Simulation of the motion of a four bar prosthetic knee mechanism fitted with a magneto-rheological damper

A comprehensive study on the application of magneto-rheological (MR) damper in above knee prosthetic is introduced. The stance phase and the swing phase were studied. The knee model was composed of four-bar prosthetic knee mechanism with an MR damper. The equations of motion, of the model, were obtained using Newton's second law at the sagittal plane. To avoid the difficulties of the inverse dynamics, due to the damper’s nonlinear characteristics, a simplified approach was suggested for the Spencer model. This simplified approach was used to calculate the input voltage to the damper to obtain the desirable control force. The input data was obtained from the experimental work by sticking surface land marks to the artificial leg and filming it during performance of the daily activity. The Stream pix, AVI Edit, and WINanalyze software were used to calculate the global kinematics, for the required surface land marks, at each frame of the recorded motion. The model results represent the voltage required for the MR damper, during most daily activities, of a person who uses this kind of an artificial leg. The results showed that the required voltage depends on displacement, velocity and calculated force.


Introduction
Walking is human's natural means to move from one place to another. Walking or gait cycle involves the motion of many of the body limbs and joints. Knee plays an important role in the gait cycle. Motion of knee joint is not that simple since it involves planar rolling and sliding motions as well as out of plane motions. Artificial limbs are used to enable amputees to function normally. Above knee artificial leg has not been so successful in the past. A concept and design of polycentric knee arthroplasty based on the biomechanics of normal knee movement was presented by Frank [1]. The diseased articular surfaces of the femoral condyles and tibia plateaus are replaced separately by prosthetic implants secured with cement. A simulation of a total knee joint replacement subjected to a single gait cycle within a knee wear simulator using finite elements was performed by Godest et al. [2]. Bennett and Appoldt [3] measured the loading on an above knee fiber-glass socket by building the socket with incorporated pressure transducers. Unfortunately, their results were accurate only for the single socket used in the experiment. This is because the fact that all modern sockets have different geometries and external loadings due to differences in the amputees. Bielefeldt and Schreck [4] found the difference in loading of four different sockets, during stance, for one patient. Their sockets were, also, built with incorporated transducers. Their results indicated that there is a general trend for the load paths and it can be gauged. Dorious [5] investigated the dynamic response of the human leg in torsion focusing on snow sking injuries. A biomechanical model of the leg system was developed and injury mechanisms to the ankle, knee and tibia were presented. The objective was to determine how Porarinsson [17] attempted to investigate the magnetic fields and braking torque in a magneto-rheological (MR) prosthetic knee. The knee used magnetic fields, MR fluid, and thin rotary blades to vary the knee resistance in real time. He pointed out that it is important to investigate the strength of the magnetic fields for different design parameters.
Hua et al. [18] introduced the functional demand of the above-knee prosthesis design. The advantages of the four-bar link mechanism and the magneto-rheological (MR) damper were analyzed in detail. The fixed position of the MR damper is optimized and a virtual proto type of knee joint was given.
The goal of the present study is to determine the input voltage required for the MR damper during most daily activities for the amputee person based on the measured displacements, velocities, accelerations and calculated forces during swing phase period.

Model of above knee prosthesis
The artificial knee was modeled as a two-dimensional sagittal plane linkage, consisting of two segments with a four bar mechanism with magneto-rheological fluid damper that simulates the action of the muscle as shown in Figs. 1 and 2. The four bar mechanism has an instantaneous center of rotation (ICR) which has a semicircular pathway located in the femoral condoyle. The instantaneous center of rotation of the socket relative to fixed shank-foot can be located at intersection point (N) of the centerlines of the posterior (EF) and anterior (CD) links.
The dimensions of the four-bar knee mechanism were obtained by using instantaneous center of rotation of shank with respect to socket for proper cosmetic appearance and are given in Fig. 2. C, D, E and F are four-bar joints, G and H are damper fixation joints, is mass center of shank segment J is ankle joint K is mass center of foot segment and AM is thigh segment. The angles and thus the angular velocities and accelerations are considered positive in the counter clockwise direction in the plane of the movement, which is essential for consistent use in subsequent kinetic analyses [19]. Table 1 shows the mass of body segments. The controller force ( ) regulates the rotation of the leg to properly prepare it to bear weight at the conclusion of swing phase. The forces in the links (FE) and (CD) pass through the knee center ( ). In deriving the dynamic equations of the motion it will be convenient to designate points of interest in the mechanism linkage. All vectors passing through the points of interest were referred to an arbitrary fixed coordinate system. The equation of motion of the artificial leg is obtained by applying Newton's second law at the agitate plane. By taking the moment about the point ( ) for both the external and effective forces: For the movements like settling on a chair and standing-off from it, the ground reaction force and chair reaction were taken into considerations. The governing equations may then be written as:  The damper model of the present study is due to Spencer model [21], see Fig. 3. The equations of this model are: , are unit vector in -direction and -direction, is damper force vector, and are mass of the shank and the foot respectively, is the Gravitational acceleration, is the vector of normal distance between damper force and point , and are mass moment of inertia of the shank and foot respectively, is the angular acceleration of the shank, and are the total linear acceleration of the mass center of the foot and the shank, , , , are linear acceleration of the shank and foot center in and direction respectively, ( − ) is the distance vector between points and , ( − ) is distance vector between points and , and are horizontal and vertical components of the vector ( − ), and are horizontal and vertical components of the vector ( − ).
The values of the MR damper parameters are listed in Table 2.

Inverse dynamic model of MR damper
It has been suggested that the force generated by the MR damper cannot be directly commanded. Only the voltage u supplied to the current driver for the MR damper can be directly changed. A clipped-optimal control algorithm proposed by Dyke et al. [23,24] is to be used based on acceleration feedback, along with MR damper to approximately generate the desired control force . The command voltage is determined based on how the damper force F is compared to the desired control force and may be written as: where, is the maximum voltage to the current driver associated with saturation of the magnetic field in the MR damper and is Heaviside step function. However, it was found that the force generated by the MR damper using the clipped algorithm oscillated about the target control force. This was primarily due to the switching of voltage at either its minimum or maximum value.
Tsang and Su [25] proposed an approximate formula based on the Bouc-Wen hysteretic model with the following two assumptions. The external displacement was much larger than the internal displacement . The evolutionary variable was assumed to be equal to its ultimate hysteretic strength (Spencer [26]), i.e.: Based on the two assumptions, the damper force is: It can be seen from Eq. (14) that the damper force is a function of the output voltage , the external displacement and the velocity . If the magnitude of the damper force is specified, the output voltage can be obtained at any instant of time: So the previous equation was putted in a simple quadratic form:

Experimental work
The walking analysis was done by placing twelve visual surface marks on the subject's body and capturing the 2-D view using a camera placed at a distance behind the subject. The goal of the experiments is to find the position, velocity and acceleration of the surface marks as illustrated in Fig. 2. One healthy male subject (20 years old, 1.8 m and 690 N weight), with no history of knee pain, participated in this experiment. After collecting the positions of the surface land marks three software's were used for extracting their global position at each frame. The First software named, Stream pix, was used to cut the desired motion, from the digital camera, by defining the start and the end of the required motion and to save this motion into a laboratory computer. The second software named AVI edit was used for saving motion pictured from the camera. The third software named WIN analyze was used to track an unlimited number of objects in Video files automatically, and to present the results in almost any desired way (x-y graphs, velocities, accelerations, etc.). The digital video camera used in this study was a JVC, GR-DVX507A. After obtaining the data, the position, velocity and acceleration for the twelve surface marks were fed to a MATLAB program, as well as, the data of the subject. Fig. 4 shows the tracing of three marks F1, F2 and F3 located on the foot of the subject as a sample on the WIN analyze software screen. Fig. 5 illustrates the main components used in collecting the two-dimensional data. The captured motion was during running. The output from the software as a sample is the displacement of the mark in two dimensional coordinates ( , ) with respect to time. The horizontal and vertical coordinates of all marks are shown in Figs. 6 and 7 respectively. Table 3 summarizes the motion during most daily activities of the subject. Some pictures of the experiments are shown from Figs. 8-15. Five-degree ramp ascent 8 Fifteen-degree ramp descent 3 Ten-degree ramp ascent 9 Twenty-degree ramp descent 4 Fifteen-degree ramp ascent 10 The stairs ascent 5 Twenty-degree ramp ascent 11 The stairs descent 6 Five-degree ramp descent

