Quanser Flexible Joint Workbook
Contents
1. Begd 2 6 and acting on the link is Q2 Bia 2 7 The torque applied at the base of the rotary arm i e at the load gear is generated by the servo motor as described by the equation ae NgK gtimkt Vin Kgkm a 2 8 See 5 for a description of the corresponding SRVO2 parameters e g such as the back emf constant km The servo damping i e friction B opposes the applied torque The flexible joint is not actuated the only force acting on the link is the damping B Again the Euler Lagrange equations is a systematic method of finding the equations of motion EOMs of a system Once the kinetic and potential energy are obtained and the Lagrangian is found then the task is to compute various derivatives to get the EOMs Co ROTFLEX Workbook Student Version i 2 1 3 Potential and Kinetic Energy Kinetic Energy Translational kinetic equation is defined as T Im 2 9 where m is the mass of the object and v is the linear velocity Rotational kinetic energy is described as 1 T ze 2 10 where J is the moment of inertia of the object and w is its angular rate Potential Energy Potential energy comes in different forms Typically in mechanical system we deal with gravitational and elastic potential energy The relative gravitational potential energy of an object is V mgAh 2 11 where m is the object mass and Ah is the change in altitude of the object from a reference point The potential en2. S X s 2 20 s2 Ub w Sie The prototype second order equation is defined s 2twns w2 where is the damping ratio and w is the natural frequency Equating the characteristic equation in 2 20 to this gives ga J and a wn w 7 Based on the measured damping ratio and natural frequency the friction or stiffness of the system is K Jur 2 21 and the viscous damping is B wnd 2 22 The natural frequency and damping ratio of a system can be found experimentally e g from a free oscillation response or frequency response 2 1 6 Power Spectrum Fourier is a way to represent a signal in terms of sinusoidals The Fourier transform or spectrum of a signal g t is denoted as G w and it shows the relative amplitudes and frequencies of the sinusoidals that are used to represent that signal 9 The Fourier transform contains both the magnitude and phase information The power spectrum of a signal is based on the magnitude of the signal Figure 2 3 for example show the power spectrum of the compound sine wave signal g t 3sin nt 2 sin 4rt 0 5 sin 10zt The spectrum shows the peaks occurring at the sine wave frequencies 1 2 and 5 Hz Similarly this can be used to find the resonant frequencies of an actual system The power of a signal is the time average of the its squared value 9 and for a continuous signal g t is defined 1 tle P lim t dt g im p 9 We want to define
3. 7 Comparison between partial state and full state feedback in step 8 in Section 3 4 4 IV CONCLUSIONS Interpret your results to arrive at logical conclusions for the following I ROTFLEX Workbook Student Version i e 1 Whether the closed loop poles are in the required location in Step 5 in Section 3 4 1 2 Whether the controller meets the specifications in Step 4 in Section 3 4 2 Full state feedback controller simu lation 3 Whether the controller meets the specifications in Step 8 in Section 3 4 3 for Full state feedback controller implementation 4 Whether the controller meets the specifications in Step 7 in Section 3 4 4 for Partial state feedback controller implementation 5 GUANSER 5 3 Tips for Report Format PROFESSIONAL APPEARANCE Has cover page with all necessary details title course student name s etc Each of the required sections is completed Procedure Results Analysis and Conclusions e Typed e All grammar spelling correct e Report layout is neat e Does not exceed specified maximum page limit if any e Pages are numbered Equations are consecutively numbered Figures are numbered axes have labels each figure has a descriptive caption e Tables are numbered they include labels each table has a descriptive caption Data are presented in a useful format graphs numerical table charts diagrams e No hand drawn sketches diagrams References are cited using correct for
4. K z which is the control used in the control algorithm O GUANSER State Feedback Control Flexible Link Plant Xda Xg X u A B C D Figure 3 2 State feedback control loop 3 3 Pre Lab Questions 1 Based on your analysis in Section 2 3 is the system stable marginally stable or unstable Did you expect the stability of the Rotary Flexibe Joint system to be as what was determined If desired use your experience with the actual device in the Modeling Laboratory in Section 2 3 2 Using the open loop poles find the characteristic equation of the system 3 Give the corresponding companion matrices A and B Do not compute the transformation matrix W this will be computed using software in the lab 4 Find the location of the two dominant poles p and p gt based on the specifications given in Section 3 1 Place the other poles at p 30 and p4 40 Finally give the desired characteristic equation 5 When applying the control u K x to the companion form it changes A B to A BK B Find the gain K that assigns the poles to their new desired location 5 GUANSER 3 4 Lab Experiments 3 4 1 Control Design 1 Run the setup_rotflex m script to load the model you found in the Modeling Experiment in Section 2 3 2 Using Matlab commands determine if the system is controllable Explain why 3 Open the d_pole_placement_student m script As shown below the companion matrices A and B fo
5. A 3 4 0 0 0 1 a A2 t Gn 1 n and 0 3 5 0 1 Define Z W TT where T is the controllability matrix defined in Equation 3 2 and B BA BA Then W tAW and W B B 3 2 4 Pole Placement If A B are controllable then pole placement can be used to design the controller Given the control law u Kz the state space in Equation 2 13 becomes Az B Kz A BK a l 5 GUANSER To illustate how to design gain K consider the following system 0 1 0 A 0 0 1 3 6 3 0 B 0 3 7 Note that A and B are already in the companion form We want the closed loop poles to be at 1 2 3 The desired characteristic equation is therefore and s 1 s 2 s 3 s 63 11s 6 3 8 For the gain K k k2 ks apply control u Kx and get 0 1 0 A KB 0 0 iL k3 3 k 1 ke 5 The characteristic equation of A KB is s kg 5 s k2 1 s k 3 3 9 Equating the coefficients between Equation 3 9 and the desired polynomial in Equation 3 8 k 3 6 pat 11 k3 5 6 Solving for the gains we find that a gain of K 9 10 1 is required to move the poles to their desired location We can generalize the procedure to design a gain K for a controllable A B system as follows Step 1 Find the companion matrices A and B Compute W TT Step 2 Compute K to assign the poles of A BK to the desired locations Applying the co
