﻿ Angular Kinematics - Biomechanical Basis of Human Movement, 3rd Edition ﻿

## Biomechanical Basis of Human Movement, 3rd Edition

### Angular Kinematics

Objectives

After reading this chapter, the student should be able to:

1. Distinguish between linear, angular, and general motion.
2. Determine relative and absolute angles.
3. Discuss the relationship among the kinematic quantities of angular distance and displacement, angular velocity, and angular acceleration.
4. Discuss the conventions for the calculation of lower extremity angles.
5. Discuss the relationship between angular and linear motion, particularly displacement, velocity, and acceleration.
6. Discuss selected research studies that have used an angular kinematic approach.
7. Solve quantitative problems that employ angular kinematic principles.

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Angular Motion

Angular motion occurs when all parts of a body move through the same angle but do not undergo the same linear displacement. The subset of kinematics that deals with angular motion is angular kinematics, which describes angular motion without regard to the causes of the motion. Consider a bicycle wheel (Fig. 9-1). Pick any point close to the center of the wheel and any point close to the edge of the wheel. The point close to the edge travels farther than the point close to the center as the wheel rotates. The motion of the wheel is angular motion.

Angular motion occurs about an axis of rotation that is a line perpendicular to the plane in which the rotation occurs. For example, the bicycle wheel spins about its axle which is its axis of rotation. The axle is perpendicular to the plane of rotation described by the rim of the wheel (Fig. 9-1).

An understanding of angular motion is critical to comprehend how one moves. Nearly all human movement involves rotation of body segments. The segments rotate about the joint centers that form their axes of rotation. For example, the forearm segment rotates about the elbow joint during flexion and extension of the elbow. When an individual moves, the segments generally undergo both rotation and translation. Sequential combinations of angular motion of multiple segments can result in linear motion of the segment end point seen in throwing and many other movements in which end point velocities are important. When the combination of rotation and translation occurs, it is described as general motion.Figure 9-2 illustrates the combination of linear and rotational motions. The gymnast undergoes translation as she moves across the ground. At the same time, she is rotating. The combination of rotation and translation is common in most human movements.

 FIGURE 9-1 A bicycle wheel as an example of rotational motion. Points A, B, and C undergo the same amount of rotation but different linear displacements, with C undergoing the greatest linear displacement. FIGURE 9-2 A gymnast completing a cartwheel as an example of general motion. The gymnast simultaneously undergoes both translation and rotation.

Measurement of Angles

Angle

An angle is composed of two lines, two planes, or a combination that intersect at a point called the vertex (Fig. 9-3). In a biomechanical analysis, the intersecting lines are generally body segments. If the longitudinal axis of the leg segment is one side of an angle and the longitudinal axis of the thigh segment is the other side, the vertex is the joint center of the knee. Angles can be determined from the coordinate points described in Chapter 8. Coordinate points describing the joint centers determine the sides and vertex of the angle. For example, an angle at the knee can

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be constructed using the thigh and leg segments. The coordinate points describing the ankle and knee joint centers define the leg segment; the coordinate points describing the hip and knee joint centers define the thigh segment. The vertex of the angle is the knee joint center.

 FIGURE 9-3 Components of an angle. Note that the lines are usually segments and the vertex of the angle is the joint center.

Definition of a segment by placing markers on the subject at the joint centers makes a technically incorrect assumption that the joint center at the vertex of the angle does not change throughout the movement. Because of the asymmetries in the shape of the articulating surfaces in most joints, one or both bones constituting the joint may displace relative to each other. For example, although the knee is often considered a hinge joint, it is not. At the knee joint, the medial and lateral femoral condyles are asymmetrical. Therefore, as the knee flexes and extends, the tibia rotates along its long axis and rotates about an axis through the knee from front to back. The location of the joint center, therefore, changes throughout any motion of the knee. The center of rotation of a joint at an instant in time is called the instantaneous joint center (Fig. 9-4). It is difficult to locate this moving axis of rotation without special techniques such as x-ray measurements. These measurements are not practical in most situations; thus, the assumption of a static instantaneous joint center must be made.

Using MaxTRAQ, import the first video file of the golfer. What joints would you use to calculate the angle at the right shoulder? What is the vertex of the angle?

 FIGURE 9-4 Instantaneous center of rotation of the knee. (Adapted from  Nordin, M., Frankel, V. H. [eds.] [1979]. Biomechanics of the Musculoskeletal System (2nd ed.). Philadelphia: Lea & Febiger. )

Units of Measurement

In angular motion, three units are used to measure angles. It is important to use the correct units to communicate the results of this work clearly and to compare values from study to study. It is also essential to use the correct units because angle measurements may be used in further calculations. The first and most commonly used is the degree. A circle, which describes one complete rotation, transcribes an arc of 360° (Fig. 9-5A) An angle of 90° has sides that are perpendicular to each other. A straight line has a 180° angle (Fig. 9-5B)

The second unit of measurement describes the number of rotations or revolutions about a circle (Fig. 9-5A) One revolution is a single 360° rotation. For example, a triple jump in skating requires the skater to complete 3.5 revolutions in the air. The skater completes a rotation of 1260°. This unit of measurement is useful in qualitative descriptions of movements in figure skating, gymnastics, and diving but is not useful in a quantitative analysis.

Although the degree is most commonly understood and the revolution is often used, the most appropriate unit for angular measurement in biomechanics is the radian.

 FIGURE 9-5 Units of angular measurement. A. Revolution. B. Perpendicular and straight lines. C. Radian.

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A radian is defined as the measure of an angle at the center of a circle described by an arc equal to the length of the radius of the circle (Fig. 9-5 C). That is:

where θ = 1 radian, s = arc of length r along the diameter, and r = radius of the circle. Because both s and r have units of length (m), the units in the numerator and denominator cancel each other out with the result that the radian is dimensionless.

In further calculations, the radian is not considered in determining the units of the result of the calculation. Degrees have a dimension and must be included in the unit of the product of any calculation. It is necessary, therefore, to use the radian as a unit of angular measurement instead of the degree in any calculation involving linear motion because the radian is dimensionless.

One radian is the equivalent of 57.3°. To convert an angle in degrees to radians, divide the angle in degrees by 57.3. For example:

To convert radians to degrees, multiply the angle in radians by 57.3. For example:

Angular measurement in radians is often determined in multiples of pi (π = 3.1416). Because 2π radians are in a complete circle, 180° may be represented as JC radians, 90° as π/2 radians, and so on.

Although the unit of angular measurement in the Systéme International d'Unités (SI) is the radian and this unit must be used in further calculations, the angular motion concepts presented in the remainder of this chapter will use the degree for ease of understanding.

Types of Angles

Absolute Angle

In biomechanics, two types of angles are generally calculated. The first is the absolute angle, which is the angle of inclination of a body segment relative to some fixed reference in the environment. This type of angle describes the orientation of a segment in space. Two primary conventions are used for calculating absolute angles. One involves placing a coordinate system at the proximal end point of the segment. The angle is then measured counterclockwise from the right horizontal. The most frequently used convention for calculating absolute angles, however, places a coordinate system at the distal end point of the segment (Fig. 9-6). The angle using this convention is also measured counterclockwise from the right horizontal. The absolute angles calculated using these two conventions are related and give comparable information. When calculating absolute angles, however, the convention used must be stated clearly. The absolute angle of a segment relative to the right horizontal is also called the segment angle.

 FIGURE 9-6 Absolute angles: The arm (a), trunk (b), thigh (c), and leg (d) of a runner.

Absolute angles are calculated using the trigonometric relationship of the tangent. The tangent is defined based on the sides of a right triangle. It is the ratio of the side opposite the angle in question and the side adjacent to the angle. The angle in question is not the right angle in the triangle. If the leg and thigh segment coordinate positions are considered, the absolute angles of both the thigh and leg segments can be calculated (Fig. 9-7).

 FIGURE 9-7 Absolute angles of the thigh and leg as defined in a coordinate system.

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To calculate the absolute leg angle, the coordinate values of the segment end points of the leg are substituted into the formula to define the tangent of the angle:

Next, the angle whose tangent is 3.23 is again determined using either the trigonometric tables (see Appendix B) or a calculator. This is called finding the inverse tangent and is written as follows:

The absolute angle of the leg, therefore, is 72.8° from the right horizontal. This orientation indicates that the leg is positioned so that the knee is farther from the vertical (y) axis of the coordinate system than the ankle. That is, the knee joint is to the right of the ankle joint (see Fig. 9-7).

Similarly, to calculate the thigh angle, the coordinate values are substituted:

Again, the angle whose tangent is -3.625 is determined as follows:

This angle is clockwise from the left horizontal because we have moved into the second quadrant with the negative x-value. To convert the angle so it is relative to the right horizontal and counterclockwise, it must be added to 180°, resulting in an absolute angle of 105.4° relative to the right horizontal (Fig. 9-8).

 FIGURE 9-8 To calculate absolute angles relative to the right horizonta requires adjustments when the orientation is such that the differences between the proximal and distal end points indicate that the segment is not in the first quadrant.

An absolute thigh angle of 105.4° in the second quadrant indicates that the thigh is oriented such that the hip joint is closer to the vertical (y) axis and above the horizontal (x) axis of the coordinate system. In this case, the thigh would be oriented with the knee to the right of the hip in this reference system. When both x and y are negative, the value is in the third quadrant, and the angle is computed counterclockwise and relative to the left horizontal, so 180° is still added to adjust the absolute angle so it is relative to the right horizontal. Finally, if there is only a negative y-value, the angle is in the fourth quadrant and taken clockwise and relative to the right horizontal, so 360° should be added to convert the absolute angle so that it is relative to the right horizontal in the counterclockwise direction.

