A survey of formal methods for determining the centre of rotation of ball joints

The determination of an accurate centre of rotation (CoR) from segment marker positions is of interest across a wide range of applications, but particularly for clinical gait analysis and for estimating the hip joint centre during surgical intervention of the knee, for limb alignment purposes. For the first time in this survey of formal methods, we classify, analyse and compare different methods (geometric, algebraic, bias compensated algebraic, and Pratt sphere fit methods, as well as the centre transformation technique, the Holzreiter approach, the helical pivot technique, the Schwartz transformation techniques, the minimal amplitude point method and the Stoddart approach) for the determination of spherical joint centres from marker position data. In addition, we propose a new method, the symmetrical CoR estimation or SCoRE, in which the coordinates of the joint centre must only remain constant relative to each segment, thus not requiring the assumption that one segment should remain at rest. For each method, 1000 CoR estimations were analysed with the application of isotropic, independent and identically distributed Gaussian noise (standard deviation 0.1cm) to each of the marker positions, to all markers on the segment simultaneously and the two in combination. For the test conditions used here, most techniques were capable of determining the CoR to within 0.3 cm, as long as the spherical range of motion (RoM) of the joint was 45 degrees or more. Under the most stringent conditions tested, however, the SCoRE was capable of best determining the CoR, to within approximately 1.2mm with a RoM of 20 degrees . The correct selection and application of these methodologies should help improve the accuracy of surgical navigation and clinical kinematic measurement.     

Bone position estimation from skin marker co-ordinates using global optimisation with joint constraints

Widespread use of gait or motion analysis in the diagnosis of patients with locomotor pathology and the subsequent planning and assessment of treatment has been limited because of its reliability, particularly in evaluating frontal and transverse plane components. This is because spatial reconstruction of the musculoskeletal system and calculation of its kinematics and kinetics via a skin marker-based multi-link model are subject to marker skin movement artefacts. Traditional methods treat each body segment separately without imposing joint constraints, resulting in apparent dislocations at joints predominantly because of skin movement artefacts. An optimisation method for the determination of the positions and orientations of multi-link musculoskeletal models from marker co-ordinates is presented. It is based on the minimisation of the weighted sum of squared distances between measured and model-determined marker positions. The model imposes joint constraints. Numerical experiments were performed to show that the new method is capable of eliminating joint dislocations and giving more accurate model position and orientation estimations. It is suggested that, with joint constraints and a global error compensation scheme, the effects of measurement errors on the reconstruction of the musculoskeletal system and subsequent mechanical analyses can be reduced globally. The proposed method minimises errors in axial rotation and ab/adduction at the joints and may extend the applicability of gait analysis to clinical problems.     

Comparison of the SCoRE and HA methods for locating in vivo the glenohumeral joint centre

A recent paper has described a new functional method, the symmetrical centre of rotation (SCoRE), for locating joint centre position [Ehrig, R.M., Taylor, W.R., Duda, G.N., Heller, M.O., 2006. A survey of formal methods for determining the centre of rotation of ball joints. Journal of Biomechanics 39 (15), 2798-2809]. For in vitro analyses, the SCoRE method showed better precision than helical axis (HA) or sphere fitting methods. Despites HA determination is very sensitive to small angular velocity, the International Society of Biomechanics has recommended to use HA for locating the glenohumeral joint centre. This paper aims at comparing the SCoRE method with the HA method for locating in vivo the glenohumeral joint centre according to the movement characteristics. Nine subjects performed 10 cycles of three different movements at two different velocities. For each test (combination of movements) the location of the centre of rotation was estimated with both methods (SCoRE and HA). Analyses focused on the 3D location of the glenohumeral joint centre and on the repeatability of location (standard deviation). This study showed that SCoRE and HA methods yielded the same GH location. Nevertheless, with SCoRE method, the location of the glenohumeral joint centre was different according to the test. This study evidenced that the SCoRE method was more precise than HA method (error of 3 mm versus 4.6 mm) and that the GH location with the SCoRE method was not affected by movements with slow velocities.     

