# ⚛  L1 Principle — Forward Dynamics

**ID:** `L1-448` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Robotics — *Robot dynamics*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** joint acceleration vector
> **📡 Carrier:** N/A · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555260

---

## 🧠 1. Introduction

**Forward Dynamics** is a **parameter-estimation problem** whose unknown lives in **joint acceleration vector** space, within the **Robot dynamics** sub-domain of **Robotics**.

Measurements consist of N/A via a **torque to acceleration** sensing mechanism.

The forward operator applies, in order: time evolution of the state; S · articulated body · aba operator; O · ode · integration operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered joint_acceleration_vector 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 ~= 80); inertia_parameter_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 · articulated body · aba → O · ode · integration → **y** (detector).

```
y = `O.ode.integration` `S.articulated_body.aba` ∂_t x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.time` | Time evolution of the state |
| `S.articulated_body.aba` | S · articulated body · aba operator |
| `O.ode.integration` | O · ode · integration operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Robotics |
| Sub domain | Robot dynamics |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | joint_acceleration_vector |
| Noise model | gaussian |
| Integration axis | time |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Existence of the recovered joint_acceleration_vector 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 ~= 80); inertia_parameter_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | joint_trajectory_RMSE_deg |
| Secondary | energy_conservation_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `parameter_estimation` · **Forward operator:** `torque_to_acceleration`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N joints | — | 6 |
| Total mass kg | kg | 10 |
| Simulation time s | s | 1 |
| Friction coefficient | — | 0.1 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N joints | — | 2 – 10 |
| Total mass kg | kg | 0.5 – 1000 |
| Simulation time s | s | 0.1 – 100 |
| Friction coefficient | — | 0.0 – 1.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 1.0 joint_trajectory_RMSE_deg

| Metric | Range |
|---|---|
| Joint trajectory rmse deg | 0.01 – 10.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **joint_trajectory_RMSE_deg**, with κ = `2000` and δ = `3`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x007249d71bb962a4a0d435aba3e4dd4602bc1650028ead53d67963ff9ca20761`
- **Chain tx hash:** `0x9a348e02c700c44ccf825997b0c15b2df5361ddd20f4150f53d8c72d555a5fa8`
- **Chain block:** `41555260`

---

## File Mapping

This bundle consists of: `L1-448.md`, `L1-448.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-448.md` | Source of truth — edit this | Human or LLM |
| `L1-448.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 `L1-448`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._