No-Jerk Skyhook Control Methods for Semiactive Suspensions

No-Jerk Skyhook Control Methods for Semiactive SuspensionsMehdi AhmadianProfessor,Fellow ASME,Center for Vehicle Systems and Safety,Department of Mechanical Engineering,Virginia Tech,Blacksburg,V A24061Xubin SongMem.ASME,Visteon Corporation,6100Mercury Dr.,Dearborn,MI48126Steve C.SouthwardMem.ASME,Lord Corporation,100Lord Drive,Cary,NCThis paper presents two alternative implementations of skyhook control,named‘‘skyhook function’’and‘‘no-jerk skyhook,’’for reducing the dynamic jerk that is often experienced with conven-tional skyhook control in semiactive suspension systems.An analysis of the relationship between the absolute velocity of the sprung mass and the relative velocity across the suspension are used to show the damping-force discontinuities that result from the conventional implementation of skyhook control.This analysis shows that at zero crossings of the relative velocity,conventional skyhook introduces a sharp increase(jump)in damping force, which,in turn,causes a jump in sprung-mass acceleration.This acceleration jump,or jerk,causes a significant reduction in iso-lation benefits that can be offered by skyhook suspensions.The alternative implementations of skyhook control included in this study offer modifications to the formulation of conventional sky-hook control such that the damping force jumps are eliminated. The alternative policies are compared to the conventional skyhook control in the laboratory,using a base-excited semiactive system that includes a heavy-truck seat suspension.An evaluation of the damping force,seat acceleration,and the electrical currents sup-plied to a magnetorheological damper,which is used for this study,shows that the alternative implementations of skyhook con-trol can entirely eliminate the damping-force discontinuities and the resulting dynamic jerks caused by conventional skyhook con-trol.͓DOI:10.1115/1.1805001͔IntroductionThe idea of skyhook control has been in existence for more than three decades.Introduced by Crosby and Karnopp͓1,2͔,sky-hook control has been studied most often for vehicle-suspension applications.Suspension systems with skyhook control draw a small amount of energy to operate a valve that adjusts the damp-ing force;therefore,they are referred to‘‘semiactive’’suspensions or dampers.The damper valve can be a mechanical element͑as is the case for mechanically adjustable dampers͒or afluid valve that takes advantage of the rheological changes of the damperfluid, such as a magnetorheological͑MR͒damper͓3,4͔.The virtues of semiactive suspensions versus conventional,pas-sive suspensions have been addressed in several past studies ͓5–11͔.Using various analytical methods,these studies have con-cluded that in nearly all cases of semiactive suspensions can more effectively reduce vibration transmission across the suspension and control the suspended body͑or sprung mass͒,as compared to passive suspensions.Some of these studies have led to different control methods for semiactive suspensions,including several variations to skyhook control,which have been reviewed by Ivers and Miller͓12͔.Although there exists a large number of studies that provide a comprehensive analysis of the skyhook control method and its variations,very few studies have provided an analysis of the dy-namic jerk that is caused by the change in damping force in sky-hook systems.This study will provide a comprehensive analysis of the dynamic jerk that is induced by skyhook control.After a brief overview of skyhook control,we will provide two alternative formulations of skyhook control for eliminating dynamic jerk.Ad-ditionally,we will provide the results of a series of dynamic tests that compare the alternative methods to conventional skyhook control for a test setup that includes a single-degree-of-freedom ͑dof͒base-excited system with a magnetorheological damper. BackgroundFor a semiactive suspension,the equations governing skyhook control can be described byͭV1V12Ͼ0F saϭG s V1V1V12Ͻ0F saϭ0(1)whereV1ϭabsolute velocity of the sprung massV12ϭrelative velocity across the suspensionF saϭskyhook damping forceG sϭa gain factor that is commonly used such that the full damping range of the skyhook damper can be usedSkyhook control,which is illustrated in Fig.1,indicates that when the absolute velocity of the sprung mass and the relative velocity across the suspension have the same sign,the damper is resisting the motion of the sprung mass and a damping force pro-portional to the sprung mass velocity is desired.Otherwise,the damping force does not resist the motion of the sprung mass and must be minimized.This logic emulates the ideal configuration of a passive damper‘‘hooked’’between the sprung mass and an imaginary‘‘sky,’’hence,the name‘‘skyhook.’’Problem StatementIn order to better highlight the dynamic jerk problem due to the skyhook control,we will evaluate a three-dimensional surface plot of Eq.