Comparing the performance and limitations of semi-active ... - CiteSeerX

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Int. J. Vehicle Systems Modelling and Testing, Vol. 2, No. 3, 2007

Comparing the performance and limitations of semi-active suspensions Xabier Carrera Akutain TECNUN (University of Navarra) Manuel de Lardizábal 13, 20018 San Sebastián, Spain Fax: (34) 943 311442 E-mail: [email protected]

Jordi Vinolas*, Joan Savall and Miguel Castro CEIT and TECNUN (University of Navarra), P.Manuel de Lardizabal 15, San Sebastián (E-20018), Spain Fax: +34 943 213076 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] *Corresponding author Abstract: The potential of a controllable suspension system is limited mainly by the damping span of the shock absorbers and the switching times of the actuators. In order to quantify the relative importance and study the sensitivity of this and other features to the constraints in the force modulation ability, this paper deals with an extensive simulation work. Several models of different complexity for controllable shock absorbers are employed, including Continuously Variable Dampers (CVD) and Discrete Stage Variable Dampers (DSVD). A full vehicle model is designed and experimentally validated with collected sensors data for this sake. Keywords: full car multi-body; semi-active suspension control strategies; damper design parameters. Reference to this paper should be made as follows: Carrera Akutain, X., Vinolas, J., Savall, J. and Castro, M. (2007) ‘Comparing the performance and limitations of semi-active suspensions’, Int. J. Vehicle Systems Modelling and Testing, Vol. 2, No. 3, pp.296–314. Biographical notes: X. Carrera Akutain received his PhD in Mechanical Engineering in 2006 from the TECNUN-University of Navarra. The title of his thesis was “Development of a semi-active suspension for an off-road vehicle”. His research interests are in the areas of vehicle design and dynamics, data-acquisition and mechatronics and he is currently employed by Tenneco Automotive in Belgium in vehicle suspension research activities. J. Vinolas is presently the Head of the Applied Mechanics Department at CEIT (Centro de Estudios e Investigaciones Técnicas) and Associate Professor at the school of Mechanical Engineering at TECNUN (University of Navarra). He received his PhD in 1991. The title of the thesis was A New Experimental Methodology for Testing Active and Conventional Active Suspensions. He has been involved in different research projects related to vehicle dynamics, noise and vibration.

Copyright © 2007 Inderscience Enterprises Ltd.

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Joan Savall is a research staff member at CEIT and he is an Assistant Professor of Machine Theory at TECNUN (University of Navarra) Engineering School. He received his PhD Degree in Mechanical Engineering from the University of Navarra in 2005. He was a Visiting Researcher in 1996 at PMA in Belgium, where he started out in mechanical design for mechatronics. He has several patents filed and pending. His research interests include mechanical design, robotics and mechatronics. M. Castro is a research student and PhD candidate in the school of Mechanical Engineering at TECNUN (University of Navarra). He received his MS Degree on Mechanical Engineering from the University of Piura (Peru) in 1995. He has been involved in several international research projects related to CAD, CAM and CAE. He has been an Assistant Professor of Computer Science since 2003 at TECNUN. His research interests are in the areas of fuzzy logic, computer aided design and simulation.

1

Introduction

Computer controlled suspensions try to reach, at a small cost penalty, a trade-off between riding comfort and vehicle handling, superior to that achievable with conventional passive suspensions. In the early 1980s, Margolis (1983) stated that semi-active systems were quite capable of providing performances approaching that of a totally active system. In order to control the unsprung weight motion or wheel hop, the ‘skyhook’ and ‘groundhook’ control theories were applied. An explanation of these control methods, which try to follow the command from an ideal fully active force generator, can be found in several previous studies (Karnopp et al., 1975; Ahmadian and Marjoram, 1989; Ivers and Miller, 1989). In addition to the studies with the popular quarter-car model, the heave and pitch modes began to be investigated in simplified inline models with promising results (Margolis, 1982). Sharp and Hassan (1986) simulated the relative performance of passive, active and on/off semi-active suspensions, introducing first-order lag dynamics with a cut-off frequency of 16 Hz for the semi-active system. In this manner, more realistic simulations were actually achieved. Jalili and Esmailzadeh (2001) presented an approach to optimal control of fully-active suspension systems, incorporating a similar actuator delay. Previously, Crolla et al. (1989) had simulated the strong influence of damping constraints, threshold delay and first order lag in vehicle ride quality in a quarter car model with on/off and CVDs, which are the most typical ones in automotive applications, providing infinite damping curves within a limited range. It was suggested that the initial threshold delay and the time constant should both remain below 10 ms in order not to compromise too much the performance improvements achievable with skyhook control. Heo et al. (2000) carried out simulations with Modified Skyhook Damping (MSD), introducing a simple damper model and including phase delay. This phase delay was shown to have an adverse effect. Nevertheless, previous experimental works of Ivers and Miller (1989) in a quarter-car test rig showed that a damper commanded by a stepper motor followed, at least partially, the results from an ideal skyhook simulation. Recently, Hong et al. (2002) demonstrated in Hardware in the Loop (HiL) ride simulations with CVDs that the semi-active system could achieve a competitive control performance by using road adaptive control laws. A prominent study is that of

