PIDM: Proprioceptive Inverse Dynamics Model

Updated 16 October 2025

Proprioceptive Inverse Dynamics Model (PIDM) is an algorithmic framework that uses internal sensor data such as joint positions and motor currents to derive control actions in robotics.
It integrates analytical models with learned neural architectures, enabling robust control, collision detection, and adaptive manipulation in diverse robotic applications.
PIDMs enhance learning efficiency and real-world performance by combining model-based estimations with data-driven residual learning and reinforcement pretraining techniques.

A Proprioceptive Inverse Dynamics Model (PIDM) is an algorithmic or analytical framework that predicts, infers, or utilizes control actions—and possibly environmental quantities—via internal proprioceptive signals (e.g., joint positions, velocities, internal force sensors, motor currents) and dynamic models, without reliance on exteroceptive sensors such as cameras. PIDMs span a spectrum from model-based controllers and adaptive estimators in soft robotics, to deep neural architectures in data-driven robot learning, encompassing both analytical and learned formulations. They are employed for scalable manipulation, collision detection, control in complex environments, and as embodiment priors for data-efficient learning. This article reviews core theory, model classes, technical architectures, validation paradigms, and the practical impacts and limitations of PIDMs across robotics domains.

1. Principles and Mathematical Foundations

PIDMs express the mapping from sequences of internal robot states to actions or environmental parameters, exploiting robot-intrinsic feedback. Typical input modalities include

Joint positions $q(t)$ , velocities $\dot{q}(t)$ , accelerations $\ddot{q}(t)$
Internal forces, torques, motor currents
Soft body flex sensor readings
Historical state/action sequences

The canonical inverse dynamics problem seeks $\tau_t$ such that for a desired motion—or in supervised imitation scenarios—the action $a_t$ causes a transition $x_{t} \rightarrow x_{t+1}$ as encoded by a dynamics model. PIDMs generalize this: for robots with proprioceptive measurement $x_t$ (and possibly history $x_{t-k : t}$ ), predict $a_t$ that produces the desired state evolution or external interaction signature. Analytical PIDMs integrate direct biomechanical and physics-based models; learned PIDMs employ function approximators—e.g. MLPs, RNNs, GPR, Transformers—mapping historical proprioception and intention into control (Fan et al., 14 Oct 2025, Tian et al., 19 Dec 2024, Reuss et al., 2022).

Key mathematical formulations:

Analytical Model-based: $y(t) = f(\xi(t)) + e(t)$ , with $\xi(t)$ (history-driven features) constructed from joint position sequences using a feature extraction matrix $R$ (Romeres et al., 2018).
Hybrid Models: $\tau_{ref} = f_{RBD}(q, \dot{q}, \ddot{q}_{des}) + f_F(\dot{q}) + f_{RNN}(q, \dot{q}, \ddot{q}_{des}, \theta_{RNN})$ merges rigid-body priors and data-driven temporal residuals (Reuss et al., 2022).
Learning-based PIDM: $I(a_t \mid x_{t-k:t+1}, a_{t-k:t-1})$ , encapsulating the mapping from a history of proprioceptive observations and actions to the required inverse control (Fan et al., 14 Oct 2025).
Transformer-based Vision-Action PIDM: $\hat{o}_{t+n} = f_{fore}(g, h_t)$ gives conditional visual foresight, $a_{t+t-1} = f_{inv}(g, h_t, \hat{o}_{t+n})$ provides inverse action solving for the predicted state (Tian et al., 19 Dec 2024).

