Use of dual Euler angles to quantify the three-dimensional joint motion and its application to the ankle joint complex

This paper presents a modified Euler angles method, dual Euler angles approach, to describe general spatial human joint motions. In dual Euler angles approach, the three-dimensional joint motion is considered as three successive screw motions with respect to the axes of the moving segment coordinate system; accordingly, the screw motion displacements are represented by dual Euler angles. The algorithm for calculating dual Euler angles from coordinates of markers on the moving segment is also provided in this study. As an example, the proposed method is applied to describe motions of ankle joint complex during dorsiflexion–plantarflexion. A Flock of Birds electromagnetic tracking device (FOB) was used to measure joint motion in vivo. Preliminary accuracy tests on a gimbal structure demonstrate that the mean errors of dual Euler angles evaluated by using source data from FOB are less than 1° for rotations and 1 mm for translations, respectively. Based on the pilot study, FOB is feasible for quantifying human joint motions using dual Euler angles approach.     

On the estimation of joint kinematics during gait.

In gait analysis, the concepts of Euler and helical (screw) angles are used to define the three-dimensional relative joint angular motion of lower extremities. Reliable estimation of joint angular motion depends on the accurate definition and construction of embedded axes within each body segment. In this paper, using sensitivity analysis, we quantify the effects of uncertainties in the definition and construction of embedded axes on the estimation of joint angular motion during gait. Using representative hip and knee motion data from normal subjects and cerebral palsy patients, the flexion-extension axis is analytically perturbed +/- 15 degrees in 5 degrees steps from a reference position, and the joint angles are recomputed for both Euler and helical angle definitions. For the Euler model, hip and knee flexion angles are relatively unaffected while the ab/adduction and rotation angles are significantly affected throughout the gait cycle. An error of 15 degrees in the definition of flexion-extension axis gives rise to maximum errors of 8 and 12 degrees for the ab/adduction angle, and 10-15 degrees for the rotation angles at the hip and knee, respectively. Furthermore, the magnitude of errors in ab/adduction and rotation angles are a function of the flexion angle. The errors for the ab/adduction angles increase with increasing flexion angle and for the rotation angle, decrease with increasing flexion angle. In cerebral palsy patients with flexed knee pattern of gait, this will result in distorted estimation of ab/adduction and rotation. For the helical model, similar results are obtained for the helical angle and associated direction cosines.(ABSTRACT TRUNCATED AT 250 WORDS)     

Determining dual Euler angles of the ankle complex in vivo using "flock of birds" electromagnetic tracking device

The dual Euler angles method has been proposed as an alternative approach to describe the general spatial human joint motion. In this study, the dual Euler angles method was applied to study the three-dimensional motion of the ankle complex. The methodology for obtaining dual Euler angles of the ankle complex was developed by using a "Flock of Birds" electromagnetic tracking device. The repeatability of the methodology was studied based on the intertester and intratester variability analysis. Finally kinematic coupling characteristics of the ankle complex during dorsiflexion-plantarflexion, eversion-inversion, and abduction-adduction were analyzed according to the parameters of the dual Euler angles.     

Thorax and pelvis kinematics during the downswing of male and female skilled golfers

Thorax and pelvis motion during the golf swing have most frequently been described for male golfers at discrete points during the swing, such as top of backswing (TBS) and ball contact (BC). Less is known about the continual motion and coordination of the thorax and pelvis throughout the downswing for either male or female golfers. The purpose of this study was to present detailed 3D kinematic profiles of thorax and pelvis motion during the downswing, and to determine if differences in kinematics exist between male and female skilled golfers. Thorax and pelvis data were collected from 19 male (26±7 years) and 19 female (25±7 years) skilled golfers (handicap ≤4) using an optical motion analysis system. 3D segment position, orientation and angular velocity were calculated, along with phase plane trajectories and thorax–pelvis separation angles. At BC males had greater pelvis posterior tilt, greater pelvis and thorax lateral tilt to the right, and less pelvis and thorax axial rotation to the left compared to females. Males achieved greater peak thorax and pelvis angular velocity, and angular velocity at BC, in the anterior–posterior and lateral tilt directions. Phase plane trajectories revealed that males and females had similar thorax lateral tilt and anterior–posterior tilt angular velocity–displacement relationships at TBS, yet by BC males had greater tilt angles and velocities compared to females. Collectively, the results suggest that male and female skilled golfers have different kinematics for thorax and pelvis motion, predominantly for lateral and anterior–posterior tilt. What might be considered optimal swing characteristics for male golfers should not be generalized to female golfers.     

