How can the manipulability of a robot arm be calculated when the kinematics are described with screw theory such as dual quaternions? These methods are sigularity-free, but what could we say about dexterity?
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A robot arm's dexterity is a function of its physical structure (arrangement of joints) and configuration (joint positions). And it can change depending on the way the arm is positioned. It has nothing to do with the mathematical representation of the arm (DH parameters vs. dual quaternions, etc).
Looking at the function for manipulability involves the Jacobian. But since this must be computed at specific joint angles, the Jacobian is just a matrix of numbers:
$$ \mathit{w}(q) = \sqrt{det(\mathbf{J}(q)\mathbf{J}^T(q)} $$
As you said, this dexterity can be calculated with a "manipulability" score. But this is just a metric. And it is not the only one. I discuss a few other metrics in my answer to this post: Evaluating the similarity of two 7 Degree of Freedom Arms
And for an example of how different arrangements of joints can lead to different manipulabilty scores, see my answer in this post: How do you arrange the dof in a robotic arm?
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$\begingroup$ could you give more information about manipulability ellipsoid ???? when we used formulation dual quaternions for kinematics... $\endgroup$– hamedmhCommented Jun 11, 2020 at 7:18