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Kitching et al. (2000a), where each component of a CVD is analysed in detail. HiL testing resulted in minimisation of Root Mean Square (RMS) body acceleration for stochastic road data and bump inputs using modified skyhook control. The hybrid control is a combination of pure skyhook and groundhook and can offer a good compromise between the sprung and unsprung masses, as tested by Goncalves and Ahmadian (2003) in a quarter car test-rig with steady state and transient inputs. The wheel hop reduction can be observed from two points of view. The first one is related to the minimisation of the road damage, caused especially by trucks. Simulation works developed with a four-degrees of freedom (dof) tractor unit model (Kitching et al., 2000b), semi-active suspension and optimal control theory led to an average 9% road damage reduction for different motorway input conditions, compared to an optimum passive suspension. The diverse road-friendly projects are very active in this area. A related relevant paper is that of Kortüm et al. (2002), in which the potential of road friendliness is proven by testing and simulations. Although several aspects of the control design were treated in a simple quarter car model, including essential non-linearities of damper, springs and tyres, the results were checked on full spatial simulations. Stochastic road profiles and transient inputs like bumps and potholes in straight line are usually employed in these studies. Additionally, the control of the wheel hop can help not only to minimise the road damage caused by trucks but, also, to increase the handling performance for all kind of road vehicles. Subsequent sections describe an approach to improve handling for ideal CVDs and for a DSVD model based on an existing shock absorber used in the real car. These discrete stage devices consist of stepper motors driving a valve that restricts the oil flow between the chambers of the shock absorber. Stepper motors have a limited number of switching positions and they are used in many applications (Chiasson and Novotnak, 1993; Kenjo, 1993); some of them are related to force modulation in semi-active suspensions (Emura and Kakizaki, 1994; Carrera Akutain et al., 2005). In order to confirm the validity of the multi-body model being used, Section 2 is dedicated to correlate experimental results of the vehicle obtained on a track with simulations in equivalent conditions. Section 3 presents the effectiveness of hybrid semi-active control and hybrid-plus extended groundhook strategies on the single-seater car when ideal actuators or CVDs with limited time response or limited damping factor range are used. Section 4 presents results when a realistic model of the current DSVDs used in the vehicle is employed instead of the simplified CVD models. The influence of the design parameters of the DSVD (time response, number of discrete positions, damping factor span) on the performance of the suspension is studied in depth.

2

Multi-body full car model

Vehicle models in previous studies range from 1-dof to multi-dof articulated heavy vehicle models. They can be classified as:

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•

Classic quarter car models (Bender, 1968; Karnopp, 1983; Sharp and Hassan, 1986; Venhovens, 1994; Soliman and Crolla, 1996; Valasek, Novak et al., 1997; Gordon and Sharp, 1998; Goncalves and Ahmadian, 2003; Giua et al., 2004). Due to their simplicity and ease of use they are widely employed in HiL simulations but only the heave mode can be studied. (Besinger et al., 1995; Hwang et al., 1998; Choi et al., 2000; Hong et al., 2002).

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Pitch-plane models (Thompson and Pearce, 1979; Margolis, 1982; Metz and Maddock, 1986; Elmadany, 1989; Krtolica and Hrovat, 1992; Kadota, 1994). Pitch attitude can be controlled.

