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RowdyWalkerSim.m
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RowdyWalkerSim.m
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function RowdyWalkerSim
% MATLAB files for the paper:
% A 3D Printed, Non-Assembly, Passive Dynamic Walking Toy: Design and Analysis
% By Christian Trevino, Joseph Galloway, Pranav A. Bhounsule
% Submitted to ASME Journal of Mechanisms and Robotics
% More information on this link: tiny.cc/pranavb
%
clc
clear all
close all
format long
robot.f = 0.9; %fraction in the body
robot.r = 5.5/100; %fixed
robot.c1 = 0.35*robot.r; %along length.
robot.w1 = 0.75/100; %below hip and backward
robot.c2 = 0.7*robot.r;
robot.w2 = robot.w1*.95;
robot.gam = atan(2.1/15.2);
robot.front_stop = -1.2/5.5;
robot.I1 = 0.00013;
robot.I2 = 0.0000030;
robot.m1 = robot.f*(157/1000);
robot.m2 = (1-robot.f)*(157/1000);
robot.g = 9.81;
robot.fps = 30;
robot.C1 = 0.005; %damping friction in the rear leg
robot.width = 0.15;
robot.front2rear = robot.gam;
robot.rear_stop = 0;
robot.rear2front = robot.gam;
robot.Tol = 1e-12; %integration tolerance
robot.Fsolve_Tol = 1e-10; %fsolve tolerance
robot.animate_x_adjust = 0.007;
robot.animate_y_adjust = -0.007;
Fontsize = 12; %for font size on plots
%%%%%% options that can be set %%%
robot.makeAVI = 0; %writes the file rowdy_walker_sim.avi
robot.animate_face = 1; %put patch object on the face
robot.animate_border = 0; %animates only the border of rowdy
robot.verbose = 0; %set to 1 to print results
steps = 4; %number of steps
%%% guess for the initial state (see Fig 5. Eqn 1 in the attached paper).
q1 = robot.gam;
u1 = -1;
z0 = [q1 u1];
%%%% fixed point %%%
%z0=[-0.019156109860067 -1.900074694074746]; %fsolve gives this root
fixed_points = 1;
if (fixed_points)
%%%% Root finding, Period one gait %%%%
options = optimset('TolFun',robot.Fsolve_Tol,'TolX',robot.Fsolve_Tol,'Display','off');
[zstar,fval,exitflag,output,jacob] = fsolve(@fixedpt,z0,options,robot);
if exitflag == 1
disp('Fixed points are');
%fval
z0=zstar;
disp(z0);
%fixed point z0 = [-0.019467427570676 -1.897577473434229];
else
error('Root finder not converged, change guess or change system parameters')
end
%%% Stability, using linearised eigenvalue %%%
J=partialder(@onestep,zstar,robot);
% disp('Jacobian is ');
% J
disp(' ');
disp('EigenValues for linearized map are');
disp(eig(J))
end
%%%%% Get data for all the steps %%%
[z,t,zz,TE] = onestep(z0,robot,steps);
Distance_sim = zz(end,5)-zz(1,5);
Time_sim = t(end);
Velocity_sim = Distance_sim/Time_sim;
disp('Parameter: Values ');
disp('-------------------------------------');
disp(['Distance: ',num2str(Distance_sim)]);
disp(['Time: ',num2str(Time_sim)]);
disp(['Velocity: ',num2str(Velocity_sim)]);
%%%% Animate result %%%
figure(1)
animate(t,zz,robot);
%%% Plot data %%%
h=figure(2);
subplot(3,1,1);
plot(t,zz(:,1),'r','LineWidth',2); ylabel('absolute angle (rad)'); hold on;
plot(t,zz(:,2),'b','LineWidth',1);
ylabel('Absolute angle (rad)','FontSize',Fontsize);
legend('front','rear','Location','Best');
grid on
subplot(3,1,2);
plot(t,zz(:,3),'r','LineWidth',2); ylabel('absolute rate (rad/s)'); hold on
plot(t,zz(:,4),'b','LineWidth',1);
ylabel('Absolute angular rate (rad/s)','FontSize',Fontsize);
xlabel('Time (s)','FontSize',Fontsize);
legend('front','rear','Location','Best');
grid on
%%% NOTE: absolute value + CCW and - is CW.
