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passivewalker.m
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% PASSIVEWALKER simulates passive ramp walkers.
% passivewalker(1) - simplest walker as a limiting case of a passive walker (Garcia et. al 1998)
% passivewalker(2) - more general passive walker with round feet.
%
% Needs ODE113, FSOLVE, INTERP1.
% If you find bugs in this code please mail,
% Pranav A. Bhounsule, [email protected]
% First version: December, 26 2009
% Updated: June 27, 2019 on R2016a (fixed Erasmode, which is depreciated)
% Updated: Dec 14, 2019 on R2019b (heelstrike equations fixed)
% The Poincare section is afer footstrike
function passivewalker(flag)
clc
close all
format long
if nargin == 0
flag = 1; %simulates simplest walker by default
end
if flag == 1
%% Garcia's simplest walker with roots for Validation
%%% Dimensions %%
%% c = COM on the leg from hip, w = COM fore-aft offset, r = radius of feet
%% M = hip mass, m = leg mass, I = leg inertia, l = leg length
%%%%% To get results close to Garcia's walker increase M %%%%%%
walker.M = 1000; walker.m = 1.0; walker.I = 0.00; walker.l = 1.0; walker.w = 0.0;
walker.c = 1.0; walker.r = 0.0; walker.g = 1.0; walker.gam = 0.009;
%%%% Initial State %%%%%
q1 = 0.2; u1 = -0.2;
q2 = 0.4; u2 = -0.3;
z0 = [q1 u1 q2 u2];
%%% Root finding will give this stable root
%zstar = [0.200161072169750 -0.199906060087682 0.400322144339512 -0.015805473227965];
else
%% More General round feet walker with roots
%%%% Dimensions %%
%% c = COM on the leg from hip, w = COM fore-aft offset, r = radius of feet
%% M = hip mass, m = leg mass, I = leg inertia, l = leg length
walker.M = 1.0; walker.m = 0.5; walker.I = 0.02; walker.l = 1.0; walker.w = 0.0;
walker.c = 0.5; walker.r = 0.2; walker.g = 1.0; walker.gam = 0.01;
%%%% Initial State %%%%%
q1 = 0.2; u1 = -0.3;
q2 = 0.4; u2 = -0.3;
z0 = [q1 u1 q2 u2];
%% Root finding will give this stable root
%zstar = [0.189472782205104 -0.239124222551699 0.378945564410209 -0.053691703909393];
%%
end
%%%%%%%%%%%%%%%%%%%%%%%%%
steps = 10; %number of steps to animate
fps = 20; %Use low frames per second for low gravity
%%%% Root finding, Period one gait %%%%
options = optimset('TolFun',1e-12,'TolX',1e-12,'Display','off');
[zstar,fval,exitflag] = fsolve(@fixedpt,z0,options,walker);
if exitflag == 1
disp('Fixed point:');
disp(zstar);
else
error('Root finder not converged, change guess or change system parameters')
end
%%% Stability, using eigenvalues of Poincare map %%%
J=partialder(@onestep,zstar,walker);
disp('EigenValues for linearized map are');
eigJ = eig(J);
for i=1:4
disp(norm(eigJ(i)));
end
%
%%%% Get data for all the steps %%%
[z,t] = onestep(zstar,walker,steps);
%%% Animate result %%%
disp('Animating...');
animate(t,z,walker,steps,fps);
%%% Plot data %%%
disp('Some plots...')
