Method
The approach presented so far is just one way of comput-ing the net forces and moments at the joints. Plagenhoef (1968, 1971) called this approach the absolute motion method of inverse dynamics because the segmental kine-frame of reference. An alternative approach outlined by Plagenhoef is the relative motion method. This method chain from an absolute frame of reference, but all other segments are referenced to moving axes that rotate with the segment. Thus, each segment’s axis, except that of method has the advantage of showing how one joint’s moment of force contributes to the moments of force of the other joints in the kinematic chain. The drawback is that the level of complexity of the analysis increases with every link added to the kinematic chain. Furthermore, the method requires the inclusion of Coriolis forces, which are forces that appear when an object rotates within a sometimes called pseudoforces—only exist because of their rotating frames of reference. From an inertial (i.e., Rarely have researchers tried to compare the two methods. However, Pezzack (1976) did compare both methods using the same coordinate data and found that the relative motion method was less accurate, especially as the kinematic chain got longer (had more segments).
For short kinematic chains (two or three segments), both methods yielded similar results. Most researchers have
FROM THE SCIENTIFIC LITERATURE
Winter, D.A. 1980. Overall principle of lower limb support during stance phase of gait. Journal of Bio-mechanics 13:923-7.
This paper presents a special way of combining the moments of force of the lower extremity during the stance phase of gait. During stance, the three moments of force—
the ankle, knee, and hip—combine to support the body and prevent collapse. The author found that by adding the three moments in such a way that the extensor moments had a positive value, the resulting “support moment” fol-lowed a particular shape. The support moment (Msupport )
Msupport hip + Mknee ankle
Notice that the negative signs for the hip and ankle moments change their directions so that an extensor
moment from these joints makes a positive contribution to the amplitude of the support moment. Figure 5.13 shows the average support moment of normal subjects and the support moment and hip, knee, and ankle moments of a 73-year-old male with a hip replacement.
This useful tool allows a clinical researcher to moni-tor a patient’s progress during gait rehabilitation. As the patient becomes stronger or coordinates the three joints more effectively, the support moment gets larger. People with one or even two limb joints that cannot adequately contribute to the support moment will still be supported by that limb if the remaining joints’ moments are large enough to produce a positive support moment.
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Figure 5.13a Averaged support moment of subjects with intact joints.
Reprinted from Journal of Biomechanics,
(continued) adopted the absolute motion method, because most
data-collection systems measure segmental kinematics with
APPLICATIONS
There are many uses for the results of an inverse dynam-ics analysis. One application sums the extensor moments of a lower extremity during the stance phase of walking
Literature). Others have used the net forces and moments in musculoskeletal models to compute the compressive loading of the base of the spine for research on lifting and low back pain (e.g., McGill and Norman 1985). An extension of this approach is to calculate the compres-sion and shear force at a joint. To do this, the researcher must know the insertion point of the active muscle acting
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Ms = + Mh − Mk − Ma
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Figure 5.13b The support moment and hip, knee, and ankle moments of a 73-year-old male with a hip replacement during the stance phase of walking.
Reprinted from Journal of Biomechanics,
across a joint and assume that there are no other active muscles.
Computing the force in a muscle requires several assumptions to prevent indeterminacy (i.e., too few equations and too many unknowns). For example, if a single muscle can be assumed to act across a joint and no other structure contributes to the net moment of force, then, if the muscle’s insertion point and line of action are known (from radiographs or estimation), the muscle force (Fmuscle )
Fmuscle = M/(r sin #) (5.3)
where M is the net moment of force at the joint, r is the distance from the joint center to the insertion point of the muscle, and # is the angle of the muscle’s line of action and the position vector between the joint center and the muscle’s insertion point. Of course, such a situ-ation rarely occurs because most joints have multiple synergistic muscles with different insertions and lines of action as well as antagonistic muscles that often act in cocontraction. By monitoring the electrical activity (with an electromyograph) of both the agonists and antagonists, one can reduce these problems, but even an inactive muscle can create forces, especially if it is stretched beyond its resting length. The contributions
of other tissues to the net moment of force can be mini-mized also as long as the motion being analyzed does not include the ends of the range of motion, when these Once the researcher has estimated the force in the muscle, the muscle stress can be computed by determin-ing the cross-sectional area. The cross-sectional areas of particular muscles can be derived from published reports or measured directly from MRI scans or radiographs.
