# Lecture 7

## Movie segment:
volley boll playin robot.

* what is Zi ?how to calculate it? : it is basically z axis or direction of angular velocity vector.
*


* Here we are finding end effector linear and angular velocity.
* angular velocity for only prismatic joints is zero for end-effector.
*

* here we have substituted Ziqdot to get below expressions..
* What is E1 , E2 .. and E1dash?
* This Jw and Jv are made up of Z1,Z2...
* Here we haven't decided any frame in which we have decided vector.
* Jv and Jw can be also simplified further if we have computed forward kinematics.
* if we use cartesian co-ordinates (instead of spherical co-ordiantes) we can directly differentiate Xp to get Jv.


* How we will do differentiation of Xp w.r.t q?
* In second slide below we have Zi in frame i then we convert it to frame zero(0) with rotation matrix.
* So basically we find oZ1, 0Z2 in last column of rotation matrix 0R1,0R2 etc.


* here he has determined J from basics i.e, from E1,E2 wala formula.


*


* Now we are simply finding translational matrix


* we need all the vectors in frame zero.
* Here we need only 3rd column of matrix.
* the red rectangles are Z1, Z2 etc.in frame zero.
* He mentioned that he will not find 2T3 then multiply with 1T2 to get 1T3 he will directly find 1T3 but he didn't explained how?..[00:51:58](https://youtu.be/6SRTAoyzC6A?t=3118) : well he was mentioning about order of multiplication i.e, go from 0 to 6 and not reverse.
* the last colomn is same from matrix 3 because we have reached end-effector.
* We simply partially derivate the last column (as it represent the position of end-effector) to get ${J}_v$
*


* Below we have got jacobian for scheinman arm


* At singularity two columns in jacobian are going to be dependent and also matrix is going to lose rank and we like to be full matrix.
# Kinematic singularity: