Ocular Dynamics and Skeletal Systems

I

ssues that are central to the modeling and analysis of a human movement system include musculotendon dynamics, the kinetics and kinematics of the biomechanical system, and the determination of neurological controls that are pertinent to a particular movement. In formulating a model for a biological control system, realism and complexity are always competing concerns. Human motion involves neurons, muscles, chemical reactions, bones, joints, and ligaments. How realistic can a model be made and still be simple enough for CONTROL SYSTEMS practical impleIN BIOLOGY mentation and analytical tractability? What features of these enormously complex mechanisms are essential to include in the model, and which may be left out? Clearly, part of the answer to these questions lies in the specific model to be analyzed and the purpose for which it is to be used. This article focuses on the dynamics and control of ocular and skeletal systems. The discussion of these systems provides insights into modeling issues that are common to the study of human movement systems. One of the first human movement systems to be studied was the ocular motor system. In 1630, Descartes [1] proposed a model of eye movement based on the principle of reciprocal innervation, a notion of paired muscular activity in which a contraction of one muscle is associated with the re-

laxation of the other. Since that time, modeling of the human ocular system and its dynamic properties have been studied extensively by neurologists, physiologists, and engineers [2]-[14]. Both practical and theoretical motivations exist for considering the ocular movement system. Clinical applications include the diagnosis and treatment of strabismus, muscle palsy, and the effects of neurological disorders. From a control-theoretic viewpoint, the ocular system is of relatively low dimension and easier to control than other neuromuscular systems. By scrutinizing the trajectories of eye movements, it is possible to infer the effects of motoneuronal activity; to deduce the central nervous system’s control strategy; and, because of the relatively small number of actuators in this system, to more systematically observe the effects of perturbations in musculotendon parameters, as well as neural controls. Models of the musculoskeletal system also provide a means for elucidating the relationships between form and function. For instance, understanding the adaptation of femoral structures to mechanical stimuli by joint and muscle loading is basic to the notion of bone remodeling, first proposed by Wolf [15]. Since the first systematic study of the muscle-bone unloading principle by Pauwels [16], there have been several investigations into the role of muscle forces on the loading conditions of bone and subsequent stress development [17]-[24]. There are several practical and clinical motivations for studying neuromuscular effects on stress development. For

The author ([email protected]) is with the Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, U.S.A. 70

0272-1708/01/$10.00©2001IEEE IEEE Control Systems Magazine

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©1995 PHOTODISC, INC.

By Lawrence Schovanec

instance, the role of muscular activity on bone loading is an issue in the functional use of electrical stimulation for the restoration of movement to paraplegics [21]. It is argued that the safety of an open- or closed-loop functional electrical stimulation system relies for protection of the bone, as in nature, on limiting the stress development due to muscular and joint loading. There are other instances in which the role of muscular activity has been suggested as a primary source of potentially damaging stress patterns. For example, it has been hypothesized that as muscles fatigue, their effectiveness in absorbing impact is diminished, and consequently more force is transmitted directly to bone. In view of the theoretical strength limits of bone, it is surmised that stress fractures of bone, for instance, may represent primarily a lack of muscular function and coordination, with bone failure a secondary event [25]. Consequently, there is strong motivation for devising a methodology that would relate musculotendon dynamics and neurological controls to a continuum analysis of bone that includes temporal effects of both neurological control and muscular loading during a complete range of motion. Defining the loading conditions that accurately reflect the state of a human bone during in vivo activity is a formidable challenge. It is difficult to determine the forces of the individual muscles that result in an applied moment since there are typically more unknown muscle forces than can be determined from mechanical relations alone. This is referred to as the redundancy problem. To address this problem, the muscle set that contributes to a specific motion may be reduced by grouping muscles of similar function or by using electromyographic (EMG) activity as a guide in determining which muscles are active during a specific movement [26], [27]. If the problem is still indeterminate, a static optimization scheme may be employed. For these types of optimization methods, the selection of appropriate optimization criteria is somewhat arbitrary, and the schemes do not take into account musculotendon dynamics [28]-[30]. Consequently, the static optimization approach often results in discontinuous muscle force histories [31]. Another limitation associated with the inverse method is that it is not predictive in nature in that one is limited to studying motion that is produced by monitored subjects. Most approaches to a stress analysis of bone that includes muscular loading are based on muscle forces derived from static optimization methods, experimental measurements of force, or a combination of both [17], [18], [20], [22]. However, when estimates of muscle force derived by these techniques are used to estimate muscular loads in a quasi-static analysis of stress, dynamic effects due to strain rates and loading are ignored. In contrast to the inverse method, in which motion acts as the input and torques are the output, the forward or direct-dynamic approach provides the motion of the system over a

