# Ficticious forces influences what an accelerometer exactly measures?

Take two different reference frames. The first one - say $\lbrace{W}\rbrace$ - is inertial and it's fixed to the Earth surface. The second one $\lbrace{B}\rbrace$ is moving and in general is rotating and accelerated, so $\lbrace{B}\rbrace$ it is not inertial.

The equations of motion for the origin of the frame $\lbrace{B}\rbrace$ can be formulated on both frames as:

$$\sum_{i}\, ^wF_{ext} = m\, ^w\dot{v}$$ in the inertial frame and $$\sum_{i}\, ^bF_{ext} = m\, ^b\dot{v} - ^b\omega \times m ^bv$$ on the non-inertial one, where $^f x$ represents the vector $x$ expressed on the frame $f$.

Let an accelerometer to be installed over $\lbrace{B}\rbrace$ which by design subtracts gravity from its output.

I'm not sure if it is measuring the term $^b\omega \times m ^bv$ or not so, what is the accelerometer exactly giving as output?

• I'm under the impression that accelerometers measure forces in three directions, so it'd measure gravity in one direction (z) and rotational forces in the other two (x,y; assuming you're rotating in that plane). I could be wrong, though. – Kyle Kanos Mar 2 '17 at 11:04
• Thanks for the answer @KyleKanos, but I don't agree. The output of the sensors in static with the platform over a flat surface returns something like the oppossite of gravity. This is because, on rest the reaction force of the surface over the platform equals the weight force but in the oppossite direction. The problem here is, what happens in motion and if I'm measuring or not the effect of $\omega \times v$ – Jay-Si Mar 2 '17 at 11:40

## 2 Answers

I've spent some time thinking about my question and I think that i've arrived to the answer.

Accelerometers measure only the acceleration of non-fictitious external forces

To reach this conclusion I planed a series of experiments which tries to identify the effect of the term $$^b\omega \times\,^bv$$ on the accelerometer outputs for different situations. Those experiments and conclusions are assuming that there exist a reference frame at the earth surface (which actually it is not true, see the comment from uhoh under the answer)

# Experiments

• [EXP1] On static: $$\omega = 0$$ and $$v = 0$$
• [EXP2] Only angular velocity: $$\omega \neq 0$$ and $$v = 0$$ with angular velocity pointing vertically ($$\approx \parallel$$ to the gravity direction)
• [EXP3] Only angular velocity: $$\omega \neq 0$$ and $$v = 0$$ with angular velocity pointing horizontally ($$\approx \bot$$ to the gravity direction)
• [EXP4] Both angular and linear velocities being perpendicular: $$\omega \neq 0$$, $$v \neq 0$$ and $$\omega \bot v$$
• [EXP5] Both angular and linear velocities being parallel: $$\omega \neq 0$$, $$v \neq 0$$ and $$\omega \parallel v$$

# Experimental environment & description

I use a the crazyflie quadrotor's accelerometers as the experimental accelerometer. The measures are taken at $$T = 10 \textrm{ms}$$

Since I need angular velocity and a constant linear velocity I decided to not turn the quadrotor on, but instead to externally generate the angular and linear velocities.

For the angular velocity I bought a little $$\mu$$usb fan that can be connected to the phone by 2€ approx. You can see all together in action here. And for the linear velocity... well I ask my wife to drive a car by the highway at constant velocity while I was on the back-seat. All the experiments that had linear velocity the car was running at $$\approx 90 \textrm{km}/{\textrm{h}}$$.

# RESULTS

In case you want to analize the results by yourself the data are available here.

The next images, shows the accelerometers output for every experiment (**Please forgive the bad units in the labels, of course acceleration has not velocity units. They should read $$m/s^2$$ **)    ## Aclarations

The blue lines correspond to two different experiments. Since going in the car and record at the same time was a bit stressful I start recording when the quadrotor had not reach the constant angular speed or even without turning on the fan.

# Conclusions

The measures of [EXP1] reveals the value of the gravity acceleration sensed by the accelerometers.

Focusing on the measures for the [EXP4], when the angular velocity is null are similar to the static ones (the oscillations are justifiable by the car bouncing in a not (so)even terrain). When the angular velocity is stabilized the output equates the results of [EXP2].

As it can be seen the effect of, rotation has an effect on the magnitude of the acceleration. It appears both when the accelerometers on rest and when moving at constant velocity. My interpretation is that it seems to be due a centrifugal effect given a possible misalignment of the accelerometers from the rotation center. This assumption is supported by the measures on the plane (x-y) axes of the experiment [EXP2] that shows this force as $$\approx$$ constant.

In the case of experiments [EXP3] and [EXP5], the acceleration of gravity perturbs the centrifugal acceleration making an oscillation from its maximum value to its minimum. However i'm not able to justify why the maximum is not above the mean of [EXP4] and [EXP2], and the oscillation magnitude does not equals the norm on [EXP1]. I can only think that's maybe given by a too high $$\omega$$ which does not allow with the given measure frequency to do a fine sampling of the signal.

Having argued all of the above, and mainly by comparing [EXP4] and [EXP2] conclusion is that

Accelerometers measure only the acceleration of non-fictitious external forces

Any comments and arguments against this result will be very welcomed. Thanks community.

