 Then, we go for actuator selection to discuss actuator selection, typically like we have these different choices available you look at different options available and then make a choice based on that. Now, when we have a positioning requirement, then BLDC motors typically are not used ok, we will use permanent magnet DC motors or you have also stepper motors here ok. So, but we are talking of a servo. So, in servo stepper motors are not a servo kind of a drive, they are more like a open loop kind of a drive ok. So, but depending upon application you may do the choice actually here we are required to do like servo operation by using some closed loop feedback that is why we are kind of like you know constrained to kind of using permanent magnet DC motors. That is a choice that we will typically carry out and then now the question is how do you get power and torque for this motor ok. And then for this again one can see that if we estimate now like you know the mass in the system and forces that are coming then we will be able to kind of get to the torque. Now, how do you get the forces? What are the forces can you think about on this linear stage motion ok, when we want to move it by say 20 mm per second kind of a speed. See this is not very so, so we can say ok friction force of course that is there, but how do you get to know what is a force coming because of the inertia or moving elements or moving masses in the system, this is the little kind of a tricky to estimate ok. So, for that you need to know what is acceleration which we have not specified in the problem ok. So, we have specified ok I want to kind of go with the range of 60 mm and then like know. So, there so that is where like know we need to get a estimates of acceleration ok. So, I will tell you how do you get the estimates of acceleration and once you have an estimate of acceleration then acceleration is equal to mass into mass into acceleration will be the force and that force times velocity will will lead you to the power in the system ok. So, there are these friction is what one element in the system that will affect the power and inertia is another element that will affect the power ok. So, friction estimation you should have some kind of a ballpark figure estimates of the friction and that for that you can use the use the tables online you will find some information about coefficient of friction between the two surfaces in presence of some lubrication or not lubrication. All those kind of things you will use make use of that see what are the normal reactions typically from your applications coming on the system and then you can estimate mu times and like know friction force and friction torque. So, many times this friction is much higher consideration than inertia especially for the slow moving systems, slow motion systems and systems which are having this kind of a spring loaded stuff. So, there is also enhance accuracy one has to so you can see that this enhancement in accuracy or backlash consideration that comes at a cost of paying like what cost you pay is like no extra power that you have to get into the system to overcome this additional friction that will happen because of the spring loading of two elements. So, this is a important thing now to you know you need to kind of plan a trajectory for getting this estimation of power. So, we are we are on this question how do we estimate power and as I said like given the speed of 20 mm range of 60 mm 20 mm per second for the range of 60 mm we have to assume something now to kind of get to the acceleration ok we are not given directly any acceleration in the system. So, how do you do that ok we say we need to go from start to end in say 4 seconds or this is one can we can ask the application user ok what is like know how much roughly the duration in which you will go like you know like to go from one place to another place ok. So, this is somewhat important say 20 mm per second we go to 60 mm like no, but we do not know how soon we should get to 20 mm ok. So, we cannot directly jump from like no start to like 20 mm like no we need to have some acceleration to go to 20 mm per second kind of a speed ok. So, we so, we go let us say we go start to end from in 4 seconds so, we can kind of do some kind of estimation see this is taking 3 seconds to go from like no start to end with 20 mm per second speed ok. So, we want to do this operation say roughly in 3 seconds with acceleration or 4 seconds with acceleration. So, if one can kind of like see that number it will be not too far from these 3 seconds ok. So, we can plan we can make some kind of assumptions of this sort and then plan our trajectory for velocity ok. So, now if I want to go now say let us say I have chosen this 4 seconds or some these whatever seconds time I have chosen and I want to kind of get limit my maximum velocity to 20 mm per second. Then I can shape like ok what are these you know straight line rise that will give me the acceleration to go from 0 to 20 mm per second speed ok. How much should be this duration such that I when I like to complete my task in 4 seconds with this deceleration also for the same time this duration as the acceleration time duration then I reach this 60 mm ok. So, one can make such kind of a assumption and you know proceed for