%h = figure(3)
subplot(3,1,3);
plot(t,TE,'k');
xlabel('time');
ylabel('Total Energy');
grid on
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% FUNCTIONS START HERE %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%===================================================================
function zdiff=fixedpt(z0,robot)
%===================================================================
zdiff=onestep(z0,robot)-z0;
%===================================================================
function J=partialder(FUN,z,robot)
%===================================================================
pert=1e-5;
%%%% Using forward difference, accuracy linear %%%
% y0=feval(FUN,z,GL_DIM);
% for i=1:length(z)
% ztemp=z;
% ztemp(i)=ztemp(i)+pert;
% J(:,i)=(feval(FUN,ztemp,GL_DIM)-y0) ;
% end
% J=(J/pert);
%%% Using central difference, accuracy quadratic %%%
for i=1:length(z)
ztemp1=z; ztemp2=z;
ztemp1(i)=ztemp1(i)+pert;
ztemp2(i)=ztemp2(i)-pert;
J(:,i)=(feval(FUN,ztemp1,robot)-feval(FUN,ztemp2,robot)) ;
end
J=J/(2*pert);
%===================================================================
function [z_temp,t_temp,zz,TE] = onestep(z0,robot,steps)
%===================================================================
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 = 0;
u2 = 0;
r = robot.r;
xh = 0;%-r*q1 + xp1;
vxh = -r*u1;
yh = r;
vyh = 0;
t0 = 0;
z0 = [q1 u1 q2 u2 xh vxh yh vyh];
E0 = total_energy1_front_stance(t0,z0,robot);
TE = E0;
t_temp = t0;
z_temp = [z0 E0];
zz = [z0(1) z0(1)-z0(3) z0(2) z0(2)-z0(4) z0(5) z0(7)];
if (robot.verbose && nargin==3)
disp(['start state is z = ',num2str(zz(1:4))]);
end
for i=1:steps
%%% starts in leg together and front tilted forward %%%%
%%%%%% one dof, front 2 rear %%%%%%%
E0 = total_energy1_front_stance(t0,z0,robot);
z0 = [z0,E0];
t0 = t_temp(end,:);
dt = 3;
options=odeset('abstol',robot.Tol,'reltol',robot.Tol,'events',@front2rear);
tspan = linspace(t0,t0+dt,1000);
[t_temp1, z_temp1] = ode113(@single_stance1_front,tspan,z0,options,robot);
zz1 = [z_temp1(:,1) z_temp1(:,1)-z_temp1(:,3) ...
z_temp1(:,2) z_temp1(:,2)-z_temp1(:,4)...
z_temp1(:,5) z_temp1(:,7)];
t_temp = [t_temp; t_temp1(2:end)];
if (robot.verbose && nargin==3)
disp(['front2rear, t = ',num2str(t_temp1(end))]);
disp(['Absolute state, zz = ',num2str(zz1(end,1:4))]);
end
z_temp = [z_temp; z_temp1(2:end,:)];
TE = [TE; z_temp1(2:end,9)];
zz = [zz; zz1(2:end,:)];
%%%%% two dof, rear support %%%%%%
q1 = z_temp1(end,1)-z_temp1(end,3);
u1 = z_temp1(end,2)-z_temp1(end,4);
q2 = z_temp1(end,3); %=0
u2 = z_temp1(end,4);%=0
xh = z_temp1(end,5);
vxh = z_temp1(end,6);
yh = z_temp1(end,7);
vyh = z_temp1(end,8);
z0 = [q1 u1 q2 u2 xh vxh yh vyh];
t0 = t_temp1(end);
E0 = total_energy2_rear_stance(t0,z0,robot);
z0 = [z0 E0];
dt = 3;
options=odeset('abstol',robot.Tol,'reltol',robot.Tol,'events',@front_stop);
tspan = linspace(t0,t0+dt,1000);
[t_temp2, z_temp2] = ode113(@single_stance2_rear_stance,tspan,z0,options,robot);
zz2 = [z_temp2(:,1)-z_temp2(:,3) z_temp2(:,1) ...
z_temp2(:,2)-z_temp2(:,4) z_temp2(:,2) ...
z_temp2(:,5) z_temp2(:,7)];
t_temp = [t_temp; t_temp2(2:end)];
z_temp = [z_temp; z_temp2(2:end,:)];
zz = [zz; zz2(2:end,:)];
TE = [TE; z_temp2(2:end,9)];
if (robot.verbose && nargin==3)
disp(['front_stop, t = ',num2str(t_temp2(end))]);
disp(['Absolute state, zz = ',num2str(zz2(end,1:4))]);
end
%%%%%%% one dof, rear 2 front %%
r1 = z_temp2(end,1);
v1 = z_temp2(end,2);
r2 = z_temp2(end,3);
v2 = z_temp2(end,4);
[q1,u1,q2,u2] = front_strike(r1,v1,r2,v2,robot);
xh = z_temp2(end,5);
vxh = -r*u1;
yh = z_temp2(end,7);
vyh = z_temp2(end,8);
z0 = [q1 u1 q2 u2 xh vxh yh vyh];
t0 = t_temp(end,:);
E0 = total_energy2_rear_stance(t0,z0,robot);
z0 = [z0 E0];
dt = 3;
options=odeset('abstol',robot.Tol,'reltol',robot.Tol,'events',@rear2front);
tspan = linspace(t0,t0+dt,1000);
[t_temp3, z_temp3] = ode113(@single_stance1_rear,tspan,z0,options,robot);
zz3 = [z_temp3(:,1)-z_temp3(:,3) z_temp3(:,1) ...
z_temp3(:,2)-z_temp3(:,4) z_temp3(:,2) ...
z_temp3(:,5) z_temp3(:,7)];
t_temp = [t_temp; t_temp3(2:end)];
z_temp = [z_temp; z_temp3(2:end,:)];
zz = [zz; zz3(2:end,:)];
TE = [TE; z_temp3(2:end,9)];
if (robot.verbose && nargin==3)
disp(['rear2front, t = ',num2str(t_temp3(end))]);
disp(['Absolute state, zz = ',num2str(zz3(end,1:4))]);
end
%%%%%% two dof, front support %%%%%%%
q1 = z_temp3(end,1)-z_temp3(end,3);
u1 = z_temp3(end,2);
q2 = -z_temp3(end,3);
u2 = 0;
xh = z_temp3(end,5);
vxh = z_temp3(end,6);
yh = z_temp3(end,7);
vyh = z_temp3(end,8);
z0 = [q1 u1 q2 u2 xh vxh yh vyh];
E0 = total_energy2_front_stance(t0,z0,robot);
z0 = [z0 E0];
t0 = t_temp(end,:);
dt = 3;
options=odeset('abstol',robot.Tol,'reltol',robot.Tol,'events',@rear_stop);
tspan = linspace(t0,t0+dt,1000);
[t_temp4, z_temp4] = ode113(@single_stance2_front_stance,tspan,z0,options,robot);
zz4 = [z_temp4(:,1) z_temp4(:,1)-z_temp4(:,3) ...
