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KneeExtElastic.m
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KneeExtElastic.m
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%% A MAIN FUNCTION
function [ t,F,X,V,dV,v_SEE,f_CE,v_CE,f_PEE,f_MTC,v_MTC,GX,AT,...
FL,l_SEE,l_CE,l_MTC ] = KneeExtElastic
% JUMPSIM_DISS_HP simulates a leg extension at an inclined legpress
%
% It includes a parallel-elastic element,
% a non-linear to linear serial elastic element, and
% a force-length relation.
% The Newtonian equation of motion is expressed as an ODE wich considers
% initial conditions and neuromuscular properties provided by th user.
%
% Finally tested with Matlab R2016b (Student License)
% Required licenses:
% - matlab
%
% USAGE
% Just run the file.
% Change parameters in the secton "Input" if you like.
%
% REFERENCE
% H. Penasso and S. Thaller, ?Determination of Individual Knee-Extensor
% Properties from Leg-Extensions and Parameter Identification,?
% Mathematical and Computer Modelling of Dynamical Systems,
% vol. in press, in press
%
% INPUT
% has to be defined below in section "Input"
%
% OUTPUT
% A txt-file is saved and contains the simulated data as well as
% the settings, the inital conditions and the properties of the system.
% t ... Time [s]
% F ... External force [N]
% X ... Position: distance from the prox. end of the model-thigh to
% the proximal end of the model-shank [m]
% V ... Velocitiy of the accelerated point-mass [m/s]
% dV ... Acceleration of the point-mass [m/s^2]
% v_SEE ... Velocity of the tendon-model [m/s]
% f_CE ... Values of the contraction-velocity dependent element (CE) [N]
% v_CE ... Contraction velocity of the musle (CE=MUSCLE=PEE) [m/s]
% f_PEE ... Force of passive muscle tissue PEE [N]
% f_MTC ... Force of the muscle-tendon complex (CE=MUSCLE=SEE) [N]
% v_MTC ... Velocity of the muscle-tendon complex [m/s]
% GX ... Values of the function of geometry [-]
% AT ... Values of the function of activation dynamics [-]
% FL ... Values of the force-rength relation [-]
% l_SEE ... Length of the serial elastic element [m]
% l_CE ... Length of the contractile and parallel elsatic element [m]
% l_MTC ... Length of the mucle-tendon complex [m]
%
% Literature:
% [1] Hoy, M. G., Zajac, F. E., & Gordon, M. E. (1990). A
% musculoskeletal model of the human lower extremity: The effect
% of muscle, tendon, and moment arm on the moment-angle
% relationship of musculotendon actuators at the hip, knee, and
% ankle. Journal of Biomechanics, 23(2), 157?169.
% doi:10.1016/0021-9290(90)90349-8
% [2] Im, H., Goltzer, O., & Sheehan, F. (2015). The effective quadriceps
% and patellar tendon moment arms Relative to the Tibiofemoral Finite
% Helical Axis. Journal of Biomechanics.
% doi:10.1016/j.jbiomech.2015.04.003
% [3] Van Eijden, T., Kouwenhoven, E., Verburg, J., & Weijs, W. (1986).
% A mathematical model of the patellofemoral joint. Journal of
% Biomechanics, 19(3), 219?229. doi:10.1016/0021-9290(86)90154-5
% [4] Van Soest, A. J., & Bobbert, M. F. (1993). The contribution of
% muscle properties in the control of explosive movements.
% Biological Cybernetics, 69(3), 195?204.
% Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/8373890
% [5] http://www.mathworks.com/matlabcentral/fileexchange/
% 11077-figure-digitizer/content/figdi.m
% [6] G?nther, M., Schmitt, S., & Wank, V. (2007). High-frequency
% oscillations as a consequence of neglected serial damping in
% Hill-type muscle models. Biological Cybernetics, 97(1), 63?79.
% doi:10.1007/s00422-007-0160-6
% [7] Haeufle, D. F. B., G?nther, M., Bayer, A., & Schmitt, S. (2014).
% Hill-type muscle model with serial damping and eccentric
% force-velocity relation. Journal of Biomechanics, 47(6), 1531?1536.
% doi:10.1016/j.jbiomech.2014.02.009
% [8] Pandy, M. G., Zajac, F. E., Sim, E., & Levine, W. S. (1990).
% An optimal control model for maximum-height human jumping.
% Journal of Biomechanics, 23(12), 1185?98.
% Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/2292598
%
% v.5.2 by Harald Penasso (16.11.2015) (cleaned up: 24.11.2016)
%% A.1 Input
% Options
t0 = 0; % Start integration at this time [s]
te = 6.5; % Latest end of integration at this time [s]
% (can be overruled by a solver stopping condition)
reltol = 1e-8; % ode45 option
maxstep = 0.005; % ode45 option
refine = 1; % ode45 option
path = {'/Users/haraldpenasso/Desktop/n/'}; % Path to final plots/results
% Initial conditions and neuromuscular properties
isoQ = 0;% true (1): simulates an isometric experiment;
% false (0): simulates a dynamic experiment.
% WARNING: The initial load determines the initial force
% at the isometric condition. add = 40 is close to reality
phi = 90; % Inclination of the plane where the point-mass moves [deg]
g_earth = 9.81; % Constant of gravity [m/s^2]
msledge = 32.5; % Partial mass #1 (mass of empty sledge: 32.5) [kg]
add = 60; % Partial mass #2 (additional weight: 40.0) [kg]
KA0 = 120; % Initial knee-extension-angel [deg]
V0 = 0; % Initial condition for ODE: first derivative of X [m/s]
dV0 = 0; % Initial condition for ODE: 2nd derivative of X [m/s^2]
t_on = 0.25; % Movement-initiation-time of actavation dynamics
% Geometry
lt = 0.43; % Length of the thigh [m]
ls = 0.43; % Length of the shank [m]
cir = 0.37; % Circumference of the knee [m]
ptl = 0.08; % Patellar tendon length [m]
% Distance: middele patella to tuberositas tibiae
% Muscle-tendon complex
% Activation dynamics
U = 12; % Rate-constant of muscle-activation-dynamics [1/s]
% Serial elastic element
ksee = 1777000; % Linear stiffness of seiral elastic element [N/m]
ns = 1; % n^th power of toe-region of the SEE (ksee x^ns)
SEE_TH = 1; % Transition point from nonlinear to linear SEE region
% Represents the perzentage (e[0,1]) of fiso
% Parallel elastic element
kpee = 0; % Stiffness of parallel elastic element [N/m]
% if kp = 0, the PEE is inactive
np = 2; % n^th power of parallel elastic element (kpee x^np)
% Contractile element
a = 7.920643997366427e3; % Hill's parameter "a" [N]
b = 0.352028622105175; % Hill's parameter "b" [m/s]
c = 9.124808590471668e3; % Hill's parameter "c" [W]
% Force-length relationship
KA_opt = 120; % Knee-ext.-angle @ geometry_funs are at opt. length [1]
WIDTH = 0.56; % Width of the force-length parabola: 0.56 [4]
% ... if set to inf the f-l relation is ineffective
l_CE_opt = 0.09; % optimal geometry_fun-length [m] [4,8]
%% A.2 Initial Calculations
% Parameter convertions
[U,a,b,c,vmax,fiso,pmax,vopt,fopt,eta,kappa]=params([U,a,b,c]);
% Serial elastic element
f_MTC_th = fiso * SEE_TH; % Force value at SEE lin to nonlin transition
dx_th = (ns*f_MTC_th)/ksee; % Change in SEE length at transition
ksnl = ksee.*ns.^(-1).*dx_th.^(1+(-1).*ns); % Value for non-lnear part
% Function of geometry
lr = cir/(2*pi); % Approximated moment-arm of the knee [m]
X0 = sqrt(lt^2+ls^2-2*lt*ptl*cos(KA0*pi/180)); % Init. position [m]
g = g_earth*sin(phi*pi/180); % Gravity relative to inclination [m/s^2]
m = msledge + add; % Total mass of sledge (Sum af mass#1 and #2) [kg]
% Initial values
[G_val0,~,dlmtc0] = geometry_fun(X0,V0,lt,ls,lr,ptl,KA_opt);
% Muscle-tendon complex
f_MTC0 = m*(dV0 + g)/G_val0; % initial force of the MTC
% Set the initial region of SEE
if f_MTC0 > f_MTC_th % linear
dx0 = (f_MTC0+ksee*dx_th*(1-1/ns))/ksee;
else % non-linear
dx0 = nthroot(f_MTC0/ksnl,ns);
end
% Initial values of force-length relation and parallel elastic element
[fL0,f_PEE0]=fl_fpee_lsee_fun(dx0,dlmtc0,ptl,l_CE_opt,WIDTH,kpee,np);
% Initial value of activation dynamics
Apre = (a.*b.*fL0.*G_val0+(-1).*c.*fL0.*G_val0).^(-1).*(b.*f_PEE0.*...