given time period as a consequence of the applied forces and given initial conditions. Solution of the direct-dynamic problem makes it possible to simulate and predict motion as a result of the forces that produce it. In a forward analysis, the torques or the muscle forces that generate the moments are the inputs and the body motion is the output. Since neural in-

The forward dynamic approach shows promise in connecting mechanisms of failure in skeletal structure with musculotendon dynamics and neural strategies.

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put activates the muscles (i.e., the actuators of the system), the true input into the system is indeed neural input. Because controls are needed for each muscle, the redundancy problem reoccurs. In the forward approach, however, muscle dynamics are incorporated into the optimization techniques used to determine the controls. When human gait is to be simulated, for instance, the dynamic optimization methods employ cost functions that usually involve both a tracking error term and a term influencing the distribution of muscle force [31], [32]. When controls for each muscle are determined, the system of differential equations for the body segments and muscle groups can be integrated forward in time to obtain the motion trajectories. In this sense, a direct-dynamic analysis is self-validating in that the specified controls do indeed result in the observed motion. A dynamic stress analysis may be carried out by including the joint torques and reaction forces as predicted by a direct-dynamic model of human movement into a stress analysis of the segmental links that represent bone. This is the approach used in [23] and [24] in which neuromusculotendon dynamics are incorporated into a model of human motion in which the human body is represented by an ensemble of articulating elastic segments. In [24], a stress analysis of the lower extremities is considered in which muscular loading is derived from a forward model of gait. The analysis is still quasi-static in that the equation of equilibrium for stresses is solved corresponding to certain fixed instances of the gait cycle. A novel hybrid parameter approach is presented in [23] that includes effects of dynamic loading while incorporating both rigid-body motions of the articulating segmental links and the elastic deformations that represent the continuum effects in the bone. The stress analysis carried out by this method not only incorporates dynamic effects due to muscular loading, but also allows for very general constitutive properties that are relevant to the accurate description of bone and connective tissue. A complete development of a direct-dynamic model needs to include a representation of the musculotendon

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complex, anatomical geometry, kinematic models and inertial characteristics of the underlying movement system, and the appropriate controls for a specific movement. In the next section, we provide an overview of musculotendon dynamics. In subsequent sections, we indicate how these dynamics are incorporated into movement systems when we revisit the problems of ocular motion and stress analysis of skeletal elements.

