Dynamical analysis of an upper limb prosthesis.

Luca, Gheorghe ; Nagy, Ramona ; Menyhardt, Karoly 等

1. INTRODUCTION

Accurate quantification of kinematics (locomotion) represents a significant requirement for the purpose of physical rehabilitation in medicine, designing of prosthesis in orthopedics, analysis and optimization of sporting disciplines, researches in bio-mechanics, humanoid robotics, etc. For this purpose, a variety of experimental measurements were done to capture the motion. But, a dynamical approach is also necessary, because of the different forces and torques that act on the prosthesis. Because of the complexity and high requirements imposed on such devices, their control system has to utilize the dynamic model. So, the control, the design, and the simulation, strongly require general dynamic models that will make prosthesis capable of handling the increasing diversity of expected tasks.

The current paper is an ongoing work and continuation of the kinematical analysis of the upper limb prosthesis within the dynamical area. Dynamic simulation is more complex because the problem needs to be further defined and more data is needed to account for the forces and torques. But, dynamics are often required to accurately simulate the actual motion of a mechanical system. Generally, kinematic simulations help evaluate motion, while dynamic simulations assists in analyzing functionallity.

Several studies were performed on the human upper limb, each approach being specific for the study at hand (Murray & Johnson, 1998), (Raikova, 1992). There is no unilateral definition for a dynamic study and basically it takes into account motions and forces to solve a task. An analogus approach can be done on different elements or systems depending on the point of interest. For example, an equivalent model can be calculated (Mendoza-Vazquez et al., 2007) without a significant decrease in accuracy and with an easier mathematical analysis, for the structure or actuator systems (Edrey et al., 2008).

2. METHOD

The current study is centered around the vibrations that appear during the prosthesis' functioning. These are undamped forced vibrations, with harmonic perturbing forces with two dregrees of freedom.

In order to accomplish the dynamical analyses it was necessary to establish a mechanical model of the upper limb made of rods moving in the same plane.

[FIGURE 1 OMITTED]

The mechanical model of the prosthesis (vibrant model) for the undamped free vibrations is considered to be composed of two jointed rods as it is shown in figure 1.

For this model, the matrix differential equation of the vibrations is:

[J]{[??]} + [k]{[theta]} = {0} (1)

The equation (1) results by writing the kinetic energy, the potential one and by applying Lagrange equations. The kinetic energy is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The potential energy is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The differential equations for the system's small oscillations are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

and the equation of the natural frequencies:

([J.sub.11] [J.sub.22] - [J.sup.2.sub.12]) [p.sup.4] - ([J.sub.11] [k.sub.22] + [J.sub.22] [k.sub.11]) [p.sup.2] + [k.sub.11] [k.sub.22] = 0 (7)

The natural frequencies are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The distribution coefficients for the natural modes of vibration are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

For the virtual model of the prosthesis, shown in figure 2, the following values were obtained for the constructive elements:

[l.sub.1] = 0.290 m

[m.sub.1] = 0.802 kg

[O.sub.1][C.sub.1] = 0.201 m

[J.sub.C1] = 3.3322725 x [10.sup.-3] kg x [m.sup.2]

[J.sub.O1] = 36.960283155 x [10.sup.-3] kg x [m.sup.2]

[l.sub.2] = 0.308 m

[O.sub.2][C.sub.2] = 0.152 m

[m.sub.2] = 0.605 kg

[J.sub.C2] = 1.924989195 x [10.sup.-3] kg x [m.sup.2]

[J.sub.O2] = 16.608753 x [10.sup.-3] kg x [m.sup.2]

Based on these values the computed results are:

[J.sub.11] = 0.087840783155 kg x [m.sup.2]

[J.sub.22] = 0.015902909195 kg x [m.sup.2]

[J.sub.12] = 0.0266684 kg x [m.sup.2]

[k.sub.11] = 3.302556 N x m

[k.sub.22] = 0.902128 N x m

so that the natural frequencies are:

[p.sub.1] = 5.116372 rad/s

[p.sub.2] = 12.883192 rad/s (9)

and the distribution coefficients:

[[mu].sub.11] = [[mu].sub.12] = 1

[[mu].sub.21] = 1.436925

[[mu].sub.22] = -2.547699 (10)

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

To avoid the resonance phenomena it is necessary for the motor angular velocity to be different from the natural frequencies of the undamped free vibrations of the system, so, at the powering of the motors, the conditions are: [omega] [not equal to] [p.sub.1] and [omega] [not equal to] [p.sub.2]

Taking into account the obtained results in the kinematical study of the mechanical model (Menyhardt et al., 2008) the most dangerous vibration model is for [p.sub.max] = 12.883192 rad/s.

3. CONCLUSION

Throughout the current research, it was possible to determine the appearance conditions for the resonance phenomena (the resonance must be avoided for the two natural vibration modes, phenomena that would induce discomfort in the functioning of the shoulder joint).

The natural frequencies were established of the undamped free vibrations of the mechanical model (eq 9).

From mechanical point of view, the studies will continue with the analyses of the upper limb prosthesis using vibrations of continuous media method for stepped rods.

4. REFERENCES

Edrey D. Ruiz-Rojas, J. L. Vazquez-Gonzalez, Ruben Alejos-Palomares, Apolo Z. Escudero-Uribe, J. Rafael Mendoza-Vazquez, (2008) Mathematical Model of a Linear Electric Actuator with Prosthesis Applications, Proceedings of the 18th International Conference on Electronics, Communications and Computers Pages 182-186, ISBN 978-0-7695-3120-5

Mendoza-Vazquez, R.; Escudero-Uribe, A.Z.; Fernandez-Mulia, (2007) Simplified Analytical Dynamic Model for a Parallel Prosthetic Elbow Engineering in Medicine and Biology Society, EMBS 2007. 29th Annual International Conference of the IEEE, Pages 3031-3034

Menyhardt K, Nagy R., Luca G. (2008) Kinematical Analysis of an Upper Limb Prosthesis, Annals of DAAM for 2008 & Proceedings of the 19th International DAAM Symposium, Pages 841-842, ISBN 978-3-901509-68-1

Murray I.A., Johnson G.R. (1998). Upper Limb Kinematics and Dynamics: the Development and Validation of a Measurement Technique, Proceedings of Fifth International Symposium on the 3D Analysis of Human Movement, Chattanooga, Tennessee.

Raikova R. (1992). "A General Approach for Modelling and Mathematical Investigation of the Human Upper Limb", Journal of Biomechanics, Vol. 25, Pages 857-867, ISSN 0021-9290