Research Article Inverse Modeling of Human Knee Joint...

Research ArticleInverse Modeling of Human Knee Joint Based onGeometry and Vision Systems for Exoskeleton Applications

Eduardo Pintildea-Martiacutenez and Ernesto Rodriguez-Leal

Graduate School of Science and Engineering Tecnologico de Monterrey Avenida E Garza Sada 2501 Sur64849 Monterrey NL Mexico

Correspondence should be addressed to Ernesto Rodriguez-Leal ernestorodriguezitesmmx

Received 17 February 2015 Accepted 2 June 2015

Academic Editor Yang Tang

Copyright copy 2015 E Pina-Martınez and E Rodriguez-LealThis is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Current trends in Robotics aim to close the gap that separates technology and humans bringing novel robotic devices in order toimprove human performance Although robotic exoskeletons represent a breakthrough in mobility enhancement there are designchallenges related to the forces exerted to the usersrsquo joints that result in severe injuries This occurs due to the fact that most of thecurrent developments consider the joints as noninvariant rotational axesThis paper proposes the use of commercial vision systemsin order to perform biomimetic joint design for robotic exoskeletons This work proposes a kinematic model based on irregularshaped cams as the joint mechanism that emulates the bone-to-bone joints in the human body The paper follows a geometricapproach for determining the location of the instantaneous center of rotation in order to design the cam contours Furthermorethe use of a commercial vision system is proposed as the main measurement tool due to its noninvasive feature and for allowingsubjects undermeasurement tomove freelyThe application of thismethod resulted in relevant information about the displacementsof the instantaneous center of rotation at the human knee joint

1 Introduction

An exoskeleton is commonly known in the literature asan external structure mechanism with joints and links cor-responding to those of the human body [1] Exoskeletonsare usually classified into two main groups (i) passivewhere the force and torques are external to the device and(ii) active where the device actuators exert the forces thatdrive the device and user Exoskeletons have applications inphysiotherapy as haptic or assistive devices [1]

Several exoskeletons have been developed in recent yearsfor example Aguirre-Ollinger et al [2] introduced a station-ary one degree of freedom (DoF) lower limb exoskeletonfor assisting knee flexion and extension exercises Kinnairdand Ferris [3] proposed a medial gastrocnemius roboticankle exoskeleton that is actuated with a pneumatic cylinderLopez et al [4] developed a lower limb robotic exoskeletonwith two DoF actuating the knee and the ankle joint TheVanderbilt powered orthosis [5] which is an active lower

limb exoskeleton with two DoF for each leg has been usedas a platform for other research projects including controlledassisted locomotion [6] and stair climbing for paraplegicusers [7] Bortole et al [8] developed a robotic exoskeletonfor poststroke patients which is used for the overgroundgait rehabilitation at the critical months after the incidentvan Ham et al [9] developed a mechanically adjustablecompliance and controllable equilibrium position actuator(MACCEPA) to actively actuate the knee joint by performingspecific rehabilitation movements

In addition to the abovementioned examples there areseveral commercial robotic exoskeleton devices in the mar-ket For example the Swiss company Hocoma [10] developedthe Lokomat which is a gait training robotic exoskeleton thatis extensively used on treadmill rehabilitation for patientswith diverse lower limb disabling conditions Hondarsquos Walk-ing Assist device [11] is a portable exoskeleton consisting ofa waist frame that connects with two thigh frames whereeach provides one DoF by using brushless DC motors

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 145734 14 pageshttpdxdoiorg1011552015145734

2 Mathematical Problems in Engineering

The Japanese company Cyberdyne is worldwide known forthe development of the HAL (acronym for hybrid assistivelimb) exoskeleton [12] which is used for either task supportrehabilitation or welfare applications The HAL robot hasbeen extensively studied for example the synchronizationcontrol that is based on the position change of the centerof ground reaction force (CoGRF) by Tsukahara et al [12]which later added a speed estimator to this control model[13] or the work presented by Hassan et al [14] that includedthe application of HAL control that is based on combiningboth lower and upper limb pose signals Furthermore theIndego exoskeleton is another good example of commercialdevices [15] that are based on the Vanderbilt exoskeleton[5] and is a powered lower limb orthosis for ground gaittrainingTheEkso fromEkso Bionics [16] has been developedby the Berkeley Robotics amp Human Engineering Laboratory[17] as an evolution of the exoskeleton robots presented byKazerooni [18] Ekso is presented as a wearable bionic suitthat enables lower limb disabled subjects to stand and walkovergroundThis device is intended for medically superviseduse providing support to lower limb paralyses patients andretraining them in the gait exercise ReWalk Robotics hasproduced the first approved FDA exoskeleton that similar toother devices (eg Indego and Ekso Bionics) includes fouractuatedDoF (hips and knees joints) and is programmedwithnormalized gait patterns which provides proper musculartraining to its users [19] Finally the REX from the RexBionics Group [20] is currently the unique device thatprovides full body weight support in equilibrium without theuse of canes

Most of the current advances of robotic exoskeletons havebeen performed by considering the joints as noninvariantrotational axis This design practice however can lead tofurther injury to the user since the human joints are complexsystems composed by bones that are driven by tendons thuspresenting motions whose rotational axes do not remainin the same location For this issue a common practiceof designers is to use Biomimetics [21] (ie the search forinspiration in nature in order to find the solution to agiven problem) for the development of exoskeletons Forinstance Mizoguchi et al [22] designed and developed amusculoskeletal humanoid robot based on the actual mus-culoskeletal arrangement on the human hips Zhu et al[23] developed a lower extremity exoskeleton design with15 DoF selecting each degree of freedom by analyzing thehuman lower limbrsquos joints as spherical joints Although theyaccepted this joints implementation as nonideal to emulatehuman joints their use was neglected due to the highdifficulty for its implementationMiranda et al [24] designedand implemented a robotic upper limb exoskeleton (elbow)with bioinspired actuators Furthermore Zhu et al [25]proposed a biomimetic knee exoskeleton focusing on theenergetic aspects of the segmented foot design in whichthey embedded a compliant joint in order to upgrade humanwalking of assistance exoskeletons

It is important to note that the majority of novel designson lower limb robotic exoskeletons have focused on theknee joint which is usually the actuated joint Other designsfocus on the actuation of the hip and ankle but in almost

L

A

O

B

w

B

A

IC

Figure 1 Instantaneous center of rotation of a mechanism

all cases those joints are mechanically designed as simplerotational axis This paper conducts a detailed study of thekinematics of the knee joint proposing the use of commercialvision systems in order to perform biomimetic joint designfor robotic exoskeletons The main contribution of thiswork is the proposition of an equivalent kinematic modelthat is based on irregular shaped cams that emulates thebone-to-bone joints in the human body whose contour isobtained by the identification of the instantaneous centerof rotation (ICR) Furthermore the use of a commercialvision system is proposed as the main measurement toolresulting in relevant information about ICR displacementsat the human knee joint This work can be applied as anapproach for the ergonomic design of exoskeleton devicesThe remainder of the paper is organized as follows Section 2presents the kinematicmodel that is based on the geometricaldetermination of the ICR of two bodies in contact Section 3presents the experimental setup that has been prepared inorder to use a commercial vision system for tracking theleg motion which includes the prototype development anddata processing Section 4 presents a detailed explanationabout the algorithms that are used to determine the ICRof the knee joint using the resulting data from the visionsystem Section 5 presents the experimental results of thiswork Finally Section 6 presents important conclusions andsuggestions of future work

2 Geometrical Model of the Knee JointBased on ICR

According to the kinematics theory of rigid bodies andconsidering planar motion the velocity of every point ina rigid body can be expressed as if this body spins on aplane about a given axis that is perpendicular to the plane[26] Furthermore this rotational axis intersects those bodiesplane on its ICR

Figure 1 illustrates the ICR or IC (instantaneous center)over planar motion of rigid bodies Note that the vectorsk119860and k

119861 which represent velocities over specific points in

that body (points 119860 and 119861 on a bar 119871) are instantaneouslyequivalent to the velocity of points 119860 and 119861 spinning arounda fixed point (ie IC) where a line intersects each point of the

Mathematical Problems in Engineering 3

Femur

Tibia

Static cam

Mobile cam

Fixed point

IC

120572

Q

120573

r

R

120574

O

(a)

IC

120572

Q

120573

O

l(120574)

(b)

Figure 2 (a) Representation of the irregular shaped bone endings as cams in contact (b) Proposed kinematic equivalent model of the kneejoint

body and the IC while at the same time being perpendicularto the velocity vectors

As discussed before in this paper most robotic exoskele-tons are mechanically modeled as single rotational jointsHowever the majority of human joints have more complexkinematic models that imply rotation and even translationmovements over different planes The knee joint is knownto have 1DoF but due to its muscles and tendon drivenrotation over a bone-to-bone contact complex rotations andtranslations are expected to happen differently to the singleplanar rotation that is commonly applied to exoskeletons

Figure 2(a) illustrates a sagittal plane view of the humanknee joint This paper focuses on the fact that both tibia andfemur maintain a close-to-tangential contact (with its tissueinterface) where the knee rotation is produced Furthermoreconsidering the femur as the reference static body the tibiarotates about the femur ending contour while the contactpoint (denoted as IC) changes its position This particularmotion in the sagittal plane could be represented as a camsystem in which one cam remains static and the other rotatesaround the static cam contour Furthermore Figure 2(b)illustrates the proposed equivalent kinematic model of theabovementioned knee joint which is represented as tworotational joints that are connected to a prismatic joint as aserial open chain

