Jacobian Implementation Ryan Keedy 5 / 23 / 2012

From Spong Jacobian is the matrix that transforms degrees of freedom rates of change into end effecter rates of change –Tracks velocity AND orientation –Always 6 rows big, DOF columns wide Upper 3 rows keep track of translation, lower three rows take care of orientation

From Spong Lower 3 rows: –In a rotation only, 2-D situation like ours, columns are simply each DOF’s local rotation matrix dotted with (0,0,1) Upper 3 rows: –Take the previous vectors and cross them with a vector running along the entirely of each joint length

Determining Joint Angles The non-square matrix can obviously not be inverted to find the joint angle velocities Multiplying by the transpose yields a singular, non-invertible matrix Turn to damped least squared paper: –Solve for J * = (J T W 1 J + W 2 ) -1 J T W 1 –Angular velocities are simply J * times x’

Considerations Because of the W matrices, result is not exactly what you want I’ve applied it incrementally, recalculating the desired translational velocity after each movement W 2 can be manipulated to alter the overall behavior/strategy of the arm

Conclusions Not as precise as the inverse kinematics solution Not sure how to integrate into planning –Based on W matrices, arm has a “mind of its own” –Solutions are “unique”