Calibration and measurement reconstruction

Transcription

1 Scuola universitaria professionale della Svizzera italiana Dipartimento Tecnologie Innovative Metrolog laborator Calibration and measurement reconstruction Scope Tasks - Calibration of a two-degree-of-freedom mechanical structure with collocated measurements and end effector measurement - Correction of the error trough modification of the collocated measurement values -Analsis of source of errors in positioning a device with collocated measurements - Positioning of the structure using collocated measurements and comparison with end effector measurements - Calculation of corrections for the collocated measurements - Integration of the corrections in the positioning device and comparison between the end effector measurements and the corrected collocated measurements Prerequisites - References - Time 4 hours Appendices - Contact Matteo Dotta

2 Metrolog laborator, Calibration and measurement reconstruction SUPSI-DTI Presentation Given is a two degree-of-freedom mechanical structure with fleible joints. Each aes is actuated b a voice-coil motor and is equipped with an optical encoder for the measurement of the displacement and. The kinematics of the sstem is such that the movement o and o of the end point (the small square platform) differs from the movement and of the two individual ais. The idea in this eperiment is to use an additional temporar measurement of o and o (performed with a laser interferometer) in order to characterize the relation between the position and of the ais and the end point position o and o. With this knowledge, functions = f ( o, o ) and = f ( o, o ) could be found to reconstruct the position and of the ais corresponding to the position ( o, o ) of the end point. As the eact form of these functions is unknown, the can be approimated with the polnomial form (here using second order polnomials) = f ( o, o ) θ + θ 2 o + θ 3 o + θ 4 o o + θ 5 2 o + θ 6 2 o = rec (.) = f ( o, o ) θ + θ 2 o + θ 3 o + θ 4 o o + θ 5 2 o + θ 6 2 o = rec (.2) where rec and rec stand for the reconstructed measurements (i.e. the estimate of what would be and for obtaining a desired o and o ). The problem is now to find the parameters θ i and θ i. This can be done with a set of measurements (at least as man measurements as parameters to be determined, but with a null degree of freedom) given b the vectors of variables,, o and o. From equation (.) rewritten as = θ + θ2 o + θ3 o + θ4 o o + θ5 2 o + θ6 o 2 +e }{{} rec where e denotes the error between and rec, the vector equation = θ + θ 2 o + θ 3 o + θ 4 o o + θ 5 2 o + θ 6 2 o + e for a set of measurements is obtained, where the power of a vector denotes here, with a small abuse of notation, the power of the elements of the vector. Finall the same equation can be rewritten in the following wa using matri/vector multiplications: θ θ 2 = [ o o o o 2 o o 2 ] θ 3 }{{} θ4 +e = Φ Θ + e Φ θ5 θ6 }{{} Θ 2 Silvano Balemi

3 SUPSI-DTI Metrolog laborator, Calibration and measurement reconstruction The approimation error e = Φ Θ can be minimized with the least square estimate Θ = Φ + giving the parameters θ i for the polnomial (.), and similarl the error e = Φ Θ can be minimized with the least square estimate Θ = Φ + giving the parameters θi for the polnomial (.2). The matri Φ + = (Φ T Φ) Φ T is the so called pseudo-inverse of Φ which can be computed in Matlab with the command pinv. Then, even without the end point measurement (under the assumption that the sstem is precise) it is possible to accuratel position the end point ( o, o ) taking the ais to the position ( rec, rec ) obtained from (.) and (.2). The objective of the present eperience is to go trough the calibration procedure, to determine the polnomial approimations of the functions f and f with least square optimizations and to verif the improvement on a real precision sstem. 2 Set-up 2. Necessar material Given are following items The mechanical sstem with two degrees of freedom shown on the front page PC with Linu RTAI, Matlab/Simulink and the Matlab librar rtcpci Compact-PCI sstem (cpci8) with a PC board, two sinusoidal encoder boards and one driver board Laser interferometer 2.2 Steps to be followed When working on the sstem the following steps have to be taken.. From the PC open a remote shell on the Compact-PCI PC with the command telnet cpci8. The username is root and the password is root 2. On both the PC and the Compact-PCI PC tpe the command loadrtai 3. Download the file smt/courses/distr calib.tgz onto the PC and uncompress it with the command tar -fvz distr calib.tgz, which creates the director calib, Silvano Balemi 3

