A Forward Dynamic Model of Gait with Application to ... - Science Direct

MATHEMATICAL COMPUTER MODELLING PERGAMON

Mathematical

and Computer

Modelling

33 (2001) 121-143 www.elsevier.nl/locate/mcm

A Forward Dynamic Model of Gait with Application to Stress Analysis of Bone Y.

DEWOODY, C. F. MARTIN* AND L. ScHovmmt Department of Mathematics, Texas Tech University Lubbock, TX 79409, U.S.A.

Abstract-A model of human gait is developed for the purpose of relating neural controls to the state of stress in a skeletal member. This is achieved by implementing a forward dynamic model of gait in which the human body is modeled as an ensemble of articulating rigid-body segments controlled by a minimal muscle set. Neurological signals 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. @ 2001 Elsevier Science Ltd. All rights reserved. Keywords-Musculotendon

dynamics,Direct dynamic, Gait simulation, Stress analysis.

1.

INTRODUCTION

To compute the stress and strain distribution in bone, one must know the loading conditions. The integrity of the calculated stress fields during a task depends not only on the accuracy of the constitutive model of bone, but also on the proper interpretation and calculation of the loading conditions to which the bone is subjected. Defining the loading conditions which accurately reflect the state of a human bone during in wiwo activity is a formidable challenge. One of the goals of this research is to develop an analytical, predictive method for acquiring appropriate dynamic loading conditions for the long bones throughout the gait cycle. A promising approach to this problem lies in the forward analysis of a mathematical model capable of simulating normal gait which incorporates neural controls and musculotendon dynamics in a stress analysis of the lower extremities. For these reasons, this work emphasizes a forward or direct dynamic approach that results in a natural flow of neural-to-muscular-to-movement events while utilizing physiologically realistic models of the musculotendon actuators that faithfully reproduce trajectories and muscle tension. If a movement system has n degrees of freedom, then the equations of motion governing the system can be written in the form

[M(6)] ti = C *Supported *Supported 003644-123.

(19,“)+ G(0) + Fm(B),

by NSF Grants ECS-9720357, DMS-9628558. by NSF Grants ECS-9720357, DUE9813056

and Texas Advanced

(1) Research

08957177/01/$ - see front matter @ 2001 Elsevier Science Ltd. All rights reserved. PII: SO895-7177(00)00234-X

Program

Typeset

Grant

by J.A&-TFX

No.

Y. DEWOODY et al.

122

where 8, e, 4 are n x 1 vectors of displacement, velocity and acceleration, [M(e)] is the n x n inertia matrix, C(~,I$ is an n x 1 vector of coriolis and centrifugal terms, G(0) is an n x 1 vector of gravitational terms, and F,(8) is an n x 1 vector of applied moments. There are essentially two approaches to analyzing such a movement system. Inverse dynamics is an approach that proceeds from known kinematic data and external forces and moments to arrive at expressions for the resultant intersegmental forces and moments. For this method, motion acts as the input and the joint torques are the output. This approach requires experimental and kinematic data and observed motions. If these variables are assumed to be known functions of time, (1) becomes an algebraic equation for &(6). It is a difficult problem however, 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 EMG activity as a guide in determining which muscles are active during a specific movement [l-3]. If the problem is still indeterminate, a static optimization scheme is employed. For these types of optimization methods, the selection of appropriate optimization criterion is somewhat arbitrary, and the schemes do not take into account musculotendon dynamics [4-6). Consequently, the static optimization approach often results in discontinuous muscle force histories [7]. Another limitation associated with the inverse method is that it is not predictive in nature in that one is limited to studying motion which is produced by monitored subjects. In contrast, 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 forward dynamics problem makes it possible to simulate and predict body segment motion as a result of the forces that produce it. In a forward analysis, the joint torques, or the muscle forces that act on the skeletal system to result in joint moments, are the inputs and the body motion is the output. This relationship is emphasized by writing equation (1) in the form 8=

[M(O)]-’(C (B,d)+ G(B) +F,(B)).

