Bayesian Calibration

Instead of finding a single 'best' value for each unknown parameter (e.g., friction coefficient, motor torque constant), Bayesian calibration represents them as probability distributions. This captures the inherent uncertainty in their true values. The result is not just a calibrated model, but a posterior distribution over all plausible models, given the data.

A core output is a quantified measure of uncertainty for both the parameters and the model's predictions. This allows engineers to ask questions like:

• What is the 95% credible interval for the link mass?
• Given sensor noise, how certain are we of the predicted end-effector force? This is critical for risk-aware sim-to-real transfer and for identifying which parameters need more precise measurement.

The method formally incorporates existing engineering knowledge through the prior distribution. For example:

• A datasheet provides a nominal motor inertia with a tolerance: this becomes an informative prior.
• A parameter must be positive (e.g., mass): this is encoded with a log-normal prior. The prior is then updated by data via Bayes' theorem to form the posterior, blending first-principles knowledge with empirical evidence.

Bayesian calibration can explicitly account for model discrepancy or structural error—the systematic mismatch between even a perfectly parameterized simulation and reality due to unmodeled dynamics. A common approach is to include a Gaussian Process or other non-parametric term that learns this residual error, preventing the physics parameters from being incorrectly adjusted to compensate for fundamental model shortcomings.

The probabilistic framework naturally supports sequential updating. As new data is collected from the real system (e.g., during initial hardware testing), the posterior distribution becomes the new prior, enabling efficient, iterative refinement. This can be guided by active learning criteria, which select the next most informative robot trajectory (excitation trajectory) to minimize parameter uncertainty fastest.

The ultimate goal is not just an accurate model, but a robust policy. Bayesian calibration enables uncertainty propagation: the posterior distribution over parameters is sampled to create an ensemble of simulation instances. A policy's performance can then be evaluated across this ensemble, providing a distribution of expected outcomes and identifying if it is robust to the remaining parameter uncertainty before real-world deployment.

System identification is the overarching discipline of constructing mathematical models of dynamic systems from measured input-output data. It encompasses both parameter estimation (finding constants) and model structure selection. Bayesian calibration is a specific, probabilistic approach within this field that treats all unknowns as probability distributions, providing not just point estimates but full uncertainty quantification.

• Core Goal: Derive a model that predicts system behavior.
• Contrast with Calibration: System ID often seeks the model and its parameters; calibration typically tunes parameters of an existing model structure.

Parameter estimation is the process of inferring the unknown, constant values within a system's mathematical model—such as mass, inertia, or friction coefficients—from observed data. Bayesian calibration is a form of probabilistic parameter estimation. Unlike frequentist methods that yield a single 'best' value, Bayesian estimation produces a posterior distribution over possible parameter values, explicitly representing estimation uncertainty.

• Key Input: Experimental data (inputs & outputs).
• Key Output: Estimates for model parameters, with confidence intervals or full distributions.

Model fidelity measures the degree to which a simulation's outputs match the real-world system's behavior. It is the ultimate benchmark for calibration success. High-fidelity models have minimal simulation bias. Bayesian calibration directly targets fidelity by adjusting parameters to reduce the discrepancy between simulated and real sensor data, and its posterior distribution quantifies the remaining model uncertainty after calibration.

• Quantified by: Fidelity metrics like Mean Squared Error (MSE) or task-specific success rates.
• Limiting Factors: Unmodeled dynamics, numerical approximations, sensor noise.

The reality gap is the performance drop observed when a simulation-trained policy is deployed on a real robot. A primary cause is the domain gap—the statistical difference between data distributions in simulation versus reality. Bayesian calibration aims to narrow the reality gap by reducing the sim-to-real transfer error. It does this by making the simulation's dynamics distribution more closely match the real world's, rather than just matching a single trajectory.

• Manifestation: Policy fails or is inefficient on real hardware.
• Bayesian Approach: Accounts for uncertainty in the gap, leading to more robust policies.

Grey-box identification is a hybrid modeling paradigm where the core model structure is derived from known physics (white-box), but certain unknown parameters or complex residual behaviors are learned from data (black-box). Bayesian calibration is perfectly suited for the grey-box approach: the physics-based equations form the forward model, and the Bayesian inference updates beliefs about the uncertain parameters. For unmodeled dynamics, a separate residual model (e.g., a Gaussian Process) can be learned within the same Bayesian framework.

• Advantage: Leverages domain knowledge while being data-informed.
• Example: A robot arm model with known kinematics but unknown joint friction and motor constants.

Quantitative validation is the rigorous, metrics-driven process of assessing a calibrated model's accuracy. After Bayesian calibration produces a posterior, validation involves testing the model on a separate dataset not used for calibration. This tests generalizability. Metrics compare simulated states (e.g., joint angles, velocities) to real-world ground truth data, requiring precise ground truth alignment for temporal and spatial synchronization.

• Purpose: Avoid overfitting to the calibration dataset.
• Bayesian Output: Validation can use the entire posterior distribution to predict a range of likely outcomes, not just a single trajectory.