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Roll-plane models (Dorling and Cebon, 1996; Dorling et al., 1996). Roll attitude can be controlled.

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Full car models. Rigid/flexible car body models with/without longitudinal/lateral dynamics (Barak and Sachs, 1985; Crolla and Abdel-Hady, 1991; Venhovens et al., 1993; Esmailzadeh and Fahimi, 1997; Kortüm et al., 2002). Heave, pitch and/or roll modes can be controlled depending on the complexity of the model.

In this work, a full car model approach has been preferred. Figure 1, shows a CAD design, very useful in the identification of mass properties and a derived 7 dof representation of the vehicle built in this work. The model is represented in SimMechanics, an extension of MATLAB/Simulink® with tools for modelling and simulating mechanical systems, with block diagrams representing bodies, connections, sensors and actuators. When considering the straight line ride, the four vertical coordinates of the unsprung masses and the three modal coordinates (heave, pitch and roll) define the system. Suspension elements were implemented as non-linear S-functions in this multi-body environment. On one hand, the stiffness of the springs was measured and continuity of first order in the force-displacement function at bump/rebound end stops was imposed, in order not to jeopardise the stability of the simulations. Preload was adjusted to that of the real vehicle. On the other hand, regarding the employed shock absorbers, they were supplied by the company APA-Kayaba and were arranged in parallel with the springs. Their characterisation was accomplished making use of a parametric damper model based on Reybrouck’s (1994) work and further developed and validated by the authors (Carrera Akutain et al., 2006). In this manner, the displacement of the suspension at the corner i would be represented as the difference between the projected coordinates of the sprung and unsprung vertical displacements in that corner ( Z Si′ − ZUi′ ). In our tests, two consecutive speed bumps of 50 mm height have been used for road testing validation (see Figure 2). The approaching speed was 30 km/h and the vehicle passes over the speed bumps without engine effort. This effect is managed in the simulations by implementing an increasing linear time delay on the rear axle with respect to the front axle. A key factor in the modelling was considering the vertical stiffness of the tyre, only effective in the case where road and tyre were in contact, which is lost as shown in Figure 2 when the wheel impacts the obstacle, thus making the study more complex. In fact, first characterisations considering perfect road-tyre contact gave unacceptable results. With the current approach, the stiffness is set to zero when the contact is lost. In the following figures, model outputs in time domain are plotted against signals of

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diverse sensors mounted on-board. The most representative ones are: displacement sensors and force transducers in each suspension corner and an accelerometer in the sprung mass (top mount of the Front Left (FL) suspension). Figure 1

Vehicle model (left) and CAD representation (right)

where x y

zC ϕ ψ

z0 zUi zSi Figure 2

Longitudinal axes of the vehicle Lateral axes of the vehicle Heave mode. Vertical displacement of the cog Pitch mode Roll mode Vertical displacement of the road input Vertical displacement of the unsprung mass of corner i Vertical displacement of the sprung mass at corner i Testing for the validation of the full car model

The dynamic stiffness of the tyres was unknown and was tuned observing differences between model outputs and collected data. A static stiffness of 90 kN/m for the four tyres was chosen. In Figures 3 and 4, the brightest lines represent the experimental displacement values (zero indicates no compression and li is the length of the ith damper in maximum extension), collected at 100 Hz sample frequency. The simulation step size was of 2 ms. The drive through the bumps and the compression of the suspension for both axles can be noticed.

Comparing the performance and limitations of semi-active suspensions Figure 3

Model validation. FL compression

Figure 4

Model validation. RL compression

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Figures 5 and 6 compare the suspension dynamic forces of the model and the experimental signals sampled at 200 Hz. Model outputs for both axles are reasonably close to the measured values, but there are some peaks in the experimental data for the rear force transducers (see Figure 6), which do not match the signals of the vehicle model. Although no definitive explanation has been found for this mismatch, it seems to be caused by some limitations of the model such as not considering either the friction on the mounts between the chassis and the wishbones, or the lateral grip of the tyre, which could have some effect as the track width slightly changes with suspension deflection.

