Probabilistically-Safe Bipedal Navigation over Uncertain Terrain via Conformal Prediction and Contraction Analysis

TL;DR

The paper tackles probabilistic safety in bipedal navigation over uncertain terrain by coupling Gaussian Process terrain modeling with Conformal Prediction to produce calibrated height intervals. It then builds contraction-based robust invariant tubes around a high-level MPC plan and augments the LIPM with a flywheel torque control to stabilize centroidal angular momentum, ensuring forward invariance under terrain disturbances. The framework delivers probabilistic safety guarantees and goal reachability across long horizons, demonstrated via MuJoCo simulations of the Digit robot across diverse terrains and confidence levels (e.g., $1-\delta$ with $\delta$ chosen for 85% and 99.5% safety). The key contribution lies in integrating CP-calibrated uncertainty with contraction theory to obtain data-driven, provably-safe, dynamically-feasible locomotion plans for rough terrain, bridging planning and low-level control under uncertainty.

Abstract

We address the challenge of enabling bipedal robots to traverse rough terrain by developing probabilistically safe planning and control strategies that ensure dynamic feasibility and centroidal robustness under terrain uncertainty. Specifically, we propose a high-level Model Predictive Control (MPC) navigation framework for a bipedal robot with a specified confidence level of safety that (i) enables safe traversal toward a desired goal location across a terrain map with uncertain elevations, and (ii) formally incorporates uncertainty bounds into the centroidal dynamics of locomotion control. To model the rough terrain, we employ Gaussian Process (GP) regression to estimate elevation maps and leverage Conformal Prediction (CP) to construct calibrated confidence intervals that capture the true terrain elevation. Building on this, we formulate contraction-based reachable tubes that explicitly account for terrain uncertainty, ensuring state convergence and tube invariance. In addition, we introduce a contraction-based flywheel torque control law for the reduced-order Linear Inverted Pendulum Model (LIPM), which stabilizes the angular momentum about the center-of-mass (CoM). This formulation provides both probabilistic safety and goal reachability guarantees. For a given confidence level, we establish the forward invariance of the proposed torque control law by demonstrating exponential stabilization of the actual CoM phase-space trajectory and the desired trajectory prescribed by the high-level planner. Finally, we evaluate the effectiveness of our planning framework through physics-based simulations of the Digit bipedal robot in MuJoCo.

Figures (4)

• Figure 1: (A) A bird's-eye view of the robot navigation path over the true rough terrain map. (B) A visualization of terrain map estimated by Gaussian Process (GP) and conformal prediction (CP). (C) The bipedal robot Digit navigates through an environment with rough terrain in our MuJoCo simulation. (D) Digit's flywheel torque control law maintains the CoM trajectory within the contraction-based robust control invariant tube on the robot with full-order dynamics.
• Figure 2: (a) The high-level planner's global dynamics is based on the Linear Inverted Pendulum Model (LIPM). For contraction analysis at the low-level, we use the Augmented LIP Model (Aug-LIPM) with flywheel torque $\tau_y$ about the CoM. (b) Sagittal phase portrait for one walking step from $q^{\rm th}$ to $(q+1)^{\rm th}$ in the local frame of the $q^{\rm th}$ footstep. The desired MPC-guided trajectory $\textbf{x}^\star(t)$ (dark blue) is designed by \ref{['eq:sagittal_cons']} given the MPC output $(x^{\rm loc}_{q},v^{\rm loc}_{q},u^f_{q})$. The true CoM trajectory from full-order dynamics $\textbf{x}(t)$ (pink) is regulated to track the desired plan via CCM control law $\tau_y^{\rm CCM}$. The RCI tube $\Omega(\textbf{x}^\star ,t)$ (light blue) is constructed around the desired trajectory; orbital energy $E$ is represented by the asymptotic slope line (orange), and tube-bound propagation is shown via the saltation matrix $\Xi$.
• Figure 3: Overall block diagram of the proposed probabilistically-safe planning and control strategy for bipedal navigation over uncertain terrain.
• Figure 4: Visualization of three large terrain profiles, T1: bumpy and rough, T2: wavy and coarse, and T3: hilly with a smoother surface (for the case study with the 85% confidence level), and two smaller terrain profiles, T4 and T5: level but rough (for the case study with the 99.5% confidence level) with sample trajectories from different runs (white indicates the start of each trajectory). For the red trajectory, we show the GP mean estimate map along with the CP confidence interval, which ensures coverage of the true terrain height at each confidence level.

Theorems & Definitions (11)

• Definition 1: Gaussian Process Regression
• Definition 2: Split Conformal Prediction
• Definition 3: Robust Control Invariant Tube
• Lemma 1: CP Safe Footstep Constraint
• proof
• Remark 1
• Lemma 2
• proof
• Lemma 3
• proof