2. Model Classes and Architectural Variants

PIDMs comprise several architectural types, each suited to particular robot complexities or operational environments:

Model Class	Input Features	Dynamics Encoding
Analytical	Joint positions, PCC	Physics equations
Derivative-Free	Position histories	Linear/structured filters
Hybrid (RBD+NN)	Positions, velocities	Rigid-body + LSTM/MLP
End-to-End Vision	RGB, proprioception	Vision Transformer + MLP
Gaussian Process	Motor current, friction	Semi-parametric GPR
RL-based PIDM	All proprioception	Modular NN encoder

Derivative-Free PIDMs eschew explicit differentiation, operating directly on position memory vectors $q(t^-)$ and extracting features via learned or structured matrices $R$ (DF, DFW, DFR, DFSR) to suppress noise amplification (Romeres et al., 2018).
Hybrid Inverse Dynamics Models leverage accurate inertial parameter estimation via LMIs and Cholesky parameterization enforced in differentiable analytical pipelines, complemented by LSTM or other memory-based components for hysteresis and partial observability (Reuss et al., 2022).
Gaussian Process Regression PIDMs extend semi-parametric inverse dynamics with additional features (e.g., error, control current) and kernel structure that separate discontinuous static friction components from kinetic ones (Alberto et al., 2019).
Transformer-based PIDMs operate over vision-language-proprioception triplets. Readout tokens ([FRS], [INV]) enabling conditional foresight for action derivation form the backbone of scalable learners (Tian et al., 19 Dec 2024).
RL Pretraining PIDMs train modular neural models mapping proprioceptive and action histories to inverse dynamics, initializing both actor and critic networks for downstream RL (Fan et al., 14 Oct 2025).

3. Data Collection, Training, and Optimization Strategies

Data-driven PIDM protocols emphasize extensive, diverse, and task-agnostic data collection for pretraining, conversion of sensor histories to structured feature vectors, and robust optimization:

Task-Agnostic Exploration: Initial data derives from unsupervised robot-environment interactions (Fan et al., 14 Oct 2025), guiding model pretraining toward embodiment-agnostic mapping of proprioceptive transitions.
Ensemble Intrinsic Reward: Intrinsic motivation leverages ensemble diversity in prediction, i.e., variance among PIDM ensemble outputs as a reward for exploration-based RL (Fan et al., 14 Oct 2025).
Supervised Pretraining: $L_1$ or MSE loss is applied between PIDM output and ground truth action (as in control-theoretic mapping from desired delta state to action) (Fan et al., 14 Oct 2025).
End-to-end Multimodal Training: Joint minimization of foresight and inverse dynamics loss ( $L_{fore}, L_{inv}$ ) balances visual, proprioceptive, and language signals for calibration on large-scale datasets (Tian et al., 19 Dec 2024).
Analytical Model Fitting: Closed-loop quadratic programming is applied for fitting analytical models under sensor constraints, as in soft arm estimation via $s = S q$ and physics-based dynamic equations (Toshimitsu et al., 2021).
Kernel Hyperparameter Marginal Likelihood: In GPR-based PIDMs, marginal likelihood substantially outperforms cross-validation for hyperparameter tuning when feature extraction matrices have many degrees of freedom (Romeres et al., 2018).

Augmentation techniques such as physical symmetry and sensor noise injection increase pretraining robustness and enable generalization to realistic deployment scenarios (Fan et al., 14 Oct 2025).

4. Validation, Performance Metrics, and Comparative Analyses

Empirical validation across PIDM implementations reveals consistent advantages in efficiency, adaptability, and accuracy:

Learning Efficiency: RL actor-critic policies warm-started by PIDM pretraining accelerate convergence by an average of 40.1% and provide 7.5% higher task performance over random initialization (Fan et al., 14 Oct 2025).
Robustness to Noise: Derivative-free PIDMs outperform numerical differentiation-based models, particularly in transient adaptation and prediction error mitigation (Romeres et al., 2018).
Real-World Manipulation: Transformer-based PIDMs (Seer) yield improvements up to 43% in real-world robotic tasks, 21% on CALVIN ABC-D simulation, and set state-of-the-art on CALVIN benchmarks (e.g. average sequence length 4.28) (Tian et al., 19 Dec 2024).
Collision Detection: Proprioceptive GPR PIDMs achieve low nMSE and robust false positive suppression in quasi-static and dynamic configurations during human-robot interaction on a UR10 industrial robot (Alberto et al., 2019).
Soft Robotics Estimation: Closed-loop optimization with embedded flex sensing produces tip trajectory errors as low as 1.27 cm and reliably estimates external load via internal signal fusion (Toshimitsu et al., 2021).

