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Three-dimensional measurement of lumbar spine kinematics for fast bowlers in cricket
Clinical Biomechanics13 (1998)574-583
Three-dimensional measurement of lumbar spine kinematics for fast bowlers in cricket A.F. Burnett,a * C.J. Barrett,b R.N. Marshall,” B.C. Elliott,d R.E. Day” aWestern Australian Institute of Sport, Mt. Claremont, Western Australia hBrian Edwards Physiotherapy Centre, Churchlands, Western Australia ‘Division qf Science and Technology, The lhziversity of Auckland, Auckland, New Zealand ‘Department of Human Movement, University of Western Australia, Nedlands, Western Australia ‘Department of Medical Physics, Royal Perth Hospital, Perth, Western Australia
Received25 September1996;accepted21 February1997
Abstract Objective. To determine whether the three-dimensional (3-D) lumbar spine kinematics for the mixed fast bowling technique differed to those of the side-on and front-on fast bowling techniques. Background. It has been previously shown that bowlers who utilise a mixed bowling technique are more likely to show lumbar spine pathology than those who bowl with either the side-on or front-on techniques. Methods. An electromagnetic device (3-Space’FastrakTM) operating at 120 Hz captured range of motion and 3-D lumbar spine kinematics during the delivery stride of 20 young high performance subjects. The trajectory of shoulder and pelvic girdle markers were simultaneously captured and these data were used to classify bowlers into either a side-on, front-on or mixed technique group. Results. No significant differences (P
Overuse injuries to fast bowlers in bricket are common. To better understand the mechanics of injury it is necessary to understand the 3-D rotations of the lumbar spine during this activity. 0 1998 Elsevier Science Ltd. All rights reserved. Keywords:
Cricket;Fastbowling; Injury; Three-dimensionalkinematics;Lumbarspine
1. Introduction
It is generally accepted that mechanical factors play an important role in the aetiology of related degenerative processes and injuries [l]. Fast bowling in cricket involves delivering the ball at between approximately 30 and 45 m s-l. In an effort to do this, the fast bowler’s trunk must flex, extend, laterally bend and *To whom correspondenceshould be addressedat: Western Australian Institute of Sport,ChallengeStadium,PO Box 139,Claremont 6010,WesternAustralia
rotate in a short period of time, whilst the body must absorb ground reaction forces as high as six times body weight. Three bowling actions exist with which a bowler can deliver the cricket ball; they are the side-on, fronton and mixed techniques. It has been found that the mixed bowling action, characterised by a counterrotation of the shoulders in the transverse plane, was related to the appearance of spinal pathology in the thoracolumbar spine [2-41. The exact cause of disc degeneration still remains controversial. Early suggestions which implied that
0268-0033/98/$19.00 + 0.000 1998ElsevierScienceLtd. All rights reserved. PII: SO268-0033(98)00026-6
A. l? Burnett et al./Clinical Biomechanics 13 (1998) 574-583
torsion alone caused disc degeneration [l] have since been refuted [5-71. It has been suggested, however, that combinations of repetitive loadings such as axial rotation and forward flexion [8,9] and high compressive forces in hyperextension [lo] may cause lumbar spine pathology. The quantitative evaluation of in vivo lumbar spinal movement has challenged researchers for years due primarily to the inaccessibility of the spine. Certainly the most direct, but invasive, method is to insert Steinmann pins into the spinous processes of the lumbar vertebrae and measure their relative displacements using three-dimensional (3-D) motion analysis [ 111. Roentgenographic means of motion measurement have limitations because exposure to ionising radiation limits the number of films that can be taken and dynamic motion can only be captured with reduced accuracy 111,121. Image-based motion analysis systems, using surface markers, have been used for lumbar motion assessment. Problems associated with measuring complex 3-D back motion with these systems include; variable marker movement within defined coordinate systems, marker crossover caused by closely positioned markers and the inherent inaccuracy of the measurement system [13,14]. Electromagnetic devices have been used to measure lumbar range of motion [15] and An et al. [16] have suggested that electromagnetic devices would be ideal for kinesiologic study. An electromagnetic device, the 3-Space@FastrakTM was therefore used to quantify the 3-D rotations of the lumbar spine in the side-on, front-on and mixed fast bowling techniques.
2. Methods
2.1. Sample Twenty right-hand male fast bowlers from the Western Australian Cricket Association fast bowling development squad, mean age 19.1 years (SD 1.4) mean height 183.2 cm (SD 5.4) mean mass 76.2 kg (SD 6.8), were recruited as subjects. This sample of subjects represented 77% of bowlers over 15 years of age who were in the state high performance squads. At the time of testing all subjects were bowling with no limitation to movement and signed a Document of Informed Consent. 2.2. Data collection The 3-Space@FastrakTM (Polhemus Navigation Sciences Division, Vermont, USA) is an electromagnetic device which measures the position and orientation of a sensor in space. The device’s source generates a low-frequenty magnetic field which is detected by the
57s
device’s sensor(s) and, from this, six degree-of-freedom data are generated. While up to four sensors may be used, this reduces the sampling frequency. As fast bowling is a highly dynamic activity, only one sensor was used in this study as a high sampling frequency was required. Standard connections linking the system electronics unit to the source and sensor were relatively short; therefore, a 15-way ribbon cable extension was connected to the existing cabling so that subjects could deliver the ball with a run-up of approximately 15 m. The cabling was lirmly taped to the subject’s left thigh. The device was linked to a 486 DX2 computer via an RS-232C serial interface. The Fastrak can provide output in several forms, with the most compact being Cardan angles [17] encoded as 16-bit binary numbers. Due to the limit of 19.2 kBaud for reliable operation of the system, this encoding was used and gave approximately 120 samples per second with specifically developed custom software. Some loss in spatial resolution was incurred with this choice. However, this was less critical than the lost temporal resolution a longer encoding would cause. The orientation of the source and sensor on the subjects defined a right-handed coordinate system with X up, Y right, Z forward. Therefore rtitations about X, Y and Z represented axial rotation ($), flexiomextension (4) and lateral bending (ti) respectively. The Cardanic sequence of rotation for the raw data was ZYX with a moving frame of reference. Several methods of attachment of the source and sensor to the subject were initially tested. This was done due to low-frequency vibration of the source being problematic. The following approach was therefore adopted. The skin of the sacral area was rubbed with alcohol and allowed to dry before further preparation. The source was attached to a moulded thermoplastic sacral plate which in turn was attached to the skin using double-sided tape, such that the top border of the plate was overlying a line connecting the subject’s posterior superior iliac spines. A Nylatex@ stretch wrap with mouldable plastic stitched to it was threaded through the sacral plate so that the attachment was firm. The wrap was tightly secured around the subject’s pelvis and a velcro band was secured to the back of the source and onto the wrap. The Ll spinous process was then identified by an experienced manipulative physiotherapist (CJB) using the following procedure. A line connecting the superior aspects of the iliac crests was drawn on the skin corresponding with either the L4 spinous process or the L415 interspace. Palpation then permitted Ll to be identified and marked with a skin pencil. A second wrap with a similar piece of mouldable plastic stitched into it was securely positioned such that its lower border lay directly over this mark. The sensor was
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A. E Burnett et al.lClinical Biomechanics 13 (1998) 574-583
attached by screwing it squarely into the mouldable plastic with two small non-metallic nut and bolts. This approach prevented the cable from moving the sensor erroneously during the fast bowling run-up and delivery by effectively ‘stiffening’ the wrap. Figure 1 shows the device attached to a subject. After subjects completed a standardised warm up, they were requested to change into a pair of cycling shorts. Each subject’s standing height and weight were recorded. For data normalisation purposes, each subject’s lumbar spine range of motion was measured in flexion, extension and for both directions in lateral bending and axial rotation. When the source and sensor were attached to the subject their axes were not coincident. Therefore, it was necessary to record a neutral position for each subject so that range of motion and kinematic data in each orthopaedic axis could be measured relative to a zero reference (for mathematical transformations, see below). For this reference measure, subjects stood upright with feet shoulder-width apart and palms facing inwards and the orientation of the sensor relative to the source was recorded. this process was repeated three times and a mean was calculated. Range of motion data were
collected over approximately 6.5 s. During this time each subject moved to the limit of motion in a given direction in response to a verbal cue and then moved back to the pre-defined neutral position. Kinematic data were collected using a five-camera VX 500 Vicon motion analysis system (Oxford Metrics, Oxford). Cameras were fitted with 25 mm lenses and operated at 50 Hz. To provide the known 3-D control points for the Direct Linear Transformation reconstruction, a calibration frame with dimensions of approximately 1.8 m x 2.1 m x 1.8 m surrounded the activity space to avoid reconstruction errors due to extrapolation [18]. Retro-reflective markers were attached to the most lateral aspects of the pelvis and the acromion processes. Preliminary testing had demonstrated the need for the cable extension from the Fastrak to be supported by an assistant during the run-up to avoid any impedance to the bowler. Subjects bowled into a mat positioned 10 m away from delivery with a marked target area which represented the normal trajectory for a full length bowled over the correct distance. The Fastrak and Vicon systems were activated almost simultaneously and data collection continued for both systems for a period of approximately 6.5 s. Figure 2 shows a bowler delivering the ball in the experimental set-up. Three successful fast bowling trials were recorded for each subject. As the Vicon system only captures the outline of retro-reflective markers and not ‘complete’ visual images, it was synchronised to a video camera operating at the same frequency. Kinematic data from the Vicon could therefore be event matched (i.e. with back foot impact, front foot impact and ball release). An assessment of the Fastrak’s accuracy was performed by comparing the device’s output to those obtained from a fabricated tri-axial protractor whose axes were orthogonal. The Fastrak’s source was mounted on a base attached to the protractor while the sensor was mounted on the horizontal axis of the triaxial protractor which depicted flexion/extension was then rotated in lo” graduations from -50 to 60” with the Fastrak measuring each position. Similarly, rotations about the remaining axes were measured between -50 and 50” with varying degrees of rotation about the other axes. By comparing the values indicated by the protractor against those measured by the Fastrak, it was evident that measurements from the Fastrak became less accurate when displaced from zero. However, the values were accurate within 0.5” when rotations about the remaining axes were present. 2.3. Data reduction and transformation
Fig . 1. The 3-Space”FastrakTM attached to a subject.
