Finite centroid and helical axis estimation from noisy landmark measurements in the study of human joint kinematics

Recent work on joint kinematics indicates that the finite centroid (centre of rotation) and the finite helical axis (axis of rotation, screw axis, twist axis) are highly susceptible to measurement errors when they are experimentally determined from landmark position data. This paper presents an analytical model to describe these effects, under isotropic conditions for the measurement errors and for the spatial landmark distribution. It appears that the position and direction errors are inversely proportional to the rotation magnitude, and that they are much more error-prone than the relatively well-determined rotation and translation magnitudes. Furthermore, the direction and rotation magnitude errors are inversely proportional to the landmark distribution radius, and the position and translation magnitude errors are minimal if the mean position of the landmarks coincides with the centroid or helical axis. For the planar centroid, the use of rigid-body constraints results in considerable precision improvement relative to the classical, finite Reuleaux method for centroid reconstruction. These analytical results can be used to define suitable measurement configurations, and they are used in this paper to explain experimental results on R?ntgenphotogrammetrically acquired in vitro wrist joint movement.     

A finite helical axis as a landmark for kinematic reference of the knee

Reference coordinates based on the finite helical axis for flexion of the knee from 0 to 90 deg are proposed. Six degree-of-freedom tracking allows the use of such a helical axis as a kinematic landmark for knee motion representation. Data from five human subjects in vivo are presented as a path of finite helical axes for flexion of the knee from 20 to 80 deg. The finite helical axis rotates by an average of 11.4 deg, the centrode translates an average of 19.8 mm, and the total axial translation averages 0.1 mm during flexion from 20 to 80 deg. Error due to the transducer was measured on a fixed-pivot pendulum and found to be 1.0 deg and 1.9 mm rms for the helical axis orientation and position, respectively, and 0.1 mm for the axial translation. Reproducibility and soft tissue effects on the measurements were repeatable to 4.0 deg and 2.7 mm rms in orientation and position, respectively, and 0.1 mm for the axial translations. Soft tissue errors averaged 4.9 deg and 3.6 mm in position and orientation, and 0.3 mm in the axial translations.     

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)     

Effects of gleno-humeral joint centre mislocation on gleno-humeral kinematics and kinetics

The gleno-humeral (GH) rotation centre is typically estimated using predictive or functional methods, however these methods may lead to location errors. This study aimed at determining a location error threshold above which statistically significant changes in the values of kinematic and kinetic GH parameters occur. The secondary aims were to quantify the effects of the direction of mislocation (X, Y or Z axis) of the GH rotation centre on GH kinematic and kinetic parameters.

Shoulder flexion and abduction movements of 11 healthy volunteers were recorded using a standard motion capture system (Vicon, Oxford Metrics Ltd, Oxford, UK), then GH kinematic and kinetic parameters were computed. The true position of the GH rotation centre was determined using a low dose x-ray scanner (EOS? imaging, France) and this position was transferred to the motion data. GH angles and moments were re-computed for each position of the GH rotation centre after errors of up to ± 20?mm were added in increments of ± 5?mm to each axis. The three-dimensional error range was 5?mm to 34.65?mm.

GH joint angle and moment values were significantly altered from 10?mm of three-dimensional error, and from 5?mm of error on individual axes. However, errors on the longitudinal and antero-posterior axes only caused very small alterations of GH joint angle and moment values respectively. Future research should develop methods of GH rotation centre estimation that produce three-dimensional location errors of less than 10?mm to reduce error propagation on GH kinematics and kinetics.     

Minimizing errors associated with calculating the location of the helical axis for spinal motions

One of the more common comparative tools used to quantify the motion of the vertebral joint is the orientation and position of the (finite) helical axis of motion as well as the amount of translation along, and rotation about, this axis. A survey of recent studies that utilize the helical axis of motion to compare motion before and after total disc replacement reveals a lack of concern for the relative errors associated with this metric. Indeed, intrinsic algorithmic and experimental errors that arise when interpreting motion tracking data can easily lead to a misinterpretation of the changes caused by replacement disc devices. While previous studies examining these errors exist, most have overlooked the errors associated with the determination of the location of the helical axis and its intersection with a chosen plane. The purpose of the study presented in this paper was to evaluate the sensitivity and reliability of the helical axis of motion as a comparative tool for kinematically evaluating spinal prostheses devices. To this end, we simulated a typical spine biomechanics testing experiment to investigate the accuracy of calculating the helical axis and its associated parameters using several popular algorithms. The resultant data motivated the development of a new algorithm that is a hybrid of two existing algorithms. The improved accuracy of this hybrid method made it possible to quantify some of the changes to the kinematics of a spinal unit that are induced by distinct placements of a total disc replacement.     

