JOINTS, BIOMECHANICS OF George (Yiorgos) Papaioannou, Ph.D. and *Yener N. Yeni, Ph.D.

Transcription

1 JOINTS, BIOMECHANICS OF George (Yiorgos) Papaioannou, Ph.D. and *Yener N. Yeni, Ph.D.

2 JOINTS, BIOMECHANICS OF Yiorgos Papaioannou, Ph.D. and *Yener N. Yeni, Ph.D. George Papaioannou Ph.D Assistant Professor Director of Advanced Biomechanics Laboratory Department of Biomedical Engineering School of Engineering The Catholic University of America Rm 123 Pangborn 621 Michigan Ave NE Washington DC Phone: (202) Fax: (202) E_mail: Yener N. Yeni, Ph.D. Head, Section of Biomechanics Bone and Joint Center Henry Ford Hospital 2799 West Grand Boulevard Detroit, Michigan, USA *For correspondence: Yener N. Yeni, Ph.D. Head, Section of Biomechanics Bone and Joint Center Henry Ford Hospital 2799 West Grand Boulevard, E&R 2015 Detroit, Michigan, USA Phone: (313) Fax: (313) E_mail:

3 ABSTRACT Bones in the skeleton join to form segments or links which provide for the attachment of muscles, ligaments, tendons etc. Altogether, these systems of bones and soft tissues, called joints, produce movement for the organism. Joint biomechanics traditionally is a division of biomechanics that studies the effect of forces on the joints of living organisms. Like all other organs, joints are subject to trauma and disease. Development of treatments, devices and exercise regimes for healthy joints, as well as learning from the great variety of naturally existing joints for the purpose of nature-inspired technology, requires a fundamental understanding of the mechanics of joints and joint tissues, and their interaction with the underlying biological mechanisms. At the current state of the art, the definition of joint biomechanics can be broadened to encompass these complex interactions and the relevant technology. It is the purpose of this chapter to introduce a current understanding of joint biomechanics with an emphasis on the structure, kinematics, kinetics, modeling and joint stability of human joints. A detailed consideration of the mechanics of artifical joints and that of the tissues forming a joint is left for other chapters. Keywords: Joints, biomechanics, joint mechanics, structure, kinematics, kinetics, modeling, joint stability, motion, joint health, experimental analysis, human joints, joint tissues. CONTENTS 1. Introduction 1.1 Articular anatomy, joint types and their function 1.2 Articular cartilage 1.3 Effects of motion and external loading on joints 2. Kinematics of joints 2.1 General comments 2.2 Characterization of the generic mechanical joint system- Terminology and definitions 2.3 Degrees of freedom 2.4 Planar motion 2.5 Instantaneous Center of Rotation-ICR 2.6 Analytical methods Data collection Data Analysis Coordinate Systems and Transformation Translation in three-dimensional space Rotations about the coordinate Axes Combined rotations as a result of a sequence of rotations Euler and Bryant-Cardan angles Parameters of the motion of a body observed in a laboratory coordinate system 3. Kinetics of joints 3.1 Equations of motion 3.2 Motion and forces on diarthroidal joints 4. Mathematical and mechanical models of joints 4.1 Assessment of mechanical factors associated with joint degeneration-limitations and future work

4 4.2 From experimental to advanced theoretical analysis in joint mechanics 4.3 Theoretical analysis of joint mechanics 4.4 Surface modeling 4.5 The joint distribution problem Phenomenological joint models Anatomical models The reduction method The optimization method The forward analysis Finite element analysis (FEA) of human joints Towards Patient-Specific and Task dependent Morphological FE models 5. Joint Stability 5.1 The hip joint 5.2 The knee joint 5.3 The foot structure; the ankle joint 5.4 The spine 5.5 The shoulder 5.6 The elbow 5.7 The wrist 6. Overview 7. References FIGURES Figure 1. Basic structure and components of a synovial joint (also called diarthroses). Figure 2. Zones of articular cartilage (from F. Nelson (1)) Figure 3. Points S and S as well as Q and Q lie on the arcs of circles around the center of rotation ICR (used synonymously with CR after section 2.5). If lines SS and QQ are bisected perpendicularly, the center of rotation CR is located at the intersection of these perpendicular bisectors. This construction assumes that the perpendicular bisectors are differently orientated but a special case arises if the bisectors are identically oriented. Then the points S, Q and the center of rotation ICR lie on a straight line. Figure 4. Changing the coordinate systems, transformation of point coordinates from one coordinate system to another. Figure 5. A rigid body (shoebox) moves parallel to itself. The radius vectors from O to P and from O to P are designated by r and r so that r = r + t where t is the difference vector. Figure 6. Rotation about the z-axis of the coordinate system. Figure 7. General rotation composed of three partial rotations. The first rotation according to the Bryant-Cardan convention (above). The first of the general rotations using Euler as the selection of the axes and angles of rotation (below). Figure 8. Motion of a rigid body: minimum configuration of three reference points in order to determine the parameters describing the spatial motion. Figure 9. Interpretation of the motion as helical motion. Motion of a rigid body and interpretation of the general motion as helical motion (Chasles Theorem). The necessary number of reference points is fixed on a rigid body (not shown here). From the spatial location of the reference points in the initial and final state, the locations of the geometric centers S(r s ) and