Trunk, thigh, leg, and foot segmental end points for both the touchdown and the toe-off in walking are graphically illustrated in Figure 9-9. The corresponding calculations of

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the absolute angles shown in Table 9-1 use the conventions discussed previously to convert all angles so they are taken counterclockwise with respect to the right horizontal. For example, the leg orientation in touchdown results in a negative x-position and positive y-position, so 180° is added to the final angle computation to make it relative to the right horizontal. In the case of toe-off, however, both x and y are positive, so there is no adjustment. Likewise, the foot orientation in touchdown and toe-off result in both negative x and y, placing it in the third quadrant, where 180° is again added. These adjustments provide a consistent reference for the computation of the absolute angles.

 FIGURE 9-9 By plotting the segmental end-points and creating a stick figure, the similarities or differences in position can be clearly observed. The differences in right foot touch down (black) and right foot toe off (red) phases of a walking gait are apparent. See Appendix C Frame 1 and Frame 76, respectively.

Using MaxTRAQ, import the first video file of the woman walking (representing right heel contact). Digitize the right shoulder, right greater trochanter, right knee, and the right ankle. Calculate the absolute angles of the trunk, thigh, and leg.

Relative Angle

The other type of angle calculated in biomechanics is the relative angle (Fig. 9-10A) This is the angle between longitudinal axes of two segments and is also referred to as the joint angle or the intersegmental angle. A relative angle (e.g., the elbow angle) can describe the amount of flexion or extension at the joint. Relative angles, however, do not describe the position of the segments or the sides of the angle in space. If an individual has a relative angle of 90° at the elbow and that angle is maintained, the arm may be in any of a number of positions (Fig. 9-10B).

Relative angles can be calculated using the law of cosines. This law, simply a more general case of the Pythagorean theorem, describes the relationship between the sides of a triangle that does not contain a right angle. For our purposes, the triangle is made up of the two segments B and C and a line, A, joining the distal end of one segment to the proximal end of the other (Fig. 9-11).

 FIGURE 9-10 A. Relative elbow angle. B. The same relative elbow angle with the arm and forearm in different positions.

In Figure 9-11, the coordinate points for two segments describing the thigh and the leg are given. To calculate the relative angle at the knee (θ), the lengths a, b, and c would be calculated using the Pythagorean relationship.

TABLE 9-1 Absolute Angle Calculation for Touchdown and Toe-off in Walking

 Frame Trunkx=(Shoulderx to Greater Trochanterx) Trunky= (Shouldery to Great Trochantery) Absolute Angle=Arctan (y/x) Thighx=(Greater - Trochanterx to Kneex) Thighy=(Greater Trochantery to Kneey) 1 -3.95 523.08 -89.57°+180°=90.43° 210.15 317.14 76 10.92 532.10 88.82° -14.76 368.95 Absolute Angle=Arctan (y/x) Legx=(Kneex to Anklex) Legy = (Kneey to Ankley) Absolute Angle= Arctan (y/x) Footx=(Heelx to Metx) Footy=(Heelyto Mety) Absolute Angle=Arctan (y/x) =-56.47° 180° -113.03 377.88 =-73.35° 180° 181.92 -67.95 =20.48°+180° =123.53° =106.65° =200.48° =87.71° + 180°=92.29° 313.49 218.66 =34.90° -51.20 169.39 =73.18° + 180°=106.82°

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 FIGURE 9-11 Coordinate points describing the hip, knee, and ankle joint center and the relative angle of the knee (u).

The next step is to substitute these values in the law of cosines equation and solve for the cosine of the angle 9.

To find the angle θ, the angle whose cosine is -0.833 can be determined using either trigonometric tables (see Appendix B) or a calculator with trigonometric functions. This process, known as finding the inverse cosine or arcos, is written as follows:

Therefore, the relative angle at the knee is 146.4°. In this case, the knee is slightly flexed (180° representing full extension).

Using MaxTRAQ, import the third video file of the woman walking (representing midstance). Digitize the right iliac crest, right greater trochanter, right knee, and the right ankle. Calculate the relative angles of the hip and knee.

A relative angle can be calculated from the absolute values to obtain a result similar to computations using the law of cosines. The relative angle between two segments can be calculated by subtracting the absolute angle of the distal segment from the proximal segment. In the example using the thigh and lower leg, the following calculation is another option:

In clinical situations, the relative angle is most often calculated because it provides a more practical indicator of function and joint position. In quantitative biomechanical analyses, however, absolute angles are calculated more often than relative angles because they are used in a number of subsequent calculations. Regardless of the type of angle calculated, however, a consistent frame of reference must be used.

Unfortunately, many coordinate systems and systems of defining angles have been used in biomechanics, resulting in difficulty comparing values from study to study. Several organizations, such as the Canadian Society of Biomechanics and the International Society of Biomechanics, have standardized the representation of angles to provide consistency in biomechanics research, especially in the area of joint kinematics.

Lower Extremity Joint Angles

In discussing the angle of a joint such as the knee or ankle, it is imperative that a meaningful representation of the action of the joint be made. A special use of absolute angles to compute joint angles is very useful for clinicians and others interested in joint function. Lower extremity joint angles can be calculated using the absolute angles similar to the procedure described previously. A system of lower extremity joint angle conventions was presented by Winter (36). These lower extremity angle definitions are for use in a (two-dimensional) 2D sagittal plane analysis only. In Winter's system, digitized points describing the trunk, thigh, leg, and foot are used to calculate the absolute angles of each (Fig. 9-12). From these absolute angles, joint angles can be computed. In such a biomechanical analysis, it is assumed that a right side sagittal view is being analyzed. That is, the right side of the subject's body is closest to the camera and is considered to be in the x-y plane.

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 FIGURE 9-12 Definition of the sagittal view absolute angles of the trunk, thigh, leg, and foot. (After Winter, D. A. [1987]. The Biomechanics and Motor Control of Gait.Waterloo, Ontario, Canada: University of Waterloo Press.)

Hip Angle

Based on the absolute angles of the trunk and the thigh calculated, the hip angle is:

In this scheme, if the hip angle is positive, the action at the hip is flexion; but if the hip angle is negative, the action is extension. If the angle is zero, the thigh and the trunk are aligned vertically in a neutral position. For example, the hip joint angle representing flexion and extension for the touchdown phase in walking (Table 9-1) would be:

The joint angle of 33.1° at touchdown indicates that the thigh is flexing at the hip joint. In a human walking at a moderate pace, the hip angle oscillates ±35° about 0°; in running, the hip angle oscillates ±45°.

Knee Angle

Using the absolute angle of the thigh and the leg, the knee joint angle is defined as:

In human locomotion, the knee angle is always positive (i.e., in some degree of flexion), and it usually varies from 0 to 50° throughout a walking stride and from 0 to 80° during a running stride. Because the knee angle is positive, the knee is always in some degree of flexion. If the knee angle gets progressively greater, the knee is flexing. If it gets progressively smaller, the knee is extending. A zero knee angle is a neutral position, and a negative angle indicates a hyperextension of the knee. The knee angle for the touchdown phase in the walking example (Table 9-1) is:

Ankle Angle

The ankle angle is calculated using the absolute angles of the foot and the leg:

This may seem more complicated than the other lower extremity joint angle calculations. Without adding 90°, the ankle angle would oscillate about 90°, making interpretation of it difficult. Adding 90° makes the ankle angle oscillate about 0°. Thus, a positive angle represents dorsiflexion, and a negative angle represents plantar flexion.

The ankle angle for the touchdown phase in the walking example (Table 9-1) is:

This value indicates that the ankle is in plantarflexion at touchdown. The ankle angle generally oscillates ±20° during a natural walking stride and ±35° during a running stride. Lower extremity angles calculated for a walking stride using Winter's convention are presented in Figure 9-13.

Using MaxTRAQ, import the first video file of the woman walking (representing right heel contact). Digitize the right shoulder, right greater trochanter, right knee, and the right ankle. Calculate the absolute angles of the trunk, thigh, and leg (this was done in a previous assignment). Using these absolute angles, calculate the hip and knee angles according to Winter (36).

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 FIGURE 9-13 Graphs of the hip (A), knee (B), and ankle (C) angles during walking.

Joint angles calculated using the relative angle approach (law of cosines) and the same angles calculated from the absolute angles using Winter's (36) convention have exactly the same clinical meaning. In the relative angle approach, the joint angle that is calculated is the included angle between the two segments. Using the absolute angle approach, the joint angle that is calculated is the difference between the two segment angles. The interpretation of these angles is exactly the same. Both types of angles are presented in Figure 9-14.

 FIGURE 9-14 Joint angle representations using the relative angle and absolute angle calculations. Angles a, c, and e are calculated from the absolute angles, and angles b, d, and f use the relative angle calculations. Both representations are the same.

Rearfoot Angle

Another lower extremity angle that is often calculated in biomechanical analyses is the rearfoot angle. The motion of the subtalar joint in a two-dimensional analysis is considered to be in the frontal plane. The rearfoot angle represents the motion of the subtalar joint. The rearfoot angle thus approximates calcaneal eversion and calcaneal inversion in the frontal plane. Calcaneal eversion and inversion are among the motions in the pronation and supination action of the subtalar joint. In the research literature, calcaneal eversion is often measured to evaluate pronation, and calcaneal inversion is measured to determine supination.

