Research Article Enhanced Dynamic Model of Pneumatic ...downloads.hindawi.com/journals/aaa/2015/906126.pdfdynamic model of pneumatic arti cial muscle with applica-tion to PAM-based

Research ArticleEnhanced Dynamic Model of Pneumatic Muscle Actuator withElman Neural Network

Alexander Hošovskyacute Jaacuten Pite8 and Kamil Cidek

Faculty of Manufacturing Technologies with a Seat in Presov Technical University of Kosice Bayerova 1 080 01 Presov Slovakia

Correspondence should be addressed to Alexander Hosovsky alexanderhosovskytukesk

Received 16 October 2014 Accepted 22 January 2015

Academic Editor Mathiyalagan Kalidass

Copyright copy 2015 Alexander Hosovsky et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

To make effective use of model-based control system design techniques one needs a good model which captures systemrsquos dynamicproperties in the range of interest Here an analytical model of pneumatic muscle actuator with two pneumatic artificial musclesdriving a rotational joint is developed Use of analytical model makes it possible to retain the physical interpretation of the modeland the model is validated using open-loop responses Since it was considered important to design a robust controller based on thismodel the effect of changed moment of inertia (as a representation of uncertain parameter) was taken into account and comparedwith nominal case To improve the accuracy of the model these effects are treated as a disturbance modeled using the recurrent(Elman) neural network Recurrent neural network was preferred over feedforward type due to its better long-term predictioncapabilities well suited for simulation use of the model The results confirm that this method improves the model performance(tested for five of themeasured variables joint anglemuscle pressures andmuscle forces) while retaining its physical interpretation

1 Introduction

Pneumatic artificial muscle (PAM) is a special type of actua-tor that converts energy of pressurized air within the muscleinto mechanical energy in the form of its contraction Theactuator has some interesting properties that make it attrac-tive for certain specific applications which might requirepowerful actuators with clean operation As such it remainsone of the few nonconventional actuators that were trans-formed from a laboratory prototype into fully fledged reliableand viable commercial product (eg FESTOfluidicmuscles)The issue of PAM modeling was pivotal in its research sincethe very beginning and it resulted in several different modeltypes which more or less characterize the modelingapproaches in all works regarding PAMs namely geometricalmodel empirical model and phenomenological model [1]The geometrical model represents a traditional approachto PAM modeling which relies on geometrical parametersthat can be difficult to measure accurately This modelingapproach was used in enhanced form in [2 3] where thedependence of fiber length on muscle pressure and thefriction between fibers themselves were taken into accountResearchers in [4] presented a detailed analytical model of

PAMderived using its geometrical andmechanical propertieswithout need for any experimentally determined parametersA different approach was chosen in [5] where a dynamicphenomenologicalmodel of PAMwas derived using the anal-ogy with Hillrsquos model of biological muscle In [6] a combinedempirical-geometric model of PAM based on the nonlinearspring-damper analogy intended for a robotic applicationwas developed Solely empirical model was derived in [7] andthis was used in [8] for the analysis of mechanical propertiesof PAM in mechatronic systems Some works included thedevelopment of PAM model within the task of model-basedcontrol design where the particular approach selected forPAM modeling differed based on the application needs andexperiment conditions In [9] an empirical PAM model withimproved force modeling was presented as a part of work oncascaded control concept for two degrees of freedom pneu-matic muscle robot Similar problem of model-based controldesign for PAM-based manipulator control was dealt within [10] where researchers used feedforward compensation ofactuator hysteresis based on the Preisach hysteresis modelBased on the empirical model of PAM researchers in [11]developed a state-space model used for a model-based non-linear control algorithm design An interesting approach is

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015 Article ID 906126 16 pageshttpdxdoiorg1011552015906126

2 Abstract and Applied Analysis

presented in [12] where a novel model based on Bouc-Wendifferential equation was used to describe the hystereticbehavior of PAM actuator and on the basis of this model anew cascaded control concept was tested

Our previous research concentrated on developing adynamic model of pneumatic artificial muscle with applica-tion to PAM-based actuator modeling with one degree offreedom (one rotational joint) driven by a pair of FESTOfluidic muscles The basic dynamic model of this systemintended for industrial applications was developed in [13]where preliminary experiments in open-loopmode were car-ried out In addition to this works [14 15] were dedicated tothe analysis of fundamental static and dynamic properties ofPAM-based actuator and its operating modes In [16] theanalytical model of PAM-based actuator developed in [13]was used formodel-based adaptive fuzzy control design usinggenetic algorithms Also in [17] the properties of pneumaticartificial muscle were analyzed from mathematical point ofview with their description given in the form of nonlineardifferential equations

Our enhanced dynamic model is based on the analyticalmodel derived in [13] where mechanical part of the PAMmodel is treated as a parallel combination of nonlinear springand nonlinear damper in accordancewith [6] It is shown thatthe model itself in general is capable of capturing the natureof actuatorrsquos dynamics yet its accuracy suffers from quite anumber of simplifications made in the description of rathercomplex physical phenomena For the comparison of modeland actuator responses we selected five variables for whichappropriate sensors were available and which were consid-ered important in view of the further research joint anglemuscle pressures and muscle forces It was also consideredimportant to take into account the effect of varying momentof inertia which is in general unpredictable and proposedcontrol system should be robust enough to provide satisfac-tory performance under this condition Results of open-loopmodemodel validation helped to uncover deviations betweenthe actuator and model performance originating from theunmodeled effects in physical description of the system withthe magnitude of their manifestation (at least in terms ofoscillations detected in the measured responses) being pro-portional to the moment of inertia represented by a loadweight attached to the actuator arm In view of these effectsrsquonature it was proposed here to use powerful approximationcapabilities of neural networks to tackle the issue of complexphysical phenomena description It has to be noted that eventhough it might have been possible to abandon analyticalapproach to PAM-based actuatormodeling completely it wasdeemed advantageous to retain it and onlymake use of neuralnetwork in combination with primary (analytical) model Inthis way the physical interpretation of the model was pre-served and unmodeled effects whichwere to be approximatedby neural network were treated as external disturbance Asto the neural network type recurrent network was preferredover the series-parallel (nonlinear autoregressive model withexogenous input NARX) configuration due to its bettersuitability for simulation tasksThemodel was intended to beused for control system design where no input in the formof actual variable value (as in series-parallel configuration

without feedback) was expected The results confirmed use-fulness of this approach in predicting actuatorrsquos dynamiccharacteristics even in closed-loop mode where unmodeledeffects caused larger deviations of the model performancecompared to the actuator performance

2 Materials and Methods

The main object of interest was pneumatic muscle-basedactuator with a pair of pneumatic artificial muscles in antag-onistic connection joined with a chain led through a sprocket(Figure 1)Themuscles themselves were fixed to a metal plateforming a rigid base at the upper part of the actuator Thesprocket was welded to a steel shaft (visible in the side view ofthe actuator in Figure 1) which passed through supportingcolumns with bearings A rotating arm was attached to oneend of the shaft to which a load weight (representing varyingmoment of inertia) could be fixed

From physical point of view PAM-based actuator isa multidomain system consisting of electrical pneumaticand mechanical part (Figure 2) The electrical part of thesystem (denoted by EPAM electropneumatic actionmodule)represents the power module needed for valve control andcontains four power transistors (BUZ11) together with apower supply As the effect of this part on the relevantdynamic characteristics of the system was considered negli-gible it was not included in modeling process

By neglecting the electrical part the system model couldbe divided into two parts pneumatic part represented by acompressor and two 33 on-off valves and mechanical partrepresented by two parallel connections of nonlinear springand nonlinear damper and rotational joint In Figure 2 all thevariables between which the dependencies expressed in theform of differential equations had to be determined as a partofmodeling process are shownThe interactions among thesevariables characterize the physical phenomena taking placewithin the system during its operation from the inflow ofpressurized air into the muscles to the transfer of force actionof the muscles to a load The principle of operation of theexamined actuator which is important for developing itsdynamic model can be clarified using Figure 3

The operation of actuator always starts from the set-up position in which both muscles are inflated to the samepressure (550 kPa in our experiments) and are at the sameinitial length denoted by 119897

0 This corresponds to position 1 in

Figure 3 where equilibrium with the highest force occursStarting from this point only one of themuscles is active (iebeing deflated) in one half of the joint movement range andthe muscles reverse their roles with arm passing through thezero position The joint movement is stopped whenever theforce equilibrium is found (corresponding to consecutiveequilibrium points 1 2 3 ) which happens at differentlength of themuscles (longer for activemuscle up to nominallength) and different pressures (lower for activemuscle downto atmospheric pressure when it is completely deflated) It isof note that the actuator operates well below the upper forcelimit for both muscles (1200N for muscles with force com-pensator) due to a relatively large clearance between the

Abstract and Applied Analysis 3

(a)

(b)

Figure 1 Mechanical configuration of pneumatic muscle actuator with two muscles in antagonistic connection joined through a chain gear((a) front view and (b) side view)

1 3

2

1 3

2

EPAM

Valves

Compressor

Rotatingarm

Pneumatic part Mechanical part

V1V2V3V4

Pc

Q2

Q1

120573 120596 120576J

P1 V1

P2 V2

l1 1 a1 F1

l2 2 a2 F2

minus+

Figure 2 Physical representation of PAM-based actuator with modeled parts being shown in grey color (in the scheme 119875119888is compressor

pressure [Pa]119876119894is volume flow rate [m3sdotsminus1] 119875

119894is muscle pressure [Pa]119881

119894is muscle volume [m3] 119897

119894is muscle length [m] V

119894is muscle velocity

[msdotsminus1] 119886119894is muscle acceleration [msdotsminus2] 119894 is muscle index 120573 is joint angle [deg] 120596 is joint angular velocity [radsdotsminus1] and 120576 is joint angular

acceleration [radsdotsminus2])


200018001600140012001000

800600400200

3 21

0

0kPa100 kPa200 kPa300kPa

400kPa500kPa600kPa

l0 Δlmax

F1F2

F3

l (m)

F(N

)

Figure 3 Static characteristics (force-length relationship) of PAM-based actuator Equilibrium points are located at the intersectionsof force curves for given muscle pressures As is shown in the figureone of the muscles remains inflated to the initial pressure and is notactive in one half of the operating range of rotational jointmovement(in the figure 119865 is muscle force 119897 is muscle length 119897

0is initial muscle

length Δ119897max is maximum range of muscle length change and 119875 ismuscle pressure)

chain and the sprocket in deflated state of both muscles(giving initial contraction of approximately 14)

21 Derivation of Analytical Model of Pneumatic Muscle Actu-ator As was mentioned above a dynamic model of pneu-matic muscle actuator consists of mechanical and pneumaticpart The former one is based on the mechanical model ofpneumatic artificial muscle represented as a parallel connec-tion of nonlinear spring and nonlinear damper [6] Accord-ing to this model PAM can be modeled as a nonlinear mass-spring-damper system described by the following nonlineardifferential equation

120585 = (1

119898) [119865119864minus 119865119899119889( 120594 119875) minus 119865

119899119904(120594 119875)] (1)

where 120585 is muscle displacement [m] 119898 is moved mass [kg]119865119864is external force [N]119865

119899119889is nonlinear damper force [N]119865

119899119904

is nonlinear spring force [N] 120594 is muscle contraction [-] and119875 is absolute muscle pressure [Pa]

Nonlinear spring force term (119865119899119904) depends on both

muscle contraction andmuscle pressure and its approximateshape can be estimated from Figure 3This termwas approxi-mated using the fifth-order polynomial (using Surface FittingTool in Matlab) based on the data read from the force-lengthrelationship given in [18] (shown in Figure 4)

The approximation equation could be expressed in thefollowing form

119865119899119904(119875 120594) =

119897

sum

119896=0

119899

sum

119898=0

119886119896119898119875119896120594119898 (2)

where 119886119896119898

is polynomial coefficients Totally 329 points(blue dots in Figure 4) taken from the experimental force-length curves were used for the function approximation with

0 5 10 15 20 25 30

12345670

5001000150020002500

Contraction ()Pressure (kPa)

Forc

e (N

)

Figure 4 Muscle force function approximation using fifth-orderpolynomial Blue circles correspond to experimental data givenin [18] Even though it is not shown in the figure the forceapproximator used saturation function at its output to limit the forceinto (0 1200)N interval

Table 1 Values of polynomial coefficients used for approximationof nonlinear spring term (all other coefficients were equal to zero)

Coefficient Value11988600

minus9084

11988610

minus7963

11988601

6508

11988620

235

11988611

minus231

11988602

115

11988630

minus1883

11988621

minus07284

11988612

3715

11988603

minus2282

11988640

0062

11988631

0081

11988622

minus0145

11988613

minus0185

11988604

2088

11988650

minus735119890 minus 5

11988641


11988632


11988623

199119890 minus 3

11988614


11988605


the values of polynomial coefficients shown in Table 1 Theresulting RMSE obtained for the spring force predictor using(2) was 44416 and coefficient of determination (1198772) was09912

The second term in brackets on the right side of (2)corresponds to the force of (nonlinear) damper This term iscommonly proportional to the velocity of mass-spring-damper system through the damping coefficient In ourmodel we use a nonlinear damper the force of which dependsnot only on the velocity but also on muscle pressure (as waspresented in [6]) This can be expressed as follows

119865119899119889(119875 120594) = minus120574119875 120594 (3)


where 120574 is modified damping coefficient [msdots] 119875 is musclepressure [Pa] and 120594 is muscle velocity [msdotsminus1] Due to theantagonistic connection of the muscles one muscle can beconsidered an agonist and the other one an antagonist whichhelps to explain the role of external force term in (2) (119865

119864) as

being equal to the force of opposing muscleDerivation of the pneumatic part of the model was based

on a simplified description of the dynamics of fluid flowwithin the system The differential equation describing thedynamics of muscle pressure can be derived from Boyle-Mariottersquos ideal gas law [6 19]

11987501198810= 119875119881119898 (4)

119889

119889119905(11987501198810

119881119898

) = 1198750(0119881119898minus 1198981198810

1198812119898

) = 1198750

0

119881119898

minus 119875119898

119898

119881119898

(5)

where 1198750is atmospheric pressure [Pa] 119881

0is volume of pres-

surized air within themuscle [m3]119881119898ismuscle volume [m3]

and 119875 is muscle pressure [Pa] In order to use (5) in actuatormodel one has to define the terms 119881

119898 119898 and

0 The last

term actually represents the volume flow of pressurized airthrough the valve which can be modeled using a standardequation given for example in [19]

0=

1198751119862radic

1198790

1198791

radic1 minus (11987521198751minus 120595

1 minus 120595)

2

if 11987521198751

gt 120595

1198751119862radic

1198790

1198791

if 11987521198751

le 120595

(6)

where 1198751is absolute upstream pressure [Pa] 119862 is sonic con-

ductance [m3sdotsminus1sdotPaminus1] 1198752is absolute downstream pressure

[Pa] 1198790is ambient air temperature at reference conditions

[K] 1198791is upstream pressure [K] and 120595 is critical ratio [-]

The muscle volume generally depends on geometricalparameters of the muscle but it can be approximated usinga polynomial in the following form

119881119898(120594) =

119898

sum

119896=0

119887119896120594119896 (7)

where 119887119896is 119896th coefficient of the approximation polynomial

In our case 119898 = 3 (third-order polynomial approximation)with the following values of polynomial coefficients 119887

3=

minus11times10minus4 1198872= minus18times10

minus6 1198871= 48times10

minus5 and 1198870= 79times10

minus6After taking time derivative of (7) for119898 = 3 we get

119898(120594 120594) = 3119887

31205942120594 + 21198872120594 120594 + 119887

1120594 (8)

Up to this point only the model of one pneumatic artificialmuscle has been derived The actuator itself uses two pneu-matic muscles working as an opposing pair From the kine-matic point of view actuator consists of one degree of free-dom in the form of a rotational joint and it was therefore nec-essary to develop themodel of joint dynamics driven by a pairof muscles described previously Of note is also the fact thatthe actuator frame was placed in horizontal direction allow-ing for a simpler model with neglected effect of gravity

To develop the joint dynamics model Lagrangianmechanics approach can be used Hence the followingequation holds [20]