6. in Section 2 3 1 2 Model Validation e Briefly describe the main goal of the experiment e Briefly describe loading the model in Step 6 in Section 2 3 2 e Briefly describe the model validation experiment in Step 13 in Section 2 3 2 ll RESULTS Do not interpret or analyze the data in this section Just provide the results 1 Frequency response plot from step 3 in Section 2 3 1 2 Model validation plot from step 13 in Section 2 3 2 3 Provide applicable data collected in this laboratory from Table 1 lll ANALYSIS Provide details of your calculations methods used for analysis for each of the following 1 Measured link stiffness in step 6 in Section 2 3 1 2 Model discrepancies given in step 15 in Section 2 3 2 IV CONCLUSIONS Interpret your results to arrive at logical conclusions for the following 1 How does the model compare with the actual system in step 14 of Section 2 3 2 State space model validation Q GUANSER 5 2 Template for Content Control I PROCEDURE 1 Control Design e Briefly describe the main goal of the control design e Briefly describe the control design procedure in Step 3 in Section 3 4 1 2 Simulation e Briefly describe the main goal of the simulation e Briefly describe the simulation procedure in Step 3 in Section 3 4 2 3 Full State Feedback Implementation e Briefly describe the main goal of this experiment e Briefly describe the experimental procedure in Step 7 i
7. ing Flexible Joint angle The measured response can then be used to find the natural frequency of the link q_rotflex_val mdl This Simulink model is used with QUARC to compare the Flexible Joint state space model with the measured re sponse from the actual system q_rotflex mdl Simulink file that implements a closed loop state feedback controller on the actual ROTFLEX system using QUARC Table 3 Files supplied with the SRVO2 Flexible Joint Control Laboratory 4 2 Setup for Finding Stiffness Before beginning in lab procedure outlined in Section 2 3 1 the q_rotflex_id Simulink diagram must be properly configured Follow these steps 1 Setup the SRVO2 with the Flexible Joint module as detailed in the Flexible Joint User Manual 7 2 Load the Matlab software 3 Browse through the Current Directory window in Matlab and find the folder that contains the file g_rotflex_id mal 4 Open the q_rotflex_id md Simulink diagram shown in Figure 2 4 CE ROTFLEX Workbook Student Version iC 5 Configure DAQ Ensure the HIL Initialize block subsystem is configured for the DAQ device that is installed in your system By default the block is setup for the Quanser Q8 hardware in the loop board See Reference 2 for more information on configuring the HIL Initialize block 4 3 Setup for Model Validation Before performing the in lab exercises in Section 2 3 2 the q_rotflex_val Simulink diagram and the setup_rotflex m scr
8. the power of the signal from its transform First consider the truncated signal of g t defined as ne lt lt 7 th O O wor A TTToTRLEX Wek So Vor SS Figure 2 3 Power spectrum of compound sine wave From Parseval s theorem the energy of this signal equals 9 CO 1 co Bon J gr t dt Gr w dw SS 2T J o The power of a signal can then be expressed as Py hin gem aw The power spectral density PSD function is Sg w jim GEG 2 23 Defining the signal power in terms of the PSD and considering only positive frequencies we get Er 1 TO 1 I alu Expressing this in term of Hertz we get the expression P 1 S w df 2 24 In practice signals are sampled and the algorithm is performed in discrete mode The Fast Fourier Transform FFT is a computational algorithm used to find the Fourier transform of a signal i e taking the FFT of a signal g t generates G w To find the power spectrum code similar to the following would be used Q GUANSER y fft g Sg lyl N Pg 2 Sg 1 N 2 where NV is the number of samples or length of the signal g t If g t was sampled at a frequency interval of T and had a duration of T it would have N T T samples Notice that because this is implemented in discrete mode we use the amount of samples N instead of the signal time duration T as used in the PSD expression in Equation 2 23 The power spectrum can be used to find resonant frequencies of a
9. the system This systematic method is often used for more complicated systems such as robot manipulators with multiple joints More specifically the equations that describe the motions of the servo and the link with respect to the servo motor voltage i e the dynamics will be obtained using the Euler Lagrange equation L OL Atdg Oq Qi 2 1 The variables q are called generalized coordinates For this system let g t 0 t a t 2 2 where as shown in Figure 2 2 6 t is the servo angle and a t is the flexible joint angle The corresponding velocities A6 t Oa t A Be oe Note The dot convention for the time derivative will be used throughout this document i e and da The time variable t will also be dropped from and a i e 0 0 t and a a t With the generalized coordinates defined the Euler Lagrange equations for the rotary flexible joint system are L OL erect 2 3 I 88 Q 2 3 ii L L eee 2 4 50a 8a 24 The Lagrangian of a system is defined L T V 2 5 where T is the total kinetic energy of the system and V is the total potential energy of the system Thus the Lagrangian is the difference between a system s kinetic and potential energies The generalized forces Q are used to describe the non conservative forces e g friction applied to a system with respect to the generalized coordinates In this case the generalized force acting on the rotary arm is Qi T
