# ⚛  L1 Principle — Forward Kinematics

**ID:** `L1-446` · **Status:** ⊙ Testnet (genesis catalog)

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

---

## 🧠 1. Introduction

**Forward Kinematics** is a **parameter-estimation problem** whose unknown lives in **cartesian pose vector** space, within the **Robot kinematics** sub-domain of **Robotics**.

Measurements consist of N/A via a **joint encoder to cartesian** sensing mechanism.

The forward operator applies, in order: computes eigen-pairs of a linear operator; S · chain · product operator; O · pose · cartesian operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered cartesian_pose_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 ~= 20); DH_calibration_error 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** → S · chain · product → O · pose · cartesian → **y** (detector).

```
y = `O.pose.cartesian` `S.chain.product` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.chain.product` | S · chain · product operator |
| `O.pose.cartesian` | O · pose · cartesian operator |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `E.eigensolve` | Computes eigen-pairs of a linear operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Robotics |
| Sub domain | Robot kinematics |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | cartesian_pose_vector |
| Noise model | gaussian |
| Integration axis | kinematic_chain |
| Difficulty delta | 3 |
| L dag | 2 |

## 📡 4. Measurement Model

Existence of the recovered cartesian_pose_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 ~= 20); DH_calibration_error dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | end_effector_position_RMSE_mm |
| Secondary | orientation_error_deg |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Load kg | kg | 5 |
| N joints | — | 6 |
| Encoder noise deg | deg | 0.01 |
| Calibration error mm | mm | 0.5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Load kg | kg | 0 – 100 |
| N joints | — | 2 – 10 |
| Encoder noise deg | deg | 0.001 – 0.1 |
| Calibration error mm | mm | 0.0 – 5.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 1.0 end_effector_position_RMSE_mm

| Metric | Range |
|---|---|
| End effector position rmse mm | 0.1 – 20.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **end_effector_position_RMSE_mm**, with κ = `500` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x6a143802bc19e6738567859c81ccedaa775f91be5418044bb020e4d7f7216b05`
- **Chain tx hash:** `0x47849de07531a7753e76a95981ea254becf573ec076d26599a7d730889652691`
- **Chain block:** `41555260`

---

## File Mapping

This bundle consists of: `L1-446.md`, `L1-446.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-446.md` | Source of truth — edit this | Human or LLM |
| `L1-446.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 `L1-446`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._