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.     

Comparison of three formal methods used to estimate the functional axis of rotation: an extensive in-vivo analysis performed on the knee joint

Estimating the main axis of rotation (AoR) of a human joint represents an important issue in biomechanics. This study compared three formal methods used to estimate functional AoR, namely a cylindrical fitting method, a mean helical axis transformation, and a symmetrical axis approach. These methods were tested on 106 subjects undergoing navigated total knee arthroplasty. AoR orientation in 3D and in the frontal and coronal planes provided by each method was compared to the transepicondylar axis direction. Although all the methods resulted effective, significant differences were identified among them, relatively to the orientation in 3D and in the frontal plane projection. This was probably due to the presence of secondary rotations during the first degrees of knee flexion.     

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.     

Finding motion parameters from spherical motion fields (or the advantages of having eyes in the back of your head)

A theory is developed for determining the motion of an observer given the motion field over a full 360 degree image sphere. The method is based on the fact that for an observer translating without rotation, the projected circular motion field about any equator can be divided into disjoint semicircles of clockwise and counterclockwise flow, and on the observation that the effects of rotation decouple around the three equators defining the three principal axes of rotation. Since the effect of rotation is geometrical, the three rotational parameters can be determined independently by searching, in each case, for a rotational value for which the derotated equatorial motion field can be partitioned into 180 degree arcs of clockwise and counterclockwise flow. The direction of translation is also obtained from this analysis. This search is two dimensional in the motion parameters, and can be performed relatively efficiently. Because information is correlated over large distances, the method can be considered a pattern recognition rather than a numerical algorithm. The algorithm is shown to be robust and relatively insensitive to noise and to missing data. Both theoretical and empirical studies of the error sensitivity are presented. The theoretical analysis shows that for white noise of bounded magnitude M, the expected errors is at worst linearly proportional to M. Empirical tests demonstrate negligible error for perturbations of up to 20% in the input, and errors of less than 20% for perturbations of up to 200%.     

Coordination of thumb joints during opposition

Thumb opposition plays a vital role in hand function. Kinematically, thumb opposition results from composite movements from multiple joints moving in multiple directions. The purpose of this study was to examine the coordination of thumb joints during opposition tasks. A total of 15 female subjects with asymptomatic hands were studied. Three-dimensional angular kinematics of the carpometacarpal (CMC), metacarpophalangeal (MCP) and interphalangeal (IP) joints were obtained by a marker-based motion analysis system. Thumb opposition revealed coordination among joints in a specific direction (inter-joint coordination) and among different directions within a joint (intra-joint coordination). In particular, linear couplings existed between the flexion and pronation at the CMC joint, and between the flexion of the CMC joint and flexion of the MCP joint. Principal component analysis showed that only two principal components adequately represented the thumb opposition data of seven movement directions. A term functional degrees of freedom by virtue of principal component analysis was proposed to uncover the extent of movement coordination in functional tasks.     

Talocrural and talocalcaneal joint kinematics and kinetics during the stance phase of walking

The purpose of this investigation was to study the kinematics and kinetics of the joints between the leg and calcaneus during the stance phase of walking. The talocrural and talocalcaneal joints were each assumed to act as monocentric single degree of freedom hinge joints. Motion at one joint was defined by the relative rotation of a point on the opposing joint. The results, based upon the gait of three subjects, showed that the hinge joint assumption may be reasonable. A discrepancy in the kinematics was shown between the talocrural joint rotation and its commonly assumed sagittal plane representation, especially during initial flatfoot. This discrepancy is due to the fact that the sagittal plane rotation is created by the combined rotations of the talocrural and talocalcaneal joints. The talocalcaneal joint showed a peak 25-30 Nm supinatory moment at 80% of stance. The talocrural joint moment was qualitatively similar to the commonly measured sagittal plane moment, but the present results show that the sagittal plane moment overpredicted the true moment by 6-22% due to the two-dimensional assumption.     

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.     