5 S (r s), as well as the rotation matrix D are determined. The change in location of the respective geometric centers is described by t s = r s - r s. The axis of rotation n and the angle of rotation ϕ can be determined from the rotation matrix. The translation t p in direction of the helical axis is defined by projection from n and t s. The location of the axis of rotation relative to the initial and final position of the object (segment) is set by points A and M. M is the midpoint of the line SS. The vector f directed from M to A is perpendicular to n and t s. The radius R and the radius vector r A can be calculated by means of the unit vectors e, n, and f (adopted from Brinckmann et al., 2000)(2). Figure 10. Relationship between the free body diagram and the link-segment model. Each segment is broken at the joints, and the reaction forces and moments of force acting at each joint are shown (adapted from Winter, 1988)(3). Figure 11. Example applications showing dynamic in vivo tibio-femoral bone surface motion during human one-legged hopping (from Anderst et al. (4)). Figure 12. Articular surface matching (femoral condyle on top and tibial plateau below) using geometrical objects (Papaioannou, 2004) (5). Figure 13. All the sagittal view measured parameters in the study of femoral head congruity (5). Figure 14 a, b, c Forward and inverse analysis driven mathematical models, a) the Strathclyde model animated using SIMM Musculographics-Motion Analysis Software from Jonkers et al., 2002 (6) and Papaioannou et al., 2000 (7) b) Force distribution method-inverse dynamics, the vector of bony contact force is shown on the tibial plateau, from Papaioannou et al., 2000 (7, 8) c) the Delft forward and Inverse shoulder model (9). Figure 15: a) Knee FE Model components shown in different color-implicit formulation (10) b) the knee FE model during simulation of single leg-hopping-explicit formulation (11). Figure 16: Tibial plateau pressure at 10 and 30 degrees flexion during single legged hopping (11). Figure 17. The axode describing the knee joint motion (from Mow et al., 2000 (12)). Figure 18. The VICON motion analysis skin marker placement on the foot and ankle joint ( and the Wayne State University Ankle joint model (10, 13)). Figure 19. The Wayne State University full body impact model and the spine model (14). TABLES Table 1. Synarthroses Fibrous Joints Table 2. Amphiarthroses Cartilaginous Joints Table 3. Diarthroses Synovial Joints Table 4. Characteristics of major human joints Table 5. Femoral medial and lateral compartment measurements and ratios

6 1. Introduction The human skeleton is a system of bones joined together to form segments or links. These links are movable and provide for the attachment of muscles, ligaments, tendons etc. to produce movement. The junction of two or more bones is called an articulation. There are a great variety of joints even within the human body and a multitude of types among living organisms that use exo- and endoskeletons to propel. Articulation can be classified according to function, position, structure and degrees of freedom for movement they allow etc. Joint biomechanics is a division of biomechanics that studies the effect of forces on the joints of living organisms Articular anatomy, joint types and their function Anatomic and structural classification of joints typically results in three major categories, according to the predominant tissue or design supporting the articular elements together, that is, joints are called fibrous, cartilaginous, or synovial.

7 Figure 1. Basic structure and components of a synovial joint (also called diarthroses). Synovial joints are cavitated. In general two rigid skeletal segments are brought together by a capsule of connective tissue and several other specialized tissues, that form a cavity. The joints of the lower and upper limbs are mainly synovial since these are the most mobile joints. Mobility varies considerably and a number of subcategories are defined based on the specific shape or architecture and topology of the surfaces involved (e.g. planar, saddle, ball and socket) and on the types of movement permitted (e.g. flexion and extension, medial and lateral rotation) (table 1). The basic structural characteristics that define a synovial joint can be summarized in four features: a fibrous capsule that forms the joint cavity, a specialized articular cartilage covering the articular surfaces, a synovial membrane lining the inner surface of the capsule which also secretes a special lubricating fluid, the synovial fluid. Additional supportive structures in synovial joints include discs, menisci, labra, fat pads, tendons and ligaments. Cartilaginous joints are also solid and are more commonly known as synchondroses and symphyses, a classification based on the structural type of cartilage that intervenes between the articulating parts (table 2). This cartilage is hyaline and fibrocartilage for synchondroses and symphyses, respectively. Synchondroses allow very little movement as in the case of the rib cage that contributes to the ability of this area to expand with respiration. Most symphyses are permanent; those of sacrum and coccyx can however, degenerate with subsequent fusion between adjacent vertebral bodies as part of the normal development of these bones. Fibrous joints are solid. The binding mechanism that dominates the connectivity of the articulating elements is principally fibrous connecting tissue, although other tissue types also may be present. Length, specific arrangement and fiber density vary considerably according to the location of the joint and its functional requirements. Fibrous joints are classified in three groups: sutures, gomphoses, and syndesmoses (table 3);

8 Table 1. Diarthroses Synovial Joints Ball and socket Other names: Spheroidal; endarthroses Description: Ball-shaped head fits into concave socket Movement: Widest range of all joints; triaxial Example: Shoulder and hip joints Hinge Other name: Ginglymus Description: Spool-shaped head fits into concave surface Movement: In one plane about single axis (uniaxial); like hinged-door movement (namely, flexion and extension) Examples: Elbow, knee, ankle, and interphalangeal joints