plot(t,z(:,1),'r',t,z(:,3),'b')
xlabel('time'); ylabel('Angle (rad)');
legend('Stance Angle','Swing Angle');
title('State variables vs time for passive walker');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% FUNCTIONS START HERE %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%===================================================================
function zdiff=fixedpt(z0,walker)
%===================================================================
zdiff=onestep(z0,walker)-z0;
%===================================================================
function J=partialder(FUN,z,walker)
%===================================================================
pert=1e-5;
n = length(z);
J = zeros(n,n);
%%%% Using forward difference, accuracy linear %%%
% y0=feval(FUN,z,walker);
% for i=1:n
% ztemp=z;
% ztemp(i)=ztemp(i)+pert;
% J(:,i)=(feval(FUN,ztemp,walker)-y0) ;
% end
% J=(J/pert);
%%% Using central difference, accuracy quadratic %%%
for i=1:n
ztemp1=z; ztemp2=z;
ztemp1(i)=ztemp1(i)+pert;
ztemp2(i)=ztemp2(i)-pert;
J(:,i)=(feval(FUN,ztemp1,walker)-feval(FUN,ztemp2,walker)) ;
end
J=J/(2*pert);
%===================================================================
function [z,t]=onestep(z0,walker,steps)
%===================================================================
M = walker.M; m = walker.m; I = walker.I;
l = walker.l; c = walker.c; w = walker.w;
r = walker.r; g = walker.g; gam = walker.gam;
flag = 1;
if nargin<2
error('need more inputs to onestep');
elseif nargin<3
flag = 0; %send only last state, for root finder and jacobian
steps = 1;
end
q1 = z0(1);
u1 = z0(2);
q2 = z0(3);
u2 = z0(4);
%%%% Derived variables %%%%
TE = 1/2*m*(((-l*cos(q1)-r)*u1-u1*(-c*cos(q1)+w*sin(q1)))^2+(-l*sin(q1)*u1+u1*(c*sin(q1)+w*cos(q1)))^2)+1/2*m*(((-l*cos(q1)-r)*u1-(u1-u2)*(-c*cos(q1-q2)+w*sin(q1-q2)))^2+(-l*sin(q1)*u1+(u1-u2)*(c*sin(q1-q2)+w*cos(q1-q2)))^2)+1/2*M*((-l*cos(q1)-r)^2*u1^2+l^2*sin(q1)^2*u1^2)+1/2*I*(u1^2+(u1-u2)^2)+2*m*g*cos(gam)*r+2*m*g*l*cos(gam-q1)-m*g*c*cos(gam-q1)-m*g*w*sin(gam-q1)+2*m*g*sin(gam)*r*q1-m*g*c*cos(gam-q1+q2)-m*g*w*sin(gam-q1+q2)+M*g*cos(gam)*r+M*g*l*cos(gam-q1)+M*g*sin(gam)*r*q1;
xp1 = 0;
xh = -l*sin(q1) - r*q1 + xp1;
vxh = (-l*cos(q1)-r)*u1;
yh = l*cos(q1) + r;
vyh = -l*sin(q1)*u1;
z0 = [q1 u1 q2 u2 TE xh vxh yh vyh];
t0 = 0;
dt = 5; %might need to be changed based on time taken for one step
time_stamps = 100;
t_ode = t0;
z_ode = z0;
for i=1:steps
options=odeset('abstol',1e-13,'reltol',1e-13,'events',@collision);
tspan = linspace(t0,t0+dt,time_stamps);
[t_temp, z_temp] = ode113(@single_stance,tspan,z0,options,walker);
zplus=heelstrike(t_temp(end),z_temp(end,:),walker);
z0 = zplus;
t0 = t_temp(end);
%%%%% Ignore time stamps for heelstrike and first integration point
t_ode = [t_ode; t_temp(2:end)];