getting the acceleration. Once you have acceleration one can get to the force is equal to mass into acceleration ok that. So, that is the maximum acceleration maximum force that will that your system will see although during the range of this 20 mm constant speed there is no you know force other than the friction force that will be seen ok. Though no inertia force will appear because you are now moving at constant speed ok. So, these are the kind of you know sense of this kind of a feel of how one can one has to kind of think and get to the estimates of power ok. So, this is the power which is corresponding to inertia and then we can estimate the power corresponding to friction losses and then we get the total power in the system ok. So, this is a typical kind of a exercise one would do for any mechatronic system selection of sensors and actuators. So, now let us see this with the example of 2R manipulator a little bit more because there are some kind of a variations that will happen here based on the because the kinematic relationships will be different ok. So, we will talk briefly about that and then we will close this chapter ok. So, now this is a 2R manipulator that you have done modeling for and if this is a problem that is given to you that we like to go from point say starting point PS to some end point PF along the straight line trajectory ok. That is what is our aim for this some kind of a straight line I have to kind of go along. So, how do I carry out planning of this trajectory and then get further estimates of you know the sensor resolution. So, so I am given that I have to go by the straight line trajectory for some kind of assembly line operation and or some kind of some operation ok. And I am given that I have been while on the trajectory I can be off the trajectory by some delta amount ok. So, that is what is given as a user specification which needs to get translated into sensor specification eventually ok. So, I want to go along this trajectory and I cannot be off the trajectory more than some delta some delta plus minus delta amount. And then how do I kind of go about like you know getting the sensor and actuator specifications ok. So, you assume some kind of inertia for these elements and thing like that. So, so this will be typically iterative process to like say if you are designing the manipulator itself like know it will be quite a few parameters that are going to be unknown to begin with. And so, so you will start off with assuming some kind of a what power figure values or some kind of you keep them as a variables and then you keep on kind of simulating and changing. So, so typically what you will do is for such a you know iterative process is to create some kind of excel sheets in which like you can these you know some of these parameters as input to the to the input block of the excel part and then you estimate some outputs and then like know you see those numbers and based on the numbers you start modifying the inputs to kind of get to like know what specification that you would like to achieve further. So, that that is one of the ways one can do nicely. So, you making use of the excel sheets to do this kind of a design calculations little bit iteratively. And you might have seen in your normal like know mechanical design process also those kind of excel sheets nowadays are available from many vendors or you know some of the manufacturers catalogs. So, this is our job to estimate now like know going from this to this now. So, first thing we should we will do is like know we should so the crux of this all is how do you plan the trajectory ok. So, as we saw in the in the last example also once we plan a trajectory lot of things will will estimate and we will get clear. So, now how do you plan for such a trajectory? So, now say let us assume that ok I want to go from this initial position to final position in some time t ok or in some say 5 seconds or something like that ok. Then how do I kind of you know plan my trajectories ok. So, can you think of what is what comes to your mind at this point? Just put it on the paper and then let us move on ok. So, for this kind of a trajectory what is the first I need to kind of get some equation for this straight line right. So, we will get the equation of a line first to say starting from point. So, you know from your basics of geometry like you can write this equation not a big problem. So, you are if you are given x p as a position then corresponding to that if I want to be on this straight line or y p position will be given by this equation ok even first like you know starting point and end point I am having this kind of a relationship to go if I know x p, but I do not know x p right is not sufficient right. How do I plan now? Now what is a thing is to plan for x p now ok then I my y p plan will be coming out of this relationship. So, I need to plan for you know how my trajectory for x p will evolve in time. So, we may say ok so, can you think about how do you plan for now x p variable what you should be desired x p variable. So, that I go from like you know this point to this point in say sometime t or 4 5 seconds for example and 5 seconds I want to go from this point to this point and I know this coordinates and other stuff. So, how do I plan for this kind of a