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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.

The design of the contact profiles of spur gears with involute teeth is well developed, with industry standards that set values for pressure angle and ratios for virtually all dimensions. This is not the case for bevel gears, where involute teeth, although known for several years, haven’t been adopted by industry. As a result, no standards for involute bevel gears are available. In the case of hypoid gears, owing their name to the pitch surfaces, the definition of involute surfaces is still to be seen. This state of affairs motivated the authors to look for alternative means of synthesizing the contact profiles of interest. Reported in this paper is the application of the Camus Theorem to the synthesis of the contact profiles of spur (planar)- and bevel (spherical)-gear teeth. It is shown that, upon using the planar or, correspondingly, the spherical logarithmic spiral as auxiliary centrode, the contact profiles of spur and bevel gears with involute teeth can be obtained within a unified framework.

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.

A Knee-Ankle-Foot orthosis (KAFO) is used as a supportive device by individuals with lower limb disability. A type of KAFO that allows knee flexion-extension is prescribed for people who need knee stability in the transverse and frontal planes. In such an orthosis, mimicking the human knee motion is vital to avoid relative motion (called pistoning) between the limb and the orthosis. A four-bar mechanism, owing to its polycentric nature, simplicity and ease of fabrication can provide a customizable, biomimetic solution. This paper presents an improved and robust optimization approach to synthesize a four-bar mechanism to closely mimic the anatomical knee motion. The reference human knee centrode is obtained from literature. A genetic algorithm is used for optimal synthesis of the fourbar mechanism. Results show that the average error between the reference centrode and the centrode of the synthesized four-bar mechanism is very small (0.2 mm). Thus, the synthesized crossed four-bar linkage can reproduce better anthropomorphic characteristics of the knee joint. The methodology can be used for the design of customized orthotic knee joints for KAFOs and knee braces.

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.

The prosthetic knee is the essential rehabilitation device for the above-knee amputees to recover from mobility impairments. This paper presents an optimal design of a geared-five bar (GFB) prosthetic knee considering the motion interference between its mechanism and internal actuating area. To ensure that the GFB prosthetic knee can provide the amputees with the expected knee centrode while there is no internal motion interference during knee flexion, the motion interference is solved by the separating axis theorem and added into the optimization process as the penalty function. The final optimization result shows that the centrode of the GFB prosthetic knee is close to the ideal one and the internal cavity for the actuation is guaranteed, which will be beneficial to the realization of the miniaturization of the powered geared five-bar prosthetic knee.

The generation of the ordinate surfaces whirl attached to a centrode (e.g. cylindrical or helical gears teeth generation) with a hob mill is an efficient method, also recommended in machining the teeth of slitting saw cutter with high dimensions. Profiling the hob mill conjugated to the disc-tool (slitting saw cutter) teeth, depending on its tooth profile, is a specific problem. There are known both analytical methods to profile the hob mill – e.g. the Gohman method (which uses an intermediary surface – the reference rack-gear), and graphical methods (developed in AutoCAD). A profiling method, which uses the CATIA environment facilities in graphical design, laying on the principle of helical motion de-composition (Nikolaev theorem) has been recently suggested. In this paper, we present a method for hob mill profiling, in graphical form, based on a complementary theorem (the Minimum distance method). The reference rack-gear, conjugated to the disc-tool teeth, is firstly profiled and then, by starting from it, the peripheral surface of the hob mill for machining the slitting saw cutter teeth is determined.

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.

By analyzing the skeleton of the kangaroo's tibia and femur, a movement is assumed. According to this hypothesis, the centrode of the shank relative to the thigh is calculated. On this basis, a special geared five-bar mechanism is proposed, and the motion analysis and optimization calculation of the mechanism are carried out. The geared five-bar bionic joint mechanism is optimized to mimics the features of the kangaroo knee movement. Finally, the 3D model design and object making are completed, and the result shows that the geared five-bar mechanism can well simulate the motion characteristics of kangaroo 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.

