Kinematics

Kinematics is the study of the geometric and time-dependent aspects of motion without analyzing the forces causing the motion. Kinematic analysis is used to relate displacement, velocity, acceleration, and time. To study kinematics in an organized manner, it is common to classify the motion as translational, rotational, or general. Translational motion occurs when a straight line drawn between two points on the body remains in the same direction during the motion. Translational motion can be either rectilinear motion (if the paths are straight lines) or curvilinear (if the paths are curved lines). Rotational motion occurs when the points on the body move in a circular path around an axis of rotation. The angular motion occurs about a central line known as the axis of rotation, which lies perpendicular to the plane of motion. The third class of motion is called general motion, or displacement, which occurs if a body undergoes both translational and rotational motion simultaneously.

Kinematics can be analyzed in two-dimensional (2-D) or three-dimensional (3-D) space. When all points of a rigid body move parallel to a plane, the motion is referred to as planar motion. This motion can be thought of as 2-D motion. 3-D motion is the more general type of rigid body motion. This motion requires six independent parameters to describe the general type of motion. These parameters are called “degrees of freedom” or the number of independent coordinates in a coordinate system that is required to completely specify the position of an object in space. A rigid body in space has a maximum of six degrees of freedom: three translations (expressed by linear coordinates) and three rotations (expressed by angular coordinates). The general movement of an object is defined by a vector quantity that is a combination of both linear and angular displacement. Velocity is the time-related change of displacement. Linear velocity is expressed in units of length per time (m/sec). Angular velocity is expressed in units of angular measure per time (rad/sec). Because velocities reflect vector quantities, both magnitude and direction must be specified. Acceleration is the time rate of change of velocity. Linear acceleration is expressed in units of length per time squared (m/sec²). Angular acceleration is the time rate of change in angular velocity (rad/sec²). Accelerations are also vector quantities, and both magnitude and direction must be specified.

Kinematic techniques have been used to study body movements in both 2-D and 3-D space. The human body is typically modeled as a number of interconnected rigid body segments (Figure 6-2). A coordinate system is affixed to each rigid body segment in order to establish an anatomic coordinate system. External markers are used to define orthogonal coordinate systems whose axes define the position of these body segments. Joint motion is then described as the relative motion of the distal body segment with respect to the proximal body segment. Limb segments are assumed to undergo angular displacement during human movement. However, more sophisticated analyses will also quantify the linear displacement that limb segments may undergo. These measures of relative segmental angles have been used to describe human walking and other activities of daily living.^2–7

Figure 6-2 Body-fixed reflective markers used for establishing anatomic coordinate systems. A, Video camera measurement systems calculate the location of external markers placed on the body segments and aligned with specific bony landmarks. B, A body-fixed external coordinate system is then computed from three or more markers on each body segment. Body-fixed external coordinate systems are shown for the thigh, T_T, and shank, S_T. The coordinate system is shown for the three orthogonal directions (e.g., x, y, z). So, the body-fixed thigh coordinate system is designated T_Tx, T_Ty, and T_Tz. C, Using a subject-specific calibration converts the external coordinate system to an anatomic coordinate system through the identification of anatomic landmarks (e.g., the medial and lateral femoral condyles and medial lateral malleoli). The anatomical coordinate system for the thigh is then designated T_Ax, T_Ay, and T_Az.

(Reproduced from Kaufman KR: Objective assessment of posture and gait. In Bronstein AM, Brandt T, Marjorie H, editors: Clinical disorders of balance, posture, and gait, Oxford, England, 2004, A Hodder Arnold Publication.)

As an example, knee motion in the sagittal plane can be characterized throughout the gait cycle (Figure 6-3). At heel strike, the knee is nearly fully extended (knee flexion of 5 degrees). During midstance, the knee flexes to about 15 degrees, which occurs at 15% of the gait cycle. The knee joint is brought back into extension by midstance. At 50% of the gait cycle, opposite foot contact occurs. The weight is shifted to the opposite limb, and the knee begins flexing. Toe-off occurs at 60% of the gait cycle. Peak knee flexion of 60 degrees occurs during the swing phase. The knee motion can be described as two flexion waves, each starting in relative extension, progressing into flexion, then returning again to extension. The first flexion wave, or stance phase knee flexion, acts as a shock absorber to aid weight acceptance. This curve peaks in early stance at opposite foot-off. The mechanical source for this shock absorber is the eccentrically contracting quadriceps muscles. The second flexion wave is necessary in order to clear the foot in early swing phase. The knee is rapidly flexed beginning just after heel rise to a maximum in swing phase just as the swinging foot passes the opposite limb.

Figure 6-3 Knee sagittal plane motion throughout the gait cycle. A positive value indicates knee flexion. For the first 60% of the gait cycle, the leg is in stance. The leg is in swing phase for the remainder of the gait cycle.