119889

119889119905(120597119871

120597 119902119894

) minus120597119871

120597119902119894

= 120591119894 (9)

where 119871 is Lagrangian and 119871 = 119870119864minus 119875119864119870119864is kinetic energy

119875119864is potential energy 119902

119894is 119894th generalized variable and 120591

119894is

119894th generalized force In case we are dealing with a rotationaljoint generalized variable corresponds to a joint angle (120573) andgeneralized force corresponds to a torque (119879) developed by ajoint actuator (in our case a pair of muscles) Thus we get

119902 = [120573] 119902 = [ 120573] 120591 = [119879] 119870 =1

2119869 1205732 (10)

Having the actuator placed in horizontal direction resulted inthe fact that potential energy term in Lagrangian could be setto zero because the reference point can be chosen arbitrarilyand only changes in 119875

119864are relevant [21] Taking this fact into

account we can write the following

119889

119889119905(120597 ((12) 119869 120573

2)

120597 120573) minus

120597 ((12) 119869 1205732)

120597120573= 119879 (11)

119869 120573 = 119879 minus 119879119871 where 119879 = (119865

1minus 1198652) 119903 (12)

where11986511198652aremuscle forces [N] 119903 is sprocket diameter [m]

and 119879119871is load torque [Nsdotm] It is also of note that the friction

term in (12) was neglected

22 Calculation of Moment of Inertia for Given Loads As wasmentioned in the introduction the effect of varying momentof inertia was supposed to be taken into account whendesigning a controller for actuator Different moment ofinertia was represented by a detachable weight that could befixed to the actuator armThe moment of inertia for the loadweights was calculated using the following formula forcompound objects

119869tot =119898

sum

119896=1

119869119896 (13)

where 119869tot is total moment of inertia and 119869119896is moment of

inertia for 119896th object This formula can be used on the condi-tion that all particular objects have the same rotational axisIn our case the compound object consists of a steel shaft withlength 119871

119878 diameter 119863

119878 and mass 119872

119878and a steel arm with

length 119871119860 diameter 119863

119860 and mass 119872

119860and both objects

rotate around the shaftrsquos longitudinal axis All three particularobjects (shaft arm and load weight) were treated as solidcylinders for which moments of inertia can be calculatedusing standard formulas (as in eg [22])

119869119878=1

21198721198781198772

119878119869119860=1

41198721198601198772

119860+1

31198721198601198712

119860 (14)

where 119869119878is shaft moment of inertia [kgsdotm2]119872

119878is shaft mass

[kg] 119877119878is shaft radius [m] 119869

119860is arm moment of inertia

[kgsdotm2] 119877119860is arm radius [m] and 119871

119860is arm length [m]


LS

DSMS

DA LA

MA

JS

JA

ML

dPAR

Figure 5 Arm and shaft dimensions for moment of inertia calculations (in the scheme 119871119878is shaft length [m]119863

119878is shaft diameter [m]119872

119878is

shaft mass [kg] 119871119860is arm length [m]119863

119860is arm diameter [m]119872

119860is armmass [kg] 119869

119878is shaft moment of inertia [kgsdotm2] 119869

119860is armmoment

of inertia [kgsdotm2]119872119871is load weight [kg] and 119889PAR is parallel axis distance [m])

Table 2 Mechanical parameters of actuator used for moment ofinertia calculation In the table 120588 is steel density 119869

119899is calculated

moment of inertia for no load and 119872119871is load weight (all other

variables as described in Figure 5)

Parameter Value Unit120588 7850 kgsdotmminus3

119863119878

375 mm119871119878

285 mm119872119878

247 kg119863119860

1813 mm119871119860

272 mm119872119860

055 kg119889PAR 2011 mm119872119871

334 kg119869119899

0014 kgsdotm2

1198691

0155 kgsdotm2

To calculate the moment of inertia of the actuator withthe load weight attached the parallel axis theorem in thefollowing form was used

119869PAR = 119869 +1198721198892

PAR (15)

where 119869 is moment of inertia calculated for rotation axis pass-ing through the centre of gravity and 119889PAR is parallel distancebetween the actual rotation axis and rotation axis passingthrough the centre of gravity All dimensions relevant tothe calculations of moment of inertia (together with theillustration of a parallel axis distance for the load weight) areshown in Figure 5 Values of given parameters and calculatedmoment of inertia can be found in Table 2

23 Elman Network for Unmodeled Dynamics IdentificationThemodel presented above neglects the description of a num-ber of effects associated with complex phenomena that occurin the system This resulted in deviations of the model

responses from the actuator responses whether in steady-state parts (more significant for nominal moment of inertia)or transient parts (more significant for increased moment ofinertia) To improve the model performance it is proposedhere to use the powerful approximation capabilities of neuralnetworksThe network itself should act as a disturbance termidentifier where disturbance term incorporates all the effectsof unmodeled dynamics that cause the deviation of modelperformance from the performance of the actuator To put itmathematically one can resort to the expression of basicequation for the dynamics of 119899-link serial manipulator [23]

119899

sum

119895=1

119863119894119895(119902) 119902119895+

119899

sum

119896=1

119899

sum

119898=1

119867119894119896119898

119902119896119902119898+ 119866119894= 119876119894

119867119894119895119896=

119899

sum

119895=1

119899

sum

119896=1

(120597119863119894119895

120597119902119896

minus1

2

120597119863119895119896

120597119902119894

)

(16)

where 119902 is generalized variable 119863119894119895is elements of inertial

matrix 119867119894119896119898

is velocity coupling vector 119866119894is gravity vector

and 119876119894is generalized force vector This equation can be

rewritten in the more compact vector form as follows

D (q) q +H (q q) + G (q) = Q (17)

According to [21] this equation can be extended with thedisturbance term denoted by 120591

119875which includes for instance

the effects of unmodeled dynamics Thus the extended formof (17) can be expressed as follows

D (q) q +H (q q) + F (q) + G (q) + 120591P = Q (18)

where F(q) is friction term and 120591P is disturbance term Thebasic principle behind the use of neural network for distur-bance term identification is shown in Figure 6 The schemedepicted in the figure consists of the actuator in open-loopmode the analytical model of the actuator and a neuralnetwork The difference between measured actuator output


Model

PAMactuator

Neuralnetwork

yia

yim

120591P

P

e120591

V

Um

Figure 6 Proposed identification scheme for the estimation ofdisturbance term 120591

119875using a neural network In the scheme V is

valve control signal vector 119910119894119886is 119894th actuator output variable 119910119894

119898is

119894th model output variable 120591119875is disturbance term 120591

119875is disturbance

term estimation 119890120591is disturbance term estimation error and Um is

neural network input vector

variable (119910119886) and model output variable (119910

119898) corresponds to

the magnitude of model error caused by unknown distur-bance effects expressed in the formof disturbance term (120591

119875mdash

wedrop the vector denomination in bold becausewe treat thisvariable as scalar)Themain role of the neural network was toestimate this term based on the information from the modelfed as an input vectorUm This estimation is denoted by 120591

119875in

the scheme and the difference between the measured distur-bance term and its estimation (119890

120591) was used as an error signal

which should have been minimized by the network learningalgorithm

It was decided to prefer recurrent neural network over thepurely feedforward type due to its better long-term memorycapabilities [24] These capabilities mainly stem from itsinternal memory representation through the delay feedbackfrom the hidden layer back to the input (Elman neuralnetwork Figure 7(b)) The input into the context layer isactually one-step prediction of the state vector x(119896+1) whichis further processed by the output linear layer to obtain one-step output vector prediction y(119896+1) (Figure 7(a)) To expressthe relevant relationships mathematically one can first definestate and output equations for Elman network

x (119896 + 1) = f (CWx (119896) + IWu (119896) + b(1))

119910 (119896) = LWx (119896) + 119887(2)(19)

where x(119896) is 119899-dimensional state vector u(119896) is 119898-dimensional input vector b(1) is hidden layer bias vector 119887(2)is output layer bias (only one neuron was used in the outputlayer) CW is context layer weight matrix LW is hiddenlayer weight matrix IW is input layer weight matrix f isnonlinear activation function vector in hidden layer and 119896 is119896th sample

All the hidden layer neurons used the same activationfunction (hyperbolic tangent) in the following form

119891 (119909) = tanh (119909) = 119890119909minus 119890minus119909

119890119909 + 119890minus119909 (20)

By using (19) we can express the disturbance term estimationusing Elman network as

120591119875= LW119891 (CWx (119896) + IWu (119896) + b(1)) + 119887(2) + 119890

120591 (21)

where 119890120591is disturbance term estimation error

Since we assumed offline identification task the strategyused for network learning was epochwise training withweight updates after the presentation of the whole sampleset In order to use conventional backpropagationmethod fornetwork training it is necessary to unfold the network in timeso that a new layer with updated state is added for each timestep Then the network is trained in two passes forward andbackward to obtain the input state and output data and tocalculate local gradients using the following formula [25]

120575119895(119896) = minus

120597119864tot (1198960 1198961)

120597119897119895 (119896)

(22)

for all 119895 isin I and 1198960lt 119896 le 119896

1 In this formula 119895 is index of

neuron for which desired response is computed 119864tot(1198960 1198961)is total error function for time period from 119896

0to 1198961 and 119897

119895(119896)

is induced local field for 119895th neuron The local gradients arecalculated backwards starting from the last time step 119896

1and

then using this value in computation of local gradient(s) inprevious time step using the following formula

120575119895(119896)

=

1198911015840(119897119895(119896)) 119890119895(119896) for 119896 = 119896

1

1198911015840(119897119895 (119896))

[

[

119890119895 (119896) + sum

119895isinI

119908119895119898120575119896

sdot (119896 + 1)]

]

for 1198960lt 119896 lt 119896

1

(23)

where 1198911015840(119897119895(119896)) is derivation of activation function of 119895th

neuron and 119908119895119896is weight between 119895th and 119898th neuron The

weights are then updated as follows

Δ119908119895119894= minus120578

120597119864tot (1198960 1198961)

120597119908119895119894

= 120578

1198961

sum

119896=1198960+1

120575119895 (119896) 119909119894 (119896 minus 1) (24)

where 120578 is learning rate and 119909119894(119896 minus 1) is input applied to 119894th

synapse of 119895th neuron in time step 119896Actual network training was carried out using the

Levenberg-Marquardt algorithm with Bayesian regulariza-tion to improve the generalization properties of resultingneural model Thus the update of whole parameter vector120579 (containing all weights and biases) had the following form[26 27]

120579new = 120579current minus [J119879J + 120583I]

minus1

J119879e where

J119879J asymp H J119879e = g(25)


NHL LHL

(a) (b)

Input

Context layer

IW LW

CW

x(k + 1) y(k + 1)u(k)

y

x1

x2

xn

u1u2

um

zminus1

zminus1

zminus1

b(1)n

b(1)2

b(1)1

b(2)1

sum

sum

sum

sum

zminus1I

Figure 7 State diagram of Elman network (a) and network structure (b) where u(119896) is input vector x(119896 + 1) is state vector y(119896 + 1) is outputvector NHL is nonlinear hidden layer LHL is linear hidden layer 119911minus1 is unit delay IW is input layer weight matrixCW is context layer weightmatrix LW is hidden layer weight matrix 119887(119896)

119897is 119897th neuron bias in 119896th layer and 119910 is network output

where J is Jacobianmatrix for network error with respect to 120579I is identity matrixH is Hessian matrix g is gradient vectorand

e = 119890119894(120579) = 119905

119894minus 119910119894

119894 = 1 119899 (26)

is error vector containing error values for all 119899 samples in dataset The parameter 120583 in (25) allows altering the direction ofmovement between the steepest descent direction and New-ton direction and its value is modified in every step accordingto the success of this step with regard to the change inerror function value Its limit value was set to 120583 = 1119890

10 andbreaching this limit was the main criterion for networktraining convergence

3 Results and Discussion

In the first step the responses of the model derived inSection 21 were compared to actuator responses as a part ofmodel validation process The variables used in this processwere selected based on the availability of sensors for theirmeasurement These included joint angle 120573 [deg] left muscleforce 119865

1[N] right muscle force 119865

2[N] left muscle pressure

1198751[kPa] and right muscle pressure 119875

2[kPa]Themuscles are

denoted by left and right when viewed from front in uprightposition of the actuator To estimate how well the modelresponse matches the response of the actuator mean squarederror (MSE) criterion was used in the following form

MSE = 1

119899

119899

sum

119896=1

(119910119896minus 119910119896)2 (27)

where 119899 is number of samples 119910119896is actuator variable in 119896th

time step and 119910119896is model variable in 119896th time step In order

to decrease the level of noise in analog signals from pressureand force transducers a simplemoving average (MA)methodwas used

MA =sum119898

119894=1119910119894

119898 (28)

where 119910119894is 119894th component of filtered signal and 119898 is time

window for filtering (119898 = 21)To compare how well the model matches the responses

of the actuator excitation signal shown in Figure 8 wasused This signal contained several randomly chosen valvecontrol pulses which corresponded to random changes ofjoint angle within its operating range It can be observedthat the pulse sequences are paired (thus forming two sets ofdistinctive sequences) in order to ensure smooth transitionof the joint angle through the reference (zero) position Sinceit was considered important to take into account the effectof changing moment of inertia as a representation of varyingsystem parameter the test was also carried out for 11-multipleof nominal moment of inertia calculated in Section 22 (119869 =11119869119899) Initial (relative)muscle pressurewas selected to be119901

0=

550 kPa at which the muscles contracted by approximately14 (these values were also set in the model for simulation)The results of both tests expressed in the quantitative formare shown in Table 3 In this table the mean squared error forgiven test responses calculated using (27) is shown

Qualitatively the results can be evaluated using Figure 9(119869 = 119869

119899) and Figure 10 (119869 = 11119869

119899) where the comparison

of responses between the model and the actuator for allvariables can be seen If one inspects all the responses severalaspects can be observed in relation to the performance ofboth model and actuator At first the model in generalcaptures the salient points of the actuatorrsquos dynamics in both


0 1 2 3 4 5 6 7 8 9 100

051

Time (s)

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

V1

V2

V3

V4

Figure 8 Excitation signal used in model validation The pulsesshown in figure correspond to logic signals used for actuator control(0 is closed valve 1 is open valve and 119881

1 1198812 1198813 and 119881

4are valves)

cases When the joint angle for nominal case is considered(Figure 9) it can be seen that up to time of 2 seconds theresponses are quite similarwhile from that time on themodelpredicts higher values of joint angleThis ismore pronouncedin steady-state part of the response with the maximumdifference of 678∘ at 65 seconds The situation is similarfor all other variables (muscle forces and muscle pressures)where the first part of the model response matches theactuator response quite well with subsequent accumulationof the prediction errorThis should not come as a surprise asthe dynamics of the joint angle depends on the torque beingdetermined by the difference in muscle forces and muscleforces themselves depend on the muscle pressure (as definedin Section 21) The deviations of the model responses fromthe actuator responses expressed through MSE in Table 3represent the accumulation of errors caused by modelinginaccuracies and simplificationsmdashcompressor pressure droperrors in force function approximation errors caused bysimplifications of modeling equations and so forth Still theresponses of the model might offer usable insight into thedynamics of the actuator

In Figure 10 the responses for 11-multiple of nominalmoment of inertia are shown It is readily observable that incontrast to the responses with nominal moment of inertiathe oscillations are now much more pronounced Similar tothe previous case after 2 seconds the model predicts highervalues of joint angle with generally lower amplitude ofoscillations

The responses of muscle force in actuator are alsomarkedly oscillatory with the amplitude of these oscillationsbeing on average higher than predicted by the model On theother hand muscle pressure responses in actuator do not

Table 3 Results of all the tests for given moments of inertiacalculated as mean squared error between both the actuator andthe model response (120573 is joint angle [deg] 119875

1is left muscle relative

pressure [kPa] 1198752is right muscle relative pressure 119865

1is left muscle

force [N] and 1198652is right muscle force [N])

Variable MSE119869 = 119869119899

MSE119869 = 11119869

119899

120573 10024 1029171198651 76632 572861198652 45618 300371198751 22989 180751198752 51602 39199

show significant oscillations however it has to be noted thatthe pressure sensors were placed approximately 1 meter downthe pressure line from themuscles and this could have causedmore dampened responses Quantitative evaluation of theresults for 119869 = 11119869

