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I have been working with the Velocity Obstacles concept. Recently, I came across a probabilistic extension of this and couldn't understand the inner workings.

Source: Recursive Probabilistic Velocity Obstacles for Reflective Navigation http://www.morpha.de/download/publications/FAW_ASER03_Kluge.pdf

Shape Uncertainty Formulation

What does the equation at the bottom and the top mean? Vij is the relative velocity of agent i to agent j. ri & ci and rj & cj are their respective radius and centers.

Update: (based on Dale M's answer) What does inf(ri + rj) and sup(ri + rj) mean? Does it mean that I should define a function that goes from 1 to 0 from inf to sup? And if not, then how do I calculate the value of PCC at any given point?

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The statement (not an equation):

$$PCC_{i,j}:\mathbb{R}^2\to [0,1]$$

means that $PCC_{i,j}$ is a function from the 2 dimensional plane (the floor presumably) to the closed interval $[0,1]$ representing the probability of a collision from no chance $0$ to certainty $1$.

The equation states that the probability of a collision is the same as the probability that the sum of the radii of the objects $r_i$ and $r_j$ is greater than the minimum of the initial distance between the centres $c_i-c_j$ plus the future (relative) position of their centres $\mu v_{ij}$ over all future time $/mu$. That is, if the size of the bodies is bigger than the distance between them in the future then they will be coincident in space-time - i.e. they crash.

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  • $\begingroup$ Thank you, that helps a lot. I guess the "min" threw me off because ci - cj + uVij will only have one value at given time instant. So I was wondering why the min. Follow up question: What does inf(ri + rj) and sup(ri + rj) mean? Does it mean that I should define a function that goes from 1 to 0 from inf to sup? And if not, then how do I calculate the value of PCC at any given point? $\endgroup$
    – simplename
    Commented Oct 21, 2014 at 17:33
  • $\begingroup$ en.wikipedia.org/wiki/Infimum_and_supremum - in this context it is the least and greatest possible value of the sum of the 2 radii. $\endgroup$
    – Dale M
    Commented Oct 22, 2014 at 4:17

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