Medium

One Step of a Constant-Velocity Kalman Filter

~20 mincode completion

Implement kalman_step(x, P, z, dt, q, r) that runs one predict step and one update step and returns a list:

[[p, v],        #  
 [P00, P01],    #  , row 0
 [P10, P11]]    #  , row 1

x is a length-2 array, P is a array, and z, , q, r are floats. Use NumPy; no filtering libraries.

Why this matters

This loop is the core of radar tracking and of sensor fusion. A perimeter air-defense system runs one of these per track, many times a second, and the covariance it maintains is what decides which new plot belongs to which track. Fusing a second sensor, such as a camera bearing or a second radar, is just another update step with its own and : the filter weighs each sensor by how much it trusts it. The same equations sit inside every GPS and IMU fusion loop and every self-driving object tracker.

Examples

The worked example

Input
kalman_step([0, 10], [[4, 0], [0, 1]], 12, 1, 0.5, 4)
Output
[[11.123288, 10.273973], [2.246575, 0.547945], [0.547945, 1.328767]]

Half-second radar sweep, correlated prior, target closing

Input
kalman_step([100, -3], [[2, 0.5], [0.5, 1]], 98.9, 0.5, 1, 1)
Output
[[98.793776, -2.887137], [0.73444, 0.282158], [0.282158, 0.950207]]

Huge measurement noise: a wild plot barely moves the estimate

Input
kalman_step([50, 5], [[1, 0], [0, 1]], 500, 1, 0.1, 1000000000)
Output
[[55.000001, 5], [2.025, 1.05], [1.05, 1.1]]

Hints

Hint 1

Use a matrix product rather than nested loops, and check which operand transposes.

Hint 2

Do not forget to process noise q in predict. That step is easy to skip.

Requirements

  • x: Prior mean [position, velocity], shape (2,)

  • P: Prior covariance, shape (2, 2)

  • z: Position measurement

  • : Time step in seconds

  • q: Process noise (acceleration variance)

  • r: Measurement noise variance

  • Return [[p, v], [P00, P01], [P10, P11]]: posterior mean and covariance.

Constraints

  • Allowed library: NumPy only

  • Time limit: 200 ms, Memory: 64 MB

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Where this shows up

~20 min

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Python
import numpy as np

def kalman_step(x, P, z, dt, q, r) -> list:
    """
    One predict + update step of a constant-velocity Kalman filter.

    Args:
        x:  Prior mean [position, velocity], shape (2,)
        P:  Prior covariance, shape (2, 2)
        z:  Position measurement
        dt: Time step in seconds
        q:  Process noise (acceleration variance)
        r:  Measurement noise variance

    Returns:
        [[p, v], [P00, P01], [P10, P11]]: posterior mean and covariance.
    """
    x = np.asarray(x, dtype=float)
    P = np.asarray(P, dtype=float)
    # Predict: x- = F x, P- = F P F^T + Q
    # Update:  y = z - H x-, S = H P- H^T + r, K = P- H^T / S
    # YOUR CODE HERE
    pass
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