119899shows that the relative errors for this case

are lower than for 119869 = 119869119899 which can be explained by the more

pronounced oscillations and the way in which mean absoluteerror is calculated

The responses shown in Figures 9 and 10 reveal theinaccuracies of derived analytical model but the informationcontained in them is hardly useful for neural network trainingand validation due to its lack of excitation richness which isvery important in experimental modeling In order to obtaindata better suited for disturbance identification purposespseudobinary random sequences depicted in Figure 11 wereused

Using the signals shown in Figure 11 as input to boththe actuator and the model it was possible to obtain suf-ficiently representative data sets which could be used inneural network training process Each of the training datasets contained 80000 samples (sampling period was set to0003 s) and another 5000 samples not included in thesesets were used as test data for checking the generalizationcapabilities of trained networks The number of neurons wasselected to be 15 and the maximum number of iterationswas limited to 1000 It is of note that not all of the networktrainings necessarily converged to local optimum within thisnumber of iterations (the training was considered to haveconverged if 120583 ge 1010)

The results of neural network training process can beevaluated using Table 4 In this table both normalized andunnormalized values of MSE criterion are used The reasonfor this distinction is the fact that in case of the trainingBayesian regularization was used which performs best whendata are normalized while in case of testing unnormalizedMSE offers better guess of the overall error in terms ofoperating range for a given variable In addition to theresulting errors the table shows also convergence status for allnetwork trainings Thus one can readily see that in fourcases the network training did not converge and the resultsprobably could have been improved if the training continued

On the other hand the number of maximum iterations(1000) was considered sufficient to give the results close to


0

10

20

30Jo

int a

ngle

(deg

)

ActuatorModel

0 15 3 45 6 75 9Time (s)

minus20

minus10

(a)

0 15 3 45 6 75 9100150200250300350400450500550600

Time (s)

Left

mus

cle fo

rce (

N)

ActuatorModel

(b)

200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 15 3 45 6 75 9Time (s)

(c)

150200250300350400450500550600

Mus

cle p

ress

ure (

kPa)

Actuator leftModel left

Actuator rightModel right

0 15 3 45 6 75 9Time (s)

(d)

Figure 9 Comparison of model and actuator performance for the excitation signal shown in Figure 8 with nominal moment of inertia ((a)joint angle response (b) left muscle force response (c) right muscle force response and (d) muscle pressure response for both muscles)

the optimum and also the final values of training parametersin nonconverged trainings did not imply that large changesin resulting MSE could have been anticipated Moreover itshould be noted that even for converged trainings the result-ing MSE was very likely only a local optimum and differentvalue could have been obtained for a training restarted withnew initial weights

The composition ofUm vector (the number and selectionof input variables) was found using trial and error approachAs can be observed from Table 4 it consists of the same inputvariables for given type of a variable (eg force or pressure)with the one exception in rightmuscle force (119869 = 11119869

119899) where

an additional variable (ie joint angle) had to be used toobtain better enhanced model performance One of the mostimportant results of the presented approach can be illustratedthrough comparison of the resulting MSE on test data set forenhanced and nonenhanced model (given in the third andthe fourth column of Table 4) These results can be evaluatedalso visually using Figures 12 and 13 where the responsesof enhanced and nonenhanced model for 119869 = 119869

119899and 119869 =

11119869119899are shownThis evaluation can be further complemented

by the inspection of error histograms obtained for the testdata set using enhanced model shown in Figure 14 Theaddition of error histograms should help to better evaluate thecontribution of neural network to improvement of the model

accuracy as well as pointing to possibly problematic parts ofthe identification process

It can be seen in Figure 12 (119869 = 119869119899) that the most sig-

nificant error for joint angle prediction using nonenhancedmodel occurs in steady-state parts of the responses Using theenhanced model it was possible to reduce MSE to 108 ofits original value According to the histogram (Figure 14(a))the higher number of instances (gt300) occurs for errors in therange of minus2∘ to minus05∘ and most error instances are negativemeaning that the model predicts very slightly higher valuescompared to the actuator The performance of the enhancedmodel for the force prediction is comparable (Figures 12(b)and 12(c)) In these cases the error reduction for left muscleand right muscle was to 152 and 158 respectively As canbe seen from the figure the nonenhancedmodel captures thenature of the force dynamics in both cases but its errors areas high as 150N in some cases Error histograms reveal thatmost error instances also have negative sign meaning thatthe model predicts higher values This fact may be causedby the composition of training and test data sets andor theresults of neural network training In case of muscle pres-sure prediction the error reduction for enhanced modelwas to 19 (left muscle) and 182 (right muscle) Theanalytical model generally predicts the higher values ofmuscle pressures compared to the actuator By comparing


0 500 1000 1500 2000 2500 3000

05

1015202530

Time (s)

Join

t ang

le (d

eg)

ActuatorModel

minus20minus15minus10minus5

(a)

150200250300350400450500550

Left

mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(b)

150200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(c)

200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 500 1000 1500 2000 2500 3000Time (s)

(d)

Figure 10 Comparison of model and actuator performance for the excitation signal shown in Figure 8 with 11-multiple of moment of inertia((a) joint angle response (b) left muscle force response (c) right muscle force response and (d) muscle pressure response in both muscles)

Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)

Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)

PRBS signal for J = Jn

PRBS signal for J = 11Jn

Figure 11 Excitation (pseudorandom binary sequence) signals usedfor training and test data acquisition for neural network training andvalidation (different signals were used for nominal and 11-multiplemoments of inertia)

respective error histograms (Figures 14(d) and 14(e)) it canbe concluded that most instances of the error occur in therange from minus3 kPa to 12 kPa for leftmuscle and from minus45 kPato 14 kPa which is very good result given the operating rangeof this variable

As was illustrated in the first part of this section thechange of nominal moment of inertia affected the actuatorrsquosdynamics in the way that was not accurately described in theoriginal analytical model This can be further demonstratedusing Figure 13 where the responses for test excitation signalwith 119869 = 11119869

119899are shown In case of joint angle prediction

MSE value was reduced to 547 of its original value It canbe observed that the analytical model does not describe lesspronounced oscillations of the transient parts of the responsewhich can be significant later in closed-loop tests Theremaining errors of the enhanced model are (similar to thenominal case) mostly in the range of minus25∘ to minus05∘ Theresponses ofmuscle forces for 11-multiple of nominalmomentof inertia probably show the most complex dynamic patternsof all measured responses Even in this case the networksreduced MSE to 254 (left muscle) and to 54 (right mus-cle)These responses reveal the intricate interplay of musclesrsquodynamics and structural dynamics of the actuator whichgive rise to such complex dynamic patternsWhen force errorhistograms (Figures 14(g) and 14(h)) are checked one can


Table 4 The results of neural network training for 119869 = 119869119899and 119869 = 11119869

119899 In the table MSE

119899is normalized mean squared error MSE is mean

squared error EM is enhanced model M is model Um is neural network input vector 120573119898is model joint angle 120596

119898is model angular velocity

120576119898is model angular acceleration 119865

1119898is left model muscle force 119865

2119898is right model muscle force 119875

1119898is left model muscle pressure and 119875

2119898

is right model muscle pressure

Variable MSE119899

trainingMSE

testing EMMSE

testing M Convergence Um

Joint angle 119869 = 119869119899 000522 1607 148896 Yes (444 it) 120573

119898 120596119898 120576119898

Left muscle force 119869 = 119869119899 0004 105925 69601 No 119865

1119898 1198652119898 120596119898 120576119898

Right muscle force 119869 = 119869119899 000688 83412 52785 Yes (164 it) 119865

1119898 1198652119898 120596119898 120576119898

Left muscle pressure 119869 = 119869119899 000595 102757 54016 Yes (88 it) 119875

1119898 1198752119898 120596119898 120576119898

Right muscle pressure 119869 = 119869119899 0006 113651 62475 Yes (66 it) 119875

1119898 1198752119898 120596119898 120576119898

Joint angle 119869 = 11119869119899 000237 4146 80721 No 120573

119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898

Right muscle force 119869 = 11119869119899 0009 316135 58577 No 119865

1119898 1198652119898 120596119898 120576119898

120573119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)

minus20

minus40

minus60

Actuator

ModelEnhanced model

Enhanced model errorModel error

(a)

0

100

200

300

400

Left

mus

cle fo

rce (

N)

ActuatorEnhanced modelModel


minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0

100

200

300

400

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0100200300400500600

Mus

cle p

ress

ure (

kPa)

minus1000 15 3 45 6 75 9 105 12 135 15

Time (s)

Actuator leftEnhanced model leftModel left

Actuator rightEnhanced model rightModel right

(d)

Figure 12 Comparison of model enhanced model and actuator generalization performance (test on unseen data with nominal of momentof inertia ((a) joint angle response (b) left muscle force response (c) right muscle force response and (d) muscle pressure response in bothmuscles))


0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)

ActuatorModelEnhanced model


minus20

minus40

minus60

minus80

(a)

0100200300400500

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0100200300400500

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0 15 3 45 6 75 9 105 12 135 15Time (s)

0100200300400500600

Mus

cle p

ress

ure (

kPa)



minus100

(d)

Figure 13 Comparison ofmodel enhancedmodel and actuator generalization performance (test on unseen data with 11-multiple of nominalmoment of inertia ((a) joint angle response (b) left muscle force response (c) right muscle force response and (d) muscle pressure responsein both muscles))

see that most error instances are distributed within a similarrange (approximately 15∘ to both sides of the zero errors) withvery low number of instances of larger errors occurring forright muscle force

This corresponds with the need for incorporation ofadditional variable into Um to achieve similar values of MSEfor right muscle force neural network and corroborates theasymmetry of actuatorrsquos operation The error in muscle pres-sure prediction was reduced to 259 (left muscle) and 176(right muscle) This time the oscillations are also present inmuscle pressure responses and these are captured very wellfor both muscles Most error instances occur in the range ofminus45 kPa to 10 kPa for left muscle and minus10 kPa to 15 kPa forright muscle The results shown in histograms support thehypothesis of complex dynamic phenomena which takeplacewithin the systemduring its operation since the numberof instances of enhanced model errors in larger ranges ishigher compared to the nonenhanced model Moreover theasymmetry of error distribution for different variables aswell as their different absolute ranges observed in histogramscan be linked to the complexity of actuator behavior whichfurther justifies the need for powerful approximators likeneural networks

It is easily observed that the nonenhanced dynamicmodel can be used for the basic description of the actuatorrsquosoperation yet its accuracy is compromised due to a numberof simplifications made during the model development Thiswas even more significant when the moment of inertia waschanged and the effect of unmodeled dynamics was magni-fiedThe results confirm that its combinationwith a recurrentneural network can be a viable solution to obtain an accuratedynamic model of the PAM-based actuator with retainedphysical interpretation Since the main purpose of this modelis simulation (in contrast to prediction where an output froma real plant allows for themodel output correction) recurrentneural networks seem to be a better option compared tonetwork without internal memory

4 Conclusion

In this work an enhanced dynamic model of one-DOFpneumaticmuscle actuator was developed One of the criticalpoints of selected approach was to retain the physical inter-pretation of the resulting model so that it could still offer aninsight into the physics of the system As a part of this work


0

100

200

300

400

500

600

700Error histogram with 20 bins

Insta

nces

Errors

Zero errors

minus368

minus3386

minus3093

minus2799

minus2506

minus2212

minus1919

minus1625

minus1332

minus1038

minus07446

minus04511

minus01576

072

31

016

131

160

41

897

013

590

4295

(a)

0100200300400500600700800900

1000

Error histogram with 20 bins

Insta

nces

203

85

162

828

511

41

145

317

66

minus4169

minus3856

minus3544

minus3232

minus2919

minus2607

minus2295

minus1982

minus167

minus1358

minus1045

minus7332

minus4208

minus1085

Errors

Zero errors

(b)

0

200

400

600

800

1000

1200

1400

1600

Insta

nces

258

75

742

889

712

05

152

118

36

215

2

minus3843

minus3527

minus3212

minus2896

minus2581

minus2265

minus195

minus1634

minus1319

minus1003

minus6877

minus3722

minus05676

Errors

Zero errors


(c)


0

100

200

300

400

500

600

700

800

Insta

nces

051

363

427

634

19

255

121

715

08 18

209

123

82

267

429

65

minus2571

minus228

minus1988

minus1697

minus1406

minus1114

minus8228

minus5314

minus24

Errors

Zero errors

(d)

0

100

200

300

400

500

600

700

800

900

Insta

nces

308

694

610

81

146

818

54

224

126

27

minus4717

minus4331

minus3944

minus3558

minus3171

minus2784

minus2398

minus2011

minus1625

minus1238

minus8516

minus4651

minus07852

Errors

Zero errors


(e)


0

100

200

300

400

500

600

700

800

Insta

nces

003

394

046

650

899

133

2

minus6887

minus6454

minus6022

minus5589

minus5157

minus4724

minus4291

minus3859

minus3426

minus2994

minus2561

minus2129

minus1696

minus1264

minus08311

minus03986

Errors

Zero errors

(f)

Figure 14 Continued


0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327

(j)Figure 14 Error histograms of enhanced model for test data set (5000 samples) In the figure (a) is joint angle (119869 = 119869

119899) (b) is left muscle

force (119869 = 119869119899) (c) is right muscle force (119869 = 119869

119899) (d) is left muscle pressure (119869 = 119869

119899) (e) is right muscle pressure (119869 = 119869

119899) (f) is joint angle

(119869 = 11119869119899) (g) is left muscle force (119869 = 11119869

119899) (h) is right muscle force (119869 = 11119869

119899) (i) is left muscle pressure (119869 = 11119869

119899) and (j) is right muscle

pressure (119869 = 11119869119899)

an analytical dynamic model of the actuator based on thenonlinear mass-spring-damper analogy was derived and ver-ified using the responses of real actuator Despite its capabilityto capture the salient points of the systemrsquos dynamics theaccuracy suffered from a number of simplifications that weremade during the development of the analytical model Thiswas even more profound for changing the nominal momentof inertia which was considered important for the develop-ment of a robust controller We proposed to use a recurrent(Elman) neural network in the role of the identifier ofdisturbance term which included all the effects causing thedeviation of the model performance from the actuatorperformance It was shown that the enhanced model offeredvery good improvement of the accuracy of original analyticalmodel while keeping its physical interpretation

Some limitations of the method can be linked to standardlimitation of the conventional neural network training like itsonly local optimality It is therefore possible to concentrate onthe use of global optimization methods (for instance hybridmetaheuristic methods) in further research In addition tothis it is also important to test the performance of thisapproach when used in closed-loop mode since this willdirectly affect the results of model-based controller design

Conflict of Interests

Authors declare that there is no conflict of interests regardingthe publication of this paper


Acknowledgment

The research work is supported by the Project of the Struc-tural Funds of the EU Operational Programme Researchand Development Measure 22 Transfer of knowledge andtechnology from research and development into practiceTitle of the project is Research and Development of Non-conventional ArtificialMuscles-Based Actuators ITMS code26220220103

References

[1] E Kelasidi G Andrikopoulos G Nikolakopoulos and SManesis ldquoA survey on pneumatic muscle actuators modelingrdquoJournal of Energy and Power Engineering vol 6 no 9 pp 1442ndash1452 2012

[2] S Davis N Tsagarakis J Canderle and D G CaldwellldquoEnhanced modelling and performance in braided pneumaticmuscle actuatorsrdquo International Journal of Robotics Researchvol 22 no 3-4 pp 213ndash227 2003

[3] S Davis and D G Caldwell ldquoBraid effects on contractile rangeand friction modeling in pneumatic muscle actuatorsrdquo Interna-tional Journal of Robotics Research vol 25 no 4 pp 359ndash3692006

[4] MDoumit A Fahim andMMunro ldquoAnalyticalmodeling andexperimental validation of the braided pneumaticmusclerdquo IEEETransactions on Robotics vol 25 no 6 pp 1282ndash1291 2009

[5] J L Serres Dynamic Characterization of a Pneumatic MuscleActuator and Its Application to a Resistive Training DeviceWright State University Dayton Ohio USA 2009

