Proprioceptive-only State Estimation for Legged Robots with Set-Coverage Measurements of Learned Dynamics

1. Problem Statement

Legged robots require accurate state estimation for autonomous operation in unstructured environments. While exteroceptive sensors (LiDAR, cameras) are informative, they fail in degraded conditions (dust, darkness, low texture). Proprioceptive sensors (IMU, joint encoders) are robust but lack direct measurements of body velocity and position.

Recent approaches use learned dynamics models (neural networks) to map proprioceptive history to body velocity "pseudo-measurements," fusing them with IMU data via Extended Kalman Filters (EKF). However, these methods rely on a Gaussian noise assumption for the prediction errors.

• The Failure Mode: In real-world scenarios with limited training data, models suffer from epistemic uncertainty. The resulting prediction errors are often biased, multi-modal, or heavy-tailed, violating the Gaussian assumption.
• Consequence: When an EKF fuses these non-Gaussian errors as if they were Gaussian, the filter becomes overconfident, leading to inconsistent estimates and eventual divergence (drift).

2. Methodology

The authors propose a framework that replaces the Gaussian noise assumption with Set-Coverage Statements derived from Uncertainty Quantification (UQ) tools, integrated into an Invariant Extended Kalman Filter (IEKF).

A. Learned Dynamics Observer

• Architecture: A Gated Recurrent Unit (GRU) network is trained to predict body-frame velocity ($\tilde{v}_I$) from a window of joint angles, velocities, and torques.
• Training: Supervised learning on ground-truth trajectories. The network outputs a mean prediction but does not assume the error distribution is Gaussian.

B. Set-Coverage Uncertainty Representation

Instead of modeling error as a probability density function (PDF), the method characterizes error using a calibrated set-coverage statement:
$$P(|e_t| \leq \epsilon) \geq \gamma$$
Where:

• $e_t$ is the prediction error.
• $\epsilon$ is a calibrated bound vector.
• $\gamma$ is a desired confidence level (e.g., 0.85).
• Key Advantage: This statement is distribution-free. It holds true regardless of whether the error is Gaussian, biased, or multi-modal, provided the calibration data is representative. The bounds $\epsilon$ are computed using Conformal Prediction on a separate calibration dataset.

C. Integration into IEKF (The Core Innovation)

The challenge is updating a Gaussian state estimate using a set-constraint rather than a likelihood function.

1. Formulation: The set-constraint is mapped to the error state space ($\delta X_t$), defining a feasible region $C_t$ (a box in the projected output space).
2. Optimization: The update seeks a posterior distribution that:
   • Assigns at least probability mass $\gamma$ to the set $C_t$.
   • Minimizes the Kullback-Leibler (KL) divergence from the prior Gaussian distribution.
3. Closed-Form Solution: The authors prove (Theorem 1) that the optimal solution involves piecewise rescaling of the prior density inside and outside the set $C_t$.
4. Gaussian Approximation: Since the resulting posterior is non-Gaussian, the method performs moment matching (calculating the mean and covariance of the truncated distribution) to recover a Gaussian representation for the next filter step.
5. Efficiency: To make this computationally feasible for real-time use, the integration is performed in a reduced 3D output space (body velocity) rather than the full 15D error state space, then "lifted" back to the full state.

3. Key Contributions

1. Novel Framework: The first proprioceptive-only state estimation framework for legged robots that models learned dynamics uncertainty via set-coverage statements rather than Gaussian assumptions.
2. Practical Algorithm: A computationally efficient method to fuse set-constrained measurements with a recursive Gaussian filter (IEKF) via KL-minimal projection and moment matching. It runs in ~0.07 ms per update on a standard laptop CPU.
3. Robustness: The method explicitly handles non-Gaussian, biased, and multi-modal errors without requiring retraining or complex particle filters.

4. Experimental Results

The method was validated in Monte Carlo simulations and on two real-world quadruped datasets (Vision60 and Spot).

• Simulation:

  • Under Gaussian noise, the proposed method had slightly higher RMSE than a standard EKF (due to the weaker set-constraint vs. full likelihood) but maintained consistency.
  • Under misspecified non-Gaussian/biased noise, the standard EKF diverged (NEES $\approx$ 100, indicating inconsistency), while the set-coverage method remained consistent (NEES $\approx$ 2–4) and competitive in accuracy.

• Real-World Experiments:

  • Spot Dataset: In challenging terrains (stairs, overpasses, sharp turns), standard proprioceptive baselines (Leg IEKF, Learned Vel IEKF with Gaussian assumption) accumulated significant drift. The proposed method remained stable and accurate, often outperforming perception-based baselines (GaRLILEO) on specific sequences.
  • Vision60 Dataset: The method achieved the lowest absolute position errors among all proprioceptive baselines, effectively correcting vertical drift seen in standard kinematic approaches.
  • Consistency: The method maintained high consistency (low NEES) across all test cases, whereas Gaussian baselines frequently failed.

5. Significance

This work addresses a critical bottleneck in deploying learning-based state estimators on robots: the gap between training assumptions (Gaussian) and real-world deployment (non-Gaussian/Biased errors).

By decoupling the uncertainty representation from the specific shape of the error distribution, the authors enable the use of powerful learned dynamics models in safety-critical, long-duration autonomous missions where sensor degradation or domain shift is inevitable. The approach provides a rigorous, computationally lightweight path to robust, drift-free navigation using only proprioception.