Flexible Joint
Contents
1. se ct ein n a na nl Figure 3 Flexible Joint Stationary Figure 4 Flexible Joint Moving Figure 3 above depicts the joint at rest while Figure 4 depicts the joint at a given a Notice that spring 1 has been compressed and spring 2 has been stretched with respect to the joint s stationary state We shall begin by deriving the lengths of each spring in Figure 4 L r Rsinw L Rcos d F L r Rsin a L L Rcosa d y yY 2 2 L VLi iy 3 1 2 2 L VL tL Next we will derive the force acting on each spring F K L L F F K L L F F F Er F F Ly Ix 1 ly 17 1 1 3 2 F F Da F F Ly 2x 27 24 27 2 2 F is the restoring force on each spring The springs supplied come pre loaded meaning they will not stretch before the force F is applied Page 4 Revision 01 Referring back to Figure 4 we can see that F amp F2are both acting on the Arm anchor point The point where both springs are attached to the arm By simple inspection we can see that the x components are opposing each other while the y components are in the same direction leaving us with F gta oe Ix F F F 3 3 y 2y ly We now have 2 forces acting on the anchor point that will bring the arm back to its original position These 2 forces will cause a torque about the joint We know that the torque restoring Moment is equal to the cross product of the radius R and each force resulting in the followin2. Once you determine the best set of parameters that meet the specified requirements you are ready to test your controller on the actual plant Make sure you have properly documented how you obtained your final set of parameters in a table much like the Iteration Table and clearly show how the requirements have been met You may now stop the simulation 4 2 Part Il Implementing the Controller After verifying the calculated controller gains it is time to implement the controllers on the actual system In the same working directory open a Simulink model called q_SRV02_Flexible_Joint m 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 d Theta deg 0 d gt pha deg LU a Measured Gama deg Matrix Gain SRVO02 ROTFLEX Fp Square Setpoint Degrees to Wave Amplitude Radians deg A fo Jo Simulated Gama Matrix Gain Rotary Mux Flexible Joint Figure 6 Flexible Joint Controller Figure 6 above depicts the Rotary Flexible Joint controller we have developed for this experiment Notice that both the actual system and an exact simulation are running in parallel thus allowing us to compare the actual and simulated results Before Building this controller make sure that the state feedback gain k is set according to your final
3. Page 8 Revision 01 Under the same directory open a Simulink model called s_SRV0O2_Flexible_Joint mdl This model is a simulation of the Flexible Joint system with a feedback law u kx The gain vector k is set by the LQR function Before beginning the simulation open a Matlab script file called teration_RotFlex m Keep this script open as you will be executing it repetitively You may save amp run this file by pressing the F5 key Run this script and then proceed to the Simulink model and start the simulation You should have 3 scopes open that are displaying alpha a theta 0 and gamma 6 a Every time the Iteration_RotFlex m script runs you will see the new response in the simulation as well as a step response of the arm tip position 6 a in a MATLAB figure If you right click on this figure and choose Characteristics gt Rise Time the rise time of the step response will be shown Switching back to your MATLAB command window you should see the calculated gain vector k as well as the corresponding closed loop eigenvalues poles E The default Q amp R values are Q diag 10 100 1 1 amp R 1 as you can see in the Iteration script You should notice that the largest value in Q is qz This should be of no surprise as the emphasis of the controller should be primarily on keeping a as close to 0 as possible Therefore the largest element of Q should be the one that is associated with a At this point
4. state feedback controller using LQR The following exercises are meant to illustrate how useful this technique is in suppressing the arm vibrations In the same working directory open the g SRVO2 ROTFLEX_Vibration mdl model Build and run this model This model alternates between the full state feedback controller you just designed and a PD controller with no deflection feedback k 2 amp k 4 are set to O every other period By observing the plant you should notice how well the full state feedback controller effectively eliminates all arm vibrations To further demonstrate the effects of full state feedback on the arm deflection we will send a chirp signal input a sinusoid with increasing frequency to both controllers Figure 9 shows the arm position a when there is no deflection notice the resonant frequency Figure 10 shows the arm position when there is full state feedback Notice how the resonant peak is completely removed and the vibrations are minimal using the full state feedback Both plots were obtained by a variable frequency sine wave with amplitude of 10 Arm Deflection Te T T T 10 n Deflection degrees m 4 4 4 4 4 4 4 1 4 O 4 6 a 10 I2 14 16 18 20 Time s Figure 9 Sine Sweep Response without deflection feedback Arm Deflection T x r Deflection degrees m 4 4 4 4 4 4 4 4 4 O r 4 S 3 10 oe 14 16 15 20 Time s Figure 10 Sine Sweep Response wi