Hip joint centre location: An ex vivo study

The human hip joint is normally represented as a spherical hinge and its centre of rotation is used to construct femoral anatomical axes and to calculate hip joint moments. The estimate of the hip joint centre (HJC) position using a functional approach is affected by stereophotogrammetric errors and soft tissue artefacts. The aims of this study were (1) to assess the accuracy with which the HJC position can be located using stereophotogrammetry and (2) to investigate the effects of hip motion amplitude on this accuracy. Experiments were conducted on four adult cadavers. Cortical pins, each equipped with a marker cluster, were implanted in the pelvis and femur, and eight skin markers were attached to the thigh. Recordings were made while an operator rotated the hip joint exploiting the widest possible range of motion. For HJC determination, a proximal and a distal thigh skin marker cluster and two recent analytical methods, the quartic sphere fit (QFS) method and the symmetrical centre of rotation estimation (SCoRE) method, were used. Results showed that, when only stereophotogrammetric errors were taken into account, the analytical methods performed equally well. In presence of soft tissue artefacts, HJC errors highly varied among subjects, methods, and skin marker clusters (between 1.4 and 38.5 mm). As expected, larger errors were found in the subject with larger soft tissue artefacts. The QFS method and the distal cluster performed generally better and showed a mean HJC location accuracy better than 10 mm over all subjects. The analysis on the effect of hip movement amplitude revealed that a reduction of the amplitude does not improve the HJC location accuracy despite a decrease of the artefact amplitude.     

Proximal placement of lateral thigh skin markers reduces soft tissue artefact during normal gait using the Conventional Gait Model

A primary source of measurement error in gait analysis is soft-tissue artefact. Hip and knee angle measurements, regularly used in clinical decision-making, are particularly prone to pervasive soft tissue on the femur. However, despite several studies of thigh marker artefact it remains unclear how lateral thigh marker height affects results using variants of the Conventional Gait Model. We compared Vicon Plug-in Gait hip and knee angle estimates during gait using a proximal and distal thigh marker placement for ten healthy subjects. Knee axes were estimated by optimizing thigh rotation offsets to minimize knee varus-valgus range during gait. Relative to the distal marker, the proximal marker produced 37% less varus-valgus range and 50% less hip rotation range (p < 0.001), suggesting that it produced less soft-tissue artefact in knee axis estimates. The thigh markers also produced different secondary effects on the knee centre estimate. Using whole gait cycle optimization, the distal marker showed greater minimum and maximum knee flexion (by 6° and 2° respectively) resulting in a 4° reduction in range. Mid-stance optimization reduced distal marker knee flexion by 5° throughout, but proximal marker results were negligibly affected. Based on an analysis of the Plug-in Gait knee axis definition, we show that the proximal marker reduced sensitivity to soft-tissue artefact by decreasing collinearity between the points defining the femoral frontal plane and reducing anteroposterior movement between the knee and thigh markers. This study suggests that a proximal thigh marker may be preferable when performing gait analysis using the Plug-in Gait model.     

Repeatability assessment of functional methods to estimate the glenohumeral joint centre

Functional methods can be used to determine the centre of rotation (CoR) of a ball-and-socket joint. The algorithms are used to locate rather the hip joint centre than the glenohumeral joint centre. The choice of the most suitable method depends especially on the intra- and inter-session repeatability of these methods. This paper aims at evaluating the intra- and inter-session repeatability of functional methods with which the glenohumeral joint rotation centre (GHRC) can be estimated in vivo. It also estimates the most suitable amplitude of functional movements. Five functional methods were tested: the algorithms of Gamage and Lasenby, bias compensation, symmetrical CoR estimation, normalisation method and helical axis. Ten subjects performed three cycles of three different movements (flexion–extension, abduction–adduction and circumduction). These movements were repeated three times with three different ranges of motion. Six subjects came back in order to evaluate the inter-session repeatability. For each test, the location of the GHRC was estimated by the five methods. The method to solve the functional problem and the range of functional movement affected the GHRC location. The results showed a good to excellent intra-session repeatability. The lowest repeatability error was found for the high amplitude whatever the methods used. The inter-session reliability was moderate. Finally, we suggest the use of functional methods with high amplitude movement in order to locate the GHRC with the best reliability.     

