Model Uncertainty

Unmodeled dynamics are physical phenomena or system behaviors absent from the mathematical model, leading to prediction errors. These are effects considered negligible, too complex, or unknown during model design.

• Examples: High-frequency structural vibrations, complex fluid-structure interactions, non-linear friction stiction, or thermal effects on material properties.
• Impact: Causes a systematic mismatch where the simulator cannot reproduce certain real-world behaviors, regardless of parameter tuning. This is a primary source of epistemic uncertainty (uncertainty from lack of knowledge).

Parameter uncertainty arises from imperfect knowledge of the numerical constants within a known model structure. Even with a correct model form, the exact values of physics parameters are often unknown or variable.

• Key Parameters: Mass, inertia tensors, center of mass location, coefficients of friction, damping, motor constants, and gearbox backlash.
• Sources: Manufacturing tolerances, wear and tear over time, or environmental conditions (e.g., temperature affecting viscosity). This is addressed through parameter estimation and system identification, but residual uncertainty always remains.

Simulation bias is systematic error introduced by the approximations and numerical methods of the simulator itself. This is uncertainty inherent to the computational tool, not the conceptual model.

• Numerical Integration: Errors from finite time-stepping (e.g., Euler vs. Runge-Kutta methods) that accumulate over long trajectories.
• Contact Modeling: Simplifications in collision detection and resolution (e.g., spring-damper penalty methods vs. complementarity formulations).
• Mesh Discretization: Inaccuracies in finite element analysis (FEA) or computational fluid dynamics (CFD) due to coarse meshing. This bias is distinct from model error and can be reduced with higher-fidelity solvers at greater computational cost.

This source stems from the imperfect translation between the simulated ideal and real hardware's noisy, delayed, and limited sensing and actuation.

• Sensing: Simulators often provide perfect, noiseless state information. Reality involves sensor noise, latency, calibration drift, and occlusions.
• Actuation: Simulators model ideal torque or position control. Real actuators have bandwidth limits, non-linear saturation, backlash, and imperfect tracking of commanded signals.
• Effect: Creates a perception-action gap. A policy trained on perfect sim-state may fail when its input is a noisy, partial observation from real sensors.

The real-world environment is non-stationary and stochastic, while simulations are often deterministic or use simplified noise models. This is a key source of aleatoric uncertainty (inherent randomness).

• Unpredictable Disturbances: Air currents, vibrations from other machinery, uneven floor textures, or changing lighting conditions.
• Variable Interactions: Properties of objects being manipulated (e.g., deformability, mass) may not be precisely known.
• Challenge: It is impossible to model every environmental variable. Techniques like domain randomization explicitly inject stochasticity into simulation to prepare policies for this uncertainty.

Model structure error occurs when the fundamental equations or architectural assumptions of the model are incorrect or overly simplified for the operating regime.

• Assumption Violations: Assuming rigid bodies when components flex, ignoring aerodynamic drag for high-speed robots, or using linear approximations for highly non-linear systems.
• Wrong Modeling Paradigm: Using a Lagrangian formulation when a system is better described by port-Hamiltonian dynamics.
• Consequence: This is a more fundamental error than parameter uncertainty. It often requires a grey-box identification or residual modeling approach, where a data-driven component corrects the physics-based model's deficiencies.

Unmodeled dynamics are physical phenomena or system behaviors not captured by the mathematical model used for simulation or control. These are a primary source of model uncertainty and lead to prediction errors.

• Examples: Complex friction (stiction), aerodynamic effects on a drone, flex in a supposedly rigid link, or thermal expansion in actuators.
• Impact: Causes a persistent reality gap that domain randomization alone may not address.
• Mitigation: Often handled via residual modeling, where a secondary data-driven model learns to predict the discrepancy.

Simulation bias is a systematic error introduced by the approximations, assumptions, and numerical methods inherent in a simulator. It causes the simulator's predictions to consistently deviate from real-world behavior.

• Sources: Simplified collision geometry, fixed integration timesteps, approximate solvers for contact dynamics, or idealized sensor models.
• Distinction: Unlike random noise, bias is directional and structured, making it a key component of model uncertainty.
• Addressing: Requires parameter calibration and quantitative validation against real-world data to identify and correct systematic offsets.

Parameter estimation is the process of inferring the unknown constant values within a system's mathematical model from observed input-output data. It directly reduces parametric model uncertainty.

• Target Parameters: Mass, inertia tensor, center of mass, viscous friction coefficients, motor torque constants.
• Methods: Often uses excitation trajectories to ensure persistent excitation. Common techniques include least-squares fitting with a dynamic regressor or Bayesian calibration.
• Pipeline: A core part of the system ID pipeline, following data collection and preceding model validation.

Bayesian calibration is a probabilistic system identification method that treats unknown model parameters as random variables. It uses Bayes' theorem to update their probability distributions based on observed data, providing a principled measure of model uncertainty.

• Output: Produces a posterior distribution over parameters, not just point estimates. The variance of this distribution quantifies residual uncertainty.
• Advantage: Naturally incorporates prior knowledge and provides uncertainty estimates that can be propagated through the simulation.
• Use Case: Ideal for grey-box identification where some physics are known but parameters are uncertain or noisy.

Residual modeling is a technique for creating a secondary, data-driven model to predict and compensate for the error between a first-principles simulation model and real-world observations. It directly addresses unmodeled dynamics.

• Process: 1. Run a physics-based simulation. 2. Measure the discrepancy (residual) between simulation output and real sensor data. 3. Train a model (e.g., neural network) to predict this residual given the system state.
• Result: A hybrid grey-box model with improved fidelity. The residual model's own uncertainty can also be quantified.
• Application: Critical for closing the reality gap in high-performance sim-to-real transfer.

Quantitative validation is the process of assessing simulation fidelity by comparing numerical outputs from the simulator against corresponding high-fidelity real-world data using statistical metrics. It measures the residual model uncertainty after calibration.

• Prerequisite: Requires ground truth alignment to synchronize simulation and real data streams.
• Metrics: Uses fidelity metrics like Mean Squared Error (MSE) for trajectories, force/torque error, or task-specific success rate differentials.
• Purpose: Provides an objective, numerical assessment of calibration error and transfer error, guiding further model improvement.