Since neural input activates the muscles, i.e., the actuators of the system, the true input into the system is indeed neural input. Because controls for each muscle are needed, the redundancy problem reoccurs. In the forward approach, muscle dynamics are incorporated into the optimization techniques used to determine the controls. When human gait is to be simulated, the dynamic optimization methods employ cost functions that usually involve both a tracking error term and a term influencing the distribution of muscle force [8]. Once controls are achieved, the system of differential equations for the body segments and the 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 controls specified do indeed result in the observed motion. In the next section, we provide an overview of musculotendon dynamics. Since the validity of the Ioading conditions relies on the ability of the model to simulate normal gait as accurately as possible, considerable attention will be paid to the development of the musculoskeletal model and the incorporation of the musculotendon actuators into the gait analysis. We then incorporate the joint reaction forces and moments derived from the gait simulation into a stress analysis of the skeletal links. 2. 2.1. Functional

MUSCULOTENDON

DYNAMICS

Properties

The muscle models utilized in this investigation are referred to as Hill-type models. A phenomenological representation of the musculotendon complex as idealized mechanical objects is

Stress Analysis of Bone

Figure

presented in Figure 1.

1. Hill type model of the musculotendon

123

complex.

This model has been shown to incorporate

enough complexity

while

remaining computationally practical [9]. The muscle of length 1, is in series and off-axis by a pennation angle Q with the tendon of length lt. The total pathlength of the musculotendon complex is denoted by lt,. The muscle is assumed to consist of two components: an active force generator and a parallel passive component. The passive component includes a parallel elastic element (Fpe) that describes the passive muscle elasticity and a damping component which corresponds to the passive muscle viscosity (B,). The model for the active contractile component is based on the generally accepted notion that the active muscle force is the product of three factors: (1) a length-tension relation fr (lm), (2) a velocity-tension relation fv(im), and (3) the activation level a(t). For multiple muscle systems, such as that used in the simulation of human gait, it is advantageous to develop curves describing the attributes of a generic muscle. This model can then be scaled with appropriate parameters to reflect the dynamics of a particular muscle. We will see that the scale parameters needed for each musculotendon group include (1) maximal isometric active muscle force F,, (2) optimal muscle length, Z,, (3) pennation angle o’, when I, = Z,, and (4) tendon slack length It,. In developing nondimensional representations for these curves the approach of [lo] is implemented and all forces and lengths are scaled as p = F/F, and im = 1,/l,,. Another quantity used to specify a muscle specific force-velocity relation is the maximum speed of shortening, defined as ~0 E 10/rC. This quantity scales time and varies for fast and slow muscles. In the case of the lower extremities, a standard value of 7, = 0.1 s is used for all muscle types.

a(t)=1.O

(a) Full activation. Figure 2. Isometric

(b) Active force scales with activation but passive force is unaffected (adapted from [lo]). force-length

relation for muscle.

Y. DEWOODY et al.

124

Muscle force is 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, Fpa(lm).

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, fr(Im). The

length at which the maximum active muscle force, F’, is developed is called the optimal muscle length, 1,. Figure 2 displays the qualitative nature of these curves. A theoretical explanation for the active fi relation is based on the microscopic nature of muscle and is explained by the Sliding Filament Theory [ll]. This theory offers an explanation for the generally accepted notion that when a muscle is completely tetanized, the active force displays a parabolic dependence on length in a nominal region, 0.5 I im I 1.5 with a maximum value of F, when 1, = 1,. At less than full activation, the forcelength

dependence is obtained by scaling the fully activated fi curve (lo].