Validation of a low-dose hybrid RSA and fluoroscopy technique: Determination of accuracy, bias and precision

Analyzing skeletal kinematics with radiostereometric analysis (RSA) following corrective orthopedic surgery allows the quantitative comparison of different implant designs. The purpose of this study was to validate a technique for dynamically estimating the relative position and orientation of skeletal segments using RSA and single plane X-ray fluoroscopy. Two micrometer-based in vitro phantom models of the skeletal segments in the hip and knee joints were used. The spatial positions of tantalum markers that were implanted into each skeletal segment were reconstructed using RSA. The position and orientation of each segment were determined in fluoroscopy images by minimizing the difference between the markers measured and projected in the image plane. Accuracy was determined in terms of bias and precision by analyzing the deviation between the applied displacement protocol and measured pose estimates. Measured translational accuracy was less than 100 microm parallel to the image plane and less than 700 microm in the direction orthogonal to the image plane. The measured rotational error was less than 1 degrees . Measured translational and rotational bias was not statistically significant at the 95% level of confidence. The technique allows real-time kinematic skeletal measurements to be performed on human subjects implanted with tantalum markers for quantitatively measuring the motion of normal joints and different implant designs.     

The use of matrices in analyzing the three-dimensional behavior of the vestibulo-ocular reflex

The vestibulo-ocular reflex rotates the eye about the axis of a head rotation at the same speed but in the opposite direction to make the visual axes in space independent of head motion. This reflex works in all three degrees of freedom: roll, pitch, and yaw. The rotations may be described by vectors and the reflex by a transformation in the form of a matrix. The reflex consists of three parts: sensory, central, and motor. The transduction of head rotation into three neural signals, which may also be described by a vector, is described by a canal matrix. The neural, motorcommand vector is transformed to an eye rotation by a muscle matrix. Since these two matrices are known, one can solve for the central matrix which gives the strength of the connections between all the vestibular neurons and all the eye-muscle motoneurons. The role of the metric tensor in these transformations is described. This method of analysis is used in three applications. A lesion may be simulated by altering the elements in any or all of the three component matrices. By matrix multiplication, the resulting abnormal behavior of the reflex can be described quantitatively in all degrees of freedom. The method is also used to directly compare the differences in brain-stem connections between humans and rabbits that accommodate the altered actions of the muscles of the two species. Finally the method allows a quantitative assessment of the changes that take place in the brainstem connections when plastic changes are induced by artificially dissociating head movements from apparent motion of the visual environment.     

A computational approach to motion perception

In this paper it is shown that the computation of the optical flow from a sequence of timevarying images is not, in general, an underconstrained problem. A local algorithm for the computation of the optical flow which uses second order derivatives of the image brightness pattern, and that avoids the aperture problem, is presented. The obtained optical flow is very similar to the true motion field — which is the vector field associated with moving features on the image plane — and can be used to recover 3D motion information. Experimental results on sequences of real images, together with estimates of relevant motion parameters, like time-to-crash for translation and angular velocity for rotation, are presented and discussed. Due to the remarkable accuracy which can be achieved in estimating motion parameters, the proposed method is likely to be very useful in a number of computer vision applications.     

Accuracy and repeatability of joint angles measured using a single camera markerless motion capture system

Markerless motion capture systems have developed in an effort to evaluate human movement in a natural setting. However, the accuracy and reliability of these systems remain understudied. Therefore, the goals of this study were to quantify the accuracy and repeatability of joint angles using a single camera markerless motion capture system and to compare the markerless system performance with that of a marker-based system. A jig was placed in multiple static postures with marker trajectories collected using a ten camera motion analysis system. Depth and color image data were simultaneously collected from a single Microsoft Kinect camera, which was subsequently used to calculate virtual marker trajectories. A digital inclinometer provided a measure of ground-truth for sagittal and frontal plane joint angles. Joint angles were calculated with marker data from both motion capture systems using successive body-fixed rotations. The sagittal and frontal plane joint angles calculated from the marker-based and markerless system agreed with inclinometer measurements by <0.5°. The systems agreed with each other by <0.5° for sagittal and frontal plane joint angles and <2° for transverse plane rotation. Both systems showed a coefficient of reliability <0.5° for all angles. These results illustrate the feasibility of a single camera markerless motion capture system to accurately measure lower extremity kinematics and provide a first step in using this technology to discern clinically relevant differences in the joint kinematics of patient populations.     