The rearfoot angle is calculated using the absolute angles of the leg and the calcaneus in the frontal plane. Two segment markers are placed on the rear of the leg to define the longitudinal axis of the leg. Two markers are also placed on the calcaneus (or the rear portion of the shoe) to define the longitudinal axis of the calcaneus (Fig. 9-15).

Researchers have reported that markers placed on the shoe rather than directly on the calcaneus do not give a true indication of calcaneal motion (30). In fact, it has been suggested that the rearfoot motion calculated when the markers are placed on the shoe is greater than when the markers are placed on the calcaneus for the same movement. Regardless of the postioning of the markers,

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they are used to calculate the absolute angles of the leg and heel; thus, the rear foot angle is:

 FIGURE 9-15 Definition of the absolute angles of the leg and calcaneus in the frontal plane. These angles are used to constitute the rearfoot angle of the right foot.

By this calculation, a positive angle represents calcaneal inversion, a negative angle represents calcaneal eversion, and a zero angle is the neutral position.

During the support phase of the gait cycle, the rearfoot, as defined by the rearfoot angle, is in an inverted position at the initial foot contact with the ground. At this instance, the rearfoot angle is positive. From that point onward until midstance, the rearfoot moves to an everted position; thus, the rearfoot angle is negative. At midstance position, the foot becomes less everted and moves to an inverted position at toe-off The rearfoot angle becomes less negative and eventually positive at toe-off Figure 9-16 is a representation of a typical rearfoot angle curve during the support phase of a running stride.

 FIGURE 9-16 A typical rearfoot angle-time graph during running. Maximum rearfoot angle is indicated.

Representation of Angular Motion Vectors

Representing angular motion vectors graphically as lines with arrows, as is the case in linear kinematics, is difficult. It is essential, however, to determine the direction of rotation in terms of a positive or negative rotation. The direction of rotation of an angular motion vector is referred to as the polarity of the vector. The polarity of an angular motion vector is determined by the right-hand rule. The direction of an angular motion vector is determined using this rule by placing the curled fingers of the right hand in the direction of the rotation. The angular motion vector is defined by an arrow of the appropriate length that coincides with the direction of the extended thumb of the right hand (Fig. 9-17). The 2D convention generally used is that all segments rotating counterclockwise from the right horizontal have positive polarity and all segments rotating in clockwise have a negative polarity.

 FIGURE 9-17 The right-hand rule used to identify the polarity of the angular velocity of a figure skater during a spin. The fingers of the right hand point in the direction of the rotation, and the right thumb points in the direction of the angular velocity vector. The angular velocity vector is perpendicular to the plane of rotation.

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Angular Motion Relationships

The relationships discussed in this chapter on angular kinematics are analogous to those in Chapter 8 on linear kinematics. The angular case is simply an analog of the linear case.

Angular Position and Displacement

The angular position of an object refers to its location relative to a defined spatial reference system. In the case of a 2D system with the y-axis representing motion vertically up and down and the x-axis representing anterior to posterior motion, angular position is described in the x-y plane. A three-dimensional (3D) system adds a third axis, z, in the medial and lateral plane. Many clinicians use planes to describe angular positioning. For example, if the axes are placed with the origin at the shoulder joint, angular position of the arm in the x-y plane would be a flexion and extension position in the y-z plane abduction and adduction, and in the x-z plane, rotation. This system works well for describing joint angles but lacks precision for describing complex movements. Absolute angles can be computed relative to a fixed reference system placed at a joint or at another fixed point in the environment. As discussed earlier, angular position can also be computed relative to a line or plane that is allowed to move. It is common to present joint angles such as those shown in Figure 9-13 to document the joint actions in a movement such as walking.

The concepts of distance and displacement in the angular case must be discerned. Consider a simple pendulum swinging in the x-y plane through an arc of 70° (Fig. 9-18). If the pendulum swings though a single arc, the angular distance is 70°, but if it swings through 1.5 arcs, the angular distance is 105°. Angular distance is the total of all angular changes measured following its exact path. As in the linear case, however, angular distance is not the same as angular displacement.

Angular displacement is the difference between the initial and final positions of the rotating object (Fig. 9-19). In the example of the pendulum, if the pendulum swings through two complete arcs, the angular displacement is zero because its final position is the same as the starting position. Angular displacement never exceeds 360° or 2π rad of rotation, but angular distance can be any value. In discussing angular displacement, it is necessary to designate the direction of the rotation. Counterclockwise rotation is considered to be positive, and clockwise rotation is negative. With a 3D reference system placed at the shoulder joint, the positive y-axis would be upward, the positive x-axis would be posterior to anterior, and the positive z-axis would be medial to lateral. The corresponding positive joint movements about these axes would be flexion/extension (about the x-axis), internal/external rotation (about the y-axis), and abduction/adduction (about the z-axis).

 FIGURE 9-18 A swinging pendulum illustrating the angular distance over 1.5 arcs of swing. FIGURE 9-19 Angular displacement is the difference between the initial position and the final position.

If the absolute angle of a segment, theta (9), is calculated for successive positions in time, the angular displacement (Δθ) is:

The polarity, or sign, of the angular displacement is determined by the sign of Δθ as calculated and may be confirmed by the right-hand rule.

Angular Velocity

Angular speed and angular velocity are analogous with linear speed and linear velocity in both definition and meaning. Angular speed is the angular distance traveled per unit of time. Angular speed is a scalar quantity and is generally not critically important in biomechanical analysis because it is not used in any further calculations.

Angular velocity, characterized by the Greek letter omega (ω), is a vector quantity that describes the time rate of change of angular position. If the measured angle is G, then the angular velocity is:

If the initial angle of a segment is 34° at time 1.25 s and the segment moves to an angle of 62° at time 1.30 s, the angular velocity would be:

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Angular speed and angular velocity are generally presented in degrees per second (°/s). If, however, as noted previously, any further computation is to be done using angular velocity, the units must be radians per second (rad/s).

In the previous example, the average angular velocity was calculated over the interval from 1.25 s to 1.30 s. According to the discussion in the previous chapter, angular velocity represents the slope of a secant on an angular position-time graph over this interval. The instantaneous angular velocity represents the slope of a tangent to an angular position-time graph and is calculated as a limit.

Angular velocity is thus the first derivative of angular position.

As in the linear case, the direction of the slope on an angle- time profile determines whether the angular velocity is positive or negative, and the steepness of the slope indicates the rate of change of angular position. If θfinal is greater than initial, ω is positive (i.e., the slope is positive), but if θfinal is less than θinitial, ω is negative (i.e., slope is negative). Both situations can be confirmed using the right-hand rule. If there is no change in the angle, the slope is zero and ω is zero.

The method used to calculate angular velocity over a series of frames of a kinematic analysis is the first central difference method. This method calculates the angular velocity at the same instant at which the data for angular position are available. For angular velocity, this formula is:

where θi is the angle at time ti Table 9-2 represents the thigh absolute angle data collected for one support phase in walking (see Appendix C). The rate of the camera was 120 frames per second, and every third frame is presented from touchdown (frame 0) to toe-off (frame 76). The time between each frame is 1/120 = 0.0083 s; thus, the time between three frames is 0.0249. Using the first central difference method, the angular velocity is calculated from the absolute angular position for each frame. After it has been calculated, the values are typically graphed to observe the pattern of motion (Fig. 9-20). The results of the calculation and graphing of angular kinematics of the thigh indicate that for most of the support phase the thigh is moving clockwise with respect to the knee joint. At heel strike, the thigh is in extreme hip flexion that is reduced as the trunk is brought over the support limb, moving the thigh clockwise (negative angular velocity). The thigh is vertically aligned in frame 39, and the trunk continues moving over the limb, forcing the thigh to continue in its clockwise rotation about the knee joint. At the end of the support phase (frame 63), the motion of the thigh reverses, and a counterclockwise movement of the thigh begins in preparation for toe-off (positive angular velocity).

Refer to the walking data in Appendix C. Using the first central difference method, calculate the angular velocity of the lower leg using the absolute segment position angles and graph the angular velocity.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity with respect to time and is symbolized by the Greek letter alpha (α).

Angular acceleration = Change in angular velocity/Change in time

For ease of understanding, biomechanists generally present their results in degrees per second squared (deg/s2), but the most appropriate unit for angular acceleration is radians per second squared (rad/s2).

As with the linear case, angular acceleration is the derivative of angular velocity and represents the slope of a line (either a secant for average angular acceleration or a tangent for instantaneous angular acceleration). If a is the slope of a secant to an angular velocity-time profile, it represents an average acceleration over a time interval. If α is the slope of a tangent, the instantaneous angular acceleration has been calculated. This also implies that the slope may be positive (ωfinalinitial), negative (ωfinalinitial) or zero (ωfinalinitial). The direction of the angular acceleration vector may be confirmed using the right-hand rule. The instantaneous angular acceleration is calculated by:

Again, in a kinematic analysis, the usual method of calculating angular acceleration is the first central difference method. The formula for angular acceleration for this method is:

where ωi. is the angular velocity at time t. Table 9-2 presents the calculated angular acceleration of the thigh for selected frames in the support phase of walking.

As in the case for linear acceleration, the sign or polarity of angular acceleration does not indicate the direction of rotation. For example, positive angular acceleration may mean increasing angular velocity in the positive direction or decreasing angular velocity in the negative direction. Also, negative angular acceleration may indicate decreasing angular velocity in the positive direction or increasing angular velocity in the negative direction. The angular position,

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velocity, and acceleration of the thigh are presented in Table 9-2, with a corresponding graph in Figure 9-20. The angular acceleration of the thigh is negative (increasing angular velocity in the negative direction) and then positive (decreasing angular velocity in the negative direction) during the portion of the support phase where the thigh angular velocity is negative. The acceleration remains positive (increasing angular velocity in the positive direction) in the later stages of support when the thigh angular velocity changes from negative to positive (direction change).

 TABLE 9-2 Calculation of Angular Position, Velocity, and Acceleration for the Thigh During the Support Phase of Walking FIGURE 9-20 Graphic representations of the thigh's absolute angle (A), angular velocity (B), and angular acceleration (C) as a function of time for the support phase of walking (data from Appendix C).

Refer to the walking data in Appendix C. Using the first central difference method, calculate and graph the angular acceleration of the leg.

Relationship Between Angular and Linear Motion

In many human movements, whereas the result of the movement is linear, the motions of the segments constituting the movement are angular. For example, a pitcher throws a baseball that travels linearly. However, the motions of the pitcher's segments resulting in the throw are rotational. For example, when it is necessary to know the linear motion of the hand, you must know that it depends on the angular motion of the segments of the upper extremity. This example suggests a mechanical relationship between linear and angular motion.

Linear and Angular Displacement

When the angular measure of an angle, the radian, is defined, it is noted that:

where θ is the angle subtended by an arc of length s that is equal to the radius of the circle. By rearranging this equation, the length of the arc can be presented as:

Suppose the forearm, with length r1, rotates about the elbow joint (Fig. 9-21). The arc described by the rotation– the distance that the wrist moves–is Δs1, and the angle is Δθ. The linear distance that the wrist travels is described as:

Therefore, the linear distance that any point on the segment moves can be described if the distance of that point to the axis of rotation and the angle through which the segment rotates are known. Suppose another point on the arm is marked as s2 with a distance of r. to the axis of rotation. The distance this point travels during the same angular motion is:

Because r1 is longer than r2, the distance traveled by s1 must be greater than s2. Thus, the most distal points on a segment travel a greater distance than points closer to the axis of rotation. The value for the expression r is called the radius of rotation and refers to the distance of a point from the axis of rotation.

 FIGURE 9-21 The relationship between linear and angular displacement.

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Consider that the change in the angle, Δθ, is very small; then the length of the arc, Δs, can be approximated as a straight line. Therefore, a relationship between angular and linear displacement can be formulated. That is, when r is the radius of rotation:

Linear displacement = Radius of rotation * Angular displacement

or

or using calculus (i.e., when dθ is very small):

For example, if the arm segment of length 0.13 m rotates about the elbow an angular distance of 0.23 radians, the linear distance that the wrist traveled is:

Δs has a unit of length m, which is the correct unit because it is a linear distance. Note that radians are dimensionless, so that the product radians times meters results in units of meters.

Linear and Angular Velocity

The relationship between linear and angular velocity is similar to the relationship between linear and angular displacement. In the example in the last section, the arm, with length r, rotates about the elbow. The linear displacement of the wrist is the product of the distance r, the radius of rotation, and the angular displacement of the segment. Differentiating this equation with respect to time:

Thus, the linear velocity of a point on a rotating body is the product of the distance of that point from the axis of rotation and the angular velocity of the body. The linear velocity vector in this expression is instantaneously tangent to the path of the object and is referred to as the tangential velocity, or vT (Fig. 9-22). That is, the linear velocity vector behaves as a tangent, touching the curved path at only one point. The linear velocity vector, therefore, is always perpendicular to the rotating segment.

For example, if the arm segment of length r = 0.13 m rotated with an angular velocity of 2.4 rad/s, the velocity of the wrist is:

 FIGURE 9-22 Tangential velocity of a rotating segment at different instants in time. The tangential velocity is perpendicular to the radius of rotation.

Linear velocity is expressed in meters per second, which results in this instance from the product of meters times radians per second because radians are dimensionless.

The relationship between linear velocity and angular velocity is a critical piece of information in a number of human movements, particularly those in which the performer throws or strikes an object. To increase the linear velocity of the ball, for example, a soccer player can either increase the angular velocity of the lower extremity segments or increase the length of the extremity by extending at the joints, or both, to gain the maximum range of a kick. For an individual, the major alternative is to increase the angular velocities of these segments. For example, Plagenhoef (23) reported foot velocities before impact of 16.33 m/s to 24.14 m/s for several types of soccer kicks for the same individual. Because the segment lengths did not change substantially, if the foot velocity changed, the angular velocity certainly must have varied for each type of kick.

In some activities, however, the length of the radius r can change. For example, in golf, the clubs have varying lengths and club head lofts according to the desired distance for the ball to travel (Fig. 9-23). For example, the two-iron is longer than the nine-iron and has a different club head loft, with the nine-iron having a greater club head loft than the two-iron. If both clubs had the same loft, a two-iron shot would go farther than a nine-iron shot, given the same angular velocity of the golf swing, as it does for most expert golfers. Golfers often use the same club but vary the length, r, by choking up on the handle, that is, gripping the club closer to the middle of the shaft. Using this technique, a golfer may swing with the same angular velocity but vary the length, thereby varying the linear velocity of the club head.

Linear and Angular Acceleration

Note that the linear velocity vector calculated from the product of the radius and the angular velocity is tangent to the curved path and can be referred to as the tangential velocity. As previously stated:

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 FIGURE 9-23 Comparison of the lengths of the shafts of golf clubs, a 2-iron (left) and a 9-iron.

If the time derivative of this expression is determined, the relationship expresses the tangential acceleration in terms of the radius of rotation and the angular acceleration. The expression of the derivative is:

where aT is the tangential acceleration, α is the radius of rotation, and a is the angular acceleration. The tangential acceleration, similar to the tangential velocity vector, is a vector tangent to the curve and perpendicular to the rotating segment (Fig. 9-24). In any activity, in which the performer spins to propel the implement (e.g., the discus throw), the purpose is to throw the object as far as possible. Therefore, to understand this activity, an understanding of tangential velocity and tangential acceleration is necessary. The time rate of change in the tangential velocity of the object along its curved path is the tangential acceleration. The peak tangential velocity is ideally reached just before the release of the object, at which time the tangential acceleration must be zero.

 FIGURE 9-24 The tangential acceleration of a swinging segment. It is perpendicular to the swinging limb.

Consider a softball pitcher using an underhand pitch; further insight into another component of linear acceleration acting during rotational movement can be gained. As the pitcher moves the arm to the point of release of the pitch, the ball follows a curved path. Because the pitcher's arm is attached to the shoulder, the ball must follow the curved path produced by the rotation of the arm. Therefore, to continue on this path, the ball moves slightly inward and slightly downward at each instant in time until the ball is released (Fig. 9-25). That is, the ball is incrementally accelerated downward and inward toward the shoulder, or the axis of rotation.

Two components of acceleration produced by the rotation of a segment have been discussed: one tangential to the path of the segment and one along the segment toward the axis of rotation. These two accelerations are necessary for the ball in the pitcher's hand to continue on its curved path. The forward movement is the result of the tangential acceleration that has been previously discussed. Acceleration toward the axis or center of rotation, however, is called centripetal acceleration (Fig. 9-26). The adjective centripetal means center seeking. Centripetal acceleration is also known as radial acceleration. Either name is correct, although for the remainder of this discussion, the term centripetal accelerationwill be used.

To derive the formula for centripetal acceleration, the resultant linear acceleration of a segment end point, such as the wrist, of a rotating segment is:

 FIGURE 9-25 The directions of the acceleration components of the wrist of a softball or baseball pitcher during the downswing of the arm to the release of the ball. The wrist is accelerated in toward the shoulder and down tangential to the path of the wrist. These two vectors are perpendicular to each other.

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 FIGURE 9-26 Tangential acceleration (aT) and centripetal acceleration (aC), which are perpendicular to each other. The tangential acceleration accelerates the segment end point downward, and the centripetal acceleration accelerates the end point toward the center of rotation. The result is motion along a curved path.

Because the segment is rotating, the linear velocity is:

Substituting this into the acceleration equation, the acceleration becomes:

If certain computational rules from calculus are applied, this equation becomes:

Because dr/dt is the linear velocity and dω/dt is the angular acceleration of the segments, this expression is:

The linear velocity, v, is equal to ωr, so the expression for the linear acceleration of the segment end point is:

or

Remember that the resultant acceleration has two components that are perpendicular to each other. This expression illustrates these two components. This explanation requires the use of vector calculus and is much more complicated in derivation than presented. The addition (+) sign in this expression means vector addition. It was previously determined that αr was the tangential acceleration so to2r is centripetal acceleration. The expression for centripetal acceleration is:

Centripetal acceleration may also be expressed in the following form as a function of the tangential velocity and the radius of rotation. That is, if v = ωr is substituted into the centripetal acceleration equation, the equation becomes:

From this expression, it can be seen that the centripetal acceleration will increase if the tangential velocity increases or if the radius of rotation decreases. For example, the usual difference between an indoor running track and an outdoor track is that the indoor track is smaller and thus has a smaller radius. If a runner attempted to maintain the same velocity around the indoor turn as on an outdoor track, the centripetal acceleration would have to be greater for the runner to accomplish the turn. Generally, the runner cannot accomplish the turn at the same velocity as outdoors, so race times on indoor tracks are somewhat slower than on outdoor tracks.