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The vertical acceleration of the FL top mount (see Figure 7) shows a good tracking of the simulated values. Although the peak values and the shape of the curves remain similar, it seems to be a slight back and forth phase shift that causes the peak signals to differ slightly, which is quite understandable considering the sensitive nature of acceleration signals. Despite the limitations noticed, the shape and correlation of the signals confirms the validity of the designed model and consequently the model is accepted as a reliable tool for the simulation of the different semi-active control setups. Figure 5

Model validation. Force through front suspensions

Figure 6

Model validation. Force through rear suspensions

Comparing the performance and limitations of semi-active suspensions Figure 7

3

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Model validation. Acceleration of the sprung mass

Comparing the effectiveness of different semi-active control setups

Diverse methods controllers have been employed for years to design suspension. Linear controllers are normally based on optimal control (LQR/LQG), the skyhook principle or robust control (H∞ and µ-synthesis). Taking into account non-linear dynamics leads to a wide range of non-linear control techniques such as multi-linear, backstepping, sliding mode, fuzzy logic, neural networks and adaptive control. Pure skyhook control (see Figure 8) is only optimal in the sense of minimising vibration transmission to the sprung body. It has the disadvantage of not considering rattle space and simulating a situation where there is no wheel damping, which leads to a massive wheel-hop resonance when compared to a traditional passive damping. Therefore, variations of the skyhook damper controller are necessary. The analogue principle may be applied to the control of the non-suspended mass. Connecting the wheel to a skyhook is termed as ‘groundhook’. It obviously presents the opposite problem, as adverse effects on body-resonance control arise (Margolis, 1983). A hybrid method combining aspects of groundhook and skyhook was discussed by Ahmadian (1997) as a potential alternative to get a more acceptable compromise for a wider range of applications. Given that the use of the relative displacement of the wheel with respect to the road profile for control purposes is not feasible in practice, the control force is usually built as a linear combination of skyhook and passive components, which takes the name of modified skyhook damping (MSD). For the simple quarter car model: FC = Csky ⋅ Z S + Cpassive ⋅ ( Z S − Zu )

where Force command as result of the skyhook law. Fc: Csky: Linear skyhook damping factor. Cpassive: Damping coefficient of the passive component.

(1)

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Z S :

Absolute vertical velocity of the sprung mass.

Zu :

Absolute vertical velocity of the unsprung mass.

Figure 8

Pure skyhook (left) and practical semi-active (right) quarter car representations

Another practical approach is to continuously employ skyhook body control for good comfort and discrete stage adaptive control modes when different road or driver inputs are detected. For instance, the excessive vibration of the wheel (detected by accelerometers) or of the suspension at the wheel-hop frequency band could be corrected. Two hybrid semi-active control strategies are compared in this section. The effectiveness of each of them is obtained by carrying out simulations in four different cases, more or less ideal. Case 1: Perfect active suspension (ideal active force generator). Case 2: Linear CVD with Csoft = 500 Ns/m and Chard = 10,000 Ns/m (very broad

damping span). Case 3: Linear CVD with Csoft = 1000 Ns/m and Chard = 3000 Ns/m. These values are more in line with what can be expected from a real controllable damper. Case 4: A first order lag of 10 ms time constant is added to Case 3. This lag equals a cut-off frequency of 15 Hz and is introduced to avoid an unrealistic instantaneous switching CVD supposition (Case 3) and to predict the performance level in a more accurate manner. The aim is to provide an estimation of performance of ‘quasi-ideal’ systems prior to more realistic analysis as those carried out in Section 4.

3.1 Hybrid semi-active control Initially, a modified hybrid control is applied. It makes use of the well known skyhook and groundhook concepts, with the addition of a passive component. Furthermore, a similar term is added by the authors in order to reduce the pitch of the body. Using the coordinates of the full car model, the controller output yields for each corner: FC′i = Csky ⋅ Z Si + Cpassive ⋅ ( Z s′i − Z u′i ) + Cground ⋅ ( Zui − Z 0′i ) ± Cpitch ⋅ ϕ

(2)