Musculotendon Dynamics Muscles are the actuators of the neuromusculoskeletal control system that produces movement. In the analysis of a control system as complex as that governing movement, a clear understanding of the physical nature of the actuators and a tractable mathematical representation of their dynamics are essential. The muscle models most commonly used in biomechanical applications are “Hill-type” models. These models have been shown to incorporate enough complexity while remaining computationally practical. Despite their widespread acceptance, alternative models are also available that vary from the simple [33] to the extremely complex [34]. Another notable alternative, due to Zahalak [35], is the distribution-moment model derived from cross-bridge theories of muscular contraction. This model, though more computationally intensive than a Hill model, provides a way to extract relatively simple state-variable models of muscle dynamics from biophysical models of contraction at the molecular level. The phemenological representation of the Hill model in Fig. 1 and the subsequent discussion follows closely the developments in [36] and [37]. In this figure, the muscle of length l m is in series and off-axis by a pennation angle α with the tendon of length l t . The total path length of the musculotendon complex is denoted by l tm . The muscle consists of active and passive components. Passive effects are usually attributed to a nonlinear elasticity, Fpe , and a damping component that corresponds to the passive muscle viscosity, Bm . The model for the active contractile component is based on the generally accepted notion that the active muscle force Fact is the product of three factors: a length-tension relation fl (l m ), a velocity-tension relation fv (l&m ), and the activation level a( t ). Full activation occurs (a( t ) = 1) when the muscle has been neurally or electrically excited long enough for activation transients to subside. Conversely, muscle is said to be inactive (a( t ) = 0) if it has been neither neurally nor electrically excited. Muscle force is easily measured at various lengths under isometric conditions to produce force-length relationships. The curve produced when muscle is not stimulated is the passive force-length curve, Fpe (l m ). When muscle is activated, the curve that results represents both passive and active contributions. The difference in these two curves is the active force-length relation, fl (l m ). The length at which the maximum active muscle force, Fo , is developed is called the optimal muscle length, l o . A theoretical explanation for the

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active fl relation is based on the microscopic nature of muscle and is explained by the Sliding Filament Theory. This theory offers an explanation for the generally accepted notion that, when the muscle is fully active, fl (l m ) displays a parabolic dependence on length in a nominal region, 0.5 l o ≤ l m ≤ 15 . l o with a maximum value of Fo whenl m = l o . At less than full activation, the force-length dependence is obtained by scaling the fully activated fl curve. Active muscle force exhibits a dependence on the rate of muscle contraction. When a muscle actively shortens, it produces less force than it would under isometric conditions. Hill was the first to quantify this result with an empirical hyperbolic relationship when muscle is shortening. In contrast to a concentric contraction, when a muscle is actively lengthening it is able to produce forces above the maximal isometric force. Experimental data reveal that this relationship is not an extension of Hill’s equation and exhibits a threshold that limits the amount of tension muscle can withstand, approximately 1.8 Fo . The fv curve that describes the force-velocity effect is also thought to scale with activation. Muscle activation, a( t ), is related to the neural input, u ( t ), by a process known as contraction dynamics. Both quantities, u ( t ) and a( t ), can be related to experimental data. In particular, u ( t ) is related to rectified EMG and a( t ) is related to filtered and rectified EMG. The process through which neural input is transformed into activation is known to be mediated through a calcium diffusion process and is represented by the first-order differential equation da( t ) 1 + dt τ act

( β + (1 − β )u( t )) a( t ) =

1 u ( t ), τ act

(1)

where β = τ act / τ deact , 0 < β < 1, and τ act , τ deact are activation and deactivation time constants that vary with fast and slow muscle. The series elastic element in Fig. 1 corresponds to the muscle tendon. A common approach to tendon dynamics (see, for example, [38]) is to assume the tendon force Ft is described by a generalized Hooke’s Law F&t = K t ( Ft )l&t .

(2)

The stiffness K t ( Ft ) is determined from the tendon stress-strain curve. In particular, tendon is typically modeled as an exponential spring for small strains and as a linear spring beyond some critical value of the strain. The second-order musculotendon dynamics also include the equation of motion for the muscle mass M m

[

]

M ml&&m = Ft cosα − cos2 α Fact + Fpe + Bml&m +

M ml&m2 tan 2 α . lm (3)

For multiple-muscle systems, it is advantageous to develop curves describing the attributes of a generic muscle.

