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Furuta (rotary) Pendulum

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Equations of motion for a pendulum mounted on rotating base are compared to SimMechanics.

The equations of motion for a rotary pendulum (furuta pendulum) is

The equations of motion for a rotary pendulum (furuta pendulum) is

tested by ode45 against a SimMechanics simulation. Actuation is in torque at the base. Equations are the full non-linear version but neglect friction and lateral inertia.

See Benjamin Seth Cazzolato and Zebb Prime, "On the Dynamics of the Furuta Pendulum,"Journal of Control Science and Engineering, Volume 2011, Article ID 528341, 8 pages, doi:10.1155/2011/528341

Created by Alan Jennings, 2012, alanjenningsohio (at) hotmail dot com

Contents

Set parameters

m1=0.2;%kg
m2=0.7; %kg
L1=1.4;%m
L2=0.5;%m
g=9.81; %m/sec^2
%create base input to excite the system, must be the same as in Simulink
%for perfict comparison
% u=@(x,t) 10.00*(t<10.01);
u_base=@(x,t) 5*sin(2*pi*t)+10; %N*m,
% x is state vector [angle(base, pen), vel(base, pen)];
%note that the Simulink has an delay for state feedback

Mod2pi=@(a) atan2(sin(a),cos(a));

Pull output from Simulink if results are in the workspace

if ~isempty(whos('simout'))
% output of Simulink is simout
t_sim=simout(:,1);
Theta1_sim   =simout(:,2); %base
Theta1d_sim  =simout(:,3);
Theta1dd_sim =simout(:,4);
Theta2_sim   =simout(:,5); %pendulum
Theta2d_sim  =simout(:,6);
Theta2dd_sim =simout(:,7);
U_sim        =simout(:,8);
Flag_Pen.SimResults=true;
x_o=simout(1,[2,5,3,6]);
else
    fprintf('Run RotPen_Test_Submitt.mdl to get SimMechanics results for comparison\n');
    Flag_Pen.SimResults=false;
    t_sim=linspace(0,1,100);
    x_o=[0,0,0,1].';
end

Set up and use ode45 to find response

MassMatrixInv=@(a2) [(m1*L1^2/3+m2*L1^2+4/3*m2*L2^2*sin(a2)^2), ...
    (-m2*L1*L2*cos(a2)); (-m2*L1*L2*cos(a2)), 4/3*m2*L2^2]^-1;
Dynamics=@(t,x) [x(3);x(4);(...
    MassMatrixInv(x(2))*([u_base(x,t)+...
    -x(3)*x(4)*4/3*m2*L2^2*(2*sin(x(2))*cos(x(2)))-x(4)^2*m2*L1*L2*sin(x(2));...
    +x(3)^2*4/3*m2*L2^2*sin(x(2))*cos(x(2))+m2*g*L2*sin(x(2))]))];

[t,Theta]=ode45(Dynamics,t_sim,x_o);

% Recalculate derivatives and control for comparison
Theta_dot=zeros(size(Theta));
U=zeros(size(t));
for angie=1:length(t)
    Theta_dot(angie,:)=Dynamics(t(angie),Theta(angie,:)).';
    U(angie)=u_base(Theta(angie,:),t(angie));
end

figure(50);clf;
subplot(2,1,1);
plot(t,atan2(sin(Theta(:,1:2)),cos(Theta(:,1:2)))*180/pi);
xlabel('t (sec)');ylabel('Angle (deg)');title('ODE45 Results');
legend('\theta_b_a_s_e','\theta_p_e_n');
subplot(2,1,2);
plot(t,Theta(:,3:4)*180/pi)
xlabel('t (sec)');ylabel('Angular Velocity (deg/sec)');

Compare results to Simulink if available

if Flag_Pen.SimResults
figure(51);clf;
plot(t,U,'b-',t_sim,U_sim,'g--');
xlabel('t (sec)');ylabel('Input (Nm)');title('Control Comparison');
legend('ode','SimMech')
%if control doesn't match, the states won't

figure(52);clf;
subplot(3,1,1);
plot(t,Mod2pi(Theta(:,1))*180/pi,'b-', ...
    t_sim,Theta1_sim(:)*180/pi,'g--')
ylabel('Angle (deg)');title('\theta_b_a_s_e')
legend('ode','SimMech')
subplot(3,1,2);
plot(t,Theta(:,3)*180/pi,'b-',...% t,Theta_dot(:,1)*180/pi,'c-x', ...
    t_sim,Theta1d_sim(:)*180/pi,'g-')
ylabel('Velocity (deg/sec)');
subplot(3,1,3);
plot(t,Theta_dot(:,3)*180/pi,'b-', ...
    t_sim,Theta1dd_sim(:)*180/pi,'g-')
xlabel('t (sec)');ylabel('Accel (deg/sec^2)');

figure(53);clf;
subplot(3,1,1);
plot(t,Mod2pi(Theta(:,2))*180/pi,'b-', ...
    t_sim,Theta2_sim(:)*180/pi,'g--')
ylabel('Angle (deg)');title('\theta_p_e_n')
legend('ode','SimMech')
subplot(3,1,2);
plot(t,Theta(:,4)*180/pi,'b-',...% t,Theta_dot(:,2)*180/pi,'c-x', ...
    t_sim,Theta2d_sim(:)*180/pi,'g-')
ylabel('Velocity (deg/sec)');
subplot(3,1,3);
plot(t,Theta_dot(:,4)*180/pi,'b-', ...
    t_sim,Theta2dd_sim(:)*180/pi,'g-')
xlabel('t (sec)');ylabel('Accel (deg/sec^2)');
end

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