z_temp4(:,2) z_temp4(:,2)-z_temp4(:,4) ...
z_temp4(:,5) z_temp4(:,7)];
t_temp = [t_temp; t_temp4(2:end)];
z_temp = [z_temp; z_temp4(2:end,:)];
zz = [zz; zz4(2:end,:)];
TE = [TE; z_temp4(2:end,9)];
if (robot.verbose && nargin==3)
disp(['rear_stop, t = ',num2str(t_temp4(end))]);
disp(['Absolute state, zz = ',num2str(zz4(end,1:4))]);
end
%%%%%%% init condition for front rocking %%%
r1 = z_temp4(end,1);
v1 = z_temp4(end,2);
r2 = z_temp4(end,3);
v2 = z_temp4(end,4);
[q1,u1,q2,u2] = rear_strike(r1,v1,r2,v2,robot);
xh = z_temp4(end,5);
vxh = -r*u1;
yh = z_temp4(end,7);
vyh = z_temp4(end,8);
z0 = [q1 u1 q2 u2 xh vxh yh vyh];
end
if flag==0
z_temp=[q1 u1];
t_temp = [];
zz = [];
end
%===================================================================
function animate(t_all,z_all,robot)
%===================================================================
fps = robot.fps;
r = robot.r;
m1 = robot.m1; c1 = robot.c1; w1 = robot.w1;
m2 = robot.m2; c2 = robot.c2; w2 = robot.w2;
gam = robot.gam;
[mm,nn] = size(z_all);
t = linspace(0,t_all(end),fps*ceil(t_all(end)));
for i=1:nn
z(:,i) = interp1(t_all,z_all(:,i),t);
end
if (robot.animate_face)
[X,Y] = rowdy_data;
scale = robot.width/(max(X) - min(X));
X = robot.animate_x_adjust+scale*X; Y = robot.animate_y_adjust+scale*Y;
N = length(X);
end
qq1 = z(:,1); qq2 = z(:,2);
xxh = z(:,5); yyh = z(:,6);
if (robot.makeAVI)
mov = VideoWriter('rowdy_walker_sim.avi');
open(mov);
end
for i=1:length(t)
q1 = qq1(i); q2 = qq2(i);
xh = xxh(i); yh = yyh(i);
x_H_com = (m1*(c1*sin(q1) - w1*cos(q1)) + m2*(c2*sin(q1 - q2) - w2*cos(q1 - q2)))/(m1 + m2);
y_H_com = -(m1*(c1*cos(q1) + w1*sin(q1)) + m2*(c2*cos(q1 - q2) + w2*sin(q1 - q2)))/(m1 + m2);
[x_H_com,y_H_com] = world_frame(gam,x_H_com,y_H_com);
com_x = xh+x_H_com;
com_y = yh+y_H_com;
[hinge_x,hinge_y] = world_frame(gam,xh,yh);
plot(hinge_x,hinge_y,'ro','Markersize',5,'Markerfacecolor','g'); hold on;
[com_x,com_y] = world_frame(gam,com_x,com_y);
plot(com_x,com_y,'ko','Markersize',20,'Markerfacecolor','k');
ramp_x_min = -1; ramp_x_max = 1;
ramp_y_min = 0; ramp_y_max = 0;
[ramp_x_min,ramp_y_min] = world_frame(gam,ramp_x_min,ramp_y_min);
[ramp_x_max,ramp_y_max] = world_frame(gam,ramp_x_max,ramp_y_max);
line([ramp_x_min ramp_x_max],[ramp_y_min ramp_y_max],'Linewidth',2,'Color','k'); %ramp
center_x = xh; center_y = yh;
tip_x = xh+r*sin(q1); tip_y = yh-r*cos(q1);
tip_x2 = xh+r*sin(q2); tip_y2 = yh-r*cos(q2);
[tip_x,tip_y] = world_frame(gam,tip_x,tip_y);
[tip_x2,tip_y2] = world_frame(gam,tip_x2,tip_y2);
th = 0.013/robot.r;
curve_x_fwd = center_x + r*sin(q1+th); curve_y_fwd = center_y - r*cos(q1+th);
curve_x2_fwd = center_x + r*sin(q1+2*th); curve_y2_fwd = center_y - r*cos(q1+2*th);
[curve_x_fwd, curve_y_fwd] = world_frame(gam,curve_x_fwd, curve_y_fwd);
[curve_x2_fwd, curve_y2_fwd] = world_frame(gam,curve_x2_fwd, curve_y2_fwd);
th = -0.013/robot.r;
curve_x_bwd = center_x + r*sin(q2+th); curve_y_bwd = center_y - r*cos(q2+th);
curve_x2_bwd = center_x + r*sin(q2+2*th); curve_y2_bwd = center_y - r*cos(q2+2*th);
[curve_x_bwd, curve_y_bwd] = world_frame(gam,curve_x_bwd, curve_y_bwd);
[curve_x2_bwd, curve_y2_bwd] = world_frame(gam,curve_x2_bwd, curve_y2_bwd);
if (robot.animate_border)
patch([hinge_x tip_x curve_x_fwd curve_x2_fwd ],...
