research article a lower extremity exoskeleton: human

Research ArticleA Lower Extremity Exoskeleton Human-MachineCoupled Modeling Robust Control Design Simulation andOverload-Carrying Experiment

Qing Guo1 Songjing Li2 and Dan Jiang3

1School of Aeronautics and Astronautics University of Electronic Science and Technology of China Chengdu 611731 China2Department of Fluid Control amp Automation Harbin Institute of Technology Harbin 150001 China3School of Mechatronics Engineering University of Electronic Science and Technology of China Chengdu 611731 China

Correspondence should be addressed to Qing Guo guoqinguestc163com

Received 4 November 2014 Revised 8 May 2015 Accepted 10 May 2015

Academic Editor Stanisław Migorski

Copyright copy 2015 Qing Guo et alThis is an open access article distributed under theCreativeCommonsAttributionLicense whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A robust 119867infin

control method and switched control algorithm for hydraulic actuator presents in human-machine coordinatedmotion to solve the motion delay of lower extremity exoskeleton After the characteristic parameters synthesis of human limband exoskeleton linkage the human-machine coupled motion model is constructed to estimate the appropriate hydraulic pressurewhich is considered as a structural uncertainty in hydraulic model Then the robust controller is designed to improve the robuststability and performance under the structural and parametric uncertainty disturbances Simulation results show that in walkingmode this robust controller can achieve a better dynamic response and aid-force efficiency than PID controller Then accordingto gait divisions of personrsquos limb motion the switched control algorithm is designed to reduce the delay of exoskeleton trackingperson Finally the experimental results show that the human-machine coordinated walk with bearing 60 kg load and squat actionwith no external load are realized effectively by this proposed method

1 Introduction

Lower extremity exoskeleton is a humanoid mechanicalequipment Since the exoskeleton can help a person bearmany loads and track a personrsquos motion the soldierrsquos loadcapacity and flexibility can be significantly improved Theexoskeleton is a complex system including 4 subsystemsmechanical structure gait sensing hydraulic control andpower supply Recently lower extremity exoskeleton can bearthe maximum load 200 lbs and realize typical action suchas walk running up and down hill or stairs squatting andtrotting [1] Since 2000 DARPA and U C Berkeley havedone many researches on exoskeleton and developed theprototypes called ldquoXOSrdquo and ldquoBLEEXrdquo [2ndash4] respectivelyLockheed has simplified the hydraulic actuator amountof BLEEX and renamed ldquoHuman Universal Load Carrierrdquo(HULC) [1] Other countries like Japan and France havedeveloped commercial or military exoskeleton called ldquoHALrdquoor ldquoHerculerdquo [5] In 2000 Atkeson et al [6] made conclusions

on the characteristics of humanoid robot sometimes themotion of machine is guided by person sometimes just theopposite especially sometimes the boundary is not clearlydivided between the two Due to the personrsquos participationthe traditional robotic program controlmethodwill cause themotional delay of exoskeleton tracking person [7]

Since 2004 the exoskeleton technology is developed inChina [8] A motor-driven exoskeleton with no external loadis designed by Naval Aeronautical Engineering Institute [9]In addition the aid-force exoskeleton with hydraulic-drivenactuator is designed to realize walk up and down stairs[10] However the human-machine coordinated motion isnot presented in aforementioned references if personrsquos gaitcharacteristic is ignored in exoskeleton control personrsquoscomfort and the aid-force effect are reduced In this paper ahuman-machine coupled model is constructed to analyze thedynamic external load This external load describes the aid-force efficiency of exoskeleton in the case of personrsquos safetyand should be compensated by exoskeleton controller

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 905761 15 pageshttpdxdoiorg1011552015905761

2 Mathematical Problems in Engineering

Due to the demands of high load efficiency and fastresponse lower extremity exoskeleton is driven by theelectrohydraulic servo system (EHSS) However EHSS hassome model uncertainties such as parametric disturbancesstructural uncertainties which are consequences of hydraulicphysical characteristics oil leakage oil temperature varia-tions and characteristics of valves near null [11]The classicalcontrol method of EHSS is PID [12] Due to lower orderand constant gains PID control is easy to be realized inengineering So the variable parameters of PID controllerare presented to suit the hydraulic dynamic characteristic[13 14] Some advanced control methods such as adaptivesliding mode [15ndash17] and backstepping controllers [18 19] areused in EHSS and get some good performance In additionthe robust controllers [20ndash22] of EHSS are presented withparameter uncertainty disturbance which are converted to119867infin

control problem solved by conventional robust analysisor linear matrix inequality The mixed sensitivity119867

infincontrol

method [23] of EHSS is presented to balance to the robuststability and response performance But the external load ofhydraulic control model is only considered as a disturbancenot to be quantitatively analyzed and described in robustuncertainty model In this paper by referring to119867

infincontrol

method aforementioned references a robust119867infincontroller is

designed to suppress structural uncertainty and parametricuncertainties which are caused by the external load andparametric disturbances Furthermore combined with theswitched control algorithm of hydraulic actuator this 119867

infin

controller can reduce the delay of exoskeleton trackingperson Simulation and experimental results validate thisproposed controller

2 Human-Machine CoupledModel Construction

Since human fast conscious is ahead of exoskeleton the onlymechanical motion model is not insufficient to analyze thesystematic dynamics So the coupled motion model needs tobe constructed to estimate the dynamic loads and the supplypower

21 Characteristic Parameters Synthesis Some characteristicparameters like mass and moment of inertia are differentbetween human limb and exoskeleton linkage So the char-acteristic parameters need be synthetic to balance 2 motionresponses activated by person and exoskeleton It meansthat human-constraint is relieved but not vanished throughequivalent multibody motion The human-constraint is asmall value in the thigh bandage which can not exceed thehuman tolerance

If the sagittal motion is considered only the rotationalmotion around the knee is shown in Figure 1 Here compo-nents 119861

1 119861

2 119861

3 119861

4 and 119861

5represent respectively human

calf mechanical calf sensing shoes human thigh andmechanical thigh

O

ltl5cltc

l4c

l3c

l3c

l1c l2c

B5B4

m1g

m2g

m3g

m4g

msg

m5g

mtg

C

B1

B2

B3

KA

Figure 1The rotationalmotion around the knee of human-machinesynthetic model

The centers of synthetic calf and thigh for human-machine coordinated motion [24] are shown as

119897119904119888=11989811198971119888 + 11989821198972119888 + 119898311989731198881198981 + 1198982 + 1198983

=11989811198971119888 + 11989821198972119888 + 11989831198973119888

119898119904

(1a)

119897119905119888=11989841198974119888 + 119898511989751198881198984 + 1198985

=11989841198974119888 + 11989851198975119888

119898119905

(1b)

According to the moment of inertia of the parallel axistheorem the moment of inertia of synthetic calf and thigharound respective center of mass is shown as

119868119904119888= [1198681119888 +1198981 (119897119904119888 minus 1198971119888)

2] + [1198682119888 +1198982 (119897119904119888 minus 1198972119888)

2]

+ [1198683119888 +1198983 (119897119904119888 minus 1198973119888)2]

(2a)

119868119905119888= [1198684119888 +1198984 (119897119905119888 minus 1198974119888)

2] + [1198685119888 +1198985 (119897119905119888 minus 1198975119888)

2] (2b)

where 119898119894is the 119894th component mass 119898

119904and 119898

119905are the total

masses of equivalent calf and thigh 1198971119888 1198972119888 and 119897

3119888are the

distances from center of mass to knee respectively 1198974119888 1198975119888are

the distances from center of mass to hip respectively and 119868119894119888

is the 119894th moment of inertia around respective center of mass

22 Human-Machine Coupled Motion Model In human-machine coordinated motion the exoskeleton linkagesshould compensate the load when human foot contacts theground So in the stance phase of walk or squat the hydraulicactuator not only realizes the exoskeleton knee rotationtracking human limb rotation but also balances the variableload In order to estimate the dynamic load on the hydraulic

Mathematical Problems in Engineering 3

Fl

O5

O4

5

4

3

2

1

1205791

1205793

120579k

Fp

O2

O3

O1

A

X

Y

FO2y

FO2x

1FO3x

FO3y

aGsy

aGsxGs

a120591s

ans120572O1

120596O1

FO1x

FO1y

TO1

TO1

1205791

1205792

FO4x

FO2x

FO2y

Fl

FO4y

a120591t

aGtx

aGty

ant

Gt

2

120572O2

120596O2

FO4x

FO3x

FO3y

FA

TA

Fp

FA

Fp

FO4y

a120591p

a120591c

anc

anp 4

120572c120596c

3TA

1205792

120572p 120596p

(a) (b)

(c) (d)

Figure 2 The synthetic model of human-machine coupled motion (linkages 1 and 2 represented the synthetic calf and thigh)

actuator the human-machine coupled motion model needsto be constructed in Figure 2 The link point ldquo119874

5rdquo between

mechanical back frame and thigh is the hip joint The anklejoint ldquo119874

1rdquo is equivalent to a hinge when human foot contacts

the ground The external load ldquo119865119897rdquo including the exoskeleton

mass and effective load hung at the back frameThe force ldquo119865119901rdquo

provided by hydraulic actuator compensates the load on theexoskeleton

The force andmoment equilibriumequations of 1 linkage(synthetic calf) are shown as

1198651198741119909 minus1198651198742119909 +1198651198743119909 = 119898119904119888

119886119866119904119909 (3a)

1198651198741119910 +1198651198742119910 minus1198651198743119910 minus119898119904119888

119892 = 119898119904119886119866119904119910 (3b)

1198791198741 +1198651198742119909119897119904119888 cos 1205791 +1198651198742119910119897119904119888 sin 1205791

minus1198651198743119909 (119897119904119888 minus 1198971) cos 1205791 minus1198651198743119910 (119897119904119888 minus 1198971) sin 1205791

minus119898119904119888119892119897119892119904sin 1205791 = minus 1198681199041198881205721198741

(3c)

119886119866119904119909

= 1198861198741119909 minus 119886119899119904 sdot cos 1205791 + 119886120591119904 sdot sin 1205791

119886119866119904119910

= 1198861198741119910 minus 119886119899119904 sdot sin 1205791 minus 119886120591119904 sdot cos 1205791

(3d)

where 119886119899119904= 119897

119892119904sdot 120596

21198741 119886120591119904 = 119897119892119904 sdot 1205721198741 1198861198741119909 = 1198861198741119910 = 0

Similarly the 2 linkage (synthetic thigh) 3 linkage(piston) and 4 linkage (cylinder) can also be analyzed asfollows

1198651198742119909 +1198651198744119909 = 119898119905

119886119866119905119909 (4a)

1198651198744119910 minus1198651198742119910 minus119898119905119888

119892minus119865load = 119898119905119886119866119905119910 (4b)

minus119865119897119897119905119888cos (120579

119896+ 1205791) + 11986511987441199091198972cos (120579119896 + 1205791)

minus 11986511987441199101198972sin (120579119896 + 1205791) +119898119905

119892119897119892119905sin120579

119896= 119868

1199051198881205721198742

(4c)

119865119901sin1205792 minus119865119860cos1205792 +1198651198743119909 = 119898119901

119886119866119901119909 (5a)


minus119865119901cos1205792 minus119865119860sin1205792 +1198651198743119910 = 119898119901

119886119866119901119910 (5b)

minus119865119860119897119903+119879

119860= 119868

119901120572119901 (5c)

minus119865119901sin1205792 +119865119860cos1205792 minus1198651198744119909 = 119898119888

119886119866119888119909 (6a)

119865119901cos1205792 +119865119860sin1205792 minus1198651198744119910 = 119898119888

119886119866119888119910 (6b)

minus119865119860(119897ℎminus 119897

119903) minus119879

119860= 119868

119888120572119888 (6c)

Some angle and angular velocity relations in Figure 2 areas follows

sin 1205791 =1198972 sin 120579119896

radic11989721 + 119897

22 minus 211989711198972 cos 120579119896

(7a)

sin 1205793 =(1198972 minus 119897119905119888) sin 120579119896

radic1198972119904119888+ (1198972 minus 119897119905119888)

2minus 2119897

119904119888(1198972 minus 119897119905119888) cos 120579119896

(7b)

1205792 = 1205793 minus 1205791 (7c)

120596119901= 120596

1198741

minus

1198972 cos 120579119896120596119896 + 1198972 sin 120579119896 (11989711198972 sin 120579119896120596119896radic11989721 + 11989722 minus 211989711198972 cos 120579119896)

cos 1205793 (11989721 + 11989722 minus 211989711198972 cos 120579119896)

(8a)

120596119888= 120596

1198742

minus


cos (120587 minus 120579119896minus 1205793) (119897

21 + 119897

22 minus 211989711198972 cos 120579119896)

(8b)

1205961198742 = 120596119896 +1205961198741 (8c)

In above equations some parameters are defined asfollows

Description of Some Motion and Geometry Parameter Con-sider the following

1198651198741119909

1198651198741119910

force from hinge 1198741to linkage 1

1198651198742119909

1198651198742119910

force from linkage 1 to linkage 21198651198743119909

1198651198743119910


1198651198744119910


119886119866119904119910

calf acceleration at the center of gravity119886119866119905119909

119886119866119905119910

thigh acceleration at the center of gravity119886120591119904 119886

119899119904 tangential and normal acceleration of calf

acceleration119886120591119905 119886119899119905 tangential and normal acceleration of thigh

119886120591119901 119886119899119901 tangential and normal acceleration of piston

119886120591119888 119886

119899119888 tangential and normal acceleration of cylin-

der119897119892119904 length between the center of gravity calf and the

ankle119897119892119905 length between the center of gravity thigh and the

knee1198971 1198972 distance between hydraulic cylinder calf thigh

connections and knee

1205961198741 120572

1198741 angular velocity and angular acceleration of

calf around the ankle1205961198742 120572


thigh around the knee120596119896 knee angular velocity

119868119901 moment of inertia of piston around the leg con-

nection point1198791198741 torque from point 119874

1

120596119901 120572


piston120596119888 120572


cylinder119865119901 radial force along the piston

119865119860 normal force perpendicular to cylinder

119879119860 torque from point 119874

3

1205791 angle between linkage 1 and vertical line

1205792 angle between piston and vertical line

1205793 angle between piston and linkage 1

120579119896 knee angle

119898119901119898

119888 piston and cylinder mass

119897ℎ total length of hydraulic actuator119897119903 length of piston

If knee angle 120579119896and angular velocity 120596

119896are given (1a)ndash

(8c) can get the analytical solution ldquo119865119901rdquo So load pressure ldquo119875

119897rdquo

of the hydraulic actuator can be computed by

119875119897=

119865119901

1205871199032119901

(9)

where ldquo119903119901rdquo is hydraulic cylinder diameter

Now the person can realize 2 typical actions walk andsquat after wearing the exoskeleton The motional rangeof knee angle is from 85∘ to 155∘ in the gait duration ofthese actions In addition the maximum external load onthe back frame of exoskeleton is 60 kg Substituting someknown parameters into (3a)ndash(9) the load pressure 119875

119897can

be computed digital simulation method shown in Figure 3So if the external Load ldquo119865

119897rdquo does not exceed 90 kg the

load pressure 119875119897will not be more than 8MPa It means

that the hydraulic supply pressure ldquo119875119904rdquo can be set as 8MPa

to appropriately compensate the external load 60 kg If theperson wears the exoskeleton with no effective load thesupply pressure is chosen as 4MPa to balance the exoskeletonmass

23 Hydraulic Model The electrohydraulic servo controlsystem is also fixed on the exoskeleton It has 2 differentcharacteristics the asymmetric hydraulic cylinder and thesingle-rod actuator The dynamics of servo valve can bedescribed as follows

119909v119906=

119870sv1205962sv

1199042 + 2120577sv120596sv119904 + 1205962sv (10)


80 90 100 110 120 130 140 150 160123456789

120579k (∘)