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where Z Si′ − Zui′ represents the ith shock absorber extension velocity and ϕ the body pitch angle variation. Apart from the typical Csky, Cground and Crelative weighting parameters, a ‘pitchhook’ Cpitch factor is introduced and multiplied by the pitch velocity. The sign of this constant is obviously opposite for front and rear axles. Previously it has been stated that the goal function would be focused on the compression of the tyres (hence it relates to the vertical dynamic tyre load), but attention has also been paid to pitch reduction. In the work of Metz and Maddock (1986), the importance of the body attitude motions for the performance of the vehicle was revealed. Beyond the aerodynamic stability effects cited, it must be noted that pitch reduction can, in addition, help to improve the confidence and safety of the driver, as well as the comfort perception. In order to summarise different targets, a simple goal performance index has been chosen, by adding the normalised contributions of wheel hop, pitch motion and acceleration of the sprung mass, each with the same penalty factor. A typical approach (Lauwerys, 2005) is making use of modal transformations from vertical accelerometer signals to calculate the force command with modal velocities, which is inversely transformed in demanded force at each corner. In the setup presented, an accelerometer was attached to the FL suspension top mount, and due to the symmetric nature of the experiment with the sleeping policeman, the same value was assumed for the front right corner. This led to a combination of corner based and modal based (signal of gyroscope) control. Simulations are carried out combining different values of the weighting parameters. The results are compared with those of the optimum passive setup, i.e., the fixed damping law of the real DSVD shock absorbers giving the best result so that it can be called ‘optimum passive’. Table 1 details the normalised RMS values of the controlled signals with respect to this optimum fixed damping law mentioned above. Simulation A is the optimum subcase for the defined performance index in each of the four cases quoted above. B is the optimum setup for reducing the compression of the FL tyre as wheel hop indicator, without paying attention to other factors, and subcase C focuses on body pitch control. These values are bold marked. The non-bold cross values are included as they reflect the mutual influence of different control gain parameters for subcases A, B or C on each other. A value of unity represents the performance of the optimal passive suspension whereas a normalised value below unity obviously means improvement of the controlled suspension with respect to the optimum passive system. Table 1 shows that the more limited the system is, the poorer the improvements of target indices (bigger normalised values) are. Obviously, the case of an active force generator, ideal in both unlimited energy supply and instantaneous reaction time, gives the best results, but an actuator like that does not exist in reality. This ideal state has been used as Case 1, since it is a good starting point to judge the relative importance of the ideal control and the practical constraints on the controllable suspension performance. The results with the CVDs of Cases 3 and 4, the latter including actuator lag dynamics, do not differ excessively, which is in agreement with previous research works where simulations with a similar cut-off frequency were carried out (Sharp and Hassan, 1986; Crolla et al., 1989). It seems to be more advantageous to broaden the force span attainable with the dampers than using very fast actuators. In any case, both features are desired, but it can not be forgotten either that quick big force changes might introduce excessive noise into the system. Pitch motion can be reduced to an acceptable level as long as the ratio of maximum to minimum damping is large enough, while the effect of the damping range on the wheel hop is less noticeable. The optimum setting of

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the different weighting parameters for the unrealistic perfect active force generator and for the more realistic cases may differ considerably. This means that employing realistic models and constraints is absolutely necessary for the development of suspension control strategies. Table 1

Optimum RMS values (hybrid control) Index

FL wheel hop

Pitch angle

A

0.872

0.809

0.357

B

1.081

0.590

1.738

C

0.968

1.019

0.245

A

0.882

0.819

0.714

B

0.941

0.699

1.282

C

0.958

0.997

0.474

A

0.898

0.820

0.813

B

0.939

0.756

1.171

C

0.917

0.891

0.700

A

0.903

0.834

0.807

B

0.931

0.765

1.055

C

0.930

0.922

0.714

Case 1

Case 2

Case 3

Case 4

3.2 Hybrid plus extended groundhook The extended groundhook is a variation proposed by Valasek et al. (1997) and has been used in several road-friendly projects for trucks (Kortüm et al., 2002). It tries to reduce virtually the stiffness of the tyre when passing through a bump, where tyre compression is unavoidable. Subsequently, this extension is combined with our previous controller: Fc′′i = Fc′i + ∆kUnsprung ⋅ ( Z ui − Z 0i ) − ∆kSprung ⋅ ( Z S′i − Z u′i )

(3)

where ∆kUnsprung refers to the tyre stiffness and ∆kSprung to the spring stiffness. A procedure analogous to Section 3 is followed to obtain the performance indexes and the results obtained are listed in Table 2. The four presented cases show that the huge benefits of controlling the wheel hop with the extended control for the ideal Case 1 (bigger than for the simple hybrid control) are sensibly reduced for the more limited case of CVD with first order lag (Case 4). Cases 2 and 3 present intermediate improvements. The conclusions for the pitch control are similar to those of Section 3. This represents an example of a very superior control strategy in an ideal model that loses its advantage when more realistic constraints are considered.