New mathematical definition and calculation of axial rotation of anatomical joints

In the field of joint kinematics, clinical terms such as internal-external, or medical-lateral, rotations are commonly used to express the rotation of a body segment about its own long axis. However, these terms are not defined in a strict mathematical sense. In this paper, a new mathematical definition of axial rotation is proposed and methods to calculate it from the measured Euler angles are given. The definition and methods to calculate it from the measured Euler angles are given. The definition is based on the integration of the component of the angular velocity vector projected onto the long axis of the body segment. First, the absolute axial rotation of a body segment with respect to the stationary coordinate system is defined. This definition is then generalized to give the relative axial rotation of one body segment with respect to the other body segment where the two segments are moving in the three-dimensional space. The well-known Codman's paradox is cited as an example to make clear the difference between the definition so far proposed by other researchers and the new one.     

How close to a pendulum is human upper limb movement during walking?

The aim of this work was to investigate how close to pendulum-like behaviour the periodic motion of the human upper limb (or upper extremity) is, during normal walking at a comfortable speed of locomotion. Twenty-five healthy young persons (males and females) participated in the experiment. Biomechanical testing was undertaken (mass and centre of mass of each segment of the total upper extremity). Participants were walking on a treadmill with a standardised velocity of 1.1 ms(-1) (comfortable speed for all of them). A video analysis system with Silicon software was used to measure the different angles of the arm and forearm. The theoretical period of motion and maximal angular velocity were computed for the centre of mass of the total upper limb from the measured phases of the arm swing and associated positional potential energies. Actual measured periods of motion, in comparison, represented a level of similarity to a lightly damped simple pendulum. Using this assumption, the "damping factor" was calculated from the ratio between theoretical and measured values. A vast majority of people exhibited an actual angular velocity exceeding the expected theoretical angular velocity calculated for a virtual pendulum of similar mass and length characteristics. This may be due to muscle forces that are contributing to the motion of the upper limb during walking rather than simple gravity force acting alone. The observed positional potential energy of the dominant limb was greater than that of the non-dominant limb for the vast majority of participants.     

A proposal for a new definition of the axial rotation angle of the shoulder joint.

The Euler/Cardan angles are commonly used to define the motions of the upper arm with respect to the trunk. This definition, however, has a problem in that the angles of both the horizontal flexion/extension and the axial rotation of the shoulder joint become unstable at the gimbal-lock positions. In this paper, a new definition of the axial rotation angle was proposed. The proposed angle was stable over the entire range of the shoulder motion. With the new definition, the neutral position of the axial rotation agreed with that in the conventional anatomy. The advantage of the new definition was demonstrated by measuring actual complex motions of the shoulder with a three-dimensional motion capture system.     

A single video camera postural assessment system to measure rotation of the shoulder during computer use

The goal of this study was to evaluate the performance of a single video camera system for measuring shoulder rotation during computer work, and to quantify the work and postural space within which the system performs optimally. Shoulder rotation angles calculated using the video system were compared with angles calculated using an active infrared LED three-dimensional motion analysis system while 10 adult volunteers simulated postures for two different trials: typical of normal computer work (freestyle) and with forced shoulder abduction (constrained). Average and absolute errors were calculated to determine the accuracy and precision of the system, respectively, for each trial, for each position, and for both the right and left hands. For the right hand, mean values for the average and absolute errors were -1 and 0 degrees, respectively. Only the absolute error increased significantly to 12 degrees for the constrained posture compared with freestyle. During normal computer work, the video system provided shoulder rotation angle values similar to those of a three-dimensional system, thus making it a viable and simple instrument to use in field studies.     

