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2
( pP ) q 1 = [sin(45 ) , [sin(45 ) , cos(45 ) , cos(45 )]]
[cos(45 ) , [0 , 0 , sin(45 )]]
=[0 , [sin(90 ) , cos(90 ) , 1]]
·
The vector component of P confirms that P is rotated to (1, 0, 1).
7.7.6 Roll, Pitch and Yaw Quaternions
Having already looked at roll, pitch and yaw rotations, we can now define
them as quaternions:
q roll = [cos( θ/ 2) , sin( θ/ 2)[0 , 0 , 1]]
q pitch = [cos( θ/ 2) , sin( θ/ 2)[1 , 0 , 0]]
q yaw = [cos( θ/ 2) , sin( θ/ 2)[0 , 1 , 0]]
(7.98)
where θ is the angle of rotation.
These quaternions can be multiplied together to create a single quaternion
representing a compound rotation. For example, if the quaternions are defined
as
q roll = [cos( roll/ 2) , sin( roll/ 2)[0 , 0 , 1]]
q pitch = [cos( pitch/ 2) , sin( pitch/ 2)[1 , 0 , 0]]
q yaw = [cos( yaw/ 2) , sin( yaw/ 2)[0 , 1 , 0]]
(7.99)
they can be concatenated to a single quaternion q :
q = q yaw q pitch q roll =[ s + x i + y j + z k ]
(7.100)
where
s =cos yaw
2
cos pitch
2
cos roll
2
+sin yaw
2
sin pitch
2
sin roll
2
sin pitch
2
cos roll
2
+sin yaw
2
cos pitch
2
sin roll
2
x =cos yaw
2
cos pitch
2
cos roll
2
sin pitch
2
sin roll
2
y =sin yaw
2
cos yaw
2
cos pitch
2
sin roll
2
sin pitch
2
cos roll
2
z =cos yaw
2
sin yaw
2
(7.101)
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