Sensor fusion

Different sensors fail differently. Wheel encoders give smooth short-term displacement but drift during slip; a camera can correct global position but may update slowly or lose features. An accelerometer reacts quickly but includes gravity and bias; a slow reference can anchor it. Fusion uses complementary information, not merely more numbers.

For two independent unbiased scalar estimates x₁ and x₂ with variances σ₁² and σ₂², inverse-variance fusion is x̂ = (x₁/σ₁² + x₂/σ₂²)/(1/σ₁² + 1/σ₂²). Its variance is 1/(1/σ₁² + 1/σ₂²). The more precise estimate receives more weight. This result depends on independence, correct uncertainty, a shared physical quantity, synchronized time, and compatible coordinates. If both estimates share the same calibration error, treating them as independent becomes overconfident.

Before combining vectors, transform them into one declared frame. A camera's right direction may not match a robot's lateral axis, and an accelerometer mounted upside down changes signs. Align timestamps or propagate older estimates forward with a model. Compute innovations—the difference between prediction and new measurement—and retain them for gating and diagnosis. The build decision may be to reject a camera update when its residual is physically implausible while continuing short-term encoder prediction and explicitly increasing uncertainty.

A common misconception is that averaging two sensors always improves accuracy. A biased sensor can pull the result away from truth; duplicated sensors exposed to the same vibration have correlated errors. Another failure is assigning tiny uncertainty because a trace looks smooth. Calibration, latency, quantization, environment, and model error all matter. Fusion should report confidence and degraded modes, not conceal disagreement. For a desk robot, stopping or slowing after prolonged localization disagreement is often sounder than trusting a polished but unsupported estimate.