͑1͒,as is shown in Fig.2.Thefigure clearly indicates that in the second and fourth quardrants,when the relative and abso-lute velocities have opposite signs,the damper force is zero.Oth-erwise it is proportional to the absolute velocity.Figure2also shows that the damping force jumps sharply at the transitions between the quardrants corresponding to the relative velocity zero crossings,which is further depicted in Fig.3.This force disconti-nuity causes acceleration jumps or jerks in various physical sys-tems,such as the single degree-of-freedom system shown in Fig. 4,which reptresents a heavy-truck seat suspension.Skyhook Function.The continuous equationC saϭI sϩe fϪeϪfI on e fϩI o f f eϪf(2)is proposed to represent the damping coefficient of the semiative suspension,which can be adjusted in real time.The constants I s and I on are such that they are small as compared to I o f f,i.e.,I sӶI o f fandI onӶI o f fContributed by the Technical Committee on Vibration and Sound for publicationin the J OURNAL OF V IBRATION AND A COUSTICS.Manuscript received November2003;final revision,February2004.Associate Editor:Lawrence A.Bergman.580ÕVol.126,OCTOBER2004Copyright©2004by ASME Transactions of the ASMEThe exponential power f represents the switching function of the semiactive suspension.In the case of skyhook control,it is a func-tion of relative and absolute velocity,i.e.,f ϭK v 1v 12(3)where K is a positive constant.In practice,the selection of the constants in Eqs.͑2͒and ͑3͒would be according to the force capabilities of the adjustible damper that is used for the semiactive suspension.Such selections,which must be done on a case-by-case basis,will have little effect on the comparison of the damper forces that will be dicussed later and,as such,will not be further elaborated on here.Figure 5shows a surface plot of the damping force with respect to the absolute and relative velocity.In Fig.5,it can be observed that smooth transition of the damping force from one quardrant to another is achieved,as compared to Fig.2.A similar difference can be observed between Figs.6and 3.Both Figs.5and 6indi-cate that the skyhook function control would cause substantially lower dynamic jerk.Based on Figs.2and 3,the approaches for reducing or elimi-nating the dynamic jerks could include:ing an analytic continuous function instead of the sky-hook control,shown in Eq.͑1͒,in order to avoid the damp-ing force discontinuity2.Modifing the skyhook control policy to enable smooth tran-sition of damper current,even when the damper is not at its true relative velocity zero crossing No-Jerk Skyhook.An alternative formulation to Eq.͑1͒for reducing dynamic jerk isC sa ϭͭK ͉V 1V 12͉V 1V 12у00V 1V 12Ͻ0(4)where K is a constant gain.This formulation is quite similar to the skyhook control shown in Eq.͑1͒,except that the dampingcoef-Fig.1Skyhook damping forceillustrationFig.2Surface plot of damper force for conventional imple-mentation of skyhook control using a ratefilterFig.3Damping force discontinuities resulting from conven-tional implementation of skyhook control using a ratefilterFig.4A single-dof base-excited system:…a …schematic model,…b …physical system in the form of a heavy-truck seat suspen-sionJournal of Vibration and Acoustics OCTOBER 2004,Vol.126Õ581ficient is a function of both absolute and relative velocity.As is shown in Figs.7and 8,including the relative velocity in skyhook control formulation eliminates damping force discontinuity and dynamic pared with Fig.2,Fig.7shows that the damp-ing force has a smooth transition between the four quadrants.The same phenomenon is observed in comparing Figs.3and 8.Of course,reducing the force discontinuity reduces dynamic jerks,as will be shown in our laboratory test results,which are included in the next sections.Test SetupThe truck-seat suspension