10. Compensator Design Matlab command Find gain K using a Matlab pole placement command and verify that the gain is the same as generated before 3 4 2 Control Simulation Using the linear state space model of the system and the designed control gain the closed loop response can be simulated This way we can test the controller and see if it satisfies the given specifications before running it on the hardware platform Experiment Setup The s_rotflex Simulink diagram shown in Figure 3 3 is used to simulate the closed loop response of the Flexible Joint using the control developed in Section 3 3 I ROTFLEX Workbook Student Version iz The Smooth Signal Generator block generates a 0 33 Hz square wave with amplitude of 1 that is passed through a Rate Limiter block to smooth the signal The Amplitude deg gain block is used to change the desired servo position command The state feedback gain K is set in the Controller gain block and is read from the Matlab workspace The Simulink State Space block reads the A B C and D state space matrices that are loaded in the Matlab workspace tC s_rotflex He File Edit wiew Simulation Format Tools QUARC Help DSS 2 pajo Nma SH ACH RB e SRV02 ROTFLEX Control Simulation X Ax Bu y CxtDu Degrees to Voltage State Space Radians Saturation gt 20 u gt gt Smooth Signal Rate Limiter G
11. EAR_COMFIG to HIGH e LOAD_TYPE to NONE e Ensure ENCODER_TYPE TACH_OPTION K_ CABLE AMP_TYPE and VMAX_DAC parameters are set according to the SRV02 system that is to be used in the laboratory e CONTROL_TYPE to MANUAL 4 4 Setup for Flexible Joint Control Simulation Before going through the control simulation in Section 3 4 2 the s_rotflex Simulink diagram and the setup_rotflex m script must be configured Follow these steps to configure the lab properly 1 Load the Matlab software 2 IMPORTANT Make sure the model you found in Section 2 3 2 is entered in ROTFLEX_ABCD eqns _student m GUANSER N ODO oO fF OQ Browse through the Current Directory window in Matlab and find the folder that contains the s_rotflex mdl file Open s_rotflex mdl Simulink diagram shown in Figure 3 3 Configure the setup_rotflex m script according to your hardware See Section 4 3 for more information Run the setup_rotflex m script Enter the stiffness Ks you found in Section 2 3 1 4 5 Setup for Flexible Joint Control Implementa tion Before beginning the in lab exercises given in Section 3 4 3 or Section 3 4 4 the q_rotflex Simulink diagram and the setup_rotflex m script must be setup Follow these steps to get the system ready for this lab a F WwW N Setup the SRVO2 with the Flexible Joint module as detailed in 7 Load the Matlab software Browse through the Current Directory window in Matlab and find th
12. Gu AN SE A Workbook 2011 Quanser Inc All rights reserved Quanser Inc 119 Spy Court Markham Ontario L3R 5H6 Canada info quanser com Phone 1 905 940 3575 Fax 1 905 940 3576 Printed in Markham Ontario For more information on the solutions Quanser Inc offers please visit the web site at http www quanser com This document and the software described in it are provided subject to a license agreement Neither the software nor this document may be used or copied except as specified under the terms of that license agreement All rights are reserved and no part may be reproduced stored in a retrieval system or transmitted in any form or by any means electronic mechanical photocopying recording or otherwise without the prior written permission of Quanser Inc ACKNOWLEDGEMENTS Quanser Inc would like to thank Dr Hakan Gurocak Washington State University Vancouver USA for his help to include embedded out comes assessment CONTENTS 1 2 O GUANSER Introduction Modeling 2 1 Background 2 2 Pre Lab Questions 2 3 Lab Experiments 2 4 Results Control Design 3 1 Specifications 3 2 Background 3 3 Pre Lab Questions 3 4 Lab Experiments 3 5 Results System Requirements 4 1 Overview of Files 4 2 Setup for Finding Stiffness 4 3 Setup for Model Validation 4 4 Setup for Flexible Joint Control Simulation 4 5 Setup for Flexible Joint Control Implementation Lab Report 5 1 Template fo
13. Open the q rotflex Simulink diagram Make sure the Manual Switch is set to the Full State Feedback upward position 4 Go to QUARC Build to build the controller 5 Go to QUARC Start to run the controller The flexible link should be tracking the default 20 degree signal 6 Stop the controller once you have obtained a representative response 7 Plot the responses from the theta deg alpha deg and Vm V scopes in a Matlab figure Similarly as described in Section 3 4 2 the response data is saved in variables data_theta data_alpha and data_vm 8 Measure the settling time and percentage overshoot of the measured servo response and the maximum link deflection Does the response satisfy the specifications given in Section 3 1 3 4 4 Implementing Partial State Feedback Control In this section the partial state feedback response of the system is assessed and compared with the full state feedback control in Section 3 4 3 1 Run the setup_rotflex m 2 Ensure the control gain you settled on in Section 3 4 3 is loaded i e gain K ROTFLEX Workbook Student Version File Edit View Simulation Format Tools QUARC Help Deas giv im ganen eee SRV02 Rotflex Control Q accelerate design 20 Pj D2R 114 0 0 0 Smooth Signal Rate Limiter Delay Start Degrees to tto Pot 1 Eno 2 Generator Amplitude 4 9 R agians 2 SRVO2 Position Sour
14. b once you know how your ROTFLExX is configured Hint The total inertia depends on where the top arm is anchored Use the parallel axis theorem to determine the inertia of the top link with respect to the pivot O GUANSER 2 3 Lab Experiments In the first part of this laboratory the stiffness of the flexible joint is determined by measuring its natural frequency In the second part the state space model is finalized and validated against actual measurements 2 3 1 Finding Stiffness To find the stiffness we need to find the natural frequency of the flexible joint This is the frequency where the link attached to the flexible joint begins to oscillate the most To find this frequency we use a Sine Sweep signal The Sine Sweep is a sine wave that goes through a range of frequencies i e from low to high We can then generate a power spectrum from the measured signal and identify the frequency with the largest amplitude the natural frequency Physical Parameters for the Lab In order to do some of the laboratory exercises you will need these values Beq 0 004N m rad s Jeg 2 08 x 107 kg m m 0 064kg L 0 298m m2 0 030 kg L 0 156m B 0 Note The equivalent viscous damping B and moment of inertia Jeq parameters are for the SRVO2 when there is no load i e the parameter found in the SRVO2 Modeling Laboratory was for servo with the disc load Experimental Setup The q_rotflex_id Simuli
