# What is a Second Order Moment Matrix?

Discussion in 'Math' started by sirch2, Jan 28, 2015.

1. ### sirch2 Thread Starter Well-Known Member

Jan 21, 2013
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I am trying to understand the paper linked below. The paper discusses a laser scanning device that can do Simultaneous Localization and Mapping (SLAM) so basically it scans the surfaces around it and works out both the position of the device and the shape of the surrounding space whilst moving continuously.

At the bottom of the left hand column of page 5 it says

In the above the "trajectory estimate" is an initial best guess at how the device has moved since the last known location

Now, assuming that I am pretty thick (it would be a fair assumption), is there a straight forward explanation of what a Second Order Moment Matrix is without a load of Matrix multiplication equations? I would be happy with text or a psuedo-code algorithm or a diagram but a screed of Matix equations would just be lost on me.

Jan 29, 2010
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3. ### sirch2 Thread Starter Well-Known Member

Jan 21, 2013
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Thanks Eric, I've read that at least a dozen times but I guess I need the bit between the first paragraph and the first equation which starts banging on about "a real-valued continuous function f(x)" - BTW I loved the whooshing sound that made as it flew over my head.

In the paper I cited they would seem to be trying to find two groups of points that represent the same part of a surface, even though the points are not the same. Anyone got a clue as to what the second order moment of a point in space is?

4. ### studiot AAC Fanatic!

Nov 9, 2007
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The moment of inertia is a second order centralised moment.

Do you understand this one?

5. ### sirch2 Thread Starter Well-Known Member

Jan 21, 2013
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Not that my actual concern is anything to do with angular momentum but I'm guessing that your point is that for a simple pendulum the moment of inertia I=mr^2 and so it becomes second order because of the r-squared? But then again perhaps not.

But I am particularly interested in how this is applied to a set of random points on a surface and yields a value identifying a part of a surface. Perhaps I should take this to a maths forum, but I am reluctant to do that since I will probably get a mathematical answer.

Jan 29, 2010
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7. ### sirch2 Thread Starter Well-Known Member

Jan 21, 2013
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like the cartoon.

That link helps a bit, thanks

8. ### studiot AAC Fanatic!

Nov 9, 2007
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No I realise you are not concerned with momentum, but step by step; I don't know what you do know about moments.

I hoped the example would be a bit familiar and reinforce the point that the first moment is a weight quantity (such as area, mass etc) times its distance from an axis
The second moment is times the square of that distance
The third moment is times the cube..

etc.
Also that whilst the axis can be anywhere, you asked for a central moment, which is about some middle or mean centreline.
The 'centreline' is the line baout which things are 'balanced'
Moments of Inertia are normally taken like this.

You may note that there is only one axis in this case. One axis corrsponsds to a single independent variable

None of the above would merit matrix methods for the summation.

However if the second moment quantity is the result of two or more independent variables then there are two or more axes.

The summation now comprises the squares of distances from two (or more ) axes and also cross producs betwen the axes.
Least squares equations are now used to calculate tha balance.
Since in your example they are a series of like points I expect the weights are all 1.

In order order to keep track of everything matrices are now invaluable.

I can give you simplified equations if you like.

9. ### sirch2 Thread Starter Well-Known Member

Jan 21, 2013
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Thanks Studiot, it really feels like I am getting somewhere with this. If it not too much trouble equations may actually help at this point. I'm guessing that it's the least squares part that find the best fit between the two sets of points?

Chris

10. ### studiot AAC Fanatic!

Nov 9, 2007
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Note where is defines the matrices in the third sheet.

11. ### sirch2 Thread Starter Well-Known Member

Jan 21, 2013
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Many thanks for that, I'll try to work my way through it.