Flexible Link
Contents
1. Figure 5 Flexible Link Controller Page 7 Revision 01 Figure 5 above depicts the Rotary Flexible Link controller we have developed for this experiment Notice that both the actual system and a simulation are running in parallel thus allowing us to compare the actual and simulated results Take note of the four slider gains in the controller These are the gains that you will be Tuning during this experiment The simulation running in parallel with the controller is of the simplified linear model The controller gains that you will be using as the base of this experiment were calculated based on this simple model Using this technique the higher order vibrations were not anticipated and that is the reason why it is necessary for the user to tune the gains in order to dampen the vibrations scope q_SR 02_Flexible_Link Measured Alpha deg File Edit Update Axis Window Background Colour E E PEN My oi diui yuyun Time Figure 6 Higher Order Vibrations Figure 6 Above clearly displays the vibrations in the link Alpha due to the higher order modes being excited The main goal of this laboratory would be to sufficiently dampen these unwanted vibrations You should first begin by preparing the following table Slider Sliderz Slider Slider Rise time Alpha Vibrations Overshoot Alp2. 2 PH KE yt KE sy FF og FS ring O46 3 7 Forming the Lagrangian PP ee ee 04 6 1K a 3 8 2 4 2 Link 2 Stiff Our 2 generalized co ordinates are and a We therefore have 2 equations 6 6L L Tr B 6 3 9 6t 50 60 am a Of OL L 0 3 10 3 Q Page 5 Revision 01 Solving Equations 3 9 amp 3 10 we are left with J eg 9 FF srm O 6 T 0 3 11 output Ba J aml H K suy a 0 3 12 Referring back to Experiment 1 Position Control we know that the output Torque on the load from the motor is yn K K V K K 9 output R m 3 13 Finally by combining equations 3 11 3 12 amp 3 13 we are left with the following state space representation of the complete system 0 1 0 0 a 0 0 0 lil 0 a 0 K sug TANK KK B Rm aX gl hal 3 y Ja J Rn J Ran amp 0 Kak sy n n K KK B Rm NaN K K Ja J arm Ja Ry J 4 Ry 3 2 Pre Lab Assignment The following laboratory was designed to give the student an insight into the various gains in a state feedback system The student is initially provided with a gain vector k that will place the link end point at a desired location but was intentionally designed to also introduce higher modes of vibrations in the flexible link The purpose of the lab is to have the students vary each gain about its initial point in an attempt to maintain the system response but dampen the higher mode v
3. you should begin by documenting your work into a lab report Included in this report should be the following i In the Pre Lab section you were asked to qualitatively describe how the system states would be effected by varying each gain Include your brief qualitative response ii In the beginning of the lab you were asked to vary the four slider gains in 5 steps forming your Iteration Table as continued from Table 1 Make sure to include your Iteration Table in this report iii After completion of the Iteration Table you were asked to begin tuning each gain to achieve the system requirements Make sure to include another table that demonstrates the steps you took in achieving the final slider values iv Once you have completely tuned the state gains be sure to include your plots of gamma and alpha These plots should look similar to Figures 7 amp 8 5 1 Post Lab Questions 1 After completing your Iteration Table you should now be fairly familiar with how the each state feedback gain effects the overall performance of the system Comment on your observations made by varying the 4 gains Did your observations agree with what you had theorized in the pre lab 2 In what ways could you improve upon the performance of the controller Is it possible to control the higher mode vibrations Explain 3 The initial gains were calculated using the LQR technique on a simple linear model of the plant What other control methods would yo