The passive force length effect exhibits an exponential behavior for small extension and is linear beyond some length. Several analytical models have been proposed for the active and passive force length curves [12]. However, in this paper, the active and passive force length curves are constructed as natural cubic splines fitted to data reported by [13] at www.kin.ucalgary.ca/isb/data/delp/delp_mus. These curves are constructed in the nondimensional forms fr(b) and fP(l,,J and then scaled by F, to provide a description for specific muscle. Figures 3 and 4 illustrate the spline approximation as well as the reported data. Active muscle force is also dependent on muscle velocity. When a muscle actively shortens, it produces less force than it would under isometric conditions. Hill [14] was the first to quantify this result with an empirical hyperbolic relationship when a muscle is shortening as

where v, = i, and shortening corresponds to v, > 0, vc denotes the maximum speed of shortening, and c is an empirically determined parameter. In contrast to a concentric contraction, when

3-

2.6 -

2-

1.5 -

l-

0.5 -

OM~ 0.8

0.9

' 1

1.1

I 1.2

I 1.3

Figure 3. Passive force-length: &,

I 1.4

vs. i,,,.

I 1.5

1.6

Stress Analysis of Bone

Figure 4. Active force-length:

&,

125

vs. L.

a muscle is actively lengthening, it is able to produce forces above the maximal isometric force. Experimental data shows that this relationship is not an extension of Hill’s equation, and there is a threshold which limits the amount of tension muscle can withstand, approximately 1.8 F,. The f,, curve is also thought to scale with activation and a qualitative representation is displayed in Figure 5. The force velocity curves that are utilized in this work are also constructed as natural

cubic splines fit to data [13] for concentric and eccentric contractions.

Due to the way the curve

is utilized in the formulation of the dynamics, it is convenient to express this relation in terms of the inverse f;‘(p). The curve is depicted along with the data in Figure 6. The output of the inverse is normalized with respect to vs. In particular, where P denotes a normalized active muscle force,

(a) Full activation

when Zm = l,,. Figure 5. Force-velocity

(b) The force-velocity relation for muscles.

scales with activation.

Y. DEWOODY et al.

126

-1

0

I 0.2

I 0.4

I 0.6

I 0.8

I 1

I 1.2

I 1.4

I 1.6

.

Figure 6. Inverse force-velocity relation for muscles: Cm vs. P.

There is evidence to suggest that total active force may be quantified as the product of the force-length and force-velocity curves [lo]. Consequently, it is convenient to visualize active force generation as a collections of surfaces obtained as a product of the nondimensionalized force velocity and force length curves scaled by activation,

Muscle activation, a(t), is related to the neural input, n(t), by a process known as contraction dynamics. Both quantities, n(t) and a(t), can be related to experimental data. In particular, n(t) is related to rectified EMG, while a(t) is related to filtered, rectified EMG [lo]. 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

where 0 < 0 < 1 and Tact is an activation time constant that varies with fast and slow muscles. The series elastic element in Figure 1 corresponds to the muscle tendon. More precisely, the tendon is assumed to behave nonlinearly under minimal extension and then to become linear with stiffness constant k, beyond a given length It, associated with a particular level of resisting force, &. A common approach to tendon dynamics is to assume a model of the form [12] @t = Kt (Ft) it, where Kt (Ft) =

kteFt + ktl, k

(2)

0 I Ft < &,

Ft 2 Ftc. 81 By integrating the above equation, the tendon force can be alternatively expressed as a function of tendon length It as

[em(he Ft&) =

(It - L))

-

11,

4, 5 4 < L 4 > kc, otherwise,

(3)

Stress Analysis of Bone

127

where lt, denotes tendon slack length. By imposing smoothness conditions on this curve and some notion of fit to experimental data, the shape and stiffness parameters can be specified. However, for our purposes, it is convenient to use a generic force-length relationship for tendon derived by a method discussed in [lo]. In particular, define tendon strain by ct = (It - Zt9)/& and normalized tendon force as pt = Ft/F,. We assume a generic force-strain curve (pt vs. E”) based on the assumptions that a nominal stress-strain curve can be formulated that represents all tendons, and that the strain in a tendon when the muscle force equals its maximal isometric value is independent of the musculotendon unit. By scaling the generic force-strain relationship by F, and Ets,a force-length function is found for a specific tendon. If we adopt the notion that the tendon behaves as an exponential spring and fit an analytical model as in equation (3) to data reported by [13], a generic force-strain relationship may be obtained in the form

Ft

(8) =

0.10377 (eglct - 1) ,

0 5 st < 0.01516,

37.526~~ - 0.26029,

0.01516 5 st < 0.1.