Fl = 0kgFl = 30kg

Fl = 60kgFl = 90kg

Pl

(MPa

)

Figure 3The load pressure 119875119897in the gait duration of walk and squat

varying load and knee angle

where 120577sv is the damping ratio 120596sv is natural frequency and119870sv is the servo valve gain

The flow equations can be linearized as follows [25]

119876119886= 119870

1119902119909v minus119870

1119888119901119886 (11a)

119876119887= 119870

2119902119909v +119870

2119888119901119887 (11b)

1198701119902= 119862

119889119908radic

2Δ119901119901

120588

1198702119902= 119862

119889119908radic

2Δ119901119877

120588

(12a)

1198701119888=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119862119889119908119909v

radic2120588Δ119901119901

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198702119888=

100381610038161003816100381610038161003816100381610038161003816

119862119889119908119909v

radic2120588Δ119901119877

100381610038161003816100381610038161003816100381610038161003816

(12b)

Δ119901119901=

1003816100381610038161003816119901119904 minus 1199011198861003816100381610038161003816 119909v ge 0

1003816100381610038161003816119901119886 minus 1199011199031003816100381610038161003816 119909v lt 0

(13a)

Δ119901119877=

1003816100381610038161003816119901119887 minus 1199011199031003816100381610038161003816 119909v ge 0

1003816100381610038161003816119901119904 minus 1199011198871003816100381610038161003816 119909v lt 0

(13b)

where 119876119886 119876

119887are 2 sides of cylinder flow 119901

119886 119901

119887are the

cylinder chamber pressures 119901119904 119901

119903are supply and return

pressures 119862119889is the valve coefficient of discharge 119908 is the

valve orifice area gradient 120588 is the oil density and 119909v is thespool valve position1198701

119902(119870

2119902) and1198701

119888(119870

2119888) are flow gains of the

servo valve and flow-pressure coefficient

If the internal and external leakages are neglected theflow-pressure continuous equation of cylinder is shown asfollows

119886=

120573119890

11988101 + 119860119886119910(119876

119886minus119860

119886119910) (14a)

119887=

120573119890

11988102 minus 119860119887119910(minus119876

119887+119860

119887119910) (14b)

11988101 = 11988102 =119881119905

2=119860119886119897ℎ

2 (15)

where120573119890is the fluid bulkmodulus119860

119886and119860

119887are the annulus

areas of 2 sides of the cylinder1198810111988102 are half-volume of thecylinder and 119910 is the cylinder position

The mechanical dynamic equation is shown as follows

119898 119910 = (119901119886119860119886minus119901

119887119860119887minus119865

119891minus119865

119901) (16)

where 119865119891is the stick-slip friction caused by the viscous damp

of oil and 119865119901is the variable load on the hydraulic actuator

If the knee angle is known the linear position of cylinder119910 can be correspondingly computed by geometric relation-ship as follows

119910 (120579119896) = radic119897

21 + 119897

22 minus 2 times 1198971 times 1198972 times cos 120579

119896 (17)

According to the analytical computation method of theload pressure 119875

119897by (9) the linear expression between 119865

119901and

119910 can be constructed as follows

119865119901(120579119896 119865119871) = 119870

119865119901(120579119896) 119910 (120579

119896) (18)

where119870119865119901represents a variable bounded scale with load and

knee angleThe stick-slip friction often includes 2 items a stick phase

119865119891static which occurs when velocity is within a small critical

velocity range and a slip friction 119865119891slip which is the same

as Coulomb friction model [26] Here the slip friction canbe estimated by simplified parameter identification methodHere the stick item is neglected so the stick-slip friction 119865

119891

can be approximately described as follows

119865119891asymp 119887 119910 (19)

where 119887 is the viscous damping coefficient of oilThe slip friction 119865

119891slip should be constructed in thesimulation model to identify the parameters 119887 The max-imum stroke of cylinder is controlled by one best PIDcontrol parameters to track the square demand The com-parison results are shown in Figure 4 The simulation resultof hydraulic position is similar to the experiment whenparameters 119887 are 100 especially the dynamic response isbetter than other conditions So the nominal value of 119887 canbe approximately identified to be used in the parametricuncertainty model in Section 3

3 Robust Controller Design

Lower extremity exoskeleton provides two features high loadbearing and fast motion So exoskeleton should have fast


0 2 4 6 8 10 12 14minus002

0

002

004

006

008

01

012

014

t (s)

y (m

)

Experimentb = 100

b = 200b = 300

Figure 4 The square response of experiment and simulationvarying the viscous damping coefficient 119887

responses but should not hinder the wearer In this sectiona robust controller is designed to improve the dynamicresponse The knee angle will be servo controlled to realizethe human-machine coordinated motion Because this EHSShas parametric disturbances and structural uncertaintiesthe robust 119867

infincontrol method is presented to satisfy the

requirement of close-loop stability and robust performanceAfter the uncertainty model of the EHSS is constructed therobust controller is designed to find the 119867

infinsuboptimal

controller

31 Uncertainty Model Construction This EHSS has someparametric uncertainties the flow gains of servo valve 119870119894

119902

(119894 = 1 2) the viscous damping coefficient 119887 the effectivebulk modulus 120573 and variable volume of 2 sides cylinder(11988101+119860

119886119910 119881

02minus119860

119887119910) and a structural uncertainty119870

119865119901from

external load These uncertainties parameters 119870119902 119864 119887 and

119870119865119901are described by upper Linear Fractional Transformation

[27 28] as follows

119872119870119902= [

[

0 119870119902

119901119870119902

119870119902

]

]

119872119864=

[[[

[

minus119901119864

1119864

minus119901119864

1119864

]]]

]

119872119887= [

0 119887

119901119887119887

]

119872119870119865119901

= [

0 1

119901119870119865119901

119870119865119901

]

(20)

119870119902= 119870

119902(1+119901

119870119902120575119870119902) minus1 le 120575

119870119902le 1 (21a)

119864 = 119864 (1+119901119864120575119864) minus1 le 120575

119864le 1 (21b)

119887 = 119887 (1+119901119887120575119887) minus1 le 120575

119887le 1 (21c)

119870119865119901= 119870

119865119901+119901

119870119865119901120575119870119865119901 (21d)

where119864 = 1198811199052120573119870

119902119864 and 119887 are the nominal values 120575

119870119902 120575119864

and 120575119887are relative variation respectively and the maximum

relative uncertainties119901119870119902119901

119864 and119901

119887can be estimated varying

some hydraulic parameters [20]If these vectors are defined the state vector 119883 =

[1199091 119909

2 119909

3 119909

4 119909

5 119909

6]119879

= [119909v v 119901119886 119901119887 119910 119910]119879 the control

input 119906(119905) which is the current of servo valve the vectorof exogenous inputs 119908(119905) = [119908

1 119908

2 119908

3 119908

4 119908

5]119879 the

output cylinder position 119910(119905) measured by encoder and theregulated output vector 119911(119905) = [119911

1 1199112 1199113 1199114 1199115 1199116]119879 then the

uncertain state-space model is constructed as follows

(119905) = 119860119883 (119905) + 1198611119882(119905) + 1198612119906 (119905) (22a)

119885 (119905) = 1198621119883 (119905) +11986311119882(119905) (22b)

119884 (119905) = 1198622119883 (119905) (22c)

where

119860 =

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

[

0 1 0 0 0 0

minus1205962v minus2120577v120596v 0 0 0 0

119870119902

119864

0 minus119870119888

119864

0 0 minus119860119886

119864

minus

119870119902

119864

0 0 minus119870119888

119864

0 1198602

119864

0 0 0 0 0 1

0 0119860119886

119898minus119860119887

119898minus

119870119865119901

119898

119887

119898

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

]

(23a)

1198611 =

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

[

0 0 0 0 00 0 0 0 0119901119870119902

119864

minus119901119864

0 0 0

minus

119901119870119902

119864

0 minus119901119864

0 00 0 0 0 0

0 0 0 minus119901119887

119898minus

119901119870119865119901

119898

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

]

(23b)


r+

minusK

u

Wp

ep

Wu

y

eu

Δ

z

w

Ghyd

Gic = Fu(Ghyd Δ)

Figure 5 The block diagram of the closed-loop system with robuststability and performance requirements

1198612 = [0 119870sv1205962v 0 0 0 0]119879 (23c)

1198621 =

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

[

119870119902

0 0 0 0 0

119870119902

119864

0 minus119870119888

119864

0 0 minus119860119886

119864

minus

119870119902

119864

0 0 minus119870119888

119864

0119860119887

119864

0 0 0 0 0 119887

0 0 0 0 1 0

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

]

(23d)

11986311 =

[[[[[[[[[[[

[

0 0 0 0 0119901119870119902

119864

minus119901119864

0 0 0

minus

119901119870119902

119864

0 minus119901119864

0 00 0 0 0 00 0 0 0 0

]]]]]]]]]]]

]

(23e)

1198622 = [0 0 0 0 1 0] (23f)

The robust stability and performance requirements of theclosed-loop system are shown in Figure 5

In Figure 5 the weighting functions 119882119901and 119882

119906are

used to reflect the relative significance of the performancerequirement The robust controller 119870 needs to be designedto optimize the robust performance 119890

119901and 119890

119906 The nominal

model 119866hyd and the interconnection model 119866ic represent therobust uncertainty model shown in (22a) (22b) and (22c)The demand and actual cylinder position 119903 119910 are the inputand output of this EHSS

32 Robust Controller Design The frequency response ofthe open-loop system with model uncertainties is shown inFigure 6 The phase margins are obviously different betweenperturbed model and nominal model which is caused bythe uncertainty parameter 119870

119865119901 Due to the presence of

119870119865119901 the stable phase of open-loop system will lead or lag

01 1 10 100628

Log

mag

nitu

de (d

B)

minus200minus150minus100

minus500

Phas

e (de

g)

Frequency (rads)

01 1 10 100628Frequency (rads)

NominalPerturbed

100

10minus2

10minus4

Figure 6 The frequency response of the open-loop system withvarying uncertainty parameters119870

119902 119864 119887 and 119870

119865119901

approximately 50 degrees in middle frequency stage (1sim628 rads)

The weight functions119882119901and119882

119906are iteratively designed

according to robust performance requirement and controlconstraint shown as follows [29]

119882119901= 095times 119904

2+ 18119904 + 5

1199042 + 8119904 + 001 (24a)

119882119906= 5times 10minus4 (24b)

The singular values of the weight function 119882119901

andthe sensitivity function of close-loop 119866

119899

119904should satisfy

119882119901119866119899

119904infinle 120583(119882

119901)120583(119866

119899

119904) lt 1 which has guaranteed the

basic control performanceThe robust controller will guarantee robust stability and

robust performance requirement which can be described inFigure 5 [30 31]

(1) 119867infin

suboptimal control problem to design a con-troller119870 satisfying the following condition

1003817100381710038171003817119865119897 (119866ic 119870)1003817100381710038171003817infin

=10038171003817100381710038171003817119865119897(119865

119906(119866hyd Δ) 119870)

10038171003817100381710038171003817infinlt 120574 (25)

where 120574 gt 1205740 1205740is theminimumvalue of 119865

119897(119866ic 119870)infin under

119870 is a stable controller

(2) Robust performance requirement the close-loop sys-tem composed of 119866ic and119870 should satisfy

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

119882119901(119868 + 119866ic119870)

minus1

119882119906119870(119868 + 119866ic119870)

minus1]

]

100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

lt 1 (26)

If the relative uncertainty Δ is from minus119868 to Δ = 119868 thesingular value of close-loop perturbed is 028 which is lessthan 1 shown in Figure 7 The close-loop system 119866

119888is stable

and its relative uncertainties Δ can be extended to 1028In Figure 8 the maximum singular value 120583 of close-loop

shows that the robust performance requirement is 14 which


Frequency (rads)

Upper boundLow bound

100

10minus1

10minus3

10minus2

100 101 10210minus110minus210minus4

120583

Figure 7 The maximum robust stability bound with varyinguncertainty parameters119870

119902 119864 119887 and 119870

119865119901

NominalUpper bound

Low bound

Frequency (rads)

100

10minus02

10minus06

10minus04

100 101 10210minus110minus2

120583

Figure 8 The maximum robust performance bound with varyinguncertainty parameters119870

119902 119864 119887 and 119870

119865119901

means that the steady tracking accuracymay not be excellentBut in middle and high frequency stage from 4 to 100 radsthe dynamic tracking performance and robustness marginmeet the requirement [32]

After iterative design the robust controller is 7 ordersTaking the actual project into account the controller shouldbe reduced into lower order by Balanced Truncation Method[33] So the controller is simplified to 4 orders as follows

119906extension = 119870 (119904) (119910 minus 119910)

=1198994119904

4+ 1198993119904

3+ 1198992119904

2+ 1198991119904 + 1198990

11988941199044 + 1198893119904

3 + 11988921199042 + 1198891119904 + 1198890

(119910 minus 119910)

(27)

where 119899119894(119894 = 0 4) 119889

119894(119894 = 0 4) are numerator and

denominator constant 119910 is demand input and 119910 is output

4 Gait Divisions of Coordinated Motion

In this section the gait phase divisions of 2 typical actionmodes are analyzed and designed According to the gaitidentification the switched control is realized to reduce

Single leg squat Up-squat

Walking cycle

The end for walk switch to a standing action

The end for squat switch to a standing action

Upper limb trunkLeft leg (thigh and shank)

Right leg (thigh and shank)

(1) (2) (3) (4) (5) (2)

(1) (6) (7) (8) (9) (2)

Figure 9 The gait subphases division of walk and squat action inhuman-machine coordinated motion

the delay of exoskeleton tracking person and improve thehuman-machine coupling effect

41 Walk Mode The walking gait law of human-machinecoordinate motion can be divided into 5 subphases shown inFigure 9 These subphases are (1) double standing (2) rightleg swing left leg stance (3) double stance with right leg frontand left leg back (4) right leg stance and left leg swing and(5) double stance with left leg front and right leg back

In order to improve the exoskeleton tracking personrsquosmotion 2 subphases (3) and (4) are merged That means ifthe subphases (3) are identified make the hydraulic cylinderof left leg retract in advance This operation realizes theexoskeleton ahead of person which helps person regulatethe center of system gravity easily in overload carryingcondition After the subphase (4) is identified by exoskeletonthe center of system gravity is steadily regulated by personSimultaneously the exoskeleton continuously realizes theright stance leg and the left swing leg The subphases (5)and (2) are similar to subphases (3) and (4) So the switchedcontrol algorithm is described in Table 1

In Table 1 119906119897and 119906

119903are control current of the of left and

right servo valve 119906max = plusmn40mA which is the maximumcurrent of servo

42 Squat Mode The squat gait law is divided into 6 sub-phases (1) double standing (6) left leg stance and right leguplift for preparation (7) left leg retraction gradually andright leg swing (8) left leg reached the minimum positionof cylinder and right knee contacting the ground whichmeans the single leg squat action ends (9) left leg extensiongradually and right leg swing and (2) left leg stance and rightleg swing which means the up-squat action ends and switchto the walking mode So the switched control algorithm isdescribed in Table 2

In gait (7) the left knee angle is retracted by fader controlThe symbol 119896 = 2 and 119905 is fader time


Table 1 The duty of exoskeleton and person in walking gaitsubphases by different switched controller form

Gait Controller Exoskeleton Person

(1) 119906119897= 119906extension

119906119903= 119906extension

Control doubleknee angle

Adapt toexoskeleton

(2) 119906119897= 119906extension119906119903= minus119906max

Control leftknee angle andretract right leg

Adapt toexoskeleton

(3) 119906119897= minus119906max

119906119903= 119906extension

Adapt to personand retract leftleg in advance

Regulate thecenter of gravity

(4) 119906119897= minus119906max

119906119903= 119906extension

Control rightknee angle andretract left leg

Adapt toexoskeleton


Adapt to personand retract rightleg in advance


Table 2The duty of exoskeleton and person in squat gait subphasesby different switched controller form




