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Roll Yaw

How to get the yaw pitch roll from 2 vectors?

Say 2 vectors are starting at the origin, Vector 1 is (0,0,-1) (meaning it's looking into the screen), and another one Vector 2 (1,1,0) (meaning it's on the xy plane looking North East. How can I get the yaw, pitch, roll that will rotate the first vector about the origin so that it looks at the same direction as vector 2? Thanks =)

For me, it is easier to start on the xy plane at due east, but we are given a point that doesn't start are the xy plane. It starts at (0, 0, -1), so I need to perform a rotation that will swing that point up onto the xy plane. The magnitude of the vector is 1 which is found by

magnitude = squareroot( ( x1 - x2 )^2 + ( y1 - y2 )^2 + ( z1 - z2 )^2 )

We are using the origin as the second point, so:

= squareroot( ( 0 - 0 )^2 + ( 0 - 0 )^2 + ( -1 - 0 )^2 )
= squareroot( 1 ) = 1

Choose a point on the xy plane that has the same magnitude away from the origin. (0, 1, 0) is due east, the first coordinate must be 0 because it is on the yz plane and the third coordinate must be 0 because it is on the xy plane.
(y, z)
vector1: (0, -1)
due east: (1, 0)

The angle between formula is:
( vector1 dot vector2 ) / (magnitude 1 multiplied by magnitude 2) = cosA

vector1 dot vector2 = 0 * 1 + -1 * 0 = 0

magnitude 1 multiplied by magnitude 2 = 1 * 1 = 1

So,

0 = cosA
A = arccos(0) = 90degrees and 270degrees, I prefer the ones in the range 0degrees to 90degrees so I am choosing 90degrees

So far we have one rotation, 90degrees along the yz plane. This rotation changes the position of vector1 to the new position of:
(0, 1, 0)

Find the magnitude of the second vector which is the final destination in this question(because when using the angle between I think it is better to have equal magnitudes).

= squareroot( ( 1 - 0 )^2 + ( 1 - 0 )^2 + ( 0 - 0 )^2 )
= squareroot( 1 + 1 + 0 ) = squareroot( 2 )

Divide each component of the second vector by that magnitude will yield the unit vector(the vector with magnitude 1).

( 1 / squareroot(2) , 1 / squareroot(2) , 0 )
Rationalize
( squareroot(2) / 2 , squareroot(2) / 2 , 0 )

Next, use the formula for finding the angle between two vector which are on the same plane(I am choosing the yz plane which means that x = 0 for both(it is ignored)). So, the points on the yz plane from the previously 3 dimensional points are:

Next, I chose to make the necessary rotation along the xy plane which means ignore the z coordinate(it is z=0). The points become:
(x, y)
(0, 1)
And the destination,
(squareroot(2) / 2, squareroot(2) / 2)

Find the angle between the two vectors(or points in this case relative to the origin),

( vector1 dot vector2 ) / (magnitude 1 multiplied by magnitude 2) = cosA

vector1 dot vector2 = 0 * squareroot(2)/2 + 1 * squareroot(2)/2
= squareroot(2)/2

magnitude 1 multiplied by magnitude 2 = 1 * 1 = 1

So,

squareroot(2)/2 = cosA
A = arccos( squareroot(2)/2 ) = 45degrees or 315degrees, again I choose the angle between 0 and 90, so 45degrees

So, it took two rotations:
90degrees about x
and
45degrees about z

Pitch, Roll, Yaw



Soyuz space capsule oddity; can someone explain this?

Last time I visited Kennedy Space Center, I noticed something peculiar about the Soyuz compared with the Appolo capsule. On Appolo, there are small boosters at 90 degree intervals to control roll, pitch, and yaw. Each little booster can fire in 4 directions. On Soyuz, there are also boosters at 90 degree intervals, but they all point straight out. I don't understand how you'd be able to control the roll of the capsule with these boosters. I asked someone at KSC, and we were going through books of drawings for about 45 minutes before we finally gave up without figuring it out. Anyone here have a clue about this?
*should be spelled, Apollo*

The Soyuz roll thrusters are at the extreme rear of the module, and have 1/10 the power of the pitch and yaw thrusters. The reason being that pitch and yaw thrusters are also used for spacecraft lateral motion during rendezvous and docking, while roll is used only for orientation.

You can see the location on the diagram below.

Tags: airplane pitch roll yaw, math, pitch, pitch roll yaw definition, pitch roll yaw order, roll, roll yaw, roll yaw coupling, roll yaw pitch, rotation, yaw

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