HT-LIP Model based Robust Control of Quadrupedal Robot Locomotion under Unknown Vertical Ground Motion

TL;DR

This work tackles robust quadrupedal locomotion on dynamic rigid surfaces with unknown vertical motions by developing a HT-LIP reduced-order model that captures hybrid continuous and discrete dynamics. A discrete-time, provably stabilizing footstep controller is formulated as a QP to compute real-time foot placements, and stability conditions are derived to guarantee asymptotic convergence under unknown DRS motions. The framework is organized into a three-layer architecture where a higher-layer HT-LIP planner outputs footstep and CoM targets, a middle layer generates feasible full-body trajectories, and a lower torque controller tracks those trajectories. Hardware experiments on a Unitree Go1 with a Motek treadmill validate robustness to aperiodic vertical surface motions and various uncertainties, demonstrating lower lateral drift and stable recovery under external pushes. The results indicate the HT-LIP based approach effectively stabilizes locomotion on dynamic surfaces and offers a practical pathway for robust legged robotics in uncertain environments.

Abstract

This paper presents a hierarchical control framework that enables robust quadrupedal locomotion on a dynamic rigid surface (DRS) with general and unknown vertical motions. The key novelty of the framework lies in its higher layer, which is a discrete-time, provably stabilizing footstep controller. The basis of the footstep controller is a new hybrid, time-varying, linear inverted pendulum (HT-LIP) model that is low-dimensional and accurately captures the essential robot dynamics during DRS locomotion. A new set of sufficient stability conditions are then derived to directly guide the controller design for ensuring the asymptotic stability of the HT-LIP model under general, unknown, vertical DRS motions. Further, the footstep controller is cast as a computationally efficient quadratic program that incorporates the proposed HT-LIP model and stability conditions. The middle layer takes the desired footstep locations generated by the higher layer as input to produce kinematically feasible full-body reference trajectories, which are then accurately tracked by a lower-layer torque controller. Hardware experiments on a Unitree Go1 quadrupedal robot confirm the robustness of the proposed framework under various unknown, aperiodic, vertical DRS motions and uncertainties (e.g., slippery and uneven surfaces, solid and liquid loads, and sudden pushes).

Figures (10)

• Figure 1: Snapshots of experiments. All experiments are under the unknown and aperiodic vertical surface motion $z_s(t)$ as shown in (a) and (b). The robot also experiences additional unknown disturbances, which include: (c) sudden pushes that result in (d) an irregular robot posture just after a push; (e) rocky surface with a peak height of 10 cm; (f) smooth glass surface; (g) solid load (36% of the robot's mass); and (h) liquid load (32% of the robot's mass).
• Figure 2: An illustration of the proposed HT-LIP model in the sagittal plane. The model describes the time-varying dynamics of the point mass (located at the CoM) under the vertical DRS displacement $z_s(t)$. It also captures the hybrid nature of legged locomotion, including both the continuous foot-swinging phase and the discrete foot-switching behavior.
• Figure 3: Illustration of the proposed hierarchical control framework. The higher layer generates the desired footstep locations. The middle layer employs a full-order kinematics model to plan physically feasible full-body trajectories. The lower-layer controller tracks the desired full-body trajectories.
• Figure 4: Illustration of the experimental setup. ①: Go1 quadruped (Unitree Robotics). ②: M-Gait treadmill (Motek Medical). ③: direction of the vertical DRS/treadmill motion $z_s(t)$ at point $S$. ④: world frame attached to the treadmill's axis of pitching. The treadmill's pitch angle at time $t$ is $\theta_s(t)$. Subplots (a) and (b) show the treadmill at its pitch angle limits.
• Figure 5: Ground-truth position trajectory of the point on the treadmill/DRS around which the robot performs the trotting gait during the unknown pitch and sway movement (HC4) of the DRS. The shaded area highlights the period during which the unknown DRS sway motion is active.

Theorems & Definitions (6)

• Theorem 1: Sufficient stability condition on S2S dynamics
• Theorem 2: Sufficient stability condition on footstep control gain
• Remark 1: Applicability of Theorems \ref{['Theorem1_SuffStabCond']} and \ref{['Theorem2_SuffStabCond_on_ContGain']}
• Theorem 3: QP-based control gain optimization
• Remark 2: Solution feasibility and optimality of the proposed QP
• Remark 3: Solving the QP in real-time