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In human binocular vision, the movement of each eye is controlled by a set of six muscles. When the eyes are fixed on an object, two things occur. First, the eye rotates so that the image of the object formed by the lens system of the eye is projected onto the fovea of the retina. This is the retinal area of greatest ocular acuity. Second, the eye lenses adjust to bring the object into focus. The studies cited here are primarily concerned with the first part of this process: how the rotation develops and how it is controlled. The brain and central nervous system process information obtained by the retina and then transmit signals to the extraocular muscles. These muscles work in three agonist-antagonist pairs to exert forces on the eye, causing it to rotate. The three muscle pairs consist of the medial and lateral recti, the superior and inferior recti, and the superior and inferior obliques. The lateral and medial recti produce primarily horizontal rotation. The superior and inferior recti work mainly to control vertical rotation. The superior oblique controls the intorsional rotation (toward the nose) of the eye, whereas the inferior oblique controls mainly extorsional rotation (toward the temple). The eye has three degrees of freedom, but experimental evidence shows that for any horizontal and vertical rotation, the amount of torsional rotation is determined by a phenomenon known as Listing’s law [8]. The bulk of eye movement studies have been restricted to horizontal motion. Fig. 2 provides a mechanical representation of the eye plant model, appropriate to horizontal motion, in which the elements of the plant are displayed as if the muscles (the medial and lateral recti) and the globe are undergoing linear motion. In this figure, J g corresponds to the inertia of the globe and the surrounding suspensory tissue is modeled by a passive elastic effect with stiffness K o and viscosity Bo . Neural controls u ( t ) that would be appropriate to a saccadic movement are illustrated and pennation effects in the musculotendon units are ignored.

This curve can then be scaled with appropriate parameters to reflect the dynamics of a particular muscle. A variety of studies have illustrated that the scale parameters needed for each musculotendon group include: • maximal isometric active muscle force Fo ; • optimal muscle length, l o ; • pennation angle α o when l m = l o ; • tendon slack length, l ts . Another quantity used to specify a muscle-specific force-velocity relation is the maximum speed of shortening,vo . For instance, it is convenient to use a generic force-length relationship for tendon derived by a method discussed in [37] in which a force-strain curve is based on the assumption that a nominal stress-strain curve can be formulated that represents all tendons. By scaling the generic force-strain relationship by Fo and l ts , a force-length function is found for a specific tendon. Curves used in the modeling of Fpe , fl , fv , and K ( Ft ) are generally developed by two methods. In the case that sufficient data are available, a natural cubic spline may be fit to the data [24], [39]. As an alternative approach, analytical expressions that capture the qualitative properties of the curves may be used. In this case, the parameters that appear in these expressions may be determined by imposing smoothness conditions in combination with a fit of experimental data (see, for example, [38]). Two state variables are required to describe the contraction dynamics of the musculotendon actuator as given by (2) and (3). For large multiple-muscle systems such as that needed to describe gait, it is desirable to reduce the system dimension. This is usually achieved by eliminating the muscle mass, but with certain attendant complications related to computation and stability. An additional consequence of eliminating muscle mass is the loss of controllability of the linearized system [38]. Additional modeling issues that arise in Hill-type actuators are amplified in the subsequent sections within the context of applications to ocular dynamics and stress analysis of segmental models.

Ocular Dynamics Although the literature on the eye is vast, a brief discussion of representative work provides insight into modeling issues of this system, as well as those aspects of musculotendon dynamics that are pertinent to ocular dynamics. Here we will focus on the contributions of homeomorphic models [2]-[14]; that is, those that incorporate elements representative of the parts of the eye positioning system and their interactions. These are to be contrasted with statistical and psychological models [40].

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lm Bm

ve t ssi en Pa pon m Co

α

Fpe

lt

Mm

tive ile Ac tract ent n n Co mpo Co

Kt(Ft ) ltm

Figure 1. Hill-type model of the musculotendon complex (adapted from [36]).