trajectory? So, then I say ok from x p perspective again from this x value here x p s value to x p f value here ok now this will be x value here I will need to go in same 5 seconds ok. So, if I know these then like you know I can plan for this x part alone to begin with and y part will be estimated from this ok. So, now I can do these ok we can I can plan say some velocity say B mm per second kind of a velocity I will achieve during my motion path and then I go again a trapezoidal kind of a form of a velocity ok. So, in say t is 5 seconds now in 5 seconds I want to go via this kind of a profile of velocity and I have under constraint that. So, I again assume that ok this is a time t 1 and then this is also like total time from here to here is t 1 so, that you are accelerating and decelerating in the same time and with that I would estimate what is a displacement that would happen and that displacement should match to my x motion which is x p s minus x p f ok. So, that is total x motion that should happen ok. So, I will plan this velocity profile first and then like I will estimate ok what is the acceleration or like what is the speed I have number I can choose to kind of achieve the final you know the displacement ok. Once I know this I will get y coordinate from the equation that I have seen in the previous slide. So, is this again sufficient no you can see that ok this is not yet sufficient because I just know now my x desired and y desired values for this point p, but they need to get translated to this q 1 and q 2 ok. So, for that you need to recall your kinematic relationships ok. So, you plan flows first your x desired value y desired value and now come to kinematic relationships to get q 1 and q 2 desired values ok. So, this is how like you know you recall your so, you recall all that coordinates of c g of link 2 where this much this and now if you change this l c 2 to l 2 then like know that becomes coordinates of point p ok. Use this inverse of this relationship to get to the values of q 1 and q 2 ok. So, you see that typically these relationships are not linear ok they have some kind of a non-linearity in them and that will have some kind of a implications ok. So, for example, for given x p and y p there will be two possibilities for q 1 and q 2 solutions ok. One is on this side you can see and other solution will be on the other side like you know the link going like that and like not coming like that ok. Symmetrics about a straight line going from the origin to the point p ok the symmetric solution on the other side ok. So, one has to consider like you know that part in doing the inverse cam computation so, that you do not suddenly kind of go for some of the positions on the other side and some positions on the on one side and then you will have a trouble in executing that kind of a trajectory ok. So, from x p and y p you get to q 1 and q 2 now as a plant directories and once you have this plant directories available one can see now if there is a data change in the resolution of q 1 or q 2 how much will be the change or like you know deviation from the straight line that is going to happen and now one can see it is not very difficult to see that if I go in the same straight line here or at some other point in the workspace ok these relationships are going to change ok. So, there are like some more considerations that will happen on the way. So, what are the now like no resolutions on q 1 and q 2 so, given in this kind of a trajectory plan. So, I want to go to final position p f within some say some accuracy say 10 micron 50 micron whatever micron accuracy then correspondingly what is like you know q 1 and q 2 resolutions that I will need to have. So, that is the kind of question that we will need to think about. So, we will discuss little bit more in the in the discussion part again, but these are the things one need to think about say given specifications on the resolution on q 1 and q 2 how do you kind of get to the deviation on the from the straight line path or reverse problem ok. So, based on that one can take a call on what is the resolution that is required on the sensors ok and then one can also see that you know if I if I choose this p s coordinate starting point at some different locations ok say it is more closer to the to the origin or this motor ok. Then the kind of variation I would get for q 1 and q 2 would be different than range of motion that we get for q 1 and q 2 will be different from if I start from some other point. So, where to choose you know this starting point from if you have a choice ok if you are given some kind of a choice for this starting point where you will choose it from ok. Those are kind of thinking that you need to do and for that it is not a very trivial thing it has to one has to kind of do some simulations and see the workspace of this manipulator and see which are the good areas in the workspace to have this operation done. Typically one of the considerations is that if you are like you know very close to this straight line position singular position ok and that we have talked about. So, when q 1 and q 2 are q 2 is 0 ok. So, only q 1 is there q 2 is 0 then this is straight line path ok and suppose now if I choose point p s at the end of that path I do not have like you know I cannot