## The Physical Plant A Kinetics and Kinematics

Kinetics refers to the forces (or torques) that generate a movement, whereas kinematics refers to a description of the motion—position, velocity, and acceleration. Kinematics and kinetics are related by Newton's second laws of motion. When the motion is restricted to a single joint, this relationship is simple:

F = ma or T = I a where Fis the force, m is the mass, and a is the object's acceleration. Similarly, for rotational motions, Tis the torque, I is the moment of inertia, and a is the angular acceleration. Now consider the problem of predicting the movement that a particular force will generate. The solution to this problem is simple because position (or angular displacement) can be found by integrating force (or torque) twice. It is also easy to compute the force needed to produce a given movement by differentiating the desired linear displacement p(t) [or the desired angular displacement 0(t)j twice.

Since it is known that neural circuits can perform the mathematical equivalents of integration and differentiation, controlling simple movements does not appear to be particularly challenging. Furthermore, it was thought that more complicated movements could be generated by simply combining several simple movements. Thus, the distinction between kinematics (movements) and kinetics (forces) becomes academic; accelerations and forces are proportional to each other.

This is not the case when a movement involves rotation at more than one joint—for example, an arm movement that involves the shoulder, the elbow, and the wrist. Consider a simple example in which the arm is in the horizontal plane and is supported against gravity. Also imagine that a single shoulder muscle is activated, generating a torque about the shoulder joint. This torque will produce an angular acceleration at the shoulder. However, if the elbow and wrist joints are free to rotate, the acceleration at the shoulder will also impart a rotation at the elbow and wrist joints. Similarly, contraction of a pure elbow or a pure wrist muscle will affect the shoulder joint, changing the motion of the humerus, the limb segment proximal to the elbow. Torque is proportional to the angular acceleration at each of the joints in a skeletal linkage; conversely, the angular acceleration at one joint is proportional to the torques generated by muscles acting at each of the joints.

The equations relating kinetics and kinematics for multijoint motion also include terms known as Cor-iolis and centripetal accelerations. These terms are related to the angular velocities of the joint motions. Furthermore, the inertia of a linkage such as the arm depends on the angles of the joints in the linkage. Thus, torque is generally related to angular acceleration, angular velocity, and angular position. This relation is illustrated in Fig. 1. The relation between angular acceleration and torque is shown to depend on several parameters: the inertia ofthe limb, its angular velocity, and its posture. Posture enters into the equation in two ways: It affects limb inertia and it also determines the gravitational loads that need to be opposed.

The crucial point that emerges from this picture is the following: Inertia, posture, and velocity all vary during a movement. Therefore, if the motion that ensues from a torque profile is to be predicted, it is necessary either to measure each of these parameters accurately or to be able to predict them accurately. (Note that in principle, a single-joint movement could be performed accurately knowing only the amplitude of the required movement because the inertia of one limb segment does not change with time.) Sensing the position and velocity of a movement involves time delays. Consequently, parameters entered into the equation of motion will be in error because they are based on the state of the system in the past. Similarly,

Limb Mechanics

Limb Mechanics

Figure 1 Relationship between kinetics (forces or torques) and kinematics (acceleration) as determined by limb mechanics. For a multijointed limb, the equations relating kinematics to kinetics include terms that depend on positional and velocity parameters, which change over time. The inertia of the limb also depends on posture and consequently also changes as the movement progresses.

Figure 1 Relationship between kinetics (forces or torques) and kinematics (acceleration) as determined by limb mechanics. For a multijointed limb, the equations relating kinematics to kinetics include terms that depend on positional and velocity parameters, which change over time. The inertia of the limb also depends on posture and consequently also changes as the movement progresses.

estimating position and velocity from the torque commands will also introduce errors arising, for example, from uncertainties in estimating the starting posture of the limb.

Gravity is ubiquitous and gravitational torques also depend on the posture of the limb but not on the speed with which the limb moves. Thus, the control of a limb movement would need to account for the effects of gravity as well. In the mid-1980s, John Hollerbach made an important observation that provides for a potential simplification of the control of movement in a gravitational environment. He showed that despite all the apparent complexity ofthe relationship between torque and angular acceleration, the equations relating these parameters scale with the speed of the movement. In other words, movements of different speeds can be generated by taking one template of torques and scaling it in amplitude and in time. Since the gravitational torques do not depend on the speed, movement control can be simplified by separating the controller into two components: a postural component that does not depend on speed and a movement component that is speed dependent. Subsequent investigations have provided experimental support for this supposition.

