Optimization Of Six Bar Knee Linkage For Stability Of Knee .
Majlesi Journal of Mechatronic SystemsVol. 1, No. 4, December 2012Optimization of Six Bar Knee Linkage for Stability of KneeProsthesisNarjes.Ghaemi1, Morteza. Dardel2, Mohammad Hassan Ghasemi3, Hassan.Zohoor4,1- M.sc student, Babol Noshirvani University of Technology, Babol, Iran.Email: ghaemi narges@yahoo.com (Corresponding author)2- Assistant Professor, Babol Noshirvani University of Technology, Babol, Iran.Email: dardel@nit.ac.ir3- Assistant Professor, Babol Noshirvani University of Technology, Babol, Iran.mhghasemi@nit.ac.ir4- Professor, Sharif University of Technology, Tehran, Iran.Email: zohoor@sharif.eduReceived July 2012Revised Sept 2012Accepted Nov.2012ABSTRACT:Loss of lower extremities has been one of the human problems in the human life. Therefore the optimal design ofhuman lower limb knee prostheses is fundamental in order to restore the lost functionality and aesthetic aspect of theamputee’s locomotion. This work presents an optimization procedure for the synthesis of a six-bar linkage for kneeprosthesis and its comparison with four-bar linkage counterpart. Different form of six bar linkages, such as Stephensonand Watt mechanisms are considered and optimized. This study shows that the performance of six bar mechanism caseI (SBM1) is better than other mechanisms.KEYWORDS: Knee prosthesis, Knee Stability, Four bar Linkage, Six bar linkage1. INTRODUCTIONProsthesis must provide what has become known asknee stability. In an artificial knee, heel contact is themost critical period of the stance phase for kneesecurity.Most knee prostheses available are divided into twolarge groups: prostheses with a single axis of rotationand polycentric prostheses. Single axis prostheses havea fixed instantaneous centre of rotation of the relativemotion between the femur and shank, are cheap butprostheses have not a good performance and do notprovide high stability in stance phase. The secondgroup, polycentric prostheses are which the position ofthe ICR changes at each flexion angle. Polycentricfour-bar knee has been studied by many researcherssuch as Sancisi, Raffaele, Castelli [1]-[3].Four-bar linkage mechanisms for the trans-femoralamputee have been widely used in the Prostheses kneefor many years and have been studied by manyscientists and scholars. Radcliffe has done usefulstudies about knee stability by changing the location ofinstantaneous centre of rotation of the relative motionbetween the femur and shank and introduced threedifferent classes of four-bar linkage prosthetic kneemechanisms, for different groups of trans-femoralamputee [1], [2]. Sancisi and Raffaele, have been38investigated about knee prosthesis’ stability andvoluntary control [3]. But although they are useful andhelpful for certain amputees, they are fitted in a limitednumber of cases.Other types of knee mechanisms for through-kneeamputees can also be found. Six-bar linkage knee-anklemechanisms (SBM) have been used in some prostheticknees in the recent years, but there are not so manyresearches about the advantage performance of the Sixbar mechanism as knee prostheses.At the University of California, Berkeley, the possibleadvantages of using six-bar linkage mechanisms havebeen investigated .This mechanism provides thepossibility of increased range of knee motion, bettercosmetic, improved stance phase stability and swingphase control as compared to four-bar designs. Theseadvantages are achieved at the expense of added weightand complexity [4]. Patil and Chakraborty designed aparticular six-bar knee-ankle mechanism to producenear normal patterns of motion during walking andsquatting [5]. The kinematic and dynamic performanceof the six-bar mechanism used in the prosthetic knee isinvestigated by Dewen Jin [6].This paper will describe the kinematics performance ofseveral types of six-bar mechanisms and discuss thedifferences criteria for four different classes of six-bar