This irregular shaped characteristic emulates a cam sys-tem behavior in which the distance between the cam axisand the contact point varies depending on the camrsquos rotationangle Consider that the ICRwill lie on the line that intersects119874 and 119876 (the reference points in the femur and tibia resp)while 120572 and 120573 represent the angles between the groundedlink and the 2nd link and between the 2nd and 3rd linkrespectively

119897 (120574) = 119877 (120574) + 119903 (1)

where 119897 represents the variable distance between cam axes 119877is the static cam contact radius at a specific 120574 value 119903 is theplanetary cam contact radius and 120574 = 120572 + 120573 The radius ratio119866 is calculated as follows

119866 (120574) =119877 (120574)

119903

(2)

and 120573 can be further expressed as

120573 = 119866 (120574) 120572 (3)

In a practical way it is necessary to have informationabout the velocity of the rigid body in order to be able tofind the ICR Nevertheless for obtaining the velocity it isnecessary to have information about the position betweentwo instants Figure 3 illustrates the determination of the ICRto the system shown in Figure 2 where 119860 and 119861 representpoints of the rigid body at instants 119894 = 1 and 119894 = 2 whileICR12 represents the rigid bodyrsquos IC between those instants

In order to complete this model it is necessary to havespecific values of 119866 and 119897 for every 120574 This would be possibleby selecting a fixed point over the cam on the femurrsquos endingand by giving a value of 120574 for every point over the irregularcontour (see Figure 2(a)) Hence 119877 will be given by thedistance between the fixed point and the point IC while 119866will be given by 120574

The next section defines the experimental setup that willbe used in this paper in order to obtain the abovementioneddata for the calculation of the ICR of the joint

3 Experimental Setup Using theVICON Vision System

This section presents the experimental procedure followedin order to develop apply and adjust a biomimetic designmethod for obtaining a mechanic model of a knee joint forrobotic exoskeletons


Bisect A1A2

BisectB1B2

B1

B2

ICR12

A1

A2

Figure 3 Geometric approach to the ICR applied to the kneejoint circular gears model Points 119860 and 119861 (illustrating rigid bodyrsquosparticles) are taken at two consecutive instants (119894 = 1 dotted lineand 119894 = 2 solid line) during movement process

This work uses the VICON vision system which is amotion capture systembased onmarker trackingThis systemconsists in a set of cameras that sense infrared light emittedby the camera and reflected on retro reflective sphericalmarkers that are positioned over the tracked moving bodyThis vision system uses homography based self-location onits set of cameras to create a static workspace in which amarker could be located Based on triangulation position theVICON system is able to locate a marker at every observedframe by at least two camerasThrough the NEXUS softwareVICON users are able to create complex models of markerconfigurations establish relationships amongst them andlabel markers in order to provide individual specific data ofposition velocity and acceleration for a given time

Figure 4 illustrates the experimental setup used in thispaper First motion data is obtained from a subject withthe objective to fit the kinematic model previously proposedHowever this model states the need of having both astatic reference and a moving body In order to fulfill thisrequirement the femoral area (thigh) is selected to be therelative static reference body by applying a markers framethat will be later transformed into a global reference frameFurthermore the tibia area (calf) is selected as the movingbody where a second markers frame will be applied Datarecords will be obtained and exported from everymarker thatismounted on the subject using theVICON system to be laterprocessed with the use of external software (LabVIEW)

Since the goal of this work is to obtain proper datato be applied on exoskeleton joints construction a reverseengineering method is used in which the rigid interfaces actas exoskeleton links (one for the thigh and the other for thecalf) and the free space over the knee acts as the black boxthat needs to be modeled Hence markers fixtures are usedas rigid interfaces in order to reach two main goals (i) toact as solid mechanical links of an exoskeleton and (ii) toproperly place markers in a useful arrangement in order to

Global reference frame

Solid body inmovement

Data record

[p9984000 p

9984001

p998400n]

Figure 4 Overall data record process applied with the VICONmotion capture system

ease the subsequent data processing The thigh markers platemust be designed in order to create a global reference frameat its markersrsquo position over a common plane Moreover thecalf markers plate should be parallel to the thigh plane andprovide the mobile markers that define the ICR

Figures 5(a) and 5(b) show the computer assisted design(CAD) and the 3D printed parts for the thigh and calfmarkers respectively For the thigh markers plate it isnecessary to have at least three markers in order to create afull reference frame however two redundant marker spacesare included in order to select the best set out of three at anygiven frame positioning For the lower limb markers plateat least two markers are required since the ICR geometricapproach is based on lines made for every mobile markerand the lines intersection but in order to obtain comparativedata at least three markers should be used Note that a totalof sevenmarker spaces are included in order to select the bestset of markers to be applied at the determination of the ICRMarkers labeled from T2 to T5 are specially selected due tothe common space between them and their arc pattern andmarker T2 is also selected as the calf angle marker MarkersT1 T6 and T7 are selected in order to proof their efficiencyat pseudorandom positions

For this experiment the VICON recordings data areprocessed with a low pass digital filter with a cutoff frequencyof 2Hz considering that the maximum angular frequency tobe reached by the subject at leg swinging will be 1Hz whilerecording

The next section presents the algorithms that have beenused in order to determine the axis motion of the knee jointusing the VICON vision system

4 Knee Joint Axis Motion Tracking

The inversemodeling of the knee joint includes the collectionof the vision system data followed by the geometrical deter-mination of the ICR by considering the system proposed in


F1

F2

F3

F5

F4

(a)

T1

T2

T3

T5

T6

T4

T7

(b)

Figure 5 CAD and 3D printed prototype of (a) thigh markers plate and (b) calf markers plate

Section 2 For this purpose the final ICR detection algorithmis divided into the following principal computing processes

(1) Point-Slope Bisectors Determination This processobtains an array of line elements (point and slope) tobe applied as bisectors for every marker and motionframe

(2) Bisectors Set Visualization Using the array created in1 lines are projected from their origin (middle point)to visualize intersections between them

(3) ICR Detection and Tracking Intersections areobtained by matching the bisector lines equations

The abovementioned computing processes are fullydetailed in the following sections

41 Point-Slope Bisectors Determination The application ofthe ICR geometric approach implies that every point that isbeing analyzed lies on the same common planeThus the firststep to be followed is to project the position of every point to asingle plane Since the data is normalized all the thigh fixturepoints are already lying on 119883-119885 plane and the calf fixturepoints are in a common plane and nearly parallel to the thigh

plane Consider that every point is projected to119883-119885 plane byeliminating its 119910-axis position component namely

119863119894=

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

[

1198751198651119879

1198751198655119879

1198751198791119879

1198751198797119879

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

]119894

119875119872=[[

[

119909

119910

119911

]]

]

(4)

where119863119894is thematrix containing a data set of point positions

at the frame 119894 Note that those point positions are values fromthe 119875119872vector which has position values at every frame for a

specific marker119872 (F1 to F5 and T1 to T7) Furthermore for


the normalized data and considering the 119910 component to beclose to zero projecting to the119883-119885 plane results in

119875119872=[[

[

119909

0119911

]]

]

997888rarr 119875119872= [

119909

119911]

(5)

The bisectors line elements (point and slope) are acquiredfor everymarker position and lie on a commonplane119863 beingthe total array of data obtained from the data arrangementprocess (12 times 3 times 119899 3D array) 119894 denotes the frame numberthat takes values from 1 to 119899 (ie the number of frames fora specific recorded session) and taking [119875

119872]119894as the position

of marker119872 over plane 119883-119885 at the motion frame 119894 the newdata set of bisectors for every marker 119861 is obtained from

119861119895=

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

[

1198871198651

1198871198655

1198871198791

1198871198797

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

]119895

119887119872= [119887119901

119879119898]

(6)

where 119861119895is the array representing a set of bisector line

elements (119887119872

where 119872 denotes the marker) at the frame 119895This frame is obtained from every pair of consecutive motionframes 119894 and 119894 minus 1 in order to apply the geometric approach tothe ICR position Thus

119898 = minus119909119894minus 119909119894minus1

119911119894minus 119911119894minus1

119887119901 = [

119901119909

119901119911

]

(7)

where 119898 is the bisector slope created from points [119875119872]119894and

[119875119872]119894minus1119909 and 119911 are components and 119887119901 is the position vector

of the middle distance point between [119875119872]119894and [119875

119872]119894minus1 119901119909

being its position along the 119909-axis and 119901119911the position along

the 119911-axis Note that 119898 will be the negative reciprocal fromthe slope created from the position change

Due to the fact that for every 119861119895set of bisectors elements

it is necessary to have 119863119894and 119863

119894minus1set of markers position

a complete 3D data array of bisectors for every marker andat every motion frame (called 119861) is obtained with a size of12 times 3 times (119899 minus 1)

In order to have angular position data of the calf that isrelative to the thigh at every frame additional line elements119887LL (calf or lower leg parallel line (LL)) are indexed after 119887

1198797

from (3) for everymotion frame Note that at the point where

T3

Lower leg

Parallel line

120579

120593

T1

T2

T3

T5

T6

T4

T7

Figure 6 Definition of 119887LL line elements Angle values 120579 and 120593(selected as example) remain constant for every calf (lower leg)position

the slope lays the reference point value 119887119901 equals the position1198751198792

in (5) at the current frame and 119898 is set to be parallel tothe tibia by design Considering a line that is parallel to thesubject tibia and crossing the 119875

1198792marker position will create

constant angles relative to any line created between 1198751198792

andany other marker at the calf fixture (see Figure 6)

Adding the last 119887LL elements to the resulting 3D array 119861results in the following representation for 119861