9 Pivot Other name: Trochoid Description: Arch-shaped surface rotates about rounded or peglike pivot Movements: Rotation: uniaxial Example: Between axis and atlas; between radius and ulna Ellipsoidal Other names: Condyloid, ovoid Description: Arch-shaped condyle fits into elliptical cavity Movements: In two planes at right angles to each other specifically, flexion, extension, abduction, and adduction; biaxial Example: Between radius and carpals

10 Saddle Other name: Reciprocal Description: Saddle-shaped bone fits into socket that is concave-convex in opposite direction; modification of condyloid joint Movements: Same kinds of movement as condyloid joint but freer; like rider in saddle; biaxial Example: Thumb, between first metacarpal and trapezium Gliding Other name: Arthroidal Description: Articulating surfaces; usually flat Movement: Gliding, a nonaxial movement Example: Between carpal bones; between sacrum and ilium (sacroiliac joints) Table 2. Amphiarthroses Cartilaginous Joints

11 Cartilaginous Other name: Synchondroses Description: The joint formed by each of the costal cartilage Movement: Bending and twisting, or slight compression Example: Between the ribs and the sternum; between carpal and tarsal bones Fibrocartilaginous Other name: Symphyses Description: Within the joint, separating the bones, is a fibrocartilaginous pad these pads, or discs, serve as shock absorbers Movement: Compression, flexion, extension, and rotation Example: Intervertebral and pubic joints Table 3. Synarthroses Fibrous Joints Sutures Other name: Description: The edges of the bones have interdigitations or grooves that fit very closely and firmly together; the connecting fibers are very short

12 Movement: None Examples: Between the flat bones of the skull Syndesmoses Other name: Ligamentous Description: Two bones, that may be widely separated tied together by ligaments; the ligaments may be in the form of cords, bands, or flat sheets Movement: None (some give) Example: Between the distal ends of the tibia and fibula Gomphosis Other name: Description: A joint in which the surfaces of bony components are adapted to each other like a peg in a hole Movement: None Example: The conical process of a tooth is inserted in the bony socket of the mandible or maxilla In addition to the obligatory components that all the synovial joints possess, several joints contain intra-articular structures. Discs and menisci are examples of such structures. They differ from one another mainly in that a disc is s circular structure that may completely subdivide a joint cavity so that it is, in reality, two joints in series whereas a meniscus is usually a crescentshaped structure that only partially subdivides the joint. Complete discs are found in the sternoclavicular joint and in the radiocarpal joint. A variety of functions have been proposed for intra-articular discs and menisci. They are normally met at locations where bone congruity is poor, and one of their main functions is to improve congruity and, therefore stability between articular surfaces. Shock absorption facilitation and combination of movements are among their likely roles. They may limit a movement or distribute the weight over a larger surface or facilitate synovial fluid circulation throughout the joint. The labrum is another intra-articular structure. In humans, this structure is only found in the glenohumeral and hip joints. They are circumferential structures attached to the rim of the glenoid and acetabular sockets. Labra are distinct from articular cartilage because they consist of fibrocartilage and they are triangular in their middle-section. Their bases are attached to the articular margins and their free apical surfaces lined by synovial membrane. Like discs, their main function is to improve fit and protect the articular margins during extremes of movement. Fat pads are localized accumulations of fat that are found around several synovial joints, although only those in the hip (acetabular pad) and the knee joint (infrapatellar pad) are named. Suggested functions for fat pads include protection of other intra-articular structures (e.g. the round ligament of the head of the femur) and serving as cushions or space-fillers thus facilitating more efficient movement throughout the entire available range. Bursae are enclosed, self-contained, flattened sacs typically with a synovial lining. They facilitate movement of musculoskeletal tissues over one another and thus are located between pairs of structures (e.g. between ligament and tendon, two ligaments, two tendons or skin and bone). Deep bursae, such as the illiopsoas bursa or the deep retrocalcaneal bursa, develop along with joints and by a similar series of events during the embryonic period. Tendons are located at the ends of many muscles and are the means by which these muscles are attached to bone or other skeletal elements. The primary structural component of tendons is type I collagen. Tendons almost exclusively operate under tensile forces.

13 Ligaments are dense bands of connective tissue that connect skeletal elements to each other, either creating (as in the case of syndesmoses) or supporting joints. According to their location they are classified as intracapsular, capsular or extracapsular. Structurally they resemble the tendons in that they consist predominantly of type I collagen. 1.2 Articular cartilage Articular cartilage, the resilient load-bearing tissue that forms the articulating surfaces of synovial joints functions through load distribution mechanism by increasing the area of contact (thereby reducing the stress) and provides these surfaces with the low friction, lubrication, and wear characteristics required for repetitive gliding motion. Biomechanically cartilage is another intra-articular absorption mechanism that dampens mechanical shocks and spreads the applied load onto subchondral bone (Figure 2). Articular cartilage should be viewed as a multiphasic material. It consists primarily of a large extracellular matrix (ECM) with a sparse population of highly specialized cells (chondrocytes) distributed throughout the tissue. The primary components of the ECM are water, proteoglycans, and collagens, with other proteins and glycoproteins present in lower amounts (15). The solid phase is comprised by this porous-permeable collagen-pg matrix filed with freely movable interstitial fluid (fluid phase) (16). A third phase is the ion phase, necessary to describe the electromechanical behaviors of the system. The structure and composition of the articular cartilage vary throughout its depth (Figure 2), from the articular surface to the subchondral bone. These differences include cell shape and volume, collagen fibril diameter and orientation, proteoglycan concentration, and water content. These all combine to provide the tissue with its unique and complex structure and mechanical properties. A fine mechanism of interstitial fluid pressurization results from the flow of interstitial fluid through the porous-permeable solid matrix which in turn defines the rate dependent load bearing response of the material. It is noteworthy that articular cartilage provides its essential biomechanical functions for 8 decades or more in most of the human synovial joints and no synthetic material performs this well as a joint surface.

