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What we do is we make a matrix and also a vector.
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We label the matrix, which is quadratic, with all the poses and all the landmarks.
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Here we assume the landmarks are distinguishable.
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Every time we make an observation, say between two poses,
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they become little additions, locally,
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in the 4 elements in the matrix defined over those poses.
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For example, if the robot moves from x0 to x1,
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and we therefore believe x1 should be the same as x0, say, plus 5,
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the way we enter this into the matrix is in two ways.
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First, 1 x0 and -1 x1--add it together should be -5.
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So we look at the equation here--x0 minus x1 equals -5.
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These are added into the matrix that starts with 0 everywhere,
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and it's a constraint that relates x0 and x1 by -5. It's that simple.
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Secondly, we do the same with x1 as positive, so we add 1 over here.
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For that, x1 minus x0 equals +5, so you put 5 over here and a -1 over here.
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Put differently, the motion constraint that relates x0 to x1 by the motion of 5
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has modified incrementally by adding values the matrix for L elements
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that fall between x0 and x1.
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We basically wrote that constraint twice.
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In both cases, we made sure the diagonal element was positive,
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and then we wrote the correspondant off-diagonal element as a negative value,
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and we added the corresponding value on the right side.
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Let me ask you a question.
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Suppose we know we go from x1 to x2 and whereas the motion over here
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was +5, say, now it's -4, so we're moving back in the opposite direction.
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What would be the new values for the matrix over here?
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I'll give you a hint.
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They only affect values that occur in the region between x1 and x2 and over here.
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Remember, these are additive.