muscle wrapping.pdf

MUSCLE WRAPPING

Muscoloskeletal geometry

1. Modeling the Paths of Muscolotendinous Actuators

2. Obstacle Set Method


Two different methods are used to model the paths of musculotendinous actuators

Straigth-line method(Jensen and Davy, 1975)

The path of a musculotendinousactuator (muscle and tendon combined) is represented by a straight line joining thecentroids of the tendon attachment sites

easy to implementX useless results when a muscle

wraps around a bone or another muscle

Centroid-line method(Jensen and Davy, 1975)

The path of the musculotendinous actuator is represented as a line passing through the locus of cross-sectional centroids of the actuator

the actuator’s line of action is represented more accurately in this way

X the centroid-line method can be difficult to apply



a) straight-line method

b) centroid-line method

The centroid-line method can be difficult to apply:

1) it may not be possible to obtain the locations of the actuator’s cross-sectional centroids for even a single position of the body

2) even if an actuator’s centroid path is known for one position of the body, it is practically impossible to determine how this path changes as body position changes








Via points at specific locations along the centroid path of the actuator

(Brand et al., 1982; Delp et al., 1990)

In this approach, the actuator’s line of action is defined by either straight-line segments or a combination of straight-line and curved-line segments between each set of via points.

The via points remain fixed relative to the bones even as the joints move, and muscle wrapping is taken into account by making the via points active or inactive, depending on the configuration of the joint

this method works quite well when a muscle spans a 1-dof hinge joint

X it can lead to discontinuities in the calculated values of moment arms when joints have more than 1 rotational dof








Via points at specific locations along the centroid path of the actuator

(Brand et al., 1982; Delp et al., 1990)

Obstacle-Set Method(Garner and Pandy, 2000)

This method idealizes each musculotendinousactuator as a frictionless elastic band that can slide freely over the bones and other actuators as the configuration of the joint changes.

The musculotendinous path is defined by a series ofstraight-line and curved-line segments joined together by via points, which may or may not be fixedrelative to the bones

• Methods Comparison

Fig.1

• Obstacle-Set Method

The locations of the attachment sitesof the muscle and the locations andorientations of the obstacles werechosen to reproduce the centroidpaths of each portion of the modeledmuscle.

Because the path of a muscle is notimproperly constrained by contactwith neighboring muscles and bones,the obstacle-set method produces:accurate estimates of muscle

moment armssmooth moment armjoint angle

curves, as illustrated in Fig.1

• Muscle moment arms

Muscles develop forces and cause rotation of the bones about a joint.

Moment arm: describes the tendency of a muscolotendinous actuatorto rotate a bone around a joint.

Two methods are commonly used to measure the moment arm of anactuator:

1. the geometric method

2. the tendon excursion method


1. The geometric method (Jensen and Davy, 1975)

• The finite center of rotation is found by x-rays, CT, or MagneticResonance Imaging

• The moment arm is found by measuring the perpendicular distancefrom the joint center to the line of action of the muscle


2. The tendon excursion method (An et al., 1983)

• The change in length of the musculotendinous actuator is measuredas a function of the joint angle

• The moment arm is obtained by evaluating the slope of the actuator-length versus joint-angle curve over the full range of joint movement

• Muscle contraction dynamics

Modeling contraction dynamics

An empirical model, proposed by A.V. Hill, is used in virtually all models ofmovement to account for the force-length and force-velocity properties of muscle(Hill, 1938).

In a Hill-type model, muscle’s force-producing properties are described by fourparameters (Zajac, 1989):

muscle’s peak isometric force, F0m

its corresponding fiber length, l0m

pennation angle, α

the intrinsic shortening velocity of muscle, vmax


• F0m is usually obtained by multiplying muscle’s physiological cross-sectional area

by a generic value of specific tension.

• Values of optimal muscle fiber length, l0m, and α, the angle at which muscle fibers

insert on tendon when the fibers are at their optimal length, are almost alwaysbased on data obtained from cadaver dissections (Freiderich and Brand, 1990).

