Control Theory Project Writeup

7/30/2019 Control Theory Project Writeup

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Control Theory - Project

Modeling and Control of 1 DOF Robot with

Flexible Link

Submitted on August 13, 2010

Barbar Moawad

102713284Brendan Dills

102724578

Juan Palacio

102352351

Lee Mitch Tome

102719640


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Given

Figure 1 shows an electro-mechanical system of a Single Joint robot model with flexible link.

Figure 1: Single Joint Robot with Flexible Link

It can be seen in the figure above that the dynamics of the robot are controlled by the torqueoutput of the armature-controlled DC motor. The mechanical component of this system is

characterized by a gear system that connects the driving shaft to the link.

Graph 1 shows the torque-speed curve of the systems DC motor.

Graph 1: Torque-Speed Curve of the Motor

It can be understood from the graph above that when the motor reaches its stall torque, it stopsspinning. It also reaches its maximum rotational speed at no-load condition.


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The following list of nomenclature will be needed in order to understand the different

annotations for different components of this project:

Ja [kgm2 ] - Armature Inertia

Da [radNms ] - Armature Damping Coefficient

Ra [ohm] - Armature ResistanceLa [sohm ] - Armature Inductance

aI[A ] - Armature Current

Va [V] - Armature VoltageTstall[Nm] - Stall Torque

noload[srad] - No-load angular velocity

JL [kgm2 ] - Load Inertia

DL [radNms ] - Load Damping Coefficient

Nm - Number of teeth of the input gear (motor gear)NL - Number of teeth of the output gear (load gear)

kL[mN] - Spring Coefficient

Table 1 shows the useful given values:

Table 1: Useful Values

The purpose of this project is: To design a controller that will monitor the robot armsdynamics. Fine tuning was made to obtain the desired output.


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The same equation applies to the link shaft except for a small variation:

()

()

In order to get rid of any confusion, link components will have the transcript m.

Part 3:The transfer function for the output shaft is the following:

The transfer function for the link shaft is as shown below:

Part 4:

Figure 2 is the block diagram for the unity feedback control system:

Figure 2: Block Diagram of the System

Part 5:

km could be found at the stall torque:

which becomes:


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a0 = den(1);a1 = den(2);a2 = den(3);a3 = kp + den(4);b1 =(a1*a2 - a0*a3)/a1;kcr = double(solve(b1));% kcr = 1.0046e+004i=sqrt(-1);eq1 = a0*(i*w)^3+a1*(i*w)^2+a2*i*w+a3+kcr;w_found = subs(subs(solve(eq1),kp,kcr),kp,kcr);w_cr = abs(w_found(1));% w_cr = 27.5503Pcr = 2*pi/w_cr;% Pcr = 0.2281

Part 8:%% Question 8%% P controllerkp1 = 0.5*kcr;% kp1 = 5.023028647379596e+03syscl_P = feedback(kp1*sys,1);step(syscl_P);S_P = stepinfo(syscl_P,'RiseTimeLimits',[0.1 0.9])

%{S_P =

RiseTime: NaNSettlingTime: NaNSettlingMin: NaNSettlingMax: NaN

Overshoot: NaNUndershoot: NaN

Peak: InfPeakTime: Inf

%}


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Graph 3: Step Response from the PI-Controller

%% PID controller

kp3 = 0.6*kcr;

% kp3 = 6.027634376855515e+03Ti3 = 0.5*Pcr;Td3 = 0.125*Pcr;ki3 = kp3/Ti3;% ki3 = 5.285953900110588e+04kd3 = kp3*Td3;% kd3 = 1.718345300944181e+02control_PID = tf([kd3 kp3 ki3], [1 0]);syscl_PID = feedback(control_PID*sys,1);step(syscl_PID);S_PID = stepinfo(syscl_PID,'RiseTimeLimits',[0.1 0.9])

%{

S_PID =

RiseTime: 0.0270SettlingTime: 0.4346SettlingMin: 0.7904SettlingMax: 1.4796

Overshoot: 47.9589Undershoot: 0

Peak: 1.4796PeakTime: 0.0775


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%}

Graph 4: Step Response from the PID-Controller

Part 9:%% Question 9 (Tuned PID controller)

kp_t = 186;ki_t = 40;kd_t = 200;control_t = tf([kd_t kp_t ki_t],[1 0]);syscl_t = feedback(control_t*sys,1)step(syscl_t)S_t = stepinfo(syscl_t,'RiseTimeLimits',[0.1 0.9])

%{S_t =

RiseTime: 0.0586SettlingTime: 0.1052SettlingMin: 0.9047SettlingMax: 1.0000

Overshoot: 0.0024Undershoot: 0

Peak: 1.0000PeakTime: 0.2253

%}


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Graph 5: Step Response from the Tuned PID-Controller

Part 10:

%% Question 10

for i=0:1:20kl = i;k=kl;num=km;den=[Jm*La La*Dm+Ra*Jm La*k+Ra*Dm+km*kb Ra*k];sys=tf(num,den);[wn,Z,P] = damp(sys);eival = imag(P(1));eivaln = imag(P(2));plot(k,eival,'bx',k,eivaln,'rx')title('Immaginary roots as the value of kl is increased')

xlabel('Value of kl')ylabel('Immaginary value')hold on

endhold off


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Graph 6: Systems Roots for 0


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There are no imaginary roots up around a 2.5 value of kL, from there and on one of the roots

increases while the other decreases in a symmetric fashion along the horizontal axis as shown in

graph 7.

Conclusion

This project proved how an electro-mechanical problem link can be controlled using computer

programming. The process was done by first finding all the useful characteristics, thendeveloping the equations of motion and the transfer functions. The transfer functions were then

used in a Matlab code in which P, PI and PID controllers were developed.

It could be seen from the three graphs that the P and PI controllers were very unstable,

witnessing an exponential increase in oscillation. The PID controller, on the other hand, showed

a greater stability and followed a desired pattern, but fine tuning was needed to make

improvements in the response. The main purpose of the tuning was to only reduce the overshoot

since the rise time was satisfying in the pre-tuned controller. The relation between kL and theEigen values was found using a for-loop that yielded two comprehensive graphs. These graphs

showed that as kL increased the Eigen values become imaginary.

As a result the steps above reflected what controlling a robotic link using Control Theory and its

tools (Matlab and Simulink) looks like.

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