Proprioceptive Forward Dynamics Module

• Proprioceptive FDM is a computational model that fuses internal sensor data and environmental inputs to predict future robot states accurately.
• It integrates learned neural surrogates with analytical models to perform rollout-based predictions for motion planning and safety-aware control.
• The module is embedded in model predictive control systems, enabling real-time trajectory corrections and efficient failure-risk estimation.

A Proprioceptive Forward Dynamics Module (proprio FDM) is a computational or analytical module that predicts the future state of a robotic system, typically fusing proprioceptive sensing (internal measurements such as joint angles, torques, actuator states) and, in advanced models, environmental context from exteroceptive sensors. Proprio FDMs underpin model-based optimal control and safety-aware motion planning in legged, wheeled, and soft robots, providing state trajectory rollouts and failure-risk estimates with high temporal fidelity. Modern proprio FDMs span learned neural surrogates for complex, platform-specific dynamics and physics-based analytical models for systems such as soft continuum manipulators (Roth et al., 27 Apr 2025, Kim et al., 2022, Toshimitsu et al., 2021).

1. Mathematical Foundations and Formulation

In the context of legged navigation, the proprio FDM learns the mapping

$$\tilde f_\theta:\{x_t, m_{t-H:t}, a_t, s_t\} \rightarrow x_{t+1} = (p_{t+1},\, r_{t+1}),$$

where $x_t$ comprises the robot's pose $p_t \in \mathrm{SE}(2)$ and a binary failure indicator $r_t \in \{0,1\}$, $a_t \in \mathbb{R}^3$ is the commanded twist, $m_{t-H:t}$ denotes a history of proprioceptive measurements (encompassing joint positions, velocities, previous actions, and base state), and $s_t$ encodes local terrain geometry as a rasterized height map (Roth et al., 27 Apr 2025). The model is typically trained to perform $n$-step rollouts, predicting both residual velocities for correction of nominal commands and future failure probabilities:

$$\tilde a_{t+k} = a_{t+k} + \Delta\tilde a_{t+k}; \quad p_{t+k+1} = p_{t+k}\oplus (\tilde{v}_x, \tilde{v}_y, \tilde{\omega})\Delta t_p,$$

with $\oplus$ denoting pose composition in $\mathrm{SE}(2)$.

For soft continuum arms, the proprio FDM is derived analytically and predicts the future joint space state $q$ and velocities $\dot q$ from actuator inputs (e.g., pneumatic pressures $p$), sensor measurements $s$, and (optionally) external forces $f_{\rm ext}$, by numerically integrating the Lagrangian equations with stiffness (K), damping (D), and input mappings (A) specific to the robot’s elastomeric structure (Toshimitsu et al., 2021).

2. Network Architectures and Computational Structure

For learning-based FDMs in legged robots, modality-specific encoders are deployed:

• Proprioceptive Streams: Histories of proprioceptive signals and past poses are processed by a GRU (2 layers, hidden size 64).
• Exteroceptive Geometry: Height map $s_t$ is rasterized and input into a 4-layer CNN ($[32, 64, 128, 256]$ channels).
• Command Sequence: Each control action is embedded by a timestep-wise MLP (hidden width 16).
• Rollout and Prediction: At each rollout step, the fused representation is updated via a GRU, with two parallel MLP heads producing the residual velocity correction and failure-risk estimates.

The network contains $\sim$1.2 million parameters and admits rollout-based prediction (parallelizable for sampling-based planning) (Roth et al., 27 Apr 2025). In contrast, soft-arm proprio FDMs construct an explicit analytical model, whereby real-time forward dynamics are integrated and fused with capacitive flex sensor data via QP-based (quadratic program) sensor fusion for drift correction and contact-force estimation (Toshimitsu et al., 2021).

3. Training Data, Losses, and Optimization Procedures

Modern learned proprio FDMs are trained on a combination of simulated and real-world datasets. For robust failure prediction and sim-to-real transfer, training sets include:

• Procedurally randomized simulation: Diverse terrain, induced failures (falls, collisions) for broad coverage of state space.
• Real-world traversals: Human-teleoperated operation in challenging outdoor environments, with ground-truth pose from RTK+total-station, capturing real dynamics, compliance, and sensor artifacts.