119895

119861119895=

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

[

1198871198651

1198871198655

1198871198791

1198871198797

119887LL

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

]119895

(8)

42 Bisectors Set Visualization Once the bisectors line ele-ments are obtained it is useful to have a visual of the mostimportant segments of these lines in order to observe theirbehavior and the possible instant location of the knee jointaxis

This previous visualization can show possible recordingerrors that were too small to be seen at the VICON recordeddata visualization

The position changes are used to obtain movementdirections (slopes) and are expected to be as minimum aspossible (in order to maintain the ICR detection error valueclose to zero) and hence minimal errors could be reflectedas major direction errors when obtaining and projectingperpendicular lines from those points to an expected ICR


position Previous bisectors visualization helps to identify thisrecording error avoiding misleading results

In order to obtain significant line segments these areprojected from their reference point 119887119901 in (7) and to theregion where the ICR is expected to be foundThis projectionis done at a useful range of distances and looking to avoidchart saturation

Line plotting is done by getting two points over the chartspace The first point will be equal to the 119887119901 vector given forevery marker while the second point is to be found using thepoint and slope line equation where

119910 = 119898 (119909minus1199091) + 1199101 (9)

A minimum distance from point to point equal to400mm is used depending on the slope value and for a setof bisector line elements 119887

119872taken from a full motion frame

set 119861119895 it is possible to find the second point by

119909 = 119901119909plusmn 400 (10)

Substituting (10) in (9) results in

119910 = 119898 (119909minus119901119909) + 119901119910 (11)

Note that the 119909 value is determined where 119909 and 119901119909define

a 400mm region that encloses the expected ICR thusdepending on this 119909 will be higher or lower than 119901

119909by 400

Also if the difference between 119910 and 119901119910takes a higher value

than 400 this second point is given by

119910 = 119901119910plusmn 400 (12)

where

119909 =

119910 minus 119901119910

119898+119901119909

(13)

and 119910 is determined in a similar manner compared to 119909 in(10) by making the expected ICR an enclosing region Thisalgorithm is repeated for every 119887

119872set of bisector line elements

from the 119861119895array in (8) and at every motion frame Figure 7

shows the resulting plot of bisector lines for a frame Also line119887LL is plotted in order to have a visualization of the leg anglerelative to the thigh (horizontal line 119910 = 0) It is importantto highlight that due to the nonfully controlled position ofthe T2 marker and the expected ICR displacement over themotion process the line whose elements are equal to 119887LL is notnecessarily expected to intersect the other lines at the sameregion

43 ICR Detection and Tracking Thefinal step to be followedfor the proper allocation of the ICR is presented in thissection At this moment four bisector lines are given due tothe markers discrimination step which brings a total of sixpossible intersections Considering low amplitude noises afinal ICR will be given by averaging those six intersectionposition values

In order to apply an intersection algorithm it is necessaryto make line equations equal between them to find a first

3500

3000

2500

2000

1500

1000

500

00

minus500

minus500

minus1000

minus1000

minus1500

minus2000

Y

X

00

100

0

500

150

0

200

0

250

0

300

0

350

0

400

0

450

0

T1

T2

T3

T4

T5

T6

T7

LL

Figure 7 Bisectors visualization plot on a single motion frame LLdotted line represents the calf angular position

axis value and then apply this value to any of those equationsagain to obtain the following Furthermore line equations areconstructed with their corresponding line elements 119887

119872from

(8) using

119910 = 119898 (119909minus119901119909) + 119901119910 (14)

Similar to the point and slope line equation the generalequation is

119910 = 119898 (119909) + 119888 (15)

119888 being the value of119910when119909 = 0 (119910-intersection) and solvingfor 119901119909and 119901

119910results in

119888 = 119901119910minus119898 (119901

119909) = 119888119872(119887119872) (16)

where119898119901119909 and119901

119910are the elements from 119887

119872 which is the set

of bisector line elements of marker119872 and is part of the fullset of bisectors elements 119861

119895that corresponds to the motion

frame 119895 Then the representation of an intersection betweentwo lines from (15) where variables 119909 and119910 are equal for bothlines is

1198981198721 (1199091198721) + 1198881198721 (1198871198721) = 1198981198722 (1199091198722) + 1198881198722 (1198871198722) (17)

where1198721 and1198722 are two different markers from the samefixture and at the samemotion frame and 119909

1198721is equal to 119909

1198722

Furthermore solving (17) for 119909 the final equation for the firstaxis value (119909) is

119909ICR = minus1198981198722 minus 1198981198721

1198881198722 (1198871198722) minus 1198881198721 (1198871198721)

(18)


and finally substituting 119909ICR in any other of the two lineequations from (15) in order to obtain the second axis value(119910) gives

119910ICR = 1198981198721 (119909ICR) + 1198881198721 (1198871198721)

= 1198981198722 (119909ICR) + 1198881198722 (1198871198722)

(19)

ICR position values as mentioned before must beobtained for every intersection (six intersections for this caseof four bisector lines) and then averaged into a single positionvector ICR

119895for every frame 119895 given by

ICR119895= [119883ICR 119884ICR]

119879 (20)

where 119883ICR and 119884ICR are the components of the averagedICR position for every marker at 119909- and 119910-axes respectivelyobtained at the same frame 119895

Also in order to have functional design information itis necessary to obtain the knee angle for every ICR locationThis way and from the same motion frame bisectors dataset 119861119895 the knee angle is obtained by using the line elements

from 119887LL and projecting a line in order to have position valuedifferences at the same line for both 119909- and 119910-axes Solvingfrom (8) these values are obtained by projecting the linecreated with 119887LL elements to the 119909-axis (119910 = 0) with

1199011199090 = minus

119888LL119898LL

(21)

while 119888LL is the 119888 component from the 119887LL line elements in(9) 119898LL is the slope of this same line and 119901

1199090 is the 119909 valueat the 119909-axis intersection Then use trigonometric functionsin order to obtain the knee angle with

Opp = 119901119910minus 0

Adj = 119901119909minus1199011199090

(22)

where Opp and Adj correspond to the 119910 and 119909 dimensionvalues respectively of the line that intersects these twopoints resulting in a 120574 angle value obtained from

120574 = tanminus1 (OppAdj) (23)

Once these values are obtained for every frame from 2to 119899 (the first frame is eliminated from the point and slopedetection) a new array of final ICR data is constructed with

119865CIR =

[[[[[[[[[

[

1198911198892

1198911198893

119891119889119899minus1

119891119889119899

]]]]]]]]]

]

119891119889119895= [ICR

119895

119879 120574119895]

(24)

where 119865CIR represents the final array of ICR data that will beapplied to the joint design while 119891119889

119895contains these values

obtained from (20) and (23) for a specific frame 119895

5 Experimental Results

The experimental design developed for this applicationresulted in a reliable tool to model the human leg behavioras two moving rigid bodies The use of an exoskeleton likemarker fixtures also facilitated the experimental applicationat the recording data process due to their mounting eas-iness

The measurement tool VICON that was used to capturemotion data over a 3D space from a subject of interest showedgood performance under correct calibration and environ-ment Performance indexes were obtained from the squarederrors of distances between markers by comparing valuesfrom design and readings from VICON for every recordedframe From the first recording iteration VICON showedsignificant noises that reached squared error value peaks thatwere up to 40mm2 producing almost useless data at themoment of developing the geometric approach Howeverminimum changes in VICON environment and recordingparameters showed major improvements in recording resultswhere the squared error values remained constantly close to5mm2 (see Figure 8)

The use of normalization showed high importance in themodel fitting process In order to obtain relative movementsof the calf about the thigh the process of normalization wasa highly reliable tool Figure 9 shows the resulting normaliza-tion of a session recording at different frames of the processthat enables the establishment of the global reference frameof the thigh fixture and the subsequent relative movement ofthe calf

Also this process showed that is preferable to make thewhole recording process in the samequadrant of theVICONrsquos3D workspace in order to simplify normalization computingprocess and the correct transformation angles selection

The markers at the thigh fixture showed an acceptablestationary body behavior while other markers move aroundthem as expected After normalization low amplitude highfrequency noises were detected along the visualization pro-cess The filtering process was implemented in order toeliminate the effect of these noises over the computation ofbisectors and performance index time series were plotted forevery recorded session (see Figure 10) resulting in importantreduction of the squared error peaks of up to 50 whencompared with the nonfiltered sessions

Figure 10 shows the performance index time series thatwere obtained before and after applying the low pass filter(2Hz cutoff frequency) The performance indexes were ameasurement of the VICON recording efficiency for specificsessions As long as the error does not increase considerablythe VICON recording is considered to be efficient andreliable for this work Also if new error peaks appear at theperformance index time series of a filtered data session thenthe filter would be considered to have a negative impact onthe data recording

As seen in Figure 10 a steady state error occurs evenbefore the filtering process is performed This is expectedto happen due to the use of 3D printing technology tomanufacture the markers frames However as long as thissteady state error remains constant the markers position


100

90

80

70

60

50

40

30

20

10

0

Am

plitu

de

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Time

(a)

100

90

80

70

60

50

40

30

20

10

0

Am

plitu

de

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Time

(b)

Figure 8 Performance indexes from the first (a) and second (b) recording iteration

Table 1 Second recording iteration A total of 9 sessions wererecorded

Session Sampling frequency (fps) Swing frequency1 200 075Hz2 200 075Hz3 200 075Hz4 200 1Hz5 200 1Hz6 200 1Hz7 300 1HzldquoGaitrdquo 200 mdashldquoFull flexionrdquo 200 mdash

given by the VICON can be considered to behave as rigidbodies which is the recording experiment objective