Squatpreparation

(7)119906119897=

119906max(1 minus 119890minus119896119905

)

119906119903= minus119906max

Retract left kneeangle gradually

Adapt toexoskeleton

(8) 119906119897= minus119906max

119906119903= minus119906max

Adapt to personand hold in the

prescriptposture


(9) 119906119897= 119906extension

119906119903= 119906extension

Control 2 kneeangles andextend 2 legs

Adapt toexoskeleton

To ensure the safety of person the squat preparation timeneed to be set as 2 or 3 sec The subphase (8) is identified bya pressure force sensor on the right kneeThe transition fromsubphase (8) to subphase (9) needs amode switch button andgait identification algorithm which is not described in thissection

5 Simulation Results

Thehydraulicmodel is simulatedwith the external load 60 kgThehydraulic parameters for simulation are shown inTable 3In order to verify the effectiveness of the proposed methodthe natural walking gait law is considered as the commandinput with 05Hz frequency The fore-half duration is stancephase for one leg with the range of knee angle from 15∘ to 30∘The afterwards duration is swing phase with range of kneeangle from 15∘ to 73∘

According to the above analysis of the load pressure119875119897 the

hydraulic supply pressure 119875119904is 8MPa and the external load

is 60 kg in simulation Some hydraulic parameters are shownin Table 3 The maximum relative uncertainties 119901

119870119902= 025

Table 3 The hydraulic parameters for simulation

Name Value UnitDamping ratio 120585sv 07 mdashNatural frequency 120596

119899628 rads

Servo valve gain 119870sv 00125 mAFluid bulk modulus 120573

1198907000 bar

Mass of hydraulic actuator119898 25 kgAnnulus areas of no-rod cylinder 119860

119886177 times 10

minus4 m2

Annulus areas of rod cylinder 119860119887

113 times 10minus4 m2

Distance between hydraulic cylindercalf connections and knee 119897

1

0322 m

Distance between hydraulic cylinderthigh connections and knee 1198972

0065 m

Return pressures 119901119903

1 barPiston stroke 119871

119903016 m

Flow-pressure coefficient 1198701

119888asymp 119870

2119888

518119890 minus 12 m3sPaNominal value of119870

1199020188 m2s

Nominal value of 119887 100 NsmNominal value of 119864 198119890 minus 14 m3paNominal value of119870

1198651199011200 Nm

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

t (s)

4 45 515202530

Knee commandRobust control

PID control

Stan

ceph

ase

Swin

gph

ase

120579lk

(∘)

Figure 10 The left knee angle response of walking gait

119901119864= 07 119901

119887= 1 and 119901

119870119865119901= 300 Here 2 control methods are

used to make the exoskeleton track the person in the walkingmode simulationThePID control parameters are designed asthe best group which satisfy the frequency domain marginThe robust119867

infincontrol method is realized by (27)

The left knee angle response of exoskeleton is shownin Figure 10 In swing phase the dynamic characteristicof PID control is same as the robust 119867

infincontrol because

there is no external load on the hydraulic actuator But instance phase the dynamic performance of robust control isbetter than PID control due to the variable compensationforce 119865

119901provided by hydraulic actuator on the exoskeleton

This external force disturbance results in the parametricuncertainty of119870

119902 119864 which reduce the stabile margin of PID

control Importantly the control current needs to balance thecompensation force 119865

119901and stick-slip friction 119865

119891in the case


0 1 2 3 4 5 6 7 8 9 10t (s)

Position commandRobust control

PID control

4 45 5034

0345035

0355

03

032

034

036

038

yl

(m)

Figure 11Theposition response of left hydraulic cylinder ofwalkinggait

0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

20

30

40

t (s)

PID controlRobust control

ul

(m)

Figure 12 The left servo valve control current of walking gait

of positional error suppression In addition the asymmetriccylinder affects the dynamic characteristic of hydraulicmodelwhen cylinder extension switches into retraction especiallyin stance phase The parametric uncertainty of 119870

119902 119864 and 119887

and structural uncertainty 119865119901are considered in the system

model construction so the robust controller can guaranteethe dynamic performance in the whole duration of walkingmode

The dynamic cylinder position of left hydraulic actuatoron exoskeleton is shown in Figure 11 It is computed by (17)after the knee angle is measured by encoder The availabledisplacement of cylinder is 007m in waking mode whichis less than the maximal stroke 016m of cylinder It meansthat only half stroke is useful in human-machine coordinatedwalk But the maximal stroke of cylinder needs to satisfy thesquat mode which requires enough length of cylinder

The control current of left servo valve is shown inFigure 12 The general trend of control current is same using2 control methods But the robust controller 119906extension ismore intense than PID control especially in stance phaseBecause the controller involves the high order item therobust controller can reduce the tracking error quickly in the

0 1 2 3 4 5 6 7 8 9 10

minus1000

0

500

1000

15001300

minus500minus600

t (s)

Com

pens

atio

n fo

rce (

N)

Robust controlPID controlFpres response

Figure 13 The aid-force effect of exoskeleton by left hydraulicactuator

case of model uncertaintyThemaximal amplitude of controlcurrent is 30mA which does not exceed the maximal servovalve control current plusmn119906max

The aid-force effect of exoskeleton by left hydraulic actu-ator is shown in Figure 13 Here the demand compensationforce 119865pres is constructed by

119865pres =

119865119901

(119879120572119904 + 1)2

stance phase

(1 minus 1(119879120572119904 + 1)2

)119865119901

swing phase(28)

where 119879120572= 005 s is defined as the transition response time

of external load 119865119897

In Figure 13 the compensation force error means theerror between 119865pres and the hydraulic output actual force119865act In stance phase of left leg the compensation force errordoes not exceed 500N by the robust controller The demandcompensation force 119865pres is 1300N approximately It meansthat the load bearing efficiency is great than 60 in stancephase of left leg Because 2 legs are alternative in walking gaitthe left hydraulic actuator needs not to compensate 119865pres inits swing phase while the hydraulic actuator on right leg ofexoskeleton is in charge of this load bearing task Thereforethe aid-force effect of simulation result can not include thehydraulic output actual force in swing phase in this figure

6 Experimental Results

The human-machine coordinated motion is realized by atester wearing the exoskeleton prototype The gait can beidentified by 4 force sensors in the sensor shoes The 2knee angles of exoskeleton are measured by 2 encoders Thehydraulic control system includes a motor a fixed pump2 servo valves 2 cylinders and an embedded controlleron the mechanical brace The net weight of prototype is30 kg In walking mode the tester can bear no more than60 kg external load In squat mode the tester can realizethe basic gait with no external load due to the safety of


0 2 4 6 8 10 12 14 16minus20

0

20

40

60

80

t (s)

Left kneeRight knee

c ef

ga b d

120579k

(∘)

Figure 14 The knee angles of 2 legs in walking mode

tester The hydraulic actuators are controlled by the robustcontrol method and the control algorithm will be switchedin different gait

61 Walking Mode with Overloading In walking mode theextension and retraction of 2 legs are alternatively realized byhydraulic actuator The walking frequency is 05Hz bearingwith 60 kg load

The dynamic characteristics of 2 knee angles are shownin Figure 14 which includes 5 subphases of walking gait Ingait (4) described by time slice 119886sim119887 (08 ssim15 s) the left legis retracted by the tester himself and the control current ofleft leg is minus119906max shown in Figure 20 In gait (5) described bytime slice 119887sim119888 (15 ssim17 s) the left leg is extended by feedbackcontrol and the right leg begins to retract in advance to helpthe tester regulate the center of gravity This ahead of controlcan make the tester move easily In gait (2) described by timeslice 119888sim119889 (17 ssim24 s) the right leg is retracted by the testerhimself like as gait (4) In gait (3) described by time slice 119889sim119890(24 ssim26 s) the right leg is extended by feedback control likegait (5)These 4 subphases are switched frequently in walkingmode If the walking mode ends the double stance phase gait(1) will arise described by time slice 119891sim119892 (14 ssim156 s) In gait(1) 2 legs are controlled by feedback controlThe external loadis compensated by 2 hydraulic actuators on the exoskeletononly in stance phase In swing phase the exoskeleton shouldhelp the tester switch the center of gravity from one leg to theother leg Otherwise the human-machine coupled effect isreduced significantly and the tester may be at loss of balance

The cylinder positions of 2 hydraulic actuators in onecycle of walking mode are shown in Figure 15 Here theextended position commands are designed as a rising stepThe servo feedback control phase of left leg is described astime slice 119887sim119890 (15 ssim26 s) But in time slice 1198871sim1198872 (155 ssim165 s) the cylinder position is still retracted after the positivecontrol current output This response delay is caused by thebandwidth limitation of the whole hydraulic control systemThe pressure of cylinder is constructed with time-consuming08 s from 0 to stable value 8MPa So the available positionresponse of left cylinder is time slice 1198872sim 119890 (165 ssim26 s)

08 1 12 14 18 2 22 24 26165 2815

032

033

034

035

036

037

038

t (s)

Cylid

ner p

ositi

on (m

)

Left commandRight command

Left actualRight actual

a b2bb1

c d1

d2

d e

Figure 15 The cylinder positions of 2 hydraulic actuators in onecycle of walking mode

0 2 4 6 8 10 12 14 16minus40minus30minus20minus10

010203040

t (s)

Con

trol c

urre

nt (m

A)

Left controlRight control

a

c e f

gb1 d1

Figure 16 The control current of 2 servo valves in walking mode

where the actual position of cylinder tracks the extendedposition commands In time slice 119886sim119887 the retracted positioncommands of left leg is designed as a fall step It is not a servocontrol phase for left leg So the control current of servo valveis minus maximal value minus119906max directly in order to help thetester retract the cylinder quicklyThe dynamic characteristicof right cylinder positions is similar to the left cylinder

The switched control currents correspond to 5 subphasesshown in Figure 16 The servo control phase of left leg istime slice 1198871sim 119890 (155 ssim26 s) in one cycle Similarly theservo control phase of right leg is starting at time 1198891 Indouble stance phase described by time slice 119891sim119892 the servocontrol value is near 0 after 2 knee angles reach the testerrsquosstance posture The dynamic pressures of 2 hydraulic actu-ators are shown in Figure 17 In stance phase the pressureis constructed from 0 to 8MPa with approximately 08 stime consumed due to bearing heavy load In swing phasethe fast time-consuming is 04 s for pressure from 8MPa to0 It reveals that personrsquos active control can be faster thanthe control response of exoskeleton if the walking speed ispossible So the switched control algorithm shown in Table 1is suitable for human-machine coordinated walk


0 2 4 6 8 10 12 14 160

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

PlPr

Figure 17 The pressures of 2 hydraulic actuators in walking mode

62 SquatMode Test According to Figure 9 6 gait subphasesof squat mode are realized by switched control algorithmshown in Table 2 The dynamic characteristics of 2 kneeangles in squat mode are shown in Figure 18 The gaitsubphases are described by time slice ℎsim119894 (0 ssim2 s) 119894sim119895 (2 ssim4 s) 119895sim119896 (4 ssim7 s) 119896sim119897 (7 ssim29 s) 119897sim119898 (29 ssim32 s) and 119898sim119899 (32 ssim35 s) respectively In gait (5) described as time sliceℎsim 119894 the double stance phase should be passed before thesquat mode begins The cylinder positions are controlled toreach the testerrsquos stance posture shown in Figure 19 In gait(6) described as time slice 119894sim119895 the right leg swings to pre-pare the squat action and the tester regulates the center ofgravity at this moment In gait (7) described as time slice119895sim119896 the left cylinder is retracted gradually by exponentialfader control shown in Table 2 So the left leg bends graduallybefore the right knee contacts the groundThe control currentis reduced from 0 to minus119906max in 3 sec shown in Figure 20 Ingait (8) described as time slice 119896 sim 119897 2 control currents areminus maximal value The exoskeleton can not compensateits mass by 2 hydraulic actuators at this phase The tester canbear the load himself through regulating the center of gravityto reduce the backward torque caused by exoskeleton massandhold in the prescript posture In gait (9) described as timeslice 119897 sim 119898 2 cylinders are extended to realize the up-squataction in 3 sec (Figure 21)The cylinder positions of 2 legs arecontrolled by robust control algorithm to reach the suitablepositions where gait (9) is switched into gait (2) After up-squat phase ends the squat mode will return to the walkingmode

63 Experimental Scene The experimental scenes of human-machine coordinated motion are shown in Figures 22 and23 In walking mode the tester can bear the external load60 kg hung on the back frame of exoskeleton When thewaking gait is switched the tester will bear a little moreload himself due to the regulation of center of gravity In thestable stance phases such as double standing or single legstance the exoskeleton can guarantee the aid-force effect bythe proposed robust controller Also the comfortable effect oftester is improved by the switched control algorithm

0 5 10 15 20 25 30 35 40

0

50

100

150

80

t (s)

Left kneeRight knee

h i j

lk

mn

minus50

120579k

(∘)

Figure 18 The knee angles of 2 legs in squat mode

0 10 20 30 40025

03

035

04

045Cy

lidne

r pos

ition

(m)


t (s)

Figure 19 The cylinder position of 2 hydraulic actuators in squatmode

010203040

Con

trol c

urre

nt (m

A)


h i j

k

m n

l

0 5 10 15 20 25 30 35 40t (s)

minus10

minus20

minus30

minus40

Figure 20 The control current of 2 servo valves in squat mode


0 10 20 30 400

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

ihj

k l

m n

PlPr

Figure 21 The pressures of 2 hydraulic actuators in squat mode

(a) Gait (1) (b) Gait (2) (c) Gait (3)

(d) Gait (4) (e) Gait (5)

Figure 22 The walking gait subphases experiment


(a) Gait (6) (b) Gait (7)

(c) Gait (8) (d) Gait (9)

Figure 23 The squat gait subphases experiment

In squat mode the tester only bears the exoskeleton massbecause of the safety of system at the present stage The testercan realize the basic gait of squat action under the help ofanother personrsquos arm especially in gait (7) and gait (9)

7 Conclusion

In this paper a human-machined coordinatedmotion controlmethod is presented to realize the aid-force of lower extremityexoskeleton This proposed method improves the robustperformance under the structural andparametric uncertaintydisturbance The positions of 2 hydraulic actuators are con-trolled to make exoskeleton track personrsquos motion by outputfeedback control Compared with conventional PID controlmethod in simulation the proposed method can achieve abetter dynamic response performance and aid-force effect instance phase of walking gait According to the gait divisions ofhuman-machined coordinated motion the switched controlalgorithm is designed to reduce the delay of exoskeletontracking personThe experimental results of 2 motion modesvalidate the effectiveness of this proposed method and gaitswitched control algorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by project of National Nat-ural Science Foundation of China (no 61305092 and no51205045) and project of China Postdoctoral Science Foun-dation (no 2013M542487) The authors would like to thankanonymous reviewers for their valuable comments and sug-gestions

References

[1] S Ackerman The Report of Production Display for ldquoHULCrdquoMade in Lockheed Martin Montgomery County Md USA2010 httpwwwlockheedmartincom

[2] Z X Wang ldquoRaytheon launched XOS2 second-generationexoskeleton devicerdquo Light Weapons pp 24ndash44 2010


[3] J R Steger A design and control methodology for humanexoskeletons [PhD dissertation] Department of MechanicalEngineering University of California Berkeley Calif USA2006

[4] J F Veneman R Ekkelenkamp R Kruidhof F C T vander Helm and H van der Kooij ldquoDesign of a series elastic-and Bowden cable-based actuation system for use as torque-actuator in exoskeleton-type trainingrdquo in Proceedings of theIEEE 9th International Conference on Rehabilitation Robotics(ICORR rsquo05) pp 496ndash499 Chicago Ill USA July 2005

[5] Verge ldquoThe report of lsquoHercule exoskeleton can help a regularhuman carry up to 220 poundsrsquordquo 2012 httpwwwthevergecom20122222815704hercule-exoskeleton-human-carry-220-pounds

