Biomechanical Basis of Human Movement

CHAPTER 9 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. Angular Motion Relationship Between Angular and Linear Motion Measurement of Angles Angle Linear and Angular Displacement Units of Measurement Linear and Angular Velocity Linear and Angular Acceleration Types of Angles Absolute Angle Angle–Angle Diagrams Relative Angle Angular Kinematics of Walking and Lower Extremity Joint Angles Running Hip Angle Knee Angle Lower Extremity Angles Ankle Angle Rearfoot Motion Rearfoot Angle Clinical Angular Changes Representation of Angular Motion Vectors Angular Kinematics of the Golf Swing Angular Motion Relationships Angular Kinematics of Wheelchair Angular Position and Displacement Propulsion Angular Velocity Angular Acceleration Summary Review Questions 337

338 SECTION III Mechanical Analysis of Human Motion Angular Motion Angular motion occurs when all parts of a body move FIGURE 9-2 A gymnast completing a cartwheel as an example of general through the same angle but do not undergo the same lin- motion. The gymnast simultaneously undergoes both translation and ear displacement. The subset of kinematics that deals with rotation. angular motion is angular kinematics, which describes angular motion without regard to the causes of the Measurement of Angles motion. Consider a bicycle wheel (Fig. 9-1). Pick any point close to the center of the wheel and any point close ANGLE to the edge of the wheel. The point close to the edge trav- els farther than the point close to the center as the wheel An angle is composed of two lines, two planes, or a combi- rotates. The motion of the wheel is angular motion. nation that intersect at a point called the vertex (Fig. 9-3). In a biomechanical analysis, the intersecting lines are gen- Angular motion occurs about an axis of rotation that is a erally body segments. If the longitudinal axis of the leg line perpendicular to the plane in which the rotation occurs. segment is one side of an angle and the longitudinal axis For example, the bicycle wheel spins about its axle which is of the thigh segment is the other side, the vertex is the its axis of rotation. The axle is perpendicular to the plane of joint center of the knee. Angles can be determined from rotation described by the rim of the wheel (Fig. 9-1). the coordinate points described in Chapter 8. Coordinate points describing the joint centers determine the sides and An understanding of angular motion is critical to com- vertex of the angle. For example, an angle at the knee can prehend 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 combi- nations of angular motion of multiple segments can result in linear motion of the segment end point seen in throw- ing and many other movements in which end point veloc- ities 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 rota- tional motions. The gymnast undergoes translation as she moves across the ground. At the same time, she is rotat- ing. The combination of rotation and translation is com- mon in most human movements. Line Angle Vertex Line FIGURE 9-1 A bicycle wheel as an example of rotational motion. Points FIGURE 9-3 Components of an angle. Note that the lines are usually seg- A, B, and C undergo the same amount of rotation but different linear ments and the vertex of the angle is the joint center. displacements, with C undergoing the greatest linear displacement.

CHAPTER 9 Angular Kinematics 339 be constructed using the thigh and leg segments. UNITS OF MEASUREMENT The coordinate points describing the ankle and knee joint centers define the leg segment; the coordinate points In angular motion, three units are used to measure angles. describing the hip and knee joint centers define the thigh It is important to use the correct units to communicate segment. The vertex of the angle is the knee joint center. the results of this work clearly and to compare values from study to study. It is also essential to use the correct units Definition of a segment by placing markers on the sub- because angle measurements may be used in further cal- ject at the joint centers makes a technically incorrect culations. The first and most commonly used is the assumption that the joint center at the vertex of the angle degree. A circle, which describes one complete rotation, does not change throughout the movement. Because of the transcribes an arc of 360Њ (Fig. 9-5A). An angle of 90Њ has asymmetries in the shape of the articulating surfaces in most sides that are perpendicular to each other. A straight line joints, one or both bones constituting the joint may dis- has a 180Њ angle (Fig. 9-5B). place relative to each other. For example, although the knee is often considered a hinge joint, it is not. At the knee joint, The second unit of measurement describes the number the medial and lateral femoral condyles are asymmetrical. of rotations or revolutions about a circle (Fig. 9-5A). One Therefore, as the knee flexes and extends, the tibia rotates revolution is a single 360Њ rotation. For example, a triple along its long axis and rotates about an axis through the jump in skating requires the skater to complete 3.5 revo- knee from front to back. The location of the joint center, lutions in the air. The skater completes a rotation of therefore, changes throughout any motion of the knee. The 1260Њ. This unit of measurement is useful in qualitative center of rotation of a joint at an instant in time is called the descriptions of movements in figure skating, gymnastics, instantaneous joint center (Fig. 9-4). It is difficult to locate and diving but is not useful in a quantitative analysis. this moving axis of rotation without special techniques such as x-ray measurements. These measurements are not practi- Although the degree is most commonly understood cal in most situations; thus, the assumption of a static and the revolution is often used, the most appropriate unit instantaneous joint center must be made. for angular measurement in biomechanics is the radian. 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 FIGURE 9-5 Units of angular measurement. A. Revolution. B. Perpendicular Nordin, M., Frankel, V. H. [eds.] [1979]. Biomechanics of the Musculoskeletal and straight lines. C. Radian. System (2nd ed.). Philadelphia: Lea & Febiger.)

