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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?

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In MATLAB use ''svd'' to find the singular vectors corresponding to zero singular values of the Jacobian. The corresponding right singular vectors describe the input directions (i.e. movement in the joint space) that result in zero movement in the task space; the corresponding left singular vectors describe the output directions (in the task space) in which no movement is possible through any combination of movements in the joint space.

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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