Analytical study of the effects of soft tissue artefacts on functional techniques to define axes of rotation

The accurate location of the main axes of rotation (AoR) is a crucial step in many applications of human movement analysis. There are different formal methods to determine the direction and position of the AoR, whose performance varies across studies, depending on the pose and the source of errors. Most methods are based on minimizing squared differences between observed and modelled marker positions or rigid motion parameters, implicitly assuming independent and uncorrelated errors, but the largest error usually results from soft tissue artefacts (STA), which do not have such statistical properties and are not effectively cancelled out by such methods. However, with adequate methods it is possible to assume that STA only account for a small fraction of the observed motion and to obtain explicit formulas through differential analysis that relate STA components to the resulting errors in AoR parameters. In this paper such formulas are derived for three different functional calibration techniques (Geometric Fitting, mean Finite Helical Axis, and SARA), to explain why each technique behaves differently from the others, and to propose strategies to compensate for those errors. These techniques were tested with published data from a sit-to-stand activity, where the true axis was defined using bi-planar fluoroscopy. All the methods were able to estimate the direction of the AoR with an error of less than 5°, whereas there were errors in the location of the axis of 30–40 mm. Such location errors could be reduced to less than 17 mm by the methods based on equations that use rigid motion parameters (mean Finite Helical Axis, SARA) when the translation component was calculated using the three markers nearest to the axis.     

Accuracy of a CT-based bone contour registration method to measure relative bone motions in the hindfoot

For measuring the in-vivo range of motion of the hindfoot, a CT-based bone contour registration method (CT-BCM) was developed to determine the three-dimensional position and orientation of bones. To validate this technique, we hypothesized that the range of motion in the hindfoot is equally, accurately measured by roentgen stereophotogrammetric analysis (RSA) as by the CT-BCM technique.Tantalum bone markers were placed in the distal tibia, talus and calcaneus of one cadaver specimen. With a fixed lower leg, the cadaveric foot was held in neutral and subsequently loaded in eight extreme positions. Immediately after acquiring a CT-scan with the foot in a position, RSA radiographs were made. Bone contour registration and RSA was performed. Helical axis parameters were calculated for talocrural and subtalar joint motion from neutral to extreme positions and between opposite extreme positions. Differences between CT-BCM and RSA were calculated.Compared with RSA, the CT-BCM data registered an overall root mean square difference (RMSd) of 0.21° for rotation about the helical axis, and 0.20 mm translation along the helical axis for the talocrural and subtalar joint and for all motions combined. The RMSd of the position and direction of the helical axes was 3.3 mm and 2.4°, respectively. The latter errors were larger with smaller helical rotations.The differences are similar to those reported for validated RSA and thus are not clinically relevant. Concluding, CT-BCM is an accurate and accessible alternative for studying joint motion, as it does not have the risk of infection and overlapping bone markers.     

Mandibular kinematics represented by a non-orthogonal floating axis joint coordinate system

There are many methods used to represent joint kinematics (e.g., roll, pitch, and yaw angles; instantaneous center of rotation; kinematic center; helical axis). Often in biomechanics internal landmarks are inferred from external landmarks. This study represents mandibular kinematics using a non-orthogonal floating axis joint coordinate system based on 3-D geometric models with parameters that are "clinician friendly" and mathematically rigorous. Kinematics data for two controls were acquired from passive fiducial markers attached to a custom dental clutch. The geometric models were constructed from MRI data. The superior point along the arc of the long axis of the condyle was used to define the coordinate axes. The kinematic data and geometric models were registered through fiducial markers visible during both protocols. The mean absolute maxima across the subjects for sagittal rotation, coronal rotation, axial rotation, medial-lateral translation, anterior-posterior translation, and inferior-superior translation were 34.10 degrees, 1.82 degrees, 1.14 degrees, 2.31, 21.07, and 6.95 mm, respectively. All the parameters, except for one subject's axial rotation, were reproducible across two motion recording sessions. There was a linear correlation between sagittal rotation and translation, the dominant motion plane, with approximately 1.5 degrees of rotation per millimeter of translation. The novel approach of combining the floating axis system with geometric models succinctly described mandibular kinematics with reproducible and clinician friendly parameters.     