340 SECTION III Mechanical Analysis of Human Motion A radian is defined as the measure of an angle at the cen- FIGURE 9-6 Absolute angles: The arm (a), trunk (b), thigh (c), and leg ter of a circle described by an arc equal to the length of the (d) of a runner. radius of the circle (Fig. 9-5C). That is: the convention used must be stated clearly. The absolute ␪ ϭ sրr ϭ 1 radian angle of a segment relative to the right horizontal is also called the segment angle. where ␪ ϭ 1 radian, s ϭ arc of length r along the diame- ter, and r ϭ radius of the circle. Because both s and r have Absolute angles are calculated using the trigonometric units of length (m), the units in the numerator and relationship of the tangent. The tangent is defined based denominator cancel each other out with the result that the on the sides of a right triangle. It is the ratio of the side radian is dimensionless. opposite the angle in question and the side adjacent to the angle. The angle in question is not the right angle in the In further calculations, the radian is not considered in triangle. If the leg and thigh segment coordinate positions determining the units of the result of the calculation. are considered, the absolute angles of both the thigh and Degrees have a dimension and must be included in the leg segments can be calculated (Fig. 9-7). unit of the product of any calculation. It is necessary, therefore, to use the radian as a unit of angular measure- ment instead of the degree in any calculation involving lin- ear motion because the radian is dimensionless. One radian is the equivalent of 57.3Њ. To convert an angle in degrees to radians, divide the angle in degrees by 57.3. For example: 72° = 1.26 rad 57.3° To convert radians to degrees, multiply the angle in radi- ans by 57.3. For example: 0.67 rad * 57.3Њ ϭ 38.4Њ Angular measurement in radians is often determined in multiples of pi (␲ ϭ 3.1416). Because 2␲ radians are in a complete circle, 180Њ may be represented as π radians, 90Њ as ␲/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 chap- ter will use the degree for ease of understanding. Types of Angles ABSOLUTE ANGLE FIGURE 9-7 Absolute angles of the thigh and leg as defined in a coordi- nate system. In biomechanics, two types of angles are generally calcu- lated. The first is the absolute angle, which is the angle of inclination of a body segment relative to some fixed refer- ence in the environment. This type of angle describes the orientation of a segment in space. Two primary conven- tions 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,