 FIGURE 9-27 The resultant linear acceleration vector (aR) comprised of the centripetal and tangential acceleration components.

Because centripetal and tangential acceleration are components of linear acceleration, they must be perpendicular to each other. The acceleration vector of these components may then be constructed. The resultant acceleration (Fig. 9-27) is computed using the Pythagorean relationship:

In computing either the tangential or the centripetal acceleration, the units of angular velocity and angular acceleration are radians per second and radians per second squared, respectively. The units of linear acceleration (meters per second squared) can result only when a radian-based unit is used in the computation.

Angle-Angle Diagrams

In most graphical presentations of human movement, usually some parameter (e.g., position, angle, velocity) is graphed as a function of time. In certain activities, such as locomotion, the motions of the segments are cyclic; that is, they are repetitive, with the end of one cycle at the beginning of the next. In these instances, an angle-angle diagrammay be useful to represent the relationship between two angles during the movement. An angle-angle diagram is the plot of one angle as a function of another angle. That is, one angle is used for the x-axis

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and one for the y-axis. In an angle-angle diagram, one angle is usually a relative angle (angle between two segments) and the other is an absolute angle (angle relative to reference frame). For the angle-angle graph to be meaningful, a functional relationship between the angles should exist (Fig. 9-28). For example, whereas in studying an individual running, the relationship between the sagittal view ankle and knee angles may be meaningful, the relationship between the elbow angle and the ankle angle may not.

One problem with this type of diagram is that time cannot be easily represented on the graph. It can be presented, however, by placing marks on the angle-angle curve to represent each instant in time at which the data were calculated. These marks are placed at equal time intervals and give an indication of the angular distance through which each joint has moved in equal time intervals. Thus, angular velocity of the movement is represented, because the farther apart the marks are on the curve, the greater the velocity of the movement. Conversely, the closer together the marks, the less the velocity (Fig. 9-29).

Angle-angle diagrams have proved useful in the examination of the relationship between the rearfoot angle and the knee angle (1,34). This relationship is based on the related anatomical motions of the subtalar and knee joints. During the support phase of gait, the knee flexes at touchdown and continues to flex until midstance. At the same time, the foot lands in an inverted position and immediately begins to evert until midstance. Both knee flexion and subtalar eversion are associated with internal tibial rotation. After midstance, the knee extends and the subtalar joint inverts. Both of these joint actions result in external tibial rotation. These actions are presented in Figure 9-29 with the knee angle expressed as a relative angle and subtalar joint inversion and eversion expressed as an absolute angle.

Figure 9-30, an angle-angle diagram presented in a paper by van Woensel and Cavanagh (34), illustrates the relationship between the knee angle and the rearfoot angle in different shoe conditions. One of the three shoes used in this study was specifically designed to force the

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runner to pronate during support, another was designed to force the runner to supinate during support, and the third pair was a neutral shoe.

 FIGURE 9-28 Angle-angle diagrams of the knee angle as a function of the thigh angle (A) and the knee angle plotted as a function of the ankle angle (B) for one complete running stride of an individual running at 3.6 m/s. TO, toe-off; FS, foot strike. (Adapted from  Williams, K. R. [1985]. Biomechanics of running. Exercise and Sports Sciences Review, 14. ) FIGURE 9-29 Angle-angle diagram of knee flexion as a function of subtalar pronation angle for an individual running at 6 min/mile on a treadmill. The dots on the curve indicate equal time intervals. (Adapted from  Bates, B. T, et al. [1978, fall]. Foot function during the support phase of running. Running, 24:29. )
 FIGURE 9-30 Knee-rearfoot angle-angle diagram of an individual in three types of running shoes. FS, foot strike. The varus shoe has a medial wedge, which mediates rearfoot pronation; the valgus shoe has a lateral wedge, which enhances rearfoot pronation; and the neutral shoe is a normal running shoe. (Adapted from  van Woensel, W., Cavanagh, P. R. [1992]. A perturbation study of lower extremity motion during running. International Journal of Sports Biomechanics, 8:30-47 ).

In many research studies, angle-angle diagrams are presented but not used in quantification of the movement. More recently, however, researchers have begun to use what is referred to as a modified vector coding technique to quantify angle-angle plots (12). This technique is used to determine the angle between each pair of contiguous points throughout a cycle (Fig. 9-31). The relative motion between these points has been used as a measure of coordination between the angles representing either segments or joints (12,24).

Using walking data in Appendix C, calculate the absolute angles of the thigh and leg for the frame denoting right foot contact and the next three frames. Using these absolute angles, calculate the relative motion between the thigh and the leg using the modified vector coding technique.

Angular Kinematics of Walking and Running

Many researchers have reported on how the lower extremity joint angles vary throughout the walking and running stride, particularly during the support portion of the stride. An angular kinematic analysis of walking and running typically includes a graphical presentation of joint actions as a function of time. Although some researchers have studied patterns of angular velocity and acceleration in both running and walking, the major focus of investigation has been on the characteristics of angular positions and displacements at critical events in the locomotion cycle. For both walking and running, the greatest range of motion occurs in the sagittal plane, and segment movements in this plane are often used to describe gait characteristics. The calculation of sagittal plane angles can be accomplished with a 2D analysis. However, movement in the other planes can be as critical to successful gait, but obtaining these angles requires a 3D analysis.

Lower Extremity Angles

Sagittal, frontal, and transverse plane joint angular kinematic patterns for walking, running, and sprinting are shown in Figure 9-32. Although there are obvious magnitude differences where the angular displacements increase with speed of locomotion, the patterns are similar across the speeds of locomotion with some temporal phasing differences. The one exception is at the ankle joint, where there is less and less plantar flexion at heel strike as locomotion speed increases until a point in very fast running at which plantar flexion may be absent (6).

 FIGURE 9-31 A representation of the modified vector coding technique. The angle between each pair of contiguous points is calculated relative to the right horizontal.

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 FIGURE 9-32 Gait analysis commonly includes a recording of angular kinematics across the gait cycle, including the support phase (percent of cycle up to the vertical line) and the swing phase (percent of cycle past the vertical line). The angular kinematic differences and similarities between walking (solid line), running (dashed line), and sprinting(dotted line) become apparent when graphed against one another (Adapted from  Novacheck, T F. [1995]. Instructional Course Lectures. Park Ridge, IL, 44:497-506. )

As contact is made with the ground in both walking and running, a loading response absorbs body weight. The angular kinematics that accompany this response are hip flexion, knee flexion, and ankle dorsiflexion. As the body continues over the foot in midstance, these movements continue until the terminal stages of stance, where there is a reversal into hip extension, knee extension, and plantarflexion.

The initial touchdown hip flexion angle for walking and running has been reported to be in the range of 35° to 40° and 45° to 50°, respectively (17,21,22). In the early phases of contact, the hip adducts in the reported range of 5° to 10° and 8° to 12° for walking and running, respectively. After touchdown, the amount of hip flexion reduces over the course of the support until toe-off, at which 0° to 3° of hip extension in walking and 3° to 5° of hip extension in running is reported (17,21,22). There is also hip movement into abduction at toe-off in the range of 2° to 5° for both walking and running. When the limb is off the ground in the swing phase, hip flexion maximum values are reported in the range of 35° to 50° for walking and 55° to 65° for running. Hip abduction in the initial portion of the swing phase is similar for walking and running, reported to be in the range of 3° to 8°. Hip adduction late in the swing phase varies more between walking and running, in the range of 0° to 5° and 5° to 15°, respectively (17,21,22).

The knee angle is flexed at touchdown and has been reported in the literature to be in the range of 10° to 15° for walking (17,21,22) and 21° to 40° (1,5,8,9,17,21,22) for running. After touchdown, the knee flexes to values ranging from 20° to 25° for walking and 38° to 60° for running, with the greater flexion occurring at faster speeds (1,2). The knee flexion movement helps to lower the body in stance. Maximal knee flexion occurs at midstance, after which the knee extends until toe-off. Full extension is not achieved at toe-off; values range from 10° to 40° in walking and 18° to 40° in running, depending on speed (4,8,17,21,22). Greater extension values at toe-off are generally associated with faster speeds. In the swing phase, knee flexion is important to shorten the swing leg before the hip flexion brings the limb forward. The range of knee flexion in walking and running is reported to be in the range of 50° to 65° and 100° to 125°, respectively (17,21,22).

Again, although the magnitude of the knee angles at these specific instants in time during the support phase of running vary, the profile of the curve does not. The profile of the knee joint angle appears to be relatively stable and immune to distortion from influences such as running shoe construction (9,34) or delayed-onset muscle soreness (10).