Also from the first set of experiments many variationsover the sampling frequencywere applied in order to establisha proper frequency range Moreover the subject underrecording was required to perform some typical motionswithout using any periodic control and using the full VICONworkspace volume available at the laboratory (close to 6m3)The resulting performance indexes for this first iteration werenot reliable given the high error peaks This resulted as wellin a second recording iteration where VICON cameras werereallocated to create a smaller workspace (close to 4m3) andexperimental parameters were controlled as shown in Table 1

A clear delimited intersection was observed during thebisectors visualization process for every recorded sessionresulting in an effective error localization tool In additionto this it has been identified that the bisectors showed anerrant behavior close to the configurations where the kneewas at the full extension and flexion points namely when theangular velocity is close to zero and the displacement is aboutto change direction Figure 11 shows an example of a framewhere the movement process lies on this region

Furthermore note that from the complete set of 12mark-ers used at the VICON recording process only 7 were usedfor the complete process As previously discussed markersF1 F2 and F5 were selected out of the five thighsrsquo markers(F1 to F5) to allocate and reorient their common plane tothe global reference frame Moreover the markers selectionprocess developed for the calf rsquosmarkers fixture resulted in thefact that the best set of markers to be considered rigid bodyrsquosparticles were markers T2 to T5

Figures 12 and 13 show the well-delimited trajectoriesthat occurred in different experimental sessions presentingrepetitive patterns that were observed before the applicationof the complete geometric approach process towards thedetermination of the ICR

Although the markers fixtures were not mounted on theexact location on the subjectrsquos leg for the different recordedsessions the results indicate that the ICR trajectories werecommonly placed on the same region relative to the thighframe This confirmed that the area that was under obser-vation corresponded to the knee joint area that consistentlypresented similar magnitudes for every session confirmingthat the ICR of the knee joint has important displacements inthis area

The resulting contours for every session were plottedshowing a normal ICR displacement Nevertheless somesessions did not present defined delimited trajectories in spiteof having similar angle values and movements in similarregions Thus these particular inequalities are attributed torecording errors for those specific sessions

Furthermore Figures 12 and 13 show the plotted contoursthat resulted from the experimental work where a total of 12swing movements (from flexion to extension or vice versa)are displayed for a common range of 120574 values from 75∘ to 125∘(plotted movement of the leg swing)

After obtaining the contours with an embedded value of120574 for any point on the curve and considering that the rangesof values are similar it is possible to develop a mechanicaldesign by placing a fixed point followed by the determinationof every value from the proposed kinematic model


Zero (00 00 00)

110000

86000

62000

38000

14000

Z

Y

X

minus60000minus36000

minus1200012000

3600060000

minus80000

minus56000

minus32000

minus8000

16000

40000

(a)

(00 00 00)

(00 00 00)

Z

Y

60000

36000

12000

minus12000

6000

X

minus60000minus3

minus3

6000minus12000

1200036000

60000minus60000

minus36000

minus12000

12000

36000

60000

Cursor 1 (00 00 00)

(b)

Figure 9 LabVIEW 3D scatter graph used to visualize preprocessed data (a) Original VICON recorded data cursor on the global zero (b)Normalized data cursor on the marker F1 which is over the global zero


100

90

80

70

60

50

40

30

20

10

0

Am

plitu

de

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time

(a)

100

90

80

70

60

50

40

30

20

10

0

Am

plitu

de

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time

(b)

Figure 10 Performance index (a) Original data before filtering (b) Filtered data

35003000250020001500100050000

minus500

minus1000

minus1500

minus2000

minus2500

minus3500

minus3000

Y

T1

T2

T3

T4

T5

T6

T7

LL

minus1000 00 1000 2000 3000 4000 5000 6000X

XY graph

(a)

35003000250020001500100050000

minus500

minus1000

minus1500

minus2000

minus2500

minus3500

minus3000

Y

T1

T2

T3

T4

T5

T6

T7

LL

minus1000 00 1000 2000 3000 4000 5000 6000X

XY graph

(b)

Figure 11 Bisectors with errant behavior (a) Fully extended leg (b) Fully flexed leg

The trajectory visualization is performed once the com-plete data set of ICR points for every frame and their paired120574 angle are acquired in a single 2D array Note howeverthat having a sole visualization of the ICR trajectories anddisplacements does not bring useful data for design purposesFigure 14 shows the resulting contour on the 119883-119884 plane thathas been obtained using the equivalent model proposed inthis paper

6 Conclusions

This paper proposes the use of commercial vision systemsin order to determine the knee joint geometrical design forrobotic exoskeletons Given the fact that most of the devicesfound in the literature are designed by considering the humanjoints as single-noninvariant rotational joints this paperproposes a kinematic model based on irregular shaped cams


350

300

250

200

150

100

50

00

minus50

minus100

400

350

300

250

200

150

100

50

00

minus50

minus100

minus150

minus200

minus250

1850 1900 1950 2000 2050 2100 2150

Extension

Y(m

m)

Y(m

m)

X (mm) X (mm)

Flexion

1700 1800 1900 2000 2100 2200

Figure 12 Resulting contour for a session recorded at 200 fps with a swing frequency of 075Hz

400

350

300

250

200

150

100

50

00

minus50

1925 1900 1975 2000 2025 20752050 2100

Extension

Y(m

m)

X (mm)

350

300

250

200

150

100

50

00

minus50

1900 1950 2000 2050 2100 2150

Y(m

m)

X (mm)

Flexion


as the jointmechanism for emulating the bone-to-bone jointsin the human bodyThe paper proposes a geometric approachfor determining the ICR location in order to design those camcontoursThe implementation ofmarkers fixtureswhere rigidframes were mounted over a subject thigh and calf showedacceptable results Reliable rigid body-like data was obtainedpresenting an average squared error of 451mm2 whichcould be attributed to manufacturing tolerances where thestandard deviation resulted in 17mm2 The vision systemsprovided a reliable measurement tool for motion tracking

presenting considerable improvements when minor changeswere made at the vision system environment (up to 5358squared error reduction compared with the first iteration)The human knee joint consistently showed important ICRdisplacements over an average area of 235mm times 348mmover the 119909- and 119910-axes respectively The geometric approachto the ICR position for every movement frame showedconsistent results even when a total of six bisectors inter-sections were identified The resulting data support that theproposed kinematic model of contacting cams with a serial


191841 101905 81869

192521 860938 825184

193215 71297 831768

1939 580283 838438

194558 465233 845187

195201 365266 852007

195855 277353 858893

196521 202349 865838

197187 14227 872837

197843 0971579 879884

198477 0662088 886974

XICR (mm) YICR (mm) 120574 (deg)

(a)

400

350

300

250

200

150

100

50

001920 1940 1960 1980 2000 206520452020 2080

Y(m

m)

X (mm)

XY graph

(b)

Figure 14 Selected trajectory visualization (a) Selected 119865ICR data segment example (b) Selected trajectory visualization over119883-119884 plane

link connection of rotational-prismatic-rotational joints fitsefficiently to the real human knee behavior Future work willfocus on applying this model for other single DoF jointsfor example elbow looking for ICR displacements in morethan 1 plane for the knee joint (sagittal plane for this paperwork) and extending the determination of the equivalentmodels that are suitable for higher DoF joints for exampleshoulder and hip Furthermore an exoskeleton prototypewillbe constructed using the model and techniques presented inthis paper

Conflict of Interests

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

Acknowledgments

This work has been supported by Consejo Nacional deCiencia y Tecnologia (CONACYT) and theNational RoboticsLaboratory at Tecnologico de Monterrey

References

[1] J C Perry J Rosen and S Burns ldquoUpper-limb poweredexoskeleton designrdquo IEEEASME Transactions on Mechatronicsvol 12 no 4 pp 408ndash417 2007

[2] G Aguirre-Ollinger J E Colgate M A Peshkin and AGoswami ldquoInertia compensation control of a one-degree-of-freedom exoskeleton for lower-limb assistance initial experi-mentsrdquo IEEE Transactions on Neural Systems and RehabilitationEngineering vol 20 no 1 pp 68ndash77 2012

[3] C RKinnaird andD P Ferris ldquoMedial gastrocnemiusmyoelec-tric control of a robotic ankle exoskeletonrdquo IEEE Transactionson Neural Systems and Rehabilitation Engineering vol 17 no 1pp 31ndash37 2009

[4] R Lopez H Aguilar-Sierra S Salazar and R Lozano ldquoModeland control of the ELLTIO with two degrees of freedomrdquoin Proceedings of the 17th International Conference on SystemTheory Control and Computing (ICSTCC rsquo13) pp 305ndash310IEEE Sinaia Romania October 2013

[5] R J Farris H A Quintero and M Goldfarb ldquoPreliminaryevaluation of a powered lower limb orthosis to aid walking inparaplegic individualsrdquo IEEE Transactions on Neural Systemsand Rehabilitation Engineering vol 19 no 6 pp 652ndash659 2011

[6] S Murray and M Goldfarb ldquoTowards the use of a lowerlimb exoskeleton for locomotion assistance in individuals withneuromuscular locomotor deficitsrdquo in Proceedings of the 34thAnnual International Conference of the IEEE Engineering inMedicine and Biology Society (EMBS rsquo12) pp 1912ndash1915 IEEESan Diego Calif USA September 2012

[7] R J Farris H A Quintero and M Goldfarb ldquoPerformanceevaluation of a lower limb exoskeleton for stair ascent anddescent with Paraplegiardquo in Proceedings of the Annual Inter-national Conference of the IEEE Engineering in Medicine andBiology Society (EMBC rsquo12) pp 1908ndash1911 IEEE San DiegoCalif USA August-September 2012

