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The local and global geometric properties of spherical coupler curves constitute spherical kinematics of spherical four-bar linkages, which can be adopted to reveal distribution characteristics of spherical coupler curves. New unified spherical adjoint approach is established in the paper to study both the local and global geometric properties in order to enrich the atlas of spherical coupler curves with geometric characteristics. Since the constraint curve of spherical four-bar linkage is a simple spherical circle and the spherical centrodes imply intrinsic properties of spherical motion of the coupler link, they are in their turn taken as the original curves in spherical adjoint approach to derive the geodesic curvature and analyze the local geometric characteristics of the spherical coupler curves. The conditions for different spherical double points, such as spherical crunodes, tacnodes and cusps of the spherical coupler curve are derived through the spherical adjoint approach. The spherical surface of the coupler link can be divided into several areas by the spherical moving centrode and the spherical tacnode's tracer curve. The points in each area trace spherical coupler curves with a specific shape. The characteristic points, which trace spherical coupler curves with cusp, geodesic inflection point, spherical Ball point, spherical Burmester point, crunode and tacnode can be readily located in the coupler link by the modelling procedure and the derived condition equations. In the end the distribution of spherical coupler curves with both local and global characteristics is elaborated. The research proposes systematic geometric properties of spherical coupler curves based on the new established approach, and provides a solid theoretical basis for the kinematic analysis and synthesis of the spherical four-bar linkages.

This paper mainly focuses on some notions of the lightlike rectifying curves and the centrodes in Minkowski 3-space. Some geometrical characteristics of the three types of lightlike curves are obtained. In addition, we obtain the conditions of the centrodes of the lightlike curves are the lightlike rectifying curves. Meanwhile, a detailed analysis between the N-type lightlike slant helices and the centrodes of lightlike curves is provided in this paper. We give the projections of the lightlike rectifying curves to the timelike planes.

A variety of medical imaging procedures, cadaver experiments, and computer models have been utilized to capture, depict, and understand the motion of the human lumbar spine. Particular interest lies in assessing the relative movement between two adjacent vertebrae, which can be represented by a temporal evolution of finite helical axes (FHA). Mathematically, this FHA evolution constitutes a seven-dimensional quantity: one dimension for the time, two for the (normalized) direction vector, another two for the (unique) position vector, as well as one for each the angle of rotation around and the amount of translation along the axis. Predominantly in the literature, however, movements are assumed to take place in certain physiological planes on which FHA are projected. The resulting three-dimensional quantity – the so-called centrode – is easily presentable but leaves out substantial pieces of available data. Here, we investigate and assess several possibilities to visualize subsets of FHA data of increasing dimensionality. Finally, we utilize an agglomerative hierarchical clustering algorithm and propose a novel visualization technique, namely the quiver principal axis plot (QPAP), to depict the entirety of information inherent to hundreds or thousands of FHA. The QPAP method is applied to flexion-extension, lateral bending, and axial rotation movements of a lumbar spine within both a reduced model as well as a complex upper body system.

Lumbar spine biomechanics during the forward-bending of the upper body (flexion) are well investigated by both in vivo and in vitro experiments. In both cases, the experimentally observed relative motion of vertebral bodies can be used to calculate the instantaneous center of rotation (ICR). The timely evolution of the ICR, the centrode, is widely utilized for validating computer models and is thought to serve as a criterion for distinguishing healthy and degenerative motion patterns. While in vivo motion can be induced by physiological active structures (muscles), in vitro spinal segments have to be driven by external torque-applying equipment such as spine testers. It is implicitly assumed that muscle-driven and torque-driven centrodes are similar. Here, however, we show that centrodes qualitatively depend on the impetus. Distinction is achieved by introducing confidence regions (ellipses) that comprise centrodes of seven individual multi-body simulation models, performing flexion with and without preload. Muscle-driven centrodes were generally directed superior–anterior and tail-shaped, while torque-driven centrodes were located in a comparably narrow region close to the center of mass of the caudal vertebrae. We thus argue that centrodes resulting from different experimental conditions ought to be compared with caution. Finally, the applicability of our method regarding the analysis of clinical syndromes and the assessment of surgical methods is discussed. Electronic supplementary material The online version of this article (10.1007/s10237-020-01382-9) contains supplementary material, which is available to authorized users.

Although many methods have been proposed in the literature for rigid-body guidance synthesis none of them establishes the position of the instant center (IC) and the centrode as a design requirement. In many cases, practical designs of mechanisms need to accurately establish the position of the IC and the centrode. Due to the lack of methods tackling this problem directly, designers try to approach it indirectly. In other words, they establish the position of the desired rigid-body to obtain the desired position of the IC. However, small errors in the position of the rigid-body may lead to significant errors in the position of the IC and a trial-and-error process is necessary to achieve accuracy. The main difficulty in implementing the centrode in a synthesis approach is how to formulate the IC conditions in an optimization method. This paper presents a synthesis approach based on optimization in which the location of the centrode is included in the objective function. The formulation developed in the paper uses exact differentiation to obtain the descent direction and can be applied to any type of planar mechanism. Multi-objective optimization is also possible, including different design criteria in the target function. Several examples are provided to demonstrate the robustness, accuracy and efficiency of the new approach.