The synchronised video records permitted the time from when the Vicon was activated until back foot
A. E Burnett et aLlClinical Biomechanics I3 (1998) 574-583
impact and release to be identified. Resulting coordinate data from the pelvic and shoulder girdle markers over these times were then converted into ASCII format and downloaded onto an IBM PC. These data were smoothed using a Butterworth second-order digital filter at a cut-off frequency of 5 Hz. Transformation of the Fastrak raw data was necessary as an alternative rotation sequence and representation system was preferred. Firstly, each data record containing three Cardan angles $, 4 and 8 were converted into the elements of their respective direction cosine matrices by the following equations:
Rz2 =
sin 8 sin 4 sin I,!J+COS 8 costi
R3, = -sin
4
Rx2 = cos I$
sin II/
R33 = cos 4 cos tj
where the direction cosine matrix R was defined as: R,, = cos 0 cos 4
RI, Rz, Rzl, Rz i Rx! I R12
R,,=cos0sin4sin$-sinOcos$
R=
R3,
RI3 = cos 0
Rz, =
sin 4 cos $+sin 0 sin I+!I
sin0 cos &
R13
R33
Secondly, in order to measure orientation relative to a zero reference the following transform was applied to the data:
VW = U?AI~[RBI where [RA] is the direction cosine matrix of the neutral data and [R,] is the direction cosine matrix of the kinematic data, while [RAITis the transpose of [RA]. Joint Coordinate System (JCS) angles described by Grood and Suntay [19] were recovered from the direction cosine matrix [R,-1.However, the formulae used to obtain these angles differed from that outlined in their paper as the Fastrak’s coordinate axes were orientated differently and further, the choice of axes for unit vectors el and e3was based on the recommendations of Cole et al. [20] ( see Appendix 1). The JCS angles p, c( and y which corresponded to lateral bending, flexion/ extension and axial rotation respectively, were calculated from [R,] by:
y=Tan-’
Fig. 2. A subject pictured during the delivery stride with the source and sensor measuring three-dimensional lumbar spine movement.
-&X3 i K-22 >
No transformation or analysis was performed on the position data as these movements are known to be negligible in the lumbar spine [21]. Using the event matching information from video, JCS angles from five samples prior to back foot impact until five samples after ball release were
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calculated. These data were then smoothed using the GCVSPL package [22]. The quintic spline option was chosen with a smoothing factor of 0.01. To allow temporal comparison across trials and subjects, kinematic data were then time-normalised as a percentage of the delivery stride using a cubic spline. Back foot impact was set to 0% while 100% depicted ball release. Kinematic data were event matched (i.e. back foot impact, minimum shoulder alignment, front foot impact and release) by interpolating at every 2% of the time range.
Biomechanics 13 (1998) 574-583
Table 1 Maximum values recorded during the delivery stride (standard deviations in parentheses) (n = 20) Group SOIFO
(n = 11) Right lateral bend (deg) Left lateral bend (deg) Flexion (deg)
2.4. Determination of in&a-subject repeatability
Extension (deg)
The Coefficient of Multiple Correlation (CMC) [23] was used to determine the intra-subject repeatability of the JCS angles representing lateral bending, flexioni extension and axial rotation during the delivery stride for the three bowling trials. The mean values of all subjects for the CMC’s for lateral bending, flexion/extension and axial rotation were 0.939 (SD 0.040) 0.907 (SD 0.059) and 0.888 (SD 0.116) respectively. Reproducibility could be influenced by both the possibility of a slightly variable movement between trials, in addition to errors due to the electromagnetic device. These results suggest that the JCS angles collected by the Fastrak during the delivery stride are reproducible between trials.
Right axial rotation (deg)
2.5. Statistical analysis
As the kinematic data were reproducible, values from the three trials were averaged to provide a single representative number for each variable. The Fastrak range of motion and kinematic data were grouped by fast bowling technique [24] using the pelvic and shoulder alignments recorded by the Vicon system. Initially, a total of 46 dependent variables were obtained from the Fastrak data. For each of these variables Cohen’s d [25], an effect size index describing the standardised mean difference, was calculated (see Tables l-3). These values have been presented to assist with subsequent power calculations. To reduce the number of dependent variables and thus improve the power of the study, a correlation matrix for the 46 variables was examined and any significant correlations (df = 18, CI= 0.05, rcrit= 0.444) were recorded. Variables that were significantly (P < 0.05) correlated and logically associated were represented by the variable most related to the bowling action and the remainder underwent no analysis for significance. Lumbar spine kinematic data at back foot impact, minimum shoulder alignment, front foot impact and release were not included in the statistical comparison as variables which represented the extremes of motion
Left axial rotation (deg)
d
Mixed (n = 9)
4.4 (5.7) 27.5
(E)
0.56
33.0
0.75
(8.1)
(6.2)
43.8 (15.4) 6.3 (12.7) 13.3
(6.6) 9.0 (4.3)
52.7 (11.8) 13.9 (12.1) 11.9 (5.5) 12.1 (6.3)
0.64 0.61 0.24 0.59
“SO/F0 indicates the side-on (SO) and front-on (FO) bowling actions.
in each orthopaedic axis were favoured. A total of 12 dependent variables remained after this procedure. Independent sample t-tests using SASTAT V6.08 (SAS Institute Inc., USA) were performed to determine whether differences existed between the sideon/front-on (these groups were combined as they are considered actions which do not cause spinal pathology) and mixed action classifications for each selected kinematic variable. Originally, the alpha level was set to 0.05 (one-tail) due to the directional hypotheses in this study, however, as multiple comparisons were being performed, to reduce the likelihood of Table 2 Peak angular velocity values in each orthopaedic axis during the delivery stride (standard deviations in parentheses) (n = 20) Group
d
SO/F@ Mixed (n=ll) (n=9) Left lateral bend peak velocity (deg SK’)
627.6 (223.9) Right lateral bend peak velocity (deg s-‘) 296.9 (172.0) Flexion peak velocity (deg SK’) 1012.0 (448.2) Extension peak velocity (deg SC’) 501.6 (265.6) Left axial rotation peak velocity (deg SC’) 462.7 (152.6) Right axial rotation peak velocity (deg s-‘) 421.6 (121.6)
822.8 0.72 (318.9) 325.1 0.17 (153.2) 1412.0 0.83 (517.0) 779.9 0.77 (449.8) 416.3 -0.25 (213.5) 485.3 0.32 (242.2)
“SO/F0 indicates the side-on (SO) and front-on (FO) bowling actions.
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A. E Burnett et aLlClinical Biomechanics 13 (1998) 574-583
making a Type I error the alpha level was adjusted using the Bonferroni procedure to CI= 0.004 (0.05/12).
3. Results
Pelvic and shoulder kinematic data were used solely to classify the subject’s bowling technique [24]. This information was not analysed further. Individual data for the lumbar spine range of motion were used for normalisation purposes. The pooled means for range of motion data (side-on/front-on and mixed groups combined) were 33.2” (right lateral bend), 32.9” (left lateral bend), 67.5” (flexion), 29.0 (extension), 15.5” (right axial rotation) and 17.2” (left axial rotation). Typical position-time histories of the JCS angles (lateral bend, tIexion/extension and axial rotation) for a front-on and mixed technique bowler are presented in Fig. 3A and 3B respectively. It has been suggested that certain combinations of movement cause spinal pathology [7-91, so it is pertinent to consider the lumbar rotations in combination. A vector was therefore derived which represented the
three orientation values for each data set. The magnitude of the resulting vector represents movement in all three axes with respect to the neutral position. It was necessary to weight each component of the vector by normalising to the range of motion of the lumbar spine in each of the three orthopaedic axes. For example, if a considerable flexion component was recorded, the vector’s magnitude would be misleading as there is more potential for movement in this orthopaedic axis. When discussing the effect size (d), 0.2, 0.5 and 0.8 depict small, medium and large effects respectively [25]. In Sections 3 and 4, only variables with the absolute value of the effect size greater than 0.7 will be discussed. JCS angles and the magnitude of the normalised vector were calculated at back foot impact, minimum shoulder alignment, front foot impact and release. Of these 16 variables, only three showed a large effect size. The greatest effect size (d = 0.99) was that for flexioniextension at front foot impact. The mean for the side-on/front-on group was -0.5” (SD 6.7) negative value indicates flexed position) while the mean for the mixed groupw as 6.5” (SD 7.5) (positive value indicating extension). The lateral bend values at
Table 3 Selected kinematic variables collected during the delivery stride. (Normalisation was performed using the mean ranges of motion for right lateral bend (RLB), left lateral bend (LLB), flexion (FX), extension (EXT), right axial rotation (RAR) and left axial rotation (LAR)). (Standard deviations are in parentheses) (n = 20) Group
Lateral bend range (deg) Flexioniextension
range (deg)
Axial rotation range (deg) Magnitude of normalised vector at maximum RLB Magnitude of normalised vector at maximum LLB Magnitude of normalised vector at maximum FX Magnitude of normalised vector at maximum EXT Magnitude of normahsed vector at maximum RAR Magnitude of normalised vector at maximum LAR Mean of absolute values of lateral bend velocities (deg s-‘) Mean of absolute values of flexioniextension
velocities (deg se ‘)
Mean of absolute values of axial rotation velocities (deg s -‘)
“SO/F0 indicates side-on (SO) and front-on (FO) bowling actions.