Experimental Set-up
The digital video camera was positioned at a suitable distance according to the required motion of the subject.
Appropriate zoom magnification was set for maximizing the portion of the motion to be captured. The environment had maximum lighting available in order to provide contrast to the surface land-marks color. Shutter speed was set at 30 frames per second to allow enough light for clear definition of the image. The data are to be fed to the inverse model. Iris settings were automatically set to correspond to the shutter speed so as to provide the brightest and clearest image.
Surface marks were secured to the points of interest on the subject as shown in Fig. 2 and then videotaped to identify the subject and to allow the software to recognize the position of the surface land-marks in two dimensions. Once the previous step was performed, the subject performed the motions as shown in Table 3.
A computer program was designed and developed using MATLAB and SIMULINK software capable of calculating the required voltage during the most daily activities of the person. Fig. 16 shows a block diagram for the software program developed and used in the present study.

Validations of the Tsang-Su model
The axial damper displacement and axial velocity during comfortable walking (swing phase) is represented by the relative motion between ends of the damper which is represented by marks H and G shown in Figs. 1, 8, and can be obtained from WIN analyze program. Table 4 shows the values of axial damper displacement ( ) and axial velocity ( ) during comfortable motion.
To examine the assumptions put forward in the Tsang-Su model, two models of MR damper (Spencer and Tsang-Su) were constructed using SIMULINK program. The input displacement to the two models at zero volts is assumed to be harmonic for simplification. The amplitude and frequency of harmonic motion is selected to be 0.008 m and 5 Hz. The selected amplitude and frequency are suitable to give the axial displacement and axial velocity of damper in the range of comfortable walking as listed in Table 4.
Fig . 17 shows the validity of the first assumption which shows that, the value of the external displacement is larger than the value of the internal displacement . Fig. 18 shows the variation of the variable of the two Models which are approximately identical to each other during the cycle. It is to be noted that the value of changed between two fixed values with sharp change at the following times 0.1 s, 0.2 s, 0.3 s and so on till the end of the cycle, and therefore, the value of the force will have a sharp change in its value taking place at the same previous time intervals. Fig. 19 shows the output forces of the two models which are approximately identical to each other during the cycle. This shows that, the MR damper model with the assumptions given above can be used in the present study.

Input data to the inverse model
The input data required for the inverse dynamic model during most daily activities are not similar. So the full data for the comfortable walking will be clarified first and the rest of the data of the movements are to follow. All data obtained from WIN analyze program are to be fed to the inverse dynamic model which was programmed in SIMULINK software.

Comfortable walking
During the swing phase of the patient's comfortable walking, the shank angle was increased with time from 1.95 to 3.18 rad. at the terminal swing in the manner shown in Fig. 20. The shank angular acceleration is seen to gradually increase from 26.1 to a maximum value of 43 rad/s 2 during the period of time of 0.18 seconds from the pre-swing and then begin to gradually decrease and reaches a value of -32 rad/s 2 during 0.9 s at the terminal swing as shown in Fig. 21. show the foot and shank linear acceleration. They start with a slight increase from pre-swing to initial swing and then start to decrease gradually until they reach the mid-swing and then start again to gradually increase until they reach a maximum value at about 90 % of the swing time. The so called evolutionary variable depends on the parameters whose adjustment allows shaping the linearity of the control during unloading and the smoothness of the transition from the pre-yield to post-yield area as well as the axial damper velocity as shown in Fig. 24. It is an indication to the axial damper velocity sign change.
The damper force as observed from the equation of motion, (see Eq. (1)) depends on inertia torque, inertia forces of shank and foot and their distance from the instantaneous center of rotation and other parameters like the weight of the shank -foot system and their distance from the instantaneous center of rotation. But from the comparison of all previous parameters it is obvious

