# Lecture 6

### Movie section
polypod: reconfiguarable robots

### instantaneous kinematics

#### - Differential motion

$\delta\theta$ is related to $\delta$x with jacobian


Explicit form:
* We are going to examine another way of doing this velocity propogation analysis rather than propagating velocities we're going to examine the structure of the kinematics of a robot and its impact on the end of factor velocities and that would lead us to something very interesting we call the explicit form of the Jacobian matrix
* i.e, we will see that each column in this matrix have an association with the specific joint.

Static forces:

* relationship btw forces and torques resulting in end-effector come from same jacobian so basically there is dual relationship btw velocities and static forces.

Joint Coordinates:

Jacobians: Direct Differentiation

Jacobian:
* $\dot{q}$ and $\dot{x}$ are connected through same jacobian as $\delta$x and $\delta$q

Example:

Stanford sheinman arm:
* we know its DH parameters which we have calculated in prievous lecture and we found out x,y,z for positions and also
* Doubt at 22:38
* Didn't understood what does this means: ( It is X axis/direction of x axis in base frame.)


His Quote:
> "" This is the x axis of the last frame that is the axis attached to the end-effactor expressed in the base frame and the second set of ops and this is the y axis and this is the z axis ""


* Now we want to find $\dot{x}_p$ for that we will differentiate ${x}_p$ with ${q}_1, {q}_2,{q}_3,{q}_4,{q}_5, {q}_6$
* and then we get:
* Now for orientation: Direction cosines

> "" Our Jacobian is not giving us the properties of the mechanism in terms of the linear velocity and angular velocity rather it is mixing the representation properties with the properties of the mechanism ""


- what does he mean by dependent on jacobian ?[29:51](https://youtu.be/fwHc0a8DMQ0?t=1791)

* how we will find ${J}_0$ ?


* What is Ep(xp) and Er(xr)?
:- derivative of euler angles are not angular velocities but they are related to angular velocity. So basically Ep(xp) matrix is a link connecting derivatives of euler angles and angular velocity.
* But how to calculate them ? Basically how he calculated this ${E}_R(x_R)$ ?
* ${E}_R({x_R})$ is a function of euler angles where as Ep(xp) is a identity matrix because derivative of cartetion co-ordinates is linear velocities.

* here he has basically have substitued $\nu \,and \,\omega$ with ${J}_v.\dot{q} \, and \,{J}_w.\dot{q}$
* Ep is assosiated with linear motion and Er is euler representation of rotation

* How to calculate Ep for cylindrical co-ordinates and spherical?
*

* we want to find v and w as a function of qdots but what does this mean ?
* This J(q) is same as J0(q)

* In this slide we are trying to find the **components** of velocity of the point w.r.t different frames But we are measuring velocity w.r.t frame A
* Origin A is moving w.r.t to origin B (or frame B) then the velocity of point P w.r.t frame B is Vp/b = Va/b + Vp/a


* points at axis of rotation will not move , so basically what is linear velocity at different points other than AoR.


*


* We need to locate p w.r.t some fixed frame then we locate point p with vector P.
* Vector Vp is orthogonal to both $\Omega$ and vector P


* Cross product operator:


* In this we find veolcity of point P in frame A ,come from Frame B than from Frame A and finally from the rotation of frame B.


#### Spatial Mechanisms

* This J(theta) is J0() or Jx() ?: It is J0() But still I don't know how we will calculate J0().


* Here $\dot{d}_{i+1}$ will appear if Frame I+1 is prismatic joint and it is velocity along ${Z}_{i+1}$
* Again here $\dot{\theta}$ is differetiation of angle btw x axis.
* How this ORn are there ?what is it ? and what he was saying about finding jacobian that we have to separate d_dot and theta_dot terms to get columns of jacobian?


#### 3R example:


* how he got this Jv matrix ?