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Kalman / EKF / UKF

Optimal recursive estimation for (nearly) linear-Gaussian systems.

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Prerequisites

Bayesian Filtering

Why it matters in robotics

The Kalman filter is the workhorse of robotic state estimation — localization, IMU fusion, and tracking. For a linear system with Gaussian noise it is the optimal estimator, fusing a motion-model prediction with noisy measurements into a single mean-and-covariance belief, automatically weighing how much to trust the prediction versus the new measurement. Its extensions carry the same idea to the nonlinear systems that dominate real robots, handling the curved, non-straight-line relationships that linear methods can't capture directly.

What to study

✓Linear KF: predict/update equations & the Kalman gain
✓Why it's the optimal linear-Gaussian estimator (MMSE)
✓EKF: linearization via Jacobians, and its failure modes
✓UKF: the unscented transform vs. EKF
✓Tuning Q and R; observability & filter divergence
✓IMM (Interacting Multiple Model): switching motion models for maneuvering targets

Where to study

Cheatsheet

Symbols used below

x

state estimate

P

state covariance (uncertainty)

F

state transition matrix (motion model)

B, u

control input matrix & control input

Q

process noise covariance

H

measurement matrix

R

measurement noise covariance

K

Kalman gain

z

measurement (sensor reading)

I

identity matrix

n

dimension of the state (used for sigma-point count)

f

the (possibly nonlinear) true motion/measurement function

Predict

$x^- = Fx + Bu$

$P^- = FPF^\top + Q$

Update

$K = P^-H^\top(HP^-H^\top + R)^{-1}$

$x = x^- + K(z - Hx^-)$

$P = (I - KH)P^-$

Kalman gain limits

As $R \to \infty$ : $K \to 0$ — trust the prediction.

As $R \to 0$ : $K \to 1$ — trust the measurement.

Q vs R, at a glance

Q (process noise) — small ⇒ tight, model-trusting. Large ⇒ wide, agile, noisier.

R (measurement noise) — shrinks or grows the gain the same way.

The Kalman filter alternates a predict step (push state + covariance through the motion model) and an update step (blend in the measurement via the gain K).

	KF	EKF	UKF
Assumption	Linear, Gaussian	Nonlinear, locally linearized	Nonlinear, no linearization
How it handles nonlinearity	N/A — exact	1st-order Taylor (Jacobians)	Sigma points through the true $f$
Optimality	Optimal (MMSE)	Approximate	Approximate, usually tighter than EKF
Cost	Lowest	Low (Jacobians)	Higher ( $2n{+}1$ sigma points)

A Gaussian pushed through a nonlinearity isn't Gaussian. The EKF approximates f by its tangent (Jacobian) at the mean; the UKF propagates sigma points through f, capturing the spread more faithfully.

Application focus

Select an application above.