Time-Varying Foot-Placement Control for Underactuated Humanoid Walking on Swaying Rigid Surfaces

Abstract

Locomotion on dynamic rigid surface (i.e., rigid surface accelerating in an inertial frame) presents complex challenges for controller design, which are essential for deploying humanoid robots in dynamic real-world environments such as moving trains, ships, and airplanes. This paper introduces a real-time, provably stabilizing control approach for underactuated humanoid walking on periodically swaying rigid surface. The first key contribution is the analytical extension of the classical angular momentum-based linear inverted pendulum model from static to swaying grounds. This extension results in a time-varying, nonhomogeneous robot model, which is fundamentally different from the existing pendulum models. We synthesize a discrete footstep control law for the model and derive a new set of sufficient stability conditions that verify the controller's stabilizing effect. Another key contribution is the development of a hierarchical control framework that incorporates the proposed footstep control law as its higher-layer planner to ensure the stability of underactuated walking. The closed-loop stability of the complete hybrid, full-order robot dynamics under this control framework is provably analyzed based on nonlinear control theory. Finally, experiments conducted on a Digit humanoid robot, both in simulations and with hardware, demonstrate the framework's effectiveness in addressing underactuated bipedal locomotion on swaying ground, even in the presence of uncertain surface motions and unknown external pushes.

Figures (18)

• Figure 1: Top: The proprietary stepping-in-place controller of the humanoid robot Digit fails to maintain stability during DRS sway at a frequency of 0.25 Hz and amplitude of 5 cm. Bottom: Our proposed framework reliably achieves stable walking under identical conditions.
• Figure 2: Illustration of the proposed hybrid ALIP-DRS model, showing both continuous and discrete components in the sagittal and frontal planes.
• Figure 3: Illustration of foot-landing time instants. $T_{k-1}^+$ and $T_k^-$ are the start and end time instants of the current walking step, respectively. The next walking step begins at $t=T_{k}^+$ and ends at $T_{k+1}^-$.
• Figure 4: Computation of the swing-foot landing location for the end of the current walking step at $T_{k}^-$. The ALIP-DRS state at the current time step $t$ ($\mathbf{x}(t)$ and $\mathbf{y}(t)$) and the nominal DRS motion profile ($x_S(t)$, $y_S(t)$) are used to compute the estimated pre-impact contact angular momentum ($L_{x,S}(T_{k}^-)$, $L_{y,S}(T_{k}^-)$) based on \ref{['equ:linear system solution']} and \ref{['equ:linear system solution lateral']}. This prediction and the corresponding user-specified desired angular momentum ($\bar{L}_{x,S}(T_{k+1}^-)$, $\bar{L}_{y,S}(T_{k+1}^-)$) are used to compute the swing-foot placement ($x_{SwC}(T_k^-)$, $y_{SwC}(T_k^-)$). Here $V_{y,2}$, $\bar{L}_{y,S}$, and $y_{SwC}$ are counterparts of $V_{x,2}$, $\bar{L}_{x,S}$, and $x_{SwC}$ for the frontal plane.
• Figure 5: Illustration of the periods of the walking step ($T_{step}$), forward DRS motion ($T_{DRS}$), and solution of the closed-loop ALIP-DRS system ($T_{sys}$). This study considers both the special case where $T_{sys}=T_{step}=T_{x,DRS}$ and the general case where $T_{sys}=N_1 T_{step} = N_2 T_{x,DRS}$ for any $N_1,N_2 \in \mathbb{N}$.

Theorems & Definitions (8)

• Remark 1: Periodic DRS sway
• Remark 2: Time-varying and nonhomogeneous ALIP-DRS
• Remark 3: Interpretation of the monodromy matrix
• Proposition 1: Stability of homogeneous system under discrete footstep control
• Theorem 1: Exponential stability of ALIP-DRS under footstep control
• Proposition 2: Continuous-phase convergence of $V_h$
• Proposition 3: Step-to-step convergence of $V_\eta$
• Theorem 2: Closed-loop stability conditions for the full-order model