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The prediction step generates a new set of states from the old set of states. The motion model of the system is used to make this best estimate of what we think the new state might be. The motion model basically uses the information about the previous state and the current control input to determine the new state. Some noise is also added for stochasticity. ...

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For a simple case, given a particle set, calculate the weighted sum of the particles by iterating over each particle $i$ in the set $N$: $$\mu_{x} = \Sigma_{I=1}^{N} w_{i} x_{i}$$ And then calculate the weighted covariance for each particle $i$ in the set $N$ and sum them up: $$P_{xx} = \Sigma_{I=1}^{N} w_{i} (x_{i}-\mu_{x})(x_{i}-\mu_{x})^{T}$$ Where $w_{i}$...

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Matching point clouds can be very tricky. It is kind of a needle-in-a-haystack type of problem when you don't have an initial guess at the correspondence. As you found, if the point clouds are very different there really isn't a great way to quantify the similarity. This holds even if the two scans are similar (or even the same!) but have very different ...

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Like edwinem mentioned, the motion model just describes how the object should move. Consider gravity: $$\ddot{y} = -g \\$$ If you wanted a motion model for position, then: $$y = y_0 + \dot{y}t + \frac{1}{2}\ddot{y}t^2 \\ y = y_0 + \dot{y}t - \frac{1}{2}gt^2 \\$$ So if you have a ball at $y_0 = 5$, does the ball go down in the next instant or does it ...

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