Optimization of Six Bar Knee Linkage for Stability of Knee Prosthesis
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1 Optimization of Six Bar Knee Linkage for Stability of Knee Prosthesis Narjes.Ghaemi 1, Morteza. Dardel 2, Mohammad Hassan Ghasemi 3, Hassan.Zohoor 4, 1- M.sc student, Babol Noshirvani University of Technology, Babol, Iran. (Corresponding author) 2- Assistant Professor, Babol Noshirvani University of Technology, Babol, Iran Assistant Professor, Babol Noshirvani University of Technology, Babol, Iran. 4- Professor, Sharif University of Technology, Tehran, Iran. Received July 2012 Revised Sept 2012 Accepted Nov.2012 ABSTRACT: Loss of lower extremities has been one of the human problems in the human life. Therefore the optimal design of human lower limb knee prostheses is fundamental in order to restore the lost functionality and aesthetic aspect of the amputee s locomotion. This work presents an optimization procedure for the synthesis of a six-bar linkage for knee prosthesis and its comparison with four-bar linkage counterpart. Different form of six bar linkages, such as Stephenson and Watt mechanisms are considered and optimized. This study shows that the performance of six bar mechanism case I (SBM1) is better than other mechanisms. 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 most critical period of the stance phase for knee security. Most knee prostheses available are divided into two large groups: prostheses with a single axis of rotation and polycentric prostheses. Single axis prostheses have a fixed instantaneous centre of rotation of the relative motion between the femur and shank, are cheap but prostheses have not a good performance and do not provide high stability in stance phase. The second group, polycentric prostheses are which the position of the ICR changes at each flexion angle. Polycentric four-bar knee has been studied by many researchers such as Sancisi, Raffaele, Castelli [1]-[3]. Four-bar linkage mechanisms for the trans-femoral amputee have been widely used in the Prostheses knee for many years and have been studied by many scientists and scholars. Radcliffe has done useful studies about knee stability by changing the location of instantaneous centre of rotation of the relative motion between the femur and shank and introduced three different classes of four-bar linkage prosthetic knee mechanisms, for different groups of trans-femoral amputee [1], [2]. Sancisi and Raffaele, have been investigated about knee prosthesis stability and voluntary control [3]. But although they are useful and helpful for certain amputees, they are fitted in a limited number of cases. Other types of knee mechanisms for through-knee amputees can also be found. Six-bar linkage knee-ankle mechanisms (SBM) have been used in some prosthetic knees in the recent years, but there are not so many researches about the advantage performance of the Sixbar mechanism as knee prostheses. At the University of California, Berkeley, the possible advantages of using six-bar linkage mechanisms have been investigated.this mechanism provides the possibility of increased range of knee motion, better cosmetic, improved stance phase stability and swing phase control as compared to four-bar designs. These advantages are achieved at the expense of added weight and complexity [4]. Patil and Chakraborty designed a particular six-bar knee-ankle mechanism to produce near normal patterns of motion during walking and squatting [5]. The kinematic and dynamic performance of the six-bar mechanism used in the prosthetic knee is investigated by Dewen Jin [6]. This paper will describe the kinematics performance of several types of six-bar mechanisms and discuss the differences criteria for four different classes of six-bar 38
