L1 ⚛ L1-447 ⊙ Testnet

Inverse Kinematics

Robotics · Robot kinematics

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⬇ L1-447.md

# ⚛ L1 Principle — Inverse Kinematics **ID:** `L1-447` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Robotics — *Robot kinematics* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** joint angle vector > **📡 Carrier:** N/A · **🌫 Noise:** deterministic > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555260 --- ## 🧠 1. Introduction **Inverse Kinematics** is a **nonlinear inverse problem** whose unknown lives in **joint angle vector** space, within the **Robot kinematics** sub-domain of **Robotics**. Measurements consist of N/A via a **cartesian to joint space inversion** sensing mechanism. The forward operator applies, in order: S · ik · jacobian pseudo inverse operator; adds a prior term that biases the solution toward smoothness/sparsity; O · gradient · task space operator. Observations are corrupted by no stochastic noise (deterministic measurement). Existence of the recovered joint_angle_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 ~= 200); singularity_proximity 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** → S · ik · jacobian pseudo inverse → O · gradient · task space → **y** (detector). ``` y = `O.gradient.task_space` `S.ik.jacobian_pseudo_inverse` x (deterministic) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `S.ik.jacobian_pseudo_inverse` | S · ik · jacobian pseudo inverse operator | | `O.gradient.task_space` | O · gradient · task space operator | **🛠 Solver components** _(used inside the solver, not in the forward equation)_: | Primitive | What it does | |---|---| | `O.regularize` | Adds a prior term that biases the solution toward smoothness/sparsity | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Robotics | | Sub domain | Robot kinematics | | Carrier | N/A | | Problem class | nonlinear_inverse | | Solution space | joint_angle_vector | | Noise model | deterministic | | Integration axis | configuration_space | | Difficulty delta | 3 | | L dag | 3 | ## 📡 4. Measurement Model Existence of the recovered joint_angle_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 ~= 200); singularity_proximity dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Deterministic sets the irreducible data-fidelity floor. | Metric | Value | |---|---| | Metric | position_error_mm | | Secondary | orientation_error_deg | ## 📏 5. Operating Range (Ω) **Center problem class:** `nonlinear_inverse` · **Forward operator:** `cartesian_to_joint_space_inversion` **Center point:** | Parameter | Unit | Value | |---|---|---| | N joints | — | 6 | | Damping factor lambda | — | 0.01 | | Target reach fraction | — | 0.8 | | Singularity distance cm | — | 10 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | N joints | — | 2 – 10 | | Damping factor lambda | — | 0.001 – 0.1 | | Target reach fraction | — | 0.5 – 1.0 | | Singularity distance cm | — | 0.0 – 50.0 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 1.0 position_error_mm | Metric | Range | |---|---| | Position error mm | 0.01 – 20.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **position_error_mm**, with κ = `5000` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x1a59a8c53db8a97ac4f9a33d9c12b13e459a3f060cddb2abcce832500fab4055` - **Chain tx hash:** `0xf092647d286e3c43a07cdb0de8242a8153306a79f64a5c82ac8d8be1f9a43395` - **Chain block:** `41555260` --- ## File Mapping This bundle consists of: `L1-447.md`, `L1-447.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-447.md` | Source of truth — edit this | Human or LLM | | `L1-447.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 `L1-447`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-447.md).
Regenerate L1-447.json so every field matches the Markdown.
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L2-447-001 Inverse Kinematics — Nominal + Mismatch Specs

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📋 JSON metadata

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  "chain_block": 41555260,
  "chain_hash": "0x1a59a8c53db8a97ac4f9a33d9c12b13e459a3f060cddb2abcce832500fab4055",
  "chain_tx_hash": "0xf092647d286e3c43a07cdb0de8242a8153306a79f64a5c82ac8d8be1f9a43395",
  "domain": "Robotics",
  "hardness_fn": {
    "delta": 3,
    "kappa": 5000,
    "metric": "position_error_mm",
    "type": "epsilon_fn"
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  "initiator_dataset": [
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      "license_hash": null,
      "name": "primary",
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    }
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  "layer": "L1",
  "observable_profile": {
    "metric": "position_error_mm",
    "regime": "Existence of the recovered joint_angle_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 ~= 200); singularity_proximity dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Deterministic sets the irreducible data-fidelity floor.",
    "secondary": "orientation_error_deg"
  },
  "physics_fingerprint": {
    "L_DAG": 3.0,
    "carrier": "N/A",
    "difficulty_delta": 3,
    "domain": "Robotics",
    "integration_axis": "configuration_space",
    "noise_model": "deterministic",
    "primitives": [
      "S.ik.jacobian_pseudo_inverse",
      "O.regularize",
      "O.gradient.task_space"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "cartesian_to_joint_space_inversion",
    "solution_space": "joint_angle_vector",
    "sub_domain": "Robot kinematics",
    "title": "Inverse Kinematics"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "cartesian_to_joint_space_inversion"
    ],
    "allowed_omega_dimensions": [
      "N_joints",
      "singularity_distance_cm",
      "damping_factor_lambda",
      "target_reach_fraction"
    ],
    "allowed_problem_classes": [
      "nonlinear_inverse"
    ],
    "center_spec": {
      "epsilon_fn_center": "1.0 position_error_mm",
      "forward_operator": "cartesian_to_joint_space_inversion",
      "input_format": "measurement_only",
      "omega": {
        "N_joints": 6,
        "damping_factor_lambda": 0.01,
        "singularity_distance_cm": 10,
        "target_reach_fraction": 0.8
      },
      "problem_class": "nonlinear_inverse"
    },
    "epsilon_bounds": {
      "position_error_mm": [
        0.01,
        20.0
      ]
    },
    "omega_bounds": {
      "N_joints": [
        2,
        10
      ],
      "damping_factor_lambda": [
        0.001,
        0.1
      ],
      "singularity_distance_cm": [
        0.0,
        50.0
      ],
      "target_reach_fraction": [
        0.5,
        1.0
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Robot kinematics",
  "title": "Inverse Kinematics"
}

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