{
  "paper_id": "robotic-world-model",
  "paper_title": "Robotic World Model: A Neural Network Simulator for Robust Policy Optimization",
  "D1": [
    {
      "id": "robotic-world-model-D1-001",
      "claim": "RWM architecture: base network is a 2-layer GRU with hidden sizes [256,256]; prediction heads are MLPs with single hidden layer of 128 units and ReLU activation, outputting Gaussian mean and standard deviation for next-observation distribution and privileged-information predictions",
      "source": "Appendix A.2.1, Table S7"
    },
    {
      "id": "robotic-world-model-D1-002",
      "claim": "RWM training hyperparameters: history horizon M=32, forecast horizon N=8, forecast decay alpha=1.0, control frequency delta_t=0.02s (50Hz); optimizer AdamW with learning rate=0.0001 and weight_decay=1e-05; batch size=1024, max training iterations=2500, random seeds=5",
      "source": "Section 4.1, Appendix A.3.1, Table S10"
    },
    {
      "id": "robotic-world-model-D1-003",
      "claim": "Training infrastructure: single NVIDIA RTX 4090 GPU with PyTorch 2.4.0 and CUDA 12.6; approximate RWM world model training time is 1 hour per random seed",
      "source": "Appendix A.3, Appendix A.3.1, Table S10"
    },
    {
      "id": "robotic-world-model-D1-004",
      "claim": "RWM pretraining configuration: 6M state transitions from simulation data induced by suboptimal policies under varied dynamics; 50 minutes pretraining on RTX 4090; followed by MBPO-PPO policy training (5 min), total 55 min for RWM+MBPO-PPO pipeline",
      "source": "Section 5, Table 1"
    },
    {
      "id": "robotic-world-model-D1-005",
      "claim": "ANYmal D (12 DOF) world model spaces: observation 45 dims (base linear vel[0:3], angular vel[3:6], projected gravity[6:9], joint pos[9:21], joint vel[21:33], joint torques[33:45]); action 12 dims (joint position targets); privileged info 8 dims (knee contact[0:4], foot contact[4:8])",
      "source": "Appendix A.1.1, Tables S2-S4"
    },
    {
      "id": "robotic-world-model-D1-006",
      "claim": "ANYmal D policy observation space: 48 dims — base linear vel[0:3], angular vel[3:6], projected gravity[6:9], velocity command[9:12], joint positions[12:24], joint velocities[24:36], last actions[36:48]",
      "source": "Appendix A.1.1, Table S5"
    },
    {
      "id": "robotic-world-model-D1-007",
      "claim": "Unitree G1 (29 DOF) world model spaces: observation 96 dims (base linear vel[0:3], angular vel[3:6], projected gravity[6:9], joint pos[9:38], joint vel[38:67], joint torques[67:96]); action 29 dims (joint position targets); privileged info 30 dims (body contact[0:26], foot height[26:28], foot velocity[28:30])",
      "source": "Appendix A.1.1, Tables S2-S4"
    },
    {
      "id": "robotic-world-model-D1-008",
      "claim": "Unitree G1 policy observation space: 99 dims — base linear vel[0:3], angular vel[3:6], projected gravity[6:9], velocity command[9:12], joint positions[12:41], joint velocities[41:70], last actions[70:99]",
      "source": "Appendix A.1.1, Table S5"
    },
    {
      "id": "robotic-world-model-D1-009",
      "claim": "MBPO-PPO configuration: 4096 parallel imagination environments, 100 imagination steps per iteration, replay buffer size |D|=1000, max training iterations=2500, step time delta_t=0.02s (50Hz), random seeds=5",
      "source": "Section 3.3, Appendix A.3.2, Table S11"
    },
    {
      "id": "robotic-world-model-D1-010",
      "claim": "PPO hyperparameters used in MBPO-PPO: learning rate=0.001, no weight decay (weight_decay=0.0), 5 learning epochs per update, 4 mini-batches, KL divergence target=0.01, discount factor gamma=0.99, clip range epsilon=0.2, entropy coefficient=0.005",
      "source": "Appendix A.3.2, Table S11"
    },
    {
      "id": "robotic-world-model-D1-011",