Comparing the performance and limitations of semi-active suspensions Table 2

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Optimum RMS values (hybrid + extended groundhook control) Index

FL wheel hop

Pitch

Case 1

0.834

0.336

0.240

Case 2

0.875

0.576

0.474

Case 3

0.891

0.713

0.696

Case 4

0.911

0.747

0.723

3.3 Results summary for both control strategies Figure 9 summarises the results previously mentioned. Cases 1–4 (B) represent the optimal wheel hop reduction with the hybrid control. These suboptimal solutions (as they are optimal for wheel-hop but not for pitch or for the global index defined) present an increase in the pitch angle RMS value for Cases 1 and 2, with respect to the passive dampers. Cases 1–4 (C) are analogues for the pitch angle reduction. Both in wheel hop (B) and in body pitch (C) reduction, Case 2 is more limited than Case 1, Case 3 more than Case 2 and Case 4 more than Case 3, although the latter two do not differ much. Results with the extended groundhook control are also included. This extension shows only a small advantage with respect to the previous hybrid Case 4B without extension. Looking back to ideal active cases 1BExt and 1CExt, the first one achieves the minimum wheel hop, while 1CExt seems to be an optimum compromise between wheel hop and pitch control. Figure 9

Wheel hop-pitch diagram. Simplified Cases 1–4, hybrid and extended

Figure 10 reflects the body pitch angle over time for Cases 1–4 and hybrid control and how the importance of a wide damping range is clearly noticeable.

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Figure 10 Pitch angle over time. Cases 1–4

4

Limitations when using a non-ideal DVSD

The DSVD shock absorbers utilised have 17 settings or damping laws, selectable with stepper motors. A detailed study of the dynamics of these actuators is presented in a previous paper (Carrera Akutain et al., 2005). One of the conclusions was that the simplest way to accurately represent the stepper motor motion was by considering a step or lag time of 4 ms for switching between consecutive positions or damping laws out of 17, which makes controlling the wheel-hop impossible. Figure 11 shows the control scheme for the FL suspension. The hybrid control command of equation (2) is generated in the same manner as in Section 3. This force command is introduced in a block where it is compared with the forces of the 17 possible damping laws with the instantaneous damper velocity. The stepper will begin to move towards the position in which the output is closest to the command force, i.e., the force command becomes a position command. Every 4 ms, the stepper is capable of reaching the position immediate closest to the current one. When this happens, the position command might obviously have already changed. Instead of selecting an integrator step size equal to the time necessary for the stepper to move from one position to the next one (Giua, 2004), a simulation step time half this actuator step size (2 ms) has been chosen for greater accuracy and motion interpolation between steady positions.

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Figure 11 Control scheme of the FL suspension

Additionally, the use of nine intermediate positions instead of the 17 (the switching step is doubled, therefore the switch between consecutive positions takes a total of 8 ms) and two end positions (64 ms between extremes, soft to hard) has been studied. Reducing the total number of positions from 17 to 9 does not significantly change the suspension’s performance, as will be shown later. If we consider that reducing the number of poles of the stepper motors to half would almost duplicate the motion speed of the motor, it would make sense to develop stepper motors with fewer poles. The effect of the stepper velocity has also been simulated subsequently, using step times of 1 ms, 2 ms, 4 ms, 8 ms and 64 ms for steppers with different number of poles. The results of Figure 12 employing the hybrid control indicate that faster actuators with the current damper help to reduce body pitch and wheel hop. Considering the standard shock absorbers, it can be observed in Figure 12 that there is no benefit from making use of the 17 damping laws instead of the intermediate nine laws, as the damping step is very narrow for 17 distinct positions. 17_4 ms and 9_8 ms are the current possible configurations. With X_Y ms the number of positions is represented by X and the step time by Y(ms). The reason for the relatively poor performance of these DSVDs for wheel hop lies both on the small damping span available and on the limited bandwidth of the actuators. With faster actuators, the ‘bandwidth’ of the controlled suspension is increased. The quotes refer to the difficulty of defining a bandwidth for a system that controls the damping, not continuously but only when command force and input velocity are of the same sign. However, for this force modulation system a bandwidth concept may not be completely wrong, and a level enough to control the wheel-hop frequency is not reached, as sufficient damping can not be provided by the controller due to saturation of the shock absorbers. At this point, an additional study is proposed, as investigating the potential performance of a suspension with dampers of higher span becomes interesting. This has led to a modification of the damper model (see ‘Modified’ in Figure 13, where curves for minimum and maximum damping are shown) in order to simulate this potential state of increased actuation area.

