Chapter 6
Investigation about penalty parameters
This chapter has been divided in two main sections: in one there is the explanation of the experiment and the results obtained; in the other one there is the construction of the Matlab model and the comparison with the results obtained from the experiment.
6.1 The
experiment
6.1.1 Presentation of the experimental setup 6.1.1.1. Aim of the experiment
As already showed in the previous chapter, the investigation about the hypertension of the leg is essential to have a more stable walk; on the other hand, the program of the penalty constraints (constraints that allow the impact of the shin against the knee in the numerical model) was explained in the chapter four. Now it is necessary to find a way to evaluate this constraints from a quantitative point of view. The idea is to make an experiment using the prostheses 3R60, and to compare the behaviour of the physical model respect to the numerical model built using Matlab. The experiment will consist in the simulation of the movement of the shin around the knee using a suitable support structure for the knee prostheses 3R60. The aim of this experiment is to obtained, as result, the evolution of the angle that the physical model forms with the vertical axes during the time. In this way the experiment represents a good means to measure the impact of the shin against the knee.
6.1.1.2. Presentation of the physical model of the leg
The physical model that will be employed in the experimental setup is shown in the following figure (Fig.6.1):
Looking at the picture, it is possible to identify each component of the physical model: the prostheses, the shin and the foot. The length and the weight of these elements have been defined with the help of the Otto-Bock Healthcare Company which has given all the physical data about them and taken from the experiments made in the previous research to understand the way of programming of the penalty constraints.[23].
The prostheses model
The special element “spring” showed in the picture 6.2 has been removed from the prostheses, to allow the rotation movement of the shin around the joint between the frame and the prostheses, rotation movement not possible with its presence. This element has the function to regulate the flexion phase of the prostheses during the walk as explained in the chapter two so that it has not influence on the impact of the shin against the prostheses.
Fig. 6-2: The special element "spring"
The weight of the prostheses is given by the sum of the weight of each component of it: the rigid bars, the special element Otto-Bock and the hinges. In this way it has been obtained:
Total weight: 2.65 Kg
The shin model
The mass of this element is not biomechanical compatible as shown in the scheme below. But this is compatible with the experiment because what is expected to create a physical model suitable for people with an amputated leg which must be as light as possible.
Biological data: Length: 0.3 m Physical data: Length: 0.315 m
Weight: 4 Kg Weight: 0.92 Kg
Inertial moment: 0.03 Kg m2 Inertial moment: 0.0076 Kg m2 Scheme 6-1: Properties of the shin
The foot model
The foot is modelled as a little bar with two cylindrical weights. In this case the biological value of it has been considered.
In the picture 6.1 it is possible to identify also the joint elements. This elements link the thigh with the prostheses and the prostheses with the shin. In this case the higher joint will link the prostheses with the support structure. Their contribute on the weight of the model is:
Weight: 0.140 Kg
Thus in the picture 6.3 it is possible to see the physical model compared to the real model built.
Fig. 6-3: Physical model and real model built
6.1.1.3. Description of the experimental setup
To allow the rotation movement of the shin around the joint of the prostheses a suitable support structure for the physical model has been built. This frame is showed in the picture 6.4. The initial position of the leg has been set using a scheme (a draw showing the initial angle) placed on the back of the physical model. Finally, with the help of a rubber rope and reference screw, it has resulted to be possible to fix the leg on the initial position (fig.6.5). The angle that the model forms with the vertical axes in this position is 30°. Looking at the picture, it is possible to see also some references (the black-white circles), placed in some particular points of the model, which will correspond to the nodes of the Matlab model. Thus the idea is to follow the evolution
during the time of one of those little circles, measuring the angle that it forms with the vertical axes.
Fig. 6-4: The support structure Fig. 6-5: The initial position
6.1.2. The measurements
The way used to obtain the measurement has been a film made using a digital camera, placed far from the structure at a distance of three meters more or less. The experiment has been run for 2 seconds, time in which there are two impacts of the shin against the knee and after which the leg remains still.