In walking, the ankle is plantarflexed in a reported range of 5° to 6° at heel strike and moves into 10° to 12° of dorsiflexion before returning to 15° to 20° of plantarflexion at toe-off. During the swing phase of walking,

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the foot continues on through 18° to 20° of plantarflex-ion and then dorsiflexes in a reported range of 2° to 5° in preparation for the next heel strike. As the speed of locomotion increases, there is less plantarflexion at heel strike until dorsiflexion is the movement occurring at touchdown. Depending on the speed, the reported range of dorsiflexion at heel strike in running is 10° to 17°, increasing to 20° to 30° in midstance and then moving to plantarflexion at toe-off (range, 10° to 20°). Plantarflexion continues into the initial phases of the swing phase (range, 15° to 30°) and then moves, as in walking, into dorsiflexion in the reported range of 10° to 15° (17,21,22).

Alterations in the angular kinematics of the lower extremity joints during both walking and running occur in response to changes in the environment. For example, when running over a noncompliant surface, some individuals make a kinematic adjustment at impact by responding with more initial knee flexion at contact (7). Walking uphill brings about a number of adjustments in the lower extremity. For example, increasing the grade from 0% to 24%, the lower extremity adjustments start at heel strike with a 22% increase in dorsiflexion, 31% more knee flexion, and 23% more hip flexion (16). Over the stance phase, there are unequal adjustments at the three lower extremity joints, with the hip joint undergoing the greatest increase in the range of motion (+ 59%), followed by the ankle (+ 20%) and an actual decrease (-12%) range of motion at the knee (16). The primary adjustment to walking downhill during the stance phase occurs at the knee joint, where there is as much as 15° more knee flexion in early stance (15). Movement adjustments in the swing phase occur at the hip and ankle, with less hip flexion and less plantarflexion.

Rearfoot Motion

A number of investigations have described the rearfoot angle during the support phase of walking and running. Excessive rearfoot motion during running has been hypothesized to cause a variety of lower extremity injuries, although little evidence directly relates excessive rearfoot motion and injury (5,20). In fact, a valid, clinical definition of excessive rearfoot motion has not yet been determined. From a functional standpoint, eversion of the calcaneus is necessary because it allows the foot to assume a flat position on the ground. Typically, maximum rearfoot angle values from -6° to -17° at midstance have been reported in the literature (5,11) for running and in the range of -9.2 to -12.9° for walking (14,26). This wide range in maximum values can be attributable to differences in the anatomical foot structure of individuals as well as the influence of footwear. It has been reported that more extreme rearfoot angles occurs at midstance in running when the subjects wear racing shoes compared with training shoes (11). More extreme rearfoot eversion angles have also been reported for runners in a shoe with a very soft midsole than in a shoe with a firmer midsole (9). Although the rearfoot angle is related in motion to the knee angle by the action of tibial rotation, it is, unlike the knee angle, highly variable and certainly can be influenced by many factors.

The simultaneous actions of these two lower extremity angles has been a topic of several investigations. Because internal tibial rotation accompanies knee flexion and subtalar joint eversion and both reach a maximum at mid-stance, the mistiming of these joint actions has been suggested as a possible mechanism for lower extremity injury (1). Hamill et al. (9) illustrated that the rearfoot angle could be changed by a running shoe with a very soft midsole, but the knee angle could not. These researchers reported that in a soft midsole running shoe, the maximum rearfoot angle occurs sooner in the support period than did maximum knee flexion. The subtalar joint also stays at this maximum while the knee begins to extend. Thus, they surmised that a twisting action may be applied to the tibia by the differential speeds at which the tibia rotated early in support and late in support. Because the tibia is a rigid structure and may be difficult to twist, the tibia may continue to rotate internally at the knee, even though it should externally rotate. This undesirable action at the knee may possibly cause knee pain in the runner. If these actions are repeated with each foot-to-ground contact and the runner has many foot-to-ground contacts, the runner may be subject to a knee injury that would prohibit training. This type of injury is often referred to as an overuse injury. It results from an accumulation of stresses rather than a single high-level, traumatic stress.

Clinical Angular Changes

Locomotion is also influenced by a variety of medical conditions and functional limitations. For example, the walking gait of an individual with Parkinson's disease usually exhibits small, quick steps and less range of motion in the lower extremity joints. A tight hip flexor (e.g., psoas muscle) in individuals with cerebral palsy can limit the hip extension during the stance phase. This causes an increase in pelvic tilt (28). Specific adjustments in the joint kinematics of individuals with hemiplegia may show a reduction in the range of motion at the knee joint, with increases in range of motion at the ankle and excessive hip and knee motion in the swing phase. Finally, individuals with injury to one limb typically compensate for pain in one limb by altering range of motion in both limbs so they can increase the time spent on the limb with no pain.

Angular Kinematics of the Golf Swing

In Chapter 8, it was pointed out that the linear speed and path of the club head are important determinants of a successful golf shot. These two linear kinematic components

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of the swing are the result of a series of angular movements; thus, the angular kinematics of the golf swing is commonly the primary focus of golf instruction and evaluation. The golf swing can be accurately described using a double pendulum model, with one link being the arm rotating about the shoulder joint and the second link being the club and wrist, where the wrist acts as a hinge about which the club rotates (18). A third connected link has been suggested between the shoulder and what is referred to as the hub axis as the body rotates about a vertical axis (31). For purposes of this introduction to the golf swing, the primary focus will be on the double pendulum characteristics of the swing.

In a golf swing, the left arm (for a right-handed golfer) sets the plane of the swing (31). The swing plane is an elliptical plane around the body that brings the club head into contact with the ball from the inside, where the face of the club ends up perpendicular to the flight path of the ball. The swing is a pivot about an axis running through the base of the neck with the head stationary (29). Many beginning golfers try to create an upright swing plane with the club brought straight back and straight forward. When the club head comes down to meet the ball with this swing plane, it is never square with the ball and puts spin on the ball at contact.

The angular positions of the club at various stages of the swing are good predictors of a successful swing. A frontal view of the golf swing provides a good perspective for evaluating club position (Fig. 9-33). In the first stage of the swing, the golfer addresses the ball. This position at the beginning of the swing should be the same as the position at impact. It establishes the arm and shoulder position that will bring the club into precise alignment for impact (29). The club and the left arm should form a straight line, and the club face should be aimed down a perpendicular line from the ball forward in a straight line (Fig. 9-34A). As the club is started in the backswing, there is an initial takeaway phase where the club head is taken back away from the ball. This is initiated with a weight shift to the rear that allows for greater range of motion at the hip and flattens the arc of the swing. A long takeaway is preferred: The club travels in a wide arc, and the wrist does not allow movement of the club until the hands are chest high. This increases the distance for the club head to travel as the shoulders are rotated farther from the target. At the end of the takeaway phase, the left arm should be horizontal to the ground, and the club should be vertical and perpendicular to the arm (Fig. 9-34B). Continuing to the top of the backswing, the upper body has rotated to allow the club to be positioned parallel to the ground again and parallel to the final target line for ball contact. The right elbow flexes at the end of the backswing to reduce the length and allow for more acceleration. The left arm continues to be straight and vertical. This position ensures that the club face will travel squarely to the ball at contact (Fig. 9-34C). From the top of the backswing, the downswing begins as the club shaft and the left arm drop in one piece to the position halfway down, where the left arm is again parallel to the ground and the club is vertical (Fig. 9-34D). Hip rotation and the legs initiate this movement as they drive forward, dropping the right shoulder and the shaft into place. The impact position should duplicate the initial address position, with the left arm and club forming a straight vertical line and the club face traveling in a straight line through the ball (Fig. 9-34E). If these angular positions can be obtained within the context of a fluid swing, the ball will travel far and accurately.

The interaction of the arm and club links are shown in the displacement, velocity, and acceleration curves in the downswing phase illustrated in Figure 9-34. The displacement of the arm segment in the downswing is 100° to 270°, and the displacement of the club relative to the arm is 50° to 175°. As the shoulder displacement increases in the early phases, the wrist angle remains constant until it uncocked in the later stages of the downswing (18). This uncocking increases dramatically 80 to 100 ms before

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impact as the club is brought in line with the hands (19). The interaction between the arm and the club segments enhances the velocity and acceleration of the club at impact. This is illustrated in the angular velocity graph, where the arm velocity moves through a range of 250°/s, increasing to 800°/s and reducing velocity to 500°/s at impact. The resulting effect on the club segment is a build in velocity from zero initially to a culminating 2300 to 4000°/s at impact (18,19). Angular accelerations of the club are minimal in the beginning of the downswing and increase rapidly to values approaching 10,000°/s/s at a point where the angular acceleration of the arm is reduced to zero and begins the negative acceleration (18).

 FIGURE 9-33 Critical angular positions in the phases representing: the address (A), take-away (B), top of backswing (C), downswing (D), and impact (E) determine the success of the golf swing. FIGURE 9-34 One model used to study the golf swing is the double pendulum. Displacement, velocity, and acceleration data for the arm (dashed line) and the club motion relative to the arm (solid line) illustrate the unique motion characteristics of each segment. (Adapted from  Milburn, P. D. [1982]. Summation of segmental velocities in the golf swing Medicine, Science in Sports and Exercise, 14:60-64. )

Using MaxTRAQ, import the first two video files of the golfer. Digitize the right shoulder, left shoulder, right elbow, left elbow, and the left wrist in each frame. Calculate the absolute angles of the upper arm and the lower arm.