SOIFO” (n = 11)
Mixed (n = 9)
31.9 (10.0) so. 1 (21.2) 22.3 (6.3) 46.3 (35.2) 133.0 (43.1) 134.7 (47.5) 75.8 (23.3) 85.5 (54.0) 111.5 (34.0) 136.8 (45.9) 260.2 (108.9) 121.6 (39.3)
41.0 (5.X) 66.6 (20.5) 24.0 (8.6) 50.X (23.9) 139.7 (21.2) 141.0 (20.7) 62.4 (19.9) 97.2 (36.0) 130.5 (59.9) 171.6 (48.0) 357.8 (120.1) 138.4 (26.8)
d
T
P
1.09
2.43
0.03
0.79
1.75
0.10
0.22
0.50
0.63
0.15
0.33
0.74
0.19
0.43
0.67
0.17
0.37
0.72
- 1.37
0.19
0.25
0.56
0.58
0.38
0.85
0.41
0.87
1.94
0.07
0.85
1.90
0.07
0.31
0.69
0.50
-0.61
580
A. E Burnett et al.lClinical
Biomechanics 13 (1998) 574-583
release for the non-mixed and mixed bowlers were -23.3” (SD 8.3) and -30.7” (SD 9.6) respectively (d = 0.83) indicating that the latter group was more laterally bent to the left when the ball was being released. The magnitude of the normalised vector at release was 110.7 (SD 28.8) and 129.5 (SD 19.8) for the side-on/front-on and mixed groups respectively (d = 0.74). Table 1 presents the maximal values recorded during the delivery stride for right and left lateral bend, flexion/extension, and right and left axial rotation. Medium effect sizes were shown between the groups for all but one variable (maximum right axial rotation). Maximum left lateral bend was the only variable with an effect size over 0.7. Variables in this table are later represented by the range of lateral bend, flexiomextension and axial rotation (see Table 3). Table 2 presents the peak JCS angular velocities recorded during the delivery stride for right and left lateral bend, fexion, extension, and right and left axial rotation. Again, these variables were not tested for significance between the two technique groups as they were significantly correlated (PcO.05) to the mean of the absolute values of angular velocity in each ortho-
REL
20 12 -4 -12 -20 -20 -36 -44 -52 -60’
” -10
’ 2
’ 14
’ . 26 38 Non-naked
(4
20
’ . ’ ’ 50 62 74 Time (percent)
’ 86
’ 98
110
REL
BFI
12 4 -4 -12 -20 -28 -36 -44 -52 -60’
-10
’ ”
2
I 14
’
’ 26
’
’ 38
Normalised
’
’ 50
’
’ 62
’
’ 74
’
’ 86
paedic axis. This latter variable provides an indication of the overall angular velocity of the lumbar spine with reference to each orthopaedic axis during the delivery stride. Flexion peak angular velocity showed a large effect size (d = 0.83) whilst medium effect sizes of 0.72 and 0.77 were recorded for left lateral bend peak angular velocity and extension peak angular velocity respectively. Table 3 presents the 12 dependent variables which were analysed for statistical significance. Amongst these are representative variables for the range of movement in each of the orthopaedic axes, the normalised vector representing three JCS angles, as well as lumbar spine angular velocity. Although there were no significant differences (P > 0.004) for any variable shown in this table, the means for the mixed technique group were higher for every variable except for the magnitude of the normalised vector at maximum extension. There was a greater, but non-significant (tlB = 2.43, p = 0.03, d = 1.09) difference for the range of lateral bend recorded during the delivery stride (mixed technique group, 41.0” (SD 5.8); side-on/front-on group, 31.9” (SD 10.0)). The means for the range of flexion/extension during the delivery stride were 50.1” (SD 21.2) (side-on/front-on) and 66.6” (SD 20.5) (mixed) which resulted in an efefct size between these two groups of 0.79. The means for the absolute values of lateral bending velocities (side-on/front-on: 136.8”s~’ (SD 45.9); mixed: 177.6” s-’ (SD 48.0) and flexioniextension (side-on/front-on: 260.2” s-’ (SD 108.9); mixed: 357.8” ss’ (SD 120.1)) showed similarly large effect sizes (d = 0.87 and 0.85 respectively). There were a total of 11 variables which displayed an effect size greater than 0.7. However, there were a number of significant correlations (P < 0.05) which existed between the variables when consideration was given to the orthopaedic axis that they occurred in. For example, in the lateral bend axis, the value at release was significantly correlated (r = - 0.85, PC 0.05) to the maximum value of left lateral bend. From examination of Fig. 3A and 3B the association is clearly demonstrated as these values almost coincide at the same point. As the maximal values are related to the hypotheses of this study, only they will be discussed. It is logical that the range of lateral bend and the mean. of the absolute values of the lateral bend velocities (r = 0.65, P < 0.05), in addition to the latter variable and the peak value of the lateral bend velocity (r = 0.84, P < 0.05) would be correlated.
’ 98
110
Time (percent)
Fig. 3. An example of a position-time graph of the Joint Coordinate System angles for (a) a front-on bowler and (B) a mixed bowler collected during the delivery stride: 0% depicts back foot impact while 100% represents ball release.
4. Discussion
To gain further insight into the mechanics of spinal pathology in fast bowlers, a 3-D kinematic description
A. E Burnett et al.lClinical
Biomechanics 13 (1998) 574-583
of the lumbar spine is necessary. To date, there has been no study which has addressed this methodologically complex issue. Prior to describing the 3-D orientation of one rigid body relative to another, one must make a decision about what representation system to adopt. In this study it was decided that a JCS [19] was preferable to either a projection or helical angle system. Although the issue of sequence dependency cannot be resolved, if investigators can agree on some standard approach then a direct comparison of 3-D data is possible. The JCS used here was that recommended by Cole et al. [20] (that is, et was selected as the flexion/extension axis of the proximal segment (sacrum) whilst e3 was embedded in the long axis of the distal segment (T12)). Any device which attempts to quantify lumbar spinal motion from the skin surface will suffer from movement of the soft tissues relative to the vertebrae [13,26]. For instance, studies which have recorded lumbar spine range of motion using the 3-Space@ IsotrakTM [27-291 have been shown to overestimate the values recorded using E-planar radiography, especially in axial rotation [21,30]. The range of motion data (Table 1) collected in this study are similar to that collected by other investigators using the Isotrak [27-291.
It was shown using the CMC [23] that the JCS angles were repeatable within trials. However, the repeatability would be significantly affected by the quality of the source’s attachment to the skin, a couple of cautionary notes should be mentioned. Firstly, care must be exercised when attaching the source to the skin. During pilot testing, sweat caused the doublesided tape to reduce the effectiveness of its attachment. As testing was conducted in high temperatures, extreme care was taken to ensure the sacral area was as dry as possible. The influence of sweating was successfully overcome in this study. Secondly, the combination of the mass of the source and the high impact forces typical during fast bowling introduced a low-frequency vibration in the signal during pilot testing. Care was therefore taken to correctly position the velcro band which stabilised the back of the source to the Nylatex wrap, secured around the pelvis. This was a successful approach in this study, however, this may not be such a critical factor in the future as technological advancements will inevitably lead to the production of a smaller source without compromising data sampling rates. Unfortunately, the power of this study was low due primarily to the availability of high performance bowlers eligible to be considered for this study. With multiple dependent variables, effect sizes will differ, but given the best case scenario, i.e. the largest effect size, the maximum power was approximately 0.37 (from Cohen [25]). To give the reader an idea of what
581
sample size may have been optimal for the research design used in this study, if the power level was set at 0.8 (a desirable level of power) a large effect was evident (d = 0.8) and c(= 0.005 (not CI= 0.004 for convenience), 38 subjects per group would have been required [25]. Unfortunately there are not 76 suitable subjects in Western Australia for this study to be run with an optimal level of power. To assist power calculations in further investigations the effect sizes for each dependent variable were presented in Tables l-3. Flexion/extension at front foot impact showed that, on average, the lumbar spine was marginally flexed for side-on/front-on bowlers whilst the mixed bowlers were 6.5” extended (d = 0.99) (Table 2). Mixed bowlers may end up with an extended lumbar spine as a result of the counter-rotation of the shoulder alignment and by attempting to look to the left of their front arm (righthanded bowlers) to sight the batsman and wicket. Adams et al. [lo] stated that the intervertebral disc can be damaged in hyperextension if simultaneously loaded by high compressive forces. Previous research has found that ground reaction forces five to six times body weight have been recorded at front foot impact [3,4,31,32]. Furthermore, it has been postulated that fast bowlers are especially predisposed to spondylolysis as the spine is alternately flexed and extended during the delivery stride [33]. At release, the magnitude of the normalised vector was non-significantly smaller (P > 0.004, d = 0.74) for the mixed group than the non-mixed group. When all three JCS angles are combined and represented as a vector, the mixed bowlers were generally further displaced from neutral than the side-on/front-on group at the point of release. The maximal amount of left lateral bend was non-significantly different between the two groups (P > 0.004, d = (1.75) although the rnaximal value for the mixed group was 5.5” greater. Moreover, the range of lateral bend was also similar (P > 0.004, d = 1.09) between the non-mixed (31.9”) and mixed (41.0”) groups (Table 3). Studies have shown that disc strain greatly increases as does the risk of injury with lateral loading or combinations of loading [9,34]. The range of flexioniextension was larger although not significantly (P > 0.004, d = 0.79) so between the two groups. Shirazi-Ad1 [9] stated that the presence of axial rotation and lateral bending during forward flexion (as would bc the case near the point of delivery) markedly increased loads on the compression facet. This could have direct implication in the etiology of bony abnormalities such as spondylolysis, as most fast bowlers show pars defects on the side contralateral to the bowling arm [35]. As the ranges of lateral bend and flexion/ extension during the delivery stride were non-significantly (P < 0.004) greater for the mixed group, it is intuitive that the angular velocity of the
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A. R Burnett et aLlClinical Biomechanics I3 (1998) 574-583
lumbar spine about these axes during the delivery stride would also be larger for the mixed bowlers. The angular velocities about the lateral bend (d = 0.87) and flexion/extension (d = 0.85) axes for the mixed bowlers were non-significantly larger for the mixed bowlers (Table 3). Marras and Mirka [36] found that as trunk velocity increased, a concomitant increase in trunk muscle co-activation was seen. This would subsequently increase loading on the spine since muscles work against each other. Furthermore, with increased trunk velocity, there is a resultant increase in lateral shear forces in the spine [37]. 5. Conclusions
The investigation of the 3-D lumbar spine kinematics of fast bowlers during the delivery stride was carried out for the first time using an electromagnetic device. From the results and within the limitations of the study it can be concluded that whilst t-test statistics produced no significant differences (P
The authors are grateful to Associate Professor Kevin Singer from the Curtin School of Physiotherapy for his generosity in lending the 3-Space@FastrakTM and Dr David Lloyd for assistance in the analysis phase.
