L1 ⚛ L1-450 ⊙ Testnet

Trajectory Optimization

Robotics · Motion planning

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⬇ L1-450.md

# ⚛ 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` --- ## 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._

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L2-450-001 Trajectory Optimization — Nominal + Mismatch Specs

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📋 JSON metadata

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    "title": "Trajectory Optimization"
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  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Motion planning",
  "title": "Trajectory Optimization"
}

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