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Disturbance observer for 2DOF robot manipulator

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HD
HD 2024-7-9
移动：Sam Chak 2024-7-18
Hello, I want to design a disturbance observer (DOB) for a 2-degree-of-freedom (2-DOF) robotic manipulator. I'm using MATLAB for simulation and implementation. Could anyone provide guidance or share a detailed example on how to model the 2-DOF robotic manipulator and implement the disturbance observer in MATLAB? Specifically, how to design and implement a DOB for this system and tips on parameter tuning and simulation in MATLAB. Also, can somone tell me if this equations for 2DOF arm are correct:
, where , , ,
From which it follows:
Any help or example code would be greatly appreciated! Thanks.
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Sam Chak
Sam Chak 2024-7-10
What is the Disturbance Observer for?
The compact equations you posted are typical 2nd-order ODEs that can be used to describe the motion of many mechanical systems. But the symbols alone cannot describe the 2-DOF Robotic Manipulator.
If the MATLAB codes are available, would you know how to design the Disturbance Observer?
Keep in mind that the Disturbance Observer alone cannot control the motion of the Robotic Manipulator.
HD
HD 2024-7-10
Thanks for your answers. I would like to set the desired path of the manipulator's tip using a disturbance observer by applying direct kinematics.

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Francisco J. Triveno Vargas
移动：Sam Chak 2024-7-18
Ran in:
@HD, inititally you can use this code:
clc
clear
close all
% Time---------------------------------------------------------------------
tf=10;
N=2000*tf;
Nt=tf/N;
t=0:tf/N:tf;
% DoF----------------------------------------------------------------------
n=2;
% Start and end points-----------------------------------------------------
xi=0;
yi=1.73;
xf=1.5;
yf=0;
% Parameters---------------------------------------------------------------
a1=1;
a2=1;
ac1=a1/2;
ac2=a2/2;
m1=2;
m2=2;
mp=0.25;
g0=9.81;
Izz1=1/12*m1*a1^2;
Izz2=1/12*m2*a2^2;
% Inverse kinematics-------------------------------------------------------
xin2=real(acos((xi^2+yi^2-a1^2-a2^2)/(2*a1*a2)));
xin1=real(atan2(yi,xi))-real(atan2(a2*sin(xin2),(a1+a2*cos(xin2))));
xin3=0;
xin4=0;
xdes2=real(acos((xf^2+yf^2-a1^2-a2^2)/(2*a1*a2)));
xdes1=real(atan2(yf,xf))-real(atan2(a2*sin(xdes2),(a1+a2*cos(xdes2))));
xdes3=0;
xdes4=0;
x_0=[xin1;xin2;xin3;xin4];
x_des=[xdes1;xdes2;xdes3;xdes4];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Solution-----------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i=1:1:length(t)
Time=t(i);
x(:,1)=x_0;
% States---------------------------------------------------------------
q1=x(1,i);
q2=x(2,i);
dq1=x(3,i);
dq2=x(4,i);
q=[q1;q2];
dq=[dq1;dq2];
% Dynamics-------------------------------------------------------------
M=[Izz1+Izz2+a1^2*m2+ac1^2*m1+ac2^2*m2+a1^2*mp+a2^2*mp+2*a1*ac2*m2*cos(q2)+2*a1*a2*mp*cos(q2),mp*a2^2+a1*mp*cos(q2)*a2+m2*ac2^2+a1*m2*cos(q2)*ac2+Izz2;...
mp*a2^2+a1*mp*cos(q2)*a2+m2*ac2^2+a1*m2*cos(q2)*ac2+Izz2,mp*a2^2+m2*ac2^2+Izz2];
g=[g0*m2*(ac2*cos(q1+q2)+a1*cos(q1))+g0*mp*(a2*cos(q1+q2)+a1*cos(q1))+ac1*g0*m1*cos(q1);...
g0*cos(q1+q2)*(ac2*m2+a2*mp)];
C=[-a1*dq2*sin(q2)*(ac2*m2+a2*mp),-a1*sin(q2)*(dq1+dq2)*(ac2*m2+a2*mp);...
a1*dq1*sin(q2)*(ac2*m2+a2*mp),0];
Df=diag([0.1,0.1]);
% Input----------------------------------------------------------------
Kp=5*eye(2);
Kd=10*eye(2);
u(:,i)=-Kp*(q-x_des(1:2))-Kd*(dq-x_des(3:4))+g;
% System---------------------------------------------------------------
f=[dq;inv(M)*(u(:,i)-C*dq-g-Df*dq)];
x(:,i+1)=x(:,i)+f*Nt;
% Error----------------------------------------------------------------
xe2=a1*cos(q1)+a2*cos(q1+q2);
ye2=a1*sin(q1)+a2*sin(q1+q2);
error(i)=sqrt((xe2-xf)^2+(ye2-yf)^2);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Plots--------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
End_Effector_error_m=error(length(t))
End_Effector_error_m = 0.0041
figure(1)
hold all;grid on;box on
plot(t,x(1,1:length(t)),'b','LineWidth',1.5)
line([0,tf],[x_des(1),x_des(1)],'LineStyle','--','Color',[1 0 0])
xlabel('t[s]')
ylabel('q_1(t)[rad]')
figure(2)
hold all;grid on;box on
plot(t,x(2,1:length(t)),'b','LineWidth',1.5)
line([0,tf],[x_des(2),x_des(2)],'LineStyle','--','Color',[1 0 0])
xlabel('t[s]')
ylabel('q_2(t)[rad]')
figure(3)
hold all;grid on;box on
plot(t,x(3,1:length(t)),'b','LineWidth',1.5)
line([0,tf],[x_des(3),x_des(3)],'LineStyle','--','Color',[1 0 0])
xlabel('t[s]')
ylabel('dq_1(t)/dt[rad/s]')
figure(4)
hold all;grid on;box on
plot(t,x(4,1:length(t)),'b','LineWidth',1.5)
line([0,tf],[x_des(4),x_des(4)],'LineStyle','--','Color',[1 0 0])
xlabel('t[s]')
ylabel('dq_2(t)/dt[rad/s]')
figure(5)
hold all;grid on;box on
plot(t,u(1,1:length(t)),'b','LineWidth',1.5)
xlabel('t[s]')
ylabel('u_1(t)')
figure(6)
hold all;grid on;box on
plot(t,u(2,1:length(t)),'b','LineWidth',1.5)
xlabel('t[s]')
ylabel('u_2(t)')
q1=x(1,:);
q2=x(2,:);
xe1=a1*cos(q1);
ye1=a1*sin(q1);
xe2=a1*cos(q1)+a2*cos(q1+q2);
ye2=a1*sin(q1)+a2*sin(q1+q2);
figure(7)
hold all
plot(xi,yi,'*r')
plot(xf,yf,'om')
plot(xe2,ye2,'-k','LineWidth',2)
for i=1:round(N/50):N
line([0;xe1(i)],[0;ye1(i)],'Color',[0 0 1])
line([xe1(i);xe2(i)],[ye1(i);ye2(i)],'Color',[1 0 0])
end
xlabel('x(t)[m]')
ylabel('y(t)[m]')
axis equal
box on
grid on
legend('start point','end point','trajectory')
figure(8)
hold all;grid on;box on
plot(t,error(1:length(t)),'b','LineWidth',1.5)
xlabel('t[s]')
ylabel('|error(t)|')
extracted from https://www.mathworks.com/matlabcentral/fileexchange/166431-pd-gravity-control-2dof-planar-robotic-manipulator
For the observer you need to linearize the nonlinear model of manipulador and use it.
Regards

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