References [l] Farfan HF, Cossette JW, Robertson GH, Wells RV, Kraus H. The effects of torsion on the lumbar intervertebral joints:the role of torsion in the production of disc degeneration. J Bone Joint Surg 1970;52-A:468-97. [2] Foster DH, John D, Elliott BC, Ackland T, Fitch K. Back injuries to young fast bowlers in cricket:a prospective study. Br J Sports Med 1989;23:150-4. [3] Elliott BC, Hardcastle PH, Burnett AF, Foster DH. The influ-
ence of fast bowling and physical factors on radiologic features in high performance young fast bowlers. Sports, Med, Training and Rehab 1992;3:113-30. [4] Burnett AF, Khangure MS, Elliott BC, Foster DH, Marshall RN, Hardcastle PH. Thoracolumbar disc degeneration in young fast bowlers in cricket:a follow-up study. Clin Biomech 1996;11:305-10. [5] Adams MA, Hutton WC. The relevance of torsion on the mechanical derangement of the lumbar spine. Spine 1981;6:241-8. [6] Gunzburg R, Hutton W, Fraser R. Axial rotation of the lumbar spine and the effect of flexion:an in vitro and in vivo biomechanical study. Spine 1991;16:22-8. [71 Shirazi-Ad1 A. Nonlinear stress analysis of the whole lumbar spine in torsion:mechanics of facet joint articulation. J Biomech 1994;27:289-99. PI Hindle RJ, Pearcy MJ. Rotational mobility of the human back in forward flexion. J Biomed Eng 1989;11:219-23. [91 Shirazi-Ad1 A. Finite-element evaluation of contact loads on facets of an L2-L3 lumbar segment in complex loads. Spine 1991;16:533-41. [lOI Adams MA, Dolan P, Hutton WC. The lumbar spine in backward bending. Spine 1988;13:1019-26. IllI Pope MH, Frymoyer JW, Krag MH. Diagnosing instability. Clin Orthop Rel Res 1992;279:60-7. WI Cholewicki J, McGill SM, Wells RP, Vernon H. Method for measuring vertebral kinematics from videofluoroscopy. Clin Biomech 1991$73-S. [I31 Pearcy MJ, Gill JM, Hindle RJ, Johnson GR. Measurement of human back movements in three-dimensions by opto-electronic devices. Clin Biomech 1992;2:199-204. 1141 Marras WS, Fathallah FA, Miller RJ, Davis SW, Mirka GA. Accuracy of a three-dimensional lumbar motion monitor for recording dynamic trunk motion characteristics. Int J Indust Ergonom 1992;9:75-87. u51 Pearcy MJ, Hindle RJ. A new method for the non-invasive three-dimensional measurement of human back movement. Clin Biomech 1989;4:73-9. WI An KN, Jacobsen MC, Berglund LJ, Chao EYS. Application of a magnetic tracking device to kinesiological studies. J Biomech 1988;21:613-20. P71 Woltring HJ. 3-D attitude representation of human joints:a standardization proposal. J Biomech 1994;27:1399-414. WI Wood GA, Marshall RN. The accuracy of DLT extrapolation in three-dimensional analysis. J Biomech 1986;19:781-5. t191 Grood ES, SUnkiy WJ. A joint coordinate system for the clinical description of three-dimensional motions:application to the knee. J Biomech Eng 1983;105:136-44. VOI Cole GK, Nigg BM, Ronsky JL, Yeadon MR. Application of the joint coordinate system to three-dimensional joint attitude and movement representation:a standardization proposal. J Biomech Eng 1993;115:344-9. VI Pearcy MJ, Portek I, Shepherd J. Three-dimensional X-ray analysis of normal movement in the lumbar spine. Spine 1984;9:294-7. PI Woltring HJ. A Fortran package for generalized, cross-validatory spline smoothing and differentiation. Adv Eng Software 1986;8:104-13. ~231 Kadaba MP, Ramakrishnan HK, Wootten ME, Gainey J, Gorton G, Cochran GVB. Repeatability of kinematic, kinetic and electromyographic data in normal adult gait. J Orthop Res 1989;7:849-60. [241 Burnett AF, Eliott BC, Marshall RN. The effect of a twelve over spell on fast bowling technique in cricket. J Sports Sci 1995;13:329-41. [25] Cohen J. Statistical Power Analysis for the Behavioural Sciences.New York:Academic Press, 1977.
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[26] Adams MA, Dolan P. Recent advances in lumbar spinal mechanics and their clinical significance. Clin Biomech 1995;10:3-19. [27] Russell P, Pearcy MJ, Unsworth A. Measurement of the range and coupled movements observed in the lumbar spine. Br J Rheumatol 1993;32:490-7. [28] Buchalter D, Parnianpour M, Viola K, Nordin M, Kahanovitz N. Three-dimensional spinal motion measurements. Part 1:A technique for examining posture and functional spinal motion. J Spinal Dis 1989;1:279-83. (291 Sutarno CG, McGill SM. Creating a normative kinematic database for 3-D movements of the lumbar spine. In:VIII Biennial Conference, Canadian Society of Biomechanics. 1994:202-3. (301 Pearcy MJ, Tibrewal SB. Axial rotation and lateral bending in the normal lumbar spine measured by three-dimensional radiography. Spine 1984;9:582-7. [31] Elliott BC, Foster DH. A biomechanical analysis of the front-on and side-on fast bowling techniques. J Hum Mov Stud 1984;10:83-94. [32] Elliott BC, Foster DH, Gray S. Biomechanical and physical factors influencing fast bowling. Aust J Sci Med Sport 1986;18:16-21. [33] Green TP, Allvey JC, Adams MA. Spondylolysis:bending of the during inferior articular processes of lumbar vertebrae simulated spinal movements. Spine 1994;19:2683-91. [34] Shirazi-Ad1 A, Ahmed AM, Shrivastava SC. Mechanical response of a lumbar motion segment in axial torque alone and combined with compression. Spine 1986;11:914-27. [35] Hardcastle P, Annear P, Foster DH, Chakera TM, McCormick C, Khangure M, Burnett A. Spinal abnormalities in young fast bowlers. .I Bone Joint Surg 1992; 74-B:421-5. [36] Marras WS, Mirka GA. A comprehensive evaluation of trunk response to asymmetric trunk motion. Spine 1992;17:318-26. [37] Marras WS, Sommerich CM. A three dimensional motion model of loads on the lumbar spine. Part II:Model validation. Human Factors 1991;33: 139-49.
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tion of the floating axis (unit vector ez) was defined as the cross product of the unit base vectors chosen as the fixed axes i.e. e2= i x J.