(4

If the strain in a tendon reaches values beyond 0.1, the tendon is known to rupture [lo]. Since such an extreme value of strain should not occur during normal locomotion, this part of the curve need not be included in our analysis. From (4), it follows that tendon stiffness defined by Kt(Ft)

= 2

is given by Kt (Ft) =

2 . +f)-$ t

t

=-.-F,

d&

4s d&t 0 5 fit 5 0.3086,

(5)

Ft 2 0.3086,

2.2.

Contraction

Dynamics

From Figure 1, it readily follows that the total force of a muscle is the sum of the passive and Muscle is known to maintain a constant volume and the active forces, F, = Fpe + Fwt + B,i,. so 1, is constant. With this observation and since

4, = It + 1, coscx, it follows that &= -imtano Ln and it

=itm--. L

cos a

The tendon dynamics may now be expressed as

(6) The equation of motion for the muscle mass is d2 (1, cos a)

M m

dt2

= Ft - [Fact + Fpe + B&a]

cosa,

Y. DEWOODY et al.

128

and with some simple manipulations, the muscle dynamics take the form

A&& = Ft cosa - cos2Q Fact + Fpe + B,i, t

+

I

kfmi: tan2 cy 1 m

.

Two state variables are required to describe the contraction dynamics of the musculotendon actuator as given by (6) and (7). For multiple muscle systems, it is desirable to reduce the system dimension. This is achieved by eliminating the muscle mass. In this case, (7) becomes a statement of force balance

Ft= F,cosa= (Foa(t)fi (t,).fv(%n)+Fofp ("m>+ B,i,)cosa. If it is assumed that the passive muscle viscosity effect is small, then i,

i, =

(Ft/‘cos4 V,

- F&

is computed as

(i;,)

= ~,f,-l Foa($fr

(

(“m>

)

’

now follows from (6) that the differential equation describing the contraction dynamics of the musculotendon is

It

where vo = 1010, i

m

=1,+-zzG L

10

’

In (R/0.10377 1+

ks

91

,

_ 0 5 Ft 5 0.3086,

(

It =

fit lt, ( COSa = (4,

1+

0.3086 5 r”t,

( - lt) &n ’

3. A DIRECT 3.1.

+ 1)

DYNAMIC

MODEL

OF GAIT

The Skeletal Model

In order to generate the loading conditions during gait, it is necessary to develop a model driven by musculotendon actuators that simulates normal gait. Although many gait models have been built, few include the complexity which is needed for realistic dynamic simulations. The approach that is adopted here builds on a model developed by [15]. The model constrains seven rigid-body segments which represent the feet, shanks, thighs, and trunk to eight degrees of freedom. All joints are assumed to be revolute having one degree of freedom with the exception of the stance hip which has two degrees of freedom. This allows the hip to ab/adduct, a condition which reduces the degree of coupling between the trunk and the swing leg. Figure 7 shows the

Stress Analysis of Bone

1---_.

129

-9

\

: J\

4

9 3 . /-s

(a) The stance hip has two DOF, joints are revolute.