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tor units provide insight into the time course of neural innervation that results in a specific movement. In particular, eye movements are generally Neuronal Signal t classified as saccadic, smooth pursuit, u2 t u1 vestibular, vergence, or optokinetic. Fig. 3 provides a qualitative repreActivation Dynamics , τ τ a1 d1 τa 2,τd 2 sentation of neural input u ( t ) that rea1(t ) sults in a saccadic, vergence, or a2(t ) Activation smooth pursuit movement. For inMuscle Dynamics stance, it is accepted as “sufficiently realistic as to be useful” [3] that the θ F act1 Fact2 rate of discharge that results in a Kt 2 Kt 1 Bm2 Bm1 saccadic movement can be described Jg by a rectangular pulse followed by a Mm1 Mm2 step. Fpe1 Fpe2 In 1954, Westheimer [2] developed Ft 2 Tendon Force Ft 1 a linear second-order approximation of eye dynamics during a saccade in which the input to the model was asBo sumed to be a step of muscle force. Orbit Viscosity, Elasticity The model worked well for 10° saccades but not for larger such moveKo ments. In addition, the model produced the unphysical predictions Figure 2. The model of the eye plant. that the saccade duration would be independent of saccade magnitude and that the peak velocity would be directly proportional to θ u(t) saccade magnitude. A more realistic representation of eye uant movement was advanced by Robinson [3]. His linear Saccade fourth-order model could simulate saccades between 5∞ t and 40∞, but the velocity profiles predicted were not physiuag cally realistic. Westheimer and Robinson recognized that t t the eye movement mechanism was inherently nonlinear, but did not explicity address this issue. A sixth-order nonlinu(t) θ ear model proposed by Cook and Stark [4], and subseuant quently modified in [6], produced realistic position, t velocity, and acceleration profiles. This Cook-Clark-Stark Smooth model addressed the nonlinear relationship between force Pursuit uag and velocity but ignored the active force-length characterist t tics of muscle. This assumption was tantamount to assumu(t) ing that the medial and lateral rectus muscles operate near θ uant the primary position that corresponds to looking straight t ahead. Their model incorporated a force-velocity dependVergence ence into the active muscle by a velocity-dependent viscosuag ity that was experimentally determined by fitting t t experimental data to Hill’s equation. The model did not inFigure 3. A qualitative representation of agonist/antagonist clude any passive viscosity and, moreover, the passive elasticity was lumped together with the nonmuscular ocular controls and corresponding movements. suspensory passive tissue. The works cited above pertain A meaningful model of the horizontal movement system only to horizontal motion and employ relatively crude can be developed for several reasons. Experimental data musculotendon dynamics, especially as compared to invesprovide values for many of the parameters that describe the tigations that involve gait and posture. In addition, these elements of the musculotendon complex. In addition, re- models do not account for nonlinear effects that occur cords of EMG activity corresponding to the firing rate of mo- when muscles act in a nontangential fashion on the eyeball u (t )

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u (t )

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Angular Velocity

Position (deg)

30 20 10 0 0

600

200 0

0.05 0.1 Time [ms]

30

Lateral Rectus

20 Medial Rectus

10 0

Activation

Tendon Force

0

10 θ (deg)

20

1

40

0

Phase Portrait

400

50

0.5 Lateral Rectus 0 0

0.05 0.1 Time [ms]

Medial Rectus 0.05 0.1 Time [ms]

Figure 4. Simulation of 10∞ saccade. 40

Eye Movement (deg)

30

20

10

0 0

100

200 Time [ms]

300

Figure 5. Saccadic eye movements from 5∞ to 40∞. Records were selected from temporal saccades on one subject as representative for each saccade (from [3]). 80

Net Muscle Force (g)

