Mechanization equations — integrating IMU to get position, velocity, attitude

WHAT is an IMU actually measuring?

The key trap is hidden right here: an accelerometer cannot sense gravity. In free fall it reads zero. So gravity must be added back in software.

The three-layer integration chain

\underbrace{\boldsymbol\omega_{ib}^b}_{\text{gyro}} \;\to\; \text{ATTITUDE } C_b^n \qquad \underbrace{\mathbf f^b}_{\text{accel}} \xrightarrow{C_b^n} \mathbf f^n \;\to\; \text{VELOCITY } \mathbf v^n \;\to\; \text{POSITION}

You must get attitude first, because you need it to rotate the accelerometer's body-frame specific force into a navigation frame before you can subtract gravity.

Layer 1 — Attitude: integrating the gyros

WHY a matrix/quaternion and not just angles? Because rotation rates do not simply add up like scalars — orientation lives on a curved space $SO(3)$ . We propagate the direction-cosine matrix (DCM) $C_b^n$ (body→nav).

Why this step? $[\boldsymbol\omega\times]\mathbf u = \boldsymbol\omega\times\mathbf u$ exactly, so stacking three columns gives the whole matrix derivative.

But the gyro measures $\boldsymbol\omega_{ib}^b$ (vs inertial space), and we want $\boldsymbol\omega_{nb}^b$ (vs the nav frame, which itself moves as Earth turns and as we travel over the curved surface):

\boldsymbol\omega_{nb}^b=\boldsymbol\omega_{ib}^b-C_n^b\big(\underbrace{\boldsymbol\omega_{ie}^n}_{\text{Earth spin}}+\underbrace{\boldsymbol\omega_{en}^n}_{\text{transport rate}}\big)

$\boldsymbol\omega_{ie}^n$ : Earth rotates at $\Omega_e\approx 7.292\times10^{-5}\,\text{rad/s}$ .
$\boldsymbol\omega_{en}^n$ (transport rate): as you fly north/east, the local "down" tilts because Earth is round.

Layer 2 — Velocity: rotate, remove gravity, correct Coriolis

Term-by-term WHY:

$C_b^n\mathbf f^b$ — rotate the felt force into nav axes (needs Layer 1).
$-(2\boldsymbol\omega_{ie}+\boldsymbol\omega_{en})\times\mathbf v$ — Coriolis + transport correction for observing motion in a rotating/curving frame.
$+\mathbf g^n$ — add gravity back (the part the accelerometer could not see). Here $\mathbf g^n$ is the plumb-bob gravity (gravitation minus centrifugal), pointing "down".

Layer 3 — Position: integrate velocity on a curved Earth

Over a sphere/ellipsoid, latitude/longitude change at rates that depend on the Earth's radii of curvature (meridian $R_M$ , transverse $R_N$ ):

For a flat-Earth / short-baseline approximation you may drop transport & curvature and just use $\dot{\mathbf r}^n=\mathbf v^n,\qquad \dot{\mathbf v}^n=C_b^n\mathbf f^b+\mathbf g^n,\qquad \dot C_b^n=C_b^n[\boldsymbol\omega_{ib}^b\times].$ That's the 80/20 core — memorize this trio; add Earth terms only for long-range navigation.

Worked Example 1 — Static IMU sanity check

An IMU sits flat & still. Accels read $\mathbf f^b=(0,0,-9.81)$ in NED body axes (Down = $+z$ ), gyros read $\mathbf 0$ .

Attitude: $\dot C=C[\mathbf 0\times]=0$ → orientation constant. Why? no rotation felt.
Velocity (flat-Earth): $\dot{\mathbf v}=C_b^n\mathbf f^b+\mathbf g^n=(0,0,-9.81)+(0,0,+9.81)=\mathbf 0$ . Why? the upward table push ( $-9.81$ in NED "Down") cancels gravity ( $+9.81$ Down). Vehicle stays put ✔.

Recall Why does a

level accelerometer read $-9.81$ on the Down axis, not $0$ ? Because the table pushes it up; specific force points opposite to that support in the Down convention. Free-fall would give $0$ .

Worked Example 2 — Constant forward acceleration, 1-D

Rocket on a track, no rotation, aligned with nav $x$ . Accel reads $f_x=5\,\text{m/s}^2$ (gravity handled on other axes).

$\dot v_x = f_x = 5$ . Why? no Coriolis (short run), no gravity on $x$ .
Integrate: $v_x(t)=5t$ . Why? constant accel.
$\dot x=v_x\Rightarrow x(t)=\tfrac12(5)t^2=2.5t^2$ . At $t=4\text{ s}$ : $x=40\text{ m}$ , $v=20\text{ m/s}$ .

Why this matters: shows the clean double-integration — but any tiny accel bias $b$ produces position error $\tfrac12 b t^2$ , growing as $t^2$ . That's INS drift.

Worked Example 3 — Attitude drift from a gyro bias

Constant gyro bias $\varepsilon=0.01\,\text{rad/s}$ about one axis, static vehicle. Attitude error $\delta\theta=\varepsilon t$ . After $10\text{ s}$ , $\delta\theta=0.1\text{ rad}\approx 5.7^\circ$ .

Now the projected gravity leaks into horizontal velocity: $\delta a\approx g\sin(\delta\theta)\approx 9.81(0.1)=0.98\,\text{m/s}^2$ . Why this step? a tilted attitude misrotates gravity, so $g$ contaminates the horizontal channel — the dominant INS error path.

