 Hello and welcome. The material that we have discussed so far has made one fundamental assumption about the nature of the trajectory that it is along a straight line. As we have already seen earlier, this is an approximation which needs to be relaxed and we need to now look at the possibilities of generating a curvilinear trajectory for the assigned mission. So, in this lecture now, we will look at one such option of generating a curvilinear trajectory. So, let us begin. So, we are going to now talk about inclined or curvilinear trajectories. Let me recall this particular picture that we had seen earlier for a typical assigned mission starting from the lift-off until the terminal point. As you can see, the flight path immediately after the lift-off starts curving in such a manner that towards the end of the mission, the velocity of the residual object is practically parallel to local horizon. So, we realize that the vertical motion or motion along a radial line is used only for a very small part of the overall mission and for the most part, assigned trajectory is inclined and curvilinear in nature. Of course, at this point it is worth noting that this is mainly due to one of the terminal constraints that is imposed by the spacecraft mission that the inclination with respect to the local horizon should be 0 or close to it which requires that a vehicle which is moving along a radial line by the time it completes the mission should be moving along a local tangent. So, obviously, you are going to require large amount of curvature while completing the assigned mission. In addition, it is worth noting that earth itself is no longer a flat surface because you are going to move along a curved path. And because of curvature of the earth, the local tangent itself changes its inclination with respect to the tangent at the launch point. So, that by the time you reach the terminal point, the amount of flight path angle changes that you are going to require are going to be significantly in excess of 90 degree. And this requires a different methodology for trajectory design and solution. The first thing that happens in this case is that the motion instead of along a straight line is in a plane described by the radial line and the tangent. And for that we need a two-dimensional model for the motion. Also, we need to realize that thrust which is the primary force for generating the velocity is generally kept along the flight path so that no amount of propellant is wasted in generating the curvature of the trajectory. And therefore, we need an alternate normal force to the velocity vector which will produce the rotation of the velocity vector and hence a curvilinear path. In order to do this, let us create a simple schematic as shown below. So, we have an instantaneous image of a rocket inclined with respect to a local x-axis and has an angle theta with respect to the local y-axis. Then we have the velocity vs along with the unit vector us along the axial direction of the vehicle and the velocity vn and the unit vector un along the direction normal to the axis or the velocity vector. The gravity we assume is to be pointing towards the negative y-direction and just to keep the picture fairly general, we introduce three additional forces, one the lift which is because of aerodynamics, fc a control force for maintaining the trajectory and finally of the thrust which is pushing the vehicle along the flight path. Now, we can easily write down the equations in the Cartesian coordinates x and y in a planar context. We can also write the equations in terms of the polar coordinates that is r and theta. But in the present context as you will notice later, we realize that the trajectory based curvilinear coordinates are the most convenient set of variables that is s and n that we are going to use. This is a coordinate system which is attached to the center of gravity of the vehicle and it is a moving coordinate system which is translating as well as rotating which obviously means that now we need to look at our equations of motion in the context of such a coordinate system and arrive at the applicable relations. So, let us first look at the expression for the velocity in the context of the curvilinear coordinate system s and n. So, let us assume a two dimensional system x and y and a curvilinear trajectory a. Let us consider a point p on the trajectory and another point p1 which is at a distance of delta s along the trajectory as delta s is assumed to be very small infinitesimal quantity p and p1 can be assumed to be close to each other so that a line joining p and p1 will be equivalent to a local tangent at p and that tangent if I divide that with delta t that is the time interval between the p and p1 what we get is a local value of the derivative of the trajectory segment delta s or what we call the local velocity which is along the tangent. Mathematically, we can now write the velocity as limit of delta s by delta t as delta t tends to 0 which is nothing but ds by dt the derivative of s and our velocity vector along the us direction is the derivative s dot into the unit vector us. So, you realize that the statement of velocity is fairly simplified if we use the trajectory based coordinate system us. Next, let us look at the acceleration model for the same motion scenario. So, we now consider the same picture and derive the expression for acceleration as follows. So, again we have the same picture but this time we assume that at point p the local tangent direction is v bar and at p1 the local tangent direction is v1 bar. We assume that there is an infinitesimal difference between these two local directions that is going from point p to p1 the velocity vector has rotated by a small angle. Now, the only way this rotation can be achieved is to introduce a velocity component which is normal to v bar at p so that at p1 the velocity vector will become v1 and this can be done through a simple velocity triangle that is shown below. So, there are two velocity components delta vx and delta vy together they generate delta v bar which is the change in the velocity and as if we take its ratio with delta t the time taken that becomes our acceleration vector at that particular point the mathematical relations are as follows. So, we write acceleration as the derivative of v vector which is the velocity now as velocity is a vector we are talking about a vector derivative. So, vector we can write as the scalar part of the velocity v and the unit vector and all of us are familiar with the chain rule of differentiation so that we can write the acceleration vector as two terms the v dot into us plus v into rate of change of the unit vector. Now, as unit vector is changing its direction the rate of change of unit vector is now related