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two_link_distributed_impedance_nonrigid.m
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close all
clear
%% robot modeling
lx = 1; lr = 0.1; g = [0 -9.81 0]; fvis = 0; fcou = 0;
rod = Cuboid([lx,lr,lr]);
Irod = rod.inertia;
dpm = {'a', lx, 'm', rod.mass, 'r', [-lx/2,0,0], 'qlim', [-pi/2, pi/2],'I', Irod,...
'B', fvis, 'Tc', [fcou -fcou]};
rob(1) = SerialLink([Revolute(dpm{:}),Revolute(dpm{:})],'name','r1','gravity',...
-g,'base',SE3([-1,0,0]/2));
rob(2) = SerialLink([Revolute(dpm{:}),Revolute(dpm{:})],'name','r2','gravity',...
-g,'base',SE3([1,0,0]/2));
%% object modeling
obj = Cuboid([lx,lx,lr]);
%% joint positions (x,y)
x0 = [-1,1.5,0;
0,1.5,0];
for i=1:2
q(i,:) = rob(i).ikine(SE3(x0(i,:)),'mask',[1,1,0,0,0,0]);
end
q(1,:) = [sum(q(1,:)),-q(1,2)];
dq = zeros(size(q));
qd = q;
%% object position (x,y,th)
Xo = [sum(x0)/size(x0,1),0,0,0];
dXo = zeros(1,6);
ro = x0-Xo(1:3);
%% visualize systems
ws = [-4 4 -4 4 -4 4];
plotopt = {'workspace', ws, 'nobase', 'notiles', 'noshading', 'noshadow', 'nowrist'};
for i=1:2
hr(i) = rob(i).plot(q(i,:),plotopt{:}); hold on
end
ho = obj.plot(Xo);
view(2);
%% object trajectory
dXd = [0.2,0,0,0,0,0];
Xd = Xo+rand(size(Xo)).*[1,1,0,0,0,1]/10;
kvo = 5; kpo = 5;
ddXo = @(Xo,dXo,Xd,dXd) -kvo*(dXo-dXd)-kpo*(Xo-Xd);
xd = x0;
hd = plot3(xd(:,1),xd(:,2),xd(:,3),'ro');
%% object impedance
Mo = blkdiag(obj.mass*eye(3),obj.inertia);
Co = @(dXo) [-obj.mass*g';skew(dXo(4:6))*obj.inertia*dXo(4:6)'];
r = @(x,Xo,i) x(i,:)-Xo(1:3);
G = @(x,Xo) [eye(3),eye(3);
skew(r(x,Xo,1)),skew(r(x,Xo,2))];
%% virtual linkage
e12 = @(x) (x(2,:)-x(1,:))/norm(x(2,:)-x(1,:));
E = @(x) [-e12(x),e12(x)]';
Kpe = 1;
f12 = @(x) Kpe*(norm(x(2,:)-x(1,:))-norm(x0(2,:)-x0(1,:)));
Fint = @(x) -(E(x)*f12(x))';
%% null-space
% Fint = @(Xo) ((eye(6)-pinv(G(Xo))*G(Xo))*[0.1,0,0,0,0,0]')';
%% command force
Fcmd = @(x,Xo,dXo,Xd) (pinv(G(x,Xo))*(Co(dXo)+Mo*ddXo(Xo,dXo,Xd,dXd)')+Fint(x)')';
%% simulation
dt = 0.05;
T = 5;
k = 0;
for t=0:dt:T
k = k+1;
% update robot
md = 1; kv = 15; kp = 25;
for i=1:2
% end-effector trajectory
J = rob(i).jacob0(q(i,:),dq(i,:)); J(3:6,:)=[];
x(i,:) = rob(i).fkine(q(i,:)).t;
dx(i,:) = J*dq(i,:)';
end
% command and internal force
Fc = Fcmd(x,Xo,dXo,Xd);
Fi = f12(x);
dxd = reshape((G(x,Xo)'*dXo')',[3,2])';
for i=1:2
% desired trajectory
xd(i,:) = x(i,:)+Fc(3*i-2:3*i)/kp;
% joint torque
Mq = rob(i).inertia(q(i,:));
Cq = rob(i).coriolis(q(i,:),dq(i,:));
Gq = rob(i).gravload(q(i,:))';
J = rob(i).jacob0(q(i,:),dq(i,:)); J(3:6,:)=[];
Jdq = rob(i).jacob_dot(q(i,:),dq(i,:)); Jdq(3:6,:)=[];
Jinv = Mq^-1*J'*(J*Mq^-1*J')^-1;
Mx = Jinv'*Mq*Jinv;
Cx1 = Jinv'*Cq*Jinv;
Cx2 = Jinv'*Mq*Jinv*Jdq;
Gx = Jinv'*Gq;
tau(i,:) = J'*(-Mx*md^-1*(kv*(dx(i,1:2)-dxd(i,1:2))+kp*(x(i,1:2)-xd(i,1:2)))'+...