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Figure 12 Wheel hop-pitch diagram. Different actuator velocities, 9–17 positions

Figure 13 Modified F-v diagram for RL shock absorber

The suboptimal values for both wheel hop and pitch angle are drawn with a single marker in Figure 14, in contrast to Figure 9, where the optimal wheel hop value is represented on the y-axis with its correspondent non-optimal pitch value and vice versa, as each suboptimal case needs its respective control parameters. Figure 14 presents the results with the current damper and the modified (M) shock absorber model functioning as two-stage or ON/OFF dampers, considering three different actuator velocities besides the real dynamics (delay of 64 ms). The current shock absorbers make a reduction of 8% in wheel hop and 11% in pitch angle from the

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optimum passive system possible. Speeding up the actuators would lead to a higher pitch reduction (up to 24%), with little improvement in wheel hop, while widening the force range (modified dampers M represented with hollow markers) of the shock absorbers presents promising results. With the normal actuator velocity, RMS pitch angle is reduced by 17% and the wheel hop by up to 23%. Again, with faster actuators, the values are improved up to 36% and 27%, respectively. Figure 14 Wheel hop-pitch diagram. Two-stage damper, real (2_X) and modified (2M_X). Different actuator velocities

In spite of the fact that this work has been focused on the wheel hop and pitch angle reduction and not on comfort issues, Figure 15 shows a noticeable reduction of the RMS acceleration in the vehicle centre of gravity. Figure 15 Body acceleration improvement. Different actuator velocities, 9–17 positions

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Conclusions

A multi-body 7 dof full car model has been designed and validated. In such an environment, hybrid skyhook and extended groundhook control strategies have been simulated. The results show that significant body-pitch motion can be reduced, while wheel hop control improvements are minor. The reasons have been investigated for further developments. It has been demonstrated that using all the 17 damping laws (discrete positions of the stepper motors) represents no advantage against using half of them and that the use of faster actuators, and especially of shock absorbers with a wider action area, should be promoted. The current shock absorbers were designed with the reference of conventional shocks for these off-road vehicles, which tend to have very soft damping. Further research should include: investigating results of considering full modal based or real corner based independent controls and deviations with respect to the current controller, the introduction of different inputs and manoeuvres for the full car model, a detailed spectral study of the measured signals on different frequency bands and last, but not least, the design of more advanced control techniques. In addition, HiL simulations could help to explore the limits of the system and go deeper in the study.

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List of notations CVD

Continuously Variable Damper

DSVD

Discrete-Stage Variable Damper

x

Longitudinal axes of the vehicle

y

Lateral axes of the vehicle

zC

Heave mode. Vertical displacement of the cog

ϕ ψ

Pitch mode Roll mode

Z0

Vertical displacement of the road input

zU

Vertical displacement of the unsprung mass of corner i

zS

Vertical displacement of the sprung mass at corner i

i

i

Z u′FL − Z s′FL + lFL

Compression of the Front Left (FL) damper

Fc

Force command as result of the skyhook law

Csky

Linear skyhook damping factor

Cpassive ( Z s′RL − Zu′RL )

Damping coefficient of the passive component

Extension velocity of the rear left damper

Case XY

Set of weighting factors optimum for the case X (1–4), for goal function/parameter Y (A/B/C)

Case XYExt

Case XY with the addition of the extension to the hybrid control