dcm2quat is returning invalid quaternions

I am using dcm2quat to convert multiple direction cosine matrices to quaternions. For several, it is working fine, but for others I am getting results that violate the fundamental constraint on quaternions--that q1²+q2²+q3²+q4²=1.

Here is one example of a DCM I'm having trouble with:

>> ACB

ACB =

    0.4970    0.8638   -0.0824
    0.8677   -0.4951    0.0435
    0.0032    0.0932    0.9956

>> q = dcm2quat(ACB)

q =

    0.7067   -0.0176    0.0303   -0.0014

>> q(1)^2+q(2)^2+q(3)^2+q(4)^2

ans =

    0.5006

This is a valid DCM. Its transpose is equal to its inverse, multiplying it by its transpose yields the identity matrix, and the sum of the squares of each row and column is 1. Why is MATLAB returning invalid quaternions?

Edit: Also tried running quat2dcm on the result from dcm2quat and got a different DCM than I started with. Something's very wrong.

Answer 1

[James Tursa]

ACB is not a valid direction cosine matrix. You can see that its determinant is not close to 1:

>> ACB = [    0.4970    0.8638   -0.0824
    0.8677   -0.4951    0.0435
    0.0032    0.0932    0.9956]
ACB =
    0.4970    0.8638   -0.0824
    0.8677   -0.4951    0.0435
    0.0032    0.0932    0.9956
>> det(ACB)
ans =
   -0.9999

Hence the problems occur when feeding this matrix into the dcm2quat function. So whatever process you are using to construct this dcm is flawed.

[Noah Gordon]

You're right, the process I was using would sometimes create left-handed coordinate systems. Couldn't find the condition the DCMs were violating, thanks!

[Ivan Lisun]

The dcm2quat function incorrectly takes the sign of cos and sin into account when calculating quaternion's parts. Simple code based on rotm2axang works fine. The code is redundant to make it clear what's going on

r2a = rotm2axang(DCM);
rot_axes = r2a(:,1:3);
q_a = cos(r2a(4)/2);
q_sin = sin(r2a(4)/2);
q_bcd=q_sin*rot_axes;
q = quaternion (q_a,q_bcd(1),q_bcd(2),q_bcd(3))

[James Tursa]

How can this possibly "work fine" on an invalid DCM matrix? If this code always produces a normalized quaternion result regardless of how bad the input might be, I don't see that necessarily as an advantage. E.g.,

DCM = rand(3); % arbitrary matrix not even close to valid rotation matrix
det(DCM)

ans = -0.2814

r2a = rotm2axang(DCM);
rot_axes = r2a(:,1:3);
q_a = cos(r2a(4)/2);
q_sin = sin(r2a(4)/2);
q_bcd=q_sin*rot_axes;
q = quaternion (q_a,q_bcd(1),q_bcd(2),q_bcd(3))

q = quaternion
      0.90233 -  0.24072i - 0.075742j -  0.34945k

norm(q)

ans = 1

Examining q, the user has no clue this is a garbage result.