Ablation studies demonstrate PIDM benefits are maximized when both actor and critic networks are jointly initialized (Fan et al., 14 Oct 2025), and probe analyses confirm intermediate layers retain strong correlation with future proprioceptive evolution.

5. Applications in Robotics and Autonomous Vehicles

PIDMs are deployed across diverse robotic disciplines, each leveraging internal sensing for robust estimation, control, and interaction:

Manipulation with Vision-Action Closure: Transformer-based PIDMs integrate visual foresight with action prediction, supporting scalable manipulation and adaptation to novel objects, lighting, and disturbances (Tian et al., 19 Dec 2024).
Articulated Object Interaction: Proprioceptive sensing models infer joint mechanisms and compliance in contact-rich manipulation (e.g., cabinets and ovens), enabling alternative approaches to vision-based systems (Lips et al., 2023).
Collision Detection: Gaussian Process PIDMs distinguish collision events solely from internal signal deviations, enhancing safety in human-collaborative robotic settings (Alberto et al., 2019).
Soft Continuum Robots: Analytical PIDM models embedded with flex sensors estimate real-time force and posture, eliminating reliance on exteroceptive motion capture (Toshimitsu et al., 2021).
Terrain-Adaptive Vehicle Control: PIDMs incorporating vertical suspension dynamics improve online estimation of terrain properties (e.g., sinkage exponent), refine trajectory predictions, and support advanced path planning via IMU integration (Buzhardt et al., 2022).
Warm-Start in RL for Motion Control: PIDMs modularly initialize RL agent policies across a range of locomotion and pedipulation tasks, demonstrating embodiment knowledge transfer and sample efficiency improvement (Fan et al., 14 Oct 2025).

6. Limitations, Open Questions, and Future Directions

Despite demonstrated robustness, several challenges and research possibilities remain:

Slip and Contact Uncertainty: Fixed grasps and slip events in proprioceptive manipulation introduce estimation errors, suggesting a need for tactile integration or adaptive regrasping strategies (Lips et al., 2023).
Noise Modelling and Feature Selection: Optimal construction and adaptation of derivative-free feature matrices $R$ require further investigation, especially regarding high-dimensional action spaces and transferability (Romeres et al., 2018).
Integration of Exteroceptive and Proprioceptive Signals: The debate remains unresolved on whether proprioceptive approaches should fully supersede vision-based methods in contact-rich environments, or be complementary (Lips et al., 2023).
Partial Observability: Recurrent architectures (LSTM, RNN) handle hysteresis and time-dependent effects, but the explicit modeling of latent states and their contribution to control stability is an active area of investigation (Reuss et al., 2022).
Scalability and Embodiment Generalization: Transformer-based PIDMs highlight scaling laws in robotic manipulation and suggest the plausible extension to cross-embodiment settings, yet systematic evaluation on heterogeneous platforms is ongoing (Tian et al., 19 Dec 2024).
Interpretability and Safety: Hybrid PIDMs enforce physical consistency via LMIs and Cholesky parametrization, but the interpretability and reliability of learned residual modules under distributional shift remain open research topics (Reuss et al., 2022).

7. Summary and Research Outlook

PIDMs represent a convergence of internal sensing, dynamic modeling, and data-driven learning. Analytical, derivative-free, kernel-based, hybrid, end-to-end trained, and RL-pretrained PIDMs each contribute technical mechanisms for internal state-based action prediction and estimation. Their capacity to improve learning efficiency, prediction accuracy, and robustness in both rigid and soft robots—the latter extending proprioceptive estimation to complex continuum morphologies—demonstrates broad utility. Future research is focused on enhancing data efficiency, uncertainty quantification, multimodal integration, and cross-embodiment applicability, potentially establishing PIDMs as foundational building blocks for autonomous, adaptable, and resilient robotic systems.