46 Kalman Filters for Sensor Fusion

analytics-ml

data

fusion

kalman

46.1 Start With the Story

Keep the Estimate and Its Doubt Together

Imagine a cart using a wheel count and a motion sensor to estimate speed. One source drifts. The other shakes. A smooth line may still be wrong if the system hides how sure it is.

Start with a state the team can name, such as speed now. Predict what the next value should be. Compare that prediction with the next reading. Give less weight to a source when its expected error is larger.

Record the estimate, time, expected error, new reading, gap, correction, and final doubt for each step. Then freeze one sensor, add a sudden jump, lose a reading, and change the rate. The output should show doubt before a person trusts it.

A simple filter depends on its model and error beliefs. It is not a truth machine. Practitioner works through tuning and a small update. Under the Hood explains gates, model limits, and failure modes that can make a calm line unsafe.

Picture an IoT team using the ideas in Kalman Filters for Sensor Fusion during a live operations review. A device has produced messy evidence, an analytic step is about to change an alert or control decision, and someone has to explain why the result should be trusted.

Read this page as that path from sensor evidence to accountable action. Start with what the system observes, keep the model or data treatment visible, and finish with the check that would convince an operator, maintainer, or auditor to act.

46.2 Kalman Filters Maintain Belief

A Kalman filter is a state estimator for systems that can be described with a model, measurements, and uncertainty. The filter keeps two things together: a best estimate of the state and a covariance that describes the uncertainty in that estimate. It predicts the next state from the model, then corrects that prediction when a measurement arrives.

For IoT sensor fusion, the important idea is not that the output is smooth. The important idea is that every fused estimate carries its uncertainty and update evidence. A temperature estimate, location estimate, or velocity estimate should record the state vector, time step, process noise, measurement noise, innovation, Kalman gain, accepted/rejected measurements, and degraded-mode status.

If the model is linear and the noise assumptions are reasonable, a Kalman filter gives a principled way to balance prediction and measurement. If those assumptions are weak, the filter still needs validation gates and review evidence.

The state vector is the contract between the physics and the software. A simple tracking filter might estimate position and velocity; a building filter might estimate temperature plus a slowly changing sensor bias. The transition model says how that state should move between samples. The observation model says how a sensor reading relates to the hidden state. Those two models are separate on purpose: a sensor may measure only one part of the state, and the filter still uses the model to carry the unmeasured parts forward.

Noise terms make the estimate reviewable instead of magical. Process noise says how much the model can be wrong between updates; measurement noise says how much the sensor can be wrong when it reports. The filter should publish those assumptions with the fused output or at least preserve them in the audit record, because changing Q or R can change whether the same measurement is accepted, rejected, or only partly trusted.

Use Figure 46.1 to follow the filter as an evidence loop rather than treating its equations as an isolated recipe.

Kalman filter evidence loop carrying both estimate and covariance through prior, prediction, measurement, innovation, gain, and correction. Residuals, covariance behaviour, Q and R assumptions, and failure evidence are checked before accepting the updated state. — Figure 46.1: A Kalman filter is reviewable when the state model, observation model, process noise, measurement noise, and update flow are visible with the fused estimate.

Trace Figure 46.1 from the prior estimate and covariance into prediction, then compare the measurement with that prediction to form the innovation. The innovation covariance determines the Kalman gain, which controls the correction and the updated uncertainty carried into the next cycle. Watch the Q and R assumptions alongside the residual: they express process and measurement uncertainty, and a persistent mismatch is diagnostic evidence. This closes the chapter's running loop between physical assumptions, sensor quality, state belief, and reviewable failure checks.

State

The hidden variables being estimated, such as temperature, position, velocity, or bias.

Covariance

The uncertainty attached to the current state estimate and relationships between state dimensions.

Prediction

The model-based step that advances the state and usually increases uncertainty.

Correction

The measurement update that uses innovation and gain to adjust the state and reduce uncertainty.

