Thinking About Accelerometers and Gravity

by Dave Redell, LUNAR #322

There have been several incarnations on r.m.r (rec.models.rockets-see note at end of this article) of the time-worn "pendulum-in-the rocket" fallacy. As usual, these involve plumb bobs, mercury switches, accelerometers and so on. Below, I outline the way of viewing this question that I personally have found most useful. An article on this subject may seem like beating a dead horse, but a couple of email side conversations I've had suggest to me that even folks who understand the problem might find the following to be of at least some interest.

The basic puzzlement typically goes something like this:

When my rocket is sitting on the pad, the accelerometer senses gravity just fine, but the minute it lifts off, you say it can no longer do so. It's the same gravity and the same accelerometer, so how can this be?

There have been a number of explanations posted on rmr about this, including some quite long ones, even with ASCII illustrations. Like many such problems, this one can seem simpler or more complicated depending on how you look at it. For me, the clearest way to see what is going on is to ignore the innards of the accelerometer (e.g. the forces on the reference mass, etc.) and analyze its black-box performance as an acceleration detector, which is, after all, what "accelerometer" should mean.

So consider an ideal 3-axis accelerometer that reports a single acceleration vector (direction + magnitude) in an arbitrary direction. From this viewpoint, the right way to describe its behavior with respect to gravity is simple:

An accelerometer never senses gravitational acceleration.

or more specifically:

An accelerometer is a device that senses deviation from freefall.

Below, I first attempt to motivate this view and then use it to explore the question of when you can-and can't-use an accelerometer as a "tilt sensor"-i.e. to determine the direction of "down".

What accelerometers sense

If we drop the accelerometer in a gravitational field (without air, etc.), it reads zero, no matter what its orientation. It is accelerating downward and yet reading zero, so it clearly isn't sensing acceleration due to gravity. Or, in the terms cited above, it is correctly sensing that the deviation from freefall is zero.

So we put the accelerometer on a tabletop. It will sense an acceleration of 1.0G straight up. Since freefall would be 1.0G straight down, the accelerometer is again correctly sensing the deviation from freefall, which is an upward acceleration vector of 1.0G, due to the normal force exerted by the tabletop.

Now, we tilt the table at an angle A from horizontal. The accelerometer begins to slide, accelerating down the (frictionless) tabletop at a rate of 1.0G*sin(A). Does the accelerometer reading match this? No: it shows an acceleration vector of 1.0G*cos(A), perpendicular to the tabletop. This is exactly the deviation from freefall, once again provided by the normal force of the tabletop.

If we put our accelerometer in a rocket, what does it measure? The rocket accelerates based on the vector sum of thrust, drag and gravity. The accelerometer registers only thrust plus drag, since that is exactly the deviation from freefall. It ignores the acceleration due to gravity.

What about inside an airplane-why is that different? In steady level flight, the situation is equivalent to the level tabletop: the deviation from freefall is provided by the 1.0G lift from the wings that keeps the airplane from falling down, so the accelerometer registers 1.0G, straight up. Other maneuvers can result in a wide range of accelerometer readings, all based on non-gravitational forces (basically lift, drag, and thrust).

So how can an accelerometer detect the direction and/or strength of gravity?

Since the accelerometer does not respond to gravitational acceleration, it never detects gravity directly. Since it detects deviation from freefall, its reading can sometimes be used to infer the properties of local gravity if we utilize other knowledge of the situation that we may have. Imagine that we are in a little box with the accelerometer, and all we know is the reading that it is returning. Based on that knowledge alone, we can never determine the direction or strength of gravity. But now, consider injecting selected other knowledge-in particular, knowledge that relates our deviation from freefall to the local gravity vector.

For example, if we know that we are stationary on the tabletop, we can infer both the direction and the strength of the gravitational field: It is exactly equal and opposite to the accelerometer reading.

At the other extreme, if we are in freefall, the accelerometer reads zero and we can't infer anything about the direction or strength of gravity. The only way we can learn anything is to open the window and look outside, taking note of the passing scenery to deduce which way we are falling. The accelerometer is no help at all.

Not too surprisingly, the intermediate case of the angled table is also intermediate in what we can infer. For example, suppose we know that we are sliding down the angled table top. If we also know either the angle of the table top or strength of local gravity, we can use simple trig to determine the other. Given the angle, we know that "down" lies somewhere on the surface of a cone that we can compute, but we can't say any more than that. In other words, compared with the level table top, we can't infer quite as much about local gravity given the equivalent extra knowledge.

Inside the rocket, we can sense drag and thrust. These determine our deviation from freefall, but since they have no particular relation to gravity, we can't use the accelerometer reading to find "down". Of course, if we already know the relation of the deviation vector to gravity, we can find "down". For example, if we know that after 15 seconds, our rocket will be headed straight down at zero thrust, we can say that the accelerometer will register a drag vector that points straight up. But to deduce this, we already used our knowledge that the rocket was headed down, so the accelerometer reading isn't adding any new information about this. We could "locate down" just as well by simply pointing toward the nose of the rocket.

This basic idea of extra knowledge is fundamental to inferring the gravity vector from the output of an accelerometer. For example, when we observe an airplane in steady flight, we might think that a passenger could use an accelerometer (or plumb bob or whatever) to determine "down". But from within the plane (without looking out the window) the passenger doesn't have enough knowledge to do this. The plumb bob may point straight at the floor, but the passenger still can't tell whether the plane is level, in a banked turn, or upside down at the top of a loop.

Anyway, I hope this rather long message hasn't bored everyone. If the above makes the problem seem simple and the answer seem obvious-well, that's just what I had in mind.

(Ed. Rec.models.rockets is a Usenet newsgroup oriented towards discussions and topics related to non-professional rocketry of all types. All questions, comments, and ongoing discussions related to non-professional rocketry are welcome. The rmr newsgroup contains a wealth of information about our hobby. The address of the rec.models.rockets archive is