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I am reading about singularities and I know that those are configurations which result in the same end-effector position and orientation. I also read that joint velocities get very high near that singularity but I don't understand why.
I get it that they have to be very high when going through that singularity because the arm has to be in multiple poses at once, but I don't know why the speed is so high if it is just near a singularity.
Please help me understand.
First of all, singularities are not configurations that have the same end-effector position and orientation. Those configurations are inverse kinematic (IK) solutions to that end-effector pose (position and orientation).
The formal definition of singularities is the configurations that the Jacobian loses its rank. At such configurations, the manipulator may lose its degrees-of-freedom (DOF), i.e., it may not be able to move its end-effector in some certain direction.
Let's consider the relation $$\dot{x} = J(q)\dot{q},$$ where $x \in \mathbf{R}^6$ is a vector representing an end-effector pose, $q \in \mathbf{R}^n$ is a robot configuration, $J(q)$ is the Jacobian at the given configuration, and $n$ is the number of DOFs.
For the sake of argument, consider when $n=6$, i.e., $J(q)$ is a square matrix. At a singular configuration $q^\ast$, the Jacobian loses its rank and therefore $\det(J) = 0$. Looking back to the previous relation, you may see that no matter how large the joint velocity is, it does not contribute anything to the end-effector velocity.
Let's consider this from a mathematical point of view. From the singular-value decomposition of $J(q)$, one has $$\dot{x} = (U\Sigma V^{-1})\dot{q},$$ where $U$ and $V$ are some orthogonal matrices and $\Sigma$ is a diagonal matrix whose entries are singular values (well, not the same singular as what we are discussing). With some algebraic manipulation, one has $$\dot{x}' = \Sigma \dot{q}',$$ where $\dot{x}' = U^{-1}\dot{x}$ and $\dot{q}' = V^{-1}\dot{q}$. Those mapping by $U^{-1}$ and $V^{-1}$ can be seen as coordinate transformations.
At the vicinity of a singular configuration, the Jacobian nearly loses its rank. The singular values (i.e., the entries of $\Sigma$) are very small. Therefore, to attain a certain $\dot{x}'$ (that is, to be able to move in some direction), you would require a very large $\dot{q}'$.
Note, however, that after all, the high joint velocity behavior near a singular configuration is a result of using the relation $\dot{x} = J\dot{q}$ to calculate $\dot{q}$ from a given $\dot{x}$. As @Ugo pointed out in his comment, there exists methods to get around this behavior.