G_val0+(-1).*b.*g.*m);
%% A.3 Filenames of output
if add >= 99 % At tis line, it alters only the filename of the output;
add = 99; % Adds information of mass#2 to the saved filename
end % WARNING: files using add > 99 will have identical filenames!
id = {['psim00g',num2str(add),'v01']}; % Filename of output files
%% A.4 Check initial settings
if KA0 < 60
error('The initial knee-extension-angle is to small (intAK < 60?)')
end
if a/fiso > 1
error('The curvature aof the FV is > 1')
end
if pmax/c > 0.7
error('The efficiency is to big (eta > 0.7)')
end
if pmax/c < 0.05
error('The efficiency is small (eta < 0.05)')
end
if te <= t_on
error('The contraction must begin within [t0,te) (te > t_on)')
end
%% A.5 ODE solving
opts=odeset('MaxStep',maxstep,'RelTol', reltol,'Events',@events,...
'Refine',refine); % ode45 solver settings
[t,x] = ode45(@model_fun,[t0,te],[X0,V0,dV0],opts); % ... Solve ODE
%% A.6 Results
X = x(:,1); % Position data [m]
V = x(:,2); % Velocity data [m]
dV = x(:,3); % Acceleration data [m/s^2]
%% A.7 Calculate the values of sub-systems
% External Force
F = m*x(:,3)+m*g; % External Force [N]
% Geometrical Function # d/dt Geometical Fun # Changes in length of MTC
[GX,dG,dlmtc] = geometry_fun(X,V,lt,ls,lr,ptl,KA_opt);
% Force of the muscle (CE+PEE) == force of the MTC == force of the SEE
f_MTC = m*(dV + g)./GX;
% Velocity of the MTC
v_MTC = GX.*V;
% Length change of the SEE
dx = zeros(size(f_MTC)); % Pre-allocation only
% Linear region
dx(f_MTC>=f_MTC_th) = (f_MTC(f_MTC>f_MTC_th)+ksee.*dx_th.*...
(1-1./ns))./ksee;
% Non-linear region
dx(f_MTC<f_MTC_th) = nthroot(abs(f_MTC(f_MTC<=f_MTC_th)./ksnl),ns);
% Force-length relation # Force of PEE # length of SEE # length of CE
[FL,f_PEE,l_SEE,l_CE]=fl_fpee_lsee_fun(dx,dlmtc,ptl,l_CE_opt,...
WIDTH,kpee,np);
% Force of the CE
f_CE = (f_MTC-f_PEE);
% Activation dynamics
AT = ES_fun(t,t_on,Apre,U)';
% Velocity of the CE == velocity of the PEE == velocity of the muscle
v_CE = c./(((f_MTC-f_PEE)./(AT.*FL))+a)-b;
% Velocity of the SEE
v_SEE = v_MTC - v_CE;
% Length of the MTC
l_MTC = l_CE + l_SEE;
%% A.8 Final plots
fig = figure('units','normalized','outerposition',[0 0 1 1]);
% CE/PEE/MTC force vs. time
subplot(4,3,[1,4])
plot(t,f_CE,'or-')
hold on
plot(t,f_PEE,'om-')
hold on
plot(t,f_MTC,'og-')
xlabel('time [s]')
ylabel('force [N]')
title('Internal forces')
leg = legend('F_{CE}','F_{PEE}','F_{MTC}','Location','Best');
set(leg,'FontSize',14);
grid on
% Geometry vs. time
subplot(4,3,[2,5])
plot(t,GX,'oc-')
title('Function of geometry')
xlabel('time [s]')
ylabel('ratio [-]')
grid on
% Activation dynamics vs. time
subplot(4,3,3)
plot(t,AT,'oc-')
xlabel('time [s]')
ylabel('ratio [-]')
ylim([0,1.1])
title('Activation dynamics')
grid on
% Force-length dynamics vs. time
subplot(4,3,6)
plot(t,FL,'.k-')
xlabel('time [s]')
ylabel('ratio [-]')
ylim([min(FL)*.99,1.01])
leg = legend('fl','Location','Best');
set(leg,'FontSize',14);
title('Force-length dynamics')
grid on
% External force vs. time
subplot(4,3,[7,10])
plot(t,F,'ob-')
xlabel('time [s]')
ylabel('force [N]')
title('External force')
grid on
% External position X vs. time
subplot(4,3,8)
plot(t,X,'ob-');
ylabel('position [m]')
ylim([min(X)*0.99,max(X)*1.01])
title('Position and Lengths')
leg = legend('X','Location','Best');
set(leg,'FontSize',14);
grid on
% MTC-/CE-/SEE lengths vs. time
subplot(4,3,11)
plot(t,l_MTC,'og-')
hold on
plot(t,l_CE,'or-')
hold on
plot(t,l_SEE,'oy-')
xlabel('time [s]')
ylabel('position [m]')
ylim([min([l_CE;l_SEE])*0.95,max(l_MTC)*1.05])
leg = legend('l_{MTC}','l_{CE}','l_{SEE}','Location','Best');
set(leg,'FontSize',14);
grid on
% External/MTC/CE/SEE velocities vs. time
subplot(4,3,[9,12])
plot(t,V,'ob-')
hold on
plot(t,v_MTC,'og-')
hold on
plot(t,v_CE,'or-')
hold on
plot(t,v_SEE,'oy-')
xlabel('time [s]')
ylabel('velocity [m/s]')
title('Velocities')
leg=legend('V_{EXT}','V_{MTC}','V_{CE}','V_{SEE}','Location','Best');
set(leg,'FontSize',14);
grid on
% Make all text in the figure to size 14 and bold
figureHandle = gcf;
set(findall(figureHandle,'type','text'),'fontSize',18,...
