Quaternion to Rotation Matrix problem

Question

i have a "simple" (i think) problem that's making me troubles finding the right solution.

I have a device on my right wrist with a RH coordinate system (z down) that gives me a Quaternion with his current local orientation and linear accelerations values. From these values i have to create a vector of Accelerations with Gravity.

Coordinate system: Link

(Device and Smartwatch's coordinate systems matches.)

Final values needs to be compared with Smartwatch's values on the same wrist.

I found an easy and working solution to this problem that is: i take the Quaternion of my device's orientation, i rotate the gravity vector (0,0,10) in this position and then add the gravity vector to my Linear acceleration vector.

To add it i converted my quaternion to a Rotation matrix as follows:

1 - 2*(qy^2 + qz^2)   2*(qx*qy + qz*qw)   2*(qx*qz - qy*qw)
2*(qx*qy - qz*qw)   1 - 2*(qx^2 + qz^2)   2*(qy*qz + qx*qw)
2*(qx*qz + qy*qw)   2*(qy*qz - qx*qw)   1 - 2*(qx2 + qy2)

And then i did a matrix multiplication with my g vector.

Problem is if i stand still my device measures an acceleration with gravity vector like this:

(10, -2, 0.5)

while the Smartwatch measues:

(10, 2, -0.5)

I am pretty sure there is a fix for this signs missmatch in my rotation matrix, but i can't find it... Can you help me? Or how can i get the right Rotation matrix from my Quaternion? On the web there are tons of variations of this conversion.

I found this one that doesn't give me an error where other convertions do: while X-Axe is pointing downwards (standing still with the hand on my leg) i measure [10,0,0] if then i roll (X-Axe Rotation) 90 deg. clockwise the Acceleration with gravity vector changes to [0,10,0], like if my X-Axe and Y-Axe were replacing each others, but in fact X-Axe is the only one facing down even if i rotate my wirst.

Answer 1

The unit quaternion derived rotation matrix as described in the Quaternions and spatial rotation Wikipedia article is $$\mathbf{R} = \left[ \begin{matrix} 1 - 2 (q_y^2 + q_z^2) & 2 (q_x q_y - q_z q_r) & 2 (q_x q_z + q_y q_r) \\ 2 (q_x q_y + q_z q_r) & 1 - 2 (q_x^2 + q_z^2) & 2 (q_y q_z - q_x q_r) \\ 2 (q_x q_z - q_y q_r) & 2 (q_y q_z + q_x q_r) & 1 - 2 (q_x^2 + q_y^2) \\ \end{matrix} \right]$$ where the unit quaternion $\mathbf{q} = q_r + q_x \mathbf{i} + q_y \mathbf{j} + q_z \mathbf{k}$.