The three-dimensional kinematics and flexibility characteristics of the human ankle and subtalar joints--Part I: Kinematics

The in-vitro, three dimensional kinematic characteristics of the human ankle and subtalar joint were investigated in this study. The main goals of this investigation were: 1) To determine the range of motion of the foot-shank complex and the associated range of motion of the ankle and subtalar joints; 2) To determine the kinematic coupling characteristics of the foot-shank complex, and 3) To identify the relationship between movements at the ankle and subtalar joints and the resulting motion produced between the foot and the shank. The tests were conducted on fifteen fresh amputated lower limbs and consisted of incrementally displacing the foot with respect to the shank while the motion of the articulating bones was measured through a three dimensional position data acquisition system. The kinematic analysis was based on the helical axis parameters describing the incremental displacements between any two of the three articulating bones and on a joint coordinate system used to describe the relative position between the bones. From the results of this investigation it was concluded that: 1) The range of motion of the foot-shank complex in any direction (dorsiflexion/plantarflexion, inversion/eversion and internal rotation/external rotation) is larger than that of either the ankle joint or the subtalar joint.; 2) Large kinematic coupling values are present at the foot-shank complex in inversion/eversion and in internal rotation/external rotation. However, only a slight amount of coupling was observed to occur in dorsiflexion/plantarflexion.; 3) Neither the ankle joint nor the subtalar joint are acting as ideal hinge joints with a fixed axis of rotation.; 4) Motion of the foot-shank complex in any direction is the result of rotations at both the ankle and the subtalar joints. However, the contribution of the ankle joint to dorsiflexion/plantarflexion of the foot-shank complex is larger than that of the subtalar joint and the contribution of the subtalar joint to inversion/eversion is larger than that of the ankle joint.; 5) The ankle and the subtalar joints have an approximately equal contribution to internal rotation/external rotation movements of the foot-shank complex.     

Validation of a new method for finding the rotational axes of the knee using both marker-based roentgen stereophotogrammetric analysis and 3D video-based motion analysis for kinematic measurements

In a previous paper, we reported the virtual axis finder, which is a new method for finding the rotational axes of the knee. The virtual axis finder was validated through simulations that were subject to limitations. Hence, the objective of the present study was to perform a mechanical validation with two measurement modalities: 3D video-based motion analysis and marker-based roentgen stereophotogrammetric analysis (RSA). A two rotational axis mechanism was developed, which simulated internal-external (or longitudinal) and flexion-extension (FE) rotations. The actual axes of rotation were known with respect to motion analysis and RSA markers within ± 0.0006 deg and ± 0.036 mm and ± 0.0001 deg and ± 0.016 mm, respectively. The orientation and position root mean squared errors for identifying the longitudinal rotation (LR) and FE axes with video-based motion analysis (0.26 deg, 0.28 m, 0.36 deg, and 0.25 mm, respectively) were smaller than with RSA (1.04 deg, 0.84 mm, 0.82 deg, and 0.32 mm, respectively). The random error or precision in the orientation and position was significantly better (p=0.01 and p=0.02, respectively) in identifying the LR axis with video-based motion analysis (0.23 deg and 0.24 mm) than with RSA (0.95 deg and 0.76 mm). There was no significant difference in the bias errors between measurement modalities. In comparing the mechanical validations to virtual validations, the virtual validations produced comparable errors to those of the mechanical validation. The only significant difference between the errors of the mechanical and virtual validations was the precision in the position of the LR axis while simulating video-based motion analysis (0.24 mm and 0.78 mm, p=0.019). These results indicate that video-based motion analysis with the equipment used in this study is the superior measurement modality for use with the virtual axis finder but both measurement modalities produce satisfactory results. The lack of significant differences between validation techniques suggests that the virtual sensitivity analysis previously performed was appropriately modeled. Thus, the virtual axis finder can be applied with a thorough understanding of its errors in a variety of test conditions.     

Measurement of the screw-home motion of the knee is sensitive to errors in axis alignment

Measurements of joint angles during motion analysis are subject to error caused by kinematic crosstalk, that is, one joint rotation (e. g., flexion) being interpreted as another (e.g., abduction). Kinematic crosstalk results from the chosen joint coordinate system being misaligned with the axes about which rotations are assumed to occur. The aim of this paper is to demonstrate that measurement of the so-called "screw-home" motion of the human knee, in which axial rotation and extension are coupled, is especially prone to errors due to crosstalk. The motions of two different two-segment mechanical linkages were examined to study the effects of crosstalk. The segments of the first linkage (NSH) were connected by a revolute joint, but the second linkage (SH) incorporated gearing that caused 15 degrees of screw-home rotation to occur with 90 degrees knee flexion. It was found that rotating the flexion axis (inducing crosstalk) could make linkage NSH appear to exhibit a screw-home motion and that a different rotation of the flexion axis could make linkage SH apparently exhibit pure flexion. These findings suggest that the measurement of screw-home rotation may be strongly influenced by errors in the location of the flexion axis. The magnitudes of these displacements of the flexion axis were consistent with the inter-observer variability seen when five experienced observers defined the flexion axis by palpating the medial and lateral femoral epicondyles. Care should be taken when interpreting small internal-external rotations and abduction-adduction angles to ensure that they are not the products of kinematic crosstalk.     

