Code Atlas: 3D Kalman State Filter

Diátaxis Type: Code Atlas Record | ID: ATLAS-ALG-001 | Owner: diabetic.dsp.kalman

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1. Mathematical Formulation

Tracks physiological continuous kinematic state \(\mathbf{x} = [g, v, a]^T\):

\[\mathbf{x}_k = \mathbf{F} \mathbf{x}_{k-1} + \mathbf{w}_{k-1}, \quad \mathbf{w} \sim \mathcal{N}(0, \mathbf{Q})\]

\[z_k = \mathbf{H} \mathbf{x}_k + v_k, \quad v \sim \mathcal{N}(0, R)\]

State Transition Matrix (\(\mathbf{F}\))

\[\mathbf{F} = \begin{bmatrix} 1 & \Delta t & \frac{1}{2}\Delta t^2 \\ 0 & 1 & \Delta t \\ 0 & 0 & 1 \end{bmatrix}\]

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2. Complexity & Numerical Invariants

• Computational Complexity: \(O(1)\) constant time per update cycle.
• Space Complexity: \(O(1)\) fixed matrix storage (\(3 \times 3\)).
• Degenerate Covariance Guard:

  # ponytail: pinv fallback for degenerate covariance
  try:
      S_inv = np.linalg.inv(S)
  except np.linalg.LinAlgError:
      S_inv = np.linalg.pinv(S)