2 linkage mechanisms for fitting to amputees and compare improvement and weakening performance of the knees with four-bar linkage knee. To specify a mechanism to simulate a pre-specified ankle trajectory it is necessary to obtain the length and initial orientations of linkages. For this purpose an optimization procedure is proposed. From this optimization procedure the length and initial orientations are obtained such that total error between pre-specified and simulated ankle trajectories to be as minimum as possible. To obtain the relativistic kinematic, due to the nonlinear equations in the six-bar mechanism, a new method is presented. To achieve natural walking motion, the hip and ankle flexion, and ankle trajectory are necessary; the angles can be calculated by solving the inverse kinematics [7]. And also in this paper is briefly discussed about the stability of the knee by changing the location of instantaneous center. In the next section, we discuss about kinematic analysis of Mechanisms. 1- Inverse kinematic analysis of knee Bar Mechanism In this section at first kinematic analysis of four and six bar knee mechanism are presented. Known four bar and six bar mechanisms are single degree of freedom mechanism, in which one of the linkage attached to the ground is input. For inverse kinematics analysis the locus of a point on the coupler is know and angles and positions of other linkages will be determined. In using these mechanisms as an artificial knee, with considering the motion with respect to hip, a two degree of freedom mechanism will be obtained. Hence the hip and ankle flexion are necessary to solve the inverse kinematics of knee for given ankle trajectory. In continue at first inverse kinematics of knee mechanism for four bar knee and then different form of six bar mechanism will be presented Inverse kinematic analysis of knee with fourbar mechanism A schematic of the four bar knee linkage to simulate human locomotion is shown in Fig.1. For this mechanism the length of thigh is assumed to be known, and unknown parameters are length of different links of four bar mechanism, length of shank and initial orientation of links fixed to the thigh and shank. Fig. 12. Schematic of a knee with four bar knee linkage. For obtaining the inverse kinematics of the mechanism is it necessary to obtain the angles of different links of the mechanism in terms of position of the couple point of and known angles of and. For this purpose it is necessary to write loop closure equation of the mechanism. For Fig.1 this equation in vector form is as follows: (1) All angles are measured relative to horizontal axis in counter clockwise direction. With separating real and imaginary parts of Eq.(1), we have cos cos cos cos (2) cos sin sin sin sin (3) sin From these equations angles and can be obtained as follows: sin (4) 2 sin 2 where : (5) tan (6) With this known value of link s angles, the position of ankle point is determined as: cos cos cos cos cos (7) 39
3 sin sin sin (8) sin sin Knee with four-bar mechanism (FBM) shown in Fig. 1, has 9 unknown geometrical parameters, which can be obtained through an optimization procedure described in the next section. These unknown parameters are:,,,,,,,,] (9) 1.2. Inverse kinematic analysis of knee with six-bar mechanism Different types of six bar linkages are known in machine theory. Among them six bar revolute joint mechanism known as Watt and Stephenson are familiar. Compared with four-bar mechanisms, six-bar mechanisms have much more design parameters. The excess links provide more flexibility in design and with them we can obtain some requirement which cannot be obtained by mechanism with lower links. Four different types of six bar linkages mechanism are shown in Fig. 2. In following kinematic analysis of each of these mechanisms are presented. As we will see, inverse kinematics of six bar linkage cannot be obtained in a similar procedure given for four bar knee mechanism. Hence a simple method will be presented for solving the inverse kinematics analysis of these six bar linkage, which can be extended to more complex mechanism too. In similar to the four bar knee case, the known parameter and angles are and and and other angles and parameters are unknown. Fig. 2. Schematics of knee with different six bar linkages The loop closure equations for different cases of Fig. 2 to obtain the angles of different links are: I: 0, (10) 0 0, 0 0, 0 IV: 0, 0 which can be written in real form as: (11) (12) (13) I: cos cos cos cos 0 sin sin sin sin 0 cos cos cos cos 0 sin sin sin sin 0 (14) 40