cos TX = K.e2(a= flexion) cosy= k.e,(y = twist) As in Grood and Suntay’s derivation, the above equations can be used to derive the magnitude of a and y; however, their sign is not yet determined, therefore:
71 = sin a (2 1
e,.Z=cos - -a
e2j=cos
--y ’ ( 2
1
=siny
and the magnitude of trunk adduction is found as follows: sin ,4= JOi@= ab/adduction) e,cosp=ixJ Given the above equations, the rotation matrix can be evaluated as outlined below:
Appendix 1
Zi = (J x K)i = (i x J).K = Kez co@ = cos cxcos fl
Calculation of the Joint Coordinate System angles differed to that presented by Grood and Suntay [19] as the orientation of the Fastrak’s axes (X up, Y right, Z forward) differed to that described for the right knee (Z up, X right, Y forward). A derivation of the rotation matrix R based on Grood and Suntay’s first principles method is presented below. If 1, J, K are unit base vectors in the sacral system and i,j, k are unit base vectors in the T12 system and let R =XZ+YJ+ZK describe a point with respect to the sacral system, then the vector r =xi+yj+zk describing the same point with respect to the T12 system is given by: R = [R]u+t where t is the vector which locates the T12 origin with respect to the sacral origin and [R] is the 3 x 3 rotation matrix. The e1 axis was selected such that flexion/extension was defined in the proximal system (i.e. J is the sacral system) and e3 was imbedded in the long axis of the distal segment (i.e. i in the T12 system). The orienta-
Ji = sin p J.k = J.(i x j) = (J x i).j = -je, cos /?= -sin y cos /I Ki = (I x J)i = Z*(J x i) = -Z*e2 cos p = -sin CIcos fi Ji = J.(k x i) = k*(i x J) = k.e2cos fl= cos y co@ Therefore, the three angles can be calculated from R by the following two argument tangent functions:
\
K22
1
Three-dimensional measurement of lumbar spine kinematics for fast bowlers in cricket A.F. Burnett,a * C.J. Barrett,b R.N. Marshall,” B.C. Elliott,d R.E. Day” aWestern Australian Institute of Sport, Mt. Claremont, Western Australia hBrian Edwards Physiotherapy Centre, Churchlands, Western Australia ‘Division qf Science and Technology, The lhziversity of Auckland, Auckland, New Zealand ‘Department of Human Movement, University of Western Australia, Nedlands, Western Australia ‘Department of Medical Physics, Royal Perth Hospital, Perth, Western Australia
Received25 September1996;accepted21 February1997
Abstract Objective. To determine whether the three-dimensional (3-D) lumbar spine kinematics for the mixed fast bowling technique differed to those of the side-on and front-on fast bowling techniques. Background. It has been previously shown that bowlers who utilise a mixed bowling technique are more likely to show lumbar spine pathology than those who bowl with either the side-on or front-on techniques. Methods. An electromagnetic device (3-Space’FastrakTM) operating at 120 Hz captured range of motion and 3-D lumbar spine kinematics during the delivery stride of 20 young high performance subjects. The trajectory of shoulder and pelvic girdle markers were simultaneously captured and these data were used to classify bowlers into either a side-on, front-on or mixed technique group. Results. No significant differences (P
Overuse injuries to fast bowlers in bricket are common. To better understand the mechanics of injury it is necessary to understand the 3-D rotations of the lumbar spine during this activity. 0 1998 Elsevier Science Ltd. All rights reserved. Keywords:
Cricket;Fastbowling; Injury; Three-dimensionalkinematics;Lumbarspine
1. Introduction
It is generally accepted that mechanical factors play an important role in the aetiology of related degenerative processes and injuries [l]. Fast bowling in cricket involves delivering the ball at between approximately 30 and 45 m s-l. In an effort to do this, the fast bowler’s trunk must flex, extend, laterally bend and *To whom correspondenceshould be addressedat: Western Australian Institute of Sport,ChallengeStadium,PO Box 139,Claremont 6010,WesternAustralia
rotate in a short period of time, whilst the body must absorb ground reaction forces as high as six times body weight. Three bowling actions exist with which a bowler can deliver the cricket ball; they are the side-on, fronton and mixed techniques. It has been found that the mixed bowling action, characterised by a counterrotation of the shoulders in the transverse plane, was related to the appearance of spinal pathology in the thoracolumbar spine [2-41. The exact cause of disc degeneration still remains controversial. Early suggestions which implied that
0268-0033/98/$19.00 + 0.000 1998ElsevierScienceLtd. All rights reserved. PII: SO268-0033(98)00026-6
A. l? Burnett et al./Clinical Biomechanics 13 (1998) 574-583
torsion alone caused disc degeneration [l] have since been refuted [5-71. It has been suggested, however, that combinations of repetitive loadings such as axial rotation and forward flexion [8,9] and high compressive forces in hyperextension [lo] may cause lumbar spine pathology. The quantitative evaluation of in vivo lumbar spinal movement has challenged researchers for years due primarily to the inaccessibility of the spine. Certainly the most direct, but invasive, method is to insert Steinmann pins into the spinous processes of the lumbar vertebrae and measure their relative displacements using three-dimensional (3-D) motion analysis [ 111. Roentgenographic means of motion measurement have limitations because exposure to ionising radiation limits the number of films that can be taken and dynamic motion can only be captured with reduced accuracy 111,121. Image-based motion analysis systems, using surface markers, have been used for lumbar motion assessment. Problems associated with measuring complex 3-D back motion with these systems include; variable marker movement within defined coordinate systems, marker crossover caused by closely positioned markers and the inherent inaccuracy of the measurement system [13,14]. Electromagnetic devices have been used to measure lumbar range of motion [15] and An et al. [16] have suggested that electromagnetic devices would be ideal for kinesiologic study. An electromagnetic device, the 3-Space@FastrakTM was therefore used to quantify the 3-D rotations of the lumbar spine in the side-on, front-on and mixed fast bowling techniques.
2. Methods
2.1. Sample Twenty right-hand male fast bowlers from the Western Australian Cricket Association fast bowling development squad, mean age 19.1 years (SD 1.4) mean height 183.2 cm (SD 5.4) mean mass 76.2 kg (SD 6.8), were recruited as subjects. This sample of subjects represented 77% of bowlers over 15 years of age who were in the state high performance squads. At the time of testing all subjects were bowling with no limitation to movement and signed a Document of Informed Consent. 2.2. Data collection The 3-Space@FastrakTM (Polhemus Navigation Sciences Division, Vermont, USA) is an electromagnetic device which measures the position and orientation of a sensor in space. The device’s source generates a low-frequenty magnetic field which is detected by the
57s
device’s sensor(s) and, from this, six degree-of-freedom data are generated. While up to four sensors may be used, this reduces the sampling frequency. As fast bowling is a highly dynamic activity, only one sensor was used in this study as a high sampling frequency was required. Standard connections linking the system electronics unit to the source and sensor were relatively short; therefore, a 15-way ribbon cable extension was connected to the existing cabling so that subjects could deliver the ball with a run-up of approximately 15 m. The cabling was lirmly taped to the subject’s left thigh. The device was linked to a 486 DX2 computer via an RS-232C serial interface. The Fastrak can provide output in several forms, with the most compact being Cardan angles [17] encoded as 16-bit binary numbers. Due to the limit of 19.2 kBaud for reliable operation of the system, this encoding was used and gave approximately 120 samples per second with specifically developed custom software. Some loss in spatial resolution was incurred with this choice. However, this was less critical than the lost temporal resolution a longer encoding would cause. The orientation of the source and sensor on the subjects defined a right-handed coordinate system with X up, Y right, Z forward. Therefore rtitations about X, Y and Z represented axial rotation ($), flexiomextension (4) and lateral bending (ti) respectively. The Cardanic sequence of rotation for the raw data was ZYX with a moving frame of reference. Several methods of attachment of the source and sensor to the subject were initially tested. This was done due to low-frequency vibration of the source being problematic. The following approach was therefore adopted. The skin of the sacral area was rubbed with alcohol and allowed to dry before further preparation. The source was attached to a moulded thermoplastic sacral plate which in turn was attached to the skin using double-sided tape, such that the top border of the plate was overlying a line connecting the subject’s posterior superior iliac spines. A Nylatex@ stretch wrap with mouldable plastic stitched to it was threaded through the sacral plate so that the attachment was firm. The wrap was tightly secured around the subject’s pelvis and a velcro band was secured to the back of the source and onto the wrap. The Ll spinous process was then identified by an experienced manipulative physiotherapist (CJB) using the following procedure. A line connecting the superior aspects of the iliac crests was drawn on the skin corresponding with either the L4 spinous process or the L415 interspace. Palpation then permitted Ll to be identified and marked with a skin pencil. A second wrap with a similar piece of mouldable plastic stitched into it was securely positioned such that its lower border lay directly over this mark. The sensor was
576
A. E Burnett et al.lClinical Biomechanics 13 (1998) 574-583
attached by screwing it squarely into the mouldable plastic with two small non-metallic nut and bolts. This approach prevented the cable from moving the sensor erroneously during the fast bowling run-up and delivery by effectively ‘stiffening’ the wrap. Figure 1 shows the device attached to a subject. After subjects completed a standardised warm up, they were requested to change into a pair of cycling shorts. Each subject’s standing height and weight were recorded. For data normalisation purposes, each subject’s lumbar spine range of motion was measured in flexion, extension and for both directions in lateral bending and axial rotation. When the source and sensor were attached to the subject their axes were not coincident. Therefore, it was necessary to record a neutral position for each subject so that range of motion and kinematic data in each orthopaedic axis could be measured relative to a zero reference (for mathematical transformations, see below). For this reference measure, subjects stood upright with feet shoulder-width apart and palms facing inwards and the orientation of the sensor relative to the source was recorded. this process was repeated three times and a mean was calculated. Range of motion data were
collected over approximately 6.5 s. During this time each subject moved to the limit of motion in a given direction in response to a verbal cue and then moved back to the pre-defined neutral position. Kinematic data were collected using a five-camera VX 500 Vicon motion analysis system (Oxford Metrics, Oxford). Cameras were fitted with 25 mm lenses and operated at 50 Hz. To provide the known 3-D control points for the Direct Linear Transformation reconstruction, a calibration frame with dimensions of approximately 1.8 m x 2.1 m x 1.8 m surrounded the activity space to avoid reconstruction errors due to extrapolation [18]. Retro-reflective markers were attached to the most lateral aspects of the pelvis and the acromion processes. Preliminary testing had demonstrated the need for the cable extension from the Fastrak to be supported by an assistant during the run-up to avoid any impedance to the bowler. Subjects bowled into a mat positioned 10 m away from delivery with a marked target area which represented the normal trajectory for a full length bowled over the correct distance. The Fastrak and Vicon systems were activated almost simultaneously and data collection continued for both systems for a period of approximately 6.5 s. Figure 2 shows a bowler delivering the ball in the experimental set-up. Three successful fast bowling trials were recorded for each subject. As the Vicon system only captures the outline of retro-reflective markers and not ‘complete’ visual images, it was synchronised to a video camera operating at the same frequency. Kinematic data from the Vicon could therefore be event matched (i.e. with back foot impact, front foot impact and ball release). An assessment of the Fastrak’s accuracy was performed by comparing the device’s output to those obtained from a fabricated tri-axial protractor whose axes were orthogonal. The Fastrak’s source was mounted on a base attached to the protractor while the sensor was mounted on the horizontal axis of the triaxial protractor which depicted flexion/extension was then rotated in lo” graduations from -50 to 60” with the Fastrak measuring each position. Similarly, rotations about the remaining axes were measured between -50 and 50” with varying degrees of rotation about the other axes. By comparing the values indicated by the protractor against those measured by the Fastrak, it was evident that measurements from the Fastrak became less accurate when displaced from zero. However, the values were accurate within 0.5” when rotations about the remaining axes were present. 2.3. Data reduction and transformation
Fig . 1. The 3-Space”FastrakTM attached to a subject.