      "claim": "Velocity tracking reward temperature factors: sigma_v_xy=0.25 for linear velocity tracking reward, sigma_omega_z=0.25 for angular velocity tracking reward; these scale parameters control the width of the exponential reward kernel",
      "source": "Appendix A.1.2"
    },
    {
      "id": "robotic-world-model-D1-012",
      "claim": "ANYmal D reward weights (Table S6): w_v_xy=1.0, w_omega_z=0.5, w_v_z=-2.0, w_omega_xy=-0.05, w_q_tau=-2.5e-5, w_ddq=-2.5e-7, w_da=-0.01, w_f_a=0.5, w_c=-1.0, w_g=-5.0, w_f_c=0.0, w_q_d=0.0",
      "source": "Appendix A.1.2, Table S6"
    },
    {
      "id": "robotic-world-model-D1-013",
      "claim": "Unitree G1 reward weights (Table S6): w_v_xy=1.0, w_omega_z=0.5, w_v_z=-2.0, w_omega_xy=-0.05, w_q_tau=-2.5e-5, w_ddq=-2.5e-7, w_da=-0.05, w_f_a=0.0, w_c=-1.0, w_g=-5.0, w_f_c=1.0, w_q_d=-1.0",
      "source": "Appendix A.1.2, Table S6"
    },
    {
      "id": "robotic-world-model-D1-014",
      "claim": "MLP baseline architecture: 2 hidden layers of 256 units each with ReLU activation; trained autoregressively with history horizon M=32 and forecast horizon N=8",
      "source": "Appendix A.2.2, Table S8"
    },
    {
      "id": "robotic-world-model-D1-015",
      "claim": "RSSM baseline architecture: GRU with hidden size 256, 2 layers, latent dimension 64, 32 categorical latent classes, with prior network; trained with teacher forcing by default",
      "source": "Appendix A.2.2, Table S8"
    },
    {
      "id": "robotic-world-model-D1-016",
      "claim": "Transformer baseline architecture: decoder-only, dimension 64, 8 attention heads, 2 layers, context length 32, sinusoidal positional encodings",
      "source": "Appendix A.2.2, Table S8"
    },
    {
      "id": "robotic-world-model-D1-017",
      "claim": "MBPO-PPO policy and value function networks: both use MLP with 3 hidden layers of 128 units each and ELU activation",
      "source": "Appendix A.2.3, Table S9"
    },
    {
      "id": "robotic-world-model-D1-018",
      "claim": "Table 1 computational comparison (reward values excluded per HARD EXCLUDE rule): Model-free PPO baseline on high-fidelity simulator uses 250M state transitions, 10 min training on RTX 4090; MBPO-PPO step inference time 1 ms; RWM+MBPO-PPO total 55 min (50 min pretraining + 5 min policy training)",
      "source": "Section 5, Table 1"
    },
    {
      "id": "robotic-world-model-D1-019",
      "claim": "Robot collision and failure count during online learning on hardware: more than 20 failures on average per online learning run, motivating simulation-based training to avoid hardware damage",
      "source": "Appendix A.4.4"
    },
    {
      "id": "robotic-world-model-D1-020",
      "claim": "Robot platforms: ANYmal D quadruped with 12 degrees of freedom; Unitree G1 humanoid with 29 degrees of freedom; both operating at 50 Hz control frequency (delta_t=0.02s)",
      "source": "Section 4, Section 4.1"
    }
  ],
  "D2": [
    {
      "id": "robotic-world-model-D2-001",
      "claim": "Autoregressive Observation Prediction (Eq 1): o'_{t+k} ~ p_phi(· | o_{t-M+k:t}, o'_{t+1:t+k-1}, a_{t-M+k:t+k-1}). At each step k of the N-step forecast, the model samples the next observation from a learned Gaussian conditioned on M historical obs-action pairs and its own k-1 prior predictions.",
      "source": "Section 3.2, Eq 1"
    },
    {
      "id": "robotic-world-model-D2-002",
      "claim": "Multi-Step Autoregressive Training Loss (Eq 2): L = (1/N) sum_{k=1}^{N} alpha^k [L_o(o'_{t+k}, o_{t+k}) + L_c(c'_{t+k}, c_{t+k})]. The model minimizes discounted observation and privileged-info prediction errors over N forecast steps; alpha^k decays later steps, L_o and L_c measure prediction discrepancy.",
      "source": "Section 3.2, Eq 2"
    },
    {
      "id": "robotic-world-model-D2-003",