z_ode = [z_ode; z_temp(2:end,:)];
end
z = zplus(1:4);
if flag==1
z=z_ode;
t=t_ode;
end
%===================================================================
function zdot=single_stance(t,z,walker)
%===================================================================
q1 = z(1); u1 = z(2);
q2 = z(3); u2 = z(4);
xh = z(6); vxh = z(7);
yh = z(8); vyh = z(9);
M = walker.M; m = walker.m; I = walker.I;
l = walker.l; c = walker.c; w = walker.w;
r = walker.r; g = walker.g; gam = walker.gam;
Th=0; %external hip torque, if needed
M11 = -2*w^2*m-2*I+2*m*l*c*cos(q2)+2*m*w*l*sin(q2)-2*m*c^2-2*m*l^2-M*l^2+2*m*l*c-2*m*r^2-M*r^2+2*m*r*c*cos(q1-q2)-2*m*r*w*sin(q1-q2)-2*M*r*l*cos(q1)-4*m*r*l*cos(q1)+2*m*r*c*cos(q1)-2*m*r*w*sin(q1);
M12 = w^2*m+I-m*l*c*cos(q2)-m*w*l*sin(q2)+m*c^2-m*r*c*cos(q1-q2)+m*r*w*sin(q1-q2);
M21 = m*w*l*sin(q2)+m*l*c*cos(q2)-m*r*w*sin(q1-q2)+m*r*c*cos(q1-q2)-m*c^2-w^2*m-I;
M22 = w^2*m+m*c^2+I;
RHS1 = -2*m*r*u1*u2*c*sin(q1-q2)-2*m*r*u1*u2*w*cos(q1-q2)+m*r*u1^2*w*cos(q1)+m*r*u1^2*c*sin(q1)-2*m*r*l*sin(q1)*u1^2+M*g*sin(gam)*r+2*m*g*sin(gam)*r+m*r*u2^2*w*cos(q1-q2)+m*r*u2^2*c*sin(q1-q2)+m*r*u1^2*w*cos(q1-q2)+m*r*u1^2*c*sin(q1-q2)-M*r*l*sin(q1)*u1^2+M*g*l*sin(gam-q1)+2*m*g*l*sin(gam-q1)-m*g*c*sin(gam-q1)+m*g*w*cos(gam-q1)-m*g*c*sin(gam-q1+q2)+m*g*w*cos(gam-q1+q2)-2*m*l*u1*u2*w*cos(q2)-m*l*u2^2*c*sin(q2)+2*m*l*u1*u2*c*sin(q2)+m*l*u2^2*w*cos(q2);
RHS2 = -m*g*c*sin(gam-q1+q2)+m*g*w*cos(gam-q1+q2)-Th-m*l*u1^2*w*cos(q2)+m*l*u1^2*c*sin(q2);
MM = [M11 M12;
M21 M22];
RHS = [RHS1; RHS2];
X = MM \ RHS;
ud1 = X(1);
ud2 = X(2);
DTE = -ud1*I*u2+2*ud1*m*u1*r^2+m*u1*r*u2^2*c*sin(q1-q2)+m*u1*r*u2^2*w*cos(q1-q2)-m*u2^2*l*u1*c*sin(q2)+u2*m*g*c*sin(gam-q1+q2)-u2*m*g*w*cos(gam-q1+q2)+2*ud1*I*u1+ud2*I*u2+m*u2^2*l*u1*w*cos(q2)+2*ud1*m*u1*c^2+ud2*m*u2*c^2-ud2*I*u1+ud1*m*u2*c*l*cos(q2)+ud1*m*u2*w*l*sin(q2)-2*ud1*m*u1*l*c*cos(q2)-2*ud1*m*l*u1*w*sin(q2)-m*u2*u1^2*w*l*cos(q2)+m*u2*u1^2*c*l*sin(q2)+2*ud1*m*u1*w^2+ud2*m*u2*w^2+ud2*m*u1*l*c*cos(q2)+ud1*M*l^2*u1-ud2*m*u1*w^2-ud1*m*u2*c^2-ud2*m*u1*c^2+2*ud1*m*l^2*u1-ud1*m*u2*w^2-2*ud1*m*l*u1*c+2*ud1*M*l*cos(q1)*u1*r+4*ud1*m*l*cos(q1)*u1*r-2*ud1*m*u1*r*c*cos(q1)+2*ud1*m*u1*r*w*sin(q1)-2*ud1*m*u1*r*c*cos(q1-q2)-2*m*u1^3*r*l*sin(q1)+m*u1^3*r*c*sin(q1)+m*u1^3*r*w*cos(q1)+m*u1^3*r*c*sin(q1-q2)+m*u1^3*r*w*cos(q1-q2)-2*m*u1^2*r*u2*c*sin(q1-q2)-2*m*u1^2*r*u2*w*cos(q1-q2)-M*u1^3*r*l*sin(q1)+2*u1*m*g*l*sin(gam-q1)-u1*m*g*c*sin(gam-q1)+u1*m*g*w*cos(gam-q1)+2*u1*m*g*sin(gam)*r+ud2*m*l*u1*w*sin(q2)+ud1*M*u1*r^2-u1*m*g*c*sin(gam-q1+q2)+u1*m*g*w*cos(gam-q1+q2)+u1*M*g*l*sin(gam-q1)+u1*M*g*sin(gam)*r+2*ud1*m*u1*r*w*sin(q1-q2)+ud1*m*u2*c*cos(q1-q2)*r-ud1*m*u2*w*sin(q1-q2)*r+ud2*m*u1*r*c*cos(q1-q2)-ud2*m*u1*r*w*sin(q1-q2);
axh = l*sin(q1)*u1^2+(-l*cos(q1)-r)*ud1;
ayh = -l*cos(q1)*u1^2-l*sin(q1)*ud1;
zdot = [u1 ud1 u2 ud2 ...