The paper resumes the traditional hypotheses for the noncircular mating centrodes design, based on the definition of i) the driving centrode geometry ii) the transmission ratio variation and iii) the rotational motion of the driven centrode. Each procedure is transferred within the gear theory and developed considering both specific gear data, such as gear modulus, gear number of teeth and gear number of rotations during a rotational period. Original codes are developed in Php amd MyQSL to illustrate gears conjugate pitch curves geometry and their kinematics for external rolling.

Tools which generate by enveloping using the rolling method may be profiled using various methods. The substitutive circles family method is a complementary method developed based a specifically theorem, in which is determined a family of circles associated with the blank’s centrode, family which envelop the profile to be generate. The method assumes the determination of the circles family, transposed in the rolling process between the blank and tool centrodes. In this paper is proposed an algorithm for curling surfaces in enveloping, associated with a pair of rolling circular centrodes. The graphical algorithm is based on the representation of the circles family enveloped the blank’s profile. It is generated the circles family transposed on the centrode associated with the gear shaped cutter and is determined a new position of contact points with the blank. The assembly of these points forms the profile of the gear shaped cutter. The numerical data proof the proposed method quality.

Cold rolling is an advanced splinterless processing method, which has been applied to process many parts, like spur gears, helical gears, spline shaft and ballscrew, etc. Ballscrew which determines the motion characteristics has been used as key transmission part of ballscrew pair. According to differential geometry and meshing theory, the mathematic model of ballscrew manufactured by cold rolling is built, and the normal section equation of ballscrew is derivated in this paper. The cold forming principle of ballscrew is investigated; the meshing model, the contact line and theoretical errors of the ballscrew are constructed. The objective moving laws between roller and workpiece are revealed by analyzing the meshing model and the contact line, the processing method of cold-rolling is considered as the generating method which can lay the foundation for roller design by analyzing forming principle. By analyzing the profile error and the error of the actual helix of the centrode of ball, the error generation causes are discovered, and the calculation methods of the profile error and the spiral error of ballscrew are proposed. By analyzing the error evolution and taking a certain type of ballscrew for example, the results indicate that the profile error is brought by roller designing, and the spiral error is caused by discontinuous contact.

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.

We propose unification of various geometric design approaches, available in the open literatures and published research papers, for profile of star rotor used in rotary piston machines type hydrostatic units, namely pumps and motors. The star rotor profile has cycloidal root. We consider three basic approaches. These approaches are otherwise self-sufficient for profile selection purpose although the end formulae are different in relation to the common parametric terms. A closer scrutiny reveals that the initial selection of co-ordinate systems, profile generating centrode pairs and the location of the generating point lead to such differences. Our unification provides infusion and integration of three different paradigms that we have considered. This investigation is aimed at helping researchers develop a better understanding design approaches of such machines.

[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 coupler motion of a planar four-bar mechanism can be reproduced by rotating the mechanism's moving centrode over its fixed centrode. For spatial four-bar mechanisms, coupler motion can be replicated by moving the mechanism's moving axode over its fixed axode in a screw motion – a combination of simultaneous rotations and translations. This study presents a model to calculate the fixed and moving axodes of the revolute–revolute–spherical–spherical (RRSS) mechanism – one of the most basic spatial four-bar mechanisms. The axodes are useful in producing the contact surfaces for a cam system to replicate RRSS coupler motion.

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.