      "claim": "Imagination Action Generation (Eq 3): a'_{t+k} ~ pi_theta(· | o'_{t+k}), where o'_{t+k} is predicted autoregressively via Eq 1. During MBPO-PPO imagination rollouts, the policy generates actions recursively conditioned on the world model's predicted observations.",
      "source": "Section 3.3, Eq 3"
    },
    {
      "id": "robotic-world-model-D2-004",
      "claim": "MBPO-PPO Policy Optimization (Algorithm 1): Iterative MBRL loop: (1) collect real data with current policy, (2) train world model autoregressively on replay buffer, (3) initialize imagination agents from real observations, (4) roll out imagined trajectories using world model, (5) update policy with PPO on imagined data",
      "source": "Section 3.3, Algorithm 1"
    },
    {
      "id": "robotic-world-model-D2-005",
      "claim": "RWM Dual-Autoregressive GRU Architecture: Two-level autoregressive mechanism: (1) Inner AR: sequentially processes each of the M historical observation-action pairs through GRU, updating hidden state autoregressively; (2) Outer AR: feeds N predicted future observations back as inputs for subsequent forecast steps. GRU hidden shape (256, 256) with ReLU MLP heads (128) predicting Gaussian mean/std for observations and privileged info.",
      "source": "Section 3.2, Appendix A.2.1, Table S7, Figure S6"
    },
    {
      "id": "robotic-world-model-D2-006",
      "claim": "Training Data Construction: Sliding Window: For trajectory D = {(o_t, a_t, c_t)}_{t=1}^{T}, construct windows W_j = (X_j, Y_j) for j in [0, T-(M+N)), where context X_j = (o_{j:j+M-1}, a_{j:j+M-1}) provides M historical obs-action pairs, and targets Y_j = (o_{j+M:j+M+N-1}, c_{j+M:j+M+N-1}) for N-step prediction. Autoregressive training per Eq.(1): o'_{t+k} = p_phi(·|o_{t-M+k:t}, o'_{t+1:t+k-1}, a_{t-M+k:t+k-1}), optimized via Eq.(2): L = (1/N) Σ_{k=1}^{N} α^k [L_o(o'_{t+k}, o_{t+k}) + L_c(c'_{t+k}, c_{t+k})], with default α=1.0, M=32, N=8.",
      "source": "Section 3.2, Eq.(1)-(2)"
    },
    {
      "id": "robotic-world-model-D2-007",
      "claim": "Linear Velocity Tracking Reward (xy-plane): r_{v_xy} = w_{v_xy} exp(-||c_xy - v_xy||^2_2 / sigma_{v_xy}^2), sigma=0.25. Exponential reward for matching commanded base linear velocity in xy-plane via L2 distance scaled by temperature factor.",
      "source": "Appendix A.1.2"
    },
    {
      "id": "robotic-world-model-D2-008",
      "claim": "Angular Velocity Tracking Reward (yaw/z-axis): r_{omega_z} = w_{omega_z} exp(-||c_z - omega_z||^2_2 / sigma_{omega_z}^2), sigma=0.25. Exponential reward matching commanded yaw angular velocity via L2 distance scaled by temperature.",
      "source": "Appendix A.1.2"
    },
    {
      "id": "robotic-world-model-D2-009",
      "claim": "Linear Velocity z Penalty: r_{v_z} = w_{v_z} v_z^2, where v_z denotes base vertical velocity. Quadratic penalty on base vertical velocity to discourage jumping and bouncing.",
      "source": "Appendix A.1.2"
    },
    {
      "id": "robotic-world-model-D2-010",
      "claim": "Angular Velocity xy Penalty: r_{omega_xy} = w_{omega_xy} ||omega_xy||^2_2, where omega_xy denotes base roll and pitch velocity. Quadratic penalty on roll/pitch angular velocity to maintain stable base orientation.",
      "source": "Appendix A.1.2"
    },
    {
      "id": "robotic-world-model-D2-011",
      "claim": "Joint Torque Penalty: r_{q_tau} = w_{q_tau} ||tau||^2_2, where tau denotes joint torques. Quadratic penalty on joint torques to encourage energy-efficient motions.",
      "source": "Appendix A.1.2"
    },
    {
      "id": "robotic-world-model-D2-012",
      "claim": "Joint Acceleration Penalty: r_{ddq} = w_{ddq} ||ddq||^2_2, where ddq denotes joint acceleration. Quadratic penalty on joint accelerations to encourage smooth motions.",
      "source": "Appendix A.1.2"