4 Metrolog laborator, Calibration and measurement reconstruction SUPSI-DTI 4. Go into the director calib just uncompressed, open Matlab on the PC with the command matlab -nojvm (which starts matlab without the Java virtual machine), 5. Run the command base id in Matlab to identif the transfer function (it uses data id.dat and data id.dat), run the command base ctr to design the controller and run the command scan to define the scanning trajector. 6. Open and run the simulink model scan.mdl (see figure.) to generate a theoretical trajector (stored in a workspace variable scan data). Tpe [o,o,,,r,r]=plot traj(scan data,ts) to plot the desired trajector, the movement of the optical encoders and the resulting end point trajector given be the laser interferometer (see figure.2). The variables [o,o,,,r,r] returned b the function give onl the reference points (see circles, and stars on figure.2) and not all the points of the trajector. o o o o Trajector ctr_base Controller 2dof fleure laser interferometer scan_data Figure.: Simulink file scan.mdl for the generation of the nominal trajector 7. Open the simulink model scan RT.mdl (see figure.3). The difference with the file scan.mdl (figure.) consists in the fact that instead of the mathematical model of the 2DOF structure the block 2dof fleure now defines the interfaces with the real mechanical sstems and the block laser interferometer defines the interfaces for the interferometer measurements. Further, the controller and the plant are enabled onl after a freeze time to get correct initial conditions. With the command Tools/Real-Time Workshop/Build Model from the menu, compile the simulink model to generate the eecutable file scan RT. 8. Using ftp, download the eecutable file scan RT from the PC onto the Compact- PCI PC 9. On the Compact-PCI PC, make the file scan RT eecutable with the command chmod + scan RT 4 Silvano Balemi

5 SUPSI-DTI Metrolog laborator, Calibration and measurement reconstruction Reference (black ), optical encoders (blue. ) and actual end point trajector (red) ais [m] ais [m] 0 3 Figure.2: Simulated trajectories. The encoder measurements (blue) follow the reference, the end point trajector (red) doesn t. o o Trajector freeze controller freeze position i i Controller o o laser interferometer sfun_rtai_scope 2dof fleure Figure.3: Simulink file scan RT.mdl for the generation of the nominal trajector on the real sstem 3 Assignment tasks. Based on the simulation data, find the approimate functions = f ( o, o ) and = f ( o, o ) using a fourth order polinomial interpolation in function of the variables o and o. The best parameters have to be determined with a least square optimization. 2. The functions f and f must be implemented in Simulink. Use a block of Silvano Balemi 5

6 Metrolog laborator, Calibration and measurement reconstruction SUPSI-DTI the form shown in figure.4 where Theta indicates the parameters of the interpolation polnomials obtained in the previous point. o o 2 [rec,rec] = f(o,o) o rec o f rec Theta parameters Theta rec 2 rec Figure.4: Implementation of the approimations of f and f. The Simulink librar block to be used is the so-called Embedded MATLAB Function. Cop scan.mdl to calib.mdl and introduce the obtained block at the appropriate place in calib.mdl. Finall rename the name of the scope variable written to the workspace from scan data to calib data 3. Run calib.mdl to simulate the behavior of the sstem with the reconstructed variables. Tpe [o,o,,,r,r]=plot traj(calib data,ts) to plot the desired trajector, the movement of the optical encoders and the resulting end point trajector given be the laser interferometer What can ou conclude? 4. On the CPCI platform tpe the command scan RT -w to prepare for eecution (in wait state) the application previousl compiled from scan RT.mdl. In a shell of the PC start the monitor program rtailab with the command rtailab. Arm the monitor program to store 20 seconds of data into the file scan datart.dat and start the real-time application with the blue arrow ke. Wait for 20 seconds, then load the data just measured into the workspace with the command load scan datart.dat which defines the variable scan datart accordingl. Tpe [o,o,,,r,r]=plot traj(scan datart,ts) to plot the desired trajector, while the movement of the optical encoders and the resulting end point trajector given be the laser interferometer. Verif that the optical encoder measurements (blue) follow the desired trajector, the laser interferometer measurements (red) do not. Calculate the RMS deviation of the selected end points from the desired position. 6 Silvano Balemi

7 SUPSI-DTI Metrolog laborator, Calibration and measurement reconstruction 5. Like in step approimate the functions = f ( o, o ) and = f ( o, o ) with a fourth order polnomial in function of the variables o and o (now,, o, and o come from measurements). 6. Cop scan RT.mdl to calib RT.mdl and introduce the two reconstruction blocks at the appropriate place in calib RT.mdl like done previousl in step 2 for calib.mdl. With the command Tools/Real-Time Workshop/Build Model from the menu, compile the simulink model to generate the eecutable file calib RT. 7. Using ftp, download the eecutable file calib RT from the Desktop PC onto the Compact-PCI PC. On the Compact-PCI PC, make the file calib RT eecutable with the command chmod + calib RT and prepare it for eecution with the command calib RT -w (starting in wait state). In a shell of the PC start the monitor program rtailab with the command rtailab. Arm the monitor program to store 20 seconds of data into the file calib datart.dat and start the real-time application with the blue arrow ke. Wait for 20 seconds, then load the data into the workspace with the command load calib datart.dat which defines the variable calib datart accordingl. Tpe [o,o,,,r,r]=plot traj(calib datart,ts) to plot the desired trajector, the movement of the optical encoders and the resulting end point trajector given be the laser interferometer. Verif that the laser interferometer measurements (red) follow the desired trajector, the optical encoder measurements (blue) do not. Calculate the RMS deviation of the selected end points from the desired position. Silvano Balemi 7