{
 "artifacts": null,
 "category": "regularization",
 "description": "Add a separately identified constant or slowly varying force channel to a neural dynamics model, analogous to the paper's online gravity-trim and mass adaptation. This isolates persistent low-frequency prediction bias from state-dependent dynamics, reducing repeated corrective action by the planner and avoiding expensive network retraining after a systematic shift.",
 "formulas_latex": [
  "$$m\\dot v=-mge_3+fRe_3,$$",
  "$$\\hat{x}_{k+1}=\\hat{F}_{\\theta}(x_k,u_k)+Gb_k,\\qquad e_k=x_{k+1}-\\hat{F}_{\\theta}(x_k,u_k)-Gb_{k-1},$$",
  "$$K_k=\\frac{P_{k-1}G}{\\lambda+G^{\\top}P_{k-1}G},\\qquad b_k=b_{k-1}+K_k^{\\top}e_k,\\qquad P_k=\\lambda^{-1}\\left(I-K_kG^{\\top}\\right)P_{k-1}.$$",
  "$$N_{\\mathrm{eff}}\\approx\\frac{1}{1-\\lambda},\\qquad \\mathbb{E}\\left[\\|b_k-b_{\\star}\\|^2\\right]\\text{ decays after a bias step and approaches a noise-dependent floor that increases as }\\lambda\\text{ decreases}.$$"
 ],
 "id": 3101,
 "implementation": "1. Integration point: insert a low-dimensional trim head between a neural dynamics predictor and its MPC or model-based RL interface. Select $G$ from physically plausible persistent channels, such as vertical acceleration, horizontal force, or actuator torque offsets. Initially use a fixed selector and do not backpropagate through the trim estimator. 2. Pseudocode: compute neural prediction $\\tilde{x}_{k+1}=\\hat{F}_{\\theta}(x_k,u_k)$; calculate $e_k=x_{k+1}-\\tilde{x}_{k+1}-Gb_{k-1}$; compute $K_k$, $b_k$, and $P_k$ using the displayed equations; return $\\tilde{x}_{k+1}+Gb_k$ to the planner; optionally reset or inflate $P_k$ when a regime-change detector fires. Add a penalty $\\gamma\\|b_k\\|^2$ only when large biases are physically implausible. 3. Computed from the mechanism: the force-balance structure, innovation, RLS gain, and covariance. Estimated empirically: $G$, initial $P_0$, forgetting factor $\\lambda$, and whether the trim should be constant or input-dependent. 4. First cheap experiment: train a neural Pendulum or CartPole model, inject a constant unknown force, and abruptly change its value halfway through evaluation. Compare neural-only, neural-plus-trained-global-bias, and neural-plus-RLS-trim models on one-step prediction and 20-step open-loop rollout. The predicted signature is geometric post-switch innovation reduction with recovery time proportional to $N_{\\mathrm{eff}}\\approx1/(1-\\lambda)$. Fit the decay curve and require its time constant to agree with the RLS prediction within 20 percent. Lower $\\lambda$ should adapt faster but produce a measurable higher stationary variance; accept the method only if it reduces post-switch bias by at least 50 percent without increasing pre-switch prediction variance by more than 20 percent.",
 "math_summary": "The vertical quadrotor equation contains a persistent acceleration contribution from gravity and thrust: $\\dot v=-ge_3+(f/m)Re_3$. Model uncertainty in mass, gravity, or actuator calibration as a low-dimensional bias $b_k$ injected through a known selector matrix $G$. The adaptive predictor is $\\hat{x}_{k+1}=\\hat{F}_{\\theta}(x_k,u_k)+Gb_k$. Given innovation $e_k=x_{k+1}-\\hat{F}_{\\theta}(x_k,u_k)-Gb_{k-1}$, estimate $b_k$ with RLS using covariance $P_k$ and forgetting factor $\\lambda$. The persistent-bias channel is useful because it has a slower and simpler structure than the full nonlinear residual; after a step change in the true bias, the estimator should converge geometrically until it reaches a measurement-noise floor.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "statistics",
  "optimization"
 ],
 "ml_areas": [
  "world-model",
  "rl",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2609.03351",
  "arxiv_url": "https://arxiv.org/abs/2609.03351",
  "summary_what_math_gives_to_ml": "The paper contributes an adaptive predictive-control mechanism based on sparse first-, second-, and third-order Taylor dictionaries, row-wise recursive least-squares identification, variable-rate forgetting, and a Jacobian held fixed over each prediction horizon. Its transferable asset is a cheap local model that adapts rapidly after payload, actuator, or environment changes while preserving a simple finite-horizon predictor. A strong neural-network transfer is to combine a global neural world model with an online Taylor residual model for model-based reinforcement learning or MPC. A second transfer is to identify persistent force or acceleration biases in a separate adaptive trim channel rather than forcing the neural model to relearn them.",
  "title": "Taylor-Informed Predictive Cost Adaptive Control for Quadrotors with Online Gravity-Trim Adaptation",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 3,
  "novelty": 5,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "accuracy",
  "sample-efficiency"
 ],
 "title": "Adaptive Constant-Force Trim Channel",
 "url": "https://synthcore.org/idea/3101/adaptive-constant-force-trim-channel",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}