 I will just give you some glimpses here for you know generalizing some part this Lagrange formulation for electrical domain also. So, this is the way kinetic energy and potential energies are defined in the in the electrical domain. So, if you use this and for a circuit and like know do independently for the circuit you will get this you know your circuit expressions whatever expressions you get in Koff's laws ok, but the difficult part is when you have this electromechanical coupling between the two domains as we saw in the motor model then things are little different ok. So, then you see here when we talk of this inductance here L i square term is only the self-inductance we usually consider ok, when we are we are used to consider in the self-inductance only, but when we talk of or motor it will not have not only have a self-inductance in fact, the self-inductance part we say d i by d t term L d i by d t term many times we ignore that term ok. So, the main important term in that case is is going to be your coupling or back M F giving term ok. So, what is the term that will give you back M F based on like know. So, that is a mutual inductance kind of a term that we need to consider and how it is to be considered we will discuss separately when we talk of little bit more about the actuators and motor centering. So, we will not get into that, but you can give a thought to that based on our expressions for back M F how this mutual inductance can be considered in the kinetic energy. So, that finally, when I derive Lagrange formulation that time I get appropriate expression for equations of dynamics which is matching our Newton's formulation dynamics ok. So, one can kind of like you know see this application that opens up like you know if you use this kind of expressions that opens up your Lagrange formulation application to this electrical or mechanical domain systems as well ok. So, we will consider only this kind of a inductance part later where we have to see we have to understand some of some more fundamentals about the electromechanical parts before we you know get delve into this. So, you need to revise some of your electromagnetics ok the magnetic circuits and things fundamentals of that that sort to kind of get to really know how do we consider these in different different kind of cases ok. So, we will do that later. So, we will close this today's part by defining these two problems ok. One is for the regulation problem in control. So, these are the definitions for a future kind of a control discussions that we will do pretty soon now ok. So, these regulation problems are the problems where you are given this like the starting point and the end point final point. So, P s and P final are given ok. So, now in I am showing it with the example of these two R manipulator, but in general this can be for one hour also ok. For the theta it is given as theta 1 like you know initial and theta 1 final ok. It can be as simple as that ok. So, you are given these two points in terms of the vector state of degrees of freedom ok. So, it can be also the the general what you say not generalize for it, but their derivatives also you can give you can be given a full state of system at initial point and full state at the final point. And then you are required to to go from like initial state to the final state. So, we typically consider for these manipulator kind of system that initially it is starting at 0 velocity from the starting point and like know it ends with the 0 velocity. So, typically the the velocities of start and end points are 0. So, that may give you the generalize coordinate velocities also to start and end to be 0. So, you can see that the this problem does not give you any information how do you want to move from here to here ok. Or in other words the time expression for these thetas to go from here to here ok. The desired time like know I do not care like know whether theta 2 in between is is one value or the other value ok. It can be any taking any value in between, but I want like know the finally, the expression should go I mean the value should match ok. So, finally when it goes here like the point p should fall on the p f ok. So, we are not bothered about this path to go from here to here ok. When I say path it becomes like a time trajectory for theta 2. So, we are not bothered about like know what trajectory of time we follow for theta 2 and theta 1 to go from this place to this place ok. So, that those kind of problems are termed as a regulation problems ok. We are just interested in the initial position and final position that is it ok. So, one can now see to go from here to here like know we just evaluate at this position what are the theta 1 theta 2 and when it goes at this position what are the theta 1 theta 2. And between now this so say this is theta 1 in S and theta 1 f and then like know we have similarly theta 2 S and theta 2 f and then we say ok for going from there to there I just need to kind of try my theta 1 from theta 1 S to theta 1 f and theta 2 from theta 2 S to theta 2 f and I am my job is done ok. I then like know I say ok I will first move theta 1 to that position theta 1 f position and then after that I move theta 2 to there who cares I mean you can move then together you can move them you know one after other or whatever, but you can take it to the final. So, that is what is a regulation problem. So, you can typically they will for saving time you will move them together like you know to start both the motors simultaneously and then you will reach the final point, but you are not bothered like know what is a path it takes to go from here to here ok. So, you have found like this what I was talking about here that you are finding this initial and final paths for the initial and final values for the generalized coordinates and command your motors to go either simultaneously or in one after other or whatever ways to go from starting to the final point ok. So, in now this like how do you kind of this command your motors. So, this is like know topic of further discussion for control. So, we will see how we can control like you know the ok, but in regulation problem is when you want to go from one position to the other final position or one state of generalized coordinates to other state of generalized coordinates without bothering what are the in between states ok. So, that is our regulation problem ok. So, we will further we will discuss this further when we talk start talking about control very soon and then other kind of a problem that we need to understand is a tracking problem ok. So, in tracking case you actually want that ok my point p should go from this position to this position along say circular path or along a straight line something like that we if we give, then what it amounts to defining the desired values of theta 1 and theta 2 is they should be at exactly some place in the along the path in time ok. So, do not confuse like know if you have single link manipulator what is a tracking problem there, there it is a path is always going to be circular. So, how can I define tracking problem? No, no it is not a it is not that it is like you know the values of theta which should be dependent upon the time in between ok. So, I want to go from this point to say for example, for single link forget about this link if I want to move from this point to say some other point on the circular path there also I can move like you know slowly there or fast there or say starting acceleration and then slowing down and go there like that I can have many different time trajectories for going from theta 1 S to theta 1 F only for single link manipulator ok. So, that also is a valid tracking problem ok although I am moving on the circular path only because there is no only possibility for this single link manipulator. When you talk of two link manipulator it is little easier to understand because there are multiple paths that can be possible which you can see ok. So, the see what we are seeing here is like in a space kind of a variation of the path, but time variation of the path is not seen when you when you kind of you know look at this to our kind of a manipulator case or one single link manipulator case I would say ok. So, now for doing this you can have multiple paths possible. So, say let us say I am defining like you know this path along which I want to go. So, not circular path, but I want to go along by a straight line kind of a path. Now the moment I say that then I have to define at every point on this path then theta 1 and theta 2 gets defined ok they will have some kind of a specific value only ok. So, and then we should have that this theta 2 at this point of time theta 1 should be this and theta 2 should be this for to be for my point to be on the path ok. That kind of a definitions would come. So, we would in general get a definition for theta 2 in terms of time and theta 1 in terms of time to go from theta 1 s to theta 1 f this starting and final positions are still same, but now I have to define a time trajectory for theta 1 and time trajectory for theta 2 to make sure that I move along this path alone ok I do not kind of deviate from this path ok. So, that is what is a tracking problem ok. So, this is called a trajectory tracking kind of a problem and this problem has then explicit time dependence because theta 2 desired is a function of time here ok or theta 1 desired is function of time here ok both are a functions of time ok and in the previous case theta 1 desired was just a final value of like final value theta 1 f ok and theta 2 desired was theta 2 f I did not have any like consideration for like you know what happens between like once from 0 to or from initial position to the final position ok. So, that is difference between the regulation problem and the tracking problem ok. The tracking problem will have a explicit time dependence, regulation problem will not have explicit time dependence ok. So, these are the steps you will typically follow as I said earlier. Use for getting this theta desired trajectory you use inverse kinematics of the manipulator ok. At inverse kinematics you will get from kinematic relationships and again you will have to kind of use some control to do this motion along the path ok. So, these two problems are there in the in typically in the control domain and we will see how we can now make use of the fundamentals from the control domain to get to these problems ok. I think we will stop here for now.