Kalman Filtering

State Vector ($x$)

$$x_k=\left[\begin{array}{c}{p}_x\\ p_y\\ p_z\\ v_x\\ v_y\\ v_z\end{array}\right]$$

• $p$: Position in North, East, and Up
• $v$: Velocity, or speed in each direction
• The radar usually only measures position, $p$
• It does not directly measure the velocity vector, $v$. It only measures radial Doppler
• The Kalman Filter infers the velocity, $v$, over time by observing changes in $p$

The Kalman Filter operates in a recursive loop: Time Update, or Predict, and Measurement Update, or Correct.

Phase A: Physics Model

I have not checked the sensor yet, but based on Newton's Law, the drone should be here.

$${\hat{x}}_{k\mid k-1}=F{\hat{x}}_{k-1}+B{u}_{k-1}$$

• $F$ is the State Transition Matrix. If velocity is 10 m/s and $\Delta t$ is 0.1 s, position moves by 1 m
• $P$ is the Covariance Matrix, or the "Uncertainty Bubble"
• Every time we predict, the bubble gets bigger because uncertainty grows. We assume that wind might have pushed the drone

Drone Context

If the radar misses a scan because the drone flies behind a billboard, the Predict step keeps running. The system "hallucinates" the drone continuing on its path. This is called "Coasting."

Phase B: Measurement

• The radar returns a raw detection

$$z_k=H{x}_k+v_k$$

• $z_k$ is the actual coordinate from the radar, such as Range: 1000 m and Azimuth: 45 degrees
• $R$ is the Measurement Noise from the manufacturer specifications: "This radar is accurate to"

Phase C: Updating Using the Kalman Filter

• The filter compares the prediction with the measurement
• If they do not match, the Kalman Filter decides which one to trust

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

• If measurement noise $R$ is small and $K$ is high, the filter chooses the radar, or Good Radar
• If radar noise $R$ is high and $K$ is low, the filter ignores the radar and trusts the physics model, or Bad Radar

Q vs R

• This is the hardest part of C-UAS tracking
• $Q$ in Kalman Filter code is a parameter that tells the filter, "How much wind or unpredictability should I expect?"
• The prediction step in the Kalman Filtering process has an added $Q$ at every time step

Increasing Q

• Prediction covariance, $P$, becomes huge
• The output tracks raw sensor measurements closely, with fast response and high jitter

Decreasing Q

• Prediction covariance, $P$, is small
• The output ignores sensor spikes, with a smooth track and slow response or lag

The reason this is hard is that you do not know who the enemy is.

• If you tune $Q$ for a DJI Phantom, with low $Q$ and a smooth flyer, and an FPV Racing Drone attacks you:
  • The Racing Drone zigs left
  • Your filter, which expects smooth motion, thinks the "zig" is only radar noise and ignores it
  • Result: Your jammer shoots at where the drone was, instead of where it is. You miss
• If you tune $Q$ for a Racing Drone, with high $Q$ and erratic movement, and a bird flies by:
  • The bird flutters slightly
  • Your filter reacts to every flutter
  • Result: The track on the screen looks like a mess of scribbles, making the operator dizzy

Importance of choosing Q

• State estimation of the drone, or predictive capability, is important to jam the drone
• If the drone is traveling in a certain direction, we cannot simply jam it at its current location
• The time lag from radar processing, network transmission, and motor movement results in a miss