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I think it is likely that you are confusing two uses of the word “singular”: The singular values of a matrix as found via singular value decomposition. The singular configurations (or singularities) that occur when the Jacobian loses rank.

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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 ...

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The Jacobian is a 6x6 matrix and there are multiple ways it can be derived. The concept of the Jacobian has its roots in Mathematics, where it is not stricly tied to any problem, it is just a matrix of partial derivates of an equation system. As such it is obtained as:  J = \frac{\partial f}{\partial \theta} = \begin{bmatrix} \frac{\partial f_{\text{1}...

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If I understand correctly, your end effector is at pose A, you want to move it to pose B, and you have Jacobian pseudo-inverse control set up so that you can specify a pose velocity $v$ and get a corresponding joint velocity $\omega$ that will produce that pose velocity. There are several ways to produce the $v$ vector. All of them start by defining a time ...

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