UNLABELLED Theoretical considerations help define the requirements for an apparatus that is to localize the instantaneous helical axes (IHA) of axial rotations of lumbar segments. RESULT Since the range of axial rotation of an L3/4 segment is in only approximately +/- 1.5 degrees the rotational angle intervals have to be smaller than 0.3 degree with an resolution less than 0.03 degree in order to be able to determine the loci of the IHAs. For the first time in vitro measurements are presented that satisfy this requirement. The data prove that the guidance by the artt. zygapophysiales critically influence the possible positions of the IHA. Comparatively, ligaments and intervertebral disk play a marginally role. During axial torques Tz the IHAs lie dorsal to the intervertebral disk and migrate from one joint to the other depending on axial rotation (length of migration: approximately 3-4 cm). The IHAs lie almost parallel to the axial torque vector. When the joints are removed the IHA is stationary and almost perpendicular to the intervertebral disk and intersects the disk's central region. The screw inclination (pitch) of the instantaneous screw movement is proportional to the rotational angle. Therefore, depending on the direction of rotation, one obtains left or right handed screw movements. This means: axial torsional load leads to an increase in thickness of the intervertebral disk. During preloads that produces extensions the fixed centrodes (paths of axis migration) of intact segments are dorsally beaten out, whereas during flexional loads they are ventrally beaten out. Then, the IHAs migrate through the canalis vertebralis. By the concept "dimeric link chain" the different shapes of the fixed centrode are traced back to the morphology of curvature of the articulating surfaces. The measurements suggest the hypothesis that the distinct nonlinearity of the load displacement curves (s-shape of alpha = alpha (Tz) funktion is an affection of IHA migration. Comparatively, the influence of ligaments can be neglected. The measurements suggest the hypothesis that the hysteresis of the load displacement curves (neutral zone) is an artefact that does not appear in vivo. Altogether, the experiments prove that the loci of the IHAs are determined by the interplay of preload, structure of the applied force system and morphology of curvature of the articulating surfaces. By that the possibility is clinically given to calculate the multitude of possible movements as function of muscle activity when in the individual the shape and the position of the articulating surfaces are measured in vivo (by NMR-methods e.g.). A physically based classification of pathological cases seems to be possible.

We review the concept of instantaneous center of rotation and its loci, the moving centrode and the fixed centrode, for the plane motion of an arbitrarily shaped rigid body. Then we recall a simple example from the literature. Finally, we present two new solutions pertaining to the motion of an elliptical lamina on a fixed plane, and of a stick moving on an oblique corner.

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.

A detailed analysis of the path traversed by the rear wheels of the bus is useful in highway design, and in the placement of curbs at intersections. In fact, when a bus turns a corner, the rear wheel could move into the path of a passing motorist or pedestrian by generating an hazard situation, as in the cases of traffic circles and changing lanes. Of course, the same problems regard long trucks, semi-trailer trucks and articulated vehicles, but also bicycles (tandem) and motorbikes. In particular, the path of the rear wheels of a bus or a long vehicle in general, is a tractrix or equi-tangential curve, which is different by that traced by the front wheels. The planar motion of the chassis of the vehicle can be represented by means of the centrodes, which are traced by the instant center of rotation on the ground (fixed centrode) and on the moving plane that is attached to the vehicle (moving centrode), respectively. Thus, the pure-rolling motion of the moving centrode that is represented by the axis of the rear wheel, on the fixed centrode (catenary) can reproduce the planar motion of the chassis. This paper deals with the formulation of a suitable algorithm for the kinematic analysis of the planar motion of vehicles that travel corners, traffic circles and changing lanes. The tractrix is considered as the involute of the fixed centrode (catenary), which takes also the rule of evolute. The inflection and the cuspidal or return circles, along with the Euler-Savary equation and the Aronhold Theorems, give other useful information on the planar motion of the chassis.

STUDY DESIGN: Human cadaveric cervical spines were subjected to nondestructive repeated-measures flexibility tests before, during, and after unilateral facet dislocation. OBJECTIVES: To assess the relative stability and kinematics of the spine in the normal condition, after unilateral facet lock without bony injury and after reducing (unlocking) the facet dislocation. SUMMARY OF BACKGROUND DATA: Experimental unilateral facet dislocation has been studied only when associated with extensive damage to disc, bone, or both. The relative stability of the normal, locked, and unlocked cervical spine has not been studied. METHODS: Unilateral facet dislocation with minimal associated injury was created by slowly applying axial torque while specimens were bent laterally and flexed. Before and after injury, nondestructive torques were applied about each anatomic axis, while three-dimensional specimen motion was recorded stereophotogrammetrically. RESULTS: Unilateral facet dislocation was created with a mean axial torque of 10.2 +/- 2.7 Nm. After locking, range of motion, lax zone, and stiff zone were significantly reduced relative to normal (P < 0.05). After unlocking, the same parameters were significantly increased relative to normal. The position of the axis of rotation shifted when the facet was locked and the centrode elongated after the facet was unlocked. CONCLUSIONS: Unilateral facet dislocation without fracture can be created with moderate loads when axial torque is applied while the facet is distracted. The spine is stable while the articular masses are locked unilaterally; however, the motion segment becomes overtly unstable after the facet dislocation is reduced.