with a magnetorheological damper,shown in Fig.4,is used for laboratory evaluation of the methods that were suggested earlier for reducing jerk.The test setup that was used for this study is shown schematically in Fig.9,and further documented by Song ͓13͔and Reichert ͓14͔.The seat is excited at the base by a hydraulic actuator with a force capacity of 2000lb.The sprung mass is supported on a suspension that con-sists of an airspring and an adjustable magnetorheological ͑MR ͒damper.The airspring has a linear spring constant within the range of the relative displacement that was used in our tests.The MR damper force is directly proportional to the current supplied to the damper.An MR damper was used instead of amechanically adjustable damper because of its availability,ease of use,and the growing trend of using such dampers for suspension applications.A standard input profile known as ISO Class 2Excitation ͑also known as ‘‘ISO2Excitation’’͒,documented in ͓15͔,was used for exciting the system.The absolute velocity needed for skyhook control was derived from an accelerometer on the sprung mass.The relative velocity across the suspension was estimated from the relative displacements across the suspension,which was mea-sured by a rotary sensor that was mounted at the cross-pivot of the scissor suspension,for the truck-seat suspension in Fig.3.Test ResultsThe test results are shown in Figs.10–13.These figures show the results of three different implementation of the skyhook con-trol that are described below,using the terminology that we have selected for this study:1.‘‘Rate Skyhook,’’is the implementation of the skyhook con-trol in Eq.͑1͒,using a rate filter that is able to estimate velocity across the suspension from relativedisplacementFig.5Surface plot of damper force for implementation of sky-hook control using skyhookfunctionFig.6Elimination of damping force discontinuties due to us-ing skyhookfunction Fig.7Surface plot of no-jerk skyhook control damperforceFig.8Elimination of damping-force discontinuities due to us-ing no-jerk skyhook formulation582ÕVol.126,OCTOBER 2004Transactions of the ASMEmeasurement,as described earlier.This is the implementa-tion of skyhook control frequently adopted for vehicle sus-pensions and similar systems.2.‘‘Skyhook Function,’’described by Eqs.͑2͒and ͑3͒,is one of the two alternative implementations of skyhook control suggested in this study.3.‘‘No-jerk Skyhook,’’is the other suggested alternative implementation of skyhook control,described by Eq.͑4͒.First,we investigate the currents to the MR damper as shown in Figs.11–13.It is of no surprise that the skyhook function is quite similar to the no-jerk skyhook.Both control policies have a smooth transition of the current to the MR damper from the off to the on states because the new modified skyhook control policies take advantage of the fact that at least one velocity changes smoothly from the on to the off state ͑or vice versa ͒.They gener-ate a smooth dynamic response,as shown in Figs.10and 12.These figures show that both the no-jerk skyhook and skyhook function-control policies cause a relatively smooth dynamic re-sponse because they eliminate the damping-force discontinuity.ConclusionsTwo different implementations of skyhook control,named ‘‘skyhook function’’and ‘‘no-jerk skyhook,’’were suggested for reducing the dynamic jerk that is often experienced with conven-tional skyhook control in suspension systems.Through an analysis of the relationship between the absolute velocity of the sprung mass and the relative velocity across the suspension,we showed the source of force discontinuity in conventional implementation of skyhook control,called ‘‘rate skyhook,’’for the purpose of this study.This analysis showed that at the zero crossings of the rela-tive velocity,rate skyhook introduces a jump in the damping force that causes a jump in sprung mass acceleration.This acceleration jump,or dynamic jerk,causes a significant reduction in isolation benefits that can be offered by skyhook suspensions.Two alterna-tive implementations of skyhook control included in the study offer modifications to the formulation of conventional skyhook control such that the damping-force jumps are eliminated.One alternative policy introduced a continuous function that can be used in place of the conventional formulation of skyhook control,which is described