Real-Time Safe Bipedal Robot Navigation using Linear Discrete Control Barrier Functions

TL;DR

This work develops a unified, safe path and gait planning framework that can be evaluated online in real-time, allowing the robot to navigate clustered environments while sustaining stable locomotion.

Abstract

Safe navigation in real-time is an essential task for humanoid robots in real-world deployment. Since humanoid robots are inherently underactuated thanks to unilateral ground contacts, a path is considered safe if it is obstacle-free and respects the robot's physical limitations and underlying dynamics. Existing approaches often decouple path planning from gait control due to the significant computational challenge caused by the full-order robot dynamics. In this work, we develop a unified, safe path and gait planning framework that can be evaluated online in real-time, allowing the robot to navigate clustered environments while sustaining stable locomotion. Our approach uses the popular Linear Inverted Pendulum (LIP) model as a template model to represent walking dynamics. It incorporates heading angles in the model to evaluate kinematic constraints essential for physically feasible gaits properly. In addition, we leverage discrete control barrier functions (DCBF) for obstacle avoidance, ensuring that the subsequent foot placement provides a safe navigation path within clustered environments. To guarantee real-time computation, we use a novel approximation of the DCBF to produce linear DCBF (LDCBF) constraints. We validate the proposed approach in simulation using a Digit robot in randomly generated environments. The results demonstrate that our approach can generate safe gaits for a non-trivial humanoid robot to navigate environments with randomly generated obstacles in real-time.

Figures (10)

• Figure 1: Safe navigation planning is tested in MuJoCo with the Digit robot.
• Figure 2: Representation of state transition of the LIP model in the global frame. The state $\mathbf{x}_{k}$ in step $k$ (shown in cyan) evolves to the state $\mathbf{x}_{k+1}$ (shown in orange) when stepping at point $(f_{x_k}, f_{y_k})$ and with a turning rate of $\omega_k$.
• Figure 3: An illustration of the LIP-MPC formulation when $N=3$. The planner first estimates the state at the end of the current step, $\mathbf{x}_0$, given the current instantaneous state, $\mathbf{x}_{cur}$, and the stance foot position contained in $\mathbf{u}_{cur}$. The turning rates in the subsequent steps are pre-computed, where the stepping positions $f_k$ will be optimized by the LIP-MPC.
• Figure 4: a) A function $F(\vec{x})$ represents the outline of an obstacle. The safe path corresponds to the condition $F(\vec{x}) \geq 0$, where $\vec{x}$ represents a position in the map. b) The first step to linearize the DCBF condition is to generate the convex function $C(\vec{x})$ that contains $F(\vec{x})$. c) We use this convex function to consider the robot position $\vec{x}_r$ and find the half-plane formed by the closest point $\vec{c}$ and its normal vector, representing the linear approximation of the safe region.
• Figure 5: Left: The closest point $\vec{c}$ to the robot is a vertex of the convex hull. Right: The closest point $\vec{c}$ lies within an edge of the convex hull. In both cases, the unit normal vector is calculated as the normalization of the line connecting the closest point to the robot position.

Theorems & Definitions (4)

• Proposition
• proof
• Proposition
• proof