Anatomical terms refer to the body locally whether it is at rest or in motion. Since we are also concerned with how we move, we need to address human motion [3 ] . We will describe how we move by examining the degrees of freedom of our motion and the means for providing such motion by our joints. We will see that our arms and legs are constructed in a very clever manner . Because joints involve motion between bones, we will need to refer to the anatomy of the skeletal system, as in Fig. 1.2.

Think of a degree of freedom (DOF) of motion as a coordinate needed to describe that type of motion. If you want to relocate an object, you are generally interested in changing its center of mass and its angular orientation. You may want to change its center of mass from an (x, y, z) of (0, 0, 0) to (a, b, c). Because three coordinates are needed to describe this change, there are three “translational” degrees of freedom. Similarly, you can change the angular orientation of the object about the x, y, and z axes, by changing the angles this object can be rotated about these three axes: , , and , respectively. So, there are also three rotational degrees of freedom. (Sometimes, these three independent rotations are defined differently, by the three Eulerian angles, which will not be introduced here.)

These six (three plus three) degrees of freedom are independent of each other. Keeping your fingers rigid as a fist, you should be able to change independently either the x, y, z, , , and of your fist by moving your arms in different ways. You should try to change the x, y, and z of your fist, while keeping , , and fixed. Also, try changing the , , and of your fist, while keeping its x, y, and z constant.

We would like each of our arms and legs to have these six degrees of freedom. How does the body do it? It does it with joints, also known as articulations. Two types of articulations, fibrous (bones joined by connective tissue) and cartilaginous (bones joined by cartilage) joints, can bend only very little. One type of fibrous joint is the suture joint that connects the bony plates of the skull. These plates interdigitate through triangle-like tooth patterns across more compliant seams with fibers, so this joint can bear and transmit loads, absorb energy, and provide flexibility for growth, respiration, and locomotion [17, 18]. There is a joint cavity between the articulating bones in synovial joints . Only these synovial joints have the large degree of angular motion needed for motion. As seen in Fig. 1.3, in synovial joints cartilage layers on the ends of opposing bones are contained in a sac containing synovial fluid . The coefficient of friction in such joints is lower than any joints made by mankind. (More on this later.)

There are several types of synovial joints in the body, each with either one, two, or three degrees of angular motion. Each has an analog with physical objects, as seen in Fig. 1.4. For example, a common door hinge is a model of one degree of angular freedom. Universal joints, which connect each axle to a wheel in a car, have two angular degrees of freedom. A ball-and-socket joint has three independent degrees of angular motion. The water faucet in a shower is a ball-and-socket joint. The balls and sockets in these joints are spherical. Condyloid or ellipsoidal joints are ball-and-socket joints with ellipsoidal balls and sockets. They have only two degrees of freedom because rotation is not possible about the axis emanating from the balls. A saddle joint, which looks like two saddles meshing into one another, also has two degrees of angular motion. Other examples are shown in Fig. 1.4.

Now back to our limbs. Consider a leg with rigid toes. The upper leg bone (femur) is connected to the hip as a ball-and-socket joint (three DOFs) (as in the song “Dry Bones” aka “Them Bones” in which “The hip bone is connected to the thigh bones, ….” The knee is a hinge (one DOF). The ankle is a saddle joint (two DOFs). This means that each leg has six degrees of angular motion, as needed for complete location of the foot. Of course, several of these degrees of freedom have only limited angular motion.

Now consider each arm, with all fingers rigid. The upper arm (humerus) fits into the shoulder as a ball-and-socket joint (three DOFs). The elbow is a hinge (one DOF). The wrist is an ellipsoidal joint (two DOFs). That makes six DOFs. The leg has these six DOFs, but the arm has one additional DOF, for a total of seven. This additional DOF is the screwdriver type motion of the radius rolling on the ulna (Figs. 1.2, 2.​7, and 2.​8), which is a pivot with 1 DOF. With only six DOFs you would be able to move your hand to a given x, y, z, position in only one way. With the additional DOF you can do it in many ways, as is seen for the person sitting in a chair in Fig. 1.5. There are many more degrees of freedom available in the hand, which enable the complex operations we perform, such as holding a ball . Figure 1.6 shows the bones of the hand, and the associated articulations and degrees of freedom associated with the motion of each finger.

We can also see why it is clever and good engineering that the knee hinge divides the leg into two nearly equal sections and the elbow hinge divides the arm into two nearly equal sections. In the two-dimensional world of Fig. 1.7 this enables a greater area (volume for 3D) to be covered than with unequal sections.

In preparation for our discussion of statics and motion of the body, we should consider the building blocks of human motion. There are four types of components: bones, ligaments, muscles, and tendons. Each has a very different function and mechanical properties. Bones are often lined with hyaline (high’u-lun) articular cartilage at the synovial joints . Ligaments hold bones together. Muscles, in particular skeletal muscles, are the motors that move the bones about the joints. (There is also cardiac muscle—the heart—and smooth muscle—of the digestive and other organs.) Tendons connect muscles to bones . Muscles are connected at points of origin and insertion via tendons; the points of insertion are where the “action” is. Figure 1.8 shows several of the larger muscles in the body, along with some of the tendons.

Muscles work by contraction only, i.e., only by getting shorter. Consequently, to be able to move your arms one way and then back in the opposite direction, you need pairs of muscles on the same body part for each opposing motion . Such opposing pairs, known as “antagonists,” are very common in the body.