14 Zones I, II,II and IV tidemark Subchondral bone Figure 2. Zones of articular cartilage (from F. Nelson (1)) The frictional characteristics between two surfaces sliding over each other are significantly influenced by the topography of the given surfaces. Anatomical shape changes affect the way in which loads are transmitted across joints, altering the lubrication mode in that joint and, thus, the physiologic state of cartilage. Articular surfaces are relatively rough, compared to machined bearing surfaces, at the microscopic level. The natural surfaces are surprisingly much rougher than joint replacement prostheses. The mean of the surface roughness for articular cartilage ranges from 1 to 6 µm, while the metal femoral head of a typical artificial hip has a value of approximately µm indicating that the femoral head is apparently much smoother. Topographic features on the joint surfaces are characterized normally by primary anatomic contours, secondary roughness (less than 0.5 mm in diameter and less than 50 µm deep), tertiary hollows on the order of 20 to 45 µm deep; and, finally, quaternary ridges 1 to 4 µm in diameter and 0.1 to 0.3 µm deep. Scanning electron micrographs of arthritic cartilage usually depict a large degree of surface irregularity and anomalous micro-topography. These surface irregularities have profound effects on the lubrication mechanism. They accelerate the effects of friction and the rate of degradation of the articular cartilage. The types of joint surface interactions vary greatly between different joints in the body, different animals, between different size animals of the same species, different genders and different ages. For example, the human hip joint is a deep congruent ball and socket joint (where the cartilage thickens peripherally at the acetabulum); this differs greatly from the bicondylar nature of the distal femur in the knee joint, and the saddle shape of the carpometacarpal joint in the thumb. The degree of shape matching between the various bones and articulating cartilage surfaces composing a joint is a major factor affecting the distribution of stresses in the cartilage and subchondral bone. 1.3 Effects of motion and external loading on joints The articular joint is viewed as an organ with complicated mechanisms of memory and adaptation that accommodates changes in its function. Joint loading results in motion and the

15 couple load/motion is required to maintain normal adult articular cartilage composition, structure, and mechanical properties. The type, intensity, and frequency of loading necessary to maintain normal articular cartilage vary over a broad range. The intensity or frequency of loading should not exceed or fall below these necessary levels, since this will disturb the balance between the processes of synthesis and degradation. Changes in the composition and microstructure of cartilage will result. Reduced joint loading, as has been observed in cases of immobilization by casting or external fixation, leads to atrophy or degeneration of the cartilage. The changes affect both the contact and noncontact areas. Changes in the noncontact areas resulting from rigid immobilization include fibrillation, decreased proteoglycan content and synthesis, and altered proteoglycan conformation, such as a decrease in the size of aggregates and amount of aggregate. Normal nutritive transport to cartilage from the synovial fluid by means of diffusion and convection has been diminished, resulting in these changes. Increased joint loading, either through excessive use, increased magnitudes of loading, or impact, also may affect articular cartilage. Catabolic effects can be induced by a single-impact or repetitive trauma, and may serve as the initiating factor for progressive degenerative changes. Osteoarthritis, a joint disease of epidemic proportions in the western world, is characterized by erosive cartilage lesions, cartilage loss and destruction, subchondral bone sclerosis and cysts, and large osteophyte formation at the margins of the joint (17). Moderate running exercise may increase the articular cartilage proteoglycan content and compressive stiffness, decrease the rate of fluid flux during loading, and increase the articular cartilage thickness in skeletally immature animals. However, no significant changes in articular cartilage mechanical properties were observed in dogs in response to lifelong increased activity that did not include high impact or torsional loading of their joints. Disruption of the intraarticular structures, such as menisci or ligaments, will alter the forces acting on the articular surface in both magnitude and areas of loading. The resulting joint instability is associated with profound and progressive changes in the biochemical composition and mechanical properties of articular cartilage. In experimental animal models, responses to transection of the anterior cruciate ligament or meniscectomy have included fibrillation of the cartilage surface, increased hydration, changes in the proteoglycan content, reduced number and size of proteoglycan aggregates, joint capsule thickening, and osteophyte formation. It seems likely that some of these changes result from the activities of the chondrocytes, because their rates of synthesis of matrix components, breakdown of matrix components, and secretion of proteolytic enzymes are all increased. In vitro studies have shown that loading of the cartilage matrix can cause all of these mechanical, electric, and physicochemical events, but thus far it has not been clearly demonstrated which signals are most important in stimulating the anabolic and catabolic activity of the chondrocytes. A holistic physicochemical and biomechanical model of cartilage function in health and disease remains a challenge in the scientific community. 2. Kinematics of joints 2.1 General comments: Mechanical analysis can refer to kinetics (forces) and/or kinematics (movement), with kinetics being the cause and kinematics the result. Mechanical analysis can develop models proceeding from forces to movements or vice versa. The analysis that starts from the cause (force) is called direct or forward dynamics and produces a defined set of forces that caused the unique movement. This approach has one solution and hence is deterministic. Starting from the

