L1 ⚛ L1-451 ⊙ Testnet

Rigid Body Dynamics Simulation

Robotics · Multibody simulation

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⬇ principle.md ⬇ principle.json

Fork L1-451

⬇ L1-451.md ⬇ L1-451.json

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You are helping me submit a MODIFICATION of L1-451 (Rigid Body Dynamics Simulation) to the PWM Protocol — a Principle (L1) artifact.

I will paste a Markdown template (or the current L1-451.md).
1. Rewrite the Markdown so the science is correct and clearly explained for my change.
2. Regenerate the sibling L1-451.json so EVERY field matches the Markdown.
3. Keep the schema in the "File Mapping" footer at the bottom of the MD.
4. Keep the parent reference unchanged unless I ask otherwise.
Rules: the Markdown is the source of truth; use SI units; do NOT invent benchmark scores.
Output each file in its own fenced code block tagged with the filename.

Here is my template:
[PASTE THE .md HERE]

⚙ Forward Model

y = `S.quaternion.integration` ∂_t x + n,    n ~ 𝒩(0, σ²)

world state x (6DOF state trajectory)

→

Time derivative ∂_t

→

S · quaternion · integration `S.quaternion.integration`

→

observation y (N/A detector)

Noise: additive Gaussian noise

Solver / auxiliary components (1)

O.lyapunov.energy_based — O · lyapunov · energy based operator

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-451.md

# ⚛ L1 Principle — Rigid Body Dynamics Simulation **ID:** `L1-451` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Robotics — *Multibody simulation* > **🎯 Problem class:** parameter estimation · **🧮 Solution space:** 6DOF state trajectory > **📡 Carrier:** N/A · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555278 --- ## 🧠 1. Introduction **Rigid Body Dynamics Simulation** is a **parameter-estimation problem** whose unknown lives in **6DOF state trajectory** space, within the **Multibody simulation** sub-domain of **Robotics**. Measurements consist of N/A via a **euler equations quaternion** sensing mechanism. The forward operator applies, in order: time evolution of the state; S · quaternion · integration operator; O · lyapunov · energy based operator. Observations are corrupted by additive Gaussian noise. Existence of the recovered 6DOF_state_trajectory is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 50); inertia_tensor_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor. ## ⚙ 2. Forward Model Physical chain: **x** → Time derivative → S · quaternion · integration → **y** (detector). ``` y = `S.quaternion.integration` ∂_t x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `D.time` | Time evolution of the state | | `S.quaternion.integration` | S · quaternion · integration operator | **🛡 Analytical properties** _(used inside the solver, not in the forward equation)_: | Primitive | What it does | |---|---| | `O.lyapunov.energy_based` | O · lyapunov · energy based operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Robotics | | Sub domain | Multibody simulation | | Carrier | N/A | | Problem class | parameter_estimation | | Solution space | 6DOF_state_trajectory | | Noise model | gaussian | | Integration axis | time | | Difficulty delta | 3 | | L dag | 2.8 | ## 📡 4. Measurement Model Existence of the recovered 6DOF_state_trajectory is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 50); inertia_tensor_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor. | Metric | Value | |---|---| | Metric | quaternion_attitude_error_deg | | Secondary | position_RMSE_m | ## 📏 5. Operating Range (Ω) **Center problem class:** `parameter_estimation` · **Forward operator:** `euler_equations_quaternion` **Center point:** | Parameter | Unit | Value | |---|---|---| | Mass kg | kg | 1 | | Integration dt ms | ms | 1 | | Inertia ratio j1 j3 | — | 2 | | Simulation duration s | s | 10 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | Mass kg | kg | 0.01 – 1000 | | Integration dt ms | ms | 0.01 – 10.0 | | Inertia ratio j1 j3 | — | 0.5 – 10.0 | | Simulation duration s | s | 0.1 – 1000 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 0.05 quaternion_attitude_error_deg | Metric | Range | |---|---| | Quaternion attitude error deg | 0.001 – 1.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **quaternion_attitude_error_deg**, with κ = `1000` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x0a97a72b25a7cf9e0031ad387c5c63375fe14178d6d2be65ca95d011c1b751ba` - **Chain tx hash:** `0xb2e8ff78dc69b1cf650f8e068c85d6a6c8ddf7ebfd3bb6628c5645b480b18502` - **Chain block:** `41555278` --- ## File Mapping This bundle consists of: `L1-451.md`, `L1-451.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-451.md` | Source of truth — edit this | Human or LLM | | `L1-451.json` | Structured metadata for the registry | LLM regenerates from the sections above | **Prompt for your LLM after editing this Markdown:** > Read the attached Markdown. Regenerate the sibling `.json` so every field matches. > Preserve the schema documented in the rows above. > Output each file in its own fenced code block tagged with the filename. > Output only the JSON object. _This Markdown was auto-synthesized from the catalog row for `L1-451`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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L2-451-001 Rigid Body Dynamics Simulation — Nominal + Mismatch Specs

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📋 JSON metadata

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  "artifact_id": "L1-451",
  "chain_block": 41555278,
  "chain_hash": "0x0a97a72b25a7cf9e0031ad387c5c63375fe14178d6d2be65ca95d011c1b751ba",
  "chain_tx_hash": "0xb2e8ff78dc69b1cf650f8e068c85d6a6c8ddf7ebfd3bb6628c5645b480b18502",
  "domain": "Robotics",
  "hardness_fn": {
    "delta": 3,
    "kappa": 1000,
    "metric": "quaternion_attitude_error_deg",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "quaternion_attitude_error_deg",
    "regime": "Existence of the recovered 6DOF_state_trajectory is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 50); inertia_tensor_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.",
    "secondary": "position_RMSE_m"
  },
  "physics_fingerprint": {
    "L_DAG": 2.8,
    "carrier": "N/A",
    "difficulty_delta": 3,
    "domain": "Robotics",
    "integration_axis": "time",
    "noise_model": "gaussian",
    "primitives": [
      "D.time",
      "S.quaternion.integration",
      "O.lyapunov.energy_based"
    ],
    "problem_class": "parameter_estimation",
    "sensing_mechanism": "euler_equations_quaternion",
    "solution_space": "6DOF_state_trajectory",
    "sub_domain": "Multibody simulation",
    "title": "Rigid Body Dynamics Simulation"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "euler_equations_quaternion"
    ],
    "allowed_omega_dimensions": [
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      "inertia_ratio_J1_J3",
      "integration_dt_ms",
      "simulation_duration_s"
    ],
    "allowed_problem_classes": [
      "parameter_estimation"
    ],
    "center_spec": {
      "epsilon_fn_center": "0.05 quaternion_attitude_error_deg",
      "forward_operator": "euler_equations_quaternion",
      "input_format": "measurement_only",
      "omega": {
        "inertia_ratio_J1_J3": 2.0,
        "integration_dt_ms": 1.0,
        "mass_kg": 1.0,
        "simulation_duration_s": 10.0
      },
      "problem_class": "parameter_estimation"
    },
    "epsilon_bounds": {
      "quaternion_attitude_error_deg": [
        0.001,
        1.0
      ]
    },
    "omega_bounds": {
      "inertia_ratio_J1_J3": [
        0.5,
        10.0
      ],
      "integration_dt_ms": [
        0.01,
        10.0
      ],
      "mass_kg": [
        0.01,
        1000
      ],
      "simulation_duration_s": [
        0.1,
        1000
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Multibody simulation",
  "title": "Rigid Body Dynamics Simulation"
}

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