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IMU Sensor

An IMU sensor (Inertial Measurement Unit) is a device that measures a robot or object's motion and orientation using:

Sensor Measures Unit
Accelerometer Linear acceleration m/s²
Gyroscope Angular velocity (rotational speed) rad/s
Magnetometer (optional) Magnetic field for heading (compass) µT
  • 6 DOF IMU, combining a 3-axis accelerometer and a 3-axis gyroscope.
  • 9 DOF IMU, adding a 3-axis magnetic compass.
  • 10 DOF IMU, which adds a barometer for estimating the sensor’s altitude.
Rotation Meaning Greek symbol Axis
Roll rotation about X \(\phi\) (phi) X
Pitch rotation about Y \(\theta\) (theta) Y
Yaw rotation about Z \(\psi\) (psi) Z

Accelerometer

An accelerometer is a sensor that measures acceleration — how fast something speeds up, slows down, or changes direction — along one or more axes.

  • It cannot distinguish gravity from acceleration during motion
  • It cannot measure change in yaw (z axis) Yaw is rotation around the gravity axis, and an accelerometer only measures gravity—so yaw doesn’t change what it sees
  • At rest the accelerometer measures gravity, not “zero”.
  • Tilt it: gravity is split across axes
  • tilt angles : pitch and roll

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\[\phi = \text{atan2}(a_y, a_z)\]
\[\theta = \text{atan2}(-a_x, \sqrt{a_y^2 + a_z^2})\]
  • \(\phi\) → roll
  • \(\theta\) → pitch
  • \(\psi\) → yaw (not observable from accel)
Rotation Gravity moves in Formula
Roll YZ plane atan2(ay, az)
Pitch XZ plane atan2(-ax, √(ay²+az²))
Yaw no change not measurable

Demo: calc pitch

pitch drawing

\[a_x = g \sin(\theta)\]
\[a_z = g \cos(\theta)\]

Extract pitch angle

from

\[a_x = g \sin(\theta)\]

divide both by g

\[\frac{a_x}{g} = \sin(\theta)\]

inverse sine

\[\theta = \sin^{-1}\left(\frac{a_x}{g}\right)\]

more general formula

  • roll != 0
\[\theta = atan2(-a_x,\sqrt{a_y^2 + a_z^2})\]

Demo: calc roll

  • gravity changes only in the Y–Z plane

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\[A_z = 1g \cdot \cos(\phi)\]
\[A_y = 1g \cdot \sin(\phi)\]

roll

\[Ay_{out} = 1g \sin(\phi)\]

divide by g

\[\sin(\phi) = \frac{Ay_{out}}{1g}\]

inverse sin

\[\phi = \sin^{-1}\left(\frac{Ay_{out}}{1g}\right)\]

more general formula

\[\phi = atan2(Ay, Az)\]

because it handles:

  • sign correctly
  • angles beyond ±90°
  • numerical stability.

Gyroscope

Gyro read angular velocity (red/sec) meaning "how fast am i rotation right now around each axis", to get orientation we integrate over time

basic idea

\[\text{roll}(t) = \text{roll}(t-\Delta t) + \omega_x \cdot \Delta t\]

Gyro drift

Gyro have bias

\[\omega_{measured} = \omega_{true} + b + noise\]

for example

  • Bias = 0.05°/s
  • After 60s → 3° error
  • After 10 min → 30° error

Orientation

Sensor What it measures Orientation info you get Absolute reference Main strengths Main limitations Typical role in fusion
Accelerometer Linear acceleration + gravity Roll & Pitch (tilt) ✅ Yes (gravity) No drift, simple, stable when still Fails during motion, noisy, no yaw Long-term tilt correction
Gyroscope Angular velocity (°/s) Roll, Pitch, Yaw (via integration) ❌ No Smooth, fast response, works in motion Drift due to bias, needs integration Short-term orientation propagation
Magnetometer Earth magnetic field Yaw (heading) ✅ Yes (north) Fixes yaw drift, global heading Disturbed by metal/electric fields Long-term yaw correction

To read and watch