We now return to our brief review of terminology, this time to describe the angular motion of joints . It is not surprising that these come in opposing pairs (Figs. 1.9 and 1.10) as supplied by antagonist muscles . When the angle of a 1D hinge, such as the elbow, increases it is called extension and when it decreases it is flexion. When you rotate your leg away from the midline of your body, it is abduction, and when you bring is closer to the midline, it is adduction. When you rotate a body part about its long axis it is called rotation. The screwdriver motion in the arm is pronation (a front facing hand rotates towards the body) or supination (away from the body), and so supination is the motion of a right hand screwing in a right-handed screw (clockwise looking distally from the shoulder to the hand) and pronation is that of a right hand unscrewing a right-handed screw (counterclockwise looking distally from the shoulder to the hand). Examples of the rotation axes for the synovial joints used in these opposing motions are given in Fig. 1.11.

One example of opposing motion is the motion of the arm (Fig. 1.12 ) . The biceps brachii have two points of origin and are inserted on the radius (as shown in Fig. 2.​10 below). When they contract, the radius undergoes flexion about the pivot point in the elbow . The triceps brachii have three points of origin, and a point of insertion on the ulna. They are relaxed during flexion. During extension they contract, while the biceps brachii are relaxed. This is an example of a lever system about a pivot point. (This is really a pivot axis normal to the plane of the arm, as is illustrated in Fig. 1.11a for a hinge joint.)

A second place where there is such opposing motion is the eye. The three types of opposite motion in each eye (monocular rotations) are shown in Fig. 1.13. During adduction the eye turns in to the midline, while during abduction it turns out. The eyeball can also undergo elevation (eye rotating upward, or supraduction) or depression (eyeball rotating downward, or infraduction). Less common is the rotation of the eyeball about an axis normal to the iris, in opposing intorsion (incycloduction) or extorsion (excycloduction) motions . There are three pairs of opposing muscles per eye, each attached to the skull behind the eye, that control these motions (Fig. 1.14 , Table 1.3). However, of these three pairs, only one is cleanly associated with only one of these pairs of opposing motions. Adduction occurs with the contraction of the medial rectus muscle, while abduction occurs when the lateral rectus contracts. The primary action of the superior rectus is elevation, while that of the opposing inferior rectus is depression. The primary action of the superior oblique is also depression, while that of the opposing inferior oblique is also elevation. These last two pairs of muscles have secondary actions in adduction/abduction and intorsion/extorsion that depend on the position of the eye . Binocular vision requires coordinated motion of the three opposing muscle pairs in both eyes, as described in Table 1.4.

We will often, but not always, model humans assuming numerical values for mass, height, etc. of a “standard” human, a 70 kg man with parameters similar to those in Table 1.5. The distributions of body heights and weights, and of course their averages, differ in different regions and change with time. For example, heights of western European men increased by 8–17 cm over the last two centuries, with much of these increases being very recent, with typical rates of 1–2 cm/decade [13].

We will need details of human anatomy in some cases, and these are provided for the standard human now and in subsequent chapters as needed. We will also need to use the findings of anthropometry, which involves the measurement of the size, weight, and proportions of the human body. Of particular use will beanthropometric data, such as those in Table 1.6 and Fig. 1.15, which provide the lengths of different anatomical segments of the “average” body as a fraction of the body height H.

Table 1.7 gives the masses (or weights) of different anatomical parts of the body as fractions of total body mass (or equivalently, total body weight ). The mass and volume of body segments are determined on cadaver body segments, respectively by weighing them and by measuring the volume of water displaced for segments immersed in water. (This last measurement uses Archimedes’ Principle, described in Chap. 7.) The average density of different body segments can then be determined, as in Table 1.7. The volumes of body segments of live humans can be measured by water displacement (Problem 1.39) and then their masses can be estimated quite well by using these cadaver densities. Whole body densities of live humans can be measured using underwater weighing, as is described in Problem 1.43 . A relation for average body density is given below in (1.3). This is closely related to determining the percentage of body fat, as is presented below in (1.4) and (1.5).

The normalized distances of the segment center of mass from both the proximal and distal ends of a body segment are given in Table 1.8. (Problems 1.45 and 1.46 explain how to determine the center of mass of the body and its limbs.) The normalized radius of gyration of segments about the center of mass, the proximal end, and the distal end are presented in Table 1.9. (The radius of gyration provides a measure of the distribution of mass about an axis, as described in (3.​28) and Fig. 3.​24b. Problem 3.​16 describes how the radii of gyration in Table 1.9 are related.)

Note that all of the data in these different tables are not always consistent with each other because of the variations of sources and the different ranges of subjects and methods used for each table.

We have a tremendous range of mobility in our articular joints, but not as much as in the idealized joints in Fig. 1.4. The average ranges of mobility in people are given in Table 1.10 for the motions depicted in Fig. 1.16, along with the standard deviations about these values. (For normal or Gaussian distributions with an average, A, and standard deviation, SD, about 68% of all values are between A SD and A SD.) Three degrees of freedom are given for the shoulder and hip, two for the wrist and the foot (listed separately as foot and ankle), and one each for the elbow and forearm. The knee, as idealized above, has one DOF, but two are listed here: the flexion in a 1 D hinge and also some rotation of the upper and lower leg about the knee.