Kinematics of a lower limb--kinematical analysis.

Vancu, Alexandru-Emil ; Dolga, Valer ; Dolga, Lia 等

1. INTRODUCTION

The lower limb is an important part of the human body (Pons, 2008; Dolga, 2008). In order to develop diverse attachments for the limb, one needs to understand the kinematics of the member (Dinn et al., 1970; Perry 1992; Morris, 1973; Oberg, 1974). A proper mathematical description of the movement states proficiency in designing and constructing the required attachment.

There are different modes to analyse the limb kinematics. One of them uses markers applied on the wanted limb or on a very appropriate one (size & proportions). A high-speed camera takes pictures in frames of a second to express the markers' spatial position (Holden et al., 1997; Winter, 1982); then the kinematics is studied. The more markers are considered, the more accurate the calculus will be. The major trouble with this method is the appearance of noise in the images, which can disturb the result (Winter et al., 1974). These pictures can be filtered, to gain a flawless series of images. One of the analyzing methods adds more cameras and then evaluates the results and eliminates the obvious false ones (Pezzak et al., 1977). Also one can take the gained data adding it to a table where one can get the kinematical data. The results gained with this calculating method will be used in future developments of human lower-limb ortheses. The new orthesis shall be made of ultra light materials. Weight is a major problem of the available orthesis and for this the pacient doesn't recover that fast because he can not use it for a very long time.

Fig. 1a shows an integration of a gyroscopic sensorial element with an acceleration element, to estimate the rotation angle and Fig. 1b the mixture of actuators and climate sensors.

The attachments can very from stiff ones (a wooden foot), to complex robotical systems (exoskeleton guided by the nerve terminals). The latters help people who suffered an accident or some disease, but they are also used in military or as powerenhacements for lifting higher weight or reaching higher speeds. With the help of medical researchers, one can develop even ortheses attached to the nervous system.

2. KINEMATICS FOR THE LOWER LIMB

Given the coordinate data from anatomical markers at either end of a limb segment, it is an easy step to calculate the absolute angle of that segment in space. It is not necessary that the two markers be at the extreme ends of the limb segment, so long as they are in line with the long-bone axis.

[FIGURE 1 OMITTED]

3. KINEMATICS FOR THE LOWER LIMB

Given the coordinate data from anatomical markers at either end of a limb segment, it is an easy step to calculate the absolute angle of that segment in space. It is not necessary that the two markers be at the extreme ends of the limb segment, so long as they are in line with the long-bone axis.

Fig. 2 sketches a leg with seven anatomical markers in a four-segment three-joint system. Markers 1 and 2 define the thigh in the sagittal plane. Note that by convention all angles are measured in a counter clockwise direction starting with the horizontal equal to 0[degrees]. Thus [[theta].sub.43] is the angle of the leg in space:

[[theta].sub.43] = arctan[([y.sub.3] - [y.sub.4])/([x.sub.3] - [x.sub.4])] (1) or, in more general notation;

[[theta].sub.ij] = arctan [([y.sub.j] - [y.sub.i])/([x.sub.j] - [x.sub.i])] (2)

[FIGURE 2 OMITTED]

These segment angles are absolute in the defined spatial reference system. It is therefore quite easy to calculate the joint angles from the angles of the two adjacent segments.

Limb angles in the spatial reference system are positively defined counterclockwise from the horizontal. Angular velocities and accelerations are positive in a counterclockwise direction in the plane of movement, which is essential for consistent use in subsequent kinetic analyses. Convention for joint angles (that are relative) is subject to wide variations among researchers, but the settlement must be elucidated.

4. JOINT ANGLES

Each joint has a convention for describing its magnitude and polarity. For example, when the knee is fully extended, it is described as 0[degrees] flexion, and when the leg moves in a posterior direction relative to the thigh, the knee is said to be in flexion. In terms of the absolute angles described previously:

knee angle = [[theta].sub.k] = [[theta].sub.21] - [[theta].sub.43] (3)

If [[theta].sub.21] > [[theta].sub.43], the knee is flexed; if [[theta].sub.21] < [[theta].sub.43], the knee is extended.

