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IMU Overview

What Is an IMU

An IMU (Inertial Measurement Unit) is a sensor module that measures the acceleration and angular velocity of an object. It is one of the core sensors for robot navigation, attitude estimation, and motion control, found in virtually all mobile robot platforms.

Sensor Components

Accelerometer

Principle: Based on a MEMS spring-mass system. When the sensor accelerates, the proof mass displaces due to inertia, and the displacement is detected via capacitance changes.

\[ F = ma \quad \Rightarrow \quad a = \frac{F}{m} = \frac{k \cdot \Delta x}{m} \]

Where:

  • \(k\) is the spring stiffness
  • \(\Delta x\) is the proof mass displacement
  • \(m\) is the proof mass

Measurement: Specific force, i.e., gravity + motion acceleration:

\[ \mathbf{f} = \mathbf{a} - \mathbf{g} \]

Effect of Gravity

An accelerometer at rest measures the gravitational acceleration \(g \approx 9.81 \, \text{m/s}^2\). When using an accelerometer for navigation, the gravity component must be subtracted from the measurement, which requires precise knowledge of the sensor's attitude.

Gyroscope

Principle: Based on the Coriolis effect. A vibrating proof mass in a rotating reference frame experiences a Coriolis force:

\[ \mathbf{F}_c = -2m(\boldsymbol{\omega} \times \mathbf{v}) \]

Where:

  • \(m\) is the vibrating mass
  • \(\boldsymbol{\omega}\) is the angular velocity
  • \(\mathbf{v}\) is the vibration velocity

MEMS gyroscopes keep a proof mass vibrating in one direction. When rotation exists, the Coriolis force produces displacement in the perpendicular direction, detected via capacitance to obtain angular velocity.

Measurement: Three-axis angular velocity \((\omega_x, \omega_y, \omega_z)\), in deg/s or rad/s.

Magnetometer

Principle: Uses the Hall effect or magnetoresistive effect to measure the Earth's magnetic field direction.

Measurement: Three-axis magnetic field strength, used to compute heading (Yaw).

Limitations:

  • Interfered by ferromagnetic materials (hard iron/soft iron errors)
  • Indoor magnetic fields are non-uniform
  • Requires calibration (ellipsoid fitting)

IMU Configuration Classification

Type Sensor Combination Measurable Quantities Representative Products
6-axis Accelerometer + Gyroscope Acceleration, angular velocity MPU6050, BMI088
9-axis Accelerometer + Gyroscope + Magnetometer Acceleration, angular velocity, heading BNO055, MPU9250
10-axis 9-axis + Barometer Above + altitude estimation BMP388 combo

IMU Noise Model

IMU accuracy is affected by multiple noise sources. Understanding the noise model is crucial for sensor fusion.

Main Noise Types

graph TD
    A[IMU Noise Sources] --> B[Deterministic Errors]
    A --> C[Random Errors]

    B --> B1[Bias<br>Constant offset]
    B --> B2[Scale Factor Error]
    B --> B3[Cross-Axis Coupling]

    C --> C1[Angle Random Walk ARW<br>White noise]
    C --> C2[Bias Instability]
    C --> C3[Rate Random Walk RRW<br>Low-frequency drift]

Gyroscope Noise Model

Continuous-time model:

\[ \tilde{\omega}(t) = \omega(t) + b_g(t) + n_g(t) \]

Where:

  • \(\tilde{\omega}(t)\) is the measurement
  • \(\omega(t)\) is the true angular velocity
  • \(b_g(t)\) is a slowly drifting bias
  • \(n_g(t)\) is Gaussian white noise, \(n_g \sim \mathcal{N}(0, \sigma_g^2)\)

Bias modeled as a random walk:

\[ \dot{b}_g(t) = n_{bg}(t), \quad n_{bg} \sim \mathcal{N}(0, \sigma_{bg}^2) \]

Accelerometer Noise Model

\[ \tilde{a}(t) = a(t) + b_a(t) + n_a(t) \]

Similar structure to the gyroscope, but acceleration is integrated twice to obtain position, so errors accumulate faster:

\[ \delta p(t) \propto \frac{1}{2} b_a \cdot t^2 + \frac{1}{6} \sigma_a \cdot t^{5/2} \]

Drift Problem in Inertial Navigation

Position error in pure inertial navigation (IMU only) grows with the square of time. Consumer-grade MEMS IMUs produce meter-level drift within seconds, so they must be fused with other sensors (GPS, LiDAR, vision).

