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The effect of differing Cardan angle sequences on three dimensional lumbo-pelvic angular kinematics during running

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Abstract

The variability in the three dimensional (3D) lumbo-pelvic angular kinematic patterns during running when using differing Cardan angle sequences was quantified. Data for four able-bodied subjects running on a treadmill at 4.0 m/s were captured using a motion analysis system with six cameras operating at 200 Hz. The adjusted coefficient of multiple correlation was used to compare graphical waveforms whilst the maximum root mean square of the differences was used to express the magnitude of any discrepancy in absolute units. Minimal qualitative differences were found between the various sequences. Quantitative differences between each of the Cardan angle sequences were not found to exceed 7.0° and 2.8° for the lumbar spine and pelvic rotations respectively. It was concluded that different Cardan angle sequences were not found to substantially affect typical 3D lumbo-pelvic angular kinematic patterns during running.

Introduction

A popular method of describing the 3D angular orientation of a ‘moving’ set of right-handed coordinate axes (frame) {B} with respect to a ‘fixed’ frame {A} is through the use of three independent angles known as Cardan angles that are obtained by an ordered sequence of rotations (Fig. 1). Each rotation can be described as occurring about the axes of frame {B} [1]. Frame {B}, initially beginning coincident with frame {A}, is rotated about YB by an angle α, then rotated about a ‘rotated’ XB (XB′) by an angle β, and finally rotated about a ‘twice-rotated’ ZB (ZB″) by an angle γ. These three sequential rotations about the axes of the moving frame {B} can be represented by the notation RyRx′→Rz″.

Cardan angles have been applied by numerous researchers in the past to describe both in vitro and in vivo 3D spinal [2], [3], [4], [5], [6], [7], [8] and pelvic motion [9], [10]. However, it is difficult to compare the results from the various studies due to inconsistencies in the implemented Cardan angle sequence. As previously noted, Cardan angles are sequence dependent. This is because multiplication of the matrices used to represent rotations is not commutative [1]. Consequently, for a given motion, different sequences will result in different values for the derived angles [11], [12], [13], [14], [15], [16].

Previous studies have demonstrated discrepancies between differing Cardan angle sequences when measuring relatively large isolated movements of the spine such as forward bending, lateral bending and axial rotation [17], [18]. For spinal rotations of 30°, the various Cardan angle sequences diverge significantly from one another [17], [19]. However, for spinal rotations of 10°, changing the Cardan angle sequence yields differences of only 2° [19]. These results suggest that for typical amplitudes of lumbo-pelvic movement during normal walking, deviations between Cardan angle sequences will be insignificant. In situations where the amplitudes of these movements are much greater, such as pathological gait (e.g. spina bifida [7]) or increasing speeds of locomotion (e.g. running [20]), it is feasible that more significant deviations may become apparent. To our knowledge there have been no published studies that have specifically assessed the degree of variability that occurs in the 3D lumbo-pelvic angular kinematic patterns during running with differing Cardan angle sequences.

The Cardan angle sequence RyRx′→Rz″ [21], or the equivalent non-orthogonal joint coordinate system (JCS) [12], [22], has been recommended for describing 3D lower limb joint motion based upon its uniformity with established anatomical definitions. The International Society of Biomechanics have proposed this sequence as the generic standard for describing relative orientation, and the RyRz′→Rx″ sequence for describing global orientation [23]. It is also important that if kinematic data are to be useful in a clinical context then the description of 3D motion must be based upon definitions that are consistent with the standard terminology used by the practitioner [13]. The RyRx′→Rz″ sequence has certainly proven to be capable of providing results in accordance with clinical evaluation when measuring the predominantly quasi-planar movements of the lower limb joints during walking and running [24].

Lumbo-pelvic motion during function does not behave in such a quasi-planar manner. It is therefore not surprising that controversy exists with respect to the recommended Cardan angle sequence for the 3D kinematic description of such movements. Several researchers [17], [19], [25] have provided various anatomical and clinical criteria as justification for a proposed ideal Cardan angle sequence for describing 3D spinal or pelvic motion, however, all have differed with respect to their recommended sequence. Given this discrepancy, the aim of the present study was to quantify the variability that occurs in the 3D lumbo-pelvic angular kinematic patterns during running when using the RyRx′→Rz″ Cardan angle sequence compared with the five alternative sequences.

Section snippets

Subjects

Approval was obtained from The University of Melbourne and the Australian Institute of Sport Ethics committees. A cohort of four able-bodied male subjects, who had an average age of 27 yr (range 21–34 yr), height of 180 cm (range 175–185 cm) and body mass of 78 kg (range 76–82 kg), were voluntarily recruited. All subjects were recreational runners who usually ran greater than 20 km/week, were not injured at the time of testing and had previous treadmill running experience. As all subjects

Results

All Cardan angle sequences were found to produce very similar qualitative kinematic descriptions of the 3D lumbo-pelvic rotations during running. The CMC values for the comparison of the graphical waveforms of the five alternative Cardan angle sequences with the RyRx′→Rz″ sequence were at least 0.924 for all rotations (Table 1).

The quantitative differences between each of the five alternative Cardan angle sequences when compared to the RyRx′→Rz″ sequence were also generally found to be quite

Discussion

The most common methods utilised in kinematic research for describing the 3D orientation between two defined frames are the Cardan angle [9], [21] or non-orthogonal JCS method [22], the projection angle method [40], [41] and the helical angle (attitude vector) method [15]. Each parameterisation has its respective advantages and disadvantages which has lead to spirited discussions in the biomechanics community regarding the method of choice (see Appendix B from Woltring [15]). The sequence

Conclusion

Different Cardan angle sequences were not found to substantially affect typical 3D lumbo-pelvic angular kinematic patterns during running. Due to differences in the recommendations of previous investigators, an accepted standard Cardan angle sequence for describing 3D lumbo-pelvic angular kinematics is currently unable to be identified.

Acknowledgements

This research was funded by a grant from the Australian Research Council. The support provided by L. Philpot of the Australian Institute of Sport and D. Hopkins of Nike Australia was greatly appreciated.

References (43)

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