Overview Knowledge Check

46.3 Tune Prediction vs Measurement

In the scalar case, the Kalman gain shows the trust balance directly. A high gain means the measurement pulls the estimate strongly. A low gain means the model prediction dominates. Process noise Q controls how much uncertainty is added during prediction; measurement noise R controls how much the filter trusts the sensor. Both should be based on observed behavior and retested after sensor, firmware, sampling, or deployment changes.

Worked example: one scalar temperature update
previous estimate x: 20.0 deg C
previous variance P: 0.50
process noise Q: 0.10
measurement z: 21.2 deg C
measurement variance R: 0.40
state model F: 1
measurement model H: 1

predict:
x_pred = 20.0
P_pred = P + Q = 0.50 + 0.10 = 0.60

innovation:
y = z - x_pred = 21.2 - 20.0 = 1.2 deg C
S = P_pred + R = 0.60 + 0.40 = 1.00

Kalman gain:
K = P_pred / S = 0.60 / 1.00 = 0.60

update:
x_new = x_pred + K * y = 20.0 + 0.60 * 1.2 = 20.72 deg C
P_new = (1 - K) * P_pred = 0.40 * 0.60 = 0.24

Interpretation:
The estimate moves toward the measurement but does not copy it.
The variance drops from 0.60 predicted to 0.24 after the measurement update.

Parameter

What It Means

If Too Small

If Too Large

Process noise: uncertainty added because the model is imperfect.

Estimate can lag real changes and reject valid motion.

Estimate can follow noisy measurements too closely.

Measurement noise: uncertainty assigned to the sensor reading.

Filter can overtrust a noisy or faulty sensor.

Filter can ignore useful measurements and drift with the model.

Current state uncertainty carried by the filter.

Filter may be overconfident and slow to correct.

Filter may be unstable or overly sensitive to measurements.

Gate

Rule deciding whether an innovation is plausible enough to update.

Valid measurements may be rejected during real changes.

Faulty measurements may corrupt the fused state.

Practitioner Knowledge Check

46.4 Innovation Gating Protects Estimates

The innovation is the difference between what the sensor reports and what the filter predicted the sensor should report. Large innovations can be real changes, bad tuning, stale timestamps, miscalibration, sensor faults, or a model that no longer fits the operating mode. Production filters should treat the innovation as evidence, not just a number inside the update equation.

A common scalar gate compares the innovation squared with its expected variance. If the normalized innovation is too large, the system can reject the measurement, downweight it, publish a degraded mode, or trigger a sensor-health review. This connects Kalman filtering to the broader fusion contract: accepted and rejected inputs must be visible.

Covariance is also a design surface, not just a matrix in the code. If a sensor becomes stale, the predicted covariance should usually grow because the state is less certain. If two measurements share a calibration source, the filter should not treat them as fully independent evidence. If an innovation gate rejects several readings in a row, the output record should show whether the system is holding the model prediction, using a fallback sensor, or stopping an action because observability is weak.

Residual review closes the loop. Engineers can plot innovations over time and compare them with the assumed noise model. Persistent positive residuals can reveal bias; residuals that spike after firmware or enclosure changes can reveal a timing or calibration problem; residuals that are always smaller than expected can indicate overestimated noise. That review evidence is what keeps Q, R, gates, and covariance from becoming untested constants.

Worked example: normalized innovation gate
predicted state: 12.0 m
predicted variance P: 0.64
measurement z: 14.0 m
measurement variance R: 0.36

innovation:
y = z - prediction = 14.0 - 12.0 = 2.0 m

innovation variance:
S = P + R = 0.64 + 0.36 = 1.00

normalized innovation squared:
NIS = y^2 / S = 4.0 / 1.00 = 4.0

decision example:
If the configured one-dimensional gate is 3.84, this measurement is outside
the gate. Quarantine it, publish the predicted state with degraded evidence,
and record the rejection reason for review.

Innovation

Measurement minus predicted measurement; the first signal that model and sensor disagree.

Gate

A plausibility rule that prevents a single suspect reading from corrupting the fused state.

Retest

Required after tuning changes, sensor replacement, firmware updates, or changed operating modes.

