# ⚛  L1 Principle — Trajectory Optimization

**ID:** `L1-450` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Robotics — *Motion planning*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** optimal trajectory
> **📡 Carrier:** N/A · **🌫 Noise:** deterministic
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555278

---

## 🧠 1. Introduction

**Trajectory Optimization** is a **nonlinear inverse problem** whose unknown lives in **optimal trajectory** space, within the **Motion planning** sub-domain of **Robotics**.

Measurements consist of N/A via a **trajectory collocation optimization** sensing mechanism.

The forward operator applies, in order: a fixed-point or gradient iteration on the unknown; O · nlp · ipopt solver operator; S · warm start · previous solution operator.

Observations are corrupted by no stochastic noise (deterministic measurement). Existence of the recovered optimal_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 ~= 2000); dynamic_obstacle_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Deterministic sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → O · nlp · ipopt solver → S · warm start · previous solution → **y** (detector).

```
y = `S.warm_start.previous_solution` `O.nlp.ipopt_solver` x    (deterministic)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `O.nlp.ipopt_solver` | O · nlp · ipopt solver operator |
| `S.warm_start.previous_solution` | S · warm start · previous solution operator |

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

| Primitive | What it does |
|---|---|
| `O.iter` | A fixed-point or gradient iteration on the unknown |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Robotics |
| Sub domain | Motion planning |
| Carrier | N/A |
| Problem class | nonlinear_inverse |
| Solution space | optimal_trajectory |
| Noise model | deterministic |
| Integration axis | time_motion |
| Difficulty delta | 5 |
| L dag | 4.5 |

## 📡 4. Measurement Model

Existence of the recovered optimal_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 ~= 2000); dynamic_obstacle_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Deterministic sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | trajectory_feasibility_score |
| Secondary | cost_optimality_ratio |

## 📏 5. Operating Range (Ω)

**Center problem class:** `nonlinear_inverse` · **Forward operator:** `trajectory_collocation_optimization`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N dof | — | 7 |
| N obstacles | — | 5 |
| T horizon s | s | 2 |
| Joint vel limit rad s | s | 3 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N dof | — | 2 – 10 |
| N obstacles | — | 0 – 50 |
| T horizon s | s | 0.1 – 10.0 |
| Joint vel limit rad s | s | 0.5 – 10.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.90 trajectory_feasibility_score

| Metric | Range |
|---|---|
| Trajectory feasibility score | 0.5 – 1.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **trajectory_feasibility_score**, with κ = `100000.0` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x38eb01210eabcb3a5332e1ed7af46f1d3174e9b47b5c86b0e58a61c232a60444`
- **Chain tx hash:** `0xbabdd5d7bfcb62e05846f35f2b2727783978de136f9fd3e13e333a533a3581c7`
- **Chain block:** `41555278`

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## File Mapping

This bundle consists of: `L1-450.md`, `L1-450.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-450.md` | Source of truth — edit this | Human or LLM |
| `L1-450.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 `L1-450`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._