[6] C G Atkeson J G Hale F Pollick et al ldquoUsing humanoidrobots to study human behaviorrdquo IEEE Intelligent Systems andTheir Applications vol 15 no 4 pp 46ndash55 2000

[7] Q Guo H Zhou and D Jiang ldquoCoordinated control methodof the lower extremity exoskeleton based on human elec-tromechanical couplingrdquo in Social Robotics 4th InternationalConference ICSR 2012 Chengdu China October 29ndash31 2012Proceedings vol 7621 of Lecture Notes in Computer Science pp650ndash659 Springer Berlin Germany 2012

[8] Q Li M Kong Z Du L Sun and D Wang ldquoInteractiverehabilitation exercise control strategy for 5-DOF upper limbrehabilitation armrdquo Chinese Journal of Mechanical Engineeringvol 44 no 9 pp 169ndash176 2008

[9] Z Y Yang Y S ZhangW J Gu et al ldquoBone clothing sensitivityamplification controlrdquo Computer Simulation vol 27 no 1 pp177ndash180 2010

[10] H Cao X W Meng Z Y Ling et al ldquoTwo-legged robotexoskeleton plantar pressure measurement systemrdquo Journ ofSensors and Actuators vol 23 no 3 pp 326ndash330 2010

[11] R Fales and A Kelkar ldquoRobust control design for a wheelloader using 119867

infinand feedback linearization based methodsrdquo

ISA Transactions vol 48 no 3 pp 312ndash320 2009[12] G P Liu and S Daley ldquoOptimal-tuning PID controller design

in the frequency domain with application to a rotary hydraulicsystemrdquo Control Engineering Practice vol 7 no 7 pp 821ndash8301999

[13] I Boiko ldquoVariable-structure PID controller for level processrdquoControl Engineering Practice vol 21 no 5 pp 700ndash707 2013

[14] M Moradi ldquoSelf-tuning PID controller to three-axis stabi-lization of a satellite with unknown parametersrdquo InternationalJournal of Non-Linear Mechanics vol 49 pp 50ndash56 2013

[15] O Cerman and P Husek ldquoAdaptive fuzzy sliding mode controlfor electro-hydraulic servo mechanismrdquo Expert Systems withApplications vol 39 no 11 pp 10269ndash10277 2012

[16] C Guan and S X Pan ldquoAdaptive sliding mode control ofelectro-hydraulic systemwith nonlinear unknown parametersrdquoControl Engineering Practice vol 16 no 11 pp 1275ndash1284 2008

[17] D H Sha V B Bajic and H Y Yang ldquoNew model and slidingmode control of hydraulic elevator velocity tracking systemrdquoSimulation Practice and Theory vol 9 no 6ndash8 pp 365ndash3852002

[18] C W Wang Z X Jiao S Wu and Y X Shang ldquoNonlinearadaptive torque control of electro-hydraulic load system withexternal active motion disturbancerdquo Mechatronics vol 24 no1 pp 32ndash40 2014

[19] I Ursu A Toader A Halanay and S Balea ldquoNew stabilizationand tracking control laws for electrohydraulic servomecha-nismsrdquo European Journal of Control vol 19 no 1 pp 65ndash802013

[20] V Milic Z Situm and M Essert ldquoRobust 119867infin

positioncontrol synthesis of an electro-hydraulic servo systemrdquo ISATransactions vol 49 no 4 pp 535ndash542 2010

[21] F L Weng Y C Ding and M K Tang ldquoLPV model-basedrobust controller design of electro-hydraulic servo systemsrdquoProcedia Engineering vol 15 pp 421ndash425 2011

[22] X Song Y Wang and Z Sun ldquoRobust stabilizer design forlinear time-varying internal model based output regulation andits application to an electrohydraulic systemrdquo Automatica vol50 no 4 pp 1128ndash1134 2014

[23] R Fales andAKelkar ldquoRobust control design for awheel loaderusing mixed sensitivity H-infinity and feedback linearizationbased methodsrdquo in Proceedings of the American Control Confer-ence (ACC rsquo05) pp 4381ndash4386 Portland Ore USA June 2005

[24] S Kawamura and M Svinin Advances in Robot Control FromEveryday Physics to Human-like Movements Springer BerlinGermany 2006

[25] A Akers M Gassman and R Smith Hydraulic Power SystemAnalysis Taylor amp Francis Group 2006

[26] A Alleyne and R Liu ldquoA simplified approach to force controlfor electro-hydraulic systemsrdquoControl Engineering Practice vol8 no 12 pp 1347ndash1356 2000

[27] W-M Lu K M Zhou and J C Doyle ldquoStabilization ofuncertain linear systems an LFT approachrdquo IEEE Transactionson Automatic Control vol 41 no 1 pp 50ndash65 1996

[28] M K H Fan A L Tits and J C Doyle ldquoRobustness inthe presence of mixed parametric uncertainty and unmodeleddynamicsrdquo IEEE Transactions on Automatic Control vol 36 no1 pp 25ndash38 1991

[29] D W Gu P H Petkov and M M Konstantinov Robust controldesign with MATLAB Springer Press 2005

[30] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard 119867

2and 119867

infincontrol prob-

lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[31] G J Balas and J C Doyle ldquoRobustness and performance trade-offs in control design for flexible structuresrdquo IEEE Transactionson Control Systems Technology vol 2 no 4 pp 352ndash361 1994

[32] A Packard M K H Fan and J C Doyle ldquoA power method forthe structured singular valuerdquo in Proceedings of the 27th IEEEConference on Decision and Control pp 2132ndash2137 Chicago IllUSA December 1988

[33] W Gawronski ldquoBalanced systems and structures reductionassignment and perturbationsrdquo in Control and Dynamic Sys-tems vol 54 pp 373ndash415 1992

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


Due to the demands of high load efficiency and fastresponse lower extremity exoskeleton is driven by theelectrohydraulic servo system (EHSS) However EHSS hassome model uncertainties such as parametric disturbancesstructural uncertainties which are consequences of hydraulicphysical characteristics oil leakage oil temperature varia-tions and characteristics of valves near null [11]The classicalcontrol method of EHSS is PID [12] Due to lower orderand constant gains PID control is easy to be realized inengineering So the variable parameters of PID controllerare presented to suit the hydraulic dynamic characteristic[13 14] Some advanced control methods such as adaptivesliding mode [15ndash17] and backstepping controllers [18 19] areused in EHSS and get some good performance In additionthe robust controllers [20ndash22] of EHSS are presented withparameter uncertainty disturbance which are converted to119867infin

control problem solved by conventional robust analysisor linear matrix inequality The mixed sensitivity119867

infincontrol

method [23] of EHSS is presented to balance to the robuststability and response performance But the external load ofhydraulic control model is only considered as a disturbancenot to be quantitatively analyzed and described in robustuncertainty model In this paper by referring to119867

infincontrol

method aforementioned references a robust119867infincontroller is

designed to suppress structural uncertainty and parametricuncertainties which are caused by the external load andparametric disturbances Furthermore combined with theswitched control algorithm of hydraulic actuator this 119867

infin

controller can reduce the delay of exoskeleton trackingperson Simulation and experimental results validate thisproposed controller

2 Human-Machine CoupledModel Construction

Since human fast conscious is ahead of exoskeleton the onlymechanical motion model is not insufficient to analyze thesystematic dynamics So the coupled motion model needs tobe constructed to estimate the dynamic loads and the supplypower

21 Characteristic Parameters Synthesis Some characteristicparameters like mass and moment of inertia are differentbetween human limb and exoskeleton linkage So the char-acteristic parameters need be synthetic to balance 2 motionresponses activated by person and exoskeleton It meansthat human-constraint is relieved but not vanished throughequivalent multibody motion The human-constraint is asmall value in the thigh bandage which can not exceed thehuman tolerance

If the sagittal motion is considered only the rotationalmotion around the knee is shown in Figure 1 Here compo-nents 119861

1 119861

2 119861

3 119861

4 and 119861

5represent respectively human

calf mechanical calf sensing shoes human thigh andmechanical thigh

O

ltl5cltc

l4c

l3c

l3c

l1c l2c

B5B4

m1g

m2g

m3g

m4g

msg

m5g

mtg

C

B1

B2

B3

KA

Figure 1The rotationalmotion around the knee of human-machinesynthetic model

The centers of synthetic calf and thigh for human-machine coordinated motion [24] are shown as

119897119904119888=11989811198971119888 + 11989821198972119888 + 119898311989731198881198981 + 1198982 + 1198983

=11989811198971119888 + 11989821198972119888 + 11989831198973119888

119898119904

(1a)

119897119905119888=11989841198974119888 + 119898511989751198881198984 + 1198985

=11989841198974119888 + 11989851198975119888

119898119905

(1b)

According to the moment of inertia of the parallel axistheorem the moment of inertia of synthetic calf and thigharound respective center of mass is shown as

119868119904119888= [1198681119888 +1198981 (119897119904119888 minus 1198971119888)

2] + [1198682119888 +1198982 (119897119904119888 minus 1198972119888)

2]

+ [1198683119888 +1198983 (119897119904119888 minus 1198973119888)2]

(2a)

119868119905119888= [1198684119888 +1198984 (119897119905119888 minus 1198974119888)

2] + [1198685119888 +1198985 (119897119905119888 minus 1198975119888)

2] (2b)

where 119898119894is the 119894th component mass 119898

119904and 119898

119905are the total

masses of equivalent calf and thigh 1198971119888 1198972119888 and 119897

3119888are the

distances from center of mass to knee respectively 1198974119888 1198975119888are

the distances from center of mass to hip respectively and 119868119894119888

is the 119894th moment of inertia around respective center of mass

22 Human-Machine Coupled Motion Model In human-machine coordinated motion the exoskeleton linkagesshould compensate the load when human foot contacts theground So in the stance phase of walk or squat the hydraulicactuator not only realizes the exoskeleton knee rotationtracking human limb rotation but also balances the variableload In order to estimate the dynamic load on the hydraulic


Fl

O5

O4

5

4

3

2

1

1205791

1205793

120579k

Fp

O2

O3

O1

A

X

Y

FO2y

FO2x

1FO3x

FO3y

aGsy

aGsxGs

a120591s

ans120572O1

120596O1

FO1x

FO1y

TO1

TO1

1205791

1205792

FO4x

FO2x

FO2y

Fl

FO4y

a120591t

aGtx

aGty

ant

Gt

2

120572O2

120596O2

FO4x

FO3x

FO3y

FA

TA

Fp

FA

Fp

FO4y

a120591p

a120591c

anc

anp 4

120572c120596c

3TA

1205792

120572p 120596p

(a) (b)

(c) (d)



5rdquo between







1198651198741119909 minus1198651198742119909 +1198651198743119909 = 119898119904119888

119886119866119904119909 (3a)

1198651198741119910 +1198651198742119910 minus1198651198743119910 minus119898119904119888

119892 = 119898119904119886119866119904119910 (3b)

1198791198741 +1198651198742119909119897119904119888 cos 1205791 +1198651198742119910119897119904119888 sin 1205791


minus119898119904119888119892119897119892119904sin 1205791 = minus 1198681199041198881205721198741

(3c)

119886119866119904119909


119886119866119904119910


(3d)

where 119886119899119904= 119897

119892119904sdot 120596

21198741 119886120591119904 = 119897119892119904 sdot 1205721198741 1198861198741119909 = 1198861198741119910 = 0


1198651198742119909 +1198651198744119909 = 119898119905

119886119866119905119909 (4a)

1198651198744119910 minus1198651198742119910 minus119898119905119888

119892minus119865load = 119898119905119886119866119905119910 (4b)

minus119865119897119897119905119888cos (120579

119896+ 1205791) + 11986511987441199091198972cos (120579119896 + 1205791)

minus 11986511987441199101198972sin (120579119896 + 1205791) +119898119905

119892119897119892119905sin120579

119896= 119868

1199051198881205721198742

(4c)

119865119901sin1205792 minus119865119860cos1205792 +1198651198743119909 = 119898119901

119886119866119901119909 (5a)



119886119866119901119910 (5b)

minus119865119860119897119903+119879

119860= 119868

119901120572119901 (5c)


119886119866119888119909 (6a)

119865119901cos1205792 +119865119860sin1205792 minus1198651198744119910 = 119898119888

119886119866119888119910 (6b)

minus119865119860(119897ℎminus 119897

119903) minus119879

119860= 119868

119888120572119888 (6c)


sin 1205791 =1198972 sin 120579119896

radic11989721 + 119897

22 minus 211989711198972 cos 120579119896

(7a)

sin 1205793 =(1198972 minus 119897119905119888) sin 120579119896

radic1198972119904119888+ (1198972 minus 119897119905119888)

2minus 2119897

119904119888(1198972 minus 119897119905119888) cos 120579119896

(7b)

1205792 = 1205793 minus 1205791 (7c)

120596119901= 120596

1198741

minus


cos 1205793 (11989721 + 11989722 minus 211989711198972 cos 120579119896)

(8a)

120596119888= 120596

1198742

minus


cos (120587 minus 120579119896minus 1205793) (119897

21 + 119897

22 minus 211989711198972 cos 120579119896)

(8b)

1205961198742 = 120596119896 +1205961198741 (8c)



1198651198741119909

1198651198741119910


1198651198742119909

1198651198742119910


1198651198743119910


1198651198744119910


119886119866119904119910


119886119866119905119910





119886120591119888 119886






1205961198741 120572







1

120596119901 120572


piston120596119888 120572





3




120579119896 knee angle

119898119901119898






119897rdquo


119875119897=

119865119901

1205871199032119901

(9)



119897can







119909v119906=

119870sv1205962sv

1199042 + 2120577sv120596sv119904 + 1205962sv (10)


80 90 100 110 120 130 140 150 160123456789

120579k (∘)

Fl = 0kgFl = 30kg

Fl = 60kgFl = 90kg

Pl

(MPa

)





119876119886= 119870

1119902119909v minus119870

1119888119901119886 (11a)

119876119887= 119870

2119902119909v +119870

2119888119901119887 (11b)

1198701119902= 119862

119889119908radic

2Δ119901119901

120588

1198702119902= 119862

119889119908radic

2Δ119901119877

120588

(12a)

1198701119888=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119862119889119908119909v

radic2120588Δ119901119901

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198702119888=

100381610038161003816100381610038161003816100381610038161003816

119862119889119908119909v

radic2120588Δ119901119877

100381610038161003816100381610038161003816100381610038161003816

(12b)

Δ119901119901=

1003816100381610038161003816119901119904 minus 1199011198861003816100381610038161003816 119909v ge 0

1003816100381610038161003816119901119886 minus 1199011199031003816100381610038161003816 119909v lt 0

(13a)

Δ119901119877=

1003816100381610038161003816119901119887 minus 1199011199031003816100381610038161003816 119909v ge 0

1003816100381610038161003816119901119904 minus 1199011198871003816100381610038161003816 119909v lt 0

(13b)

where 119876119886 119876


119886 119901

119887are the





119902(119870

2119902) and1198701

119888(119870




119886=

120573119890

11988101 + 119860119886119910(119876

119886minus119860

119886119910) (14a)

119887=

120573119890

11988102 minus 119860119887119910(minus119876

119887+119860

119887119910) (14b)

11988101 = 11988102 =119881119905

2=119860119886119897ℎ

2 (15)


119886and119860




119898 119910 = (119901119886119860119886minus119901

119887119860119887minus119865

119891minus119865

119901) (16)




119910 (120579119896) = radic119897

21 + 119897


119896 (17)



119901and


119865119901(120579119896 119865119871) = 119870

119865119901(120579119896) 119910 (120579

119896) (18)






119891


119865119891asymp 119887 119910 (19)






0 2 4 6 8 10 12 14minus002

0

002

004

006

008

01

012

014

t (s)

y (m

)