First, we study rectifying curves via the dilation of unit speed curves on the unit sphere $S^{2}$ in the Euclidean space $\mathbb E^3$. Then we obtain a necessary and sufficient condition for which the centrode $d(s)$ of a unit speed curve $\alpha(s)$ in $\mathbb E^3$ is a rectifying curve to improve a main result of \cite{cd05}. Finally, we prove that if a unit speed curve $\alpha(s)$ in $\mathbb E^3$ is neither a planar curve nor a helix, then its dilated centrode $\beta(s)=\rho(s) d(s)$, with dilation factor ${\rho}$, is always a rectifying curve, where $\rho$ is the radius of curvature of $\alpha$.

This paper presents the mechanical design, dynamics analysis and ankle trajectory analysis of a prosthetic knee joint using the geared five-bar mechanism. Compared with traditional four-bar or six-bar mechanisms, the geared five-bar mechanism is better at performing diverse movements and is easy to control. This prosthetic knee joint with the geared five-bar mechanism is capable of fine-tuning its relative instantaneous center of rotation and ankle trajectory. The centrode of this prosthetic knee joint, which is mechanically optimized according to the centrode of human knee joint, is better in the bionic performance than that of a prosthetic knee joint using the four-bar mechanism. Additionally, the stability control of this prosthetic knee joint during the swing and stance phase is achieved by a motor. By adjusting the gear ratio of this prosthetic knee joint, the ankle trajectories of both unilateral and bilateral amputees show less deviations from expected than that of the four-bar knee joint.

Planar kinematics has been studied systematically based on centrodes, however axodes are underutilized to set up the curvature theories in spherical and spatial kinematics. Through a spherical adjoint approach, an axode-based theoretical system of spherical kinematics is established. The spherical motion is re-described by the adjoint approach and vector equation of spherical instant center is concisely derived. The moving and fixed axodes for spherical motion are mapped onto a unit sphere to obtain spherical centrodes, whose kinematic invariants totally reflect the intrinsic property of spherical motion. Based on the spherical centrodes, the curvature theories for a point and a plane of a rigid body in spherical motion are revealed by spherical fixed point and plane conditions. The Euler-Savary analogue for point-path is presented. Tracing points with higher order curvature features are located in the moving body by means of algebraic equations. For plane-envelope, the construction parameters are obtained. The osculating conditions for plane-envelope and circular cylindrical surface or circular conical surface are given. A spherical four-bar linkage is taken as an example to demonstrate the spherical adjoint approach and the curvature theories. The research proposes systematic spherical curvature theories with the axode as logical starting-point, and sets up a bridge from the centrode-based planar kinematics to the axode-based spatial kinematics.

Se presenta un nuevo par de engranajes no circulares para obtener pequeñas aceleraciones del eje. Las formas de los engranajes pueden ser controladas dependiendo de la máxima aceleración y suavidad de los contornos requeridos. Se hace una comparación con engranajes elípticos y sinusoidales. Los resultados muestran que, para engranajes de dos lóbulos, los valores máximos y mínimos de los radios polares y de la relación de transmisión son iguales para los perfiles desarrollados y los sinusoidales, pero difieren para los elípticos. En contraste, los valores máximos de la aceleración angular, aceleración tangencial y ángulo de presión difieren. Se concluye que los nuevos perfiles proporcionan pequeñas aceleraciones angulares y tangenciales y menores ángulos de presión, excepto que los elípticos pueden exhibir menores aceleraciones tangenciales y ángulos de presión para grandes variaciones de la velocidad. Consecuentemente, las cargas y esfuerzos podrían ser menores. Esto podría resultar en sistemas más compactos. This article presents a new noncircular gear pair to obtain small shaft accelerations. The centrode contours may be controlled depending on the required maximum acceleration and smoothness of the centrodes. A comparison among elliptical, sinusoidal, and the new gears is provided. Results show that, for two-lobule gears, the maximum and minimum polar radii and gear ratios are the same for the new and sinusoidal profiles but differ for the elliptical ones. Conversely, there are significant differences in the maximum angular acceleration, tangential acceleration, and pressure angle. It is concluded that the novel gears provide not only small shaft accelerations, but also small tangential accelerations and pressure angles, and it is excepted that the elliptical gears may exhibit lower tangential accelerations and pressure angles for large values of the angular speed alternating component. Consequently, shaft and tooth loads and stresses may be lower for the new gears. This may result in more compact systems.

