Why is the state four-dimensional?

The filter tracks x, y and the velocities vₓ, vᵧ. Position-only sensors still inform velocity through the constant-velocity dynamics and the Kalman update.

What happens if R is much smaller than Q?

The filter trusts measurements and the estimate hugs the noisy dots, so the ellipse shrinks but the track can look jittery. Large R does the opposite: smoother tracks with more lag.

Kalman Filter 2D Tracking

4-state constant-velocity tracker: noisy (x,y) measurements, process acceleration noise Q, measurement variance R, and a live ≈2σ covariance ellipse.

About this model

A constant-velocity Kalman filter estimates planar position and velocity from noisy (x, y) measurements. Process noise is modeled as white acceleration, so the discrete Q matrix couples position and velocity uncertainty. The live amber ellipse is the ≈2σ contour of the position block of P; comparing estimate RMSE with raw measurement RMSE shows how Q and R trade lag against jitter.

Who it's for: Estimation, robotics, control, signal processing, aerospace tracking, and applied probability courses.

Key terms

Kalman filter
Covariance ellipse
Process noise Q
Measurement noise R
Constant-velocity model
Innovation

How it works

A 4-state constant-velocity Kalman filter tracks a moving target in the plane from noisy position measurements. The amber ellipse is the filter’s ≈2σ position uncertainty from the covariance matrix P; tune process acceleration noise Q and measurement variance R to see lag versus jitter.

Key equations

x = [x, y, vₓ, vᵧ]ᵀ · F constant-velocity · z = [x, y]ᵀ + v

Predict: x̂⁻ = F x̂, P⁻ = F P Fᵀ + Q

Update: K = P⁻ Hᵀ (H P⁻ Hᵀ + R)⁻¹, x̂ = x̂⁻ + K(z − H x̂⁻)

Frequently asked questions

Why is the state four-dimensional?: The filter tracks x, y and the velocities vₓ, vᵧ. Position-only sensors still inform velocity through the constant-velocity dynamics and the Kalman update.
What happens if R is much smaller than Q?: The filter trusts measurements and the estimate hugs the noisy dots, so the ellipse shrinks but the track can look jittery. Large R does the opposite: smoother tracks with more lag.