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.     

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.     

Comparing non-invasive scapular tracking methods across elevation angles,planes of elevation and humeral axial rotations

Altered scapular motions premeditate shoulder impingement and other musculoskeletal disorders. Divergent experimental conditions in previous research precludes rigorous comparisons of non-invasive scapular tracking techniques. This study evaluated scapular orientation measurement methods across an expanded range of humeral postures. Scapular medial/lateral rotation, anterior/posterior tilt and protraction/retraction was measured using an acromion marker cluster (AMC), a scapular locator, and a reference stylus. Motion was captured using reflective markers on the upper body, as well as on the AMC, locator and stylus. A combination of 5 arm elevation angles, 3 arm elevation planes and 3 arm axial rotations was examined. Measurement method interacted with elevation angle and plane of elevation for all three scapular orientation directions (p < 0.01). Method of measurement interacted with axial rotation in anterior/posterior tilt and protraction/retraction (p < 0.01). The AMC had strong agreement with the reference stylus than the locator for the majority of humeral elevations, planes and axial rotations. The AMC underestimated lateral rotation, with the largest difference of ∼2° at 0° elevation. Both the locator and AMC overestimated posterior tilt at high arm elevation by up to 7.4°. Misestimations from using the locator could be enough to potentially obscure meaningful differences in scapular rotations.     

A new non-orthogonal decomposition method to determine effective torques for three-dimensional joint rotation

This paper describes a new non-orthogonal decomposition method to determine effective torques for three-dimensional (3D) joint rotation. A rotation about a joint coordinate axis (e.g. shoulder internal/external rotation) cannot be explained only by the torque about the joint coordinate axis because the joint coordinate axes usually deviate from the principal axes of inertia of the entire kinematic chain distal to the joint. Instead of decomposing torques into three orthogonal joint coordinate axes, our new method decomposes torques into three "non-orthogonal effective axes" that are determined in such a way that a torque about each effective axis produces a joint rotation only about one of the joint coordinate axes. To demonstrate the validity of this new method, a simple internal/external rotation of the upper arm with the elbow flexed at 90 degrees was analyzed by both orthogonal and non-orthogonal decomposition methods. The results showed that only the non-orthogonal decomposition method could explain the cause-effect mechanism whereby three angular accelerations at the shoulder joint are produced by the gravity torque, resultant joint torque, and interaction torque. The proposed method would be helpful for biomechanics and motor control researchers to investigate the manner in which the central nervous system coordinates the gravity torque, resultant joint torque, and interaction torque to control 3D joint rotations.     

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 joint coordinate system proposal for the study of the trapeziometacarpal joint kinematics

The International Society of Biomechanics (ISB) has recommended a standardisation for the motion reporting of almost all human joints. This study proposes an adaptation for the trapeziometacarpal joint. The definition of the segment coordinate system of both trapezium and first metacarpal is based on functional anatomy. The definition of the joint coordinate system (JCS) is guided by the two degrees of freedom of the joint, i.e. flexion-extension about a trapezium axis and abduction-adduction about a first metacarpal axis. The rotations obtained using three methods are compared on the same data: the fixed axes sequence proposed by Cooney et al., the mobile axes sequence proposed by the ISB and our alternative mobile axes sequence. The rotation amplitudes show a difference of 9 degrees in flexion-extension, 2 degrees in abduction-adduction and 13 degrees in internal-external rotation. This study emphasizes the importance of adapting the JCS to the functional anatomy of each particular joint.     

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.     

Functional knee axis based on isokinetic dynamometry data: Comparison of two methods, MRI validation, and effect on knee joint kinematics