16 movement the analysis is called inverse dynamics. In this case an infinite number of combinations of individual forces acting on the system can be the causes of the same unique movement, which makes the inverse dynamics approach not deterministic. The simplest and most essential system of mechanical formulations for explaining and describing motion is the Newton s second law. More advanced techniques include the Lagrange, d Alembert and Hamilton s methods. In general all these methods start by describing equations of motion for a rigid body for translation, rotation or combinations of them for both two and three-dimensional space. If the model assumes that the articulated segments that create the articulation are modeled as rigid bodies the remaining task is to calculate the relative motion between the two segments by applying graphics or joint kinematic analysis. Kinematics is the study of the movements of rigid structures, independent of the forces that might be involved. Two types of movement, translation (linear displacement) and rotation (angular displacement), occur within 3 orthogonal planes; that is, movement has 6 degrees of freedom. Humans belong to the vertebrate portion of the phylum chordata, and as such possess a bony endoskeleton that includes a segmented spine and paired extremities. Each extremity is composed of articulated skeletal segments linked together by connective tissue elements and surrounded by skeletal muscle. Motion between skeletal segments occurs at joints. Most joint motion is minimally translational and primarily rotational. The deviation from absolute rotatory motion may be noted by the changes in the path of a joint s instantaneous center of rotation. These paths have been measured for most of the joints in the body and vary only slightly from true arcs of rotation. For human motion to be effective, not only must a comparatively rigid segment rotate its position relative to an adjacent segment, but many adjacent limb movements must interact as well. Whether the hand is trying to write or the foot must be lifted high enough to clear an obstacle on the ground, the activity is achieved via coordinated movements of multiple limb segments. To provide for the greatest possible function of an extremity, the proximal joint must have the widest range of motion to position the limb in space. This joint must allow for rotatory motions of large degrees in all 3 planes about all 3 axes. A means is also provided to translate the limb, so that an extremity can function at all locations within its global range. Rotational motion of the elbow and knee joints allows such overall changes as adjacent limb segments move. Finally, to fine-tune the use of this mechanism with respect to the extremities, for their functional purposes, the hand and foot are required to have a vast amount of movement about all 3 axes, although the rigid segments are relatively small. Such movement requires the presence of relatively universal joints at the terminal aspect of each extremity. 2.2 Characterization of the general mechanical joint system- Terminology and definitions The displacement of a point is simply the difference between its position after a motion and its position before that motion. It can be represented by a three-dimensional vector drawn from the initial position of the point to its final position. The components of the displacement vector will be the changes in the coordinates of the point s position from measurement in the reference coordinate system. It is apparent that not only the positions but also displacements measured are relative to some reference. Rigid body (RB) displacements are more complicated than point displacements since for a rigid body a displacement is a change in its position relative to some reference, but more than three parameters are needed to describe it. Two simple types of rigid-

17 body displacement can be described: translation and rotation. An important property of pure translation of a RB is that the displacement vectors of all points in the body are identical and are non-zero. In pure rotation of a RB, although points in the body experience nonzero displacements, one point in that body experiences zero displacement. In addition to that rule Euler s theorem shows that in pure rotation all points along a particular line through that undisplaced point also experience zero displacement. This line is also known as the axis of rotational displacement. Chasles Theorem further states that any displacement of a RB can be accomplished by a translation along a line parallel to the axis of rotation that is defined by Euler s theorem plus a rotation about that same parallel axis. Simply that suggests that any displacement in 3D is equivalent to the motion of a nut, representing the body, on an appropriate stationary screw that was centered on the line described above. Indeed, it can be shown that any displacement in three dimensions (3D) is equivalent to a translation plus a rotation. 2.3 Degrees of freedom The biological organisms capable of propelling themselves through different media consist of more than one rigid body. A system consisting of a 3D reference frame and an isolated rigid body in space has six degrees of freedom (DOF). To describe the position of each body relative to the ground reference frame it would be necessary to use six parameters, so for two unconnected rigid bodies twelve parameters would be necessary. The system consisting of these two unconnected bodies and the fixed ground reference would have twelve degrees of freedom. The human/animal body consists of a combination of suitably connected bodies. The connections-joints between the bodies serve to constrain the motions of the bodies so that they are not free to move with what would otherwise be six degrees of freedom for each body. Therefore, we can define the number of DOF that the joint removes as the number of degrees of constraint that it provides. It can be shown that every time a joint is added to a system, the number of degrees of freedom in that system is reduced by the number of degrees of constraint provided by that joint. This suggests the following generic formula for the calculation of the degrees of freedom of a system: F = 6(L-1)-5J 1 4J 2 3J 3 2J 4 J 5 where F = is the number of degrees of freedom in the system of connected joints L= is the number of joints in the system, including the ground joint (which has no degrees of freedom) and J n = the number of joints having n degrees of freedom each. Table 4 contains a description of the major joints in the human body along with the segmentsbones that they articulate, their respective type DOF and type/range of motion they provide. Table 4. Characteristics of major human joints Joint Bones Type DOF Type of motion Shoulder Humerus-Scapula Diarthrosis (spheroidal) 3 Flexion Extension Abduction Adduction Rotation Elbow Humerus-ulna Diarthrosis (ginglymus) 2 Flexion Extension Rotation Range of Motion (deg) Complete