DTE vxh axh vyh ayh]';
%===================================================================
function [gstop, isterminal,direction]=collision(t,z,walker)
%===================================================================
M = walker.M; m = walker.m; I = walker.I;
l = walker.l; c = walker.c; w = walker.w;
r = walker.r; g = walker.g; gam = walker.gam;
q1 = z(1); q2 = z(3);
gstop = -q2 + 2*q1;
if (q2>-0.05) %allow legs to pass through for small hip angles (taken care in real walker using stepping stones)
isterminal = 0;
else
isterminal=1; %ode should terminate is conveyed by 1, if you put 0 it goes till the final time u specify
end
direction=-1; % The t_final can be approached by any direction is indicated by the direction
%===================================================================
function zplus=heelstrike(t,z,walker)
%===================================================================
r1 = z(1); v1 = z(2);
r2 = z(3); v2 = z(4);
xh = z(6); yh = z(8);
q1 = r1 - r2;
q2 = -r2;
M = walker.M; m = walker.m; I = walker.I;
l = walker.l; c = walker.c; w = walker.w;
r = walker.r; g = walker.g; gam = walker.gam;
M11 = 2*I + m*((r - c*cos(q1) + l*cos(q1) + w*sin(q1))^2 + (c*sin(q1) + w*cos(q1) - l*sin(q1))^2) + M*(l^2 + r^2 + 2*l*r*cos(q1)) + m*((c*sin(q1 - q2) - l*sin(q1) + w*cos(q1 - q2))^2 + (r + l*cos(q1) - c*cos(q1 - q2) + w*sin(q1 - q2))^2);
M12 = c*l*m*cos(q2) - c^2*m - m*w^2 - I + l*m*w*sin(q2) + c*m*r*cos(q1 - q2) - m*r*w*sin(q1 - q2);
M21 = I + c^2*m + m*w^2 - c*l*m*cos(q2) - l*m*w*sin(q2) - c*m*r*cos(q1 - q2) + m*r*w*sin(q1 - q2);
M22 = - I - c^2*m - m*w^2;
RHS1 = 2*I*v1 - I*v2 + 2*c^2*m*v1 - c^2*m*v2 + 2*m*r^2*v1 + 2*m*v1*w^2 - m*v2*w^2 + M*r^2*v1 - 2*c*l*m*v1 + c*l*m*v2 + M*l^2*v1*cos(r2) + 2*l^2*m*v1*cos(r2) + M*l*r*v1*cos(r1 - r2) + 2*m*r*v1*w*sin(r1) - 2*c*m*r*v1*cos(r1 - r2) + c*m*r*v2*cos(r1 - r2) + 2*l*m*r*v1*cos(r1 - r2) + 2*m*r*v1*w*sin(r1 - r2) - m*r*v2*w*sin(r1 - r2) + M*l*r*v1*cos(r1) - 2*c*l*m*v1*cos(r2) - 2*c*m*r*v1*cos(r1) + 2*l*m*r*v1*cos(r1);
RHS2 = I*v1 + m*((v1*(c*sin(r1) + w*cos(r1)) - l*v1*sin(r1))*(c*sin(r1) + w*cos(r1)) - (v1*(r + l*cos(r1)) - v1*(c*cos(r1) - w*sin(r1)))*(c*cos(r1) - w*sin(r1)));
MM = [M11 M12;
M21 M22];
RHS = [RHS1; RHS2];
X = MM \ RHS;
u1 = X(1);
u2 = X(2);
TE = 1/2*m*(((-l*cos(q1)-r)*u1-u1*(-c*cos(q1)+w*sin(q1)))^2+(-l*sin(q1)*u1+u1*(c*sin(q1)+w*cos(q1)))^2)+1/2*m*(((-l*cos(q1)-r)*u1-(u1-u2)*(-c*cos(q1-q2)+w*sin(q1-q2)))^2+(-l*sin(q1)*u1+(u1-u2)*(c*sin(q1-q2)+w*cos(q1-q2)))^2)+1/2*M*((-l*cos(q1)-r)^2*u1^2+l^2*sin(q1)^2*u1^2)+1/2*I*(u1^2+(u1-u2)^2)+2*g*m*cos(gam)*r+2*m*g*l*cos(gam-q1)-m*g*c*cos(gam-q1)-m*g*w*sin(gam-q1)+2*g*m*sin(gam)*r*q1-m*g*c*cos(gam-q1+q2)-m*g*w*sin(gam-q1+q2)+g*M*cos(gam)*r+M*g*l*cos(gam-q1)+g*M*sin(gam)*r*q1;
vxh = (-l*cos(q1)-r)*u1;
vyh = -l*sin(q1)*u1;
zplus = [q1 u1 q2 u2 TE xh vxh yh vyh];
%===================================================================
function animate(t_all,z_all,walker,steps,fps)
%===================================================================
%%%% Interpolate linearly using fps %%%%%
z_all_plot = [z_all(:,1) z_all(:,3) z_all(:,6) z_all(:,8)];