Experimentb = 100

b = 200b = 300





infinsuboptimal

controller


119902


119886119910 119881

02minus119860


119865119901from



[27 28] as follows

119872119870119902= [

[

0 119870119902

119901119870119902

119870119902

]

]

119872119864=

[[[

[

minus119901119864

1119864

minus119901119864

1119864

]]]

]

119872119887= [

0 119887

119901119887119887

]

119872119870119865119901

= [

0 1

119901119870119865119901

119870119865119901

]

(20)

119870119902= 119870

119902(1+119901

119870119902120575119870119902) minus1 le 120575

119870119902le 1 (21a)

119864 = 119864 (1+119901119864120575119864) minus1 le 120575

119864le 1 (21b)

119887 = 119887 (1+119901119887120575119887) minus1 le 120575

119887le 1 (21c)

119870119865119901= 119870

119865119901+119901

119870119865119901120575119870119865119901 (21d)

where119864 = 1198811199052120573119870


119870119902 120575119864



119864 and119901



[1199091 119909

2 119909

3 119909

4 119909

5 119909

6]119879

= [119909v v 119901119886 119901119887 119910 119910]119879 the control


1 119908

2 119908

3 119908

4 119908

5]119879 the


1 1199112 1199113 1199114 1199115 1199116]119879 then the


(119905) = 119860119883 (119905) + 1198611119882(119905) + 1198612119906 (119905) (22a)

119885 (119905) = 1198621119883 (119905) +11986311119882(119905) (22b)

119884 (119905) = 1198622119883 (119905) (22c)

where

119860 =

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

[

0 1 0 0 0 0

minus1205962v minus2120577v120596v 0 0 0 0

119870119902

119864

0 minus119870119888

119864

0 0 minus119860119886

119864

minus

119870119902

119864

0 0 minus119870119888

119864

0 1198602

119864

0 0 0 0 0 1

0 0119860119886

119898minus119860119887

119898minus

119870119865119901

119898

119887

119898

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

]

(23a)

1198611 =

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

[

0 0 0 0 00 0 0 0 0119901119870119902

119864

minus119901119864

0 0 0

minus

119901119870119902

119864

0 minus119901119864

0 00 0 0 0 0

0 0 0 minus119901119887

119898minus

119901119870119865119901

119898

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

]

(23b)


r+

minusK

u

Wp

ep

Wu

y

eu

Δ

z

w

Ghyd

Gic = Fu(Ghyd Δ)


1198612 = [0 119870sv1205962v 0 0 0 0]119879 (23c)

1198621 =

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

[

119870119902

0 0 0 0 0

119870119902

119864

0 minus119870119888

119864

0 0 minus119860119886

119864

minus

119870119902

119864

0 0 minus119870119888

119864

0119860119887

119864

0 0 0 0 0 119887

0 0 0 0 1 0

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

]

(23d)

11986311 =

[[[[[[[[[[[

[

0 0 0 0 0119901119870119902

119864

minus119901119864

0 0 0

minus

119901119870119902

119864

0 minus119901119864

0 00 0 0 0 00 0 0 0 0

]]]]]]]]]]]

]

(23e)

1198622 = [0 0 0 0 1 0] (23f)



119906are


119901and 119890

119906 The nominal





01 1 10 100628

Log

mag

nitu

de (d

B)


minus500

Phas

e (de

g)

Frequency (rads)


NominalPerturbed

100

10minus2

10minus4


119902 119864 119887 and 119870

119865119901





119882119901= 095times 119904

2+ 18119904 + 5

1199042 + 8119904 + 001 (24a)

119882119906= 5times 10minus4 (24b)



119899


119882119901119866119899

119904infinle 120583(119882

119901)120583(119866

119899




(1) 119867infin


1003817100381710038171003817119865119897 (119866ic 119870)1003817100381710038171003817infin

=10038171003817100381710038171003817119865119897(119865

119906(119866hyd Δ) 119870)

10038171003817100381710038171003817infinlt 120574 (25)


119897(119866ic 119870)infin under



100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

119882119901(119868 + 119866ic119870)

minus1

119882119906119870(119868 + 119866ic119870)

minus1]

]

100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

lt 1 (26)


119888is stable




Frequency (rads)


100

10minus1

10minus3

10minus2


120583


119902 119864 119887 and 119870

119865119901

NominalUpper bound

Low bound

Frequency (rads)

100

10minus02

10minus06

10minus04

100 101 10210minus110minus2

120583


119902 119864 119887 and 119870

119865119901



119906extension = 119870 (119904) (119910 minus 119910)

=1198994119904

4+ 1198993119904

3+ 1198992119904

2+ 1198991119904 + 1198990

11988941199044 + 1198893119904

3 + 11988921199042 + 1198891119904 + 1198890

(119910 minus 119910)

(27)

where 119899119894(119894 = 0 4) 119889






Walking cycle





(1) (2) (3) (4) (5) (2)

(1) (6) (7) (8) (9) (2)





In Table 1 119906119897and 119906








(1) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton



Adapt toexoskeleton

(3) 119906119897= minus119906max

119906119903= 119906extension



(4) 119906119897= minus119906max

119906119903= 119906extension


Adapt toexoskeleton








Squatpreparation

(7)119906119897=

119906max(1 minus 119890minus119896119905

)

119906119903= minus119906max


Adapt toexoskeleton

(8) 119906119897= minus119906max

119906119903= minus119906max


prescriptposture


(9) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton







119870119902= 025



119899628 rads


1198907000 bar


119886177 times 10

minus4 m2




1

0322 m


0065 m



119903016 m


119888asymp 119870

2119888


1199020188 m2s


1198651199011200 Nm

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

t (s)

4 45 515202530


PID control

Stan

ceph

ase

Swin

gph

ase

120579lk

(∘)


119901119864= 07 119901

119887= 1 and 119901












119891in the case


0 1 2 3 4 5 6 7 8 9 10t (s)


PID control

4 45 5034

0345035

0355

03

032

034

036

038

yl

(m)


0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

20

30

40

t (s)


ul

(m)



119902 119864 and 119887





0 1 2 3 4 5 6 7 8 9 10

minus1000

0

500

1000

15001300

minus500minus600

t (s)

Com

pens

atio

n fo

rce (

N)





119865pres =

119865119901

(119879120572119904 + 1)2

stance phase

(1 minus 1(119879120572119904 + 1)2

)119865119901

swing phase(28)







0 2 4 6 8 10 12 14 16minus20

0

20

40

60

80

t (s)

Left kneeRight knee

c ef

ga b d

120579k

(∘)






08 1 12 14 18 2 22 24 26165 2815

032

033

034

035

036

037

038

t (s)

Cylid

ner p

ositi

on (m

)



a b2bb1

c d1

d2

d e



010203040

t (s)

Con

trol c

urre

nt (m

A)


a

c e f

gb1 d1





0 2 4 6 8 10 12 14 160

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

PlPr




0 5 10 15 20 25 30 35 40

0

50

100

150

80

t (s)

Left kneeRight knee

h i j

lk

mn

minus50

120579k

(∘)


0 10 20 30 40025

03

035

04

045Cy

lidne

r pos

ition

(m)


t (s)


010203040

Con

trol c

urre

nt (m

A)


h i j

k

m n

l

0 5 10 15 20 25 30 35 40t (s)

minus10

minus20

minus30

minus40



0 10 20 30 400

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

ihj

k l

m n

PlPr










7 Conclusion




Acknowledgments


References



































2and 119867

infincontrol prob-












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Fl

O5

O4

5

4

3

2

1

1205791

1205793

120579k

Fp

O2

O3

O1

A

X

Y

FO2y

FO2x

1FO3x

FO3y

aGsy

aGsxGs

a120591s

ans120572O1

120596O1

FO1x

FO1y

TO1

TO1

1205791

1205792

FO4x

FO2x

FO2y

Fl

FO4y

a120591t

aGtx

aGty

ant

Gt

2

120572O2

120596O2

FO4x

FO3x

FO3y

FA

TA

Fp

FA

Fp

FO4y

a120591p

a120591c

anc

anp 4

120572c120596c

3TA

1205792

120572p 120596p

(a) (b)

(c) (d)



5rdquo between







1198651198741119909 minus1198651198742119909 +1198651198743119909 = 119898119904119888

119886119866119904119909 (3a)

1198651198741119910 +1198651198742119910 minus1198651198743119910 minus119898119904119888

119892 = 119898119904119886119866119904119910 (3b)

1198791198741 +1198651198742119909119897119904119888 cos 1205791 +1198651198742119910119897119904119888 sin 1205791


minus119898119904119888119892119897119892119904sin 1205791 = minus 1198681199041198881205721198741

(3c)

119886119866119904119909


119886119866119904119910


(3d)

where 119886119899119904= 119897

119892119904sdot 120596

21198741 119886120591119904 = 119897119892119904 sdot 1205721198741 1198861198741119909 = 1198861198741119910 = 0


1198651198742119909 +1198651198744119909 = 119898119905

119886119866119905119909 (4a)

1198651198744119910 minus1198651198742119910 minus119898119905119888

119892minus119865load = 119898119905119886119866119905119910 (4b)

minus119865119897119897119905119888cos (120579

119896+ 1205791) + 11986511987441199091198972cos (120579119896 + 1205791)

minus 11986511987441199101198972sin (120579119896 + 1205791) +119898119905

119892119897119892119905sin120579

119896= 119868

1199051198881205721198742

(4c)

119865119901sin1205792 minus119865119860cos1205792 +1198651198743119909 = 119898119901

119886119866119901119909 (5a)



119886119866119901119910 (5b)

minus119865119860119897119903+119879

119860= 119868

119901120572119901 (5c)


119886119866119888119909 (6a)

119865119901cos1205792 +119865119860sin1205792 minus1198651198744119910 = 119898119888

119886119866119888119910 (6b)

minus119865119860(119897ℎminus 119897

119903) minus119879

119860= 119868

119888120572119888 (6c)


sin 1205791 =1198972 sin 120579119896

radic11989721 + 119897

22 minus 211989711198972 cos 120579119896

(7a)

sin 1205793 =(1198972 minus 119897119905119888) sin 120579119896

radic1198972119904119888+ (1198972 minus 119897119905119888)

2minus 2119897

119904119888(1198972 minus 119897119905119888) cos 120579119896

(7b)

1205792 = 1205793 minus 1205791 (7c)

120596119901= 120596

1198741

minus


cos 1205793 (11989721 + 11989722 minus 211989711198972 cos 120579119896)

(8a)

120596119888= 120596

1198742

minus


cos (120587 minus 120579119896minus 1205793) (119897

21 + 119897

22 minus 211989711198972 cos 120579119896)

(8b)

1205961198742 = 120596119896 +1205961198741 (8c)



1198651198741119909

1198651198741119910


1198651198742119909

1198651198742119910


1198651198743119910


1198651198744119910


119886119866119904119910


119886119866119905119910





119886120591119888 119886






1205961198741 120572







1

120596119901 120572


piston120596119888 120572





3




120579119896 knee angle

119898119901119898






119897rdquo


119875119897=

119865119901

1205871199032119901

(9)



119897can







119909v119906=

119870sv1205962sv

1199042 + 2120577sv120596sv119904 + 1205962sv (10)


80 90 100 110 120 130 140 150 160123456789

120579k (∘)

Fl = 0kgFl = 30kg

Fl = 60kgFl = 90kg

Pl

(MPa

)





119876119886= 119870

1119902119909v minus119870

1119888119901119886 (11a)

119876119887= 119870

2119902119909v +119870

2119888119901119887 (11b)

1198701119902= 119862

119889119908radic

2Δ119901119901

120588

1198702119902= 119862

119889119908radic

2Δ119901119877

120588

(12a)

1198701119888=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119862119889119908119909v

radic2120588Δ119901119901

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198702119888=

100381610038161003816100381610038161003816100381610038161003816

119862119889119908119909v

radic2120588Δ119901119877

100381610038161003816100381610038161003816100381610038161003816

(12b)

Δ119901119901=

1003816100381610038161003816119901119904 minus 1199011198861003816100381610038161003816 119909v ge 0

1003816100381610038161003816119901119886 minus 1199011199031003816100381610038161003816 119909v lt 0

(13a)

Δ119901119877=

1003816100381610038161003816119901119887 minus 1199011199031003816100381610038161003816 119909v ge 0

1003816100381610038161003816119901119904 minus 1199011198871003816100381610038161003816 119909v lt 0

(13b)

where 119876119886 119876


119886 119901

119887are the





119902(119870

2119902) and1198701

119888(119870




119886=

120573119890

11988101 + 119860119886119910(119876

119886minus119860

119886119910) (14a)

119887=

120573119890

11988102 minus 119860119887119910(minus119876

119887+119860

119887119910) (14b)

11988101 = 11988102 =119881119905

2=119860119886119897ℎ

2 (15)


119886and119860




119898 119910 = (119901119886119860119886minus119901

119887119860119887minus119865

119891minus119865

119901) (16)




119910 (120579119896) = radic119897

21 + 119897


119896 (17)



119901and


119865119901(120579119896 119865119871) = 119870

119865119901(120579119896) 119910 (120579

119896) (18)






119891


119865119891asymp 119887 119910 (19)






0 2 4 6 8 10 12 14minus002

0

002

004

006

008

01

012

014

t (s)

y (m

)

Experimentb = 100

b = 200b = 300





infinsuboptimal

controller


119902


119886119910 119881

02minus119860


119865119901from



[27 28] as follows

119872119870119902= [

[

0 119870119902

119901119870119902

119870119902

]

]

119872119864=

[[[

[

minus119901119864

1119864

minus119901119864

1119864

]]]

]

119872119887= [

0 119887

119901119887119887

]

119872119870119865119901

= [

0 1

119901119870119865119901

119870119865119901

]

(20)

119870119902= 119870

119902(1+119901

119870119902120575119870119902) minus1 le 120575

119870119902le 1 (21a)

119864 = 119864 (1+119901119864120575119864) minus1 le 120575

119864le 1 (21b)

119887 = 119887 (1+119901119887120575119887) minus1 le 120575

119887le 1 (21c)

119870119865119901= 119870

119865119901+119901

119870119865119901120575119870119865119901 (21d)

where119864 = 1198811199052120573119870


119870119902 120575119864



119864 and119901



[1199091 119909

2 119909

3 119909

4 119909

5 119909

6]119879

= [119909v v 119901119886 119901119887 119910 119910]119879 the control


1 119908

2 119908

3 119908

4 119908

5]119879 the


1 1199112 1199113 1199114 1199115 1199116]119879 then the


(119905) = 119860119883 (119905) + 1198611119882(119905) + 1198612119906 (119905) (22a)

119885 (119905) = 1198621119883 (119905) +11986311119882(119905) (22b)

119884 (119905) = 1198622119883 (119905) (22c)

where

119860 =

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

[

0 1 0 0 0 0

minus1205962v minus2120577v120596v 0 0 0 0

119870119902

119864

0 minus119870119888

119864

0 0 minus119860119886

119864

minus

119870119902

119864

0 0 minus119870119888

119864

0 1198602

119864

0 0 0 0 0 1

0 0119860119886

119898minus119860119887

119898minus

119870119865119901

119898

119887

119898

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

]

(23a)

1198611 =

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

[

0 0 0 0 00 0 0 0 0119901119870119902

119864

minus119901119864

0 0 0

minus

119901119870119902

119864

0 minus119901119864

0 00 0 0 0 0

0 0 0 minus119901119887

119898minus

119901119870119865119901

119898

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

]

(23b)


r+

minusK

u

Wp

ep

Wu

y

eu

Δ

z

w

Ghyd

Gic = Fu(Ghyd Δ)


1198612 = [0 119870sv1205962v 0 0 0 0]119879 (23c)

1198621 =

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

[

119870119902

0 0 0 0 0

119870119902

119864

0 minus119870119888

119864

0 0 minus119860119886

119864

minus

119870119902

119864

0 0 minus119870119888

119864

0119860119887

119864

0 0 0 0 0 119887

0 0 0 0 1 0

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

]