[hinge_y tip_y curve_y_fwd curve_y2_fwd ],...
[255,255,255]/255,'EdgeColor','k'); %front leg
patch([hinge_x tip_x2 curve_x_bwd curve_x2_bwd ],...
[hinge_y tip_y2 curve_y_bwd curve_y2_bwd ],...
[255,255,255]/255,'EdgeColor','k');
%[255,140,0]/255,'EdgeColor',[255,140,0]/255);
else
patch([hinge_x tip_x curve_x_fwd curve_x2_fwd ],...
[hinge_y tip_y curve_y_fwd curve_y2_fwd ],...
[241, 90, 34]/255,'EdgeColor',[241, 90, 34]/255); %front leg
patch([hinge_x tip_x2 curve_x_bwd curve_x2_bwd ],...
[hinge_y tip_y2 curve_y_bwd curve_y2_bwd ],...
[255,140,0]/255,'EdgeColor',[255,140,0]/255);
%[255,140,0]/255,'EdgeColor',[255,140,0]/255);
end
if (robot.animate_face)
R = rotation(q1);
for ii=1:N
pos_face = [R(1,1); R(1,2)]*X(ii) + [R(2,1); R(2,2)]*Y(ii);
x_face(ii) = xh + pos_face(1); y_face(ii) = yh+pos_face(2);
[x_face(ii),y_face(ii)] = world_frame(gam,x_face(ii),y_face(ii));
end
if (robot.animate_border)
patch(x_face,y_face,[255,255,255]/255,'EdgeColor','k');
else
patch(x_face,y_face,[241, 90, 34]/255,'EdgeColor',[255,255,255]/255);
end
end
% %%%%% draw hinge, tip_x, and tip_x2
plot(hinge_x,hinge_y,'ko','Markersize',3,'MarkerFacecolor','k');
plot(tip_x,tip_y,'ko','Markersize',3,'MarkerFacecolor','k');
plot(tip_x2,tip_y2,'ko','Markersize',3,'MarkerFacecolor','k');
axis('equal');
axis([-0.08 0.21 -0.05 0.1]);
if (robot.makeAVI)
axis off
set(gcf,'Color',[1,1,1])
currFrame = getframe;
writeVideo(mov,currFrame);
end
pause(0.05);
hold off;
end
if (robot.makeAVI)
close(mov);
end
%===================================================================
function R = rotation(theta)
%===================================================================
c = cos(theta); s = sin(theta);
R = [c s; -s c];
%===================================================================
function [x,y] = world_frame(theta,x,y)
%===================================================================
c = cos(theta); s = sin(theta);
R = [c s; -s c];
orig_vec = [x;y];
new_vec = R*orig_vec;
x = new_vec(1); y = new_vec(2);
%===================================================================
function [gstop, isterminal,direction]=front2rear(t,z,robot)
%===================================================================
q1 = z(1);
gstop = q1 - robot.front2rear;% gam;
isterminal = 1; %terminate integration
%q1 goes from - to 0 to +
%q1-gam goes from - to +
direction = 1;
%===================================================================
function [gstop, isterminal,direction]=rear2front(t,z,robot)
%===================================================================
q1 = z(1);
q2 = z(3);
gstop = (q1-q2) - robot.rear2front;% gam;
isterminal = 1; %terminate integration
% % q1 goes from + to 0 to -
% % q1-gam goes from + to -
direction = -1;
%===================================================================
function [gstop, isterminal,direction]=front_stop(t,z,robot)
%===================================================================
q2 = z(3);
gstop = q2 - robot.front_stop;%
isterminal = 1; %terminate integration
%q3 goes from 0 to -
%q1-robot.rear2front goes from + to -
direction = -1;
%===================================================================
function [gstop, isterminal,direction]=rear_stop(t,z,robot)
%===================================================================
q2 = z(3);
gstop = q2 - robot.rear_stop;%
isterminal = 1; %terminate integration
%q2 goes from 0 to -
%q1-robot.rear2front goes from + to -
direction = -1;
%===================================================================
function zdot=single_stance1_front(t,z,robot) %PAB (DONE)
%===================================================================
q1 = z(1); u1 = z(2);
q2 = z(3); u2 = z(4);
xh = z(5); vxh = z(6);
yh = z(7); vyh = z(8);
m1 = robot.m1; m2 = robot.m2; g = robot.g; gam = robot.gam;
r = robot.r; c2 = robot.c2; w2 = robot.w2; c1 = robot.c1; w1 = robot.w1;
I1 = robot.I1; I2 = robot.I2;
MM = 2*c1*m1*r*cos(q1) - I2 - c1^2*m1 - c2^2*m2 - m1*r^2 - m2*r^2 - m1*w1^2 - m2*w2^2 - I1 + 2*m1*r*w1*sin(q1) + 2*c2*m2*r*cos(q1 - q2) + 2*m2*r*w2*sin(q1 - q2);
RHS = g*m1*r*sin(gam) - g*m2*w2*cos(gam - q1 + q2) - c2*g*m2*sin(gam - q1 + q2) + g*m2*r*sin(gam) - c1*g*m1*sin(gam - q1) - g*m1*w1*cos(gam - q1) + c1*m1*r*u1^2*sin(q1) - m1*r*u1^2*w1*cos(q1) + c2*m2*r*u1^2*sin(q1 - q2) - m2*r*u1^2*w2*cos(q1 - q2);
X = MM \ RHS;
ud1 = X(1);
ud2 = 0;
axh = -r*ud1;
ayh = 0;
DTE = u1*(I1*ud1 + I2*ud1 + c1^2*m1*ud1 + c2^2*m2*ud1 + m1*r^2*ud1 + m2*r^2*ud1 + m1*ud1*w1^2 + m2*ud1*w2^2 - c2*g*m2*sin(gam - q1 + q2) - g*m2*w2*cos(gam - q1 + q2) + g*m1*r*sin(gam) + g*m2*r*sin(gam) - c1*g*m1*sin(gam - q1) - g*m1*w1*cos(gam - q1) - 2*m1*r*ud1*w1*sin(q1) - 2*c2*m2*r*ud1*cos(q1 - q2) - 2*m2*r*ud1*w2*sin(q1 - q2) + c1*m1*r*u1^2*sin(q1) - m1*r*u1^2*w1*cos(q1) + c2*m2*r*u1^2*sin(q1 - q2) - m2*r*u1^2*w2*cos(q1 - q2) - 2*c1*m1*r*ud1*cos(q1));
zdot = [u1 ud1 u2 ud2 ...