nn = size(z_all_plot,2);
total_frames = round(t_all(end)*fps);
t = linspace(0,t_all(end),total_frames);
z = zeros(total_frames,nn);
for i=1:nn
z(:,i) = interp1(t_all,z_all_plot(:,i),t);
end
%%%% Now animate the results %%%%%%%
clf
M = walker.M; m = walker.m; I = walker.I;
l = walker.l; c = walker.c; w = walker.w;
r = walker.r; g = walker.g; gam = walker.gam;
mm = size(z,1);
min_xh = min(z(:,3)); max_xh = max(z(:,3));
dist_travelled = max_xh - min_xh;
camera_rate = dist_travelled/mm;
window_xmin = -1*l; window_xmax = 1*l;
window_ymin = -0.1; window_ymax = 1.1*(l+r);
axis('equal')
axis([window_xmin window_xmax window_ymin window_ymax])
axis off
set(gcf,'Color',[1,1,1])
%%%%%%
lines_for_feet = 4; %no of straight lines to represent a foot.
counter = 2 + 2*lines_for_feet; % total number of segments needed for animation. 2 legs and other for feet
th = 0.25; %%%% angle subtended by each straight line segment of foot
%%% creat object for hinge %%%%%
hingepic=line('xdata',0,'ydata',0, 'marker','.','markersize',[20], 'color','black');
%%%% create object for legs and feet %%%%
barref = [0 0; 0 -1]; %%% bar along negative y-axis
y = [0;-1]; %%% vector along negative y
O = [0; 0]; %%%% origin
%%%% legs in red %%%
for p = 1:2
barpic(p)=line('xdata',barref(1,:),'ydata',barref(2,:),'linewidth', 2,'color','red');
end
%%% feet in blue %%%
for p = 3:counter
barpic(p)=line('xdata',barref(1,:),'ydata',barref(2,:),'linewidth', 2,'color','blue');
end
%%%% create ramp %%%%
rampref=[min_xh-1 max_xh+l ; 0 0];
%%%% Draw ramp %%%%%%%%%%
line('xdata',rampref(1,:),'ydata',rampref(2,:),'linewidth', 1,'color','black');
moviescaling = 1; % slow down factor
delay =floor(moviescaling); %delay per frame in .001 secs
for i=1:mm
%%%% Put some delay if needed %%%%%%%%%%
% for j=1:100, log(1:delay*17); end %delay for graphics.
% %the number in this expression
% %is machine dependent.
% %The LOG is just something
% %to keep the machine busy.
q1 = z(i,1); q2 = z(i,2);
xh = z(i,3); yh = z(i,4);
window_xmin = window_xmin + camera_rate;
window_xmax = window_xmax + camera_rate;
axis('equal')
axis([window_xmin window_xmax window_ymin window_ymax])
%%% hinge coordinates
hinge=[xh; yh];
%%% leg coordinates
A = [q1 -(q2-q1)];
for p = 1:2
bar(:,:,p) = [hinge, hinge] + (l+r)*R(A(p))*barref;
center(:,:,p) = hinge + l*R(A(p))*y; %%% center of each circle on foot
end
%%%% feet coordinates
%%% angle subtended by arc at the center %%%
%%% This is for lines_for_feet = 4 and need changes if more lines
%%% are needed to represent a foot. Size of this matrix is lines_for_feet*2
B = [-th -2*th; 0 -th; th 0; 2*th th];
incr = 3;
for p=1:2
for q=1:4
C = A(p) + B(q,:);
bar(:,:,incr) = [center(:,:,p), center(:,:,p)] + r*R(C(1))*[O,y] + r*R(C(2))*[y, O];
incr = incr + 1;
end
end
%%% animate now
set(hingepic,'xdata',hinge(1),'ydata',hinge(2));
for p=1:counter
set(barpic(p),'xdata',bar(1,:,p),'ydata',bar(2,:,p));
end
drawnow
end
%===================================================================
function rotation = R(A)
%===================================================================
rotation = [cos(A) -sin(A); sin(A) cos(A)];