Majlesi Journal of Mechatronic Systemslinkage mechanisms for fitting to amputees andcompare improvement and weakening performance ofthe knees with four-bar linkage knee. To specify amechanism to simulate a pre-specified ankle trajectoryit is necessary to obtain the length and initialorientations of linkages. For this purpose anoptimization procedure is proposed. From thisoptimization procedure the length and initialorientations are obtained such that total error betweenpre-specified and simulated ankle trajectories to be asminimum as possible.To obtain the relativistic kinematic, due to thenonlinear equations in the six-bar mechanism, a newmethod is presented. To achieve natural walkingmotion, the hip and ankle flexion, and ankle trajectoryare necessary; the angles can be calculated by solvingthe inverse kinematics [7]. And also in this paper isbriefly discussed about the stability of the knee bychanging the location of instantaneous center.In the next section, we discuss about kinematic analysisof Mechanisms.1- Inverse kinematic analysis of knee BarMechanismIn this section at first kinematic analysis of four and sixbar knee mechanism are presented. Known four bar andsix bar mechanisms are single degree of freedommechanism, in which one of the linkage attached to theground is input. For inverse kinematics analysis thelocus of a point on the coupler is know and angles andpositions of other linkages will be determined. In usingthese mechanisms as an artificial knee, withconsidering the motion with respect to hip, a twodegree of freedom mechanism will be obtained. Hencethe hip and ankle flexion are necessary to solve theinverse kinematics of knee for given ankle trajectory.In continue at first inverse kinematics of kneemechanism for four bar knee and then different form ofsix bar mechanism will be presented.1.1. Inverse kinematic analysis of knee with fourbar mechanismA schematic of the four bar knee linkage to simulatehuman locomotion is shown in Fig.1. For thismechanism the length of thigh is assumed to be known,and unknown parameters are length of different links offour bar mechanism, length of shank and initialorientation of links fixed to the thigh and shank.Vol. 1, No. 4, December 2012Fig. 12. Schematic of a knee with four bar kneelinkage.For obtaining the inverse kinematics of the mechanismis it necessary to obtain the angles of different links ofthe mechanism in terms of position of the couple pointand. For thisofand known angles ofpurpose it is necessary to write loop closure equation ofthe mechanism. For Fig.1 this equation in vector formis as follows:(1)All angles are measured relative to horizontal axis incounter clockwise direction. With separating real andimaginary parts of Eq.(1), we have(2)coscoscoscoscossinsinsinsin(3)sinFrom these equations anglesas follows:sinsinandcan be obtained(4)2(5)2where :tan(6)With this known value of link’s angles, the position ofankle point is determined as:coscoscos(7)coscos39
Majlesi Journal of Mechatronic SystemssinsinsinsinsinVol. 1, No. 4, December 2012(8)Knee with four-bar mechanism (FBM) shown in Fig. 1,has 9 unknown geometrical parameters, which can beobtained through an optimization procedure describedin the next section. These unknown parameters are:(9), , , , , , , , ]1.2. Inverse kinematic analysis of knee with six-barmechanismDifferent types of six bar linkages are known inmachine theory. Among them six bar revolute jointmechanism known as Watt and Stephenson arefamiliar. Compared with four-bar mechanisms, six-barmechanisms have much more design parameters. Theexcess links provide more flexibility in design and withthem we can obtain some requirement which cannot beobtained by mechanism with lower links. Four differenttypes of six bar linkages mechanism are shown in Fig.2. In following kinematic analysis of each of thesemechanisms are presented. As we will see, inversekinematics of six bar linkage cannot be obtained in asimilar procedure given for four bar knee mechanism.Hence a simple method will be presented for solvingthe inverse kinematics analysis of these six bar linkage,which can be extended to more complex mechanismtoo.In similar to the four bar knee case, the knownparameter and angles areandandandother angles and parameters are unknown.Fig. 2. Schematics of knee with different six barlinkagesThe loop closure equations for different cases of Fig. 2to obtain the angles of different links are:I:0,(10)0II:0,0III:(11)0,(12)0IV:0,(13)0which can be written in real form sinsin040(14)