6.1.2.1. The film
The camera does 25 photos every second following the European standard. Thus the film has been shared into all its frame during the time of the experiment. Finally 50 photos has been analyzed using a graphic software. In the picture 6.6. there is the frame corresponding to the beginning of the experiment, in which the circle taken as reference point for the measurement of the angle, can be observed. The calculus of the position of this point has been made manually for every photos using the trigonometry theory: in fact the angle has been obtained from the previous calculus of the position of that point in the picture through the measurement of its position along the X and Y axes (in pixel), respect to the fixed point, around which the model rotate. The fixed point is the first point of the prostheses that links it with the joint. Measuring
an angle, allows to avoid the problems connected with the no correspondence between the real unit of measure used (meter) and the unit of measure used in the graphic analysis (pixel). At the end this choice has resulted to be correct.
Fig. 6-6: The point taken as reference
6.1.2.2. Analysis of the results
Using this method, the evolution of the angle, during the time, that the point chosen forms with the vertical axes, has been found. The time step is Dt= 0.04 sec.
according to the properties of the digital camera used. In the scheme 6.2 it is possible to observe how the measures have been taken, in the scheme 6.3 the values of the angle during the time are shown and in the scheme 6.4, the relative graph is displayed. The graph shows that the measurement seems to be correct: in fact, as the experiment demonstrates, two impacts of the shin against the prostheses are visible before the end of it. The negative values of the angle are right: in fact, looking at the picture 6.7, it is possible notice that the angle formed by the point chosen with the vertical axes is negative when there is the impact, according to the reference system used. It is possible to observe also that the during the impact, there are vibrations of the frame which have a big influence upon the behaviour of the model after the first impact. Due to these vibrations, the impact time is bigger, respect to the reality. For this reason only the first impact of the shin against the prostheses will be studied.
Fig. 6-7: The impact angle
Looking at the figure, the angle is given by:
L x y x L 2 2 arcsin = + ⎛ ⎞ θ = ⎜ ⎟ ⎝ ⎠
N° of the frames X (pixel) Y (pixel) L (pixel) q (degree) t (sec.) Dt = 0.04 sec. 1 21 73 75,96052 16,04900479 0 2 20 72 74,72617 15,524111 0.040 3 19 73 75,43209 14,58891873 0.080 4 16 74 75,70997 12,20046873 0,12 5 13 74 75,13322 9,96380419 0,16 6 8 75 75,42546 6,088528154 0,2 7 4 75 75,10659 3,052882515 0,24 8 -3 75 75,05998 -2,290610043 0,28 9 -7 74 74,33034 -5,40379136 0,32 10 -13 73 74,1485 -10,09750438 0,36 11 -13 74 75,13322 -9,96380419 0,4 12 -10 74 74,67262 -7,696051722 0,44 13 -4 75 75,10659 -3,052882515 0,48 14 0 75 75 0 0,52 15 4 75 75,10659 3,052882515 0,56 16 7 75 75,32596 5,332158882 0,6 17 8 75 75,42546 6,088528154 0,64 18 9 75 75,53807 6,842773413 0,68 19 9 74 74,54529 6,934348901 0,72 20 9 75 75,53807 6,842773413 0,76 21 8 75 75,42546 6,088528154 0,8 22 6 76 76,23647 4,513988458 0,84 23 3 75 75,05998 2,290610043 0,88 24 1 75 75,00667 0,763898461 0,92 25 -3 75 75,05998 -2,290610043 0,96 25 -6 75 75,23962 -4,57392126 1 27 -9 74 74,54529 -6,934348901 1,04 28 -12 74 74,96666 -9,211026541 1,08 29 -12 74 74,96666 -9,211026541 1,12 30 -11 74 74,8131 -8,455027677 1,16 31 -8 75 75,42546 -6,088528154 1,2 32 -8 75 75,42546 -6,088528154 1,24 33 -5 75 75,16648 -3,814074834 1,28 34 -3 75 75,05998 -2,290610043 1,32 35 -2 75 75,02666 -1,527525442 1,36 36 -1 75 75,00667 -0,763898461 1,4 37 0 75 75 0 1,44 38 0 75 75 0 1,48 39 0 75 75 0 1,52 40 -1 75 75,00667 -0,763898461 1,56 41 -2 75 75,02666 -1,527525442 1,6 42 -3 75 75,05998 -2,290610043 1,64 43 -3 75 75,05998 -2,290610043 1,68 44 -4 75 75,10659 -3,052882515 1,72 45 -5 75 75,16648 -3,814074834 1,76 46 -6 75 75,23962 -4,57392126 1,8 47 -7 75 75,32596 -5,332158882 1,84 48 -8 75 75,42546 -6,088528154 1,88 49 -8 75 75,42546 -6,088528154 1,92 50 -8 75 75,42546 -6,088528154 1,96 51 -8 75 75,42546 -6,088528154 2