'fontWeight','bold')
% Add additional force-elongation plot of the SEE
% Place second set of axes on same plot
if ~isoQ
handaxes2 = axes('Position', [0.45,0.673,0.102,0.234]);
else
handaxes2 = axes('Position', [0.512,0.734,0.102,0.171]);
end
plot(1e3*dx(f_MTC>=f_MTC_th),1e-3*f_MTC(f_MTC>=f_MTC_th),'xg-',...
1e3*dx(f_MTC<f_MTC_th),1e-3*f_MTC(f_MTC<f_MTC_th),'or-')
grid on
set(handaxes2, 'Box', 'off')
xlabel('\Delta l_{SEE} [mm]','FontSize',14)
ylabel('f_{MTC} [kN]','FontSize',14)
title('SEE')
% Save figue as FIG and PDF
set(findall(fig,'type','text'),'fontSize',18,'fontWeight','bold')
set(fig, 'PaperPosition', [0 0 48 38]);
set(fig, 'PaperSize', [48 38]);
saveas(fig,[path{:},id{:},'.fig'],'fig')
saveas(fig,[path{:},id{:},'.pdf'],'pdf')
%% A.9 Write settings, initials, and ode45 results to txt-file
% Header and data-matrix
header = {'Time [s]','Fz [N]','X [m]', 'V [m/s]',...
'A [m/s^2]','V_SEE [m/s]','F_CE [N]','V_CE [m/s]','F_PEE [N]',...
'F_MTC [N]','V_MTC [m/s]','GeomFun [-]','dGeomFun [-]',...
'ActFun [-]','force-length []','l_SEE [m]','l_CE [m]','l_MTC [m]'};
data = [t,F,X,V,dV,v_SEE,f_CE,v_CE,f_PEE,f_MTC,v_MTC,GX,dG,AT,...
FL,l_SEE,l_CE,l_MTC];
% Open file
fid = fopen([path{:},id{:},'.txt'],'w');
% Write preamble
fprintf(fid, '%s', ['This file contains data from a simulated leg'...
' extension using ',mfilename,'.m (by Harald Penasso).']);
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'ID:');
fprintf(fid, '%s', id{:});
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'Date of simulation [yyyy-mm-dd-HH-MM-SS]:');
fprintf(fid, '%s', datestr(now,'yyyy-mm-dd-HH-MM-SS'));
fprintf(fid, '\n');
fprintf(fid, '%s', '#');
% Options
fprintf(fid, '\n');
fprintf(fid, '%s', 'SOLVER AND SOLVER OPTIONS');
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'Saved to:');
fprintf(fid, '%s', path{:});
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'MaxStep [-]:');
fprintf(fid, '%s', num2str(maxstep,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'RelTol [-]:');
fprintf(fid, '%s', num2str(reltol,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'Refine [-]:');
fprintf(fid, '%s', num2str(refine,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 't0_sim [s]:');
fprintf(fid, '%s', num2str(t0,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'te_sim [s]:');
fprintf(fid, '%s', num2str(te,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'Iso:');
fprintf(fid, '%s', num2str(isoQ,1));
fprintf(fid, '\n');
fprintf(fid, '%s', '#');
% Initials
fprintf(fid, '\n');
fprintf(fid, '%s', 'ENVIRONMENT AND INITIAL CONDITIONS');
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'Inclination angle [deg]:');
fprintf(fid, '%s', num2str(phi,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'Constant of gravity [m/s^2]:');
fprintf(fid, '%s', num2str(g_earth,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'Resulting gravity [m/s^2]:');
fprintf(fid, '%s', num2str(g,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'Mass of sledge [kg]:');
fprintf(fid, '%s', num2str(msledge,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'Additional load [kg]:');
fprintf(fid, '%s', num2str(add,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'Total mass [kg]:');
fprintf(fid, '%s', num2str(m,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'Initial Knee-Ext-Angle [deg]:');
fprintf(fid, '%s', num2str(KA0,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'X0 [m]:');
fprintf(fid, '%s', num2str(X0,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'V0 [m/s]:');
