‫Welcome back.
‫So we have just covered an enormous section in how to derive the kinematics and dynamics, fundamental
‫equations we learned about rotation and transfer matrices in order to connect and be able to transfer
‫information between the inertial and body frames.
‫And then I showed you how the Newton oilor formulation in the body frame for a six degree of freedom
‫system is achieved.
‫It was a big section, however, it was also a necessary one.
‫Without that foundation, you would not fully understand the Oves plan model.
‫And so now I am going to derive a specific plan model for our USV.
‫Using the results from the previous section, you will see that you have already done the hard part.
‫Applying stuff from the previous section to our USV will be very easy.
‫So let's get onto it.
‫In the end of the last section, we had a Newton oilor formulation that looked like this, and then
‫we explored an alternative form for it, which looks like this, this equation is the exact same equation
‫like this first one.
‫The only thing is that it is written in a different form.
‫So this matrix M in the B frame, it was this matrix here, this entire one, then this vector here
‫would be this one.
‫Then this vector here would be this entire thing here.
‫And the biggest difference was the second term that appeared because the body frame rotated with respect
‫to the inertia frame.
‫So we took this term here and we converted it into this term here and now we are going to take this
‫alternative form and work with that.
‫Now, in principle, our goal is the same that we had when we worked with the vehicle in case of a vehicle,
‫we went from the equations of motion.
‫That looked like this.
‫To the states based representation that looked like this in red.
‫If you remember then the A, B, C, D, E and F.
‫Constance were composed of many variables related to the vehicle, and since all those variables were
‫themselves constant, including X Dot and I in the B frame, then D, A, B, C, D, E, F variables
‫were also fixed values.
‫And that's why we call them constants.
‫Not that they don't have to be constant, they are constant in our case, thanks to assuming that our.
‫X double dot or our longitudinal acceleration was zero meters per second squared.
‫In other words, our longitudinal velocity was constant and that's why our A, B, C, D, e, f variables
‫were constants.
‫We learned that state's basic equations are first order differential equations because you only have
‫there and input Delta states, why dad and BPCI that.
‫And the first time derivative of the states, why double dot inside, double dot y double that can be
‫rewritten in this form here and BPCI double dot in this form here.
‫And when you take that first order differential equation and you express the time derivative of the
‫states as a function of states themselves and the inputs like this.
‫Why double date as a function of why that side dot and Delta equals something and then double that as
‫a function of why dot?
‫Cidade and Delta equals something else.
‫Then you have a state based equation.
‫In general variables, it would look like this if you assign X one to Y dot, X to topside that and
‫then you to Delta.
‫Then this is how it would look like like this.
‫So even though you have a double time derivative here, then don't confuse that thing, because remember
‫that other states, they already have one time derivative.
‫So when you have a double time derivative here, then the difference is only one derivation with respect
‫to time, and that's why it's a first order differential equation.
‫If instead of why, that you would simply have why and then here you had why double that, then it would
‫not be a first order differential equation, then it would be a second order differential equation,
‫especially if you had another term, like, for example, why that here?
‫So you would have y y dot and then y double dot.
‫That wouldn't be the first order differential equation and therefore it wouldn't be a state based equation.
‫A state's basic equation is a first order differential equation where you have the states, the inputs
‫and then the time derivatives of the states and you express the time derivatives of the states as a
‫function of the states and the inputs.
‫And now we want to have the same thing in our USV case.
‫This is our Newton oilor formulation, the alternative form of ID Lambda B that has three net force
‫elements in it affects F, Y and Z and it has three net moment elements in it and X and Y and then Mzee.
‫And remember, these are all net values, meaning that each element here is a summation of forces in
‫their respective dimensions or moments in their respective dimensions.
‫Then this is our state vector.
‫In this state, Victor has six elements.
‫Three for translational velocity in the body frame.
‫You've N.W. meters per second and three, four angular velocities in the body frame, so P, Q and R.
‫Radians per second, this vector looks like this, and then the time derivative of our states is this
‫vector here.
‫That's how it looks like.
‫The first three elements here are translational acceleration in the body frame and then the last three
‫are the angular acceleration in the body frame.
‫And so what we want to do is essentially the same thing.
‫We want to express the state time derivatives in terms of their states and inputs.
‫So in other words, we want to put our Newton oilor equations of motion in the state's space form.
‫So the data vector B equals everything else.
‫And so you have an exercise now you have this equation here, this alternative Newton oilor form.
‫And so your task now is to take this entire form and you have to put it in the same space form.
‫Now, don't bother with specifics.
‫This is a one minute exercise and you can do it in two or three operations.
‫I just want you to do it before I show it to you in the next video.
‫So give it a try.
‫Thank you very much.
‫And I see you in the next video.