15. ce Ku State Feedback SRVOZ Flexible Joint High Gain Observer ha oe Fi ull State Feedback iia Tagan Partial State Feedback Figure 3 5 q rotflex Simulink diagram used the model 3 Open the q rotflex Simulink diagram Make sure the Manual Switch is set to the Partial State Feedback down ward position 4 Go to QUARC Start to run the controller The link should be tracking the default 20 degree square wave 5 Stop the controller once you have obtained a representative response 6 As in Section 3 4 3 attach a Matlab figure representing the SRVO02 angle and flexible link angle response as well as the input voltage 7 Measure the settling time and percentage overshoot of the measured servo response and the maximum link deflection Does the response satisfy the specifications given in Section 3 1 8 Examine the difference between the partial state feedback PSF response and the full state feedback FSF response Explain why PSF control behaves this way by looking at the q_rotflex Simulink diagram Q ROTFLEX Workbook Student Version v1 0 GQGUANSER 3 5 Results Fill out Table 2 with your answers from your control lab results both simulation and implementation Description Symbol Value Units Pre Lab Questions Desired poles DP Companion Gain K Simulation Control Design Transformation Matrix W Co
16. d by changing the mounting position of the shorter top arm The main bottom arm which is connected to the pivot has a length of L and a mass of m The length and mass of the top link is L and mz The distance between the pivot and the middle of the top arm which can be changed is denoted by the variable diz The moment of inertia of the entire link is specified by J and it changes depending on the position of the top arm See the Rotary Flexible Joint User Manual in 7 for the values of these parameters The deflection angle of the link is denoted as a and increases positively when rotated CCW Yo L2 e N a S 7 Figure 2 1 Rotary Flexible Joint Angles The flexible joint system can be represented by the diagram shown in Figure 2 2 Our control variable is the input servo motor voltage Vm This generates a torque 7 at the load gear of the servo that rotates the base of the link The viscous friction coefficient of the servo is denoted by Beg This is the friction that opposes the torque being applied at the servo load gear The friction acting on the link is represented by the viscous damping coefficient B Finally the flexible joint is modeled as a linear spring with the stiffness Ks 7 0 a 4 4 HE Val UB KA By Figure 2 2 Rotary Flexible Joint Model 5 GUANSER 2 1 2 Finding the Equations of Motion Instead of using classical mechanics the Lagrange method is used to find the equations of motion of
17. device in parallel we can verify whether the dynamic model which drives the simulation accurately represents our system Experimental Setup The q_rotflex_val Simulink diagram shown in Figure 2 6 applies either a step or pulse input to both the Quanser Flexible Joint hardware and to the Flexible Joint model and reads the servo and link angles The SRV0O2 Flexible Joint subsystem contains the QUARC blocks that interface to the actual hardware The Simulink State Space block reads the A B C and D state space matrices that are loaded in the Matlab workspace This model outputs the deflection angle of the link Q GUANSER tC q_rotflex_val File Edit Yiew Simulation Format Tools QUARC Help Dee TRS yi Extena v R i De BEE Pot 1 Enc 2 oooo a E SRVO2 Position Source Signal y Generator AmPlitude 09 Ay itch Transport Manual Switch Daisy SRVO2 Flexible Joint R2D State Space theta theta deg dto deg R2D alpha alpha deg tad to deg ode1 Figure 2 6 q_rotflex_val Simulink diagram used validate the model IMPORTANT Before you can conduct this experiment you need to make sure that the lab files are configured according to your system setup If they have not been configured already then go to Section 4 3 to configure the lab files first 1 Run the setup_rotflex m 2 When prompted enter the stiffness you found in Section 2 3 1 This is saved to the Matlab var
18. e folder that contains the q_rotflex mdl file Open the q_rotflex madl Simulink diagram shown in Figure 3 5 Configure DAQ Ensure the HIL Initialize block in the SRV02 Flexible Joint subsystem is configured for the DAQ device that is installed in your system By default the block is setup for the Quanser Q8 hardware in the loop board See Reference 2 for more information on configuring the HIL Initialize block Configure Sensor The position of the SRV02 load shaft can be measured using the potentiometer or the encoder Set the Pos Src Source block in q_rotflex as shown in Figure 3 5 as follows e 1 to use the potentiometer e 2 to use to the encoder Note that when using the potentiometer there will be a discontinuity Configure setup script Set the parameters in the setup_rotflex m script according to your system setup See Section 4 3 for more details Run the setup_rotflex m script 5 LAB REPORT This laboratory contains two groups of experiments namely 1 Modeling the Quanser Rotary Flexible Joint system and 2 State feedback control For each experiment follow the outline corresponding to that experiment to build the content of your report Also in Section 5 3 you can find some basic tips for the format of your report 5 1 Template for Content Modeling I PROCEDURE 1 Finding Stiffness e Briefly describe the main goal of the experiment e Briefly describe the experiment procedure in Step 3