[6] T Kerscher J Albiez J M Zollner and R Dillmann ldquoEval-uation of the dynamic model of fluidic muscles using quick-releaserdquo in Proceedings of the International Conference onBiomedical Robotics and Biomechatronics pp 637ndash642 PisaItaly February 2006

[7] K C Wickramatunge and T Leephakpreeda ldquoEmpirical mod-eling of pneumatic artificial musclerdquo in Proceedings of the Inter-national Multi Conference of Engineers and Computer Scientistspp 1726ndash1730 Hong Kong 2009

[8] K C Wickramatunge and T Leephakpreeda ldquoStudy onmechanical behaviors of pneumatic artificial musclerdquo Interna-tional Journal of Engineering Science vol 48 no 2 pp 188ndash1982010

[9] A Hildebrandt O Sawodny R Neumann and A HartmannldquoCascaded control concept of a robot with two degrees offreedom driven by four artificial pneumatic muscle actuatorsrdquoin Proceedings of the American Control Conference vol 1 pp680ndash685 IEEE Portland Ore USA June 2005

[10] F Schreiber Y Sklyarenko G Runge and W SchumacherldquoModel-based controller design for antagonistic pairs of fluidicmuscles in manipulator motion controlrdquo in Proceedings ofthe 17th International Conference on Methods and Models inAutomation and Robotics (MMAR rsquo12) pp 499ndash504 Miedzyz-drojie Poland August 2012

[11] S V Krichel O Sawodny and A Hildebrandt ldquoTrackingcontrol of a pneumaticmuscle actuator using one servovalverdquo inProceedings of the American Control Conference (ACC rsquo10) pp4385ndash4390 Baltimore Md USA July 2010

[12] J Zhong J Fan Y Zhu J Zhao andWZhai ldquoOne nonlinear pidcontrol to improve the control performance of a manipulatoractuated by a pneumatic muscle actuatorrdquoAdvances inMechan-ical Engineering vol 2014 Article ID 172782 19 pages 2014

[13] A Hosovsky andMHavran ldquoDynamicmodeling of one degreeof freedom pneumatic muscle-based actuator for industrialapplicationsrdquo Tehnicki Vjesnik vol 19 no 3 pp 673ndash681 2012

[14] M Balara and M Tothova ldquoStatic and dynamic properties ofthe pneumatic actuator with artificialmusclesrdquo in Proceedings ofthe IEEE 10th Jubilee International Symposium on IntelligentSystems and Informatics (SISY rsquo12) pp 577ndash581 Subotica SerbiaSeptember 2012

[15] M Tothova J Pitelrsquo and J Borzıkova ldquoOperating modes ofpneumatic artificial muscle actuatorrdquo Applied Mechanics andMaterials vol 308 pp 39ndash44 2013

[16] A Hosovsky J N Marcincin J Pitelrsquo J Borzıkova and KZidek ldquoModel-based evolution of a fast hybrid fuzzy adaptivecontroller for a pneumatic muscle actuatorrdquo International Jour-nal of Advanced Robotic Systems vol 9 no 56 pp 1ndash11 2012

[17] A Macurova and S Hrehova ldquoSome properties of the pneu-matic artificial muscle expressed by the nonlinear differentialequationrdquo Advanced Materials Research vol 658 pp 376ndash3792013

[18] FESTO Fluidic Muscle DMSPMAS datasheet 2008 httpwwwfestocomrepen corpassetspdfinfo 501 enpdf

[19] P Beater Pneumatic Drives System Design Modelling andControl Springer New York NY USA 2007

[20] S H Zak Systems and Control Oxford University Press NewYork NY USA 2003

[21] F L Lewis D M Dawson and C T Abdallah Robot Motionand Control Marcel Dekker New York NY USA 2nd edition2004

[22] Hyperphysicsmdashcalculation of moments of inertia httphyperphysicsphy-astrgsueduhbasemihtml

[23] R N JazarTheory of Applied Robotics Springer New York NYUSA 2nd edition 2010

[24] S Samarasinghe Neural Networks for Applied Sciences andEngineering Auerbach Publications Boca Raton Fla USA 1stedition 2006

[25] S Haykin Neural NetworksmdashA Comprehensive FoundationPearson Prentice Hall Delhi India 2nd edition 2005

[26] J-S R Jang C-T Sun and E Mizutani Neuro-Fuzzy and SoftComputingmdashA Computational Approach to Learning andMachine Intelligence Prentice Hall Upper Saddle River NJUSA 1st edition 1997

[27] B H Demuth M Beale and M T Hagan Neural NetworkDesign PWS Publishing Boston Mass USA 1st edition 1996

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


presented in [12] where a novel model based on Bouc-Wendifferential equation was used to describe the hystereticbehavior of PAM actuator and on the basis of this model anew cascaded control concept was tested

Our previous research concentrated on developing adynamic model of pneumatic artificial muscle with applica-tion to PAM-based actuator modeling with one degree offreedom (one rotational joint) driven by a pair of FESTOfluidic muscles The basic dynamic model of this systemintended for industrial applications was developed in [13]where preliminary experiments in open-loopmode were car-ried out In addition to this works [14 15] were dedicated tothe analysis of fundamental static and dynamic properties ofPAM-based actuator and its operating modes In [16] theanalytical model of PAM-based actuator developed in [13]was used formodel-based adaptive fuzzy control design usinggenetic algorithms Also in [17] the properties of pneumaticartificial muscle were analyzed from mathematical point ofview with their description given in the form of nonlineardifferential equations

Our enhanced dynamic model is based on the analyticalmodel derived in [13] where mechanical part of the PAMmodel is treated as a parallel combination of nonlinear springand nonlinear damper in accordancewith [6] It is shown thatthe model itself in general is capable of capturing the natureof actuatorrsquos dynamics yet its accuracy suffers from quite anumber of simplifications made in the description of rathercomplex physical phenomena For the comparison of modeland actuator responses we selected five variables for whichappropriate sensors were available and which were consid-ered important in view of the further research joint anglemuscle pressures and muscle forces It was also consideredimportant to take into account the effect of varying momentof inertia which is in general unpredictable and proposedcontrol system should be robust enough to provide satisfac-tory performance under this condition Results of open-loopmodemodel validation helped to uncover deviations betweenthe actuator and model performance originating from theunmodeled effects in physical description of the system withthe magnitude of their manifestation (at least in terms ofoscillations detected in the measured responses) being pro-portional to the moment of inertia represented by a loadweight attached to the actuator arm In view of these effectsrsquonature it was proposed here to use powerful approximationcapabilities of neural networks to tackle the issue of complexphysical phenomena description It has to be noted that eventhough it might have been possible to abandon analyticalapproach to PAM-based actuatormodeling completely it wasdeemed advantageous to retain it and onlymake use of neuralnetwork in combination with primary (analytical) model Inthis way the physical interpretation of the model was pre-served and unmodeled effects whichwere to be approximatedby neural network were treated as external disturbance Asto the neural network type recurrent network was preferredover the series-parallel (nonlinear autoregressive model withexogenous input NARX) configuration due to its bettersuitability for simulation tasksThemodel was intended to beused for control system design where no input in the formof actual variable value (as in series-parallel configuration

without feedback) was expected The results confirmed use-fulness of this approach in predicting actuatorrsquos dynamiccharacteristics even in closed-loop mode where unmodeledeffects caused larger deviations of the model performancecompared to the actuator performance

2 Materials and Methods

The main object of interest was pneumatic muscle-basedactuator with a pair of pneumatic artificial muscles in antag-onistic connection joined with a chain led through a sprocket(Figure 1)Themuscles themselves were fixed to a metal plateforming a rigid base at the upper part of the actuator Thesprocket was welded to a steel shaft (visible in the side view ofthe actuator in Figure 1) which passed through supportingcolumns with bearings A rotating arm was attached to oneend of the shaft to which a load weight (representing varyingmoment of inertia) could be fixed

From physical point of view PAM-based actuator isa multidomain system consisting of electrical pneumaticand mechanical part (Figure 2) The electrical part of thesystem (denoted by EPAM electropneumatic actionmodule)represents the power module needed for valve control andcontains four power transistors (BUZ11) together with apower supply As the effect of this part on the relevantdynamic characteristics of the system was considered negli-gible it was not included in modeling process

By neglecting the electrical part the system model couldbe divided into two parts pneumatic part represented by acompressor and two 33 on-off valves and mechanical partrepresented by two parallel connections of nonlinear springand nonlinear damper and rotational joint In Figure 2 all thevariables between which the dependencies expressed in theform of differential equations had to be determined as a partofmodeling process are shownThe interactions among thesevariables characterize the physical phenomena taking placewithin the system during its operation from the inflow ofpressurized air into the muscles to the transfer of force actionof the muscles to a load The principle of operation of theexamined actuator which is important for developing itsdynamic model can be clarified using Figure 3

The operation of actuator always starts from the set-up position in which both muscles are inflated to the samepressure (550 kPa in our experiments) and are at the sameinitial length denoted by 119897

0 This corresponds to position 1 in

Figure 3 where equilibrium with the highest force occursStarting from this point only one of themuscles is active (iebeing deflated) in one half of the joint movement range andthe muscles reverse their roles with arm passing through thezero position The joint movement is stopped whenever theforce equilibrium is found (corresponding to consecutiveequilibrium points 1 2 3 ) which happens at differentlength of themuscles (longer for activemuscle up to nominallength) and different pressures (lower for activemuscle downto atmospheric pressure when it is completely deflated) It isof note that the actuator operates well below the upper forcelimit for both muscles (1200N for muscles with force com-pensator) due to a relatively large clearance between the


(a)

(b)


1 3

2

1 3

2

EPAM

Valves

Compressor

Rotatingarm


V1V2V3V4

Pc

Q2

Q1

120573 120596 120576J

P1 V1

P2 V2

l1 1 a1 F1

l2 2 a2 F2

minus+










200018001600140012001000

800600400200

3 21

0


400kPa500kPa600kPa

l0 Δlmax

F1F2

F3

l (m)

F(N

)


0is initial muscle




120585 = (1

119898) [119865119864minus 119865119899119889( 120594 119875) minus 119865

119899119904(120594 119875)] (1)



119899119904





119865119899119904(119875 120594) =

119897

sum

119896=0

119899

sum

119898=0

119886119896119898119875119896120594119898 (2)

where 119886119896119898


0 5 10 15 20 25 30

12345670

5001000150020002500


Forc

e (N

)




minus9084

11988610

minus7963

11988601

6508

11988620

235

11988611

minus231

11988602

115

11988630

minus1883

11988621

minus07284

11988612

3715

11988603

minus2282

11988640

0062

11988631

0081

11988622

minus0145

11988613

minus0185

11988604

2088

11988650


11988641


11988632


11988623

199119890 minus 3

11988614


11988605




119865119899119889(119875 120594) = minus120574119875 120594 (3)



119864) as



11987501198810= 119875119881119898 (4)

119889

119889119905(11987501198810

119881119898

) = 1198750(0119881119898minus 1198981198810

1198812119898

) = 1198750

0

119881119898

minus 119875119898

119898

119881119898

(5)


0is volume of pres-



119898 119898 and

0 The last


0=

1198751119862radic

1198790

1198791


1 minus 120595)

2

if 11987521198751

gt 120595

1198751119862radic

1198790

1198791

if 11987521198751

le 120595

(6)






119881119898(120594) =

119898

sum

119896=0

119887119896120594119896 (7)



3=





119898(120594 120594) = 3119887

31205942120594 + 21198872120594 120594 + 119887

1120594 (8)



119889

119889119905(120597119871

120597 119902119894

) minus120597119871

120597119902119894

= 120591119894 (9)




119894is


119902 = [120573] 119902 = [ 120573] 120591 = [119879] 119870 =1

2119869 1205732 (10)




119889

119889119905(120597 ((12) 119869 120573

2)

120597 120573) minus

120597 ((12) 119869 1205732)

120597120573= 119879 (11)

119869 120573 = 119879 minus 119879119871 where 119879 = (119865

1minus 1198652) 119903 (12)





119869tot =119898

sum

119896=1

119869119896 (13)



119878 diameter 119863

119878 and mass 119872



119860 and mass 119872



119869119878=1

21198721198781198772

119878119869119860=1

41198721198601198772

119860+1

31198721198601198712

119860 (14)


119878is shaft mass






LS

DSMS

DA LA

MA

JS

JA

ML

dPAR



119878is



119860is armmass [kg] 119869


119860is armmoment



119899is calculated




119863119878

375 mm119871119878

285 mm119872119878

247 kg119863119860

1813 mm119871119860

272 mm119872119860

055 kg119889PAR 2011 mm119872119871

334 kg119869119899

0014 kgsdotm2

1198691

0155 kgsdotm2


119869PAR = 119869 +1198721198892

PAR (15)




119899

sum

119895=1

119863119894119895(119902) 119902119895+

119899

sum

119896=1

119899

sum

119898=1

119867119894119896119898

119902119896119902119898+ 119866119894= 119876119894

119867119894119895119896=

119899

sum

119895=1

119899

sum

119896=1

(120597119863119894119895

120597119902119896

minus1

2

120597119863119895119896

120597119902119894

)

(16)


matrix 119867119894119896119898




D (q) q +H (q q) + G (q) = Q (17)




D (q) q +H (q q) + F (q) + G (q) + 120591P = Q (18)



Model

PAMactuator

Neuralnetwork

yia

yim

120591P

P

e120591

V

Um




119898is








119875mdash


119875in





x (119896 + 1) = f (CWx (119896) + IWu (119896) + b(1))

119910 (119896) = LWx (119896) + 119887(2)(19)



119891 (119909) = tanh (119909) = 119890119909minus 119890minus119909

119890119909 + 119890minus119909 (20)


120591119875= LW119891 (CWx (119896) + IWu (119896) + b(1)) + 119887(2) + 119890

120591 (21)



120575119895(119896) = minus

120597119864tot (1198960 1198961)

120597119897119895 (119896)

(22)




0to 1198961 and 119897

119895(119896)


1and


120575119895(119896)

=

1198911015840(119897119895(119896)) 119890119895(119896) for 119896 = 119896

1

1198911015840(119897119895 (119896))

[

[

119890119895 (119896) + sum

119895isinI

119908119895119898120575119896

sdot (119896 + 1)]

]

for 1198960lt 119896 lt 119896

1

(23)




Δ119908119895119894= minus120578

120597119864tot (1198960 1198961)

120597119908119895119894

= 120578

1198961

sum

119896=1198960+1

120575119895 (119896) 119909119894 (119896 minus 1) (24)





minus1

J119879e where

J119879J asymp H J119879e = g(25)


NHL LHL

(a) (b)

Input

Context layer

IW LW

CW

x(k + 1) y(k + 1)u(k)

y

x1

x2

xn

u1u2

um

zminus1

zminus1

zminus1

b(1)n

b(1)2

b(1)1

b(2)1

sum

sum

sum

sum

zminus1I




e = 119890119894(120579) = 119905

119894minus 119910119894

119894 = 1 119899 (26)










MSE = 1

119899

119899

sum

119896=1

(119910119896minus 119910119896)2 (27)




MA =sum119898

119894=1119910119894

119898 (28)




0=



119899) and Figure 10 (119869 = 11119869




0 1 2 3 4 5 6 7 8 9 100

051

Time (s)

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

V1

V2

V3

V4


1 1198812 1198813 and 119881

4are valves)







1is left muscle


Variable MSE119869 = 119869119899

MSE119869 = 11119869

119899

120573 10024 1029171198651 76632 572861198652 45618 300371198751 22989 180751198752 51602 39199










0

10

20

30Jo

int a

ngle

(deg

)

ActuatorModel

0 15 3 45 6 75 9Time (s)

minus20

minus10

(a)

0 15 3 45 6 75 9100150200250300350400450500550600

Time (s)

Left

mus

cle fo

rce (

N)

ActuatorModel

(b)

200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 15 3 45 6 75 9Time (s)

(c)

150200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 15 3 45 6 75 9Time (s)

(d)




119899) where


119899and 119869 =







0 500 1000 1500 2000 2500 3000

05

1015202530

Time (s)

Join

t ang

le (d

eg)

ActuatorModel


(a)

150200250300350400450500550

Left

mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(b)

150200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(c)

200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 500 1000 1500 2000 2500 3000Time (s)

(d)


Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)

Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)


