CHAPTER 9 Angular Kinematics 341 To calculate the absolute leg angle, the coordinate val- ) ues of the segment end points of the leg are substituted into the formula to define the tangent of the angle: () tan ␪leg ϭ ydistal Ϫ yproximal/xdistal Ϫ xproximal FIGURE 9-8 To calculate absolute angles relative to the right horizontal ϭ yknee Ϫ yankle/xknee Ϫ xankle requires adjustments when the orientation is such that the differences ϭ 0.51 Ϫ 0.09/1.22 Ϫ 1.09 between the proximal and distal end points indicate that the segment is not in the first quadrant. ϭ 0.42/0.13 180Њ, resulting in an absolute angle of 105.4Њ relative to ϭ 3.23 the right horizontal (Fig. 9-8). Next, the angle whose tangent is 3.23 is again determined An absolute thigh angle of 105.4Њ in the second quad- using either the trigonometric tables (see Appendix B) or rant indicates that the thigh is oriented such that the hip a calculator. This is called finding the inverse tangent and joint is closer to the vertical (y) axis and above the horizon- is written as follows: tal (x) axis of the coordinate system. In this case, the thigh would be oriented with the knee to the right of the hip in ␪leg ϭ tanϪ1 3.23 this reference system. When both x and y are negative, the ϭ 72.8Њ value is in the third quadrant, and the angle is computed counterclockwise and relative to the left horizontal, so 180Њ The absolute angle of the leg, therefore, is 72.8Њ from the is still added to adjust the absolute angle so it is relative to right horizontal. This orientation indicates that the leg is the right horizontal. Finally, if there is only a negative y- positioned so that the knee is farther from the vertical (y) value, the angle is in the fourth quadrant and taken clock- axis of the coordinate system than the ankle. That is, the wise and relative to the right horizontal, so 360Њ should be knee joint is to the right of the ankle joint (see Fig. 9-7). added to convert the absolute angle so that it is relative to the right horizontal in the counterclockwise direction. Similarly, to calculate the thigh angle, the coordinate values are substituted: Trunk, thigh, leg, and foot segmental end points for both the touchdown and the toe-off in walking are graphically tan ␪thigh ϭ yhip Ϫ yknee/xhip Ϫ xknee illustrated in Figure 9-9. The corresponding calculations of ϭ 0.80 Ϫ 0.51/1.14 Ϫ 1.22 ϭ 0.29/Ϫ0.08 ϭ Ϫ3.625 Again, the angle whose tangent is –3.625 is determined as follows: ␪thigh ϭ tanϪ1 Ϫ3.625 ϭ Ϫ74.58Њ 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 1600 1400 1200 1000 mm 800 600 FIGURE 9-9 By plotting the segmental end- 400 points and creating a stick figure, the simi- 200 larities or differences in position can be clearly observed. The differences in right 0 foot touch down (black) and right foot toe off (red) phases of a walking gait are 0 500 1000 1500 apparent. See Appendix C Frame 1 and Frame 76, respectively. mm

342 SECTION III Mechanical Analysis of Human Motion 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 neg- ative x-position and positive y-position, so 180Њ 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 ori- entation in touchdown and toe-off result in both negative x and y, placing it in the third quadrant, where 180Њ is again added. These adjustments provide a consistent reference for the computation of the absolute angles. 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 FIGURE 9-10 A. Relative elbow angle. B. The same relative elbow angle with the arm and forearm in different positions. The other type of angle calculated in biomechanics is the relative angle (Fig. 9-10A). This is the angle between the sides of a triangle that does not contain a right angle. longitudinal axes of two segments and is also referred to For our purposes, the triangle is made up of the two seg- as the joint angle or the intersegmental angle. A relative ments B and C and a line, A, joining the distal end of one angle (e.g., the elbow angle) can describe the amount of segment to the proximal end of the other (Fig. 9-11). flexion or extension at the joint. Relative angles, however, do not describe the position of the segments or the sides In Figure 9-11, the coordinate points for two segments of the angle in space. If an individual has a relative angle describing the thigh and the leg are given. To calculate the of 90Њ at the elbow and that angle is maintained, the arm relative angle at the knee (␪), the lengths a, b, and c would may be in any of a number of positions (Fig. 9-10B). be calculated using the Pythagorean relationship. 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 TA B L E 9 - 1 Absolute Angle Calculation for Touchdown and Toe-off in Walking Frame Trunkx ϭ (Shoulderx to Trunky ϭ (Shouldery Absolute Angle ϭ Thighx ϭ (Greater Thighy ϭ (Greater 1 Greater Trochanterx) to Great Trochantery) Arctan (y/x) Trochanterx to Kneex) Trochantery to Kneey) Ϫ210.15 317.14 76 Ϫ3.95 523.08 ϭ Ϫ89.57Њ ϩ 180Њ ϭ 90.43Њ Ϫ14.76 368.95 10.92 532.10 ϭ 88.82Њ Absolute Legx ϭ Legy ϭ Absolute Footx ϭ Footy ϭ Absolute Angle ϭ (Kneex to (Kneey to Angle ϭ (Heelx to (Heely to Angle ϭ Arctan (y/x) Anklex) Ankley) Arctan (y/x) Metx) Mety) Arctan (y/x) 377.88 Ϫ67.95 ϭ Ϫ56.47Њ ϩ 180Њ Ϫ113.03 ϭ Ϫ73.35Њ ϩ 180Њ Ϫ181.92 ϭ 20.48Њ ϩ 180Њ ϭ 123.53Њ 218.66 ϭ 106.65Њ 169.39 ϭ 200.48Њ ϭ Ϫ87.71Њ + 180Њ 313.49 ϭ 34.90Њ Ϫ51.20 ϭ Ϫ73.18Њ ϩ 180Њ ϭ 92.29Њ ϭ 106.82Њ