19. enerator Amplitude deg High Gain Observer Full State Feedback Lo iia 2 0 2 0 Partial State Feedback Figure 3 3 s_rotflex Simulink diagram used to simulate the state feedback control IMPORTANT Before you can conduct this experiment you need to make sure that the lab files are configured according to your system setup If they have not been configured already go to Section Section 4 4 to configure the lab files first Make sure the model you found in Section 2 3 2 is entered in ROTFLEX ABCD eqns_student m and the d_pole placement_student m calculates the control gain 1 Run setup_rotpen m Ensure the gain K you found in Section 3 4 1 is loaded 2 In s_rotflex make sure the Manual Switch is set to the Full State Feedback upward position 3 Run s_rotflex to simulate the closed loop response with this gain The response in the scopes shown in Figure 3 4 were generated using an arbitary gain Plot the simulated response of the servo link and motor input voltage obtained using your obtained gain K in a Matlab figure and attach it to your report Note When the simulation stops the last 10 seconds of data is automatically saved in the Matlab workspace to the variables data_theta data_alpha and data_Vm The time is stored in the data_theta 1 vector the de sired and measured rotary arm angles are saved in the data_theta 2 and data_theta 3 arrays the pendulum an
20. ergy of an object that rises from the table surface i e the reference up to 0 25 meter is Ah 0 25 0 0 25 and the energy stored is V 0 25mg The equation for elastic potential energy i e the energy stored in a spring is V KAn 2 12 where K is the spring stiffness and Az is the linear or angular change in position If an object that is connected to a spring moves from in its initial reference position to 0 1 m then the change in displacement is Ax 0 1 0 0 1 and the energy stored equals V 0 005K 2 1 4 Linear State Space Model The linear state space equations are t Ax Bu 2 13 and y Cr Du 2 14 where z is the state u is the control input A B C and D are state space matrices For the Rotary Flexible Joint system the state and output are defined r 0a 2 15 and y ex x 2 16 In the output equation only the position of the servo and link angles are being measured Based on this the C and D matrices in the output equation are 1000 d 1 0 J eng and i D o 2 18 The velocities of the servo and link angles can be computed in the digital controller e g by taking the derivative and filtering the result though a high pass filter Q GUANSER 2 1 5 Finding Second Order System Parameters Consider a second order system described by J Bi Kr 0 2 19 Assuming the initial conditions z 07 ao and 07 0 the Laplace transform of Equation 2 19 is
21. g DER 46S PS mma Fak oO Time offset 0 Figure 2 5 Typical Flexible Joint Sine Sweep Response 3 After the controller stops i e after 15 sec the data is automatically saved in the Matlab workspace to the variable data_alpha The time is stored in data_alpha 1 vector and the link angle is stored the data_alpha 2 vector Plot the response in a Matlab figure 4 Using the sweep response data and Matlab commands generate a plot of the power spectrum The back ground theory and pseudo code to find the power spectrum was given in Section 2 1 6 You can also use any available Matlab examples to help you out Show the commands you used and the plot generated Hint When using the discrete Matlab FFT command fft x n the size of n should be base of 2 Use the nextpow2 function to find this size For example if your signal size is 250 samples long this will return 8 which gives n 28 256 5 Find the natural frequency of the link Because the damping in relatively low assume the damped natural frequency which is being measured is equivalent to the undamped natural frequency 6 Find the total moment of inertia of the link J using the equation you developed in the Exercise 8 in Section 2 2 The corresponding value of diz is given in the Rotary Flexible Joint User Manual 7 Once the inertia is computed find the stiffness of the flexible joint Ks 2 3 2 Model Validation By running a simulation and the actual
22. gle is stored the data_alpha 2 vector and the control input is in the data_vm 2 structure 4 Measure the settling time and percentage overshoot of the simulated servo response the maximum link de flection and the voltage used Does the response satisfy the specifications given in Section 3 1 3 4 3 Control Implementation In this experiment the servo position is controlled while minimizing the link deflection using the control found in Section 3 4 1 Measurements will then be taken to ensure that the specifications are satisfied Experiment Setup The q_rotflex Simulink diagram shown in Figure 3 5 is used to run the state feedback control on the Quanser Flexible Joint system The SRV02 Flexible Joint subsystem contains QUARC blocks that interface with the DC motor and sensors of the system The feedback developed in Section 3 4 1 is implemented using a Simulink Gain block IMPORTANT Before you can conduct this experiment you need to make sure that the lab files are configured according to your system setup If they have not been configured already then go to Section 4 5 to configure the lab files first 1 Run the setup_rotflex m Q GUANSER theta deg 50 0 Time offset 0 Time offset 0 b Flexible Link Angle vm V DAR s6 5 s 8a a Time offset 0 c Voltage Figure 3 4 Default Simulated Closed Loop Response 2 Ensure the controller you found in Section 3 4 1 is loaded i e gain K 3
23. heta 1 the simulated servo angle is in data_theta 2 and the measured SRV02 angle is in data_theta 3 Similarly for the flexible joint response saved in data_alpha Plot the re sponse in a Matlab figure 14 How well does your model represent the actual system We want a model that is fairly representative but having said that keep in mind that no model is perfect This is just a quick test to see how well your model represents the actual device As shown in figures 2 7a and 2 7b the simulation does not match the measured response perfectly Q GUANSER 15 Give at least one reason why the model does not represent the system accurately 16 In Matlab find the open loop poles i e eigenvalues of the system using the state space matrix A that is loaded Note These will be required for a pre lab question in Section 3 3 2 4 Results Fill out Table 1 with your answers from your modeling lab results both simulation and implementation Description Symbol Value Unit Finding Stiffness Natural frequency Wn rad s Stiffness K N m rad Model Validation State Space Matrix A State Space Matrix B State Space Matrix C State Space Matrix D Open loop poles OL Table 1 Results 9 GUANSER 3s CONTROL DESIGN 3 1 Specifications The time domain requirements are Specification 1 Servo angle 4 settling time t lt 0 5 s Specification 2 Servo angle percen