2119898


Variable MSE119899

trainingMSE

testing EMMSE



119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

Joint angle 119869 = 11119869119899 000237 4146 80721 No 120573

119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898

120573119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)

minus20

minus40

minus60

Actuator

ModelEnhanced model


(a)

0

100

200

300

400

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0

100

200

300

400

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0100200300400500600

Mus

cle p

ress

ure (

kPa)

minus1000 15 3 45 6 75 9 105 12 135 15

Time (s)



(d)



0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)



minus20

minus40

minus60

minus80

(a)

0100200300400500

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0100200300400500

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0 15 3 45 6 75 9 105 12 135 15Time (s)

0100200300400500600

Mus

cle p

ress

ure (

kPa)



minus100

(d)





4 Conclusion



0

100

200

300

400

500

600


Insta

nces

Errors

Zero errors

minus368

minus3386

minus3093

minus2799

minus2506

minus2212

minus1919

minus1625

minus1332

minus1038

minus07446

minus04511

minus01576

072

31

016

131

160

41

897

013

590

4295

(a)

0100200300400500600700800900

1000


Insta

nces

203

85

162

828

511

41

145

317

66

minus4169

minus3856

minus3544

minus3232

minus2919

minus2607

minus2295

minus1982

minus167

minus1358

minus1045

minus7332

minus4208

minus1085

Errors

Zero errors

(b)

0

200

400

600

800

1000

1200

1400

1600

Insta

nces

258

75

742

889

712

05

152

118

36

215

2

minus3843

minus3527

minus3212

minus2896

minus2581

minus2265

minus195

minus1634

minus1319

minus1003

minus6877

minus3722

minus05676

Errors

Zero errors


(c)


0

100

200

300

400

500

600

700

800

Insta

nces

051

363

427

634

19

255

121

715

08 18

209

123

82

267

429

65

minus2571

minus228

minus1988

minus1697

minus1406

minus1114

minus8228

minus5314

minus24

Errors

Zero errors

(d)

0

100

200

300

400

500

600

700

800

900

Insta

nces

308

694

610

81

146

818

54

224

126

27

minus4717

minus4331

minus3944

minus3558

minus3171

minus2784

minus2398

minus2011

minus1625

minus1238

minus8516

minus4651

minus07852

Errors

Zero errors


(e)


0

100

200

300

400

500

600

700

800

Insta

nces

003

394

046

650

899

133

2

minus6887

minus6454

minus6022

minus5589

minus5157

minus4724

minus4291

minus3859

minus3426

minus2994

minus2561

minus2129

minus1696

minus1264

minus08311

minus03986

Errors

Zero errors

(f)

Figure 14 Continued


0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327











pressure (119869 = 11119869119899)






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a)

(b)


1 3

2

1 3

2

EPAM

Valves

Compressor

Rotatingarm


V1V2V3V4

Pc

Q2

Q1

120573 120596 120576J

P1 V1

P2 V2

l1 1 a1 F1

l2 2 a2 F2

minus+










200018001600140012001000

800600400200

3 21

0


400kPa500kPa600kPa

l0 Δlmax

F1F2

F3

l (m)

F(N

)


0is initial muscle




120585 = (1

119898) [119865119864minus 119865119899119889( 120594 119875) minus 119865

119899119904(120594 119875)] (1)



119899119904





119865119899119904(119875 120594) =

119897

sum

119896=0

119899

sum

119898=0

119886119896119898119875119896120594119898 (2)

where 119886119896119898


0 5 10 15 20 25 30

12345670

5001000150020002500


Forc

e (N

)




minus9084

11988610

minus7963

11988601

6508

11988620

235

11988611

minus231

11988602

115

11988630

minus1883

11988621

minus07284

11988612

3715

11988603

minus2282

11988640

0062

11988631

0081

11988622

minus0145

11988613

minus0185

11988604

2088

11988650


11988641


11988632


11988623

199119890 minus 3

11988614


11988605




119865119899119889(119875 120594) = minus120574119875 120594 (3)



119864) as



11987501198810= 119875119881119898 (4)

119889

119889119905(11987501198810

119881119898

) = 1198750(0119881119898minus 1198981198810

1198812119898

) = 1198750

0

119881119898

minus 119875119898

119898

119881119898

(5)


0is volume of pres-



119898 119898 and

0 The last


0=

1198751119862radic

1198790

1198791


1 minus 120595)

2

if 11987521198751

gt 120595

1198751119862radic

1198790

1198791

if 11987521198751

le 120595

(6)






119881119898(120594) =

119898

sum

119896=0

119887119896120594119896 (7)



3=





119898(120594 120594) = 3119887

31205942120594 + 21198872120594 120594 + 119887

1120594 (8)



119889

119889119905(120597119871

120597 119902119894

) minus120597119871

120597119902119894

= 120591119894 (9)




119894is


119902 = [120573] 119902 = [ 120573] 120591 = [119879] 119870 =1

2119869 1205732 (10)




119889

119889119905(120597 ((12) 119869 120573

2)

120597 120573) minus

120597 ((12) 119869 1205732)

120597120573= 119879 (11)

119869 120573 = 119879 minus 119879119871 where 119879 = (119865

1minus 1198652) 119903 (12)





119869tot =119898

sum

119896=1

119869119896 (13)



119878 diameter 119863

119878 and mass 119872



119860 and mass 119872



119869119878=1

21198721198781198772

119878119869119860=1

41198721198601198772

119860+1

31198721198601198712

119860 (14)


119878is shaft mass






LS

DSMS

DA LA

MA

JS

JA

ML

dPAR



119878is



119860is armmass [kg] 119869


119860is armmoment



119899is calculated




119863119878

375 mm119871119878

285 mm119872119878

247 kg119863119860

1813 mm119871119860

272 mm119872119860

055 kg119889PAR 2011 mm119872119871

334 kg119869119899

0014 kgsdotm2

1198691

0155 kgsdotm2


119869PAR = 119869 +1198721198892

PAR (15)




119899

sum

119895=1

119863119894119895(119902) 119902119895+

119899

sum

119896=1

119899

sum

119898=1

119867119894119896119898

119902119896119902119898+ 119866119894= 119876119894

119867119894119895119896=

119899

sum

119895=1

119899

sum

119896=1

(120597119863119894119895

120597119902119896

minus1

2

120597119863119895119896

120597119902119894

)

(16)


matrix 119867119894119896119898




D (q) q +H (q q) + G (q) = Q (17)




D (q) q +H (q q) + F (q) + G (q) + 120591P = Q (18)



Model

PAMactuator

Neuralnetwork

yia

yim

120591P

P

e120591

V

Um




119898is








119875mdash


119875in





x (119896 + 1) = f (CWx (119896) + IWu (119896) + b(1))

119910 (119896) = LWx (119896) + 119887(2)(19)



119891 (119909) = tanh (119909) = 119890119909minus 119890minus119909

119890119909 + 119890minus119909 (20)


120591119875= LW119891 (CWx (119896) + IWu (119896) + b(1)) + 119887(2) + 119890

120591 (21)



120575119895(119896) = minus

120597119864tot (1198960 1198961)

120597119897119895 (119896)

(22)




0to 1198961 and 119897

119895(119896)


1and


120575119895(119896)

=

1198911015840(119897119895(119896)) 119890119895(119896) for 119896 = 119896

1

1198911015840(119897119895 (119896))

[

[

119890119895 (119896) + sum

119895isinI

119908119895119898120575119896

sdot (119896 + 1)]

]

for 1198960lt 119896 lt 119896

1

(23)




Δ119908119895119894= minus120578

120597119864tot (1198960 1198961)

120597119908119895119894

= 120578

1198961

sum

119896=1198960+1

120575119895 (119896) 119909119894 (119896 minus 1) (24)





minus1

J119879e where

J119879J asymp H J119879e = g(25)


NHL LHL

(a) (b)

Input

Context layer

IW LW

CW

x(k + 1) y(k + 1)u(k)

y

x1

x2

xn

u1u2

um

zminus1

zminus1

zminus1

b(1)n

b(1)2

b(1)1

b(2)1

sum

sum

sum

sum

zminus1I




e = 119890119894(120579) = 119905

119894minus 119910119894

119894 = 1 119899 (26)










MSE = 1

119899

119899

sum

119896=1

(119910119896minus 119910119896)2 (27)




MA =sum119898

119894=1119910119894

119898 (28)




0=



119899) and Figure 10 (119869 = 11119869




0 1 2 3 4 5 6 7 8 9 100

051

Time (s)

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

V1

V2

V3

V4


1 1198812 1198813 and 119881

4are valves)







1is left muscle


Variable MSE119869 = 119869119899

MSE119869 = 11119869

119899

120573 10024 1029171198651 76632 572861198652 45618 300371198751 22989 180751198752 51602 39199










0

10

20

30Jo

int a

ngle

(deg

)

ActuatorModel

0 15 3 45 6 75 9Time (s)

minus20

minus10

(a)

0 15 3 45 6 75 9100150200250300350400450500550600

Time (s)

Left

mus

cle fo

rce (

N)

ActuatorModel

(b)

200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 15 3 45 6 75 9Time (s)

(c)

150200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 15 3 45 6 75 9Time (s)

(d)




119899) where


119899and 119869 =







0 500 1000 1500 2000 2500 3000

05

1015202530

Time (s)

Join

t ang

le (d

eg)

ActuatorModel


(a)

150200250300350400450500550

Left

mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(b)

150200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(c)

200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 500 1000 1500 2000 2500 3000Time (s)

(d)


Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)

Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)


















2119898


Variable MSE119899

trainingMSE

testing EMMSE



119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

Joint angle 119869 = 11119869119899 000237 4146 80721 No 120573

119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898

120573119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)

minus20

minus40

minus60

Actuator

ModelEnhanced model


(a)

0

100

200

300

400

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0

100

200

300

400

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0100200300400500600

Mus

cle p

ress

ure (

kPa)

minus1000 15 3 45 6 75 9 105 12 135 15

Time (s)



(d)



0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)



minus20

minus40

minus60

minus80

(a)

0100200300400500

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0100200300400500

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0 15 3 45 6 75 9 105 12 135 15Time (s)

0100200300400500600

Mus

cle p

ress

ure (

kPa)



minus100

(d)





4 Conclusion



0

100

200

300

400

500

600


Insta

nces

Errors

Zero errors

minus368

minus3386

minus3093

minus2799

minus2506

minus2212

minus1919

minus1625

minus1332

minus1038

minus07446

minus04511

minus01576

072

31

016

131

160

41

897

013

590

4295

(a)

0100200300400500600700800900

1000


Insta

nces

203

85

162

828

511

41

145

317

66

minus4169

minus3856

minus3544

minus3232

minus2919

minus2607

minus2295

minus1982

minus167

minus1358

minus1045

minus7332

minus4208

minus1085

Errors

Zero errors

(b)

0

200

400

600

800

1000

1200

1400

1600

Insta

nces

258

75

742

889

712

05

152

118

36

215

2

minus3843

minus3527

minus3212

minus2896

minus2581

minus2265

minus195

minus1634

minus1319

minus1003

minus6877

minus3722

minus05676

Errors

Zero errors


(c)


0

100

200

300

400

500

600

700

800

Insta

nces

051

363

427

634

19

255

121

715

08 18

209

123

82

267

429

65

minus2571

minus228

minus1988

minus1697

minus1406

minus1114

minus8228

minus5314

minus24

Errors

Zero errors

(d)

0

100

200

300

400

500

600

700

800

900

Insta

nces

308

694

610

81

146

818

54

224

126

27

minus4717

minus4331

minus3944

minus3558

minus3171

minus2784

minus2398

minus2011

minus1625

minus1238

minus8516

minus4651

minus07852

Errors

Zero errors


(e)


0

100

200

300

400

500

600

700

800

Insta

nces

003

394

046

650

899

133

2

minus6887

minus6454

minus6022

minus5589

minus5157

minus4724

minus4291

minus3859

minus3426

minus2994

minus2561

minus2129

minus1696

minus1264

minus08311

minus03986

Errors

Zero errors

(f)

Figure 14 Continued


0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327











pressure (119869 = 11119869119899)






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










200018001600140012001000

800600400200

3 21

0


400kPa500kPa600kPa

l0 Δlmax

F1F2

F3

l (m)

F(N

)


0is initial muscle




120585 = (1

119898) [119865119864minus 119865119899119889( 120594 119875) minus 119865

119899119904(120594 119875)] (1)



119899119904





119865119899119904(119875 120594) =

119897

sum

119896=0

119899

sum

119898=0

119886119896119898119875119896120594119898 (2)

where 119886119896119898


0 5 10 15 20 25 30

12345670

5001000150020002500


Forc

e (N

)




minus9084

11988610

minus7963

11988601

6508

11988620

235

11988611

minus231

11988602

115

11988630

minus1883

11988621

minus07284

11988612

3715

11988603

minus2282

11988640

0062

11988631

0081

11988622

minus0145

11988613

minus0185

11988604

2088

11988650


11988641


11988632


11988623

199119890 minus 3

11988614


11988605




119865119899119889(119875 120594) = minus120574119875 120594 (3)



119864) as



11987501198810= 119875119881119898 (4)

119889

119889119905(11987501198810

119881119898

) = 1198750(0119881119898minus 1198981198810

1198812119898

) = 1198750

0

119881119898

minus 119875119898

119898

119881119898

(5)


0is volume of pres-



119898 119898 and

0 The last


0=

1198751119862radic

1198790

1198791


1 minus 120595)

2

if 11987521198751

gt 120595

1198751119862radic

1198790

1198791

if 11987521198751

le 120595

(6)






119881119898(120594) =

119898

sum

119896=0

119887119896120594119896 (7)



3=





119898(120594 120594) = 3119887

31205942120594 + 21198872120594 120594 + 119887

1120594 (8)



119889

119889119905(120597119871

120597 119902119894

) minus120597119871

120597119902119894

= 120591119894 (9)




119894is


119902 = [120573] 119902 = [ 120573] 120591 = [119879] 119870 =1

2119869 1205732 (10)




119889

119889119905(120597 ((12) 119869 120573

2)

120597 120573) minus

120597 ((12) 119869 1205732)

120597120573= 119879 (11)

119869 120573 = 119879 minus 119879119871 where 119879 = (119865

1minus 1198652) 119903 (12)





119869tot =119898

sum

119896=1

119869119896 (13)



119878 diameter 119863

119878 and mass 119872



119860 and mass 119872



119869119878=1

21198721198781198772

119878119869119860=1

41198721198601198772

119860+1

31198721198601198712

119860 (14)


119878is shaft mass






LS

DSMS

DA LA

MA

JS

JA

ML

dPAR



119878is



119860is armmass [kg] 119869


119860is armmoment



119899is calculated




119863119878

375 mm119871119878

285 mm119872119878

247 kg119863119860

1813 mm119871119860

272 mm119872119860

055 kg119889PAR 2011 mm119872119871

334 kg119869119899

0014 kgsdotm2

1198691

0155 kgsdotm2


119869PAR = 119869 +1198721198892

PAR (15)




119899

sum

119895=1

119863119894119895(119902) 119902119895+

119899

sum

119896=1

119899

sum

119898=1

119867119894119896119898

119902119896119902119898+ 119866119894= 119876119894

119867119894119895119896=

119899

sum

119895=1

119899

sum

119896=1

(120597119863119894119895

120597119902119896

minus1

2

120597119863119895119896

120597119902119894

)

(16)


matrix 119867119894119896119898




D (q) q +H (q q) + G (q) = Q (17)




D (q) q +H (q q) + F (q) + G (q) + 120591P = Q (18)



Model

PAMactuator

Neuralnetwork

yia

yim

120591P

P

e120591

V

Um




119898is








119875mdash


119875in





x (119896 + 1) = f (CWx (119896) + IWu (119896) + b(1))

119910 (119896) = LWx (119896) + 119887(2)(19)



119891 (119909) = tanh (119909) = 119890119909minus 119890minus119909

119890119909 + 119890minus119909 (20)


120591119875= LW119891 (CWx (119896) + IWu (119896) + b(1)) + 119887(2) + 119890

120591 (21)



120575119895(119896) = minus

120597119864tot (1198960 1198961)

120597119897119895 (119896)

(22)




0to 1198961 and 119897

119895(119896)


1and


120575119895(119896)