18 (radius) Radioulnar Superior radius-ulna Diarthrosis (trochoid) 1 Pronation Supination Wrist Radius-carpal Diarthrosis (condyloid) 2 Flexion Extension Radial deviation Ulnar deviation Circumduction Complete Metacarpalphalangeal Metacarpalphalanges Diarthrosis (condyloid) 2 Flexion Extension Radial deviation Ulnar deviation Finger Interphalanges Diarthrosis (ginglymus) 1 Flexion Extension Thumb First metacarpalcarpal Diarthrosis (reciprocal) 2 Flexion Extension Abduction Adduction Circumduction Complete Hip Femur-acetabulum Diarthrosis (spheroidal) 3 Flexion Extension Abduction Adduction Medial rotation Lateral rotation Circumduction Complete Knee Tibia-femur Diarthrosis (ginglymus) 2 Flexion Extension Medial rotation Lateral rotation Ankle Tibia-fibula-talus Diarthrosis (ginglymus) 1 Flexion Extension Intertarsal Tarsals Diarthrosis (arthroidal) 2 Gliding Limited motion Metatarsalphalangeal Metatarsalsphalanges Diarthrosis (condyloid) 2 Flexion Extension Abduction Adduction Limited Interphalangeal Phalanges Diarthrosis (arthroidal) 1 Flexion Extension 90 0 Tibio-fibular Distal tibia-fibula Synarthrosis 0 Slight Give (syndesmosis) movement Skull Cranial Synarthrosis (suture) 0 No movement Sterno-costal Ribs-sternum Amphiarthrosis (synchondrosis) 0 Slight movement Sacroiliac Sacrum-ilium Amphiarthrosis 0 No movement Elastic (synchondrosis) Intervertebral Cervical vertebrae Thoracic vertebrae Diarthrosis (arthroidal) Diarthrosis (arthroidal) 3 3 Flexion Extension Lateral flexion Axial rotation Flexion Extension Lateral flexion Axial rotation

19 Lumbar vertebrae Atlas axis Diarthrosis (arthroidal) Diarthrosis (trochoid) 3 1 Flexion Extension Lateral flexion Axial rotation Pivoting motion Planar motion Some human joints move predominantly in one plane (e.g. the knee joint) in which case the motion can be approximated and analyzed by graphical methods. Here the rotation is characterized by the motion of all points on concentric cycles with an identical angle of rotation around the undisplaced center of rotation (CR). The CR may be located inside and outside of the boundaries of the rotating body. The most common graphical method for the calculation is the so called bisection technique. If the initial and final states of the body are known, the position of the center or rotation and the angle of rotation may be reconstructed (Figure 3). Figure 3. Points S and S as well as Q and Q lie on the arcs of circles around the center of rotation ICR (used synonymously with CR after section 2.5). If lines SS and QQ are bisected perpendicularly, the center of rotation CR is located at the intersection of these perpendicular bisectors. This construction assumes that the perpendicular bisectors are differently orientated but a special case arises if the bisectors are identically oriented. Then the points S, Q and the center of rotation ICR lie on a straight line. 2.5 Instantaneous Center of Rotation-ICR When a 2D body is rotating without translation, for example, a rotating stationary bicycle gear, any marked point P on the body may be observed to move in a circle about a fixed point called the axis of rotation or center of rotation. When a rigid body is both rotating and translating, for example, the motion of the femur during gait, its motion at any instant of time can be described as rotation around a moving center of rotation. The location of this point at any instant, termed the instantaneous center of rotation (ICR), is determined by finding the point which, at that instant, is not translating. Then by definition, at that instant, all points on the rigid body are rotating about the ICR. For practical purposes the ICR is determined by noting the paths traveled