The synchronised video records permitted the time from when the Vicon was activated until back foot
A. E Burnett et aLlClinical Biomechanics I3 (1998) 574-583
impact and release to be identified. Resulting coordinate data from the pelvic and shoulder girdle markers over these times were then converted into ASCII format and downloaded onto an IBM PC. These data were smoothed using a Butterworth second-order digital filter at a cut-off frequency of 5 Hz. Transformation of the Fastrak raw data was necessary as an alternative rotation sequence and representation system was preferred. Firstly, each data record containing three Cardan angles $, 4 and 8 were converted into the elements of their respective direction cosine matrices by the following equations:
Rz2 =
sin 8 sin 4 sin I,!J+COS 8 costi
R3, = -sin
4
Rx2 = cos I$
sin II/
R33 = cos 4 cos tj
where the direction cosine matrix R was defined as: R,, = cos 0 cos 4
RI, Rz, Rzl, Rz i Rx! I R12
R,,=cos0sin4sin$-sinOcos$
R=
R3,
RI3 = cos 0
Rz, =
sin 4 cos $+sin 0 sin I+!I
sin0 cos &
R13
R33
Secondly, in order to measure orientation relative to a zero reference the following transform was applied to the data:
VW = U?AI~[RBI where [RA] is the direction cosine matrix of the neutral data and [R,] is the direction cosine matrix of the kinematic data, while [RAITis the transpose of [RA]. Joint Coordinate System (JCS) angles described by Grood and Suntay [19] were recovered from the direction cosine matrix [R,-1.However, the formulae used to obtain these angles differed from that outlined in their paper as the Fastrak’s coordinate axes were orientated differently and further, the choice of axes for unit vectors el and e3was based on the recommendations of Cole et al. [20] ( see Appendix 1). The JCS angles p, c( and y which corresponded to lateral bending, flexion/ extension and axial rotation respectively, were calculated from [R,] by:
y=Tan-’
Fig. 2. A subject pictured during the delivery stride with the source and sensor measuring three-dimensional lumbar spine movement.
-&X3 i K-22 >
No transformation or analysis was performed on the position data as these movements are known to be negligible in the lumbar spine [21]. Using the event matching information from video, JCS angles from five samples prior to back foot impact until five samples after ball release were
578
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calculated. These data were then smoothed using the GCVSPL package [22]. The quintic spline option was chosen with a smoothing factor of 0.01. To allow temporal comparison across trials and subjects, kinematic data were then time-normalised as a percentage of the delivery stride using a cubic spline. Back foot impact was set to 0% while 100% depicted ball release. Kinematic data were event matched (i.e. back foot impact, minimum shoulder alignment, front foot impact and release) by interpolating at every 2% of the time range.
Biomechanics 13 (1998) 574-583
Table 1 Maximum values recorded during the delivery stride (standard deviations in parentheses) (n = 20) Group SOIFO
(n = 11) Right lateral bend (deg) Left lateral bend (deg) Flexion (deg)
2.4. Determination of in&a-subject repeatability
Extension (deg)
The Coefficient of Multiple Correlation (CMC) [23] was used to determine the intra-subject repeatability of the JCS angles representing lateral bending, flexioni extension and axial rotation during the delivery stride for the three bowling trials. The mean values of all subjects for the CMC’s for lateral bending, flexion/extension and axial rotation were 0.939 (SD 0.040) 0.907 (SD 0.059) and 0.888 (SD 0.116) respectively. Reproducibility could be influenced by both the possibility of a slightly variable movement between trials, in addition to errors due to the electromagnetic device. These results suggest that the JCS angles collected by the Fastrak during the delivery stride are reproducible between trials.
Right axial rotation (deg)
2.5. Statistical analysis
As the kinematic data were reproducible, values from the three trials were averaged to provide a single representative number for each variable. The Fastrak range of motion and kinematic data were grouped by fast bowling technique [24] using the pelvic and shoulder alignments recorded by the Vicon system. Initially, a total of 46 dependent variables were obtained from the Fastrak data. For each of these variables Cohen’s d [25], an effect size index describing the standardised mean difference, was calculated (see Tables l-3). These values have been presented to assist with subsequent power calculations. To reduce the number of dependent variables and thus improve the power of the study, a correlation matrix for the 46 variables was examined and any significant correlations (df = 18, CI= 0.05, rcrit= 0.444) were recorded. Variables that were significantly (P < 0.05) correlated and logically associated were represented by the variable most related to the bowling action and the remainder underwent no analysis for significance. Lumbar spine kinematic data at back foot impact, minimum shoulder alignment, front foot impact and release were not included in the statistical comparison as variables which represented the extremes of motion
Left axial rotation (deg)
d
Mixed (n = 9)
4.4 (5.7) 27.5
(E)
0.56
33.0
0.75
(8.1)
(6.2)
43.8 (15.4) 6.3 (12.7) 13.3
(6.6) 9.0 (4.3)
52.7 (11.8) 13.9 (12.1) 11.9 (5.5) 12.1 (6.3)
0.64 0.61 0.24 0.59
“SO/F0 indicates the side-on (SO) and front-on (FO) bowling actions.
in each orthopaedic axis were favoured. A total of 12 dependent variables remained after this procedure. Independent sample t-tests using SASTAT V6.08 (SAS Institute Inc., USA) were performed to determine whether differences existed between the sideon/front-on (these groups were combined as they are considered actions which do not cause spinal pathology) and mixed action classifications for each selected kinematic variable. Originally, the alpha level was set to 0.05 (one-tail) due to the directional hypotheses in this study, however, as multiple comparisons were being performed, to reduce the likelihood of Table 2 Peak angular velocity values in each orthopaedic axis during the delivery stride (standard deviations in parentheses) (n = 20) Group
d
SO/F@ Mixed (n=ll) (n=9) Left lateral bend peak velocity (deg SK’)
627.6 (223.9) Right lateral bend peak velocity (deg s-‘) 296.9 (172.0) Flexion peak velocity (deg SK’) 1012.0 (448.2) Extension peak velocity (deg SC’) 501.6 (265.6) Left axial rotation peak velocity (deg SC’) 462.7 (152.6) Right axial rotation peak velocity (deg s-‘) 421.6 (121.6)
822.8 0.72 (318.9) 325.1 0.17 (153.2) 1412.0 0.83 (517.0) 779.9 0.77 (449.8) 416.3 -0.25 (213.5) 485.3 0.32 (242.2)
“SO/F0 indicates the side-on (SO) and front-on (FO) bowling actions.
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making a Type I error the alpha level was adjusted using the Bonferroni procedure to CI= 0.004 (0.05/12).
3. Results
Pelvic and shoulder kinematic data were used solely to classify the subject’s bowling technique [24]. This information was not analysed further. Individual data for the lumbar spine range of motion were used for normalisation purposes. The pooled means for range of motion data (side-on/front-on and mixed groups combined) were 33.2” (right lateral bend), 32.9” (left lateral bend), 67.5” (flexion), 29.0 (extension), 15.5” (right axial rotation) and 17.2” (left axial rotation). Typical position-time histories of the JCS angles (lateral bend, tIexion/extension and axial rotation) for a front-on and mixed technique bowler are presented in Fig. 3A and 3B respectively. It has been suggested that certain combinations of movement cause spinal pathology [7-91, so it is pertinent to consider the lumbar rotations in combination. A vector was therefore derived which represented the
three orientation values for each data set. The magnitude of the resulting vector represents movement in all three axes with respect to the neutral position. It was necessary to weight each component of the vector by normalising to the range of motion of the lumbar spine in each of the three orthopaedic axes. For example, if a considerable flexion component was recorded, the vector’s magnitude would be misleading as there is more potential for movement in this orthopaedic axis. When discussing the effect size (d), 0.2, 0.5 and 0.8 depict small, medium and large effects respectively [25]. In Sections 3 and 4, only variables with the absolute value of the effect size greater than 0.7 will be discussed. JCS angles and the magnitude of the normalised vector were calculated at back foot impact, minimum shoulder alignment, front foot impact and release. Of these 16 variables, only three showed a large effect size. The greatest effect size (d = 0.99) was that for flexioniextension at front foot impact. The mean for the side-on/front-on group was -0.5” (SD 6.7) negative value indicates flexed position) while the mean for the mixed groupw as 6.5” (SD 7.5) (positive value indicating extension). The lateral bend values at
Table 3 Selected kinematic variables collected during the delivery stride. (Normalisation was performed using the mean ranges of motion for right lateral bend (RLB), left lateral bend (LLB), flexion (FX), extension (EXT), right axial rotation (RAR) and left axial rotation (LAR)). (Standard deviations are in parentheses) (n = 20) Group
Lateral bend range (deg) Flexioniextension
range (deg)
Axial rotation range (deg) Magnitude of normalised vector at maximum RLB Magnitude of normalised vector at maximum LLB Magnitude of normalised vector at maximum FX Magnitude of normalised vector at maximum EXT Magnitude of normahsed vector at maximum RAR Magnitude of normalised vector at maximum LAR Mean of absolute values of lateral bend velocities (deg s-‘) Mean of absolute values of flexioniextension
velocities (deg se ‘)
Mean of absolute values of axial rotation velocities (deg s -‘)
“SO/F0 indicates side-on (SO) and front-on (FO) bowling actions.