Evolution of the angle of the node 9 -15 -10 -5 0 5 10 15 20 0 0,12 0,24 0,36 0,48 0, 6 0,72 0,84 0,96 1,08 1, 2 1,32 1,44 1,56 1,68 1, 8 1,92 time (sec.) angl e ( d egree)
Scheme 6.4: The graph of the evolution of the angle
6.1.2.3. Different problems 6.1.2.3.1. Quality of the images
In order to obtain clear images some solutions have been chosen:
• the camera has been placed at a distance of 3 meters and a little bit of zoom was used to avoid to have distortion of the images.
• Two additional table lamps have been fixed in the frame to make the experiment area as lighter as possible.
Anyway after analysing the film and its frames, it has been noticed that only for one point the images have resulted clear enough. For this point (fig.6.6), it has been possible to examine all the frame with a good accuracy.
6.1.2.3.2. Stability of the frame
To obtain a more stable support structure, a lot of metal bars have been assembled together as showed in the picture 6.4, previously shown. Besides, the bar, through which the physical model has been fixed, the central one, has been considerably made heavy by placing on it two big weights (25 kg for each one) in order to reduce the vibration’s effect on the frame, caused by the impact.
6.1.2.3.3. Reliability of the initial position results
The picture 6.8 shows the physical model and the Matlab model. From the analysis of the picture it appears clear that the initial position fixed for the physical model is different from the initial position on which the Matlab model will be placed. This, because, there is no correspondence between the real element of the prostheses and the correspondent element of the Matlab model and this is the reason why the first angle measured is not 30° but 16,04900479° as the scheme 6.3 shows. It is possible to see this angle in the picture 6.8 (the angle J).
Fig. 6-8: The physical model and the Matlab model
Anyway the measurement of the initial condition has been made analyzing the first frame of the film. Since this value does not represent a variation, but rather an absolute value, it is necessary to verify its accuracy. This verification has been made putting on the frame some references as the picture 6.9 shows and measuring the same angle manually. The result has been: J= 14,3239°. Thus, the error done with the analysis of the frame has been: error= 1,72510479°.
As consequence, all the values taken from the analysis of the film have been modified according to the error done. The evolution of the angle definitive is showed in the scheme 6.5.
Fig. 6-9: The measurements of the initial condition
Evolution of the angle of the node 9 corrected -15 -10 -5 0 5 10 15 20 0 0,12 0,24 0,36 0,48 0,6 0,72 0,84 0,96 1,08 1,2 1,32 1,44 1,56 1,68 1, 8 1,92 time (sec) an gle (d egr e e ) Scheme 6.5
6.2. The
simulation
6.2.1 The Matlab model used: analysis of Data file
The Matlab model has been structured in order to represent the physical model of the experiment. First of all, the special element “spring” (fig.6.2) has been removed from the model and, all the nodes have been redefined. Then, besides the prostheses scheme, also the scheme of the joint (between the prostheses and the frame), the scheme of the shin and finally the scheme of the foot, have been set. The foot is modelled by a punctual mass, applied to the extremity of the shin element. All these changes have been applied working on the Data file. The fig.6.10 shows the Matlab model.