Angular Kinematics of Wheelchair Propulsion

Angular kinematic characteristics of the trunk and joint actions at the shoulder, elbow, and wrist are the focus of many investigations of wheelchair propulsion. Both linear and angular kinematics are constrained because the hand must follow the rim (32). Differences in hand position on the rim as well as different seat positions and other adjustments, however, can considerably alter the angular kinematics. A stick figure illustrating the sagittal angular positions of the arm, forearm, and hand segments during wheelchair propulsion is shown in Figure 9-35. The angular positions are shown for various stages in the event at a rim contact position that is -15° with respect to top dead center continuing on through +60° in 15° increments.

The range of motion in the elbow and shoulder joints has been reported to be an average of 55° to 62° of elbow flexion and extension, 60° to 65° of shoulder flexion and extension, 20° of shoulder abduction and adduction, 36° of shoulder internal and external rotation, 35° of wrist flexion and extension, and 68° to 72° of wrist ulnar and radial flexion (13,25). There is also a reported approximate 37° of pronation and supination (3). The trunk contributes to wheelchair propulsion via flexion in the propulsive phase and extension in the recovery phase after hand release (33). Angular velocity and acceleration during wheelchair propulsion have not been studied extensively, but reported values approach 300°/s for elbow extension, even at slow speeds (1.11 m/s) (35).

If propulsion is made with a lever rather than on the hand rim, the angular kinematics change, requiring more elbow range of motion, less shoulder extension, more shoulder rotation, and more shoulder abduction (13). Likewise, speed-dependent changes are seen in angular

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displacement. It has been reported that with increased speed of propulsion, the trunk displacement increases and shoulder displacement decreases (27). Propelling up a slope also influences angular kinematics, resulting in greater trunk displacement and an increase in arm displacement at the shoulder. Finally, seat adjustments influence angular kinematics, depending on the direction and level of alteration (13).

 FIGURE 9-35 Angular positions of the upper extremity during wheelchair propulsion at 1.11 m/s. (Adapted from  Van der Helm, F. C. T., Veeger, H. E. J. [1996]. Quasi-static analysis of muscle forces in the shoulder mechanism during wheelchair propulsion. Journal of Biomechanics, 29:39-52. )

Using MaxTRAQ, import the video files of the individual in the wheelchair. Digitize the right shoulder, right elbow, and right wrist for five frames after the initial propulsion phase. Calculate the absolute angles of the upper arm and the lower arm, the angular velocity, and the angular acceleration. Note that the time between frames is 0.0313 s.

Summary

Nearly all purposeful human movement involves the rotation of segments about axes passing through the joint centers; therefore, knowledge of angular kinematics is necessary to understand human movement. Angles may be measured in degrees, revolutions, or radians. If the angular measurement is to be used in further calculations, the radian must be used. A radian is equal to 57.3°.

Angles may be defined as relative and absolute, and both may be used in biomechanical investigations. A relative angle measures the angle between two segments but cannot determine the orientation of the segments in space. An absolute angle measures the orientation of a segment in space relative to the right horizontal axis placed at the distal end of the segment. How segment angles are defined must be clearly stated when presenting the results of any biomechanical analysis.

The kinematic quantities of angular position, displacement, velocity, and acceleration have the same relationship to each other as their linear analogs. Thus, angular velocity is calculated using the first order central difference method as follows:

Likewise, angular acceleration is defined as:

The techniques of differentiation and integration apply to angular quantities as well as linear quantities. Angular velocity is the first derivative of angular position with respect to time and angular acceleration is the second derivative. The concept of the slope of a secant and a tangent also applies in the angular case to distinguish between average and instantaneous quantities. Integration implies the area under the curve. Thus, the area under a velocity-time curve is the average angular displacement, and the area under an acceleration-time curve is the average angular velocity.

It is difficult to represent angular motion vectors in the manner in which linear motion vectors were represented. The right-hand rule is used to determine the direction of the angular motion vector. Generally, rotations that are counterclockwise are positive, and clockwise rotations are negative.

Sagittal view lower extremity angles were defined in this chapter using a system suggested by Winter (36). In this convention, ankle, knee, and hip angles were defined using the absolute angles of the foot, leg, thigh, and pelvis segments. The rearfoot angle measures the relative motion of the leg and the calcaneus in the frontal plane and is calculated from the absolute angles of the calcaneus and the leg.

There is a relationship between linear and angular motion. Comparable quantities of the two forms of motion may be related when the radius of rotation is considered. The linear velocity of the distal end of a rotating segment is called the tangential velocity and is calculated as:

where ω is the angular velocity of the segment and r is the length of the segment. The derivative of the tangential velocity, the tangential acceleration is:

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Equation Review for Angular Kinematics

 Purpose Given Formula Relative angle between two segments using the law of cosines Length of segment a and b and distance between end of a and b (length c) θ=arccos(b2+c2-a2/2 * b* c) Absolute angle Endpoints – Horizontal and vertical components θ=arctan (ydistal-yproximal-xdistal-xproximal) Calculate position Starting position relative to origin, constant velocity (zero acceleration), and time θ=θInitial+ωinitialt Calculate position Starting position at origin, constant velocity (zero acceleration), and time θ=ωinitialt Calculate position Initial velocity, time, constant acceleration θ=ωit+1/2αt2 Calculate position Initial velocity =Zero, time, constant acceleration θ=1/2αt2 Calculate final displacement Final angular velocity; constant angular acceleration θ=ω2/2*α Calculate average velocity Displacement and time ω=θ2-θ1/t2-t1 Calculate average velocity Initial and final velocity ω=ωinitial+ωfinal/2 Calculate final velocity Initial velocity, constant acceleration, and time ωf=ωinitial+αt Calculate final velocity Starting velocity = Zero, constant acceleration, and time ω=αt Calculate final velocity Velocity at time = Zero, constant acceleration, initial position relative to origin, final position ω=√ωinitial2+2α(θ-θini tial) Calculate final velocity Initial velocity = Zero, constant acceleration, initial and final position ωf2=2αθ Calculate acceleration Final velocity and displacement ω=√2α(θ-θinitial) Calculate average acceleration Velocity and time α=ω2-ω1/t2-t1 Calculate average acceleration Displacement, time α=2tH/t2 Calculate time Displacement, constant acceleration t=√2θ/α Calculate linear distance Radius, angular displacement S=rθ Calculate linear velocity (tangential) Radius, angular velocity V=rω Calculate linear acceleration (tangential) Radius, angular acceleration a=rα Centripetal acceleration Radius, angular velocity ac=w2r Centripetal acceleration Radius, tangential linear velocity ac=v2/r

where α is the angular acceleration of the rotating segment. The other component of the linear acceleration of the end point of the rotating segment is the centripetal or radial acceleration. This is expressed as:

The tangential and centripetal acceleration components are perpendicular to each other.

A useful tool in biomechanics is the presentation angular motion in angle-angle diagrams. These diagrams generally present angles of joints that are anatomically functionally related. Time can be presented only indirectly on this type of graph, however. Recently, angle-angle plots have been quantified using a modified vector coding technique (12).

Review Questions

True or False

1. ____A thigh segment with an angular velocity of -3.2 rad/s is extending
2. ____Choking up on a bat in baseball is done to increase the angular velocity of the bat.
3. ____With increased speed of running, there is a concomitant increase in the amount of plantarflexion at heel strike
4. ____Angular acceleration of the golf club is at the maximum level when the left arm's angular acceleration is zero.
5. ____All points on a thigh flexing through 75° undergo a similar amount of rotation
6. ____Linear acceleration toward the axis of rotation in a rotating lever is termed centripetal

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1. ___The relative angle between the trunk and the arm abducted out to the side is 90°.
2. ___Trunk extension accompanies shoulder flexion and elbow extension in wheelchair propulsion
3. ___The absolute angle of the thigh when standing upright is 0°.
4. ___Walking uphill results in an increase in motion at the hip, knee, and ankle joints, and walking downhill results in a decrease in motion
5. ___A radian is an angle formed when the length of the radius is equal to the length of the arc.
6. ___There is minimal motion at the wrist joint during wheelchair propulsion
7. ___Linear displacement of a point on a lever is a function of the length of the radius and the angular velocity.
8. ___A vertex is the angle between two segments
9. ___Positive angular acceleration indicates that counterclockwise movement is occurring
10. ___Maximum rearfoot angle occurs at heel strike in running
11. ___The law of cosines provides the tool for computing relative angles between two segments not at a right angle to each other.
12. ___In human locomotion, the knee joint angle is always in flexion at contact in the support phase.
13. ___Angle-angle diagrams usually plot a relative angle against an absolute angle.
14. ___Absolute angles are always computed with respect to the right horizontal
15. ___Most movements in running occur in the frontal plane
16. ___The swing plane for golf is upright.
17. ___The initiation of the downswing in golf occurs with the change in angle of the wrists
18. ___The units for a radian are degrees
19. ___Angular velocity is the slope of secant to an angle-time curve.

Multiple Choice

Questions 1-4: A golf club 1.01 m long completes a downswing in 0.25 s through a range of 180°. Assume a uniform angular velocity.

