L1 ⚛ L1-449 ⊙ Testnet

Inverse Dynamics

Robotics · Robot dynamics

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⬇ L1-449.md

# ⚛ L1 Principle — Inverse Dynamics **ID:** `L1-449` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Robotics — *Robot dynamics* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** joint torque vector > **📡 Carrier:** N/A · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555277 --- ## 🧠 1. Introduction **Inverse Dynamics** is a **linear inverse problem** whose unknown lives in **joint torque vector** space, within the **Robot dynamics** sub-domain of **Robotics**. Measurements consist of N/A via a **trajectory to torque computation** sensing mechanism. The forward operator applies, in order: time evolution of the state; S · regressor · inertia parameters operator; O · least squares · param id operator. Observations are corrupted by additive Gaussian noise. Existence of the recovered joint_torque_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 ~= 50); acceleration_estimation_noise 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** → Time derivative → S · regressor · inertia parameters → O · least squares · param id → **y** (detector). ``` y = `O.least_squares.param_id` `S.regressor.inertia_parameters` ∂_t x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `D.time` | Time evolution of the state | | `S.regressor.inertia_parameters` | S · regressor · inertia parameters operator | | `O.least_squares.param_id` | O · least squares · param id operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Robotics | | Sub domain | Robot dynamics | | Carrier | N/A | | Problem class | linear_inverse | | Solution space | joint_torque_vector | | Noise model | gaussian | | Integration axis | kinematic_chain | | Difficulty delta | 3 | | L dag | 2.8 | ## 📡 4. Measurement Model Existence of the recovered joint_torque_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 ~= 50); acceleration_estimation_noise dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor. | Metric | Value | |---|---| | Metric | torque_prediction_RMSE_Nm | | Secondary | parameter_identification_RMSE | ## 📏 5. Operating Range (Ω) **Center problem class:** `linear_inverse` · **Forward operator:** `trajectory_to_torque_computation` **Center point:** | Parameter | Unit | Value | |---|---|---| | N joints | — | 6 | | N trajectory samples | — | 1000 | | Acceleration noise db | dB | -20 | | Excitation bandwidth hz | Hz | 2 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | N joints | — | 2 – 10 | | N trajectory samples | — | 100 – 10000 | | Acceleration noise db | dB | -40 – -10 | | Excitation bandwidth hz | Hz | 0.5 – 10.0 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 2.0 torque_prediction_RMSE_Nm | Metric | Range | |---|---| | Torque prediction rmse nm | 0.1 – 20.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **torque_prediction_RMSE_Nm**, with κ = `1000` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x05dca4b066ca2181478e5dde661fff6d429d658391f8d205a7ccbd4316bcd3d8` - **Chain tx hash:** `0x1e99291f16a45bd235ea8ad95190dcb336d08df6075a294179eb3a1572cc29ae` - **Chain block:** `41555277` --- ## File Mapping This bundle consists of: `L1-449.md`, `L1-449.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-449.md` | Source of truth — edit this | Human or LLM | | `L1-449.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 `L1-449`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-449.md).
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L2-449-001 Inverse Dynamics — Nominal + Mismatch Specs

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📋 JSON metadata

{
  "artifact_id": "L1-449",
  "chain_block": 41555277,
  "chain_hash": "0x05dca4b066ca2181478e5dde661fff6d429d658391f8d205a7ccbd4316bcd3d8",
  "chain_tx_hash": "0x1e99291f16a45bd235ea8ad95190dcb336d08df6075a294179eb3a1572cc29ae",
  "domain": "Robotics",
  "hardness_fn": {
    "delta": 3,
    "kappa": 1000,
    "metric": "torque_prediction_RMSE_Nm",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "torque_prediction_RMSE_Nm",
    "regime": "Existence of the recovered joint_torque_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 ~= 50); acceleration_estimation_noise dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.",
    "secondary": "parameter_identification_RMSE"
  },
  "physics_fingerprint": {
    "L_DAG": 2.8,
    "carrier": "N/A",
    "difficulty_delta": 3,
    "domain": "Robotics",
    "integration_axis": "kinematic_chain",
    "noise_model": "gaussian",
    "primitives": [
      "D.time",
      "S.regressor.inertia_parameters",
      "O.least_squares.param_id"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "trajectory_to_torque_computation",
    "solution_space": "joint_torque_vector",
    "sub_domain": "Robot dynamics",
    "title": "Inverse Dynamics"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "trajectory_to_torque_computation"
    ],
    "allowed_omega_dimensions": [
      "N_joints",
      "excitation_bandwidth_Hz",
      "acceleration_noise_dB",
      "N_trajectory_samples"
    ],
    "allowed_problem_classes": [
      "linear_inverse"
    ],
    "center_spec": {
      "epsilon_fn_center": "2.0 torque_prediction_RMSE_Nm",
      "forward_operator": "trajectory_to_torque_computation",
      "input_format": "measurement_only",
      "omega": {
        "N_joints": 6,
        "N_trajectory_samples": 1000,
        "acceleration_noise_dB": -20,
        "excitation_bandwidth_Hz": 2.0
      },
      "problem_class": "linear_inverse"
    },
    "epsilon_bounds": {
      "torque_prediction_RMSE_Nm": [
        0.1,
        20.0
      ]
    },
    "omega_bounds": {
      "N_joints": [
        2,
        10
      ],
      "N_trajectory_samples": [
        100,
        10000
      ],
      "acceleration_noise_dB": [
        -40,
        -10
      ],
      "excitation_bandwidth_Hz": [
        0.5,
        10.0
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Robot dynamics",
  "title": "Inverse Dynamics"
}

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