Fig. 6-10: The Matlab model
The model has been constrained in correspondence of the node one, in order to block the element one (the joint element); thus the translation, along x and y axis, and the rotation, around z axis, are equal to zero for every node of this element.
Then the physical characteristics of the elements (the mass, the inertial moment and the length), defined in the previous paragraph, have been entered in the file. Particular attention has been given to the initial condition of the model and to the penalty constraint.
6.2.1.1 The initial condition
According to the initial position of the physical model in the experiment, the initial condition of the correspondent element of the Matlab model, has been defined. So the value of the initial angle (J= 14,3239°), found in the experiment, has been programmed for the node 9 (fig. 6.10, if there are two-three numbers near the nodes, this means that two-three elements are linked together in the same hinge) or better for the element 6, of which the node 9 represents the inferior extremity. The calculus of the position of the other elements has been made manually, looking at the frame corresponding to the initial position. Anyway the program makes a control on the initial position of the model in order to verify that all the kinematic constraints are satisfied. In this way the program has fixed the position of the reference element (element 6) on the angle equal to J= 14,1979° as the picture 6.11 shows. This difference is very little, thus it has been possible to ignore this gap.
6.2.1.2. The update of the penalty constraint
These constraints allow the impact of the shin against the knee, as explained in the chapter 3 and 4. Anyway this case is different from that one correspondent to the Ballistick walk model because of the presence of the prostheses. So it’s necessary to update them. The equation of the constraints has to be written considering that the shin is integral with the prostheses, there is no relative displacement between the prostheses and the shin. For this reason the virtual damper and the spring will be applied on the prostheses, and in particular on the element 6. So the constraint equation is:
impact
7 4 0
θ − θ − θ = (6.1)
where, - J7 = the angle of the node 7 - J4 = the angle of the node 4
- Jimpact = the angle on which there is the impact
This choice is justified because, looking at the film, the impact happens when the metal bar, correspondent to the element 6 of the model, knocks against the black element of the prostheses, the protective cap. In the model, this means that there is the impact, when the angle that the element 6 forms with the vertical axis, is equal to
Jimpact. This impact value of the angle has been taken directly from the experiment’s
results and it is equal to: Jimpact= - 11,82260917°. Putting this value on the program of the constraints, the model has simulated the impact in a correct way, or better, similar to the experiment; in fact it has been possible to see the “rebound effect” of the shin against the knee. Before this change, it had been not possible.
6.2.2. Analysis of the simulation
In a previous research the value of the penalty coefficients has been fixed on:
t t K N m C N m 1000 / 1000 sec/ = = (6.2) with the value of the friction’s coefficient of the hinges equal to: h= 0.001 Nsec/rad.
Anyway the evolution of the angle of the node 9,fig.6.12, shows that these value are wrong. In fact after the activation of the constraints, there is “no rebound effect” of the shin; it seems that the leg does not come back after the impact. This means that probably the value of the coefficients must not be the same and in particular, that the value of the damper coefficient must be higher than the spring coefficient.
Fig. 6-12: Evolution of the angle
Looking at the picture, it is possible to notice that the time of the impact, and thus the time of the activation of the constraints, is smaller than that one of the experiment. In fact the impact time in the experiment is equal to: t = 0.36 sec (look at the scheme
6.3), while in the simulation it is equal to: t = 0.305 sec. The fig.6.13 shows the evolution, during the time, of the equation 6.1. The result is a graph on which it is possible to observe correspondent time to a violation of the penalty constraints ( thus probably hypertension of the leg), and the time of their activation and deactivation.
