Quaternion concatenation is noncommutative. That is

q_a * q_b ? q_b * q_a

However

q^-1 * q = q * q^-1 = I_q

where q^-1 is the inverse of q and I_q is the identity quaternion. VQS concatenation is also noncommutative:

T_{A_B} * T_{B_C} ? T_{B_C} * T_{A_B}

Where T_{A_B} represents a VQS transformation. Now, we find the inverse of T_{A_B} like so:

T_{A_B}^-1 = T_{B_A}

My question is, is the concatenation of a VQS with its inverse commutative? Ie, is the following statement correct?

T_{A_B} * T_{B_A} = T_{B_A} * T_{A_B} = I_VQS

Where I_VQS is the identity VQS. With the implementation I’m using I’m finding

T ^-1 * T = I_VQS, whereas

T * T ^-1 ? I_VQS

This seems incorrect; both sould return I_VQS.

EDIT:

Here is the implementation of VQS Inverse and concatenation functions I'm using:

//--------------------------------------------------------------------------------
//		Concatenation
//--------------------------------------------------------------------------------
VQS VQS::operator*(const VQS& rhs) const
{
	VQS result;

	//Combine translation vectors
	result.v = q.Rotate(rhs.v) * s + v;

	//Combine quaternions
	result.q = q * rhs.q;

	//Combine scales
	result.s = s * rhs.s;

	//Return result
	return result;

}	//End: VQS::operator*()


//--------------------------------------------------------------------------------
//		Returns inverse VQS
//--------------------------------------------------------------------------------
VQS Inverse(const VQS& other)
{
	VQS temp;

	//Inverse scale
	temp.s = 1.0f / other.s;

	//Inverse quaternion
	temp.q = Inverse(other.q);

	//Inverse vector
	temp.v = temp.q.Rotate(-other.v) * temp.s;

	return temp;

}	//End: Inverse()

Yes, you should get the identity in both cases. Can you post a specific example where you don't get that? Simple numbers would be best, so we can do the computations by hand...

Ok good idea. Here's an example that might help. Incidentally, the issue it seems appears to be with the Inverse(VQS) function. Here's a test case:

T = [(-3.2, 1.6, -10.3) , (-0.52, 0.68, 0.49, 0.32) , 2]

Inverse( Inverse(T) ) = [(-4.0, 2.5, -9.9) , (-0.52, 0.68, 0.49, 0.32) , 2]

Inverse( Inverse(T) ) should return T. The scalar and quaternion components are correct, however the translation vector is off. It looks like the method to find the inverse translation component might be incorrect.

SOLVED.

Solution was to ensure all quaternions were normalized, thus representing rotations.

Just to let you know just because your quaternions are normalized does not mean they represent rotation. Quaternion can be used for other things other than rotation.

In the graphics world, it is used mostly for that. But in reality they can be used for other things other than rotation.