1. What is the average angular velocity of the club?
3. 727.27s
5. 6597s
6. What is the linear distance moved by the end of the club?
7. 181.8 m
8. 172.6 m
9. 2.45 m
10. 3.17 m
11. What is the tangential velocity of the end of the club?
12. 12.70 m/s
13. 13.56 m/s
14. 11.59 m/s
15. 3.20 m/s
16. What is the tangential acceleration of the end of the club?
17. 101.53 m/s2
18. 100.52 m/s2
19. 50.78 m/s2
20. 48.34 m/s2
21. During an elbow flexion exercise, the relative angle at the elbow is 22° at 0.3 s and 155° at 0.62 s. What is the angular velocity of the elbow?
22. 443.97s
23. -443.97s
24. -415.67s
25. 415.67s
26. If the angular velocity at 0.19 s is 3.2 rad/s and 1.2 rad/s at 0.27 s, what is the average angular acceleration over this time interval?
27. -25 r/s2
28. 25 r/s2
29. 7.41 r/s2
30. -7.41 r/s2
31. If a skater is rotating during a spin at a constant angular velocity of 2.5 rad/s for 1.1 s, what is the angular acceleration?
32. 2.27 r/s2
33. -2.27 r/s2
34. 0 r/s2
35. No movement
36. An object is spinning at 4 rev/s and starts decelerating at 0.9 rev/s2. How long will it take the object to stop?
37. 3.6 s
38. 3.89 s
39. 4.44 s
40. 4.72 s
41. The kicking phase of a soccer kick takes place over 0.096 s and includes hip flexion (23°) and knee extension (29°). If the thigh is 0.61 m long and the leg 0.51 m long, what is the tangential velocity of the foot resulting from these movements?
42. 2.55 m/s
43. 2.69 m/s
44. 5.24 m/s
45. 5.55 m/s
46. If a discus thrower rotates through the last 45° of a turn in 0.11 s and the distance from the axis of rotation to the center of the discus is 1.22 m, calculate the average linear velocity of the discus
47. 499.09 m/s
48. 502.10 m/s
49. 8.71 m/s
50. 9.02 m/s
51. An object is spinning at a constant angular velocity of 13.9 rad/s. If it spins for 1.9 s, what is the angular distance traveled?

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1. A hammer thrower releases the hammer after reaching an angular velocity of 14.9 rad/s. If the hammer is 1.6 m from the shoulder joint, what is the linear velocity at release?
2. 853.77 cm/s
3. 845.34 cm/s
4. 23.84 m/s
5. 21.79 m/s
6. Calculate the relative angle at the knee and the absolute angles of the leg given the following positions in degree: hip (2.128, 1.891), knee (2.122, 1.642), ankle (1.897, 1.210).
7. Abs = 88.62°; rel = 153.9°
8. Abs = 88.62°; rel = 62.48°
9. Abs = 62.48°; rel = 153.9°
10. Abs = 62.48°; rel = 26.14°
11. During the support phase of walking, the absolute angle of the thigh has the following angular velocities
 Frame Time (s) Angular Velocity (rad/s) 38 0.6167 1.033 39 0.6333 1.511 40 0.6500 1.882 41 0.6667 2.190

Calculate the angular acceleration at frame 40

5. An ice skater rotating around a vertical axis increases in angular velocity from 4507s to 6107s in 2.3 seconds. Find the angular acceleration
6. 2657s2
8. 69.577s2
10. A baseball bat is swing through 140° in 0.7 s. What is the angular acceleration?
11. 200.07/s2
12. 189.17/s2
13. 564.37/s2
14. 571.47/s2
15. Angular velocity of a rotating object changes from 6507s to 4957s in 1.2 seconds. What is the angular acceleration?
16. -131.207/s2
17. 131.207/s2
18. -129.177/s2
19. 129.177/s2
20. A cyclist completes 3.2 cycle revolutions in 1.4 s. What is the angular velocity in radians per second?
21. 2.29
22. 822.86
23. 14.36
24. 8.34
25. An individual's arm segment is 0.18 m long and has an angular velocity of 1177s. What is the tangential velocity of the wrist?
26. 21.06 m/s
27. 2.04 m/s
28. 0.75 m/s
29. 0.38 m/s
30. A diver accelerates through a somersault at 9507s. What is the angular displacement over 0.15 s?
31. 10.69°
32. 6333.3°
33. 142.5°
34. 21.38°
35. An individual is running around a turn with an 11-m radius at 3.75 m/s. What is the runner's centripetal acceleration?
36. 32.2 m/s2
37. 25.2 m/s2
38. 0.34 m/s2
39. 1.28 m/s2
40. The final angular velocity of a golf swing was 3707s with a constant angular acceleration of 5807s2. How far did the club rotate?
41. 210°
42. 118°
43. 156°
44. 213°
45. If the linear velocity of a club head in the golf swing is 50 m/s at impact, what was the angular velocity if the club is 0.95 m long?
48. 52.67s
49. 47.57s
50. A baseball bat is rotating at 350 rpm at contact. What is the angular velocity in radians per second?
51. 2198.9
52. 36.7
53. 6.1
54. 131937.2
55. In the release phase of bowling, the bowler's arm and forearm travel through 0.23 and 0.27 radians, respectively, in 0.03 seconds. If the length of the arm is 0.8 m and the forearm 0.5 m, what is the tangential velocity of the ball resulting from these two movements?
56. 10.64 m/s
57. 14.34 m/s
58. 21.67 m/s
59. 16.67 m/s

References

1. Bates, B. T., et al. (1978). Foot function during the support phase of running. Running, 24:29.
2. Bates, B. T., et al. (1979). Functional variability of the lower extremity during the support phase of running. Medicine and Science in Sports and Exercise, 11(4):328-331.
3. Boniger, M. L., et al. (1997). Wrist biomechanics during two speeds of wheelchair propulsion: An analysis using a local coordinate system. Archives Physical Medicine Rehabilitation, 78:364-372.
4. Cavanagh, P. R., et al. (1977). A biomechanical comparison of good and elite distance runners. In P. Milvy (Ed.). The Marathon: Physiological, Medical, Epidemiological, and Psychological Studies.New York: New York Academy of Science, 328-345.

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2. Czerniecki, J. M. (1988). Foot and ankle biomechanics in walking and running. American Journal of Physical Medicine and Rehabilitation, 67:246-252.
3. Dixon, S. J., et al. (2000). Surface effects on ground reaction forces and lower extremity kinematics in running. Medicine and Science in Sports and Exercise, 32:1919-1926.
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12. Lange, G. W., et al. (1996). Electromyographic and kinematic analysis of graded treadmill walking and the implications for knee rehabilitation. Journal of Orthopedic and Sports Physical Therapy, 23:294-301.
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14. Milburn, P. D. (1982). Summation of segmental velocities in the golf swing. Medicine and Science in Sports and Exercise, 14:60-64.
15. Neal, R J., Wilson, B. D. (1985). 3D kinematics and kinetics of the golf swing. International Journal of Sports Biomechanics, 1:221-231.
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17. Novacheck, T F. (1995). Walking, running, and sprinting: A three dimensional analysis of kinematics and kinetics. Instructional Course Lectures, 44:497-506.
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20. Pollard, C. D., Heiderscheit, B. C, Van Emmerik, R E. A., Hamill, J. Gender differences during an unanticipated cutting maneuver. Journal of Applied Biomechanics21:143-152, 2005.
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Glossary

Glossary

Absolute Angle

Angle of a segment as measured from the right horizontal that describes the orientation of the segment in space.

Angle

A figure formed by two lines meeting at a point, the vertex.

Angle-Angle Diagram

A graph in which the angle of one segment is plotted as a function of the angle of another segment.

Angular Acceleration

The change in angular velocity per unit time.

Angular Displacement

The difference between the final angular position and the initial angular position of a rotating body.

Angular Distance

The total of all angular changes of a rotating body.

Angular Kinematics

The description of angular motion, including angular positions, angular velocities, and angular accelerations, without regard to the causes of the motion.

Angular Motion

Motion about an axis of rotation in which different regions of the same object do not move through the same distance in the same time.

Angular Speed

The angular distance traveled divided by the time over which the angular motion occurred.

Angular Velocity

The time rate of change of angular displacement.

Axis of Rotation

The point about which a body rotates.

Centripetal Acceleration

The component of the linear acceleration directed toward the axis of rotation.

Degree

A unit of angular measurement: 1/360 of a revolution.

General Motion

Motion that involves both translation and rotation.

Instantaneous Joint Center

The center of rotation of a joint at any instant in time.

Joint Angle

Angle between two segments that is relative and does not change with body orientation.

Law of Cosines

The general case of the Pythagorean theorem:

a2 = b2 + c2 – 2ab cos A

where a is the length of the side opposite angle A and b and c are the lengths of the other two sides in a triangle.

Overuse Injury

An injury caused by continual low-level stress on a body.

Polarity

The direction of rotation designated as positive or negative.

See Centripetal Acceleration.

The measure of an angle at the center of a circle described by an arc equal to the length of the radius of the circle (1 rad = 57.3°).

The linear distance from the axis of rotation to a point on the rotating body.

Relative Angle

The angle formed by the longitudinal axes of two adjacent segments whose vertex is at the joint.

Revolution

A unit of measurement that describes one complete cycle of a rotating body.

Right-Hand Rule

The convention that designates the direction of an angular motion vector; the fingers of the right hand are curled in the direction of the rotation and the right thumb points in the direction of the vector.

Rotation

A motion that occurs when not all parts of an object undergo the same displacement.

Segment Angle

The angle of the segment with respect to the right horizontal that is absolute and that changes according to orientation of the body.

Tangent

The ratio of the side opposite an angle to the adjacent angle in a right triangle.

Tangential Acceleration

The change in linear velocity per unit time of a body moving along a curved path.

Tangential Velocity

The change in linear position per unit time of a body moving along a curved path.

Vertex

The intersection of two lines that form an angle.

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