An eight-week golf-specific exercise program improves physical characteristics, swing mechanics, and golf performance in recreational golfers

The purpose of this study was to determine the effects of an 8-week golf-specific exercise program on physical characteristics, swing mechanics, and golf performance. Fifteen trained male golfers (47.2 +/- 11.4 years, 178.8 +/- 5.8 cm, 86.7 +/- 9.0 kg, and 12.1 +/- 6.4 U.S. Golf Association handicap) were recruited. Trained golfers was defined operationally as golfers who play a round of golf at least 2-3 times per week and practice at the driving range at least 2-3 times per week during the regular golf season. Subjects performed a golf-specific conditioning program 3-4 times per week for 8 weeks during the off-season in order to enhance physical characteristics. Pre- and posttraining testing of participants included assessments of strength (torso, shoulder, and hip), flexibility, balance, swing mechanics, and golf performance. Following training, torso rotational strength and hip abduction strength were improved significantly (p < 0.05). Torso, shoulder, and hip flexibility improved significantly in all flexibility measurements taken (p < 0.05). Balance was improved significantly in 3 of 12 measurements, with the remainder of the variables demonstrating a nonsignificant trend for improvement. The magnitude of upper-torso axial rotation was decreased at the acceleration (p = 0.015) and impact points (p =0.043), and the magnitude of pelvis axial rotation was decreased at the top (p = 0.031) and acceleration points (p = 0.036). Upper-torso axial rotational velocity was increased significantly at the acceleration point of the golf swing (p = 0.009). Subjects increased average club velocity (p = 0.001), ball velocity (p = 0.001), carry distance (p = 0.001), and total distance (p = 0.001). These results indicate that a golf-specific exercise program improves strength, flexibility, and balance in golfers. These improvements result in increased upper-torso axial rotational velocity, which results in increased club head velocity, ball velocity, and driving distance.     

Interpolation of segment Euler angles can provide a robust estimation of segment angular trajectories during asymmetric lifting tasks

Video-based field methods that estimate L5/S1 net joint moments from kinematics based on interpolation in the sagittal plane of joint angles alone can introduce a significant error on the interpolated joint angular trajectory when applied to asymmetric dynamic lifts. Our goal was to evaluate interpolation of segment Euler angles for a wide range of dynamic asymmetric lifting tasks using cubic spline methods by comparing the interpolated values with the continuous measured ones. For most body segments, the estimated trajectories of segment Euler angles have less than 5° RMSE (in each dimension) with 5-point cubic spline interpolation when there is no measurement error of interpolation points. Sensitivity analysis indicates that when the measurement error exists, the root mean square error (RMSE) of estimated trajectories increases. Comparison among different lifting conditions showed that lifting a load from a high initial position yielded a smaller RMSE than lifting from a low initial position. In conclusion, interpolation of segment Euler angles can provide a robust estimation of segment angular trajectories during asymmetric lifting when measurement error of interpolation points can be controlled at a low level.     

A simple method to obtain consistent and clinically meaningful pelvic angles from euler angles during gait analysis

Clinical gait analysis usually describes joint kinematics using Euler angles, which depend on the sequence of rotation. Studies have shown that pelvic obliquity angles from the traditional tilt-obliquity-rotation (TOR) Euler angle sequence can deviate considerably from clinical expectations and have suggested that a rotation-obliquity-tilt (ROT) Euler angle sequence be used instead. We propose a simple alternate approach in which clinical joint angles are defined and exactly calculated in terms of Euler angles from any rotation sequence. Equations were derived to calculate clinical pelvic elevation, progression, and lean angles from TOR and ROT Euler angles. For the ROT Euler angles, obliquity was exactly the same as the clinical elevation angle, rotation was similar to the clinical progression angle, and tilt was similar to the clinical lean angle. Greater differences were observed for TOR. These results support previous findings that ROT is preferable to TOR for calculating pelvic Euler angles for clinical interpretation. However, we suggest that exact clinical angles can and should be obtained through a few extra calculations as demonstrated in this technical note.     