Cx1*dx(i,1:2)'+Cx2+Gx+(eye(2)-Mx*md^-1)*Fc(3*i-2:3*i-1)');
ddq(i,:) = rob(i).accel(q(i,:),dq(i,:),tau(i,:)-Fc(3*i-2:3*i-1)*J);
end
dq = dq+dt*ddq;
q = q+dt*dq;
for i=1:2
rob(i).animate(q(i,:));
end
set(hd,'xdata',x(:,1),'ydata',x(:,2),'zdata',x(:,3));
% update object
ddXo = (Mo^-1*(G(x,Xo)*Fc'-Co(dXo)))';
dXo = dXo+dt*ddXo;
Xo = Xo+dt*dXo;
obj.animate(Xo);
% update trajectory
Xd = Xd+dt*dXd;
drawnow
% save data
t_data(k) = t;
Fcmd_data(k,:) = Fc;
Fint_data(k,:) = Fi;
tau1_data(k,:) = tau(1,:);
tau2_data(k,:) = tau(2,:);
Xo_err(k,:) = Xo-Xd;
x1_err(k,:) = rob(1).fkine(q(1,:)).t'-(SO3.rpy(Xo(4:6))*ro(1,:)')'-Xo(1:3);
x2_err(k,:) = rob(2).fkine(q(2,:)).t'-(SO3.rpy(Xo(4:6))*ro(2,:)')'-Xo(1:3);
end
figure
subplot(2,4,1)
plot(t_data,Fcmd_data(:,1),t_data,Fcmd_data(:,2),t_data,Fcmd_data(:,3));
axis([0 5 -.5 .5])
legend('$F_x$','$F_y$','$F_z$','Interpreter','latex')
title('robot 1 command force')
subplot(2,4,2)
plot(t_data,Fcmd_data(:,4),t_data,Fcmd_data(:,5),t_data,Fcmd_data(:,6));
axis([0 5 -.5 .5])
legend('$F_x$','$F_y$','$F_z$','Interpreter','latex')
title('robot 2 command force')
subplot(2,4,3)
plot(t_data,tau1_data(:,1),t_data,tau1_data(:,2));
axis([0 5 -.5 .5])
legend('$\tau_1$','$\tau_2$','Interpreter','latex')
title('robot 1 torque')
subplot(2,4,4)
plot(t_data,tau2_data(:,1),t_data,tau2_data(:,2));
axis([0 5 -.5 .5])
legend('$\tau_1$','$\tau_2$','Interpreter','latex')
title('robot 2 torque')
subplot(2,4,5)
plot(t_data,Xo_err(:,1),t_data,Xo_err(:,2),t_data,Xo_err(:,6));
axis([0 5 -.5 .5])
legend('$x$','$y$','$\theta$','Interpreter','latex')
title('object position error')
subplot(2,4,6)
plot(t_data,x1_err(:,1),t_data,x1_err(:,2));
axis([0 5 -.02 .02])
legend('$x$','$y$','Interpreter','latex')
title('robot 1 end-effector error')
subplot(2,4,7)
plot(t_data,x2_err(:,1),t_data,x2_err(:,2));
axis([0 5 -.02 .02])
legend('$x$','$y$','Interpreter','latex')
title('robot 2 end-effector error')
subplot(2,4,8)
plot(t_data,Fint_data(:,1));
axis([0 5 -.02 .02])
legend('$f_{12}$','Interpreter','latex')
title('internal force')