once it has undergone a sufficiently large rotation. An additional geometric nonlinear effect is due to the fact that the paths of ocular muscles may be constrained by orbital connective tissue. In particular, the stability of muscle paths during large ocular rotations is due in part to their passage through extraocular muscle pulleys [41]. In [10] and [11], a model of 3-D ocular motion is presented that treats the orientation of the eye as the output of a velocity-position integrator. Roughly speaking, the output of the system is a signal converted to torque, in contrast to an approach in which musculotendon forces generate moments in response to neural stimulation from the central nervous system. The results of [12] extend the vector integrator model to include the effects of muscle pulleys. In contrast to the approach of [10]-[12], the 3-D studies in [13], [14], and [42] emphasize the incorporation of general nonlinear musculotendon dynamics, as reviewed in the previous section, though the later study is restricted to a static analysis. Simulations of a 10° saccade from [13] are illustrated in Fig. 4. The model, based on the inclusion of muscle mass and general nonlinear musculotendon dynamics, provides excellent agreement with the experimental data reported by [3] and illustrated in Figs. 5 and 6. In particular, the approach of [13] and [14] appears to offer a certain sense of robustness in that the model provides for good predictions of trajectories, phase portraits, and tendon forces over a fairly large range of saccadic movements. In addition, 3-D simulations of saccades carried out in [13] support the claim that horizontal eye movement may be accurately modeled by including only the medial and lateral rectus muscles, although inclusion of the six extraocular muscles provides predictions of trajectories and phase portraits that more accurately reproduce the empirical data reported in [3]. It has been suggested that the oculomotor system operates in an open-loop mode during a saccade due to insufficient time for information to pass from the retina and muscle proprioreceptors to the central nervous system (CNS) [7]. On the other hand, other movements, such as smooth pursuit, most likely involve feedback. Since feedback from muscles is provided by spindles (position-derivative) and the Golgi tendon organs (force), it is important to construct system models that produce accurate predictions of tendon force to investigate these closed-loop systems. Several issues in ocular motion remain to be addressed. Arguments are made in [43] that certain components of the musculotedon unit are modulated by activation. For example, it is suggested that the tendon stiffness in (2) is a function not only of tendon force, but also of activation. More recently, additional motivation is provided in [44] for considering that parameters in the ocular system and the coordination of agonist-antagonist pairs are affected by neural input. Other topics that require attention include the role of nonlinear geometric effects, such as pulley constraints, on the dynamical movement of the eye. The determination of

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40° 35° 30° 25° 20° 15° 10° 5°

40

20

0 0

100

200 Time [ms]

300

Figure 6. Tension recorded for various saccadic movements of 5∞ to 40∞. Shading on the first three records represents the range of 18 records over three subjects (from [3]).