The laws of motion relating kinematics and kinetics have other implications. First, the direction of force and the direction of movement generally do not coincide. Consider the following example. Ask a subject to exert an isometric force with the hand in some particular direction against a resistance, and then release the resistance. The hand will begin to move in a direction that need not coincide with the direction in which the force was exerted. Second, because of the interactions between the motions of the various limb segments, the sign of the torque at a particular joint need not be the same as the sign of the angular acceleration at that joint. For example, elbow extension may require elbow flexor torque. Consequently, eccentric contractions (i.e., activation of muscles that lengthen during a movement) are not uncommon. Movements depart in the correct direction, with the activation of the appropriate muscles predicted on the basis of mechanics. Accordingly, the central nervous system (CNS) must take these factors into account in specifying which muscles should be activated.

B. Muscles

The mechanical properties of muscles also need to be taken into account by the CNS. Since the work ofA. V.

Hill in the 1930s, it has been appreciated that muscle force depends on muscle length (the length-tension relation) and on the rate at which muscle length changes (the force-velocity curve). Tendon compliance also affects the dynamical relation between muscle activation and force generation, and it is an important factor in mechanical models of muscle, such as those developed by Felix Zajac. It is also well recognized that the moment arm of a muscle can change with changes in the limb configuration. Since muscle torque is equal to muscle force times moment arm, changes in a muscle's moment arm will also affect the amount of torque generated by a specific level of muscle activation.

Most muscles generate torque about more than one axis. For example, biceps brachii acts as shoulder flexor and as an elbow flexor and supinator. Similarly, anterior deltoid acts as a shoulder flexor and adductor. One can always define a torque axis, a vector defining the mechanical action of that muscle. The question arises, do these actions depend on the posture of the limb? In other words, does the torque axis remain fixed as limb posture changes? If the answer is the affirmative, one can imagine two possibilities: (i) The torque axis remains fixed relative to the orientation of the distal segment, and (ii) it remains fixed relative to the proximal segment. For example, considering deltoid, one could imagine that the mechanical action was fixed relative to the trunk (proximal segment) or that it was fixed relative to the humerus (the distal segment). In fact, the answer is neither, and torque axes depend on limb posture in a manner that is intermediate to the two possibilities just outlined. The axis moves along with the distal limb segment, but not by as much as the limb segment moves. The postural dependence of the mechanical actions of muscle reinforces the conclusion drawn in the previous section: The CNS needs to take into account posture and rate of change of posture in planning and regulating movement.

Although the postural dependence of the mechanical actions of muscles is not negligible, it need not be overly complicated. For example, for shoulder muscles, simple linear models relating the directions of the torque axes to the orientation of the humerus could adequately account for the experimental data. Muscles are customarily modeled by assuming that their forces are exerted in a straight line from an effective origin to an effective insertion. (The effective origins and insertions need not coincide with the anatomical origins and insertions.) These ''rubber band models'' are indeed adequate to account for the actions of shoulder muscles, even those that have widely distrib uted origins such as the heads of deltoid. Thus, it appears that fairly simple models can be used to account for the mechanical actions of muscles and for their postural dependence.

As noted previously, the initial conception of movement regulation was that for any particular movement, there were two distinct groups of muscles —agonists and antagonists. Muscles within these groups would act as synergists (i.e., they would be activated synchronously). Initially, it was also thought that this grouping would hold for all movements [e.g., a classification of muscles as flexors and extensors (or gravity vs antigravity)], and that synergies would be hardwired. This viewpoint was subsequently modified to account for the fact that a pair of muscles could be synergists for one movement and antagonists for another. Furthermore, the need for synchronization was also abandoned and it was posited that a group of muscles belonging to a synergy could be activated in a fixed, sequential order. In this concept of "flexible synergies,'' the relative timing of the activation of different muscles would be fixed, at least for a broad class of movements.