as a bistate function in terms of the products of the absolute velocity of the sprung mass and the suspensionrela-Fig.9Implementation of skyhook control on a physical sys-tem representing a single-dof base-excited system with a semi-activesuspensionFig.10Sprung mass accelerations caused by different sky-hook control formulations in response to a 1.45Hz pure-tone baseexcitation Fig.11Comparison of electrical currents supplied to a mag-netorheological damper in a semiactive suspension in re-sponse to a 1.45Hz pure-tone baseexcitationFig.12Sprung mass accelerations caused by different sky-hook control formulations in response to a broadband …ISO2…base excitationJournal of Vibration and Acoustics OCTOBER 2004,Vol.126Õ583tive velocity.Another alternative policy maintains the bistate for-mulation of the conventional skyhook control,but introduces the relative velocity of the suspension in the force-switching formu-lation such that the force discontinuities are eliminated.The two alternative methods were compared with rate skyhook control,in terms of the relationship between the damper force and the relative and absolute velocities of the system in order to show the elimination of the force discontinuity at the zero crossings of the relative velocity across the suspension.Additionally,the alter-native policies were compared with conventional skyhook control using a laboratory setup that included a heavy-truck seat suspen-sion representing a base-excited single-dof system.An evaluation of the seat acceleration,as well as the electrical current supplied to the magnetorheological damper that was used for this study,clearly showed the elimination of jerk with the alternative sky-hook controls.References͓1͔Karnopp,D.C.,and Cosby,M.J.,1974,‘‘System for Controlling the Trans-mission of Energy Between Spaced Members,’’U.S.Patent 3,807,678.͓2͔Crosby,M.J.,and Karnopp,D.C.,1973,‘‘The Active Damper,’’Shock Vib.Bull.,43.͓3͔Carlson,J.D.,Catanzarite,D.M.,and Clair,K.A.St.,1995,‘‘CommercialMagneto Rheological Fluid Devices,’’International Conference On Electro-Rheological,Magneto rheological Suspensions and Associated Technology ,Sheffield.͓4͔Carlson,J.D.,and Chrzan,M.J.,1994,‘‘Magneto Rheological Fluid Damp-ers,’’Patent No.5,277,281.͓5͔Ahmadian,M.,and Marjoram,R.H.,1989,‘‘Effects of Passive and Semi-active Suspensions on Body and Wheelhop Control,’’mercial Veh.,98,pp.596–604.͓6͔Ahmadian,M.,and Marjoram,R.H.,1989,‘‘On the Development of a Simu-lation Model for Tractor Semitrailer Systems With Semiactive Suspensions,’’Proc.of the Special Joint Symposium on Advanced Technologies ,1989ASME Winter Annual Meeting,San Francisco,DSC13,ASME,New York.͓7͔Chalasani,R.M.,1986,‘‘Ride Performance Potential of Active SuspensionSystems—Part II:Comprehensive Analysis Based on a Full-Car Model,’’Pro-ceedings of the 1986ASME Winter Annual Meeting ,Los Angeles,ASME,New York.͓8͔Chalasani,R.M.,1986,‘‘Ride Performance Potential of Active SuspensionSystems—Part I:Simplified Analysis Based on a Quarter-Car Model,’’Pro-ceedings of the 1986ASME Winter Annual Meeting ,Los Angeles,ASME,New York.͓9͔Ahmadian,M.,1999,‘‘On the Isolation Properties of Semiactive Dampers,’’J.Vib.Control,5͑2͒,pp.217–232.͓10͔Krasnicki,E.J.,1980,‘‘Comparison of Analytical and Experimental Resultsfor a Semiactive Vibration Isolator,’’Shock Vib.Bull.,50.͓11͔Miller,L.R.,1988,‘‘An approach to Semiactive Control of Multiple-Degree-of-Freedom Systems,’’Ph.D.thesis,North Carolina State University,Raleigh,NC.͓12͔Ivers,D.E.,and Miller,L.R.,1991,‘‘Semiactive Suspension Technology:AnEvolutionary View,’’ASME Advanced Automotive Technologies ,DE-40,Book No.H00719-1991,ASME,New York,pp.327–346.͓13͔Song,X.,1999,‘‘Design of Adaptive Vibration Control Systems With Appli-cation to Magneto-Rheological Dampers,’’Ph.D.Dissertation,Virginia Poly-technic Institute and State University,Blacksburg,V A.͓14͔Reichert,B.A.,1997,‘‘Application of Magneto Rheological Dampers forVehicle Seat Suspensions,’’M.S.thesis,Virginia Polytechnic Institute and State University,Blacksburg,V A.͓15͔International Standards Organization,1994,‘‘Earth-Moving Machinery—Laboratory Evaluation of Operator Seat Vibration,’’Standard 7096,ISO.Fig.13Comparison of electrical currents supplied to a mag-netorheological damper in a semiactive suspension in re-sponse to a broadband …ISO2…base excitation584ÕVol.126,OCTOBER 2004Transactions of the ASME。