research article vibration attenuation of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 242476 13 pageshttpdxdoiorg1011552013242476

Research ArticleVibration Attenuation of Magnetorheological Landing GearSystem with Human Simulated Intelligent Control

X M Dong1 and G W Xiong2

1 State Key Laboratory of Mechanical Transmission Chongqing University Chongqing 400044 China2 School of International Business Sichuan International Studies University Chongqing 400031 China

Correspondence should be addressed to X M Dong xm dong126com

Received 8 August 2013 Accepted 13 October 2013

Academic Editor Sebastian Anita

Copyright copy 2013 X M Dong and G W XiongThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

Due to the short duration of impulsive impact of an aircraft during touchdown a traditional landing gear can only achieve limitedperformance In this study a magnetorheological (MR) absorber is incorporated into a landing gear system an intelligent controlalgorithm a human simulated intelligent control (HSIC) is proposed to adaptively tune the MR absorber First a two degree-of-freedom (DOF) dynamic model of a landing gear system featuring an MR absorber is constructed The control model ofan MR damper is also developed After analyzing the impact characteristic during touchdown an HSIC is then formulated Agenetic algorithm is adopted to optimize the control parameters of HSIC Finally a numerical simulation is performed to validatethe proposed damper and the controller considering the varieties of sink velocities and sprung masses The simulations underdifferent scenarios show that the landing gear system based on the MR absorber can greatly reduce the peak impact load of sprungmass within the stroke The biggest improvement of the proposed controller is over 40 compared to that of skyhook controllerFurthermore HSIC exhibits better adaptive ability and strong robustness than skyhook controller under various payloads and sinkvelocities

1 Introduction

The very important design issue of landing gear systems withan adaptive ability of impact energy mitigation and simulta-neous generation of minimal deceleration on the protectedaircraft structure is an intensive research topic To compassall impact scenarios (ie a wide range of flight parameterssuch as ground friction coefficient aircraft overall weightattitude and sink velocity) a traditionally passive landinggear has to be employed to meet the requirement of the mostheavy dynamic excitation or harsh environmental loading Asa result it cannot avoid the highly redundant structure andtrade-off performance To attenuate the landing impact trans-mitted to aircraft some active or semiactive types of landinggear systems have been suggested in aircrafts One of themostattractive candidates to formulate an adaptive landing gear isto use magnetorheological (MR) fluid absorber because of itsfast response characteristic to magnetic field compact sizeand hence wide control bandwidth [1ndash3]

The MR technology brings us salient features in allevi-ating the underdesirable impact load on aircrafts and pilotson one hand On the other hand the significant hysteresiticnonlinearity of the MR absorbers as well as dynamic modeluncertainties of the landing gear system (eg variable sinkvelocities and payloads) and especially short time impact(typically a touchdown ranging from 50 and 200ms) providesyet a significant challenge in choosing a suitable controlstrategy

So far many control methods have been investigatedby many researchers in recent years through numericalsimulations or simple experimental studies to deal with thechallenge There are two types of control strategies One isthat control strategy is formulated based on energy balanceFor example Mikułowski and Jankowski [4] investigatedtwo control strategies (semiactive and active) to mitigatethe peak force transferred to the aircraft structure Thesemiactive control strategy is formulated based on a presetorifice area for different touchdown scenarios Ghiringhelli

2 Mathematical Problems in Engineering

[5] discussed the possibilities of the semiactive control oflanding gears According to the energy balance the optimalcontrol force is calculated The researches [4ndash7] indicate thatthe control method based on energy balance is simple andeffective and easy to understand for the single degree oflanding gear system When higher degree dynamic landinggear system or practical case is applied the control strategymay have difficulty in calculating the energy sometimes itis even impossible considering the many uncertainties andnonlinearity in a practical landing gear system

The other type is the nonlinear control strategies consid-ering the nonlinearity and uncertainty of the system Choiand Wereley [3] investigated the feasibility and effectivenessof electrorheological (ER) and MR fluid-based landing gearsystems on attenuating dynamic load and vibration usingsliding mode control Wang and Carl [8] proposed thefuzzy logic control for the semiactive landing gear systemnumerical results showed that the maximal structural loadcan be reduced with the semiactive landing gear Howeverthe control idea needs many sensors to sense the statesof landing gear system which will increase the cost andcomplexity of system

Consequently the main purpose of this study is to finda simple and effective control strategy to improve the ridecomfort and stability of landing gear system simultaneouslythe control can potentially be employed in a practical caseIn the prior work [3 9] we have already investigated alanding gear system featuring MR fluids and demonstratedits feasibility and effectiveness on attenuating dynamics loadand vibration due to the landing impact As a continuationof the prior work this study will propose a new controlstrategy a human simulated intelligent control (HSIC) forthe landing gear system The control strategy has beensuccessfully applied in the MR semiactive suspension system[10 11] Many numerical and actual road tests show theeffectiveness in reducing the vibration of vehicle suspensionand improving the ride comfort

To accomplish this goal a two-DOF dynamic model ofaircraft incorporated with an MRF-based landing gear isfirstly constructed to simulate the course of touchdown Afteranalyzing the response of a classic touchdown the HSICis then proposed Subsequently the numerical simulation isperformed to validate the effectiveness of the control strategyunder different sink speeds and masses In comparison theskyhook control is also formulated Finally a great deal ofnumerical simulations considering variable masses and sinkvelocities are performed to evaluate the effectiveness of theproposed control algorithm

2 Modeling and Control Formulation ofLanding Gear System Featured MR Absorber

21 MR Absorber In the work [3] the authors proposed onetype of controllableMR absorber based on flow-mode whichis schematically shown in Figure 1 The absorber consistsof working and gas compensation reservoirs The workingreservoir is separated into two chambers by the piston inwhich an MR fluid is filled There is an annual valve across

Pneumatic reservoir

Floating piston

Piston rod

Colis

Figure 1 Schematic diagram of an MR absorber [3]

the piston with a winding magnetic coil By the motion of thepiston the MR fluid will flow from one chamber to the otherIn the absence of external magnetic field the MR fluid canfreely flow like linear viscous fluid whereas the rheology ofthe MR fluid will undergo significantly controllable changeswhile applying a magnetic field To compensate the bulkdifference occupied by the piston rod during compressionor rebound course the pressured gas is filled in the gascompensation reservoirs

Typically the damping force of an MR absorber can beexpressed as a combination of viscous damping force andcoulomb damping force [10]

119865 = 119888119904V + 119865MR sign (V) (1)

where 119888119904denotes the damping constant due to viscous damp-

ing V is the relative velocity of MR absorber or piston rodrelative to the absorber body 119865MR represents the controllablecoulomb damping force which is a function of externalmagnetic field or electric current and sgn() indicates the signfunction

22 Two-DOF Landing Gear Systemwith anMRAbsorber Toformulate control strategy a suitable dynamic model needsfirstly to be constructed In this study one of two DOFdynamic models of a landing gear system is adopted inFigure 2 [9] In this model the landing object is treated asrigid body which is slightly different to the actual case Theinternal characteristics (stiffness and damping properties) areneglected in this study although they are very important forthe final landing scenario [7] However the simple model issufficient to formulate the control strategy and themodel candescribe main dynamic behaviors of a landing gear duringtouchdown

In this figure the sprung mass 119898119904represents the mass

of aircraft body and 119898119906denotes the mass of tire The MR

absorber is employed between sprung mass and tire it is alsoin parallel with a coil spring

Mathematical Problems in Engineering 3

ms

ks cs

zs

zu

fus

FMR

mu

Figure 2 Dynamic modeling of an MR landing gear

The motion equations of the landing gear system can bewritten as

119898119904119904minus 119898119904119892 + 119896119904(119911119904minus 119911119906) + 119888119904(119904minus 119906) + 119865MR + 119871

119891= 0

119898119906119906minus 119898119906119892 minus 119896119904(119911119904minus 119911119906) minus 119888119904(119904minus 119906) minus 119865MR + 119891

119906119904= 0

(2)

where 119892 is the gravitation acceleration 119911119904 119904 and

119904rep-

resent the absolute displacement velocity and accelerationof sprung mass respectively 119911

119906 119906 and

119906represent the

absolute displacement velocity and acceleration of unsprungmass respectively and 119871

119891is the lift force and

119871119891= 119897119892 (119898

119904+ 119898119906) (3)

Here 119897 is the lift factor and is chosen to be 0667 [7] 119891119906119904

describes the nonlinear spring force of a tire given by 119903119911119899119906 In

this study 119903 = 480000 and 119899 = 1365 [12] V0represents the

initial sink velocityIf the state of the landing gear is set as x = [119911

119904 119904 119911119906 119906]119879

and control input is 119906 = [119865MR] then (2) can be expressed as

x = Ax + B119906 + f (x) (4)

where

A =

[[[[[[[

[

0 1 0 0

minus119896119904

119898119904

minus119888119904

119898119904

119896119904

119898119904

119888119904

119898119904

0 0 0 1

119896119904

119898119906

119888119904

119898119906

minus119896119904

119898119906

minus119888119904

119898119906

]]]]]]]

]

B =

[[[[[[

[

0

1

119898119904

0

minus1

119898119906

]]]]]]

]

f (x) =

[[[[[[[

[

0

119892 minus

119871119891

119898119904

0

119892 +119891119906119904

119898119906

]]]]]]]

]

(5)

3 Formulation of Controller

In (4) there are significant nonlinearity of MR absorber anduncertainties such as the mass of sprung mass and the initial

Actual acceleration

Desired acceleration

Time (s)

Acce

lera

tion

(ms

2)

10

5

0

minus5

minus10

minus150 02 04 06 08 1 12 14 16 18 2

Figure 3 A classic acceleration time history of sprung mass

sink velocity which result in the difficulty of formulating asimple and effective control strategy In our prior work [9ndash11] we investigated the HSIC in reducing the vibration ofMR suspension system through simulation and road testTheresults show that the control strategy is effective not only insteady dynamic course but also in transient dynamic courseInspired by our prior work the control idea is applied to thelanding gear system in this study

The HSIC algorithm inspired by the excellent controlexperience of an expert is proposed by Zhou and Baiearly in 1983 [13] With many years of development thetheory has been successfully in many perplexing processesand plants (such as delay system multivariable system andnonlinear system) The main and important characteristicof the theory consists of characteristic identification andmultimode control which simulate the humanrsquos intuitionreasoning namely from identification to decision and fromdecision to operation [11 14]

Generally speaking the design procedure of applying thetheory includes the analysis of controlled plant to obtain thedesired phase portrait and the design of controller In thefollowing the two steps will be discussed in order

31 Determination of Desired Portrait of Motion As a heli-copter landing on ground a classic acceleration time his-tory of sprung mass is shown in Figure 3 There are twomain control tasks for the landing gear system One is toreduce the maximum deceleration or impact forceThe otheris to shorten the adjusting time Obviously the dampingrequirement is different for the different control tasks Asingle control strategy is difficult to implement the twocontrol tasks simultaneously Therefore it is necessary toapply different control model through simulating expertsrsquocontrol experience

For the first control task it is helpful to consider themanrsquosreaction to a jump To avoid the injure caused by the bigimpact from the land a person has to bend his leg as soonas possible which inspires us to make full use of stroke ofMR absorber Moreover a human body is more sensitive to


Figure 4The desired phase portrait of displacement and velocity ofsprung mass

acceleration which means that it is better for maintaining aconstant deceleration