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or damage in skeletal members. Since abnormalities in muscle function or innervation can more easily be incorporated into a direct dynamic model, this approach provides insight into the role of neural control in skeletal injury. In [24], a forward dynamic model of gait is implemented in which the human body is modeled as an ensemble of articulating rigid-body segments controlled by a minimal muscle set. Neurological signals derived by dynamic optimization methods act as the input into musculotendon dynamics. From the muscular forces, the joint moments and resulting motion of the segmental model are derived. At fixed moments in the gait cycle, the joint torques and joint reaction forces are incorporated into an equilibrium analysis of the segmental elements, modeled as elastic bodies undergoing biaxial bending. The approach to gait simulation that is adopted in [24] builds on a model developed by Yamaguchi [31]. The Musculotendon Dynamics and Stress model constrains seven rigid-body segments that represent Analysis in Skeletal Structures Few studies have focused on the role of individual muscles in the feet, shanks, thighs, and trunk to eight degrees of freestress development in bone. A promising approach to this dom. Ten musculotendon units are incorporated into the problem lies in a forward or direct dynamic analysis that simu- model, five on each leg. On the stance leg, the relevant muscle lates human motion and incorporates the interaction of the ac- groups are the soleus, gastrocnemius, vasti, gluteus medius tive and passive structures of the body [23], [24]. An objective and minimus, and the iliopsoas. The swing leg utilizes the of this work is to relate musculotendon dynamics and neuro- dorsiflexors, hamstring, vasti, gluteus medius and minimus, logical controls to a continuum analysis of bone to show how and the iliopsoas. Thus, seven different musculotendon the effects of muscular control bear upon predictions of failure groups need to be specified in this model. When a direct-dynamic analysis of gait is to be simulated, controls for the musculotendon actuators must be derived. Developing these controls constitutes one of the more difficult aspects of a forward analysis. Some form of dynamic opT timization is usually used in developing the controls. The ∧ controls used in the gait simulation of [24] are derived n 1 q1 through a two-phase process: 1) a coarse formulation based on a dynamic optimization scheme and 2) fine-tuning via trial and error. The coarse controls are derived, as in [31], by ~ u2 ( x1 , employing a cost function that consists of an error-tracking t) ∧ term which penalizes deviations from a desired trajectory b1 ∧ and a term related to muscle fatigue [30]. The nominal gait ∧ b2 n2 trajectory is specified according to data recorded in [45]. Once crude activation controls are formulated, simulations are run, and these controls are fine-tuned by ad hoc methods to meet the specific needs of the model. Figure 7. Single link arm with load. The approach to stress analysis in [24] is based on equilibrium consider−3 ations by “freezing” the gait model at 4 10 100 fixed instants in time and regarding the 80 2 segmental elements as linear isotropic Joint Angle Strain elastic beams undergoing biaxial de60 0 q1 formation. The applied loads for the (deg) 40 −2 stress analysis are due to the joint re0.4 action forces and the muscle forces, as 20 Length (cm) 0.2 derived from the gait analysis. The 0 0 0.5 02 1 1.5 0 0.5 1 1.5 2 components of the stress tensor are Time [s] Time [s] calculated in terms of these internal forces and moments. Results show Figure 8. Ballistic simulation: Joint angle q1 (see Fig. 7) and strain along the length of that muscular loads have a significant the arm. neural controls specific to a particular movement and the mechanisms by which the central nervous system switches from one mode of eye movement to another poses many interesting questions. Beyond any clinical application that may be derived from studies of ocular dynamics, there are compelling reasons for addressing the role of musculotendon dynamics and neural control in the context of the eye system. As alluded to earlier, the low dimensionality of this system provides a more tractable setting in which to examine different modeling assumptions and control-theoretic issues that have application to larger human movement systems. Gait and posture are examples of more complicated human movement systems and will be addressed in the next section.

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effect on the stress distribution that includes a general re- length of the link. As expected, the arm settles into an equilibrium position corresponding to hanging straight down, duction of bending moments throughout the gait cycle. In [23], modeling of articulated elastic segments, cou- and the only measurable strain in the bone occurs when the pled to muscle dynamics, is accomplished using a motion is impeded by the soft constraints due to the passive nonholonomic hybrid parameter projection method [46], joint effects that keep the arm from hyperextending. When a general n-link system is linearized about a nomi[47]. A set of field and boundary equations and the discrete coordinate ordinary differential equations are provided by nal trajectory, one obtains a system that is controllable and this method. These equations are minimal in the sense that observable (with feedback only from the tendon organs and the dynamics are projected on the constraint-free manifold muscle spindles) [38]. Moreover, it can be shown that the of the generalized speed space and thus require no alge- same feedback control that stabilizes the linear system stabraic side conditions. bilizes the nonlinear system about a nominal trajectory. For purposes of illustration, the results presented here are for the fore10−5 8 6 arm, in which case the relevant system 6 is shown in Fig. 7. The arm and load are 4 Tip Displacement Joint Angle 4 shown in an exaggerated deformed u2(cm) 2 state so that the coordinates can be 2 q (deg) clearly seen. The coordinate frame 1 0 0 n$ 1 , n$ 2 is the Newtonian frame. Frame −2 b$1 , b$2 rides with the undeformed state of −4 −2 the beam (body 1). The domain of the 0.2 1.15 1 0.05 0 0.2 1.15 1 0.05 0 beam is one dimensional, and the indeTime [s] Time [s] pendent coordinate is x 1 , measured 500 4 10−3 from the root of the beam along the 400 2 beam’s undeformed neutral axis. The Strain 300 loading torque due to tendon forces 0 Biceps Force and passive joint effects is designated (N) 200 −2 asT. The coordinate measuring angular 0.4 100 position of the beam is q 1( t ). The deLength 0 0.2 (cm) Triceps flection of the beam is given by −100 0 00.2 0.15 0.1 0.05 u~2( x 1 , t ) b$2 (simple flexure). The tip 0.2 1.15 1 0.05 0 Time [s] Time [s] mass is the second body, and it is allowed both mass and mass inertia (relative to point b at the tip of the beam). Figure 9. Simulation with the muscle length control strategy. Gravity acts along the n$ 2 direction. 10−5 8 6 The equations of motion and the field equations are derived from an ex5 6 Tip Displacement tended D’Alembert-type principle that Joint Angle 4 4 is implemented with a computer algeu2(cm) 3 bra system. These integral-differential 2 q1 (deg) 2 equations are then recast in a weak for0 1 mulation and solved by a two-mode 0 −2 Hermite polynomial approximation for 0.2 1.15 1 0.05 0 0.2 1.15 1 0.05 0 the displacement. The resulting kineTime [s] Time [s] matic differential equations of motion 400 4 10−3 are coupled to the control equations 2 300 via the torques that reflect the comStrain bined moment due to the muscle 0 200 Biceps Force forces and the joint passive effects. −2 (N) 100 0.4 Numerical illustrations for ballistic Length 0 and active simulations are presented. 0.2 (cm) Triceps In the first case, the forearm is released −100 0 0 0.05 0.2 0.15 0.1 0.2 1.15 1 0.05 0 at rest from the horizontal position Time [s] Time [s] with a load of 5 kg. Fig. 8 shows the joint angle q 1 and the strain along the Figure 10. Simulation for optimal control with full state feedback. August 2001

