‫Welcome back.
‫In this video, I would like to briefly discuss one matter from the previous course in this core series,
‫the lateral control of the vehicle.
‫Let's look again and how we formed the lateral control equations of motion.
‫They looked like this.
‫Here they were.
‫This was your net force in the lateral direction, and since we see lateraled, and that means that
‫we were in the body frame and this was our net moment in the body frame.
‫And so we equated the net force to this and the net moment to this.
‫These are dynamics, equations into the you add up all the forces in the lateral direction and you equate
‫them to the mass times acceleration in the lateral direction.
‫And you do the same with the moments and you equate the net moment to mass movement of inertia times.
‫The angular acceleration, the mass remains constant in time, so mass dot equals zero kilograms per
‫second.
‫In other words, the vehicle's mass does not change in time.
‫And the same thing is true for the mass movement of inertia.
‫It is constant in time as well, because just like in the Jones case, we wrote down these equations
‫in the body frame.
‫And that's why this mass movement of inertia will not change as we rotate our vehicle.
‫Now, in the oven case, we made a distinction between rotating about the inertial Z axis for which
‫we use the angular velocity side dot radians per second, and about the body frame Z axis for which
‫we used our radians per second.
‫We had to make this distinction because the directions of the big Z and the small Z, the inertial and
‫the body frames the axis, their directions were not guaranteed to be in the same direction.
‫That's because we worked in 3D in the vehicle case.
‫We only worked in 2D.
‫And the direction of all the stations in our 2D world was either out of the screen like this or into
‫the screen like this, according to the right hand rule, and that is true for rotating about the inertial
‫ze axis and the body frames the axis.
‫And therefore we forgot about rotating about the inertial frame Z axis and we used BPCI that radians
‫per second to describe the angular velocity about the body frame, the Z axis.
‫There was no need to describe angular velocity about the inertial frame, the Z axis.
‫We did not have to do it.
‫We only had to describe our rotation about the body frames, the axis.
‫But OK, if you remember then we wrote down our lateral acceleration like this, and then I explained
‫to you that we had two terms here.
‫The first term showed how the lateral velocity changed with time.
‫But then you also had this centripetal acceleration that you had to take into account.
‫I said that this y double dot is the term that describes the lateral acceleration, assuming that the
‫body frame was not rotating with respect to the inertia frame, the body frame that we attach to the
‫vehicle.
‫And then the second term was there, which was the centripetal acceleration which made the object and
‫therefore the body frame rotate.
‫Now, that was one way of showing it a more simplified way.
‫However, now that you know about rotating frames and you know that if you work in the body frame,
‫you have to consider that.
‫Now, I can give you a clearer and a more formal explanation as to why this acceleration term has these
‫two terms here.
‫So let's look at it.
‫In our vehicle case, we only considered the forces in the lateral direction.
‫So these are your two forces here in the lateral direction.
‫Let's for a moment consider both the longitudinal and lateral direction.
‫And remember, we still work in the body frame and not in the inertia frame and the body frame is attached
‫to the vehicle.
‫This is here in purple Y and X.
‫So our equations of motion are the following.
‫The net force in the body frame equals mass times, the dot vector in the body frame and also the net
‫moment in the body frame equals your inertia tensor in the body frame times the omega dot vector in
‫the body frame and that was your angular acceleration vector.
‫Since we are in 2D, there are no forces in the Z direction, nor there are velocities in the Z direction.
‫That is why the Z component of the force vector is zero Newtons and the Z component of the velocity
‫vector is zero meters per second.
‫In addition, the only rotation that occurs happens about the Z axis.
‫That is why our angular velocity vector has only one non-zero element, and that element is about the
‫body from the axis.
‫So that would be our side, that radians per second.
‫And the inertia tensor becomes mass moment of inertia about the Z axis, about the body from Z axis.
‫And we also assume that the vehicle is mass symmetric about the body from Z axis.
‫So our equations of motion in 2D are the following.
‫You have the two force components here in the X and Y direction.
‫I and J.
‫And remember, we're in the body frame equals mass times.
‫You have X double dot times I which was the time change of the longitudinal velocity plus then you have
‫the longitudinal velocity times I dot and then plus the time derivative of the lateral velocity times
‫J plus the lateral velocity itself times J dot.
‫You know that your eye dot and J dot can be calculated like this with the cross product where you take
‫the angular velocity vector and cross it with I and J unit vector respectively.
‫For IDOT and Jadot you can rewrite this entire thing like this where this part here would be your idot
‫and this part here would be your J Dot.
‫You can move this X dot here and you can move this Y dot here.
‫And so you can rewrite this equation like this, you can factor out your angle of velocity vector and
‫you can just put it here and then you will cross it with this vector here x dot I plus Y that J.
‫Instead of using unit vectors you can just rewrite this vector like this, which is the same thing equals
‫mass times.
‫Then this part you can rewrite like this plus and now this part here you can rewrite it like this here.
‫Our angular velocity vector still has three elements, but then the rotation about the body frame X
‫and Y axis didn't exist.
‫So we only had our side dot and then we crossed it with another vector, which was this one.
‫And we can rewrite it like this.
‫We can have X dot here, we can have Y dot here and then we can have zero here.
‫So it would be like X the I plus Y that J plus zero K like this, but since zero times K equals zero,
‫then we don't have to write it.
‫And so I close the parentheses and now we have to find this cross product.
