From: Balaji Ragupathy
To: Martin Baker
Sent: Tuesday, February 19, 2002 11:04 AM
Subject: Singularity in rotational matrix

Hello Martin,

I have gone through your explanations on the Euler angles and it's
mathematics. It is very good and very easy to understand the concepts
the way that you have provided.

In NASA Standard Airplane section, R1, R2 and R3 matrix was combined and given
the trnaformational matrix, can you please explain how do we know where the singularity
occurs. If singularity occurs, how do we overcome using euler angles.

Thanking You,

Regards,
R.Balaji

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From: "Martin Baker"
To: "Balaji Ragupathy"
Subject: Re: Singularity in rotational matrix
Date: 20 February 2002 19:35

Hi R.Balaji

I think the singularities can be found by taking the derivative, the matrix
that relates the angular velocities (pitch,roll,yaw) to the angles is:

-sin(attitude)	0	1
sin(heading)*cos(attitude)	cos(heading)	0
cos(heading)*cos(attitude)	-sin(heading)	0

 

when attitude = +PI/2 or -PI/2 the first and third columns become similar so
a singularity is reached.

The problem can be reduced by choosing an angle sequence which does not have
a singularity in the working range, for instance this singularity only
occurs when the aeroplane is going straight up or down, which might not be
on the normal operational parameters of most aircraft. I don't know how to
overcome the singularity issue using euler angles completely, I guess we
need to use quaternions to avoid the problem.

Martin