7

while all other

(b) Front view showing pelvis list. Stance angles are specified with respect to the inertial frame, ii, while the swing angles are respect to the titled trunk reference frame, d’ (adapted

Figure 7. 3-D, eight DOF

from [15]).

model showingsegmentangle definitions.

generalized coordinates used to describe the configuration of the body. Joint angles q1, q2, and q3 are measured with respect to the horizontal or transverse plane and the rotation of these joints occur about an sxes parallel to &. Movement of the stance leg is confined to the sagittal plane, but due to the extra degree of freedom granted to the stance hip, the swing leg and trunk can also move in the frontal or coronal plane through pelvic list. Joint angle q4 tilts the trunk as well as the swing leg about an axis parallel to 81 = dl. Joint angles q5 through qs are measured in the tilted plane and these rotations occur about an axis parallel to a$. This musculoskeletal model represents a normal male with mass totaling 76 kilograms. Segmental dimensions and inertial parameters used in the model are presented in [16]. A key element of the model is that only one-half (14%, approximately left-toe-off, to 62%, approximately left-footflat) of the gait cycle is simulated in this analysis. The complete gait cycle can be reconstructed under the assumption of bilateral symmetry. This assumption simplifies the modeling in that the stance leg is always the stance leg and the swing leg is always the swing leg. As a result, muscles in the swing and stance leg can vary according to the function of that leg in the simulation. With this assumption, it is also valid to have the stance toe constrained to the ground which eliminates one more degree of freedom. This simplification requires fewer muscles to be modeled and reduces the system dimension. The range of each joint angle is limited to normal ranges through the use of ligamentous constraints. If the joint angle stays within a nominal range, then the effects of the passive structure are minimal, but when the nominal range is exceeded, the passive torques grow exponentially.

Y. DEWOODY et al.

130

The general form of the passive moments is given by M pass = kl e-h(~-~z)

_ k3 e-w~l-~)

_ &j.

Here 8 is the joint angle measured in radians, 8, is the joint velocity measured in radians per second and M,,, is measured in Newton-meters. Note that & < 19< 81 represents a nominal range for that joint. Initial estimates of the parameters kj, $, and c were taken from [8] and then modified in the course of validating the model.

The inclusion of the passive term (-cd)

is vital since the passive viscosity was excluded from the musculotendon model when mass was eliminated. A list of the passive parameters and definitions of the joint angles in terms of the generalized coordinates is given in [16]. Although the stance toe is constrained throughout the simulation, it is necessary to incorporate additional constraints in order to prevent the stance heel and swing foot from penetrating the “ground” and to eliminate excess sliding of the swing foot during double support. These additional constraints are considered to be soft [17]. The vertical ground reaction forces which act on the heels of both the stance and swing leg are modeled as highly-damped, stiff linear springs F normal =

zheel 2 0,

0, 1 - (1.5 x 105) zheel - (1 x 103) zheel,

zheel < 0,

where zheel is the height of the heel above the ground. On the swing heel, an additional frictional force applied in a direction parallel to the ground in the sagittal plane is used to prevent excess sliding. This force is proportional to the normal force applied to the swing heel,

zh~lswingL 0,

0,

F’riction =

-0.5 IFnormal (zheel,,i,J

,

zheelgwing < 0.

Once flat-foot is achieved by the swing leg, a counterclockwise torque is applied to prevent the foot from penetrating the ground. The torque is modeled as a damped, torsional spring which resists plantarflexion of the foot. This torque is given by T=

0,

zh~lswing > 0 or d > 0,

-56966 - 38.198,

zheelswrng2 0 and 6 < 0,

where b is the angle the bottom of the foot makes with the ground. These soft constraints allow the same model to be used during the single and double support phases of gait. If the ground had been modeled as a “hard” constraint, then one would lose a degree of freedom on the swing leg when the toe touches the ground. This would mean two models would be needed to simulate this phase of gait, and a switch between the two systems would occur at heel strike. The model that is developed here incorporates the same muscle sets ss those utilized in [15]. As a result, ten musculotendon units are incorporated into our model, five on each leg. On the stance leg, the relevant muscle groups are the soleus, gastrocnemius, vasti, gluteus medius and minimus, and the iliopsoas. The swing leg utilizes the dorsiflexors, hamstring, vasti, gluteus medius and minimus, and the iliopsoas. Thus, seven different musculotendon groups need to be specified in this model. The constituent muscles composing each musculotendon group, and the parameters which are used to distinguish the dynamics of each particular musculotendon unit, that is, tendon slack length, optimal muscle length, maximal isometric force, and pennation angle, are listed in [16]. In order to incorporate the musculotendon actuators into the dynamics, it is necessary to place the muscles geometrically on the body segment so that the length of the musculotendon, Itm, and the velocity of the musculotendon, vtm, can be derived as a functions of the state variables, i.e., the generalized coordinates, qd. The attachment of the musculotendon to bone is specified by defining an origin or proximal attachment, and an insertion or distal attachment. Effective origins and

