# Unachievable motions because of singular Jacobian

This is the Jacobian of a robot arm (IBB IRB 120) with already specified joint angles (0,0,0,0,0,0). This Jacobian is singular for this configuration.

J =
[       0,    17/50,    7/100, 0,      0, 0]
[ 187/500,        0,        0, 0,      0, 0]
[       0, -187/500, -187/500, 0, -9/125, 0]
[       0,        0,        0, 1,      0, 1]
[       0,        1,        1, 0,      1, 0]
[       1,        0,        0, 0,      0, 0]


What motions are unachievable with this specific configuration? How to find that out conveniently with MATLAB?

The Jacobian defines the relation between infinitesimal displacements $dx$ in the task space and infinitesimal displacements $dq$ in the joint space: $$dx = \mathbf{J} dq$$ One way to calculate unachievable directions of (infinitesimal) motion from the Jacobian is to use it's singular value decomposition: $$\mathbf{J} = U \Sigma V^T$$ The left singular vector matrix $U$ and the right singular vector matrix $V$ are orthogonal and span the (infinitesimal / local linearized) task and joint spaces respectively; the diagonal matrix of singular values $\Sigma$ determines the scaling between the individual input and output directions.
Unachievable motions are scaled versions of singular vectors corresponding to zero singular value(s). A right singular vector $V$, corresponding to a zero singular value, implies that this input direction (i.e. movement in the joint space) will result in zero movement in the task space. A left singular value $U$, corresponding to a zero singular value, implies that (due to the zero scaling factor) movement in this output direction is not possible.
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