SLAM D 04
This is a modal window.
Das Video konnte nicht geladen werden, da entweder ein Server- oder Netzwerkfehler auftrat oder das Format nicht unterstützt wird.
Formale Metadaten
| Titel |
| |
| Serientitel | ||
| Anzahl der Teile | 76 | |
| Autor | ||
| 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. | |
| Identifikatoren | 10.5446/49002 (DOI) | |
| Herausgeber | ||
| Erscheinungsjahr | ||
| Sprache |
Inhaltliche Metadaten
| Fachgebiet | |
| Genre |
SLAM and path planning55 / 76
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
00:00
DatensatzMatrizenrechnungAusdruck <Logik>ErwartungswertMoment <Mathematik>Konstruktor <Informatik>MultiplikationsoperatorVarianzSigma-AlgebraKovarianzfunktionComputeranimation
00:48
DatensatzErwartungswertSigma-AlgebraQuadratzahlMultiplikationsoperatorComputeranimation
00:57
VarianzMatrizenrechnungErwartungswertNeuroinformatikComputeranimation
01:02
EigenwertproblemMatrizenrechnungSigma-AlgebraFehlermeldungMultiplikationsoperatorComputeranimation
01:14
EigenwertproblemVektorraumMultiplikationsoperatorNeuroinformatikComputeranimation
01:29
EigenwertproblemVektorraumMultiplikationsoperatorKlasse <Mathematik>ZweiRechter WinkelComputeranimation
01:49
MultiplikationsoperatorComputeranimation
01:54
EllipseFehlermeldungRichtungRechter WinkelGraphfärbungComputeranimation
02:10
RauschenVariableGeradeKette <Mathematik>VerschlingungMultipliziererZusammenhängender GraphElement <Gruppentheorie>SoftwareentwicklerEllipseFehlermeldungMoment <Mathematik>Computeranimation
02:45
VarianzEllipseMultiplikationsoperatorNormalverteilungURNComputeranimation
Transkript: Englisch(automatisch erzeugt)
00:00
Let's find this out. We're looking for the covariance matrix, which is the expected value of C minus the expected value of C times C minus the expected value of C transposed. And so C is as we have defined just xy, which is by the construction of our lever arm x and 2x. So this is because our lever arm looked like that. Here was our motor and here was the other joint and this was L and this was also L.
00:26
So every movement here at this joint is doubled at that joint. And so we knew that x was normal distributed with expectation 0 and variance sigma x squared. And so from that we see that the expectation of x was of course 0.
00:41
And so from that it follows that the expectation of C is 0, 0. And so the formula above here becomes very simple. Sigma now is just the expectation of C C transposed, which is the same as x2x times x2x as a row vector, which is simply x squared, 2x squared and 4x squared.
01:02
And this is symmetric. But the expectation of x squared is the variance of x. So this simply becomes sigma x squared times the matrix 1, 2, 2, 4. And so to compute the error ellipse, we just compute the eigenvectors of this matrix and we will now not care about sigma x squared. So the eigenvectors of 1, 2, 2, 4. And if you compute those you will obtain vector 1 is 1, 2.
01:26
Because 1, 2, 2, 4 times 1, 2 is 1 times 1 plus 2 times 2 and 1 times 2 plus 2 times 4, which is 5, 10. And so this is 5 times 1, 2. And so the eigenvalue w1 is 5.
01:41
Because if I put in this vector, I get out 5 times the vector. And so the second eigenvector is minus 2, 1. And this leads to the following, minus 2 plus 2 and minus 4 plus 4. And so this is 0, which is 0 times minus 2, 1. And so we get our error ellipse as follows. Our v1 is 1 to the right and 2 up.
02:02
v2 is 2 to the left and 1 up. And so this is perpendicular. Our error ellipse is indeed stretched in the v1 direction. w2 is 0. So it is indeed a degenerated ellipse, which is just this line. And this reflects the fact that by assuming that this link here is a perfect multiplier.
02:22
The two variables will be perfectly coupled. So if there is some error in x, then there will be just double the error in y with no added noise. In reality, I will be unable to produce these mechanical components perfectly. And so in reality, my lever arm will add some noise which will lead then to an error ellipse like that.
02:41
But as we have set up the problem, it is indeed a degenerated ellipse. Now let me ask you the following. Now for this mechanical setup, which we just had. With our ellipse being degenerated and x being normal distributed like that. y being normal distributed like that. What is the variance in y? Is it the same as in x? Is it 2 times the variance in x?
03:00
Or 4 times the variance in x? Or 5 times the variance in x? Now what is the correct solution? A, B, C or D?