Stress Analysis of Bone

131

Table 1. Origins and insertions of the musculotendon groups. X, Y, and 2 axes correspond to pelvic, femoral, tibial, and foot coordinates system (see Figure 8). The foot reference frames is the same as the tibia1 reference frame at anatomical position.

Musculotendon

Iliopsoss

Point

X

Y

z

Reference

(m)

(m)

(m)

Frame

0,

0.0075

0.1350

-0.0400

pelvic

0,

0.0260

0.0293

-0.0042

pelvic

L

-0.0180

0.3351

0.0116

femoral

Gluteus Medius

0

-0.0155

0.0785

0.0076

pelvic

& Minimus

I

-0.0159

0.3873

0.0589

femoral

Hamstring

0,

-0.0409

I,

-0.0170

Vssti

Dorsiflexors

0.3800

-0.0140

pelvic

0.0073

tibia1

0.0205

femoral

0l.S

0.0106

0.2026

La

0.0170

0.3930

-0.0006

tibia1

0.0071

-0.0073

femoral

-0.0203

Gastrocnemius

Soieus

-0.0455

-0.0368 Oa

-0.0268

1,

-0.0365

0,

-0.0155

Oe

0.0259

1,

0.1035

-0.0429 0.2467 -0.0428 0.2175 0.0117 -0.0520

0.0028

foot

0.0006

tibia1

0.0056

foot

0.0134

tibia1

-0.0093 0.000

tibia1 foot

effective insertions are specified when the straight path from the actual origin to actual insertion passes through bone or out of anatomical range during certain body configurations. Origins and insertions are specified with respect to coordinate systems which are directed along the bones and are fixed with respect to the foot, shank, thigh, and pelvis. The origin and insertion points for the seven muscle groups used in this analysis are given in Table 1. The data for the origins, insertions, and pathways were taken from [l&19]. The total length and velocity of the musculotendon complex, which is needed as input into the musculotendon dynamics, can be derived for most muscles through vector addition. In this case, the length of the musculotendon is given by

where 0 and I refer to origin and insertion, and the subscripts e and a refer to actual and effective coordinates. The velocity of the musculotendon is then just the time derivative of the length. Musculotendon forces were incorporated into the body dynamics as joints torques. When a musculotendon spans a joint, a torque is realized across that joint. The joint torque due to musculotendons which do not span the knee is computed using standard vector cross product methods. In this case, the moment 32 acting on the proximal segment is defined as

Here p’ is a vector from the joint to a point on the line of action of the musculotendon. The line of action is defined by the unit vector Ox/(flI’( e e , and the tension in the musculotendon

132

Y. DEWOODY et al.

XP

Figure 8. Definitions for the foot, tibial, femoral, and pelvic frames.

complex, Ft, is produced according to the muscle dynamics (8). The sign in equation (9) depends on whether the musculotendon acts to extend or flex the joint it spans. The cross product,

se iii+jw-----

I02 I’

represents the effective moment arm associated with the musculotendon at a certain body configuration (see Figure 9). Vector addition works well for all muscles which do not span the knee. The complication which arises for those musculotendons which do span the knee (vasti, hamstring, and gastrocnemius) occurs because of the complexity of the knee joint. In short, as the knee flexion angle varies, this produces both a change in the location of the knee joint center and in the location of the patella. Since this complexity is not accounted for in the simple segmental model formulated here, an alternative to the vector method is implemented for the knee. We define the effective moment arms of the vasti, hamstring, and gastrocnemius according to curves formulated in (71.

