[IONLAB Lectures] Quaternion Kinematics

Transcript

all right so today what we want to establish is a relation between angular velocities and the rate to which our quaternion changes so just a small reminder we know that we can use quaternion through quaternal multiplication to establish the rotational vectors so this expression over here and this will give me exactly the same Vector but on a different basis so for instance if this is a inertial fret if this is a frame I and this is a frame B this would be the quaternion that takes the rotation from from frame I to frame B and you can find that that Vector through this hormone as we know from quaternal multiplication and how quaternions decode rotation however if this pattern if this Frame B is rotating with respect to frame I in times of this Frame B and this Frame B depends on time with respect to some frame I what you will have is that this pattern is not a fixed number it's not a fixed set of four numbers but instead the disquaternion changes in time so both of those changes in time and we want to understand how this evolution is done we know that angular velocities is what describe those rates and therefore we should expect that there is a relation between the angular velocity and the quaternal dawn when quaternion dot is really the derivative that you would think you can take here all the quaternion uh components and you take the derivative one by one so when I talk about the quadrillion derivative in time this is what I'm talking about is the each one of those quaternal components differentiated with the static to time and we want to know how they how would that be now we can think about this so before going into that derivative of of this itself I just want to make a small notice of quaternion um on composition of rotations so for instance if we want to rotate from a Time D and then later we have the same B Ora on a Time t plus delta T that means there exists a rotation here a quaternion that takes from this time T to time t plus delta T this means that and the way we recall this is not by an addition of the quaternity it should be by composition by doing this formula once again so for instance if we want to add a q add this guy over here what we want to do really is to we have this Vector over here zero v i and we will rotate it from uh to B with QT right and later we want to add to rotate it even more by delta T so we will add here a Delta Q so this would be our Delta q that we are adding later and this is our Delta Q inverse over here and as we go so we can compose alternative rotations by the quaternal product so this would be the composition of rotations anytime and notice that they are done by uh by a quaternal multiplication instead so we need so the derivative but the derivative here is really what you would expect it's just the normal differentiation I mean when I think of a q Dot what I'm really thinking is on the limit when time goes to zero of Q plus delta T minus Q of t divided by delta T but this is not exactly a rotation because quaternity just do not work that way the way they compose rotations is through multiplication so let's apply that inside uh this will be the limit delta T from zero now this the Q at t plus delta T is really the Q of t with this Delta Q now now minus Q of T divided by delta T notice that my limit is taking it moves back to delta T So this Q can go outside the limit and what we will have is QT times the limit and this times it's really a quadrian times through Delta Q minus 1 divided by T and this one is the unit pattern so this is just our normal one that we multiply this is the quaternion um is the identity by multiplication so having that this is the part that we really want and notice that Delta Q then we can go back to now how Delta Q corresponds to a rotation and this will be how it decodes rotation is recorded like so this so if you have this is how quaternion relates to a Delta angle and the axis of rotation and for very small Titans since this is a limiting uh condition what we have over here is that this cosine of Delta minus 1 will be for the in the limiting case this will be one in the limiting case this will be the instantaneous axis of rotation and the sign of theta for very limiting angles will become like so and then we add this to the Limit over here this minus one will cancel this one over here and we will have and at this limit in the end of the day will be Q of t alternative multiplied by minus one will be then zero over here this n stays over there divided by these two and delta T over delta T will give you the angular velocity and this will give us what we wanted which is this angular velocity in the very beginning so this is our Omega and therefore what we have is that Q Dot is equal to [Music] 2 Q one half of Q multiplied by Omega by zero Omega so this is the final formula that we want and I just want to give one more property about this formula in particular because this can be written in multiple ways another way that this can be done you can go through the multiplication here and that will give you Q well this is autonium multiplication as we know it so this will be let me just make them a very explicit over here times zero and Omega perhaps here we can also see those three as one unit that I will call Q vector okay this just for notation and if we go through the math what we will have I'll let you guys do this as an exercise we will have a matrix here that is zero minus Omega transpose Omega and minus Omega this is the Matrix that performs a cross product procure okay this part over here can be written sorry this cube is really the quadrant this can be written as that and if you call this m this is another way of seeing the same equation huh but if you call this n notice that m is a skew symmetric meaning that if you take M transpose M transpose will be the transpose of this Matrix so V is zero over here we will have this guy over there so it will be Omega transpose over here you will have that guy falling over here this will be minus Omega transpose and this Matrix is skill symmetric itself so you will have uh Omega I suppose remember that the cross product is excusing is an excuse symmetric Matrix so this but in the other in the end of the day notice that this is nothing else than minus n this is what we call skill symmetric Matrix means that when you take the transpose you get the minus of it and that's extremely interesting for differential equations because of the following if you take the norm of a pattern right so take the norm squared let me quaternion and I want to know how the norm itself this is a number this is a real number and I want to know what happens to the derivative of this number in time so the derivative of that number well we can compute it this the norm of a quaternion can be written in Vector form as Q transpose Q right the norm squared so if you take that derivative this will give us two Q trans transpose times Q dot this is the derivative of of two vectors Q dot now we know how to compute it so this will give me QQ transpose times Q dot which is uh one half so this has to go over here one half of this in times Q itself and now I just want to analyze this a little bit further what is that so uh there is a number still this is a real number and any real number of course since that's just a number every number is equal to its transpose right so I'm expecting Q transpose and Q to be equal to Q transpose and Q transpose that's simply by the fact that this is a real number therefore if you take the transpose of the product is the product of the of the of the the transpose of the product is the product of the transpose in order reverse right so this would be Q transpose times M transpose times Q so this is what we had but for this specific case M transpose is equal to minus n so we will have minus Q transpose m q so this is equal to that what I can do is take this part and take it to the left side so this will be two Q transpose mq will be equal to zero and therefore Q m q is equal to zero which means that this product over here is actually uh zero so this entire thing is zero and and therefore this derivative here it's zero and that's a very interesting property which kind of makes uh a very comfortable environment for us to work with which means because since here I put the just a phase notice that I put the phase of the quaternion inside the limit so I'm taking the derivative of the phase but this could really be any quaternion because this differential equation here as it is what we put here we checked what was the quadrillion of this different the the norm of the quaternion assuming this quaternity respects this differential equation and it does the storm should not evolve in time okay so this is a constant and as soon as you put any quaternion through this differential equation the con the the the amplitude of that quaternity will not change only its phase so that goes to say again that we can use non-winetary quaternals for rotation and as soon as you put them in this differential equation it will drive it in a way that you want to change its Norm it will change it only its phase and normally a lot of people will just use the the the unitary because they suffice for rotations what I would like you to think a little bit what else can be done if you also use the amplitude and you evolve this is actually interesting for optimization applications as well when you don't need to restrict yourself to unitary patterns that can bring you some interesting properties and I'll let you think about that but until next time goodbye