[8] M Bortole A del Ama E Rocon J C Moreno F Brunetti andJ L Pons ldquoA robotic exoskeleton for overground gait rehabili-tationrdquo in Proceedings of the IEEE International Conference onRobotics and Automation (ICRA rsquo13) pp 3356ndash3361 May 2013

[9] R van Ham B Vanderborght M van Damme B Verrelstand D Lefeber ldquoMACCEPA The mechanically adjustablecompliance and controllable equilibrium position actuatorfor lsquoControlled Passive Walkingrsquordquo in Proceedings of the IEEEInternational Conference on Robotics and Automation (ICRArsquo06) pp 2195ndash2200 May 2006


[10] HOCOMA LokomatmdashHocoma 2014 httpwwwhocomacomproductslokomat

[11] Honda HondamdashWalk Assist And Mobility Devices 2014httpcorporatehondacominnovationwalk-assist

[12] A Tsukahara Y Hasegawa and Y Sankai ldquoGait supportfor complete spinal cord injury patient by synchronized leg-swing with HALrdquo in Proceedings of the IEEERSJ InternationalConference on Intelligent Robots and Systems (IROS rsquo11) pp 1737ndash1742 September 2011

[13] A Tsukahara Y Hasegawa K Eguchi and Y Sankai ldquoRestora-tion of gait for spinal cord injury patients usingHALwith inten-tion estimator for preferable swing speedrdquo IEEE Transactions onNeural Systems and Rehabilitation Engineering vol 23 no 2 pp308ndash318 2015

[14] M Hassan H Kadone K Suzuki and Y Sankai ldquoExoskeletonrobot control based on cane and body joint synergiesrdquo inProceedings of the 25th IEEERSJ International Conference onRobotics and Intelligent Systems (IROS rsquo12) pp 1609ndash1614October 2012

[15] Indego IndegomdashPowering People Forward Parker Indego 2014httpwwwindegocomindegoenhome

[16] Ekso-Bionics Ekso BionicsmdashExoskeleton wearable robot forpeople with paralysis from SCI or stroke 2014 httpwwweksobionicscomekso

[17] Berkeley Exoskeletons Berkeley Robotics amp Human Engineer-ing Laboratory 2014 httpbleexmeberkeleyeduresearchexoskeleton

[18] H Kazerooni ldquoExoskeletons for human power augmentationrdquoin Proceedings of the IEEE IRSRSJ International Conference onIntelligent Robots and Systems (IROS rsquo05) pp 3120ndash3125 August2005

[19] R Robotics ReWalk 2014 httpwwwrewalkcom[20] Rex Bionics Group Rex BionicsmdashStep into the Future 2014

httpwwwrexbionicscom[21] J F V Vincent ldquoBiomimeticsmdasha reviewrdquo Proceedings of the

Institution of Mechanical Engineers vol 223 no 8 pp 919ndash9392009

[22] H Mizoguchi Y Asano T Izawa et al ldquoBiomimetic designand implementation of muscle arrangement around hip jointfor musculoskeletal humanoidrdquo in Proceedings of the IEEEInternational Conference on Robotics and Biomimetics (ROBIOrsquo11) pp 1819ndash1824 December 2011

[23] Y Zhu J Cui and J Zhao ldquoBiomimetic design and biomechan-ical simulation of a 15-DOF lower extremity exoskeletonrdquo inProceedings of the IEEE International Conference onRobotics andBiomimetics (ROBIO rsquo13) pp 1119ndash1124 December 2013

[24] A B W Miranda A Y Yasutomi C Souit and A Forner-Cordero ldquoBioinspired mechanical design of an upper limbexoskeleton for rehabilitation and motor control assessmentrdquoin Proceedings of the 4th IEEE RAS amp EMBS InternationalConference on Biomedical Robotics andBiomechatronics (BioRobrsquo12) pp 1776ndash1781 June 2012

[25] J Zhu Q Wang Y Huang and L Wang ldquoAdding compliantjoints and segmented foot to bio-inspired below-knee exoskele-tonrdquo in Proceedings of the IEEE International Conference onRobotics and Automation (ICRA rsquo11) pp 605ndash610 IEEE Shang-hai China May 2011

[26] F P Beer E R Johnston and W E Clausen ldquoCinematicade cuerpos rıgidosrdquo in Mecanica Vectorial para IngenierosDinamica 8th edition 2007

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


The Japanese company Cyberdyne is worldwide known forthe development of the HAL (acronym for hybrid assistivelimb) exoskeleton [12] which is used for either task supportrehabilitation or welfare applications The HAL robot hasbeen extensively studied for example the synchronizationcontrol that is based on the position change of the centerof ground reaction force (CoGRF) by Tsukahara et al [12]which later added a speed estimator to this control model[13] or the work presented by Hassan et al [14] that includedthe application of HAL control that is based on combiningboth lower and upper limb pose signals Furthermore theIndego exoskeleton is another good example of commercialdevices [15] that are based on the Vanderbilt exoskeleton[5] and is a powered lower limb orthosis for ground gaittrainingTheEkso fromEkso Bionics [16] has been developedby the Berkeley Robotics amp Human Engineering Laboratory[17] as an evolution of the exoskeleton robots presented byKazerooni [18] Ekso is presented as a wearable bionic suitthat enables lower limb disabled subjects to stand and walkovergroundThis device is intended for medically superviseduse providing support to lower limb paralyses patients andretraining them in the gait exercise ReWalk Robotics hasproduced the first approved FDA exoskeleton that similar toother devices (eg Indego and Ekso Bionics) includes fouractuatedDoF (hips and knees joints) and is programmedwithnormalized gait patterns which provides proper musculartraining to its users [19] Finally the REX from the RexBionics Group [20] is currently the unique device thatprovides full body weight support in equilibrium without theuse of canes

Most of the current advances of robotic exoskeletons havebeen performed by considering the joints as noninvariantrotational axis This design practice however can lead tofurther injury to the user since the human joints are complexsystems composed by bones that are driven by tendons thuspresenting motions whose rotational axes do not remainin the same location For this issue a common practiceof designers is to use Biomimetics [21] (ie the search forinspiration in nature in order to find the solution to agiven problem) for the development of exoskeletons Forinstance Mizoguchi et al [22] designed and developed amusculoskeletal humanoid robot based on the actual mus-culoskeletal arrangement on the human hips Zhu et al[23] developed a lower extremity exoskeleton design with15 DoF selecting each degree of freedom by analyzing thehuman lower limbrsquos joints as spherical joints Although theyaccepted this joints implementation as nonideal to emulatehuman joints their use was neglected due to the highdifficulty for its implementationMiranda et al [24] designedand implemented a robotic upper limb exoskeleton (elbow)with bioinspired actuators Furthermore Zhu et al [25]proposed a biomimetic knee exoskeleton focusing on theenergetic aspects of the segmented foot design in whichthey embedded a compliant joint in order to upgrade humanwalking of assistance exoskeletons

It is important to note that the majority of novel designson lower limb robotic exoskeletons have focused on theknee joint which is usually the actuated joint Other designsfocus on the actuation of the hip and ankle but in almost

L

A

O

B

w

B

A

IC

Figure 1 Instantaneous center of rotation of a mechanism

all cases those joints are mechanically designed as simplerotational axis This paper conducts a detailed study of thekinematics of the knee joint proposing the use of commercialvision systems in order to perform biomimetic joint designfor robotic exoskeletons The main contribution of thiswork is the proposition of an equivalent kinematic modelthat is based on irregular shaped cams that emulates thebone-to-bone joints in the human body whose contour isobtained by the identification of the instantaneous centerof rotation (ICR) Furthermore the use of a commercialvision system is proposed as the main measurement toolresulting in relevant information about ICR displacementsat the human knee joint This work can be applied as anapproach for the ergonomic design of exoskeleton devicesThe remainder of the paper is organized as follows Section 2presents the kinematicmodel that is based on the geometricaldetermination of the ICR of two bodies in contact Section 3presents the experimental setup that has been prepared inorder to use a commercial vision system for tracking theleg motion which includes the prototype development anddata processing Section 4 presents a detailed explanationabout the algorithms that are used to determine the ICRof the knee joint using the resulting data from the visionsystem Section 5 presents the experimental results of thiswork Finally Section 6 presents important conclusions andsuggestions of future work

2 Geometrical Model of the Knee JointBased on ICR

According to the kinematics theory of rigid bodies andconsidering planar motion the velocity of every point ina rigid body can be expressed as if this body spins on aplane about a given axis that is perpendicular to the plane[26] Furthermore this rotational axis intersects those bodiesplane on its ICR

Figure 1 illustrates the ICR or IC (instantaneous center)over planar motion of rigid bodies Note that the vectorsk119860and k

119861 which represent velocities over specific points in

that body (points 119860 and 119861 on a bar 119871) are instantaneouslyequivalent to the velocity of points 119860 and 119861 spinning arounda fixed point (ie IC) where a line intersects each point of the


Femur

Tibia

Static cam

Mobile cam

Fixed point

IC

120572

Q

120573

r

R

120574

O

(a)

IC

120572

Q

120573

O

l(120574)

(b)






119897 (120574) = 119877 (120574) + 119903 (1)


119866 (120574) =119877 (120574)

119903

(2)


120573 = 119866 (120574) 120572 (3)







Bisect A1A2

BisectB1B2

B1

B2

ICR12

A1

A2







Data record

[p9984000 p

9984001

p998400n]









F1

F2

F3

F5

F4

(a)