The accuracy of measuring glenohumeral motion with a surface humeral cuff

Conclusions about normal and pathologic shoulder motion are frequently made from studies using skin surface markers, yet accuracy of such sensors representing humeral motion is not well known. Nineteen subjects were investigated with flock of birds electromagnetic sensors attached to transcortical pins placed into the scapula and humerus, and a thermoplastic cuff secured on the arm. Subjects completed two repetitions of raising and lowering the arm in the sagittal, scapular and coronal planes, as well as shoulder internal and external rotation with the elbow at the side and abducted to 90°. Humeral motion was recorded simultaneously from surface and bone fixed sensors. The average magnitude of error was calculated for the surface and bone fixed measurements throughout the range of motion. ANOVA tested for differences across angles of elevation, raising and lowering, and differences in body mass index. For all five motions tested, the plane of elevation rotation average absolute error ranged from 0-2°, while the humeral elevation rotation average error ranged from 0-4°. The axial rotation average absolute error was much greater, ranging from 5° during elevation motions to approaching 30° at maximum excursion of internal/external rotation motions. Average absolute error was greater in subjects with body mass index greater than 25. Surface sensors are an accurate way of measuring humeral elevation rotations and plane of elevation rotations. Conversely, there is a large amount of average error for axial rotations when using a humeral cuff to measure glenohumeral internal/external rotation as the primary motion.     

New least squares solutions for estimating the average centre of rotation and the axis of rotation.

A new method is proposed for estimating the parameters of ball joints, also known as spherical or revolute joints and hinge joints with a fixed axis of rotation. The method does not require manual adjustment of any optimisation parameters and produces closed form solutions. It is a least squares solution using the whole 3D motion data set. We do not assume strict rigidity but only that the markers maintain a constant distance from the centre or axis of rotation. This method is compared with other methods that use similar assumptions in the cases of random measurement errors, systematic skin movements and skin movements with random measurement noise. Simulation results indicate that the new method is superior in terms of the algorithm used, the closure of the solution, consistency and minimal manual parameter adjustment. The method can also be adapted to joints with translational movements.     

Orientation and location of the finite helical axis of the equine forelimb joints

To reduce anatomically unrealistic limb postures in a virtual musculoskeletal model of a horse's forelimb, accurate knowledge on forelimb joint constraints is essential. The aim of this cadaver study is to report all orientation and position changes of the finite helical axes (FHA) as a function of joint angle for different equine forelimb joints. Five horse cadaver forelimbs with standardized cuts at the midlevel of each segment were used. Bone pins with reflective marker triads were drilled into the forelimb bones. Unless joint angles were anatomically coupled, each joint was manually moved independently in all three rotational degrees of freedom (flexion–extension, abduction–adduction, internal–external rotation). The 3D coordinates of the marker triads were recorded using a six infra-red camera system. The FHA and its orientational and positional properties were calculated and expressed against joint angle over the entire range of motion using a finite helical axis method. When coupled, joint angles and FHA were expressed in function of flexion–extension angle. Flexion–extension movement was substantial in all forelimb joints, the shoulder allowed additional considerable motion in all three rotational degrees of freedoms. The position of the FHA was constant in the fetlock and elbow and a constant orientation of the FHA was found in the shoulder. Orientation and position changes of the FHA over the entire range of motion were observed in the carpus and the interphalangeal joints. We report FHA position and orientation changes as a function of flexion–extension angle to allow for inclusion in a musculoskeletal model of a horse to minimize calculation errors caused by incorrect location of the FHA.     

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.     