    },
    {
      "id": "robotic-world-model-D2-013",
      "claim": "Action Rate Penalty: r_{da} = w_{da} ||a' - a||^2_2, where a' and a denote previous and current actions. Quadratic penalty on L2 distance between consecutive actions for smooth control signals.",
      "source": "Appendix A.1.2"
    },
    {
      "id": "robotic-world-model-D2-014",
      "claim": "Feet Air Time Bonus: r_{f_a} = w_{f_a} t_{f_a}, where t_{f_a} is the total time feet spend in the air per step. Bonus proportional to air time, encouraging dynamic gaits with flight phases.",
      "source": "Appendix A.1.2"
    },
    {
      "id": "robotic-world-model-D2-015",
      "claim": "Undesired Contacts Penalty: r_c = w_c c_u, where c_u denotes the count of undesired contacts (e.g., knee contacts, body-ground contacts). Penalty proportional to the count of undesired body/limb contacts.",
      "source": "Appendix A.1.2"
    },
    {
      "id": "robotic-world-model-D2-016",
      "claim": "Flat Orientation Reward: r_g = w_g g_{xy}^2, where g_{xy} denotes xy-components of projected gravity. Quadratic penalty on projected gravity xy to keep the robot base level (upright).",
      "source": "Appendix A.1.2"
    },
    {
      "id": "robotic-world-model-D2-017",
      "claim": "Foot Clearance Reward: r_{f_c} = w_{f_c} h_{f_c}, where h_{f_c} denotes the clearance height of swing feet above ground. Bonus proportional to swing foot clearance height.",
      "source": "Appendix A.1.2"
    },
    {
      "id": "robotic-world-model-D2-018",
      "claim": "Joint Deviation Penalty: r_{q_d} = w_{q_d} ||q - q_0||_1, where q_0 denotes default joint position. L1 penalty on deviation from default pose to maintain natural joint configuration.",
      "source": "Appendix A.1.2"
    },
    {
      "id": "robotic-world-model-D2-019",
      "claim": "Total Reward Aggregation: r = r_{v_xy} + r_{omega_z} + r_{v_z} + r_{omega_xy} + r_{q_tau} + r_{ddq} + r_{da} + r_{f_a} + r_c + r_g + r_{f_c} + r_{q_d}. Total per-step reward sums all individual terms with respective weights (Table S6); weights differ between ANYmal D and Unitree G1.",
      "source": "Appendix A.1.2, Table S6"
    },
    {
      "id": "robotic-world-model-D2-020",
      "claim": "Termination Prediction via Privileged Information Head: RWM predicts privileged information (including contacts) that encodes base-ground contact events; MBPO-PPO treats predicted base-ground contacts as episode termination signals during imagination, affecting PPO's GAE return computation and value function targets",
      "source": "Section 3.3, Appendix A.4.3"
    },
    {
      "id": "robotic-world-model-D2-021",
      "claim": "MBPO-PPO PPO Update on Imagined Data: PPO clipped surrogate objective: L^CLIP(theta) = E[min(r_t * A_t, clip(r_t, 1-epsilon, 1+epsilon) * A_t)] where r_t = pi_theta(a_t|s_t) / pi_old(a_t|s_t), applied to imagination rollout data (Schulman et al., 2017, 'Proximal Policy Optimization Algorithms'); policy and value networks are MLPs (128,128,128) with ELU activation; PPO uses clipping, entropy bonus, and KL divergence target",
      "source": "Section 3.3, Appendix A.2.3, Appendix A.3.2, Table S11"
    }
  ],
  "D3": [
    {
      "id": "robotic-world-model-D3-001",
      "claim": "Autoregressive Trajectory Prediction Evaluation: Collect trajectories from ANYmal D hardware under velocity-tracking policy (50 Hz control). Train RWM with history horizon M=32, forecast horizon N=8, architecture Table S7, training params Table S10. Evaluate autoregressive prediction fidelity: compare predicted vs ground-truth state trajectories over extended rollouts beyond training forecast horizon. Visualize alignment for all observed variables (Fig 3a, Fig 1 bottom).",
      "source": "Section 4.1, Appendix A.2.1, Appendix A.3.1"
    },
    {