SOIFO” (n = 11)
Mixed (n = 9)
31.9 (10.0) so. 1 (21.2) 22.3 (6.3) 46.3 (35.2) 133.0 (43.1) 134.7 (47.5) 75.8 (23.3) 85.5 (54.0) 111.5 (34.0) 136.8 (45.9) 260.2 (108.9) 121.6 (39.3)
41.0 (5.X) 66.6 (20.5) 24.0 (8.6) 50.X (23.9) 139.7 (21.2) 141.0 (20.7) 62.4 (19.9) 97.2 (36.0) 130.5 (59.9) 171.6 (48.0) 357.8 (120.1) 138.4 (26.8)
d
T
P
1.09
2.43
0.03
0.79
1.75
0.10
0.22
0.50
0.63
0.15
0.33
0.74
0.19
0.43
0.67
0.17
0.37
0.72
- 1.37
0.19
0.25
0.56
0.58
0.38
0.85
0.41
0.87
1.94
0.07
0.85
1.90
0.07
0.31
0.69
0.50
-0.61
580
A. E Burnett et al.lClinical
Biomechanics 13 (1998) 574-583
release for the non-mixed and mixed bowlers were -23.3” (SD 8.3) and -30.7” (SD 9.6) respectively (d = 0.83) indicating that the latter group was more laterally bent to the left when the ball was being released. The magnitude of the normalised vector at release was 110.7 (SD 28.8) and 129.5 (SD 19.8) for the side-on/front-on and mixed groups respectively (d = 0.74). Table 1 presents the maximal values recorded during the delivery stride for right and left lateral bend, flexion/extension, and right and left axial rotation. Medium effect sizes were shown between the groups for all but one variable (maximum right axial rotation). Maximum left lateral bend was the only variable with an effect size over 0.7. Variables in this table are later represented by the range of lateral bend, flexiomextension and axial rotation (see Table 3). Table 2 presents the peak JCS angular velocities recorded during the delivery stride for right and left lateral bend, fexion, extension, and right and left axial rotation. Again, these variables were not tested for significance between the two technique groups as they were significantly correlated (PcO.05) to the mean of the absolute values of angular velocity in each ortho-
REL
20 12 -4 -12 -20 -20 -36 -44 -52 -60’
” -10
’ 2
’ 14
’ . 26 38 Non-naked
(4
20
’ . ’ ’ 50 62 74 Time (percent)
’ 86
’ 98
110
REL
BFI
12 4 -4 -12 -20 -28 -36 -44 -52 -60’
-10
’ ”
2
I 14
’
’ 26
’
’ 38
Normalised
’
’ 50
’
’ 62
’
’ 74
’
’ 86
paedic axis. This latter variable provides an indication of the overall angular velocity of the lumbar spine with reference to each orthopaedic axis during the delivery stride. Flexion peak angular velocity showed a large effect size (d = 0.83) whilst medium effect sizes of 0.72 and 0.77 were recorded for left lateral bend peak angular velocity and extension peak angular velocity respectively. Table 3 presents the 12 dependent variables which were analysed for statistical significance. Amongst these are representative variables for the range of movement in each of the orthopaedic axes, the normalised vector representing three JCS angles, as well as lumbar spine angular velocity. Although there were no significant differences (P > 0.004) for any variable shown in this table, the means for the mixed technique group were higher for every variable except for the magnitude of the normalised vector at maximum extension. There was a greater, but non-significant (tlB = 2.43, p = 0.03, d = 1.09) difference for the range of lateral bend recorded during the delivery stride (mixed technique group, 41.0” (SD 5.8); side-on/front-on group, 31.9” (SD 10.0)). The means for the range of flexion/extension during the delivery stride were 50.1” (SD 21.2) (side-on/front-on) and 66.6” (SD 20.5) (mixed) which resulted in an efefct size between these two groups of 0.79. The means for the absolute values of lateral bending velocities (side-on/front-on: 136.8”s~’ (SD 45.9); mixed: 177.6” s-’ (SD 48.0) and flexioniextension (side-on/front-on: 260.2” s-’ (SD 108.9); mixed: 357.8” ss’ (SD 120.1)) showed similarly large effect sizes (d = 0.87 and 0.85 respectively). There were a total of 11 variables which displayed an effect size greater than 0.7. However, there were a number of significant correlations (P < 0.05) which existed between the variables when consideration was given to the orthopaedic axis that they occurred in. For example, in the lateral bend axis, the value at release was significantly correlated (r = - 0.85, PC 0.05) to the maximum value of left lateral bend. From examination of Fig. 3A and 3B the association is clearly demonstrated as these values almost coincide at the same point. As the maximal values are related to the hypotheses of this study, only they will be discussed. It is logical that the range of lateral bend and the mean. of the absolute values of the lateral bend velocities (r = 0.65, P < 0.05), in addition to the latter variable and the peak value of the lateral bend velocity (r = 0.84, P < 0.05) would be correlated.
’ 98
110
Time (percent)
Fig. 3. An example of a position-time graph of the Joint Coordinate System angles for (a) a front-on bowler and (B) a mixed bowler collected during the delivery stride: 0% depicts back foot impact while 100% represents ball release.
4. Discussion
To gain further insight into the mechanics of spinal pathology in fast bowlers, a 3-D kinematic description
A. E Burnett et al.lClinical
Biomechanics 13 (1998) 574-583
of the lumbar spine is necessary. To date, there has been no study which has addressed this methodologically complex issue. Prior to describing the 3-D orientation of one rigid body relative to another, one must make a decision about what representation system to adopt. In this study it was decided that a JCS [19] was preferable to either a projection or helical angle system. Although the issue of sequence dependency cannot be resolved, if investigators can agree on some standard approach then a direct comparison of 3-D data is possible. The JCS used here was that recommended by Cole et al. [20] (that is, et was selected as the flexion/extension axis of the proximal segment (sacrum) whilst e3 was embedded in the long axis of the distal segment (T12)). Any device which attempts to quantify lumbar spinal motion from the skin surface will suffer from movement of the soft tissues relative to the vertebrae [13,26]. For instance, studies which have recorded lumbar spine range of motion using the 3-Space@ IsotrakTM [27-291 have been shown to overestimate the values recorded using E-planar radiography, especially in axial rotation [21,30]. The range of motion data (Table 1) collected in this study are similar to that collected by other investigators using the Isotrak [27-291.
It was shown using the CMC [23] that the JCS angles were repeatable within trials. However, the repeatability would be significantly affected by the quality of the source’s attachment to the skin, a couple of cautionary notes should be mentioned. Firstly, care must be exercised when attaching the source to the skin. During pilot testing, sweat caused the doublesided tape to reduce the effectiveness of its attachment. As testing was conducted in high temperatures, extreme care was taken to ensure the sacral area was as dry as possible. The influence of sweating was successfully overcome in this study. Secondly, the combination of the mass of the source and the high impact forces typical during fast bowling introduced a low-frequency vibration in the signal during pilot testing. Care was therefore taken to correctly position the velcro band which stabilised the back of the source to the Nylatex wrap, secured around the pelvis. This was a successful approach in this study, however, this may not be such a critical factor in the future as technological advancements will inevitably lead to the production of a smaller source without compromising data sampling rates. Unfortunately, the power of this study was low due primarily to the availability of high performance bowlers eligible to be considered for this study. With multiple dependent variables, effect sizes will differ, but given the best case scenario, i.e. the largest effect size, the maximum power was approximately 0.37 (from Cohen [25]). To give the reader an idea of what
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sample size may have been optimal for the research design used in this study, if the power level was set at 0.8 (a desirable level of power) a large effect was evident (d = 0.8) and c(= 0.005 (not CI= 0.004 for convenience), 38 subjects per group would have been required [25]. Unfortunately there are not 76 suitable subjects in Western Australia for this study to be run with an optimal level of power. To assist power calculations in further investigations the effect sizes for each dependent variable were presented in Tables l-3. Flexion/extension at front foot impact showed that, on average, the lumbar spine was marginally flexed for side-on/front-on bowlers whilst the mixed bowlers were 6.5” extended (d = 0.99) (Table 2). Mixed bowlers may end up with an extended lumbar spine as a result of the counter-rotation of the shoulder alignment and by attempting to look to the left of their front arm (righthanded bowlers) to sight the batsman and wicket. Adams et al. [lo] stated that the intervertebral disc can be damaged in hyperextension if simultaneously loaded by high compressive forces. Previous research has found that ground reaction forces five to six times body weight have been recorded at front foot impact [3,4,31,32]. Furthermore, it has been postulated that fast bowlers are especially predisposed to spondylolysis as the spine is alternately flexed and extended during the delivery stride [33]. At release, the magnitude of the normalised vector was non-significantly smaller (P > 0.004, d = 0.74) for the mixed group than the non-mixed group. When all three JCS angles are combined and represented as a vector, the mixed bowlers were generally further displaced from neutral than the side-on/front-on group at the point of release. The maximal amount of left lateral bend was non-significantly different between the two groups (P > 0.004, d = (1.75) although the rnaximal value for the mixed group was 5.5” greater. Moreover, the range of lateral bend was also similar (P > 0.004, d = 1.09) between the non-mixed (31.9”) and mixed (41.0”) groups (Table 3). Studies have shown that disc strain greatly increases as does the risk of injury with lateral loading or combinations of loading [9,34]. The range of flexioniextension was larger although not significantly (P > 0.004, d = 0.79) so between the two groups. Shirazi-Ad1 [9] stated that the presence of axial rotation and lateral bending during forward flexion (as would bc the case near the point of delivery) markedly increased loads on the compression facet. This could have direct implication in the etiology of bony abnormalities such as spondylolysis, as most fast bowlers show pars defects on the side contralateral to the bowling arm [35]. As the ranges of lateral bend and flexion/ extension during the delivery stride were non-significantly (P < 0.004) greater for the mixed group, it is intuitive that the angular velocity of the
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A. R Burnett et aLlClinical Biomechanics I3 (1998) 574-583
lumbar spine about these axes during the delivery stride would also be larger for the mixed bowlers. The angular velocities about the lateral bend (d = 0.87) and flexion/extension (d = 0.85) axes for the mixed bowlers were non-significantly larger for the mixed bowlers (Table 3). Marras and Mirka [36] found that as trunk velocity increased, a concomitant increase in trunk muscle co-activation was seen. This would subsequently increase loading on the spine since muscles work against each other. Furthermore, with increased trunk velocity, there is a resultant increase in lateral shear forces in the spine [37]. 5. Conclusions
The investigation of the 3-D lumbar spine kinematics of fast bowlers during the delivery stride was carried out for the first time using an electromagnetic device. From the results and within the limitations of the study it can be concluded that whilst t-test statistics produced no significant differences (P
The authors are grateful to Associate Professor Kevin Singer from the Curtin School of Physiotherapy for his generosity in lending the 3-Space@FastrakTM and Dr David Lloyd for assistance in the analysis phase.