5. you should be varying the components of Q amp R and making note of their effect on the closed loop system characteristics rise time overshoot a range By the end of this exercise you should have a table of the form q q2 q q4 r Rise time ms Overshoot Alpha Range 10 100 1 1 1 771 0 lt a lt 4 100 100 1 1 1 137 9 32 13 lt a lt 13 10 1 0 0 0 1 104 15 2 20 lt a lt 20 Table 1 teration Table Sample Entries Table 1 above shows 3 different entries into the Iteration Table that you should be constructing In total there should be 25 entries in the table vary each parameter in 5 steps while holding the other parameters constant For an idea of what values to simulate refer to the following table for suggested ranges Parameter q q2 q q4 r Range 0 lt q lt 500 0 lt q2 lt 2000 0 lt q lt 10 0 lt qs lt 20 0 1 lt r lt 10 Table 2 Suggested Parameter Ranges As you proceed with your iterations take note of the closed loop eigenvalues poles as you vary each parameter By the end of your 25 iterations you should have a qualitative understanding of how each parameter affects the closed loop poles Now that your Iteration Table is complete look back at the controller requirements of this lab while cross referencing your newly formed table to determine what parameters would result in meeting the specifications Page 9 Revision 01
6. SRV02 Series Rotary Experiment 4 Flexible Joint Student Handout SRV02 Series Q Rotary Experiment 4 cg ene Flexible Joint Student Handout 1 Objectives The objective in this experiment is to design a state feedback controller for the rotary flexible joint module which allows you to command a desired tip angle position The controller should eliminate the arm 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 joint system To linearize the model about an operating point To dampen the arm vibrations with full state feedback 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 SRVO2 Servo plant 1 Quanser ROTFLEX Rotary Flexible Joint Module 1 PC equipped with the required software as stated in the WinCon user manual The required configuration of this experiment is the SRV02 in the High Gear configuration along with a ROTFLEX Rotary Flexible Joint module as well as a UPM 2405 1503 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
7. WinCon to control 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 Joint User Manual Page 2 Revision 01 3 Mathematical Model Figure 1 below depicts the Flexible Joint module coupled to the SRVO2 plant in the correct configuration The Module is attached to the SRVO2 load gear by two thumbscrews The Main Arm is attached to the module body by two identical springs thus resulting in the flexible joint Figure 2 shows the different anchor points on the body and the arm resulting in many configurations of the module Figure 1 Flexible Joint Module Body al Anchor B Points C Main Arm Extra Arm Anchor Points Arm 1 Anchor 2 Points 3 SOF Figure 2 Flexible Joint Illustration The following table is a list of the nomenclature used is the following illustrations and derivations Symbol Description Symbol Description R Distance from joint to arm anchor L1 L2 Lengths of Spring 1 amp 2 d Distance from joint to body anchor Fy F2 Forces on Spring 1 amp 2 r Fixed distance r 3 18 cm k Spring Stiffness 0 Servo load gear angle radians L Unstretched Spring Length a Arm Deflection radians M Restoring Moment Page 3 Revision 01 Arm Anchor Point Body Anchor Point Body Anchor Point Joint Joint
8. ains and any re iterative calculations made if any 5 1 Post Lab Questions 1 After completing your Iteration Table you should now be fairly familiar with how to setup the Q amp R matrices to achieve the optimal controller What intuitive reasoning on choosing the Q amp R matrices have you learnt from the completion of this lab What are the important considerations when using the LQR technique in calculating the full state feedback gains 2 Having an insight into the location of the closed loop poles as you varied your parameters would you have chosen a pole placement technique to control this plant and meet the required specifications Are there any other control strategies that you believe could be used to in this lab 3 In the final section of lab you were shown how the full state feedback control could effectively eliminate arm vibrations When the deflection feedback was turned off k 2 amp k 4 set to 0 there was a resonant frequency at which the arm oscillations were at a maximum refer to Figure 9 Given only the model of the plant at what frequency did this resonant peak occur Remember you may only use the system state space Page 13 Revision 01 matrices A B C amp D to determine the resonant frequency Hint Use the Matlab functions SS amp Bode to calculate this frequency Do not forget to set k 2 amp k 4 to 0 4 How closely did your measured and simulated responses match Did your initial calcu
9. design 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 measured and simulated together and another for gamma measure and simulated together Page 10 Revision 01 WARNING If at any point the system is not behaving as expected make sure to immediately press STOP on the WinCon server If at any time you hear a high frequency hum from the system this is an indication that the gains are too high and you need to re calculate your controller with a larger R How close do the simulated and measured responses match By first inspection would you say that the controller you designed has met the requirements The following two figures show the measured and simulated response of gamma Figure 7 and alpha Figure 8 As you can clearly see all the design objectives have been met Figure 8 Measured amp Simulated Alpha Plot Page 11 Revision 01 If your measured values do not meet the requirements for instance if the measured alpha exceeds 10 you will have to re iterate your controller and try again Remember to first simulate before implementing the new controller on the actual plant When your controller meets the required specifications make sure to print your plots They should look similar to Figures 7 amp 8 You have now successfully designed a full