4. running the file by the name Setup _SRV0O2_Exp5 m This MATLAB script file will setup all the specific system parameters and will set the system state space matrices A B C amp D You are now ready to begin the laboratory The MATLAB LQR function returns a set of calculated gains based on the system matrices A amp B and the design matrices Q amp R The initial gains that you have been given They should be displayed in the MATLAB window were calculated using the LQR method Under the same directory open a Simulink model called q_SRV0O2_Flexible_Link mdl This model has the I O connection blocks linking to the physical plant as well as a simulated block to compare real and simulated results SRVO2 Experiment 5 FLEXGAGE Rotary Flexible Link State Feedback Controller LQR Experiment vs Simulation Run Setup_SRV02_Exp5 R2D al us lic 1 gt sie gt Slider 1 aot cai So 30 2R ska 1 gt i pi H Square Setpoint Degrees to Slider 2 Wave Amplitude Radians 3 1 l Measured deg Gama deg Slider 4 1 gt j Slider 4 Measured Theta deg E FLEXGAGE 12D gt Simulated Theta R2D al i Simulated Alpha m F Matrix P al Gain Rotary Mux p Flexible Link Simulated Gama Model
5. SRV02 Series Rotary Experiment 5 Innovate EOUEATE Flexible Link Student Handout SRV02 Series Q Rotary Experiment 5 SLO EEE EEEE Flexible Link Student Handout 1 Objectives The objective in this experiment is to tune a state feedback controller for the rotary flexible link module The final controller should allow the user to command a desired tip angle position The controller should eliminate the link s vibrations while maintaining a fast response Upon completion of the exercise you should have experience in the following How to mathematically model the Flexible Link system To linearize the model about an operating point To dampen the arm vibrations by tuning the controller To design and simulate a WinCon controller for the system How to design an optimal LQR controller 2 System Requirements To complete this Lab the following hardware is required 1 Quanser UPM 2405 1503 Power Module or equivalent 1 Quanser MultiQ PCI MQ3 or equivalent 1 Quanser SRV02 E T servo plant 1 Quanser FLEXGAGE Rotary Flexible Link Module 1 PC equipped with the required software as stated in the WinCon user manual The required configuration of this experiment is the SRV02 E T in the High Gear configuration with a UPM 2405 power module and a suggested gain cable of 1 It is assumed that the student has successfully completed Experiment 0 of the SRV02 and is familiar in using WinCon to con
6. ha Range Gamma 0 5 1 1 1 Slower More 0 3 5 lt a lt 3 5 0 75 1 1 1 Slower Unchanged 0 6 lt a lt 6 1 1 1 1 Same Unchanged 0 8 lt a lt 8 1 5 1 1 1 Faster Less 15 00 10 lt a lt 10 2 1 1 1 Faster Less 35 00 12 lt a lt 12 Table 1 Iteration Table Sample Entries Page 8 Revision 01 Table 1 shows the 5 iterations of Slider You will be responsible to complete the above table with 5 iterations for each of the other sliders For each iteration make sure to set all other sliders to their default value of 1 For the Rise Time and Alpha Vibrations columns only a quantitative observation will suffice Be sure to compare each observation to the default set where each slider is set to 1 The goal of this excerise is to notice the effects of each gain on the individual system parameters You may now proceed to Build the controller through the WinCon menu After the code has compiled start the controller through WinCon and open up two scopes one for alpha and another for gamma WARNING If at any point the system is not behaving as expected make sure to immediately press STOP on the WinCon server You may now begin to vary each of the slider gains in the 5 steps mentioned above Once you have completed the Iteration Table you should have a good understanding of the effects each gain has on the system response and should now proceed to tune each gain Hint You should be
7. ibrations In doing so the student should begin to develop a more intuitive understanding of the various state feedback gains The state feedback law u kx is implemented in this laboratory The controller is initialize with 4 gains k4 k2 Ks k4 and the student is to vary each parameter in 5 steps 0 5k 0 75k k 1 5k 2k For the pre lab what effect on the 2 state variables 8 amp a would you expect as you vary each gain Only a qualitative answer is required Page 6 Revision 01 4 In Lab Procedure The rotary flexible Link is an ideal experiment intended to model a flexible Link on a robot or any linkage system This experiment is also useful in the study of vibration analysis and resonance The purpose of the lab is to tune the state feedback gains in order to dampen the higher modes of vibrations while maintaining an acceptable system response The first task upon entering the laboratory is to familiarize yourself with the system The arm deflection signal a should be connected to encoder channel 1 and the servomotor s position signal 8 should be connected to encoder channel 0 Analog Output channel 0 should be connected to the UPM Amplifier and from the amplifier to the input of the servomotor This system has one input Vm and two outputs 8 amp a You are now ready to begin the lab Launch MATLAB from the computer connected to the system Under the SRV02_Exp5_ Flexible Link directory begin by