• I think it's great when people take the time to post an answer to their own question. Happy to hear that you did not actually turn on the Crazyflie quadrotor while inside a car moving down a highway! By the way, the Earth's surface isn't so inertial; near the equator $\omega^2 r$ is about 0.034 m/s^2, or about 0.35% of gravity. Weather, ocean currents, ballistic projectiles, etc. are profoundly affected by this. – uhoh Sep 24 '17 at 5:27

Accelerometers measure compression... well; more precisely, they measure compression indirectly, by measuring the displacement. Note that manufacturers calibrate accelerometers for Earth (see IEEE 1293-1998).

## Physics

The basics of how traditional accelerometers work is as follows: You have a housing that is attached to the object on which you want to measure acceleration. Inside the housing exists an internal body that can move. This internal body (known as proof mass or seismic mass) is constrained to movement on a single axis, attached via a spring to the housing frame. Friction dampens the movement of the internal body.

It is common to install three perpendicular accelerometers to be able to measure acceleration in three dimensions. If you need to measure rotation, you add a Gyroscope.

Note: In modern accelerometers, the spring is the internal body itself.

When object that has the accelerometer attached moves, the internal body lags behind the housing due to inertia, and engages in a damped simple harmonic motion. If we can measure the displacement, we should be able to calculate the force applied to the system.

It follows from simple sum of forces:

$m\frac{{d}_{2}}{{dt}^{2}}(x) = {F}_{applied} - {F}_{dampening} - {F}_{spring}$

Where $m$ is the mass of the internal body (which is known) and $x$ is the displacement of the internal body (which is what we measure).

$m\frac{{d}_{2}}{{dt}^{2}}(x) + {F}_{dampening} + {F}_{spring} = {F}_{applied}$

$=>$

$m\frac{{d}_{2}}{{dt}^{2}}(x) + c\frac{d}{dt}(x) + kx = {F}_{applied}$

Where $c$ is the dampening coefficient and $k$ is the spring constant.

Consider that the accelerometer has no way to distinguish if the applied force is due to acceleration or gravitation. In fact, it has no way to sense Earth... instead the conversion (analog or digital) from the measurements of the displacement of the internal body to the acceleration presented to the user (which may require an integrating converter) has a correction applied that is calibrated to yield 0 at rest on Earth and is tested on known conditions.

Furthermore, the accelerometer uses a small (negligible) mass. Thanks to that, and due to its calibration, the output is a measurement of proper acceleration.

This is where the accelerometer differs from the seismometer: the seismometer uses a large mass and a lose spring, while the accelerometer uses a small mass and a stiff spring. Thus, the seismometer is better to measure small movements, and should not be disturbed as to no compromise the data. Furthermore, the intention of the seismometer is to measure the amplitude of the seismic waves, so it does not require the same computations.

## Engineering

The detail of how the measurement works change depending on the type of accelerometer:

• A mechanical accelerometer (Accelerograph) measures the displacement of the internal body. Mechanical accelerometers use a pen, mechanically attached to the internal body; to mark on paper... the paper has grades depending on the calibration of the accelerometer.
• A capacitive accelerometer measures the distance between capacitor plates being pushed the internal body. As you know, the capacitance is inversely proportional to the distance. The capacitance is be measured by charging and discharging the capacitor with a known current and measuring the change in voltage.
• A piezoelectric accelerometer measures the deformation of a piezoelectric crystal (such as quartz) being compressed by the internal body. Piezoelectric crystals when compressed generate measurable voltage.
• A piezo-resistive accelerometer measures the deformation of a piezo-resistive material (such as silicon) being compressed by the internal body. Piezo-resistive materials when compressed change their impedance. To measure the impedance, use ohms law, combined with a known voltage and measuring the current, or with a known current and measuring the voltage.
• A inductance accelerometer measures the displacement of the internal body... by making the internal body of a ferromagnetic material and having it move inside of a coil. Due to the presence of the internal body, which serve as core for the coil, the impedance of the coil changes.

On the engineering aspect, modern accelerometers, it often falls back to voltage measurements. An accelerometer for human use will convert the measurement of the voltage (with any calibration corrections applied) to be displayed.

Note: There are no ideal current sources. That is, that having a known current is a myth.

• This is a wonderfully detailed answer to a different question than was asked. – Floris Mar 2 '17 at 14:17
• @Floris really? Well, shame on me, I guess. I see OP talks about the equations of motion in two different frames of reference, the accelerometer can't know about another frame of reference! - I answered the question on the title. – Theraot Mar 2 '17 at 14:33
• Thanks for the detailed explanation Thearot. I thought that I knew the principles under accelerometers, but your explanation helps me a lot to fix and make clear some ideas. However as @Floris points out it does not help me to figure out what the real answer is. It will be of very help if you could do some appreciation. Thanks again – Jay-Si Mar 2 '17 at 15:36
• Thanks for this answer; although it doesn't address the body of the question head-on, it certainly does address the title; What's an accelerometer exactly measuring? which is what will bring some people here (like me). – uhoh Sep 24 '17 at 5:09
• @uhoh, thanks for the comment, and sorry for the late reply. After thinking on what you said I decided to modify the title of the question to clarify the issue. Thanks for pointing out the lack of focus – Jay-Si Nov 20 '19 at 8:57