3.5.15 · D5Guidance, Navigation & Control (GNC)
Question bank — IMU — integrated accelerometer + gyroscope
These items target the ideas behind the IMU, its two sensors, the strapdown chain, and why it drifts. Nothing here needs a calculator — that's the D3/D4 job.
Notation & conventions used on this page
Before the traps, we lock down every symbol and the frame convention. If any of these feels new, this section is the definition — nothing here is assumed from elsewhere.
True or false — justify
An accelerometer directly measures the acceleration of the vehicle.
False. It measures specific force — the spring push per unit mass — so gravity is always baked in and must be added back before you get true .
A stationary accelerometer sitting on a table reads zero.
False. With it reads , magnitude m/s²; since is down, this vector points up, sensing the table's normal force, not the (zero) net acceleration.
A perfectly free-falling accelerometer reads zero on all axes.
True. In free fall , so — no spring force acts on the proof mass; this is exactly the elevator/weightlessness case.
You can find "which way is down" from a gyroscope alone.
False. A gyro only senses rate of turning; the direction of gravity comes from the accelerometer at rest, whose reading points opposite to down.
Integrating the gyro's angular rate once gives you orientation.
True in principle — drags each body axis by the spin and so integrates rate into attitude — but any bias leaks in as a growing angle error .
A gyroscope measures the vehicle's heading angle directly.
False. It measures angular velocity (rad/s); heading is only obtained by integrating that rate, which is why a heading offset accumulates over time.
If a stationary IMU is tilted, its accelerometer output changes even though it isn't accelerating.
True. The fixed gravity reaction just points differently within the tilted body axes, so the three readings redistribute while the magnitude stays .
Position error from a constant accelerometer bias grows linearly with time.
False. A constant integrated twice gives — quadratic — which is why even a tiny offset becomes large after a minute.
Gyro-tilt position error grows even faster than accelerometer-bias error over long times.
True. The gyro bias tilts the frame (), leaking a gravity ramp; integrating twice gives — cubic beats quadratic eventually.
A high-quality IMU can navigate accurately for hours with no external help.
False. Errors grow as () and (), so even excellent units drift meters within a minute and must be fused (Sensor Fusion & Kalman Filter) with external references.
The rotation matrix and gravity vector are irrelevant if you only want relative motion.
False. Without rotating into the world with and subtracting , a merely-tilted still sensor looks accelerating, corrupting even relative tracks.
Spot the error
"To get position, just integrate accelerometer output twice — calculus guarantees ."
The output is specific force , not ; you must first rotate it with and add gravity back (), or a resting sensor "accelerates" at .
"The IMU reads acceleration in the world frame, so no rotation is needed."
It reads in its own body frame (Reference Frames — Body vs World); world-frame acceleration needs using the current attitude.
"A resting tilted sensor gives that rotates back to something other than , proving it's moving."
For a still sensor , so always; getting anything else means you used a wrong , not that the box moved.
"Since gravity is constant, we can subtract it in the body frame before rotating."
Gravity is constant in the world frame ; in the tilted body frame its components change with attitude, so you must rotate to world first, then subtract world gravity. (Concretely: tilt and a still box reads ; only after does it collapse back to .)
"The skew matrix is just a fancy way to write a dot product."
It encodes the cross product (see Figure s02), which produces the perpendicular velocity of rigid-body rotation — a dot product would give a scalar, not motion.
"A 0.01°/s gyro bias is negligible — it's basically zero."
Through the cubic this produces tens of meters of position error in one minute, so it is decisively not negligible.
"We can skip the gyro and just use the accelerometer to know orientation while moving."
While accelerating, — the motion term contaminates the gravity direction, so you can no longer trust it as "down"; the gyro-driven fills this gap.
Why questions
Why does an accelerometer report a vector pointing up when it's not accelerating?
Because it senses the spring/normal force holding the proof mass against gravity: leaves when , and points opposite to down (up).
Why must gyroscope data be processed before accelerometer data in the strapdown chain?
The attitude is needed to know where the accelerometer's body axes point in the world, so you can rotate correctly and add gravity in the right direction (Figure s03).
Why is the cross product (not some other operation) the natural language of rotation?
A point at on a spinning body moves with velocity — perpendicular to both axis and radius (Figure s02) — which is exactly rigid rotation (Rotation Matrices & Quaternions).
Why does hold — geometrically?
Each column of is a body axis seen in the world; over the spin drags that axis by , and packing all three drags into one matrix gives exactly .
Why does gyro-tilt position error scale as while accelerometer bias scales as ?
The gyro bias makes tilt grow linearly, leaking a gravity ramp (); one integration gives velocity error and a second gives position error — one power more than a constant .
Why is an IMU called "inertial" and self-contained?
It relies only on internal proof masses and rate sensors reacting to motion, needing no external signal — which is also why it drifts, since it has nothing to correct against.
Why do we fuse an IMU with GPS or a camera instead of using better sensors alone?
No sensor is bias-free (), and integration turns any residual bias into unbounded / growth; an external absolute reference bounds it via a Kalman filter.
Why is Dead Reckoning with an IMU trustworthy for seconds but not minutes?
Over short spans the drift terms (, ) are tiny, but their steep growth quickly dominates, so IMUs are superb short-term and hopeless long-term.
Edge cases
What does the gyro read for a body moving in a perfectly straight line at constant speed?
Zero angular rate () — translation with no turning produces no gyro signal, even though the body is clearly moving.
What does the accelerometer read for that same straight, constant-speed motion?
Just (as if at rest), because ; constant velocity is indistinguishable from standstill to a proof mass, since no net force loads its spring.
What does an ideal accelerometer read at the exact top of a projectile's arc?
Zero on all axes — the whole flight is free fall (), so and the proof mass is unloaded (no spring force) at every instant, apex included.
Can an accelerometer distinguish "parked on a hill" from "accelerating on flat ground"?
Not by itself — a tilt redistributes across the body axes, and a horizontal acceleration can produce the same ; only the gyro-tracked (or fusion) resolves the ambiguity.
What happens to attitude if the gyro momentarily reads exactly zero while the body actually spins?
The integrated freezes and lags the true orientation; that attitude error then mis-points in every later step, so inherits a leaked-gravity error.
At the equator vs. the pole, does a perfect stationary IMU read identical gravity?
Not quite — Earth's spin and shape make slightly smaller and add tiny rotation-rate terms at the equator, so the resting differs subtly and ultra-precise units must model it.
What is the reading of a hypothetical zero-bias, zero-noise IMU dropped in deep space, far from any mass?
Both sensors read zero — no spring force (no gravity or contact, so ) and no rotation () — leaving dead reckoning to preserve whatever initial velocity you seeded.
Recall Quick self-test
The three drift growth rates you must never confuse ::: angle error (gyro bias ), position-from-accel (accel bias ), position-from-tilt (gyro-tilt leaking gravity). The single sentence that fixes most IMU mistakes ::: attitude first → rotate to world with → add gravity back → integrate twice.