For the initial sink velocity V0 we assume the deceleration

to be kept constant desired 119886119889 The motion velocity is

119904= V0minus 119886119889119879 (6)

where 119879 is the time when the MR absorber reaches themaximum stroke 119904max and is given by

119879 = radic2119904max119886119889

(7)

Substituting (7) into (6) and integrating gives

119911119904= int

119879

0

119904119889119905 = V

0119879 minus

1

21198861198891198792

= minus1

2119886119889

(119904)2

+1

2119886119889

(V0)2

(8)

From (8) the desired phase portrait of displacement andvelocity of sprung mass in time domain is shown in Figure 4The blue light is a quadratic function curve Before reachingthe maximum displacement of sprung mass a constantdeceleration needs to be maintained To achieve the mostdesirable control performance the maximum should alwaysnearly be equal to the maximum stroke of the MR absorber

The calculated or estimated desired deceleration is veryimportant for improving control performance Consideringthe nonlinearity and uncertainty of landing gear systemthe method cannot be used anymore In this study we givean effective and simple method to estimate the infimum ofdesired acceleration if assuming that the desired decelerationcan be guaranteed and the effective stroke equals the maxi-mum stroke of theMR absorberThe desired deceleration canbe calculated from (9) as

119886119889=

V20

2119904max (9)

A genetic algorithm can be employed to optimize the decel-eration as an alternative

As mentioned earlier the main control purpose is tomaintain a constant deceleration during touchdown There-fore we can divide the course of touchdown into two phasesin time domain One is to reduce the maximum impactacceleration the other is to avoid a big overshoot It isobvious that different controlmodals are desirable for the twocontrol phases Considering the effectiveness and potentialapplication the acceleration feedback control is applied forreducingmaximum impact accelerationwhereas the skyhookcontrol is adopted for reducing a big overshoot becauseskyhook control is very effective in vibration attenuation insemiactive vibration control Two control modes needs onlyto sense the acceleration and velocity signal which is easilyapplied in practical case

According to the above discussion it is found thatthe acceleration feedback control and skyhook control maybe effective in different phases How to coordinate themand optimize control parameters will be discussed in thefollowing

32 Development of HSIC Since the HSIC theory wasproposed the theory has advanced by Li and Wang [14]using schema theory The term schema is originated fromcognitive science which refers to a mental structure weuse to organize and simplify our knowledge of the worldaround us The schema-based HSIC is a multilevel hierar-chical structure information processing system Each controlmode is made up of several mode units and several controlmodes Every control unit consists of direct control levelparameter correction level and task adjustment level Theentire running control level forms amotor sensory intelligentschema (SMIS)

321 Running Control Level

(1) Sensed Schema

Definition 1 (sensed schema) It is an intelligent modulewhich can be formulated by an agentrsquos repeated learning andaccumulating control experiences and its structure is shownin Figure 5

In this figure the input information set 119877 includes thestate signal from the plant itself or external environmentThe state signal is the basis of selecting control action Inthis study 119877 = 119905

119904 119906 isin Σ

3 consists of the control timeinformation and sink velocities of sprungmass and unsprungmass

Similar to mankindrsquos reaction to itself state or externalenvironment the input informationwill be sensed and sent tothe block of characteristic identification The function of theblock of characteristic identification is to classify the sensedinput information The outputs of the block of characteristicidentification are some characteristic elements which willclassify the state space into different control regions Theset of all characteristic elements are named as characteristicprimitive set 119876

119876 = 1199021 1199022 1199023 1199024119879

(10)


Figure 5 Structure of sensed schema

in which 1199021= 119905 if 119905 le 119879 119902

2= 119905 if 119905 gt 119879 119902

3= 119904(119904minus 119906) if

119904(119904minus 119906) ge 0 and 119902

4= 119904(119904minus 119906) if 119904(119904minus 119906) lt 0

Combining the elements of the characteristic primitiveset 119876 through a logic operator otimes characteristic modes canbe formulated otimes is defined as 119870 otimes 119876 = cap

44

119894=1119895=1119902119894119896119894119895 119896119894119895is an

element of 119870 and can be valued as minus1 0 and 1 which meanstaking negative zeros and positive Operational symbol isan ldquoandrdquo relation between two characteristic primitives Acharacteristic mode is one kind of division of the systemrsquosdynamic information space Σ according to the solution goalof the control problem and difference in control specifica-tions which is corresponding to the selection of the relationmatrix For example the HSIC can recognize that the landinggear system is in the phase of crash and the control modal ofacceleration will be automatically activated if 119896

1= 1 0 1 0

In this study 119870 = [

1 0 1 0

1 0 0 1

0 1 1 0

0 1 0 1

] All characteristic modes will be

combined into a characteristic mode set Φ which is also theoutput of the sensed schema

The final output is the characteristic model set and can begotten as

Φ = 119870 otimes 119876 = 1205931| 119905 le 119879

119904(119904minus 119906) ge 0

1205932| 119905 le 119879

119904(119904minus 119906) lt 0

1205933| 119905 gt 119879

119904(119904minus 119906) ge 0

1205934| 119905 gt 119879

119904(119904minus 119906) lt 0

(11)

In summary the mapping relation of the sensed schemabetween the input information space and the output

characteristic mode set can be simply described by a five-element-group with order

119878119875= ⟨119877119876119870 otimesΦ⟩ (12)

The output of the sensed schema will be the input of thefollowing associated schema

(2) Motion Schema

Definition 2 (motion schema) It is a stereotype embracedcontrol strategy based on outside information of system anditself inner states and its structure is shown in Figure 6

The input information of the motion schema is the sameas that of sensed schema The physical sense of the motionschema is in the fact that it simulates how to adopt someactual control modes according to the sensing informationlike mankind The function of the block of construct is toadaptively select the required feedback signal of the controlmode

The selection of control modes is dependent on thecontrol tasks Usually there are some traditional controlmodes such as proportional control primitive 119896

119901119890 differential

control primitive 119896119889119890 integral control primitive 119896

119894int 119890119889119905 and

keeping control primitive 119896ℎsum119899

119894=1119890119898119894 According to the above

analysis the acceleration feedback and skyhook control maybe a better choice in this study Therefore the control modeprimitive set 119875 can be written as 119875 = [119901

1 1199012]119879

isin Σ2 where

1199011denotes the acceleration feedback of minus119896

119886(119886119889minus 119904) and 119901

2is

the skyhook control mode of 119896sky119904


Input R

OutputAcceleration

feedback

Skyhookcontrol

MR landing

gear system

Feedback

Asso

ciat

edsc

hem

a

Fusio

nop

erat

ion

Con

struc

t

Con

trol t

ime

info

rmat

ion

Control modeprimitive set P

Control mode set Ψ

Motion schema SM

Sens

edsc

hem

a

Con

trol

mod

eu1

Con

trol

mod

eu2

Con

trol

mod

eu4

payload

mass

Initi

al si

nkve

loci

ty o

runsprung

mass

Velo

city

of

Velo

city

of

spru

ng m

ass

Figure 6 Structure of motion schema

The block of fusion operation is to logically combine thecontrol modes There is an operation relation between thecontrol mode primitive set and the control mode set

Ψ =

[[[

[

1199061

1199062

1199063

1199064

]]]

]

= 119871119875 =

[[[[[[[[[[[[[

[

2

sum

119895=1

1198971119895119901119895

2

sum

119895=1

1198972119895119901119895

2

sum

119895=1

1198973119895119901119895

2

sum

119895=1

1198974119895119901119895

]]]]]]]]]]]]]

]

=

[[[

[

minus119896119886(119886119889minus 119904)

0

119896sky1199040

]]]

]

(13)

in which 119897119894119895is an element of 119871 and can be valued as ndash1

0 and 1 which denote taking negative zero and positive

respectively In this study119871 = [

1 0

0 0

0 1

0 0

]The zero output is due to

the semiactive features of the MR damper The control modeset is connected to the MR landing gear system through agroup of logic switchThe logic rules will be generated by theassociated schema

For brevity the motion schema can be formulated as afive-element group with order

119878119872= ⟨119877 119875 119871 Ψ 119880⟩ (14)

(3) Associated Schema

Definition 3 (associated schema) It is used to coordinate therelation between sensed schema and motion schema whichsimulates human beingrsquos intuition reasoning and decisionprocess The structure is shown in Figure 7

In this figure once the sensed schema andmotion schemaare obtained through the block of computation the function

mapping relation among the characteristic mode set andcontrol mode set is determined by the block of intuitionreasoning and decisionmakingThe physical sense is that onemankind has a double mapping when doing something Hefirstly needs to judge qualitatively the object and then adoptsquantitatively control behavior The output of the block ofintuition reasoning and decisionmaking is some logic valueswhich will activate the corresponding the control mode if thelogic switch is true

Generally the function of the association schema is toobtain production rules ldquoIF Then rdquo The process can beexpressed as

Ω Φ 997888rarr Ψ (15)

where Ω = 1205961 1205962 1205963 1205964 120596119894= if 120593

119894then 120595

119894 isin Σ

4

(qualitativemapping) andΨ 119877 rarr 119880Ψ = [1205951120595212059531205954]119879

(quantitative mapping)

322 ParameterAdjustment Level It is obvious that theHSICwith the fixed control parameter cannot be able adaptivelyto meet the requirement of all impact scenarios of landinggear system Therefore the control parameter 119896

119886in (13) is

dependent on the sink velocity and mass of sprung massTherelation will be determined by genetic algorithm in Section 4

323 Control Structure of HSIC After designing the sensedschema motion schema and associated schema the entirecontrol structure ofHSIC can be combined as a three-elementgroup with order (which is shown in Figure 8)

119878KG = ⟨119878119901 119878119872 119878119860⟩ (16)


Input RSensed schema SP

Characteristicmode 1205931



Controlmode

Controlmode

Controlmode

Intu

ition

reas

onin

g an

d

Com

puta

tion

Com

puta

tion

Output

Motion schema SM

payload

mass

Initi

al si

nkve

loci

ty o

rC

ontro

l tim

ein

form

atio

nunsprung

mass

Velo

city

of

Velo

city

of

spru

ng m

ass

1205951

1205952

1205954

deci

sion

mak

ing

Figure 7 Structure of associated schema

ms

ks cs

mu

HSIC controller

Parameteradjustment

level

Runningcontrol

levelLanding

gearsystem

SA

SMSP

ad kaFMR

Figure 8 The HSIC controller structure-based SMIS for MRlanding gear system

According to above discussion the final control law canbe simply written as in Algorithm 1

4 Control Parameter Optimization

Control parameters of the HSIC that need to be determinedinitially include 119886

119889and 119896

119886according to the impact scenario

of landing gear system Usually the course of determinationfor those threshold value parameters needs much trial orerrorTherefore in this paper a genetic algorithm is proposedto tune the parameters through the Matlab 70Simulink InSection 31 the issue of desired deceleration is discussed andthe value can be calculated simply from (9) To estimate theaccurate value of desired deceleration the genetic algorithmis also used to search the optimal deceleration In summarythere are three control parameters 119896

119886 119896sky and 119886

119889of HSIC

that needed to be optimized Figure 9 gives the optimal proce-dure of control parametersThe objective of optimization is toachieve the minimal deceleration of sprung mass As a result

IF 119905 le 119879

IF 119904(119904minus 119906) ge 0

119865MR = minus119896119886(119886119889minus 119904)

Else119865MR = 0

ElseIF 119904(119904minus 119906) ge 0

119865MR = 119896sky119904Else

119865MR = 0

End

Algorithm 1

the fitness of genetic algorithm is evaluated by the maximumdeceleration during each simulation of dynamic model Thenumerical time is set to 2 seconds for each simulation

