# ⚛  L1 Principle — Inverse Dynamics

**ID:** `L1-449` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Robotics — *Robot dynamics*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** joint torque vector
> **📡 Carrier:** N/A · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555277

---

## 🧠 1. Introduction

**Inverse Dynamics** is a **linear inverse problem** whose unknown lives in **joint torque vector** space, within the **Robot dynamics** sub-domain of **Robotics**.

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

The forward operator applies, in order: time evolution of the state; S · regressor · inertia parameters operator; O · least squares · param id operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered joint_torque_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 ~= 50); acceleration_estimation_noise 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 · regressor · inertia parameters → O · least squares · param id → **y** (detector).

```
y = `O.least_squares.param_id` `S.regressor.inertia_parameters` ∂_t x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.time` | Time evolution of the state |
| `S.regressor.inertia_parameters` | S · regressor · inertia parameters operator |
| `O.least_squares.param_id` | O · least squares · param id operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Robotics |
| Sub domain | Robot dynamics |
| Carrier | N/A |
| Problem class | linear_inverse |
| Solution space | joint_torque_vector |
| Noise model | gaussian |
| Integration axis | kinematic_chain |
| Difficulty delta | 3 |
| L dag | 2.8 |

## 📡 4. Measurement Model

Existence of the recovered joint_torque_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 ~= 50); acceleration_estimation_noise dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | torque_prediction_RMSE_Nm |
| Secondary | parameter_identification_RMSE |

## 📏 5. Operating Range (Ω)

**Center problem class:** `linear_inverse` · **Forward operator:** `trajectory_to_torque_computation`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N joints | — | 6 |
| N trajectory samples | — | 1000 |
| Acceleration noise db | dB | -20 |
| Excitation bandwidth hz | Hz | 2 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N joints | — | 2 – 10 |
| N trajectory samples | — | 100 – 10000 |
| Acceleration noise db | dB | -40 – -10 |
| Excitation bandwidth hz | Hz | 0.5 – 10.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 2.0 torque_prediction_RMSE_Nm

| Metric | Range |
|---|---|
| Torque prediction rmse nm | 0.1 – 20.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **torque_prediction_RMSE_Nm**, with κ = `1000` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x05dca4b066ca2181478e5dde661fff6d429d658391f8d205a7ccbd4316bcd3d8`
- **Chain tx hash:** `0x1e99291f16a45bd235ea8ad95190dcb336d08df6075a294179eb3a1572cc29ae`
- **Chain block:** `41555277`

---

## File Mapping

This bundle consists of: `L1-449.md`, `L1-449.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-449.md` | Source of truth — edit this | Human or LLM |
| `L1-449.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 `L1-449`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._