Axial stress ($)
sectional area. For pennate types of muscles in which the force is assumed to act along the line of the muscle,
$ = FmuscleA, where Fmuscle is the muscle force in newtons and A is the cross-sectional area in square meters. The units of stress are called pascals (Pa), but because of the large magnitudes, kilopascals (kPa) are usually used. Of course, true stress on the directly measuring the actual muscle force.
The following sections present and discuss the pat-terns of planar lower-extremity joint moments during walking and running. These movements and others such as stair ascent and descent often can be analyzed two-dimensionally because their principal motions occur primarily in the sagittal plane. The convention
FROM THE SCIENTIFIC LITERATURE
McGill, S., and R.W. Norman. 1985. Dynamically and statically determined low-back moments during lifting. Journal of Biomechanics 18:877-85.
This study presents a method for computing the compres-sive load at the L4-L5 joint from data collected on the motion of the upper body during a manual lifting task.
Three different methods for computing the net moments of force at L4-L5 were compared. A conventional dynamic analysis was performed using planar inverse dynamics to compute the net forces and moments at the shoulder and neck, from which the net forces and moments were cal-culated for the lumbar end of the trunk (L4-L5). Second, a static analysis was done by zeroing the accelerations of the segments. A third, quasi-dynamic approach assumed a static model but used dynamic information about the load. Once the lumbar net forces and moments were cal-culated, it was assumed that a “single equivalent muscle”
was responsible for producing the moments of force and that the effective moment arm of this muscle was 5 cm.
The magnitude of the compressive force (Fcompress ) on the L4-L5 joint was then computed by measuring the angle of the lumbar spine (actually, the trunk), #. The equation used was
Fcompress= M
r + Fycos + Fxsin (5.4)
where M is the net moment of force at the L4-L5 joint, r is the moment arm of the single equivalent extensor muscle across L4-L5 (5 cm), (Fx, Fy )
L5 joint, and # is the angle of the trunk (the line between L4-L5 and C7-T1).
The researchers showed that there were “statistically sig-the three methods for sig-the pattern and peak values of sig-the net moment of force at L4-L5. In general, the dynamic method yielded larger peak moments than the static approach but smaller values than the quasi-dynamic method. Compari-sons of a single subject’s lumbar moment histories and the averaged histories of all subjects showed that each approach produced very different patterns of activity. These compari-sons also showed that although the static approach produced compressive loads that were less than the 1981 National Institute for Occupational Health and Safety (NIOSH) lift-ing standard, the more accurate dynamic model produced loads greater than the “maximum permissible load.” The quasi-dynamic approach produced even higher compres-sive loads. Clearly, then, one should apply the most accurate method to obtain realistic conclusions.
This is in agreement with the engineering standard that mechanical actions that lengthen a system are positive (positive strain) and actions that shorten the system
positive. Therefore, these two moments are usually presented as the negative of what is calculated.
Walking
Joint moments during walking have many typical fea-ankle, knee, and hip joint moments. These data are pre-sented as percentages of the gait cycle; the vertical line at 60% of the cycle represents toe-off, and the 0% and 100%
axes, the joint moments are presented in newton meters.
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Figure 5.14 Moments of force at the (a) ankle, (b) knee, and (c) hip during normal level walking. HC = heel-contact; TO = toe-off.
Sometimes these values are scaled to the percentage of body mass or percentage of body mass times leg length to assist in comparing different subjects. We keep these data in newton meters for continuity with the preceding examples. In general, joint moments do scale up and down with body size. They also change magnitude with the speed of movement.
a condition known clinically as foot-slap. Although 15 graph, this peak is nonetheless a very common feature of normal walking. Thereafter, we see a large plantar
to effect push-off. As this peak diminishes, the limb becomes unloaded. As toe-off occurs, we again see a although small, is important because it lifts the toes out dysfunction have a problem with their toes catching the ground at this part of the gait cycle. For the rest of the swing phase, the ankle moment is near zero.
b, there are four distinct peaks of the knee moment. The largest peak during stance is extensor, the limb is loaded and thus the extensor moment acts to prevent collapse of the limb. Note that this peak occurs slightly earlier than the peak of the ankle moment. Often leg is pulled through the remainder of the stance. During the knee that occurs because the lower extremity is being swung forward from the hip. Without this peak, the knee walking velocities. The second swing-phase peak is a knee reaches full extension.
Figure 5.14c shows that the hip moment tends to have
the lower extremity forward. Then, similar to the knee moment, the hip moment has an extensor peak of about before heel-contact.
