Perceptive Mixed-Integer Footstep Control for Underactuated Bipedal Walking on Rough Terrain

TL;DR

The paper tackles the challenge of dynamic, underactuated bipedal walking on rough, discontinuous terrain by integrating a real-time Model Predictive Footstep Control (MPFC) that optimizes discrete foothold selection, footstep positions, ankle torque, and timing within a short horizon. A key contribution is Stable Steppability Segmentation (S3), a temporally consistent perception pipeline that classifies safe terrain and generates convex foothold polygons online, enabling MIQP-based online planning. The approach is validated on the Cassie biped through outdoor experiments, showing sub-10 ms MPFC solve times, robust walking over stairs, curbs, and uneven terrain, and superior temporal consistency of S3 compared to plane segmentation baselines. Together, the full-stack MPFC+S3 framework provides perceptive, dynamic, underactuated walking capabilities on real-world rough terrain with real-time performance and robust perception.

Abstract

Traversing rough terrain requires dynamic bipeds to stabilize themselves through foot placement without stepping in unsafe areas. Planning these footsteps online is challenging given non-convexity of the safe terrain, and imperfect perception and state estimation. This paper addresses these challenges with a full-stack perception and control system for achieving underactuated walking on discontinuous terrain. First, we develop model-predictive footstep control (MPFC), a single mixed-integer quadratic program which assumes a convex polygon terrain decomposition to optimize over discrete foothold choice, footstep position, ankle torque, template dynamics, and footstep timing at over 100 Hz. We then propose a novel approach for generating convex polygon terrain decompositions online. Our perception stack decouples safe-terrain classification from fitting planar polygons, generating a temporally consistent terrain segmentation in real time using a single CPU thread. We demonstrate the performance of our perception and control stack through outdoor experiments with the underactuated biped Cassie, achieving state of the art perceptive bipedal walking on discontinuous terrain. Supplemental Video: https://youtu.be/JK16KJXJxi4

Figures (20)

• Figure 1: The bipedal robot Cassie walks up and down brick steps using the perception and control framework developed in this paper. Left: the physical robot and steps. Middle: an elevation map of the steps. Right: a convex decomposition of the safe terrain. Our MPC footstep planner constrains the center of Cassie's foot to a convex polygon foothold for each planned footstep. These convex footholds are generated online via Stable Steppability Segmentation, our novel terrain segmentation approach designed for temporal consistency of the safe terrain classification.
• Figure 2: The perception and control stack proposed in this paper to achieve underactuated walking over discontinuous terrain. Our perception stack (A) generates convex polygon foothold constraints for MPFC, a mixed-integer MPC style footstep planner (B). MPFC sends the next footstep, step timing adaptation, and ankle torque plan to a low-level operational-space-control process (C) which performs kHz level torque control.
• Figure 3: The ALIP model assumes that the robot's CoM is restricted to a virtual plane above the terrain. The states of the ALIP model are the horizontal CoM positions, and the angular momentum of the robot about the horizontal axes.
• Figure 4: Top: Key MPFC decision variables and constraints for a horizon of 2 stance phases. $x_{c}$ is the current ALIP state, $u$ is ankle torque applied during the current stance phase, $x_{0}$ is the ALIP state at the end of the current stance phase, and $x_{1}$ is the ALIP state at the end of the next stance phase. The current stance foot position, $p_{0}$, is unconstrained, and subsequent footsteps are constrained to lie in either $\mathcal{P}_{j}$ or $\mathcal{P}_{q}$ using integer variables. Bottom: Relationship between the nominal stance phases and the MPFC gait timing optimization. The initial stance duration is adjusted continuously by optimizing over the remaining stance time, $T$.
• Figure 5: To enforce the planarity assumption of the ALIP, we use OSC to drive Cassie's CoM to a virtual plane defined by current and upcoming stance foot positions.