10. g M RxXF RF sin tt 2 a RF cosx M RxF RF sin 27 a RF sin x 3 4 M M M Roosa F F Rsna F F We now have our restoring moment M We will be modeling the flexible joint as a simplified spring with the following dynamic equation M K gig amp Figure 5 below shows the simplified model that will be used for the flexible joint Since M is non linear we can get a linear 1 order linear estimate of the joint stiffness Kseit M ig evaluated atx 0 Y 6a For a complete and detailed derivation of the linear joint stiffness please refer to the Maple file K_Stiff_Linear mws The final derivation as obtained from maple is 2 Dd Rr F D d DLd Rr L K 3 5 stiff p Side View Spring torque stiff amp Figure 5 Simplified Model for System Dynamics Page 5 Revision 01 3 1 Deriving The System Dynamic Equations Now that we have developed a linear model for the joint 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 V P E L 2 Spring gt sug 3 6 Kinetic Energy The Kinetic Energies in the system arise from the moving hub and flexible arm TAKE pyt KE gm 5I ag FSF yy O48 3 7 Forming the Lagrangian pe a E 2 Oa eon a re ee amO amp 3E supt 3 8 Our 2 ge
11. lated gain k work on the actual plant the first time or did you have to re iterate your design Page 14 Revision 01
12. ms toolbox 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_Exp4_Flexible Joint directory begin by running the file by the name Setup _SRV02_Exp4 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 design process 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 In this section of the Lab you will begin the iterative design process by varying Q amp R and taking notice on the effect those changes have on the simulated system response and the closed loop poles eigen values For the purpose of this lab we will fix the Q matrix to be only diagonal This will allow you to vary 4 parameters for Q q1 q2 43 q4 and one parameter for R r R in this case is scalar as there is only 1 input and therefore 1 control signal qg 0 0 0 ge 9 OL pn 0 0 q 0 0 0 04
13. neralized co ordinates are 8 and a We therefore have 2 equations 5 6L L a SiN 50 oo ume Bea 3 9 SL 8L OL eal tals lees 3 10 6t da 6a Solving Equations 3 9 amp 3 10 we are left with J OFF ym OF G T BO 3 11 output eq J sm lOAQAK pyy a 0 3 12 Referring back to Experiment 1 Position Control we know that the output Torque on the load from the motor is nn KK V KK output R m 3 13 Page 6 Revision 01 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 0 1 0 0 b 0 0 0 1 0 0 a 1 0 Stiff 1 01 K K pK B Rm X 1 7 K K y Ja J Rn J Rn i 2 _ x 0 K sap Jat T am Np N K K Ka tB RM x NaN K K J aii JR J Rin 3 2 Pre Lab Notes The purpose of the lab is to design a state feedback controller that will place the tip of the arm 8 a at a given command The controller must also meet the following criteria The controller s response to a step input should have a maximum rise time of 250 ms There should be a maximum of 5 overshoot The Arm deflection should never exceed 10 10 lt a lt 10 The following lab will be divided into two parts There will first be a design and simulation section that will entail a number of iterations and simulations of the controller In this section the student will be required to design a full state feedback con
14. th Full State Feedback Page 12 Revision 01 5 Post Lab Question and Report Upon completion of the lab you should begin by documenting your work into a lab report Included in this report should be the following i In Part of the lab you were asked to vary 5 parameters 5 steps for each for a total of 25 entries Make sure to include your Iteration Table in this report ii As you filled out the Iteration Table you were asked to qualitatively take note on the location of the closed loop poles E as you varied each parameter You should include in this report a brief description on what effects the variation of each parameter had on the poles ex poles went further to the left imaginary components became larger etc iii With the controller requirements in mind you were asked to determine the optimal parameters to achieve those specifications Include your design steps and all iterations used in determining the final controller iv After implementing your designed controller on the real plant in Part II of the lab include your plots of gamma and alpha These plots should be look similar to Figures 7 amp 8 If you had to re iterate your design after implementation include ALL plots from each iteration V In Part Il you implemented your controller on the physical plant Comment on the performance of your controller on the actual system as opposed to the simulated model Vi Make sure to include your final controller g
15. troller using the LQR method to calculate the gains The student should come into the lab with a theoretical understanding of LQR as well as a functional understanding of MATLAB Part Il of the laboratory will consist of implementing the final controller on the physical plant SRVO2 Flexible Joint as well as comparing the performance of the full state feedback controller as opposed to a controller with feedback only from the servo motor This section will also entail some frequency analysis on both closed loop models so an understanding of natural frequencies and resonance will also be required Page 7 Revision 01 4 In Lab Procedure The rotary flexible joint is an ideal experiment intended to model a flexible joint on a robot or spacecraft This experiment is also useful in the study of vibration analysis and resonance The purpose of the lab is to design a state feedback controller that will place the tip of the arm 8 a at a given command The controller must also meet the following criteria The controller s response to a step input should have a maximum rise time of 250 ms There should be a maximum of 5 overshoot The Arm deflection should not exceed 10 10 lt a lt 10 4 1 Parti Design amp Simulation The first part of this lab will be to design a state feedback controller that will meet the required specifications The method of calculating the feedback gains will be the LQR function in MATLAB s control syste
Download Pdf Manuals
Related Search