Fig. 6-13: The activation and deactivation of the constraints
It is possible to note that the constraints are activated and they remain active until the end of the simulation.
6.2.3. Comparison between the results obtained from Matlab model and the experiment
To make the Matlab model’s response similar to the one of the experiment, the impact parameters have been changed in the model. The idea has been to increase the value of the spring coefficient and in the same time to decrease the value of the damper coefficient, with the aim to demonstrate the different influence of the coefficients on the model. In this way, at first, the damper coefficient has been changed in the range [1000 , 250] and then in the range [10 , 0], while the spring coefficient has been changed in the range [1000 , 15000]. The results are visible in the pictures 6.14 and 6.15.
Fig. 6-15: Evolution of the angle
From the analysis of the graphs, it seems clear that the damper has a big influence on the behaviour of the model, more than the spring. In fact as the picture 6.14 shows, also with a big value of the spring coefficient, a value of the damper, equal to Ct = 250 Nsec/m, is enough to damp “the rebound effect” of the shin against the knee ( the line green of the graph). Then the deactivation of the constraints is considerably late respect to the experiment. This means that a too high value of Ct cause a big impact time. On the other hand, looking at the picture 6.15, with a smaller damper coefficient, the impact time is similar to that one of the experiment. Changing this value in the range [10 , 0], the angle, after the impact, increases until the ideal value, correspondent to the system’s response without friction. The dark blue line of the graph, shows a good response of the model, given that the angle after the impact is almost equal to that one of the experiment. Nevertheless, the time of the activation of the constraints and thus the impact time, is smaller respect to that one of the experiment and the slope of the graph is different, before and after the impact. To modify the slope of the graph, it is necessary to work about the friction coefficient of the hinges too. For this reason it is decided to consider the damper value equal to zero and the spring value, variable, together with the friction coefficient of the hinge.
In the next paragraph the influence of the friction of the hinges on the model is demonstrated.
6.2.4. The influence of the friction coefficient of the hinge
In the picture 6.16 it is possible to observe the influence of the friction coefficient of the hinges, considering the value of the spring coefficient and of the damper coefficient respectevely equal to Kt = 15000 N/m, Ct = 0 Nsec/m.
Fig. 6-16: Evolution of the angle
From the analysis of the graphs it is clear that the friction coefficient of the hinges defines the slope of the curves before and after the impact and in this way, it changes the impact time of the system. In particular friction values included in the range [0.01 , 0.1] should be correct, adjusting subsequently the value of the spring. A first big result could be achieved: the damper value should be equal to zero. In the next paragraphs other tests on the model will be done, with the aim to demonstrate the validity of this discovery.
6.2.5. Analysis of the dissipation energy
Last tests on the model have detected the influence of the hinge friction. Looking at the evolution of the energy of the system, a strange result is emerged. In the picture 6.17 the evolution of the energy is showed, with the values of the coefficients equal to:
Kt = 11000 N/m Ct = 0
h = 0.07 Nsec/rad
Fig. 6-17: Evolution of the energy of the system
The evolution of the total energy, obtained by the sum the kinetic and potential energy of the system, is correct: in fact when there is the impact, a little decrease of the energy is noted. This fact is reasonable because, at the impact time, the kinetic energy is equal to zero, given that the system is in a stationary condition. On the
contrary, the dissipation function increases immediately at the impact time. The expression of the dissipation function is given by:
D=7/200*(td4-td8)^2+7/200*(td5-td12)^2+7/200*(td16-td9)^2+7/200*(td13-td21)^2 +7/200*(td19-td22)^2+7/200*(td13-td25)^2+7/200*(td17-td26)^2
where every terms of the equation correspond to the contribute of every hinge to the dissipation energy. Analysing this function at the impact time, this sudden increase is caused by the difference between td13 (θ ) and td21 (13 θ ), and the difference 21 between td13 and td25 (θ ). 25 θ corresponds to the ddl of the extremity’s node of 13 the special element Otto-bock (look at the picture 6.10). Thus probably, when there is the impact, and the activation of the constraints, there is also a reaction of the special element Otto-bock on the node 13 that cause an increase on the dissipation energy. This effect happens only for this value of the stiffness, in fact, choosing a lowest value of Kt, similar to 1000 N/m, there is not this development. To the ends of the calculation of the impact coefficient, this effect does not influence the behaviour of the model. Anyway it is important to understand as the behaviour of the special element Otto-bock influences the model. At last, another test on the model has been done, changing the value of the stiffness of the special element (which is fixed on the value: KOtto-Bock = 130 x 103N/m) in a bigger value (KOtto-Bock = 130 x 106N/m). The result obtained has demonstrated that the stiffness value has not influence on the model.