Majlesi Journal of Mechatronic inIII:cos0sin0cossinsincoscoscoscossinVol. 1, No. 4, December :coscos(16)cos0sin0coscoscos(17)sinsinsinsin0The ankle point position coordinates for different casesof Fig. 2 1)coscoscoscosSolving together of Eqs.(10 , 14), Eqs.(11, 15), Eqs.(12,16) and Eqs.(13, 17) in a similar method applied to thefour bar linkage is possible, but is very difficult toobtain and useless. Hence a different procedure forobtaining the unknown angles is presented in continuo,which can be applied to more complex problems.Unknown parameters which must be obtained throughan optimization procedure to simulate pre-specifiedknee trajectory for different case of six bar linkagemechanism sinsincossincoscoscos00sinIII:cosIII:IV:(22), , , , , , ,,,,, , , ,(23), , , , , , , , ,,, , , ,(24), , , , ], , , , , , , , ,,, , , , , , , , ,, , , ](25)2.1.Solution procedure for solving inversekinematics of complex mechanisms and robotsForward and inverse kinematics problems of robots andmechanism are in terms of sin or cos of differentangles. These terms maybe in multiplied or power formsuch as cos cos or cos. With using triangularidentities such expressions can be expressed in terms of,linear terms of cos or sin such as cossin, cos. With using Taylor expansioneach of these terms can be expressed as follows:(26)cosΔsincosΔcos 2(27)sinsinΔcosΔsin 2whereis the value for the unknown variable at stepk’th. For small values of Δ, we have41
Majlesi Journal of Mechatronic SystemsVol. 1, No. 4, December 2012(28)coscosΔsin(29)sinΔcossinWith substituting Eqs. (28-29) in loop closure equationfor mechanisms or robots, with known value of ,Δcan be obtained.The presented approach to solving inverse kinematicsanalysis of mechanism or robots is applied to theproposed six bar knee mechanism. Hence we ossinsin(33)and inII:III:sincossincos(32)2.2.Optimization ProcedureOptimization is the act of obtaining the best resultunder given circumstances.The ultimate goal of all such decisions is either tominimize the effort required or to maximize the desiredbenefit. Since the effort required or the benefit desiredin any practical situation can be expressed as a functionof certain decision variables, optimization can bedefined as the process of finding the conditions thatgive the maximum or minimum value of a function [8].The purpose of the optimization procedure is to find a4BM or 6BM which follows the experimental kneemotion as better as possible and, at the same time, andshould be designed in order to restore the stabilityduring walking. Meantime, the dimensions of linksshould be within an acceptable range.Here, the objective function value F(X) (the function tominimize) can be obtained as the sum of the squareddistances between the i-th experimental position of the,) at the i-th experimentalreference point (flexion angle and the respective position of the coupler), computedpoint P at the same flexion angle ( ,
Majlesi Journal of Mechatronic SystemsVol. 1, No. 4, December 2012according to the equations. So the optimizationproblem is expressed asMin(35)where n is the number of selected points in a gait cycle,21.Results and DiscussionThe reference data for the simulation of ankle kneemotion are given in [7]. After the optimization, thedesign parameters for different mentioned kneemechanism are obtained. These results are shown inTable 1. The dimensions are in millimeters.Table 1. Design parameters obtained from optimizationprocedure for different mechanismsmechanism Design parametersFour-bar55.5, 48, 46, 54, 399, 1, 47, 1.5, 4.1X [54, 63, 63.5, 71, 87, 92, 74.5, 79, 386,Six-bar(1)53, 78, 3.18, 0.04, 0.53, 0.37]Six-bar(2)Six-bar(3)Six-bar(4)X [45, 53, 40, 40, 57, 20, 70, 38, 19, 28,361.5, 0.22, 2.5, 3.14, 0.7]X [29, 39.5, 30.5, 34, 20, 11, 10 ,8, 9.5, 7,8,390, 0.17, 1.03, 0.4, 2.67]X [48, 42, 45, 61, 52, 31, 86, 21, 21, 361,2.9,. 