8. ng in the following equation a 3 1 To get a complete model of a flexible link is beyond the scope of this laboratory In controlling the tip of the link it is sufficient to use a simplified model that will adequately describe the motion of the endpoint Figure 4 below depicts the simplified model Side View Spring torque stiff amp Figure 4 Simple Flexible Link Model From Figure 4 the equation of a rotary spring is J rin TTK sup amp 3 2 Page 4 Revision 01 To get an estimation of the the modeled stiffness Ksur we fixed the clamped end of the link and set an initial condition of alpha and measured the link s damped natural frequency With any given oscillation system we have the following relation amp Wo 3 3 Combining equations 3 2 amp 3 3 we are left with the following expression K O I rin 3 4 Were the Link is modeled as a rod rotating about its endpoint with moment of inertia ML Link 3 3 5 3 1 Deriving The System Dynamic Equations Now that we have developed a simple model for the Link the system dynamic equations can be obtained using the Euler Lagrange formulation We obtain the Potential and Kinetic energies in our system as Potential Energy The only potential energy in the system is in the spring 1 2 V P E ping 5K sug amp 3 6 Kinetic Energy The Kinetic Energies in the system arise from the moving hub and flexible link 1 2 1
9. trol the plant through Simulink It is also assumed that all the sensors and actuators are connected as per dictated in the SRVO2 User Manual and the Rotary Flexible Link User Manual Page 2 Revision 01 3 Mathematical Model Figure 2 below depicts the Flexible Link module coupled to the SRVO2 plant in the correct configuration The Module is attached to the SRV02 load gear by two thumbscrews The Link is firmly attached to the module at its base Figure 1 depicts the strain gage mounted at the base of the link The gage is calibrated to output 1 volt per 1 inch of tip deflection as Figure 1 Strain Gage Figure 2 SRV02 w FlegGage Module The following table is a list of the nomenclature used is the following illustrations and derivations Symbol Description L Length of Flexible Link m Mass of Flexible Link K_Gage Strain Gage Calibration factor 1 Volt Inch 6 Servo load gear angle radians a Arm Deflection radians D Link End point Deflection Arc Length We Link s Damped Natural Frequency Experimentally Calculated Jhink Modeled Link Moment of Inertia Page 3 Revision 01 Jhub Yi Xi Figure 3 Schematic of Flexible Link Module Figure 3 depicts the flexible link at a given deflection resulting in an end point displacement of D The stain gage is calibrated such that it outputs 1 volt per 1 inch deflection of D resulti
10. u have chosen to more accurately control this plant Page 11 Revision 01
11. using your table as a guide to pick the best set of slider settings to achieve the system requirements Your final slider values should result in a response with A fast rise time should be similar to the default No Alpha Vibrations very minimal at the least 0 Overshoot on Gamma Alpha should not exceed 10 As you are tuning the gains to achieve the system requirements make sure to fill out another table similar to your Iteration Table to document the steps taken at achieving your final slider values that best meet the system requirements Make sure to try a variety of combinations as there will be a few different configurations that will meet the requirements The following plots on the next page are of Alpha and Gamma with the Slide gains set to their optimum values in meeting the controller requirements Once you have tuned the gains to their final values your plots should look very similar to Figure 7 amp Figure 8 Page 9 Revision 01 B scope q_SR 02_Flexible_Link Measured Alpha deg File Edit Update Axis window Time Figure 7 Alpha Plot Notice the Reduced Vibrations M scope q_SR 02_Flexible_Link Measured Gama deg File Edit Update Axis Window Background Colour Text Colour Text Font Time Figure 8 Gamma Plot Notice the Fast Response without Overshoot Page 10 Revision 01 5 Post Lab Questions and Report Upon completion of the lab
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