Lab Report A763319

09PEC028 ANDVACED METHODS OF ANALYSIS IN BIOMECHANICS

A synchronized quasi-static analysis of a drop jump and a squat jump

Author: A763319

Word Count: 1478

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A synchronized quasi-static analysis of a drop jump and a squat jump

Toni S. Evitts

Loughborough University, Biomechanics Laboratory. Department of Sport, Exercise and Health Studies. Loughborough University, Loughborough, Leicestershire, LE11 3TU, ENGLAND.

A B S T R A C T

Quai-static analysis uses kinematic and kinetic data to estimate joint moments during the execution of sporting movements. Joint moments are approximated by considering static loads, but ignoring inertial loads. Static loads, gravitational and external loads, are calculated using ground reaction force and position data from digitisation. Inertial loads, angular acceleration, angular velocity, centre of mass and weight of segments are not considered. The assumption that the body is static and that inertial load exert negligible effects results in inaccurate values. Other sources of error, systematic error and random error, further amplify inaccuracy of calculated joint moments. Sources of error include, recording kinematic data, digitisation error, joint centre approximation, and multi-planar movement. Application of the estimated joint moments in interpreting real-life movements is limited as; effects of other joints are not considered, sporting movements often involve movement in more than one plane, no differentiation between bilateral limbs or individual muscle can be made, and co-contraction and friction are not accounted for. There are many sources of error when estimating joint moments using quasi-static analysis and therefore care should be taken to firstly reduce systematic error and secondly in this interpretation of data due to the shortcomings of the model of human movement employed.

Key words: quasi-static analysis, moment, static load, inertial load.

1. Introduction

THE MEASUREMENT or estimation of human joint forces and moments is a topic of interest in the study of human movement. Joint forces and moments are generated by the muscles and passive soft tissues around the joint to control body segmental movement. During movement, joint forces and moments are not only used to counter the gravitational and external loads (static loads) due to interaction between the body and the environment, but also to counter the inertial loads of the body segments, which are related to the segmental angular acceleration and segmental CoM (Wu and Ladin, 1996). The effects of velocity, acceleration of segments are ignored in quasi-static analysis; segment weight can also be omitted.The static load during can be calculated by measuring the ground reaction forces, and moments from force data and position data of joints from digitisation.The joint forces and moments can be approximated by static load, assuming that the inertial load is negligible. This approximation simplifies the inverse dynamics process. The errors in joint load calculation caused by eliminating the inertial load have been evaluated (Bresler and Frankel, 1950; Cappozzo et al., 1980; Wells, 1981). The purpose of this paper is to evaluate the quasi-static analysis of a drop jump and a squat jump, methods used to find jump height of a Drop Jump (DJ) and Counter-movement jump (CMJ) using kinetic data are also considered. Assumptions underling the methods, and limitations will be discussed.

2. Methods

A maximal vertical DJ was performed by one male subject and a maximal vertical CMJ by another. All participants gave their informed consent in accordance with the policy statement of Loughborough University. Seven reflective markers

were placed on the right side of the body, on the glenohumeral-joint (shoulder), humeroulnar-joint (elbow), radiocarpal-joint (wrist), greater-trochanter (hip), lateral femoral-condyle (knee), lateral-malleolus (ankle), and fifth-metatarsal (toe). Ground reaction forces and moments of force were collected with a Kistler piezoelectric force plate, sampled at 1000 Hz and recorded using Bioware software. A video-recorder (50 Hz) and digitising software (SiliconCoach) was used to collect and process kinematic data. The camera was set up perpendicular to the sagittal plane of motion to capture the movement. Timing lights were used to synchronise video and force data (20 light emitting diodes illuminated subsequently at 0.001-sec intervals, at some point after video recorded had commenced; the following equation was used to synchronize data: Time zero in force data

Joint centres were digitised using the reflective markers as well as the vertex of the cranium.