fprintf(fid, '%s', num2str(V0,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'dV0 [m/s^2]:');
fprintf(fid, '%s', num2str(dV0,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'Innervation t0 [s]:');
fprintf(fid, '%s', num2str(t_on,6));
fprintf(fid, '\n');
fprintf(fid, '%s', '#');
% Subject
fprintf(fid, '\n');
fprintf(fid, '%s', 'SUBJECT-PARAMETERS');
% Function of geometry
fprintf(fid, '\n');
fprintf(fid, '%s', 'LEG-PARAMETERS');
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'lo [m]:');
fprintf(fid, '%s', num2str(lt,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'lu [m]:');
fprintf(fid, '%s', num2str(ls,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'lr [m]:');
fprintf(fid, '%s', num2str(lr,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'ku [m]:');
fprintf(fid, '%s', num2str(ptl,6));
fprintf(fid, '\n');
fprintf(fid, '%s', '#');
% Muscle
fprintf(fid, '\n');
fprintf(fid, '%s', 'MUSCLE-TENDON');
fprintf(fid, '\n');
fprintf(fid, '%s', '*');
fprintf(fid, '\n');
fprintf(fid, '%s', '*ACTIVATION DYNAMICS');
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'U [1/s]:');
fprintf(fid, '%s', num2str(U,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'Apre [1/s]:');
fprintf(fid, '%s', num2str(Apre,6));
fprintf(fid, '\n');
fprintf(fid, '%s', '*');
fprintf(fid, '\n');
fprintf(fid, '%s', '*FORCE-VELOCITY RELATION');
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'a [N]:');
fprintf(fid, '%s', num2str(a,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'b [m/s]:');
fprintf(fid, '%s', num2str(b,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'c [W]:');
fprintf(fid, '%s', num2str(c,6));
fprintf(fid, '\n');
fprintf(fid, '%s', '*');
fprintf(fid, '\n');
fprintf(fid, '%s', '*SERIAL ELASTIC ELEMENT');
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'k_see_lin [N/m]:');
fprintf(fid, '%s', num2str(ksee,9));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'k_see_nonlin [N/m^n_see]:');
fprintf(fid, '%s', num2str(ksnl,9));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'n_see_nonlin []:');
fprintf(fid, '%s', num2str(ns,9));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'QT length = ku [m]:');
fprintf(fid, '%s', num2str(ptl,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'lin_to_nonlin_TH [%]:');
fprintf(fid, '%s', num2str(SEE_TH,6));
fprintf(fid, '\n');
fprintf(fid, '%s', '*');
fprintf(fid, '\n');
fprintf(fid, '%s', '*PARALLEL ELASTIC ELEMENT');
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'k_pee [N/m^n_pee]:');
fprintf(fid, '%s', num2str(kpee,9));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'n_pee []:');
fprintf(fid, '%s', num2str(np,9));
fprintf(fid, '\n');
fprintf(fid, '%s', '*');
fprintf(fid, '\n');
fprintf(fid, '%s', '*FORCE-LENGTH RELATIONSHIP');
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'KA_l_CE_opt [deg]:');
fprintf(fid, '%s', num2str(KA_opt,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'WIDTH []:');
fprintf(fid, '%s', num2str(WIDTH,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'l_CE_opt [m]:');
fprintf(fid, '%s', num2str(l_CE_opt,6));
fprintf(fid, '\n');
fprintf(fid, '%s', '#');
fprintf(fid, '\n');
fprintf(fid, '%s', 'CONVERTED FORCE-VELOCITY-PARAMETERS');
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'vmax [m/s]:');
fprintf(fid, '%s', num2str(vmax,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'fiso [N]:');
fprintf(fid, '%s', num2str(fiso,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'pmax [W]:');