The real number encoding is adopted in this paper Thelength of chromosome is determined by the number of thecontrol parameters If (119903

1 1199032 1199033) is the optimal solution the

initial chromosomes can be obtained by

119903119894= 119897119894+ 120573 (119906

119894minus 119897119894) (17)

where 119897119894and 119906

119894are the lower limit and the upper limit

of optimized parameters respectively 120573 is the proportionalfactor

To improve control performance the fitness function ischosen as the maximum impact deceleration The crossoveroperators used here are one-cut-point crossover by convexcrossoverThemutation operator used in this paper is convexcombination The parameters used in the simulation of


Selection

Crossover

Mutation

Reach the maximumgeneration or meet

convergent condition

Output

Yes

No

Simulation

Initialparameters

Fitnesscalculation

SimulationFitnesscalculation

Figure 9 The optimal procedure of control parameters

the genetic algorithm are population size = 50 mutationprobability = 03 and crossover probability = 01 As anillustration Figure 10 shows the convergent course of thedesired deceleration for the sink velocity of 2ms and sprungmass of 800 kg It can be found that the convergent is veryfast

To determine the specific function relations betweendesired deceleration or acceleration feedback gain and sinkvelocity as well as sprung mass many simulations are per-formed considering all kinds of combinatorial sink velocityas well as sprung mass The results are shown in Figures11 and 12 From Figure 11 it can be found that the desireddecelerationwill growwith the increment of sink velocity anddecrease slightlywith the increment of sprungmass Figure 12gives the relation of desired acceleration feedback gain withsink velocity and sprung mass It can be seen that the sprungmass has bigger effect on the acceleration feedback gain thanthe sink velocity

To reduce the complexity of computation the linearsurface fitting method is adopted in this study The simple

0 10 20 30 40 50115

1155

116

1165

117

1175

118

1185

Generation

a d(m

s2)

Idea

l dec

eler

atio

n

Figure 10 The convergent procedure of fitness value with genera-tions

400600

8001000

1200

1

2

3

40

10

20

30

40

a d(m

s2 )

Initial sink velocity 0 (ms) Sprung mass (kg)

ms

Des

ired

idea

l dec

eler

atio

n

Figure 11The relation of desired decelerationwith sink velocity andsprung mass

400600

8001000

1200

1

2

3

40

Initial sink velocity 0 (ms)

2000

4000

6000

8000

Des

ired

acce

lera

tion

feed

back

gai

n

Sprung mass (kg)ms

ka(N

(ms

2))



specific function relations can be evaluated and expressed asfollows

119886119889= 779V

0minus 00086119898

119904+ 31536 (18)

119896119886= 574V

0+ 623119898

119904+ 59244 (19)

Themethod of online identifications of V0and119898

119904is studied by

some researchers [15] Once the values of V0and119898

119904are deter-

mined the feedback gain can be easily calculated Figures 13and 14 give the results after surface fitting According to thesefigures or (18)-(19) the control parameters can be adjustedadaptively with the change of impact scenarios

5 Numerical Simulation and Discussion

To evaluate the control performance and adaptive abilityof MR landing gear system we constructed the numericalmodel based on MatlabSimulink The system and controlparameters employed in this study are given in Table 1

In order to figure out the sensitivity of the desired decel-eration value on the control performance thirteen differentdeceleration values are applied in the control numericalexperiment The results are shown in Figure 15 Among allthose deceleration values point A represents the evaluateddeceleration value according to (18) It can be found that allsituations can achieve good control performance It is notedthat the best control performance is obtained not at point Abut at the point B due to the linear surface fitting error of (18)Nevertheless the results indicate that the HSIC has strongrobust considering the difficulty in determining the value ofdesired deceleration in real environment

To simulate the different scenarios of touchdown themass of sprungmass can be increased or reduced so does thesink velocity of landing gear system For comparison purposethe skyhook control law is also adopted and its control gainis also optimized by the genetic algorithm Consider

119906 = 119896sky119904 if

119904(119904minus 119906) gt 0

0 if 119904(119904minus 119906) le 0

(20)

Once the set value of desired deceleration equals or fallsinto a small region nearby the optimal deceleration thecontrol performance of HSIC will be significantly improvedcompared to that of acceleration feedback control skyhookcontrol or passive which is shown in Figure 16 From thisfigure it can be seen that the maximum impact accelerationalmost maintains the value of the desired deceleration Thereduction of maximum acceleration of HSIC is over 11compared to that of passive caseThemaximum impact forcecan also be greatly reduced which means that the discomfortcaused by the strong impact will be significantly alleviatedduring touchdown In addition the control strategy needsalmost the same effective stroke of the skyhook controlAlthough the required stroke of the skyhook control issmaller than that of the passive the impact acceleration iseven bigger than that of the passive This implies that theskyhook controller is good at reducing the damper strokebut is not good at reducing the acceleration reduction for the

0

10

20

30

40

a d(m

s2)

1

2

3

4Initial sink velocity

0 (ms)400

600800

10001200

Sprung mass (kg)ms

Fitte

d id

eal d

ecel

erat

ion

Figure 13The relation of fitted ideal deceleration with sink velocityand sprung mass after surface fitting

400600

8001000

1200

1

2

3


0 (ms)

8000

7000

6000

5000

4000

3000

Sprung mass (kg)ms

ka(N

(ms

2))

Fitte

d ac

cele

ratio

n fe

edba

ck g

ain

Figure 14The relation of fitted acceleration feedback gain with sinkvelocity and sprung mass after surface fitting

Table 1 System and control parameters

Quantity Symbol Value UnitsMass of sprung mass 119898

119904800 kg

Mass of unsprung mass 119898119906

153 kgSpring stiffness of coil 119896

11990480000 Nm

Damping constant of MRabsorber 119888

1199044500 Nsm

Maximum controllabledamping force of absorber 119865MRmax 5000 N

touchdown of the landing gear system From this figure it canbe found that although the acceleration feedback control canreduce the maximum impact acceleration a big oscillationappears after the initial touchdown Therefore the controlstrategy is not suitable for the whole touchdown courseand is not compared to the HSIC and the skyhook controlanymore At last it is noted that the HSIC can also achievethe improvement of vibration of unsprung mass


Table 2 Maximum acceleration reduction compared to passive case

Sprung mass Strategy Sink rate1ms 2ms 3ms 4ms

500 kg Skyhook 566 minus291 minus619 minus771HSIC 2715 1119 707 503

700 kg Skyhook 904 285 minus005 minus154HSIC 3879 1788 1153 821

800 kg Skyhook 808 808 789 774HSIC 1797 1186 932 744

900 kg Skyhook 997 520 263 124HSIC 4159 2094 1293 927

1100 kg Skyhook 1018 634 403 273HSIC 4143 2191 1361 987

Table 3 Maximum stroke reduction compared to passive case


500 kg Skyhook 954 926 895 875HSIC 1329 789 629 538

700 kg Skyhook 849 843 820 804HSIC 1772 1113 897 728

800 kg Skyhook 965 425 153 009HSIC 4110 1976 1242 890

900 kg Skyhook 773 778 762 748HSIC 1888 1230 922 747

1100 kg Skyhook 715 727 715 703HSIC 1771 1242 927 762

minus14 minus12 minus10 minus8 minus6 minus4 minus2 011

115

12

125

13

135

14

145Passive

Max

imum

acce

lera

tion

(ms

2)

A B

C

Variable desired deceleration ad (ms2)

Figure 15 Sensitive analysis of desired deceleration on the controlperformance

There are two gains of skyhook applied in this studyOne is the 1200Nsm which is chosen to make both skyhookcontrol and HISC keep almost the same stroke value so thatthey can compare to each other in terms of the improvementof impact acceleration The corresponding results are shown

in Figure 16The other is 2500Nsm which is chosen in orderto achieve the best improvement of impact decelerationCorrespondingly the results are given in Figure 17 and Tables1 and 2 In addition the feedback control strategy during thewhole crash course is also adopted in comparison

The comparison results of different sink velocities andmasses are demonstrated in Figure 16 and Tables 2 and 3Theresults show that the maximum deceleration will grow withthe increment of sink velocities or decrement of masses TheHSIC exhibits the better adaptive ability In some cases theskyhook control cannot achieve the reduction of impact It isnoted that theHSIC and skyhook control can achieve the bestcontrol performance when the mass of sprung mass changesto 1100 kg for different sink velocities At this situation thecontrol performance of HSIC is also prior to that of skyhookcontrol

6 Conclusion

In this study the semiactive control of a landing gearsystem featured with a MR absorber during touchdown hasbeen investigated An intelligent control strategy humansimulated intelligent control (HSIC) is proposedThe geneticalgorithm is employed to optimize the control parametersof HSIC Numerical simulations considering the variety ofsink velocities and masses have been performed to check


0 05 1 15 2minus15

minus10

minus5

0

5

10

15

Time (s)

Acce

lera

tion

(ms

2)

Acceleration feedbackSkyhook controlHSIC

Off-control

(a) Vibration acceleration of sprung mass (V0= 2ms)


0 01 02 03 04 05minus150

minus100

minus50

0

50

100

Time (s)

Acce

lera

tion

(ms

2)

Off-control

(b) Vibration acceleration of unsprung mass (V0= 2ms)


0 05 1 15 2minus005

0

005

01

015

02

025

03

Time (s)

Stro

ke (m

)

Off-control

(c) Displacement of MR absorber (V0= 2ms)

minus4000

minus2000

0

2000

4000

6000

Dam

ping

forc

e (N

)

0 05 1 15 2

Time (s)

Acceleration feedbackSkyhook control

HSIC

(d) Controllable damping force

0 005 01 015 02 025 03minus1

minus05

0

05

1

15

times104


Off-control

Impa

ct fo

rce (

N)

Stroke (m)

(e) Relation of impact force and stroke

Figure 16 The control performance comparison with sink velocity of 2ms and mass of sprung mass of 800 kg


Skyhook controlHSIC

0

10

20

30

40

50

1 15 2 25 3 35 4

Initial sink velocity (ms)

Off-control

Max

imum

impa

ct ac

cele

ratio

n (m

s2 )

1100 kg

800 kg

500kg

(a) Maximum impact acceleration

1 15 2 25 3 35 4005

01

015

02

025

03

035

04

045

Skyhook controlHSIC


Max

imum

effec

tive s

troke

(m)

1100 kg

800 kg

500kg

Off-control

(b) Maximum stroke of MR absorber

Figure 17 The control performance comparison with different sink velocities and masses

the effectiveness and adaptive ability of the proposed controlalgorithm It can be concluded from the numerical resultsthat

(1) compared to passive or based skyhook control systemlanding gear system the control strategy can effec-tively reduce the maximum impact acceleration andmaintain theminimum change of acceleration duringtouchdown simultaneously mains minimum stroke

(2) the control strategy performs very well and exhibitsstrong robustness in different situations such as vari-able sink velocities and sprung masses

(3) the control strategy can easily be applied in practicallanding gear system

In the future themore precise model ofMR absorber willbe incorporated and then experimental research will also beperformed to check the actual control performance

Acknowledgments

This research is supported financially by the National NaturalScience Foundation of China (Project nos 51275539 and60804018) the Fundamental Research Funds for the CentralUniversities (CDJZR12110058 and CDJZR13135553) and theProgram for New Century Excellent Talents in University(NCET-13-0630) Dr Y T Choi and Professor N M Wereleyin University of Maryland provided the data of theMR shockabsorber and gave some helpful suggestions in improving thepaper These supports are gratefully acknowledged

