‫Welcome back.
‫So we have now seen how to derive and apply the rotation matrices and how to obtain the transfer matrix.
‫Let's now see how we can apply the transfer matrix along with the rotation matrix.
‫In our specific situation here, you can see the inertial frame and then the body frame and you can
‫see the linear velocities of the drone in the direction of the body frame axis you the N.W. in meters
‫per second.
‫And you can also see the angular velocity of the drone, or in other words, about the body frame axes
‫that are attached to the drone PKU.
‫You are in radians per second.
‫Earlier we defined a vector for the drones position.
‫That vector was the big gamma and you had here X, Y, Z.
‫This is the position vector for the inertial frame and we could also define a vector for the oilor angles,
‫which was theta superscript e arrow.
‫And it looked like this Phi Theta and PSI.
‫We can combine them in one vector like this.
‫You have big gamma here and you have big theta here.
‫We transpose it and let's call it Epsilon.
‫We can also write it out like this big X, big Y, big Z and then Phi Beta and PSI.
‫And then we transpose it because actually it's a column vector, not a row vector.
‫But I just want to say some space, if we take a time derivative of this epsilon vector and let's put
‫arrows here as well to indicate that it's a vector, then it looks like this.
‫All the elements here will become the time derivatives of these variables.
‫These are the linear or the translational velocities in the inertial frame, which are in meters per
‫second.
‫And these are angular velocities in the inertia frame or the time derivatives of the oil or angles which
‫are radians per second.
‫Earlier, we also defined the vector for the linear velocities in the body frame, which was this huge
‫VW and that was the vector, and we also had the angular velocities in the body frame that looked like
‫this.
‫And so we can also combine them in one vector like this or like this, where, again, this part is
‫in meters per second and this part is in radians per second.
‫The only difference between this vector and this vector is that they're measured in different reference
‫frames.
‫We also saw how to connect the linear velocities in the inertial frame to the linear velocities in the
‫body frame using this rotation matrix product or let's just call it R and we followed the convention
‫are some small Z, Y, X, meaning that we first rotated about the Z axis, then Y axis and then X axis
‫using the moving body frame approach or the oilor angle approach.
‫And in the same way we can connect the oilor time derivatives and the body frame angular velocities
‫like this using the transfer matrix.
‫And in fact you can write all that even in a more compact way.
‫You can define a matrix that we call J that looks like this.
‫You have rotation matrix here.
‫You have transfer matrix here.
‫Now the rotation matrix is a three by three matrix.
‫Right.
‫And so is the transfer matrix.
‫So this entire matrix will be a six by six matrix, which means that you need a three by three zero
‫matrix here that looks like this.
‫And also here, that's how this three by three zero matrix looks like.
‫And then using this J matrix, you can have this kind of relationship epsilon, dot, vector equals
‫J matrix and then the vector here.
‫You can also write it out like this.
‫And now you see why this J matrix looks like this, because if you write out this entire equation,
‫then you see that this rotation matrix only affects you, VW and the transfer matrix only affects P,
‫Q, R and then R times you.
‫VW will give you these values here, these ones and T times P Q are will give you these values here,
‫which is exactly what you want.