CHAPTER 9 Angular Kinematics 343 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. FIGURE 9-11 Coordinate points describing the hip, knee, and ankle joint A relative angle can be calculated from the absolute val- center and the relative angle of the knee (u). ues to obtain a result similar to computations using the law of cosines. The relative angle between two segments a = 2(xh - xa)2 + (yh - ya)2 can be calculated by subtracting the absolute angle of the = 2(1.14 - 1.09)2 + (0.80 - 0.09)2 distal segment from the proximal segment. In the exam- = 20.0025 + 0.5041 ple using the thigh and lower leg, the following calcula- ϭ 0.71 tion is another option: b = 2(xh - xk)2 + (yh - yk)2 ␪relative ϭ ␪absolute thigh Ϫ ␪absolute leg = 2(1.14 - 1.22)2 + (0.80 - 0.51)2 = 20.0064 + 0.0841 ␪relative ϭ Ϫ74.58Њ Ϫ 72.8Њ ϭ 0.30 ␪relative ϭ 147.4Њ c = 2(xk - xa)2 + (yk - ya)2 = 2(1.22 - 1.09)2 + (0.51 - 0.09)2 In clinical situations, the relative angle is most often cal- culated because it provides a more practical indicator of = 20.0169 + 0.1764 function and joint position. In quantitative biomechanical ϭ 0.44 analyses, however, absolute angles are calculated more often than relative angles because they are used in a num- The next step is to substitute these values in the law of ber of subsequent calculations. Regardless of the type of cosines equation and solve for the cosine of the angle ␪. angle calculated, however, a consistent frame of reference must be used. a2 ϭ b2 ϩ c2 Ϫ 2 * b * c * cos ␪ cos ␪ ϭ b2 ϩ c2 Ϫ a2/2 * b * c Unfortunately, many coordinate systems and systems of cos ␪ ϭ 0.302 ϩ 0.442 Ϫ 0.712/2 * 0.30 * 0.44 defining angles have been used in biomechanics, resulting in difficulty comparing values from study to study. Several cos ␪ ϭ Ϫ0.833 organizations, such as the Canadian Society of Biomechanics and the International Society of Biomechanics, have stan- To find the angle ␪, the angle whose cosine is –0.833 can dardized the representation of angles to provide consistency be determined using either trigonometric tables (see in biomechanics research, especially in the area of joint Appendix B) or a calculator with trigonometric functions. kinematics. This process, known as finding the inverse cosine or arcos, is written as follows: Lower Extremity Joint Angles ␪ ϭ cosϪ1 Ϫ0.833 In discussing the angle of a joint such as the knee or ankle, ␪ ϭ 146.4Њ it is imperative that a meaningful representation of the action of the joint be made. A special use of absolute Therefore, the relative angle at the knee is 146.4Њ. In this angles to compute joint angles is very useful for clinicians case, the knee is slightly flexed (180Њ representing full and others interested in joint function. Lower extremity extension). 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 biome- chanical analysis, it is assumed that a right side sagittal view is being analyzed. That is, the right side of the sub- ject’s body is closest to the camera and is considered to be in the x–y plane.