References

[1] D C Batterbee N D Sims R Stanway and Z WolejszaldquoMagnetorheological landing gear 1 A design methodologyrdquo

Smart Materials and Structures vol 16 no 6 pp 2429ndash24402007

[2] DC BatterbeeND Sims R Stanway andMRennison ldquoMag-netorheological landing gear 2 Validation using experimentaldatardquo Smart Materials and Structures vol 16 no 6 pp 2441ndash2452 2007

[3] Y-T Choi and N M Wereley ldquoVibration control of a landinggear system featuring electrorheologicalmagnetorheologicalfluidsrdquo Journal of Aircraft vol 40 no 3 pp 432ndash439 2003

[4] G Mikułowski and Ł Jankowski ldquoAdaptive landing gear opti-mum control strategy and potential for improvementrdquo Shockand Vibration vol 16 no 2 pp 175ndash194 2009

[5] G L Ghiringhelli ldquoEvaluation of a landing gear semi- activecontrol system for complete aircraft landingrdquoAerotecnicaMissilie Spazio vol 83 no 6 pp 21ndash31 2004

[6] G L Ghiringhelli ldquoTesting of semiactive landing gear controlfor a general aviation aircraftrdquo Journal of Aircraft vol 37 no 4pp 606ndash616 2000

[7] G M Mikułowski and J Holnicki-Szulc ldquoAdaptive landinggear conceptmdashfeedback control validationrdquo Smart Materialsand Structures vol 16 no 6 pp 2146ndash2158 2007

[8] X M Wang and U Carl ldquoFuzzy control of aircraft semi-active landing gear systemrdquo in Proceedings of the 37th AerospaceSciences Meeting and Exhibit pp 1ndash11 1999

[9] X Dong M Yu C Liao and W Chen ldquoAdaptive semiactivebumper for aircraftrdquo Journal of Southwest Jiaotong Universityvol 44 no 5 pp 748ndash752 2009

[10] X M Dong M Yui Z Li C Liao andW Chen ldquoA comparisonof suitable control methods for full vehicle with four MRdampers part I formulation of control schemes and numericalsimulationrdquo Journal of Intelligent Material Systems and Struc-tures vol 20 no 7 pp 771ndash786 2009

[11] X-M Dong Z-S Li M Yu C-R Liao and W-M ChenldquoHuman simulated intelligent control and its application inmagneto-rheological suspensionrdquo Control Theory and Applica-tions vol 27 no 2 pp 245ndash253 2010


[12] P J Reddy V T Nagaraj and V Ramamurti ldquoAnalysis of asemi-active levered suspension landing gear system with someparametric studyrdquo Journal of Dynamic Systems Measurementand Control Transactions of the ASME vol 218 no 3 pp 218ndash224 1984

[13] Q Zhou and J Bai ldquoAn intelligent controller of novel designrdquo inProceedings ofTheMulti-National Instrument Conference Part 1pp 137ndash149 Shanghai China 1983

[14] Z Li and G Wang ldquoSchema theory and human simulatedintelligent controlrdquo in Proceedings of the IEEE on Robotics ampIntelligent Systems and Signal Processing pp 524ndash530 Chang-sha China 2003

[15] K Sekula Cz Graczykowski and J Holnicki-Szulc ldquoOn-lineimpact load identificationrdquo Shock and Vibration vol 20 pp123ndash141 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


[5] discussed the possibilities of the semiactive control oflanding gears According to the energy balance the optimalcontrol force is calculated The researches [4ndash7] indicate thatthe control method based on energy balance is simple andeffective and easy to understand for the single degree oflanding gear system When higher degree dynamic landinggear system or practical case is applied the control strategymay have difficulty in calculating the energy sometimes itis even impossible considering the many uncertainties andnonlinearity in a practical landing gear system

The other type is the nonlinear control strategies consid-ering the nonlinearity and uncertainty of the system Choiand Wereley [3] investigated the feasibility and effectivenessof electrorheological (ER) and MR fluid-based landing gearsystems on attenuating dynamic load and vibration usingsliding mode control Wang and Carl [8] proposed thefuzzy logic control for the semiactive landing gear systemnumerical results showed that the maximal structural loadcan be reduced with the semiactive landing gear Howeverthe control idea needs many sensors to sense the statesof landing gear system which will increase the cost andcomplexity of system

Consequently the main purpose of this study is to finda simple and effective control strategy to improve the ridecomfort and stability of landing gear system simultaneouslythe control can potentially be employed in a practical caseIn the prior work [3 9] we have already investigated alanding gear system featuring MR fluids and demonstratedits feasibility and effectiveness on attenuating dynamics loadand vibration due to the landing impact As a continuationof the prior work this study will propose a new controlstrategy a human simulated intelligent control (HSIC) forthe landing gear system The control strategy has beensuccessfully applied in the MR semiactive suspension system[10 11] Many numerical and actual road tests show theeffectiveness in reducing the vibration of vehicle suspensionand improving the ride comfort

To accomplish this goal a two-DOF dynamic model ofaircraft incorporated with an MRF-based landing gear isfirstly constructed to simulate the course of touchdown Afteranalyzing the response of a classic touchdown the HSICis then proposed Subsequently the numerical simulation isperformed to validate the effectiveness of the control strategyunder different sink speeds and masses In comparison theskyhook control is also formulated Finally a great deal ofnumerical simulations considering variable masses and sinkvelocities are performed to evaluate the effectiveness of theproposed control algorithm

2 Modeling and Control Formulation ofLanding Gear System Featured MR Absorber

21 MR Absorber In the work [3] the authors proposed onetype of controllableMR absorber based on flow-mode whichis schematically shown in Figure 1 The absorber consistsof working and gas compensation reservoirs The workingreservoir is separated into two chambers by the piston inwhich an MR fluid is filled There is an annual valve across

Pneumatic reservoir

Floating piston

Piston rod

Colis

Figure 1 Schematic diagram of an MR absorber [3]

the piston with a winding magnetic coil By the motion of thepiston the MR fluid will flow from one chamber to the otherIn the absence of external magnetic field the MR fluid canfreely flow like linear viscous fluid whereas the rheology ofthe MR fluid will undergo significantly controllable changeswhile applying a magnetic field To compensate the bulkdifference occupied by the piston rod during compressionor rebound course the pressured gas is filled in the gascompensation reservoirs

Typically the damping force of an MR absorber can beexpressed as a combination of viscous damping force andcoulomb damping force [10]

119865 = 119888119904V + 119865MR sign (V) (1)

where 119888119904denotes the damping constant due to viscous damp-

ing V is the relative velocity of MR absorber or piston rodrelative to the absorber body 119865MR represents the controllablecoulomb damping force which is a function of externalmagnetic field or electric current and sgn() indicates the signfunction

22 Two-DOF Landing Gear Systemwith anMRAbsorber Toformulate control strategy a suitable dynamic model needsfirstly to be constructed In this study one of two DOFdynamic models of a landing gear system is adopted inFigure 2 [9] In this model the landing object is treated asrigid body which is slightly different to the actual case Theinternal characteristics (stiffness and damping properties) areneglected in this study although they are very important forthe final landing scenario [7] However the simple model issufficient to formulate the control strategy and themodel candescribe main dynamic behaviors of a landing gear duringtouchdown

In this figure the sprung mass 119898119904represents the mass

of aircraft body and 119898119906denotes the mass of tire The MR

absorber is employed between sprung mass and tire it is alsoin parallel with a coil spring


ms

ks cs

zs

zu

fus

FMR

mu




119891= 0


119906119904= 0

(2)


119904rep-


119906 119906 and

119906represent the



119871119891= 119897119892 (119898

119904+ 119898119906) (3)





119904 119904 119911119906 119906]119879


x = Ax + B119906 + f (x) (4)

where

A =

[[[[[[[

[

0 1 0 0

minus119896119904

119898119904

minus119888119904

119898119904

119896119904

119898119904

119888119904

119898119904

0 0 0 1

119896119904

119898119906

119888119904

119898119906

minus119896119904

119898119906

minus119888119904

119898119906

]]]]]]]

]

B =

[[[[[[

[

0

1

119898119904

0

minus1

119898119906

]]]]]]

]

f (x) =

[[[[[[[

[

0

119892 minus

119871119891

119898119904

0

119892 +119891119906119904

119898119906

]]]]]]]

]

(5)



Actual acceleration


Time (s)

Acce

lera

tion

(ms

2)

10

5

0

minus5

minus10

minus150 02 04 06 08 1 12 14 16 18 2












119904= V0minus 119886119889119879 (6)


119879 = radic2119904max119886119889

(7)


119911119904= int

119879

0

119904119889119905 = V

0119879 minus

1

21198861198891198792

= minus1

2119886119889

(119904)2

+1

2119886119889

(V0)2

(8)



119886119889=

V20

2119904max (9)






(1) Sensed Schema



119904 119906 isin Σ



119876 = 1199021 1199022 1199023 1199024119879

(10)



in which 1199021= 119905 if 119905 le 119879 119902

2= 119905 if 119905 gt 119879 119902

3= 119904(119904minus 119906) if

119904(119904minus 119906) ge 0 and 119902

4= 119904(119904minus 119906) if 119904(119904minus 119906) lt 0


44

119894=1119895=1119902119894119896119894119895 119896119894119895is an


1= 1 0 1 0


1 0 1 0

1 0 0 1

0 1 1 0

0 1 0 1




Φ = 119870 otimes 119876 = 1205931| 119905 le 119879

119904(119904minus 119906) ge 0

1205932| 119905 le 119879

119904(119904minus 119906) lt 0

1205933| 119905 gt 119879

119904(119904minus 119906) ge 0

1205934| 119905 gt 119879

119904(119904minus 119906) lt 0

(11)



119878119875= ⟨119877119876119870 otimesΦ⟩ (12)


(2) Motion Schema






119894int 119890119889119905 and




1 1199012]119879

isin Σ2 where


119886(119886119889minus 119904) and 119901

2is



Input R

OutputAcceleration

feedback

Skyhookcontrol

MR landing

gear system

Feedback

Asso

ciat

edsc

hem

a

Fusio

nop

erat

ion

Con

struc

t

Con

trol t

ime

info

rmat

ion


Control mode set Ψ

Motion schema SM

Sens

edsc

hem

a

Con

trol

mod

eu1

Con

trol

mod

eu2

Con

trol

mod

eu4

payload

mass

Initi

al si

nkve

loci

ty o

runsprung

mass

Velo

city

of

Velo

city

of

spru

ng m

ass



Ψ =

[[[

[

1199061

1199062

1199063

1199064

]]]

]

= 119871119875 =

[[[[[[[[[[[[[

[

2

sum

119895=1

1198971119895119901119895

2

sum

119895=1

1198972119895119901119895

2

sum

119895=1

1198973119895119901119895

2

sum

119895=1

1198974119895119901119895

]]]]]]]]]]]]]

]

=

[[[

[

minus119896119886(119886119889minus 119904)

0

119896sky1199040

]]]

]

(13)




1 0

0 0

0 1

0 0




119878119872= ⟨119877 119875 119871 Ψ 119880⟩ (14)






Ω Φ 997888rarr Ψ (15)

where Ω = 1205961 1205962 1205963 1205964 120596119894= if 120593

119894then 120595

119894 isin Σ

4




119886in (13) is



119878KG = ⟨119878119901 119878119872 119878119860⟩ (16)