vxh axh vyh ayh DTE]';
%===================================================================
function zdot=single_stance1_rear(t,z,robot) %PAB (DONE)
%===================================================================
q1 = z(1); u1 = z(2);
q2 = z(3); u2 = z(4);
xh = z(5); vxh = z(6);
yh = z(7); vyh = z(8);
m1 = robot.m1; m2 = robot.m2; g = robot.g; gam = robot.gam;
r = robot.r; c2 = robot.c2; w2 = robot.w2; c1 = robot.c1; w1 = robot.w1;
I1 = robot.I1; I2 = robot.I2;
MM = 2*c2*m2*r*cos(q1) - I2 - c1^2*m1 - c2^2*m2 - m1*r^2 - m2*r^2 - m1*w1^2 - m2*w2^2 - I1 + 2*m2*r*w2*sin(q1) + 2*c1*m1*r*cos(q1 - q2) + 2*m1*r*w1*sin(q1 - q2);
RHS = g*m1*r*sin(gam) - g*m1*w1*cos(gam - q1 + q2) - c1*g*m1*sin(gam - q1 + q2) + g*m2*r*sin(gam) - c2*g*m2*sin(gam - q1) - g*m2*w2*cos(gam - q1) + c2*m2*r*u1^2*sin(q1) - m2*r*u1^2*w2*cos(q1) + c1*m1*r*u1^2*sin(q1 - q2) - m1*r*u1^2*w1*cos(q1 - q2);
X = MM \ RHS;
ud1 = X(1);
ud2 = 0;
axh = -r*ud1;
ayh = 0;
DTE = u1*(I1*ud1 + I2*ud1 + c1^2*m1*ud1 + c2^2*m2*ud1 + m1*r^2*ud1 + m2*r^2*ud1 + m1*ud1*w1^2 + m2*ud1*w2^2 - c1*g*m1*sin(gam - q1 + q2) - g*m1*w1*cos(gam - q1 + q2) + g*m1*r*sin(gam) + g*m2*r*sin(gam) - c2*g*m2*sin(gam - q1) - g*m2*w2*cos(gam - q1) - 2*m2*r*ud1*w2*sin(q1) - 2*c1*m1*r*ud1*cos(q1 - q2) - 2*m1*r*ud1*w1*sin(q1 - q2) + c2*m2*r*u1^2*sin(q1) - m2*r*u1^2*w2*cos(q1) + c1*m1*r*u1^2*sin(q1 - q2) - m1*r*u1^2*w1*cos(q1 - q2) - 2*c2*m2*r*ud1*cos(q1));
zdot = [u1 ud1 u2 ud2 ...
