SLAM D 14
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| Lizenz | CC-Namensnennung - keine kommerzielle Nutzung - keine Bearbeitung 3.0 Deutschland: Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nicht-kommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. | |
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SLAM and path planning45 / 76
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00:00
KorrelationsfunktionMultiplikationsoperatorResultanteNeuroinformatikWärmeübergangRauschenTermGamecontrollerRechter WinkelTopologieRelativitätstheorieMatrizenrechnungAnalogieschlussTransportproblemDerivation <Algebra>AggregatzustandFunktionalWeg <Topologie>VarianzOrdnung <Mathematik>VorhersagbarkeitSigma-AlgebraKlasse <Mathematik>Physikalisches SystemRobotikNichtlineares GleichungssystemPartielle DifferentiationKovarianzmatrixDimensionsanalyseComputeranimation
01:44
Video GenieBitratePrimidealDerivation <Algebra>Jensen-MaßThetafunktionZusammenhängender GraphPartielle DifferentiationSinusfunktionTermProdukt <Mathematik>TeilbarkeitBruchrechnungMultiplikationsoperatorMereologieFunktionalNichtlineares GleichungssystemTrigonometrische FunktionVorzeichen <Mathematik>SchlüsselverwaltungMetropolitan area networkRuhmasseKlasse <Mathematik>ZweiComputeranimation
03:47
Zusammenhängender GraphCliquenweiteDerivation <Algebra>Jensen-MaßCASE <Informatik>Nichtlineares GleichungssystemMailing-ListeAbgeschlossene MengeZellularer AutomatComputeranimation
04:36
Partielle DifferentiationSchnittmengeNichtlineares GleichungssystemZusammenhängender GraphRechenschieberCASE <Informatik>Derivation <Algebra>Elektronische PublikationTaskComputeranimation
04:52
KontrollstrukturMathematikThetafunktionCodeJensen-MaßKomponente <Software>MatrizenrechnungDerivation <Algebra>Elektronische PublikationTaskGamecontrollerCASE <Informatik>MAPFunktionalDatensatzXML
05:08
KontrollstrukturJensen-MaßThetafunktionCodeDerivation <Algebra>VariableNotepad-ComputerMatrizenrechnungAggregatzustandCASE <Informatik>MatrizenrechnungFunktionalDatensatzXML
05:23
Analytische MengeVariableKontrollstrukturCodeThetafunktionGleitkommarechnungDerivation <Algebra>Notepad-ComputerMultiplikationsoperatorWorkstation <Musikinstrument>HochdruckDifferenteAnalytische MengeFunktionalCASE <Informatik>DifferentialXML
Transkript: Englisch(automatisch erzeugt)
00:00
Now let's have a look at where we are. So our prediction step was as follows. Our mu was computed by g and our sigma prediction was computed by g sigma g transposed plus our system noise r. And so we have set up this equation and we also computed g which was the derivative with respect to the state. So we know how to do this and this and this is the previous sigma.
00:23
So we are done with that. What remains is this. Now this is a 3x3 matrix which results from the noise of our control. And so similar to here we compute our RT from the noise in our control given by the covariance matrix of our control multiplied from the left and right by a matrix V where V is
00:43
the derivative of g with respect to the control. So see the analogy between this and that where this construct transports the variance of the old state to the variance of the new state. And this construct transports the variance of the control to the new state.
01:01
And in order to write that out we will have RT is 3t times the variance of the left control the variance of the right control times the transposed of b. So those variances capture the inexactness of our movement of the left and right track of the robot assuming that there is no correlation between the two.
01:21
So if you look at the dimensions this is 3 times 3. Now this as we see is 2 times 2. So this must be 3 times 2. And this is the transposed matrix so it's 2 times 3. And so indeed this V matrix looks like this. So it is indeed 3 times 2. The first column being the partial derivatives with respect to L.
01:41
The second being with respect to R. Now let's compute those derivatives. So our function g, just g1, g2, g3 is x, y, theta plus those terms. And now we have to compute the partial derivative of g1 with respect to L. Now there is no L in those equations because it is hidden in R.
02:02
So R was L divided by alpha and alpha was R minus L divided by W. So we see that R equals L W divided by R minus L. And so the term R plus W half, this is L W divided by R minus L plus W half. Which is the same as W half times R plus L divided by R minus L.
02:24
So let's go on here. We have to compute the partial derivative with respect to L of the first component. Which is x plus this term here, which we just computed, times the sine of theta prime. Where we will just set theta prime is theta plus alpha minus the sine of theta.
02:42
And so the derivative of x with respect to L is zero. Whereas here we have the L in this term and we also have the L in the alpha hidden here. So we have to apply the product rule. So the derivative of the first factor is 1 divided by the denominator squared times
03:00
R minus L plus R plus L times this part unmodified plus the first part unmodified times the derivative of the second part, which is the cosine, times the derivative of theta prime with respect to L. So this is the derivative of alpha with respect to L, which is minus 1 divided by W.
03:21
So overall we obtain W R divided by R minus L squared times the sine of theta prime minus sine of theta minus, now this comes from here, R plus L divided by 2 R minus L times the cosine of theta prime. So this is our partial derivative of the first component of G with respect to L.
03:44
And we will have to do six of those. Let me just write down all those for you. So this is what we just computed. And for the second component we have, for the third component, this is quite simple. Minus 1 divided by the width. Now the same for the derivatives with respect to R.
04:03
So we get a minus here and a plus here. And finally, the derivative of the third component with respect to R is 1 divided by the width. And so remember this is for R not equal to L and the theta prime which is used here is theta plus alpha.
04:21
Now unfortunately we'll have to do all this for the case R equals L as well. And so for every component we will have to find out what happens if alpha goes to zero and so theta prime goes to theta. So let me just give you those equations. So in the case R equals L we will have to use this set of equations with a partial
04:44
derivative of the third component of G with respect to L and R as the same as on the previous slide. So now let's program this. So I prepared this SLAM7C control derivative question file for you. And so essentially your task is the same as in the previous case only that the derivative
05:02
of G you have to take is now with respect to control meaning to L and R and not with respect to the state. And so this is the function you'll have to fill out. Again there's two cases for R not equal to L and for R equal to L and you'll have to return a matrix which is three rows times two columns.
05:23
And down here in the main function everything is set up as in the previous case. This time the numeric differentiation is with respect to L and R and it will compare this to your analytic solution and will bring out the difference. So now please program this.