Flashcards

What does an accelerometer physically measure?

Specific force (non-gravitational contact acceleration), not gravity; static reading is

+g

opposite the support.

Why must attitude be computed before velocity in mechanization?

Accelerometer force is in body axes; you need

C_b^n

to rotate it into the nav frame before subtracting/adding gravity.

DCM propagation equation

\dot C_b^n = C_b^n[\boldsymbol\omega_{nb}^b\times]

, the skew-symmetric cross-product matrix of the body rate.

Full nav-frame velocity mechanization equation

\dot{\mathbf v}^n=C_b^n\mathbf f^b-(2\boldsymbol\omega_{ie}^n+\boldsymbol\omega_{en}^n)\times\mathbf v^n+\mathbf g^n

What is transport rate

\boldsymbol\omega_{en}^n

The rotation of the local nav frame as the vehicle moves over the curved Earth surface.

Why does the term

2\boldsymbol\omega_{ie}\times\mathbf v

appear?

Coriolis correction for expressing motion in the rotating Earth/nav frame.

Geodetic latitude rate

\dot\phi=v_N/(R_M+h)

— arc-rate over meridian radius of curvature.

Why does raw double-integration of accel fail when static?

It integrates the

+g

specific-force bias, producing fake motion; gravity must be added back in nav frame first.

How does a gyro bias

\varepsilon

corrupt position?

Attitude error

\varepsilon t

tilts the gravity vector, injecting

\approx g\varepsilon t

horizontal accel → error grows like

t^3

Flat-Earth core mechanization trio

\dot C_b^n=C_b^n[\boldsymbol\omega_{ib}^b\times]

\dot{\mathbf v}^n=C_b^n\mathbf f^b+\mathbf g^n

\dot{\mathbf r}^n=\mathbf v^n

Recall Feynman: explain to a 12-year-old

Imagine you're blindfolded in a car. You can feel pushes when it speeds up, brakes, or turns — but you can't feel the road passing. To guess where you are, you first figure out which way you're facing by tracking every turn (that's the gyro/attitude part). Then you use the pushes you felt to work out how fast you're going, remembering that a push while sitting still is just gravity holding you in the seat — so you subtract that out. Add up your speed over time and you know how far you moved. Tiny mistakes in "which way you're facing" make gravity sneak into your speed guess, so after a while you drift off — that's why cars also use GPS to keep the guess honest.

Connections

Strapdown vs Gimbaled INS — strapdown keeps sensors on the body, so mechanization does the "virtual gimbal" in software.
Direction Cosine Matrix and Quaternions — the attitude representation being propagated.
Coriolis and Centrifugal Effects — origin of the $2\boldsymbol\omega_{ie}\times\mathbf v$ term.
Gravity Model and Geoid — supplies $\mathbf g^n$ .
Kalman Filter for INS-GNSS Integration — corrects the $t^2/t^3$ drift derived here.
Radii of Curvature of the Earth Ellipsoid — $R_M,R_N$ in position rates.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, IMU sirf do cheezein "feel" karta hai: gyro se rotation rate aur accelerometer se specific force. Yahan sabse bada point yeh hai ki accelerometer gravity ko sense NAHI karta — jab woh table par rakha ho tab bhi $+g$ dikhata hai kyunki table use upar dhakka de raha hai, aur free fall mein zero dikhata hai. Isliye mechanization equations mein gravity ko software se wapas add karna padta hai.

Kaam ka order fixed hai: pehle attitude ( $C_b^n$ matrix) nikalo gyro ko integrate karke, kyunki accelerometer ka force body frame mein hai aur usse nav frame mein rotate karne ke liye attitude chahiye. Phir $C_b^n f^b$ karke gravity add karo aur Coriolis/transport terms subtract karo — yeh mil gaya velocity rate. Velocity ko integrate karo toh position aa jayegi (curved Earth par latitude/longitude ke liye radius se divide karna padta hai).

Sabse zaroori intuition drift ka hai. Agar accelerometer mein chhota sa bias $b$ ho, toh position error $\frac12 b t^2$ ki tarah badhta hai. Aur agar gyro mein bias ho, toh attitude thoda tilt ho jaata hai, jisse gravity horizontal channel mein leak karti hai — error $t^3$ ki speed se badhta hai! Isiliye pure INS long time tak accurate nahi rehta, aur GPS/Kalman filter ke saath combine karna padta hai.

Yaad rakhne ka tareeka: A-V-P (Attitude, Velocity, Position) — hamesha isi order mein, aur gravity tabhi hataao jab "down" kaunsa hai woh pata ho. Exam ke liye flat-Earth trio ( $\dot C=C[\omega\times]$ , $\dot v=C f^b+g$ , $\dot r=v$ ) ratt lo — yahi 80/20 core hai.

Mechanization equations — integrating IMU to get position, velocity, attitude

WHAT is an IMU actually measuring?

The three-layer integration chain

Layer 1 — Attitude: integrating the gyros

Layer 2 — Velocity: rotate, remove gravity, correct Coriolis

Layer 3 — Position: integrate velocity on a curved Earth

Worked Example 1 — Static IMU sanity check

Worked Example 2 — Constant forward acceleration, 1-D

Worked Example 3 — Attitude drift from a gyro bias

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Guidance, Navigation & Control (GNC)

Connections