Controlmode

Controlmode

Controlmode

Intu

ition

reas

onin

g an

d

Com

puta

tion

Com

puta

tion

Output

Motion schema SM

payload

mass

Initi

al si

nkve

loci

ty o

rC

ontro

l tim

ein

form

atio

nunsprung

mass

Velo

city

of

Velo

city

of

spru

ng m

ass

1205951

1205952

1205954

deci

sion

mak

ing


ms

ks cs

mu

HSIC controller

Parameteradjustment

level

Runningcontrol

levelLanding

gearsystem

SA

SMSP

ad kaFMR





119889and 119896



119886 119896sky and 119886

119889of HSIC


IF 119905 le 119879

IF 119904(119904minus 119906) ge 0

119865MR = minus119896119886(119886119889minus 119904)

Else119865MR = 0

ElseIF 119904(119904minus 119906) ge 0

119865MR = 119896sky119904Else

119865MR = 0

End

Algorithm 1





119903119894= 119897119894+ 120573 (119906

119894minus 119897119894) (17)

where 119897119894and 119906





Selection

Crossover

Mutation



Output

Yes

No

Simulation

Initialparameters

Fitnesscalculation






0 10 20 30 40 50115

1155

116

1165

117

1175

118

1185

Generation

a d(m

s2)

Idea

l dec

eler

atio

n


400600

8001000

1200

1

2

3

40

10

20

30

40

a d(m

s2 )


ms

Des

ired

idea

l dec

eler

atio

n


400600

8001000

1200

1

2

3

40


2000

4000

6000

8000

Des

ired

acce

lera

tion

feed

back

gai

n

Sprung mass (kg)ms

ka(N

(ms

2))




119886119889= 779V

0minus 00086119898

119904+ 31536 (18)

119896119886= 574V

0+ 623119898

119904+ 59244 (19)


119904is studied by


119904are deter-






119906 = 119896sky119904 if

119904(119904minus 119906) gt 0

0 if 119904(119904minus 119906) le 0

(20)


0

10

20

30

40

a d(m

s2)

1

2

3


0 (ms)400

600800

10001200

Sprung mass (kg)ms

Fitte

d id

eal d

ecel

erat

ion


400600

8001000

1200

1

2

3


0 (ms)

8000

7000

6000

5000

4000

3000

Sprung mass (kg)ms

ka(N

(ms

2))

Fitte

d ac

cele

ratio

n fe

edba

ck g

ain




119904800 kg



11990480000 Nm


1199044500 Nsm








800 kg Skyhook 808 808 789 774HSIC 1797 1186 932 744

900 kg Skyhook 997 520 263 124HSIC 4159 2094 1293 927

1100 kg Skyhook 1018 634 403 273HSIC 4143 2191 1361 987



500 kg Skyhook 954 926 895 875HSIC 1329 789 629 538

700 kg Skyhook 849 843 820 804HSIC 1772 1113 897 728

800 kg Skyhook 965 425 153 009HSIC 4110 1976 1242 890

900 kg Skyhook 773 778 762 748HSIC 1888 1230 922 747

1100 kg Skyhook 715 727 715 703HSIC 1771 1242 927 762


115

12

125

13

135

14

145Passive

Max

imum

acce

lera

tion

(ms

2)

A B

C






6 Conclusion



0 05 1 15 2minus15

minus10

minus5

0

5

10

15

Time (s)

Acce

lera

tion

(ms

2)


Off-control



0 01 02 03 04 05minus150

minus100

minus50

0

50

100

Time (s)

Acce

lera

tion

(ms

2)

Off-control



0 05 1 15 2minus005

0

005

01

015

02

025

03

Time (s)

Stro

ke (m

)

Off-control


minus4000

minus2000

0

2000

4000

6000

Dam

ping

forc

e (N

)

0 05 1 15 2

Time (s)


HSIC


0 005 01 015 02 025 03minus1

minus05

0

05

1

15

times104


Off-control

Impa

ct fo

rce (

N)

Stroke (m)




Skyhook controlHSIC

0

10

20

30

40

50

1 15 2 25 3 35 4


Off-control

Max

imum

impa

ct ac

cele

ratio

n (m

s2 )

1100 kg

800 kg

500kg


1 15 2 25 3 35 4005

01

015

02

025

03

035

04

045

Skyhook controlHSIC


Max

imum

effec

tive s

troke

(m)

1100 kg

800 kg

500kg

Off-control








Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ms

ks cs

zs

zu

fus

FMR

mu




119891= 0


119906119904= 0

(2)


119904rep-


119906 119906 and

119906represent the



119871119891= 119897119892 (119898

119904+ 119898119906) (3)





119904 119904 119911119906 119906]119879


x = Ax + B119906 + f (x) (4)

where

A =

[[[[[[[

[

0 1 0 0

minus119896119904

119898119904

minus119888119904

119898119904

119896119904

119898119904

119888119904

119898119904

0 0 0 1

119896119904

119898119906

119888119904

119898119906

minus119896119904

119898119906

minus119888119904

119898119906

]]]]]]]

]

B =

[[[[[[

[

0

1

119898119904

0

minus1

119898119906

]]]]]]

]

f (x) =

[[[[[[[

[

0

119892 minus

119871119891

119898119904

0

119892 +119891119906119904

119898119906

]]]]]]]

]

(5)



Actual acceleration


Time (s)

Acce

lera

tion

(ms

2)

10

5

0

minus5

minus10

minus150 02 04 06 08 1 12 14 16 18 2












119904= V0minus 119886119889119879 (6)


119879 = radic2119904max119886119889

(7)


119911119904= int

119879

0

119904119889119905 = V

0119879 minus

1

21198861198891198792

= minus1

2119886119889

(119904)2

+1

2119886119889

(V0)2

(8)



119886119889=

V20

2119904max (9)






(1) Sensed Schema



119904 119906 isin Σ



119876 = 1199021 1199022 1199023 1199024119879

(10)



in which 1199021= 119905 if 119905 le 119879 119902

2= 119905 if 119905 gt 119879 119902

3= 119904(119904minus 119906) if

119904(119904minus 119906) ge 0 and 119902

4= 119904(119904minus 119906) if 119904(119904minus 119906) lt 0


44

119894=1119895=1119902119894119896119894119895 119896119894119895is an


1= 1 0 1 0


1 0 1 0

1 0 0 1

0 1 1 0

0 1 0 1




Φ = 119870 otimes 119876 = 1205931| 119905 le 119879

119904(119904minus 119906) ge 0

1205932| 119905 le 119879

119904(119904minus 119906) lt 0

1205933| 119905 gt 119879

119904(119904minus 119906) ge 0

1205934| 119905 gt 119879

119904(119904minus 119906) lt 0

(11)



119878119875= ⟨119877119876119870 otimesΦ⟩ (12)


(2) Motion Schema






119894int 119890119889119905 and




1 1199012]119879

isin Σ2 where


119886(119886119889minus 119904) and 119901

2is



Input R

OutputAcceleration

feedback

Skyhookcontrol

MR landing

gear system

Feedback

Asso

ciat

edsc

hem

a

Fusio

nop

erat

ion

Con

struc

t

Con

trol t

ime

info

rmat

ion


Control mode set Ψ

Motion schema SM

Sens

edsc

hem

a

Con

trol

mod

eu1

Con

trol

mod

eu2

Con

trol

mod

eu4

payload

mass

Initi

al si

nkve

loci

ty o

runsprung

mass

Velo

city

of

Velo

city

of

spru

ng m

ass



Ψ =

[[[

[

1199061

1199062

1199063

1199064

]]]

]

= 119871119875 =

[[[[[[[[[[[[[

[

2

sum

119895=1

1198971119895119901119895

2

sum

119895=1

1198972119895119901119895

2

sum

119895=1

1198973119895119901119895

2

sum

119895=1

1198974119895119901119895

]]]]]]]]]]]]]

]

=

[[[

[

minus119896119886(119886119889minus 119904)

0

119896sky1199040

]]]

]

(13)




1 0

0 0

0 1

0 0




119878119872= ⟨119877 119875 119871 Ψ 119880⟩ (14)






Ω Φ 997888rarr Ψ (15)

where Ω = 1205961 1205962 1205963 1205964 120596119894= if 120593

119894then 120595

119894 isin Σ

4




119886in (13) is



119878KG = ⟨119878119901 119878119872 119878119860⟩ (16)






Controlmode

Controlmode

Controlmode

Intu

ition

reas

onin

g an

d

Com

puta

tion

Com

puta

tion

Output

Motion schema SM

payload

mass

Initi

al si

nkve

loci

ty o

rC

ontro

l tim

ein

form

atio

nunsprung

mass

Velo

city

of

Velo

city

of

spru

ng m

ass

1205951

1205952

1205954

deci

sion

mak

ing


ms

ks cs

mu

HSIC controller

Parameteradjustment

level

Runningcontrol

levelLanding

gearsystem

SA

SMSP

ad kaFMR





119889and 119896



119886 119896sky and 119886

119889of HSIC


IF 119905 le 119879

IF 119904(119904minus 119906) ge 0

119865MR = minus119896119886(119886119889minus 119904)

Else119865MR = 0

ElseIF 119904(119904minus 119906) ge 0

119865MR = 119896sky119904Else

119865MR = 0

End

Algorithm 1





119903119894= 119897119894+ 120573 (119906

119894minus 119897119894) (17)

where 119897119894and 119906





Selection

Crossover

Mutation



Output

Yes

No

Simulation

Initialparameters

Fitnesscalculation






0 10 20 30 40 50115

1155

116

1165

117

1175

118

1185

Generation

a d(m

s2)

Idea

l dec

eler

atio

n


400600

8001000

1200

1

2

3

40

10

20

30

40

a d(m

s2 )


ms

Des

ired

idea

l dec

eler

atio

n


400600

8001000

1200

1

2

3

40


2000

4000

6000

8000

Des

ired

acce

lera

tion

feed

back

gai

n

Sprung mass (kg)ms

ka(N

(ms

2))




119886119889= 779V

0minus 00086119898

119904+ 31536 (18)

119896119886= 574V

0+ 623119898

119904+ 59244 (19)


119904is studied by


119904are deter-






119906 = 119896sky119904 if

119904(119904minus 119906) gt 0

0 if 119904(119904minus 119906) le 0

(20)


0

10

20

30

40

a d(m

s2)

1

2

3


0 (ms)400

600800

10001200

Sprung mass (kg)ms

Fitte

d id

eal d

ecel

erat

ion


400600

8001000

1200

1

2

3


0 (ms)

8000

7000

6000

5000

4000

3000

Sprung mass (kg)ms

ka(N

(ms

2))

Fitte

d ac

cele

ratio

n fe

edba

ck g

ain




119904800 kg



11990480000 Nm


1199044500 Nsm








800 kg Skyhook 808 808 789 774HSIC 1797 1186 932 744

900 kg Skyhook 997 520 263 124HSIC 4159 2094 1293 927

1100 kg Skyhook 1018 634 403 273HSIC 4143 2191 1361 987



500 kg Skyhook 954 926 895 875HSIC 1329 789 629 538

700 kg Skyhook 849 843 820 804HSIC 1772 1113 897 728

800 kg Skyhook 965 425 153 009HSIC 4110 1976 1242 890

900 kg Skyhook 773 778 762 748HSIC 1888 1230 922 747

1100 kg Skyhook 715 727 715 703HSIC 1771 1242 927 762


115

12

125

13

135

14

145Passive

Max

imum

acce

lera

tion

(ms

2)

A B

C






6 Conclusion



0 05 1 15 2minus15

minus10

minus5

0

5

10

15

Time (s)

Acce

lera

tion

(ms

2)