24. iable Ks Enter link stiffness Ks 3 Depending on your stiffness the Matlab prompt should generate a gain similarly as shown below this gain is generated for a Ks of 1 K 0 0 0 0 cls_poles 0 0 0 0 This means the script ran correctly 4 In Matlab open the M File called ROTFLEXE_ABCD_eqns_student m The script has the following state space matrices entered A 001 0 0001 0 500 5 0 0 750 5 O B 0 0 500 500 C zeros 2 4 iw ll zeros 2 1 ROTFLEX Workbook Student Version 5 Enter the state space matrices you found in Section 2 2 for A B C and D In Matlab the stiffness and link moment of inertia are defined as Ks and JI and the SRVOZ2 inertia and damping are denoted Jeq and Beg 6 Run the ROTFLEX_ABCD_eqns_student m script to load the state space matrices in the Matlab workspace Show the numerical matrices that are displayed in the Matlab prompt 7 The input of the state space model you found in Section 2 2 is the torque acting at the servo load gear i e at the pivot of the flexible joint However we do not control torque directly we control the servo input voltage Recall the voltage torque relationship given in Equation 2 8 in Section 2 1 2 In the System Model section of the setup_rotflex m script the actuator dynamics are added to your state space matrices with the code Ao A Bo B B eta_g Kg eta_m kt Rm Bo A 3 3 Ao 3 3 Bo 3 eta_g Kg 2 eta_m kt km R
25. ipt must be configured Follow these steps to get the system ready for this lab 1 Setup the SRVO2 with the Flexible Joint module as detailed in 7 2 Load the Matlab software 3 Browse through the Current Directory window in Matlab and find the folder that contains the QUARC ROTFLEX file g_rotflex_val mdl 4 Open the q_rotflex_val mdl Simulink diagram shown in Figure 2 6 5 Configure DAQ Ensure the HIL Initialize block in the SRV0O2 Flexible Joint subsystem is configured for the DAQ device that is installed in your system By default the block is setup for the Quanser Q8 hardware in the loop board See Reference 2 for more information on configuring the HIL Initialize block 6 Configure Sensor The position of the SRVO2 load shaft can be measured using either the potentiometer or the encoder Set the Pos Src Source block in g_rotflex_val as shown in Figure 2 6 as follows e 1 to use the potentiometer e 2 to use to the encoder Note that when using the potentiometer there will be a discontinuity 7 Configure Input Set the Manual Switch to the DOWN position for a step input or the UP position for a square signal 8 Open the setup_rotflex m file This is the setup script used for the ROTFLEX Simulink models 9 Configure setup script When used with the Flexible Joint the SRV0O2 has no load i e no disc or bar and has to be in the high gear configuration Make sure the script is setup to match this setup e EXT_G
26. le minimizing link deflection that is introduced by the flexible joint 3 2 1 Stability The stability of a system can be determined from its poles 10 e Stable systems have poles only in the left hand plane e Unstable systems have at least one pole in the right hand plane and or poles of multiplicity greater than 1 on the imaginary axis e Marginally stable systems have one pole on the imaginary axis and the other poles in the left hand plane The poles are the roots of the system s characteristic equation From the state space the characteristic equation of the system can be found using det sI A 0 3 1 where det is the determinant function s is the Laplace operator and 7 the identity matrix These are the eigenvalues of the state space matrix A CO ROTFLEX Workbook Student Version i 3 2 2 Controllability If the control input u of a system can take each state variable x where i 1 n from an initial state to a final state then the system is controllable otherwise it is uncontrollable 10 Rank Test The system is controllable if the rank of its controllability matrix T B AB A B AB 3 2 equals the number of states in the system rank T n 3 3 3 2 3 Companion Matrix If A B are controllable and B is n x 1 then A is similar to a companion matrix 1 Let the characteristic equation of A be 1 s ans a Then the companion matrices of A and B are 0 1 0 0 0 0 0 0
27. m A 4 3 Ao 4 3 Bo 4 xeta_g Kg 2 eta_m kt km Rm 8 Run the setup_rotflex m script again so your Flexible Joint model is based on DC motor voltage 9 In the q_rotflex_val Simulink diagram go to QUARC Build to build the QUARC controller 10 Make sure the area around the Flexible Joint experiment is clear Set the Manual Switch to the downward position to feed a Step input 11 Go to QUARC Start to run the q_rotflex_val controller Typical scope responses are shown in Figure 2 7 The theta deg scope displays the simulated servo angle in yellow and the measured angle in purple Similarly the alpha deg scope shows the simulated link angle in yellow and the measured angle in purple theta deg alpha deg DAR E 458 OSS ABB OOAR Time offset 0 a Servo Angle b Flexible Link Angle Figure 2 7 Typical Flexible Joint model validation response in scopes 12 If your simulation and measured response match go to the next step If they do not then there is an issue with your model Here are some issues to investigate e Ensure the state space model was enterred properly in the script e The stiffness K found in Section 2 3 1 is not correct Review your calculations or redo the experiment e Review your model derivation in Section 2 2 e g might be a mistake in solving the EOMs 13 The theta deg and alpha deg scopes save their data to the variables data theta and data_alpha For data_theta the time is in data_t
28. mat REFERENCES 1 Bruce Francis Ece1619 linear systems course notes university of toronto 2001 2 Quanser Inc 3 Quanser Inc 4 Quanser Inc 5 Quanser Inc 6 Quanser Inc 7 Quanser Inc 8 Quanser Inc QUARC User Manual SRV02 QUARC Integration 2008 QUARC Installation Guide 2009 SRV02 User Manual 2009 QUARC Compatibility Table 2010 SRV02 Rotary Flexible Joint User Manual 2011 SRV02 Rotary Flexible Link User Manual 2011 9 B P Lathi Modern Digital and Analog Communication Systems Oxford University Press Inc 3rd edition 1998 10 Norman S Nise Control Systems Engineering John Wiley amp Sons Inc 2008 5 GUANSER