The convention for the ankle is slightly different in that 90[degrees] between the leg and the foot is boundary between plantar flexion and dorsi flexion. Therefore,

ankle angle = [[theta].sub.a] = [[theta].sub.43] - [[theta].sub.65] + 90[degrees] (4)

If [[theta].sub.a] is positive, the foot is plantar flexed; If [[theta].sub.a] is negative, the foot is dorsi flexed.

5. VELOCITY CALCULATION

One assumes that the raw displacement data have been suitably smoothed by digital filtering and a set of smoothed coordinates and angles to operate upon is available. To calculate the velocity from displacement data, all that is needed is to take the finite difference. For instance, to find the velocity in the x direction, one calculates [DELTA]x / [DELTA]t, where [DELTA]x = [x.sub.i+1] - [x.sub.i] and [DELTA]t is the time between adjacent samples [x.sub.i+1] and [x.sub.i].

The velocity calculated in this manner does not represent the velocity at either of the sample times. Rather, it represents the velocity of a point in time halfway between the two samples. This can result in errors later on when the study tries to relate the velocity-derived information to displacement data, and both results do not occur at the same point in time. A way around this problem is to calculate the velocity and accelerations on the basis of 2 [DELTA]t rather than [DELTA]t. Thus the velocity at the [i.sup.th] sample is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

6. ACCELERATION CALCULATION

Similarly to the previous exposed considerations, the acceleration can be determined:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The equationship (6) requires displacement data from samples i + 2 and i - 2; thus a total of five successive data points go into the acceleration.

An alternative and slightly better calculation of acceleration uses only three successive data coordinates and operates with the calculated velocities halfway between sample times:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

and:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

7. CONCLUSION

The kinematics has an important function in regulating data for stabilizing a system with feedback or optimal control of it.

It is necessary to calculate the start moment of the movement when the heel doesn't have anymore contact with the floor, the landing of the heel, the maximum height of the heel above ground level during swing. When does this occur during the swing phase? The study shall consider the lowest displacement of the heel marker during stance as an indication of ground level. The analysis shall also determine the vertical displacement of the toe marker when it reaches its lowest point in late stance and compare that with the lowest point during swing, and thereby determine how much toe clearance took place.

If the movement of the body can be calculated, one can reproduce it easily with the same movement features.

By adding a small computer able to do the kinematical and dynamical calculus of the lower limb, one can develop other better and safer attachments for the leg.

In the future recovery will be much easier for patients who are able to use modern attachments. The main purpose is to help people in need and we can do this with the help of modern technology.

8. REFERENCES

Dinn, D. F.;Winter, D.A. & Trenholm, B.G. (1970). CINTEL Computer Interface for Television, IEEE Trans. Comput., Vol.19, No.11, 1091-1095

Dolga, V. et al. (2008). Complex mechatronics systems for medical applications ("SMART"), National Romanian Research Grant N. 72197/2008

Holden, J. P., et al. (1997). Surface movement errors in shank kinematics and knee kinetics during gait, Gait & Posture, Vol.5, Issue 3, June 1997, 217-227

Morris, J. R.W. (1973). Accelerometry--A Technique for the Measurement of Human Body Movements, Journal of Biomechanics, Vol.6, No.6, 729-736

Oberg, K. (1974). Mathematical Modeling of Human Gait: An Application of the SELSPOT System, In: Biomechanics, Vol. II., R. C. Nelson and C. A. Moorehouse, Ed. (University Park Press, Baltimore,), 79-84, Baltimore, U.S.A.

Perry, J. (1992). Gait analysis: normal and pathological function, SLACK Incorporated, ISBN 1-55642-192-3, Thorofare, New Jersey

Pezzack, J. C., Norman, R.W. & Winter, D.A. (1977). An Assessment of Derivative Determining Techniques Used for Motion Analysis, Journal Biomechanics, Vol.10, No. 5-6, 377-382

Pons, J. L. (2008). Wearable Robots: Biomechatronic Exoskeletons, John Wiley & Sons, Ltd., ISBN 978-0-47051294-4, West Sussex

Winter, D. A, Sidwall, H.G. & Hobson, D.A. (1974). Measurement and Reduction of Noise in Kinematics of Locomotion, Journal of Biomechanics, Vol. 7, No. 2, 157-159

Winter, D. A. (1982). Camera Speeds for Normal and Pathological Gait Analyses, Medical and Biological Engineering and Computing, Vol. 20, No. 4, 408-412, ISSN 0140-0118