fprintf(fid, '%s', num2str(pmax,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'vopt [m/s]:');
fprintf(fid, '%s', num2str(vopt,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'fopt [N]:');
fprintf(fid, '%s', num2str(fopt,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'efficiency [-]:');
fprintf(fid, '%s', num2str(eta,6));
fprintf(fid, '\n');
fprintf(fid, '%s\t', 'curvature [-]:');
fprintf(fid, '%s', num2str(kappa,6));
fprintf(fid, '\n');
fprintf(fid, '%s', '#');
% Header
fprintf(fid, '\n');
fprintf(fid, '%s\t', header{:});
fprintf(fid, '\n');
% Write data
for i=1:length(data) % Write one column at each i-th iteration
fprintf(fid, '%20.10g\t', data(i,:));
fprintf(fid, '\n');
end
% Close file
fclose(fid);
%% A.10 Final message
disp(' ')
disp(['Done. I saved the results to', path{:},id{:},'.txt'])
disp(' ')
%% B NESTED-FUNCTIONS
%% B.1 THE MODEL
function [ X ] = model_fun(t,Y)
% MODEL_FUN contains the model-equation of the leg-extension task
%
% This funcion is build in order to be send to an ode-solver
%
% INPUT
% t ... Time [s]
% Y ... Y-value of the actual step to solve the next iteration step
% 1st column: Position [m]
% 2nd column: Velocity [m/s]
% 3nd column: Acceleration [m/s^2]
%
% OUTPUT
% X ... Numerical value of derivative of the ODE
% 1st column: Position [m]
% 2nd column: Velocity [m/s]
% 3nd column: Acceleration [m/s^2]
% Manipulations & error-checking before the contraction is initiatd (t_on)
if t<=t_on % Time befor the contraction is initiated
% Check for initial concentric movement:
if geometry_fun(Y(1),Y(2),lt,ls,lr,ptl,KA_opt) * V0 > vmax
error('Initial contraction-velocity > vmax');
end
% Check if the initial weight can be held
if m*g > geometry_fun(X0, V0,lt,ls,lr,ptl,KA_opt) * (c/...
(geometry_fun(X0, V0,lt,ls,lr,ptl,KA_opt)*V0 + b)-a)
error('Subject cannot hold the weight at the initial position');
end
end
% Model-eqautions
% MTC-force pependant changes of the SEE properties
% Calculate value of function of geometry
[GX,dG,dlmtc] = geometry_fun(Y(1),Y(2),lt,ls,lr,ptl,KA_opt);
% Calculate muscle-tendon complex force
f_MTC = abs(m*(Y(3) + g)/GX); % abs: Am Ende werden winzige negative
% [ WARNING: abs() is needed to eliminate negligeable negative values
% before solver termination ]
% Elongation of the SEE
if f_MTC < f_MTC_th % non-linear part
dx = nthroot(f_MTC/ksnl,ns);
else % linear part
dx = (f_MTC+ksee*dx_th*(1-1/ns))/ksee;
end
% Calculate values of force-length relation and PEE
[FL,f_PEE] = fl_fpee_lsee_fun(dx,dlmtc,ptl,l_CE_opt,WIDTH,kpee,np);
% Calculate value of activation dynamics
AT = ES_fun(t,t_on,Apre,U);
% Calculate actual position and velocity
% Fix X if the isometric conditon is selected
if isoQ
X(1,1) = 0;
X(2,1) = 0;
else
X(1,1) = Y(2);
X(2,1) = Y(3);
end
% Calculate actual acceleration
if f_MTC < f_MTC_th % non-linear part
X(3,1) = GX.^(-1).*dG.*(g+Y(3))+(-1).*dx_th.^(1+(-1).*ns).*ksee...
.*m.^(-1).*GX.*(dx).^((-1)+ns).*(a.*AT.*FL.*GX+(-1).*f_PEE.*...
GX+m.*(g+Y(3))).^(-1).*(AT.*FL.*GX.*(a.*b+(-1).*c+a.*GX.*...
Y(2))+(b+GX.*Y(2)).*((-1).*f_PEE.*GX+m.*(g+Y(3))));
else % linear part
X(3,1) = GX.^(-1).*(dG.*(g+Y(3))+ksee.*m.^(-1).*GX.^2.*((-1).*b+...
GX.*((-1).*Y(2)+c.*AT.*FL.*(a.*AT.*FL.*GX+(-1).*f_PEE.*GX+...
m.*(g+Y(3))).^(-1))));
end
end % <<< function (model_fun) end
%% B.1.1 FUNCTION NESTED TO MAIN FUNCTION
% STOP INTEGRATION OF ODE45 AT EVENT DETECTED
function [value,isterminal,direction] = events(~,x)
% EVENTS locates events during the integration of the ODE and aborts the
% integration if the condition(s) are met. An "odeset(...)" function.