This paper compares geometry-based knee axes of rotation (transepicondylar axis and geometric center axis) and motion-based functional knee axes of rotation (fAoR). Two algorithms are evaluated to calculate fAoRs: Gamage and Lasenby's sphere fitting algorithm (GL) and Ehrig et al.'s axis transformation algorithm (SARA). Calculations are based on 3D motion data acquired during isokinetic dynamometry. AoRs are validated with the equivalent axis based on static MR-images. We quantified the difference in orientation between two knee axes of rotation as the angle between the projection of the axes in the transversal and frontal planes, and the difference in location as the distance between the intersection points of the axes with the sagittal plane. Maximum differences between fAoRs resulting from GL and SARA were 5.7° and 15.4mm, respectively. Maximum differences between fAoRs resulting from GL or SARA and the equivalent axis were 5.4°/11.5mm and 8.6°/12.8mm, respectively. Differences between geometry-based axes and EA are larger than differences between fAoR and EA both in orientation (maximum 10.6°).and location (maximum 20.8mm). Knee joint angle trajectories and the corresponding accelerations for the different knee axes of rotation were estimated using Kalman smoothing. For the joint angles, the maximum RMS difference with the MRI-based equivalent axis, which was used as a reference, was 3°. For the knee joint accelerations, the maximum RMS difference with the equivalent axis was 20°/s(2). Functional knee axes of rotation describe knee motion better than geometry-based axes. GL performs better than SARA for calculations based on experimental dynamometry.     

A joint coordinate system proposal for the study of the trapeziometacarpal joint kinematics

The International Society of Biomechanics (ISB) has recommended a standardisation for the motion reporting of almost all human joints. This study proposes an adaptation for the trapeziometacarpal joint.

The definition of the segment coordinate system of both trapezium and first metacarpal is based on functional anatomy. The definition of the joint coordinate system (JCS) is guided by the two degrees of freedom of the joint, i.e. flexion–extension about a trapezium axis and abduction–adduction about a first metacarpal axis. The rotations obtained using three methods are compared on the same data: the fixed axes sequence proposed by Cooney et al., the mobile axes sequence proposed by the ISB and our alternative mobile axes sequence. The rotation amplitudes show a difference of 9° in flexion–extension, 2° in abduction–adduction and 13° in internal–external rotation.

This study emphasizes the importance of adapting the JCS to the functional anatomy of each particular joint.     

Sensitivity of the knee joint kinematics calculation to selection of flexion axes

Various flexion axes have been used in the literature to describe knee joint kinematics. This study measured the passive knee kinematics of six cadaveric human knee specimens using two widely accepted flexion axes; transepicondylar axis and the geometric center axis. These two axes were found to form an angle of 4.0 degrees +/- 0.8 degrees. The tibial rotation calculated using the transepicondylar axis was significantly different than the rotation obtained using the geometric center axis for the same knee motion. At 90 degrees of flexion, the tibial rotation obtained using the transepicondylar axis was 4.8 degrees +/- 9.4 degrees whereas the rotation recorded using the geometric center axis at the same flexion angle was 13.8 degrees +/- 10.2 degrees. At 150 degrees of knee flexion, the rotations obtained from the transepicondylar and the geometric center axes were 7.2 degrees +/- 5.7 degrees and 19.9 degrees +/- 6.9 degrees, respectively. The data suggest that a clear definition of the flexion axis is necessary when reporting knee joint kinematics.     

A kinematic model of the human ankle

The spatial gross motion of the foot with respect to the shank is modelled as rotations about two fixed ankle axes: the upper ankle rotation axis (plantarflexion/dorsiflexion) and the subtalar rotation axis (inversion/eversion). The positions of the axes are determined by externally visible bony landmarks of the lower leg and are measured for a living subject. The model input data are the plantarflexion/dorsiflexion and inversion/eversion rotation angles; the model output is a 4 × 4 transformation matrix which quantitatively describes the relative position of a foot coordinate system with respect to a shank coordinate system.     

Rotational flexibility of the human knee due to varus/valgus and axial moments in vivo