• vmax is often assumed to be muscle independent; for example, simulations ofjumping (Pandy et al., 1990), pedaling (Raasch et al., 1997), and walking(Anderson and Pandy, 2001b) assume a value of vmax = 10 s-1 for all muscles,which models the summed effect of slow, intermediate, and fast fibers (Zajac,1989).


Tendon is usually represented as elastic (Pandy et al., 1990; Anderson and Pandy,1993).

Even though force varies nonlinearly with a change in length as tendon is stretchedfrom its resting lS

T length, a linear force-length curve is sometimes used (Andersonand Pandy, 1993).

This simplification will overestimate the amount of strain energy stored in tendon,but the effect on actuator performance is not likely to be significant, becausetendon force is small in the region where the force-length curve is nonlinear.


For the actuator shown in Fig.2, musculotendon dynamics is described by a single, nonlinear, differential equation that relates musculotendon force (FMT), musculotendon length (lMT), musculotendon shortening velocity (vMT), and muscle activation (am) to the time rate of change in musculotendon force:

ḞMT = f(FMT, lMT, vMT, am), 0≤am≤

1, (1)

Given values of FMT, lMT, vMT, and am at one instant in time, Eq. (1) can be integrated numerically to find musculotendon force at the next instant.


Fig.2 Schematic diagram of a model commonly used to simulate musculotendon actuation.


Each musculotendon actuator isrepresented as a three-elementmuscle in series with an elastictendon.

The mechanical behavior of muscle isdescribed by:

• a Hill-type contractile element (CE)that models muscle’s force-length-velocity property

• a series elastic element (SEE) thatmodels muscle’s active stiffness

• a parallel-elastic element (PEE) thatmodels muscle’s passive stiffness.


The instantaneous length of theactuator is determined by the lengthof the muscle, the length of thetendon, and the pennation angle ofthe muscle.

In this model the width of the muscleis assumed to remain constant asmuscle length changes.

1. Ricostruzione Mesh 3D

2. Confronto lunghezza elica-arco cilindro

a) Cambio sistema di riferimento

close all

clear all

clc

A=[0,0,0;0,0,0;0,0,0];

tol=10^(-14);

a0=[1;1;1]

b0=[1;2;1]

c0=[0;2;3]

a1=[2.612372436;0.387627564;1.5]

b1=[2.862372436;1.137627564;2.11237243

6]

c1=[3.337117307;-

0.337117307;3.724744872]

% a0=[1;1;7]

% b0=[4;7;1]

% c0=[7;10;10]

% a1=[7;1;7]

% b1=[3;9;6]

% c1=[10;10;13]

d=([a1-a0,b1-b0,c1-c0])'

u=inv(d)*[1;1;1]

uu=u/(sqrt((u(1))^2+(u(2))^2+(u(3))^2)

u_emi=[0,-uu(3),uu(2);uu(3),0,-uu(1);-

uu(2),uu(1),0]

h1=u_emi*(a0-b0)

h2=u_emi*(a1-b1)

c=(((h1)')*h2)/((norm(h1))*(norm(h2)))

theta=acos(c)

A=[(uu(1)^2)*(1-c)+c,

(uu(1)*uu(2))*(1-c)-uu(3)*sin(theta),

(uu(1)*uu(3))*(1-c)+uu(2)*sin(theta);

(uu(1)*uu(2))*(1c)+uu(3)*sin(theta),

(uu(2)^2)*(1-c)+c,(uu(2)*uu(3))*(1-c)

uu(1)*sin(theta);

(uu(1)*uu(3))*(1-c)-uu(2)*sin(theta),

(uu(2)*uu(3))*(1-c)+uu(1)*sin(theta),

(uu(3)^2)*(1-c)+c]


b) Calcolo parametri elica

close all

clear all

clc

% x1=0.02236068

% y1=0.01118034

% z1=0.10

% x2=0.00928477

% y2=0.02321192

% z2=0.15

% t1=atan(y1/x1)

% t2=atan(y2/x2)

% c=((z1-z2)/(t1-t2))