Estimation of the axis of a screw motion from noisy data--a new method based on Plücker lines

The problems of estimating the motion and orientation parameters of a body segment from two n point-set patterns are analyzed using the Plücker coordinates of a line (Plücker lines). The aim is to find algorithms less complex than those in conventional use, and thus facilitating more accurate computation of the unknown parameters. All conventional techniques use point transformation to calculate the screw axis. In this paper, we present a novel technique that directly estimates the axis of a screw motion as a Plücker line. The Plücker line can be transformed via the dual-number coordinate transformation matrix. This method is compared with Schwartz and Rozumalski [2005. A new method for estimating joint parameters from motion data. Journal of Biomechanics 38, 107-116] in simulations of random measurement errors and systematic skin movements. Simulation results indicate that the methods based on Plücker lines (Plücker line method) are superior in terms of extremely good results in the determination of the screw axis direction and position as well as a concise derivation of mathematical statements. This investigation yielded practical results, which can be used to locate the axis of a screw motion in a noisy environment. Developing the dual transformation matrix (DTM) from noisy data and determining the screw axis from a given DTM is done in a manner analogous to that for handling simple rotations. A more robust approach to solve for the dual vector associated with DTM is also addressed by using the eigenvector and the singular value decomposition.     

Can one angle be simply subtracted from another to determine range of motion in three-dimensional motion analysis?

To determine the range of motion of a joint between an initial orientation and a final orientation, it is convenient to subtract initial joint angles from final joint angles, a method referred to as the vectorial approach. However, for three-dimensional movements, the vectorial approach is not mathematically correct. To determine the joint range of motion, the rotation matrix between the two orientations should be calculated, and angles describing the range of motion should be extracted from this matrix, a method referred to as the matrical approach. As the matrical approach is less straightforward to implement, it is of interest to identify situations in which the vectorial approach leads to insubstantial errors. In this study, the vectorial approach was compared to the matrical approach, and theoretical justification was given for situations in which the vectorial approach can reasonably be used. The main findings are that the vectorial approach can be used if (1) the motion is planar (Woltring HJ. 1994. 3-D attitude representation of human joints: a standardization proposal. J Biomech 27(12): 1399–1414), (2) the angles between the final and the initial orientation are small (Woltring HJ. 1991. Representation and calculation of 3-D joint movement. Hum Mov Sci 10(5): 603–616), (3) the angles between the initial orientation of the distal segment and the proximal segment are small and finally (4) when only one large angle occurs between the initial orientation of the distal segment and the proximal segment and the angle sequence is chosen in such a way that this large angle occurs on the first axis of rotation. These findings provide specific criteria to consider when choosing the angle sequence to use for movement analysis.     

Effects of Aging on Arm Swing during Gait: The Role of Gait Speed and Dual Tasking

Healthy walking is characterized by pronounced arm swing and axial rotation. Aging effects on gait speed, stride length and stride time variability have been previously reported, however, less is known about aging effects on arm swing and axial rotation and their relationship to age-associated gait changes during usual walking and during more challenging conditions like dual tasking. Sixty healthy adults between the ages of 30–77 were included in this study designed to address this gap. Lightweight body fixed sensors were placed on each wrist and lower back. Participants walked under 3 walking conditions each of 1 minute: 1) comfortable speed, 2) walking while serially subtracting 3’s (Dual Task), 3) walking at fast speed. Aging effects on arm swing amplitude, range, symmetry, jerk and axial rotation amplitude and jerk were compared between decades of age (30–40; 41–50; 51–60; 61–77 years). As expected, older adults walked slower (p = 0.03) and with increased stride variability (p = 0.02). Arm swing amplitude decreased with age under all conditions (p = 0.04). In the oldest group, arm swing decreased during dual task and increased during the fast walking condition (p<0.0001). Similarly, arm swing asymmetry increased during the dual task in the older groups (p<0.004), but not in the younger groups (p = 0.67). Significant differences between groups and within conditions were observed in arm swing jerk (p<0.02), axial rotation amplitude (p<0.02) and axial jerk (p<0.001). Gait speed, arm swing amplitude of the dominant arm, arm swing asymmetry and axial rotation jerk were all independent predictors of age in a multivariate model. These findings suggest that the effects of gait speed and dual tasking on arm swing and axial rotation during walking are altered among healthy older adults. Follow-up work is needed to examine if these effects contribute to reduced stability in aging.     