The loss minimized across a rollout of length $n$ incorporates pose regression, failure-risk cross-entropy, and a stop-loss penalizing pose drift after a detected failure:

$$\mathcal{L} = \epsilon_{\rm pose}\mathcal{L}_{\rm pose} + \epsilon_{\rm risk}\mathcal{L}_{\rm risk} + \epsilon_{\rm stop}\mathcal{L}_{\rm stop}.$$

Optimization employs AdamW with scheduled learning rate and weight decay; pretraining is performed in simulation (up to 1.2M parameters on a single GPU in 8 hours), followed by fine-tuning on a mixed sim-real buffer (Roth et al., 27 Apr 2025).

In analytical modules for continuum robots, parameters are identified through excitation experiments (isolated actuation, motion capture, polynomial fit for $(q, \dot q, \ddot q)$) and least-squares fitting to observed input-output trajectories (Toshimitsu et al., 2021). Sensor fusion and contact-force estimation are solved in real-time via off-the-shelf QP solvers.

4. Integration with Model Predictive Control and Planning

The proprio FDM is embedded directly into sampling-based model predictive planners. In zero-shot MPPI (Model Predictive Path Integral) planning:

1. Generate $C$ candidate command sequences via stochastic perturbation of the current plan.
2. For each candidate, roll out predicted future states and failures via the FDM.
3. Compute a reward combining terminal pose proximity and accumulated risk penalty.
4. Weight candidates by a softmax over rewards and update the nominal plan accordingly.
5. After several iterations, execute the first action in the updated plan (Roth et al., 27 Apr 2025).

This architecture enables heuristic-free, long-horizon planning, eliminating hand-designed traversability cost maps. For soft continuum arms, real-time forward prediction and sensor-corrected models enable proprioceptive closed-loop control in the absence of exteroceptive feedback (Toshimitsu et al., 2021).

5. Empirical Results and Platform-Specific Outcomes

Metric	Proprio FDM Performance (Roth et al., 27 Apr 2025)	Analytical Proprio FDM (Toshimitsu et al., 2021)
Final-step position error	41.3% lower vs. LiDAR-FDM baseline	Tip RMSE: 1.27–1.96 cm (3 s, 30 cm sweep)
Failure/collision-risk F1	$>89\%$ (all), $>0.85$ (3D terrain)	Contact-force: linear w/ true mass, $0.3\times$ scaling
Planning success rate	88.3% (2D), 73.8% (3D)	–
Throughput (inference+planning)	7 Hz (2048 rollouts, Jetson Orin)	QP+integration: $>$500 Hz control loop
Cost map tuning	None required	None; parameters physically interpretable

Conditioning FDMs on full proprioceptive state histories and exteroceptive geometry dramatically improves long-range state prediction (up to 70.6% error reduction vs. non-perceptive models), with robust risk detection maintained across challenging terrains and environments. Analytical proprio FDMs demonstrate decimeter-to-centimeter accuracy for soft arm manipulation, with closed-loop sensor fusion enabling operation in occluded or uninstrumented settings.

6. Comparative Context and Methodological Significance

Proprio FDMs represent a departure from traditional robot dynamics models, which relied either on rigid-body analytical dynamics or coarse heuristics for traversability and risk. By learning the complex, platform-specific mapping from internal and external state histories to future outcomes, FDMs bypass the manual cost-map engineering steps and enable safe, generalizable navigation with minimal per-environment adaptation (Roth et al., 27 Apr 2025, Kim et al., 2022). For highly compliant systems, analytical FDMs combined with dense proprioceptive feedback achieve model-consistent estimation and planning without exteroceptive sensing, a critical advantage in unstructured or occluded environments (Toshimitsu et al., 2021).

A key distinction from prior frameworks is the explicit prediction of both long-horizon future poses and failure/collision probability, enabling planners to reject unsafe trajectories at the sample level before execution. This suggests that future advances may further exploit joint representation learning and multimodal fusion to extend FDM efficacy to more complex platforms and tasks, including those with rich contact interactions and partial observability.