll mode phase portrait with respect to passive, active roll control and
5. Integrated control strategy In general, the controlled responses are affected by the characteristics of an actuating system. Typically, the bandwidth of the actuating system causes some degradations of control performance (Darling, Dorey, & Ross-Martin, 1992). Previous experimental results in Section 4 are also affected by ARC system module characteristics. In order to improve on this problem, a high response actuating system or mechanism can be used. From a practical point of view, this is more costly. Another method is to
ð27Þ
Fig. 11. Schematic diagram of integrated control strategy.
H.-J. Kim / Control Engineering Practice 19 (2011) 820–827
0.01
6. Conclusion
Roll Angle (rad)
0 -0.01 -0.02 -0.03 -0.04 -0.05
0
0.5
1
1.5 Time (sec)
2
2.5
3
Fig. 12. Comparison of roll responses in J-turn maneuver. (Dashed: uncontrolled, dotted: ARC only and solid: integrated control.)
0.15 0.1 Roll Angular Vel. (rad/sec)
827
0.05 0 -0.05 -0.1 -0.15 -0.2 -0.045 -0.04 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 Roll Angle (rad)
0
Fig. 13. Comparison of roll mode phase portraits. (Dashed: uncontrolled, dotted: ARC only and solid: integrated control.)
integrated control case, respectively. This figure shows that control performance can be achieved both in a steady-state and in a transient region with a smoother response with respect to uncontrolled cases.
In this paper, an investigation of an active roll motion control system has been presented. A designed controller, which has taken into account the influence of parameter perturbation, was applied to a full vehicle model with nonlinear tire characteristics. An enhancement in roll behavior under loading patterns, speed and maneuvering disturbance conditions was achieved in the simulation work. In order to evaluate the feasibility as an actual system in a car, the electrically actuated roll control system (having a high compliance to external road excitation during inoperative periods) was devised. Through experimental works using the hardware-in-the-loop technique, the control performance was investigated. Finally, as an option of a car equipped with a semi-active suspension system, the integrated control method has been presented and it has been shown that this control strategy is feasible for providing improved steadystate and transient behaviors with respect to passive vehicle conditions. References Bakker, E., Pacejka, H. B., & Linder, L. (1989). A new tire model with an application in vehicle dynamics studies. SAE paper 890087 (pp. 83–95). Darling, J., Dorey, R. E., & Ross-Martin, T. J. (1992). A low cost active anti-roll suspension for passenger cars. Transactions of the ASME, Journal of Dynamic Systems, Measurement and Control, 114, 599–605. Darling, J., & Hickson, L. R. (1998). An Investigation of a roll control suspension hydraulic actuation system. Transactions of the ASME, Fluid Power Systems and Technology, 5, 49–54. Darling, J., & Ross-Martin, T. J. (1997). A theoretical investigation of a prototype active roll control system. Proceedings of Institute of Mechanical Engineers, Part D, Journal of Automobile Engineering, 211, 3–12. Hrovat, D. (1993). Applications of optimal control to advanced automotive suspension design. Transactions of the ASME, Journal of Dynamic Systems, Measurement and Control, 328–342. Kim, H. J., Yang, H. S., & Park, Y. P. (2002). Robust roll control of a vehicle; experimental study using hardware-in-the-loop setup. Proceedings of Institute of Mechanical Engineers, Part D, Journal of Automobile Engineering, 216, 1–9. Kim, H. J., & Park, Y. P. (2002). Hybrid attitude control in steering maneuver using ARC Hil setup. Control Engineering Practice, 10(12), 1339–1345. Li, H., & Goodall, R. M. (1999). Linear and non-linear skyhook damping control laws for active railway suspensions. Control Engineering Practice, 7(7), 843–850. Lin, R. C., Cebon, D., & Cole, D. J. (1996). Optimal roll control of a single-unit lorry. Proceedings of Institute of Mechanical Engineers, Part D, Journal of Automobile Engineering, 210, 45–55. Poussot-Vassal, C., Sename, O., Dugard, L., Gaspar, P., Szabo, Z., & Bokor, J. (2008). A new semi-active suspension control strategy through LPV technique. Control Engineering Practice, 16(12), 1519–1534. Priyandoko, G., Mailah, M., & Jamaluddin, H. (2009). Vehicle active suspension system using skyhook adaptive neuro active force control. Mechanical Systems and Signal Processing, 23, 855–868. Yagiz, N., & Hacioglu, Y. (2008). Backstepping control of a vehicle with active suspensions. Control Engineering Practice, 16(12), 1457–1467. Yoshimura, T., Nakaminami, K., Kurimoto, M., & Hino, J. (1999). Active suspension of passenger cars using linear and fuzzy-logic controls. Control Engineering Practice, 7(1), 41–47.