Knee ligamentous injuries persist in the sport of Alpine skiing. To better understand the load mechanisms which lead to injury, pure varus/valgus and pure axial moments were applied both singly and in combination to the right knees of six human test subjects. The corresponding relative knee rotations in three degrees of freedom were measured. Knee flexion angles for each test subject were 15 and 60 degrees for the individual moments and 60 degrees for the combination moments. For both knee flexion angles the hip flexion angle was 0 degrees. Leg muscles were quiescent and axial force was minimal during all tests. Tables of data include sample statistics for each of four flexibility parameters in each loading direction. Data were analyzed statistically to test for significant differences in flexibility parameters between the test conditions. In flexing the knee from 15 to 60 degrees, the resulting knee rotations under single moments depended upon flexion angle with varus, valgus, and internal rotations increasing significantly. Also, rotations were different depending on load direction; varus rotation was significantly different and greater than valgus rotation at both flexion angles. Also external rotation was significantly different and greater than internal at 15 degrees flexion, but not at 60 degrees flexion. Coupled rotations under single moments were also observed. Applying pure varus/valgus moments resulted in coupled external/internal rotations which were inconsistent and hence not significant. Applying pure axial moments resulted in consistent and hence significant varus/valgus rotations; an external axial moment induced varus rotation and an internal axial moment induced valgus rotation. For combination moments, varus/valgus rotations decreased significantly from those rotations at similar load levels in the single moment studies. Also, a varus moment significantly increased external rotation and a valgus moment significantly decreased internal rotation. These differences indicate significant interaction between corresponding load combinations. These results suggest that load interaction is a potentially important phenomenon in knee injury mechanics.     

Comparison of different calculations of three-dimensional joint kinematics from video-based system data

When skin-fixed marker trajectories are used to calculate 3D joint kinematics, the measurement errors (i.e. the difference between the trajectories of the external markers and those of the skeleton) influence to some extent the accuracy of the results, depending both on the calculation method and on the axes about which the rotations are expressed. The purpose of this paper is to compare several expressions of joint angular variations. Two kinematic concepts are used to calculate the changes in the orientation of the distal segment versus the proximal one: the first method consists of computing the components of the spatial attitude vector, the second one deals with the determination of elementary rotations about successive axes. For each of these methods, two sets of three axes are tested to express the results: the axes forming the reference frame affixed to the body segment adjacent to the joint (named fixed axes), and a set consisting of a first axis belonging to the proximal segment, a third axis belonging to the distal segment and a second (floating) axis defined as the cross-product between the two other ones (named mobile axes). To compare these four distinct expressions on the knee joint, numerical simulations of perturbed skin marker trajectories are performed, based on experimental data recorded by a Motion Analysis system during a normal gait cycle. A significant difference is pointed out only for the internal–external rotation angle, for which the best expression — from the viewpoint of sensitivity to experimental errors — is obtained using the components of the attitude vector in a segment-embedded reference frame.     

Kinematics of human arm reconstructed from spatial tracking system recordings

The kinematics of the human arm in terms of angles of rotations in the joints is reconstructed from the spatial tracking system (Fastrack() Polhemus) recordings. The human arm is modeled by three rigid bodies (the upper arm, the forearm and the hand) with seven degrees of freedom (three in the shoulder, two in the elbow and two in the wrist). Joint geometry parameters (orientations of the axes relative to the arm segments, the angles and the distances between the axes) have been calculated on the basis of passive rotations in the joints. The calculated parameters have been used to solve the direct kinematics problem for the reaching movements in different directions. The difference between calculated and recorded positions and accelerations of the hand has been used to assess the accuracy of the proposed method of kinematics reconstruction. The error analysis showed that spatial tracking system recordings and human arm kinematics reconstruction could reliably be used to accurately analyze multijoint movement in humans.     

The influence of axial rotation upon interception of solar radiation by plant leaves

Axial rotation of a plant leaf about the midrib is an important factor which permits effective maximization or minimization of the interception of direct solar radiation. Through use of matrix rotations of coordinate systems, the cosine of the angle of incidence, a measure of the fraction of solar radiation intercepted by the leaf, is described as a function of angles which specify the relative orientation of the leaf to the sun. All of these angles can be measured directly except for the axial rotation angle, which may be calculated from more easily measured quantities. Computer modeling and theoretical calculations demonstrate that relatively large amounts of average axial rotation permit a plant to reduce the cosine of angle of incidence of each flat leaf to zero, resulting in no interception of direct solar radiation, while smaller amounts of average axial rotation allow a plant to intercept as much as 80% of direct radiation, even if leaves are distributed uniformly over 360° of azimuth.     