6.2.6. Evaluation of the penalty parameters together with the hinge’s friction coefficient
In this last paragraph the model has been tested again to adjust the value of the spring’s stiffness and of the hinges’ friction. Two main tests have been done, both of them with the value of Ct equal to zero: in one of them the value of spring’s stiffness is equal to Kt = 10000 N/m and the hinges’ friction coefficient variable in the range [0.05 , 0.08]; in the other one it has been programmed Kt = 1000 N/m and the friction coefficient variable in the range [0.06 , 0.07]. The results are visible in the pictures 6.18 and 6.19.
Fig. 6-19: Evolution of the angle
The obtained results show how both the values of the spring could be correct: in fact the slope of the graph, before and after the impact is similar to that one of the experiment and with values of the hinge’s friction coefficient equal to 0.065 Nsec/rad, the leg comes back more or less in the same way of the experiment. Anyway there is a difference: the impact time, using Kt = 10000 N/m, is smaller than that one of the experiment; on the contrary, using Kt = 1000 N/m the impact time is almost the same of the experiment. In fact, looking at the black line, (Kt = 10000 N/m and h= 0.065 Nsec/rad) on the picture 6.18, there is the activation of the constraint at the time t = 0.332 sec and the correspondent deactivation at the time t = 0.343 sec (thus the correspondent impact time, approximated, is given by the average value of them, t = 0.3375 sec) while in the experiment, the impact time is equal to t = 0.36 sec. On the contrary, using Kt = 1000 N/m and h= 0.065 Nsec/rad, the activation of the constraints is at the time t = 0.331 sec and the correspondent deactivation at the time t = 0.37 sec (thus, like before, the approximated impact time is given by the average value of them, t = 0.35 sec). The drawback using this smaller value of the stiffness, is a penetration of the constraint, due to the elastic component of the constraint.
Finally two tests have been done on the model to obtain better results. In the following pictures it is possible to analyze the evolution of the angle, the evolution of the constraints (the function 6.1 during the time), the evolution of the total energy and of the dissipation energy relative to a model programmed with Kt = 800-900 N/m, h = 0.0625-0.065 Nsec/rad and with Ct = 0.
Fig. 6-20: Evolution of the angle
Fig. 6-21: The evolution of the constraints
Looking at the pictures, it seems clear that the model’s response is good from all points of view: the behaviour of the model is really similar to that one of the experiment and it is stable. The response relative to the energy is also correct: in fact at every impact, there is a loss of energy as showed in the picture 6.22.
6.2. Conclusion
The research’s study has been pursued following two principal ways in order to solve the problem of the impact of the shin against the knee. Thus, at first an experiment has been made to simulate the natural movement of the shin around the knee using the prostheses 3R60 and then, a Matlab model that represents the experiment, has been built. By comparison the results obtained from the experiment with those ones coming from the Matlab model, a big result has been obtained: in the model the contribute of the virtual damper during the impact is equal to zero and the value of the stiffness of the virtual spring can be fixed around the value of Kt = 1000 N/m. Also the contribute due to the hinges’ friction coefficient of the prostheses have been analyzed: this value can be fixed around the value of h = 0.06-0.07 Nsec/rad.