4 cos cos cos cos 0 sin sin sin sin 0 cos cos cos cos 0 sin sin sin sin 0 (15) cos cos cos cos cos cos sin cos sin sin sin sin (20) cos cos cos cos 0 sin sin sin sin 0 cos cos cos cos cos 0 sin sin sin sin sin 0 (16) cos cos cos cos 0 sin sin sin sin 0 IV: cos cos cos (17) cos cos 0 sin sin sin sin sin 0 The ankle point position coordinates for different cases of Fig. 2 are: I: cos cos cos cos cos cos cos sin sin sin sin sin sin sin cos cos cos cos cos sin sin sin sin sin (18) (19) IV: cos cos cos cos cos sin cos cos cos cos (21) Solving together of Eqs.(10, 14), Eqs.(11, 15), Eqs.(12, 16) and Eqs.(13, 17) in a similar method applied to the four bar linkage is possible, but is very difficult to obtain and useless. Hence a different procedure for obtaining the unknown angles is presented in continuo, which can be applied to more complex problems. Unknown parameters which must be obtained through an optimization procedure to simulate pre-specified knee trajectory for different case of six bar linkage mechanism are: I: (22),,,,,,,,,,,,,, (23),,,,,,,,,,,,,, (24),,,,,,,,,,,,,,] IV: (25),,,,,,,,,,,,] 2.1. Solution procedure for solving inverse kinematics of complex mechanisms and robots Forward and inverse kinematics problems of robots and mechanism are in terms of sin or cos of different angles. These terms maybe in multiplied or power form such as cos cos or cos. With using triangular identities such expressions can be expressed in terms of linear terms of cos or sin such as cos, sin, cos. With using Taylor expansion each of these terms can be expressed as follows: cos cos Δ sin (26) Δ cos 2 sin sin Δ cos (27) Δ sin 2 where is the value for the unknown variable at step k th. For small values of Δ, we have 41
5 cos cos Δ sin (28) sin sin Δ cos (29) With substituting Eqs. (28-29) in loop closure equation for mechanisms or robots, with known value of, Δ can be obtained. The presented approach to solving inverse kinematics analysis of mechanism or robots is applied to the proposed six bar knee mechanism. Hence we have: sin I: sin cos cos 0 sin 0 cos sin 0 θ cos 0 θ sin sin θ cos cos θ (30) cos cos sin sin cos cos sin sin cos cos sin sin cos cos sin sin sin sin cos cos sin 0 θ cos 0 θ sin sin θ cos cos θ (31) cos cos sin sin cos cos sin sin cos cos sin sin cos cos sin sin sin sin cos cos (32) sin 0 cos 0 sin 0 θ cos 0 θ sin sin θ cos cos θ cos cos sin sin cos cos cos sin sin sin cos cos sin sin cos cos sin sin sin sin cos cos 0 sin 0 cos sin 0 cos 0 sin sin IV: cos cos (33) cos cos sin sin cos cos cos sin sin sin cos cos sin sin cos cos cos sin sin sin and finally: (34) 2.2. Optimization Procedure Optimization is the act of obtaining the best result under given circumstances. The ultimate goal of all such decisions is either to minimize the effort required or to maximize the desired benefit. Since the effort required or the benefit desired in any practical situation can be expressed as a function of certain decision variables, optimization can be defined as the process of finding the conditions that give the maximum or minimum value of a function [8]. The purpose of the optimization procedure is to find a 4BM or 6BM which follows the experimental knee motion as better as possible and, at the same time, and should be designed in order to restore the stability during walking. Meantime, the dimensions of links should be within an acceptable range. Here, the objective function value F(X) (the function to minimize) can be obtained as the sum of the squared distances between the i-th experimental position of the reference point (, ) at the i-th experimental flexion angle and the respective position of the coupler point P at the same flexion angle (, ), computed 42