3. Analysis

The human body was modelled as planar, rigid-body system comprising of seven segments linked by frictionless, hinge joints (Fig.1). Quasi-static analysis of the joint position data and force data was used to find joint moments. The co-ordinates of each joint position was corrected to give its position relative to the centre of the force plate (joint coordinates (xj,yj) – centre of force plate coordinates (xc,yc). Reference points in the plane of movement were used to scale from pixels to real life distance in metres. Quasi-static analysis of these positions was used to find the reaction moment and hence joint moment at each joint.

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Figure 1 – model of human body and axis of data collection

Fy and Fz data for the two jumps was used along with digitised data to estimate joint moments. During the ground contact phase of the jumps the moments at the joints were estimated by ignoring the inertial contributions, and the weights of the segments. The moment was computed by taking the product of the magnitude of the ground reaction force vector (Fz) and the perpendicular distance between the joints position (d) and the ground reaction force vector originating from the centre of pressure (Ay). Position of joint centres from digitised data were corrected relative to the centre of the force plate and calibrated to give real life values in cm. Force data was synchronized with video data using LED timing lights. Reaction moments were calculated by

where Ay is position of centre of pressure in relation to centre of force plate in the sagital plane, px is joint position in the sagittal plane, Fz is vertical force data(transverse plane), Fy is force data in the sagittal plane, and py is joint position in the transverse direction. Joint moment is the reaction moment multiplied by minus one. Joint moments were calculated for ankle, knee and hip and the sum of the modulus of these values for each joint was used to find total support moment, i.e.

Where Am, Km and Hm are values of joint moment for the ankle, knee and hip respectively.

Kinetic data was used to analyse each trial using 4 methods; 1 method using double numerical integration (DI), 2 methods based on vertical takeoff velocity as calculated from the force platform (SI & I-M), and 1 method based on time of flight (FT). An overview of these methods is presented in Appendix 1.

4. Results

Jump heights calculated using the four aforementioned methods for the Counter movement- and drop-jump are detailed in Table 1.

Jump height cmCounter

movement jump

Drop-jump

FT 38.2 44.3

DI 36.0 3010

I-M 36.4 165

SI 36.4 327Table 1 - Jump heights (change in CoM height) for counter movement- and drop-jump calculated using four methods:

flight time (FT), double integration (DI), impulse-momentum (I-M) and single integration (SI).

Quasi-static analysis of the CMJ and DJ are summarised in Fig.2 and Fig.3 respectively. Negative values denote clockwise and positive, anticlockwise, angular motion.

Figure 2 – Joint moments and total support moment during a counter movement-jump.

Figure 3 - Joint moments and total support moment during a drop-jump.

The maximum moment at each joint and in each jump is detailed in Table 2 and total moments are presented in table 3.

Max Joint moment NmCounter

movement jump

Drop-jump

Ankle 552.7 633.6

Knee -615.5 -279.7

Hip 753.6 -350.0

Support moment 1572.0 1011.3

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Table 2 - Maximum joint moment at each joint during a counter movement- and the drop-jump.

Total moment (Absolute) NmCounter

movement jump

Drop-jump

Ankle 12383.6 (12404.2)

7856.0 (7876.9)

Knee -7740.4 (8255.3)

2290.7 (3501.8)

Hip 8088.5 (10051.1)

3374.6 (6606.7)

Support moment (30710.6) (17985.5)

Table 3 - Total support and absolute total moment at each joint and total absolute support moment during a counter

movement- and a drop-jump.