draw the straight line very easily I have to get out of this singular position and then maybe I will be able to kind of move along the straight line which is inside the workspace. So, this notion of workspace of this motion of point p workspace is basically all the set of points p which can be reachable by whatever the link lengths that are given here and for whatever angles for q 1 and q 2 considering like you know that they can move total 360 degree possibility. So, those are kind of considerations that one would need to give for sensor specifications ok. And when you come to extruder sizing we need to go back to this modeling part to some extent. If at all we want to get into you remember this model that we have derived for the toward manipulator. So, from this plan of q 1 and q 2 to decrease one can get to know this is say suppose this is q double dot and q dot all these things are desired values then I can get what directly the torque which is required to kind of achieve that ok. So, that is based on only like no inertial torsional friction is considered in this equation yet. So, if it is in a vertical plane this gravity component will be dominant and the inertia components depending upon how fast we want to kind of go the inertia components may be dominant. So, both C and D are kind of inertial terms in the system ok. So, I will substitute this desired trajectories in here to get estimate of like you know the torque is one way of doing things or one can kind of say that in the similar manner that we have seen what is the maximum torque that is maximum acceleration that is coming into the system and you know estimating like no more kind of a conservative forces estimations of the forces coming ok. And see now this matrices or inertia terms are dependent upon q in general. So, where you will choose to move you know that starting point p s you will have that of you know variations that can be possible in the terms of D q matrix and correspondingly like your estimates of you know the inertial torques. So, once you have estimated like you know some kind of a torque here you can kind of consider that. So, this is like coming as a tau 1 and tau 2 torque in first motor and the second motor. Then you so by substituting like you know your desired trajectories then you use that with the with the velocity to torque into velocity as a power estimation for the motor. And this torque will give you like maximum torque you know along the trajectory where it is happening and based on that you can add some factor of safety and you can add friction also to estimate like you know some kind of a torque for your motors in respective joints ok. So, now one can ok. So, this is about you know sizing the actuator in the same kind of a fashion as we did for the case of. So, all this you know actuator sizing considerations will need you to plan a trajectory that is a bottom line here. You need to plan this trajectory to kind of get to this. So, for example, here we had just said this q 1 and q 2 will be obtained based on this you know plan for x p to move along the along the straight line ok. This is a kind of a so you will have similar kind of a diagram for q 1 and q 2 again ok. So, yeah so, planning the trajectory is an important step in many many mechatronics systems I would say ok. For whatever kind of a motion that you would like to carry out you will need to have this kind of a motion plan to to get the get estimates of the specifications ok. So, that is what is a bottom line. And now I would suggest like you know you think about this problem because if you want to they can draw a circle instead of just moving along a straight line I would like to draw a circle here ok. And I have given some accuracy and some speed of drawing circle ok. So, speed of 10 mm per second is is what I have been given. So, the end effector here should be moving along this circle ok by a speed of 10 mm per second ok. Then in that case how would you plan this trajectory is a question ok. The circle equation something will be given then how do you kind of plan now the further thing ok. So, you think about this issue that ok I want to move the my velocity along the circle should be 10 mm per second how can I plan such a kind of a trajectory ok. So, think about ok. So, is this right right choice for the coordinate system for the planning or not you know you didn't think about that ok. So, I will I will not I will stop here we will see if there are any I mean you know we will take up this again if at all needs to be in the discussion class part ok. So, this is this is how like now you can have now make this kind of a concept generalized. So, from these two examples first we have seen like a linear stage which is for single degree of freedom kind of a case and now this is 2R manipulator 2 degree of freedom case this can be now generalized to n degree of freedom kind of a system to think about. So, for that you will need your equations of motion in place typically ok. Otherwise you will get too much into two conservative estimates might be happening if you do not really do the modeling completely then your estimates you can still kind of give actuators which are now they can function, but they are like know maybe too far more than what is required ok. So, I think we will stop here.