A MATHEMATICAL MODEL TO DETERMINE THE TORQUE FOR A PROSTHETIC LEG-LAGRANGIAN EQUATION

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Volume 6 No. 5 7, 5-56 ISSN: 3-88 (printed version); ISSN: 34-3395 (on-line version) url: http://www.ijpam.

1 Volume 6 No. 5 7, 5-56 ISSN: 3-88 (printed version); ISSN: (on-line version) url: A MATHEMATICAL MODEL TO DETERMINE THE TORQUE FOR A PROSTHETIC LEG-LAGRANGIAN EQUATION ijpam.eu Y. Kalyan Chakravarthy, Mohammed Shabnam Sultana, Dr. A. Srinath 3 G.R.S. Naga Kumar 4,,3,4 Electrical and Electronics Engineering, K L University, Guntur, Andhra Pradesh - 55, India. kalyan_me@kluniversity.in, shabbu6@gmail.com, 3 srinath@kluniverisyt.in, 4 naga3ee@kluniversity.in Abstract: Development of the prosthetic is not enough these days, after the introduction of actuation systems into the prosthetics. For actuation purpose, one must have sound knowledge of actuation parameters, principles of actuators, concepts of feedback system as of basics in the beginning. Considering a few human subjects, actuation data pre-requisites are taken and noted, which are later developed and utilized for determination of actuator input to have the desired actuation in terms of torque. On observing the developed results, they are analyzed to determine the suitable actuator. For the development of the parameters, the data acquired is processed through mathematical equations using MATLAB tool for precision and reliability. Keywords: Gait, Torque, Knee angle, Mathematical modelling, Lagrangian,, Matlab.. Introduction. Prosthetics Replacement or artificial support to a human limb is known as prosthetic. The study of prosthetics is called as prosthesis. Prosthetics and prosthesis happen to date back to B.C, where primarily knights and kings were given the privilege to have the sophisticated prosthetics at the time[]. Whereas the common people had made use of regularly available tools and supports like wood, bars which are still used for immediate prosthesis till date. Prosthetics these days are more than that of the passive prosthetics available previously. The new age prosthetics are known for their advancement in being active prosthetics and of various actuation systems. While concentrating on the lower limb prosthetics, transtibial [] amputations are less likely to need actuation systems when compared to the transfemoral [3] amputation. Hence, transfemoral amputation prosthetics are mainly concentrated for developing actuation systems in reference to the knee mechanism especially. It is the major joint in the amputee that requires more attention and detail for prosthetic development. Thus, helping the amputated to have an effective day work and daily life, in spite of having the prosthetic.. Gait Analysis.. Gait: It is the sstudy of the human s walking pattern. Further giving the scope for gait analysis and hence deriving the required data for the purpose of actuating prosthetics... Gait analysis: The study of human locomotion, by the virtue of limb movements. These movements are studied to understand the pattern of limb movement, pattern of walking for prosthetics, providing the primary data for development of actuation system...3. Gait Cycle: It is the time elapsed between the consecutive contacts of the foot with the ground during the human locomotion of the leg. The observation of the cycle is mainly classified into two phases, namely Stance phase and swing phase as illustrated in Fig.. The time spent in the gait cycle can be divided into single support and double support stages. Single support stage is only when one limb makes and lies in contact with the surface, where as 5

2 the double stage is when both the limbs are in contact with the surface.[4][5] Where both, the knee joint and hip joint are at their extreme positions in the matters of movement and limb ready to make the heel strike to begin another gait cycle, moving into double support from single support and into stance phase from swing phase. These have been illustrated in the Fig., Fig. 3 below. Figure. Sub Classification of gait cycle..3.. Stance phase: The duration where first the heel touches the ground and toe of the same leg takes off. It is where the weight of the body acts on the foot[6]. It contributes to 6% of the gait cycle and sub divided into 5 stages further Heel strike: The contact point by the limb by making the heel to touch the surface Loading response: After heel strike, the weight of the body is shifted on to the heel struck limb, making it from a single support to double support Mid stance: The time of the gait cycle where the tibia s position is perpendicular to the supporting surface Terminal stance: The time of the stance, where the heel takes off the surface Swing phase: The duration covered between the toe off and the following heel strike of a limb. It constitutes of 3 stages contributing to the rest 4% of gait cycle.[6]..3.. Initial swing: Begins after the toe takes off the surface at the end of pre-swing in stance phase. Where the movement start with the hip joint alone keeping the knee joint ideal Mid swing: Where the knee joint also comes into movement in parallel with the hip joint Terminal swing: Figure. Model Leg Figure 3. Gait cycle..4. Gait Terminologies: There exist 7 terminologies of gait, categorized into distance variables and time variables Distance variables: Stride length: The length covered between the consecutive contacts of the same heel during the gait cycle Step length: The length covered from heel strike of one limb and the heel strike of the other limb Degree of toe out: The angle made by an imaginary line passing through the second toe and the heel of the foot, with the foot progression line in a gait cycle. It happens to be that angle of toe out decreases with the increase in walking speed. Normal degree of toe 5

