{
  "method": {
    "name": "Polynomial Representations for Interaction-Structured Motor Control",
    "construction": "u = W1 x + b1; v = W2 x + b2; psi2(x) = u + s2 (u elementwise-multiply v)",
    "default_degree": 2,
    "integration": "Replaces the first affine state/proprioceptive projection and retains downstream policy interfaces, native learning objectives, sensing, action interface, and low-level controller.",
    "modalities": "Locomotion: available proprioception, commands, observation/action history. Manipulation: proprioceptive pathway; RGB/language use existing encoders.",
    "message": "Learn physical interactions from existing observations, rather than prescribing physical equations or adding force inputs.",
    "scale_note": "Interaction scales are learned jointly by default; setting all scales to zero recovers an affine representation. Specific robustness training recipe uses a scheduled warm-up and fixed scale, so avoid an unconditional learned-scale claim across every experiment."
  },
  "figure2_summary": [
    {
      "label": "Walking",
      "metric": "24-second survival (%)",
      "baseline": 51.0,
      "prism": 92.5,
      "gain_percentage_points": 41.5,
      "source": "Table 1 / Figure 2",
      "baseline_definition": "MLP baseline in Humanoid-Gym",
      "protocol": "Five training seeds"
    },
    {
      "label": "Manipulation",
      "metric": "Mean task success (%)",
      "baseline": 63.5,
      "prism": 67.6,
      "gain_percentage_points": 4.1,
      "source": "Table 2 / Figure 2",
      "baseline_definition": "ACT across three ManiSkill3 tasks",
      "protocol": "One training seed per method/task, final 100K checkpoint, 500 attempts per task/method; five equal-size evaluation groups"
    },
    {
      "label": "Multi-task",
      "metric": "LIBERO mean success (%)",
      "baseline": 62.0,
      "prism": 63.82,
      "display_prism": 63.8,
      "gain_percentage_points": 1.82,
      "source": "Table 4 / Figure 2",
      "baseline_definition": "SmolVLA",
      "protocol": "Three training seeds, aligned 80K checkpoints, 40 tasks, 50 rollouts/task (2000 per checkpoint)"
    },
    {
      "label": "Unseen dynamics: flat",
      "metric": "Mean 60-second survival (%) over six out-of-range settings",
      "baseline": 78.3,
      "prism": 92.3,
      "gain_percentage_points": 14.0,
      "source": "Section 4.2 / Figure 2",
      "baseline_definition": "Parameter-matched MLP in Humanoid-Gym",
      "protocol": "Fixed trained checkpoints without retraining; six settings across friction, robot mass/inertia, and payload outside the respective training support"
    },
    {
      "label": "Unseen dynamics: rough",
      "metric": "Mean 20-second supported survival (%) over six out-of-range settings",
      "baseline": 80.0,
      "prism": 87.1,
      "gain_percentage_points": 7.1,
      "source": "Section 4.2 / Figure 2",
      "baseline_definition": "Standard MLP in mjlab on random rough terrain",
      "protocol": "Fixed trained checkpoints without retraining; six settings across friction, robot mass/inertia, and payload outside the respective training support"
    },
    {
      "label": "Drop recovery",
      "metric": "Recovery success (%)",
      "baseline": 40.9,
      "prism": 52.3,
      "gain_percentage_points": 11.4,
      "baseline_count": 18,
      "prism_count": 23,
      "attempts_per_method": 44,
      "source": "Table 6 / Figure 2",
      "baseline_definition": "ACT",
      "protocol": "Same 44 validated induced-drop states; final 100K checkpoints; five-second horizon; failures included"
    },
    {
      "label": "Real robot: tape pickup",
      "metric": "Hardware task success (%)",
      "baseline": 68.0,
      "prism": 94.0,
      "gain_percentage_points": 26.0,
      "baseline_count": 34,