Fusion of radiostereometric analysis data into computed tomography space: application to the elbow joint

Improvement of joint prostheses is dependent upon information concerning the biomechanical properties of the joint. Radiostereometric analysis (RSA) and electromagnetic techniques have been applied in previous cadaver and in vivo studies on the elbow joint to provide valuable information concerning joint motion axes. However, such information is limited to mathematically calculated positions of the axes according to an orthogonal coordinate system and is difficult to relate to individual skeletal anatomy. The aim of this study was to evaluate the in vivo application of a new fusion method to provide three-dimensional (3D) visualization of flexion axes according to bony landmarks. In vivo RSA data of the elbow joint's flexion axes was combined with data obtained by 3D computed tomography (CT). Results were obtained from five healthy subjects after one was excluded due to an instable RSA marker. The median error between imported and transformed RSA marker coordinates and those obtained in the CT volume was 0.22 mm. Median maximal rotation error after transformation of the rigid RSA body to the CT volume was 0.003 degrees . Points of interception with a plane calculated in the RSA orthogonal coordinate system were imported into the CT volume, facilitating the 3D visualization of the flexion axes. This study demonstrates a successful fusion of RSA and CT data, without significant loss of RSA accuracy. The method could be used for relating individual motion axes to a 3D representation of relevant joint anatomy, thus providing important information for clinical applications such as the development of joint prostheses.     

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.     

Minimizing errors during in vitro testing of multisegmental spine specimens: considerations for component selection and kinematic measurement

Apparatus-induced artifacts may invalidate standard spine testing protocols. Kinematic measurements may be compromised by the configuration of motion capture equipment. This study has determined: (1) the influence of machine design (component friction) on in vitro spinal kinetics; (2) the sensitivity of kinematic measurements to variations in the placement of motion capture markers. A spinal loading simulator has been developed to dynamically apply pure bending moments (three axes) with or without a simultaneous compressive preload. Two linear slider types with different friction coefficients, one with caged ball bearings and one with high-precision roller bearings on rails, were mounted and specimen response compared in sequential tests. Three different optoelectronic marker cluster configurations were mounted on the specimen and motion data was captured simultaneously from all clusters during testing. A polymer tube with a uniform bending stiffness approximately equivalent to a polysegmental lumbar spine specimen was selected to allow reproducible behavior over multiple tests. The selection of sliders for linear degrees of freedom had a marked influence on parasitic shear forces. Higher shear forces were recorded with the caged-bearing design than with the high-precision rollers and consequently a higher moment was required to achieve a given rotation. Kinematic accuracy varied with each marker configuration, but in general higher accuracy was achieved with larger marker spacings and situations where markers moved predominantly parallel to the camera's imaging plane. Relatively common alternatives in the mechanical components used in an apparatus for in vitro spine testing can have a significant influence on the measured kinematic and kinetics. Low-magnitude parasitic shear forces due to friction in sliders induces a linearly increasing moment along the length of the specimen, precluding the ideal of pure moment application. This effect is compounded in polysegmental specimens. Kinematic measurements are highly sensitive to marker design and placement, despite equivalent absolute precision of individual marker measurements, however marker configurations can be designed to minimize errors related to spatial distribution and system bias.     

Phase plane modeling of leg motion

Phase plane analysis of dynamical systems, in which variables are plotted against their time derivatives, has been recently emphasized as a general method for reconstructing system dynamics from data. The purpose of this experiment was to develop a model of leg movement in a stepping task using the phase plane approach. In this model, the leg is represented as a three-body linkage and the motion of the leg is assumed to be planar with four degrees of freedom. Experimental data was collected on one subject stepping six times, using a two dimensional videomotion analysis system with reflective markers placed on the lower limb joints. A computer program able to solve the equations of motion and compute the state of the system for a given task was implemented. This computer program was written to generate the motion of the leg for a given task using inverse kinematics and a preplanned foot path. Foot trajectories with cycloidal, constant acceleration/deceleration and sinusoidal velocity profiles were studied. From the results, an attempt was made to identify the variables which are measured and to determine the motion characteristics during stepping. The preliminary results support the concept of a hierarchical control structure with openloop control during normal operation. During routine activity there is no direct intervention of the Central Nervous System (CNS). The results support the existence of preprogramming and provide a starting point for the study of the development of control in multiarticulate movements.