3 out is 7 o. The pictorial illustration has been shown below, in Fig. 4. Figure 4. Step Length of the human..4. Time Variables: Single limb support time: The duration of time spent in single support during the swing phase Double limb support time: The duration of time spent for double support during the stance phase Cadence: It is the number of steps observed for a human to cover in a unit time Cycle time: Using cycle time puts the calculation even better an clear when replaced with cadence. Speed (m/s) = stride length (m)/cycle time (s) Speed: Distance covered in a given time scaled down to distance covered per unit time. The relation between cadence, step length and speed is given by the following formula. Average speed / velocity = Step length* cadence.3 Kinematics and Kinetics: Kinematics is the study of movement, displacement of a body. Kinetics is the study of effect, causing the movement of the body or the subject under the influence..4 Potentiometer: Potentiometer is used to determine the angles formed during the gait cycle, for considering human subjects. For this, it is placed on the knee joint with required software and hardware console, to capturing the data in dynamic conditions of the gait cycle. Through this gait timings and respective angles involved are noted down for the phases and positions of gait, as explained previously in the discussion.[7][8][6].5 Angle Capturing Techniques: Angles involved in joints can be obtained through various practices available. A few such are by using accelerometer, potentiometer, using MEMS, etc. In this discussion, we are making use of potentiometer for data capturing.[9].6 Anthropometric: It is the study of human limb dimensions, for the development of artificial supports or limbs, prosthetics. It describes the link lengths to be followed for a taken set of subjects, as lengths differ from person to person and area to area. []. Modeling Of Prosthetic Leg Anthropometric data has been absorbed from the existing study, done by research personnel till date. Using the data like thigh length, shank length, foot length are utilized for development of virtual 3D model, with the help of drawing and design tools like solid works. Sample material is considered for properties. The model is used to analyze with analysis tools like Ansys etc. The material properties and mechanical parameters are iterated till the FOS of the prosthetic is sound enough. After finalizing the design, model mass is considered to be the prosthetic weight, to be carried forward for future use in the discussion. 3. Kinetic Analysis The evaluation of kinetic parameters is known as kinetic analysis. For this a mathematical model is developed which resembles like a two-link manipulator. For this purpose physical laws like lagrangian mechanics and Newton mechanics are applied. Which are derived from energy conservation equations and force balance equations respectively. The kinetic energy equation bounding for link- is [] = + = + = Eq. Its potential energy is bounded by = sin Eq.3 53

4 Where, g is the magnitude due to gravity, in ve y- axis direction. The kinetic energy equation for that of link- is bounded by = + = ( + ) Eq. The potential energy equation for link- is = + Eq.4 The lagrangian equation, is L=K-P = + Eq.5 Through substitution and simplification it has been derived that = ) Eq.6 The Lagrange-Euler formulation for links gives the torques of the -link planar manipulator. As both the joints are revolute, the generalized torques, which represent the actual joint torques can be written as τ = m + m L + ml + ml LC θ m L + L LC θ m L L S θ θ m L LS θ 3 + m + m glc + m glc τ m = L + L LC θ + ml θ ml LS θ + mglc Eq.7 After solving the above equations the final torques of Eq(8) and Eq(9) are obtained,! = " Ѳ# +" +$ +% Eq.8! = " Ѳ# +" +$ +% Eq.9 M = m + m L + ml + ml LC 3 3 M = M = m L + L LC 3 M = ml 3 H = ml LS θ θ ml LS θ H = ml LS θ G = m + m LC + mlc g 3 G = mlc g These coefficients are defined as M = effective inertia, ii M = effective coupling inertia, ij H i = centrifugal and acceleration forces In order to simplify and make use of the machines, a mathematical model has been developed to make it easy to operate and avoid the fuss of manual work. The MATLAB/SIMULINK model is as illustrated in the Fig.5 with the help of above mentioned Torque equations in Eq.8, Eq.9. Figure 5. Mathematical modeling by Matlab Torque(N-m) Velocity (rad/sec) accelaration(rad/sec ) Torque Analysis on Person Time (sec) Figure 6. Torque analysis for subject velo(:,) velo(:,) acc(:,) acc(:,) torque(:,) torque(:,) 54