Despite the Bennett loop's modest appearance as merely a skew, 4-hinge, 4-bar chain and the great attention which has been accorded it over almost a century, the axodes of its motion have not previously been delineated. That task is carried out here, and there is also established the relationship between the ruled surface and the corresponding centrode of the linkage's planar form. The occasion is taken to determine as well the fixed axode of the Schatz loop's central coupler.

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.

The kinematics of the distal radioulnar joint were examined using the method of centrode analysis in vivo. A group of normal subjects was studied along with a group of rheumatoid arthritis patients who had distal radioulnar joint involvement. Serial computed tomographic scans were obtained through the distal radioulnar joint of nine subjects (10 wrists) in varying degrees of pronation and supination. For normal subjects, the movement that occurs in a stable distal radioulnar joint is not erratic and the center of rotation lies within a well-defined area. When subjects with rheumatoid arthritis were analyzed, it was determined that early in the disease process bone erosions may occur in the sigmoid notch of the distal radius. When the joint contour in this region is maintained, the kinematics are not markedly altered. Erosion involving the dorsal border of the sigmoid notch of the radius is associated with subluxation of the distal radioulnar joint. The ulna becomes dorsally positioned relative to the radius, and significant alteration in the kinematics of the distal radioulnar joint occurs.

Based on a differential geometry approach, the basic equations for the coupler curve of the four-bar linkage are concisely derived. The geometric properties of coupler curves for the crank-rocker linkage are analyzed, and the distribution law of various shapes of coupler curves is revealed. The moving centrode, Ball's curve and the self-tangent curve divide the coupler plane into several areas. The points in each of these areas trace a specific shape of curves. Therefore, any shape of coupler curves with a cusp, inflection point, Ball's point, crunode, tacnode or oval shape, etc. for the first time, could be readily located.

The system of the anterior (a) and posterior (p) cruciate ligaments and their distances between attachments to the femur (f) and tibia (t), respectively, as found in the knee joint of tetrapods is considered as a crossed four-bar linkage. Starting from a flat or curved tibial articulating surface the shape of the femoral articulating surfaces in principle can be derived (Huson, 1974; Menschik, 1974). The point of intersection (S) of the cruciate ligaments is the instantaneous centre of rotation and describes a curve during knee angulation; the centrode. Two centrodes are distinguished: S1 with the tibia stationary and S2 with the femur stationary. During leg bending the centrodes roll over each other, without sliding. Four series of knee-joint simulations were made. In series A, the femoral bar (distance between cruciate ligaments) was varied, in series B , the anterior cruciate ligament (so the length of a ligament). In series C and D, combinations of bar lengths were varied. The maximum leg rotation range was calculated from the lengths of the bars of the cruciate ligament system for each of the series A … D . Also the leg rotation range for which the knee joint is mechanically stabilized by the cruciate ligaments was calculated. The stabilization criterium chosen was that a cruciate ligament may not become perpendicular (>78·5°) to the articulating surfaces. It was found that the cruciate ligaments alone cannot stabilize the knee joint adequately over the whole required range of leg movement. External structures are required to obtain full stabilization. As the femoral and tibial articulating surfaces never lie at the centrodes, considerable sliding occurs between them. It is suggested that a uniformly distributed sliding is essential for lubrication and so for a proper knee-joint functioning.

It has been proposed that the restraining structures of an individual lumbar segment may become weakened during the early stages of degeneration. The process results in a loss of stiffness in the restraints, compromising their constraining and restoring forces. The effected level would offer less resistance to an applied load and it may displace through excessive range or display abnormal movement patterns. Erratic centrode patterns have been found in cadaveric segments with early disc degeneration. Abnormal ratios between segment translation and rotation have been detected in a group of ten patients with degenerative disc disease. Early attempts to detect abnormal motion radiologically have proved unreliable. Two new techniques are being researched. Some clinicians consider that an unstable lumbar segment can be detected by perceiving abnormal compliance during manual passive accessory tests. These perceptions have been defined biomechanically in terms of an increased neutral zone in the segments...