% k=z1-((z1-z2)/(t1-t2))*t1

% x1=-0.01345541

% y1=0.02107017

% z1=0.16494454

% x2=0.02491081

% y2=0.0021099

% z2=0.11129092

% t1=atan(y1/x1)

% t2=atan(y2/x2)

% c=((z1-z2)/(t1-t2))

% k=z1-((z1-z2)/(t1-t2))*t1

% x1=1.94448291*10^(-3)

% y1=14.87343223*10^(-3)

% z1=66.39059567*10^(-3)

% x2=-14.02379309*10^(-3)

% y2=-5.32289651*10^(-3)

% z2=64.55098222*10^(-3)

x2= 14.02379309*(10^(-3))

y2= -5.32289651*(10^(-3))

z2= 63.44430452*(10^(-3))

x1= 1.94448292*(10^(-3))

y1= 14.87343223*(10^(-3))

z1= 76.86365141*(10^(-3))

t1=atan(y1/x1)

t2=atan(y2/x2)

c=((z1-z2)/(t1-t2))

k=z1-((z1-z2)/(t1-t2))*t1


c) Confronto lunghezza

close all

clear all

clc

a=1;

b=-1;

c=2;

d=-3;

theta = 0:0.1:pi/2;

plot3(cos(theta),sin(theta),-

(a*cos(theta)+b*sin(theta)+d)/c)

view(135,30)

grid on

box on

f = @(theta) sqrt(sin(theta).^2 +

cos(theta).^2 + ((-

a*sin(theta)+b*cos(theta))/(c)).^2);

len1 = integral(f,0,pi/2)

hold on

x1=1

y1=0

z1=1

x2=0

y2=1

z2=2

t1=atan(y1/x1)

t2=atan(y2/x2)

c=((z1-z2)/(t1-t2))

k=z1-((z1-z2)/(t1-t2))*t1

r = 1;

t = t1:0.01:t2;

x = r*cos(t);

y = r*sin(t);

z = c*t+k;

figure(1)

plot3(x, y, z);

grid on

xlabel('x'); ylabel('y');

title('Circula helix');

f = @(t) sqrt((sin(t)).^2 +

(cos(t)).^2 + (c).^2);

len2 = integral(f,t1,t2)

len1

len2

diff1=(abs(len1-len2))/(len2)

diff=diff1*100

3. Confronto wrapping sfera-cilindro

Errore relativo sfera-lunghezza esatta, cilindro-lunghezza esatta

3. Confronto wrapping sfera-cilindro

Lunghezza esatta

4. Problema della retta d’azione

5. Soluzione proposta

Bisogna distinguere tra:

I. il problema della determinazione del raggio del cilindro/sferaottimo, che meglio approssima l’andamento della lunghezza delsistema muscolo-tendine precedentemente calcolato come lalunghezza minima che unisce il punto di Origine e di Inserzione

II. il problema della determinazione del numero di sferette dautilizzare nella simulazione dinamica al fine di ottenere valoriomogenei in termini di forza di reazione su ciascuna molla, inquanto l’attenzione è sulla compenetrazione del cilindro da partedella retta d’azione che collega le singole sferette


I. il problema della determinazione del raggio del cilindro/sferaottimo, che meglio approssima l’andamento della lunghezza delsistema muscolo-tendine precedentemente calcolato come lalunghezza minima che unisce il punto di Origine e di Inserzione

Confrontare i risultati ottenuti con raggi differenti, trovare il raggio che minimizza l’errore relativo, proporre un criterio che

possa essere replicabile su ciascun modello patient-specific


II. il problema della determinazione del numero di sferette dautilizzare nella simulazione dinamica al fine di ottenere valoriomogenei in termini di forza di reazione su ciascuna molla, inquanto l’attenzione è sulla compenetrazione del cilindro da partedella retta d’azione che collega le singole sferette

e.g. è possibile utilizzare una sola sferetta per i segmenti con angolo relativo di 180 gradi, si rendono necessarie 4-5-6-7-8

sferette per un angolo di 45 gradi

muscle wrapping.pdf

Documents

straight line

actuators line of action

actuators centroid path

curvedline segments

path changes

muscle wraps

single position

body position changes1