Development of a computational model for astronaut reorientation

The ability to model astronaut reorientations computationally provides a simple way to develop and study human motion control strategies. Since the cost of experimenting in microgravity is high, and underwater training can lead to motions inappropriate for microgravity, these techniques allow for motions to be developed and well-understood prior to any microgravity exposure. By including a model of the current space suit, we have the ability to study both intravehicular and extravehicular activities. We present several techniques for rotating about the axes of the body and show that motions performed by the legs create a greater net rotation than those performed by the arms. Adding a space suit to the motions was seen to increase the resistance torque and limit the available range of motion. While rotations about the body axes can be performed in the current space suit, the resulting motions generated a reduced rotation when compared to the unsuited configuration.     

A measure for the angle between projections based on the extent of correlation between corresponding central sections

A pre-condition for the ab initio assignment of Euler angles to a set of projections from an asymmetric object is that at least three of the available projections correspond to rotations about different axes. For symmetric objects this condition may be relaxed. There are some applications of single-particle electron microscopy, such as the reconstruction of filamentous macromolecular assemblies, where all available projections more-or-less correspond to rotations about a common rotation axis making it difficult to satisfy this condition. Here, a method has been developed to overcome this problem, based on the fact that the correlation between two central sections of the Fourier transform of a compact object will not be limited to an infinitesimal central line but will have a finite extent, which is related to the angle between the corresponding projections. Projections from model filaments, with different degrees of rotational symmetry about the long axis, have been used to test the methodology. The results show that angle determination is robust down to signal-to-noise ratios as low as 2 and that, in general, the error decreases as the degree of symmetry increases. The method has been used to assign angles to a set of negatively stained muscle thick filament projections to obtain an initial 3D reconstruction. The main features of the projections are seen to be faithfully reproduced in the reprojections from the reconstruction. A real-space adaptation of this method is also discussed.     

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.     

An anatomy-based coordinate system for the description of the kinematic displacements in the human knee

The effects and interaction of the anatomical displacements in the human knee are a prerequisite to an accurate assessment and communication of the kinematic data. For the kinematic information to be used to improve diagnosis and treatment, and for better prosthetic design and installation, there must be clear, concise, and universal definitions of the displacements. In general, the displacements are defined as three translations and three rotations. In this paper, anatomic landmarks on the femur and on the tibia are used to define the locations and orientations of the six displacement axes; i.e. three translational and three rotational displacement axes. The most commonly accepted kinematic representation of the knee joint, in the literature, is a special geometry three-cylindric open chain in which the axes of the cylindric joints are defined according to the rotational displacement axes. The sequentially adjacent joint axes are assumed to not only intersect but to intersect at right-angles. The open chain permits a total of six degrees of freedom between a Cartesian reference frame attached to the femur and a Cartesian reference frame attached to the tibia. In this paper, the three rotational axes are shown to be skewed and off-set from each other, therefore, a three-cylindric open chain with skewed joint axes is proposed to measure the six displacements between the two reference frames. The authors believe that the proposed open chain is the most general to date and provides a more realistic representation of the displacements in the knee. To illustrate the significance of the reference frames on the interpretation of measured data, the anterior/posterior drawer is plotted against per cent gait cycle for three existing open chains and the proposed open chain.     

A kinematic model for calculating cup alignment error during total hip arthroplasty

Reduced range of motion, prosthetic impingement, and joint dislocation can all result from misalignment of the acetabular component (i.e. cup alignment) in patients undergoing total hip arthroplasty. Most methods for acetabular component alignment are designed to provide 45-50 degrees abduction and 15-25 degrees of operative anteversion (also known as flexion) with respect to the anterior pelvic plane coordinate system. Yet in most cases, this coordinate system is not assigned properly, due to differences in patient anatomy and improper positioning in the operating room. This misalignment can result in an error in the cup alignment, which can cause the above-mentioned consequences. This work presents a complete mathematical formulation for the analysis of the inaccuracies related to the anterior pelvic plane axes (APPA) definition and their effect on final cup orientation. We do this by introducing a method taken from Kinematics of Mechanisms, and by representing the errors in the APPA as three concurrent axes of rotation, followed by the version and abduction rotations which are defined relative to the previous rotations. We also present a sensitivity analysis of the results by introducing differential changes between sequential coordinate frames, which simulates the errors in the APPA and their effect on cup orientation. Finally, we demonstrate a computational method which provides corrected version and abduction angles to achieve the desired cup orientation, given that the actual measurement errors are known.