5 Torque(N-m) Figure 7. Torque analysis for subject Velocity (rad/sec) Accelaration(rad/sec ) Torque(N-m) Velocity (rad/se) Accelaration(rad/sec ) Torwue Analysis on Person Time (Sec) Figure 8. Torque analysis for subject 3 velo(:,) velo(:,) acc(:,) acc(:,) torque(:,) torque(:,) Torque Analysis on Person Time (Sec) From the Fig. 6,Fig. 7,Fig. 8, the torque, velocity acceleration of prosthetic leg has observed for different subjects under same operating conditions like Constant velocity of km/hr. 4. Conclusion Using the anthropometric data and preliminary data required for determination of toques, to actuate the considered prosthetic leg, generalized torque equations have been derived with the help of mathematical modeling through lagrangian equations using MATLAB/SIMULINK environment. The equations stand good for modified parameters as well, because the torques to be attained are entirely dependent on the input parameters and this will be helpful for the indigenous development of prosthetic leg, for Indian amputee. Giving better development in prosthetic industries by obtaining the toques required for actuation, as depicted in Fig. 6, Fig. 7, Fig.8. Acknowledgements The authors are grateful to the support of K L University FIST sponsored Advance prototyping and Manufacturing lab SR/FST/ETI-37/(C). Mechanical properties of prosthetic limbs: adapting to the patient., J. Rehabil. Res. Dev., vol. 38, no. 3, pp ,. [3]F. F. Klijnstra, Control and Implementation of a transfemoral prosthesis for walking at different speeds, no. 8,. [4]M. W. Whittle, Gait analysis: an introduction, Library (Lond)., vol. 3, pp.,. [5]S. K. Au, J. Weber, and H. Herr, Powered ankle-foot prosthesis improves walking metabolic economy, IEEE Trans. Robot., vol. 5, no., pp. 5 66, 9. [6]MohamedArif.N, Y.udhayakumar,Inbarasan.G, " Design Of High Frequency Earthing System Used For Gas Insulated Substation, International Innovative Research Journal of Engineering and Technology, vol., no., pp , 6. [7]S. Au, M. Berniker, and H. Herr, Powered ankle-foot prosthesis to assist level-ground and stair-descent gaits, Neural Networks, vol., no. 4, pp , 8. [8]Karthikeyan.T, Balakumar.A, Sathyanarayanan.M, Senthilkumar.R, Sugavanam.K. R, " Design of PSO Based KY Boost Converter to Reduce Output Ripple Voltage," International Innovative Research Journal of Engineering and Technology, vol., pp [9]T. J. Yeh, M. J. Wu, T. J. Lu, F. K. Wu, and C. R. Huang, Control of McKibben pneumatic muscles for a power-assist, lower-limb orthosis, Mechatronics, vol., no. 6, pp ,. []A. Furse, W. Cleghorn, and J. Andrysek, Improving the gait performance of nonfluid-based swing-phase control mechanisms in transfemoral prostheses, IEEE Trans. Biomed. Eng., vol. 58, no. 8, pp ,. []S. J. Abbass and G. Abdulrahman, Kinematic analysis of human gait cycle, vol. 6, no., pp. 8, 4. []R. Borjian, Design, Modeling, and Control of an Active Prosthetic Knee, 8. [3]R.K.Mittal text book publications. References []K. Norton, A brief History of Prosthetics, InMotion, vol. 7, no. 7, pp. 3, 7. []G. K. Klute, C. F. Kallfelz, and J. M. Czerniecki, 55

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A MODIFIED DYNAMIC MODEL OF THE HUMAN LOWER LIMB DURING COMPLETE GAIT CYCLE

A MODIFIED DYNAMIC MODEL OF THE HUMAN LOWER LIMB DURING COMPLETE GAIT CYCLE Int. J. Mech. Eng. & Rob. Res. 2013 S M Nacy et al., 2013 Research Paper ISSN 2278 0149 www.ijmerr.com Vol. 2, No. 2, April 2013 2013 IJMERR. All Rights Reserved A MODIFIED DYNAMI MODEL OF THE HUMAN LOWER