      "prism_count": 47,
      "attempts_per_method": 50,
      "source": "Table 7 / Figure 2",
      "baseline_definition": "ACT on physical SO-101",
      "protocol": "Same 50 demonstrations; 50 hardware rollouts/method; lift tape roll at least 5cm and maintain that height for 3s"
    }
  ],
  "table1_locomotion": {
    "source": "Table 1",
    "uncertainty": "Mean and standard deviation across five training seeds",
    "survival_horizon_s": 24,
    "columns": [
      "method",
      "reported_params_M",
      "episode_length_mean",
      "episode_length_sd",
      "linear_error_mean",
      "linear_error_sd",
      "yaw_error_mean",
      "yaw_error_sd",
      "survival_percent_mean",
      "survival_percent_sd"
    ],
    "rows": [
      [
        "MLP baseline",
        0.926,
        1340.8,
        39,
        0.4607,
        0.011,
        0.2817,
        0.008,
        51.0,
        3.7
      ],
      [
        "Larger MLP",
        1.321,
        1349.9,
        29,
        0.4738,
        0.028,
        0.3086,
        0.01,
        52.25,
        2.9
      ],
      [
        "PRISM",
        1.321,
        2233.4,
        23,
        0.2099,
        0.008,
        0.1221,
        0.005,
        92.5,
        2.4
      ]
    ],
    "parameter_note": "Main-table counts must not be called actor-only. Appendix Table 12 actor-only counts are 527K, 922K, 922K. Larger MLP and PRISM match at reported precision."
  },
  "table2_manipulation": {
    "source": "Table 2 / Appendix A.2",
    "uncertainty": "Mean and sample SD across five equal-size evaluation groups of 100 attempts for one trained checkpoint; not training-seed variability",
    "attempts_per_task_per_method": 500,
    "horizon_control_steps": 100,
    "horizon_s": 5,
    "columns": [
      "method",
      "StackCube_mean",
      "StackCube_sd",
      "TwoRobotStackCube_mean",
      "TwoRobotStackCube_sd",
      "G1TransportBox_mean",
      "G1TransportBox_sd",
      "average"
    ],
    "rows": [
      [
        "ACT",
        58.8,
        6.3,
        47.0,
        3.2,
        84.8,
        4.9,
        63.5
      ],
      [
        "PRISM",
        68.8,
        5.8,
        48.2,
        2.8,
        85.8,
        5.1,
        67.6
      ]
    ],
    "success_definition": "Native info.success at any executed tick; all scheduled attempts, including runtime errors, remain in denominator"
  },
  "table3_bfm_zero": {
    "source": "Table 3",
    "protocol": "Reference-motion tracking, aligned checkpoints after 9.6M environment steps, one training seed",
    "columns": [
      "method",
      "params_M",
      "nominal_tracking_EMD"
    ],
    "rows": [
      [
        "Baseline",
        32.109,
        1.104
      ],
      [
        "Larger model",
        49.26,
        1.09
      ],
      [
        "PRISM",
        32.665,
        1.05
      ]
    ]
  },
  "table4_smolvla": {
    "source": "Table 4",
    "protocol": "LIBERO, aligned 80K-update checkpoints, mean and SD over three training seeds",
    "columns": [
      "method",
      "params_M",
      "success_mean",
      "success_sd",
      "gain_percentage_points"
    ],
    "rows": [
      [
        "Baseline",
        450.046,
        62.0,
        1.67,
        0.0
      ],
      [
        "Larger model",
        456.245,
        62.57,
        3.14,
        0.57
      ],
      [
        "PRISM",
        451.925,
        63.82,
        2.59,
        1.82
      ]
    ]
  },
  "table5_bfm_dynamics": {
    "source": "Table 5 / Appendix B.2",
    "protocol": "Fixed checkpoints; friction set to0.20 or link masses scaled1.15; no retraining",
    "columns": [
      "method",
      "nominal_EMD",
      "low_friction_EMD",
      "increased_mass_EMD",
      "mean_EMD"
    ],
    "rows": [
      [
        "Baseline",
        1.104,
