# Lecture 12




* Kinetic energy for link i


* how we will find the ${I}_{ci}$?


* jacobian releted to end-effector
* Now we have found the mass matrices
* mass matrics is basically jacobian transpose jacobian scaled by the mass factor


* in the second slide for ${J}_{vi}$ we only find partial derivative upto ${q}_i$ and leave other columns zero.
* Also in $J_w$ we do similar thing.
* What is this mass matrix mean?:what $m_{11}$ represent? it represent effective inertia percieved at each of the joints that is what each of the diagonal element represents.
* what does he meant by percived effective inertia?
* $m_{nn}$ is constant why?: because it is inertia of last joint and there is no joint after that :laughing:
* $m_{11}$ will not be fucntion of joint likewise $m_{22}$ will be function of all joints after the 2nd joint.
* if we add an joint to an existing arm it will add extra column and row to the existing block.
* now $m_{12} , m_{13}$ basically represents coupling btw joint 1,2 ; 1,3 respectively.
* $m_{12}$ is representing the coupling between the acceleration of joint 2 on joint 1 $m_{21}$ is the opposite joint 1 on joint 2
* also if we imagine what happens when we multiple q1,q2 to first row, we basically get the torque of first joint
* so as far as joints are fixed it doesn't create any coupling forces but when joints are loose they starts coupling on each other.
* as object has to have a mass but $m_{12}$ will be always positive therefore $\dot{q}$ has to be zero, which implies that joint will be fixed.
*


* here $M_{q1}$ means all the elements are partial derivative wrt $q_1$
* where as $m_{111}$ means partial derivative of $m_{11}$ wrt to the q1
* why is he writing $m_{121} - m_{121}$?


* what is cristoffel symbols?
* is $J_{v1}^T$ column of jacobian ?


* torque is r*force than why is he multiplying mg with jacobian?
*


* Now in the mass matrix we are using parallel axis theorem where we are adding ${m_1l_1}^2$ to inertia at COM.
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