Key Noise Parameters

Parameter Symbol Unit (Gyro) Unit (Accel) Meaning
Angle/Velocity Random Walk ARW / VRW deg/sqrt(h) m/s/sqrt(h) White noise intensity
Bias Instability BI deg/h ug Minimum detectable bias change
Bias Repeatability Turn-on bias deg/h mg Bias variation per power-on
Scale Factor Error SF error ppm ppm Proportional measurement error

Allan Variance Analysis

Allan variance is the standard method for characterizing IMU noise, identifying noise components by computing variance at different averaging times \(\tau\).

\[ \sigma^2(\tau) = \frac{1}{2(N-1)} \sum_{k=1}^{N-1} \left( \bar{y}_{k+1} - \bar{y}_k \right)^2 \]

Where \(\bar{y}_k\) is the average of the \(k\)-th interval of length \(\tau\).

Allan Variance Curve Interpretation

On a \(\log \sigma\) vs. \(\log \tau\) plot, different noise types correspond to different slopes:

Noise Type Slope Characteristic
Quantization noise -1 Appears at short \(\tau\)
Angle Random Walk (ARW) -1/2 White noise, read at \(\tau = 1s\)
Bias Instability (BI) 0 Minimum point of the curve
Rate Random Walk (RRW) +1/2 Appears at long \(\tau\)
Rate ramp +1 Deterministic trends like temperature drift 
import numpy as np
import allantools

# Compute Allan variance from IMU static data
# data: static samples from one gyroscope axis
# rate: sampling rate (Hz)
(taus, adevs, errors, ns) = allantools.oadev(
    data, rate=rate, data_type="freq"
)

# ARW: Allan deviation at tau=1
# BI: Minimum Allan deviation value

Sensor Fusion Pipeline

graph LR
    A[Accelerometer] --> D[Attitude Estimation]
    B[Gyroscope] --> D
    C[Magnetometer] --> D

    D --> E[Complementary Filter<br>or EKF]

    E --> F[Roll, Pitch, Yaw]
    F --> G[Fusion with External Sensors]

    G --> G1[GPS -> Position]
    G --> G2[LiDAR -> SLAM]
    G --> G3[Camera -> VIO]
    G --> G4[Wheel Odometry -> Velocity]

Attitude Representation

Representation Parameters Advantages Disadvantages
Euler angles 3 Intuitive Gimbal lock
Rotation matrix 9 No singularities Parameter redundancy
Quaternion 4 No singularities, computationally efficient Less intuitive
Rotation vector/axis-angle 3 Minimal parameterization Singular near 0 degrees

Quaternion update:

\[ \mathbf{q}_{k+1} = \mathbf{q}_k \otimes \begin{bmatrix} 1 \\ \frac{1}{2}\boldsymbol{\omega} \Delta t \end{bmatrix} \]

IMU Grade Classification

Grade Gyro Bias Stability Representative Product Application Price
Consumer >10 deg/h MPU6050, BMI160 Smartphones, remotes $1-5
Industrial 1-10 deg/h BMI088, ADIS16465 Drones, robots $10-100
Tactical 0.1-1 deg/h HG1120, STIM300 Missiles, high-precision navigation $1K-10K
Navigation <0.01 deg/h HG9900 Inertial navigation systems $50K+
Strategic <0.001 deg/h Ring laser gyro Submarines, ICBMs $100K+

References

  • Quaternion kinematics for the error-state Kalman filter - Joan Sola
  • Strapdown Inertial Navigation Technology - Titterton & Weston
  • Bosch BMI088/BNO055 Datasheets
  • InvenSense MPU6050 Datasheet
  • Allan Variance Analysis Tutorial: IEEE Std 952-1997

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