        1.582,
        1.121,
        1.269
      ],
      [
        "Larger model",
        1.09,
        1.589,
        1.114,
        1.264
      ],
      [
        "PRISM",
        1.05,
        1.548,
        1.073,
        1.224
      ]
    ]
  },
  "table6_recovery": {
    "source": "Table 6 / Appendix B.3",
    "attempts_per_method": 44,
    "horizon_s": 5,
    "columns": [
      "method",
      "recovered_count",
      "recovery_percent",
      "capped_recovery_time_s",
      "terminal_placement_error_mm"
    ],
    "rows": [
      [
        "ACT",
        18,
        40.9,
        3.38,
        141.7
      ],
      [
        "ACT + MCC-Sensorless",
        6,
        13.6,
        4.5,
        123.5
      ],
      [
        "ACT + MCC-Oracle",
        15,
        34.1,
        3.71,
        105.4
      ],
      [
        "PRISM",
        23,
        52.3,
        3.11,
        96.2
      ]
    ],
    "diagnostic_definition": "Descriptive means across all44attempts, including failures; capped time assigns5s to unsuccessful attempts; placement uses common stacking goal",
    "mcc_definition": "Simulation adaptations: sensorless corrects translation commands using a force proxy; oracle execution wrapper uses simulator contact force",
    "regrasp_percent": {
      "ACT": 68.2,
      "PRISM": 75.0
    },
    "scope": "Controlled-drop pilot distinct from ordinary task success and official CroSTAta recovery evaluation; first44valid states from fixed candidate order, not selected by policy outcome",
    "removed_metrics": [
      "Peak contact force",
      "SD across44states",
      "Smoothness",
      "Position/orientation tracking errors from nominal experiments"
    ]
  },
  "table7_hardware": {
    "source": "Table 7",
    "demonstrations": 50,
    "attempts_per_method": 50,
    "columns": [
      "method",
      "success_count",
      "early_joint_path_degrees_mean",
      "early_joint_path_sd",
      "command_state_error_degrees_mean",
      "command_state_error_sd"
    ],
    "rows": [
      [
        "ACT",
        34,
        31.1,
        2.9,
        27.4,
        2.9
      ],
      [
        "PRISM",
        47,
        38.5,
        1.5,
        24.3,
        4.2
      ]
    ],
    "path_window_s": 3
  },
  "table8_probes": {
    "source": "Table 8",
    "protocol": "Frozen representations, linear probes, held-out rollout windows, horizon5; simulator targets for analysis only",
    "columns": [
      "target",
      "baseline_MSE",
      "PRISM_MSE",
      "baseline_PCC",
      "PRISM_PCC"
    ],
    "rows": [
      [
        "Slip velocity",
        0.711,
        0.7,
        0.501,
        0.549
      ],
      [
        "Joint power",
        0.328,
        0.282,
        0.82,
        0.848
      ],
      [
        "Contact impulse",
        0.301,
        0.241,
        0.829,
        0.861
      ],
      [
        "Contact work proxy",
        0.362,
        0.319,
        0.755,
        0.78
      ]
    ],
    "interpretation": "Better linear predictability; probes alone do not establish causal use of the response variables"
  },
  "figure4_nominal_distinctions": {
    "flat": {
      "baseline": 84.3,
      "prism": 96.6,
      "horizon_s": 60
    },
    "rough": {
      "baseline": 90.1,
      "prism": 97.1,
      "horizon_s": 20
    },
    "note": "These nominal values are distinct from six-setting out-of-range averages and24-secondTable1walking. Archived98/82/61roughnumbers are not currentFigure4values."
  },
  "source": {
    "title": "PRISM: Polynomial Representations for Interaction-Structured Motor Control",
    "authors": [
      "SeungHyun Lee",
      "Stella X. Yu"
    ],
    "manuscript_date": "2026-10-13",
    "source_pdf_sha256": "e40ba9ddf644c1b8205efa59581ac2a9bd8b4e9296b071d1b043a2826ac2dd32"
  }
}