‫So what I'm going to do, I'm going to take this class product and I'm going to calculate it here.
‫And the way you calculate it is like this.
‫So you had your I j and then K and then here I would have zero zero and then bpci dot and then here
‫I would have x dot, y dot and zero.
‫And let's see what we're going to get if we take our AI unit vector, then here you would have this
‫part from which you had to take the determinant.
‫So it would be zero times zero minus sign the times y dot.
‫So you would end up with this term here like this and then you would have your minus J and then now
‫you would consider these elements and these elements and it would be zero times zero minus sign the
‫times x dot.
‫Like this, and since you have two minuses here, it will become a plus and then you will have plus
‫K, but then you can already see your two by two determined will be zero.
‫So it would be zero times Y, not minus zero times X dot.
‫So this will be zero.
‫So you can just say that this term is zero.
‫That means this entire thing now equals mass times, this vector plus and now this thing here can be
‫rewritten in this form here.
‫Now, we said that in our vehicle case when we did our lateral control and we were in a straight road
‫and we only wanted to change our lanes, then we made an assumption that our longitudinal velocity was
‫constant.
‫Therefore, the time derivative of our longitudinal velocity was zero meters per second squared.
‫Now, if you really wanted to be picky, then you should consider this term here.
‫Of course.
‫However, we neglected it.
‫Why?
‫Well, if you look at our situation, then in our case we had a car on a straight road attempting to
‫only change lanes.
‫So that was our car here on a straight road.
‫We wanted to perform this maneuver in I think it was seven seconds, our longitudinal velocity was,
‫I think around 20 meters per second.
‫And now if you think about the lateral velocity, then you can imagine that in this case, the lateral
‫velocity would be extremely small compared to the longitudinal velocity.
‫So you could really say that.
‫Why that is a lot smaller than X dot.
‫So this term would be very small.
‫And also our maneuver was very gradual because we only wanted to go from one lane to another lane on
‫a straight road and all that had to happen, let's say, in seven seconds.
‫And that means that we didn't have to do any sudden turns.
‫So we could also say that our side that was small, our angular velocity was small.
‫And so in order to simplify our lives, we would like to get rid of those small, annoying terms that
‫contribute very little to precision, but make our equations more complicated.
‫And so we assume that this term here is also zero meters per second squared.
‫That means that our net force in the longitudinal direction was zero Newtons and we are only left with
‫this second row where you only had the lateral forces and the lateral acceleration.
‫So in other words, and I'm going to write it here just to squeeze the final piece of information here.
‫So if I force the net force in the lateral direction equals mass times and then you have your Y double
‫dot, which goes here and then plus and then you have your side up times X dot.
‫So it would be here and you see that this is your lateral acceleration.
‫So you have the same result that you had in the previous course.
‫Only now you can truly, mathematically justify it using the rotational frames since you're working
‫in the body frame and your body frame rotates with respect to the inertia frame, you have two terms
‫in the lateral acceleration.
‫So y double that is the lateral acceleration, assuming that the body frame is not rotating.
‫And then the second term X that time side, that corrects for the fact that the body frame is rotating
‫with respect to the inertia frame and that correction is the centripetal acceleration.
‫If you had worked in the inertia frame, then you would not have had this correction factor.
‫But then you must have considered forces and accelerations in the X and Y direction.
‫Your forces in the X direction would be like this.
‫That would be your net force in the direction and then in the Y direction.
‫It would be like this.
‫And then your accelerations would have been also in the X direction and also in the Y direction.
‫But since we worked in the body frame, we could only consider our forces in one direction.
‫But we had this second term because we had to take into account that the body frame was rotating with
‫respect to the inertia frame.
‫Finally, let's consider our moment equation.
‫So this was our net moment here and let's write it down like this.
‫So the net moment that we had in the body frame equals and now you had your math moment of inertia about
‫the body frame, the Z axis, and we followed the same derivation.
‫So you had your side double dot and now you have your K unit vector and it's only a K unit vector because
‫you're only rotating about the Z axis, right?
‫You're only rotating about the body from Z axis and therefore you only have this K here, but then.
‫We should also right here upside down at times like that, so you already know that K dot equals the
‫angular velocity vector cross K.
‫That means that this entire thing, you can write it like this double dot, K plus side at times and
‫then the angular velocity across K and then you can take this side dot and you can put it here and that
‫would be your K here.
‫Now if you want to find this, then it would look like this.
‫So your angular velocity would only have this zero, the zero and then you would have side up here.
‫And then you would cross it with zero zero and upside that which is the same vector.
‫Now, of course, you can methodically compute this cross product, however, if you look at it, it
‫is the same vector and even if it wasn't, but if it was parallel, if these two vectors, if they were
‫parallel, then what did we say about cross products?
‫You need to have a perpendicular component of one vector with respect to the other one.
‫Then you have a non-zero cross product.
‫If both vectors are parallel to each other, then they're cross product is zero.
‫And so you can see that not only these two vectors are the same vectors, but then they are also parallel
‫to each other.
‫Therefore, this entire thing is zero radians per second squared.
‫So this second term will disappear.
‫And so you're left with a net moment in the body frame equals the mass moment of inertia about the body
‫from Z axis times, BPCI double dot times, the unit vector K.
‫And there you go.
‫Now you see how all these equations for the vehicle's lateral control were developed.
‫In the next video, we will go back to our other e-mail.
‫Thank you very much.
‫And see you next on.