(23d)

11986311 =

[[[[[[[[[[[

[

0 0 0 0 0119901119870119902

119864

minus119901119864

0 0 0

minus

119901119870119902

119864

0 minus119901119864

0 00 0 0 0 00 0 0 0 0

]]]]]]]]]]]

]

(23e)

1198622 = [0 0 0 0 1 0] (23f)



119906are


119901and 119890

119906 The nominal





01 1 10 100628

Log

mag

nitu

de (d

B)


minus500

Phas

e (de

g)

Frequency (rads)


NominalPerturbed

100

10minus2

10minus4


119902 119864 119887 and 119870

119865119901





119882119901= 095times 119904

2+ 18119904 + 5

1199042 + 8119904 + 001 (24a)

119882119906= 5times 10minus4 (24b)



119899


119882119901119866119899

119904infinle 120583(119882

119901)120583(119866

119899




(1) 119867infin


1003817100381710038171003817119865119897 (119866ic 119870)1003817100381710038171003817infin

=10038171003817100381710038171003817119865119897(119865

119906(119866hyd Δ) 119870)

10038171003817100381710038171003817infinlt 120574 (25)


119897(119866ic 119870)infin under



100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

119882119901(119868 + 119866ic119870)

minus1

119882119906119870(119868 + 119866ic119870)

minus1]

]

100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

lt 1 (26)


119888is stable




Frequency (rads)


100

10minus1

10minus3

10minus2


120583


119902 119864 119887 and 119870

119865119901

NominalUpper bound

Low bound

Frequency (rads)

100

10minus02

10minus06

10minus04

100 101 10210minus110minus2

120583


119902 119864 119887 and 119870

119865119901



119906extension = 119870 (119904) (119910 minus 119910)

=1198994119904

4+ 1198993119904

3+ 1198992119904

2+ 1198991119904 + 1198990

11988941199044 + 1198893119904

3 + 11988921199042 + 1198891119904 + 1198890

(119910 minus 119910)

(27)

where 119899119894(119894 = 0 4) 119889






Walking cycle





(1) (2) (3) (4) (5) (2)

(1) (6) (7) (8) (9) (2)





In Table 1 119906119897and 119906








(1) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton



Adapt toexoskeleton

(3) 119906119897= minus119906max

119906119903= 119906extension



(4) 119906119897= minus119906max

119906119903= 119906extension


Adapt toexoskeleton








Squatpreparation

(7)119906119897=

119906max(1 minus 119890minus119896119905

)

119906119903= minus119906max


Adapt toexoskeleton

(8) 119906119897= minus119906max

119906119903= minus119906max


prescriptposture


(9) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton







119870119902= 025



119899628 rads


1198907000 bar


119886177 times 10

minus4 m2




1

0322 m


0065 m



119903016 m


119888asymp 119870

2119888


1199020188 m2s


1198651199011200 Nm

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

t (s)

4 45 515202530


PID control

Stan

ceph

ase

Swin

gph

ase

120579lk

(∘)


119901119864= 07 119901

119887= 1 and 119901












119891in the case


0 1 2 3 4 5 6 7 8 9 10t (s)


PID control

4 45 5034

0345035

0355

03

032

034

036

038

yl

(m)


0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

20

30

40

t (s)


ul

(m)



119902 119864 and 119887





0 1 2 3 4 5 6 7 8 9 10

minus1000

0

500

1000

15001300

minus500minus600

t (s)

Com

pens

atio

n fo

rce (

N)





119865pres =

119865119901

(119879120572119904 + 1)2

stance phase

(1 minus 1(119879120572119904 + 1)2

)119865119901

swing phase(28)







0 2 4 6 8 10 12 14 16minus20

0

20

40

60

80

t (s)

Left kneeRight knee

c ef

ga b d

120579k

(∘)






08 1 12 14 18 2 22 24 26165 2815

032

033

034

035

036

037

038

t (s)

Cylid

ner p

ositi

on (m

)



a b2bb1

c d1

d2

d e



010203040

t (s)

Con

trol c

urre

nt (m

A)


a

c e f

gb1 d1





0 2 4 6 8 10 12 14 160

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

PlPr




0 5 10 15 20 25 30 35 40

0

50

100

150

80

t (s)

Left kneeRight knee

h i j

lk

mn

minus50

120579k

(∘)


0 10 20 30 40025

03

035

04

045Cy

lidne

r pos

ition

(m)


t (s)


010203040

Con

trol c

urre

nt (m

A)


h i j

k

m n

l

0 5 10 15 20 25 30 35 40t (s)

minus10

minus20

minus30

minus40



0 10 20 30 400

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

ihj

k l

m n

PlPr










7 Conclusion




Acknowledgments


References



































2and 119867

infincontrol prob-












Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119886119866119901119910 (5b)

minus119865119860119897119903+119879

119860= 119868

119901120572119901 (5c)


119886119866119888119909 (6a)

119865119901cos1205792 +119865119860sin1205792 minus1198651198744119910 = 119898119888

119886119866119888119910 (6b)

minus119865119860(119897ℎminus 119897

119903) minus119879

119860= 119868

119888120572119888 (6c)


sin 1205791 =1198972 sin 120579119896

radic11989721 + 119897

22 minus 211989711198972 cos 120579119896

(7a)

sin 1205793 =(1198972 minus 119897119905119888) sin 120579119896

radic1198972119904119888+ (1198972 minus 119897119905119888)

2minus 2119897

119904119888(1198972 minus 119897119905119888) cos 120579119896

(7b)

1205792 = 1205793 minus 1205791 (7c)

120596119901= 120596

1198741

minus


cos 1205793 (11989721 + 11989722 minus 211989711198972 cos 120579119896)

(8a)

120596119888= 120596

1198742

minus


cos (120587 minus 120579119896minus 1205793) (119897

21 + 119897

22 minus 211989711198972 cos 120579119896)

(8b)

1205961198742 = 120596119896 +1205961198741 (8c)



1198651198741119909

1198651198741119910


1198651198742119909

1198651198742119910


1198651198743119910


1198651198744119910


119886119866119904119910


119886119866119905119910





119886120591119888 119886






1205961198741 120572







1

120596119901 120572


piston120596119888 120572





3




120579119896 knee angle

119898119901119898






119897rdquo


119875119897=

119865119901

1205871199032119901

(9)



119897can







119909v119906=

119870sv1205962sv

1199042 + 2120577sv120596sv119904 + 1205962sv (10)


80 90 100 110 120 130 140 150 160123456789

120579k (∘)

Fl = 0kgFl = 30kg

Fl = 60kgFl = 90kg

Pl

(MPa

)





119876119886= 119870

1119902119909v minus119870

1119888119901119886 (11a)

119876119887= 119870

2119902119909v +119870

2119888119901119887 (11b)

1198701119902= 119862

119889119908radic

2Δ119901119901

120588

1198702119902= 119862

119889119908radic

2Δ119901119877

120588

(12a)

1198701119888=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119862119889119908119909v

radic2120588Δ119901119901

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198702119888=

100381610038161003816100381610038161003816100381610038161003816

119862119889119908119909v

radic2120588Δ119901119877

100381610038161003816100381610038161003816100381610038161003816

(12b)

Δ119901119901=

1003816100381610038161003816119901119904 minus 1199011198861003816100381610038161003816 119909v ge 0

1003816100381610038161003816119901119886 minus 1199011199031003816100381610038161003816 119909v lt 0

(13a)

Δ119901119877=

1003816100381610038161003816119901119887 minus 1199011199031003816100381610038161003816 119909v ge 0

1003816100381610038161003816119901119904 minus 1199011198871003816100381610038161003816 119909v lt 0

(13b)

where 119876119886 119876


119886 119901

119887are the





119902(119870

2119902) and1198701

119888(119870




119886=

120573119890

11988101 + 119860119886119910(119876

119886minus119860

119886119910) (14a)

119887=

120573119890

11988102 minus 119860119887119910(minus119876

119887+119860

119887119910) (14b)

11988101 = 11988102 =119881119905

2=119860119886119897ℎ

2 (15)


119886and119860




119898 119910 = (119901119886119860119886minus119901

119887119860119887minus119865

119891minus119865

119901) (16)




119910 (120579119896) = radic119897

21 + 119897


119896 (17)



119901and


119865119901(120579119896 119865119871) = 119870

119865119901(120579119896) 119910 (120579

119896) (18)






119891


119865119891asymp 119887 119910 (19)






0 2 4 6 8 10 12 14minus002

0

002

004

006

008

01

012

014

t (s)

y (m

)

Experimentb = 100

b = 200b = 300





infinsuboptimal

controller


119902


119886119910 119881

02minus119860


119865119901from



[27 28] as follows

119872119870119902= [

[

0 119870119902

119901119870119902

119870119902

]

]

119872119864=

[[[

[

minus119901119864

1119864

minus119901119864

1119864

]]]

]

119872119887= [

0 119887

119901119887119887

]

119872119870119865119901

= [

0 1

119901119870119865119901

119870119865119901

]

(20)

119870119902= 119870

119902(1+119901

119870119902120575119870119902) minus1 le 120575

119870119902le 1 (21a)

119864 = 119864 (1+119901119864120575119864) minus1 le 120575

119864le 1 (21b)

119887 = 119887 (1+119901119887120575119887) minus1 le 120575

119887le 1 (21c)

119870119865119901= 119870

119865119901+119901

119870119865119901120575119870119865119901 (21d)

where119864 = 1198811199052120573119870


119870119902 120575119864



119864 and119901



[1199091 119909

2 119909

3 119909

4 119909

5 119909

6]119879

= [119909v v 119901119886 119901119887 119910 119910]119879 the control


1 119908

2 119908

3 119908

4 119908

5]119879 the


1 1199112 1199113 1199114 1199115 1199116]119879 then the


(119905) = 119860119883 (119905) + 1198611119882(119905) + 1198612119906 (119905) (22a)

119885 (119905) = 1198621119883 (119905) +11986311119882(119905) (22b)

119884 (119905) = 1198622119883 (119905) (22c)

where

119860 =

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

[

0 1 0 0 0 0

minus1205962v minus2120577v120596v 0 0 0 0

119870119902

119864

0 minus119870119888

119864

0 0 minus119860119886

119864

minus

119870119902

119864

0 0 minus119870119888

119864

0 1198602

119864

0 0 0 0 0 1

0 0119860119886

119898minus119860119887

119898minus

119870119865119901

119898

119887

119898

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

]

(23a)

1198611 =

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

[

0 0 0 0 00 0 0 0 0119901119870119902

119864

minus119901119864

0 0 0

minus

119901119870119902

119864

0 minus119901119864

0 00 0 0 0 0

0 0 0 minus119901119887

119898minus

119901119870119865119901

119898

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

]

(23b)


r+

minusK

u

Wp

ep

Wu

y

eu

Δ

z

w

Ghyd

Gic = Fu(Ghyd Δ)


1198612 = [0 119870sv1205962v 0 0 0 0]119879 (23c)

1198621 =

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

[

119870119902

0 0 0 0 0

119870119902

119864

0 minus119870119888

119864

0 0 minus119860119886

119864

minus

119870119902

119864

0 0 minus119870119888

119864

0119860119887

119864

0 0 0 0 0 119887

0 0 0 0 1 0

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

]

(23d)

11986311 =

[[[[[[[[[[[

[

0 0 0 0 0119901119870119902

119864

minus119901119864

0 0 0

minus

119901119870119902

119864

0 minus119901119864

0 00 0 0 0 00 0 0 0 0

]]]]]]]]]]]

]

(23e)

1198622 = [0 0 0 0 1 0] (23f)



119906are


119901and 119890

119906 The nominal





01 1 10 100628

Log

mag

nitu

de (d

B)


minus500

Phas

e (de

g)

Frequency (rads)


NominalPerturbed

100

10minus2

10minus4


119902 119864 119887 and 119870

119865119901





119882119901= 095times 119904

2+ 18119904 + 5

1199042 + 8119904 + 001 (24a)

119882119906= 5times 10minus4 (24b)



119899


119882119901119866119899

119904infinle 120583(119882

119901)120583(119866

119899




(1) 119867infin


1003817100381710038171003817119865119897 (119866ic 119870)1003817100381710038171003817infin

=10038171003817100381710038171003817119865119897(119865

119906(119866hyd Δ) 119870)

10038171003817100381710038171003817infinlt 120574 (25)


119897(119866ic 119870)infin under



100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

119882119901(119868 + 119866ic119870)

minus1

119882119906119870(119868 + 119866ic119870)

minus1]

]

100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

lt 1 (26)


119888is stable




Frequency (rads)


100

10minus1

10minus3

10minus2


120583


119902 119864 119887 and 119870

119865119901

NominalUpper bound

Low bound

Frequency (rads)

100

10minus02

10minus06

10minus04

100 101 10210minus110minus2

120583


119902 119864 119887 and 119870

119865119901



119906extension = 119870 (119904) (119910 minus 119910)

=1198994119904

4+ 1198993119904

3+ 1198992119904

2+ 1198991119904 + 1198990

11988941199044 + 1198893119904

3 + 11988921199042 + 1198891119904 + 1198890

(119910 minus 119910)

(27)

where 119899119894(119894 = 0 4) 119889






Walking cycle





(1) (2) (3) (4) (5) (2)

(1) (6) (7) (8) (9) (2)





In Table 1 119906119897and 119906








(1) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton



Adapt toexoskeleton

(3) 119906119897= minus119906max

119906119903= 119906extension



(4) 119906119897= minus119906max

119906119903= 119906extension


Adapt toexoskeleton








Squatpreparation

(7)119906119897=

119906max(1 minus 119890minus119896119905

)

119906119903= minus119906max


Adapt toexoskeleton

(8) 119906119897= minus119906max

119906119903= minus119906max


prescriptposture


(9) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton







119870119902= 025



119899628 rads


1198907000 bar


119886177 times 10

minus4 m2




1

0322 m


0065 m



119903016 m


119888asymp 119870

2119888


1199020188 m2s


1198651199011200 Nm

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

t (s)

4 45 515202530


PID control

Stan

ceph

ase

Swin

gph

ase

120579lk

(∘)


119901119864= 07 119901

119887= 1 and 119901












119891in the case


0 1 2 3 4 5 6 7 8 9 10t (s)


PID control

4 45 5034

0345035

0355

03

032

034

036

038

yl

(m)


0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

20

30

40

t (s)


ul

(m)



119902 119864 and 119887





0 1 2 3 4 5 6 7 8 9 10

minus1000

0

500

1000

15001300

minus500minus600

t (s)

Com

pens

atio

n fo

rce (

N)





119865pres =

119865119901

(119879120572119904 + 1)2

stance phase

(1 minus 1(119879120572119904 + 1)2

)119865119901

swing phase(28)







0 2 4 6 8 10 12 14 16minus20

0

20

40

60

80

t (s)

Left kneeRight knee

c ef

ga b d

120579k

(∘)






08 1 12 14 18 2 22 24 26165 2815

032

033

034

035

036

037

038

t (s)

Cylid

ner p

ositi

on (m

)



a b2bb1

c d1

d2

d e



010203040

t (s)

Con

trol c

urre

nt (m

A)


a

c e f

gb1 d1





0 2 4 6 8 10 12 14 160

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

PlPr




0 5 10 15 20 25 30 35 40

0

50

100

150

80

t (s)

Left kneeRight knee

h i j

lk

mn

minus50

120579k

(∘)


0 10 20 30 40025

03

035

04

045Cy

lidne

r pos

ition

(m)


t (s)


010203040

Con

trol c

urre

nt (m

A)


h i j

k

m n

l

0 5 10 15 20 25 30 35 40t (s)

minus10

minus20

minus30

minus40



0 10 20 30 400

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

ihj

k l

m n

PlPr










7 Conclusion




Acknowledgments


References



































2and 119867

infincontrol prob-












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










80 90 100 110 120 130 140 150 160123456789

120579k (∘)