Comparison of two methods for in vivo estimation of the glenohumeral joint rotation center (GH-JRC) of the patients with shoulder hemiarthroplasty

Determination of an accurate glenohumeral-joint rotation center (GH-JRC) from marker data is essential for kinematic and dynamic analysis of shoulder motions. Previous studies have focused on the evaluation of the different functional methods for the estimation of the GH-JRC for healthy subjects. The goal of this paper is to compare two widely used functional methods, namely the instantaneous helical axis (IHA) and symmetrical center of rotation (SCoRE) methods, for estimating the GH-JRC in vivo for patients with implanted shoulder hemiarthroplasty. The motion data of five patients were recorded while performing three different dynamic motions (circumduction, abduction, and forward flexion). The GH-JRC was determined using the CT-images of the subjects (geometric GH-JRC) and was also estimated using the two IHA and SCoRE methods. The rotation centers determined using the IHA and SCoRE methods were on average 1.47±0.62 cm and 2.07±0.55 cm away from geometric GH-JRC, respectively. The two methods differed significantly (two-tailed p-value from paired t-Test ～0.02, post-hoc power ～0.30). The SCoRE method showed a significant lower (two-tailed p-value from paired t-Test ～0.03, post-hoc power ～0.68) repeatability error calculated between the different trials of each motion and each subject and averaged across all measured subjects (0.62±0.10 cm for IHA vs. 0.43±0.12 cm for SCoRE). It is concluded that the SCoRE appeared to be a more repeatable method whereas the IHA method resulted in a more accurate estimation of the GH-JRC for patients with endoprostheses.     

Comparison of hierarchical and six degrees-of-freedom marker sets in analyzing gait kinematics

The objective of this study was to determine how marker spacing, noise, and joint translations affect joint angle calculations using both a hierarchical and a six degrees-of-freedom (6DoF) marker set. A simple two-segment model demonstrates that a hierarchical marker set produces biased joint rotation estimates when sagittal joint translations occur whereas a 6DoF marker set mitigates these bias errors with precision improving with increased marker spacing. These effects were evident in gait simulations where the 6DoF marker set was shown to be more accurate at tracking axial rotation angles at the hip, knee, and ankle.     

Error propagation from kinematic data to modeled muscle-tendon lengths during walking

Kinematic data from 3D gait analysis together with musculoskeletal modeling techniques allow the derivation of muscle-tendon lengths during walking. However, kinematic data are subject to soft tissue artifacts (STA), referring to skin marker displacements during movement. STA are known to significantly affect the computation of joint kinematics, and would therefore also have an effect on muscle-tendon lengths which are derived from the segmental positions. The present study aimed to introduce an analytical approach to calculate the error propagation from STA to modeled muscle-tendon lengths. Skin marker coordinates were assigned uncorrelated, isotropic error functions with given standard deviations accounting for STA. Two different musculoskeletal models were specified; one with the joints moving freely in all directions, and one with the joints constrained to rotation but no translation. Using reference kinematic data from two healthy boys (mean age 9 y 5 m), the propagation of STA to muscle-tendon lengths was quantified for semimembranosus, gastrocnemius and soleus. The resulting average SD ranged from 6% to 50% of the normalized muscle-tendon lengths during gait depending on the muscle, the STA magnitudes and the musculoskeletal model. These results highlight the potential impact STA has on the biomechanical analysis of modeled muscle-tendon lengths during walking, and suggest the need for caution in the clinical interpretation of muscle-tendon lengths derived from joint kinematics.     

Kinematic models of the upper limb joints for multibody kinematics optimisation: An overview

Soft tissue artefact (STA), i.e. the motion of the skin, fat and muscles gliding on the underlying bone, may lead to a marker position error reaching up to 8.7 cm for the particular case of the scapula. Multibody kinematics optimisation (MKO) is one of the most efficient approaches used to reduce STA. It consists in minimising the distance between the positions of experimental markers on a subject skin and the simulated positions of the same markers embedded on a kinematic model. However, the efficiency of MKO directly relies on the chosen kinematic model. This paper proposes an overview of the different upper limb models available in the literature and a discussion about their applicability to MKO.The advantages of each joint model with respect to its biofidelity to functional anatomy are detailed both for the shoulder and the forearm areas. Models capabilities of personalisation and of adaptation to pathological cases are also discussed. Concerning model efficiency in terms of STA reduction in MKO algorithms, a lack of quantitative assessment in the literature is noted. In priority, future studies should concern the evaluation and quantification of STA reduction depending on upper limb joint constraints.     