T1

T2

T3

T5

T6

T4

T7

(b)









119863119894=

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

[

1198751198651119879

1198751198655119879

1198751198791119879

1198751198797119879

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

]119894

119875119872=[[

[

119909

119910

119911

]]

]

(4)






119875119872=[[

[

119909

0119911

]]

]

997888rarr 119875119872= [

119909

119911]

(5)




119861119895=

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

[

1198871198651

1198871198655

1198871198791

1198871198797

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

]119895

119887119872= [119887119901

119879119898]

(6)


elements (119887119872



119911119894minus 119911119894minus1

119887119901 = [

119901119909

119901119911

]

(7)




119872]119894minus1 119901119909








1198797


T3

Lower leg

Parallel line

120579

120593

T1

T2

T3

T5

T6

T4

T7








119895

119861119895=

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

[

1198871198651

1198871198655

1198871198791

1198871198797

119887LL

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

]119895

(8)








119910 = 119898 (119909minus1199091) + 1199101 (9)




119909 = 119901119909plusmn 400 (10)


119910 = 119898 (119909minus119901119909) + 119901119910 (11)



119909by 400



119910 = 119901119910plusmn 400 (12)

where

119909 =

119910 minus 119901119910

119898+119901119909

(13)







3500

3000

2500

2000

1500

1000

500

00

minus500

minus500

minus1000

minus1000

minus1500

minus2000

Y

X

00

100

0

500

150

0

200

0

250

0

300

0

350

0

400

0

450

0

T1

T2

T3

T4

T5

T6

T7

LL



119872from

(8) using

119910 = 119898 (119909minus119901119909) + 119901119910 (14)


119910 = 119898 (119909) + 119888 (15)


119910results in

119888 = 119901119910minus119898 (119901

119909) = 119888119872(119887119872) (16)

where119898119901119909 and119901






1198981198721 (1199091198721) + 1198881198721 (1198871198721) = 1198981198722 (1199091198722) + 1198881198722 (1198871198722) (17)


1198721is equal to 119909

1198722


119909ICR = minus1198981198722 minus 1198981198721

1198881198722 (1198871198722) minus 1198881198721 (1198871198721)

(18)



119910ICR = 1198981198721 (119909ICR) + 1198881198721 (1198871198721)

= 1198981198722 (119909ICR) + 1198881198722 (1198871198722)

(19)



ICR119895= [119883ICR 119884ICR]

119879 (20)




1199011199090 = minus

119888LL119898LL

(21)



Opp = 119901119910minus 0

Adj = 119901119909minus1199011199090

(22)




119865CIR =

[[[[[[[[[

[

1198911198892

1198911198893

119891119889119899minus1

119891119889119899

]]]]]]]]]

]

119891119889119895= [ICR

119895

119879 120574119895]

(24)













100

90

80

70

60

50

40

30

20

10

0

Am

plitu

de

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Time

(a)

100

90

80

70

60

50

40

30

20

10

0

Am

plitu

de

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Time

(b)














Zero (00 00 00)

110000

86000

62000

38000

14000

Z

Y

X


minus1200012000

3600060000

minus80000

minus56000

minus32000

minus8000

16000

40000

(a)

(00 00 00)

(00 00 00)

Z

Y

60000

36000

12000

minus12000

6000

X

minus60000minus3

minus3

6000minus12000

1200036000

60000minus60000

minus36000

minus12000

12000

36000

60000

Cursor 1 (00 00 00)

(b)



100

90

80

70

60

50

40

30

20

10

0

Am

plitu

de

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time

(a)

100

90

80

70

60

50

40

30

20

10

0

Am

plitu

de

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time

(b)


35003000250020001500100050000

minus500

minus1000

minus1500

minus2000

minus2500

minus3500

minus3000

Y

T1

T2

T3

T4

T5

T6

T7

LL

minus1000 00 1000 2000 3000 4000 5000 6000X

XY graph

(a)

35003000250020001500100050000

minus500

minus1000

minus1500

minus2000

minus2500

minus3500

minus3000

Y

T1

T2

T3

T4

T5

T6

T7

LL

minus1000 00 1000 2000 3000 4000 5000 6000X

XY graph

(b)



6 Conclusions



350

300

250

200

150

100

50

00

minus50

minus100

400

350

300

250

200

150

100

50

00

minus50

minus100

minus150

minus200

minus250

1850 1900 1950 2000 2050 2100 2150

Extension

Y(m

m)

Y(m

m)

X (mm) X (mm)

Flexion

1700 1800 1900 2000 2100 2200


400

350

300

250

200

150

100

50

00

minus50

1925 1900 1975 2000 2025 20752050 2100

Extension

Y(m

m)

X (mm)

350

300

250

200

150

100

50

00

minus50

1900 1950 2000 2050 2100 2150

Y(m

m)

X (mm)

Flexion





191841 101905 81869

192521 860938 825184

193215 71297 831768

1939 580283 838438

194558 465233 845187

195201 365266 852007

195855 277353 858893

196521 202349 865838

197187 14227 872837

197843 0971579 879884

198477 0662088 886974


(a)

400

350

300

250

200

150

100

50

001920 1940 1960 1980 2000 206520452020 2080

Y(m

m)

X (mm)

XY graph

(b)





Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Femur

Tibia

Static cam

Mobile cam

Fixed point

IC

120572

Q

120573

r

R

120574

O

(a)

IC

120572

Q

120573

O

l(120574)

(b)






119897 (120574) = 119877 (120574) + 119903 (1)


119866 (120574) =119877 (120574)

119903

(2)


120573 = 119866 (120574) 120572 (3)







Bisect A1A2

BisectB1B2

B1

B2

ICR12

A1

A2







Data record

[p9984000 p

9984001

p998400n]









F1

F2

F3

F5

F4

(a)

T1

T2

T3

T5

T6

T4

T7

(b)









119863119894=

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

[

1198751198651119879

1198751198655119879

1198751198791119879

1198751198797119879

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

]119894

119875119872=[[

[

119909

119910

119911

]]

]

(4)






119875119872=[[

[

119909

0119911

]]

]

997888rarr 119875119872= [

119909

119911]

(5)




119861119895=

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

[

1198871198651

1198871198655

1198871198791

1198871198797

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

]119895

119887119872= [119887119901

119879119898]

(6)


elements (119887119872



119911119894minus 119911119894minus1

119887119901 = [

119901119909

119901119911

]

(7)




119872]119894minus1 119901119909








1198797


T3

Lower leg

Parallel line

120579

120593

T1

T2

T3

T5

T6

T4

T7








119895

119861119895=

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

[

1198871198651

1198871198655

1198871198791

1198871198797

119887LL

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

]119895

(8)








119910 = 119898 (119909minus1199091) + 1199101 (9)




119909 = 119901119909plusmn 400 (10)


119910 = 119898 (119909minus119901119909) + 119901119910 (11)



119909by 400



119910 = 119901119910plusmn 400 (12)

where

119909 =

119910 minus 119901119910

119898+119901119909

(13)







3500

3000

2500

2000

1500

1000

500

00

minus500

minus500

minus1000

minus1000

minus1500

minus2000

Y

X

00

100

0

500

150

0

200

0

250

0

300

0

350

0

400

0

450

0

T1

T2

T3

T4

T5

T6

T7

LL



119872from

(8) using

119910 = 119898 (119909minus119901119909) + 119901119910 (14)


119910 = 119898 (119909) + 119888 (15)


119910results in

119888 = 119901119910minus119898 (119901

119909) = 119888119872(119887119872) (16)

where119898119901119909 and119901






1198981198721 (1199091198721) + 1198881198721 (1198871198721) = 1198981198722 (1199091198722) + 1198881198722 (1198871198722) (17)


1198721is equal to 119909

1198722


119909ICR = minus1198981198722 minus 1198981198721

1198881198722 (1198871198722) minus 1198881198721 (1198871198721)

(18)



119910ICR = 1198981198721 (119909ICR) + 1198881198721 (1198871198721)

= 1198981198722 (119909ICR) + 1198881198722 (1198871198722)

(19)



ICR119895= [119883ICR 119884ICR]

119879 (20)




1199011199090 = minus

119888LL119898LL

(21)



Opp = 119901119910minus 0

Adj = 119901119909minus1199011199090

(22)




119865CIR =

[[[[[[[[[

[

1198911198892

1198911198893

119891119889119899minus1

119891119889119899

]]]]]]]]]

]

119891119889119895= [ICR

119895

119879 120574119895]

(24)













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(b)














Zero (00 00 00)

110000

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minus1200012000

3600060000

minus80000

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minus8000

16000

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(a)

(00 00 00)

(00 00 00)

Z

Y

60000

36000

12000

minus12000

6000

X

minus60000minus3

minus3

6000minus12000

1200036000

60000minus60000

minus36000

minus12000

12000

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60000

Cursor 1 (00 00 00)

(b)



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Y

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minus1000 00 1000 2000 3000 4000 5000 6000X

XY graph

(a)

35003000250020001500100050000

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minus3500

minus3000

Y

T1

T2

T3

T4

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minus1000 00 1000 2000 3000 4000 5000 6000X

XY graph

(b)



6 Conclusions



350

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400

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00

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minus250

1850 1900 1950 2000 2050 2100 2150

Extension

Y(m

m)

Y(m

m)

X (mm) X (mm)

Flexion

1700 1800 1900 2000 2100 2200


400

350

300

250

200

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minus50

1925 1900 1975 2000 2025 20752050 2100

Extension

Y(m

m)

X (mm)

350

300

250

200

150

100

50

00

minus50

1900 1950 2000 2050 2100 2150

Y(m

m)