29. n Section 3 4 3 4 Partial State Feedback Implementation e Briefly describe the main goal of this experiment e Briefly describe the experimental procedure in Step 6 in Section 3 4 4 ll RESULTS Do not interpret or analyze the data in this section Just provide the results 1 Matrices from Step 3 in Section 3 4 1 Find transformation matrix Response plot from step 3 in Section 3 4 2 Full state feedback controller simulation Response plot from step 7 in Section 3 4 3 for Full state feedback controller implementation Response plot from step 6 in Section 3 4 4 for Partial state feedback controller implementation a fF W N Provide applicable data collected in this laboratory from Table 2 lll ANALYSIS Provide details of your calculations methods used for analysis for each of the following 1 Controllability of system in Step 2 in Section 3 4 1 Closed loop poles in Step 5 in Section 3 4 1 Matlab commands used to generate the control gain in Step 6 in Section 3 4 1 gt OO N Settling time percent overshoot link deflection and input voltage Step 4 in Section 3 4 2 Full state feedback controller simulation 5 Settling time percent overshoot link deflection and input voltage in Step 8 in Section 3 4 3 Full state feedback controller implementation 6 Settling time percent overshoot link deflection and input voltage in Step 7 in Section 3 4 3 Partial state feedback controller implementation
30. nk diagram shown in Figure 2 4 is used to find the natural frequency of the flexible joint The QUARC blocks are used to interface with the DC motor and encoders of the system The Sine Sweep block generates a sine wave that goes from a low frequency to a high frequency for a predetermined duration For more information about the QUARC software see Reference 3 This model outputs the deflection angle of the link P q_rotflex_id He File Edit view Simulation Format Tools QUARC Help D H8 is Exem v Be RB BEE Quarc accelerate design 25 P D2R Amplitude deg Degrees to Radians SRVOZ Flexible Joint xe theta rad to deg P R2D alpha alpha deg tad to deg Figure 2 4 q_rotflex_id Simulink diagram used to find flexible joint stiffness IMPORTANT Before you can conduct this experiment you need to make sure that the lab files are configured according to your system setup If they have not been configured already then you need to go to Section 4 2 to configure the lab files first 1 In the g_rotflex_id Simulink diagram go to QUARC Build to build the QUARC controller 2 Go to QUARC Start to run the controller The SRVO2 tracks a sine sweep that goes from 1 Hz to 5 Hz The alpha deg Scope should be reading a response similarly as shown in Figure 2 5 Note that the controller is set to run for 15 seconds alpha de
31. ntrol Gain K Closed loop poles CLP Simulation Closed Loop System Maximum deflection almaz deg Maximum voltage Valas V Implementation Control Gain K Maximum deflection ala deg Maximum voltage Volani V Table 2 Results 4 SYSTEM REQUIREMENTS Required Software e Microsoft Visual Studio e Matlab with Simulink Real Time Workshop and the Control System Toolbox e QUARC 2 1 or later See the QUARC software compatibility chart at 6 to see what versions of MS VS and Matlab are compatible with your version of QUARC and for what OS Required Hardware e Data acquisition DAQ card that is compatible with QUARC This includes Quanser Hardware in the loop HIL boards such as Q2 USB Q8 USB QPID QPIDe and some National Instruments DAQ devices e g NI USB 6251 NI PCle 6259 For a full listing of compliant DAQ cards see Reference 2 e Quanser SRV02 ET rotary servo e Quanser Rotary Flexible Joint attached to SRVO2 e Quanser VoItPAQ X1 power amplifier or equivalent Before Starting Lab Before you begin this laboratory make sure e QUARC is installed on your PC as described in 4 e The QUARC Analog Loopback Demo has been ran successfully e SRV02 Rotary Flexible Joint and amplifier are connected to your DAQ board as described Reference 8 O GUANSER 4 1 Overview of Files File Name Description Flexible Joint User Manual pdf This manual describes the hardware of
32. ntrol law u K to the general system given in Equation 3 4 0 1 0 0 0 0 0 0 Az z 3 10 0 0 tee 0 1 a ky ag ka Gn 1 kn 1 An kn Step 3 Find K KW to get the feedback gain for the original system A B Remark It is important to do the K gt K conversion Remember that A B represents the actual system while the companion matrices A and B do not 3 2 5 Desired Poles The rotary inverted pendulum system has four poles As depicted in Figure 3 1 poles p and p are the complex conjugate dominant poles and are chosen to satisfy the natural frequency wn and damping ratio specifications given in Section 3 1 Let the conjugate poles be pi 0 jwa 3 11 and p2 0 jwa 3 12 where o Cwn and wg wn y 1 is the damped natural frequency The remaining closed loop poles p3 and pa are placed along the real axis to the left of the dominant poles as shown in Figure 3 1 3 A Py a TLRS TIS EAP ASEM E JO ee O a xX x gt 0 gt R Pa Ps o o NE jo P2 Figure 3 1 Desired closed loop pole locations 3 2 6 Feedback Control The feedback control loop that in Figure 3 2 is designed to stabilize the servo to a desired position 04 while mini mizing the deflection of the flexible link The reference state is defined Tda Oa 00 0 3 13 and the controller is u K aaq 2 3 14 Note that if z4 0 then u
33. r Content Modeling 5 2 Template for Content Control 5 3 Tips for Report Format 1 INTRODUCTION The objective of this experiment is to control the position of the servo while minimizing the motions the flexible rotary link Topics Covered e Modeling the Rotary Flexible Joint using Lagrange e Find the linear state space model of the system e Do some basic model validation e Design a state feedback controller using Pole Placement PP e Simulate the closed loop flexible joint system e Implement the designed controller on the device e Compare the simulated and measured closed loop results e Assess the behaviour of implementing a partial state feedback controller Prerequisites In order to successfully carry out this laboratory the user should be familiar with the following e Basics of Simulink e Transfer function fundamentals e State space modeling e g obtaining state equations from a set of differential equations e SRV02 QUARC Integration Laboratory 3 in order to be familiar using QUARC with the SRVO2 2 MODELING 2 1 Background 2 1 1 Model The Rotary Flexible Joint model is shown in Figure 2 1 The base of the module is mounted on the load gear of the SRV02 system The servo angle 0 increases positively when it rotates counter clockwise CCW The servo and thus the link turn in the CCW direction when the control voltage is positive i e Vm gt 0 The total length of the link can be varie
