‫Welcome back.
‫So how do we find IDOT, Jadot and Cadart remember that I, J and K are unit vectors.
‫They're magnetites don't change.
‫They are always one.
‫The only thing that changes is the direction of the eye J and K when the body frame rotates with respect
‫to the inertia frame like this.
‫So this is your inertial frame here in white.
‫In red you have your body frame and then yellow you have the unit vectors in the body frame and in purple.
‫This W b, it's the general rotation of this body frame.
‫And you can see that this general rotation doesn't happen about any particular axis here.
‫That means that all unit vectors change IDOT, Jadot and Cadart because it's a general rotation that
‫rotates all the body frame axis with respect to the inertia frame and caution.
‫Don't confuse DWB with W in the V B vector because remember this vector was U V and W transposed metres
‫per second.
‫This is the translational motion of the body frame or the drone in the body from Z direction.
‫However, this guy here, this is an angular velocity vector in the body frame.
‫In other words, we can write it down like this.
‫W in the body frame equals P, Q and R transpose radians per second.
‫So I will always add a B to this guy here.
‫OK, but now in order to find don't jadot and K Dot, let's zoom in a little bit.
‫Let's look at this body frame rotation more closely.
‫So this is our body frame here in red and in yellow.
‫You have the unit vectors I, J and K, and now we're going to rotate our body frame about this purple
‫axis and let's call this purple axis l that is just a random axis that we choose to rotate about, which
‫doesn't correspond neither with X, Y or Z.
‫And let's say that we rotate about this purple axis by an angle, the alpha radiance, which is the
‫differential angle and infinitesimally small angle.
‫We have rotated our body frame about this purple axis by an infinitesimally small angle.
‫Since this purple angle doesn't correspond neither to X, Y or Z axis, it means that if we rotate about
‫it, then there will be a change in the direction of I, J and K, and there will only be a change in
‫the direction, not in magnitude.
‫The magnitude of I will always be one.
‫And the same thing is true for J and four K, I, J and K there you need the vectors.
‫So they do have a direction, but they're magnitudes by definition are always one, and therefore in
‫our case only that the direction of these unit vectors will change.
‫And so by rotating about this purple axis, this eye unit vector will go from here to here.
‫Let's call it I at T plus T, so that's a differential change in time.
‫So after a very small time, your eye unit vector will move from here to here.
‫And the same thing happens with the J unit vector and K unit vector.
‫They all rotate a little bit like this.
‫That means that this vector here.
‫Light blue is D I is the change of the eye vector right, first your eye vector was here, now it's
‫here.
‫Therefore this is the change.
‫You get it like this d I equals the eye unit vector at T plus the T minus the unit vector at TI.
‫And so when you do this vector subtraction then you will get your di the same thing, you will get your
‫D.J., which is the infinitesimal change of your vector, and here you will have your DKA, the change
‫of your K unit vector and all these changes are vector changes.
‫And now we want to relate our unit vector changes with our actual unit vectors.
‫And with this, the alpha radiance, the angle that we have rotated about this purple axis that we called
‫hat in so we can write D A equals D Alpha Cross.
‫I also d.j equals the Alpha Cross, J and K equals the Alpha Cross K y.
‫Well, it makes sense actually.
‫If you take for example, di equals the Alpha Cross I so only the component of the Alpha and that is
‫perpendicular to the unit vector ie is multiplied by EI and the result is D I and DS perpendicular to
‫both I and D Alpha.
‫Also, if you use your right hand rule and you first point your fingers towards the alpha and then towards
‫eye, then your thump points towards the eye and the same logic applies to Jay and D.J. and Kay and
‫Dick.
‫So you have these three equations.
‫And now what we are going to do, we're going to divide both sides by DETI, also here and here, the
‫differential time change.
‫Then this guy becomes I dot, this guy becomes Jadot and this guy becomes K dot, because that's what
‫this point above means.
‫It means time derivative.
‫It's just another notation.
‫And so since the Alpha was in Radiance, then the Alpha, the tea is in radians per second and so this
‫the Alpha deti is nothing else but our angular velocity vector in the body frame about this.
‫L had access, therefore I that equals w.
‫B vector cross i j dot equals W. b. vector cross J and K dot equals w. b vector cross.
‫OK, so there you go.
‫Now you know what your idot Jadot and K Dot are.