Accelerometers, gyroscopes and IMUs

Why it matters

Every drone, phone, balancing robot, vehicle stability system and machine-condition monitor relies on accelerometers and gyroscopes to know how it is moving and which way is up. These sensors are cheap and tiny, but their raw outputs contain gravity, bias and noise, so a mechatronics engineer must know what they really measure and how errors grow when their signals are integrated.

Key ideas

• Seismic (spring–mass–damper) principle. Almost every accelerometer is a proof mass m on a spring of stiffness k with damping c inside a case. When the case accelerates, the mass lags and the spring deflects. Below the natural frequency the deflection is proportional to acceleration: x = m·a / k = a / ω_n². So a stiff, light design (high ω_n) gives wide bandwidth but small deflection, i.e. low sensitivity. This bandwidth–sensitivity trade-off is the central design choice.
  • Used well below ω_n (with ζ ≈ 0.6–0.7 for a flat response) the instrument is an accelerometer. Used well above ω_n (soft spring, heavy mass) the mass stays nearly still in space and the instrument measures displacement (a seismometer/vibrometer). This links directly to the second-order dynamic response from the measurement-characteristics topic.
• What an accelerometer actually reads: specific force. It senses the force the case must apply to the proof mass, so it reads f = a − g (vectors). A sensor lying still on a table reads +1 g (9.81 m/s²) upwards on its vertical axis; in free fall it reads zero. This is why a static accelerometer can measure tilt from the gravity components, and why it cannot by itself separate gravity from real motion.
• Types of accelerometer.
  • Piezoelectric – the proof mass squeezes a piezoelectric crystal, giving charge q = S_q·a. Very wide bandwidth (Hz to tens of kHz), rugged, ideal for vibration and shock. Charge leaks away, so it cannot measure steady (DC) acceleration or tilt; it needs a charge amplifier.
  • Capacitive MEMS – a silicon proof mass with comb fingers moves between fixed electrodes, changing a differential capacitance. Responds down to DC, so it measures tilt as well as motion; small, cheap, low power. Used in phones, wearables, airbag sensors.
  • Piezoresistive / strain-gauge – gauges on the suspension beam; responds to DC, good for crash and shock testing.
  • Servo (force-balance) – feedback drives a coil to hold the mass at null; the coil current measures acceleration. Most accurate (navigation, seismic), most expensive.
• Gyroscopes measure angular velocity (rate).
  • Mechanical – a spinning rotor keeps its angular momentum H; an applied rate Ω about a perpendicular axis needs a gyroscopic torque T = H·Ω, which a rate gyro measures with a restraining spring.
  • MEMS vibratory (Coriolis) – a proof mass is driven to vibrate at a few kHz to tens of kHz with velocity v. Rotation at rate Ω about a perpendicular axis produces a Coriolis force F = 2·m·Ω·v on the third axis; the resulting sense-axis motion is picked up capacitively and demodulated. Cheap and small but with larger bias and drift.
  • Optical (ring-laser and fibre-optic) – use the Sagnac effect: light travelling with and against the rotation has a path difference proportional to Ω. No moving parts, very low drift, used in aircraft and missiles.
• Integration and drift. Angle comes from integrating rate, θ = ∫ω dt, so a constant gyro bias b gives an angle error that grows linearly with time. Position comes from integrating acceleration twice, so a constant accelerometer bias gives a position error that grows with t². Random noise integrates into a random walk. This is why low-cost IMUs cannot do dead reckoning for more than seconds without correction.
• IMU and sensor fusion. An IMU combines a 3-axis accelerometer and a 3-axis gyroscope (6-DOF); adding a 3-axis magnetometer gives 9-DOF. The gyro is accurate over short times but drifts; the accelerometer gives a drift-free tilt reference but is noisy and corrupted by motion; the magnetometer gives heading but suffers from nearby iron. A complementary filter (high-pass on the gyro angle, low-pass on the accelerometer angle) or a Kalman filter fuses them. An AHRS outputs attitude; an INS also integrates to velocity and position, usually aided by GPS or wheel odometry.
• Key specifications. Range (±g or °/s), sensitivity (mV/g, LSB/g), bandwidth, noise density (µg/√Hz, (°/s)/√Hz), bias and bias stability, scale-factor error, cross-axis sensitivity and temperature drift. Datasheet values vary widely by grade; take them from the manufacturer's sheet.

Formulas

ω_n = √(k / m), f_n = ω_n / (2π)

• k suspension stiffness (N/m), m proof mass (kg), ω_n natural frequency (rad/s), f_n (Hz). Usable flat range of an accelerometer is roughly up to f_n/5 to f_n/3 for ζ ≈ 0.7.

x = m·a / k = a / ω_n²

• x proof-mass deflection (m), a acceleration (m/s²). Valid for input frequencies well below ω_n.

(C₁ − C₂) / (C₁ + C₂) = x / d, with C₁ = ε·A / (d − x), C₂ = ε·A / (d + x)

• differential capacitive pick-off: d nominal gap (m), A electrode area (m²), ε permittivity (F/m). The ratio is exactly linear in x.

q = S_q · a, V = q / C

• piezoelectric: q charge (C), S_q charge sensitivity (C per m/s², often quoted in pC/g), C total capacitance (F). Dynamic only.

θ = arcsin(a_x / g) (single axis), θ = atan2(a_x, a_z) (two axes)

• tilt θ from the horizontal reference, using static readings a_x, a_z (m/s²); g = 9.81 m/s². Valid only when the sensor is not accelerating.