5. Discussion

Quasi-static analysis relies on a number of assumptions in order to function. One assumption is that ground reaction force dominates movement determination; the degree to which this is true determines a proportion of the error introduced. Another assumption is that the body is in a static position; this is seldom true in sporting movements, such as the jumps analysed here. The implications of these assumptions are discussed below.The quasi-static analysis approach does not take into account internal forces and therefore only reflects the gravitational and external reactions between the body and the environment. This may introduce error in the results especially if fast movements are analysed; the effect of the inertia/forces and moments becomes more important as the speed of locomotion increases. Eliminating these forces also may not be appropriate for analysis of proximal joints, i.e. hip; and where the shear components of the force and moment are of interest. (Ladin and Wu, 1996)Quasi-static analysis may be beneficial is practical application due to its simplicity (no complicated kinematic transducers, inverse dynamics algorithms). However, the accuracy in which total joint loads can be approximated by static loads only needs to be considered (Callis, 1997). One source of error may be that; although the body segments have small masses and moments of inertia, the large accelerations they experience during some sports movements result in inertial forces and moments that are similar to, or greater than, the static loads, hence discounting these will give inaccurate results. It may be appropriate to ignore the inertial load if there is only interest in determining the order of magnitude of joint forces and moments under certain conditions. However, the inertial loads, especially the inertial forces, contribute greatly in the non-dominant directions (i.e. in the coronal plane). Furthermore, the inertial force is more important at the proximal joint than at the distal joint (Ladin and Wu, 1996).However one advantage of using quasi-static analysis is that by only considering static loads, computation of derivatives from inherently noisy displacement data is avoided, eliminating influence

of error in derivative values (Lanshammer, 1982) on moments.The resultant force vector is assumed to act in the same direction on all segments; effects of other joints are not considered. As each joint is calculated in isolation the results are essentially approximations. Also, if the axis of motion is not parallel to the camera then error will occur. Movement in the plane perpendicular to the camera shot is not quantified, hence 3D analysis may be required to analyse actions with significant movement in such plane. Factors such as co-contraction of antagonistic muscle groups and friction are not considered, as joints assumed freely rotational pin joints. A source of error in the method used here is that joint centres are not accurately depicted by the reflective markers, and their position may be distorted for example by muscle deformation during contraction. This is an example of systematic error. Other sources of systematic error that should be considered include; error in digitising, and the natural vibration frequency, sample frequency (≥1000Hz necessary), cut-off frequency (≥580Hz is acceptable), Street at al. (2001), and calibration error of force plate. Care should be taken to minimise effects of systematic error. Also force data is not corrected hence introducing further error.As opposite limbs are assumed contributing equally and simultaneously, it is not possible to examine the contribution of individual limbs.Contact time of the drop jump is shorter than the drop jump. Can not establish which individual muscles are acting at each joint or what proportion of the total torque they are producing.

6. References

[1] WU, G., LADIN, Z., 1996. Limitations of quasi-static estimation of human joint loading during locomotion. Medical and Biological Engineering and Computing, 34: 472-476.

[2] BRESLER B., FRANKEL J. P., 1950. The forces and moments in the leg during level walking. Transactions of the American Society of Mechanical Engineers, 72: 27-36.

[3] CAPPOZZO, A., LEO T., and CORTESl, S. S., 1980. A polycentric knee-ankle mechanism for above knee prostheses. Journal of Biomechanics, 13, (3): 231-239.

[4] WELLS, R. P., 1981. The projection of the ground reaction force as a predictor of internal joint moments. Bull. Prosth. Res, 18, (1): 15-19.

[5] CHALLIS, J.H., 1997. Producing physiologically realistic individual muscle force estimations by imposing constraints when using optimization techniques. Medical Engineering & Physics, 19 (3):253-261.

[6] LANSHAMMER, H., 1982. On precision limits for derivatives numerically calculated from noisy data', Journal of Biomechanics, 15, (6): 459-470.

[7] STREET, G., MCMILLAN, S., BOARD, W., RASMUSSEN, M., HENEGHAM, M.J., 2001. Sources of error in determining countermovement jump height with impulse method. Journal of Applied Biomechanics, 17:43-54.

[8] ARAGÓN-VARGAS, L. F., 1997. Kinesiological factors in vertical jump performance: Differences among individuals. Journal of Applied Biomechanics, 13, 24–44.