Off-control



0 01 02 03 04 05minus150

minus100

minus50

0

50

100

Time (s)

Acce

lera

tion

(ms

2)

Off-control



0 05 1 15 2minus005

0

005

01

015

02

025

03

Time (s)

Stro

ke (m

)

Off-control


minus4000

minus2000

0

2000

4000

6000

Dam

ping

forc

e (N

)

0 05 1 15 2

Time (s)


HSIC


0 005 01 015 02 025 03minus1

minus05

0

05

1

15

times104


Off-control

Impa

ct fo

rce (

N)

Stroke (m)




Skyhook controlHSIC

0

10

20

30

40

50

1 15 2 25 3 35 4


Off-control

Max

imum

impa

ct ac

cele

ratio

n (m

s2 )

1100 kg

800 kg

500kg


1 15 2 25 3 35 4005

01

015

02

025

03

035

04

045

Skyhook controlHSIC


Max

imum

effec

tive s

troke

(m)

1100 kg

800 kg

500kg

Off-control








Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119904= V0minus 119886119889119879 (6)


119879 = radic2119904max119886119889

(7)


119911119904= int

119879

0

119904119889119905 = V

0119879 minus

1

21198861198891198792

= minus1

2119886119889

(119904)2

+1

2119886119889

(V0)2

(8)



119886119889=

V20

2119904max (9)






(1) Sensed Schema



119904 119906 isin Σ



119876 = 1199021 1199022 1199023 1199024119879

(10)



in which 1199021= 119905 if 119905 le 119879 119902

2= 119905 if 119905 gt 119879 119902

3= 119904(119904minus 119906) if

119904(119904minus 119906) ge 0 and 119902

4= 119904(119904minus 119906) if 119904(119904minus 119906) lt 0


44

119894=1119895=1119902119894119896119894119895 119896119894119895is an


1= 1 0 1 0


1 0 1 0

1 0 0 1

0 1 1 0

0 1 0 1




Φ = 119870 otimes 119876 = 1205931| 119905 le 119879

119904(119904minus 119906) ge 0

1205932| 119905 le 119879

119904(119904minus 119906) lt 0

1205933| 119905 gt 119879

119904(119904minus 119906) ge 0

1205934| 119905 gt 119879

119904(119904minus 119906) lt 0

(11)



119878119875= ⟨119877119876119870 otimesΦ⟩ (12)


(2) Motion Schema






119894int 119890119889119905 and




1 1199012]119879

isin Σ2 where


119886(119886119889minus 119904) and 119901

2is



Input R

OutputAcceleration

feedback

Skyhookcontrol

MR landing

gear system

Feedback

Asso

ciat

edsc

hem

a

Fusio

nop

erat

ion

Con

struc

t

Con

trol t

ime

info

rmat

ion


Control mode set Ψ

Motion schema SM

Sens

edsc

hem

a

Con

trol

mod

eu1

Con

trol

mod

eu2

Con

trol

mod

eu4

payload

mass

Initi

al si

nkve

loci

ty o

runsprung

mass

Velo

city

of

Velo

city

of

spru

ng m

ass



Ψ =

[[[

[

1199061

1199062

1199063

1199064

]]]

]

= 119871119875 =

[[[[[[[[[[[[[

[

2

sum

119895=1

1198971119895119901119895

2

sum

119895=1

1198972119895119901119895

2

sum

119895=1

1198973119895119901119895

2

sum

119895=1

1198974119895119901119895

]]]]]]]]]]]]]

]

=

[[[

[

minus119896119886(119886119889minus 119904)

0

119896sky1199040

]]]

]

(13)




1 0

0 0

0 1

0 0




119878119872= ⟨119877 119875 119871 Ψ 119880⟩ (14)






Ω Φ 997888rarr Ψ (15)

where Ω = 1205961 1205962 1205963 1205964 120596119894= if 120593

119894then 120595

119894 isin Σ

4




119886in (13) is



119878KG = ⟨119878119901 119878119872 119878119860⟩ (16)






Controlmode

Controlmode

Controlmode

Intu

ition

reas

onin

g an

d

Com

puta

tion

Com

puta

tion

Output

Motion schema SM

payload

mass

Initi

al si

nkve

loci

ty o

rC

ontro

l tim

ein

form

atio

nunsprung

mass

Velo

city

of

Velo

city

of

spru

ng m

ass

1205951

1205952

1205954

deci

sion

mak

ing


ms

ks cs

mu

HSIC controller

Parameteradjustment

level

Runningcontrol

levelLanding

gearsystem

SA

SMSP

ad kaFMR





119889and 119896



119886 119896sky and 119886

119889of HSIC


IF 119905 le 119879

IF 119904(119904minus 119906) ge 0

119865MR = minus119896119886(119886119889minus 119904)

Else119865MR = 0

ElseIF 119904(119904minus 119906) ge 0

119865MR = 119896sky119904Else

119865MR = 0

End

Algorithm 1





119903119894= 119897119894+ 120573 (119906

119894minus 119897119894) (17)

where 119897119894and 119906





Selection

Crossover

Mutation



Output

Yes

No

Simulation

Initialparameters

Fitnesscalculation






0 10 20 30 40 50115

1155

116

1165

117

1175

118

1185

Generation

a d(m

s2)

Idea

l dec

eler

atio

n


400600

8001000

1200

1

2

3

40

10

20

30

40

a d(m

s2 )


ms

Des

ired

idea

l dec

eler

atio

n


400600

8001000

1200

1

2

3

40


2000

4000

6000

8000

Des

ired

acce

lera

tion

feed

back

gai

n

Sprung mass (kg)ms

ka(N

(ms

2))




119886119889= 779V

0minus 00086119898

119904+ 31536 (18)

119896119886= 574V

0+ 623119898

119904+ 59244 (19)


119904is studied by


119904are deter-






119906 = 119896sky119904 if

119904(119904minus 119906) gt 0

0 if 119904(119904minus 119906) le 0

(20)


0

10

20

30

40

a d(m

s2)

1

2

3


0 (ms)400

600800

10001200

Sprung mass (kg)ms

Fitte

d id

eal d

ecel

erat

ion


400600

8001000

1200

1

2

3


0 (ms)

8000

7000

6000

5000

4000

3000

Sprung mass (kg)ms

ka(N

(ms

2))

Fitte

d ac

cele

ratio

n fe

edba

ck g

ain




119904800 kg



11990480000 Nm


1199044500 Nsm








800 kg Skyhook 808 808 789 774HSIC 1797 1186 932 744

900 kg Skyhook 997 520 263 124HSIC 4159 2094 1293 927

1100 kg Skyhook 1018 634 403 273HSIC 4143 2191 1361 987



500 kg Skyhook 954 926 895 875HSIC 1329 789 629 538

700 kg Skyhook 849 843 820 804HSIC 1772 1113 897 728

800 kg Skyhook 965 425 153 009HSIC 4110 1976 1242 890

900 kg Skyhook 773 778 762 748HSIC 1888 1230 922 747

1100 kg Skyhook 715 727 715 703HSIC 1771 1242 927 762


115

12

125

13

135

14

145Passive

Max

imum

acce

lera

tion

(ms

2)

A B

C






6 Conclusion



0 05 1 15 2minus15

minus10

minus5

0

5

10

15

Time (s)

Acce

lera

tion

(ms

2)


Off-control



0 01 02 03 04 05minus150

minus100

minus50

0

50

100

Time (s)

Acce

lera

tion

(ms

2)

Off-control



0 05 1 15 2minus005

0

005

01

015

02

025

03

Time (s)

Stro

ke (m

)

Off-control


minus4000

minus2000

0

2000

4000

6000

Dam

ping

forc

e (N

)

0 05 1 15 2

Time (s)


HSIC


0 005 01 015 02 025 03minus1

minus05

0

05

1

15

times104


Off-control

Impa

ct fo

rce (

N)

Stroke (m)




Skyhook controlHSIC

0

10

20

30

40

50

1 15 2 25 3 35 4


Off-control

Max

imum

impa

ct ac

cele

ratio

n (m

s2 )

1100 kg

800 kg

500kg


1 15 2 25 3 35 4005

01

015

02

025

03

035

04

045

Skyhook controlHSIC


Max

imum

effec

tive s

troke

(m)

1100 kg

800 kg

500kg

Off-control








Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











in which 1199021= 119905 if 119905 le 119879 119902

2= 119905 if 119905 gt 119879 119902

3= 119904(119904minus 119906) if

119904(119904minus 119906) ge 0 and 119902

4= 119904(119904minus 119906) if 119904(119904minus 119906) lt 0


44

119894=1119895=1119902119894119896119894119895 119896119894119895is an


1= 1 0 1 0


1 0 1 0

1 0 0 1

0 1 1 0

0 1 0 1




Φ = 119870 otimes 119876 = 1205931| 119905 le 119879

119904(119904minus 119906) ge 0

1205932| 119905 le 119879

119904(119904minus 119906) lt 0

1205933| 119905 gt 119879

119904(119904minus 119906) ge 0

1205934| 119905 gt 119879

119904(119904minus 119906) lt 0

(11)



119878119875= ⟨119877119876119870 otimesΦ⟩ (12)


(2) Motion Schema






119894int 119890119889119905 and




1 1199012]119879

isin Σ2 where


119886(119886119889minus 119904) and 119901

2is



Input R

OutputAcceleration

feedback

Skyhookcontrol

MR landing

gear system

Feedback

Asso

ciat

edsc

hem

a

Fusio

nop

erat

ion

Con

struc

t

Con

trol t

ime

info

rmat

ion


Control mode set Ψ

Motion schema SM

Sens

edsc

hem

a

Con

trol

mod

eu1

Con

trol

mod

eu2

Con

trol

mod

eu4

payload

mass

Initi

al si

nkve

loci

ty o

runsprung

mass

Velo

city

of

Velo

city

of

spru

ng m

ass



Ψ =

[[[

[

1199061

1199062

1199063

1199064

]]]

]

= 119871119875 =

[[[[[[[[[[[[[

[

2

sum

119895=1

1198971119895119901119895

2

sum

119895=1

1198972119895119901119895

2

sum

119895=1

1198973119895119901119895

2

sum

119895=1

1198974119895119901119895

]]]]]]]]]]]]]

]

=

[[[

[

minus119896119886(119886119889minus 119904)

0

119896sky1199040

]]]

]

(13)




1 0

0 0

0 1

0 0




119878119872= ⟨119877 119875 119871 Ψ 119880⟩ (14)






Ω Φ 997888rarr Ψ (15)

where Ω = 1205961 1205962 1205963 1205964 120596119894= if 120593

119894then 120595

119894 isin Σ

4




119886in (13) is



119878KG = ⟨119878119901 119878119872 119878119860⟩ (16)






Controlmode

Controlmode

Controlmode

Intu

ition

reas

onin

g an

d

Com

puta

tion

Com

puta

tion

Output

Motion schema SM

payload

mass

Initi

al si

nkve

loci

ty o

rC

ontro

l tim

ein

form

atio

nunsprung

mass

Velo

city

of

Velo

city

of

spru

ng m

ass

1205951

1205952

1205954

deci

sion

mak

ing


ms

ks cs

mu

HSIC controller

Parameteradjustment

level

Runningcontrol

levelLanding

gearsystem

SA

SMSP

ad kaFMR





119889and 119896



119886 119896sky and 119886

119889of HSIC


IF 119905 le 119879

IF 119904(119904minus 119906) ge 0

119865MR = minus119896119886(119886119889minus 119904)