X (mm)

Flexion





191841 101905 81869

192521 860938 825184

193215 71297 831768

1939 580283 838438

194558 465233 845187

195201 365266 852007

195855 277353 858893

196521 202349 865838

197187 14227 872837

197843 0971579 879884

198477 0662088 886974


(a)

400

350

300

250

200

150

100

50

001920 1940 1960 1980 2000 206520452020 2080

Y(m

m)

X (mm)

XY graph

(b)





Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Bisect A1A2

BisectB1B2

B1

B2

ICR12

A1

A2







Data record

[p9984000 p

9984001

p998400n]









F1

F2

F3

F5

F4

(a)

T1

T2

T3

T5

T6

T4

T7

(b)









119863119894=

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

[

1198751198651119879

1198751198655119879

1198751198791119879

1198751198797119879

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

]119894

119875119872=[[

[

119909

119910

119911

]]

]

(4)






119875119872=[[

[

119909

0119911

]]

]

997888rarr 119875119872= [

119909

119911]

(5)




119861119895=

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

[

1198871198651

1198871198655

1198871198791

1198871198797

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

]119895

119887119872= [119887119901

119879119898]

(6)


elements (119887119872



119911119894minus 119911119894minus1

119887119901 = [

119901119909

119901119911

]

(7)




119872]119894minus1 119901119909








1198797


T3

Lower leg

Parallel line

120579

120593

T1

T2

T3

T5

T6

T4

T7








119895

119861119895=

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

[

1198871198651

1198871198655

1198871198791

1198871198797

119887LL

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

]119895

(8)








119910 = 119898 (119909minus1199091) + 1199101 (9)




119909 = 119901119909plusmn 400 (10)


119910 = 119898 (119909minus119901119909) + 119901119910 (11)



119909by 400



119910 = 119901119910plusmn 400 (12)

where

119909 =

119910 minus 119901119910

119898+119901119909

(13)







3500

3000

2500

2000

1500

1000

500

00

minus500

minus500

minus1000

minus1000

minus1500

minus2000

Y

X

00

100

0

500

150

0

200

0

250

0

300

0

350

0

400

0

450

0

T1

T2

T3

T4

T5

T6

T7

LL



119872from

(8) using

119910 = 119898 (119909minus119901119909) + 119901119910 (14)


119910 = 119898 (119909) + 119888 (15)


119910results in

119888 = 119901119910minus119898 (119901

119909) = 119888119872(119887119872) (16)

where119898119901119909 and119901






1198981198721 (1199091198721) + 1198881198721 (1198871198721) = 1198981198722 (1199091198722) + 1198881198722 (1198871198722) (17)


1198721is equal to 119909

1198722


119909ICR = minus1198981198722 minus 1198981198721

1198881198722 (1198871198722) minus 1198881198721 (1198871198721)

(18)



119910ICR = 1198981198721 (119909ICR) + 1198881198721 (1198871198721)

= 1198981198722 (119909ICR) + 1198881198722 (1198871198722)

(19)



ICR119895= [119883ICR 119884ICR]

119879 (20)




1199011199090 = minus

119888LL119898LL

(21)



Opp = 119901119910minus 0

Adj = 119901119909minus1199011199090

(22)




119865CIR =

[[[[[[[[[

[

1198911198892

1198911198893

119891119889119899minus1

119891119889119899

]]]]]]]]]

]

119891119889119895= [ICR

119895

119879 120574119895]

(24)













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(a)

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0 200 400 600 800 1000 1200 1400 1600 1800 2000

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(b)














Zero (00 00 00)

110000

86000

62000

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14000

Z

Y

X


minus1200012000

3600060000

minus80000

minus56000

minus32000

minus8000

16000

40000

(a)

(00 00 00)

(00 00 00)

Z

Y

60000

36000

12000

minus12000

6000

X

minus60000minus3

minus3

6000minus12000

1200036000

60000minus60000

minus36000

minus12000

12000

36000

60000

Cursor 1 (00 00 00)

(b)



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0 200 400 600 800 1000 1200 1400 1600 1800 2000Time

(a)

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0 200 400 600 800 1000 1200 1400 1600 1800 2000Time

(b)


35003000250020001500100050000

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Y

T1

T2

T3

T4

T5

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LL

minus1000 00 1000 2000 3000 4000 5000 6000X

XY graph

(a)

35003000250020001500100050000

minus500

minus1000

minus1500

minus2000

minus2500

minus3500

minus3000

Y

T1

T2

T3

T4

T5

T6

T7

LL

minus1000 00 1000 2000 3000 4000 5000 6000X

XY graph

(b)



6 Conclusions



350

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minus50

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minus250

1850 1900 1950 2000 2050 2100 2150

Extension

Y(m

m)

Y(m

m)

X (mm) X (mm)

Flexion

1700 1800 1900 2000 2100 2200


400

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250

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1925 1900 1975 2000 2025 20752050 2100

Extension

Y(m

m)

X (mm)

350

300

250

200

150

100

50

00

minus50

1900 1950 2000 2050 2100 2150

Y(m

m)

X (mm)

Flexion





191841 101905 81869

192521 860938 825184

193215 71297 831768

1939 580283 838438

194558 465233 845187

195201 365266 852007

195855 277353 858893

196521 202349 865838

197187 14227 872837

197843 0971579 879884

198477 0662088 886974


(a)

400

350

300

250

200

150

100

50

001920 1940 1960 1980 2000 206520452020 2080

Y(m

m)

X (mm)

XY graph

(b)





Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










F1

F2

F3

F5

F4

(a)

T1

T2

T3

T5

T6

T4

T7

(b)









119863119894=

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

[

1198751198651119879

1198751198655119879

1198751198791119879

1198751198797119879

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

]119894

119875119872=[[

[

119909

119910

119911

]]

]

(4)






119875119872=[[

[

119909

0119911

]]

]

997888rarr 119875119872= [

119909

119911]

(5)




119861119895=

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

[

1198871198651

1198871198655

1198871198791

1198871198797

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

]119895

119887119872= [119887119901

119879119898]

(6)


elements (119887119872



119911119894minus 119911119894minus1

119887119901 = [

119901119909

119901119911

]

(7)




119872]119894minus1 119901119909








1198797


T3

Lower leg

Parallel line

120579

120593

T1

T2

T3

T5

T6

T4

T7








119895

119861119895=

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

[

1198871198651

1198871198655

1198871198791

1198871198797

119887LL

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

]119895

(8)








119910 = 119898 (119909minus1199091) + 1199101 (9)




119909 = 119901119909plusmn 400 (10)


119910 = 119898 (119909minus119901119909) + 119901119910 (11)



119909by 400



119910 = 119901119910plusmn 400 (12)

where

119909 =

119910 minus 119901119910

119898+119901119909

(13)







3500

3000

2500

2000

1500

1000

500

00

minus500

minus500

minus1000

minus1000

minus1500

minus2000

Y

X

00

100

0

500

150

0

200

0

250

0

300

0

350

0

400

0

450

0

T1

T2

T3

T4

T5

T6

T7

LL



119872from

(8) using

119910 = 119898 (119909minus119901119909) + 119901119910 (14)


119910 = 119898 (119909) + 119888 (15)


119910results in

119888 = 119901119910minus119898 (119901

119909) = 119888119872(119887119872) (16)

where119898119901119909 and119901






1198981198721 (1199091198721) + 1198881198721 (1198871198721) = 1198981198722 (1199091198722) + 1198881198722 (1198871198722) (17)


1198721is equal to 119909

1198722


119909ICR = minus1198981198722 minus 1198981198721

1198881198722 (1198871198722) minus 1198881198721 (1198871198721)

(18)



119910ICR = 1198981198721 (119909ICR) + 1198881198721 (1198871198721)

= 1198981198722 (119909ICR) + 1198881198722 (1198871198722)

(19)



ICR119895= [119883ICR 119884ICR]

119879 (20)




1199011199090 = minus

119888LL119898LL

(21)



Opp = 119901119910minus 0

Adj = 119901119909minus1199011199090

(22)




119865CIR =

[[[[[[[[[

[

1198911198892

1198911198893

119891119889119899minus1

119891119889119899

]]]]]]]]]

]

119891119889119895= [ICR

119895

119879 120574119895]

(24)













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(a)

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0 200 400 600 800 1000 1200 1400 1600 1800 2000

Time

(b)














Zero (00 00 00)

110000

86000

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Z

Y

X


minus1200012000

3600060000

minus80000

minus56000

minus32000

minus8000

16000

40000

(a)

(00 00 00)

(00 00 00)

Z

Y

60000

36000

12000

minus12000

6000

X

minus60000minus3

minus3

6000minus12000

1200036000

60000minus60000

minus36000

minus12000

12000

36000

60000

Cursor 1 (00 00 00)

(b)



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90

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0

Am

plitu

de

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time

(a)

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0 200 400 600 800 1000 1200 1400 1600 1800 2000Time

(b)


35003000250020001500100050000

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minus3000

Y

T1

T2

T3

T4

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T7

LL

minus1000 00 1000 2000 3000 4000 5000 6000X

XY graph

(a)

35003000250020001500100050000

minus500

minus1000

minus1500

minus2000

minus2500

minus3500

minus3000

Y

T1

T2

T3

T4

T5

T6

T7

LL

minus1000 00 1000 2000 3000 4000 5000 6000X

XY graph

(b)



6 Conclusions



350

300

250

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150

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00

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00

minus50

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minus250

1850 1900 1950 2000 2050 2100 2150

Extension

Y(m

m)

Y(m

m)

X (mm) X (mm)