to a rotational rate theta dot and the local velocity v. So, this is a locally circular motion concept so that the normal velocity is normal to the tangent direction and v theta dot represents that particular normal velocity except and together with v theta dot we get normal acceleration. So, now, we have two acceleration components as which is v dot that is along the velocity direction and we have another acceleration an which is normal to the velocity direction. So, we realize that in the context of the trajectory coordinates s and n the vehicle now has two accelerations one along the trajectory and one normal to the trajectory. Of course, as we have mentioned earlier we will assume that the lift is 0 because the vehicle is flying with zero angle of attack and that we do not need an FC for rotating the velocity vector. So, we assume both these to be 0 and with that we write down now the equations of equilibrium. So, the first equation of equilibrium that is AS acceleration along the trajectory direction is nothing but dv by dt the scalar component of acceleration and this is not is nothing but the difference between the thrust and a component of gravity in the direction opposite to thrust that is g tilde cos theta. In addition to this now we have a normal acceleration an which is v into d theta by dt and this should be equal to the normal component of the gravity in the s and n coordinate system that is g tilde sin theta. Here v and theta are treated as trajectory parameters. Now, as we see from these two equations the velocity solution is now function of the thrust of course but angle theta. So, it depends now on the angle theta. You can readily see that if I put theta equal to 0 degree which corresponds to the motion along a radial line the normal equilibrium disappears and we recover our basic rectilinear motion equation. But more importantly what we realize from the second equation that the solution for theta which is nothing but the angle by which the velocity vector rotates as it moves along the trajectory is controlled by the gravitational term g sin theta. Which means that gravity acceleration alone is now responsible for the turning of the vehicle of course the v is also an important parameter. Here it is worth noting that conceptually we now have a set of two coupled nonlinear differential equations for actually three unknowns the velocity the mass or the dm by dt and the flight path angle theta. But we have only two equations in three unknowns. So, from our basic understanding of solution of such equations we note that if I specify any one of these I can solve for the other two. Which means now I have a requirement that I must specify either velocity or burn profile or a flight path angle profile and this gives us an important insight into the solution itself that in as much as that these equations can be solved either as a trajectory design problem where I specify a burn rate and solve for v and theta or as a vehicle design problem where I specify a trajectory either in terms of theta or in terms of v and solve explicitly for either m or dm by dt which will enable such a flight path. However it is worth noting that the problem is complex because we only have an initial point which is the launch point and terminal conditions which we are going to talk about later which can be met from many different profile geometries. So, obviously there are multiple solutions possible which means depending upon how I specify my m or m dot I am going to get various combinations of v and theta which will be valid trajectories and similarly for multiple values of theta or v I will be able to solve for a large number of burn rates or m which will be solutions further we also note that these equations are time varying and all linear. So, most of such solutions are going to be numerical in nature as close form analytical solutions are not going to be feasible in a general case. However as we have been mentioning from the beginning in the early design stage particularly when we are looking at vehicle sizing close form analytical solutions are found to be very useful and we also note that in case we can get such close form solutions we can get a quick assessment of a large number of concepts and can carry out a trade-off study to choose a few of the better performing concepts for more detailed investigations. Of course you should realize that such things are going to be possible only under simplifying assumptions that limit their applicability. So, that is the flip side of simplified solutions they provide quick assessment but then there is a certain amount of approximation or constraint that we include for arriving at such solutions. So, what are those kind of solutions that one can think of? There are many possibilities that one can think of which would generate close form solutions but among such possibilities there are three which are simple, elegant and also have practical utility. In this regard it is worth noting that most launch vehicles will use some kind of control mechanism during its ascent mission and therefore it is possible to assume that we would have a control to either maintain one of the three variables that is either d theta by dt or t by m or v. You will find that by keeping one of these as constants it is possible for us to arrive at close form solutions which have lot of practical utility. While we will look at these in more detail in the next lecture it is worth noting here that constant d theta by dt generally helps us to manage the mission time better. You will see how it happens. Similarly when we specify constant t by m that helps us to manage the structural mass better as t by m represents an indicator of the forward acceleration that the vehicle is undergoing. Lastly the constant velocity case commonly helps us to manage the overall propellant mass better. To summarize the gravity based rotation of the velocity vector is a convenient way of achieving the desired trajectory profile while minimizing the propellant. Further we have also established that simplified scenarios can help in arriving at close form solutions that aid in quick performance assessment. Hi, so in this lecture segment we have seen the basic idea of a trajectory which is curviliner in nature and is driven only by the gravitational force so that we can manage our propellant better and we have also seen that use of curviliner coordinate systems SNN helps in writing down the equations in a fairly compact and simplified manner. We have also noted that using simplifying assumptions we can obtain close form solutions for couple of cases which will help us to generate the trajectories quickly and with practical value. We will do this in our next lecture so bye see you in the next lecture and thank you.