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By watching this video which explains how to calculate the classic Denavit–Hartenberg parameters of a kinematic chain, I was left with the impression that the parameter $r_i$ (or $a_i$) will always be positive.

Is this true? If not, could you give examples where it could be negative?

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I think this is true. Although I don't have a formal proof. I haven't read the original Denavit–Hartenberg paper, but this paper: "Lipkin 2005: A Note on Denavit-Hartenberg Notation in Robotics" explains the 3 main DH parameter conventions and how they differ. It breaks down each of the 4 parameters into vector forms. Specifically:

$$ a_2 = \vec{{O}_1 {O}_2} \cdot {x}_2 $$ Where $\vec{{O}_1 {O}_2}$ is the vector between the two frames. The dot product can be negative, but only if the angle is greater than 90 degrees. But because of the strict way you construct the frames, I don't think that can happen.

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  • $\begingroup$ While a length is alway positive I think you define a link frame such that the distance to the next link frame was $-a$ or $-d$. $\endgroup$
    – Peter Corke
    Oct 19, 2019 at 12:26
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Parameter "r" or "a" is the link length of a robot's link which is a physical entity. And that can never be negative.

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