How to get the rotation between accelerometer's axis and motion vector?

Question

I'm working on a Pedestrian Navigation System on Android. I am currently trying to get the rotation matrix between the 3-axis accelerometer referential and the motion that is applied to the device.

Let's say you are walking straight forward with the device in your hand, but the Y axis of the accelerometer (thus the device's Y axis) is not oriented in the same direction as the one you're heading yourself (basically you're holding the device in an awkward way). Then if I want to apply distances (based on step detection), it would be wrong to apply them to the accelerometer referential (which I know the orientation) : it has to be rotated accordingly to the motion heading.

This is why I would like to know if you could enlighten me about a method to compute the accelerometer readings to turn them into rotation angles (or a matrix). Such a method has to avoid integrating the acceleration even once, as the error is abyssal on cheap accelerometers (otherwise it would be pretty easy to perform I think).

EDIT : Magnetometer and Gyroscope help you to find the device's orientation even when stationnary, but doesn't allow you to know in which direction the device is moving. I have the first one, and I'm searching for the second. Basically :

Human Referential (motion direction) -> Device Referential (or accelerometer referential) -> Earth Referential

and i'm searching the way to find the rotation matrix to compute distances from HR to DR, which I would then apply to the rotation matrix I found to go from DR to ER.

Answer 1

I found an article which might be useful, here is the link.
The Idea is based upon acquiring the accelerometer data in the 3 axis, (which is an n times 3 matrix) and finding the PCA (principle component analysis) of the matrix. The PCA is the 3 vectors with the highest energy of the matrix).
Idealy, the main vector (with the highest energy) direction is upwards, and the second is the heading direction. You can read it all in the article. I tried implementing the algorithm in matlab, the result is o.k. (not great) Hope you can do better (I would like to hear about good results).

Hope this helps

Ariel