Flexion

1700 1800 1900 2000 2100 2200


400

350

300

250

200

150

100

50

00

minus50

1925 1900 1975 2000 2025 20752050 2100

Extension

Y(m

m)

X (mm)

350

300

250

200

150

100

50

00

minus50

1900 1950 2000 2050 2100 2150

Y(m

m)

X (mm)

Flexion





191841 101905 81869

192521 860938 825184

193215 71297 831768

1939 580283 838438

194558 465233 845187

195201 365266 852007

195855 277353 858893

196521 202349 865838

197187 14227 872837

197843 0971579 879884

198477 0662088 886974


(a)

400

350

300

250

200

150

100

50

001920 1940 1960 1980 2000 206520452020 2080

Y(m

m)

X (mm)

XY graph

(b)





Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119875119872=[[

[

119909

0119911

]]

]

997888rarr 119875119872= [

119909

119911]

(5)




119861119895=

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

[

1198871198651

1198871198655

1198871198791

1198871198797

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

]119895

119887119872= [119887119901

119879119898]

(6)


elements (119887119872



119911119894minus 119911119894minus1

119887119901 = [

119901119909

119901119911

]

(7)




119872]119894minus1 119901119909








1198797


T3

Lower leg

Parallel line

120579

120593

T1

T2

T3

T5

T6

T4

T7








119895

119861119895=

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

[

1198871198651

1198871198655

1198871198791

1198871198797

119887LL

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

]119895

(8)








119910 = 119898 (119909minus1199091) + 1199101 (9)




119909 = 119901119909plusmn 400 (10)


119910 = 119898 (119909minus119901119909) + 119901119910 (11)



119909by 400



119910 = 119901119910plusmn 400 (12)

where

119909 =

119910 minus 119901119910

119898+119901119909

(13)







3500

3000

2500

2000

1500

1000

500

00

minus500

minus500

minus1000

minus1000

minus1500

minus2000

Y

X

00

100

0

500

150

0

200

0

250

0

300

0

350

0

400

0

450

0

T1

T2

T3

T4

T5

T6

T7

LL



119872from

(8) using

119910 = 119898 (119909minus119901119909) + 119901119910 (14)


119910 = 119898 (119909) + 119888 (15)


119910results in

119888 = 119901119910minus119898 (119901

119909) = 119888119872(119887119872) (16)

where119898119901119909 and119901






1198981198721 (1199091198721) + 1198881198721 (1198871198721) = 1198981198722 (1199091198722) + 1198881198722 (1198871198722) (17)


1198721is equal to 119909

1198722


119909ICR = minus1198981198722 minus 1198981198721

1198881198722 (1198871198722) minus 1198881198721 (1198871198721)

(18)



119910ICR = 1198981198721 (119909ICR) + 1198881198721 (1198871198721)

= 1198981198722 (119909ICR) + 1198881198722 (1198871198722)

(19)



ICR119895= [119883ICR 119884ICR]

119879 (20)




1199011199090 = minus

119888LL119898LL

(21)



Opp = 119901119910minus 0

Adj = 119901119909minus1199011199090

(22)




119865CIR =

[[[[[[[[[

[

1198911198892

1198911198893

119891119889119899minus1

119891119889119899

]]]]]]]]]

]

119891119889119895= [ICR

119895

119879 120574119895]

(24)













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6 Conclusions



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1900 1950 2000 2050 2100 2150

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Flexion





191841 101905 81869

192521 860938 825184

193215 71297 831768

1939 580283 838438

194558 465233 845187

195201 365266 852007

195855 277353 858893

196521 202349 865838

197187 14227 872837

197843 0971579 879884

198477 0662088 886974


(a)

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001920 1940 1960 1980 2000 206520452020 2080

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(b)





Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119910 = 119898 (119909minus1199091) + 1199101 (9)




119909 = 119901119909plusmn 400 (10)


119910 = 119898 (119909minus119901119909) + 119901119910 (11)



119909by 400



119910 = 119901119910plusmn 400 (12)

where

119909 =

119910 minus 119901119910

119898+119901119909

(13)







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119872from

(8) using

119910 = 119898 (119909minus119901119909) + 119901119910 (14)


119910 = 119898 (119909) + 119888 (15)


119910results in

119888 = 119901119910minus119898 (119901

119909) = 119888119872(119887119872) (16)

where119898119901119909 and119901






1198981198721 (1199091198721) + 1198881198721 (1198871198721) = 1198981198722 (1199091198722) + 1198881198722 (1198871198722) (17)


1198721is equal to 119909

1198722


119909ICR = minus1198981198722 minus 1198981198721

1198881198722 (1198871198722) minus 1198881198721 (1198871198721)

(18)



119910ICR = 1198981198721 (119909ICR) + 1198881198721 (1198871198721)

= 1198981198722 (119909ICR) + 1198881198722 (1198871198722)

(19)



ICR119895= [119883ICR 119884ICR]

119879 (20)




1199011199090 = minus

119888LL119898LL

(21)



Opp = 119901119910minus 0

Adj = 119901119909minus1199011199090

(22)




119865CIR =

[[[[[[[[[

[

1198911198892

1198911198893

119891119889119899minus1

119891119889119899

]]]]]]]]]

]

119891119889119895= [ICR

119895

119879 120574119895]

(24)













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6 Conclusions



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Flexion





191841 101905 81869

192521 860938 825184

193215 71297 831768

1939 580283 838438

194558 465233 845187

195201 365266 852007

195855 277353 858893

196521 202349 865838

197187 14227 872837

197843 0971579 879884

198477 0662088 886974


(a)

400

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001920 1940 1960 1980 2000 206520452020 2080

Y(m

m)

X (mm)

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(b)





Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119910ICR = 1198981198721 (119909ICR) + 1198881198721 (1198871198721)

= 1198981198722 (119909ICR) + 1198881198722 (1198871198722)

(19)



ICR119895= [119883ICR 119884ICR]

119879 (20)




1199011199090 = minus

119888LL119898LL

(21)



Opp = 119901119910minus 0

Adj = 119901119909minus1199011199090

(22)




119865CIR =

[[[[[[[[[

[

1198911198892

1198911198893

119891119889119899minus1

119891119889119899

]]]]]]]]]

]

119891119889119895= [ICR

119895

119879 120574119895]

(24)













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6 Conclusions



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1900 1950 2000 2050 2100 2150

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X (mm)

Flexion





191841 101905 81869

192521 860938 825184

193215 71297 831768

1939 580283 838438

194558 465233 845187

195201 365266 852007

195855 277353 858893

196521 202349 865838

197187 14227 872837

197843 0971579 879884

198477 0662088 886974


(a)

400

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200

150

100

50

001920 1940 1960 1980 2000 206520452020 2080

Y(m

m)

X (mm)

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(b)





Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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6 Conclusions



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1900 1950 2000 2050 2100 2150

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X (mm)

Flexion





191841 101905 81869

192521 860938 825184

193215 71297 831768

1939 580283 838438

194558 465233 845187

195201 365266 852007

195855 277353 858893

196521 202349 865838

197187 14227 872837

197843 0971579 879884

198477 0662088 886974


(a)

400

350

300

250

200

150

100

50

001920 1940 1960 1980 2000 206520452020 2080

Y(m

m)

X (mm)

XY graph

(b)





Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Zero (00 00 00)

110000

86000

62000

38000

14000

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minus1200012000

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Cursor 1 (00 00 00)

(b)



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T1

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(b)



6 Conclusions



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1900 1950 2000 2050 2100 2150

Y(m

m)

X (mm)

Flexion





191841 101905 81869

192521 860938 825184

193215 71297 831768

1939 580283 838438

194558 465233 845187

195201 365266 852007

195855 277353 858893

196521 202349 865838

197187 14227 872837

197843 0971579 879884

198477 0662088 886974


(a)

400

350

300

250

200

150

100

50

001920 1940 1960 1980 2000 206520452020 2080

Y(m

m)

X (mm)

XY graph

(b)





Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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(a)

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T1

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(b)



6 Conclusions



350

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Flexion

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250

200

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100

50

00

minus50

1900 1950 2000 2050 2100 2150

Y(m

m)

X (mm)

Flexion





191841 101905 81869

192521 860938 825184

193215 71297 831768

1939 580283 838438

194558 465233 845187

195201 365266 852007

195855 277353 858893

196521 202349 865838

197187 14227 872837

197843 0971579 879884

198477 0662088 886974


(a)

400

350

300

250

200

150

100

50

001920 1940 1960 1980 2000 206520452020 2080

Y(m

m)

X (mm)

XY graph

(b)





Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










350

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100

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00

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1900 1950 2000 2050 2100 2150

Y(m

m)

X (mm)

Flexion





191841 101905 81869

192521 860938 825184

193215 71297 831768

1939 580283 838438

194558 465233 845187

195201 365266 852007

195855 277353 858893

196521 202349 865838

197187 14227 872837

197843 0971579 879884

198477 0662088 886974


(a)

400

350

300

250

200

150

100

50

001920 1940 1960 1980 2000 206520452020 2080

Y(m

m)

X (mm)

XY graph

(b)





Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










191841 101905 81869

192521 860938 825184

193215 71297 831768

1939 580283 838438

194558 465233 845187

195201 365266 852007

195855 277353 858893

196521 202349 865838

197187 14227 872837

197843 0971579 879884

198477 0662088 886974


(a)

400

350

300

250

200

150

100

50

001920 1940 1960 1980 2000 206520452020 2080

Y(m

m)

X (mm)

XY graph

(b)





Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Inverse Modeling of Human Knee Joint...

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Transcript of Research Article Inverse Modeling of Human Knee Joint...