20 by two points, S and Q, on the object in a very short period of time to S and Q. The paths SS and QQ will be perpendicular to lines connecting them to the ICR because they approximate, over short periods, tangents to the circles describing the rotation of the body around the ICR at that instant. Perpendicular bisectors to these two paths will intersect at the instantaneous axis of rotation. If the ICR is considered to be a point on a moving body its path on the fixed body is called a fixed centrode. If the ICR is considered to be a point on a fixed body, its path on the moving body is called the moving centrode. Although in principle two objects may move relative to one another in any combination of rotation and translation, diarthroidal joint surfaces are constrained in their relative motion. The articular surface geometry, the ligamentous restraints, and the action of muscles spanning the joint are the main constraining systems. In general, joint surface separation (or gappingproximal-distal) and impaction are small compared to overall joint motion. Mechanically, when surfaces are adjacent to each other they may move relative to each other in either sliding or rolling contact. In rolling contact, the contacting points on the two surfaces have zero relative velocity, that is, no slip. Rolling and sliding contacts occur together when the relative velocity at the contact point is not zero. The instant center will then lie between the geometric center and the contact point. All diarthroidal joint motion consists of both rolling and sliding motion. In the hip and shoulder, sliding motion predominates over rolling motion. In the knee, both rolling and sliding articulation occur simultaneously. These simple concepts affect the design of total joint prostheses. For example, some total knee replacements have been designed for implantation while preserving the posterior cruciate ligament, which appears to help maintain the normal kinematics of rolling and sliding in the knee. Other knee prostheses substitute for ligament control of kinematics by alterations in articular surface contour through constraining congruity. 2.6 Analytical methods Simple kinematic analysis of pure planar translations and rotation or combinations of the two as well as complicated 3D analysis of a rigid body requires the positional information of a minimum of three noncolinear points to describe this motion uniquely. If the position of three points at two instants is known, the displacement from one position to another may be interpreted as translation, rotation or both. Therefore, the first task is to continually monitor the positions of three points on each rigid body. This analysis is conveniently divided into data collection and data analysis Data collection A constant challenge for the experimental motion analyst is the collection of accurate spatial displacement kinematics of a joint. Several methods have been employed. A review is presented here. Video and digital optical motion capture (tracking) systems offer state-of-the-art, high resolution, accurate motion capture options to acquire, analyze and display three dimensional motion data. The systems are integrated with analog data acquisition systems to enable simultaneous acquisition (1-300 Hz) of force plate and electromyographic data. Clinically validated software

21 analysis packages are used to analyze and display kinematic, kinetic and electromyographic data in forms that are easy to interpret. The major components of a video and digital motion capture system are the cameras, the controlling hardware modules, the software to analyze and present the data, and the host computer to run the software. These systems are designed to be flexible, expandable (from 3 cameras to up to 200 cameras in motion analysis tracking for Hollywood animation movies) and easy to integrate into any working environment. This system collects and processes coordinate data in the least amount of time and requires minimal operator intervention. This system uses motion capture cameras to rapidly acquire three-dimensional coordinate positions from reflective markers placed on subjects. Illuminating strobes with differing wavelengths are used to track the spatial displacement (between 1mm to 10mm resolution) of spherical reflective markers attached to the subject s skin at appropriately chosen locations, preferably on bony landmarks on the human body to minimize skin movement. They can be infrared, visible red or near-infrared strobes to fit the lighting conditions of the capture environment. Also the lenses can be of fixed or variable focal length for total adaptability. Images are processed within the optical capture cameras where markers are identified and coordinates are calculated before being transferred to the computers. After the completion of the movement the system provides 3D coordinate and kinematic data. The disadvantages of the system include the skin movement error whose effect is more prominent (3 cm error) at high movement speeds. These high-speed motion tasks (impact biomechanics for example) are handled by high speed cine cameras with data acquisition rates several orders of magnitude greater than clinical motion analysis systems. The processing method which is almost real time uses combinations of skin markers (minimum three at each segment) to produce coordinate systems for each segment and eventually describe intersegmental relative motion or relate all the different segment motions to the laboratory fixed-coordinate system. Recently methods employing clusters of markers have shown to somewhat reduce the skin marker artifact but are yet to be adopted in the clinical practice. A more accurate method (<1mm translation and up to 1000 Hz) is the cineradiographical method, which employs an X-ray machine and uses special cameras for capture of sequences of the digital radiographs. Additional to accuracy these systems directly access the in-vivo skeletal kinematics so that the resulting analysis can be directly related to bony landmarks. Radiation issues, magnification and distortion factors are some drawbacks that can be overcome by appropriate image analysis techniques. This method is however prone to occlusion errors when two segments overlap and simultaneously cross the field of view of the X-ray source. Stereosystems with more than one X-ray sources can limit this artifact. A biplane radiographic system consists of two x-ray generators and two image intensifiers optically coupled to synchronized high-speed video cameras that can be configured in a custom gantry to enable a variety of motion studies. The system can be set up with various set-up modes, e.g. a 60º interbeam angle, an x- ray source to object distance of 1.3 m and an object to intensifier distance of 0.5 m. Images are acquired with the generators in continuous radiographic mode (typically 100 ma, 90 kvp). The video cameras are electronically shuttered to reduce motion blur. Short (0.5 s) sequences are recorded to minimize radiation exposure. X-ray exposure and image acquisition are controlled by an electronic timer/sequencer to capture only the desired phase of movement. CODA is an acronym of Cartesian Opto-electronic Dynamic Anthropometer, a name first coined in 1974 to give a working title to an early research instrument developed at Loughborough University, United Kingdom. The 3D capability is an intrinsic characteristic of