34. r the model are automatically found denoted as Ac and Bc in Matlab Characteristic equation s 4 a_4 s 3 a_3 s 2 a_2 s a_l a poly A Companion matrices Ac Bc Ac 0100 0010 0001 a 5 a 4 a 3 a 2 Bc 0 0 0 1 Controllability T 0 Controllability of companion matrices Tc 0 Transformation matrices W 0 In order to find the gain K we need to find the transformation matrix W TT note T is denoted as Tc in Matlab Modify the d_pole_placement_student m script to calculate the controllability matrix T the companion controllabilty matrix Tc the inverse of Tc and W Show your completed script and the resulting T Te Tet and W matrices 4 Enter the companion gain K you found in the pre lab as Kc in d_pole_placement_student m and modify it to find gain K using the transformation detailed in Section 3 2 Run the script again to calculate the feedback gain K and record its value in Table 2 5 Evaluate the closed loop poles of the system i e the eigenvalues of A Bk Record the closed loop poles of the system when using the gain K calculated above Have the poles been placed to their desired locations If not then go back and re investigate your control design until you find a gain that positions the poles to the required location 6 In the previous exercises gain K was found manually through matrix operations All that work can instead be done using a pre defined
35. system One common technique is feeding a sine sweep or chirp signal to the system and measuring the corresponding output response By taking its FFT and finding the spectrum the user will see various peaks These peaks represent the sine wave frequencies that describe the system 2 2 Pre Lab Questions 1 Energy is stored in the flexible joint springs as it rotates by an angle of a see Figure 2 1 Find the potential energy of the flexible joint using the parameters described in Section 2 1 2 Find the total kinetic energy of the system contributed by the rotary servo 6 and the rotation of the link a Use the parameters shown in Figure 2 2 3 Compute the Lagrangian of the system 4 Find the first Euler Lagrange equation given in 2 3 Keep the equations in terms of applied torque 7 i e not in terms of DC motor voltage Also make sure your equations follow the general form J Bt Kz u 5 Find the second Euler Lagrange Equation 2 4 6 Find the equations of motion 6 f1 0 0 a a 7 and f2 0 0 a T Assume the viscous damping of the link is negligible i e B 0 7 Given state x defined in Equation 2 15 find the linear state space matrices A and B 8 Express the moment of inertia of the rotary arm in terms of the length and masses given in Figure 2 1 As already mentioned the total arm length varies depending on where the short arm is mounted on top of the long arm The inertia can be computed in the la
36. tage overshoot PO lt 5 Specification 3 Maximum link angle deflection a lt 12 5 deg Specification 4 Maximum control effort voltage Vm lt 10 V Desired closed loop poles e Damping ratio 0 6 e Natural frequency wn 20 rad s e Non dominant poles p3 30 p4 40 Design a feedback controller that places the poles of the closed loop system to the desired locations indicated The servo settling time overshoot link deflection and control effort time domain requirements are to be satisfied when the servo is tracking a 20 degree angle square wave Note for the settling time specifications the servo angle must reach 4 of its final steady state angle in the alloted time For instance if the measured steady state servo angle is 20 degrees then it must settle within 20 0 8 degrees in 0 5 seconds 3 2 Background In Section 2 2 we found a linear state space model that represents the Rotary Flexible Joint system This model is used to investigate the stability properties of the Flexible Joint system in Section 3 2 1 In Section 3 2 2 the notion of controllability is introduced The procedure to transform matrices to their companion form is described in Section 3 2 3 Once in their companion form it is easier to design a gain according to the pole placement principles which is discussed in Section 3 2 4 Lastly Section 3 2 6 describes the state feedback control used to control the servo position whi
37. the Rotary Flexi ble Joint system and explains how to setup and wire the system for the experiments Flexible Joint Workbook Student pdf This laboratory guide contains pre lab questions and lab experiments demonstrating how to design and implement a position controller on the Quanser SRVO2 Flexible Joint plant using QUARC setup_rotflex m The main Matlab script that sets the SRV02 motor and sensor parameters the SRV02 configuration dependent model parameters and the Flexible Joint i e rotflex sen sor parameters Run this file only to setup the laboratory config _srv02 m Returns the configuration based SRV02 model specifica tions Rm kt km Kg eta g Beq Jeq and etam the sensor calibration constants K POT K ENC and K_TACH and the amplifier limits VMAX_AMP and IMAX_AMP config _rotflex m Returns the Flexible Joint model inertia Jj and viscous damping B ROTFLEX_ABCD_egns_student m Contains the incomplete state space A B C and D ma trices These are used to represent the Flexible Joint sys tem calc_conversion_constants m Returns various conversions factors d_pole_placement_student m Use this script to find the feedback control gain K s_rotflex mdl Simulink file that simulates the Flexible Joint system when using a full or partial state feedback control q_rotflex_id mdl When ran with QUARC this Simulink model feed a Sine Sweep signal to the servo and measures the correspond
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