Stress Analysis of Bone

133

Figure 9. Muscle pathway and effective moment arm. These curves define the moment arms as functions of the knee flexion angle (Figure 10). joint moments produced by these musculotendons which span the knee are given by

G = fFt.

The

m, (0,))

where Ft is calculated from (8) and m, is the effective moment arm as a function of the knee flexion angle, Of. The length of these musculotendons is calculated by integration of the moment arm as in [19]. In this method, the relationship between the length of the musculotendon, effective moment arm, and the joint angle is given by d L, Vtm = dt

= m, (Bf) z.

Consequently, three more differential equations are added to the system. 0.045

I

0.04

-

0.035

-

I

Figure 10. Effective moment arm of vssti, hamstring, gastrocnemius knee flexion fIf.

,

as a function of

the

134

Y. DEWOODY

I

0.035

et al.

1

I

0.03 -

0.02

0.05

I 0

20

I

, 40

60

I

I 50

I 100

I

0.035 -

0.03 -

o.ow -

0.02

I 0

20

I 40

I 50

50

100

Figure 10. (cont.)

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 optimization is usually utilized in developing the controls. The controls used in the gait simulations of this paper are derived through a two-phase process: (1) a coarse formulation based on a dynamic optimization scheme, and (2) a finetuning via trial-and-error. The coarse controls are derived ss in [15] by employing a cost function that consists of an error tracking term which penalizes deviations from a desired trajectory and a term related to muscle

135

Stress Analysis of Bone {Activation} { Soleus 1

I

1

{Dorsiflexor 1

0.8

0.8 I

0.6 0.4

?t/

0.1

0.2 0.3 0.4

Os2M 0.1

0.2 0.3 0.4

{Hamstring}

1, {Gastrocnemius] 1 0.8 I 0.6 0.4 0.21

/---

r

0.1 0.2 0.3 0.4

0.1 0.2 0.3 0.4

{Vasti}

{Vasti)

1

11 0.8

0.8

0.6

0.6

0.4

I:://

om211rIzl0.1

0.1

0.2 0.3 0.4

\/ 0.2 0.3 0.4

{Iliopsoas}

{Iliopsoas] 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2 :

0.2 . 0.1 0.2 0.3 0.4

0.1 0.2 0.3 0.4

{Glyteus Medius & Minimus}

{Gl;teus Medius & Minimus)

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 L--___L 0.1 0.2 0.3 0.4

0.1

0.2 0.3 0.4

Figure11. Controls utilized in the gait simulation. fatigue(41,

In equation (lo), z[, w,,r are elements of the state variable, r? = (ql, 42,. . . ,Ql, &,43,. . . ) (refer to Figure 7), and Fo and Al are the force and physiological cross-sectional area of muscle 1. The nominal gait trajectory is described by 2l,&s, and is specified according to data recorded in [20]. Once crude activation controls are formulated, simulations were run and these controls

Y.

136

DEWOODY et al.

were fine-tuned by ad-hoc methods to meet the specific needs of our model. The final formulation of the control laws derived for the model varied slightly from those of [15]. The resulting muscle activation is illustrated in Figure 11. 3.2.