Measuring three-dimensional knee kinematics under torsional loading

Excessive knee joint laxity is often used as an indicator of joint disease or injury. Clinical assessment devices are currently limited to anterior–posterior drawer measurements, while tools used to measure movement in the remaining degrees of freedom are either invasive or prone to soft tissue artefact. The objective of this work was, therefore, to develop a methodology whereby in vivo knee joint kinematics could be measured in three dimensions under torsional loading while still maintaining a non-invasive procedure. A device designed to administer a subject-normalized torque in the transverse plane of the knee was securely fastened to the outer frame of an open magnetic resonance imaging (MRI) magnet. Low resolution 3D T1-weighted images (6.25 mm slice thickness) were generated by the 0.2 Tesla MRI scanner in less than 3 min while the joint was under load. The 3D image volume was then shape-matched to a high resolution image volume (1.56 mm slice thickness) scanned in a no-load position. Three-dimensional rotations and translations of the tibia with respect to the femur were calculated by comparing the transformation matrices before and after torque was applied. Results from six subjects showed that this technique was repeatable over five trials with the knee in extended and flexed positions. Differences in range of rotation were shown between subjects and between knee positions, suggesting that this methodology has sufficient utility for further application in clinical studies.     

Influence of joint constraints on lower limb kinematics estimation from skin markers using global optimization

In order to obtain the lower limb kinematics from skin-based markers, the soft tissue artefact (STA) has to be compensated. Global optimization (GO) methods rely on a predefined kinematic model and attempt to limit STA by minimizing the differences between model predicted and skin-based marker positions. Thus, the reliability of GO methods depends directly on the chosen model, whose influence is not well known yet.This study develops a GO method that allows to easily implement different sets of joint constraints in order to assess their influence on the lower limb kinematics during gait. The segment definition was based on generalized coordinates giving only linear or quadratic joint constraints. Seven sets of joint constraints were assessed, corresponding to different kinematic models at the ankle, knee and hip: SSS, USS, PSS, SHS, SPS, UHS and PPS (where S, U and H stand for spherical, universal and hinge joints and P for parallel mechanism). GO was applied to gait data from five healthy males.Results showed that the lower limb kinematics, except hip kinematics, knee and ankle flexion–extension, significantly depend on the chosen ankle and knee constraints. The knee parallel mechanism generated some typical knee rotation patterns previously observed in lower limb kinematic studies. Furthermore, only the parallel mechanisms produced joint displacements.Thus, GO using parallel mechanism seems promising. It also offers some perspectives of subject-specific joint constraints.     

An ultrasound based non-invasive method for the measurement of intrinsic foot kinematics during gait

Soft tissue artefact (STA) and marker placement variability are sources of error when measuring the intrinsic kinematics of the foot. This study aims to demonstrate a non-invasive, combined ultrasound and motion capture (US/MC) technique to directly measure foot skeletal motion. The novel approach is compared to a standard motion capture protocol. Fourteen participants underwent instrumented barefoot analysis of foot motion during gait. Markers were attached to foot allowing medial longitudinal arch angle and navicular height to be determined. For the US/MC technique, the navicular marker was replaced by an ultrasound transducer which was secured to the foot allowing the skeletal landmark to be imaged. Ultrasound cineloops showing the location of the navicular tuberosity during the walking trials were synchronised with motion capture measurements and markers mounted on the probe allowed the true position of the bony landmark to be determined throughout stance phase. Two discrete variables, minimum navicular height and maximum MLA angle, were compared between the standard and US/MC protocols. Significant differences between minimum navicular height (P=0.004, 95% CI (1.57, 6.54)) and maximum medial longitudinal arch angle (P=0.0034, 95% CI (13.8, 3.4)) were found between the measurement methods. The individual effects of STA and marker placement error were also assessed. US/MC is a non-invasive technique which may help to provide more accurate measurements of intrinsic foot kinematics.     

On intrinsic equivalences of the finite helical axis,the instantaneous helical axis,and the SARA approach. A mathematical perspective

Accurate determination of joint axes is essential for understanding musculoskeletal function. Whilst numerous algorithms to compute such axes exist, the conditions under which each of the methods performs best remain largely unknown. Typically, algorithms are evaluated for specific conditions only limiting the external validity of conclusions regarding their performance. We derive exact mathematical relationships between three commonly used algorithms for computing joint axes from motion data: finite helical axes (FHA), instantaneous helical axes (IHA) and SARA (symmetrical axis of rotation approach), including relationships for an extension to the mean helical axes methods that facilitate determining joint centres and axes. Through the derivation of a sound mathematical framework to objectively compare the algorithms we demonstrate that the FHA and SARA approach are equivalent for the analysis of two time frames. Moreover, we show that the position of a helical axis derived from the IHA using positional data is affected by a systematic error perpendicular to the true axis direction, whereas the axis direction is identical to those computed with either the FHA or SARA approach (true direction). Finally, with an appropriate choice of weighting factors the mean FHA (MFHA) method is equivalent to the Symmetrical Centre of Rotation Estimation (SCoRE) algorithm for determination of a Centre of Rotation (CoR), and similarly, equivalent to the SARA algorithm for determination of an Axis of Rotation (AoR). The deep understanding of the equivalences between methods presented here enables readers to choose numerically efficient, robust methods for determining AoRs and CoRs with confidence.     

