Foundations — IMU — integrated accelerometer + gyroscope

Before you can read a single equation in the parent note, you need the alphabet those equations are written in. This page introduces every symbol and idea, in an order where each one is built only from the ones before it. Nothing is assumed. If a smart 12-year-old reads from line one, they should never hit a wall.

1. Vectors — an arrow with a length and a direction

The very first thing the parent note writes is ${\mathbf{f}}_b$, in bold. Bold letters are not ordinary numbers. A plain number like $9.81$ is just a size. A bold letter is a vector: an arrow that has both a length (how much) and a direction (which way).

Why the IMU needs this. A push has a direction — you can be shoved sideways or downward. Gravity points a specific way (down). "Which way is up" is a direction. None of these can be a single number; they all must be arrows. So the whole subject is written in vectors.

2. Axes and the right-hand rule — naming the three directions

To say "along the $x$-axis" we need three fixed reference directions that don't move relative to each other. We call them $x$, $y$, $z$.

To avoid mirror-image confusion (is "up" $+z$ or $-z$?), engineers fix the handedness with the right-hand rule: point your right index finger along $x$, middle finger along $y$, and your thumb points along $z$. This same hand also tells you the positive spin direction later, so learn it once here.

Why the IMU needs this. The sensor reports three numbers per reading. Without agreed axes, "the first number" is meaningless. Every ${\mathbf{\omega }}_b=({\omega }_x,{\omega }_y,{\omega }_z)$ in the parent note is "spin about $x$, spin about $y$, spin about $z$."

3. The three things a number can measure here: force, acceleration, velocity, position

The parent note freely mixes $\mathbf{f}$, $\mathbf{a}$, $\mathbf{v}$, $\mathbf{p}$, $\mathbf{g}$. These are four different kinds of quantity. Confusing them is the #1 error in the whole topic.

Why the IMU needs this. The whole navigation trick is a chain: turn what you feel ($\mathbf{f}$) into true acceleration ($\mathbf{a}$), add it up into velocity ($\mathbf{v}$), add that up into position ($\mathbf{p}$). Four different quantities, four different meanings.

4. Rate of change and the dot — what $\dot{R}$ and "$\int$" mean

The parent note writes ${\dot{R}}_b^w$, ${\dot{\mathbf{v}}}_w$, and integral signs $\int$. These are the two halves of the same idea.

Why the IMU needs this. A gyro gives a rate ($\dot{}$ of angle). An accelerometer feeds the rate of rate of position. To get orientation and position, the box must integrate — add up the little pieces. Both symbols are unavoidable.

5. Reference frames — the body frame vs the world frame

The subscripts $b$ and $w$ (as in ${\mathbf{f}}_b$, ${\mathbf{p}}_w$, $R_b^w$) are the trickiest idea here, so we slow right down.

Why the IMU needs this. The sensor speaks "body." The navigation answer must be "world." Translating between the two is the single job of the rotation object $R_b^w$ (next section). Skip this translation and a merely-tilted, motionless sensor will look like it's flying off — exactly the trap in Worked Example 2 of the parent note.

6. The rotation matrix $R_b^w$ — the translator between frames

Why the IMU needs this. This object is built by integrating the gyro, and it's what lets us rotate the felt push ${\mathbf{f}}_b$ into world acceleration. It is the hinge of the whole method.

7. Angular velocity $\mathbf{\omega }$ and the cross product $\times$

Why the IMU needs this. Turning a gyro's spin-rate into "how the orientation changes" is the cross product, packaged as the skew matrix $[\mathbf{\omega }{]}_{\times }$ in the parent note.

8. Radians and degrees — measuring angle honestly

The parent uses both $30^{\circ }$ and rad/s, and converts $0.01°/\text{s}=1.75\times 10^{-4}$ rad/s.

9. Bias and error — why nothing is perfect

The parent introduces $b_g$ (gyro bias) and ${\epsilon }_a$ (accel error).

How these foundations feed the topic

Equipment checklist

Test yourself — you are ready for the parent note only if each reveal matches what you'd say.