% Condition for abort is returned in "value" and is true if exists:
% 01 negative acceleration
% 02 knee-angle above 179 deg
% Conditions 01 and 02
value = [m*x(3)+m*g,...
acos( (lt^2+ls^2-x(1)^2)/(2*lt*ls) )*180/pi-179];
isterminal = [1,1]; % stop the integration
direction = [0,0];
end % <<< nested function (events) end
% NESTED FUNCTION END #####################################################
%##########################################################################
end % <<< main function (JumpSim_DISS_HP) end
%% C SUB-FUNCTIONS
%% C.1 FUNCT. OF GEOM. & FORCE-LENGTH RELATIONS & PARALLEL ELASTIC ELEM.
function [G,dG,dlMTC]=geometry_fun(X,dX,lt,ls,cir,ptl,KA_opt)
% GEOMETRY_FUN calculates geometrical relations between internal
% (MTC) and external force values. Credit: Prof. Kappel, jumpsens.m(!),
% which was extedned by the derivative of the geometrical realtion and the
% length-change of the muscle-tendon complex (MTC)
%
% INPUT
% X ... Actual position data [m]
% dX ... Velocity data [m/s]
% lt ... Length of the thigh [m]
% ls ... Length of the shank [m]
% cir ... Moment arm of the knee [m]
% ptl ... Distance: middele patella to tuberositas tibiae [m]
% KA_opt ... Optimal knee-ext.-angle [deg] [1]
% (The angel where opt. fascicle length is reached)
%
% OUTPUT
% G ... Ratio of gemetrical relation of the model-leg [-]
% dG ... d/dt GEOM [-]
% dlMTC ... Change of MTC rlative to its length at optimal
% knee-angle [m]
% Simplifications, re-naming
ko = lt;
r = cir;
% Calculation
[G,dG,dlMTC] = Gfuncvec(X);
G(X>lt+ls) = 0;
%% C.1.1 NESTED FUNCTIONS FROM PROF. KAPPEL, JUMPSENS.M
function [G,dG,dl_MTC]=Gfuncvec(x)
% Calculate actual knee-angle
sigma=acos((lt^2+ls^2-x.^2)/(2*lt*ls));
% Calculate the half angle (beta) between model muscle and model
% patellar-tendon direction at each knee-angle
beta=betafunc(sigma);
% Beta at optimal knee-angle, where it is assumed that the fascicle
% length is optimal length too
b_opt = betafunc(KA_opt*pi/180);
% Length of the model muscle-tendon-complex @ 120 deg Knee-Ext.-A [1]
l_MTC_opt = sqrt(lt^2+r^2-2*lt*r*cos(pi-b_opt-...
asin((r*sin(b_opt))/lt)));
% Actual length of the MTC
l_MTC = sqrt(lt^2+r^2-2*lt*r*cos(pi-beta-asin((r*sin(beta))/lt)));
% Change of model-muscle length due to knee extension
dl_MTC = l_MTC_opt-l_MTC;
% 1st derivative of the function of geometry
dG = dG_fun(x,dX,sigma,beta);
% Final value of the function of geometry
G=r*x.*sin(beta)./(lt*ls*sin(sigma));
%% C.1.1.1 NESTED #1
function beta=betafunc(sigma)
s = sigma; M = length(s); beta = zeros(M,1);
for k=1:M
rho=s(k);
options=optimset('MaxFunEvals',12000,'MaxIter',10000);
beta(k)=fminbnd(@bfunc,0,pi,options);
end
%% C.1.1.1.1 NESTED #2
function w=bfunc(v) % v = beta (w = alpha)
w=(2*v+asin(r*sin(v)/ko)+asin(r*sin(v)/ptl)-rho)^2;
end % <<< function (bfunc) end
end % <<< function (betafunc) end
%% C.1.2 FUNCTION TO CALC. THE 1ST DERIVATIVE OF THE FUNCTION OF GEOMETRY
function [ dG ] = dG_fun(X,dX,alpha,beta)
% DG_FUN calculates the 1st derivative of the geometry function,
% including the derivative of the empiric ratio
%
% INPUT:
% X ... Position data [m]
% dX ... Velocity data [m/s]
% alpha ... Knee-angle [rad]
% beta ... Angle at the patella [rad]
% +++ Uses anthropometric lengths from its parent-function
%
% OUTPUT
% dG ... Value of the 1st derivative of the geometrical function
% These variables are set to disablede a functionality that was removed
% from the version
Ratio = 1;
dRatio = 0;
% Derivative of the function of geometry
dG = (lt.^(-1).*cir.*ls.^(-1).*csc(alpha).*(sin(beta).*X.*dRatio...
+sin(beta).*Ratio.*dX+2.*lt.^(-1).*ls.^(-1).*((-1).*...
cot(alpha).*sin(beta)+cos(beta).*(2+cos(beta).*...
(ko.*cir.*(ko.^2+(-1).*cir.^2.*sin(beta).^2).^(-1).*...