344 SECTION III Mechanical Analysis of Human Motion KNEE ANGLE Using the absolute angle of the thigh and the leg, the knee joint angle is defined as: ␪knee ϭ ␪thigh absolute Ϫ ␪leg absolute In human locomotion, the knee angle is always positive (i.e., in some degree of flexion), and it usually varies from 0 to 50Њ throughout a walking stride and from 0 to 80Њ 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 flex- ing. 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: ␪knee ϭ ␪thigh Ϫ ␪leg ϭ 123.53Њ Ϫ 106.65Њ ϭ 16.88Њ FIGURE 9-12 Definition of the sagittal view absolute angles of the trunk, ANKLE ANGLE thigh, leg, and foot. (After Winter, D. A. [1987]. The Biomechanics and Motor Control of Gait. Waterloo, Ontario, Canada: University of The ankle angle is calculated using the absolute angles of Waterloo Press.) the foot and the leg: HIP ANGLE ␪ankle joint angle ϭ ␪leg Ϫ ␪foot ϩ 90Њ Based on the absolute angles of the trunk and the thigh This may seem more complicated than the other lower calculated, the hip angle is: extremity joint angle calculations. Without adding 90Њ, the ankle angle would oscillate about 90Њ, making inter- ␪hip ϭ ␪thigh absolute Ϫ ␪trunk absolute pretation of it difficult. Adding 90Њ makes the ankle angle oscillate about 0Њ. Thus, a positive angle represents dorsi- In this scheme, if the hip angle is positive, the action at flexion, and a negative angle represents plantarflexion. the hip is flexion; but if the hip angle is negative, the action is extension. If the angle is zero, the thigh and the The ankle angle for the touchdown phase in the walk- trunk are aligned vertically in a neutral position. For ing example (Table 9-1) is: example, the hip joint angle representing flexion and extension for the touchdown phase in walking (Table 9-1) ␪ankle ϭ ␪leg Ϫ ␪foot ϩ 90Њ would be: ϭ 106.65Њ – 200.48Њ ϩ 90Њ ␪hip ϭ ␪thigh Ϫ ␪trunk ϭ 123.53Њ Ϫ 90.43Њ ϭ Ϫ3.83Њ ϭ 33.1Њ This value indicates that the ankle is in plantarflexion at touchdown. The ankle angle generally oscillates Ϯ20Њ dur- The joint angle of 33.1Њ at touchdown indicates that the ing a natural walking stride and Ϯ35Њ during a running thigh is flexing at the hip joint. In a human walking at a stride. Lower extremity angles calculated for a walking stride moderate pace, the hip angle oscillates Ϯ35Њ about 0Њ; in using Winter’s convention are presented in Figure 9-13. running, the hip angle oscillates Ϯ45Њ. 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).

CHAPTER 9 Angular Kinematics 345 a b c d e f 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 calcula- tions. Both representations are the same. FIGURE 9-13 Graphs of the hip (A), knee (B), and ankle (C) angles REARFOOT ANGLE during walking. Another lower extremity angle that is often calculated in Joint angles calculated using the relative angle biomechanical analyses is the rearfoot angle. The motion approach (law of cosines) and the same angles calculated of the subtalar joint in a two-dimensional analysis is con- from the absolute angles using Winter’s (36) convention sidered to be in the frontal plane. The rearfoot angle rep- have exactly the same clinical meaning. In the relative resents the motion of the subtalar joint. The rearfoot angle approach, the joint angle that is calculated is the angle thus approximates calcaneal eversion and calcaneal included angle between the two segments. Using the inversion in the frontal plane. Calcaneal eversion and absolute angle approach, the joint angle that is calculated inversion are among the motions in the pronation and is the difference between the two segment angles. The supination action of the subtalar joint. In the research lit- interpretation of these angles is exactly the same. Both erature, calcaneal eversion is often measured to evaluate types of angles are presented in Figure 9-14. pronation, and calcaneal inversion is measured to deter- mine supination. The rearfoot angle is calculated using the absolute angles of the leg and the calcaneus in the frontal plane. Two seg- ment 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,

346 SECTION III Mechanical Analysis of Human Motion Representation of Angular Motion Vectors FIGURE 9-15 Definition of the absolute angles of the leg and calcaneus Representing angular motion vectors graphically as lines in the frontal plane. These angles are used to constitute the rearfoot with arrows, as is the case in linear kinematics, is difficult. angle of the right foot. It is essential, however, to determine the direction of rota- tion in terms of a positive or negative rotation. The direc- they are used to calculate the absolute angles of the leg tion of rotation of an angular motion vector is referred to and heel; thus, the rear foot angle is: as the polarity of the vector. The polarity of an angular motion vector is determined by the right-hand rule. The ␪rearfoot ϭ ␪leg Ϫ ␪calcaneus direction of an angular motion vector is determined using this rule by placing the curled fingers of the right hand in By this calculation, a positive angle represents calcaneal the direction of the rotation. The angular motion vector is inversion, a negative angle represents calcaneal eversion, defined by an arrow of the appropriate length that coin- and a zero angle is the neutral position. cides with the direction of the extended thumb of the right hand (Fig. 9-17). The 2D convention generally used During the support phase of the gait cycle, the rearfoot, is that all segments rotating counterclockwise from the as defined by the rearfoot angle, is in an inverted position right horizontal have positive polarity and all segments at the initial foot contact with the ground. At this instance, rotating in clockwise have a negative polarity. 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 nega- tive 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. FIGURE 9-17 The right-hand rule used to identify the polarity of the Maximum rearfoot angle is indicated. 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 vec- tor is perpendicular to the plane of rotation.