Results and discussions
The results and their discussions are to be made for the different walking conditions starting with the case of comfortable walking. The results are to deal with the voltage obtained from the simulation and are to be fed to the MR damper in order to give the prescribed motion and appropriate damping force during the wing time inspects. It is clear that this voltage depends on the required displacement, acceleration and the damping force.
Starting by the comfortable walking zone, Fig. 48 shows the required damper voltage which decreases and then increases gradually in the beginning of the cycle following the trend of the damper input force.
It is noted that irrespective of the local sudden change in the voltage appearing at approximately 0.47 s of swing time the voltage continues to increase gradually and decreases again till the end of the swing time. This shows that the effect of ubrupt change in the voltage has no considerable effect on the damping force. This may be attributed to the low rate of momentum transfer due to the smallness of the masses involved.
The damper voltage required during a ramp ascent is shown in Fig. 49. It can be seen that from the beginning of the swing time till a value of 0.325 s, where the same ubsrupt change in damper velocity takes place that, the voltage follows the trend of the required damper force behavior. From the ubrupt sudden increase in voltage till the end of the period the voltage decreases gradually until it reaches an asymptotic value of -0.425 V.  The abrupt rise in voltage may be attributed to the larger average rate of momentum required during a ramp descent and is in general in qualitative agreement with that of ramp ascent as can be seen from Fig. 50.
It is seen that disregarding the small zone ahead of swing time 0.035 s the voltage follows the same trend as the required average damping input force.
The required damper voltage is also shown in Figs. 51 and 52 for cases of stairs ascent and descent respectively. The qualitative agreement on the behavior of the voltage for both cases is clear and in general follows the trend of the required damper forces as can be seen from Figs. 43 and 47.
From the foregoing discussion it can be seen from the results of the required voltage that it follows in general the same trend as that of the input damper force irrespective of the type of motion. Attia E. M. suggested the model and planned for the experimental work, performed the dynamic equation of the model, analyzed the data from experimental work, constructed and arranged the manuscript to be ready for publication. Abd El-Naeem M. A, performed the dynamic equation of MR damper model and its inverse, take the pictures during the experimental work by camera, constructed the computer programs, collected the references. El-Gamal H. A, revised the text especially the results and discussion, performed the conclusions. Awad T. H., revised the text and computer programs, commented on computer program block diagram. Mohamed K. T., revised the dynamic equation of the model, revised the text and references, commented on the design of experimental work.

Conclusions
This study introduced a dynamic model of artificial knee for the amputee person. The knee is formulated from a four bar mechanism connected by a Magneto -rheological damper to control the motion of the amputee person during most daily activities. Also the present work studies the input voltage required for the MR damper based on the displacement, velocity, acceleration and force during swing phase period. The value of input voltage is important for the damping force and moment to achieve the realistic and comfortable motion or any activity for the amputee person. Such a model can be used for improving the mechanical design of the artificial knee, both by suggesting improvements of the MR damper itself and by understanding the relation between an applied voltage and the time dependent damping force of the knee during motion.
From the experimental work and analysis of the presented results the following conclusions are drawn, 1) The voltage required for MR damper to cope with most daily activities is always following the trend of the damping force.
2) The variation of the voltage over the swing time for comfortable motion case is relatively small and has a maximum of order 1.6 V which is tractable by simple control units.
3) For the case of ramp ascent and decent the maximum change in voltage is of order 1.5 V for both cases and close to the comfortable motion case.
4) The variation of the voltage for stairs ascent and descent over the swing time does not exceed the order of 2.5 V for both cases and as far as the case of control in concerned it matches with the other cases of motion considered.