[9] DOWLING, J. J., & VAMOS, L. 1993. Identification of kinetic and temporal factors related to vertical jump performance. Journal of Applied Biomechanics, 9, 95–110.

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APPENDIX 1 – JUMP HEIGHT ESTIMATION METHODS

TF method used constant acceleration equation

where t = time of flight from force data, u = velocity at takeoff calculated by using v=u+at (v=0). Force data was used in two different ways to find velocity at take-off (Vto). Firstly Vto was obtained using the impulse-momentum (I-M) method from instantaneous vertical velocity-time curve; which was in turn calculated integrating propulsive force (Fzp), Fz minus corrected body weight, with respect to time (impulse) and normalizing by body mass according to:

where tto-n is some time just before takeoff and t0 is the beginning of data collection when the subject is standing on the force plate, then

where Fzp is propulsive force, obtained from subtracting body weight from the vertical ground reaction force; t0 is the beginning of data collection, and tto is the instant of takeoff. Change in CoM vertical position can be found by

The single integration (SI) method Instantaneous velocity at time of takeoff (IVto), taken from the instantaneous vertical velocity versus time curve, was also be used to calculate change in vertical CoM position:

DI method used corrected force data to calculate acceleration, then numerical integration using the trapezium rule to find velocity, then displacement. Firstly force data was corrected so total impulse ≈ 0. Correction factor was calculated by total impulse divided by total time of force recording (T), and subtracted from Fzp; this process was reiterated until total impulse was negligible.

Acceleration was calculated from this corrected force data.

Integration of acceleration data using the trapezium method gives velocity

Where an is instant acceleration at a given time interval and an+1 is instant acceleration at the subsequent time interval, t is the time interval between data points, and Vn-1 is the previous

Velocity value calculated, i.e. using an-1 and an. Displacement is calculated by again integrating this data using the trapezium rule

The maximum value of S produced give the greatest change of vertical CoM displacement from starting position; hence jump height from take off is

Where Smax is maximum displacement and Sto is displacement at time of take-off. As it is not possible to obtain a correction factor using the drop-jump data an alternative method was used to calculate jump height. Jump impulse was calculated by integration of the force versus time curve between time of take-off and touch-down, minus the weight impulse:

Where tc is contact time, tto is time at take-off, and ttd is time at touch-down. Take-off velocity is calculated by

where Vtd is the velocity of the CoM just before touchdown. Change in CoM position is therefore

FT, SI and I-M do not take into consideration the displacement of the CoM prior to takeoff; therefore may not be appropriate to compare jump performance between individuals. Although, previous researchers have found that the major contribution to vertical jump height differences among participants comes from VTO, whereas the relative takeoff height is very similar from one participant to another (Aragón-Vargas, 1997). DJ may be the most useful method as it can take into consideration displacement of CoM prior to take off, i.e. total CoM displacement relative to the starting standing position. From Table 1 it can be seen that all four methods give reliable results for CMJ although FT gives a slightly higher value than the other three. This can be explained as follows; the calculation of jump height using the FT method has been criticized in the literature because some of the assumptions involved are not correct (Dowling & Vamos, 1993). One clear limitation is that this method assumes that the time to peak height of CoM is equal to one-half of the time in the air, which is only true if the participant takes off and lands with the body in the same position. In reality body position at landing can be crouched.2, hence time down would be longer than time up. This results in an overestimation of the distance from takeoff to peak.Data from the DJ gives somewhat more inconsistent results. FT is the only realistic value obtained. Limitations of the other four methods

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mean they are not suitable for calculating jump height of DJs. Problems arise as it is not possible to correct the force data or to take into account corrected body mass as it is when considering CMJ data. This results in error in force data being magnified by integration, producing nonsense displacement values. I-M method can not be performed as it is in CMJ analysis as take-off impulse can not be calculated. Although the adapted method is theoretically correct the uncorrected force data, and lack of consideration for the effect of weight results in a meaningless outcome.

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