Inertial sensor based method for identifying spherical joint center of rotation

The miniaturized wireless inertial measurement unit (IMU) technology and algorithms presented herein promote rapid and accurate predictions of the center-of-rotation (CoR) for ball/spherical joints. The algorithm improves upon existing IMU-based methods by directly utilizing the measured acceleration and angular velocity provided by the IMU to deduce the CoR in lieu of relying on error-prone velocity and position estimates. Results demonstrate that this new method resolves the position of the CoR to within a 3 mm sphere of the true CoR determined by a precision coordinate measuring machine. Such accuracy may render this method attractive for broad use in field, laboratory and clinical settings requiring non-invasive and rapid estimates of joint CoR.     

Introducing a new staircase design to quantify healthy knee function during stair ascent and descent

The design, manufacture and validation of a new free standing staircase for motion analysis measurements are described in this paper. The errors in vertical force measurements introduced when the stairs interface with a force plate (FP) are less than 0.6%. The centre of pressure error introduced is less than 0.7 mm compared to the error from the FP. The challenges of introducing stair gait into a clinical trial with a limited number of FPs and time limitations for assessment sessions are addressed by introducing this cost effective solution.

The staircase was used in a study to measure non-pathological knee function of 10 subjects performing stair ascent and descent. The resulting knee kinematics and knee joint moments are in agreement with previous studies. The kinematic and joint moment profiles provide a normative range, which will be useful in future studies for identifying alterations in joint function associated with pathology and intervention.     

Kalman smoothing improves the estimation of joint kinematics and kinetics in marker-based human gait analysis

We developed a Kalman smoothing algorithm to improve estimates of joint kinematics from measured marker trajectories during motion analysis. Kalman smoothing estimates are based on complete marker trajectories. This is an improvement over other techniques, such as the global optimisation method (GOM), Kalman filtering, and local marker estimation (LME), where the estimate at each time instant is only based on part of the marker trajectories. We applied GOM, Kalman filtering, LME, and Kalman smoothing to marker trajectories from both simulated and experimental gait motion, to estimate the joint kinematics of a ten segment biomechanical model, with 21 degrees of freedom. Three simulated marker trajectories were studied: without errors, with instrumental errors, and with soft tissue artefacts (STA). Two modelling errors were studied: increased thigh length and hip centre dislocation. We calculated estimation errors from the known joint kinematics in the simulation study. Compared with other techniques, Kalman smoothing reduced the estimation errors for the joint positions, by more than 50% for the simulated marker trajectories without errors and with instrumental errors. Compared with GOM, Kalman smoothing reduced the estimation errors for the joint moments by more than 35%. Compared with Kalman filtering and LME, Kalman smoothing reduced the estimation errors for the joint accelerations by at least 50%. Our simulation results show that the use of Kalman smoothing substantially improves the estimates of joint kinematics and kinetics compared with previously proposed techniques (GOM, Kalman filtering, and LME) for both simulated, with and without modelling errors, and experimentally measured gait motion.     

Propagation of soft tissue artifacts to the center of rotation: A model for the correction of functional calibration techniques

This paper presents a mathematical model for the propagation of errors in body segment kinematics to the location of the center of rotation. Three functional calibration techniques, usually employed for the gleno-humeral joint, are studied: the methods based on the pivot of the instantaneous helical axis (PIHA) or the finite helical axis (PFHA), and the “symmetrical center of rotation estimation” (SCoRE). A procedure for correcting the effect of soft tissue artifacts is also proposed, based on the equations of those techniques and a model of the artifact, like the one that can be obtained by double calibration. An experiment with a mechanical analog was performed to validate the procedure and compare the performance of each technique. The raw error (between 57 and 68 mm) was reduced by a proportion of between 1:6 and less than 1:15, depending on the artifact model and the mathematical method. The best corrections were obtained by the SCoRE method. Some recommendations about the experimental setup for functional calibration techniques and the choice of a mathematical method are derived from theoretical considerations about the formulas and the results of the experiment.