Globographic visualisation of three dimensional joint angles

Three different methods for describing three dimensional joint angles are commonly used in biomechanics. The joint coordinate system and Cardan/Euler angles are conceptually quite different but are known to represent the same underlying mathematics. More recently the globographic method has been suggested as an alternative and this has proved particularly attractive for the shoulder joint. All three methods can be implemented in a number of ways leading to a choice of angle definitions. Very recently Rab has demonstrated that the globographic method is equivalent to one implementation of the joint coordinate system. This paper presents a rigorous analysis of the three different methods and proves their mathematical equivalence. The well known sequence dependence of Cardan/Euler is presented as equivalent to configuration dependence of the joint coordinate system and orientation dependence of globographic angles. The precise definition of different angle sets can be easily visualised using the globographic method using analogues of longitude, latitude and surface bearings with which most users will already be familiar. The method implicitly requires one axis of the moving segment to be identified as its principal axis and this can be extremely useful in helping define the most appropriate angle set to describe the orientation of any particular joint. Using this technique different angle sets are considered to be most appropriate for different joints and examples of this for the hip, knee, ankle, pelvis and axial skeleton are outlined.     

The finite displacements vector's method: an application to the scoliotic spine

This paper presents a vectorial method to directly obtain the components of the screw displacement between two positions of a body in a three-dimensional space (position of the helical axis of motion, rotation around this axis and translation along it). This method can be applied either to the case of a bone, moving with respect to the reference frame, or to the case of the relative motion of a joint; it gives exact formulae even if the displacements are finite; it generalizes the results (already published) obtained for finite displacements in the plane. The involved computation is easy, and the use of this method deals with only a small magnification of experimental errors. The technique of a screw displacement is applied to the vertebral segments of a scoliolic spine. The necessary data is taken from a couple of in-vivo X-rays. The goals of this study are: first, to describe the shape of the spine at each step of its evolution and second, to quantify the evolution in time of any segment of the spine between two states.     

Basic laws of blood screw motion in human common carotid arteries

The basic laws of blood screw motion in common carotid arteries in people were determined by means of modern ultrasound techniques for the first time. 92 healthy adults, aged 18-30, were examined. The blood flow in the middle one-third of common carotid arteries was registered by means of Color Doppler Imaging and impulse Doppler with the help of ultrasound Medison 8000EX scanner by linear transducer of 5-9 MHz. The steady registration of blood screw motion in both common carotid arteries in Color Doppler Imaging regimen was observed in 54.3 % of cases. The direction of screw stream rotation in most cases (54%) was multi-directed: in the right common carotid artery it was right, in the left common carotid artery--left (48%), and in 6% of cases it was reverse. For 46% of cases blood rotation in both common carotid arteries was one-directed (26%--right, 20%--left). The velocity parameters of rotation component of blood motion were determined, maximum velocity being 19.68 +/- 5.84 cm/sec, minimum--4.57 +/- 2.89 cm/sec, average--7.48 +/- 2.49 cm/sec, angular--10.7 +/- 2.49 sec(-1). The rated velocity of blood cells motion in screw motion with regard of screw current lines to the vessel vertical axis makes up from 158.67 +/- 32.79 to 224.39 +/- 46.37 cm/sec.     

Accuracy and precision of simpler and lower-cost technologies to measure the initial lean angle,step length and step velocity for forward lean releases

Current technologies to measure the maximum forward lean angle, step length and velocity in a clinical setting are neither simple nor cheap. Therefore, the purpose of this study was to determine the accuracy and precision of four live and one post-processing measurement methods compared to the 3D motion analysis gold standard. Twelve healthy younger adults recovered balance, after being released from six randomly ordered forward initial lean angles, using four different live measurement methods: LabVIEW, load cell, inclinometer and protractor. The initial lean angle, step length and velocity were also calculated in post-processing using 2D video analysis and 3D motion analysis. The LabVIEW method was the most accurate and precise, followed by the protractor, inclinometer and load cell methods. The load cell method was the most complex, followed by the LabVIEW, inclinometer and protractor methods. The LabVIEW method was the most expensive, followed by the load cell, inclinometer and protractor methods. Video analysis was sufficiently accurate and precise, equal in complexity and much less expensive than the gold standard. Simpler and lower-cost technologies to measure the initial lean angle, step length and velocity are sufficiently accurate and precise (live: protractor, post-processing: video analysis) to potentially use in a clinical setting.