Knee joint secondary motion accuracy improved by quaternion-based optimizer with bony landmark constraints

Skin marker-based motion analysis has been widely used in biomechanical studies and clinical applications. Unfortunately, the accuracy of knee joint secondary motions is largely limited by the nonrigidity nature of human body segments. Numerous studies have investigated the characteristics of soft tissue movement. Utilizing these characteristics, we may improve the accuracy of knee joint motion measurement. An optimizer was developed by incorporating the soft tissue movement patterns at special bony landmarks into constraint functions. Bony landmark constraints were assigned to the skin markers at femur epicondyles, tibial plateau edges, and tibial tuberosity in a motion analysis algorithm by limiting their allowed position space relative to the underlying bone. The rotation matrix was represented by quaternion, and the constrained optimization problem was solved by Fletcher's version of the Levenberg-Marquardt optimization technique. The algorithm was validated by using motion data from both skin-based markers and bone-mounted markers attached to fresh cadavers. By comparing the results with the ground truth bone motion generated from the bone-mounted markers, the new algorithm had a significantly higher accuracy (root-mean-square (RMS) error: 0.7 ± 0.1 deg in axial rotation and 0.4 ± 0.1 deg in varus-valgus) in estimating the knee joint secondary rotations than algorithms without bony landmark constraints (RMS error: 1.7 ± 0.4 deg in axial rotation and 0.7 ± 0.1 deg in varus-valgus). Also, it predicts a more accurate medial-lateral translation (RMS error: 0.4 ± 0.1 mm) than the conventional techniques (RMS error: 1.2 ± 0.2 mm). The new algorithm, using bony landmark constrains, estimates more accurate secondary rotations and medial-lateral translation of the underlying bone.     

Determination of rigid body registration marker error from edge error

Registration markers affixed to rigid bodies (fixed to bone as opposed to skin) are commonly used when tracking 3D rigid body motion. The measured positions of registration markers are subject to unavoidable errors, both systematic and non-systematic. Prior studies have investigated the error propagated to such derived properties as rigid body positions and helical axes, while others have focused on the error associated with a specific position tracking system under restricted conditions. Theoretical and simulation-based error propagation requires knowledge of the variation due to individual registration markers; however, the variation in registration marker position measurement has previously been either assumed or determined from static cases. The objective of this paper is the introduction of a method for determining individual marker variation irrespective of change in rigid body position or motion by utilizing the distances between the markers (edge lengths), which are invariant under rotation and translation. Simulations were used to validate and characterize the introduced technique, demonstrating that the predictions improve with greater edge length and additional markers, converge on reference values where the edge length is at least 4 times the magnitude of the maximum vertex variation, and that under ideal conditions the confidence interval about the predicted variation is within 7% of the maximum variation associated with that marker set. The introduced technique was tested on the results of a motion tracking experiment to demonstrate the wide disparity in vertex variation between static and non-static measurements of the same registration markers, where the non-static variation exceeded the static variation by an average factor of 12.7.     

In vivo determination of the Achilles tendon moment arm in three-dimensions

Two-dimensional methods have been applied to determine the Achilles tendon moment arm in previous studies, although the talocrural joint rotates in three-dimension. The purpose of this study was to develop a method for determining the Achilles tendon moment arm in three-dimensions (3DMA). A series of sagittal ankle images were obtained at ankle positions of -20°, -10° (dorsiflexed position), 0° (neutral position), +10°, +20°, and +30° (plantarflexed position). The talocrural joint axis was determined as the finite helical axis of the ankle joint over 20° of displacement, and the 3DMA was determined as the shortest distance from the talocrural joint axis to the line of action of the Achilles tendon force. The corresponding 2DMA was determined with the center of rotation method using the images captured on the sagittal plane passing through the mid-point of the medio-lateral width of the tibia. The 3DMA ranged from 35 to 41 mm across various ankle positions and was, on average, 11 mm smaller than 2DMA. The difference between the two measures was attributable primarily to the deviations of the talocrural joint axis from the anatomical medio-lateral direction. The deviations on the coronal plane (21.4±20.7°) and on the transverse planes (14.8±22.6°) accounted for the errors of 1.3 mm and 3.0 mm, respectively. In addition, selecting either a medially or laterally misaligned sagittal-plane image for determining the 2DMA gave rise to error by 3.5 mm. The remaining difference was accounted for by the random measurement error.     

