Is one accelerometer enough for detecting tilt/orientation of a rigid body?

Question

I posted this question earlier in physics.stackexchange, but I think this is actually the best place for it.

I’m not familiar with sensor hardware, and I don’t really know how an accelerometer works. But I do know that from a mathematical point of view, you wouldn’t be able to uniquely determine the orientation of a solid object knowing only the acceleration of one of its points.

Indeed, assuming known initial conditions and acceleration function of one point, say point X, one can solve the second order ODE and get the position function for that point, but that wouldn’t determine the position of the rest of the points in the solid object, because the object could perform a rotation centered on point X, leaving it unchanged. Despite all that, I have seen several claims that modern mobile phones benefit from the inclusion of accelerometer sensor in such a way that enables them to, for example, determine the orientation of the screen. I don’t understand how an IPhone would know to what position the screen was tilted. Consider the following example where the red dot indicates the location of the accelerometer sensor.

Suppose acceleration data was processed and the motion of the sensor was determined as the red arc shown in the pictures below. How one could distinguish from the two depicted motions? Notice that the final orientation of the device is different.

Please note that by one accelerometer, I mean one that would be able to register accelerations in all 3 dimensions. That way, upon integration I would get the spacial (3D) trajectory of the point. I believe the issue I describe here holds even in such scenario.

Thanks for reading!

Answer 1

The accelerometer allows the device to know which way is up and which way is down. So when the display is vertical, it is totally trivial to orient the display correctly. If the display is on a level table, and then rotated slowly, the accelerometer is basically useless. Some devices also have either a three axis magnetometer or at least one angular rate sensor (gyro) or both. These can help with detecting rotational motion that the accelerometer is not good at detecting. So if the display is vertical, there is no problem. If it is then tilted to a horizontal plane, the accelerometer is out of the picture.

But if the device has a chip gyro or magnetometer, it can detect a quick rotation and cause the display to rotate also. Most of the time this is correct (the quick rotation was caused by the user turning the display while horizontal), but sometimes it is an annoyance (maybe you turned a corner quickly, and the display then suddenly went sideways).

In theory having two accelerometers with some baseline between them would allow you to detect angular rotation, but I don't think that is done in practice, and I don't think it would actually work, because it would require a level of precision that consumer accelerometers don't have. The chip gyro gives you much more bang for your buck. For total dead reckoning, you basically need a 3-axis accelerometer and a 3-axis gyro or angular rate sensor. Very precise electronic dead reckoning devices exist, but they are expensive, and subject to export control in the USA (because they can be used in missile guidance systems).

Answer 2

If you are only interested in determining 'up' or rather 'down'relative to the body frame of your device, after it has completed a series of rotations, then triaxially mounted accelerometers that measure to zero frequency and are sensitive enough to resolve to fractions of the gravitational acceleration will work. And in this case the math is fairly simple, you simply resolve the Euler angles from the three accelerometer signals when motion stops.

But tracking rotations moment to moment can be much more complicated, and as mkeith suggested in his answer, you would require a triad of rate gyros to complement your accelerometer triad as well as more complicated mathematics to fuse the measurements.