# ⚛  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`

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## 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._