      "id": "robotic-world-model-D3-002",
      "claim": "Noise Robustness Evaluation: Apply Gaussian noise perturbations to BOTH observations and actions during autoregressive rollouts. Compare RWM vs MLP baseline (both trained autoregressively with same M=32, N=8). Measure relative prediction error e across increasing forecast steps at multiple noise levels. Yellow=RWM, grey=MLP (Fig 3b). Purpose: test model stability when predictions deviate from training distribution.",
      "source": "Section 4.2"
    },
    {
      "id": "robotic-world-model-D3-003",
      "claim": "Generality Across Robotic Environments: Compare RWM-AR, RWM-TF, MLP, RSSM (GRU-based latent dynamics), and Transformer (decoder-only) across diverse robotic tasks: manipulation, quadruped locomotion, and humanoid locomotion. All models receive same context; baselines trained with teacher forcing (default implementation); RWM-TF serves as ablation. Measure relative autoregressive prediction error e (Fig 4). Training params per Table S8 and A.2.2.",
      "source": "Section 4.3, Appendix A.2.2, Table S8"
    },
    {
      "id": "robotic-world-model-D3-004",
      "claim": "MBPO-PPO Policy Learning and Hardware Transfer: Train goal-conditioned velocity tracking policy for ANYmal D and Unitree G1 using MBPO-PPO in RWM imagination (Table S11 params). Compare against SHAC (first-order gradient through world model) and DreamerV3 (latent-space actor-critic). Monitor model error e and policy reward r during training (Fig 5). Zero-shot deploy learned policy on ANYmal D and Unitree G1 hardware; evaluate robustness under external disturbances (impacts, terrain). Policy obs/action: A.1.1, architecture: Table S9, rewards: A.1.2.",
      "source": "Section 4.4, Appendix A.1.1, A.1.2, A.3.2, A.4.3"
    },
    {
      "id": "robotic-world-model-D3-005",
      "claim": "Dual-Autoregressive Mechanism Ablation: Sweep combinations of history horizon M and forecast horizon N. Measure relative autoregressive prediction error e (left heatmap, Fig S8). Measure training time (right heatmap, Fig S8). Include N=1 (teacher-forcing) as baseline for both metrics. Identify optimal trade-off: moderate M and N balance accuracy vs computational cost. Key finding: M=32, N=8 achieves strong performance with manageable training time.",
      "source": "Appendix A.4.1, Figure S8"
    },
    {
      "id": "robotic-world-model-D3-006",
      "claim": "RWM World Model Standalone Training Protocol: Construct training data by sliding window of size M+N over collected trajectories. Input: M historical (obs, action) pairs. Predict N future steps autoregressively via Eq 1. Loss: discounted sum of observation error + privileged-info error over N steps (Eq 2) with decay alpha=1.0. Apply reparameterization for end-to-end gradient propagation through autoregressive predictions. Optimizer: Adam with lr=1e-4, weight decay=1e-5, batch size 1024, max 2500 iterations, 5 seeds. Hardware: NVIDIA RTX 4090, PyTorch 2.4.0, CUDA 12.6. Approx 1 hour per seed.",
      "source": "Section 3.2, Appendix A.3, A.3.1, Table S10"
    },
    {
      "id": "robotic-world-model-D3-007",
      "claim": "RWM Pretraining and Online Fine-Tuning Pipeline: Phase 1 (Pretraining): Pretrain RWM on 6M state transitions from simulation data collected by suboptimal policies under varied dynamics — provides warm start, avoids overfitting on limited online data, and prevents chaotic imagined rollouts from immature policies. Phase 2 (MBPO-PPO): Learn policy from scratch purely in RWM imagination, with RWM fine-tuned on single-environment online dataset (1 env, akin to real-world constraints). Phase 3 (Collision Handling per A.4.3): Terminate rollouts and reset environment on base-ground contact; train RWM privileged-info head to predict contacts/terminations; MBPO-PPO treats predicted terminations as episode-ending events for PPO return/value computation. Note: pretraining only needed for locomotion tasks (discontinuous dynamics); manipulation tasks skip Phase 1.",