6 according to the equations. So the optimization problem is expressed as Min (35) where n is the number of selected points in a gait cycle, 21. Results and Discussion The reference data for the simulation of ankle knee motion are given in [7]. After the optimization, the design parameters for different mentioned knee mechanism are obtained. These results are shown in Table 1. The dimensions are in millimeters. Table 1. Design parameters obtained from optimization procedure for different mechanisms mechanism Design parameters Four-bar 55.5, 48, 46, 54, 399, 1, 47, 1.5, 4.1 Six-bar(1) X = [54, 63, 63.5, 71, 87, 92, 74.5, 79, 386, 53, 78, 3.18, 0.04, 0.53, 0.37] Six-bar(2) X =[45, 53, 40, 40, 57, 20, 70, 38, 19, 28, 361.5, 0.22, 2.5, 3.14, 0.7] Six-bar(3) X =[29, 39.5, 30.5, 34, 20, 11, 10,8, 9.5, 7,8, 390, 0.17, 1.03, 0.4, 2.67] Six-bar(4) X =[48, 42, 45, 61, 52, 31, 86, 21, 21, 361, 2.9,. 3,. -23] The comparison of the generated trajectory of the ankle joint with expected is shown in Fig. 3. The mean square errors for four-bar linkage, six-bar linkage case I, sixbar case II, six-bar case III, six-bar case IV and six-bar case V are Err 0.038%, Err 0.16%, Err 0.51%, Err 0.86% and Err 0.36%, respectively. The comparison of the generated trajectory of the Instant center with is shown in Fig. 4, shows that location of the Instant center in the six-bar mechanism1 is more elevated than four-bar mechanism and this makes that SBM 1 is more stable than FBM. In Fig. 4c, the most interesting feature of this centrode is the almost constant height of the instant centers. According to Fig. 4b, this mechanism is much variation in the instant center. The comparison of the generated trajectory of the Instant center with is shown in Fig. 4 shows that location of the Instant center in the six-bar mechanism1 is more elevated than four-bar mechanism and this makes that SBM 1 is more stable than FBM. The comparison of the generated trajectories of the ankle joints shown in Figs. 3, shows that the performance of the SBM1 is better than SBM4, performance of the SBM4 is better than SBM2 and the performance of the SBM2 is better than SBM3. 43
7 (b) Fig 3. The expected trajectory of the ankle joint and realized trajectory by four-bar and six bar linkage. (c) (a) (d) 44
8 (e) Fig 4 a: Trajectory of the Instant center by four-bar mechanism b: six-bar case I, c: six-bar case II, d: sixbar case III, e: six-bar case IV. Conclusion and discussion The kinematic performance of the several different mechanisms such as four-bar linkage and six-bar linkage are shown in above figures. And are compared improvement and weakening performance of the sixbar knees with four-bar linkage knee. Meantime a new method is presented for solving nonlinear equation of the six-bar mechanism which eases to obtain kinematic relations. The comparison of the trajectory of the ankle joint in swing phase of the six-bar linkage knee with that of a four-bar knee mechanism shows that six-bar linkage knee has better performance than four-bar knee mechanism. Also the comparison between various sixbar mechanism shows that the performance of six bar mechanism case I is better than other mechanisms. REFERENCES [1] Charles W Radcliffe and ME. Deg, Biomechanics of Knee Stability Control. with Four-Bar Prosthetic Knees, presented at the Proc. ISPO Australia Annual Meeting, [2] C.W. Radcliffe, Four-bar linkage prosthetic knee mechanisms: kinematics, alignment and prescription criteria, Prosthetics and orthotics international, vol. 18, no. 3, pp , [3] Nicola Sancisi., Raffaele Caminati.,and Vincenzo Parenti-Castelli., Optimal Four-Bar Linkage for the Stability and the Motion of the Human Knee Prostheses, presented at the Atti del XIX CONGRESSO dell'associazione Italiana di Meccanica Teorica e Applicata. Ancona, pp. 1-10, [4] K.O Berg, Knee mechanisms for through-knee prostheses, Prosthetics and orthotics international, pp , [5] J. K. Chakraborty and K. M. Patil, A new modular six-bar linkage trans-femoral prosthesis for walking and squatting Prosthetics and Orthotics International, pp , 1994 [6] Dewen Jin., Ruihong Zhang., HO Dimo., Rencheng Wang., and Jichuan Zhang., Kinematic and dynamic performance of prosthetic knee joint using six-bar mechanism, Journal of Rehabilitation Research, pp , [7] K. H. Low, Subject-oriented overground walking pattern generation on a rehabilitation robot based on foot and pelvic trajectories Procedia IUTAM 2, pp , [8] Singiresu S. Rao., Engineering Optimization Theory and Practice, Fourth Edition. John Wiley & Sons, Inc., Hoboken, New Jersey, Published simultaneously in Canada,
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