The hip moment is the most variable of the three lower-extremity joint moments. The foot moment tends to be the least variable because the segment is con-strained by the ground. The hip, in contrast, not only is responsible for lower-extremity control but also has task, given that the torso accounts for at least two-thirds of the body weight of an average individual. Winter and
Sienko (1988) showed that the movement of the torso this regard, the hip moment becomes somewhat hard to interpret during stance.
Running
Lower-extremity joint moments during running are -lar to their analogues in walking. The most prominent differences are their magnitudes. Running is a more forceful activity than walking; thus, just as GRFs are larger during running, so, too, are the joint moments.
Furthermore, the stance phase (HC to TO) is a smaller percentage of the running cycle than during walking.
a) has a the swing-phase (TO to second HC) ankle moment is near zero. The ankle moment at heel-contact varies slightly depending on running style. Runners with a
not have this peak because there is no need to control foot-slap.
The stance-phase knee moment (Figure 5.15b), like the ankle moment, consists primarily of a single extensor
the knee rapidly before the lower extremity is swung forward. During the swing, there is an extensor phase in phase before heel-contact to slow the leg.
c) is extensor for much
lower extremity forward. Then there is an extensor action to slow the thigh before heel-contact.
SUMMARY
This chapter focused heavily on proper technique for inverse dynamics problems. This focus is necessary simply because the technique clearly has many steps -aged to practice such problems until they can solve them without referring to this book. Students should be able to draw an FBD for any segment, construct the three equations of motion, and solve them.
Students are also encouraged to be mindful of the limitations of joint moments. Joint moments are only a summary representation of human effort, one step more advanced than the information obtained from a force plate. Joint moments are not an end-all statement of human kinetics; rather, they are convenient standards
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Figure 5.15 Moments of force at the (a) ankle and (b) knee during normal level running. HC = heel-contact;
TO = toe-off.
for evaluating the relative efforts of different joints and actions must use either electromyography (chapter 8), musculoskeletal models (chapter 9), or both. It is noteworthy that the great Russian scientist Nikolai Bernstein (see Bernstein 1967) refers to moments but in fact preferred to use segment accelerations as overall
representations of segmental efforts. It is hard to argue
can develop a relative basis for the magnitudes of joint moments.
The problem with analyzing limbs segmentally is that it can promote the misconception that each joint is controlled independently. We stated that two-joint muscles are poorly treated with this method. In addi-tion, the individual segments of an extremity interact.
For example, we discussed the fact that thigh moments affect the movement of the leg. Therefore, in terms of joint moments, we know that a hip moment will affect the thigh; however, we do not establish the resulting effect
on the leg. Many studies have noted the importance of intersegmental coordination (see, for example, Putnam 1991); in fact, Bernstein cited the use of segment
interac -ment. Clinicians are also beginning to recognize such effects in various populations, particularly in people with amputation. Systemic analyses such as the Lagrangian approach, outlined in chapter 10, may be preferable for these situations.
SUGGESTED READINGS
Biomechanical Analysis of Funda-mental Human Movements. Champaign, IL: Human Kinetics.
Nigg, B.M., and W. Herzog. 1999. Biomechanics of the Musculo-Skeletal System
Fundamentals of Biomechanics. 3rd ed. New York: Van Nostrand Reinhold.
squares method for inverse dynamic analysis. Computer Methods in Biomechanics and Biomedical Engineering 11:1, 3-9.
Biomechanics and Motor Control of Human Movement. 4th ed. Toronto: Wiley.
Kinetics of Human Motion. Cham-paign, IL: Human Kinetics.
131 instance, physicists have yet to identify any atomic or subatomic particle that corresponds to a basic unit, understanding energy is that it takes many forms. Matter itself is one form of energy, which Einstein was able to quantify with his most famous equation, E = mc2, but this energy is only manifest when the matter itself is torn apart. Other forms of energy include nuclear, electrical, thermal (heat), solar, light, chemical, and, the one of greatest interest to biomechanists, mechanical.
In this chapter, we are mainly concerned with mechanical energy, and so we
examine the concepts of energy, work, and the laws of thermodynamics;
introduce the concepts of conservation of mechanical energy and a conservative force;
outline methods for directly measuring mechanical work, called direct ergometry;
outline methods for indirectly quantifying mechani-cal work, including the use of results from inverse dynamics; and
examine the relationship between work and the cost
of doing work, called .