Extended Kalman Filter (EKF) Step Calculator

Simulate one complete EKF predict-update cycle — see Kalman gain, innovation, updated state estimate, and updated covariance for any combination of process noise and measurement noise.

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⚙️ Input Parameters

Prior State (before predict step)

State Estimate x̂ (m)

Current best position estimate

Covariance P

Current uncertainty (σ²)

Predict Step

Process Noise Q

Model uncertainty per step. Higher Q = less trust in prediction

Update Step

Measurement z (m)

Sensor reading this step

Measurement Noise R (σ²)

GPS: 1–25 | LiDAR: 0.001–0.01

📊 Results

📊

Enter prior state, covariance, process noise, and measurement to simulate one complete EKF predict-update cycle.

Updated State Estimate x̂

—m

Predicted covariance P⁻ = P + Q—

Kalman Gain K = P⁻/(P⁻+R)—

Innovation z − x̂ (residual)—

State correction K × innovation—

Updated covariance P = (1−K)P⁻—

Understanding the EKF: The Engine Behind AV Localisation

What This Calculates

One complete EKF cycle. The predict step propagates state and grows covariance by Q (how much the real world can deviate from your model). The update step computes Kalman gain K — the optimal weight to apply to the measurement’s innovation — then adjusts the state estimate and reduces covariance. Covariance always decreases after a valid measurement: the filter learned something.

When to Use It

Understanding how Q and R values affect filter behaviour before tuning
Teaching EKF/Kalman filter concepts to students or team members
Verifying manual EKF calculations during filter design
Understanding why high-R (noisy sensor) gets low Kalman gain
Initialising Q and R matrices for GPS/IMU/LiDAR fusion filters

Formula

PREDICT: P⁻ = P + QUPDATE: K = P⁻ / (P⁻ + R) innovation = z − x̂ x̂_new = x̂ + K × innovation P_new = (1 − K) × P⁻K → 0: trust prediction more K → 1: trust measurement more

Built by Dr. Dilip Kumar Limbu · Co-Founder, MooVita · Former Principal Scientist, A*STAR · UDHY Engineering Tools

FAQs on xtended Kalman Filter (EKF) Step Calculator

1. What is an Extended Kalman Filter and how is it used in robotics?

An EKF is an algorithm for optimal state estimation in systems with Gaussian noise. It maintains a probability distribution (mean + covariance) over the true system state and updates this distribution each time a prediction is made or a measurement is received. In robotics and AVs, EKFs fuse GPS, IMU, wheel encoders, and LiDAR into a single optimally-weighted state estimate. The "Extended" part means the filter handles nonlinear system models by linearising them at the current estimate.

2. What is Kalman gain and what does its value mean?

Kalman gain K (always between 0 and 1) is the optimal weighting applied to the measurement's innovation. K → 0 means the filter strongly trusts the prediction over the measurement (the sensor is very noisy relative to model uncertainty). K → 1 means the filter strongly trusts the measurement (the sensor is very precise relative to model uncertainty). K = 0.5 means equal weighting. A GPS measurement with K = 0.9 means the filter almost completely trusts the GPS reading for this step.

3. How do I choose process noise Q and measurement noise R?

Process noise Q represents how much the true state can deviate from your motion model between updates — set it based on actual process variability. For a vehicle at approximately constant speed, Q_position is small; for a pedestrian with unpredictable motion, Q is large. Measurement noise R represents sensor accuracy — set it to sensor variance (σ² from datasheet or empirical testing). The ratio Q/R matters more than absolute values. If the filter lags behind truth, Q is too low; if it's too noisy, Q is too high.

4. Why does covariance always decrease after an EKF measurement update?

The updated covariance P = (1 − K) × P⁻, and since K is always between 0 and 1, P_updated is always less than P⁻. Receiving any valid measurement always reduces uncertainty because you know more about the state after observing it than before. The amount of decrease depends on K: a high-K measurement (precise sensor) reduces uncertainty dramatically; a low-K measurement (noisy sensor) reduces it only slightly.

5. What is the difference between a Kalman filter and an EKF?

A standard Kalman filter assumes linear system dynamics and linear measurement models. An EKF handles nonlinear dynamics by computing Jacobian matrices (first-order Taylor expansion) of the nonlinear functions at the current estimate, then applying standard Kalman equations to the linearised versions. For a ground robot with (x, y, θ) state and non-linear trigonometric motion equations, the EKF linearises at each step. When nonlinearity is severe, the linearisation error can cause divergence — motivating the UKF or particle filter.

6. When should I use EKF vs particle filter vs UKF for AV localisation?

Use EKF when system dynamics are mildly nonlinear, state space is continuous, and computational efficiency matters — GPS/IMU fusion with a smooth vehicle dynamics model is the most common AV localisation case. Use UKF when nonlinearities are stronger than EKF handles well (bearings-only navigation). Use a particle filter when the state distribution is multi-modal (localisation against a map where multiple hypotheses are plausible) — global localisation often requires particle filters. For production AV systems with large state spaces and real-time requirements, EKF or Factor Graph Optimisation are the practical choices.