3,. -23]The comparison of the generated trajectory of the anklejoint with expected is shown in Fig. 3. The mean squareerrors for four-bar linkage, six-bar linkage case I, sixbar case II, six-bar case III, six-bar case IV and six-bar0.038% , Err0.16%,case V are ErrErr0.51%, Err0.86% and Err0.36%, respectively.The comparison of the generated trajectory of theInstant center with is shown in Fig. 4, shows thatlocation of the Instant center in the six-bar mechanism1is more elevated than four-bar mechanism and thismakes that SBM 1 is more stable than FBM.In Fig. 4c, the most interesting feature of this centrodeis the almost constant height of the instant centers.According to Fig. 4b, this mechanism is much variationin the instant center.The comparison of the generated trajectory of theInstant center with is shown in Fig. 4 shows thatlocation of the Instant center in the six-bar mechanism1is more elevated than four-bar mechanism and thismakes that SBM 1 is more stable than FBM.The comparison of the generated trajectories of theankle joints shown in Figs. 3, shows that theperformance of the SBM1 is better than SBM4,performance of the SBM4 is better than SBM2 and theperformance of the SBM2 is better than SBM3.43
Majlesi Journal of Mechatronic SystemsVol. 1, No. 4, December 2012(b)Fig 3. The expected trajectory of the ankle joint andrealized trajectory by four-bar and six bar linkage.(c)(a)(d)44
Majlesi Journal of Mechatronic SystemsVol. 1, No. 4, December 2012REFERENCES[1] Charles W Radcliffe and ME. Deg, “Biomechanics of[2][3](e)Fig 4 a: Trajectory of the Instant center by four-barmechanism b: six-bar case I, c: six-bar case II, d: sixbar case III, e: six-bar case IV.Conclusion and discussionThe kinematic performance of the several differentmechanisms such as four-bar linkage and six-barlinkage are shown in above figures. And are comparedimprovement and weakening performance of the sixbar knees with four-bar linkage knee. Meantime a newmethod is presented for solving nonlinear equation ofthe six-bar mechanism which eases to obtain kinematicrelations. The comparison of the trajectory of the anklejoint in swing phase of the six-bar linkage knee withthat of a four-bar knee mechanism shows that six-barlinkage knee has better performance than four-bar kneemechanism. Also the comparison between various sixbar mechanism shows that the performance of six barmechanism case I is better than other mechanisms.[4][5][6][7][8]Knee Stability Control. with Four-Bar ProstheticKnees”, presented at the Proc. ISPO Australia AnnualMeeting, 2003.C.W. Radcliffe, “Four-bar linkage prosthetic kneemechanisms:kinematics,alignmentandprescription criteria”, Prosthetics and orthoticsinternational, vol. 18, no. 3, pp. 159–173, 1994.Nicola Sancisi., Raffaele Caminati.,and VincenzoParenti-Castelli., “Optimal Four-Bar Linkage forthe Stability and the Motion of the Human KneeProstheses”, presented at the Atti del XIXCONGRESSO dell'Associazione Italiana di MeccanicaTeorica e Applicata. Ancona, pp. 1-10, 2009.K.O Berg, “Knee mechanisms for through-kneeprostheses”, Prosthetics and orthotics international,pp. 107-112, 1983.J. K. Chakraborty and K. M. Patil, “A new modularsix-bar linkage trans-femoral prosthesis forwalking and squatting” Prosthetics and OrthoticsInternational, pp. 98-108, 1994Dewen Jin., Ruihong Zhang., HO Dimo., RenchengWang., and Jichuan Zhang., “Kinematic anddynamic performance of prosthetic knee jointusing six-bar mechanism”, Journal of RehabilitationResearch, pp. 39-48, 2003.K. H. Low, “Subject-oriented overground walkingpattern generation on a rehabilitation robot basedon foot and pelvic trajectories” Procedia IUTAM 2,pp. 109-127, 2011.Singiresu S. Rao., Engineering Optimization Theoryand Practice, Fourth Edition. John Wiley & Sons, Inc.,Hoboken, New Jersey, Published simultaneously inCanada, 2009.45
KEYWORDS: Knee prosthesis, Knee Stability, Four bar Linkage, Six bar linkage 1. INTRODUCTION Prosthesis must provide what has become known as knee stability. In an artificial knee, heel contact is the
using a knee-stability diagram (Figure 3(b)).8,9 Like the four-bar knee linkage, the polycentric knee is widely adopted by amputees because the knee flexion angle increases in which the ICM assumes a series of positions (Figure 3).8 The four-bar prosthetic knee with elevated ICM has been available for many years and has the general appearance
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