Else119865MR = 0

ElseIF 119904(119904minus 119906) ge 0

119865MR = 119896sky119904Else

119865MR = 0

End

Algorithm 1





119903119894= 119897119894+ 120573 (119906

119894minus 119897119894) (17)

where 119897119894and 119906





Selection

Crossover

Mutation



Output

Yes

No

Simulation

Initialparameters

Fitnesscalculation






0 10 20 30 40 50115

1155

116

1165

117

1175

118

1185

Generation

a d(m

s2)

Idea

l dec

eler

atio

n


400600

8001000

1200

1

2

3

40

10

20

30

40

a d(m

s2 )


ms

Des

ired

idea

l dec

eler

atio

n


400600

8001000

1200

1

2

3

40


2000

4000

6000

8000

Des

ired

acce

lera

tion

feed

back

gai

n

Sprung mass (kg)ms

ka(N

(ms

2))




119886119889= 779V

0minus 00086119898

119904+ 31536 (18)

119896119886= 574V

0+ 623119898

119904+ 59244 (19)


119904is studied by


119904are deter-






119906 = 119896sky119904 if

119904(119904minus 119906) gt 0

0 if 119904(119904minus 119906) le 0

(20)


0

10

20

30

40

a d(m

s2)

1

2

3


0 (ms)400

600800

10001200

Sprung mass (kg)ms

Fitte

d id

eal d

ecel

erat

ion


400600

8001000

1200

1

2

3


0 (ms)

8000

7000

6000

5000

4000

3000

Sprung mass (kg)ms

ka(N

(ms

2))

Fitte

d ac

cele

ratio

n fe

edba

ck g

ain




119904800 kg



11990480000 Nm


1199044500 Nsm








800 kg Skyhook 808 808 789 774HSIC 1797 1186 932 744

900 kg Skyhook 997 520 263 124HSIC 4159 2094 1293 927

1100 kg Skyhook 1018 634 403 273HSIC 4143 2191 1361 987



500 kg Skyhook 954 926 895 875HSIC 1329 789 629 538

700 kg Skyhook 849 843 820 804HSIC 1772 1113 897 728

800 kg Skyhook 965 425 153 009HSIC 4110 1976 1242 890

900 kg Skyhook 773 778 762 748HSIC 1888 1230 922 747

1100 kg Skyhook 715 727 715 703HSIC 1771 1242 927 762


115

12

125

13

135

14

145Passive

Max

imum

acce

lera

tion

(ms

2)

A B

C






6 Conclusion



0 05 1 15 2minus15

minus10

minus5

0

5

10

15

Time (s)

Acce

lera

tion

(ms

2)


Off-control



0 01 02 03 04 05minus150

minus100

minus50

0

50

100

Time (s)

Acce

lera

tion

(ms

2)

Off-control



0 05 1 15 2minus005

0

005

01

015

02

025

03

Time (s)

Stro

ke (m

)

Off-control


minus4000

minus2000

0

2000

4000

6000

Dam

ping

forc

e (N

)

0 05 1 15 2

Time (s)


HSIC


0 005 01 015 02 025 03minus1

minus05

0

05

1

15

times104


Off-control

Impa

ct fo

rce (

N)

Stroke (m)




Skyhook controlHSIC

0

10

20

30

40

50

1 15 2 25 3 35 4


Off-control

Max

imum

impa

ct ac

cele

ratio

n (m

s2 )

1100 kg

800 kg

500kg


1 15 2 25 3 35 4005

01

015

02

025

03

035

04

045

Skyhook controlHSIC


Max

imum

effec

tive s

troke

(m)

1100 kg

800 kg

500kg

Off-control








Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Input R

OutputAcceleration

feedback

Skyhookcontrol

MR landing

gear system

Feedback

Asso

ciat

edsc

hem

a

Fusio

nop

erat

ion

Con

struc

t

Con

trol t

ime

info

rmat

ion


Control mode set Ψ

Motion schema SM

Sens

edsc

hem

a

Con

trol

mod

eu1

Con

trol

mod

eu2

Con

trol

mod

eu4

payload

mass

Initi

al si

nkve

loci

ty o

runsprung

mass

Velo

city

of

Velo

city

of

spru

ng m

ass



Ψ =

[[[

[

1199061

1199062

1199063

1199064

]]]

]

= 119871119875 =

[[[[[[[[[[[[[

[

2

sum

119895=1

1198971119895119901119895

2

sum

119895=1

1198972119895119901119895

2

sum

119895=1

1198973119895119901119895

2

sum

119895=1

1198974119895119901119895

]]]]]]]]]]]]]

]

=

[[[

[

minus119896119886(119886119889minus 119904)

0

119896sky1199040

]]]

]

(13)




1 0

0 0

0 1

0 0




119878119872= ⟨119877 119875 119871 Ψ 119880⟩ (14)






Ω Φ 997888rarr Ψ (15)

where Ω = 1205961 1205962 1205963 1205964 120596119894= if 120593

119894then 120595

119894 isin Σ

4




119886in (13) is



119878KG = ⟨119878119901 119878119872 119878119860⟩ (16)






Controlmode

Controlmode

Controlmode

Intu

ition

reas

onin

g an

d

Com

puta

tion

Com

puta

tion

Output

Motion schema SM

payload

mass

Initi

al si

nkve

loci

ty o

rC

ontro

l tim

ein

form

atio

nunsprung

mass

Velo

city

of

Velo

city

of

spru

ng m

ass

1205951

1205952

1205954

deci

sion

mak

ing


ms

ks cs

mu

HSIC controller

Parameteradjustment

level

Runningcontrol

levelLanding

gearsystem

SA

SMSP

ad kaFMR





119889and 119896



119886 119896sky and 119886

119889of HSIC


IF 119905 le 119879

IF 119904(119904minus 119906) ge 0

119865MR = minus119896119886(119886119889minus 119904)

Else119865MR = 0

ElseIF 119904(119904minus 119906) ge 0

119865MR = 119896sky119904Else

119865MR = 0

End

Algorithm 1





119903119894= 119897119894+ 120573 (119906

119894minus 119897119894) (17)

where 119897119894and 119906





Selection

Crossover

Mutation



Output

Yes

No

Simulation

Initialparameters

Fitnesscalculation






0 10 20 30 40 50115

1155

116

1165

117

1175

118

1185

Generation

a d(m

s2)

Idea

l dec

eler

atio

n


400600

8001000

1200

1

2

3

40

10

20

30

40

a d(m

s2 )


ms

Des

ired

idea

l dec

eler

atio

n


400600

8001000

1200

1

2

3

40


2000

4000

6000

8000

Des

ired

acce

lera

tion

feed

back

gai

n

Sprung mass (kg)ms

ka(N

(ms

2))




119886119889= 779V

0minus 00086119898

119904+ 31536 (18)

119896119886= 574V

0+ 623119898

119904+ 59244 (19)


119904is studied by


119904are deter-






119906 = 119896sky119904 if

119904(119904minus 119906) gt 0

0 if 119904(119904minus 119906) le 0

(20)


0

10

20

30

40

a d(m

s2)

1

2

3


0 (ms)400

600800

10001200

Sprung mass (kg)ms

Fitte

d id

eal d

ecel

erat

ion


400600

8001000

1200

1

2

3


0 (ms)

8000

7000

6000

5000

4000

3000

Sprung mass (kg)ms

ka(N

(ms

2))

Fitte

d ac

cele

ratio

n fe

edba

ck g

ain




119904800 kg



11990480000 Nm


1199044500 Nsm








800 kg Skyhook 808 808 789 774HSIC 1797 1186 932 744

900 kg Skyhook 997 520 263 124HSIC 4159 2094 1293 927

1100 kg Skyhook 1018 634 403 273HSIC 4143 2191 1361 987



500 kg Skyhook 954 926 895 875HSIC 1329 789 629 538

700 kg Skyhook 849 843 820 804HSIC 1772 1113 897 728

800 kg Skyhook 965 425 153 009HSIC 4110 1976 1242 890

900 kg Skyhook 773 778 762 748HSIC 1888 1230 922 747

1100 kg Skyhook 715 727 715 703HSIC 1771 1242 927 762


115

12

125

13

135

14

145Passive

Max

imum

acce

lera

tion

(ms

2)

A B

C






6 Conclusion



0 05 1 15 2minus15

minus10

minus5

0

5

10

15

Time (s)

Acce

lera

tion

(ms

2)


Off-control



0 01 02 03 04 05minus150

minus100

minus50

0

50

100

Time (s)

Acce

lera

tion

(ms

2)

Off-control



0 05 1 15 2minus005

0

005

01

015

02

025

03

Time (s)

Stro

ke (m

)

Off-control


minus4000

minus2000

0

2000

4000

6000

Dam

ping

forc

e (N

)

0 05 1 15 2

Time (s)


HSIC


0 005 01 015 02 025 03minus1

minus05

0

05

1

15

times104


Off-control

Impa

ct fo

rce (

N)

Stroke (m)




Skyhook controlHSIC

0

10

20

30

40

50

1 15 2 25 3 35 4


Off-control

Max

imum

impa

ct ac

cele

ratio

n (m

s2 )

1100 kg

800 kg

500kg


1 15 2 25 3 35 4005

01

015

02

025

03

035

04

045

Skyhook controlHSIC


Max

imum

effec

tive s

troke

(m)

1100 kg

800 kg

500kg

Off-control








Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Controlmode

Controlmode

Controlmode

Intu

ition

reas

onin

g an

d

Com

puta

tion

Com

puta

tion

Output

Motion schema SM

payload

mass

Initi

al si

nkve

loci

ty o

rC

ontro

l tim

ein

form

atio

nunsprung

mass

Velo

city

of

Velo

city

of

spru

ng m

ass

1205951

1205952

1205954

deci

sion

mak

ing


ms

ks cs

mu

HSIC controller

Parameteradjustment

level

Runningcontrol

levelLanding

gearsystem

SA

SMSP

ad kaFMR





119889and 119896



119886 119896sky and 119886

119889of HSIC


IF 119905 le 119879

IF 119904(119904minus 119906) ge 0

119865MR = minus119896119886(119886119889minus 119904)

Else119865MR = 0

ElseIF 119904(119904minus 119906) ge 0

119865MR = 119896sky119904Else

119865MR = 0

End

Algorithm 1





119903119894= 119897119894+ 120573 (119906

119894minus 119897119894) (17)

where 119897119894and 119906





Selection

Crossover

Mutation



Output

Yes

No

Simulation

Initialparameters

Fitnesscalculation






0 10 20 30 40 50115

1155

116

1165

117

1175

118

1185

Generation

a d(m

s2)

Idea

l dec

eler

atio

n


400600

8001000

1200

1

2

3

40

10

20

30

40

a d(m

s2 )


ms

Des

ired

idea

l dec

eler

atio

n


400600

8001000

1200

1

2

3

40


2000

4000

6000

8000

Des

ired

acce

lera

tion

feed

back

gai

n

Sprung mass (kg)ms

ka(N

(ms

2))




119886119889= 779V

0minus 00086119898

119904+ 31536 (18)

119896119886= 574V

0+ 623119898

119904+ 59244 (19)


119904is studied by


119904are deter-






119906 = 119896sky119904 if

119904(119904minus 119906) gt 0

0 if 119904(119904minus 119906) le 0

(20)


0

10

20

30

40

a d(m

s2)

1

2

3


0 (ms)400

600800

10001200

Sprung mass (kg)ms

Fitte

d id

eal d

ecel

erat

ion


400600

8001000

1200

1

2

3


0 (ms)