(1+(-1).*ko.^(-2).*cir.^2.*sin(beta).^2).^(1/2)+ptl.*cir.*...
(ptl.^2+(-1).*cir.^2.*sin(beta).^2).^(-1).*(1+(-1).*ptl.^...
(-2).*cir.^2.*sin(beta).^2).^(1/2))).^(-1)).*Ratio.*X.^2.*...
((-1).*lt.^(-2).*ls.^(-2).*((lt.^2+(-1).*ls.^2).^2+(-2).*...
(lt.^2+ls.^2).*X.^2+X.^4)).^(-1/2).*dX));
end % <<< function (dG_fun) end
end % <<< function (Gfunvec) end
end % <<< function (geometry_fun) end
%% C.2 FUNCTION OF FORCE-LENGTH RELATIONSHIP & PEE
function [ fl, f_PEE, l_SEE, l_CE ] = ...
fl_fpee_lsee_fun(dx,dlmtc,QPTL,l_CE_opt,WIDTH,k_pee,n_pee)
% FL_FUN calculates a ratio of the force-length relationship based
% on [4]. According to [4,8] the optimal fascicel-length is 0.09 m and
% according to [1] is reached at 120 ? knee-extension-angle.
%
% INPUT:
% dx ... length-change of the SEE [m]
% dlmtc ... length-change of the MTC relative to its length @ optimal
% knee-flexion-angle [1] [m]
% QPTL ... length of the merged quadriceps-to-tibia tendon [m]
% l_CE_opt ... optimal length of the CE [4,8] [m]
% WIDTH ... Width of the force-length parabola [m] [4]
% k_pee ... length-change of the SEE [N/m^pee]
% n_pee ... n-th power of the PEE [-]
%
% OUTPUT:
% fl ... ratio of the force-length relationship [-]
% f_PEE ... force produced by the parallel elastic element [N]
% l_SEE ... length of the serial elastic element [m]
% l_CE ... lengh of the contractile element [m]
% Actual length of the SEE
l_SEE = QPTL+dx;
% Actual length of the CE
l_CE = l_CE_opt-dx-dlmtc;
% Calculate the force-length relation [4]
if min(l_CE)>=(1-WIDTH)*l_CE_opt && min(l_CE)<=(1+WIDTH)*l_CE_opt
c = -1/WIDTH^2;
fl = c*(l_CE./l_CE_opt).^2-2*c*(l_CE./l_CE_opt)+c+1;
else
error('The CE is out of bounds!')
end
% Calculate the force of the parallel elsatic element: lpee == lce [6,7]
f_PEE = (k_pee.*(l_CE-l_CE_opt).^n_pee);
f_PEE(l_CE<l_CE_opt) = 0;
end % <<< function (fl_fpee_lsee_fun) end
%% C.3 FUNCTION OF ACTIVATION DYNAMICS
function [ A ] = ES_fun(t, t_on, Apre, U )
% ES_FUN calculates the activation dynamics of the muscle
%
% INPUT:
% t ... time vector [s]
% t_on ... time when the contraction starts [s]
% Apre ... constant pre-activation level [1/s]
% U ... property of muscle activation [1/s]
%
% OUTPUT:
% A ... ratio [0,1] of muscle activation dynamics
% Simplifications (to preserve full functionality)
nmax = 1;
umax = U;
% Calculate it
A = zeros(1,length(t));
A(t<t_on) = Apre;
A(t>=t_on) = exp(1).^((-1).*exp(1).^((-1).*(t(t>=t_on)-t_on).*U).*...
U.^(-1).*((-1).*Apre+umax+exp(1).^((t(t>=t_on)-t_on).*U).*...
(Apre+((-1)+(t(t>=t_on)-t_on).*U).*umax))).*(Apre+(-1).*nmax)+nmax;
end % <<< function (ES_fun) end
%% C.4 PROPERTY CONVERSION FUNCTION
function[U,a,b,c,vmax,fiso,pmax,vopt,fopt,eta,kappa]=params(Par)
% PARAMS is a convenient muscle-properties conversion function
%
% INPUT
% Par ... vector containing the properties [U,a,b,c]
%
% OUTPUT
% returns the complete set of parameters based on the input provided
U = Par(1);
a = Par(2);
b = Par(3);
c = Par(4);
vmax = c/a-b;
fiso = c/b-a;
pmax = a*b+c-2*sqrt(a*b*c);
vopt = sqrt((b*c)/a)-b;
fopt = sqrt((a*c)/b)-a;
eta = pmax/c;
kappa = a/fiso;
end % <<< function (params) end