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Consequently, linear quadratic regulator (LQR) methods may be employed, as in [36], to examine optimal control strategies that correspond to regulating, for example, muscle length, joint position, or tendon stiffness. The effect of these various control strategies on stress and strain development may then be explored using the direct dynamic, hybrid parameter approach [48]. For instance, simulations in Figs. 9 and 10 display the results that correspond to a perturbation of the arm from a horizontal position. In this case, the linear system is time invariant, and one can compute optimal feedback controllers that correspond to a given proposed neural control strategy. As in [36], from the feedback gain matrix one gains insight into how the proprioceptors and tendon organs in the biceps and triceps project onto the motoneuron pools that control the limb. Illustrations in these figures show the joint angle, the tip displacement, and the tendon forces that correspond to correcting a small perturbation of about 5∞. The strategies that are compared in Figs. 9 and 10 correspond to regulating the joint angle and full state feedback control. The latter strategy is tantamount to regulating joint angles and muscle forces and lengths. One observes that, with full state feedback, muscle forces required to correct the perturbation are smaller and there is less overshoot in the joint response. The strain energy in the arm is less for full state feedback, although there is no significant difference in the maximum strains. The point to be emphasized by this simple example is that, although the LQR approach is most likely a gross simplification of strategies employed by the human body, the methods discussed here provide a means of connecting neural control with attendant consequences in skeletal structure. The application of this approach shows promise in connecting mechanisms of failure in skeletal structure with musculotendon dynamics and neural strategies.

Acknowledgments This work was supported by NSF grants ECS-9720357, ANI-9906090, and DMS-9977197.

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[48] A. Barhorst and L. Schovanec, “Optimal motor control strategies and hybrid approach to stress analysis in skeletal systems,” in Proc. 2001 American Control Conf., Arlington, VA, to be published.

Lawrence Schovanec received the B.S. degree from Phillips University in 1975, the M.S. degree from Texas A&M University in 1977, and the Ph.D. degree from Indiana University in 1982, all in mathematics. He has been on the faculty of Texas Tech University since 1982 except for time spent as a visitor at Texas A&M University and as a Research Fellow at the Air Force Research Laboratory, Edwards AFB. He is now Professor and Chair of the Department of Mathematics and Statistics at Texas Tech University. His research interests are in control-theoretic aspects of biological systems and solid mechanics, with an emphasis on fracture in elastic and viscoelastic media.

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