Fl = 0kgFl = 30kg

Fl = 60kgFl = 90kg

Pl

(MPa

)





119876119886= 119870

1119902119909v minus119870

1119888119901119886 (11a)

119876119887= 119870

2119902119909v +119870

2119888119901119887 (11b)

1198701119902= 119862

119889119908radic

2Δ119901119901

120588

1198702119902= 119862

119889119908radic

2Δ119901119877

120588

(12a)

1198701119888=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119862119889119908119909v

radic2120588Δ119901119901

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198702119888=

100381610038161003816100381610038161003816100381610038161003816

119862119889119908119909v

radic2120588Δ119901119877

100381610038161003816100381610038161003816100381610038161003816

(12b)

Δ119901119901=

1003816100381610038161003816119901119904 minus 1199011198861003816100381610038161003816 119909v ge 0

1003816100381610038161003816119901119886 minus 1199011199031003816100381610038161003816 119909v lt 0

(13a)

Δ119901119877=

1003816100381610038161003816119901119887 minus 1199011199031003816100381610038161003816 119909v ge 0

1003816100381610038161003816119901119904 minus 1199011198871003816100381610038161003816 119909v lt 0

(13b)

where 119876119886 119876


119886 119901

119887are the





119902(119870

2119902) and1198701

119888(119870




119886=

120573119890

11988101 + 119860119886119910(119876

119886minus119860

119886119910) (14a)

119887=

120573119890

11988102 minus 119860119887119910(minus119876

119887+119860

119887119910) (14b)

11988101 = 11988102 =119881119905

2=119860119886119897ℎ

2 (15)


119886and119860




119898 119910 = (119901119886119860119886minus119901

119887119860119887minus119865

119891minus119865

119901) (16)




119910 (120579119896) = radic119897

21 + 119897


119896 (17)



119901and


119865119901(120579119896 119865119871) = 119870

119865119901(120579119896) 119910 (120579

119896) (18)






119891


119865119891asymp 119887 119910 (19)






0 2 4 6 8 10 12 14minus002

0

002

004

006

008

01

012

014

t (s)

y (m

)

Experimentb = 100

b = 200b = 300





infinsuboptimal

controller


119902


119886119910 119881

02minus119860


119865119901from



[27 28] as follows

119872119870119902= [

[

0 119870119902

119901119870119902

119870119902

]

]

119872119864=

[[[

[

minus119901119864

1119864

minus119901119864

1119864

]]]

]

119872119887= [

0 119887

119901119887119887

]

119872119870119865119901

= [

0 1

119901119870119865119901

119870119865119901

]

(20)

119870119902= 119870

119902(1+119901

119870119902120575119870119902) minus1 le 120575

119870119902le 1 (21a)

119864 = 119864 (1+119901119864120575119864) minus1 le 120575

119864le 1 (21b)

119887 = 119887 (1+119901119887120575119887) minus1 le 120575

119887le 1 (21c)

119870119865119901= 119870

119865119901+119901

119870119865119901120575119870119865119901 (21d)

where119864 = 1198811199052120573119870


119870119902 120575119864



119864 and119901



[1199091 119909

2 119909

3 119909

4 119909

5 119909

6]119879

= [119909v v 119901119886 119901119887 119910 119910]119879 the control


1 119908

2 119908

3 119908

4 119908

5]119879 the


1 1199112 1199113 1199114 1199115 1199116]119879 then the


(119905) = 119860119883 (119905) + 1198611119882(119905) + 1198612119906 (119905) (22a)

119885 (119905) = 1198621119883 (119905) +11986311119882(119905) (22b)

119884 (119905) = 1198622119883 (119905) (22c)

where

119860 =

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

[

0 1 0 0 0 0

minus1205962v minus2120577v120596v 0 0 0 0

119870119902

119864

0 minus119870119888

119864

0 0 minus119860119886

119864

minus

119870119902

119864

0 0 minus119870119888

119864

0 1198602

119864

0 0 0 0 0 1

0 0119860119886

119898minus119860119887

119898minus

119870119865119901

119898

119887

119898

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

]

(23a)

1198611 =

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

[

0 0 0 0 00 0 0 0 0119901119870119902

119864

minus119901119864

0 0 0

minus

119901119870119902

119864

0 minus119901119864

0 00 0 0 0 0

0 0 0 minus119901119887

119898minus

119901119870119865119901

119898

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

]

(23b)


r+

minusK

u

Wp

ep

Wu

y

eu

Δ

z

w

Ghyd

Gic = Fu(Ghyd Δ)


1198612 = [0 119870sv1205962v 0 0 0 0]119879 (23c)

1198621 =

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

[

119870119902

0 0 0 0 0

119870119902

119864

0 minus119870119888

119864

0 0 minus119860119886

119864

minus

119870119902

119864

0 0 minus119870119888

119864

0119860119887

119864

0 0 0 0 0 119887

0 0 0 0 1 0

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

]

(23d)

11986311 =

[[[[[[[[[[[

[

0 0 0 0 0119901119870119902

119864

minus119901119864

0 0 0

minus

119901119870119902

119864

0 minus119901119864

0 00 0 0 0 00 0 0 0 0

]]]]]]]]]]]

]

(23e)

1198622 = [0 0 0 0 1 0] (23f)



119906are


119901and 119890

119906 The nominal





01 1 10 100628

Log

mag

nitu

de (d

B)


minus500

Phas

e (de

g)

Frequency (rads)


NominalPerturbed

100

10minus2

10minus4


119902 119864 119887 and 119870

119865119901





119882119901= 095times 119904

2+ 18119904 + 5

1199042 + 8119904 + 001 (24a)

119882119906= 5times 10minus4 (24b)



119899


119882119901119866119899

119904infinle 120583(119882

119901)120583(119866

119899




(1) 119867infin


1003817100381710038171003817119865119897 (119866ic 119870)1003817100381710038171003817infin

=10038171003817100381710038171003817119865119897(119865

119906(119866hyd Δ) 119870)

10038171003817100381710038171003817infinlt 120574 (25)


119897(119866ic 119870)infin under



100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

119882119901(119868 + 119866ic119870)

minus1

119882119906119870(119868 + 119866ic119870)

minus1]

]

100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

lt 1 (26)


119888is stable




Frequency (rads)


100

10minus1

10minus3

10minus2


120583


119902 119864 119887 and 119870

119865119901

NominalUpper bound

Low bound

Frequency (rads)

100

10minus02

10minus06

10minus04

100 101 10210minus110minus2

120583


119902 119864 119887 and 119870

119865119901



119906extension = 119870 (119904) (119910 minus 119910)

=1198994119904

4+ 1198993119904

3+ 1198992119904

2+ 1198991119904 + 1198990

11988941199044 + 1198893119904

3 + 11988921199042 + 1198891119904 + 1198890

(119910 minus 119910)

(27)

where 119899119894(119894 = 0 4) 119889






Walking cycle





(1) (2) (3) (4) (5) (2)

(1) (6) (7) (8) (9) (2)





In Table 1 119906119897and 119906








(1) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton



Adapt toexoskeleton

(3) 119906119897= minus119906max

119906119903= 119906extension



(4) 119906119897= minus119906max

119906119903= 119906extension


Adapt toexoskeleton








Squatpreparation

(7)119906119897=

119906max(1 minus 119890minus119896119905

)

119906119903= minus119906max


Adapt toexoskeleton

(8) 119906119897= minus119906max

119906119903= minus119906max


prescriptposture


(9) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton







119870119902= 025



119899628 rads


1198907000 bar


119886177 times 10

minus4 m2




1

0322 m


0065 m



119903016 m


119888asymp 119870

2119888


1199020188 m2s


1198651199011200 Nm

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

t (s)

4 45 515202530


PID control

Stan

ceph

ase

Swin

gph

ase

120579lk

(∘)


119901119864= 07 119901

119887= 1 and 119901












119891in the case


0 1 2 3 4 5 6 7 8 9 10t (s)


PID control

4 45 5034

0345035

0355

03

032

034

036

038

yl

(m)


0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

20

30

40

t (s)


ul

(m)



119902 119864 and 119887





0 1 2 3 4 5 6 7 8 9 10

minus1000

0

500

1000

15001300

minus500minus600

t (s)

Com

pens

atio

n fo

rce (

N)





119865pres =

119865119901

(119879120572119904 + 1)2

stance phase

(1 minus 1(119879120572119904 + 1)2

)119865119901

swing phase(28)







0 2 4 6 8 10 12 14 16minus20

0

20

40

60

80

t (s)

Left kneeRight knee

c ef

ga b d

120579k

(∘)






08 1 12 14 18 2 22 24 26165 2815

032

033

034

035

036

037

038

t (s)

Cylid

ner p

ositi

on (m

)



a b2bb1

c d1

d2

d e



010203040

t (s)

Con

trol c

urre

nt (m

A)


a

c e f

gb1 d1





0 2 4 6 8 10 12 14 160

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

PlPr




0 5 10 15 20 25 30 35 40

0

50

100

150

80

t (s)

Left kneeRight knee

h i j

lk

mn

minus50

120579k

(∘)


0 10 20 30 40025

03

035

04

045Cy

lidne

r pos

ition

(m)


t (s)


010203040

Con

trol c

urre

nt (m

A)


h i j

k

m n

l

0 5 10 15 20 25 30 35 40t (s)

minus10

minus20

minus30

minus40



0 10 20 30 400

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

ihj

k l

m n

PlPr










7 Conclusion




Acknowledgments


References



































2and 119867

infincontrol prob-












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12 14minus002

0

002

004

006

008

01

012

014

t (s)

y (m

)

Experimentb = 100

b = 200b = 300





infinsuboptimal

controller


119902


119886119910 119881

02minus119860


119865119901from



[27 28] as follows

119872119870119902= [

[

0 119870119902

119901119870119902

119870119902

]

]

119872119864=

[[[

[

minus119901119864

1119864

minus119901119864

1119864

]]]

]

119872119887= [

0 119887

119901119887119887

]

119872119870119865119901

= [

0 1

119901119870119865119901

119870119865119901

]

(20)

119870119902= 119870

119902(1+119901

119870119902120575119870119902) minus1 le 120575

119870119902le 1 (21a)

119864 = 119864 (1+119901119864120575119864) minus1 le 120575

119864le 1 (21b)

119887 = 119887 (1+119901119887120575119887) minus1 le 120575

119887le 1 (21c)

119870119865119901= 119870

119865119901+119901

119870119865119901120575119870119865119901 (21d)

where119864 = 1198811199052120573119870


119870119902 120575119864



119864 and119901



[1199091 119909

2 119909

3 119909

4 119909

5 119909

6]119879

= [119909v v 119901119886 119901119887 119910 119910]119879 the control


1 119908

2 119908

3 119908

4 119908

5]119879 the


1 1199112 1199113 1199114 1199115 1199116]119879 then the


(119905) = 119860119883 (119905) + 1198611119882(119905) + 1198612119906 (119905) (22a)

119885 (119905) = 1198621119883 (119905) +11986311119882(119905) (22b)

119884 (119905) = 1198622119883 (119905) (22c)

where

119860 =

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

[

0 1 0 0 0 0

minus1205962v minus2120577v120596v 0 0 0 0

119870119902

119864

0 minus119870119888

119864

0 0 minus119860119886

119864

minus

119870119902

119864

0 0 minus119870119888

119864

0 1198602

119864

0 0 0 0 0 1

0 0119860119886

119898minus119860119887

119898minus

119870119865119901

119898

119887

119898

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

]

(23a)

1198611 =

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

[

0 0 0 0 00 0 0 0 0119901119870119902

119864

minus119901119864

0 0 0

minus

119901119870119902

119864

0 minus119901119864

0 00 0 0 0 0

0 0 0 minus119901119887

119898minus

119901119870119865119901

119898

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

]

(23b)


r+

minusK

u

Wp

ep

Wu

y

eu

Δ

z

w

Ghyd

Gic = Fu(Ghyd Δ)


1198612 = [0 119870sv1205962v 0 0 0 0]119879 (23c)

1198621 =

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

[

119870119902

0 0 0 0 0

119870119902

119864

0 minus119870119888

119864

0 0 minus119860119886

119864

minus

119870119902

119864

0 0 minus119870119888

119864

0119860119887

119864

0 0 0 0 0 119887

0 0 0 0 1 0

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

]

(23d)

11986311 =

[[[[[[[[[[[

[

0 0 0 0 0119901119870119902

119864

minus119901119864

0 0 0

minus

119901119870119902

119864

0 minus119901119864

0 00 0 0 0 00 0 0 0 0

]]]]]]]]]]]

]

(23e)

1198622 = [0 0 0 0 1 0] (23f)



119906are


119901and 119890

119906 The nominal





01 1 10 100628

Log

mag

nitu

de (d

B)


minus500

Phas

e (de

g)

Frequency (rads)


NominalPerturbed

100

10minus2

10minus4


119902 119864 119887 and 119870

119865119901





119882119901= 095times 119904

2+ 18119904 + 5

1199042 + 8119904 + 001 (24a)

119882119906= 5times 10minus4 (24b)



119899


119882119901119866119899

119904infinle 120583(119882

119901)120583(119866

119899




(1) 119867infin


1003817100381710038171003817119865119897 (119866ic 119870)1003817100381710038171003817infin

=10038171003817100381710038171003817119865119897(119865

119906(119866hyd Δ) 119870)

10038171003817100381710038171003817infinlt 120574 (25)


119897(119866ic 119870)infin under



100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

119882119901(119868 + 119866ic119870)

minus1

119882119906119870(119868 + 119866ic119870)

minus1]

]

100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

lt 1 (26)


119888is stable




Frequency (rads)


100

10minus1

10minus3

10minus2


120583


119902 119864 119887 and 119870

119865119901

NominalUpper bound

Low bound

Frequency (rads)

100

10minus02

10minus06

10minus04

100 101 10210minus110minus2

120583


119902 119864 119887 and 119870

119865119901



119906extension = 119870 (119904) (119910 minus 119910)

=1198994119904

4+ 1198993119904

3+ 1198992119904

2+ 1198991119904 + 1198990

11988941199044 + 1198893119904

3 + 11988921199042 + 1198891119904 + 1198890

(119910 minus 119910)

(27)

where 119899119894(119894 = 0 4) 119889






Walking cycle





(1) (2) (3) (4) (5) (2)

(1) (6) (7) (8) (9) (2)





In Table 1 119906119897and 119906








(1) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton



Adapt toexoskeleton

(3) 119906119897= minus119906max

119906119903= 119906extension



(4) 119906119897= minus119906max

119906119903= 119906extension


Adapt toexoskeleton








Squatpreparation

(7)119906119897=

119906max(1 minus 119890minus119896119905

)

119906119903= minus119906max


Adapt toexoskeleton

(8) 119906119897= minus119906max

119906119903= minus119906max


prescriptposture


(9) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton







119870119902= 025



119899628 rads


1198907000 bar


119886177 times 10

minus4 m2




1

0322 m


0065 m



119903016 m


119888asymp 119870

2119888


1199020188 m2s


1198651199011200 Nm

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

t (s)

4 45 515202530


PID control

Stan

ceph

ase

Swin

gph

ase

120579lk

(∘)


119901119864= 07 119901

119887= 1 and 119901












119891in the case


0 1 2 3 4 5 6 7 8 9 10t (s)


PID control

4 45 5034

0345035

0355

03

032

034

036

038

yl

(m)


0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

20

30

40

t (s)


ul

(m)



119902 119864 and 119887





0 1 2 3 4 5 6 7 8 9 10

minus1000

0

500

1000

15001300

minus500minus600

t (s)

Com

pens

atio

n fo

rce (

N)





119865pres =

119865119901

(119879120572119904 + 1)2

stance phase

(1 minus 1(119879120572119904 + 1)2

)119865119901

swing phase(28)