=

1198911015840(119897119895(119896)) 119890119895(119896) for 119896 = 119896

1

1198911015840(119897119895 (119896))

[

[

119890119895 (119896) + sum

119895isinI

119908119895119898120575119896

sdot (119896 + 1)]

]

for 1198960lt 119896 lt 119896

1

(23)




Δ119908119895119894= minus120578

120597119864tot (1198960 1198961)

120597119908119895119894

= 120578

1198961

sum

119896=1198960+1

120575119895 (119896) 119909119894 (119896 minus 1) (24)





minus1

J119879e where

J119879J asymp H J119879e = g(25)


NHL LHL

(a) (b)

Input

Context layer

IW LW

CW

x(k + 1) y(k + 1)u(k)

y

x1

x2

xn

u1u2

um

zminus1

zminus1

zminus1

b(1)n

b(1)2

b(1)1

b(2)1

sum

sum

sum

sum

zminus1I




e = 119890119894(120579) = 119905

119894minus 119910119894

119894 = 1 119899 (26)










MSE = 1

119899

119899

sum

119896=1

(119910119896minus 119910119896)2 (27)




MA =sum119898

119894=1119910119894

119898 (28)




0=



119899) and Figure 10 (119869 = 11119869




0 1 2 3 4 5 6 7 8 9 100

051

Time (s)

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

V1

V2

V3

V4


1 1198812 1198813 and 119881

4are valves)







1is left muscle


Variable MSE119869 = 119869119899

MSE119869 = 11119869

119899

120573 10024 1029171198651 76632 572861198652 45618 300371198751 22989 180751198752 51602 39199










0

10

20

30Jo

int a

ngle

(deg

)

ActuatorModel

0 15 3 45 6 75 9Time (s)

minus20

minus10

(a)

0 15 3 45 6 75 9100150200250300350400450500550600

Time (s)

Left

mus

cle fo

rce (

N)

ActuatorModel

(b)

200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 15 3 45 6 75 9Time (s)

(c)

150200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 15 3 45 6 75 9Time (s)

(d)




119899) where


119899and 119869 =







0 500 1000 1500 2000 2500 3000

05

1015202530

Time (s)

Join

t ang

le (d

eg)

ActuatorModel


(a)

150200250300350400450500550

Left

mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(b)

150200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(c)

200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 500 1000 1500 2000 2500 3000Time (s)

(d)


Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)

Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)


















2119898


Variable MSE119899

trainingMSE

testing EMMSE



119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

Joint angle 119869 = 11119869119899 000237 4146 80721 No 120573

119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898

120573119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)

minus20

minus40

minus60

Actuator

ModelEnhanced model


(a)

0

100

200

300

400

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0

100

200

300

400

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0100200300400500600

Mus

cle p

ress

ure (

kPa)

minus1000 15 3 45 6 75 9 105 12 135 15

Time (s)



(d)



0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)



minus20

minus40

minus60

minus80

(a)

0100200300400500

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0100200300400500

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0 15 3 45 6 75 9 105 12 135 15Time (s)

0100200300400500600

Mus

cle p

ress

ure (

kPa)



minus100

(d)





4 Conclusion



0

100

200

300

400

500

600


Insta

nces

Errors

Zero errors

minus368

minus3386

minus3093

minus2799

minus2506

minus2212

minus1919

minus1625

minus1332

minus1038

minus07446

minus04511

minus01576

072

31

016

131

160

41

897

013

590

4295

(a)

0100200300400500600700800900

1000


Insta

nces

203

85

162

828

511

41

145

317

66

minus4169

minus3856

minus3544

minus3232

minus2919

minus2607

minus2295

minus1982

minus167

minus1358

minus1045

minus7332

minus4208

minus1085

Errors

Zero errors

(b)

0

200

400

600

800

1000

1200

1400

1600

Insta

nces

258

75

742

889

712

05

152

118

36

215

2

minus3843

minus3527

minus3212

minus2896

minus2581

minus2265

minus195

minus1634

minus1319

minus1003

minus6877

minus3722

minus05676

Errors

Zero errors


(c)


0

100

200

300

400

500

600

700

800

Insta

nces

051

363

427

634

19

255

121

715

08 18

209

123

82

267

429

65

minus2571

minus228

minus1988

minus1697

minus1406

minus1114

minus8228

minus5314

minus24

Errors

Zero errors

(d)

0

100

200

300

400

500

600

700

800

900

Insta

nces

308

694

610

81

146

818

54

224

126

27

minus4717

minus4331

minus3944

minus3558

minus3171

minus2784

minus2398

minus2011

minus1625

minus1238

minus8516

minus4651

minus07852

Errors

Zero errors


(e)


0

100

200

300

400

500

600

700

800

Insta

nces

003

394

046

650

899

133

2

minus6887

minus6454

minus6022

minus5589

minus5157

minus4724

minus4291

minus3859

minus3426

minus2994

minus2561

minus2129

minus1696

minus1264

minus08311

minus03986

Errors

Zero errors

(f)

Figure 14 Continued


0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327











pressure (119869 = 11119869119899)






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119864) as



11987501198810= 119875119881119898 (4)

119889

119889119905(11987501198810

119881119898

) = 1198750(0119881119898minus 1198981198810

1198812119898

) = 1198750

0

119881119898

minus 119875119898

119898

119881119898

(5)


0is volume of pres-



119898 119898 and

0 The last


0=

1198751119862radic

1198790

1198791


1 minus 120595)

2

if 11987521198751

gt 120595

1198751119862radic

1198790

1198791

if 11987521198751

le 120595

(6)






119881119898(120594) =

119898

sum

119896=0

119887119896120594119896 (7)



3=





119898(120594 120594) = 3119887

31205942120594 + 21198872120594 120594 + 119887

1120594 (8)



119889

119889119905(120597119871

120597 119902119894

) minus120597119871

120597119902119894

= 120591119894 (9)




119894is


119902 = [120573] 119902 = [ 120573] 120591 = [119879] 119870 =1

2119869 1205732 (10)




119889

119889119905(120597 ((12) 119869 120573

2)

120597 120573) minus

120597 ((12) 119869 1205732)

120597120573= 119879 (11)

119869 120573 = 119879 minus 119879119871 where 119879 = (119865

1minus 1198652) 119903 (12)





119869tot =119898

sum

119896=1

119869119896 (13)



119878 diameter 119863

119878 and mass 119872



119860 and mass 119872



119869119878=1

21198721198781198772

119878119869119860=1

41198721198601198772

119860+1

31198721198601198712

119860 (14)


119878is shaft mass






LS

DSMS

DA LA

MA

JS

JA

ML

dPAR



119878is



119860is armmass [kg] 119869


119860is armmoment



119899is calculated




119863119878

375 mm119871119878

285 mm119872119878

247 kg119863119860

1813 mm119871119860

272 mm119872119860

055 kg119889PAR 2011 mm119872119871

334 kg119869119899

0014 kgsdotm2

1198691

0155 kgsdotm2


119869PAR = 119869 +1198721198892

PAR (15)




119899

sum

119895=1

119863119894119895(119902) 119902119895+

119899

sum

119896=1

119899

sum

119898=1

119867119894119896119898

119902119896119902119898+ 119866119894= 119876119894

119867119894119895119896=

119899

sum

119895=1

119899

sum

119896=1

(120597119863119894119895

120597119902119896

minus1

2

120597119863119895119896

120597119902119894

)

(16)


matrix 119867119894119896119898




D (q) q +H (q q) + G (q) = Q (17)




D (q) q +H (q q) + F (q) + G (q) + 120591P = Q (18)



Model

PAMactuator

Neuralnetwork

yia

yim

120591P

P

e120591

V

Um




119898is








119875mdash


119875in





x (119896 + 1) = f (CWx (119896) + IWu (119896) + b(1))

119910 (119896) = LWx (119896) + 119887(2)(19)



119891 (119909) = tanh (119909) = 119890119909minus 119890minus119909

119890119909 + 119890minus119909 (20)


120591119875= LW119891 (CWx (119896) + IWu (119896) + b(1)) + 119887(2) + 119890

120591 (21)



120575119895(119896) = minus

120597119864tot (1198960 1198961)

120597119897119895 (119896)

(22)




0to 1198961 and 119897

119895(119896)


1and


120575119895(119896)

=

1198911015840(119897119895(119896)) 119890119895(119896) for 119896 = 119896

1

1198911015840(119897119895 (119896))

[

[

119890119895 (119896) + sum

119895isinI

119908119895119898120575119896

sdot (119896 + 1)]

]

for 1198960lt 119896 lt 119896

1

(23)




Δ119908119895119894= minus120578

120597119864tot (1198960 1198961)

120597119908119895119894

= 120578

1198961

sum

119896=1198960+1

120575119895 (119896) 119909119894 (119896 minus 1) (24)





minus1

J119879e where

J119879J asymp H J119879e = g(25)


NHL LHL

(a) (b)

Input

Context layer

IW LW

CW

x(k + 1) y(k + 1)u(k)

y

x1

x2

xn

u1u2

um

zminus1

zminus1

zminus1

b(1)n

b(1)2

b(1)1

b(2)1

sum

sum

sum

sum

zminus1I




e = 119890119894(120579) = 119905

119894minus 119910119894

119894 = 1 119899 (26)










MSE = 1

119899

119899

sum

119896=1

(119910119896minus 119910119896)2 (27)




MA =sum119898

119894=1119910119894

119898 (28)




0=



119899) and Figure 10 (119869 = 11119869




0 1 2 3 4 5 6 7 8 9 100

051

Time (s)

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

V1

V2

V3

V4


1 1198812 1198813 and 119881

4are valves)







1is left muscle


Variable MSE119869 = 119869119899

MSE119869 = 11119869

119899

120573 10024 1029171198651 76632 572861198652 45618 300371198751 22989 180751198752 51602 39199










0

10

20

30Jo

int a

ngle

(deg

)

ActuatorModel

0 15 3 45 6 75 9Time (s)

minus20

minus10

(a)

0 15 3 45 6 75 9100150200250300350400450500550600

Time (s)

Left

mus

cle fo

rce (

N)

ActuatorModel

(b)

200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 15 3 45 6 75 9Time (s)

(c)

150200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 15 3 45 6 75 9Time (s)

(d)




119899) where


119899and 119869 =







0 500 1000 1500 2000 2500 3000

05

1015202530

Time (s)

Join

t ang

le (d

eg)

ActuatorModel


(a)

150200250300350400450500550

Left

mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(b)

150200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(c)

200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 500 1000 1500 2000 2500 3000Time (s)

(d)


Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)

Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)


















2119898


Variable MSE119899

trainingMSE

testing EMMSE



119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

Joint angle 119869 = 11119869119899 000237 4146 80721 No 120573

119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898

120573119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)

minus20

minus40

minus60

Actuator

ModelEnhanced model


(a)

0

100

200

300

400

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0

100

200

300

400

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0100200300400500600

Mus

cle p

ress

ure (

kPa)

minus1000 15 3 45 6 75 9 105 12 135 15

Time (s)



(d)



0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)



minus20

minus40

minus60

minus80

(a)

0100200300400500

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0100200300400500

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0 15 3 45 6 75 9 105 12 135 15Time (s)

0100200300400500600

Mus

cle p

ress

ure (

kPa)



minus100

(d)





4 Conclusion



0

100

200

300

400

500

600


Insta

nces

Errors

Zero errors

minus368

minus3386

minus3093

minus2799

minus2506

minus2212

minus1919

minus1625

minus1332

minus1038

minus07446

minus04511

minus01576

072

31

016

131

160

41

897

013

590

4295

(a)

0100200300400500600700800900

1000


Insta

nces

203

85

162

828

511

41

145

317

66

minus4169

minus3856

minus3544

minus3232

minus2919

minus2607

minus2295

minus1982

minus167

minus1358

minus1045

minus7332

minus4208

minus1085

Errors

Zero errors

(b)

0

200

400

600

800

1000

1200

1400

1600

Insta

nces

258

75

742

889

712

05

152

118

36

215

2

minus3843

minus3527

minus3212

minus2896

minus2581

minus2265

minus195

minus1634

minus1319

minus1003

minus6877

minus3722

minus05676

Errors

Zero errors


(c)


0

100

200

300

400

500

600

700

800

Insta

nces

051

363

427

634

19

255

121

715

08 18

209

123

82

267

429

65

minus2571

minus228

minus1988

minus1697

minus1406

minus1114

minus8228

minus5314

minus24

Errors

Zero errors

(d)

0

100

200

300

400

500

600

700

800

900

Insta

nces

308

694

610

81

146

818

54

224

126

27

minus4717

minus4331

minus3944

minus3558

minus3171

minus2784

minus2398

minus2011

minus1625

minus1238

minus8516

minus4651

minus07852

Errors

Zero errors


(e)


0

100

200

300

400

500

600

700

800

Insta

nces

003

394

046

650

899

133

2

minus6887

minus6454

minus6022

minus5589

minus5157

minus4724

minus4291

minus3859

minus3426

minus2994

minus2561

minus2129

minus1696

minus1264

minus08311

minus03986

Errors

Zero errors

(f)

Figure 14 Continued


0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327











pressure (119869 = 11119869119899)






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










LS

DSMS

DA LA

MA

JS

JA

ML

dPAR



119878is



119860is armmass [kg] 119869


119860is armmoment



119899is calculated




119863119878

375 mm119871119878

285 mm119872119878

247 kg119863119860

1813 mm119871119860

272 mm119872119860

055 kg119889PAR 2011 mm119872119871

334 kg119869119899

0014 kgsdotm2

1198691

0155 kgsdotm2


119869PAR = 119869 +1198721198892

PAR (15)




119899

sum

119895=1

119863119894119895(119902) 119902119895+

119899

sum

119896=1

119899

sum

119898=1

119867119894119896119898

119902119896119902119898+ 119866119894= 119876119894

119867119894119895119896=

119899

sum

119895=1

119899

sum

119896=1

(120597119863119894119895

120597119902119896

minus1

2

120597119863119895119896

120597119902119894

)

(16)


matrix 119867119894119896119898




D (q) q +H (q q) + G (q) = Q (17)




D (q) q +H (q q) + F (q) + G (q) + 120591P = Q (18)



Model

PAMactuator

Neuralnetwork

yia

yim

120591P

P

e120591

V

Um




119898is








119875mdash


119875in





x (119896 + 1) = f (CWx (119896) + IWu (119896) + b(1))

119910 (119896) = LWx (119896) + 119887(2)(19)



119891 (119909) = tanh (119909) = 119890119909minus 119890minus119909

119890119909 + 119890minus119909 (20)


120591119875= LW119891 (CWx (119896) + IWu (119896) + b(1)) + 119887(2) + 119890

120591 (21)



120575119895(119896) = minus

120597119864tot (1198960 1198961)

120597119897119895 (119896)

(22)




0to 1198961 and 119897

119895(119896)


1and


120575119895(119896)

=

1198911015840(119897119895(119896)) 119890119895(119896) for 119896 = 119896

1

1198911015840(119897119895 (119896))

[

[

119890119895 (119896) + sum

119895isinI

119908119895119898120575119896

sdot (119896 + 1)]

]

for 1198960lt 119896 lt 119896

1

(23)




Δ119908119895119894= minus120578

120597119864tot (1198960 1198961)

120597119908119895119894

= 120578

1198961

sum

119896=1198960+1

120575119895 (119896) 119909119894 (119896 minus 1) (24)





minus1

J119879e where

J119879J asymp H J119879e = g(25)


NHL LHL

(a) (b)

Input

Context layer

IW LW

CW

x(k + 1) y(k + 1)u(k)

y

x1

x2

xn

u1u2

um

zminus1

zminus1

zminus1

b(1)n

b(1)2

b(1)1

b(2)1

sum

sum

sum

sum

zminus1I




e = 119890119894(120579) = 119905

119894minus 119910119894

119894 = 1 119899 (26)










MSE = 1

119899

119899

sum

119896=1

(119910119896minus 119910119896)2 (27)




MA =sum119898

119894=1119910119894

119898 (28)




0=



119899) and Figure 10 (119869 = 11119869




0 1 2 3 4 5 6 7 8 9 100

051

Time (s)

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

V1

V2

V3

V4


1 1198812 1198813 and 119881

4are valves)