The complexity of kinematic analysis increases substantially when going from planar analysis to 3-D analysis. The complexity of the analysis arises from the technical difficulty that large rigid body rotations cannot be treated as vectors and, hence, do not obey the vectorial principles of transformation, independence, and interchangeability of operations. For finite spatial rotation, the sequence of rotations is extremely important and must be specified for a unique description of joint motion.^8,9 For the same amount of rotation, different final orientations will result from different sequences of rotation (Figure 6-4). However, with proper selection and definition of the axes of rotation between two bony segments, it is possible to make finite rotation sequence independent or communitive.^8,9 The concept of Eulerian angles has been adopted in the field of orthopedic biomechanics to unify the definition of finite spatial rotation. In the selection of reference axes, one axis is fixed to the stationary segment and another axis is fixed to the moving segment (Figure 6-5). In the knee joint, for example, the flexion/extension angle, Φ, occurs about a medial-laterally directed axis defined by a line connecting the medial and lateral femoral condyles. The axial rotation angle, ψ, is measured about an axis defined by the line along the shaft of the tibia. The third axis, also defined as the floating axis, is orthogonal to the other two axes and defines abduction/adduction, Θ. These rotations match the Eulerian angle description and are thought to be performed in such a way as to bring the moving segment from the reference orientation into the current orientation. The advantage of using this system for description of the spatial rotation of anatomic joints is that the angular rotations do not have to be referred back to the neutral position of the joint because the rotation sequence can be independent. Thus the measurement can be easily obtained and related to anatomic structures.

Figure 6-4 This example illustrates the sequence dependence of rigid body motion. The object undergoes two rotations about the X-axis and Z-axis. The sequence of these rotations differs in the top and bottom rows. The result is that the final orientation of the object is different.

Figure 6-5 Description of knee joint motion using Eulerian angle system. An axis, e_ϕ, fixed to the distal femur defines flexion/extension motion, ϕ. An axis, e_ψ, fixed to the proximal tibia along its anatomic axis defines internal external rotation, ψ. A floating axis, e_θ, orthogonal to the other two axes is used to measure abduction-adduction, θ. The rotation matrix, [T], between the proximal and distal joint segment is then given by the multiplication of the three rotation matrices associated with the rotations around the three axes (i.e., [T] = [T_ψ] [T_θ] [T_ϕ]).

(Reproduced with permission from Chao EYS: Justification of triaxial goniometer for the measurement of joint rotation, J Biomech 13:989–1006, 1980.)

A complete analysis of total joint movement (i.e., six degrees of freedom) can be obtained using markers embedded in the bone^10,11 or dual fluoroscopic imaging techniques.^12,13 This general unconstrained movement in 3-D space requires the description of three translations and three rotations to fully describe joint motion. The most commonly used analytic method for description of six degrees-of-freedom displacement of a rigid body is the screw displacement axis (SDA).^14–16

Kinetics

Kinetics is the study of the forces that cause motion of a rigid body. When there are unbalanced forces or moments acting on a rigid body, it is under a nonequilibrium, or dynamic, condition, resulting in motion. Understanding the kinetics of human movement provides a fundamental understanding of the musculoskeletal system. Before one can begin to analyze the forces during human movement, some basic definitions and assumptions must be made.

The key quantities in kinetics are force, moment, and torque. Forces represent an interaction between two bodies. Forces can be contact forces (bodies touching each other) or field forces (bodies separated by a distance, such as gravitational, electric, or magnetic forces). Forces are represented by vectors. Vectors are composed of four components: magnitude, direction, sense, and position (also called point of application). According to Newton’s Second Law, force is any action that tends to change the state of rest or state of motion of a body to which it is applied. Forces are represented by vectors. The vector may be resolved into several component forces, usually along specified mutually perpendicular coordinate axes. Conversely, forces can be summed using vectorial addition. A moment represents the turning, twisting, or rotational effect of a force. A moment is a vector. A moment is defined as the product of the force and the perpendicular distance between the line of action of the force and the axis of rotation of the motion that the force produces (Figure 6-6). Its magnitude is the force times the perpendicular distance to the axis of rotation. The direction of the moment is along the axis of rotation (or potential rotation) and thus perpendicular to the plane in which the twisting force is applied. The moment arm, the distance used to calculate the moment, is the shortest distance from the force action line to the actual or potential pivot point of the system, regardless of the state of motion. Skeletal motions are the result of moments applied by muscles that cross the joints on which they act. Moments of a force about an axis measure the tendency of the force to impart to the body a motion of rotation about a fixed axis. A torque is a special type of moment that results when a pair of forces that have equal magnitude, parallel lines of action, and opposite senses act on a body (Figure 6-7). The magnitude of the torque is Fd, where d is the perpendicular distance between the two forces. The resultant force is zero because the two forces are equal and oppose each other.

Figure 6-6 The moment (M) of a force (F) about a point is equal to the force times the moment arm (d), which is the perpendicular distance between the point and the line of action of the force. The moment arm may (A) or may not (B) be the distance along the limb segment to the axis of rotation. The moment arm is always the shortest distance between the line of force application and the axis of rotation.