F_c = 2 · m · Ω · v

• Coriolis force (N) on mass m (kg) moving at velocity v (m/s) perpendicular to rotation rate Ω (rad/s).

v_peak = 2π · f_d · X_d

• peak drive velocity (m/s) of a mass vibrating with amplitude X_d (m) at drive frequency f_d (Hz).

θ_err = b_g · t, x_err = ½ · b_a · t²

• b_g gyro bias (°/s or rad/s), b_a accelerometer bias (m/s²), t time (s).

θ_k = α·(θ_{k−1} + ω·Δt) + (1 − α)·θ_acc, α = τ / (τ + Δt)

• complementary filter: Δt sample period (s), τ crossover time constant (s), θ_acc tilt from the accelerometer.

Worked examples

Example 1 (standard) – MEMS accelerometer element. Proof mass m = 2 × 10⁻⁹ kg, suspension stiffness k = 2 N/m, differential capacitor gap d = 2 µm. Find the natural frequency, the deflection under 1 g, the differential capacitance ratio, and a reasonable usable bandwidth.

1. ω_n = √(k/m) = √(2 / 2 × 10⁻⁹) = √(10⁹) = 31 623 rad/s.
2. f_n = 31 623 / 2π = 5033 Hz.
3. x = m·a/k = 2 × 10⁻⁹ × 9.81 / 2 = 9.81 × 10⁻⁹ m (about 9.8 nm).
4. (C₁ − C₂)/(C₁ + C₂) = x/d = 9.81 × 10⁻⁹ / 2 × 10⁻⁶ = 0.0049, i.e. 0.49 % per g.
5. Usable flat range ≈ f_n/5 ≈ 1000 Hz.

Answer: f_n ≈ 5.03 kHz, x ≈ 9.8 nm per g, ratio ≈ 0.49 % per g, bandwidth ≈ 1 kHz. The nanometre deflection explains why MEMS read-out electronics must resolve attofarad changes.

Example 2 (GATE level) – tilt, drift and Coriolis force. A stationary IMU reads a_x = 3.0 m/s², a_y = 0, a_z = 9.33 m/s². Its gyro has a constant bias of 0.01 °/s and its accelerometer a bias of 0.01 m/s². The MEMS gyro's 1 × 10⁻⁹ kg proof mass is driven at 20 kHz with 10 µm amplitude. (a) Find the tilt angle. (b) Find the heading error after 10 min of pure gyro integration. (c) Find the position error after 60 s of double-integrating the accelerometer. (d) Find the peak Coriolis force for a rotation rate of 100 °/s.

1. Check magnitude: √(3.0² + 9.33²) = 9.80 m/s² ≈ g, so the IMU is static and tilt is valid.
2. θ = atan2(a_x, a_z) = atan2(3.0, 9.33) = 17.8°.
3. θ_err = b_g·t = 0.01 °/s × 600 s = 6.0°.
4. x_err = ½·b_a·t² = 0.5 × 0.01 m/s² × (60 s)² = 18 m.
5. v_peak = 2π·f_d·X_d = 2π × 20 000 × 10 × 10⁻⁶ = 1.257 m/s.
6. Ω = 100 × π/180 = 1.745 rad/s; F_c = 2·m·Ω·v = 2 × 10⁻⁹ × 1.745 × 1.257 = 4.39 × 10⁻⁹ N.

Answer: (a) 17.8°, (b) 6.0°, (c) 18 m, (d) ≈ 4.4 nN. Even a tiny 1 mg-class bias gives 18 m of error in one minute, which is why INS outputs must be corrected by GPS or other aiding.

Common mistakes

• Forgetting that an accelerometer at rest reads 1 g, not zero, and that in free fall it reads zero.
• Using a piezoelectric accelerometer to measure tilt or slowly varying acceleration – it has no DC response.
• Saying a gyroscope measures angle. A rate gyro measures angular velocity; angle comes from integration and drifts.
• Mixing °/s and rad/s, or g and m/s², in drift and Coriolis calculations.
• Computing tilt from an accelerometer while the body is accelerating – the motion is wrongly read as tilt.
• Treating the drift of a double-integrated accelerometer as linear in time; it grows with t².
• Operating a seismic sensor near its natural frequency, where the response is amplified and phase-shifted.

For GATE ME

Expect conceptual MCQs on what each sensor measures, which type responds to DC, the seismic-instrument principle (accelerometer below ω_n, displacement pick-up above ω_n) and the role of each sensor in an IMU. Numericals typically ask for natural frequency and static deflection of a spring–mass sensor, tilt from gravity components, angle obtained by integrating a rate, or error growth from bias. Practise second-order response, unit conversions between g, m/s², °/s and rad/s, and simple integration of constant rates and accelerations.

Quick check

1. What does a capacitive MEMS accelerometer lying flat and still on a table read on its vertical axis?
2. Why can a piezoelectric accelerometer not be used as an inclinometer?
3. A gyro bias of 0.05 °/s is integrated for 2 min. What is the angle error?
4. A spring–mass accelerometer has k = 50 N/m and m = 0.5 g. What is its natural frequency in Hz?
5. Which IMU sensor corrects long-term heading drift, and what can disturb it?

Answers: 1. About +9.81 m/s² (1 g); 2. Its charge output leaks away, so it has no DC (static) response; 3. 0.05 × 120 = 6°; 4. ω_n = √(50/0.0005) = 316.2 rad/s, so f_n ≈ 50.3 Hz; 5. The magnetometer; nearby iron, motors and currents distort it.