1is left muscle


Variable MSE119869 = 119869119899

MSE119869 = 11119869

119899

120573 10024 1029171198651 76632 572861198652 45618 300371198751 22989 180751198752 51602 39199










0

10

20

30Jo

int a

ngle

(deg

)

ActuatorModel

0 15 3 45 6 75 9Time (s)

minus20

minus10

(a)

0 15 3 45 6 75 9100150200250300350400450500550600

Time (s)

Left

mus

cle fo

rce (

N)

ActuatorModel

(b)

200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 15 3 45 6 75 9Time (s)

(c)

150200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 15 3 45 6 75 9Time (s)

(d)




119899) where


119899and 119869 =







0 500 1000 1500 2000 2500 3000

05

1015202530

Time (s)

Join

t ang

le (d

eg)

ActuatorModel


(a)

150200250300350400450500550

Left

mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(b)

150200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(c)

200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 500 1000 1500 2000 2500 3000Time (s)

(d)


Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)

Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)


















2119898


Variable MSE119899

trainingMSE

testing EMMSE



119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

Joint angle 119869 = 11119869119899 000237 4146 80721 No 120573

119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898

120573119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)

minus20

minus40

minus60

Actuator

ModelEnhanced model


(a)

0

100

200

300

400

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0

100

200

300

400

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0100200300400500600

Mus

cle p

ress

ure (

kPa)

minus1000 15 3 45 6 75 9 105 12 135 15

Time (s)



(d)



0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)



minus20

minus40

minus60

minus80

(a)

0100200300400500

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0100200300400500

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0 15 3 45 6 75 9 105 12 135 15Time (s)

0100200300400500600

Mus

cle p

ress

ure (

kPa)



minus100

(d)





4 Conclusion



0

100

200

300

400

500

600


Insta

nces

Errors

Zero errors

minus368

minus3386

minus3093

minus2799

minus2506

minus2212

minus1919

minus1625

minus1332

minus1038

minus07446

minus04511

minus01576

072

31

016

131

160

41

897

013

590

4295

(a)

0100200300400500600700800900

1000


Insta

nces

203

85

162

828

511

41

145

317

66

minus4169

minus3856

minus3544

minus3232

minus2919

minus2607

minus2295

minus1982

minus167

minus1358

minus1045

minus7332

minus4208

minus1085

Errors

Zero errors

(b)

0

200

400

600

800

1000

1200

1400

1600

Insta

nces

258

75

742

889

712

05

152

118

36

215

2

minus3843

minus3527

minus3212

minus2896

minus2581

minus2265

minus195

minus1634

minus1319

minus1003

minus6877

minus3722

minus05676

Errors

Zero errors


(c)


0

100

200

300

400

500

600

700

800

Insta

nces

051

363

427

634

19

255

121

715

08 18

209

123

82

267

429

65

minus2571

minus228

minus1988

minus1697

minus1406

minus1114

minus8228

minus5314

minus24

Errors

Zero errors

(d)

0

100

200

300

400

500

600

700

800

900

Insta

nces

308

694

610

81

146

818

54

224

126

27

minus4717

minus4331

minus3944

minus3558

minus3171

minus2784

minus2398

minus2011

minus1625

minus1238

minus8516

minus4651

minus07852

Errors

Zero errors


(e)


0

100

200

300

400

500

600

700

800

Insta

nces

003

394

046

650

899

133

2

minus6887

minus6454

minus6022

minus5589

minus5157

minus4724

minus4291

minus3859

minus3426

minus2994

minus2561

minus2129

minus1696

minus1264

minus08311

minus03986

Errors

Zero errors

(f)

Figure 14 Continued


0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327











pressure (119869 = 11119869119899)






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Model

PAMactuator

Neuralnetwork

yia

yim

120591P

P

e120591

V

Um




119898is








119875mdash


119875in





x (119896 + 1) = f (CWx (119896) + IWu (119896) + b(1))

119910 (119896) = LWx (119896) + 119887(2)(19)



119891 (119909) = tanh (119909) = 119890119909minus 119890minus119909

119890119909 + 119890minus119909 (20)


120591119875= LW119891 (CWx (119896) + IWu (119896) + b(1)) + 119887(2) + 119890

120591 (21)



120575119895(119896) = minus

120597119864tot (1198960 1198961)

120597119897119895 (119896)

(22)




0to 1198961 and 119897

119895(119896)


1and


120575119895(119896)

=

1198911015840(119897119895(119896)) 119890119895(119896) for 119896 = 119896

1

1198911015840(119897119895 (119896))

[

[

119890119895 (119896) + sum

119895isinI

119908119895119898120575119896

sdot (119896 + 1)]

]

for 1198960lt 119896 lt 119896

1

(23)




Δ119908119895119894= minus120578

120597119864tot (1198960 1198961)

120597119908119895119894

= 120578

1198961

sum

119896=1198960+1

120575119895 (119896) 119909119894 (119896 minus 1) (24)





minus1

J119879e where

J119879J asymp H J119879e = g(25)


NHL LHL

(a) (b)

Input

Context layer

IW LW

CW

x(k + 1) y(k + 1)u(k)

y

x1

x2

xn

u1u2

um

zminus1

zminus1

zminus1

b(1)n

b(1)2

b(1)1

b(2)1

sum

sum

sum

sum

zminus1I




e = 119890119894(120579) = 119905

119894minus 119910119894

119894 = 1 119899 (26)










MSE = 1

119899

119899

sum

119896=1

(119910119896minus 119910119896)2 (27)




MA =sum119898

119894=1119910119894

119898 (28)




0=



119899) and Figure 10 (119869 = 11119869




0 1 2 3 4 5 6 7 8 9 100

051

Time (s)

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

V1

V2

V3

V4


1 1198812 1198813 and 119881

4are valves)







1is left muscle


Variable MSE119869 = 119869119899

MSE119869 = 11119869

119899

120573 10024 1029171198651 76632 572861198652 45618 300371198751 22989 180751198752 51602 39199










0

10

20

30Jo

int a

ngle

(deg

)

ActuatorModel

0 15 3 45 6 75 9Time (s)

minus20

minus10

(a)

0 15 3 45 6 75 9100150200250300350400450500550600

Time (s)

Left

mus

cle fo

rce (

N)

ActuatorModel

(b)

200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 15 3 45 6 75 9Time (s)

(c)

150200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 15 3 45 6 75 9Time (s)

(d)




119899) where


119899and 119869 =







0 500 1000 1500 2000 2500 3000

05

1015202530

Time (s)

Join

t ang

le (d

eg)

ActuatorModel


(a)

150200250300350400450500550

Left

mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(b)

150200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(c)

200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 500 1000 1500 2000 2500 3000Time (s)

(d)


Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)

Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)


















2119898


Variable MSE119899

trainingMSE

testing EMMSE



119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

Joint angle 119869 = 11119869119899 000237 4146 80721 No 120573

119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898

120573119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)

minus20

minus40

minus60

Actuator

ModelEnhanced model


(a)

0

100

200

300

400

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0

100

200

300

400

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0100200300400500600

Mus

cle p

ress

ure (

kPa)

minus1000 15 3 45 6 75 9 105 12 135 15

Time (s)



(d)



0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)



minus20

minus40

minus60

minus80

(a)

0100200300400500

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0100200300400500

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0 15 3 45 6 75 9 105 12 135 15Time (s)

0100200300400500600

Mus

cle p

ress

ure (

kPa)



minus100

(d)





4 Conclusion



0

100

200

300

400

500

600


Insta

nces

Errors

Zero errors

minus368

minus3386

minus3093

minus2799

minus2506

minus2212

minus1919

minus1625

minus1332

minus1038

minus07446

minus04511

minus01576

072

31

016

131

160

41

897

013

590

4295

(a)

0100200300400500600700800900

1000


Insta

nces

203

85

162

828

511

41

145

317

66

minus4169

minus3856

minus3544

minus3232

minus2919

minus2607

minus2295

minus1982

minus167

minus1358

minus1045

minus7332

minus4208

minus1085

Errors

Zero errors

(b)

0

200

400

600

800

1000

1200

1400

1600

Insta

nces

258

75

742

889

712

05

152

118

36

215

2

minus3843

minus3527

minus3212

minus2896

minus2581

minus2265

minus195

minus1634

minus1319

minus1003

minus6877

minus3722

minus05676

Errors

Zero errors


(c)


0

100

200

300

400

500

600

700

800

Insta

nces

051

363

427

634

19

255

121

715

08 18

209

123

82

267

429

65

minus2571

minus228

minus1988

minus1697

minus1406

minus1114

minus8228

minus5314

minus24

Errors

Zero errors

(d)

0

100

200

300

400

500

600

700

800

900

Insta

nces

308

694

610

81

146

818

54

224

126

27

minus4717

minus4331

minus3944

minus3558

minus3171

minus2784

minus2398

minus2011

minus1625

minus1238

minus8516

minus4651

minus07852

Errors

Zero errors


(e)


0

100

200

300

400

500

600

700

800

Insta

nces

003

394

046

650

899

133

2

minus6887

minus6454

minus6022

minus5589

minus5157

minus4724

minus4291

minus3859

minus3426

minus2994

minus2561

minus2129

minus1696

minus1264

minus08311

minus03986

Errors

Zero errors

(f)

Figure 14 Continued


0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327











pressure (119869 = 11119869119899)






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










NHL LHL

(a) (b)

Input

Context layer

IW LW

CW

x(k + 1) y(k + 1)u(k)

y

x1

x2

xn

u1u2

um

zminus1

zminus1

zminus1

b(1)n

b(1)2

b(1)1

b(2)1

sum

sum

sum

sum

zminus1I




e = 119890119894(120579) = 119905

119894minus 119910119894

119894 = 1 119899 (26)










MSE = 1

119899

119899

sum

119896=1

(119910119896minus 119910119896)2 (27)




MA =sum119898

119894=1119910119894

119898 (28)




0=



119899) and Figure 10 (119869 = 11119869




0 1 2 3 4 5 6 7 8 9 100

051

Time (s)

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

V1

V2

V3

V4


1 1198812 1198813 and 119881

4are valves)







1is left muscle


Variable MSE119869 = 119869119899

MSE119869 = 11119869

119899

120573 10024 1029171198651 76632 572861198652 45618 300371198751 22989 180751198752 51602 39199










0

10

20

30Jo

int a

ngle

(deg

)

ActuatorModel

0 15 3 45 6 75 9Time (s)

minus20

minus10

(a)

0 15 3 45 6 75 9100150200250300350400450500550600

Time (s)

Left

mus

cle fo

rce (

N)

ActuatorModel

(b)

200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 15 3 45 6 75 9Time (s)

(c)

150200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 15 3 45 6 75 9Time (s)

(d)




119899) where


119899and 119869 =







0 500 1000 1500 2000 2500 3000

05

1015202530

Time (s)

Join

t ang

le (d

eg)

ActuatorModel


(a)

150200250300350400450500550

Left

mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(b)

150200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(c)

200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 500 1000 1500 2000 2500 3000Time (s)

(d)


Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)

Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)


















2119898


Variable MSE119899

trainingMSE

testing EMMSE



119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

Joint angle 119869 = 11119869119899 000237 4146 80721 No 120573

119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898

120573119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)

minus20

minus40

minus60

Actuator

ModelEnhanced model


(a)

0

100

200

300

400

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0

100

200

300

400

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0100200300400500600

Mus

cle p

ress

ure (

kPa)

minus1000 15 3 45 6 75 9 105 12 135 15

Time (s)



(d)



0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)



minus20

minus40

minus60

minus80

(a)

0100200300400500

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0100200300400500

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0 15 3 45 6 75 9 105 12 135 15Time (s)

0100200300400500600

Mus

cle p

ress

ure (

kPa)



minus100

(d)





4 Conclusion



0

100

200

300

400

500

600


Insta

nces

Errors

Zero errors

minus368

minus3386

minus3093

minus2799

minus2506

minus2212

minus1919

minus1625

minus1332

minus1038

minus07446

minus04511

minus01576

072

31

016

131

160

41

897

013

590

4295

(a)

0100200300400500600700800900

1000


Insta

nces

203

85

162

828

511

41

145

317

66

minus4169

minus3856

minus3544

minus3232

minus2919

minus2607

minus2295

minus1982

minus167

minus1358

minus1045

minus7332

minus4208

minus1085

Errors

Zero errors

(b)

0

200

400

600

800

1000

1200

1400

1600

Insta

nces

258

75

742

889

712

05

152

118

36

215

2

minus3843

minus3527

minus3212

minus2896

minus2581

minus2265

minus195

minus1634

minus1319

minus1003

minus6877

minus3722

minus05676

Errors

Zero errors


(c)


0

100

200

300

400

500

600

700

800

Insta

nces

051

363

427

634

19

255

121

715

08 18

209

123

82

267

429

65

minus2571

minus228

minus1988

minus1697

minus1406

minus1114

minus8228

minus5314

minus24

Errors

Zero errors

(d)

0

100

200

300

400

500

600

700

800

900

Insta

nces

308

694

610

81

146

818

54

224

126

27

minus4717

minus4331

minus3944

minus3558

minus3171

minus2784

minus2398

minus2011

minus1625

minus1238

minus8516

minus4651

minus07852

Errors

Zero errors


(e)


0

100

200

300

400

500

600

700

800

Insta

nces

003

394

046

650

899

133

2

minus6887

minus6454

minus6022

minus5589

minus5157

minus4724

minus4291

minus3859

minus3426

minus2994

minus2561

minus2129

minus1696

minus1264

minus08311

minus03986

Errors

Zero errors

(f)

Figure 14 Continued


0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327











pressure (119869 = 11119869119899)






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 100

051

Time (s)

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

005

1

0 1 2 3 4 5 6 7 8 9 10

V1

V2

V3

V4


1 1198812 1198813 and 119881

4are valves)







1is left muscle


Variable MSE119869 = 119869119899

MSE119869 = 11119869

119899

120573 10024 1029171198651 76632 572861198652 45618 300371198751 22989 180751198752 51602 39199










0

10

20

30Jo

int a

ngle

(deg

)

ActuatorModel

0 15 3 45 6 75 9Time (s)

minus20

minus10

(a)

0 15 3 45 6 75 9100150200250300350400450500550600

Time (s)

Left

mus

cle fo

rce (

N)

ActuatorModel

(b)

200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 15 3 45 6 75 9Time (s)

(c)

150200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 15 3 45 6 75 9Time (s)

(d)




119899) where


119899and 119869 =







0 500 1000 1500 2000 2500 3000

05

1015202530

Time (s)

Join

t ang

le (d

eg)

ActuatorModel


(a)

150200250300350400450500550

Left

mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(b)

150200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(c)

200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 500 1000 1500 2000 2500 3000Time (s)

(d)


Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)

Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)


















2119898


Variable MSE119899

trainingMSE

testing EMMSE



119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

Joint angle 119869 = 11119869119899 000237 4146 80721 No 120573

119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898

120573119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)

minus20

minus40

minus60

Actuator

ModelEnhanced model


(a)

0

100

200

300

400

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0

100

200

300

400

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0100200300400500600

Mus

cle p

ress

ure (

kPa)

minus1000 15 3 45 6 75 9 105 12 135 15

Time (s)



(d)



0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)



minus20

minus40

minus60

minus80

(a)

0100200300400500

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0100200300400500

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0 15 3 45 6 75 9 105 12 135 15Time (s)

0100200300400500600

Mus

cle p

ress

ure (

kPa)



minus100

(d)





4 Conclusion



0

100

200

300

400

500

600


Insta

nces

Errors

Zero errors

minus368

minus3386

minus3093

minus2799

minus2506

minus2212

minus1919

minus1625

minus1332

minus1038

minus07446

minus04511

minus01576

072

31

016

131

160

41

897

013

590

4295

(a)

0100200300400500600700800900

1000


Insta

nces

203

85

162

828

511

41

145

317

66

minus4169

minus3856

minus3544

minus3232

minus2919

minus2607

minus2295

minus1982

minus167

minus1358

minus1045

minus7332

minus4208

minus1085

Errors

Zero errors

(b)

0

200

400

600

800

1000

1200

1400

1600

Insta

nces

258

75

742

889

712

05

152

118

36

215

2

minus3843

minus3527

minus3212

minus2896

minus2581

minus2265

minus195

minus1634

minus1319

minus1003

minus6877

minus3722

minus05676

Errors

Zero errors


(c)


