600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001CalibrationCalibrate (vt) : 1. to determine the caliber of (as a thermometer tube); 2. to determine, rectify, or mark the gradations of (as a thermometer tube); 3. to standardize (as a measuring instrument) by determining the deviation from a standard so as to ascertain the proper correction factors; 4. ADJUST, TUNE600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001CalibrationRealityMeasuring deviceModel of Reality+-Error600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001CalibrationRealityActuation deviceModel of Reality+-Error600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Basic Techniques• Parameter Estimation• Mapping the space600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Parameter Estimation• Compare observed system performance to reference standard (“ground truth”)• Compute parameters of mathematical model that minimizes residual error.RealitySystemModel of reality+-ErrorParameters Calibration process600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Pointing device calibrationFRpiii=( , )pt (unknown)ppivot600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Pointing device calibrationFRpiii=( , )pt (unknown)ppivot600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Pointing device calibrationFRpiii=( , ) tpppivot600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Pointing device calibrationitipivot+=RpppMMFRpiii=( , )pt (unknown)ppivot600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Pointing device calibrationtjpivotj≅−−pRIppMMMMMMFRpiii=( , )pt (unknown)ppivot600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Linear Parameter Estimationp f q qp f q qf q qf q q qfnom nijnq qfqqqJ= == +≈ +∂∂LNMMMMOQPPPPLNMMMOQPPP≡ +( ) [ ,...( )( ) ( )( ) ( ) where ] are parametersT*11∆∆∆∆O ML LM OM600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Linear Parameter Estimation•Given f(q), and a set of observations pk* corresponding to nominal parameter values qk*, solve the least squares problem MMMMJk k kfq q p f q( ) ( )*LNMMMOQPPP≈ −LNMMMOQPPP∆600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Example: 2 link robot calibration*kp1a2a600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Example: 2 link robot calibration*kp1a2a600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Example: 2 link robot calibrationpp=++LNMMMOQPPP= +=+ + + + + ++ + + + + +LNMMMOQPPPa aa aa a a aa a a a1 1 2 121 1 2 1212 1 21 1 1 1 2 2 12 1 21 1 1 1 2 2 12 1 200sin sin( )cos cos( )( )sin( ) ( )sin( )( cos( ) ( )cos( )*θθθ θθ θ θθ θ θ θ θθ θ θ θ θ where ∆ ∆ ∆ ∆ ∆∆ ∆ ∆ ∆ ∆600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Example: 2 link robot calibrationp f q ffqfqfqfq pyk kk kk kk k k k ka aa aa aa aaaa Rot= = =++LNMMMOQPPP= +∂∂∂∂∂∂∂∂LNMMMMOQPPPPLNMMMMOQPPPP≈ −( ) ( , , , )sin sin( )cos cos( )( ) ( ) ( ) ( )( ,, ,, ,*,1 2 1 21 1 2 121 1 2 1212 1 21 2 1 212121 10θ θθ θθ θθ θ θθ θθθθ where so we solve the least squares problemMMMMMMMMM∆∆∆∆kka Rot)( , ),+LNMOQPLNMMMMMOQPPPPP2 1y θM600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Example: 2 link robot calibrationHeresoJa aa aa aa aaakk k k k kk k k k kk k k k kk k k k kfq( )sin sin (cos cos ) coscos cos (sin sin ) cossin sin (cos cos ) coscos cos (sin sin ) cos, , , , ,, , , , ,, , , , ,, , , , ,=+− + −LNMMMOQPPP+− + −LNMMMMOQPPPPθ θ θ θ θθ θ θ θ θθ θ θ θ θθ θ θ θ θθθ1 12 1 1 12 2 121 12 1 1 12 2 121 12 1 1 12 2 121 12 1 1 12 2 121210 0 0 0MM∆∆∆∆21 1 2 121 1 2 12LNMMMMOQPPPP≈− +− +LNMMMMOQPPPPMMx a az a ak k kk k k*, ,*, ,sin sin( )cos cos( )θ θθ θ600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Example: Robodoc Wrist Calibration• Basic robot had very accurate calibration • Custom wrist was less accurate• Crucial goal was to determine position of cutter tip CutterCalibration post600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Kinematic ModelppRzxvtoolwristdistal=++•+(,)()θθα44∆vRxRyvvdistalcc=•++(,)[(,)()]βθθ55∆∆600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001LinearizationppRRvvpost wrist c c≈+++[ ( )] .....4 5∆600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Linear Least Squares• Most commonly used method for parameter estimation• Many numerical libraries• See the web site• Here is a quick reviewMicrosoft PowerPoint Presentation600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Example: Undistorted fluoroscope calibration600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Calibration if no distortion (version 1){ }01Assume no distortion. For the moment alsoassume that you have N point calibrationfeatures (e.g., small steel balls) at knownpositions ,, relative to the detector.Assume further that the pointsN−aaL{ }01 create imagesat corresponding points ,, on the detector. Estimate the position of the x-raysource relative to the detector N−ddsL600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001ApproachskdkaCalibration objectobjFDetector600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Projection of a point featuresda()ssλλµµ=−+=s=(a-d)+d(a-d)•(s-d)=(a-d)•(a-d)dass(a-)•(d-s)(a-)•(a-s)600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Approachsda()()()()000zzyyzzxxyyxxskews−×−=−•=−×=×−×=×−−−−=×−−adsd0adaddadddaddaadaddasaddaad600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Approachsda00001111Solve least squares problem ()()xyzNNNNskewskew−−−−−×≅−×sadadssadadMMMM600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001What if pose of calibration object is imprecisely known?• This is a hairier problem, but solvable• In fact, it makes a great homework assignment ….600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Mapping the space• Compare observed system performance to reference standard (“ground truth”)• Interpolate residual errorsRealitySystemModel of reality+-ErrorLookup TableModel Correction600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001Example: Fluoroscope