vxh axh vyh ayh DTE]';
%===================================================================
function zdot=single_stance2_rear_stance(t,z,robot) %PAB (DONE)
%===================================================================
q1 = z(1); u1 = z(2);
q2 = z(3); u2 = z(4);
xh = z(5); vxh = z(6);
yh = z(7); vyh = z(8);
m1 = robot.m1; m2 = robot.m2; g = robot.g; gam = robot.gam;
r = robot.r; c2 = robot.c2; w2 = robot.w2; c1 = robot.c1; w1 = robot.w1;
C1 = robot.C1;
I1 = robot.I1; I2 = robot.I2;
M11 = 2*c2*m2*r*cos(q1) - I2 - c1^2*m1 - c2^2*m2 - m1*r^2 - m2*r^2 - m1*w1^2 - m2*w2^2 - I1 + 2*m2*r*w2*sin(q1) + 2*c1*m1*r*cos(q1 - q2) + 2*m1*r*w1*sin(q1 - q2);
M12 = I1 + c1^2*m1 + m1*w1^2 - c1*m1*r*cos(q1 - q2) - m1*r*w1*sin(q1 - q2);
M21 = c1*m1*r*cos(q1 - q2) - c1^2*m1 - m1*w1^2 - I1 + m1*r*w1*sin(q1 - q2);
M22 = I1 + c1^2*m1 + m1*w1^2;
RHS1 = C1*u1-m1*((c1*sin(q1 - q2) - w1*cos(q1 - q2))*(u1 - u2)^2*(c1*cos(q1 - q2) - r + w1*sin(q1 - q2)) - (c1*cos(q1 - q2) + w1*sin(q1 - q2))*(c1*sin(q1 - q2) - w1*cos(q1 - q2))*(u1 - u2)^2) - m2*(u1^2*(c2*sin(q1) - w2*cos(q1))*(c2*cos(q1) - r + w2*sin(q1)) - u1^2*(c2*cos(q1) + w2*sin(q1))*(c2*sin(q1) - w2*cos(q1))) - g*m2*(sin(gam)*(c2*cos(q1) - r + w2*sin(q1)) - cos(gam)*(c2*sin(q1) - w2*cos(q1))) - g*m1*(sin(gam)*(c1*cos(q1 - q2) - r + w1*sin(q1 - q2)) - cos(gam)*(c1*sin(q1 - q2) - w1*cos(q1 - q2)));
RHS2 = 0*u2+g*m1*(cos(gam)*(c1*sin(q1 - q2) - w1*cos(q1 - q2)) - sin(gam)*(c1*cos(q1 - q2) + w1*sin(q1 - q2)));
MM = [M11 M12;
M21 M22];
RHS = [RHS1; RHS2];
X = MM \ RHS;
ud1 = X(1);
ud2 = X(2);
axh = -r*ud1;
ayh = 0;
DTE = I1*u1*ud1 - I1*u1*ud2 - I1*u2*ud1 + I2*u1*ud1 + I1*u2*ud2 + c1^2*m1*u1*ud1 - c1^2*m1*u1*ud2 - c1^2*m1*u2*ud1 + c1^2*m1*u2*ud2 + c2^2*m2*u1*ud1 + m1*r^2*u1*ud1 + m2*r^2*u1*ud1 + m1*u1*ud1*w1^2 - m1*u1*ud2*w1^2 - m1*u2*ud1*w1^2 + m1*u2*ud2*w1^2 + m2*u1*ud1*w2^2 - c2*g*m2*u1*sin(gam - q1) - g*m2*u1*w2*cos(gam - q1) + c2*m2*r*u1^3*sin(q1) - m2*r*u1^3*w2*cos(q1) - c1*g*m1*u1*sin(gam - q1 + q2) + c1*g*m1*u2*sin(gam - q1 + q2) + c1*m1*r*u1^3*sin(q1 - q2) - g*m1*u1*w1*cos(gam - q1 + q2) + g*m1*u2*w1*cos(gam - q1 + q2) - m1*r*u1^3*w1*cos(q1 - q2) + g*m1*r*u1*sin(gam) + g*m2*r*u1*sin(gam) - m1*r*u1*u2^2*w1*cos(q1 - q2) + 2*m1*r*u1^2*u2*w1*cos(q1 - q2) - 2*c2*m2*r*u1*ud1*cos(q1) - 2*m2*r*u1*ud1*w2*sin(q1) - 2*c1*m1*r*u1*ud1*cos(q1 - q2) + c1*m1*r*u1*ud2*cos(q1 - q2) + c1*m1*r*u2*ud1*cos(q1 - q2) - 2*m1*r*u1*ud1*w1*sin(q1 - q2) + m1*r*u1*ud2*w1*sin(q1 - q2) + m1*r*u2*ud1*w1*sin(q1 - q2) + c1*m1*r*u1*u2^2*sin(q1 - q2) - 2*c1*m1*r*u1^2*u2*sin(q1 - q2);
zdot = [u1 ud1 u2 ud2 ...
vxh axh vyh ayh DTE]';
%===================================================================
function zdot=single_stance2_front_stance(t,z,robot) %PAB (DONE)
%===================================================================
q1 = z(1); u1 = z(2);
q2 = z(3); u2 = z(4);
xh = z(5); vxh = z(6);
yh = z(7); vyh = z(8);
m1 = robot.m1; m2 = robot.m2; g = robot.g; gam = robot.gam;
r = robot.r; c2 = robot.c2; w2 = robot.w2; c1 = robot.c1; w1 = robot.w1;
I1 = robot.I1; I2 = robot.I2;
M11 = 2*c1*m1*r*cos(q1) - I2 - c1^2*m1 - c2^2*m2 - m1*r^2 - m2*r^2 - m1*w1^2 - m2*w2^2 - I1 + 2*m1*r*w1*sin(q1) + 2*c2*m2*r*cos(q1 - q2) + 2*m2*r*w2*sin(q1 - q2);