References [l] Farfan HF, Cossette JW, Robertson GH, Wells RV, Kraus H. The effects of torsion on the lumbar intervertebral joints:the role of torsion in the production of disc degeneration. J Bone Joint Surg 1970;52-A:468-97. [2] Foster DH, John D, Elliott BC, Ackland T, Fitch K. Back injuries to young fast bowlers in cricket:a prospective study. Br J Sports Med 1989;23:150-4. [3] Elliott BC, Hardcastle PH, Burnett AF, Foster DH. The influ-
ence of fast bowling and physical factors on radiologic features in high performance young fast bowlers. Sports, Med, Training and Rehab 1992;3:113-30. [4] Burnett AF, Khangure MS, Elliott BC, Foster DH, Marshall RN, Hardcastle PH. Thoracolumbar disc degeneration in young fast bowlers in cricket:a follow-up study. Clin Biomech 1996;11:305-10. [5] Adams MA, Hutton WC. The relevance of torsion on the mechanical derangement of the lumbar spine. Spine 1981;6:241-8. [6] Gunzburg R, Hutton W, Fraser R. Axial rotation of the lumbar spine and the effect of flexion:an in vitro and in vivo biomechanical study. Spine 1991;16:22-8. [71 Shirazi-Ad1 A. Nonlinear stress analysis of the whole lumbar spine in torsion:mechanics of facet joint articulation. J Biomech 1994;27:289-99. PI Hindle RJ, Pearcy MJ. Rotational mobility of the human back in forward flexion. J Biomed Eng 1989;11:219-23. [91 Shirazi-Ad1 A. Finite-element evaluation of contact loads on facets of an L2-L3 lumbar segment in complex loads. Spine 1991;16:533-41. [lOI Adams MA, Dolan P, Hutton WC. The lumbar spine in backward bending. Spine 1988;13:1019-26. IllI Pope MH, Frymoyer JW, Krag MH. Diagnosing instability. Clin Orthop Rel Res 1992;279:60-7. WI Cholewicki J, McGill SM, Wells RP, Vernon H. Method for measuring vertebral kinematics from videofluoroscopy. Clin Biomech 1991$73-S. [I31 Pearcy MJ, Gill JM, Hindle RJ, Johnson GR. Measurement of human back movements in three-dimensions by opto-electronic devices. Clin Biomech 1992;2:199-204. 1141 Marras WS, Fathallah FA, Miller RJ, Davis SW, Mirka GA. Accuracy of a three-dimensional lumbar motion monitor for recording dynamic trunk motion characteristics. Int J Indust Ergonom 1992;9:75-87. u51 Pearcy MJ, Hindle RJ. A new method for the non-invasive three-dimensional measurement of human back movement. Clin Biomech 1989;4:73-9. WI An KN, Jacobsen MC, Berglund LJ, Chao EYS. Application of a magnetic tracking device to kinesiological studies. J Biomech 1988;21:613-20. P71 Woltring HJ. 3-D attitude representation of human joints:a standardization proposal. J Biomech 1994;27:1399-414. WI Wood GA, Marshall RN. The accuracy of DLT extrapolation in three-dimensional analysis. J Biomech 1986;19:781-5. t191 Grood ES, SUnkiy WJ. A joint coordinate system for the clinical description of three-dimensional motions:application to the knee. J Biomech Eng 1983;105:136-44. VOI Cole GK, Nigg BM, Ronsky JL, Yeadon MR. Application of the joint coordinate system to three-dimensional joint attitude and movement representation:a standardization proposal. J Biomech Eng 1993;115:344-9. VI Pearcy MJ, Portek I, Shepherd J. Three-dimensional X-ray analysis of normal movement in the lumbar spine. Spine 1984;9:294-7. PI Woltring HJ. A Fortran package for generalized, cross-validatory spline smoothing and differentiation. Adv Eng Software 1986;8:104-13. ~231 Kadaba MP, Ramakrishnan HK, Wootten ME, Gainey J, Gorton G, Cochran GVB. Repeatability of kinematic, kinetic and electromyographic data in normal adult gait. J Orthop Res 1989;7:849-60. [241 Burnett AF, Eliott BC, Marshall RN. The effect of a twelve over spell on fast bowling technique in cricket. J Sports Sci 1995;13:329-41. [25] Cohen J. Statistical Power Analysis for the Behavioural Sciences.New York:Academic Press, 1977.
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[26] Adams MA, Dolan P. Recent advances in lumbar spinal mechanics and their clinical significance. Clin Biomech 1995;10:3-19. [27] Russell P, Pearcy MJ, Unsworth A. Measurement of the range and coupled movements observed in the lumbar spine. Br J Rheumatol 1993;32:490-7. [28] Buchalter D, Parnianpour M, Viola K, Nordin M, Kahanovitz N. Three-dimensional spinal motion measurements. Part 1:A technique for examining posture and functional spinal motion. J Spinal Dis 1989;1:279-83. (291 Sutarno CG, McGill SM. Creating a normative kinematic database for 3-D movements of the lumbar spine. In:VIII Biennial Conference, Canadian Society of Biomechanics. 1994:202-3. (301 Pearcy MJ, Tibrewal SB. Axial rotation and lateral bending in the normal lumbar spine measured by three-dimensional radiography. Spine 1984;9:582-7. [31] Elliott BC, Foster DH. A biomechanical analysis of the front-on and side-on fast bowling techniques. J Hum Mov Stud 1984;10:83-94. [32] Elliott BC, Foster DH, Gray S. Biomechanical and physical factors influencing fast bowling. Aust J Sci Med Sport 1986;18:16-21. [33] Green TP, Allvey JC, Adams MA. Spondylolysis:bending of the during inferior articular processes of lumbar vertebrae simulated spinal movements. Spine 1994;19:2683-91. [34] Shirazi-Ad1 A, Ahmed AM, Shrivastava SC. Mechanical response of a lumbar motion segment in axial torque alone and combined with compression. Spine 1986;11:914-27. [35] Hardcastle P, Annear P, Foster DH, Chakera TM, McCormick C, Khangure M, Burnett A. Spinal abnormalities in young fast bowlers. .I Bone Joint Surg 1992; 74-B:421-5. [36] Marras WS, Mirka GA. A comprehensive evaluation of trunk response to asymmetric trunk motion. Spine 1992;17:318-26. [37] Marras WS, Sommerich CM. A three dimensional motion model of loads on the lumbar spine. Part II:Model validation. Human Factors 1991;33: 139-49.
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tion of the floating axis (unit vector ez) was defined as the cross product of the unit base vectors chosen as the fixed axes i.e. e2= i x J.
cos TX = K.e2(a= flexion) cosy= k.e,(y = twist) As in Grood and Suntay’s derivation, the above equations can be used to derive the magnitude of a and y; however, their sign is not yet determined, therefore:
71 = sin a (2 1
e,.Z=cos - -a
e2j=cos
--y ’ ( 2
1
=siny
and the magnitude of trunk adduction is found as follows: sin ,4= JOi@= ab/adduction) e,cosp=ixJ Given the above equations, the rotation matrix can be evaluated as outlined below:
Appendix 1
Zi = (J x K)i = (i x J).K = Kez co@ = cos cxcos fl
Calculation of the Joint Coordinate System angles differed to that presented by Grood and Suntay [19] as the orientation of the Fastrak’s axes (X up, Y right, Z forward) differed to that described for the right knee (Z up, X right, Y forward). A derivation of the rotation matrix R based on Grood and Suntay’s first principles method is presented below. If 1, J, K are unit base vectors in the sacral system and i,j, k are unit base vectors in the T12 system and let R =XZ+YJ+ZK describe a point with respect to the sacral system, then the vector r =xi+yj+zk describing the same point with respect to the T12 system is given by: R = [R]u+t where t is the vector which locates the T12 origin with respect to the sacral origin and [R] is the 3 x 3 rotation matrix. The e1 axis was selected such that flexion/extension was defined in the proximal system (i.e. J is the sacral system) and e3 was imbedded in the long axis of the distal segment (i.e. i in the T12 system). The orienta-
Ji = sin p J.k = J.(i x j) = (J x i).j = -je, cos /?= -sin y cos /I Ki = (I x J)i = Z*(J x i) = -Z*e2 cos p = -sin CIcos fi Ji = J.(k x i) = k*(i x J) = k.e2cos fl= cos y co@ Therefore, the three angles can be calculated from R by the following two argument tangent functions:
\
K22
1