0 2 4 6 8 10 12 14 16minus20

0

20

40

60

80

t (s)

Left kneeRight knee

c ef

ga b d

120579k

(∘)






08 1 12 14 18 2 22 24 26165 2815

032

033

034

035

036

037

038

t (s)

Cylid

ner p

ositi

on (m

)



a b2bb1

c d1

d2

d e



010203040

t (s)

Con

trol c

urre

nt (m

A)


a

c e f

gb1 d1





0 2 4 6 8 10 12 14 160

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

PlPr




0 5 10 15 20 25 30 35 40

0

50

100

150

80

t (s)

Left kneeRight knee

h i j

lk

mn

minus50

120579k

(∘)


0 10 20 30 40025

03

035

04

045Cy

lidne

r pos

ition

(m)


t (s)


010203040

Con

trol c

urre

nt (m

A)


h i j

k

m n

l

0 5 10 15 20 25 30 35 40t (s)

minus10

minus20

minus30

minus40



0 10 20 30 400

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

ihj

k l

m n

PlPr










7 Conclusion




Acknowledgments


References



































2and 119867

infincontrol prob-












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










r+

minusK

u

Wp

ep

Wu

y

eu

Δ

z

w

Ghyd

Gic = Fu(Ghyd Δ)


1198612 = [0 119870sv1205962v 0 0 0 0]119879 (23c)

1198621 =

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

[

119870119902

0 0 0 0 0

119870119902

119864

0 minus119870119888

119864

0 0 minus119860119886

119864

minus

119870119902

119864

0 0 minus119870119888

119864

0119860119887

119864

0 0 0 0 0 119887

0 0 0 0 1 0

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

]

(23d)

11986311 =

[[[[[[[[[[[

[

0 0 0 0 0119901119870119902

119864

minus119901119864

0 0 0

minus

119901119870119902

119864

0 minus119901119864

0 00 0 0 0 00 0 0 0 0

]]]]]]]]]]]

]

(23e)

1198622 = [0 0 0 0 1 0] (23f)



119906are


119901and 119890

119906 The nominal





01 1 10 100628

Log

mag

nitu

de (d

B)


minus500

Phas

e (de

g)

Frequency (rads)


NominalPerturbed

100

10minus2

10minus4


119902 119864 119887 and 119870

119865119901





119882119901= 095times 119904

2+ 18119904 + 5

1199042 + 8119904 + 001 (24a)

119882119906= 5times 10minus4 (24b)



119899


119882119901119866119899

119904infinle 120583(119882

119901)120583(119866

119899




(1) 119867infin


1003817100381710038171003817119865119897 (119866ic 119870)1003817100381710038171003817infin

=10038171003817100381710038171003817119865119897(119865

119906(119866hyd Δ) 119870)

10038171003817100381710038171003817infinlt 120574 (25)


119897(119866ic 119870)infin under



100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

119882119901(119868 + 119866ic119870)

minus1

119882119906119870(119868 + 119866ic119870)

minus1]

]

100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

lt 1 (26)


119888is stable




Frequency (rads)


100

10minus1

10minus3

10minus2


120583


119902 119864 119887 and 119870

119865119901

NominalUpper bound

Low bound

Frequency (rads)

100

10minus02

10minus06

10minus04

100 101 10210minus110minus2

120583


119902 119864 119887 and 119870

119865119901



119906extension = 119870 (119904) (119910 minus 119910)

=1198994119904

4+ 1198993119904

3+ 1198992119904

2+ 1198991119904 + 1198990

11988941199044 + 1198893119904

3 + 11988921199042 + 1198891119904 + 1198890

(119910 minus 119910)

(27)

where 119899119894(119894 = 0 4) 119889






Walking cycle





(1) (2) (3) (4) (5) (2)

(1) (6) (7) (8) (9) (2)





In Table 1 119906119897and 119906








(1) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton



Adapt toexoskeleton

(3) 119906119897= minus119906max

119906119903= 119906extension



(4) 119906119897= minus119906max

119906119903= 119906extension


Adapt toexoskeleton








Squatpreparation

(7)119906119897=

119906max(1 minus 119890minus119896119905

)

119906119903= minus119906max


Adapt toexoskeleton

(8) 119906119897= minus119906max

119906119903= minus119906max


prescriptposture


(9) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton







119870119902= 025



119899628 rads


1198907000 bar


119886177 times 10

minus4 m2




1

0322 m


0065 m



119903016 m


119888asymp 119870

2119888


1199020188 m2s


1198651199011200 Nm

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

t (s)

4 45 515202530


PID control

Stan

ceph

ase

Swin

gph

ase

120579lk

(∘)


119901119864= 07 119901

119887= 1 and 119901












119891in the case


0 1 2 3 4 5 6 7 8 9 10t (s)


PID control

4 45 5034

0345035

0355

03

032

034

036

038

yl

(m)


0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

20

30

40

t (s)


ul

(m)



119902 119864 and 119887





0 1 2 3 4 5 6 7 8 9 10

minus1000

0

500

1000

15001300

minus500minus600

t (s)

Com

pens

atio

n fo

rce (

N)





119865pres =

119865119901

(119879120572119904 + 1)2

stance phase

(1 minus 1(119879120572119904 + 1)2

)119865119901

swing phase(28)







0 2 4 6 8 10 12 14 16minus20

0

20

40

60

80

t (s)

Left kneeRight knee

c ef

ga b d

120579k

(∘)






08 1 12 14 18 2 22 24 26165 2815

032

033

034

035

036

037

038

t (s)

Cylid

ner p

ositi

on (m

)



a b2bb1

c d1

d2

d e



010203040

t (s)

Con

trol c

urre

nt (m

A)


a

c e f

gb1 d1





0 2 4 6 8 10 12 14 160

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

PlPr




0 5 10 15 20 25 30 35 40

0

50

100

150

80

t (s)

Left kneeRight knee

h i j

lk

mn

minus50

120579k

(∘)


0 10 20 30 40025

03

035

04

045Cy

lidne

r pos

ition

(m)


t (s)


010203040

Con

trol c

urre

nt (m

A)


h i j

k

m n

l

0 5 10 15 20 25 30 35 40t (s)

minus10

minus20

minus30

minus40



0 10 20 30 400

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

ihj

k l

m n

PlPr










7 Conclusion




Acknowledgments


References



































2and 119867

infincontrol prob-












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Frequency (rads)


100

10minus1

10minus3

10minus2


120583


119902 119864 119887 and 119870

119865119901

NominalUpper bound

Low bound

Frequency (rads)

100

10minus02

10minus06

10minus04

100 101 10210minus110minus2

120583


119902 119864 119887 and 119870

119865119901



119906extension = 119870 (119904) (119910 minus 119910)

=1198994119904

4+ 1198993119904

3+ 1198992119904

2+ 1198991119904 + 1198990

11988941199044 + 1198893119904

3 + 11988921199042 + 1198891119904 + 1198890

(119910 minus 119910)

(27)

where 119899119894(119894 = 0 4) 119889






Walking cycle





(1) (2) (3) (4) (5) (2)

(1) (6) (7) (8) (9) (2)





In Table 1 119906119897and 119906








(1) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton



Adapt toexoskeleton

(3) 119906119897= minus119906max

119906119903= 119906extension



(4) 119906119897= minus119906max

119906119903= 119906extension


Adapt toexoskeleton








Squatpreparation

(7)119906119897=

119906max(1 minus 119890minus119896119905

)

119906119903= minus119906max


Adapt toexoskeleton

(8) 119906119897= minus119906max

119906119903= minus119906max


prescriptposture


(9) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton







119870119902= 025



119899628 rads


1198907000 bar


119886177 times 10

minus4 m2




1

0322 m


0065 m



119903016 m


119888asymp 119870

2119888


1199020188 m2s


1198651199011200 Nm

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

t (s)

4 45 515202530


PID control

Stan

ceph

ase

Swin

gph

ase

120579lk

(∘)


119901119864= 07 119901

119887= 1 and 119901












119891in the case


0 1 2 3 4 5 6 7 8 9 10t (s)


PID control

4 45 5034

0345035

0355

03

032

034

036

038

yl

(m)


0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

20

30

40

t (s)


ul

(m)



119902 119864 and 119887





0 1 2 3 4 5 6 7 8 9 10

minus1000

0

500

1000

15001300

minus500minus600

t (s)

Com

pens

atio

n fo

rce (

N)





119865pres =

119865119901

(119879120572119904 + 1)2

stance phase

(1 minus 1(119879120572119904 + 1)2

)119865119901

swing phase(28)







0 2 4 6 8 10 12 14 16minus20

0

20

40

60

80

t (s)

Left kneeRight knee

c ef

ga b d

120579k

(∘)






08 1 12 14 18 2 22 24 26165 2815

032

033

034

035

036

037

038

t (s)

Cylid

ner p

ositi

on (m

)



a b2bb1

c d1

d2

d e



010203040

t (s)

Con

trol c

urre

nt (m

A)


a

c e f

gb1 d1





0 2 4 6 8 10 12 14 160

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

PlPr




0 5 10 15 20 25 30 35 40

0

50

100

150

80

t (s)

Left kneeRight knee

h i j

lk

mn

minus50

120579k

(∘)


0 10 20 30 40025

03

035

04

045Cy

lidne

r pos

ition

(m)


t (s)


010203040

Con

trol c

urre

nt (m

A)


h i j

k

m n

l

0 5 10 15 20 25 30 35 40t (s)

minus10

minus20

minus30

minus40



0 10 20 30 400

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

ihj

k l

m n

PlPr










7 Conclusion




Acknowledgments


References



































2and 119867

infincontrol prob-












Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(1) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton



Adapt toexoskeleton

(3) 119906119897= minus119906max

119906119903= 119906extension



(4) 119906119897= minus119906max

119906119903= 119906extension


Adapt toexoskeleton








Squatpreparation

(7)119906119897=

119906max(1 minus 119890minus119896119905

)

119906119903= minus119906max


Adapt toexoskeleton

(8) 119906119897= minus119906max

119906119903= minus119906max


prescriptposture


(9) 119906119897= 119906extension

119906119903= 119906extension


Adapt toexoskeleton







119870119902= 025



119899628 rads


1198907000 bar


119886177 times 10

minus4 m2




1

0322 m


0065 m



119903016 m


119888asymp 119870

2119888


1199020188 m2s


1198651199011200 Nm

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

t (s)

4 45 515202530


PID control

Stan

ceph

ase

Swin

gph

ase

120579lk

(∘)


119901119864= 07 119901

119887= 1 and 119901












119891in the case


0 1 2 3 4 5 6 7 8 9 10t (s)


PID control

4 45 5034

0345035

0355

03

032

034

036

038

yl

(m)


0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

20

30

40

t (s)


ul

(m)



119902 119864 and 119887





0 1 2 3 4 5 6 7 8 9 10

minus1000

0

500

1000

15001300

minus500minus600

t (s)

Com

pens

atio

n fo

rce (

N)





119865pres =

119865119901

(119879120572119904 + 1)2

stance phase

(1 minus 1(119879120572119904 + 1)2

)119865119901

swing phase(28)







0 2 4 6 8 10 12 14 16minus20

0

20

40

60

80

t (s)

Left kneeRight knee

c ef

ga b d

120579k

(∘)






08 1 12 14 18 2 22 24 26165 2815

032

033

034

035

036

037

038

t (s)

Cylid

ner p

ositi

on (m

)



a b2bb1

c d1

d2

d e



010203040

t (s)

Con

trol c

urre

nt (m

A)


a

c e f

gb1 d1





0 2 4 6 8 10 12 14 160

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

PlPr




0 5 10 15 20 25 30 35 40

0

50

100

150

80

t (s)

Left kneeRight knee

h i j

lk

mn

minus50

120579k

(∘)


0 10 20 30 40025

03

035

04

045Cy

lidne

r pos

ition

(m)


t (s)


010203040

Con

trol c

urre

nt (m

A)


h i j

k

m n

l

0 5 10 15 20 25 30 35 40t (s)

minus10

minus20

minus30

minus40



0 10 20 30 400

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

ihj

k l

m n

PlPr










7 Conclusion




Acknowledgments


References



































2and 119867

infincontrol prob-












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 10t (s)


PID control

4 45 5034

0345035

0355

03

032

034

036

038

yl

(m)


0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

20

30

40

t (s)


ul

(m)



119902 119864 and 119887





0 1 2 3 4 5 6 7 8 9 10

minus1000

0

500

1000

15001300

minus500minus600

t (s)

Com

pens

atio

n fo

rce (

N)





119865pres =

119865119901

(119879120572119904 + 1)2

stance phase

(1 minus 1(119879120572119904 + 1)2

)119865119901

swing phase(28)







0 2 4 6 8 10 12 14 16minus20

0

20

40

60

80

t (s)

Left kneeRight knee

c ef

ga b d

120579k

(∘)






08 1 12 14 18 2 22 24 26165 2815

032

033

034

035

036

037

038

t (s)

Cylid

ner p

ositi

on (m

)



a b2bb1

c d1

d2

d e



010203040

t (s)

Con

trol c

urre

nt (m

A)


a

c e f

gb1 d1





0 2 4 6 8 10 12 14 160

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

PlPr




0 5 10 15 20 25 30 35 40

0

50

100

150

80

t (s)

Left kneeRight knee

h i j

lk

mn

minus50

120579k

(∘)


0 10 20 30 40025

03

035

04

045Cy

lidne

r pos

ition

(m)


t (s)


010203040

Con

trol c

urre

nt (m

A)


h i j

k

m n

l

0 5 10 15 20 25 30 35 40t (s)

minus10

minus20

minus30

minus40



0 10 20 30 400

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

ihj

k l

m n

PlPr










7 Conclusion




Acknowledgments


References



































2and 119867

infincontrol prob-












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12 14 16minus20

0

20

40

60

80

t (s)

Left kneeRight knee

c ef

ga b d

120579k

(∘)






08 1 12 14 18 2 22 24 26165 2815

032

033

034

035

036

037

038

t (s)

Cylid

ner p

ositi

on (m

)



a b2bb1

c d1

d2

d e



010203040

t (s)

Con

trol c

urre

nt (m

A)


a

c e f

gb1 d1





0 2 4 6 8 10 12 14 160

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

PlPr




0 5 10 15 20 25 30 35 40

0

50

100

150

80

t (s)

Left kneeRight knee

h i j

lk

mn

minus50

120579k

(∘)


0 10 20 30 40025

03

035

04

045Cy

lidne

r pos

ition

(m)


t (s)


010203040

Con

trol c

urre

nt (m

A)


h i j

k

m n

l

0 5 10 15 20 25 30 35 40t (s)

minus10

minus20

minus30

minus40



0 10 20 30 400

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

ihj

k l

m n

PlPr










7 Conclusion




Acknowledgments


References



































2and 119867

infincontrol prob-












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12 14 160

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

PlPr




0 5 10 15 20 25 30 35 40

0

50

100

150

80

t (s)

Left kneeRight knee

h i j

lk

mn

minus50

120579k

(∘)


0 10 20 30 40025

03

035

04

045Cy

lidne

r pos

ition

(m)


t (s)


010203040

Con

trol c

urre

nt (m

A)


h i j

k

m n

l

0 5 10 15 20 25 30 35 40t (s)

minus10

minus20

minus30

minus40



0 10 20 30 400

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

ihj

k l

m n

PlPr










7 Conclusion




Acknowledgments


References



































2and 119867

infincontrol prob-












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 400

2

4

6

8

10

t (s)

Pres

sure

(MPa

)

ihj

k l

m n

PlPr










7 Conclusion




Acknowledgments


References



































2and 119867

infincontrol prob-












Volume 2014




Journal of











Journal of


Function Spaces






Algebra














7 Conclusion




Acknowledgments


References



































2and 119867

infincontrol prob-












Volume 2014




Journal of











Journal of


Function Spaces






Algebra









































2and 119867

infincontrol prob-












Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a lower extremity exoskeleton: human

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