3.5.13 · D5Guidance, Navigation & Control (GNC)
Question bank — Inertial navigation — accelerometer measures non-gravitational specific force
True or false — justify
An accelerometer measures acceleration.
False. It measures specific force — the contact force per unit mass the spring supplies. Acceleration is only recovered after the computer adds back.
A perfectly still accelerometer on a bench reads zero.
False. At rest , so upward (). The spring is actively pushing up to hold the proof mass against gravity, and that push is the reading.
In free fall the accelerometer reads pointing down.
False. In free fall the only force is gravity, which the spring cannot feel, so and . It reads exactly zero — weightlessness.
Two accelerometers, one at rest on Earth and one in a rocket in deep space accelerating at , read the same thing.
True. Both read along the "up" axis and cannot be told apart — this is the Equivalence Principle made physical: gravity and acceleration produce identical specific force.
Doubling the proof mass doubles the reading.
False. The reading is force per unit mass, . A heavier mass needs proportionally more spring force, so the ratio — and hence the reading — is unchanged.
If you know only the sensor output, you can integrate it twice to get position.
False. Raw output is , not . You must form first, otherwise the missing integrates into a error. See Dead Reckoning and Error Drift.
An accelerometer can directly sense the presence of a gravitational field.
False. Gravity pulls the proof mass and the case equally, so the spring never fights it. Gravity is invisible to the sensor; it must be supplied by a gravity model from outside.
Spot the error
"Gravity is a contact force on the proof mass, so it shows up in the reading."
Gravity is a body force acting on every atom of the mass and the case alike — not a contact force. Only contact forces (spring, electrostatic feedback) are read, so gravity is excluded.
"At rest the spring reads because gravity points down."
The spring reads the force it supplies, which is upward to hold the mass. So (up), not . The reading is opposite to gravity's direction.
", so subtract gravity to get true acceleration."
Sign error. From we get — you add the modelled gravity, not subtract it.
"Because is a constant , we can add it once and forget about it."
varies with position (altitude, latitude, the J2 flattening). This makes the loop coupled: you need position to know , and to update position — see Gravity Model (WGS-84 / J2).
"An astronaut in orbit feels no gravity, which is why the accelerometer reads zero."
There is strong gravity in orbit (it's what curves the path). The accelerometer reads zero because the craft is in free fall — everything accelerates together, so no contact force is needed. Absence of weight, not absence of gravity.
"The accelerometer measures , so the navigation equation needs no gyroscope."
is measured in the moving body frame; to add (known in the world frame) you must know the device's orientation. That comes from the gyroscopes and strapdown equations.
Why questions
Why does the accelerometer name mislead people?
Because it outputs units of and shares them with acceleration, but the quantity is specific force — equal to true acceleration only when (deep space).
Why can't a single accelerometer distinguish "at rest on Earth" from "accelerating up in space"?
Both give identical specific force . No local measurement can separate uniform gravity from uniform acceleration — this is precisely the Equivalence Principle.
Why must the derivation start in an inertial frame?
Newton's Second Law () only holds without fictitious forces in an inertial frame; in a rotating/accelerating frame you'd need extra pseudo-force terms to get the clean .
Why does forgetting gravity make position error grow so fast?
A missing constant in acceleration integrates once to in velocity and again to in position — a quadratic runaway that reaches metres within a couple of seconds.
Why is the "add gravity back" step done by the computer rather than the sensor?
The sensor physically cannot feel gravity (equal pull on mass and case), so is information the hardware never has. Only a stored model plus known position can supply it.
Why does a pilot in a coordinated banked turn feel "heavier"?
The seat's contact force must both support weight and provide centripetal acceleration, so — the accelerometer reads exactly that increased magnitude.
Edge cases
What does the accelerometer read at the very top of a projectile's arc (thrown ball, engine off)?
Zero throughout the flight, top included — the ball is in free fall the entire time, so and regardless of its velocity being momentarily horizontal.
What does a horizontal-axis accelerometer read when the device sits still and level?
Zero on that axis — gravity has no horizontal component, and the horizontal contact force needed to keep the still mass in place is zero. Only the vertical axis reads .
What if the accelerometer is in deep space, far from any mass, drifting at constant velocity?
It reads exactly zero — and , so . Here (and only here) "zero reading" genuinely means "no acceleration."
What does it read in deep space while a thruster fires steadily?
The thrust magnitude per unit mass: (since ). This is the one regime where the sensor output equals true acceleration.
What happens to the reading right at the instant a supporting cable is cut?
It jumps discontinuously from (supported) to (free fall) the moment the contact force vanishes — the sensor tracks the sudden loss of spring force, not any change in gravity.
On a rotating platform (spin about a vertical axis), what does a radially-mounted accelerometer at radius read?
It reads the centripetal specific force pointing inward, because the contact force must curve the proof mass's path — even though the platform's centre isn't translating.
Connections
- Equivalence Principle — why rest-on-Earth and accelerating-in-space are indistinguishable.
- Newton's Second Law — the inertial-frame starting point for .
- Strapdown Inertial Navigation System — where gets integrated.
- Gyroscope and Attitude Determination — supplies the orientation needed to add .
- Gravity Model (WGS-84 / J2) — the source of the you add back.
- Dead Reckoning and Error Drift — the runaway when gravity is mishandled.