Resolving Pitch and Yaw

Date: 02/10/2003 at 23:42:53
From: Corey
Subject: Resolving Pitch and Yaw

How do you find the resultant of Pitch and Yaw? Is there an equation?

I consider them as vectors, and I think I'm supposed to find the root
mean square. 

As far as I know, the resultant of pitch and yaw is equal to wobble, 
no? Is there an equation to find this? Is it Root Sum Square (square 
root of pitch angle squared and yaw angle squared)? The application is 
the mating of two bodies in space.

Date: 02/11/2003 at 13:03:09
From: Doctor Douglas
Subject: Re: Resolving Pitch and Yaw

Hi, Corey,

Thanks for submitting your question to the Math Forum.

Typically in vector math, the resultant is the sum of the vectors.

If I think of myself standing in an airplane and facing forward, a 
pitch is a rotation about an axis that is horizontal (say wingtip to 
wingtip) and a yaw is a rotation that is about an axis that is up and 
down.  The sum of these vectors will be a vector that say points from 
my left foot to my right ear. This is the vector about which the plane 
rotates if the plane carries out a simultaneous pitch and roll.

I've never heard of "wobble" so precisely defined.

I am not sure what the exact application is (in math terms), but I 
will make a guess and hope that it suits your needs. If it doesn’t, 
feel free to write back and we can discuss this further.

If you are looking for the resultant angle of the combined pitch and 
yaw, and these angles are all *small*, then it is possible to use the 
root-sum-square formula on the angles to find the approximate 
magnitude of the displacement angle. If the angles are not small, one 
has to be careful.

Let’s say that a particular spacecraft axis is initially pointing 
along +z, and that the pitch and yaw are rotations about the +x and +y 
axis, respectively. Now let’s say that the object first makes a yaw of 
Ry degrees and then a pitch of Rx degrees. One question we can ask is 
what is the angle that the spacecraft (originally pointing along z) 
now makes with the actual z-axis after the yaw-then-pitch procedure.  
Call this angle "alpha".

Clearly after the yaw procedure but before the pitch, the angle is 
just Ry. Let the initial direction of the spacecraft in (x,y,z) 
coordinates be (0,0,1). Then, if we can determine the direction to 
which the spacecraft rotates, we can compute the angle it makes with 
the original (0,0,1) axis by taking the inverse cosine of the dot 
product, because A.B = |A||B|cos(q), where A and B are two 3-vectors 
and q is the angle between them. We will conveniently work with 
vectors such that |A|=|B|=1.

   (0,0,1)               spacecraft initially

   (sin(Ry),0,cos(Ry))   after yaw (rotate by Ry about y-axis)
                         angle that this vector makes with +z is 
                            given by arccos(cos(Ry)) = Ry.  
                            This makes sense.

Now, we perform a pitch rotation – rotate by Rx about the x-axis:
 
   (sin(Ry),cos(Ry)sin(Rx),cos(Ry)cos(Rx))     reduces to previous 
                                               when Rx=0

Note that the length of this vector is 1.0 (it should be, since it is 
the rotation of something that started with length 1.0).
 
Now, what is the angle alpha after all this?  Taking the dot 
product with the initial direction of +z=(0,0,1), we get

   0 + 0 + cos(Ry)cos(Rx) = cos(alpha)

so the final formula for alpha is 

   alpha = arccos[cos(Ry)cos(Rx)].

As an example, if Ry = 5 deg and Rx = 6 deg, alpha = 7.804 deg. This 
is close, but not quite the same as the rms formula:  

   rms = sqrt(Rx^2 + Ry^2) = sqrt(25 + 36) = sqrt(61) = 7.810 deg

If the angles are larger (say Ry=45 deg and Ry=56 deg), then alpha = 
66.708 deg and the rms formula gives 71.84 deg. As an extreme example, 
you can consider Rx=Ry=90 deg, and convince yourself that the combined 
yaw-pitch ends up at 90 degrees from the starting position. The 
formula for alpha given above gets this correct, but the rms formula 
(which gives 90 deg * sqrt(2)) cannot be right.  

You can see that the rms formula is a reasonable formula for small
angles Rx and Ry, but becomes less accurate for larger and larger
pitch/yaw combinations. I believe that if one expresses the formulas 
above in terms of Taylor series near Rx=0 and Ry=0, we will be able to 
see where the two formulas differ, although I have not done this 
analysis.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/