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Sensor Fusion: Combining Intelligence

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Why Sensor Fusion?

No single sensor is perfect. Each has strengths and weaknesses — fusion combines them to get the best estimate. Think of it like a team of experts: the accelerometer is great at detecting gravity but noisy during motion, the gyroscope is clean during rotation but drifts over time, and the magnetometer provides heading but gets confused near metal.

Sensor Fusion ArchitectureAccelerometer3-axis, 100HzGyroscope3-axis, 1kHzMagnetometer3-axis, 50HzGPSPosition, 10HzAHRSAttitude & HeadingReference SystemMadgwick / MahonyKalman / EKFEKFExtended Kalman FilterPosition + VelocityOrientationRoll, Pitch, YawAngular velocityNavigationPosition, VelocityWaypoint trackingFlightControllerPID loopsMissionPlannerPath planningSensor Data RatesAccel: 100-1000HzGyro: 1000-8000HzMag: 50-100HzGPS: 1-20Hz

Complementary Filter

The simplest fusion approach. It trusts the gyroscope for fast changes (high-pass) and the accelerometer for slow/steady changes (low-pass).

Madgwick Filter

The Madgwick filter is the most popular choice for drone AHRS. It uses gradient descent to minimize the error between predicted and measured orientations.

Madgwick Filter: Prediction vs CorrectionGyro Integrationq_est = q_est + 0.5 × q_est ⊗ ω × dtFast, accurate short-termDrifts over time without correctionGradient Descent∇f = Jᵀ × f(q_est, a, m)Corrects orientation errorUses accel + mag as referenceFused Quaternionq_final = q_gyro + β × ∇f × dtβ = filter gain (0.01–0.1)Complementary: Simple, low CPU | Madgwick: Good balance | Kalman: Best accuracy, high CPU

Kalman Filter: Full Mathematical Framework

The Kalman filter maintains a state estimate and its uncertainty, optimally combining predictions with measurements.

Kalman Filter: Complete CycleState Vector x̂[φ, θ, ψ, b_gx, b_gy, b_gz]Angles + gyro biases6-state EKFPREDICTx̂⁻ = f(x̂, u)P⁻ = F·P·Fᵀ + QGyro inputUPDATEK = P⁻·Hᵀ·(H·P⁻·Hᵀ+R)⁻¹x̂ = x̂⁻ + K·(z − h(x̂⁻))Accel + Mag correctionKalman Filter MatricesFState transition (predicts next state from current)HMeasurement model (maps state to measurement)QProcess noise (how uncertain is our model?)RMeasurement noise (how noisy are sensors?)PError covariance (estimation uncertainty)KKalman gain (how much to trust measurement)State estimate (the output!)zMeasurement vector (sensor readings)

Hands-On Project: 6-State EKF Implementation

Filter Comparison

FilterCPUAccuracyTuningBest For
ComplementaryVery LowGoodSimple (α)Basic drones
MadgwickLowVery GoodModerate (β)Most applications
MahonyLowGoodModerate (Kp, Ki)Outdoor flight
EKFMediumExcellentComplexProfessional systems

Key Takeaways

  • Sensor fusion combines multiple sensors for better estimates than any single sensor
  • Complementary filter: simplest, good for basic applications
  • Madgwick filter: best balance of performance and simplicity for most drones
  • Kalman/EKF: most accurate but complex to tune and implement
  • Always calibrate sensors before fusion — garbage in, garbage out
  • Gyro bias estimation is critical for long-term accuracy
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