# Lecture 8
## movie segment
Automatic Parrallel parking

## Kinematic Singularty:
* Well i didn't understood his explaination of singularity using that..
* rank of matrix decreases if there is singularity.
* here we will get deteminate as (-------)qdot and we simply need to equate it to zero.
* Why determinate of J is independent of frame ?:in second slide below if we take determinant both side , (othogonal matrix has determinant equal to one.) we get second statment below.
*


* When Theta2 approches zero or pi or n*pi joint approches singularity.
*


* At singularity i.e, when Theta2 equal to zero we get below given J1 matrix.
* What Theta1 they have assumed to get that final J1? ( In first slide below) : It is J0 but still doubt not resolved)[27:28](https://youtu.be/XrNdB4k5kUk?t=1648)
* We are looking at the displacement by $\delta x \,and\, \delta y$ when we make small displacment $\delta \theta$
* The two column are dependent and therfore matrix has rank = 1.
* How $\delta \theta$ equal to zero mathematically(Didn't understood it well)?
* When there is small $\delta x \,and\, \delta y$ their is no displacment along x axis therefore $\delta x$ = 0 and almost all displacment is along tangent direction and therefore almost all is along y direction.(This all is in frame 1)
* This is actually J0 and also in formula above it.
* Now if we want to find the $\delta \theta$ required for small displacement $\Delta X$ we get inverse of jacobian.( Here is it a inverse kinematics ?) (second slide)
*

* Now we get appoximation of Delta_q1 and Delta_q2.
* i didn't understood the graph (So basically Delta_q1 is change in angle q1 then why the change in angle be infinite as angle gets smaller it feels to me quite contradicting)?
*

* As we are having only two degrees of freedom we can analyze only ${J}_v$ instead of whole Jv and Jw.
* Now for 3 DOF arm in ${2}^{nd}$ slide we get Jacobian as below:


* How is he reducing J matrix to 3 rows he has elemenated 3 rows of zeros(i didn't understood the maths part)?
*


*


* When ${\theta}_5$ goes to zero we get below matrix where column 4 and 6 are identical and rank of matrix becomes 5.
* ${V}_e$ is end effector velocity .


* We have written it in reverse order because now we can write it in matrix form using Identity.
* Here we are using cross product operator i.e, ${\hat P}_{ne}$.
* The ${J}_n$ (is defined in frame n) is related to ${J}_e$ with matrix.
* Now if Jn is in frame zero then the P_hat opereator should be also in frame zero.
* How we will convert P operator from nth frame to 0th frame?ye yahi hai jo mujhe samajh nahi raha hai assignment ke 1st question mai bhi usne yahi kiya tha.: We use here similarity transform i.e, we premultiply and postmultiply. Would like to get further clarification ....[48:08](https://youtu.be/XrNdB4k5kUk?t=2928)
*


* how this second matrix came?


*


* He has not explained anthing here would require more clarification.: Here he has used product operator formula to get $\hat P$ from P
## Resolved Motion Rate control:
* so here we know $\theta$ then we find $\delta x$ then with it $\delta \theta$ then we find increment $\theta$ with which then we move the robot. Is it enough to know the desired angle to move robot like hoe it works ?
*

* This motion rate control might work for small displacement otherwise it gives lots of errors.

## Static Forces:

* We are going to find relation btw forces excerted by the end effectors and torques produced at the joints.
*

* first equation represent sort of a Jacobian relating velocities V and Omega or theta dot and the second relation is representing sort of transfers of the Jacobian
### Fundamental equations in kinematics of robotics:
* These two eauations are fundamentas in kinematics in robot: $\nu = J\dot{\theta}\,and \,\tau = {J}^T F$.
*



* for The arm to be in static equilibrium summation of forces on each link = 0 as well as summation of moments at a point = 0;


*


* Here we need to project forces along axes.
* There are 3 ways to calculate jacobian velocity propogation, Jacobian transpose can be calculated with back propogation and expicitly calculate by analyising the structure.
### virtual work principal:
* internal forces can be ignored and only active forces are taken into account.
* Here ${\tau}^T\delta q$ represents dot product btw $\tau \, and \, \delta q$


### Example(Static forces):