M12 = I2 + c2^2*m2 + m2*w2^2 - c2*m2*r*cos(q1 - q2) - m2*r*w2*sin(q1 - q2);
M21 = c2*m2*r*cos(q1 - q2) - c2^2*m2 - m2*w2^2 - I2 + m2*r*w2*sin(q1 - q2);
M22 = I2 + c2^2*m2 + m2*w2^2;
RHS1 = - m2*((c2*sin(q1 - q2) - w2*cos(q1 - q2))*(u1 - u2)^2*(c2*cos(q1 - q2) - r + w2*sin(q1 - q2)) - (c2*cos(q1 - q2) + w2*sin(q1 - q2))*(c2*sin(q1 - q2) - w2*cos(q1 - q2))*(u1 - u2)^2) - m1*(u1^2*(c1*sin(q1) - w1*cos(q1))*(c1*cos(q1) - r + w1*sin(q1)) - u1^2*(c1*cos(q1) + w1*sin(q1))*(c1*sin(q1) - w1*cos(q1))) - g*m1*(sin(gam)*(c1*cos(q1) - r + w1*sin(q1)) - cos(gam)*(c1*sin(q1) - w1*cos(q1))) - g*m2*(sin(gam)*(c2*cos(q1 - q2) - r + w2*sin(q1 - q2)) - cos(gam)*(c2*sin(q1 - q2) - w2*cos(q1 - q2)));
RHS2 = g*m2*(cos(gam)*(c2*sin(q1 - q2) - w2*cos(q1 - q2)) - sin(gam)*(c2*cos(q1 - q2) + w2*sin(q1 - q2)));
MM = [M11 M12;
M21 M22];
RHS = [RHS1; RHS2];
X = MM \ RHS;
ud1 = X(1);
ud2 = X(2);
axh = -r*ud1;
ayh = 0;
DTE = I1*u1*ud1 + I2*u1*ud1 - I2*u1*ud2 - I2*u2*ud1 + I2*u2*ud2 + c1^2*m1*u1*ud1 + c2^2*m2*u1*ud1 - c2^2*m2*u1*ud2 - c2^2*m2*u2*ud1 + c2^2*m2*u2*ud2 + m1*r^2*u1*ud1 + m2*r^2*u1*ud1 + m1*u1*ud1*w1^2 + m2*u1*ud1*w2^2 - m2*u1*ud2*w2^2 - m2*u2*ud1*w2^2 + m2*u2*ud2*w2^2 - c1*g*m1*u1*sin(gam - q1) - g*m1*u1*w1*cos(gam - q1) + c1*m1*r*u1^3*sin(q1) - m1*r*u1^3*w1*cos(q1) - c2*g*m2*u1*sin(gam - q1 + q2) + c2*g*m2*u2*sin(gam - q1 + q2) + c2*m2*r*u1^3*sin(q1 - q2) - g*m2*u1*w2*cos(gam - q1 + q2) + g*m2*u2*w2*cos(gam - q1 + q2) - m2*r*u1^3*w2*cos(q1 - q2) + g*m1*r*u1*sin(gam) + g*m2*r*u1*sin(gam) - m2*r*u1*u2^2*w2*cos(q1 - q2) + 2*m2*r*u1^2*u2*w2*cos(q1 - q2) - 2*c1*m1*r*u1*ud1*cos(q1) - 2*m1*r*u1*ud1*w1*sin(q1) - 2*c2*m2*r*u1*ud1*cos(q1 - q2) + c2*m2*r*u1*ud2*cos(q1 - q2) + c2*m2*r*u2*ud1*cos(q1 - q2) - 2*m2*r*u1*ud1*w2*sin(q1 - q2) + m2*r*u1*ud2*w2*sin(q1 - q2) + m2*r*u2*ud1*w2*sin(q1 - q2) + c2*m2*r*u1*u2^2*sin(q1 - q2) - 2*c2*m2*r*u1^2*u2*sin(q1 - q2);
zdot = [u1 ud1 u2 ud2 ...
vxh axh vyh ayh DTE]';
%===================================================================
function TE = total_energy1_front_stance(t,z,robot) %PAB
%===================================================================
m1 = robot.m1; c1 = robot.c1; w1 = robot.w1;
m2 = robot.m2; c2 = robot.c2; w2 = robot.w2;
r = robot.r; g = robot.g; gam = robot.gam;
I1 = robot.I1; I2 = robot.I2;
for i=1:length(t)
q1 = z(i,1); u1 = z(i,2);
q2 = z(i,3); %u2 = z(i,4);
TE(i,1) = (I1*u1^2)/2 + (I2*u1^2)/2 + (m2*(u1^2*(c2*sin(q1 - q2) - w2*cos(q1 - q2))^2 + (r*u1 - u1*(c2*cos(q1 - q2) + w2*sin(q1 - q2)))^2))/2 + (m1*(u1^2*(c1*sin(q1) - w1*cos(q1))^2 + (r*u1 - u1*(c1*cos(q1) + w1*sin(q1)))^2))/2 - g*m2*(cos(gam)*(c2*cos(q1 - q2) - r + w2*sin(q1 - q2)) - sin(gam)*(q1*r - c2*sin(q1 - q2) + w2*cos(q1 - q2))) + g*m1*(sin(gam)*(q1*r - c1*sin(q1) + w1*cos(q1)) - cos(gam)*(c1*cos(q1) - r + w1*sin(q1)));
end
%===================================================================
function TE = total_energy2_front_stance(t,z,robot) %PAB (DONE)
%===================================================================
m1 = robot.m1; c1 = robot.c1; w1 = robot.w1;
m2 = robot.m2; c2 = robot.c2; w2 = robot.w2;
r = robot.r; g = robot.g; gam = robot.gam;
I1 = robot.I1; I2 = robot.I2;
for i=1:length(t)
q1 = z(i,1); u1 = z(i,2);
q2 = z(i,3); u2 = z(i,4);
TE(i,1) = (m2*(((c2*cos(q1 - q2) + w2*sin(q1 - q2))*(u1 - u2) - r*u1)^2 + (c2*sin(q1 - q2) - w2*cos(q1 - q2))^2*(u1 - u2)^2))/2 + (I1*u1^2)/2 + (I2*(u1 - u2)^2)/2 + (m1*(u1^2*(c1*sin(q1) - w1*cos(q1))^2 + (r*u1 - u1*(c1*cos(q1) + w1*sin(q1)))^2))/2 - g*m2*(cos(gam)*(c2*cos(q1 - q2) - r + w2*sin(q1 - q2)) - sin(gam)*(q1*r - c2*sin(q1 - q2) + w2*cos(q1 - q2))) + g*m1*(sin(gam)*(q1*r - c1*sin(q1) + w1*cos(q1)) - cos(gam)*(c1*cos(q1) - r + w1*sin(q1)));
end
%===================================================================
function [TE,KE_m1,KE_m2] = total_energy2_rear_stance(t,z,robot) %PAB (DONE)
%===================================================================
m1 = robot.m1; c1 = robot.c1; w1 = robot.w1;
m2 = robot.m2; c2 = robot.c2; w2 = robot.w2;
r = robot.r; g = robot.g; gam = robot.gam;
I1 = robot.I1; I2 = robot.I2;
for i=1:length(t)
q1 = z(i,1); u1 = z(i,2);
q2 = z(i,3); u2 = z(i,4);
TE(i,1) = (m1*(((c1*cos(q1 - q2) + w1*sin(q1 - q2))*(u1 - u2) - r*u1)^2 + (c1*sin(q1 - q2) - w1*cos(q1 - q2))^2*(u1 - u2)^2))/2 + (I2*u1^2)/2 + (I1*(u1 - u2)^2)/2 + (m2*(u1^2*(c2*sin(q1) - w2*cos(q1))^2 + (r*u1 - u1*(c2*cos(q1) + w2*sin(q1)))^2))/2 - g*m1*(cos(gam)*(c1*cos(q1 - q2) - r + w1*sin(q1 - q2)) - sin(gam)*(q1*r - c1*sin(q1 - q2) + w1*cos(q1 - q2))) + g*m2*(sin(gam)*(q1*r - c2*sin(q1) + w2*cos(q1)) - cos(gam)*(c2*cos(q1) - r + w2*sin(q1)));
end
KE_m2(i,1) = (I2*u1^2)/2 + (m2*(u1^2*(c2*sin(q1) - w2*cos(q1))^2 + (r*u1 - u1*(c2*cos(q1) + w2*sin(q1)))^2))/2;
KE_m1(i,1) = (I1*(u1 - u2)^2)/2 + (m1*(((c1*cos(q1 - q2) + w1*sin(q1 - q2))*(u1 - u2) - r*u1)^2 + (c1*sin(q1 - q2) - w1*cos(q1 - q2))^2*(u1 - u2)^2))/2;
%===================================================================
function [q1,u1,q2,u2] = front_strike(r1,v1,r2,v2,robot) %PAB (DONE)
%===================================================================
m1 = robot.m1; c1 = robot.c1; w1 = robot.w1;
m2 = robot.m2; c2 = robot.c2; w2 = robot.w2;
r = robot.r;
I1 = robot.I1; I2 = robot.I2;
%g = robot.g; gam = robot.gam;
q1 = r1;
q2 = r2;
u2 = 0;
Mh11 = I1 + I2 + c1^2*m1 + c2^2*m2 + m1*r^2 + m2*r^2 + m1*w1^2 + m2*w2^2 - 2*c2*m2*r*cos(q1) - 2*m2*r*w2*sin(q1) - 2*c1*m1*r*cos(q1 - q2) - 2*m1*r*w1*sin(q1 - q2);
RHSh1 = I2*v1 + I1*(v1 - v2) + m1*(((c1*cos(r1 - r2) + w1*sin(r1 - r2))*(v1 - v2) - r*v1)*(c1*cos(r1 - r2) - r + w1*sin(r1 - r2)) + (c1*sin(r1 - r2) - w1*cos(r1 - r2))^2*(v1 - v2)) + m2*(v1*(c2*cos(r1) - r + w2*sin(r1))^2 + v1*(c2*sin(r1) - w2*cos(r1))^2);
u1 = Mh11 \ RHSh1;
%===================================================================
function [q1,u1,q2,u2] = rear_strike(r1,v1,r2,v2,robot) %PAB (DONE)
%===================================================================
m1 = robot.m1; c1 = robot.c1; w1 = robot.w1;
m2 = robot.m2; c2 = robot.c2; w2 = robot.w2;
r = robot.r;
I1 = robot.I1; I2 = robot.I2;
q1 = r1;
q2 = r2;
u2 = 0;
Mh11 = I1 + I2 + c1^2*m1 + c2^2*m2 + m1*r^2 + m2*r^2 + m1*w1^2 + m2*w2^2 - 2*c1*m1*r*cos(q1) - 2*m1*r*w1*sin(q1) - 2*c2*m2*r*cos(q1 - q2) - 2*m2*r*w2*sin(q1 - q2);
RHSh1 = I1*v1 + I2*(v1 - v2) + m2*(((c2*cos(r1 - r2) + w2*sin(r1 - r2))*(v1 - v2) - r*v1)*(c2*cos(r1 - r2) - r + w2*sin(r1 - r2)) + (c2*sin(r1 - r2) - w2*cos(r1 - r2))^2*(v1 - v2)) + m1*(v1*(c1*cos(r1) - r + w1*sin(r1))^2 + v1*(c1*sin(r1) - w1*cos(r1))^2);
u1 = Mh11 \ RHSh1;