      "source": "Appendix A.4.3, Section 5"
    },
    {
      "id": "robotic-world-model-D3-008",
      "claim": "SHAC Baseline Evaluation: Run SHAC (first-order gradient-based policy optimization through world model) for velocity tracking on ANYmal D and Unitree G1. Compare model error e and policy reward r against MBPO-PPO and DreamerV3 (Fig 5). Evaluate on discontinuous legged locomotion dynamics where contact pattern changes cause gradient inaccuracy. SHAC uses same RWM world model but different policy optimization method.",
      "source": "Section 4.4, Figure 5"
    },
    {
      "id": "robotic-world-model-D3-009",
      "claim": "DreamerV3 Baseline Evaluation: Run DreamerV3 (latent-space RSSM dynamics model + actor-critic) for velocity tracking on ANYmal D and Unitree G1. Compare model error e and policy reward r against MBPO-PPO and SHAC (Fig 5). Evaluate impact of DreamerV3's shorter planning horizons on long-horizon dependencies in stochastic environments. DreamerV3 uses its own world model architecture, not RWM.",
      "source": "Section 4.4, Figure 5"
    },
    {
      "id": "robotic-world-model-D3-010",
      "claim": "Model-Free PPO Baseline Comparison (Table 1): Train standard PPO directly on high-fidelity Isaac Lab simulator (no world model, unlimited sim interactions). Budget: 250M state transitions, 10 min training on RTX 4090. Compare real tracking reward (quantitative), training time, and sample efficiency against RWM+MBPO-PPO pipeline (6M pretraining transitions + 55 min total). Purpose: contextualize MBRL vs model-free trade-off.",
      "source": "Section 5, Table 1"
    }
  ],
  "D4": [
    {
      "id": "robotic-world-model-D4-001",
      "claim": "Collision/Termination Handling Protocol (step ordering): (1) Detect base-ground contact during rollout as failure signal; (2) Terminate rollout and reset environment; (3) Train RWM privileged-information prediction head to explicitly predict contact/termination events; (4) During MBPO-PPO imagination rollouts, when termination is predicted by the privileged-info head, treat as episode-ending event; (5) Episode termination affects PPO's GAE return computation and value function targets (truncated returns)",
      "source": "Appendix A.4.3"
    },
    {
      "id": "robotic-world-model-D4-002",
      "claim": "RWM Pretraining and Fine-Tuning Pipeline Ordering: Phase 1 (if locomotion task): Pretrain RWM on 6M state transitions from simulated suboptimal-policy data under varied dynamics; Phase 2: Learn policy from scratch purely in RWM imagination via MBPO-PPO; Phase 3: Fine-tune RWM using single-environment online dataset generated during policy learning; Branch: For manipulation tasks, skip Phase 1 (pretraining unnecessary without discontinuous dynamics). Rationale: pretraining prevents severe overfitting on limited online data and chaotic imagined rollouts from immature policies",
      "source": "Appendix A.4.3, Section 5"
    },
    {
      "id": "robotic-world-model-D4-003",
      "claim": "Dual-Autoregressive Prediction Sequence (RWM forward pass): Step 1 (Inner AR): Sequentially process each of the M historical observation-action pairs through the GRU, updating hidden state autoregressively after each step. Step 2 (Outer AR): For k = 1 to N: predict o'_{t+k} from current GRU hidden state via MLP heads (Gaussian mean/std for observations + privileged info), then feed (o'_{t+k}, a_{t+k}) as input to GRU for the next prediction step k+1. The dashed arrows in Fig S6 denote the sequential autoregressive prediction flow",
      "source": "Section 3.2, Figure S6, Appendix A.2.1"
    }
  ]
}