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Coordinate Dependence of Variability Analysis
Figure 1
Sketch of a two-joint arm reaching to a target line as a simple example for a redundant task.
The target line is defined in extrinsic coordinates, x, y. A: Illustration of absolute angle definitions of the shoulder and elbow joint, 
and 
, with respect to the shoulder axis. The relation between joint angles and extrinsic hand coordinates is given by the equations where l1 and l2 refer to the respective segment lengths, both 40 cm in the simulation. B: Relative joint angle definitions, 
and 
. C: Simulated data displayed in the space of the relative joint angles, 
and 
. The color code denotes deviations from the target line, with darker colors referring to larger distances. The curved white line denotes the UCM, the set of solutions for which the end-effector is exactly on the target line. The dashed straight line denotes the nullspace of the Jacobian matrix evaluated at the mean of the data distribution, providing a linear approximation that is tangent to the UCM at that point. The data are aligned with the direction of the nullspace, i.e. show structure. D: The same simulated data displayed in the space of the absolute joint angles, 
and 
. The same data are now not aligned with the direction of the nullspace, i.e. do not show structure. E: The same set of data displayed in extrinsic hand space, in which it was generated to have an isotropic random distribution with its mean centered on the target line.
doi: https://doi.org/10.1371/journal.pcbi.1000751.g001