[EN] We propose to model planar movements between two human segments by means of rolling-without-slipping kinematic pairs. We compute the path traced by the instantaneous center of rotation (ICR) as seen from the proximal and distal segments, thus obtaining the fixed and moving centrodes, respectively. The joint motion is then represented by the rolling-without-slipping of one centrode on the other. The resulting joint kinematic model is based on the real movement and accounts for nonfixed axes of rotation; therefore it could improve current models based on revolute pairs in those cases where joint movement implies displacement of the ICR. Previous authors have used the ICR to characterize human joint motion, but they only considered the fixed centrode. Such an approach is not adequate for reproducing motion because the fixed centrode by itself does not convey information about body position. The combination of the fixed and moving centrodes gathers the kinematic information needed to reproduce the position and velocities of moving bodies. To illustrate our method, we applied it to the flexion-extension movement of the head relative to the thorax. The model provides a good estimation of motion both for position variables (mean R pos=0.995) and for velocities (mean R vel=0.958). This approach is more realistic than other models of neck motion based on revolute pairs, such as the dual-pivot model. The geometry of the centrodes can provide some information about the nature of the movement. For instance, the ascending and descending curves of the fixed centrode suggest a sequential movement of the cervical vertebrae. © 2010 Elsevier Ltd., This work was funded by the Spanish Government and co-financed by EU FEDER funds (Grants DPI2006-14722-C02-01, DPI2006-14722-C02-02, DPI2009-13830-C02-01, DPI2009-13830-C02-02 and Ramon y Cajal contract to JAG).

The machining of ordered curl of surfaces, associated to a centrode, is completed in very good yield and precision conditions with hob mill tool. In this case, the issue is to determine the enwrapping worm of the ordered curl of profiles, as a fundament for hob mill tool's construction. Based on the "generating trajectories" method, an algorithm for profiling the generating hob mill for machining the trilobed rotor of a peristaltic pump is presented. A graphical example, developed in CATIA design environment, shows the profiling technique.

Because the disc and facets work together to constrain spinal kinematics, changes in the instant axis of rotation associated with disc degeneration or disc replacement may adversely influence risk for facet overloading and arthritis. The relationships between L5/S1 segmental kinematics and facet forces are not well defined, since previous studies have separated investigations of spinal motion and facet force. The goal of this cadaveric biomechanical study was to report and correlate a measure of intervertebral kinematics (the centrode, or the path of the instant axis of rotation) and the facet forces at the L5/S1 motion segment while under a physiologic combination of compression and anterior shear loading. Twelve fresh-frozen human cadaveric L5/S1 joints (age range 50-64 years) were tested biomechanically under semi-constrained conditions by applying compression plus shear forces in several postures: neutral, and 3 degrees and 6 degrees of flexion, extension and lateral bending. The experimental boundary conditions imposed compression and shear representative of in vivo conditions during upright stance. The 3-D instantaneous axis of rotation (IAR) was calculated between two consecutive postures. The facet joint force was simultaneously measured using thin-film sensors placed between both facet surfaces. Variations of IAR location and facet force during motion were analyzed. During flexion and extension, the IAR was oriented laterally. The IAR intersection with the mid-sagittal plane moved cephalad relative to S1 endplate during flexion (P=0.010), and posterior during extension (P=0.001). The facet force did not correlate with posture (P=0.844). However, changes in the facet force between postures did correlate with IAR position: higher IAR's during flexion correlated with lower facet forces and vice versa (P=0.04). During lateral bending, the IAR was oblique relative to the main plane of motion and translated parallel to S1 endplate, toward the side of the bending. Overall, the facet force was increased on the ipsilateral side of bending (P=0.002). The IAR positions demonstrate that the L5 vertebral body primarily rotates forward during flexion (IAR close to vertebral body center) and rotates/translates backward during extension (IAR at or below the L5/S1 intervertebral disc). In lateral bending, the IAR obliquity demonstrated coupling with axial torsion due to resistance of the ipsilateral facet.

This paper deals with a new mechanism, the generalized Oldham coupling., This is a turning block double-slider crank mechanism with circular-arc-sliders and is a cyclical varying speed coupling. Type of a cyclical varying speed motion is decided by form of circular-arc-sliders. The motion of the intermediate element disk is similar to that of the “hula-hoop”, but more complicated. This paper aims at analyzing the motion of the intermediate element of the generalized Oldham coupling. First an instantaneous center of rotation of the intermediate element is found, and the fixed centrode is obtained as the locus of this center in the fixed coordinate system. Secondly the moving centrode is also obtained as the locus of this center in the moving coordinate system attached on the intermediate element. The motion of intermediate element is expressed as the motion that this moving centrode rolls on the fixed centrode.

In this paper motion as a combination of translation of the base point O and rotation about axis Z of a plane mechanism. The equations of the fixed centrode (B), the trajectory (T) and the movable centrode (R) are obtained. Three inverse problems are studied: if two of these curves are known, find the third one. Implicit differential equations are obtained. Numerical methods to solve these implicit differential equations are presented.