This paper presents a novel modeling framework for contact detection and estimation of contact location on multi-backbone continuum robots. The paper presents modified kinematics for constrained continuum robots and introduces the concept of Fixed Centrode Deviation (FCD) for continuum robots. It is shown that the change in fixed centrode locus may be used for detection of contact and for contact location estimation. An alternative method for contact detection using Joint Force Deviation (JFD) is investigated and a lower bound for contact detectability is derived while considering uncertainties in joint forces. An estimation framework for the location of contact is presented based on FCD and the constrained kinematics Jacobian of the continuum robot. These methods are validated by simulation and experiments. This framework may be used for enhancing the safety of robotic surgical slaves and for exploration of unknown environments such as in scenarios of search and rescue.

For a planar motion of a body, there exists an instantaneous center of zero velocity or what is known as the centrode . In three-dimensions, the same is represented by the instantaneous screw axis. In two dimensions, however, there is a geometric method of construction the instant center by using the velocity vectors of two points of the body. The instant center is the point of intersection of the lines perpendicular to the two velocity vectors. This type of construction, however, did not exist for the instantaneous screw axis. In this paper, we present such a geometric construction for the instantaneous screw axis using line geometry.Copyright © 2007 by ASME

Total disc replacement is a novel approach for dynamically stabilizing a painful intervertebral segment. While this approach is gaining popularity, and several types of implants are used, the effect of disc arthroplasty on lumbar biomechanics has not been widely reported. Consequently, beneficial or adverse effects of this procedure may not be fully realized, and data for kinematic optimization are unavailable.To characterize kinematic and load transfer modifications at L5/S1 secondary to joint replacement.A human cadaveric biomechanical study in which the facet forces and instant axes of rotation (IAR) were measured for different spinal positions under simulated weightbearing conditions before and after total disc replacement at L5/S1 using semiconstrained (3 degrees of freedom [DOF]; Prodisc) and unconstrained (5 DOF; Charité) articulated implants.Twelve radiographically normal human cadaveric L5/S1 joints (age range 45-64 years) were tested before and after disc replacement using Prodisc II implants (Spine Solutions, Paoli, PA) in six specimens and SB Charité III (JohnsonJohnson, New Brunswick, NJ) in six other specimens. Semiconstrained fixtures in combination with a servo-hydraulic materials testing system subjected the test specimens to a physiologic combination of compression and anterior shear. Multiple intervertebral positions were studied and included up to 6 degrees of flexion, extension, and lateral bending. The IAR was calculated for every 3-degree intervals, and the force through the facet joints was simultaneously measured using flexible intra-articular sensors. Data were analyzed using repeated-measures analysis of variance.During flexion/extension, the average IAR positions and directions were not significantly modified by implantation with the exception that the IAR was higher relative to S1 end plate with the Charité (p=.028). The IAR had a vertically oriented centrode throughout flexion/extension with the Prodisc (p.001) and the Charité (p.016). The centrode tended to be greater with the Prodisc. There was a trend that the facet force was decreased throughout flexion/extension for the Prodisc; however, this was statistically significant only at 6 degrees extension (27%, p=.013). In lateral bending, the IAR was significantly modified by Prodisc replacement, with a decreased inclination relative to S1 end plate, (ie, increased coupled axial rotation). While the IAR moved in the horizontal plane toward the side of bending, this effect was more pronounced with the Prodisc. The ipsilateral facet force was significantly increased in 6 degrees lateral bending with the Charité (85%; p=.001).The degree of constraint affects post-implantation kinematics and load transfer. With the Prodisc (3 DOF), the facets were partially unloaded, though the IAR did not match the fixed geometrical center of the UHMWPE. The latter observation suggests joint surface incongruence is developed during movement. With the Charité (5 DOF), the IAR was less variable, yet the facet forces tended to increase, particularly during lateral bending. These results highlight the important role the facets play in guiding movement, and that implant constraint influences facet/implant synergy. The long-term consequences of the differing kinematics on clinically important outcomes such as wear and facet arthritis have yet to be determined.

The use of the instant center as a reference point in angular momentum theorems has the advantage that unknown contact forces do not appear in the equations of motion. However, care must be taken to distinguish the point as being on the body or the space centrode.