0

100

200

300

400

500

600

700

800

Insta

nces

051

363

427

634

19

255

121

715

08 18

209

123

82

267

429

65

minus2571

minus228

minus1988

minus1697

minus1406

minus1114

minus8228

minus5314

minus24

Errors

Zero errors

(d)

0

100

200

300

400

500

600

700

800

900

Insta

nces

308

694

610

81

146

818

54

224

126

27

minus4717

minus4331

minus3944

minus3558

minus3171

minus2784

minus2398

minus2011

minus1625

minus1238

minus8516

minus4651

minus07852

Errors

Zero errors


(e)


0

100

200

300

400

500

600

700

800

Insta

nces

003

394

046

650

899

133

2

minus6887

minus6454

minus6022

minus5589

minus5157

minus4724

minus4291

minus3859

minus3426

minus2994

minus2561

minus2129

minus1696

minus1264

minus08311

minus03986

Errors

Zero errors

(f)

Figure 14 Continued


0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327











pressure (119869 = 11119869119899)






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

10

20

30Jo

int a

ngle

(deg

)

ActuatorModel

0 15 3 45 6 75 9Time (s)

minus20

minus10

(a)

0 15 3 45 6 75 9100150200250300350400450500550600

Time (s)

Left

mus

cle fo

rce (

N)

ActuatorModel

(b)

200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 15 3 45 6 75 9Time (s)

(c)

150200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 15 3 45 6 75 9Time (s)

(d)




119899) where


119899and 119869 =







0 500 1000 1500 2000 2500 3000

05

1015202530

Time (s)

Join

t ang

le (d

eg)

ActuatorModel


(a)

150200250300350400450500550

Left

mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(b)

150200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(c)

200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 500 1000 1500 2000 2500 3000Time (s)

(d)


Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)

Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)


















2119898


Variable MSE119899

trainingMSE

testing EMMSE



119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

Joint angle 119869 = 11119869119899 000237 4146 80721 No 120573

119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898

120573119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)

minus20

minus40

minus60

Actuator

ModelEnhanced model


(a)

0

100

200

300

400

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0

100

200

300

400

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0100200300400500600

Mus

cle p

ress

ure (

kPa)

minus1000 15 3 45 6 75 9 105 12 135 15

Time (s)



(d)



0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)



minus20

minus40

minus60

minus80

(a)

0100200300400500

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0100200300400500

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0 15 3 45 6 75 9 105 12 135 15Time (s)

0100200300400500600

Mus

cle p

ress

ure (

kPa)



minus100

(d)





4 Conclusion



0

100

200

300

400

500

600


Insta

nces

Errors

Zero errors

minus368

minus3386

minus3093

minus2799

minus2506

minus2212

minus1919

minus1625

minus1332

minus1038

minus07446

minus04511

minus01576

072

31

016

131

160

41

897

013

590

4295

(a)

0100200300400500600700800900

1000


Insta

nces

203

85

162

828

511

41

145

317

66

minus4169

minus3856

minus3544

minus3232

minus2919

minus2607

minus2295

minus1982

minus167

minus1358

minus1045

minus7332

minus4208

minus1085

Errors

Zero errors

(b)

0

200

400

600

800

1000

1200

1400

1600

Insta

nces

258

75

742

889

712

05

152

118

36

215

2

minus3843

minus3527

minus3212

minus2896

minus2581

minus2265

minus195

minus1634

minus1319

minus1003

minus6877

minus3722

minus05676

Errors

Zero errors


(c)


0

100

200

300

400

500

600

700

800

Insta

nces

051

363

427

634

19

255

121

715

08 18

209

123

82

267

429

65

minus2571

minus228

minus1988

minus1697

minus1406

minus1114

minus8228

minus5314

minus24

Errors

Zero errors

(d)

0

100

200

300

400

500

600

700

800

900

Insta

nces

308

694

610

81

146

818

54

224

126

27

minus4717

minus4331

minus3944

minus3558

minus3171

minus2784

minus2398

minus2011

minus1625

minus1238

minus8516

minus4651

minus07852

Errors

Zero errors


(e)


0

100

200

300

400

500

600

700

800

Insta

nces

003

394

046

650

899

133

2

minus6887

minus6454

minus6022

minus5589

minus5157

minus4724

minus4291

minus3859

minus3426

minus2994

minus2561

minus2129

minus1696

minus1264

minus08311

minus03986

Errors

Zero errors

(f)

Figure 14 Continued


0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327











pressure (119869 = 11119869119899)






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 500 1000 1500 2000 2500 3000

05

1015202530

Time (s)

Join

t ang

le (d

eg)

ActuatorModel


(a)

150200250300350400450500550

Left

mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(b)

150200250300350400450500550

Righ

t mus

cle fo

rce (

N)

ActuatorModel

0 500 1000 1500 2000 2500 3000Time (s)

(c)

200250300350400450500550600

Mus

cle p

ress

ure (

kPa)



0 500 1000 1500 2000 2500 3000Time (s)

(d)


Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)

Time (s)0 30 60 90 120 150 180 210 240

0

05

1

15

minus05

Am

plitu

de (mdash

)


















2119898


Variable MSE119899

trainingMSE

testing EMMSE



119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

Joint angle 119869 = 11119869119899 000237 4146 80721 No 120573

119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898

120573119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)

minus20

minus40

minus60

Actuator

ModelEnhanced model


(a)

0

100

200

300

400

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0

100

200

300

400

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0100200300400500600

Mus

cle p

ress

ure (

kPa)

minus1000 15 3 45 6 75 9 105 12 135 15

Time (s)



(d)



0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)



minus20

minus40

minus60

minus80

(a)

0100200300400500

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0100200300400500

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0 15 3 45 6 75 9 105 12 135 15Time (s)

0100200300400500600

Mus

cle p

ress

ure (

kPa)



minus100

(d)





4 Conclusion



0

100

200

300

400

500

600


Insta

nces

Errors

Zero errors

minus368

minus3386

minus3093

minus2799

minus2506

minus2212

minus1919

minus1625

minus1332

minus1038

minus07446

minus04511

minus01576

072

31

016

131

160

41

897

013

590

4295

(a)

0100200300400500600700800900

1000


Insta

nces

203

85

162

828

511

41

145

317

66

minus4169

minus3856

minus3544

minus3232

minus2919

minus2607

minus2295

minus1982

minus167

minus1358

minus1045

minus7332

minus4208

minus1085

Errors

Zero errors

(b)

0

200

400

600

800

1000

1200

1400

1600

Insta

nces

258

75

742

889

712

05

152

118

36

215

2

minus3843

minus3527

minus3212

minus2896

minus2581

minus2265

minus195

minus1634

minus1319

minus1003

minus6877

minus3722

minus05676

Errors

Zero errors


(c)


0

100

200

300

400

500

600

700

800

Insta

nces

051

363

427

634

19

255

121

715

08 18

209

123

82

267

429

65

minus2571

minus228

minus1988

minus1697

minus1406

minus1114

minus8228

minus5314

minus24

Errors

Zero errors

(d)

0

100

200

300

400

500

600

700

800

900

Insta

nces

308

694

610

81

146

818

54

224

126

27

minus4717

minus4331

minus3944

minus3558

minus3171

minus2784

minus2398

minus2011

minus1625

minus1238

minus8516

minus4651

minus07852

Errors

Zero errors


(e)


0

100

200

300

400

500

600

700

800

Insta

nces

003

394

046

650

899

133

2

minus6887

minus6454

minus6022

minus5589

minus5157

minus4724

minus4291

minus3859

minus3426

minus2994

minus2561

minus2129

minus1696

minus1264

minus08311

minus03986

Errors

Zero errors

(f)

Figure 14 Continued


0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327











pressure (119869 = 11119869119899)






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















2119898


Variable MSE119899

trainingMSE

testing EMMSE



119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

Joint angle 119869 = 11119869119899 000237 4146 80721 No 120573

119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898


1119898 1198652119898 120596119898 120576119898

120573119898


1119898 1198752119898 120596119898 120576119898


1119898 1198752119898 120596119898 120576119898

0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)

minus20

minus40

minus60

Actuator

ModelEnhanced model


(a)

0

100

200

300

400

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0

100

200

300

400

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0100200300400500600

Mus

cle p

ress

ure (

kPa)

minus1000 15 3 45 6 75 9 105 12 135 15

Time (s)



(d)



0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)



minus20

minus40

minus60

minus80

(a)

0100200300400500

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0100200300400500

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0 15 3 45 6 75 9 105 12 135 15Time (s)

0100200300400500600

Mus

cle p

ress

ure (

kPa)



minus100

(d)





4 Conclusion



0

100

200

300

400

500

600


Insta

nces

Errors

Zero errors

minus368

minus3386

minus3093

minus2799

minus2506

minus2212

minus1919

minus1625

minus1332

minus1038

minus07446

minus04511

minus01576

072

31

016

131

160

41

897

013

590

4295

(a)

0100200300400500600700800900

1000


Insta

nces

203

85

162

828

511

41

145

317

66

minus4169

minus3856

minus3544

minus3232

minus2919

minus2607

minus2295

minus1982

minus167

minus1358

minus1045

minus7332

minus4208

minus1085

Errors

Zero errors

(b)

0

200

400

600

800

1000

1200

1400

1600

Insta

nces

258

75

742

889

712

05

152

118

36

215

2

minus3843

minus3527

minus3212

minus2896

minus2581

minus2265

minus195

minus1634

minus1319

minus1003

minus6877

minus3722

minus05676

Errors

Zero errors


(c)


0

100

200

300

400

500

600

700

800

Insta

nces

051

363

427

634

19

255

121

715

08 18

209

123

82

267

429

65

minus2571

minus228

minus1988

minus1697

minus1406

minus1114

minus8228

minus5314

minus24

Errors

Zero errors

(d)

0

100

200

300

400

500

600

700

800

900

Insta

nces

308

694

610

81

146

818

54

224

126

27

minus4717

minus4331

minus3944

minus3558

minus3171

minus2784

minus2398

minus2011

minus1625

minus1238

minus8516

minus4651

minus07852

Errors

Zero errors


(e)


0

100

200

300

400

500

600

700

800

Insta

nces

003

394

046

650

899

133

2

minus6887

minus6454

minus6022

minus5589

minus5157

minus4724

minus4291

minus3859

minus3426

minus2994

minus2561

minus2129

minus1696

minus1264

minus08311

minus03986

Errors

Zero errors

(f)

Figure 14 Continued


0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327











pressure (119869 = 11119869119899)






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 15 3 45 6 75 9 105 12 135 15

020406080

Time (s)

Join

t ang

le (d

eg)



minus20

minus40

minus60

minus80

(a)

0100200300400500

Left

mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(b)

0100200300400500

Righ

t mus

cle fo

rce (

N)



minus100

minus2000 15 3 45 6 75 9 105 12 135 15

Time (s)

(c)

0 15 3 45 6 75 9 105 12 135 15Time (s)

0100200300400500600

Mus

cle p

ress

ure (

kPa)



minus100

(d)





4 Conclusion



0

100

200

300

400

500

600


Insta

nces

Errors

Zero errors

minus368

minus3386

minus3093

minus2799

minus2506

minus2212

minus1919

minus1625

minus1332

minus1038

minus07446

minus04511

minus01576

072

31

016

131

160

41

897

013

590

4295

(a)

0100200300400500600700800900

1000


Insta

nces

203

85

162

828

511

41

145

317

66

minus4169

minus3856

minus3544

minus3232

minus2919

minus2607

minus2295

minus1982

minus167

minus1358

minus1045

minus7332

minus4208

minus1085

Errors

Zero errors

(b)

0

200

400

600

800

1000

1200

1400

1600

Insta

nces

258

75

742

889

712

05

152

118

36

215

2

minus3843

minus3527

minus3212

minus2896

minus2581

minus2265

minus195

minus1634

minus1319

minus1003

minus6877

minus3722

minus05676

Errors

Zero errors


(c)


0

100

200

300

400

500

600

700

800

Insta

nces

051

363

427

634

19

255

121

715

08 18

209

123

82

267

429

65

minus2571

minus228

minus1988

minus1697

minus1406

minus1114

minus8228

minus5314

minus24

Errors

Zero errors

(d)

0

100

200

300

400

500

600

700

800

900

Insta

nces

308

694

610

81

146

818

54

224

126

27

minus4717

minus4331

minus3944

minus3558

minus3171

minus2784

minus2398

minus2011

minus1625

minus1238

minus8516

minus4651

minus07852

Errors

Zero errors


(e)


0

100

200

300

400

500

600

700

800

Insta

nces

003

394

046

650

899

133

2

minus6887

minus6454

minus6022

minus5589

minus5157

minus4724

minus4291

minus3859

minus3426

minus2994

minus2561

minus2129

minus1696

minus1264

minus08311

minus03986

Errors

Zero errors

(f)

Figure 14 Continued


0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327











pressure (119869 = 11119869119899)






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

100

200

300

400

500

600


Insta

nces

Errors

Zero errors

minus368

minus3386

minus3093

minus2799

minus2506

minus2212

minus1919

minus1625

minus1332

minus1038

minus07446

minus04511

minus01576

072

31

016

131

160

41

897

013

590

4295

(a)

0100200300400500600700800900

1000


Insta

nces

203

85

162

828

511

41

145

317

66

minus4169

minus3856

minus3544

minus3232

minus2919

minus2607

minus2295

minus1982

minus167

minus1358

minus1045

minus7332

minus4208

minus1085

Errors

Zero errors

(b)

0

200

400

600

800

1000

1200

1400

1600

Insta

nces

258

75

742

889

712

05

152

118

36

215

2

minus3843

minus3527

minus3212

minus2896

minus2581

minus2265

minus195

minus1634

minus1319

minus1003

minus6877

minus3722

minus05676

Errors

Zero errors


(c)


0

100

200

300

400

500

600

700

800

Insta

nces

051

363

427

634

19

255

121

715

08 18

209

123

82

267

429

65

minus2571

minus228

minus1988

minus1697

minus1406

minus1114

minus8228

minus5314

minus24

Errors

Zero errors

(d)

0

100

200

300

400

500

600

700

800

900

Insta

nces

308

694

610

81

146

818

54

224

126

27

minus4717

minus4331

minus3944

minus3558

minus3171

minus2784

minus2398

minus2011

minus1625

minus1238

minus8516

minus4651

minus07852

Errors

Zero errors


(e)


0

100

200

300

400

500

600

700

800

Insta

nces

003

394

046

650

899

133

2

minus6887

minus6454

minus6022

minus5589

minus5157

minus4724

minus4291

minus3859

minus3426

minus2994

minus2561

minus2129

minus1696

minus1264

minus08311

minus03986

Errors

Zero errors

(f)

Figure 14 Continued


0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327











pressure (119869 = 11119869119899)






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

100

200

300

400

500

600

700


Insta

nces

098

895

697

104

115

11

198

224

53

292

433

95

386

543

36

minus4609

minus4138

minus3668

minus3197

minus2726

minus2255

minus1784

minus1314

minus8427

minus3719

Errors

Zero errors

(g)

0

200

400

600

800

1000

1200


Insta

nces

053

98

239

159

423

64

313

439

04

467

4

minus9956

minus9186

minus8416

minus7646

minus6876

minus6106

minus5336

minus4566

minus3796

minus3026

minus2256

minus1486

minus7161

Errors

Zero errors

(h)

0100200300400500600700800900

1000

Insta

nces

Errors

165

44

646

763

710

63

136

216

61

196

225

925

59

285

831

57

345

637

55


minus1929

minus1629

minus133

minus1031

minus732

minus4329

minus1337

Zero errors

(i)


0100200300400500600700800900

1000In

stanc

es

Errors

273

678

710

84

149

189

623

01

270

731

13

351

839

24

Zero errors

minus3784

minus3378

minus2972

minus2567

minus2161

minus1755

minus135

minus944

minus5384

minus1327











pressure (119869 = 11119869119899)






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Enhanced Dynamic Model of Pneumatic ...downloads.hindawi.com/journals/aaa/2015/906126.pdfdynamic model of pneumatic arti cial muscle with applica-tion to PAM-based