Dynamic

Simulations

For this study, the algebraic manipulation program AUTOLEV (211 was utilized to derive the This program implements Kane’s method as a

equations of motion for the gait simulation.

means of deriving Euler’s equations of motion. A detailed discussion of the procedure by which the equations are derived is given in [16]. It suffices here to point out that AUTOLEV generates the equations of motion as well as FORTRAN or C code for integrating these equation forward in time using a fourth-order Runge Kutta scheme. Necessary initial data to run a simulation includes the initial segmental orientations (~1, . . . , qg), the initial angular speeds (41,. . . , &), the initial force in each musculotendon, and the initial lengths of the musculotendons. Precise values of this data are not readily available and were approximated by several methods. For instance, the initial force in each musculotendon unit was found by running simulations to find the steady state muscle force achieved by the initial control when the motion of the model was constrained. Initial lengths of the musculotendon were estimated so that the muscle maintained an appropriate length throughout the gait cycle. A summary of initial conditions that were utilized in running gait simulations can be found in [16]. A standard means of comparing and validating gait simulations is achieved by displaying the joint torques which drive the system and the resulting joint trajectories. Figure 12 depicts the standard definitions of the joint angles. Notice that these joint angles are distinct from the

Degrees

Dew-s

50 40 30

Degrees

(Ankle

Angle)

I~nee

Angle1

q;

{Hip Angle]

% of Gait

Figure 12. Joint angle definitions.

Figure 13. Joint trajectories.

Stress Analysis

of Bone

137

{Normalized Force) {Soleus}

{Dorsiflexors}

1 0.8

1. 0.8

0.6 0.4

0.6 0.4

0.2 !Yzz!z 0.1 0.2 0.3 0.4

o-2-

0.1 0.2 0.3 0.4

{Gastrocnemius}

{Hamstring}

1 0.8

9 0.8

0.6

0.6.

0.4

0.4

“*“U

0.2 B 0.1 0.2 0.3 0.4

0.1 0.2 0.3 0.4

Wasti.1

{Vasti)

1

1,

0.8

0.8

0.6

0.6

0.4 0.2 L-

0.4 0.2. K

0.1 0.2 0.3 0.4 {Iliospoas}

{Iliopsoas}

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2 I

0.2 L-

0.1 0.2 0.3 0.4

{G$.uteusMedius & Minimus}

0.1 0.2 0.3 0.4

{G:uteus Medius & Minimus}

0.8

0.8

0.6 0.4 0.2 b

0.6 q_

0.1 0.2 0.3 0.4

-

0.1 0.2 0.3 0.4

/

0.1 0.2 0.3 0.4

Figure 14. Normalized force histories for ewh musculotendon. The left column refers to stance muscle, while the right column refers to the swing side muscles.

generalized coordinates used in the analysis. Figure 13 displays the resulting joint trajectories in the simulation. These trajectories provide good qualitative agreement with the trajectories reported by other researchers (see, for example, [20]). Joint torques and power trajectories, which reflect the rate at which the muscles and tendons are expending and absorbing energy, were in general accordance with the data reported by other researchers [6,8].

Y. DEWOODY et al.

138

Figure 14 reports the normalized force realized in the tendons of all ten musculotendon actuators. These normalized forces, F,/F,, were obtained through the dynamics in (8) and are utilized in specifying the loading conditions for the stress analysis.

Additional

ditions needed for the stress analysis were obtained by appropriately coordinates so that joint reaction forces could be determined.

4. THE The stress distribution considerations elements

STRESS

boundary con-

constraining

generalized

ANALYSIS

in the long bones of the lower extremities

is calculated from equilibrium

by ‘freezing’ the gait model at a iixed instant in time and regarding the segmental

as linear isotropic elastic bodies.

The method that is implemented

here follows the

approach given in [22]. The applied loads for the stress analysis are the joint reaction forces and the muscle forces as calculated

in the gait analysis. The joint reaction forces and joint moments

due to musculature effects are resolved into forces and moments at each cross section of the tibia. The components of the stress tensor are calculated in terms of these internal forces and moments. The relevant geometry is indicated in Figure 15. Assign to the ith cross section a local set of orthogonal axes parallel to the unit vectors &, vi, & with the origin at the center of gravity. The following measures of anthropometric data for each segment for each segmental link must be collected or approximated. Let Ai denote the area of the ith cross section, (Ic)i, (I,)i the statical moments of inertia around the