Fallback

Degraded-state publishing, last-good hold, alternate sensor use, or actuation stop when observability is weak.

Failure Mode

Symptom

Likely Cause

Control

Lag

Estimate follows real changes too slowly.

Q too small, model too rigid, or valid motion outside the assumed envelope.

Retune Q, segment by operating mode, and validate against step changes.

Jitter

Estimate copies measurement noise.

Q too large, R too small, or missing sensor-health checks.

Re-estimate R, add gates, and compare against raw sensor noise tests.

Divergence

Estimate drifts away from reviewed reality.

Wrong model, clock offset, calibration bias, or unhandled fault.

Reset state, isolate faulty inputs, and rerun calibration and timing checks.

Overconfidence

Published uncertainty is small while errors are large.

Underestimated Q/R, duplicated evidence, or missing process disturbance.

Audit residuals, inflate covariance, and track shared-evidence assumptions.

Under-the-Hood Knowledge Check

46.5 Summary

Kalman filters combine a predictive model with measurements while carrying uncertainty through every step. The prediction step advances the state and increases uncertainty with process noise. The update step compares the measurement with the prediction, computes an innovation and gain, adjusts the state, and reduces uncertainty when the measurement is accepted. Practical IoT use requires explicit Q/R tuning evidence, timestamp alignment, calibration checks, innovation gates, rejected-input records, degraded modes, and retest triggers.

Key Takeaway

A Kalman filter is trustworthy only when the fused state, covariance, tuning assumptions, innovation evidence, accepted or rejected measurements, and fallback behavior are visible together.

46.6 See Also

Fusion Architectures

Place Kalman state estimates within centralised, hierarchical, or distributed fusion patterns.

Fusion Best Practices

Use calibration, time alignment, gates, and degraded modes around the estimator.

Particle Filters

Compare Kalman assumptions with sampling-based filters for nonlinear or non-Gaussian states.

Complementary IMU Fusion

Contrast covariance-based correction with lightweight embedded orientation fusion.

46.7 Constant-Velocity and Navigation Evidence

Inspect Figure 46.2 to make the hidden state, observed quantity, and uncertainty path explicit before tuning a filter.

One-dimensional cart with position and velocity, F constant-velocity matrix, H position-only observation row, Q process noise, R measurement noise, predicted covariance, and corrected state. — Figure 46.2: Constant-velocity Kalman model for a one-dimensional cart with state transition, observation, process noise, measurement noise, covariance, and correction panels.

Read Figure 46.2 from PREDICT, where the F matrix couples position and velocity, to OBSERVE, where H = [1 0] measures position only. PREDICTED COVARIANCE and CORRECTED STATE show that the filter carries uncertainty as well as a state value.

Next, inspect Figure 46.3 to turn the algebra into a visible uncertainty cycle and compare what prediction and measurement each contribute.

Kalman predict-update sequence with prior Gaussian, widened propagated Gaussian, measurement likelihood, Kalman gain balance, and narrowed posterior Gaussian. — Figure 46.3: Five Kalman distribution panels showing prior, propagation, measurement, gain, and posterior Gaussian beliefs.

Follow PRIOR to PROPAGATE in Figure 46.3 to see covariance widen, then compare MEASURE with POSTERIOR to see the accepted fix reduce uncertainty. The GAIN panel names the model-versus-sensor balance rather than hiding it behind a smoothed trace.

Finally, inspect Figure 46.4 for a state-design failure that repeated position fixes cannot solve by themselves.

Three aligned timelines: constant true heading, gyro-integrated estimate curving under bias, and intermittent GPS position fixes, with a warning to add gyro bias to the state. — Figure 46.4: Aligned true-heading, gyro-integrated, and GPS-fix timelines showing an unmodelled gyro bias bending the path after every position correction.

Compare TRUE HEADING with GYRO INTEGRAL in Figure 46.4. The GPS POSITION FIXES repair position snapshots, but the STATE-DESIGN FAILURE panel explains why the next segment bends again until gyro bias becomes an estimated state.