CHAPTER 9 Angular Kinematics 347 Angular Motion Relationships The relationships discussed in this chapter on angular kine- FIGURE 9-19 Angular displacement is the difference between the initial matics are analogous to those in Chapter 8 on linear kine- position and the final position. matics. The angular case is simply an analog of the linear case. discussing angular displacement, it is necessary to designate ANGULAR POSITION AND DISPLACEMENT the direction of the rotation. Counterclockwise rotation is considered to be positive, and clockwise rotation is nega- The angular position of an object refers to its location rela- tive. With a 3D reference system placed at the shoulder tive to a defined spatial reference system. In the case of a 2D joint, the positive y-axis would be upward, the positive x- system with the y-axis representing motion vertically up and axis would be posterior to anterior, and the positive z-axis down and the x-axis representing anterior to posterior would be medial to lateral. The corresponding positive joint motion, angular position is described in the x–y plane. movements about these axes would be flexion/extension A three-dimensional (3D) system adds a third axis, z, in the (about the x-axis), internal/external rotation (about the medial and lateral plane. Many clinicians use planes to y-axis), and abduction/adduction (about the z-axis). describe angular positioning. For example, if the axes are placed with the origin at the shoulder joint, angular position If the absolute angle of a segment, theta (␪), is calcu- of the arm in the x–y plane would be a flexion and extension lated for successive positions in time, the angular displace- position in the y–z plane abduction and adduction, and in ment (⌬␪) is: the x–z plane, rotation. This system works well for describ- ing joint angles but lacks precision for describing complex ⌬␪ ϭ ␪final Ϫ ␪initial movements. Absolute angles can be computed relative to a fixed reference system placed at a joint or at another fixed The polarity, or sign, of the angular displacement is deter- point in the environment. As discussed earlier, angular posi- mined by the sign of ⌬␪ as calculated and may be con- tion can also be computed relative to a line or plane that is firmed by the right-hand rule. allowed to move. It is common to present joint angles such as those shown in Figure 9-13 to document the joint actions ANGULAR VELOCITY in a movement such as walking. Angular speed and angular velocity are analogous with The concepts of distance and displacement in the angu- linear speed and linear velocity in both definition and lar case must be discerned. Consider a simple pendulum meaning. Angular speed is the angular distance traveled swinging in the x–y plane through an arc of 70Њ (Fig. 9-18). per unit of time. Angular speed is a scalar quantity and is If the pendulum swings though a single arc, the angular generally not critically important in biomechanical analy- distance is 70Њ, but if it swings through 1.5 arcs, the angu- sis because it is not used in any further calculations. lar distance is 105Њ. Angular distance is the total of all angular changes measured following its exact path. As in Angular velocity, characterized by the Greek letter the linear case, however, angular distance is not the same as omega (␻), is a vector quantity that describes the time rate angular displacement. of change of angular position. If the measured angle is ␪, then the angular velocity is: Angular displacement is the difference between the ini- tial and final positions of the rotating object (Fig. 9-19). In ␻ ϭ Change in angular position/Change in time the example of the pendulum, if the pendulum swings through two complete arcs, the angular displacement is ϭ ␪final Ϫ ␪initial/timefinal Ϫ timeinitial zero because its final position is the same as the starting position. Angular displacement never exceeds 360Њ or 2␲ ϭ ⌬␪/⌬t rad of rotation, but angular distance can be any value. In If the initial angle of a segment is 34Њ at time 1.25 s and FIGURE 9-18 A swinging pendulum illustrating the angular distance over the segment moves to an angle of 62Њ at time 1.30 s, the 1.5 arcs of swing. angular velocity would be: