Calculate how many degrees a physical device is off centre (offset) given X, Y, Z

Question

Let's say I have a physical device that I can rotate with my hands and receive X Y Z coordinates of that device. What I need to calculate is the offset - how far is the pendulum of gravity straight down. Basically how many degrees I am off centre. Is there a way of doing so? I've googled a lot of questions but so far I'm struggling.

Thanks.

Updated:

There is a physical device (sensor) which I can rotate. The device sends its X Y Z coordinates and a calculated acceleration vector in mg (mill mg). I can read this data from this device using BLE. The device should lie flat while transportation, so I need to calculate how many degrees the device is off centre (which is the laying flat on table).

Updated x2:

I need to calculate the angle between the device's centre vector and a gravity pendulum (the blue angle) on the photo.

Answer 1

First, note that accelerometer sensors tend to be noisy — giving rapidly varying values that jitter around the true measure, even when the device is at rest. So you'll likely want to gather up several measurements into a windowed average to smooth out this noise and get a more consistent orientation estimate.

Next you'll want to measure what value the device reports when lying stationary and flat. This will probably correspond to a vector along one of the axes of the sensor, whose magnitude corresponds to 1 g of acceleration. Something like (0, 1000, 0) if your sensor uses units of 1/1000th of 9.8 m/s².

Discard the magnitude to make a unit vector, like (0, 1, 0) in the example above, but keep the sign (ie. if we had (0, 0, -1234) we'd normalize it to (0, 0, -1)). Store this in your program as your "reference direction"

Now you can compute a degree offset from it like so:

direction = averageAccel 
                * 1/Sqrt(Dot(averageAccel, averageAccel))

cosine = Dot(direction, referenceDirection)

degreeError = Acos(cosine) * 180/PI

Here we're using the fact that the dot product between two unit vectors gives the cosine of the angle between them. So we can use the inverse cosine function to turn that into an angle in radians, then rescale that to change the units into degrees.