1. Continuing with Jacobian and its uses ME 4135 – Slide Set 7 R. R. Lindeke, Ph. D.

2. Connecting the  Operator to the Jacobian  Examination of the Velocity Vector:  If we consider motion to be made in UNIT TIME: dt =  t = 1  Then x dot (which is dx/dt) – is dx  Similarly for y dot, z dot, and the  ’s. They are: d y, d z and  x,  y, and  z respectively

3. These data then can build the  operator: Populate it with the outtakes from the D dot Vector – which was found from: J*Dq dot

4. Using these two ideas:  Forward Motion in Kinematics: Given Joint Velocities and Positions Find Jacobian (a function of Joint positions) & T 0 n Compute D dot, finding d i ’s and  i ’s – in unit time Use the d i ’s and  i ’s to build  With  and T 0 n compute new T 0 n Apply IKS to new T 0 n which gets new Joint Positions Which builds new Jacobian and new D dot  and so on

5. Most Common use of Jacobian is to Map Motion Singularities  Singularities are defined as: Configurations from which certain directions of motion are unattainable Locations where bounded (finite) TCP velocities may correspond to unbounded (infinite) joint velocities Locations where bounded gripper forces & torques may correspond to unbounded joint torques Points on the boundary of manipulator workspaces Points in the manipulator workspace that may be unreachable under small perturbations of the link parameters Places where a unique solution to the inverse kinematic problem does not exist (No solutions or multiple solutions)

6. Finding Singularities:  They exist wherever the Determinate of the Jacobian vanishes:  Det(J) = 0  As we remember, J is a function of the Joint positions so we wish to know if there are any combinations of these that will make the determinate equal zero  … And then try to avoid them!

7. Finding the Jacobian’s Determinate  We will decompose the Jacobian by Function: J11 is the Arm Joints contribution to Linear velocity J22 is the Wrist Joints contribution to Angular Velocity J21 is the (secondary) contribution of the ARM joints on angular velocity J12 is the (secondary) contribution of the WRIST joints on the linear velocity  Note: Each of these is a 3X3 matrix in a full function robot

8. Finding the Jacobian’s Determinate  Considering the case of the Spherical Wrist:  J12:  Of course O 3, O 4, O 5 are a single point so if we ‘choose’ to solve the Jacobian (temporally) at this (wrist center) point then J12 = 0! This really states that O n = O 3 = O 4 = O 5 (which is a computation convenience but not a ‘real Jacobian’)

9. Finding the Jacobian’s Determinate  With this simplification: Det(J) = Det(J11)  Det(J22) The device will be singular then whenever either Det(J11) or Det(J22) equals 0 These separated Singularities would be considered ARM Singularities or WRIST Singularities, respectively

10. Lets Compute the ARM Singularities for a Spherical Device  From Earlier efforts we found that:  To solve lets “Expand by Minors” along 3 rd row

11. Lets Compute the ARM Singularities for a Spherical Device After simplification: the 1 st term is zero; The second term is d 3 2 S 2 2 C 2 ; The 3 rd term is d 3 2 C 2 2 C 2

12. Lets Compute the ARM Singularities for a Spherical Device, cont. This is the ARM determinate, it would be zero whenever Cos(  2 ) = 0 (90  or 270  )