22 the design of the sensor units, equivalent to but much more accurate than the stereoscopic depth perception in normal human vision. The system is pre-calibrated for 3D measurement, which means that the lightweight sensor can be set up at a new location in a matter of minutes, without the need to recalibrate using a space-frame. Each sensor unit must be independently capable of measuring the 3D coordinates of skin markers in real-time. As a consequence, there is great flexibility in the way the system can be operated. For example, a single sensor unit can be used to acquire 3D data in a wide variety of projects, such as unilateral gait. Up to six sensor units can be used together and placed around a capture volume to give extra sets of eyes and maximum redundancy of viewpoint. This enables the system to track 360 degree movements which often occur in animation and sports applications. The calculation of the 3D coordinates of markers is done in real-time with an extremely low delay of 5 ms. Special versions of the system are available with latency shorter than 1 millisecond. This opens up many applications that require real-time feedback such as research in neuro-physiology and high quality virtual reality systems as well as tightly coupled real-time animation. It is also possible to trigger external equipment using the real-time data. The automatic intrinsic identification of markers combined with processing of all 3D co-ordinates in real-time means that graphs and stick figures of the motion and many types of calculated data can be displayed on a computer screen during and immediately after the movement occurs. The data are also immediately stored to file on the hard drive. A new concept in measuring movement disorders utilizes a unique miniature solid state gyroscope, not to be confused with gravity sensitive accelerometers. The instrument is fixed with straps directly on the skin surface of the structure whose motion is of interest. It has been successfully used to quantify: tremor (resting, posture, kinetic), rapid pronation/supination of the hand, arm swing, lateral truncal sway, leg stride, spasticity (pendulum drop test), dyskinesia and alternating dystonia. The system (Motus) senses rotational motion only and is ideal for quantifying human movement since most skeletal joints produce rotational motion. This disadvantage is outweighed by its miniature size that allows it to be of great value for certain types of studies. A different system (Gypsy Gyro) uses 18 small solid-state inertial sensors ( gyros ) to accurately measure the exact rotations of the actor s bones in real-time for motion capture. The system can easily be worn beneath normal clothing. With wireless range these systems/suits can be used to record up to 64 actors simultaneously. Another concept for 3D motion analysis is the measurement system CMS10 (Zebris) designed as a compact device for everyday use. The measurement procedure is based on the travel time measurement of ultrasonic pulses that are emitted by miniature transmitters (markers placed on the skin) to the three microphones built into the compact device. A socket for the power pack (supplied with the device) as well as the interface to a computer are located on the back of the device. The evaluation and display of the measurement data are carried out in real time. It is possible to use either a table clamp or a mobile floor stand with two joints to support the measurement system Data Analysis Coordinate Systems and Transformation In the analysis of experimental joint mechanics data, the transformation of point coordinates from one coordinate system to another is a frequent task (18, 19). A typical application of such a

23 transformation would be gait analysis data recorded in a laboratory fixed coordinate system (by means of film or video sequences) that must be converted to a reference system fixed to the skeleton of the test subject. The laboratory fixed coordinate system may be designated by xyz and the body reference system by abc (Figure 4). The location of a point S(a/b/c) in the body reference system is defined by the radius vector s =a e a + b e b + c e c. Consider the reference system to be embedded into the laboratory system. Then the radius vector r m = x m e x + y m e y + z m e z. describes the origin of the reference system in the laboratory system. The location of S(x/y/z) is now expressed by the coordinates a, b, c. The vector equation r = r m + s gives the radius vector for point S in the laboratory system (Figure 4). Employing the full notation we have: r =(x e x + y e y + z e z ) = (x m e x + y m e y + z m e z ) +(a e a + b e b + c e c ). A set of transformation equations results after some intermediate matrix algebra to describe the coordinates. Figure 4. Changing the coordinate systems, transformation of point coordinates from one coordinate system to another. S e c e z r S e b r m e a e y e x The scalar products of the unit vectors in the xyz and abc systems produce a set of nine coefficients C ij. The cosine of the angle between the coordinate axes of the two systems corresponds to the value of the scalar products. Three direction cosines define the orientation of each unit vector in one system with respect to the three unit vectors of the other system. Due to the inherent properties of orthogonality and unit length of the unit vectors, there are six constraints on the nine direction cosines, which leaves only three independent parameters describing the transformation. Employing the matrix notation of the transformation equation we have:

24 x xm c11 c12 c13 a y = y + c c c b m z zm c31 c32 c33 c In coordinate transformations the objects remain unchanged and only their location and orientation are described in a rotated and possibly translated coordinate system. If a measurement provides the relative spatial location and orientation of two coordinate systems the relative translation of the two systems and the nine coefficients C ij can be calculated. The coefficients are adequate to describe the relative rotation between the two coordinate systems Translation in three-dimensional space In translation in 3D space the rigid object moves parallel to itself (Figure 5). Pure translation in 3D space leaves the orientation of the body unchanged as in the case of pure 2D translation. R t R' t Q' Q t P' P Figure 5. A rigid body (shoebox) moves parallel to itself. The radius vectors from O to P and from O to P are designated by r and r so that r = r + t where t is the difference vector Rotations about the coordinate Axes A rotation in three-dimensional space is defined by specifying an axis and an angle of rotation (Figure 6). The axis can be described by its 3D orientation and location (20). A rotation, as does the translation explained earlier, leaves all the points on the axis unchanged; all other points move along circular arcs in planes oriented perpendicular to the axis (21).