A detailed investigation of the patterns of muscle activation for arm movements in different directions has shown that this is not the case. Although the basic "three-burst" pattern of electromyographic activity does hold for multijoint movements as well as for single-joint movements, the timing of the bursts in individual muscles is asynchronous. Furthermore, the relative timing of muscle bursts, among the various shoulder and elbow muscles, varies with the direction of movement. For any given muscle, there is an orderly progression of the timing of muscle activation with movement direction, but this pattern is different from muscle to muscle. Accordingly, it is not possible to define even pairs of muscles that are synergists, in the strict sense of this word. However, these studies did provide support for the idea introduced by Hollerbach in the sense that it was possible to identify for each muscle a "tonic" pattern of activity related to static postures and counteracting gravity and a "dynamic" component that scaled with the speed of the movement.

Thus, in general, it does not appear that control is simplified to two groups of muscles, acting as agonists or as antagonists for a given movement. Is it possible to conceive of movement regulation on a muscle-by-muscle level? It has long been known that some muscles are compartmentalized, with muscle fibers having mechanical actions that differ from compartment to compartment. Nevertheless, it was supposed that, at least within each compartment, motor units were recruited in an orderly fashion, according to the size principle, and that except for this proviso motor units would be recruited similarly and simultaneously. Recent studies do not support such a simplifying assumption. Instead, they indicate that the control and activation of motor units is effected in a distributed fashion.

This assertion is based on results of studies of the recruitment of single motor units of elbow and shoulder muscles, primarily biceps and deltoid, under isometric conditions (Fig. 2). Under isometric conditions (as well as during reaching movements), muscles generally exhibit "cosine tuning.'' When force is exerted in a particular direction, muscle activation is greatest. When the same amount of force is exerted in a different direction, the amount of muscle activity decreases in a manner that is approximated by a cosine

Figure 2 Directional tuning of motor units. (Top left) A typical experiment used to define directional tuning of motor units. The arm is maintained isometrically and force is exerted in a variety of directions. For each direction of force, firing frequency and the threshold of recruitment of single motor units are determined. (A) The typical tuning curve, with the distance from the origin to the edge of the closed curve representing firing frequency when the same amplitude of force is exerted in different directions. The arrow denotes the best direction for this motor unit. (B) The thresholds for recruitment of threemotor units. Note that the threshold for each unit is defined by a straight line (equivalent to cosine directional tuning in A), but that the slope of the line is different for each unit. The arrows show the best direction for each unit. Note that the recruitment order of these three units depends on the direction in which force is exerted.

Figure 2 Directional tuning of motor units. (Top left) A typical experiment used to define directional tuning of motor units. The arm is maintained isometrically and force is exerted in a variety of directions. For each direction of force, firing frequency and the threshold of recruitment of single motor units are determined. (A) The typical tuning curve, with the distance from the origin to the edge of the closed curve representing firing frequency when the same amplitude of force is exerted in different directions. The arrow denotes the best direction for this motor unit. (B) The thresholds for recruitment of threemotor units. Note that the threshold for each unit is defined by a straight line (equivalent to cosine directional tuning in A), but that the slope of the line is different for each unit. The arrows show the best direction for each unit. Note that the recruitment order of these three units depends on the direction in which force is exerted.

function. Thus, muscles can have a "best direction.'' (Sometimes, the tuning curve shows two peaks and thus there are two best directions.) As shown in Fig. 2A, this tuning property is also obeyed by single motor units. However, the best directions of individual motor units in one muscle do not coincide but are dispersed over a wide range of directions (Fig. 2B). Furthermore, there is no evidence for a clustering of these best directions; thus, the results cannot be explained by assuming muscle compartmentalization.

The threshold for recruitment of a motor unit is lowest when force is being exerted in that motor unit's best direction. If the best directions of motor units vary over a fairly wide range, as was found for the muscles studied so far, then there is no fixed recruitment order of motor units within a given muscle because there will be a different recruitment order for every force direction (see the threshold levels for the three motor units in Fig. 2B).

In summary, the experiments discussed in this section lead to two main conclusions. First, in order to predict the forces and torques developed by activation of any muscle, the motion and posture at each of the joints must be sensed (or predicted accurately) throughout the movement. Second, neural control of the musculature is not exerted over groups of muscles (synergies) or even at the level of individual muscles. Rather, this control is distributed, with individual motor units within a muscle having individual tuning characteristics.

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