Propagation of errors from skull kinematic measurements to finite element tissue responses

Real-time quantification of head impacts using wearable sensors is an appealing approach to assess concussion risk. Traditionally, sensors were evaluated for accurately measuring peak resultant skull accelerations and velocities. With growing interest in utilizing model-estimated tissue responses for injury prediction, it is important to evaluate sensor accuracy in estimating tissue response as well. Here, we quantify how sensor kinematic measurement errors can propagate into tissue response errors. Using previous instrumented mouthguard validation datasets, we found that skull kinematic measurement errors in both magnitude and direction lead to errors in tissue response magnitude and distribution. For molar design instrumented mouthguards susceptible to mandible disturbances, 150–400% error in skull kinematic measurements resulted in 100% error in regional peak tissue response. With an improved incisor design mitigating mandible disturbances, errors in skull kinematics were reduced to <50%, and several tissue response errors were reduced to <10%. Applying 30\(^{\circ }\) rotations to reference kinematic signals to emulate sensor transformation errors yielded below 10% error in regional peak tissue response; however, up to 20% error was observed in peak tissue response for individual finite elements. These findings demonstrate that kinematic resultant errors result in regional peak tissue response errors, while kinematic directionality errors result in tissue response distribution errors. This highlights the need to account for both kinematic magnitude and direction errors and accurately determine transformations between sensors and the skull.     

Effects of data smoothing on the reconstruction of helical axis parameters in human joint kinematics

In biomechanical joint-motion analyses, the continuous motion to be studied is often approximated by a sequence of finite displacements, and the Finite Helical Axis (FHA) or "screw axis" for each displacement is estimated from position measurements on a number of anatomical or artificial landmarks. When FHA parameters are directly determined from raw (noisy) displacement data, both the position and the direction of the FHA are ill-determined, in particular when the sequential displacement steps are small. This implies, that under certain conditions, the continuous pathways of joint motions cannot be adequately described. The purpose of the present experimental study is to investigate the applicability of smoothing (or filtering) techniques, in those cases where FHA parameters are ill-determined. Two different quintic-spline smoothing methods were used to analyze the motion data obtained with Roentgenstereophotogrammetry in two experiments. One concerning carpal motions in a wrist-joint specimen, and one relative to a kinematic laboratory model, in which the axis positions are a priori known. The smoothed and non-smoothed FHA parameter errors were compared. The influences of the number of samples and the size of the sampling interval (displacement step) were investigated, as were the effects of equidistant and nonequidistant sampling conditions and noise invariance.     

Point of optimal kinematic error: Improvement of the instantaneous helical pivot method for locating centers of rotation

This paper proposes a variation of the instantaneous helical pivot technique for locating centers of rotation. The point of optimal kinematic error (POKE), which minimizes the velocity at the center of rotation, may be obtained by just adding a weighting factor equal to the square of angular velocity in Woltring?s equation of the pivot of instantaneous helical axes (PIHA). Calculations are simplified with respect to the original method, since it is not necessary to make explicit calculations of the helical axis, and the effect of accidental errors is reduced. The improved performance of this method was validated by simulations based on a functional calibration task for the gleno-humeral joint center. Noisy data caused a systematic dislocation of the calculated center of rotation towards the center of the arm marker cluster. This error in PIHA could even exceed the effect of soft tissue artifacts associated to small and medium deformations, but it was successfully reduced by the POKE estimation.     

Helical axes of skeletal knee joint motion during running

The purpose of this study was to determine the changes in the axis of rotation of the knee that occur during the stance phase of running. Using intracortical pins, the three-dimensional skeletal kinematics of three subjects were measured during the stance phase of five running trials. The stance phase was divided into equal motion increments for which the position and orientation of the finite helical axes (FHA) were calculated relative to a tibial reference frame. Results were consistent within and between subjects. At the beginning of stance, the FHA was located at the midepicondylar point and during the flexion phase moved 20mm posteriorly and 10mm distally. At the time of peak flexion, the FHA shifted rapidly by about 10-20mm in proximal and posterior direction. The angle between the FHA and the tibial transverse plane increased gradually during flexion, to about 15 degrees of medial inclination, and then returned to zero at the start of the extension phase. These changes in position and orientation of FHA in the knee should be considered in analyses of muscle function during human movement, which require moment arms to be defined relative to a functional rotation axis. The finding that substantial changes in axis of rotation occurred independent of flexion angle suggests that musculoskeletal models must have more than one kinematic degree-of-freedom at the knee. The same applies to the design of knee prostheses, if the goal is to restore normal muscle function.