8000

7000

6000

5000

4000

3000

Sprung mass (kg)ms

ka(N

(ms

2))

Fitte

d ac

cele

ratio

n fe

edba

ck g

ain




119904800 kg



11990480000 Nm


1199044500 Nsm








800 kg Skyhook 808 808 789 774HSIC 1797 1186 932 744

900 kg Skyhook 997 520 263 124HSIC 4159 2094 1293 927

1100 kg Skyhook 1018 634 403 273HSIC 4143 2191 1361 987



500 kg Skyhook 954 926 895 875HSIC 1329 789 629 538

700 kg Skyhook 849 843 820 804HSIC 1772 1113 897 728

800 kg Skyhook 965 425 153 009HSIC 4110 1976 1242 890

900 kg Skyhook 773 778 762 748HSIC 1888 1230 922 747

1100 kg Skyhook 715 727 715 703HSIC 1771 1242 927 762


115

12

125

13

135

14

145Passive

Max

imum

acce

lera

tion

(ms

2)

A B

C






6 Conclusion



0 05 1 15 2minus15

minus10

minus5

0

5

10

15

Time (s)

Acce

lera

tion

(ms

2)


Off-control



0 01 02 03 04 05minus150

minus100

minus50

0

50

100

Time (s)

Acce

lera

tion

(ms

2)

Off-control



0 05 1 15 2minus005

0

005

01

015

02

025

03

Time (s)

Stro

ke (m

)

Off-control


minus4000

minus2000

0

2000

4000

6000

Dam

ping

forc

e (N

)

0 05 1 15 2

Time (s)


HSIC


0 005 01 015 02 025 03minus1

minus05

0

05

1

15

times104


Off-control

Impa

ct fo

rce (

N)

Stroke (m)




Skyhook controlHSIC

0

10

20

30

40

50

1 15 2 25 3 35 4


Off-control

Max

imum

impa

ct ac

cele

ratio

n (m

s2 )

1100 kg

800 kg

500kg


1 15 2 25 3 35 4005

01

015

02

025

03

035

04

045

Skyhook controlHSIC


Max

imum

effec

tive s

troke

(m)

1100 kg

800 kg

500kg

Off-control








Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Selection

Crossover

Mutation



Output

Yes

No

Simulation

Initialparameters

Fitnesscalculation






0 10 20 30 40 50115

1155

116

1165

117

1175

118

1185

Generation

a d(m

s2)

Idea

l dec

eler

atio

n


400600

8001000

1200

1

2

3

40

10

20

30

40

a d(m

s2 )


ms

Des

ired

idea

l dec

eler

atio

n


400600

8001000

1200

1

2

3

40


2000

4000

6000

8000

Des

ired

acce

lera

tion

feed

back

gai

n

Sprung mass (kg)ms

ka(N

(ms

2))




119886119889= 779V

0minus 00086119898

119904+ 31536 (18)

119896119886= 574V

0+ 623119898

119904+ 59244 (19)


119904is studied by


119904are deter-






119906 = 119896sky119904 if

119904(119904minus 119906) gt 0

0 if 119904(119904minus 119906) le 0

(20)


0

10

20

30

40

a d(m

s2)

1

2

3


0 (ms)400

600800

10001200

Sprung mass (kg)ms

Fitte

d id

eal d

ecel

erat

ion


400600

8001000

1200

1

2

3


0 (ms)

8000

7000

6000

5000

4000

3000

Sprung mass (kg)ms

ka(N

(ms

2))

Fitte

d ac

cele

ratio

n fe

edba

ck g

ain




119904800 kg



11990480000 Nm


1199044500 Nsm








800 kg Skyhook 808 808 789 774HSIC 1797 1186 932 744

900 kg Skyhook 997 520 263 124HSIC 4159 2094 1293 927

1100 kg Skyhook 1018 634 403 273HSIC 4143 2191 1361 987



500 kg Skyhook 954 926 895 875HSIC 1329 789 629 538

700 kg Skyhook 849 843 820 804HSIC 1772 1113 897 728

800 kg Skyhook 965 425 153 009HSIC 4110 1976 1242 890

900 kg Skyhook 773 778 762 748HSIC 1888 1230 922 747

1100 kg Skyhook 715 727 715 703HSIC 1771 1242 927 762


115

12

125

13

135

14

145Passive

Max

imum

acce

lera

tion

(ms

2)

A B

C






6 Conclusion



0 05 1 15 2minus15

minus10

minus5

0

5

10

15

Time (s)

Acce

lera

tion

(ms

2)


Off-control



0 01 02 03 04 05minus150

minus100

minus50

0

50

100

Time (s)

Acce

lera

tion

(ms

2)

Off-control



0 05 1 15 2minus005

0

005

01

015

02

025

03

Time (s)

Stro

ke (m

)

Off-control


minus4000

minus2000

0

2000

4000

6000

Dam

ping

forc

e (N

)

0 05 1 15 2

Time (s)


HSIC


0 005 01 015 02 025 03minus1

minus05

0

05

1

15

times104


Off-control

Impa

ct fo

rce (

N)

Stroke (m)




Skyhook controlHSIC

0

10

20

30

40

50

1 15 2 25 3 35 4


Off-control

Max

imum

impa

ct ac

cele

ratio

n (m

s2 )

1100 kg

800 kg

500kg


1 15 2 25 3 35 4005

01

015

02

025

03

035

04

045

Skyhook controlHSIC


Max

imum

effec

tive s

troke

(m)

1100 kg

800 kg

500kg

Off-control








Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119886119889= 779V

0minus 00086119898

119904+ 31536 (18)

119896119886= 574V

0+ 623119898

119904+ 59244 (19)


119904is studied by


119904are deter-






119906 = 119896sky119904 if

119904(119904minus 119906) gt 0

0 if 119904(119904minus 119906) le 0

(20)


0

10

20

30

40

a d(m

s2)

1

2

3


0 (ms)400

600800

10001200

Sprung mass (kg)ms

Fitte

d id

eal d

ecel

erat

ion


400600

8001000

1200

1

2

3


0 (ms)

8000

7000

6000

5000

4000

3000

Sprung mass (kg)ms

ka(N

(ms

2))

Fitte

d ac

cele

ratio

n fe

edba

ck g

ain




119904800 kg



11990480000 Nm


1199044500 Nsm








800 kg Skyhook 808 808 789 774HSIC 1797 1186 932 744

900 kg Skyhook 997 520 263 124HSIC 4159 2094 1293 927

1100 kg Skyhook 1018 634 403 273HSIC 4143 2191 1361 987



500 kg Skyhook 954 926 895 875HSIC 1329 789 629 538

700 kg Skyhook 849 843 820 804HSIC 1772 1113 897 728

800 kg Skyhook 965 425 153 009HSIC 4110 1976 1242 890

900 kg Skyhook 773 778 762 748HSIC 1888 1230 922 747

1100 kg Skyhook 715 727 715 703HSIC 1771 1242 927 762


115

12

125

13

135

14

145Passive

Max

imum

acce

lera

tion

(ms

2)

A B

C






6 Conclusion



0 05 1 15 2minus15

minus10

minus5

0

5

10

15

Time (s)

Acce

lera

tion

(ms

2)


Off-control



0 01 02 03 04 05minus150

minus100

minus50

0

50

100

Time (s)

Acce

lera

tion

(ms

2)

Off-control



0 05 1 15 2minus005

0

005

01

015

02

025

03

Time (s)

Stro

ke (m

)

Off-control


minus4000

minus2000

0

2000

4000

6000

Dam

ping

forc

e (N

)

0 05 1 15 2

Time (s)


HSIC


0 005 01 015 02 025 03minus1

minus05

0

05

1

15

times104


Off-control

Impa

ct fo

rce (

N)

Stroke (m)




Skyhook controlHSIC

0

10

20

30

40

50

1 15 2 25 3 35 4


Off-control

Max

imum

impa

ct ac

cele

ratio

n (m

s2 )

1100 kg

800 kg

500kg


1 15 2 25 3 35 4005

01

015

02

025

03

035

04

045

Skyhook controlHSIC


Max

imum

effec

tive s

troke

(m)

1100 kg

800 kg

500kg

Off-control








Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














800 kg Skyhook 808 808 789 774HSIC 1797 1186 932 744

900 kg Skyhook 997 520 263 124HSIC 4159 2094 1293 927

1100 kg Skyhook 1018 634 403 273HSIC 4143 2191 1361 987



500 kg Skyhook 954 926 895 875HSIC 1329 789 629 538

700 kg Skyhook 849 843 820 804HSIC 1772 1113 897 728

800 kg Skyhook 965 425 153 009HSIC 4110 1976 1242 890

900 kg Skyhook 773 778 762 748HSIC 1888 1230 922 747

1100 kg Skyhook 715 727 715 703HSIC 1771 1242 927 762


115

12

125

13

135

14

145Passive

Max

imum

acce

lera

tion

(ms

2)

A B

C






6 Conclusion



0 05 1 15 2minus15

minus10

minus5

0

5

10

15

Time (s)

Acce

lera

tion

(ms

2)


Off-control



0 01 02 03 04 05minus150

minus100

minus50

0

50

100

Time (s)

Acce

lera

tion

(ms

2)

Off-control



0 05 1 15 2minus005

0

005

01

015

02

025

03

Time (s)

Stro

ke (m

)

Off-control


minus4000

minus2000

0

2000

4000

6000

Dam

ping

forc

e (N

)

0 05 1 15 2

Time (s)


HSIC


0 005 01 015 02 025 03minus1

minus05

0

05

1

15

times104


Off-control

Impa

ct fo

rce (

N)

Stroke (m)




Skyhook controlHSIC

0

10

20

30

40

50

1 15 2 25 3 35 4


Off-control

Max

imum

impa

ct ac

cele

ratio

n (m

s2 )

1100 kg

800 kg

500kg


1 15 2 25 3 35 4005

01

015

02

025

03

035

04

045

Skyhook controlHSIC


Max

imum

effec

tive s

troke

(m)

1100 kg

800 kg

500kg

Off-control








Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 2minus15

minus10

minus5

0

5

10

15

Time (s)

Acce

lera

tion

(ms

2)


Off-control



0 01 02 03 04 05minus150

minus100

minus50

0

50

100

Time (s)

Acce

lera

tion

(ms

2)

Off-control



0 05 1 15 2minus005

0

005

01

015

02

025

03

Time (s)

Stro

ke (m

)

Off-control


minus4000

minus2000

0

2000

4000

6000

Dam

ping

forc

e (N

)

0 05 1 15 2

Time (s)


HSIC


0 005 01 015 02 025 03minus1

minus05

0

05

1

15

times104


Off-control

Impa

ct fo

rce (

N)

Stroke (m)




Skyhook controlHSIC

0

10

20

30

40

50

1 15 2 25 3 35 4


Off-control

Max

imum

impa

ct ac

cele

ratio

n (m

s2 )

1100 kg

800 kg

500kg


1 15 2 25 3 35 4005

01

015

02

025

03

035

04

045

Skyhook controlHSIC


Max

imum

effec

tive s

troke

(m)

1100 kg

800 kg

500kg

Off-control








Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Skyhook controlHSIC

0

10

20

30

40

50

1 15 2 25 3 35 4


Off-control

Max

imum

impa

ct ac

cele

ratio

n (m

s2 )

1100 kg

800 kg

500kg


1 15 2 25 3 35 4005

01

015

02

025

03

035

04

045

Skyhook controlHSIC


Max

imum

effec

tive s

troke

(m)

1100 kg

800 kg

500kg

Off-control








Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article vibration attenuation of

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