L1 ⚛ L1-446 ⊙ Testnet

Forward Kinematics

Robotics · Robot kinematics

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⬇ L1-446.md

# ⚛ 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._

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Read the attached Markdown (L1-446.md).
Regenerate L1-446.json so every field matches the Markdown.
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L2-446-001 Forward Kinematics — Nominal + Mismatch Specs

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📋 JSON metadata

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  "artifact_id": "L1-446",
  "chain_block": 41555260,
  "chain_hash": "0x6a143802bc19e6738567859c81ccedaa775f91be5418044bb020e4d7f7216b05",
  "chain_tx_hash": "0x47849de07531a7753e76a95981ea254becf573ec076d26599a7d730889652691",
  "domain": "Robotics",
  "hardness_fn": {
    "delta": 3,
    "kappa": 500,
    "metric": "end_effector_position_RMSE_mm",
    "type": "epsilon_fn"
  },
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      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
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  "layer": "L1",
  "observable_profile": {
    "metric": "end_effector_position_RMSE_mm",
    "regime": "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.",
    "secondary": "orientation_error_deg"
  },
  "physics_fingerprint": {
    "L_DAG": 2.0,
    "carrier": "N/A",
    "difficulty_delta": 3,
    "domain": "Robotics",
    "integration_axis": "kinematic_chain",
    "noise_model": "gaussian",
    "primitives": [
      "E.eigensolve",
      "S.chain.product",
      "O.pose.cartesian"
    ],
    "problem_class": "parameter_estimation",
    "sensing_mechanism": "joint_encoder_to_cartesian",
    "solution_space": "cartesian_pose_vector",
    "sub_domain": "Robot kinematics",
    "title": "Forward Kinematics"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "joint_encoder_to_cartesian"
    ],
    "allowed_omega_dimensions": [
      "N_joints",
      "calibration_error_mm",
      "encoder_noise_deg",
      "load_kg"
    ],
    "allowed_problem_classes": [
      "parameter_estimation"
    ],
    "center_spec": {
      "epsilon_fn_center": "1.0 end_effector_position_RMSE_mm",
      "forward_operator": "joint_encoder_to_cartesian",
      "input_format": "measurement_only",
      "omega": {
        "N_joints": 6,
        "calibration_error_mm": 0.5,
        "encoder_noise_deg": 0.01,
        "load_kg": 5
      },
      "problem_class": "parameter_estimation"
    },
    "epsilon_bounds": {
      "end_effector_position_RMSE_mm": [
        0.1,
        20.0
      ]
    },
    "omega_bounds": {
      "N_joints": [
        2,
        10
      ],
      "calibration_error_mm": [
        0.0,
        5.0
      ],
      "encoder_noise_deg": [
        0.001,
        0.1
      ],
      "load_kg": [
        0,
        100
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Robot kinematics",
  "title": "Forward Kinematics"
}

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